COMPARISON OF SOME METHODS FOR MODIFYING STOKES' FORMULA IN THE GOCE ERA

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Harold Harrington
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1 COPARISON OF SOE ETHODS FOR ODIFYING STOKES' FORUA IN THE GOCE ERA J. Ågre (1) ad.e. Sjöberg (1) (1) Royal Istitute of Techology, Drottig Kristias väg 3, SE 1 44 Stockholm, Swede ABSTRACT The dedicated satellite gravity missio GOCE will drastically improve our kowledge of the log to medium wavelegths of the Earth's gravity field. I order to determie the fiest details i regioal geoid determiatio, however, we still have to utilise gravity data. It is the purpose of this paper to study three modificatios of Stokes' formula umerically, usig error propagatios with simulated stadard errors for the GOCE potetial coefficiets. The methods tested are the stadard remove-compute-restore, the least squares, ad the low-degree GOCE-oly modificatios. I the latter techique it is required that oly GOCE iformatio must ifluece the determiatio of the lowest degrees. It is cocluded that of the modificatios tested, the least squares method is most suitable to be used with a GOCE satellite-oly model. This is the case also whe pessimistic weights are used for the gravity aomalies. The mai fault with the stadard remove-compute-restore method is its sesitivity to log-wavelegth errors i the gravity aomalies, while a very large trucatio error is the most serious problem for the low-degree GOCE-oly techique. 1 INTRODUCTION The GOCE satellite, to be lauched durig 6, will utilise Satellite Gravity Gradiometry (SGG) ad Satellite to Satellite Trackig i the high-low (SST-hl) mode to determie the log to medium wavelegths of the Earth s gravity field. Accordig to the performed simulatios (e.g. [5] ad [1]), the geoid commissio error is expected to be somewhere aroud 1 cm for a resolutio of 8 km (correspodig to the spherical harmoic maximum degree = ). Eve though this performace is truly astoishig, terrestrial gravity will still be eeded to determie the highest harmoics i a regioal geoid determiatio. Now, terrestrial gravity observatios are usually distributed i ihomogeeous ways ad they are ofte affected by systematic errors. As such errors are difficult to detect, there is always the risk that the high accuracy obtaied from GOCE for the lower degrees is ruied i the combied solutio. Furthermore, if it is assumed that the GOCE Earth Gravity odel (EG) is used up to the degree, for which a geoid commissio RS-error of 1 cm or so is obtaied, it is ot reasoable to believe that the EG ca be improved by terrestrial gravity aomalies. It is therefore crucial to choose a kerel modificatio that is isesitive to the logwavelegth part of the terrestrial gravity aomaly spectrum. As the maximum degree is ot too high, the size of the trucatio error is also a importat factor to be cosidered i the GOCE case. I order to avoid log-wavelegth errors i the gravity aomalies, [11] proposed the low-degree GOCE-oly modificatio. This techique is derived to filter out the gravity aomaly power below degree +1 completely. Aother method that might be suitable is the least squares modificatio [8]. It is the mai purpose of this paper to study the error propagatio of these kerel modificatios ad compare them with the stadard remove-compute-restore (r-c-r) method that utilises a umodified kerel (see e.g. [7]). The focus is maily o the aspects that are importat whe a kerel modificatio method is used together with a GOCE satellite-oly EG. It is oted that also other modificatios might be suitable, for istace the Wog ad Gore method [15], but these ivestigatios are left for the future. THE UNBIASED GEOID ESTIATOR It is assumed that the geoid is computed by meas of a remove-compute-restore type of estimator usig some modificatio of Stokes fuctio. Both the terrestrial gravity aomalies as well as the EG cotributios are corrected for the direct topographic effect, ad the idirect topographic effect is fially added to the geoid height; see e.g. [3] ad [9] for details. The topographic reductio method is left uspecified. I order to keep the formulas below as simple as possible, o explicit correctios are itroduced for the zero ad first degree topographic effects [1]. Atmospheric ad ellipsoidal correctios are also implicitly uderstood. Now, if it is assumed that oly gravity aomalies over a spherical cap σ with radius ψ are utilised, the estimator becomes: Proc. Secod Iteratioal GOCE User Workshop GOCE, The Geoid ad Oceaography, ESA-ESRIN, Frascati, Italy, 8-1 arch 4 (ESA SP-569, Jue 4)

