Airborne Gravimetry for Geoid Determination

Transcription

1 THE UNlVERSITY OF CALGARY Airborne Gravimetry for Geoid Determination by Yecai Li A DISSERTATION SUBMITTED TO TJd5 FACULTY OF GRADUATE STUDIES IN PARTIAL, FULFILLMENT OF THE REQUIREA4ENTS FOR DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF GEOMATICS ENGINEERING CALGARY, ALBERTA JULY, Yecai Li 2000

2 m*l National Library of Canada Acquisitions and Bibliographic Services Bibliotheque nationale du Canada Acquisitions et services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1 A ON4 Ottawa ON K1 A ON4 Canada Canada Your fiin Vorre reference Our M Notre refdnmce The author has granted a nonexclusive licence allowing the National Librq of Canada to reproduce, loan, distribute or sell copies of this thesis in microform, paper or electronic formats. The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts &om it may be printed or otherwise reproduced without the author's permission. L'auteur a accorde une licence non exclusive pennettant a la Bibliotheque nationale du Canada de reproduire, przter, distribuer ou vendre des copies de cette these sous la forme de microfiche/film, de reproduction sur papier ou sur format electronique. L'auteur conserve la propried du droit d'auteur qui protege cette these. Ni la these ni des extraits substantiels de celle-ci ne doivent Stre imprimes ou autrement reproduits sans son autorisation.

3 Abstract In this dissertation, the potential contribution of an airborne gravity system consisting of a strapdown inertial navigation system and differential global positioning system is investigated. Specifically, its contribution to geoid determination in terms of wavelength resolution and achievable accuracy is studied fiom a numerical and operational point of view. Gravity values in various areas with typical topographic features were used for the spectral analysis of the requirements of geoid determination on gravity data. Results indicate that a cm-geoid can be achieved if the minimum wavelength resolved on the ground is about 14 Ian in flat areas and 5 km in mountainous areas. The achievable accuracy and resolution of airborne gravimetry in geoid determination is studied through an assessment of MS and GPS errors in the spectral range of interest. The results show that the total GPSKNS induced geoid error is less than 1 cm for wavelengths between 5 lan and 100 km and less than 10 cm for wavelengths between 5 krn and 500 km. To verify these results, an airborne test was flown over a very rugged area. Airborne gravity disturbances were collected above a 100 km x 100 krn area and were downward continued to the ground level. The downward continued gravity disturbances agree with ground gravity data at the level of 3.8 mgal (la) and 2.5 rnga.1 (la) for a wavelength resolution of 10 km and 20 km, respectively. In the test area, the local airborne geoid agrees with the reference solution at the level of 1-2 cm (1 o). The regional geoid computed fiom the downward continued gravity disturbances and 30' x 30' mean gravity anomalies in the outer area agrees with the reference geoid computed fiom ground gravity data on a 5' x 5' grid at the 5 cm level (1 c). The same level of agreement exists between the reference geoid and a GPSAevellingderived geoid when fitting out the systematic trend. Thus, airborne gravimetry is capable of geoid determination at the 2 cm level for wavelengths between 10 krn and 100 km and at the 5 cm level for wavelengths above 10 km when the systematic trend is removed.

4 Acknowledgments I wish to express my deepest gratitude to my supervisor, Dr. Klaus-Peter Schwarz, for his help, support and guidance throughout the course of my Ph.D. program. His encouragement was essential for the completion of this dissertation. Many thanks also go to Dr. Michael Sideris and Dr. Michael Chapman for their help and suggestions on various aspects. Dr. Doug Phillips at IT of the university is appreciated for help running the downward continuation program under the MACI-based DEC Alpha workstation. I want to extend my appreciation to Professor Zelin Guan (Oct. 30, Nov. 10, 1999) for his teaching of my frrst course in physical geodesy, supervision of my f~st M. Sc. program, and the continuous encouragement and friendship during the eight years I worked with him at Wuhan Technical University of Surveying and Mapping, China. Special thanks go to my fellow student Mr. Gengsheng Zhang for the discussions we have had on inertial navigation systems, Mr. Alex Bruton for his help and friendship, and Dr. Craig Glemie for providing the processed airborne gravity data. Financial support for this research partly came fiom research grants of the supervisor, and partly fiom the Graduate Research Scholarship of the University of Calgary (1994, 1995, 1996, 1997), Province of Alberta Graduate Fellowship (1 995, 1997): Ralph Steinhauer Awards of Distinction (1 996), Izaak Walton Killam Memorial Scholarships (1996), and Helmut Moritz Graduate Scholarship (I 996). Research on Chapter 4 was initially sponsored by a contract between Sander Geophysics Ltd and the University of Calgary. This is sincerely appreciated. Finally, I would like to appreciate my wife Sheny Jin for her understanding and support, and our children Richard and Jessica for missing their father on numerous weekends and evenings.

5 Table of Contents. Approval Page Abstract Acknowledgments iv Table of Contents... v List of Tables... vii... List of Figures vrli Notation...., xi Chapter 1 Introduction Background Airborne Gravity Measurements Dissertation Outline... 8 Chapter 2 Requirements for Geoid Determination...* Geoid Determination with the Stokes Integral The Stokes integral..., Modification of the Stokes kernel function Geoid Deterrnination with the Hotine Integral The Hotine integral Modification of the Hotine kernel function Height Anomaly Determination with the Mo lodensky Integral Series The Molodensky integral series Relation between the Molodensky and the Stokes solution Requirements for Wavelength Resolution Requirement for an Optimal Integral Radius Summary of Results 35 Chapter 3 Airborne Gravity Systems and Their Potential Contribution System Principle Effect of INS Errors on Airborne Gravirnetry Effect of GPS Errors on Airborne Gravirnetry Effect of Combined INS/GPS Errors on Airborne Gravimetry Effect of Other Critical Factors on Airborne Gravimetry Possible Geoid Resolution of Airborne Gravimetry...,...,.,, Potential Geoid Accuracy From Airborne Gravimetry Airborne Gravimetry Flight Test Description The June 1995 test The September 1996 test Summary ofresults... 69

6 Chapter 4 Terrain Corrections and Their Effect on Airborne Gravimetry Terrain Correction for Vector Gravity Measurements The Effect of Density on Terrain Correction..., Topographic Effect on Processing Airborne Gravity Measurements Topographic Effect on Processing Airborne Gravity Disturbances Summq of Results Chapter 5 Determination of the Reference Geoid Geoid Determined from the Stokes Integral The effect of the Stokes kernel modification Comparison with GPSAevelling-derived geoid undulations Geoid Determined from the Hotine Integral Height Anomalies Determined from the Molodensky Series Effect of the Integration Radius on Geoid Determination Sumrnary ofresults Chapter 6 The Importance of Downward Continuation The Poisson Equation and its Inversion Regularization of the Inverse Poisson Integral Numerical Examples with Synthetic Gravity Anomalies The simulated point mass model data Test of upward continuation with Poisson's integral Test of downward continuation with Poisson's integral Continuation of Airborne Gravity Disturbances Comparison of airborne gravity with upward-continued reference Comparison of downward-continued airborne gravity with reference Summary of Results Chapter 7 The Use of Airborne Gravity Data for Geoid Determination Evaluation ofthe Local Geoid at Flight Level Evaluation ofthe Local Geoidonthe Ground Evaluation of the Regional Airborne Geoid on the Ground Summary ofresults Chapter 8 ConcIusions and Recommendations Conclusians Recommendations REFERENCES...~ Appendix A: Terrain attraction formulas for a mass-line topographic model and a mass-prism topographic model Appendix 8: The kernel function for the Fourier formulas with a mass-prism topographic model

7 List of Tables Statistics of the gridded free-air gravity anomalies Statistics of the residual geoid undulations Parameters of geoid undulation PSD models The parameters used in the INS PSD model PSD Model of GPS-derived Acceleration Errors The standard deviation of the time derivative of aircraft kinematic acceleration with different low-pass cut-offfiequencies Recoverable wavelength in airborne gravimetry Topographic attraction in space and the differences when using a horizontally varying density model instead of a constant one (h, = 3 km) Statistics of the Merences of the topographic attraction at altitudes of 5 km and 6 km..., 81 Statistics of gravity anomalies in mgai Statistics of the gravimetric geoid undulations...~.~~~~.~~~~.~~ Statistics of the differences between the gravimetric geoid undulations and those derived fiom GPSAevelling Statistics of the differences between Stokes and Hotine geoid undulations Statistics of gra* anomalies and topographic heights Statistics of the G,, and t;. terms for different grid spacings Statistics of the differences between the contribution of G, and c Statistics of the differences between the predicted and the GPSllevelling derived geoid undulations Statistics of geoid undulation differences due to different integration radii Statistics of gravity anomalies computed fiom the point mass model Statistics of the differences between the reference and the upward continued gravity anomalies Statistics of the differences between the reference and the downward continued gravity anomalies and their effect on geoid undulations Statistics of the differences between the upward-continued reference and airborne gravity disturbances...~ The statistics of the differences bemeen the reference and the downward continued airborne gravity disturbances The statistics of the differences between the reference and the downward continued airborne gravity disturbances (without regularization) Statistics of the geoid undulation differences propagated fiom the differences between the references and airborne gravity disturbances Statistics of the differences between the reference and the airborne geoid undulations vii

8

9 Topographic attraction in space (h = 3km) with constant density Topographic attraction differences when using constant vs varying densities PSD of the topographic attraction differences when using constant vs varying densities Cumulative ms spectrum of the topographic attraction differences due to using constant vs varying densities The vertical component of topographic attraction in space (h = 5h) in the Canadian Rockies ~ Topographic attraction at flight level and airborne gravity disturbance of the July 1995 test Gravity difference at flight level due to the remove-restore method for the topographic effect The differences between Dayl and Day2 gravity disturbances in space on 1 l x 11 crossoverpoiuts The differences between Dayl and Day2 gravity disturbances in space on a 51 x 51 grid (grid spacing: 2 km) Topographic attraction in space (h = 4350 m) The differences between Dayl and Day2 gravity disturbances in space on a 51 x 51 grid (grid spacing: 2 krn, topographic effects were removed) Gravity anomaly in the Canadian Rocky Mountains Geoid in the Canadian Rocb Mountains Geoid undulation differences due to the modification of the Stokes kernel Geographic location of the GPS/levelling benchmarks in Alberta Geoid undulation differences (Stokes's integral vs Hotine's integral) Geographic location of GP SLevelling benchmarks Illustration of area to whichthe Stokes integral is applied Geoid undulation error due to the combination of the integration inner and outer areas The exponential factor in downward continuation in the frequency domain Central profile of the downward and the upward continued results of a constant 10 mgal within a 100 x 100 bkea Gravity anomalies computed from the point mass model at h = 0 km Gravity anomalies computed from the point mass mode1 at h = 4 krn The difference between the reference and the upward-continued anomalies The difference between the reference and the downward-continued anomalies Amplitude spectrum of the gravity anomaly differences at h = 0 krn Digital topographic model below September 1996 flight profiles Gravity disturbances below September 1996 flight profiles The differences between airborne and upward-continued gravity disturbances (Day 1, 4350 rn) The differences between airborne and upward-continued gravity disturbances (Day2, 4350 m).....,, 125

10 6-12 The differences between airborne and upward-continued gravity disturbances (Day m) Differences between reference and the downward-continued airborne gravity disturbances (EGM96 and topographic effect were removed) North-south and west-east components of the amplitude spectrums of ground gravity disturbances The differences between geoid undulation computed from upward-continued gravity disturbances and those fiom airborne gravity at flight level The differences between geoid computed from ground gravity disturbances and those form downward-continued airborne gravity disturbances Illustration of combining airborne and ground gravity for geoid determination The differences between the reference and airborne geoid (Day I, Model A) The differences between the reference and airborne geoid (Day 1, Model B) The contribution of ground gravity disturbances in area b to geoid undulations in the inner area

11 1. Conventions Vectors are bold and lower case Matrices are bold and upper case A dot above a matrix or vector denotes time differentiation ccvector" refers to componenl of a vector. Vector superscripts refer to the coordinate h e the vector is represented in, e-g., Rotation matrices are specified by indices such that the transformation from the b-frame to the I-frame is given by rf = ~ : r ~ Angular velocity of, for example, the b-frame with respect to the i-frame (subscripts) coordinated in the b-frame (superscript) is given by b Wib = [a, my Or by the corresponding skew-symmetric matrix 7 0 -a_ av b nib = 2. Coordinate Frames , --ay 0.r Local Astronomic Frame (a-frame) origin: at observation point x-axis: tangent to astronomic meridian pointing north y-axis: completes a left-handed system z-axis: orthogonal to level surface at observation point 2.2 Operational Inertial Frame (i-frame) origin: at center of mass of the Earth x-axis: pointing towards mean vernal equinox y-axis: completes a right-handed system z-axis: coincident with Earth mean spin axis 2.3 Body Frame (b-frame) origin: center of mass of accelerometer proof masses, assumed to coincide with center of rotation of sensor unit x-axis: pointing to the right when looking fonvard y-axis: pointing forward longitudinally along body

12 z-axis: completes a right-handed system 2.4 Earth-Fixed Frame (e-he) origin: Earth's center of mass x-axis: orthogonal to z-axis in meridian plane of Greenwhich y-axis: completes a right-handed sysyem z-axis: parallel to Earth's mean spin axis The rotation of the e-frame with respect to the i-frame is given by To Local-Level Frame (1-frame) origin: at the origin of the sensor erne x-axis: completes a right-handed system y-axis: along the north direction of the ellipsoidd meridian z-axis: orthogonal to reference eliipsoid, pointing outwards The rotation of the 1-frame with respect to the e-frame is given by a;, = [-6 icoscp isinqlt Wherecp and h are geodetic latitude and longitude, and the dot denotes time differentiation The rotation matrix between the I-frame and the e-he is given by r I 1 -sid -simp cod cosv coshl L O coscp s~.(p j The rotation matrix between the 1-he abd the b-frame is given by cosycoscp - sin~ysinesinrp -siq~ose cosyrsintp + sinyrsin0cosq R: = sinyrcoscp + cos~sinesincp cosyrcose sinvsintp - cosysinecoscp -COS~S* sine COS~COSC~ where the three attitude angles of the body frame, roll (q), pitch( )) and yaw (v), defined as: roll about y-axis, pitch about x-axis, and yaw about z-axis. The rotation are positive counter-clockwise when view from the positive end of the respective axis. 3. Acronyms Dayl, dayl: the first day of the June 1996 airborne gravimetry test Day2, day2: the second day of the June 1996 airborne gravimetry test xii

13 Day3, day3 : the third day of the June 1996 airborne gravimefq test DEM: digital elevation model DGPS: differential global positioning system EGM96: Earth geopotential mode 1996 FIR: finite impulse response GBVP: geodetic boundary value problem GM: geopotential model GRACE: gravity recovery and climate experiment (NASA) GOCE: gravity field and steady-state ocean circuiation expiorer (ESA) W: inertial measurement unit INS: inertial navigation system LRF m: Honeywell Laseref III strapdown inertial system PSD power spectrum density RISG: rotation invariant scalar gravimetry RMS, rms: roo t-mean-square SISG: strapdown inertid scalar gravirnetry STD,std: standard deviation tc terrain correction 4. Symbols vector of acceierometer biases vector of gyro drifts time synchronization error frequency in Hz skew-symmetric matrix of specific force vector of accelerometer specific force measurements vector of accelerometer errors three components of specific force in the 1-frame three components of specific force in the e-fime magnitude of gravity gravity vector in the 1-frame gravity vector in the e-fi-ame geodetic height transformat ion matrix. fiom the b-frame to the e- frame R: transformation matrix fkom the b-frame to the 1-&me r' position vector in the e-frame 6r' position error vector in the e-frame R mean value of the Earth's radius KT, meridian radius of curvature R, prime vertical radius of curvature... Xlll

14 V' skew-symmetric matrix of vehicle velocity ve vehicle velocity vector in the e-fiame 6vc vehicle velocity error in the e-fixme v1 vehicle velocity vector in the 1-fi-ame 6v' vehicle velocity error in the I-ke v, v, v,, east, north and up components of the vehicle velocity in the 1-fime t;' vehicle acceleration in the 1-Eame d~' vehicle acceleration error in the 1-fiame dv chi &u vehicle acceleration errors in the 1-frame e n W b vector of accelerometer bias white noise Wd vector of gyro drift white noise E' attitude errors due to gyro measurement noise and initial misalignment E, E, E, attitude errors in the 1-fiame rl deflection of the vertical in east-west direction 0 pitch angle 4 deflection of the vertical in north-south direction 0 standard deviation Y yaw angle skew-symmetric matrix of the angular velocity between the b-frame and be a' el ab ib a [ ie l Oel b Wib l ie o the e-fiame skew-symmetric matrix of the angular velocity s kew-symmetric matrix of the angular velocity a,: skew-symmetric matrix of the angular velocity a*: angular vehicle rate over the ellipsoid angular velocity measured by INS gyroscopes earth rotation vector earth rotation rate frequency in radids 0 geodetic latitude h geodetic longitude when used for coordinate wavelength when used for spectral analysis 8 roll angle when used for attitude co-latitude when used for coordinate F F-' the two-dimensional Fourier transform operator the two-dimensional inverse Fourier transform operator xiv

15 gravitational constant geoid undulation geoid undtdation computed from residual gravity anomalies geoid undulation computed residual graviy anomalies in the inner area geoid undulation computed residual gravity anomalies in the outer area geoid undulation computed fkom a geopotential model the topographic indirect effect on the geoid undulation due to The second method of Helmert's condensation of the topography spectral density spectral density of airborne gravity disturbance errors spectral density of GP S errors induced airborne gravity disturbance errors spectral density of MS errors induced airborne gravity disturbance errors spectral density of airborne gravity errors induced geoid undulation errors disturbing potential of the Earth's gravity field normal gravity vector normal gravity gravity anomaly vector gravity anomaly gravity anomaly computed from a geopotential model terrain correction gravity disturbance vector gravity disturbance gravity disturbance computed fiorn a geopotential model spherical distance kernel function of the Stokes integral kernel function of the Hotine integral truncation coeecients of the Stokes integral with the original kernel function truncation coefficients of the Stokes integral with the modified kernel hction truncation coefficients of the Hotine integral with the original kernel function truncation coefficients of the Hotine integral with the modified kernel fimction nh degree Legendre polynomial height anomaly n-th term of the Molodensky integral series density of topography

16 Chapter 1 Introduction 1.1 Background Geoid determination is an important topic in geodesy, geophysics and geodynamics. The geoid can be loosely defined as the equipotential dace of the Earth's gravity field at mean sea level. Tn geodesy, it is used as the reference surface of the orthometric height system. In geophysics and geodynamics, it is used to study the deep Earth mass anomaly structure, tectonic forces, oceanic lithosphere, Earth rotation, as well as the tides, currents and sea surface topography (Vanicek and Christou, 1993). The main factors affecting the accuracy of gravimetric geoid determination are the resolution and the accuracy of the gravity measurements. Conventionally, gravity measurements have been taken point by point on the ground with a gravimeter. Such an operational procedure is time-consuming, laborious, and logistically dficult in some areas. Airborne gravimetry has the potential of solving most of these problems and of providing gravity information with a resolution of a few kilometers and significantly improved efficiency. Essentially, the same methods that are used to determine geoid unduiations from terrestrial gravity measurements can be applied to airborne measurements. However, because airborne gravimetry outputs gravity disturbances along ff ight trajectories in space, modifications are necessary to downward continue gravity measurements to a boundary surface and to formulate the geodetic boundary value problem with gravity disturbances as input.

