Geopotential numbers from GPS satellite surveying and disturbing potential model: a case study of Parana, Brazil

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1 Note: This is the accepted version of the ms /JAG available at: Journal of Applied Geodesy Geopotential numbers from GPS satellite surveying and disturbing potential model: a case study of Parana, Brazil Vagner G. Ferreira and Silvio R. C. de Freitas Abstract. This article deals with the approach to determining the geopotential numbers from Global Positioning System (GPS) satellite surveying and disturbing potential at a point on the Earth s surface. The disturbing potential was solved by the Brovar-type solution of Molodenskii s boundary value problem but in the context of the fixed problem. The solution is compatible with the techniques for smoothing of external field: remove restore techniques, residual terrain model, and use of highresolution global geopotential models. This method provides an absolute geopotential numbers related to the so-called world height system. As an example, we calculate the disturbing potential model in Southern Brazil. The general condition of the sparse and absent gravity values is the main point to be solved. The absolute and relative accuracies of the quasigeoid model were tested vs. GPS/leveling data. Based on 99 GPS/leveling points, the mean value of fitting for the quasigeoid model was estimated near m in the absolute view and 0.2 ppm in the relative view. Keywords. Quasigeoid, gravity disturbances, V. V. Brovar, height anomalies. 1. Introduction In accordance with Sansò and Vanícek (2006), for the adjustment of a precise leveling network, it is necessary to use a vertical coordinate defined in a so-called holonomic ( path-independent) height system. In this context, the natural holonomic vertical coordinate system is based on the geopotential number. To compute orthometric and normal heights, gravity is usually measured along with the leveling lines. However, since su cient gravity observations are not available, normal gravity has been used in calculating the normal geopotential numbers (also known as spheropotential numbers) as approximations of geopotential numbers. A height system defined in the holonomic context characterized above is very important for many practical purposes. In this article, we are interested in vertical datum unification to a world height system (WHS). Following Rummel (2002), the main point of the vertical datum connection is to determine the potential di erences and, in the end, the height di erences between them. The heights of the Brazilian Vertical Network (BVN) were obtained using spirit leveling dissociated from gravity information. The same problem is observed in several countries in South America; only relative normal orthometric corrections are applied in some networks (de Freitas and Blitzkow 1999). In doing it thus, provides the South American countries height system with geopotential numbers is the main obstacle to be overcome. Taking into account that the leveling observations are devoid of gravity values, we can ask Is it necessary relevel the country using spirit leveling associated with gravimetric observations? Is it possible to calculate a geoid or quasigeoid model with the precision required for such work? Regarding the first question, the current extent of leveling network in Brazil is about four times the distance around the world. If we consider counterleveling as well, this number doubles (see Figure 1). In addition, spirit leveling is a very time-consuming, tedious, and expensive operation. With regard to the second question, the geoid (or quasigeoid) is still a topic of fundamental importance for emerging nations such as Brazil. Many regions around the world require an improved gravimetric database to support very accurate geoid (or quasigeoid) modeling for the modernization of height systems. A modern height system enables the emerging technologies of Global Navigation Satellite Systems (GNSS) technology, in particular, the Global Positioning System (GPS), to obtain normal or orthometric heights. Most of the South American vertical datums are based on a determination of mean sea level (MSL) at di erent tide gauges over a varying range of time intervals and at di erent epochs (Bolivia and Paraguay are landlocked countries). Therefore, each vertical datum is referred to a particular equipotential surface, associated with the tide gauge and fixed for

2 2 V. G. Ferreira and S. R. C. de Freitas Figure 1: The Brazilian Vertical Network (BVN). Red dots show the location of bench marks with normal orthometric heights (about 63,409 bench marks). BVN data courtesy of Brazilian Institute of Geography and Statistics. a specific epoch. In general, these surfaces are not coincident with the global geoid. The ocean surface does not coincide with a surface in equilibrium (e.g., the geoid) of Earth s gravity field; the deviations are called sea surface topography (SST, also Dynamic Ocean Topography) (Torge 2001, p. 78). Recently, the International Association of Geodesy proposed in its program of activities a study of regional vertical systems and their relations to a WHS for practical applications as one of main objectives. This goal is proposed to be achieved until This proposal is in accordance with the Geocentric Reference System for the