Publication n 121 of the International Association of Hydrological Sciences Proceedings of the Anaheim Symposium, December 197G

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1 Publication n 121 of the International Association of Hydrological Sciences Proceedings of the Anaheim Symposium, December 197G DETECTABILITY OF LAND SUBSIDENCE FROM SPACE W. D. Kahn and F. O. Vonbun Goddard Space Flight Center Greenbelt, Maryland Abstract Earth surface motions on the order of one to several cm per year have been observed. These motions could be due to land subsidence (i. e. the gulf coast of Texas subsides about 5 to 10 cm year; likewise, similar values hold for portions of the Flordia coast), crustal uplift (e.g. dilatancy which is a phenomena observed to precede earthquakes primarily along thrust faults), and loading (i.e. due to large dams, etc.). Knowledge of these motions is of practical importance to government and local agencies. A possible technique of ranging from a spacecraft to a number of ground reflectors is considered for the accurate (i.e. to within fractions of cm/yr) detection of these motions. It is shown that over a six day period, assuming a 50% cloud cover, utilizing spaceborne precision ranging systems, intersite distances on the order of 5 to 25 km (dependent mostly on the beam width of the laser) can be determined in the vertical and horizontal components with errors in the 0.5 to 1.5 cm range. These errors are almost independent of ground survey errors up to 0.25 meters and orbit errors up to 200 meters. The results shown make such a system very attractive as a new tool for monitoring very small earth surface motions. INTRODUCTION In many geographic areas of the globe, extremely small motions on the earth's surface have to be studied and monitored. Land subsidence, is such a relatively slow process that may continue for several decades, as shown by Tank (1973) and Legget (1973). Subsidence may produce conditions that trigger some instantaneous event such as for example, the failure of a dam or a levee. A classical example for land subsidence is that experienced in Long Beach, California. In the latter city, an area of about 52 square km. subsided about 30 to 35 cm over a twenty year period. This corresponds to a average annual rate of subsidence of 1.75 cm/yr. A precision laser ranging system in orbit and corner cube retroreflectors distributed on the ground can be used to monitor small motions near or on large dams, major construction sites, shore facilities, and structures, etc. I. SYSTEM CONCEPT In order to observe small motions such as those that are associated with land subsidence, arrays of laser retroreflectors are to be distributed over the subsidence region as shown in Figure 1. The smallness of the motions to be observed over several years requires the use of a rather specific ranging and/or tracking systems concept. In essence, one wants to determine very accurately (cm-range) a distance 246

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3 D I or better, its changes between two ground points using satellite technology. In general, there are three major obstacles or error sources to overcome if one wishes to compute extremely accurate vector distances of 5 to 20 km, that is accurate to 0.5 to 1.5 cm from a spacecraft. These are: 1. Orbital errors of the spacecraft 2. Bias errors in the ranging system 3. Atmospheric propagation errors These can be eliminated to first order by using range differencing. Practically, this means sending one pulse from the spacecraft to two or more ground stations, subtracting these measurements and using only their "differences" to compute \ DJ, the distance between neighboring ground stations (i.e. transponders or laser corner cubes). It is thus evident by studying Figure 2 that the above mentioned error sources will not play any major role (only second order) since Ap kj = (p kj + Sp kj )- (P kj+1 + s P k] + 1 ) = Ok, ~< c k ]+ i) + (Sa, - 8 Pk ]+ i) For bias errors in the ranging system because both the j t h and (j + l) th reflectors are interrogated by the same signal (within the beam). For orbital errors and atmospheric refraction, Su,.., = 6.:;,. 4 (second order terms) r k j +1 ' k j v In summary, using range differences as the basic "measured" quantity eliminates all first order errors in the determination of the distance j D I. Using one pulse to cover more than one corner cube (or transponder) unfortunately raises a power problem. From a signal to noise ratio vantage point it is preferable to send a single beam out at a time to one ground station. This does, however, create a problem by introducing the effects of errors 1 and 2 mentioned before. For instances a two second time interval between pulses means their origin (spacecraft position) has separated in space by say 15 km. This means the orbit error introduced by those 15 kn along the orbit pass will now increase the error in the determination of D. The bias error in range, from firing to one cube and the subsequent firing to the other cube will further directly influence the determination of I D!. Thus the control of bias errors in the ranging system and the orbit error becomes an important systems design factor. A detailed analysis of this second possibility will subsequently be published. 248