2 where + c DIR R EG DIR EG DIR N = S ( ψ) g g ( g g ) dσ c ( g g ) δni π + + r 1 p σ = = (1) g is the observed terrestrial gravity aomaly at the Earth s surface, * EG g is the gravity aomaly aplace =. Here R is the mea Earth radius ad γ is the mea ormal gravity at harmoic of degree for the EG ad c R/( γ ) sea level. Furthermore, the direct topographic effect o the surface gravity aomaly is deoted by DIR g, while the. A star * is used to deote dowward cotiuatio to sea level. The idirect effect o the geoid height is give by δ NI modified Stokes fuctio is defied as k + 1 S = S s P () ( ψ ) ( ψ) ( cosψ) k k k= where ( ) Pk cosψ is the egedre polyomial of degree k ad s k are the modificatio parameters. The maximum degree of modificatio is take to be equal to or larger tha the maximum degree of the EG. We ow itroduce ε EG ad ε for the radom errors i the topographically corrected aplace S ψ is the origial Stokes fuctio, ( ) g harmoics of the dowward cotiued gravity aomaly ad the EG, respectively. The errors are assumed to be ucorrelated ad to have zero expectatio for all. It is show i [1] that the spectral form of Eq. (1) is give by * DIR g DIR EG N = c s Q ( g g + ε ) + c ( Q + s)( g g + ε ) + δn (3) I = 1 = * * where s = s wheever ad s = otherwise. The trucatio coefficiets for the modified Stokes kerel Q are defied as π k + 1 Q = S( ψ) P( cosψ) siψdψ = Q ske (4) k ψ i which Q are the trucatio coefficiets for the origial Stokes fuctio ad e k are the so-called Paul s coefficiets. Neglectig the harmoics of degree zero ad oe, the true geoid height becomes = k = DIR N = c ( g g ) + δ N (5) I 1 It is ow assumed that the error covariace fuctios for the (topographically corrected) terrestrial gravity ad the EG are homogeeous ad isotropic. The correspodig error degree variaces are deoted by σ, ad g σ. Usig c EG, for the (sigal) gravity aomaly degree variaces of the topographically reduced field, the expected global mea square error becomes [8], 1 δn = E ( N N) dσ c s Q σ, g ( s Q ) σ, EG 4π = σ = (6) + c s Q σ, g + ( s + Q ) c + c Q σ, g + ( Q ) c = = Eq. (6) is fudametal i this paper as it is used to compare differet modificatio methods. If the parameters s k are give, the expected global RS-error ca be computed for the specified values of the sigal ad oise degree variaces. The terms cotaiig σ, ad g σ are the cotributios from the errors i the terrestrial gravity data ad the EG, EG, respectively, while those parts cotaiig c represet the bias or trucatio error. It is also possible to study the cotributio for differet frequecy bads. Note that o bias is preset for the degrees up to, which is the reaso for [8] callig the estimator i Eq. (1) ubiased.

3 3 SIGNA AND NOISE DEGREE VARIANCES I this sectio the sigal ad error degree variace models that were utilised i the error propagatio studies below are preseted. The sigal degree variace model used for the topographically reduced field above degree is of Tscherig ad Rapp type [14]: + ( 1) RB c = A (7) + B R ( )( ) with the parameters chose to R = 6371 km, A = 5 mgal, B = 4 ad R R B = 3.5 km as specified i Forsberg [4]. The degree variaces obtaied by Eq. (7) agree well o average with the spectral ivestigatios of the topographically corrected gravity field i [3] ad [4]. The shape of the oise degree variace curve for the EG stemmig from GOCE was take from Fig. 4.1 i [5]. The scale was chose so that a global commissio RS-error of 1 cm is obtaied for the maximum degree =. This is a little bit more pessimistic tha the value i Table 8.4 i [5], but is i good agreemet with other simulatios (e.g. [1]). However, for the preset study, the scale of the GOCE oise degree variace model is ot very crucial. What is importat is that the shape is reasoable at the same time as the commissio error is practically egligible up to some rather high maximum degree ; cf. the itroductio. These requiremets are fulfilled by the chose model, which is illustrated i Fig. 1. It is oted that the quality i the GOCE polar gaps are likely to be worse compared to the rest of the world [13], of coarse depedig o how ad to what extet the gaps are filled. This meas that the use of a homogeeous ad isotropic error model is ot strictly justified. However, we may view our global error degree variaces as represetative of the situatio outside the polar gaps oly, where the error field is almost homogeeous ad isotropic [5] Ucorrelated Correlated GOCE x 1-3 Ucorrelated Correlated GOCE.1 3 Degree-order RS [mgal] Degree-order RS [mgal] Degree Degree Fig. 1a,b. Degree-order stadard deviatios for GOCE ad the two gravity aomaly oise models with σ = 1 mgal, illustrated usig differet scales o the horizotal ad vertical