Americas (SIRGAS) statements for establishing the South American Vertical Network. There is a tendency among South American countries to monitor and integrate the vertical networks according to SIRGAS project statements to take advantage of modern geodetic space technologies in a continental basis. A unified vertical datum in South American is important for monitoring common problems related to the environment, engineering, natural resources, land management, and cadastral surveying. In this article, we will follow, from a practical point of view, the basic equations suggested by Hofmann- Wellenhof and Moritz (2006, p. 318) in determining the geopotential numbers from GPS satellite surveying on the bench marks and a disturbing potential model. Results show the potential di erences between the actual height system (normal orthometric) and the WHS defined in the so-called holonomic system. However, since these normal-orthometric heights are referred to a local datum, these comparisons may be biased due to the e ect of localized SST and possible o set between the unknown reference for the BVN and the local quasigeoid model. 2. Experimental procedures 2.1. Geopotential numbers from disturbing potential The principal objective is to determine the disturbing potential ðt P Þ by geodetic boundary value problem (GBVP) and its relationship with the geopotential number ðc P Þ. Starting with the basic idea given by Hofmann-Wellenhof and Moritz (2006, p. 318) as

3 GPS satellite surveying and disturbing potential model 3 W P ¼ U P þ T P ; it is the geopotential required by C P ¼ W 0 W P : ð1þ ð2þ The geopotential number C P contains the physical measure of height above an equipotential surface associated with the global MSL, conventionally obtained by leveling and gravity observation along the leveling lines. The geopotential numbers are computed in a direct way from gravity data. It is more general than the geometric determination of the normal height from ellipsoidal height associated with the quasigeoid model according to the relationship H N P ¼ h P z P ð3þ where H N is the normal height, h is the ellipsoidal height, and z is the height anomaly. To transform C P from the conventional global W 0 to the regional level W j 0, the di erence between W 0 and W j 0 can be determined by GPS/leveling in selected colocation points by dw ¼ W 0 T P UP GPS C j P : ð4þ The reference ellipsoid and normal gravity field is defined by Somigliana-Pizzetti theory, and the normal gravity potential and all its derivatives can therefore be regarded as known in the exterior space to the reference ellipsoid Disturbing potential on the Earth s surface In agreement with Heck (2011), the first-order solution of Molodenskii s boundary value problem can obtained using the Brovar-type solution. However, V. V. Brovar has developed the solution of Molodenskii s problem in the context of a fixed GBVP (Brovar 1972), where the first-order solution is given as T P ¼ R ðð ðdg þ g 1 þþhðcþ ds; ð5þ 4p s where HðcÞ is Hotine s function calculated from Hofmann-Wellenhof and Moritz (2006, equation (2-368)), R is the mean radius of the Earth, s is the unit sphere, c is the angle between the points of interest P, and ds is the integral area element. The first correction g 1 is g 1 ¼ R2 2p ðð s h h P l0 3 dgds; ð6þ l 0 is the Euclidian distance between the computation point, and the integration element is computed from Hofmann-Wellenhof and Moritz (2006, equation (2-395)). Gravity disturbances dg can be calculated using (Hofmann-Wellenhof and Moritz 2006, p. 93) dg P ¼ g P g P ð7þ where g P is the gravity value of observed point P on the Earth s surface and P is the normal gravity value at point P computed from Hofmann-Wellenhof and Moritz (2006, equation (2-208)). The solutions of equations (5) and (6) are particularly well suited for fast Fourier transform technique (for details, see Sideris and Schwarz 1986; Schwarz et al. 1990). In this study, the gravimetric disturbing potential model was calculated, with the available gravity disturbance as a starting point, from which the terrain e ects and the reference field, global geopotential model (GGM), were removed, thus producing smooth gravity disturbances. The terrain effects are calculated using the residual terrain model (RTM) according to Forsberg (1984) to better account for the short wavelengths (small features) of the Earth s gravity field. The remove restore technique, in accordance with Forsberg and Tscherning (1981), can be applied on the disturbing potential, which includes the following steps: Remove step Restore step dg res ¼ dg dg GGM dg RTM T ¼ T GGM þ T res þ T RTM : ð8þ ð9þ More details about the remove restore technique can be also find in books such as Geodesy by Torge (2001, Section 6.7.2) and Physical Geodesy by Hofmann-Wellenhof and Moritz (2006, chapter 11) Data sets applied The methodology introduced in the previous sections is applied to a test area in Parana, south of Brazil, to compute the disturbing potential model. Regarding the data, to solve equation (5), an important factor is their spatial resolution. The gravity coverage that is su cient for geoid (quasigeoid) modeling depends on the accuracy required for disturbing potential. In the remove restore technique, the spherical harmonic coe cients for the GGM introduce mainly long-wavelength errors into T GGM.