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5 \ \ \ \ \ \ \ \ \ \ LASER NOISE: 5cm ORBIT ERRORS: UP TO 200 METERS SURVEY ERRORS: UP TO 0.2 METERS ZENITH ANGLE: 70 SAMPLE RATE: 1 MEAS/SEC BASELINE SEPARATION: 25km MEAN ORBIT ALT.: 400km ORBITAL INCLINATION: 50 HORIZONTAL 0.5 VERTICAL BOTH ERROR COMPONENTS ARE ALMOST LINEAR FUNCTIONS OF THE SYSTEM NOISE ONLY. (DOUBLING LASER SYSTEM NOISE DOUBLES THESE ERRORS) TIME IN ORBIT (DAYS) Figure 3. Intersite Distance Errors vs. Time in Orbit 250

6 o^ojst. ORBIT ERROR: 200 METERS SURVEY ERROR: 0.2 METERS BASELINE SEPARATION: 25km ORBIT ALT.: 400km ORBIT INCL.: 50 DAYS IN ORBIT: 6 ZENITH ANGLE: 70 SAMPLE RATE: 1 ME AS/SEC LASER SYSTEM UNCERTAINTY (cm) Figure 4. Intersite Distance Errors vs. Laser System Uncertainty 251

7 IL MATHEMATICAL MODELS 1. DIRECT BASELINE ESTIMATION USING SIMULTANEOUS RANGING The variation in D is the quantity needed to determine small (cm-range) horizontal and vertical displacements in the earth's upper crust. This quantity represents the magnitude of the vector D. The vector D is to be computed from range difference measurements which are scalar quantities. The basic measurement, slant range p, from the ground stations to the spacecraft or vice versa can be expressed in matrix notation as follows: where P = (Z T Z) 1/2 (1) Z T Z = U T U + S T S - 2S T U and (U T U) 1/2 = geocentric distance to spacecraft (S T S) 1/2 = geocentric distance to ground station By assuming that at time t k, a signal is sent out by the ranging system onboard the spacecraft and received at the j t h and (j + 1) th station. The range at each station is by (1) expressed as follows: P kj = L(U T U) k + (S T S) ] -2SjU k ] 1/2 P kj + 1 = l(^v) k + (S-r$) i + 1-2S] +1 U k ]"z (2a) (2b) with similar expressions as (2a) and (2b) resulting for times t k + x, t k+2,..., The range difference measurement is obtained by subtracting equation (2b) from (2a) which then represents the fundamental observation equation, that is: (P k] -P kj + 1 ) - t(u T U) k + (S T S) - 2SJU k ] 1/2 (3) - [(U T U) k + (S T S) j + 1-2S]' +1 U k ] 1 ' i In arriving at equation (3), the assumption made is that both stations, the j t h and (j + l) th simultaneously observe the laser pulse at time t k. Since the range difference measurement is subject to errors as all measurements are, and since D is derived from these measurements, D is also subject to errors. We estimate the errors of D due to errors in the range difference measurements (p. - p_. + ). To do so, standard linear estimation theory, as shown by Deutsch (1965), is used for a set of k observations of the same cubes. That is, by varying (3) and using first order terms of the Taylor expansion one obtains: y (kx l) ^ A(kX 6) X + 0 B (kx 3) s j + C (kx 3) S j+l + E (kx 1) ( 4 ) (6X 1) (3X 1) (3x i) 252

8 where y (k*l) v kj ''kj+l \kxi) A., s J _1_ [U, - S. J - rti _ c i (kx6) "1,. k j L Uk o. +1J "kj A 'kj+1 3 u; XQ 2 ÔU0 ( 6X 1 ) (6X1) B (kx3)s - J - [l^-spt ''kj s. - os. (3X1) L '(kx3) (kx3) ; tu k '"kj + 1 J c = r n _ s 1 T j + i J (3X1) 2. INTERSITE DISTANCE DETERMINATION AND ITS ACCURACY An estimation of the intersite distance between two ground emplaced corner cube and their errors is obtained from range difference measurements by using the standarc least squares estimation technique as shown by Deutsch (1965), Lynn (1974), and Reference 5. In order to minimize the effects due to geopotential uncertainties, multiple short orbital arcs of 1 to 1-1/2 revolutions are used. In the least squares process, estimates are to be obtained simultaneously for each orbital arc's state vecto as well as the coordinates of the corner cubes. The intersite distances and their erroi are computed by a least sqaures solution. A mathematical description of the estimatio process now follows. The generalized matrix equation which represents all the observation equations obtained from simultaneous ranging from the spacecraft to two ground emplaced laser retroreflectors reads as follows: where \ x i " 'tx (6r +6) T (6r + 6)X l + \ x x (5) r ' number of different orbital arcs for which the state vector is to be estimated r = 1, 2,... p; kj ; k. r total number of observations associated with each orbital arc. i > > > (6r -, 6) 253