17 Chapter 1: Introduction 2 Methods often used in geoid determination include least-squares collocation (Moritz, 1980), spherical harmonic expansion (Heiskanen and Moritz, 1967), integral farrnulas for the solution of Geodetic Boundary Value Problems (GBVP), such as the Stokes integral (Stokes, 1849), the Hotine integral (Hotine, 1969) and the Molodensky integral series (Molodensky et al., 1962). W e least-squares collocation works with a variety of data types, the integral formula solutions use gravity data only. Spherical harmonic expansion of gravity provides an effective way of modelling the long wavelength information of the global field in order to get a reference fieid for the Iocal approximation. The geopotential models are therefore used in all methods of precise geoid determination, independent on whether they are of the collocation or integral type. The most recent global geopotentid model has been developed up to degree and order 360 and corresponds to a set of global mean gravity data with a resolution (half-wavelength) of 30 arc-minutes (IGeS, 1997). Geoid information for the medium and shart wavelength part ofthe spectrum can be obtained by using point gravity data or small block averages within a local area using either collocation or one of the integral formulas. The Stokes integral computes geoid undulations from gravity anomalies. The Hotine integral computes geoid undulations from gravity disturbances. The Molodensky series takes the gravity anomaiies on the natural surface of the Earth as input and determines the height anomalies which can then be transformed into approximate geoid undulations. Gravity reduction is required by the Stokes theory, the Hotine theory, and collocation to reduce the measured gravity on the Earth's surface to the geoid and to remove the masses outside the geoid. Although Molodensky's approach theoretically avoids the problem of gravity reduction and mass-shifting, its practical

18 Chapter 1 : Introduction 3 implementation is restricted due to the requirement that both gravity anomalies and topographic heights be available on the same points. This would require very dense gravity measurements in mountainous areas. Numerous modifications and refinements of the integral formulas have been proposed with the goal of either improving the accuracy or enhancing the computational efficiency (Heiskanen and Moritz, 1967; Moritz, 1980; Meissl, 1971 ; Wichiencharoen, 1982; Lachapelle, 1984; Sideris, 1986; Schwarz et al., 1990; Sideris and Forsberg, 1990; Vanicek and Sjoberg, 1991 ; Li, 1993). Due to the fact that the gravity anomaly is the most convenient variable for managing terrestrial gravity measurements, most of the studies have been focused on improving the results obtained from the Stokes integral, Due to the increasing availability of high-accuracy topographic heights and airborne gravity measurements in mountainous areas, it is necessary to conduct more investigations on the Hotine integral and the Molodensky series for the purpose of high precision gravimetric geoid determination. Such as how the topography contributes to geoid determination. 1.2 Airborne Gravity Measurements The concept of airborne gravity was proposed half a century ago and the successful implementation of such a system has been the dream of geodesists and geophysicists for decades (Hammer, 1950). The first flight tests were conducted in 1960 with disappointing results mainly due to the inaccurate determination of the aircraft position and velocity (Nettleton et al., 1960). The development of the Global Positioning System (GPS) during the mid 1980s re-kindled activities in designing and operating airborne gravity systems

19 Chapter 1 : Introduction 4 (Schwarz, 1980; Hammer, 1983; Brozena and Peters, 1988; Forsberg, 1993). An airbome gravity system determines gravity disturbances along the flight trajectory by separating the dynamic acceleration of the aircraft from the measured specific force. Aircraft dynamic acceleration can be obtained by differentiating the accurate position andlor the velocity provided by measurements fkom the differential global positioning system (DGPS) (Wei et al., 1991; Czompo, 1994). Specific force can be measured in principle by one of two approaches: either by a strapdown inertial navigation system (SINS) or by a platform inertial navigation system (PINS). The strapdown system determines the orientation of the specific force sensors (three accelerometers) mathematically (Schwarz and Wei, 1993), while the platform system stabilizes a gravimeter (a precision accelerometer) in a vertical direction physically (Knickmeyer, 1990; Salychev et al., 1994). Tests of coxnmercially available airbome gravity systems with a stabilized platform, such as the LaCoste & Romberg and the Bell gravimeter, have been conducted in many areas, such as Switzerland (Klingele et al., 1994), Greenland (Brozena et al., 1997; Forsberg and Kenyon, 1994), Antarctica (Bell et al., 1991). Recent results show that airbome gravity, based on the LaCoste & Romberg gravimeter (S-99), is achieving an accuracy of about 2 mgal (RMS) for a half-wavelength resolution of 6 km, based both on crossover comparisons and independent checks with surface gravity data (Forsberg et d., 1999). A prototype airborne gravity system consisting of DGPS/SINS has been developed and tested by the University of Calgary. Compared to upward continued ground gravity data and crossover results, an accuracy of 2-3 mgal for a minimum wavelength resolution of about 8 km has been achieved (Glennie et al., 1999; Wei and Schwarz, 1998). Even though extensive studies have been conducted on evaluating the accuracy of airbome

20 Chapter 1 : Introduction 5 gravity disturbances, the use of airborne gravity for geoid determination is still at an early stage and no comprehensive study of the operational and numerical problems has been published. Airborne gravity data shows some important differences with respect to conventional terrestrial gravity data, which affects their use in geoid determination. Some of the unsolved questions are: 1) What is the minimum resolution of the geoid determined from airborne gravity and does this resolution meet the requirement of geoid determination with the desired accuracy of 1 cm? Airborne gravity data is always bandwidth limited due to the use of a low-pass filter and the limited geographical area flown. Due to the very low signal-to-noise ratio in the high frequency band, a low-pass filter has to be used in order to exclude the GPS-derived acceleration errors from the processed airborne gravity. The value of the cut-off frequency of the low-pass filter limits the minimum resolvable wavelength. Therefore, it is necessary to investigate the required minimum wavelength for geoid determination, given a specific accuracy of, for example, 1 cm. This has then to be compared to the resolvable minimum waveiength of an airborne gravity system. 2) How should the airborne gravity data be reduced from flight level to ground Ievel? Airborne gravity data is obtained along the flight trajectory in space. For geoid determination with the conventional geodetic boundary value problem, this data has to be downward continued from flight level to the boundary sdace of the GBVP, the geoid. The most common method applied to downward continuation is the use of the inverse Poisson integral. The Poisson integral is the solution of the first geodetic

21 Cha~ter 1 : Introduction 6 boundary value problem. It takes gravity on the boundary surface as input and predicts the corresponding values in outer space. In other words, it can be used directly for gravity field upward continuation (Heiskanen and Moritz, 1967). With discrete observations, the input and output can be related mathematically by linear equations. The inversion of the linear equations provides a solution for the gravity field downward continuation (Forsberg and Kenyon, 1994). Since the underlying process is unstable, numerical di&culties can be expected. This means practically that the reliability of the downward continuation depends on the grid interval selected, smoothness of the field, and the depth of continuation, i.e. the flying altitude in airborne gravimetry. Two groups of methods that can be used to stabilize downward continuation are regdarization and band-limiting the signal at flying altitude. Regularization methods to solve geodetic improperly posed problems have been discussed, for example, in Schwarz (1 979) and Rauhut (1992). For airborne gravity data, earlier simulation studies should be supplemented by real data analysis, taking into account the range of flight altitudes and the knowledge on the gravity and the error spectrum for the high frequency band. Bandlimiting the observed gravity field by means of a low-pass filter will usually stabilize the downward continuation process but will often result in a somewhat smoothed geoid. Often, the removal of the topographic attraction from the observed gravity field will eliminate most of the high frequencies, so that band-limiting will not affect geoid accuracy in a major way. The computation of topograp hic attraction or terrain correction has been extensively studied and is well understood (Nagy, 1966; Sideris, 1984; Tziavos et al., 1988; Li, 1993). Published formulas are, however, often designed for specialized applications. A unification of these formulas for general application and for improved efficiency is therefore needed.

22 Chapter 1 : Introduction 7 3) What is the overall achievable accuracy of a geoid determined from airborne gravity? What airborne gravimetry provides are gravity disturbances within a Limited area and for a limited bandwidth. By applying the Hotine integral, this information can be used directly for the determination of a local geoid with the same bandwidth limits. For regional or global gravimetric geoid determination, i-e., for a geoid without limits on the long wavelength band, the long wavelength information has to be recovered fiom existing sources of data, such as a global geopotential model or mean gravity anomalies on a much coarser grid. It is important to investigate the numerical procedures of airborne geoid determination and to examine the achievable accuracies of both the local geoid and the regional gravimetric geoid determined fkom airborne gravity. It is the objective of this thesis to study the achievable resolution and accuracy of the geoid determined from airborne gravity from a numerical and an operational point of view. To reach the objective, this research will focus on the foilowing four areas: 1) The requirements for geoid determination, such as the minimum wavelength needed for a geoid with an accuracy at the centimeter-level. 2) The potential contribution of airborne gravity to geoid determination in terms of the error characteristics, such as the potential resolution and the achievable accuracy of the geoid determined from such data. 3) The challenge of downward continuing airborne gravity data from flight level to the ground and using them together with available ground gravity. 4) The contribution of airborne gravity data to the accuracy of the local, regional and global geoid.

23 Chapter 1 : Introduction 8 13 Dissertation Outline The research is documented in eight chapters. In Chapter 2, the data requirements for precise geoid determination are investigated. The solutions &om three different geodetic boundary value problems, namely the Stokes integral, the Hotine integral and the Molodenslcy integral series, are compared and necessary modifications to the integral kernel hctions are introduced. Formulas are given with all details necessary for their subsequent implementation. Based on the typical signature of gravity anomalies in different areas, the requirements on wavelength resolution for relative geoid with an accuracy of 1 cm (rms) and 10 cm (rms) are studied. The choice of the two accuracy classes (nns values) is determined by the current situation. A 1-cm geoid is often quoted in the literature as a desired goal (ESA, 1999). A 10 cm-geoid with a resolution of about 5 arc-minute is currently achievable in well-surveyed areas in North America, Europe, and Australia and in ocean areas covered by dense altimeter profiles. However, in most of terrestrial, accuracies are much poorer and uncertainties often reach the meter level. The potential of airborne gravimetry for an improvement of the current geoid is therefore investigated both in well-surveyed and poorly surveyed regions. In Chapter 3: the principle of an airborne gravity system is described and its potential contribution to geoid determination is investigated. The mechanization of an airborne gravity system consisting of a strapdown LMU and DGPS will be used as an example. The two basic mathematical procedures for scalar gravimetry, i.e., the strapdown inertial scalar gravimetry (SISG) and the rotation invariant scalar gravimetry (MSG), will be summarized. The achievable accuracy of airborne gravity data will be analyzed in detail in terms of the spectral characteristics of different error sources, such as the GPS errors, the INS errors, and

24 Chapter 1 : Introduction 9 synchronization errors. The potentid resolution and accuracy of a local geoid detennined fkom airborne gravity will be studied based on propagated airborne gravity errors. Two series of airborne flight tests conducted in June 1995 and September 1996 by the University of Calgary will be used for numerical investigations. They are described at the end of this chapter. In Chapter 4, a set of unified terrain correction formulas is presented and the topographic effect on airborne gravity data processing is investigated. Numerical examples will be used to examine the effect of density information on the terrain correction, the topographic effect on airborne data processing and on airborne gravity data downward continuation. In Chapter 5, a reference geoid for the test area will be computed from ground gravity measurements and a detailed topographic model. It wiil be used later on to evaluate the accuracy of the gzoid detennined from airborne data. The effect of different methods and different integration strategies in local geoid determination is numerically analyzed. Comparisons are made between the gravimeeic geoid undulations computed with the Stokes integral, the Hotine integral and the Molodensky integral series, respectively, and between the gravimetric geoid and the one derived by GPSAevelIing. The effect of integral kernel modifications, the effect of the integration area size, and the combination of gravity data with different resolutions are studied and their influence on the geoid is investigated numerically. In Chapter 6, gravity field continuation with the Poisson integral is investigated and the accuracy of airborne gravity disturbances is evaluated. Regularization methods are discussed in order to recast the improperly posed inverse Poisson integral and to reduce the high frequency oscillations in the downward continued gravity data. The performance of upward and dotvnward continuation with the Poisson integral are first assessed in a noise-eee

25 Chapter 1 : Introduction 10 environment with a synthetic anomalous gravity field generated with a point mass model. The accuracy of the gravity disturbances coiiected during the September 1996 airborne gravity tests is evaluated at flight level, by comparing them with the upward continued gravity reference, and on ground, by comparing the downward continued flight data with the ground gravity measurements. In Chapter 7, geoid undulations are computed fiom the airbome gravity disturbances and are compared to the reference geoid obtained fiom ground gravity measurements. Different procedures of using airborne gravity data in geoid determination were investigated, such as computing the geoid ftom the downward continued airbome gravity disturbances alone, and by combining airborne gravity with 30' x 30' mean anomalies on the ground. Tests are also conducted by computing the undulations of the geopotential surface directly at flight level and then downward continuing them to the geoid. The current capability of airborne gravimetry for geoid determination is assessed based on the differences between the geoid undulations determined fiom airbome gravity data and those fkom tenestrial gravity measurements. In Chapter 8, conclusions and recommendations for future work are presented. Some of the findings in this dissertation have either been published in referred journals or documented in a research contract report- Since the candidate is a co-author of these publications, quotations fiom this material is not specifically identified.

26 11 Chapter 2 Requirements for Geoid Determination Geodetic boundary value problems (GBVPs) provide the fundamental solution for gravimetric geoid determination. This chapter describes the three most widely used solutions: the Stokes integral, the Hotine integral and the Molodensky integral series, and discusses their internal relations. Detailed formulas for using these solutions in practical operations are provided and will be used in the following chapters for geoid computation fiom airborne gravity data as well as the determination of the reference gravimetric geoid. Requirements on accuracy and wavelength resolution of the gravity measurements are studied based on the desired accuracy of geoid determination. These requirements will be used in the next chapter to assess whether or not current airborne gravity systems have the potential to determine the gravity field with cm-accuracy. 2.1 Geoid Determination with the Stokes Integral The Stokes integral The Stokes integral is the solution of the third geodetic boundary value problem,which uses the geoid as the boundary d ace and gravity anomalies on the geoid as boundary condition. The solution is expressed as (Stokes, 1849)

27 Chapter 2: Requirements for Geoid Determination 12 where Ag stands for gravity anomalies on the geoid, and w stands for the integration area which, theoreticaliy, should cover the whoie Earth surface. S(y) is the kernel fimction which can be expressed either as a summation of Legendre polynomials or as a closed function where yr is the spherical distance between the computation point and the running point of the integration. Due to the limitation in both area coverage and point density of the gravity measurements, the integration of the Stokes integral can only be carried within a limited area. The limitation of data coverage limits the maximum recoverable wavelength and the sparse distribution of gravity measurements limits the minimum resolvable wavelength. To compensate the limitations, high-precision gravimetric geoid is usually determined using remove-restore techniques. The long wavelength information is modelled by a reference field of a geopotential model and the short wavelength information is constructed by using a high resolution digital topographic model. Both a geopotential model computed gravity anomalies

28 Chapter 2: Requirements for Geoid Determination 13 *go and the topographic effects AgT are removed mathematically kom the observed gravity anomalies Ag on the geoid. The Stokes integral is then used to convert the reduced or residual gravity anomalies Agr to residual geoid undulations Nr. The ha1 geoid undulation N is obtained by restoring the contribution of the geopotential model reference field NGM and of the topography N, to the geoid (Moritz, 1 993; Forsberg and Madsen, 1 990; Sideris and Forsberg, 1990). This procedure can be mathematically described as Equations used for the computation of gravity anomalies AgGM and geoid unduiationsnw from a geopotential model can be found in, for example, Heiskanen and Moritz (1967). In equation (2-4), the terrain correction Ag, will be llly discussed in Chapter 4. N, represents the topographic indirect effect on the geoid undulation due to, e-g., the second method of Helmert's condensation of the topography (Wichiencharoen, 1 982) Splitting the integration area into two parts such that o = oo U Go, where oo represents the i~er area which is a spherical cap with radius yr, around the computation point, ando, represents the outer area which is the rest of the Earth surface, equation (2-1) can be

29 Chapter 2: Requirements for Geoid Determination 14 expressed as where Ag and A~~~~ stand for the &degree harmonics of the gravity anomalies and those computed fiom a geopotential model, respectively. In other words, No= defined by equation (2-1 0) represents the geoid undulation errors propagated fiom the geopotential model used in geoid computation. Not defined by equation (2-1 1) stands for the part of geoid undulation which is omitted due to the limit of maximum degree, n of a geopotential model. Consequently, Not is usually called the truncation error.

30 Chapter 2: Requirements for Geoid Determination 15 The truncation coefficients are a fimction of the spherical radius vo of the integration area and the harmonic degree n. For a given degree n, the bigger the integration area is, the smaller the truncation coefficient will be. For a given integration radius yro, it decreases to zero with the increase of harmonic degree n (Molodensky et al., 1962; Li, 1988). q (Q can be computed recursively as Both the effect of the uncertainties in a geopotential model and the truncation error can be reduced with the increase of the size of the inner integration area. This, however, cannot be implemented in practical operations, simply due to the limitation in data availability. Alternatively, a modification of the Stokes kernel hction is used to achieve better results.

31 Chaster 2: Reauiremenl for Geoid Determination Modification of the Stokes kernel function To reduce geoid errors caused by the use of a limited integration area size, numerous geodesists, starting in the early 1980s, have tried to modify the Stokes kernel function. Originally proposed by Molodensky et al. (1962), the basic idea is to improve the convergency of the truncation coeecients by modifying the Stokes kernel function (Wong and Gore, 1969; Meissl, 1971 ; Hsu and Zhu, 1979; JekeLi, 1981 ; Vanicek and Kleusberg, 1987; Featherstone et d., 1998). Assume that a global geopotential model is accurate enough to reproduce the Earth's gravity field with negligible errors for a band fiom degree 2 up to degree and order L. Then, Due to the orthogonality of the spherical harmonic functions (Keiskanen and Moritz, 1967), we have Denoting and scull = s s c~) - s '(YQ.

32 Chapter 2: Requirements for Geoid Determination 17 equation (2-7) can be identicdiy written as where, For n + k, for n = k,

33 Chapter 2: Requirements for Geoid Determination 18 with the initial values, po(urj = 1, X0,,(yrJ = cosv* +l Y Considering the fact that the lower degree terms in the spherical harmonic series represent the long wavelength information, the modification ofthe Stokes kernel as dehed in equation (2-17) acts as a high-pass filter on the kernel function. In principle, this filter should not change the numerical results if the integration covers all fiequency bands, or in the spatial domain, the integration covers the whole Earth surface. When the integration only extends over a limited area, i.e., only covers part of the frequency bands, the high-pass filter is necessary in order to avoid the effect of the low degree harmonics on the high-frequency band. Figure 2- Z shows the values of the original and the modified Stokes kernel functions as well as the Stokes series up to degree 20. Figure 2-2 shows the power spectrum of Stokes kernel functions based on the samples computed for spherical distance between O0 to Figure 2-2 indicates that the power of Stokes series from spherical degree 2 to 20 only affects the part of spectrum with wavelength longer than 16O. Therefore, if the residual gravity anomalies do not contain information on wavelengths longer than 16", Stokes series up to degree 20 will have no effect on the computed geoid undulation. This, however, is not true if the integration is carried out over a limited area.

34 Chapter 2: Requirements for Geoid Determination L-* ~-~--.---~-~~~-~ i ' L *1--* stokes kernel r------*---r SphericaI distance (degree) Figure 2-1 Stokes kernel fimction 20 : PSD of $tokes kernel C I PSD o f~tokb , t ; series (2-20) ----; I :. : : : : : : : Wavelength (degree) Figure 2-2 PSD of Stokes kernel function Figure 2-3 shows the power spectrum of Stokes series based on samples computed for spherical distances between O0 to 180' and between 0" and 1 go, respectively. Figure 2-3 indicates that Stokes series still has a considerable effect on the short wavelength part when the integration radius is 18O. Therefore, it is necessary to remove the low-degree Stokes series from the Stokes kernel when computing geoid undulations based on residual gravity anomdies within a local area.

35 Chapter 2: Requirements for Geoid Determination Wavelength (degree) Figure 2-3 Power spectrum of the Stokes series with different integration cap size 2.2 Geoid Determination with the Hotine Integral The Hotine integral The Eotine integral is the solution of the second geodetic boundary value problem which uses the geoid as the boundary surface and the gravity disturbance on the geoid as boundary condition. The solution is (Hotine, 1969) The Hotine kernel hction can be expressed by either the infinite series or by the closed expression

36 Chapter 2: Requirements for Geoid Determination 21 With the remove-restore procedure, the contribution of the residual gravity disturbances to geoid undulation can be computed with the Hotine integral as The residual gravity disturbance is defined as It is worth mentioning that, with gravity disturbances taken in space above the Earth's surface such as the case in airborne gravimetry, the Hotine integral can be directly used at the flight level to compute the undulation of the equipotential surface of the gravity field from the corresponding equipotential surface of the normal gravity field. In the same way as for the Stokes integral, equation (2-25) can also be split into two parts: N = N.+Noy (2-3 0) where

37 Chapter 2: Requirements for Geoid Determination 22 where 6g: = 6gn - tjgngm stands for the differences between the &degree harmonics of the gravity disturbances 6gn and those computed from a geopotentid model tigngm- The truncation coefficients can be evaluated recursively as (Li, 1 988) vo Ro = csc- - 1, 2 Using a geopotentiai model, the low degree harmonics in the Hotine kernel function can also be removed by modifying the kernel function.

38 Chapter 2: Requirements for Geoid Determination Modification of the Hotine kernel function Denoting and H(Y) = Hs(v) - HS(vJ9 equation (2-3 0) can be rewritten as where, the contribution of the inner zone and the outer zone, i-e., equations (2-30) and (2-3 I), become

39 Chapter 2: Requirements for Geoid Determination Height Anomaly Determination with the Molodensw Integral Series 23.1 The Molodensky integral series Molodensky et al. (1960) used the Earth surface Z as the boundary surface and gravity anomalies Ag on the Earth's surface as the data. The solution of this third geodetic boundary value problem is the Molodensky integral series (Brovar, 1964; Moritz, 1980): t;=& t';, +&+b +---, where : 2," do, n 2 2. n -2 In this equation, S(v) is the Stokes kernel function. Go = Ag are the gravity anomalies on the telluroid. The first term in equation (2-41) is mathematically identical to the Stokes integral. The next three G,-terms take the form ~2 h-h "2 = d Sm L ~, d + o G ~ ~ ~ ~ P, 1,'

40 Chapter 2: Requirements for Geoid Determination 25 ~2 h-h P ~ z d ~ l,' - --[Im + ~ ~ t ~ ~ p "3 = z;;ss,- 4n; Godo, Y I, = 2R sin- 2' c where is the spherical distance between the running element do and the computation point P, p is the total terrain inclination angle, and h and h, are the topographic height at the - running element and the computation point, respectively. In practice, lo can be taken as the horizontal distance between the computation point and running element. For more details, consult Sideris (1994). The integrals in equations (2-45) to (2-46) can be uniformly expressed as To avoid the singularity problem of the kernel hctions(yr) when y = 0, the contribution of G, at the computation point can be computed as (Schwarz et d., 1990) The singularity of the kernel functions in equations (2-45) to (2-46) at 1, = 0 can be avoided by setting it to zero at the origin. This is reasonable because the height difference (h - h$ is

41 Chapter 2: Requirements for Geoid Determination 26 zero when the running element contains the computation point. The function tg2p in equations (2-45) and (2-46) can be expressed as tg2p = tg2k + tg2py = h2+h2, h = ah/a~, h = away. X Y X Y Ln planar approximation, the partial derivative of the topographic heights can be evaluated by the Fourier transform as where i = fl and u and v are the frequency components corresponding to the latitude and longitude directions, respectively Relation between the Molodensky and the Stokes solution Moritz (1980) has shown that, under the assumption that the free-air gravity anomalies are linearly correlated with topographic elevations, the contribution of G, to the height anomaly c, and the contribution of the terrain correction c to the geoid undulation N satisfy the relation S represents the Stokes integral operator and h is the topographic height of the computation point. Due to the relation between the last term in the above equation and the topographic indirect effect expressed in equation (2-4), the following equation holds:

42 Chapter 2: Requirements for Geoid Determination 27 S(GJ = S(C) +A$ (2-53) This is the motivation to repiace S(G,) by the two terms in the right-hand-side of equation (2-53), since this overcomes the restrictions of gravity anomalies being available exactly at the same gridded topographic points as required by the integral S(G,). Wag (1993) gives another expression for the relation between S(c) and S(G,), namely where Ag is the fiee-air gravity anomaly at the computation point and, 6h2 is the sum of the zero- and the first-order spherical harmonics of h2, which can be approximated by (ibid) In theory, equation (2-54) is superior to equation (2-52) because it does not make any explicit assumptions. 2.4 Requirement for Wavelength Resolution The objective of this section is to investigate the relation between the expected accuracy of the geoid and the required wavelength resolution. This, in turn, will provide requirements for the wavelength resolution of gravity anomalies. To have a quantitative analysis, point gravity measurements obtained from the Geodetic Division of Geomatics Canada and KMS of Denmark in both flat and mountainous areas were used. Table 2-1 contains some statistics of the gridded fiee-air gravity anomalies used in the following analyses. In the table, South-

43 Chapter 2: Requirements for Geoid Determination 28 West 4, h are the latitude and longitude at the South-West corner of the area. The dimensions for the areas with grid spacing of I km are 120 km x 120 km and for the areas with grid spacing of 10 km are 1200 km x 1200 h. The distance between gravity measurements is about 1.7 km in the areas with 1-km grid spacing, and is about 9 krn in the areas with 10 km grid spacing. The areas are subdivided into areas with moderate height differences, typically less than 100 m, which are denoted as cci3at", and areas with substantial height differences, usually more than 1000 m, which are denoted as "mount". Table 2-1 Statistics of the gridded free-air gravity anomalies Grid South-West Gravity Anomaly, mgal Area Spacing $ 9 max min mean =s 0 Flat 1 1km 37-5",258.5" Flat 2 Flat 3 Flat 4 Flat 5 Mount I Mount 2 Mount 3 Mount 4 Mount 5 1km 1km 10 km 10 krn 1 km 1 km 1 km 10 km 10 km Long wavelength information is removed by subtracting the gravity anomalies computed fiom the global geopotential model EGM96 (Rapp et al., 1996) with degree and order up to 360, which is approximately equal to the maximum wavelength of the areas with a grid spacing of 1 km. Therefore, in the following discussions, the power of the gravity field is represented by the residual gravity anomalies referred to the geopotentid model EGM96. Theoretically, this residual field would contribute wavelengths of 110 km and smaller.