4 4 V. G. Ferreira and S. R. C. de Freitas Errors in the T res component depend on the coverage, density, and accuracy of the regional gravity data, dg. Short-wavelength errors are introduced in the dg RTM and T RTM components due to the spacing of the digital terrain model (DTM). For more details about the gravity data requirements, particularly, the necessary resolutions for a desired precision in geoid computation, see, e.g., Kearsley (1986) and Jekeli et al. (2009); see also the discussion on topographic reductions and aliasing e ects on gravity and the geoid based on various DTMs by Tziavos et al. (2010). The data set for the numerical investigation of this study were obtained from the Sao Paulo University gravity database for Brazil. Observed gravity data in the study area were available with varying distribution, mostly along the roads and valleys. After careful outlier detection, a total of 4104 observed gravity data points were accepted in addition to the corresponding elevation data. These gravity points are available for the region bounded by 7 in longitude and 5 in latitude (Figure 2). For the region of study, we have one gravity observation per approximately 105 km 2. How do we compute a precise geoid (quasigeoid) with this gravity coverage? For most of the gravity points, the error in their height, observed barometrically, is about 6 m; this estimate infers an error in the gravity anomaly of 1.8 mgal. This situation will not be improved upon until new gravity surveys are conducted using improved height determination techniques, such as leveling or even a combination of GPS and gravimetric geoid heights, or, for purely geodetic purposes, gravity and GPS in the context of the fixed GBVP. However, a scheme was devised to convert the orthometric heights associated with gravity values on ellipsoidal heights using a geoid model from Earth Gravitational Model 2008 (EGM2008; Pavlis et al. 2008) up to degree The EGM2008 up to degree and order 360 was used for modeling the long wavelength of gravity field. A DTM, specifically the Shuttle Radar Topography Mission (SRTM30_PLUS) (Becker et al. 2009), was used to model the short-wavelength features of the gravity field. In the context of equation (6), the EGM96 (Lemoine et al. 1998) geoid model was restored and we used ellipsoidal heights from SRTM to calculate the term g 1. RTM data (d grtm and T RTM ) are constructed from SRTM elevations and the long wavelength from the spherical harmonic model of Earth s topography (DTM2006.0; Pavlis et al. 2007) up to degree and order Results and discussions The gravimetric disturbing potential model (cf. Figure 3) was computed as a tool to support the Figure 2: Distribution of 4104 gravity points (red dots) and 97 GPS/leveling (blue dots) on the land over the test area in south Brazil.