9 The estimate of the vector T is obtained from -t range difference measurements with -i > > 1 and a priori information about the state of each orbital arc and station survey. That is: V^xo = [r^r + w^]" 1 [rnr'y + W;;T 0 ] (6) (6r +6)X (6r +6) (6r + 6)x 1 where W~ 1? A priori covariance matrix associated with system accuracy. WÇ 1 -A priori covariance matrix associated with orbit accuracy and station survey for each retroreflector. The errors associated with the estimate of Tp are obtained from the covariance matrix given by the following equations E(" T )- (^W 1!- +w;j]-* r + 6)X(6r + 6) (7) Equations (6) and (7) are now used to (a) compute the intersite distance and (b) its errors. The intersite distance, that is, the difference between the corner cube position vectors (s. - s". + 1 ) is obtained by a coordinate transformation of the vector T. This transformation reads as follows; D : '- (s. - s. +1) 3X j = v 3X ( 6r + 6 )»( 6r + 6)x i (8) where V ' L "(3X6r)! l (3X3)l (3X3) J 3X(6r+6) 0 '- a nu 11 ma t r ix 1 = the identity matrix The error in the intersite distance is now given by the covariance matrix of the intersite distances, that is: E[(. -s j+1 )(s. -s ]+1 )Tj (3X3) = {QE(TTT)QT} (3X3) (9) which upon introducing Equation (7) finally reads: EKSj -S j + 1 )^j -^]+1 ) T J (3 X3)-[ rtw "' lr + W ;^1, 3 T >(3X3) (10) Equation (10) represents the intersite distance error, and is the basic equation used for spaceborne laser ranging system error analysis. Numerical results using (10) are depicted in Figures 3 and 4. III. ERROR ANALYSIS RESULTS Figures 3 and 4 describe the results of a covariance error analysis in which the former figure provides an estimate of the precision with which the intersite vector 254

10 components can be determined for a 5-element, 25-km grid as a function of time in orbit and the latter figure relates intersite vector component precision to the spaceborne laser ranging system. For this analysis, very moderate systems errors were assumed. Although orbital errors up to 200 meters were used in the error analysis, their contribution to the intersite vector's components (i. e., vertical and horizontal) consistently remained negligible. In a similar fashion, survey errors associated with the location of the retroreflectors also do not significantly change the errors in the vertical and horizontal components of the intersite vector. In Figure 4 however, there is linear dependency indicated for both the vertical and, to a lesser extent, horizontal components of the intersite vector to the spaceborne lasers system precision. As can be seen, a 5-cm system is adequate, producing errors in the 0.7 and 1.3-cm ranges, respectively. Ground laser ranging systems having precision of ±5 cm are now operational at Goddard Space Flight Center, thus no difficulties are forseen in achieving this same level of precision for a spaceborne laser system. IV. PRACTICAL APPLICATIONS The spaceborne laser ranging system can be developed as a payload for the shuttle applications program. The typical mission duration would be 7 days with a nominally circular 400-km orbit.. Orbital inclination would be between 50 and 57 degrees. A breadboard model of the spaceborne laser system is under development at GSFC as shown by Minott (1975). Such a system can be utilized to detect small variations in the earth's upper crust in the 0.3 to 0.5-cm/yr range. Monitoring, is thus possible, of land subsidence as it occurs in the coastal regions of California, Texas, and Florida, as well as construction sites such as dams and even large buildings. V. CONCLUSIONS This paper and earlier Vonbun, Kahn, et al., have shown that a spaceborne laser ranging system can be used to determine intersite distances up to 25 km, depending only on the beamwidth or beam splitting of a spaceborne laser system within a 6-day shuttle mission to a precision 0.5 cm to 1.5 cm (assuming 50% cloud coverage). Covariance error analyses have shown that (a) orbital errors up to 200 meters, (b) survey errors up to 0.25 meters, and (c) range bias errors (many meters) do not significantly influence the accuracy with which the intersite vector can be determined. In fact, the latter error sources are second order effects. Consequently, the spaceborne laser ranging system as envisaged, has potentially many practical applications where small relative motions are to be monitored. REFERENCES Deutsch, Ralph, "Estimation Theory" Prentice Hall, Legget, T. R., "Cities and Geology", 1973, McGraw Hill, Inc. Lynn, J. J., "NAPCOV Program Documentation". Old Dominion Systems, Inc. November 1973 (revised May 1, 1974). 25,5

11 Minott, P. O., "Design of a Ground Based Laser Retroreflector Array for Space to Ground Geodey", Unpublished Report, Revised March Old Dominion Systems, Inc., "User's Guide to Data Preparation, Navigation Analysis Program (NAP 3.1)," April Tank, R. W., "Focus on Environmental Geology", Oxford University Press, Vonbun, F. O., Kahn, W. D., et al., "Spaceborne Earth Applications Ranging System SPEAR," Goddard Space Flight Center Preprint, December