44 Chapter 2: Requirements for Geoid Determination 29 Geoid undulations are computed in these areas using Stokes' integral. The integration is perform over the area covered by gravity anomalies. The statistical information on the residual geoid is given in Table 2-2. Table 2-2 Statistics of the residual geoid undulations Grid South-West Residual Geoid Undulation, m Area Spacing $ 3 max min mean nns a Flat I I km 37S0, 258.5" Flat 2 1 km 40.3 O, 270-0" Flat 3 lkm 55.5",8.2S Flat 4 10 km 35.0, 265.0" Flat 5 10 km 55.0, 255.0" Mount 1 1 km 35.0, 239.5" Mount 2 1 krn 37.0, 238.0" Mount 3 1 km 38.0, 243.0" Mount4 1Okm 35.O0,24O.O Mount 5 10 km 48.0, 230.0" The power spectral density of gravity anomalies is usually estimated by applying the Hankel transform to the corresponding covariance model of gravity anomalies (Jordan, 1972; Vassiliou and Schwarz, 1987). In this study, the power spectral density of the residual geoid undulations was estimated directly in the frequency domain based on the Fourier transform. The PSD of the residual geoid undulations in the 10 areas were estimated by means of the periodogram method. The computed two-dimensional power spectral densities are usudiy anisotropic. The 1D isotropic PSD values were obtained from the 2D anisotropic values by averaging over all azimuths with respect to different wavelength bands. To simplify the following computations, power spectral density models of the form

45 Chapter 2: Requirements for Geoid Determination 30 Sm(h) =a ha + b (2-56) had been fitted to the empirical data, where h is the wavelength in kilometers, and a and b, and a are coefficients estimated from the curve fit. The fitted model agrees with the empirical data very well for the interested wavelength band. The parameters are given in Table 2-3. Table 2-3 Parameters of geoid undulation PSD models Geoid Coefficient 2shs 120 km 120 sh km a Flat areas a b a Mount areas a b The rms value of the geoid undulations for wavelengths between h, and h, can be expressed as For simplicity, we call rms(l, h+ah) the rms spectrum and rms(;~,~,,,h) the cumulative rrns spectrum, where hi", the minimum wavelength, is equal to 2 km for the gravity anomalies used in this analysis. Figures 2-4 and 2-5 show the plots of cumulative rms spectrum rms(~,~,,h) of the residual geoid undulations.

46 Chanter 2: Reauirements for Geoid Determination 3 1 Figure 2-4 The rms spectrum of the residual geoid undulations in flat areas wavelength (km) Figure 2-5 The rms spectrum of the residual geoid unddations in mountainous areas Figures 2-4 and 2-5 indicate that, based on the data set used, in order to determine geoid undulations with an accuracy of I cm (rms), the minimum full waveiength to be inciuded is about 14 km in flat areas and 5 km in mountainous areas. For geoid estimation with an accuracy of 10 cm (rms), the minimum full wavelength is about 70 km in flat areas and 40 km in mountainous areas- It should be noted, how-ever, that these values are based on the averaged values computed from the data set used. Local variations in the direction of a large

47 Chapter 2: Requirements for Geoid Determination 32 gravity gradient may be much larger than indicated by the rms values. Conclusions based on these results should therefore be considered as an average, not a worst case, scenario. Figures 2-4 and 2-5 show a break in the rms spectrum between 110 and 120 km. This is mainly due to subtracting the geopotential model fiom the gravity data. The fact that the spectrum does not drop to zero after 120 krn indicates that the geopotential model is not a good approximation for the areas considered. This is confirmed by the large mean values in these areas. 2.5 Requirement for an Optimal Integration Radius As expressed in equation (2-9), with the aid of a spherical geopotential model, the outer zone contribution can be divided into two parts, one is due to deviation from the geopotential model (which contains the information fiom degree 2 to the maximum degree n, of the geopotential model used), the other contains the gravity field information for the degrees above n,. This part has to be excluded from the computation because of the lack of information. The error induced due to this exclusion can be expressed as

48 Chapter 2: Requirements for Geoid Determination 33 The degree variance of the gravity anomalies can be computed from the degree variance o: of a geopotentid model as 2 2 d(~g,') = ~ Yn-1) on, (2-6 1 ) For the purpose of error estimation, the buncation coefficients qn(yr,) can be approximated as (Hsu, 1984; Li, 1987) Adopting Kaula's degree variance model of the gravity anomalies (Kaula, 1979) Based on the gravity anomaly degree variances computed fiom the geopotential model EGM96, parameters in equation (2-63) are estimated as A = mgap, a = Figure 2-6 shows the EGM96 and the modelled gravity anomaly degree variances. Based on equations (2-63) and (2-64), the truncation error is about 0.65 m when the integration radius is 0, i.e., in the case when no local gravity measurements are used, and is negligible (less than rn) when the integration radius is larger than 0.5 ". Figure 2-7 shows geoid undulation errors propagated from the error degree variance of geopotentid model EGM96 with different radiuses for the integration area.

49 Chapter 2: Requirements for Geoid Determination 34 Figure 2-6 Gravity anomaly degree variances (computed vs modelled, unit: rngaiz) Figure 2-7 Geoid undulation errors propagated from the error degree variances of EGM96 when using different integration radii Figure 2-7 indicates that the EGM96 induced geoid undulation error is about 0.36 m when the integration radius is 0". Using equation (2-59) and taking into account the truncation error of 0.65 m, it can then be concluded that the overdl geoid undulation error is about 0.75 m when only the geopotential model EGM96 is used in geoid computation. It has to be

50 Chapter 2: Requirements for Geoid Determination 35 emphasized again that the EGM96 induced error varies considerably with geography and can easily rise to 1-2 m in certain areas on the globe. Figure 2-7 also shows that EGM96 induced geoid undulation error is about 0.2 m when the integration radius is 0.5". For geoid determination with an expected accuracy of 0.1 m or better, the integration radius should be larger than 1-5". or 2" when also taking into account the effect of the errors of gravity anomalies within the inner zone. 2.6 Summary of Results In this chapter, the requirements for precise geoid determination were analyzed. Statistical information on the gravity field obtained from the test areas indicates that on average acentirneter-geoid can be achieved ifthe minimum full wavelength resolved is about 24 km in flat areas and 5 km in mountainous areas. To obtain a geoid with an accuracy of 20 cm (rms), the minimum wavelength is about 70 km in flat areas and 40 krn in mountainous areas. Discrete data should be spaced at half the wavelength and much denser for a reliable geoid determination. Modification of the kernel functions are beneficial for absolute geoid determination when the integration is carried out over a limited regional area. For absolute geoid determination with an accuracy of 10 cm or better, the integration radius should be larger than 1.5, or 2". The achievable geoid accuracy is about 20 cm when the integration radius is 0.5 ".

51 Chapter 3 Airborne Gravity Systems And Their Potential Contribution In this chapter, the mechanization of airborne gravity systems that consist of a strapdown inertial navigation system (SINS) and a differential global positioning system (DGPS) will be described. Then, the effect of both SINS errors and DGPS errors on airborne gravity measurements will be analyzed- Thirdly, the potential resolution and accuracy of a geoid determined from airborne gravimetry will be investigated based on the error spectral characteristics. The potential resolution and accuracy will be compared to the values obtained in Chapter 2 from the PSD analysis to see if airborne gravimetry is capable of determining the geoid with the required resolution and accuracy. The agreement of the predicted geoid accuracy with the accuracy actually achieved by using airborne data in Chapter 7 is a strong argument for the consistency and reliability of the results presented in this thesis. Finally, a brief description of the two airborne gravity flight tests that will be used in later chapters will be given. 3.1 System Principle The principle of airborne gravimetry is based on Newton's second law of motion in the gravitational field of the Earth, which when expressed in an inertial reference kame (i) is of

52 Chapter 3: Airborne Gravity Systems 37 the form where, i is the position vector fiom the origin of the inertial reference frame to the moving object and #is the second derivative of this vector with respect to time, f is the specific force vector, and 3 is the vector of all gravitational accelerations acting on the moving object. Transforming equation (3-1) into a local-level frame (I), gravity can be expressed as (Schwarz and Wei, 1993) where, g' is the gravity vector which is the sum of the Earth's gravitational vector and the centripetal acceleration due to the Earth's rotation with respect to the inertial frame, 3' is the dynamic acceleration of the aircraft, and v' = (ve, v,, VJ' is the velocity vector. The kinematic acceleration $', velocity v' and the position of the aircraft can be determined by the global positioning system (GPS) carrier phase and phase rate measurements. Figure 3-1 Il~ustration of attitude angles and the body frame (assuming INS body frame is aligned with aircraft's body frame)

53 Chapter 3: Airborne Gravity Systems 38 Figure 3-2 Mechanization of a strapdown INS in the local-level h e The specific force measured by a set of three accelerometers of an inertial system and expressed in the system's body frame (6) is where R,' is the transformation matrix fiorn the body f?ame to the iocal-level fiarne. It is defined by the thee attitude angles, roll (cp), pitch(8) and yaw (y~), of the rotation of the body fiame to the local-level frame sequentially about the y-axis, x-axis and z-axis. As shown in Figure 3-2, the attitude can be obtained by integrating the measured angular

54 Chapter 3: Airborne Gravity Systems 39 velocities between the body h e and the inertial frame wf6 after correcting for the Earth's rotation ofe and aircraft rate dl. For a detailed description of the mechanization equation and the solution of a strapdown INS system in the local-level h e, see, for example, Schmidt (1978) and Schwarz and Wei (1994). qe and R:, are the skew-symmetric matrices of mfe, the angular velocity of the Earth's rotation with respect to the inertial kame, and dl, the angular velocity of rotation of the local-level fiame with respect to the Earth. o! and a:, can be expressed as re R, andr, stand for the meridian and prime vertical radii of curvature of the reference ellipsoid, h is the ellipsoidal height of the INS centre. The gravity vector g' is usually expressed as the sum of the normal gravity vector y' and the gravity disturbance y' is a function of position and can be computed from a normal gravity formula, or for a better approximation, fiom the coefficients of a global geopotential model (Heiskanen and Moritz, 1967). Thus, the problem of airborne gravirnetry can be described as the determination of the gravity disturbmce fiom the following expression

55 Chapter 3: Airborne Gravity Systems 40 or in scalar form, These are the fundamental equations for vector gravimetry by which the three components of the gravity disturbance vector are determined. A scalar gravity system determines the vertical component of the gravity vector only. Due to the fact that the deflections of the vertical are rarely larger than 30", the difference between the magnitude ofthe vertical component and that of the gravity vector is usually less than 0.05 mgal which is well below the accuracy requirements of almost all current applications. There are basically two mathematical procedures for scalar gravimetry. One is strapdown inertial scalar gravimetry (SISG), the other is rotation invariant scalar gravimetry (RISG)(Czompo, 1994; Wei and Schwarz, 1997). The SISG method determines the vertical component of gravity according to equation (3-9c), and the RISG method computes the magnitude of gravity according to the foilowing formula (ibid): where

56 Chapter 3: Airborne Gravity Systems 41 c = (ce,cdc-jt = (2% +l2;i)vf. (3-1 1) Figures 3-3 and 3-4 show the general procedures of the two scalar gravirnetry systems. The error model of airborne gravirnetry can be obtained by Linearizing equation (3-8) in the following way (Schwarz and Li, 1996): where d6gstands for the errors in the gravity disturbance vector Sg, &represents the attitude errors due to initial misalignment and gyro measurement noise, df represents the accelerometer noise, d+ and dv stand for errors in the computed aircraft acceleration and velocity, respectively. dyis the error in the computed normal gravity due to position error. F and V are skew-symmetric matrixes containing the components of the specific force vector f and the velocity vector v', respectively. f is the time derivative of the specific force. The last term in equation (3-12) represents the gravity error induced by the time synchronization error dt between the INS and GPS measurements. Figure 3-3 Illustration of the SISG system

57 Chapter 3: Airborne Gravity Systems 42 g INS Figure 3-4 Illustration of the RISG system Denoting

58 Chapter 3: Airborne Gravity Systems 43 equation (3-12) can be W er developed as which provides a direct relation between the airborne gravity error and the system measurement errors. 3.2 Effect of INS Errors on Airborne Gravimetry INS errors usually consist of two parts: one is the gyro error which is the main contributor to the attitude errors, and the other is the acceleration error which results in the specific force error. The effect of the attitude error on a airborne gravimetry is represented by the fust term on the right-hand-side in equation (3-1 6), i-e., which clearly indicates that the attitude error will induce errors in each component of the gravity disturbance vector when specific force exists. Therefore, constant velocity is the ideal operational environment for scalar gravimetry because it eliminates specific force tiom the horizontal channels, and makes the system fkee from the effect of attitude errors. In practical operations, horizontal acceleration is not avoidable due to, for example, the phugoid motion.

59 Chapter 3: Airborne Gravity Systems 44 Figure 3-5 shows the plot of the amplitude spectrum of the horizontal accelerations derived fiom the horizontal phugoid motion of the aircraft during a flight test (A. Bruton, 2000, personal communication). Figure 3-6 shows the cumulative rms spectrum of gravity (vertical component) errors due to the phugoid motion and an attitude error of 1 ". Figure 3-6 indicates that, when using a low-pass filter with cut-off frequency of Hz (60s), phugoid motion induced gravity error is 0.5 mgal when the horizontal attitude error is 1 ", and is 2.5 rngal if the horizontal attitude error is 5". Therefore, it is important to keep both the attitude error and horizontal acceleration as small as possible. n CI 100 i ; Frequency (Hz) Figure 3-5 Amplitude spectrum of phugoid motion induced horizontal acceleration Figure 3-6 The cumulative rms spectrum of gravity errors due to the phugoid motion and attitude misalignment

60 Chapter 3: Airborne Gravity Systems 45 During a flight mission, the specific forces and the kinematic acceleration usually vary. Figure 3-7, for example, shows the accelerations of the aircraft during a flight test in west- east direction above the Canadian Rocky Mountains performed on June In, Figure 3-8 shows the amplitude spectrum of the accelerations. Figure 3-7 Aircraft kinematic acceleration during June 1, 1995 airborne gravity test Figure 3-8 Amplitude spectrum of the kinematic acceleration (unit: mgai)

61 Chapter 3: Airborne Gravity Systems 46 The figures clearly indicate that the aircraft dynamics have different characteristics in different fiequency bands, and consequently, their effects on the gravity determination will be dieerent in different fiequency ranges. Because most practical applications require gravity information with different resolutions and area coverage, the properties of the INS induced errors will be studied in the frequency domain. For an INS with ring-laser gyros, the gyro output errors can be approximated as (Savage, 1978) where ob represents the angular velocity, K' represents the scale factor error when ib transforming the gyro impulse output to angular velocity, AC'~ represents the misalignment between the gyro physical body h e and the system body fiame as well as the mounting error of the gyroscopes, p represents the gyro bias. The total attitude error can be approximated by a constant initid attitude error, a constant gyro drift rate and dl accelerometer bias, see Britting (2971) I 1 a, sins, o,si.n2cpsinoiet sino f - c - 0: 20:

62 Chapter 3 : Airborne Gravity Systems where c = coso,r - cosost, a, is the Schuler frequency, q, cn, eu stand for the initial attitude errors, he,60,, 601, represent the gyro drifts and 6fe, 6fn, 6fu are the accelerometer biases. Denoting the power spectral density (PSD) of the gyro drift as d2 and the PSD of the g accelerometer bias as b:, the PSD of the attitude errors can be derived as., (0) = ( 1 ) 2 m2p a: tan2q (a: +-d +- o2 g2 02(m;-a) b 2, 22 a after neglecting the terms related to mi=. This simplification is reasonable because they are much smaller than those containing w,. The specific force error measured by the accelerometers of a strapdown INS can be approximated by the formula (Schwarz, 1996) where b is the accelerometer bias error, which changes slowly with time, s, and s, are the linear and quadratic scale factor errors which are a function of specific force, and n is the sensor noise which has white noise and correlated components. The specific force error can be modeled as a second-order Gauss-Markov process with the following PSD expression

63 Chapter 3: Airborne Gravity Systems 48 To demonstrate the gravity error induced by the INS attitude and specific force errors under perfect Dight condition, i-e., for zero horizontal components of the specific force, the PSD of the horizontal components of the gravity error can be expressed as where d is the variance of accelerometer colored noise, Qa represents the PSD of white a noise, ms represents the Schuler frequency, and Pa is the inverse of the correlation time of accelerometer colored noise. Table 3-1 lists the values of these parameters used in this research, which are typical for gyros and accelerometers in a navigation grade strapdown system. Figure 3-9 shows the PSD of both the horizontal component and the vertical component of the INS induced gravity errors. parameter Table 3-1 The panmeters used in the INS PSD mode1 b2 a d2 I value 100 mgd2/hz 1 0" (degh)'/hz 100 mga12 1 mga12/hz 2h I I? 2 a Qa I/P, I

64 Chapter 3: Airborne Gravity Systems 49 Figure 3-9 indicates that the error spectrum of the specific force error has more power at low fkequencies and dominates the low end of the spectrum in airborne gravimetry. It should be noted, however, that some short-term INS errors are not modeied here. They include the effects of mislevelling under horizontal accelerations, scale factor and non-orthogonality errors in high dynamics. Their omission from the error model will not affect geodetic applications, because they are mainly adding to the high-frequency error spectrum, where they are covered by GPS noise. Similarly, noise stemming from high-frequency vibrations has not been considered. Although it is a problem when estimating the gravity signal, it does not affect geoid determination. Figure 3-9 PSD of the INS induced gravity error 3.3 Effect of GPS Errors on Airborne Gravirnetry The major function of the GPS subsystem in GPS/INS airborne gravimetry is to provide a precise kinematic description of the aircraft trajectory. The observables are carrier phase and

65 Chapter 3: Airborne Gravity Systems 50 pseudorange data, typically in their double-differenced form. Accelerations can either be determined fiom GPS-derived velocity by single differentiation or from GPS-derived position by double differentiation (Schwarz et al., 1991). With existing GPS technology, the accuracy of airborne DGPS positioning with monitorremote separations in the range of km is at the level of 10 cm (1 a) using high quality receivers and reliable ambiguity initialization (Bruton et al., 1999,2000). Aircraft velocity can be determined with an accuracy of 1 cm/s horizontally and 2 cds vertically when using carrier phase observations (Cannon et d,, 1992; Wei and Sch~varz, 1984, 1995). Assuming a flight speed of 100 m/s, the gravity error can be estimated from equation (3-16), i.e., a 1 cds velocity error will generate in 0.15 mgai gravity error, and a 10 cm position error will introduce a 0.03 mgal gravity error. Considering the fact that the current accuracy expectation for the airborne gravity system is at the I mgai level, the effect of GPS position and velocity errors can be negiected. Consequently, oniy the effect of the GPS-derived acceleration error on airborne gravity has to be investigated. In general, long term biases in position do not pose a problem to acceleration recovery since, through the process of differentiating twice, any constant or linear error is eliminated. Thus, it would be expected that tropospheric errors, orbital errors, and incorrect carrier phase ambiguity fixing would have little effect on acceleration determination. High frequency errors, such as receiver phase noise and some multipath, can be reduced with low-pass filtering. However, these errors are amplified by differentiation (Dierendonck et al., 1994; Wei and Schwarz, 1995), which is believed to be the critical part affecting the achievable accuracy of airborne gravimetry, especially for the high frequency band of the spectrum.

66 Chapter 3: Airborne Gravity Systems 51 To have a quantitative description of the GPS-derived acceleration errors, a set of data collected during a flight test on April 28,1993 over Lake Ontario was used. During the test, four Trimble 4000 SSE GPS receivers were operated in C/A-code mode, two as ground monitors and two in the aircraft, and both Ll and L2 data were collected. The maximum baseline length was about 150 Ian. Ellipsoidal heights of the aircraft bs were computed fiom a combination of Ll and L2 observations. A laser altimeter was used to monitor the height hl between the aircraft and the lake surface. The geoid undulations N in Ontario are availabie with a relative accuracy of 0.5 to 3.0 ppm for distances between 20 to 200 km. These values were obtained by comparing geoid undulations derived fkom GP S/levelling and fkom gravity at about 200 benchmarks (Sideris, 1993). Assuming the orthometric height H of the lake surface as constant, which is satisfied to better than 0.5 cm, the difference can be taken as the GPS height measurement error, which in fact contains geoid variations, laser range errors, and lake surface dynamics. Figure 3-10 shows the height differences. The large spikes are caused by laser range errors due to airplane attitude changes (see Czompo [I9931 for a more detailed discussion). The symmetry between the errors in the forward and the backward run indicates the presence of either lake surface or geoid changes along the flight trajectory. Height differences in hvo periods, each with a time duration of 1800 s, were used for power spectrum estimation. The standard deviation of the height differences taken at a 2-Hz rate were 15 and 17 cm during the forward and the backward flight, respectively. The averaged PSD of the height differences is shown in Figure for frequencies fkom 1 O5 to lo-' Hz.

67 Chapter 3: Airborne Gravity Systems 52 time in second Figure 3-10 The height differences 6h for the Lake Ontario airborne test To eliminate the effect of the laser altimeter errors, a set of static data was collected with the laser altimeter. During the test, the laser altimeter head was mounted on a tripod and was pointed to a wall. The measured distance between the laser head and the wall was recorded at a 10-Hz rate and was then reduced to 2-Hz by passing it through an ideal low-pass filter. The standard deviation of the measured distances is 3 cm. The PSD of the static laser altimeter measurements is shown in Figure 3-11.