5 GPS satellite surveying and disturbing potential model 5 Figure 3: The disturbing potential model for the region of the study. Disturbing potentials are given with respect to the World Geodetic System 1984 (WGS 84) (unit is m 2 s 2 ; contour interval is 10 m 2 s 2 ). identification of datum biases in relation to global datum and of potential inconsistence in the small portion of the BVN. It was calculated under the conditions in Sections 2.2 and 2.3, however, that the gridding was made by least squares collocation using the residual gravity disturbances (band limited). To evaluate the disturbing potential model, 97 GPS/ leveling points are available, with the locations illustrated on the map in Figure 2. The problem with this analysis lies in the quality of the observations. For example, the standard deviations of the GPS heights range from a maximum of 9.1 cm and to a minimum of 2.3 cm, with an average of 5.0 cm. For most part of the BVN, leveling measurements p were carried out with a tolerance e4 mm ffiffiffi k, where k is the leveled distance in kilometers. If we take into account the error associated with normal orthometric height, these values are about the 8.9 cm. The di erence between the ellipsoidal height ðhþ and normal orthometric height ðh NO Þ is given by h BVD ¼ h H NO : The error propagation to the h BVD can be given as ð10þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s hbvd ¼ sh 2 þ s2 : H NO ð11þ We can estimate an error of about 10.4 cm for h BVD. The direct application of the relation (4) is not possible in the Brazilian context because C j P (geopotential numbers) are not available. Therefore, we can get from BVN C 0j P NO ¼ ghp ; ð12þ where C 0j NO P is the spheropotential number, HP is the normal orthometric height, and g is the mean value of the normal gravity along the normal plumb line between the level ellipsoid and the spheropotential surface UðH NO Þ. The di erence between geopotential number and the spheropotential number can be given as dc ¼ C j P C 0j P : Applying equation (13) in equation (4), dw ¼ W 0 T P U GPS P the following relation can be obtained dg ¼ W 0 T P U GPS P C 0j P dc P; C 0j P ; ð13þ ð14þ ð15þ

6 6 V. G. Ferreira and S. R. C. de Freitas Table 1: Statistics of the di erences between the GPS/ leveling ðh BVD Þ and height anomaly ðz WHS Þ and the potential di erence dg at 97 bench marks. Table 2: Statistics of the di erences between the GPS/ leveling ðh BVD Þ and geoid models (EGM2008 and MAPGEO 2010) (in meters) at 97 bench marks. Parameter Min. Max. Mean Standard deviation Statistics Model dh (m) dg (m 2 s 2 ) where dg is the potential di erence between the WHS and the local height datum, i.e., BVD. We adopt the geopotential value W 0 ¼ 62, 636, m 2 s 2 (cf. Burša et al. 2007) for a definition of WHS in equation (15). The normal gravity potential U GPS in equation (15) is computed in the point at the Earth s surface using Somigliana s formula (cf. Hofmann-Wellenhof and Moritz 2006, p. 69). Local terms h BVD and the height anomalies z WHS must correspond to the same point on the Earth s surface, and the di erences dh can be expressed as dh ¼ h BVD z WHS ¼ dg g : ð16þ The grid of disturbing potential model (Figure 3) was interpolated to the position of GPS/leveling stations (Figure 2) based on a gridding scheme such as collocation. A comparison with a pure gravimetric solution gave a 3.2 cm mean residual with 23.4 cm standard deviation (see Table 1) without reductions of systematic distortions between the references. This leads to an approximated 3.2 cm a priori estimate for the average permanent SST in the area (Figure 3). However, this estimate is biased by leveling errors, local MSL definitions, GPS satellite surveying errors, and inconsistence between the reference surface for the normal orthometric heights and the quasigeoid model. The derived parameter h BVD from GPS/leveling are used for comparison with the geoid models EGM2008 up to degree 2190 and MAPGEO 2010 (Table 2). Table 2 shows that the EGM2008 has a slightly better fit to the BVN, indicating that the gravity data in the study area (Figure 2) should be revised. The EGM2008 was used here in its functional geoid height (geoid model). The MAPGEO 2004 is the o cial geoid model provided by the Brazilian Institute of Geography and Statistics. However, the PRQ is a quasigeoid based on disturbing potential model showed in Figure 3, after applying Bruns equation. In particular, the mean of the di erences reveals MAPGEO EGM2008 PRQ Min Max Mean Standard deviation Root mean square Table 3: Comparison of GPS/leveling vs. quasigeoid model in the relative view. Parameter Min. Max. Mean Standard deviation E W (ppm) N S (ppm) positive o sets of the BVD from the three models (MAPGEO, EGM2008, and PRQ). We know that the results of GPS observations and leveling in the relative form have high accuracy. This leads us to estimate the relative accuracy for the quasigeoid model based on the di erence of the derived normal heights from combination of GPS and quasigeoid between A and B vs. height di erences from leveling to the same points. This di erence can be presented in the relative form in parts per million ( ppm): ppm ¼ ½Dh AB Dz AB Šðin mmþ ; ð17þ ½ds AB Šðin kmþ where ds is the length of the baseline. However, the choice of the pairs A and B for the calculation of the leveling di erences must comply with the resolution to which the quasigeoid model meets (ds AB b 5 km). Table 3 shows the results of fitting in relative sense for the quasigeoid model in the east west direction (E W) and in the north south direction (N S). We found that testing the geoid models in the sense of relative accuracy gives realistic information about the quality of quasigeoid model. It is worth mentioning that all height concepts are related to potential di erences only, whereas no use is made of absolute potentials. From this point of view, the disturbing potential model associated with GPS satellite surveying can be used for e cient determinations of potential di erences.