68 Cha~ter 3: Airborne Gravitv Svstems 53 How well does the difference spectrum characterize GPS height errors depends on how well the static laser spectrum represen& the kinematic test conditions. Experimental results indicate that the noise characteristics of the laser altimeter are similar for static and kinematic operation (Applied Analytics Corporation, 1992). The amplitude spectrum of the laser measurements in the lake test could be higher than in the static case because of errors in the attitude corrections to the Iaser, errors in the geoid model used in (3-26), and lake d ace change due to atmospheric or other conditions. The spectra of the first two phenomena are below the cut-off frequency of the filter and will be largely eliminated through a proper filter design. The next two effects will increase the amplitudes of the medium and Iow fiequencies and will therefore change the spectral characteristics. Thus, the laser spectrum shown in Figure could be slightly optimistic, especially in the low frequencies. This would mean that the difference spectrum is slightly pessimistic, and the resulting PSD model for the GPS errors is on the conservative side. This would agree with results pubfished by Wei et d. (1991), where the GPS error spectnun has been determined under well-controlled test conditions and seems to be flat for the medium and low frequencies. However, since the lake test represents a more realistic situation in terms of production work, the consewative PSDs of GPS height errors resulting from Figure has been used. The accuracy of GPS positioning is strongly dependent on the distance of the airborne receiver to the master station on the ground. In this test, the maximum distance to one of the ground receivers was about 150 km. If longer distances to the ground receiver have to be accepted because of operational conditions, the low-fkequency errors will increase. Since the model proposed here is on the conservative side for this part of the spectrum, it may also be representative for situations that are somewhat different from the conditions in the Lake Ontario test. The PSD differences can then be used to derive the PSD of GPS-derived

69 Chapter 3: Airborne Gravity Systems 54 acceleration errors. The latter is used for constructing a PSD model of the form The unit of S,(A) is mga12/hz. Table 3-2 contains the coefficients, and Figure 3-12 shows the plot of the PSD of airborne gravity errors due to the errors in GPS-derived kinematic accelerations. As can be seen from Figure 3-12, errors of the GPS-derived acceleration are very large at the high frequency end- This is because the carrier phase measurement noise is in generd in the high-frequency band and is amplified after double differentiating. For more discussions on GPS measurement noise and their effect on the derived kinematic acceleration, see, for instance, Kleusberg et al. (1990) and Wei et al. (1991). TabIe 3-2 PSD Model of GPS-derived Acceleration Errors Figure 3-12 PSD of the GPS induced gravity error

70 Chapter 3: Airborne Gravity Systems Effect of the Combined GPSmYS Errors on Airborne Gravimetry Since the estimated gravitational accelerations are the differences between INS specific forces and GPS-derived accelerations, which are mutually uncorrelated, the power spectral density of the gravity error should be the sum of the INS and GPS contributions, i.e., Figure 3-13 shows the plot of the PSDs of the propagated gravity errors for both the horizontal component and the vertical component. The parabolic curve in this figure indicates that the resolution of both the scalar and the vector airborne gravimetry systems is limited due to the large errors in the high frequency band, while maximum wavelength is limited due to large errors in the low frequencies for the horizontal component. Therefore, with current navigation grade strapdown inertial systems, it is not practical to measure the horizontal components of the gravity components with a meaningfid accuracy, say for example 2-3 rngal. Studies in the following sections will focus on scalar gravimetry only. Figure 3-13 PSD of airborne gravity error

71 Chapter 3: Airborne Gravity Systems Effect of Other Critical Factors on Airborne Gravimetry In addition to the actuai sensor errors, vibration noise and aircraft dynamics wiu have to be considered. If the lns and GYS sensors were in the same location on the aircraft and accurately synchronized, most of these errors would be eliminated by the differencing process. Since this is not the case, their difference will usually not tend to zero. The gravity error induced by time synchronization errors between the two systems is described by the last term on the right-hand-side of equation (3-16), which clearly indicates that this kind of gravity error depends on dt, the magnitude of the time synchronization error between the GPS and the INS systems, and aircraft dynamics f, the change of the kinematic acceleration with respect to time- Figure 3-14 shows the time derivative of the kinematic acceleration of the aircraft during an airborne gravity test performed on June lst, 1995, and Figure 3-15 shows its amplitude spectrum with respect to different fkequencies. The large magnitude of the amplitude spectrum in the high frequency band indicates that the time synchronization between the GPS and INS systems is a critical issue in airborne gravity systems, and it could be one of the factors prohibiting recovery of the gravity information in the high frequency band. Applying a low-pass filter with different cut-off frequencies to the data shown in Figure 3-14, Table 3-3 tabulates the standard deviations of the gravity errors for the case of 1 ms time synchronization error. The table indicates that, to keep the gravity error induced by a time synchronization error below 1 rnga.1, time in GPS and INS system should be well synchronized with an accuracy of better than 1 ms if a low-pass filter with cut-off frequency of 0.03 Hz is desired.

72 Chapter 3: Airborne Gravity Systems 57 Figure 3-14 Time derivative of aircraft kinematic acceleration frequency (Hz) Figure 3-15 Amplitude spectrum of time derivative of aircraft acceleration I Table 3-3 The standard deviation of the time derivative of aircraft kinematic acceleration with different low-pass cut-off frequencies The above discussion indicates that aircraft dynamics may have very serious effects on the

73 Chapter 3: Airborne Gravity Systems 58 performance of an airborne gravity system. The effect of air turbulence on aircraft motion is usually divided into longitudinal dynamics and lateral dynamics (Etkin, 1982; McRuer et al., 1973). For airborne gravimetry, especially for a scalar gravity system, the longitudinal dynamics is the main concern due to the motion in the vertical direction. The longitudinal motion is a hction of aircraft-specific parameters such as wing load, velocity, lift slope, etc. and statistical turbulence parameters like variance and correlation length, and consists of two oscillatory modes. One of these is a relatively well-damped oscillation called the "short period", and the other is a lightly damped oscillation called the ccphugoid". Their frequencies are about 0.2 Hz and Hz, respectively, for a stable transport airplane over a wide range of conditions (Etkin, 1982). Therefore, the phugoid motion has high potential effect on airborne gravimetry due to its frequency characteristics. From an operational point of view, the aircraft dynamics can be minimized with higher flight altitude and higher flight speed during stable atmospheric conditions. On the other hand, however, as discussed in the following chapters, very high flight altitudes and large flight speeds are not preferred because the gravity information is damped with altitude and the resolvable wavelength is inversely proportional to the aircraft speed. Therefore, it is important to choose the optimal operational parameters in practical applications. 3.6 Potential Geoid Resolution of Airborne Gravimetry The spectral properties of the geoidal signal at flight level can be investigated by continuing the geoid undulations upward into space. The PSD of geoid undulations in space at altitude h can be estimated as

74 Chapter 3: Airborne Gravity Systems 59 sn,o = SJVe -4xWX 3 (3-2 9) where the exponent of the e function stems from the kernel of the upward continuation integral (Poisson equation). Figures 3-16 and 3-17 show the rms spectrum of the geoid undulations versus the geoidd errors obtained by propagating airborne gravity errors with the formula where SN(AC) stands for the power spectral density of geoidal errors and Sk(AG) is the power spectral density described in Chapter 2. The rms spectrum of geoid undulations has been computed in space at heights of 0.0,0.5, 2.5, and 5.0 km, respectively. An aircraft velocity of v = 300 krn/h has been assumed. Table 3-4 summarizes the minimum wavelengths above which the rms values of the total errors are smaller than those of the geoid undulations. Although the maximum recoverable wavelength is not visible in Figures 3-16 and 3-17, the trend of the curves suggests that the geoid signal might be recoverable for profile lengths of more than 1000 kilometers, albeit with diminishing accuracy. Table 3-4 Recoverable wavelength in airborne gravimetry Area h = 0.5 km h = 2.5 krn h = 5.0 km Flat area >5km > 1Okm > 14km Mount area >4km >6km >ll km

75 Chapter 3: Airborne Gravity Systems 60 Aircraft velocity and flight altitude aftect the wavelength ranges within which gravity information can be detected. In general, lower velocity will allow the resolution of shorter wavelengths while higher velocity will extend the range of long wavelengths that can be resolved with a certain accuracy. This, however, does not mean that a low aircraft velocity will always benefit gravity field determination in airborne gravimetry (for a counterexample, see Schwarz et al. 1994). z E 1 E g 0.5 u e g wavelength in km Figure 3-16 The rms spectrum of geoid undulation in mountain areas and airborne gravity errors (v = 300 km/h) I * wavelength in km Figure 3-17 The rms spi~trum of geoid undulation in flat areas and airborne gravity errors (v = 3 00 krn/h)

76 Chapter 3: Airborne Gravity Systems 61 The effect of aircraft altitude can also be clearly seen fiom Figures 3-15 and The power of the gravity field attenuates steadily with altitude, especidly for the shortwavelength part of the spectrum. If the measurement noise is constant, this means that the signal-to-noise ratio is smaller at higher altitude, or that lower-altitude flights will in general give a better signal-to-noise ratio, especially for short wavelengths. Whether measurement noise is independent of altitude is a question that has to be firrther studied. Different atmospheric conditions influence the flight dynamics aad thus the errors at different altitudes. Table 3-4 shows that to recover geoid information with a minimum wavelength of about 10 krn in flat areas and 6 km in mountainous areas, a flight height of up to 2.5 km is a good compromise. To detect shorter wavelength information, the flight height must be reduced. Thus airborne gravimetry has the potential to provide the idormation required for centimeter-level geoid determination in both flat and mountainous areas. 3.7 Potential Geoid Accuracy From Airborne Gravimetry Figure 3-18 shows the cumulative rms error spectrum of geoid unduiations estimated fiom an airborne gravity system within different spectral windows. The minimum wavelengths are 5, 10, and 15 km, respectively, and the assumed aircraft speed is 300 kmk. This rms cumulative spectrum represents the achievable accuracy of the geoid undulations within this spectral range determined fiom airborne gravity, i.e., the low-frequency error of the geopotential model has not been modelled. As can be seen &om Figure 3-18, the use of different minimum wavelengths, fiom 5 h to 15 krn has only a small effect on the shoa wavelengths of the geoid undulations. This effect

77 Chapter 3: Airborne Gravity Systems 62 is almost negligible ifthe desired accuracy level is I cm or higher. Figure also indicates that, for wavelengths of up to 100 h, the cumulative geoid undulation error is 1 crn or less; for wavelengths up to 500 km, it is 10 cm or less; and for wavelengths up to a 1000 km, it is 30 crn or less. Correspondingly, for relative geoid determination, the accuracy is about 0.1,0.2 and 0.3 ppm for wavelengths of up to 100,500, and 1000 km, respectively. This is accurate enough to meet most geodetic requirements. wavelength in km - Figure 3-18 The cumulative rms spectrum of geoid undulation errors propagated fiom airborne gravity errors On this basis, it can be said that airborne gravimetry can contribute significantly to precise geoid determination. By recovering the gravity signal with minimum wavelengths of 5 to 10 km, the local geoid can be determined with centimeter-level accuracy in those areas where the medium and long wavelength information of the gravity field is well represented by the geopotential model. This accuracy will be available globally once data fiom the two proposed dedicated gravity satellites (GRACE and GOCE) became available (ESA, 1999). It is expected that they will resolve the global gravity field to wavelengths of 100 to 200 km.

78 Chapter 3: Airborne Gravity Systems 63 By combining satellite and airborne gravity data, centimeter-accuracy will be achievable globally. In areas such as fica and Asia, where currently the gravity field is often not well represented by the geopotential model, airborne gravimetry can be used to determine a relative geoid with an accuracy of 10 to 30 cm (rms), using a prome spacing of 15 to 25 km. In addition, current efforts for the establishment of a precise global height system are also enhanced by these techniques when dl methods are combined. The above analysis does not allow statements on optimal operational procedures. The required minimum wavelength to which the gravity signal must be resolved has to be available in both dong and across flight track directions. Statements on flight track spacing are made using the Nyquist theorem. Thus, to recover signal wavelengths of 10 km, for example, a flight track spacing of 5 km is required and the cut-off frequency of the low-pass filter should not be lower than 1/120 Hz when the flight speed is about 300 km/h. 3.8 Airborne Gravirnetry Flight Test Description In 1995 and 1996, the University of CaIgary conducted two series of airborne gravity tests in the Canadian Rocky Mountains. The digital topographic model, as shown in Figure 3-19, varies between 300 m and 3573 m with a mean value of rn and a standard deviation of 498 m. The purpose of the tests was to verify the theoreticd developments that have been conducted during the last decade within the universiw and to evaluate the potential contribution of SINSIDGPS airborne gravimetry to geoid determination. The area was chosen due to the high variability of the gravity field, and the availability of relatively dense ground gravity measurements as reference.

79 Chapter 3: Airborne Gravity Systems 64 Longitude Figure 3-19 Digital topographic model in the Canadian Rocky Mountains The June 1995 test The first test was conducted in June I, 1995 in the Canadian Rocky Mountains with an eastwest profile of 250 km. The prototype of the airborne gravity system consisted of a Honeywell LASEREF III strapdown system, which is a high performance strapdown INS with GG1342 ring laser gyros and QA2000 accelerometers, two GPS receivers with zero baseline on the airplane and 5 GPS receivers on the ground. During the test, the GPS receivers collected L1 data only. Figure 3-20 shows the flight trajectory. Figure 3-21 shows the flight altitude and the topographic profile under the trajectory. The flying altitude is about 5.5 km, i.e., about 2.5 km to 5.0 km above ground, and the average flying speed was about 430 kmh. This corresponds to a spatial resolution (half wavelength of cut-off frequency) of 5 km to 7 krn when using filter lengths between 90 and 120 seconds.

80 Chapter 3: Airborne Gravity Systems 65 The gravity field along the flight trajectory is very rough with an amplitude range of up to 150 rngal. Data collected during this flight test will be mainly used in chapter 4 for evaluating the contribution of topographic gravitation in processing airborne gravity data. For more detailed discussion about this test and the accuracy evaluation of the gravity disturbances determined fiom this flight test, see Wei and Schwarz (1996). Figure 3-20 Flight trajectory of the June 1995 airborne gravity test Figure 3-21 Flight altitude and topographic profile

81 Chapter 3 : Airborne Gravity Systems The September 1996 test On September 9,10 and 11 of 1996, three airborne gravity flight tests were conducted by the University of Calgary over the Canadian Rocky Mountains. The prototype of the airborne gravity system, as designed by the University of CaIgary, consisted of two different navigation grade strapdown systems, the Honeywell LASEREF III and the Litton 101 Flagfhip, two GPS receivers (an Ashtech Z 12 and a Trimble 4000SSI) formed a zero baseline on the airplane, and four GPS receivers on the ground stations (a NovAtel GPSCard at Calgary Airport, an Ashtech 212 and a Trimble 4000SSI at Banff, a Trimble 4000SSE at Invermere). The fist 1) was flown at an average height of 4350 m in east-west direction, the second (Day 2) was at the same height but in north-south direction, and the last (Day 3) was at an average height of 7300 m in east-west direction. The frst t o days of testing were designed to examine the effect of flight direction on the determination of gravity disturbances. The third day flight at a higher altitude was to explore the effect of flight height on the accuracy of disturbance determination. All ff ights covered the same 100 km by 100 km area. The height of the terrain in this area varies from 800 m to 3600 rn. All Longitude Figure 3-22 Flight trajectory, Day1 and Day3

82 Chapter 3: Airborne Gravity Systems 67 flights were done at night between 12 am and 6 am local time to minimize the effect of air turbulence. Average flight velocity for all three tests was 360 km/h. Figure 3-22 shows the flight trajectory of the Day 1 and the Day 3 tests and Figure 3-23 shows the flight trajectory of the Day 2 test Longitude Figure 3-23 Flight trajectory, Day2 Figure 3-24: Airborne gravity disturbances (Day 1, 4350 m) (min: max: mean: 89.7 ms: 99.1 std: 42.1 rngal) Figures 3-24,3-25 and 3-26 show the gravity disturbances determined fiom these three flight

83 Chapter 3: Airborne Gravity Systems 68 tests. The geographic coordinates corresponding to X = 0 and Y = 0 in these figures are latitude = 50.4O, longitude = 243.6O. These data will be used in chapter 5 for conducting gravity field upward and downward continuation and in chapter 6 for gravimetric geoid determination. For detailed analysis of the flight tests and data processing, see Glennie (1999). Figure 3-25 Airborne gravity disturbances (Day 2, 4350 m) (min: max: mean: 91.0 rms: 99.9 std: 41.1 mgal) Figure 3-26 Airborne gravity disturbances pay 3,7300 m) (min: 38.8 min: mean: rms: std: 35.3 mgal)

84 Chapter 3: Airborne Graviv Systems Summary of ResuIts In this chapter, the principle of airborne gravity systems consisting of a strapdown INS and DGPS systems has been reviewed and the effect of the system errors on gravity determination has been analyzed. In addition, the potential contribution of such airborne gravity systems to relative geoid determination has been studied. Based on the spectral characteristics of the gravity field and the errors in airborne gravity, spectral studies indicate that geoid undulations can be determined with a minimum wavelength of I0 krn in flat areas and 6 krn in mountainous areas when using an aircraft speed of about 300 krn/h and a flight altitude of 2.5 km or lower. Shorter wavelength information will become detectable ifthe resolution of the GPS-derived acceleration can be improved. The two main contributions of airborne gravimetry to geoid determination will be a detailed high-precision geoid in regions where the long and medium wavelengths are well represented and a global geoid of more uniform accuracy. The spectral analysis of the airborne gravity system errors indicates that accuracies of geoid undulations at the centimeter-level can be expected for the wavelength range between 5 and 100 km; 10 crn are possible in the range between 5 and 500 kaq and for wavelengths of up to 1000 km, the rms error will usually not be larger than 30 cm.

85 Chapter 4 Unified Terrain Corrections and Their Effect on Airborne Gravimetry In this chapter, a set of unified formulas for the computation of topographic attraction and/or terrain corrections has been developed, with gravity measurements taken either on the Earth's surface or in space. Detailed computational fonnulas can be found in Appendixes A and 5. The density of the topography can be either constant or varying horizontally. The digital topographic model can be either a mass-line or a mass-prism representation. Numerical examples computed in the Austrian Alps will be used to indicate the effect of varying densities on the computation of terrain corrections- Numerical examples computed in the Canadian Rocky Mountains will be used to show the effect of topographic attraction on the processing of airborne gravity measurements and airborne gravity disturbances. Formulas developed in this chapter will be used in Chapter 6 for the computation of topographic attractions to airborne gravity data before downward continuation. 4.1 Terrain Correction for Vector Gravity measurements To provide a gravity reference at flight level or to downward continue the airborne disturbances to ground level, terrain corrections or topographic gravitational attraction are

86 Chapter 4: Terrain Correction and Airborne Gravimetry 71 needed to model the high frequency components of the gravity field (Forsberg and Kenyon, 1995; Klingele and Halliday, 1995). The three components of the topographic gravitational attraction can be expressed as (Heiskanen and Moritz, 1967) where the trinal integration operator is defined as and E represents the integration area, G is the gravitational constant, (x, y, q) represent the coordinates of the computation point; r@, x, y, z) represents the distance between points (x,, yp, z,,) and (x, y, z); p, and h, denote the topographic density and the topographic height at the variable integration point (x, y), respectively, and h, represents the height of the reference surface. With different selections of h, and z,,, the above equations can be used for one of the following four cases: 1) h, = 0 and 5= hp7 topographic gravitational attraction at the Earth's surface, ta(x,, y, hd; 2) h, = 0 and %= 4, topographic gravitational attraction at a surface with constant elevation hc, ta(xp, Y,, h3; 3) h, = h, and = h,, terrain correction at the Earth's surface, tc(x,, y, hp); 4) h, = h, and % = 4, terrain correction at a surface with constant elevation b, tc(x,, y, hj

87 Chapter 4: Terrain Correction and Airborne Gravimetry 72 The topographic attraction is usually divided into two parts: one is the Bouguer correction which represents the gravitational attraction of a mass plate with thickness equa.1 to the topographic height at the computation point, and the other is the terrain correction which is the gravitational attraction of the residual terrain with respect to the Bouguer plate as illustrated in Figure 2-2 (Heiskanen and Moritz, 1967). If constant density is assumed, the Bouguer plate is usually taken as a circular plate of radius R with centre at the computation point. Consequently, the horizontal components of the gravitational attraction of the Bouguer plate are equal to zero and the vertical component can be approximated as which can be further simplified to for a Bouguer plate with infinite radius or a radius much larger than the topographic height h, and the elevation of the computation point q. With horizontally variable densities, however, all three components of the gravitation of the Bouger plate have to be computed by carrying out the integrations explicitly. For a gridded digital topographic model (DTM), the double integration over the -plane in equation (4-1) can be written as double summations,

88 Chapter 4: Terrain Correction and Airborne Gravimetry 73 where Ax and Ay represent the grid spacing, N and M represent the number of grid nodes in the X- and Y-direction, respectively. For different assumptions on the topographic model, the above equations can be fuaher modified. Appendix A contains further formulas for a mass-line topographic model and a mass-prism topographic model. For the effect of these two topographic representations on terrain corrections, see Li and Sideris (1994). Assuming that the topographic height change is smaller than the horizontal distance between the data and the computation point, the kernel functions can be expanded into a Taylor series involving powers of the height, and consequently, terrain correction formulas can be expressed as s = x, y, 2. (4-5) Realizing the fact that each term in the above series represents a two-dimensional linear convolution and assuming that the location of the computation points coincide with the grid nodes, the Fourier transform can then be used to perform the computations,

89 Chapter 4: Terrain Correction and Airborne Gravimetry c34kr1(h K*} - 4 5~1p %} - c36~1(h - c37r1(h -??$}' (4-6a) where H ' = F {~ h '1 For the mass-line topographic model, the kernel function k-'(x, y, - w) and ksi(x, Y, w) have the following expressions:

90 Chapter 4: Terrain Correction and Airborne Gravimetry 75 (2i - I)!! = 1 when i = 0 and (2i-I)!! = (2i-l)x(2i-3)x-*x3xl when i > 0. The coefficients cij and dij as well as the kernel functions kei(x, - y, w) and k:(x, y, w) for the mass-prism topographic model are provided in Appendix B. The parameters a and P are used to improve the convergence of the series (Li and Sideris, 1994) and have the following values: - 1) for topographic attraction at ground level (h, = 0, = hp), a = h and p = a, ; 2) for topographic attraction in space (h, = 0, z, = U,a = h and P = h -h; 3) for terrain correction at ground level (h, = hp, 7, = h,), a = 0 and p = oh ; and 4) for terrain correction in space (h, = h, z, = b), a = h c -h and p = h -h, C where q, is the standard deviation of the topographic heights. Appendix A contains the expressions of the kernel function kqf(x, y, w) and k '(x, y, w) for the mass-line topographic model. 4.2 The Effect of Density on Terrain Correction Numericd examples of the effect of topographic density on terrain corrections on the ground level can be found, for example, in Forsberg (1 984) and Bhaskara et al. (1 993). Results presented in this section will show the effect of using horizontally varying density for the

91 Chapter 4: Terrain Correction and Airborne Gravimetry 76 computation of the topographic correction in space, for example, at the flight level of an airborne gravity survey. A set of topograp hic data and a density model in Austria were used. The area has an extension of about 178 km in the north-south direction with a grid spacing of 348 m and about 170 lon in the west-east direction with a grid spacing of 395 m. Figure 4-1 and 4-2 show the digital topographic and the density model, respectively. Topographic gravitational attraction was computed in space on a plane with an altitude of 3 km. Figure 4-3 shows the vertical component of the topographic gravitational attraction in space with constant density (2.67g/cm3) and Figure 4-4 shows the differences between using the constant density and using the density model. Figure 4-5 shows the power spectral density of the topographic attraction differences and Figure 4-6 shows the cumulative rrns spectrum computed with the formula longitude Figure 4-1 DTM in the Austrian Alps (max: 2350, min: 114, mean: 552, a: 358m)

92 Chapter 4: Terrain Correction and Airborne Gravimetry 77 Longitude Figure 4-2 Density model in the Austrian Alps (max: 2.85, min: 2.00, mean: 2.48 g/cm3) ls Iongitnde Figure 4-3 Topographic attraction in space & = 3km) with constant density (unit: mgal)

93 Chapter 4: Terrain Correction and Airborne Gravimetry longitude Figure 4-4 Topographic attraction differences when using constant vs varying densities (unit: mgal) Figure 4-5 PSD of the topographic attraction differences when using constant vs varying densities Figure 4 6 Cumulative rms spectrum of the topographic attraction differences due to using constant vs varying densities

94 Chapter 4: Terrain Correction and Airborne Gravimetry 79 In Table 4-1, the statistical information of the topographic gravitational attraction is tabulated for constant density, as well as the differences between using the constant density model and the horizontally varying density model. The initials UD, WE and NS in TabIe 4-1 stand for the up-down, the west-east and the north-south component, respectively. Table 4-1 Topographic gravitational attraction in space and the differences when using a horizontally varying density model instead of a constant one (h, = 3 krn, unit in mga.1) Ts UD WE NS the computed values max min mean c the dii3erences max min mean Table 4-1 indicates that, in the computation area, the use of a horizontally varying density model results in a difference of 2.6 mgd (1 o) in the computed topographic attraction. As shown in Figure 4-4, the main differences occur in the South-West corner where the topography varies strongly. Here the terrain corrections change the most. Figure 4-6 indicates that the cumulative effect is smaller than 0.4 mgal (rms) for wavelengths below 10 km and is about 2.5 mgal (rms) for wavelengths below 100 km. Neglecting this effect will contribute no more than 0.1 cm (rms) to geoid determination and consequently can be tolerated. It should be noted, however, that the values in Figure 4-6 are somewhat attenuated because of the flight height of 3 km. For an altitude of a few hundred metres, the high-frequency spectrum will be considerable stronger. It is therefore safer to eliminate the effect of the known density variations before the analysis of the airborne data

95 Chapter 4: Terrain Correction and Airborne Gravimetry Topographic Effect on Processing Airborne Gravity Measurements This section will use the data collected on June 1, I995 airborne test to investigate the effect of removing the topographic attraction when processing airborne gravity data. A digital topographic model sampled on a 960 by 600 km grid under the flight area was used to compute topographic gravitation at two constant leveis in space at altitudes of 5 km and 6 km, respectively. A constant crust density of 2.67 g/cm3 is used in the computation. The grid spacing is 0.93 km in a N-S direction and 1.15 krn in a W-E direction. The digital topographic model and the statistics of the heights were shown in Figure Figure 4-7 shows the vertical component of the topographic attraction at a height of 5 km. Table 4-2 summarizes the statistical information of the topographic gravitational attraction and the differences of the topographic attraction at altitudes 5 krn and 6 km Iongitnde Figure 4-7 The vertical component of topographic attraction in space 01, = 5km) in the Canadian Rockies

96 Chapter 4: Terrain Correction and Airborne Gravimetry 81 Table 4-2 indicates that in terms of the mean difference, the vertical component of the topographic attraction changes by oniy 1 mgal for an altitude difference of I km at 5 km height, and the maximum difference is about 10 mgal. Considering the fact that, during airborne gravity surveys, the trajectory of the aircraft can usually be maintained within a 10- metre level in vertical direction, the change of the topographic attraction due to the variation of the aircraft flying altitude is negligible. In other words, the topo,pphic gravitation can be computed at the mean flight level by using the fast Fourier transform. Table 4-2 Statistics of the differences of the topographic attraction at altitudes of 5 km and 6 km (unit: mga.1) Ts U-D W-E N-S topographic gravitation (h, = 5km) max rnin mean cr differences ( h, = 5km vs = 6km) max min mean ir Figure 4-8 shows the topographic attraction along the flight trajectory and the airborne gravity disturbances. The latter has not been corrected for topographic effects. The correlation between the topographic attraction and the gravity disturbance shown in Figure 4-8 indicates that airbome gravity measurements are strongly affected by the local topography. Consequently, when downward continuing the airborne gravity disturbances to a reference level, such as mean sea level, it will be advantageous to remove the topographic gravitationd attraction fiom the airborne gravity disturbances before performing the downward continuation, and then to restore it at the reference level afterwards. This will prevent the amplification of high frequency signal components in the downward continuation.

97 Chapter 4: Terrain Correction and Airborne Gravimetry Longitude Figure 4-8 Topographic attraction at night level and airborne gravity disturbance of the June 1995 test Data collected during the 1995 airborne test was processed in two ways. First, no topographic information was used and second, the topographic effect was taken care of by using the so-called remove-restore technique. The topographic attraction is first removed fiom the INS measured specific force, then the differences between the INS specific force and the GPS kinematic acceleration are processed with a low-pass filter with a desired cutoff frequency. Finally, the topographic effect is added back to the low-pass filtered values. Figure 4-9 depicts the differences of the gravity disturbances at flight level computed with and without the use of the remove-restore method. The dotted line represents the differences before filtering and the solid line represents the resuits after a low-pass filter with cut-off frequency of Hz has been applied. The corresponding cut-off wavelength is about 10 km. The maximum difference for the unfiltered data is about 2.5 mga1 with a standard deviation (std) of about 1 mgal. After applying the low-pass filter, the standard deviation of the differences is reduced to 0.14 mgal, which is far less than the accuracies required for geoid determination and tectonics. Therefore, it is not necessary to pay special attention to the topographic effect when processing airborne gravimetry measurements for these applications.

98 Chapter 4: Terrain Correction and Airborne Gravimetry 83 Figure 4-9 Gravity difference at flight level due to the remove-restore method for the topographic effect 4.4 Topographic Effect on Processing Airborne Gravity Disturbances The objective of this section is to investigate if the topographic attraction has to be removed when processing the airborne gravity disturbances. Data used in this section are those obtained from the September 1996 airborne tests. After applying a 90s low-pass filter, the gravity disturbances have a minimum wavelength resolution of 10 km along the flight trajectory and 20 km across the flight trajectory. In other words, for Dayl, the minimum waveiength resolution is 10 krn in West-East direction and 20 km in North-South direction. For day2, the minimum wavelength resolution is 10 krn in North-South direction and 20 km in West-East direction. Gravity disturbances of Day 1 and Day2 were first gridded on 10 by 10 km grids, respectively. These grid nodes coincide with the 11 by 11 crossover points of Dayl and Day2 flight trajectories. Figure 4-10 shows the differences of the two gridded gravity disturbances on the 121 points. The standard deviation of the differences is 2.1 mgd. The gravity disturbances of Dayl and Day2 are then gridded on 51 by 51 points with a grid spacing of 2 by 2 km. Their differences are shown in Figure These differences have

99 Chapter 4: Terrain Correction and Airborne Gravimetry 84 a peak value of about 15 mgal and standard deviation of 3.8 mgal which are much bigger than those on the crossover points. This can be expected because of the different flight patterns and different resolutions in N-S and W-E directions. It is clear that the magnitude of the differences depends on the characteristics of the gravity field. The rougher the gravity field is, the bigger the difference will be. In other words, these differences can be reduced if the gravity field can be smoothed out, which brings in the procedure of topographic correction. Figure 4-10 The differences between Dayl and Day2 gravity disturbances in space, on 1 1 x 1 1 crossover points (min: -5.4 max: 5.2 std: 2.1 mgd) Figure 4-12 shows the topographic attraction at flight level. Comparing Figure 12 with Figure 1 I, we can see that the differences of the gravity disturbances are highly correlated with the topographic attraction. After removing the topographic attraction from the gravity disturbances of Dayl and Day2 respectively, the remaining residuals are gridded on 2 by 2 km grids again. Figure 4-13 shows the differences of the gridded Dayl and Day2 remaining residuals. These differences have a peak value of about 9 mgal and a standard deviation of

100 Chapter 4: Terrain Correction and Airborne Gravimetry mgai which is about 1 mgal smaller than those when the topographic effect is not removed before gridding. This procedure, as shown in Chapter 6, has significant effect when downward continuing the gravity disturbances fiom flight level to ground level. Figure The differences between Day 1 and Day2 gravity disturbances in space, on a 5 1 x 5 1 grid, grid spacing: 2 km (min: max: 10.4 std: 3-8 meal) Figure 4-12 Topographic attraction in space (h = 4350) (min: max: std: 30.8 mgal)

101 Chapter 4: Terrain Correction and Airborne Gravimetry 86 Figure 4-13 The differences between Day1 and Day2 gravity disturbances in space, on a 5 1 x 5 1 grid, grid spacing: 2 krn topographic effects were removed before gridding (min: -8.1 max: 9.3 std: 2.9 mgal) 4.5 Summary of Results Formulas developed in this chapter can be used to compute the topographic attraction or the terrain corrections for vector gravity measurements. The topography can be modeled by either the mass-line or the mass-prism presentation. Horizontally variable densities digitized on the same grid of the DTM can be used in the computation. The computation can be done either at the Earth's surface or in space at a constant altitude. With the data set used, the effect of a horizontally varying density model on the topographic gravitational attraction is about 2.6 mgai (1 a) in the vertical component. Considering the power spectral distribution and the accuracy and wavelength resolution required in different applications, the density information wili not affect the results of geoid determination.

102 Chapter 4: Terrain Correction and Airborne Gravimetry 87 Due to the high correlation between topographic attraction and gravity disturbance in space, rernove-restore of the topographic effect will be advantageous when downward continuing the airborne gravity data to sea level. It is not necessary, however, to remove and restore the topographic effect when processing airborne gravity measurements for geoid determination.

103 88 Chapter 5 Determination of the Reference Geoid To provide a reference for the geoid undulations determined &om airborne gravity, gravimetric geoid undulations were computed in an area bounded by latitudes 36" and 66" and longitudes 229" and 259". The gravity anomalies in this area, as shown in Figure 5-1, were available on a grid with a grid spacing of 5' in both directions. The white color in Figure 5-1 represents areas with no gravity measurements. The global geopotential model EGM96 was used in the remove-restore fashion to represent the long wavelength signal in the geoid determination. Some statistical information on the gravity anomalies field is given in Table 5.1. It shows that the gravity field in this area is extremely rough, but that the mean Longitude mgal Figure 5-1 Gravity anomaly in the Canadian Rocky Mountains

104 Chapter 5: Determination of the Reference Geoid 89 values for alz components are close to zero. The effect of merent methods and different data coverage on geoid determination were studied. Investigations were also made on the contribution of the kemel modification on the long wavelength geoid errors when the truncated Stokes or Hotine integral was used for local geoid determination. Gravimetric geoid undulations and their accuracy determined in this chapter will be used in Chapter 7 in evaluating the geoid determined fiom airborne gravity disturbances. Table 5-1 Statistics of gravity anomalies in mgal max min mean I-n-ls cr measured EGM96 ; residual Geoid Determined from the Stokes Integral Effect of the Stokes kernel modification Geoid undulations, as shown in Figure 5-2, were computed using the Stokes integral with both the original spherical kemel function, equation (2-3), and its modified version, equation (2-17b). The maximum degree of the Stokes series removed from the kernel function was 20. Figure 5-3 shows the differences between the geoid undulations computed with the original and with the modified Stokes kemel function. The relatively smooth pattern in Figure 5-3 implies that the modification of the Stokes kemel mainly affects the long

105 Chapter 5: Determination of the Reference Geoid 90 wavelength part, i.e., that errors in the EGM96 model are reduced by using ground gravity data. Table 5-2 summarizes the statistics of the geoid undulations and the differences due to the use of the original and the modified kernel function. As shown in Table 5-2, the geoid undulations have a mean value of 20 m which are almost the same with both the original and the modified Stokes kernel function. Table 5-2 Statistics of the gravimetric geoid undulations in metre Stokes kernel hction original, eq. 2-3 modified, eq. 2-17b original - modified max min mean mls o Longitude Figure 5-2 Geoid in the Canadian Rocky Mountains

106 Chapter 5: Determination of the Reference Geoid 91 Table 5-2 indicates that the modification of the Stokes kernel function changed the computed geoid undulations by 0-43 m in terms of the standard deviation and about 2m in terms of the range (max- min). For the geoid determination with accuracy expectation of 10 crn or better, this difference is not negligible. m Figure 5-3 Geoid undulation differences due to the modification of the Stokes kernel fimction

107 Cha~ter 5: Determination of the Reference Geoid Comparison with GPS/Levelling-derived geoid undulations To evaluate if the kernel modification improves the results, the gravimetric geoid undulations were compared with those derived from GPSAevelkg on 209 benchmarks available in this area. These benchmarks are mostly in the valleys and their geographic distribution is shown in Figure 5-4. The statistics of the differences between the gravimetric geoid undulations and those derived fiom GPSAevelling are summarized in Table 5-3. Due to factors such as the height system datum shift and long wavelength errors, there exist systematic differences between the geoid undulations derived by the Stokes integral and by GPSAevelling. This kind of systematic differences can be partially modelled and removed by fitting a plane through the geoid differences using the formula (Forsberg and Madsen, 1990) Table 5-3 Statistics of the differences between the gravimetric geoid undulations and those derived from GPS/levelling, unit in metre area kernel function before fit max min mean RMS 0 max min after fit an tilt NA original modified CA original modified S A original modified

108 Chapter 5: Determination of the Reference Geoid 93 The plane is characterized by two parameters: &uth and tilt. Admuth is the direction of the steepest gradient of the plane or the steepest ascending line in the plane. Tilt is the relative gradient of the steepest ascending line and is represented in ppm (part per million). The last two columns in Table 5-3 contain the azimuth (a) and the tilt (pprn) of the fitted planes. NA, CA and SA in the first column of Table 5-3 represent Northern Aiberta, Central Alberta and Southern Alberta, as used in Figure 5-4. Table 5-3 indicates that the use of the modified Stokes kemel reduces the differences between the computed gravirnetric geoid undulations and those derived from GPSllevelling. On the 106 benchmarks in Southern Alberta, for example, the range of the differences (max - min) is reduced by about 30 % fiom 1.14 m to 0.80 m, the standard deviation is reduced by about 35% fiom 0.23 m to On the 52 benchmarks in Central Alberta, the improvement is more than 60% in terms of both the difference and the standard deviation. Table 5-3 shows that, after fitting a plane to remove the systematic difference between the two independent solutions, geoid undulations computed with both the original Stokes kernel function and the modified Stokes kemel function and those derived from the GPSAevelling agree at the same level of accuracy. However, the use of the modified Stokes kemel function results in a smaller tilt for the fitted plane. The biggest improvement is in Central Alberta. The relative gradient of the steepest ascending line of the fitted plane is reduced by about 90% from 1.29 ppm to 0.16 ppm when the modified Stokes kernel is used. This suggests that the use of the modified Stokes kernel function can partially eliminate the long wavelength errors in the computed gravirnetric geoid undulations. The overall agreement between the gravime~c geoid undulation and those derived fiom GPSAevelling is at the level of 10 cm when the modified Stokes kernel function is used.

109 Chapter 5: Determination of the Reference Geoid 94 After fitting a plane, the agreement is at the level of 5 cm regardless whether the Stokes kernel function is modified or not. The plane fit could only be avoided if a better global geopotentid model can be developed such that the 1-cm accuracy of global geoid with a wavelength resolution of 100 km can be 1999). 5.2 Geoid Determined from the Hotine Integral Gravity disturbances in the test area were obtained fiom the gravity anomalies and the geoid undulations computed in the above section. Geoid undulations were computed with the Hotine integral on the same 5' x 5' grid nodes and compared with those computed with the Stokes integral. Figure 5-5 shows the differences and Table 5-4 summarizes the statistics of the differences. As expected from theory, geoid undulations computed with Hotine's integral agree very well with those obtained by the Stokes integral. Both the mean and Longitude Figure 5-5 Geoid undulation differences Stokes integral vs Hotine integral

110 Chapter 5: Determination of the Reference Geoid 95 standard deviation values of the differences are 0 cm. The maximum value of the differences is 2 crn. The differences are mainly locate at the south-west comer where, as shown in Figure 5- I, no gravity measurements are available. Table 5-4 Statistics of the differences between Stokes and Hotine geoid undulations, in metre max min mean RMS G Height Anomalies Determined from the Molodensky Series As stated in Chapter 1 that one of the advantages of airborne gravimetry is the capability of providing gravity data in some areas, such as very rugged rocky mountains, where are logistically difficult or even impossible with the conventional terrestrial gravity survey. With certain airborne gravity systems, such as the STAR-31 system (Wei et al., 1998), a high accurate detailed digital terrain model can be obtained at the same time as conducting airborne gravity survey. It is practically useful to investigate how a detailed digital topographic model can contribute to geoid determination. A 5O by 10" area bounded in latitude between 49O and 54O and longitude between 236" and 246" was chosen for the computation of height anomalies. A grid of 600 by 600 digital topographic heights with a grid spacing of 30" (lkm) in the North-South direction and 60" (I km) in the West-East direction was available in this area. The 9915 randomly distributed fiee-air gravity anomalies available in this area were also gridded on the 600 by 600 grid nodes. The topographic attraction was removed and restored for the gridding. Note that this resdts in a smoothed gravity model for the area.

111 Chapter 5: Determination of the Reference Geoid 96 To investigate the effect of the grid spacing on the prediction of height anomalies, the 30" by 60" gravity anomalies and topographic heights were averaged into a new set of data with a grid spacing of 5' by 5'. Table 5-5 summarizes the statistics of the data for the different grid spacings. Both the gravity anomalies and the topographic heights become much smoother, i-e., their standard deviations (o) become smaller when the grid spacing is increased fiom 30" by 60" to 5' by 5'. This is expected because the averaging procedure works like a low-pass filter, eliminating the high frequency information in the height data. However, it is more realistic in terms of the gravity data. Table 5-5 Statistics of gravity momalies and topographic heights with different grid spacings data gravity anomalies (mga1) topographic heights (m) grid spacing 30" by 6Q" 5' by 5' 30" by 60" 5' by 5' max min mean RMS G Height anomalies were computed on both the 30" by 60" and 5 * by 5 -grids with the gravity anomalies and the topographic heights. Table 5-6 gives the statistical properties of both the G, terms and the height anomaly &, terms. Table 5-6 shows that only the Go (Go) and <, (GI) terms are significant when the grid spacing is 5 ' x 5 '. When the grid spacing is 30-x 6OW, the <, (Gd and <, (G,) terms are also significant since they contribute up to 14 cm to the height anomalies. For the different grid spacings,

112 Chapter 5: Determination of the Reference Geoid 97 the contributions of Go are, on the average, of the same order of magnitude. The differences between the statistics of the term <, computed fiom the two data sets are very significant. Thus, in mountainous areas, the data used for height anomaly prediction should be on a grid with a grid spacing finer &an 5'. Table 5-6 Statistics of the G, and T, terns for different grid spacings (residual values with respect to the OSU91A geopotential model) grid size quantity Go (mgal) G2 (mga G3 (mgal) To (m) <I (m) Tz (m) r 3 (m) 30" by 60" max min mean rms ' by 5' max rnin mean rms Section 3.2 of Chapter 2 presented the detailed theoretical relation between the geoid undulation and height anomaly. To demonstrate how well the reality reflects the theoretical formulation, Table 5-7 gives the statistics between the contribution of G, to the height anomaly and that of the terrain correction c plus the first term of the topographic indirect effect. The comparison was done on the 600 by 600 grid with spacing of 30 by 60. In the table, t;, = S(G,), t, = S(c) + 6N, and s, represents the right-hand side of equation (2-61). The differences have a millrimurn value of 50 crn and standard deviations of about 10 cm, which account for up to 20% of the contribution of c or GI. These differences imply that the

113 Chapter 5: Determination of the Reference Geoid 98 assumption of a linear relationship between the 6ee-air gravity anomaly and the topographic height is ody approximately true. This assumption should therefore not be made if the desired accuracy is 10 cm or better. Table 5-7 also indicates that the use of relation (2-61) does not reduce the differences between the contributions of GI and c, i-e., equations (2-60) and (2-61) are of comparable accuracy. Both the gravimetric geoid undulations computed from the Stokes integral and the height anomalies computed &om the Molodensky series are compared with the geoid undulations derived from GPSflevelling on 50 benchmarks. Figure 5-6 shows the geographic location of the GPSAevelling benchmarks. The height anomalies were transformed into geoid undulations by the equation (Heiskanen and Moritz, 1967) where? is the mean value of normal gravity along the normal plumb line between the telluroid and the ellipsoid, AgB is the Bouguer anomaly at the computation point, and H is the orthometric height. Table 5-7 Statistics of the differences between the contribution of G, and c in metres quantity rl t I SI GI - t* il - SI t~ - SI max min mean rms G