7 GPS satellite surveying and disturbing potential model 7 4. Summary The heights of BVN were determined without observations of the modulus of gravity intensity; however, corrections based the normal gravity field were applied on the slopes obtained by spirit leveling. That is why this system is unique and incapable of being connected to the leveling networks of neighboring countries without a prior treatment. Alternative methods that allow the determination of geopotential numbers and hence the determination of heights connected to real Earth s gravity field are needed. The simple method presented in this article was suggested by Hofmann-Wellenhof and Moritz (2006) and it is based on the determination of geopotential numbers using the disturbing potential on the Earth s surface, GPS satellite surveying, and a value for the geopotential W 0. This approach allows the determination of geopotential numbers linked to a WHS as well as a local system. The evaluation of disturbing potential on the surface, in this sense, is related to the GBVP, in agreement with Van Gelederen and Rummel (2001). GBVP is the starting point for determining the Earth s gravity field. The approach to determine a geopotential number from gravity observable uses band-limited gravity disturbances combining regional gravity data, GGM, in our case, the EGM2008, and the RTM; in this sense, the SRTM plays a crucial role in geodetic sciences. The accuracy and the resolution of the geopotential numbers depend on the accuracy and resolution on the quasigeoid model as well as the GPS satellite surveying. We have applied the direct Brovar-type solution for the Molodenskii s boundary value problem to determine a new experimental quasigeoid model for the study area of Parana, south Brazil. As a result, the main obstacle to precise geoid (quasigeoid) determination in Brazil is the lack of gravity data in some regions. The location of which is not even the case, and the error of omission is still a problem. Acknowledgments The authors would like to express their sincere thanks to Prof. Bernhard Heck for the scientific discussion about the geodetic boundary value problems, to the CNPq (Brazilian Council of Research) for financial support ( process /2008-0), and to the PROBRAL project (CAPES/DAAD, process 228/06). We are very grateful to the editors and anonymous reviewers for their constructive comments and suggestions for the improvements on the manuscript. We thank Prof. Denizar Blitzkow and Prof. Nelsi C. de Sá for providing the gravity and GPS/leveling database. References Becker, J. J., Sandwell, D. T., Smith, W. H. F., Braud, J., Binder, B., Depner, J., Fabre, D., Factor, J., Ingalls, S., Kim, S.-H., Ladner, R., Marks, K., Nelson, S., Pharaoh, A., Trimmer, R., Von Rosenberg, J., Wallace, G. and Weatherall, P., Global bathymetry and elevation data at 30 arc seconds resolution: SRTM30_PLUS, Mar. 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8 8 V. G. Ferreira and S. R. C. de Freitas Schwarz, K. P., Sideris, M. G. and Forsberg, R., The use of FFT techniques in physical geodesy, Geophys. J. Int. 100 (1990), Sideris, M. G. and Schwarz, K. P., Solving Molodensky s series by fast Fourier transform techniques, J. Geodesy 60 (1986), Torge, W., Geodesy, 3 rd edition, Walter de Gruyter, Berlin, Tziavos, I., Vergos, G. and Grigoriadis, V., Investigation of topographic reductions and aliasing e ects on gravity and the geoid over Greece based on various digital terrain models, Surv. Geophys, 31 (2010), Van Gelederen, M. and Rummel, R., The solution of general geodetic boundary value problem by least squares, J. Geodesy 19 (2001), Author information Vagner G. Ferreira School of Earth Sciences and Engineering Hohai University 1st Xikang Road Nanjing, Jiangsu , China vagnergf@hhu.edu.cn Silvio R. C. de Freitas Department of Geomatics Federal University of Parana Curitiba, Brazil Received: aa Accepted: aa