114 Chapter 5: Determination of the Reference Geoid 99 Figure 5-6 Geographic location of GPSLevelling benchmarks Table 5-8 summarizes the statistics of the differences between the predicted geoid undulations and those derived fiom GPSAevelling. In the table, N, is defined as Table 5-8 Statistics of the differences between the predicted and and the GPSAevelling derived geoid undulations, in metres quantity No -% NI -NGL NZ - NGL N3 'NGL - NGL %tokes max min mean rrns ( The results show that, for the geoid undulations converted from the height anomalies, the standard deviation of the differences as compared to those derived fiom GPSAevelling are

115 Chapter 5: Determination of the Reference Geoid 100 about the same when <,, or Cj are included. The geoid undulations converted Eom Molodensky's solution and those computed from the Stokes integral agree with the GPSfievelling-derived results at the same level in terms of the standard deviation of the differences. The advantage of using the Molodensky series is that it results in smaller RMS differences and a smaller difference range (max - min). This may indicate that it is usefid to make use of height density topographic data even if the available gravity data do not have the same resolution. 5.4 Effect of the Integration Radius on Geoid Determination Theoretically, the integration of the Stokes integral should be carried over the whole surface of the Earth. Practically, the integration is always done over an area of limited size, a procedure which is both unavoidable md reasonable. It is unavoidable because of the Iack of globally distributed gravity measurements. It is reasonable because the contribution of gravity anomalies far away from the computation point is negligible due to the nature of the Stokes kernel function. In addition, the long wavelength information of the gravity field can be recovered from a global gravity geopotential model. In theory, an Earth geopotential model represented by a spherical harmonic series with a maximum degree and order 360 has a spatial resolution of about 0.5 " in both longitudinal and latitudinal directions. This means that a geopotential model contains the low and medium frequency part of the gravity field information and has a theoretical spatial resolution of about 1 O. In other words, the gravity field information for wavelengths above 1 degree, or approximately 100 h, can be recovered from a geopotential model. Integration of Stokes formula with local gravity measurements provides the high and very high fkequency information of the gravity field for

116 Chapter 5: Determination of the Reference Geoid 101 wavelengths below 1 degree. However, a geopotential model represents the optimized global solution and cannot always provide the best approximation for regional gravity modelling even for information with wavelengths longer than 1 degree. Therefore, for regional geoid determination by combining the local gravity measurements with a geopotential model, the region covered by local gravity measurements should be large enough to recover the medium wavelength information and correct the results in the geopotential model which are of considerable size in this frequency band. An obvious question is how large is "large enough" for the size of integration area in order to determine geoid undulations with a desired accuracy. To demonsmte how the integration area size and computation strategy &ect regional geoid determination, trial computations were conducted with two groups of integration combinations and different integration area sizes as illustrated in Figure 5-7. In group 1,5' x 5' mean gravity anomalies within areas of sizes of 1 " x 1 "(a), 2" x 2"@) and 5" x 5"(c) were used for Stokes integration. In group 2, the integration area was divided into an outer area and an inner area. Gravity anomalies in Figure 5-7 Illustration of area to which the Stokes integral is applied

117 Chapter 5: Determination of the Reference Geoid 102 the inner area are the 5' x 5' mean values with area size of 0 " x 0 "(d), 1 " x 1 "(e), 2" x 2"Q and 5 " x 5 "(g), respectively. In the outer area with a size of 30" x 30 ", 30' x 30' mean values were used. All areas are symmetric about the central point with latitude 51 " and longitude 244". AU gravity anomalies used in this section are measured gravity anomalies referenced to the geopotentid model EGM96 up to degree and order 360. Geoid undulations in the innermost 1 " x 1 " area were compared with the reference obtained when the inner area concides with the outer area, i-e., when 5' x 5' mean gravity anomalies are available in a 3 0" x 30 " area. Table 5-9 summarizes the statistics of the differences. Table 5-9 indicates that the use of gravity measurements on a finer grid within the inner area is important in geoid determination. As compared with the reference, the use of the 30' x 30' mean gravity anomalies only (case d) results in an undulation error of 0.29 m (lo). This error is much larger than the one for case (a) where 5' x 5' gravity anomalies are used in a small I" x I" area a b c d e f g Table 5-9 Statistics of geoid undulation differences due to the use of different integration area size, unit in metres inner area size max min mean rms c group 1 : no gravity used in outer area lox I" "x 2" Sox 5" group 2: 30' mean gravity anomalies were used in outer area OOx 0" lox 1" 2"x 2" 5"x 5"

118 Chapter 5: Determination of the Reference Geoid 103 The use of the 30' x 30' mean gravity anomalies, however, is very important in recovering the long wavelength information. Cases (a), (b) and (c) produce a systematic geoid error (mean difference) of 0.87m, and 0.29 m, respectively, in the tend 1 " x 1 " area. The systematic error is completely eliminated in cases ( ) and (g) where the 30' x 30' mean gravity anomalies were used in the outer area. Because 30' x 30' mean gravity anomalies are available worldwide, local geoid determination should always be done with a dense grid in the inner area and a coarse grid in the outer area. This is especially beneficial for geoid determination with airborne gravimetry because the flight extension is usually restricteho a small area (lo x lo to 2O x 2"). Table 5-9 implies that, when the 30' x 30' mean gravity anomalies are used in the outer area, the computed geoid undulations differ from the reference by only 0.01 m and 0.06 cm when the inner area is 2" x 2 "(case f) and 1 x 1 " (case e), respectively. Figure 5-8 shows differences between the reference and case (e) on 12 by I2 grid nodes. Figure 5-8 indicates that the relatively big differences are at the boundary of the area The standard deviation of the differences is 0.03m, m and 0-OOm when the comparison is made for the central 10 x 10, 8 x 8 and 4 x 4 grid nodes, respectively. Figure 5-8 Geoid undulation error due to the combination of the integration inner and outer areas (case e)

119 Chapter 5: Determination of the Reference Geoid Summary of Results In this chapter, the effect of using different methods and different input data for geoid determination is investigated. Numerical comparisons indicate that the modification of Stokes' kernei function by removing the low part of the spherical series can improve geoid accuracy. The geoid undulations determined between using the original and the modified Stokes kernei have a standard deviation of 0.43 rn with the test data. The use of the modified Stokes kernel function provides a geoid that agrees better with the one derived &om GPSfleveIling in terms of the rms difference values. The Molodensky series provides a similar results as the Stokes integral but a slightly smaller RMS value for the differences when compared. with GPS/levelling redts. Geoid undulations computed in section 5.1 with the modified Stokes kernel function will be used as reference for those determined from airborne gravity data in chapter 7. Geoid undulations can be determined by integrating the gravity anomalies on a finer grid in the inner area and gravity anomalies on a coarser grid in the outer zone. Because geoid undulations computed &om gravity anomalies with the Stokes integral and those fiom gravity disturbances with the Hotine integral are identical, combinations can also be made such that the inner area has gravity disturbances and outer area has gravity anomalies. Gravirnetric geoid undulations can be computed by summing up those obtained by applying the Hotine integral to the gravity disturbances within the inner area and those obtained by applying Stokes integral to the gravity anomalies in the outer area Therefore, for the purpose of geoid determination, there is no need to transform the airborne gravity disturbances to gravity anomalies.

120 Chapter 6 The Importance of Downward Continuation Gravity field upward continuation is required to provide gravity references at flight level for evaluating the achievable accuracies of airborne gravity. For geoid determination, on the other hand, gravity field downward continuation is necessary in order to bring down the gravity disturbances fiom the flight level to the ground level. In this chapter, the Poisson integral and its underlying problems will be reviewed, and the regularization methods for using the ~oi'sson integral in gravity field downward continuation will be discussed. Synthetic numerical examples are used to highiight the achievable accuracy of the Poisson integral in both downward and upward continuation of the gravity field for the cases of errorfiee measurements. The accuracies of the gravity disturbances collected during the September 1996 airborne gravity tests are evaluated at both the flight level and the ground level by comparing them with the upward continued gravity references and the ground gravity measurements, respectively. Evaluation of the effect of downward continuation on the measured airborne data is essential for the assessment of airborne gravimetry as a geoid determination method. The downward-continued airborne gravity disturbances will be used in the next chapter for gravimetric geoid determination. 6.1 The Poisson Equation and its Inversion The Poisson integral is the solution of the first geodetic boundary value problem, or the

121 Chapter 6: Downward Continuation 106 Dirichlet problem. Denoting T as the disturbing potential of the Eaah's gravity field, the gravity field continuation can be mathematically described by the first geodetic boundary value problem. With the aid of spherical harmonic functions, the solution of the Dirichlet problem is (Heiskanen and Moritz, 1967) where (r, 8, A) and (R, 8, h) are geocentric coordinates, i-e., radius, co-latitude and longitude. T,@, 0, A) is of the surface harmonic function determined from the measurements on the d ace with radius I2 where y is the spherical distance between the computation point and the integration running point, and Pn(co s y) represents sp hericd Legendre function. The convergence of the series (6-1) is guaranteed if there is no mass outside the reference surface C of radius Et- Inserting equation (6-2) into equation (6-1) and using the relations C U"P (cosy) = Jl +a2-2acosy, a < 1, n =O n - 2 nanp (cosyr) = (acosyr-a2)(l+d--2acosyr)-', a < I, n =O n equation (6-1) can be expressed as

122 Chapter 6: Downward Continuation 107 Equation( 6-5) is usually referred to as the Poisson integral or the Poisson equation and provides the solution of the upward continuation of the gravity fieid. With the aid of spherical harmonic functions, it is not difficult to prove the following relations for gravity anomalies on two spherical surfaces with dzerent radii R and r, see Heiskanen and Moritz (1 967) for details, Using the Poisson integrai and denoting gravity anomalies on the sphere of radius R as Ago, the following equation is true for gravity anomalies Ag(r,O,h) = R 2(r - R 4nr I Ag0(eJ-) do, R srsr, (J z3 For limited integration cap size a, the Cartesian XY-coordinate system is usually used instead of the geographic coordinate system. With the following approximations

123 Cha~ter 6: Downward Continuation 208 Rlr.= 1, 4~~sin R 2 r 2v R - +-(r- RI2) = (xp -x12 +(~p -y)2 +hi, and the relation equation (6-9) can be approximated as The condition r 2 R or k 2 0 for equation (6-9) and (6-10) indicates that the Poisson integral can be directly used to upward continue the gravity anomalies &om a lower IeveI dace to a tzigher Ievel surface. On the other hand, realizing the fact that both equations (6-9) and (6-10) are linear systems or linear equations if expressed as discrete summation, gravity anomalies A go at lower level surface can also be estimated ftom gravity measurements A gh at a higher level surface by solving the linear equations. In other words, equation (6-9) or (6-1 0) can also be used for the downward continuation of the gravity anomalies. It is not difficult to prove that equations (6-9) and (6-10) are also valid for gravity disturbances. With discrete gravity measurements, both equations (6-9) and (6-1 0) can be expressed as numerical summations. The computational eeciency can be significantly improved if the fast Fourier transform is used. Approximating (9 +&/2 in equation (6-7) with the mean latitude cpm of the computation area, equation (6-6) becomes

124 Chapter 6: Downward Continuation (~,k) / = h2+4rr (sin- 2h-~' 2(P-(P +sin -(co~2~~-sin 2q -9)) With the above approximation and assuming that gravity anomalies are given on equallyspaced grid points, the Poisson integration can be expressed as a two-dimensional linear convolution, and consequently, can be evaluated by using the Fourier transform. In the frequency domain, we have AGe(u,v) = c AG,,[U,V)L(U,V). For equation (6-9), i.e. when spherical coordinates are chosen, "Ge(~Y v) = F{A~cP. krj), z 21 (sir?? + sin -(cos2qm = h + 4 ~ r C P * ~ sin2~)), When the Cartesian XY-coordinate system is used, i.e., using equation (6-1 O), the quantities in equation (6-12) are

125 Chapter 6: Downward Continuation 110 L(Uu,v) = F 1-, { x,; } (6-1 4c) From equation (6-12), the Fourier transform of the gravity anomalies on the lower Level AGO@, v) can be expressed as a function of the Fourier transform of the gravity anomalies on the higher level AGh(u, v), If we use the continuous Fourier transform pair equation (6-1 0) can also be expressed in the fkequency domain as Consequently, gravity field downward continuation can be expressed as Making use of the inverse Fourier transform, the gravity anomalies on the lower level surface can be obtained, which seems to provide a solution of the gravity field downward continuation. However, a serioms difficulty with the convergence of this solution is posed by the presence of the exponential factor, which renders downward continuation a so-called improperly posed problem in physics mashed, 1974). As q increases, AGh will be amplified

126 Chapter 6: Downward Continuation 111 by larger and larger factors. This means that errors in the data will affect the accuracy with which the high fkequencies of the gravity spectrum can be resolved. In practice, wavelengths that are above a certain numerical value for the product (q h) cannot be resolved. Figure 6-1 shows the exponential factor contained in equation (6-18) with elevation differences of 1 km, 2 km, 5 km and 10 km, respectively. As can be seen fiom Figure 6-1, the high frequency components will be significantly amplified especially for a Iarge elevation difference h. This is especially dangerous ifthe gravity measurements contain the contribution of near-surface masses or high frequency noise, as in the case in airborne gravimetry. An improperly posed or ill-posed problem is characterized by at least one of the following properties (Hadamard, ; Rauhut, 1 992) : a) No solution exists because the data are inconsistent due to, e.g, measurement errors. b) No unique solution exists because the data are not sficient to determine all unknowns, such as in an under-determined system. I wavelength (krn) I I O-O1 0.1 frequency (lkrn) Figure 6-1 The exponential factor in downward continuation in the frequency domain

127 Chapter 6: Downward Continuation 112 c) The solution is unstable such that small measurement errors will result in large errors in the estimated parameters. Improperly posed problems can only be reconditioned through the introduction of some independent a priori knowledge. This is called regularization. Yet the solution can still be unstable when errors in the observations are ampiified during the data processing. A low flight altitude can effectively avoid the stability problem in downward continuing the airborne gravity disturbances. From an operational point of view, however, this is not always practical because the restriction of topography and large air turbulence near the ground. More practical approaches to stabitize the downward continuation are those that can be used at the computation stage. The first method is to remove a large part of the high fkequency components by subtracting the topographic attraction fiom the observed airborne gravity disturbances. The second method is by selecting areasonable minimum wavelength to be recovered or, for gridded input gravity data, a large enough grid interval for a given h. The increase of the grid sampling intervals must always be made by keeping the physical problem in view, i.e. in which band is the gravity information to be recovered. The third method is the use of a regularization method which in its simplest fonn is a smoothing factor which acts as a low-pass filter. 6.2 ReguIarization of the Inverse Poisson Integral Improperly posed problems usually have a stable solution, once they are regularized. Regularization approaches which involve a change of the operator have been developed to obtain well-posed solutions to improperly posed problems, such as tnlncated singular value

128 Chapter 6: Downward Continuation 113 decomposition, Tikhonov regularization, stochastic methods, projection methods, and iterative methods (Lines, 1988; Louis, 1989). Detailed theoretical comparisons and practical application of these methods for the solution of the inverse Stokes problem can be found in Rauhut (1992). To regularize the process of the gravity field downward continuation, a probabilistic approach will be discussed in the following. This method uses a priori information about the unknown function in the form of a probability distribution. The problem is in this case regularized by prescribing an ensemble of functions which is smooth enough to make the problem well-posed. Such an approach is motivated by the fact that most improperly posed problems occurring in practice are problems which involve experimental data. Ifwe describe the measuring errors in these data in a probabilistic way, it is only one further step to look for the solution in a probabilistic setting. In this way, it is often possible to include previous experience in the a priori information (Schwarz, 1979). For simplicity, rewrite the discretized form of equation (6-10) in the following way: where n are samples from a random process Z and represent the random error of measurement, f are samples taken fiom a random process F and represent the observed airborne gravity disturbances in space, g represent the gravity disturbances to be solved for on a surface below the measurement level, A is the mapping operator. We can write the stochastic equation corresponding to the sample equation (6-1 9) as

129 Chapter 6: Downward Continuation 114 E{vi) = 0, i= 1,2,3, (6-2 1) where E( - ) denotes the mathematical expectation. Denoting the correlation operator, i.e. the a priori information, as E{v,v,') = Rii, i=1,2,3, and assuming ensembles fiom G and fiom Z as independent, i-e., E(v,v,') = 0, the auto-correlation R,, of the ensemble fiom F will have the form R,, = ARIIA * + Rn. where A * is the adjoint operator (Bachman and Narici, 1966). With the following minimization condition we can get If the inverse of the operator (A *R;'A + R;') exists, then we have the estimate

130 Chapter 6: Downward Continuation 115 g = H,f, (6-27) where H- = (A * R:A + R;')-'A * R ~ I. H, is the regularization matrix (Nashed, 1974). Assuming covariance matrixes R,, and R, are diagonal matrixes with variances3 and < respectively, i.e-, R,, = $1, R, = $1, and denoting a = the regularization matrix H, becomes and the solution for the downward continuation can be expressed as Minimizing the fist term on the left hand side of equation (6-25) is identical to the classical least-squares minimization problem. Minimizing the second term guarantees the smoothness and stability of the estimates g which makes the difference between the classic least-squares adjustment and the regularization problem. The parameter cr can be regarded as a weighting

131 Chapter 6: Downward Continuation 116 factor between the two parts. Increasing the value of a means putting more weight on smoothing. On the other hand, decreasing the value of a means emphasizing the leastsquares part. Using an a that is too small may lead to an ill-conditioned equation and result in an amplification of the errors in f. 6.3 Numerical Examples with Synthetic Gravity Anomalies The Poisson integral was used in studying gravity field downward and upward continuation in an anomalous gravity field generated fkom a point mass model. The gravity anomalies computed in space fiom the point mass model were taken as the reference for the anomalies derived fkom the Poisson integral. It has to be pointed out that the non-zero mean value, or the DC value, of the input data should be removed before conducting either upward continuation or downward continuation. This step is necessary firom a theoretical point of view and is important fiom an operational point view. Theoretically, the maximum wavelength to be recovered should be no longer than the dimension of the area. With an area size of 100 km by 100 km, for example, a maximum wavelength of 100 Ism can be recovered. Any components corresponding to wavelengths longer than 100 h should be determined fkom other data sources. In geoid determination, the long wavelength components should be computed fiom a geopotential model or fiom gravity anomalies on a coarser grid within a larger area In practical operations, removing the DC value can reduce the distortions in the computed results around the edges. To illustrate this 'edge effect', a constant value of 10 mgal on a grid within an area of 100 km x 100 km was downward-continued to a lower level plane and upward-

132 Chapter 6: Downward Continuation 117 continued to a higher level plane. The continuation height difference was 4 km. Figure 6-2 shows one profile of the downward-continued results and one profile of the upwardcontinued results. The significant distortions around the edges shown in Figure 6-2 clearly indicate that the side effect of a non-zero constant value in the input data on either downward continuation or upward continuation is not negligible. X (km, Y = 50) Figure 6-2 Central profiie of the downward and the upward continued results of a constant 10 mgal within a 100 x 100 km2 area (Continuation height difference: 4k1n) The simulated point mass model data After subtracting the contribution of the geopotential model EGM96, the reduced gravity anomalies were gridded into 80 x 80 nodes with a grid spacing of 5 km in both directions, and were then modelled by three point mass layers parallel to the earth surface at depths 30 km, 20 km and 5km, respectively. Gravity anomalies were then computed from the point mass model at surfaces with elevation of 0 km and 4 km, respectively, and are shown in Figures 6-3 and 6-4. Table 6-1 summarizes the statistics of the gravity anomalies computed from the point mass model.

133 Chapter 6: Downward Continuation 118 Table 6-1 Statistics of gravity anomalies computed fkom the point mass model in mgal altitude Ag(h=O km) Ag(h=4km) min ma. mean 1lf1s std L km Figure 6-3 Gravity anomalies computed fiom the point mass model at h = 0 lan Figure 6-4 Gravity anomalies computed fkom the point mass model at h = 4km

134 Chanter 6: Downward Continuation Test of upward continuation with Poisson's integral The Poisson integral was used in upward continuing the gravity anomalies from 0 irm to 4 km. Figure 6-5 shows the differences between the gravity anomalies computed fiom the point masses and the upward continued vaiues. Table 6-2 contains the statistics of the differences. Table 6-2 shows that the RMS difference is about 0.5 mgal (lo) with a maximum difference of about 10 mgal. As can be seen fiom the figure larger differences are all located around the edges and, which is expected because of the lack of input data outside the boundary. In the central area of 350 x 350 km2, the directly computed and the upward continued gravity anomalies are almost identical. The standard deviation of the differences is only 0-04 mgd- Figure 6-5 The difference between the reference and the upward continued (0 to 4 krn) gravity anomalies Table 6-2 Statistics of the differences between the reference and the upward continued gravity anomalies area whole area: 400x400 Ian2 central area: 350x350 krn2 rnin max

135 Chapter 6: Downward Continuation 120 Figure 6-6 The difference between the reference and the downward continued (4 to 0 km) gravity anomalies wavelength (km) Figure 6-7 Amplitude spectrum of the gravity anomaly merences at h = 0 km Test of downward continuation with Poisson's integral The Poisson integral was also used in downward continuing the gravity anomalies from 4 km to 0 krn. Figure 6-6 shows the differences between the gravity anomalies computed tom the point masses and the downward continued values and Table 6-3 contains the statistics of the differences. Figure 6-7 shows the amplitude spectrum ofthe gravity anomaly dierences.

136 Chapter 6: Downward Continuation 121 t Table 6-3 Statistics of the differences between the reference and the downward continued gravity anomalies and their effect on geoid undulations area whole area: 400x400 Ian2 N.E. comer: 200x200 k d gravity differences (mgal) min max CT effect on geoid (m) rnin max G Table 6-3 indicates that the differences bemeen the reference and the downward continued vaiue has a standard deviation of 2.0 mgal and a maximum value of about 10 mgai. As can be seen from Figure 6-6, most of the big differences are located along the southern and western edges where the gravity anomdies vary the most. In the north-eastern (N. W.) corner with a size of 200 x 200 km2, the differences are much smaller and have a standard deviation of 0.8 mgal. Comparing Figure 6-6 with Figure 6-3 one finds that the magnitude of the differences is somewhat correlated with that of the gravity anomalies. Inthe frequency domain, the amplitude spectrum shown in Figure 6-7 indicates that the main differences are due to the high frequency end, corresponding to a wavelengths of about 10 km, which is the minimum resolvable wavelength due to the grid spacing of 5 km of the input data used in modei1ing the anomalous gravity field. This result agrees with the theoretical analysis conducted in the previous section and confirms that the main problem in downward continuation of the gravity field is how to correctly recover signals at the high fiequency bands. Applying the Stokes integral to the gravity anomaly differences provides the geoid undulation errors ioduced by the process of downward continuation. The statistics, as shown in Table 6-3, indicate that, with this data set, the effect of the gravity 'errors' due to the

137 Chapter 6: Downward Continuation 122 downward continuation on the geoid undulation is at the centimeter level with a standard deviation of 1 cm. This effect is negiigible if the objective is to achieve an accuracy of the geoid undulation at the 10-cm level with a minimum wavelength of 10 km. For higher accuracy expectations and a finer wavelength resolution, however, downward continuation of gravity anomalies fiom the flight level to the geoid is still a problem, especially when the flight altitude is high and the gravity disturbances determined fiom airborne gravimetry contains noise at the high frequency end. 6.4 Continuation of Airborne Gravity Disturbances In this section, the gravity disturbances collected during the September 1996 airborne gravity tests were compared with gravity references both at the flight level and on the ground. The flight tests have been described in Section 3.2. Figures 3-22 and 3-23 show the flight trajectory and Figures 3-24, 3-25 and 3-26 show the gravity disturbances determined fkom day 1, day 2 and day 3 of the airborne gravity test, respectively. The digital topographic model around the airborne gravity test area is shown in Figure The digital topographic model of a 100 lan x 100 km area directly below the flight profde is shown in Figure 6-8. The mean vaiue of the topographic height within this area is 2200m. Figure 5-1 shows the Faye gravity anomalies. The average spacing of the gravity points within this area is about 10 km. Therefore, the minimum resolvable wavelength ofthe gravity anomalies in this area is about 20 km. This resolvable minimum wavelength is about the same as the minimum wavelength of the airborne gravity disturbances at the cross track direction and is twice as that ofthe airborne gravity disturbances along the flight trajectories.

138 Chapter 6: Downward Continuation 123 Figure 6-9 shows the gravity disturbances below the fight profiles. It has to be pointed out that, due to the sparse distribution of the ground gravity measurements and the fact that most of the them were taken along a road or in the valleys within the mountains, the differences between the ground measured and the airborne gravity disturbances may only reflect their agreement but not the accuracy achieved by the airborne gravity systems. km mgal Figure 6-9 Gravify disturbances below the flight profiles

139 Chapter 6: Downward Continuation Comparison of airborne gravity disturbances with the upward-continued reference field The gravity disturbances were first upward continued by the Poisson integral to the airborne flight levels at 4350 m and 7300 m, respectively. Figures 6-10, and 6-12 show the differences between the upward continued gravity anomalies and those determined fiom airborne gravimetry on Dayl, Day2 and Day3, respectively. Table 6-4 contains the statistics of the differences. The mean differences were omitted fkom the statistics because the gravity field information of interest is that with wavelengths between 10 km and 100 Ian. km Figure 6-10 The differences between airborne and upward-continued gravity disturbances (dayl, 4350 m) Table 6-4 indicates that the airborne gravity disturbances agree with the upward-continued values at the level of mgal(1o). Comparing Figures 6-10,6-11 and 6-12 with Figure 6-8, we can see that the gravity differences of the three days have a similar pattern which follows the north-west to south-east trend of the topography. This implies that the topographic effect on the gravity field has not been completely canceled by differencing. In order to

140 Chapter 6: Downward Continuation 125 evaluate if the residual topographic effect is due to airborne gravimetry or due to the upwardcontinued reference, the resolution of the ground reference gravity has to be improved with denser ground gravity measurements. m Figure 6-11 The differences between airborne and upward-continued gravity disturbances (day2,43 50 m) mgal km Figure 6-12 The differences between airborne and upward-continued gravity disturbances (day3,7300 m)

141 Chapter 6: Downward Continuation 126 The smaller standard deviatio of the differences of Day3 than those of Dayl and Day2 is because the gravity field at the level of 7300m is much smoother than that at altitude 4350m. Table 6-4 Statistics of the differences between the upward-continued reference and the airborne gravity disturbances, unit in mgal Day1 Day2 Day3 airborne gravity disturbances min max the differences min ma. CT Comparison of downward-continued airborne gravity disturbances with ground gravity measurements To conduct downward continuation, the long wavelength information modelled by the global geopotential model EGM96 was first removed fiom the airborne gravity disturbances. Two groups of tests were used to illustrate the contribution of the topographic effect in gravity field downward continuation. In one group, the topographic attraction was subtracted fiom the airborne gravity disturbances before the downward continuation, and added back to the downward continued values afterwards. This procedure is called 'remove-restore' technique for the topographic effect in the following discussions. In the other group, the topographic effect was not explicitly computed and thus remained part of the gravity disturbances in the downward continuation procedure. The residual gravity disturbances were gridded with a spacing of 5 km. The Poisson integral with the recgularization solution of equation (6-32)

142 Cha~ter 6: Downward Continuation 127 was used to downward continue the residual gravity disturbances from flight level to the plane with the average elevation of the topographic heights. The parameter a was selected as 2. The downward continued airborne gravity disturbances were low-pass filtered with a minimum cut-off wavelengths of 10 km and 20 km, respectively. The 10 km wavelength coincides with the minimum resolvable wavelength of the bw-pass filtered airborne gravity disturbances along the flight trajectory. The 20 km cut-off wavelength, twice the flight track spacing, corresponds to the minimum resolvable wavelength perpendicular to the flight trajectory. -- The downward continued airborne gravity disturbances were compared with the ground references. Table 6-5 summarizes the statistics of the differences. Case a and e in this table represent the case where airborne gravity disturbances fiom days 1 and 2 were merged together as one set of data which was then used in conducting the downward continuation. Figure 6-13 shows the differences for the case when the topographic attraction were removerestored during the downward continuation. Table 6-5 clearly indicates that the topographic attraction should be removed fiom the airborne gravity disturbances before conducting the downward continuation. The topographic remove-restore procedure reduced both the magnitude and the standard deviation of the differences by about 70 to 80 percent. With the remove-restore topographic effect, the agreement between the reference and the downward continued airborne gravity disturbances is about 2.5 mgal (lo). If the topogcaphic attraction is removed and restored, downward continuing the combined day1 and day2 airborne data has a similar statistical performance as downward continuing

143 Chapter 6: Downward Continuation 128 dayl and day2 data separately. This is an important result for airborne gravity surveys because it considerably reduces the number of flight lines needed to cover a given area. The slightly smaller differences for the dayl data compared to the day2 data could be due to the different variations of the gravity field dong the flight trajectories of the two days. Figure shows the north-south and the west-east component of the amplitude spectrum a: day l&2 c: day2 b: day1! -. n XOun) X b > Figure 6-13 Differences between reference gravity disturbances and the downward-continued airborne gravity disturbances (minimum wavelength: 10 km. EGM96 and topographic effect are remove-restored during downward-continuation)

144 Cha~ter 6: Downward Continuation 129 of ground gravity disturbances. The figure indicates that the gravity field in the test area has a stronger amplitude spectrum in the west-east direction (across the main mountain ranges), i.e. along the trajectory of the day1 airborne test. More investigations are necessary to verify if this is true in general. The conclusion can be heipll in designing airborne flight patterns such as selecting the flight direction based on the variations of the topography. TabIe 6-5 The statistics of the differences between the reference and the downward continued airborne gravity disturbances, unit: mgal airborne data used in downward continuation a - - min. wavelength: I0 km min ma. o min. wavelength: 20 km min max G EGM96 contribution and topographic attraction were remove-restored dayl&2 b day 1 c day 2 d day 3 e day1&2 f day 1 g day 2 h day Only the EGM96 contribution was remove-restored I To illustrate the effect of regularization on downward continuation, the airborne gravity disturbances were also downward-continued to the ground with the inverse Poisson integral but no regularization was used. Both the long wavelength information of EGM96 and the

145 Chapter 6: Downward Continuation 130 topographic attraction were removed and restored. Table 6-6 contains the statistics of the differences between the ground reference and the downward-continued airborne gravity disturbances. waverength (km) 100 I0 I a a I---*----.-* * I n i iiiliiil ; ;:;;ii;l : :::::::I 0, I. I 1 1 1, I I frequency ( I h ) Figure 6-14 North-south and west-east components of the amplitude spectrum of ground gravity disturbances Table 6-6 The statistics of the differences between the reference and the downward continued airborne gravity disturbances (without regularization), unit: mgal airborne data used in downward continuation a day 1 &2 b day 1 c day 2 d day 3 min. wavelength: 10 km min max cr min. wavelength: 20 km min max

146 Chapter 6: Downward Continuation 131 Comparing Table 6-6 with Table 6-5 we can see that, for day1 and day2, the use of regularization reduced the standard deviation of the dberences by about 0.1 mgal for all cases. For day3, the use of regularization reduced the standard deviation of the differences by about 30% fiom 6.9 mgal to 4.7 mgal and about 20% from 3.5 mgal to 2.9 mgal when the minimum wavelength resolution is 10 km and 20 lan, respectively. The peak-to-peak range (max - min) of the differences is reduced fiom 56.0 mgal to 33.9 mgal when the minimum wavelength resolution is 10 km. Therefore, the use of regularization can improve the resuits of downward continuation, especially in the case of a high night altitude. The downward continued airborne gravity disturbances in the cases a, b, c, and d in Table 6-5 will be used for geoid determination in next chapter. 6.5 Summary of Results In this chapter, the Poisson integral was reviewed for the purpose of gravity field continuation. Regularization methods were discussed to stabilize the improperly posed problem inherent in the use of the inverse Poisson integral for gravity downward continuation. Synthetic studies with noise-fiee measurements indicate that the Poisson integral provided almost error-free results in the central area when upward continuing the gravity anomalies from 0 km to a level of 4 km. When downward continuing the gravity anomalies fiom 4 km to 0 km level, the Poisson integral gave a gravity anomaly error of 2.0 mgal (lo) in the whole area of 400 x 400 km2 and 0.8 mgal (lo) in a sub-area of 200 x 200 W. The effect of these gravity anomaly differences on geoid undulation is 1 cm (1~).

147 Chapter 6: Downward Continuation 132 The differences between the airborne gravity disturbances and the Poisson integral upward continued references have a standard deviation of 2.5 mgal(1 o) for the day1 and day2 tests, and 2.0 mgal (lo) for day3. The Poisson integral downward continued airborne gravity disturbances agree with the ground references at a level of 3-8 mgal (I o) when the minimum wavelength included is 10 km, and of 2.5 mgal when the minimum wavelength included is 20 km. These values include system errors, downward continuation effects, and errors in the ground reference. It has to be pointed out again that the differences between the downward-continued airborne gravity disturbances and the ground reference do not necessarily reflect the actual accuracy of airborne gravity measurements. The relatively larger differences when a minimum wavelength of 10 km is included could indicate a lack of information for this wavelength in the ground reference. Therefore, the actual accuracy of gravity disturbances determined by airborne gravimetry could be better than the differences indicate.

148 133 Chapter 7 The Use of Airborne Gravity Data for Geoid Determination The contribution of airborne gravimetry to geoid determination is evaluated by comparing the geoid undulations computed fiom airborne gravity disturbances with those fiom ground gravity measurements. The tern 'local geoid' is used in this chapter to indicate that geoid undulations were computed from gravity disturbances within the 100 km x 100 krn area only. The term 'regional geoid' means that the geoid undulations were computed by integrating gravity disturbances within the inner area on a 5' x 5' grid and in the outer area either on a 30' x 30' grid or on a 5' x 5' grid. In other words, the 'local geoid' is bandwidth limited and the 'regional geoid' has no Iirnit on the long wavelength end. The term'airborne geoid' means that airborne gravity disturbances were used in computing the geoid undulations. Ln all cases, the geopotentid model EGM96 was used to model the long wavelength component. The Hotine integral was used to convert the gravity disturbances to geoid undulations. Results in this chapter answer the main objective of the thesis "how well can the gravimetric geoid be determinated by airborne gravimetry". 7.1 Evaluation of the Local Airborne Geoid at Flight Level The evaluation was first done at flight level. Strictly speaking, what the Hotine integral provides in space is not the real geoid unddation but the deviation or the undulation of the

149 Chapter 7: Airborne Geoid Undulations 134 equipotential surface of the Earth's gravity field from the normal equipotential surface of the reference ellipsoid at the computation point. The term 'geoid undulationj is still used here for convenience. The differences between the upward-continued and the airborne gravity disturbances were converted to geoid undulation differences. Figure 7-1 shows the plot of the geoid undulation differences at flight level and Table 7-1 contains the statistics. Table 7-1 indicates that the geoid undulation differences are 2 cm (1 o) at the flight level of 4.35 km (Day1 and Dayz), and 1 cm (1 o) at the flight level with a attitude of 7.3 km pay 3). The averaged difference of Day1 and Day2 (Dayl&2) has a standard deviation of 1 cm. The smaller standard deviation value of the geoid undulation differences from Day3 is due to the fact that the gravity field and the geoid undulation are much smoother at this level. Figure 7-1 The differences between geoid undulation computed fkom upward-continued gravity disturbances and those from airborne gravity at flight level

150 Chapter 7: Airborne Geoid Undulations 135 Table 7-1 Statistics of the geoid undulation differences propagated from the differences between the gravity references and airborne gravity disturbances, unit: m airborne data set Day 1 &2 Day1 Day2 Day3 at flight level max min cr on the ground rnax min CT Evaluation of the Local Airborne Geoid on the Ground The differences between downward continued airborne gravity disturbances and the gravity reference were also converted to geoid undulations. Figure7-2 shows the plot of the geoid undulation differences and the last column of Table 7-1 contains the statistical information. Table 7-1 shows that the standard deviation of the geoid differences is 2 crn for all the data sets. The maximum value of the differences is about 5 cm. The fact that the statistics for all three days are about the same indicates that the different flight altitudes of 4.35 km and 7.3 km do not have a significant effect on geoid determination with an expected minimum wavelength resolution of 10 km. Similar statistical results were obtained when comparing the reference geoid undulations to the downward continued local airborne geoid. Geoid differences shown in Figure7-2 are rougher than those shown in Figure 7-1. This is expected due to the fact that the downward continuation amplifies the high frequency band of gravity disturbances and consequently results in a rougher gravity field.

151 Chapter 7: Airborne Geoid Undulations :. :..: ,.>._....._... _ -;...-:>: :.:. a :: A,.".:.. _ -:..::'..:.. <-'.--- ;; -'.,..- '......:.. : :. /--.:;.. -; L.,. '* ;-.>.,+.;.~:~.;..;z!:.. ::-:--,s. -:- -.:<:>&.z;z;::z:2 ;- : ??:---% :-.- ;;:;=-<5-.:. - ; -;2;:,::,., f, 0..-.;-.-2--:..:: -. :. - I..?-,.> 2, -..,.-,.:-2-: *;- =>*:;-... ~ ~.., ~. ~ ~ >!? ->-&-. <y' ;.:. ;:,.,., --ti - --y:.--km-:- : :-.- -.A. ---<:el *- -<-I ax/..---*-;-+: :...:..:.:> /.2. ;-:-Y.3..-.,..;: :...: Jx* \....-,".::5/y.; A. - \ , -*re. /-&*. _ -_._. '. - ' '. -<-% - "8K r 50' \ ,. *...*-? : --.-+~+m-+. : a * -0 <@- r%<+ - A &-'A \ - -, / +- - <- Figure 7-2 Differences between geoid undulations computed from ground gravity disturbances and those from downward continued airborne gravity disturbances 7.3 Evaluation of the Regional Airborne Geoid on the Ground For regional geoid determination, the differences between the reference geoid and the airborne geoid should have the same statistical characteristics as those in the above section if the same ground gravity data in the outer zone were used in geoid computation. More specifically, if gravity disturbances on a 5' x 5' grid in the outer zone are available, the airborne regional geoid should agree with the reference gravimetric geoid at the level of 2 cm (1 o) when the same gravity data in the outer zone are used in geoid computation. Agreement does not mean accuracy in this case because errors in the outer zone data would cancel out in the differences. The same is true when a global geopotential model is used

152 Chapter 7: Airborne Geoid Undulations 137 instead of outer zone data because errors in the coefficients of the global model will degrade the accuracy of the global solution. This may change, however, in the near fbture once the results of the two proposed dedicated gravity satellites (GRACE and GOCE) become available. If the results correspond to the predictions, a geopotential model computed from these satellite data will provide a global geoid of 1 crn accuracy or better for wavelengths of 100 km and larger (ESA, 1999; GRACE, 1998). Combination of such models with airborne gravity data wodd result in an overall accuracy of about 2 cm, Currently, in most places on the Earth, gravity data are available only on a 30' x 30' grid. From a practical point of view, it is usell to investigate the achievable accuracy of geoid undulations determined fiorn airborne gravity disturbances on a 5' x 5' grid in the inner zone and ground gravity anomalies on a 30' x 30' grid in the outer zone. The downward continued airborne gravity disturbances were used in computing the regional geoid undulations. The Hotine integral with the modified spherical kernel function was used in the computation. The inteagation area was divided into an inner and an outer area as shown in Model A in Figure 7-3. The inner area has a size of 100 km by 100 km (1 ' x 1.5 ) and is directly below the flight profiles. Within the inner area, the downward continued airborne gravity disturbances were gridded on a 5' x 5' grid with 12 samples in the latitude direction and 18 samples in the longitude direction. The outer area contains the 30' x 30' mean gravity disturbances as shown in Figure 5-1. The computed geoid undulations within the her area (will be called airborne geoid thereafter) were compared with the reference geoid as determined in section 5.2. Figure7-4 shows the geoid undulation differences and Table 7-2 shows the statistical information. Table 7-4 indicates that, within the area of 100 x 100 km', the differences between the

153 Chapter 7: Airborne Geoid Undulations 138 airborne gravimetric geoid and the reference have a peak value of about 40 cm and a standard deviation of about 8 cm. The peak value of the differences is located along the south-westem edge where the gravity field is much rougher than in other parts as can be seen from Figure 6-9. In the 90 x 90 la? area in the north-eastern comer, the differences have a peak value of aborrt 20 cm and a standard deviation of 5 cm. To illustrate the contribution of the gravity disturbances around the airborne flight area, a half degree strip of 5' x 5' ground data is added to the inner area on each side as area b. Geoid undulations were computed by integrating the gravity disturbances in area a, b and c as shown as Model B in Figure 7-3. Figure7-5 shows the plot of the geoid undulation differences and Table 7-2 shows the statistical information. Figure 7-6 shows the plot of the contribution of the ground 5' x 5' gravity disturbances within area b to geoid undulations in the inner area a: 5'xSf airborne 6g, size: 1" x 1.5" c: 30'x301 ground 8g, size: 30" x 30" b: S'x5' ground 6g, size: 2" x 2.5" L Figure 7-3 Illustration of combining airborne and ground gravity for geoid determination

154 Chapter 7: Airborne Geoid Undulations 139 Figure 7-4 The differences between the reference and airborne geoid (Day 1, Model A) Figure 7-5 The differences between the reference and airborne geoid (Dayl, Model B)

155 Chapter 7: Airborne Geoid Undulations 140 Figure 7-6 The contribution of ground gravity disturbances in area b to geoid undulations in the inner area Comparing Figure 7-5 with Figure 7-4 we can see that, in the south-west edges, the differences shown in Figure 7-5 are much smoother than those shown in Figure 7-4. Figure 7-6 indicates that the gravity disturbances within area b mainly affect the geoid undulations of the inner area around the South-Westem edges and have a negligible effect on those in the North-Eastern subarea. This is due to the rougher gravity disturbances around the South- Western edges. Table 7-2 indicates that, with the use of 5' x 5' gravity disturbances in the middle area, the differences between the reference and the airborne geoid undulations have a maximum value of about 20 cm and a standard deviation of 5 cm within the whole area of 100 krn x 100 krn. This is about the same as that within North-Eastem subarea of 90 krn x 90 km.

156 Chapter 7: Airborne Geoid Unddations 141 Table 7-2 Statistics of the differences between the reference and the airborne gravimetric geoid undulations, unit in metre comparison area airborne data used L the whole 100 x 100 km' max min a the North-East 90 x 90 km' max rnin CY airborne 5' x 5' 6g combined with ground 30' x 30' mean 6g (Model A) Day Day Day airborne 5' x 5' 6g combined with ground 5' x 5' and 30' x 30' mean 6g (Model B) Day Day Day It is worth pointing out that the differences between the reference and the airborne geoid undulations do not necessarily reflect the actual accuracy of the geoid determined fkom the airborne gravity disturbances, Comparing Table 7-2 with the 'after fit' statistics in Table 5-9 we can see that both the peak-to-peak range and the standard deviation values are very similar in the two tables. This suggests that the agreement between the reference geoid and the airborne geoid is at the same level as that between the reference geoid and the GPSAeveiling-derived geoid when the systematic trend is removed. The actual accuracy of the geoid undulations determined fkom airborne gravity disturbances could be better than 5 cm. Table 7-2 also indicates that the airborne gravimetric geoid undulations from Dayl, Day2 and Day3 have almost the same agreement with the reference geoid. This suggests that, for geoid determination with a required wavelength resolution of 10 km, data taken at a flight altitude of 7 km can produce a similar result as data at a flight altitude of 4 km.

157 Cha~ter 7: Airborne Geoid Undulations Summary of Results In this chapter, the contribution of airborne gravimetry to geoid determination is evaluated by comparing the airborne geoid with the reference computed from ground gravity measurements. The geoid undulation differences computed from the differences between the airborne gravity disturbances and the reference gravity have a standard deviation of about 1-2 cm at flight altitude and 2 cm on the ground. As compared with the reference geoid determined on a 5' x 5' grid, the differences between the reference geoid and the airborne geoid have a standard deviation of 5 cm (1 o) when the ground gravity measurements on a coarse grid of 30' x 30' are used outside the flight area The differences between the airborne geoid and the reference geoid have the same statistical characteristics as the differences between the reference geoid and the GPSAevelling-derived geoid when the difference trend is removed by a plane fit. Mainly due to the errors in a global geopotential model, the overall accuracy of a global gravimetric geoid is at the level of about 30 cm (1 o) when the trend is not removed. Therefore, the increase from o = 2 cm to a = 5 cm is most likely due to inaccuracies in the reference solution. Geoid undulations computed from downward continued gravity disturbances fiom flight altitudes of 4.35 km and 7.3 km show a similar performance when compared with the reference geoid for a minimum wavelength resolution of 10 km.

158 Chapter 8 Conclusions and Recommendations The overal1 objective of this research was to study the use of airborne gravity data for geoid determination fiom a numerical and an operational point of view. Although standard techniques of gravimetric geoid determination can be applied in general, the use of airbome gravity data also requires considerations specific to this task. Among them are: 1) Airborne gravimetry provides gravity disturbances along flight trajectories. They have to be downward continued to the boundary surface for geoid computation. 2) The resolution and accuracy of downward-continued gravity disturbances are affected by flight speed, flight altitude and the track spacing of airborne gravimetry. Special procedures such as regularization and removal of the high frequency components with a digital topographic model have to be used in processing airborne gravity disturbances. 3) Airborne gravity surveys are usuaily taken within an area of limited size. The middle and long wavelength information has to be recovered from other data sources, such as a geopotential model and ground gravity measurements on a much coarser grids. The main contribution of this dissertation is the comprehensive investigation of geoid determination from airborne gravimety with emphasis on the above three specific problems. The results obtained from the analysis of real data in an area with a very rough gravity field provide new insights that go beyond earlier simulation studies conducted by some geodesists.

159 Chapter 8: Conclusions and Recommendations Conclusions Detailed analyses and numerical evaluations have been done to deal with these special problems and the following conclusions can be drawn Eom these studies. 1) From a frequency domain analysis of the gravity spectrum, one can conclude that under ideal circumstances a centimeter-geoid can be achieved if the minimum full wavelength resolved on the ground is about 14 km in flat areas and 5 krn in mountainous areas. To obtain a geoid with an average accuracy of 10 cm (lo), the minimum fidl wavelength is about 70 km in flat areas and 40 km in mountainous areas. The above statements assume that all wavelengths above the minimum wavelength are perfectly known. 3) When applying currently available airborne gravity systems and the existing global models, the accuracy of local geoid determination is at the level of a few centimeters for wavelengths between 10 and 100 km; 10 cm is possible for wavelengths between 5 and 500 km; and for wavelengths of up to 1000 km, the rms error will usually not be larger than 30 cm. The radius of the area to be covered by airborne data should be at least 1.5' in order to achieve centimeter levei accuracy. 3) To eliminate the effects of downward continuation as much as possible, the topographic attraction has to be removed from the gravity disturbances at flying height before conducting downward continuation. In addition, some form of a regularization for the downward continuation process is necessary and the probabilistic approach is recommended. 4) For the September 1996 airborne gravity tests, the differences between the airborne gravity disturbances and the upward continued reference have a standard deviation of 2.5 mgal (I c) for Day1 and Day2 at the flight level of 4350 m, and 2.0 mgal (la) for Day3 at the flight

160 Chapter 8: Conclusions and Recommendations 145 level of 7300 m. The downward continued airborne gravity disturbances agree with the ground references at a level of 3.8 mgal (lo) when the minimum wavelength resolution is 10 km and 2.5 mgal (lo) when the rrhimum wavelength resolution is 20 km. 5) It is advantageous from an operational point of view to use the Stokes and the Hotine integral in combination, if terrestrial data and airborne data are combined, The Hotine integral is applied to the downward continued gravity disturbances and the Stokes integral is applied to the gravity anomalies in the outer area In other words, for the purpose of geoid determination, there is no need to transform airborne gravity disturbances into gravity anomalies. 6) The Iocal "geoid undulation differences converted from the differences between the upward-continued gravity reference and airborne aavity disturbances have a standard deviation of 2 cm at the flight altitude of 4350 m and of 1 cm at the flight altitude of 7300m. 7) The local geoid undulation differences obtained by integrating the difeerences between the ground gravity reference and the downward continued airborne gravity disturbances have a standard deviation of 2 cm (1 G). Similar results are obtained when directly downward continuing the 'geoid' undulation fiom flight level to the geoid. These results fully confirm the results of the spectral analysis in conclusions 1 and 2. 8) When regional geoid undulations are computed by combining the downward-continued airborne gravity disturbances with ground gravity measurements on a coarse grid (such as 30' x 30') outside the flight area, the differences between the reference geoid determined by integrating the 5' x 5' ground gravity anomalies within the 30" x 30" area and the airborne geoid have a standard deviation of 5 cm (1 a) within the 100 km x 100 km area. The

161 Chapter 8: Conclusions and Recommendations 146 merences have similar statistical characteristics as those between the reference geoid m the GPSAeveUing-derived geoid after a trend surface has been subtracted. Taking (7) into account, this would indicate that the error contribution of the outer zones on gravimetric geoid determination is much larger than the error contribution of airborne gravity. 9) Based on the above numerical results, it can be concluded that the current airborne gravity systems have the potential to determine gravimetric geoid undulations anywhere on the Earth with an accuracy of 2-5 crn (lo) if the accuracy of the global geopotential model can be improved to the level expected fiom future gravity satellite missions (1 cm for wavelengths above 100 krn). 8.2 Recommendations Although the major objective of this research has been achieved, further research is needed for the application of airborne gravirnetry to geophysical exploration. These applications require higher accuracy and higher wavelength resolution of the gravity signal. Areas of fbture research therefore include the following. 1) The improvement of the accuracy of GPS-derived acceleration. Currently, the resolvable wavelength of airborne gravimetry is mainly restricted by the high frequency errors in the GPS-derived acceleration. 2) The effect of airborne dynamic motion on the performance of the M U should be studied. By removing the modelled dynamic effect fiom the measured specific force, the performance of the low-pass filter can be improved.

162 Chapter 8: Conclusions and Recommendations 147 3) Studies should be conducted on how to utilize the different resolutions of airborne gravimetry along the flight direction and across the flight tracks in the process of downward continuation and in gravirnetric geoid determination. 4) Numerical investigations should be conducted on how the statistical information obtained during processing of INS and GPS observations can be used in downward continuation and in gravimetric geoid determination. 5) More investigations on regularization methods in downward continuing airborne gravity disturbances are necessary. This is important in stabilizing downward continuation and in possibly removing some of the restrictions requiring a low flying altitude. 6) Ground gravity measurements with finer wavelength resolution, such as 5 km, and an accuracy of 1 mgal or better are needed in order to assess the accuracy of the downwardcontinued airborne gravity disturbances, and consequently, to evaluate the accuracy of geoid undulations determined from the downward-continued airborne gravity disturbances.

163 References Applied Analytics Corporation, Experimental report: Optech 5OISB laser rangefinder, contract COO1 for Canagrav Research Ltd., Calgary, Alberta, Canada, August Argeseanu, V., 1995, Upward continuation of surface gravity anomaiy data. In: Proceedings of Airborne Gravity, IAG Symposium G4, Boulder, Colorado, July 2-14, pp Bachman, G. and L, Narici, 1966, Functional Analysis, Academic Press, New York and London. Bell, R.E., B.J. Coakley, D.D. Blankenship, S.M. Hodge, and J.M. Brozena, 1991, Airborne gravimetry from a light aircraft. The sixth International Symposium on Antarctic Earth Science, Tokyo SCAR Meeting, Japan. Bhaskara, D., M.J. Prakash and N. Rarnesh Babu, 1993, Gravity interpretation using Fourier transforms and simple geometrical modeis with exponential density contrast. Geophysics, Vol. 58, Brozena, J.M., and M-F. Peters, 1988, An airborne gravity study of eastern North Carolina, Geophysics, 53, Brozena, J.M., 199 1, The Greedand aerogeophysics project: Airborne gravity, topographic and magnetic mapping of an entir continent. International Association of Geodesy Symposium 110, S pringer-verlag, New York, USA. Brozena, J.M., and M.F. Peters, 1994, Airborne gravity measurement at NRL, paper presented at the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Department of Geomatics Engineering, The University of Calgary, Banff, Canada Brozena, J.M., G.L. Mader, and M.F. Peters, 1989, Interferometric global positioning system: Three-dimensional positioning source for airborne gravimetry, J Geophys. Res., 94@9), Brozena, J.M., M.F. Peters, and R. Salman, Arctic airborne gravity measurement program, In: Sagawa, J., H. Fujirnoto, and S. Okubo (eds), Gravity, Geoid and Marine geodesy, Proc. L4G Symposia GRAGEOMR, Tokyo 1 996, IAG Symposia, Vol 1 1 7, pp , Springer Verlag, Heidelberg, Bruton, A.M., C. Glennie and K.P. Schwarz, 1999, Differentiation for high Precision GPS velocity and acceleration determination. GPS Solutions, John Wiley & Sons, hc. Vol. 2, No. 4.

164 Bmton, A.M., K-P. Schwarz and M. Kern, 2000, Acceleration from GPS for airborne gravity: error models, processing methods and results. To be submitted to GPSSolutions, John Wiley & Sons, Inc., April Bruton, A.M., K.P. Schwarz and M. Kern, 2000, How Accurate is Kinematic DGPS? To be submitted to Geomatica, Canadian Institute of Geomatics, April Cannon, M.E., K.P. Schwarz, M. Wei, and D. Delikaraogloy 1992, A consistency test of airborne GPS using multiple monitor stations. BuIletin Geodesique, Vol. 6, pp Czompo, J., 1993, GPS accuracy test for airborne gravimetry, paper presented at the Shth International Technical Meeting of the Satellite Division of the Institute of Ncrvigution, The Institute of Navigation of the United States, Salt Lake City, Utah. Czompo, J., 1994, Airborne scalar gravimetry system errors in the spectral domain, UCGE Rep , 133 pp., Dep. of Geomatics Eng., The Univ. of Calgary, Calgary, Alberta, Canada Ektin, B., 1982, Dynamics of Flight - stability and control. John Wiley and Sons. ESA, 1999, Gravity Field and Steady-State Ocean Circulation Mission, ESA SP-1233(1), ESP Publication Division, ESTEC, Noordwij k, The Netherlands. Featherstone, W., J, Evans, J. Olliver, 1998, A Meissl-modified Vanicek and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations. JournaZ of Geodesy, Vol. 72, pp Forsberg, R., 1984, A study of terrain corrections, density anomalies and geophysical inversion methods in gruvityf;eld modelling. OSU Rep. 33 5, Department of Geodetic Science and Survey, Ohio State University. Forsberg, R. and Madsen, F., 1 990, High-precision geoid heights for GPS levelling. Proceedings of the 2"' International Symposium on Precise Positioning with the Glo bal Positioning System, pp , Sept. 3-7, Ottawa, Canada. Fonberg, R., 1993, Impact of airborne gravirnetry on geoid determination - the Green;and example. Bull. International Geoid Service, no. 2, pp , Milano. Forsberg, R., and S. Kenyon, 1994, Evaluation and downward continuation of airborne gravity data - the Greenland example, paper presented at the international Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Department of Geornatics Engineering, The University of Calgary, Banff, Canada-

165 Forsberg, R., A.V. Olesen, and K. Keller, 1999, Airborne gravity survey of the north Greenland SheIf1998, Technical Report, No. 10, National Survey and Cadastre, Denmark* Fotopoulos, G., C. Kotsakis, and M.G. Sideris, 1 999, Evaluation of Geoid Models and Their Use in Combined GPS/levelling/Geoid Height Network Ac$ustments. Technical Report No , Department of Geodesy and Geoinforrnatics, University of Stuttgart, Germany. Glennie, C.L., 1999, An analysis of airborne gravity by strapdown LVS/DGPS. Ph.D. thesis, University of Calgary, UCGE Report # 20125, Calgary. Glennie, C.L. and K.P. Schwarz, 1999, A comparison and analysis of airborne gravimetry results from two strapdown inertialldgps systems, accepted for publication by Journal of Geodesy. Glennie, C., K.P. Schwarz, A.M. Bruton, R. Forsberg, A.V. Oleson and K. Keller, 1999, A Comparison of stable platform and strapdown airborne gravity. Accepted by the Journal of Geodesy, Springer-Verlag, Berlin, Germany, December GRACE, 1998, Gravity Recoveiy and Climate Experiment: Science and Mission Requirements Document, revision A, PLD-15928, NASA's Earth System Science Pathfinder Program. Grant, F.S. and G.F. West, 1965, Interpretation Theory in Applied Geophysics. New York, McGraw Hill Book Co. Inc. Gumert, W.R., 1995, Third generation aerogravity system,paper presented at the UG Symposium on Airborne Gravity FieldDetermination, The Int. Union of Geod. and Geophys., Boulder, Colo.. Hagiwara, V., 1976, A new formula for evaluating the truncation error coefficient, Bull. Geod., Vol. 50, No. 2. Halpenny, J.F., and D.M. Darbha, 1995, Airborne gravity tests over Lake Ontario, Geophysics, 60, Hammada, Y. and K.P. Schwarz, 1997, Airborne gravimetry, model-based versus eequency-domain filtering approaches, In: M.E. Cannon, and G. Lachapelle (eds.), Proc. International Symposium on Kinematic System in Geodesy, Geomatics and Navigation, Banff, Canada, June 3-6, pp Hammer, S., 1950, Recent development in gravity prospecting. Quart. Colo. School of Mines, V. 45(4a), pp Hammer, S., 1983, Airborne gravity is here!, Geophysics, 48, Harrison, J.C., J.D. MacQueen, A.C. Rauhut, and J.Y. Cnrz, 1 995, The LCT airborne gravity system,

166 paper presented at the UG Symposium on Airborne Gravity Field Determination, The ht. Union of Geod. and Geophys., Boulder, Colo.. Heiskanen, W-A- and H. Moritz, 1967, Physical Geodesy. W.H. Freeman and Company, San Francisco. Hotine, M. 1969, Mathematical Geodesy. ESSA Monograph 2, US. Department of Commerce, Washington, D.C.. Hsu, H.T. arid Zhu, Z.W., 1979, The approximation of Stokes' function and estimation of trucation error. Paper presented at the I Th IUGG General AssernbZy, Canbema, Australia Hsu, H.T., 1984, Kernel function approximation of Stokes integral, Proceedings of Local Gravity Field Approximation, Beijing, China. IgeS - International Geoid Senice, 1997, The Earth Gravity Model EGM96: Testing Procedures at IgeS- Special Issue, Bulletin n. 6, D.I.I.A.R., Politechnico di Milano, Italy. Jekeli, C., 1981, Modifying Stokes' function to reduce the error of geoid undulation computations. J Geophys Res 86(E38): Jordan, S.K., 1972, Self-consistent statistical models for the gravity anomaly, vertical deflections, and undulation of the geoid. J. Geophys Res., 77, Kaula, W.M., 1979, Statistical and harmonic Analysis of Gravity- Army Map Service, Technical Report No. 24. Kenyon, S., 1998, The contribution of airborne and polar gravity to NIMA's global gravity modelling efforts, International Workshop 'Xirborne Gravity and the Polar Gravity Field" Kangerlussuaq, Greenland, June 2-4. Kleusberg, E., D. Peyton, and D. Wells, 1990, Airborne gravimetry and the global positioning system, paper presented at the IEEE PLANS 1990 Meeting, IEEE AES Society, Las Vegas, NV. Klingele, E.E., L. Bagnaschi, M. Halliday, M. Cocard, and H.G. Kahle, 1994, Airborne gravimetric survey of Switzerland. Eidgenossische Technische Hochschdee Zurich, Institut fk Geodasie und Photogrammetie, Bericht 239. Klingele, E.E., and M.E. Halliday, 1995, Airborne gravity survey in Switzerland, paper presented at the L4G Symposium on Airborne Gravity Field Determination, The Int. Union of Geod. and Geophys., Boulder, Cola-. Knickmeyer, E.T., 1990, Vector Gravimetry by a Combination of inertial and GPS Satellire

167 Measurements, UCSE Reports, No , Department of Surveying Engineering, The University of Calgary. Kulhanek, 0., 1 976, Introduction to Dis-tal Filtering in Geophysics, Elsevier Scientific Publishing Company. Lachapelle, G., 1984, Gravity anomdy field component estimation with integrd fodae. Proceedings of Local Gravity Field Approximation, Beijing, China Lachapelle, G., 1995, GPS: Theory and Applications. ENSU 625 Lecture Notes, Department of Geomatics Eningeering, The University of Calgary, Canada Li, Y.C. 1988, Oceanic Geoid Determination via GB VP. M-Sc thesis, Department of Geodesy, Wuhan Technical University of Surveying and Mapping, China. Li, Y.C., 1993, Optimized Specfral Geoid Determination. UCGE Report 20050, Department of Geomatics Engineering, The University of Cdgq, Canada Li, Y.C. and M.G. Sideris, 1994, Improved gravimetric terrain correction. Geophysical Journal International, Vol. 1 19, No. 3, pp Li, Y.C., M.G. Sideris and K.P. Schwarz, 1995, A numerical investigation on height anomaly prediction in mountainous areas. Bulletin Geodesique, Vol. 69, No. 3, pp Li, Y.C. and M.G. Sideris, 1997, Terrain correction for airborne and ground vector gravity measztrements in land, ocean and coastal areas with different density options. Research report submitted to Sander Geophysics Ltd., Ottawa. Li, Y.C., M.G. Sideris, and K.P. Schwarz, 1999, Unified terrain correction formulas for vector gravimetry. Accepted on September 1999 for publication by PiNSA Physical Sciences Indian National Science Academy- Lines, L.R., 1988, Inversion of geophysical data. Geophysics reprint series No. 9, Society of Exploration Geophysicists, Tulsa, USA. Louis, A., 1989, Inverse und schlechf gestellte probleme. B.G. Teubner, Stuttgart, Germany. McRuer, D., I. Ashkenas and D. Graham, 1973, Aircrafl Dynamics and Automatic Conpol. Princeton University Press, Princeton, New Jersey. Meissl, P., 1971, On the linearization of the geodetic boundary value problem. OSU Rep 152, Dept of Geodetic Science and Surveying, The Ohio State University.

168 Menke, W., 1 984, Geophysical data analysis: discrete inverse theory. Academic Press, Tronoto. Molodensky, M.S., V.F. Eremeev and M.I. Yurkina, 1960, Methods for the study of the gravitational field of the earth. Translated from Russian, Israel program for scientific translations, Jerusalem, 1962a. Moritz, H., 1980, AhancedPhysical Geodesy, Herbert Wichrnann Verlag, Karisruhe, Abacus Press, Tunbridge Wells, Kent. Moritz, H., 1 984, Geodetic Reference System h: C.C. Tscherning (ed.), The Geodesist's Handbook, Bulletin Geodesique, Vol. 58, pp Nagy, D., 1966, The prism method for terrain corrections using digital computers. Pure appl. Geophys. Vol. 63, pp Nashed, M.A., 1974, Approximated regularized solutions to improperly posed linear integral and operator equations. In Lecture Notes in Mathematics, No. 430, Berlin, New York. Nettleton L.L., L.J.B. LaCoste, and Thompson, L.G.D., 1960, Tests of an airborne gravity meter, Geophysics, 25, 1, pp Nielsen, L. and B.H. Jacobsen, 200, Integrated gravity and wide angle seismic inversion of twodimensional crustal model. Geophys. J. Int. 140, pp Olesen, A.V., R. Forsberg, and A. Gidskehaug, Airborne pvimetry using the LaCoste & Romberg gravimeter - an error analysis, In: M.E. Cannon, and G. Lachapelle (eds.), Proc. International Symposium on Kinematic System in Geodesy, Geomatics andhravigation, Bd, Canada, June 3-6, 1997, pp Oppenheirn, A.V. and R. W. Schafer, 1989, Discrete-Time Signal Processing. Prentice Hdl- Pellinen, L., 1970, Estimation and application of degree variances of gravity. Stud Geiuphys. Et Geod., 14, Peters, M.F., and J.M. Brozena, 1995, Methods to improve existing shipboard gravimeters for airborne gravimetry, paper presented at the L4G Symposium on Airborne Gravity Field Determination, The Int. Union of Geod. and Geophys., Boulder, Colo.. Rapp, R-H., , The Earth's gravity field to degree and order 180 using Seasat altimeter data, terrestrial gravity data, and other data, OSU Rep. 322, Dep. of Geod. Sci. and Sw., Ohio State Univ., Columbus, Ohio, USA. Rapp, R-H., Y.M. Wang, and N.K. Pavis, 199 1, The Ohio State 1991 geopotential and sea surface

169 topography harmonic coefficient models, OSU Rep , Dep. of Geod. Sci. and Surv., Ohio State Univ., Columbus, Ohio, USA- Rauhut, A.C., 1992, Regularization methods for the solution of the inverse Stokes problem. UCGE Reports # Department of Geomatics Engineering, The University of Calgary, Canada. Salychev, O.S., A.V. Bykovsky, V.V. Voronov, K.P. Schwaq Z. Liu, M. Wei, and J. Panenka, 1994, Determination of gravity and deflection of the vertical for geophysical applications using the ITC-2 platform, paper presented at the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Dep. Of Geomatics Eng., Univ. Of Calgary, Banff, Canada. Salychev, O.S., and V.V. Voronov, 1995, Airborne gravimetry application of the ITC-2 inertial survey system. UCGE Rep ,34 pp., Department of Geomatics Engineering, The University of Calgary, Canada. Savage, P., 1 978, Strapdown Sensors. In Strap-down Inertial Systems, AGARD Lecture Series No. 95, Advisory Group for Aerospace Research & Development, Noah Atlantic Treaty Organization. Schwarz, K.P., 1979, Geodetic improperly posed problems and their regularization. ANN0 XXXVIII - Bollettino Di Geodesia E Science Afini - N. 3. Schwarz, K.P., 1980, Gravity field approximation using inertial survey systems. The Canadian Surveyor, Vol. 34, No. 4, pp Schwarz, K.P., 1984, Data types and their spectral properties, Paper presented at International Summer School on Local Gravity FieZdApproximation, The Int. Assoc. of Geod., Beijing, China Schwarz, K.P., M.G. Sideris, and R. Forsberg, 1990, The use of FFT techniques in physical geodesy. Geophysics. J. Int. Vol. 100, F? Schwarz, K.P., 0. Colombo, G. Hein, and E.T. Knickmeyer, 1991, Requirements for airborne vector gravimetry, paper presented at the XX General Assembly of the IUGG, ht. Union of geod. And Geophys., Vienna, Austria, August Schwacz, K.P., Y.C. Li, and M. Wei, 1994, The spectral window for airborne gravity and geoid determination, paper presented at the litternational Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Dep. Of Geomatics Eng. Univ. Of Calgary,. Banff, Canada Schwarz, K.P., and M. Wei, 1993, Inertial Geodesy and I2VS/GPS Integration. Lecture Notes, Department of Geomatics Engineering, The University of Calgary, Canada, Schwarz, K.P., and M. Wei, 1 995, Some unsolved problems in airborne gravimetry, in Proceedings