axes. It is difficult to choose a represetative error model for the terrestrial gravity aomalies. Sice their quality depeds very much o locatio, the use of a homogeeous ad isotropic covariace fuctio is questioable. However, i the same way as for GOCE above, a global error model eed oly be viewed as represetative of the area over which it is used. This meas that we may view the error degree variaces as describig a global field with the same quality as is exemplified by the gravity aomalies iside the spherical cap i Eq. (1). It should be further oticed that the size of the spherical cap limits how much differet frequecies are affected by the gravity aomaly errors, but this fact should ot be reflected i the choice of error degree variaces. It is automatically take care of by the trucatio coefficiets i Eq. (6). I the simulatios below two differet oise models were utilised, which both have the variace σ distributed below the Nyquist degree (frequecy) of the gravity aomaly grid. Above this degree, the sigal degree variace model i Eq. (7) was used to model the presece of high frequecy iterpolatio errors. The first error model assumes that the observatio oise is ucorrelated, which is approximately modelled by badlimited white oise with costat degreeorder variace, σ, g /( + 1), up to the Nyquist degree; see e.g. [6]. As ca be see i Figs. 1a ad 1b, the resultig degree variaces for σ = 1 mgal seem rather optimistic compared to the GOCE-model. I order to model the situatio

4 whe systematic errors are preset, the secod error model describes correlated oise, which cotais more power i the lower portio of the spectrum. It is costructed as a mix of ucorrelated ad correlated oise, each cotaiig half the total variace, i.e. σ /. The ucorrelated part is modelled as badlimited white oise i the same way as above, but with half the variace, while the correlated part is quite arbitrarily defied with the shape idicated i Figs. 1a ad 1b, so that it cotais the other half of the variace. The reasos behid choosig this model are that the degree-order stadard deviatios should be realistic i compariso to GOCE ad that the terrestrial data should be affected by logwavelegth systematic errors that decrease with the degree. I all error propagatios preseted below, the sigal ad oise degree variace models preseted above were utilised. The gravity aomalies were assumed to be give i a equiagular 3 x3 grid, which correspods to the Nyquist degree o 36. The spherical cap was take to have the radius ψ = 5. I order to limit the ifluece of log-wavelegth errors i the gravity data, the cap should ot be too large, while a too small radius yields a large trucatio error. The chose value seems like a reasoable compromise. Furthermore, the stadard deviatio of the gravity aomaly oise was take as σ = 1 mgal, ad it was assumed that either the correlated or ucorrelated oise model costituted the truth. It should fially be metioed that the summatios to ifiity i Eq. (6) were summed to degree 18 i practice. 4 THE STANDARD REOVE-COPUTE-RESTORE ETHOD The first modificatio method to be tested was the stadard remove-compute-restore (r-c-r) method (see e.g. [7]), i which Stokes fuctio is ot modified at all. All modificatio parameters are cosequetly set to zero. The correspodig expected global RS-errors were computed by Eq. (6) usig both the ucorrelated ad the correlated gravity aomaly error models. The results are preseted i Table 1. As oted at the ed of Sect., RS-errors are preseted also for differet frequecy bads ad for the parts correspodig to the ifluece of errors i the gravity aomalies, to the errors i the EG ad to the trucatio error (or bias). Table 1. Expected global RS-errors for the stadard remove-compute-restore method. = =. Uits:[mm]. Global RS [mm] Noise model for g Ucorrelated, σ = 1 mgal Correlated, σ = 1 mgal Degrees All All Total g EG (GOCE) Bias By studyig Table 1 it ca be see that the stadard r-c-r method is very sesitive to the presece of correlatios i the gravity aomaly data. The RS cotributio is large for the degrees up to =, while the situatio improves drastically whe the observatios are ucorrelated. As it is believed that the correlated oise model is ot totally urealistic i may cases, cosiderig the well-kow fact that gravity data are ofte affected by systematic errors, it is cocluded that the simple r-c-r method is ot suitable for use together with a high-accuracy GOCE-model. It is too sesitive to log-wavelegth errors i the terrestrial gravity data. Aother problem with the stadard r-c-r method is the sigificat trucatio error. 5 THE OW-DEGREE GOCE-ONY ODIFICATION I order to block the ifluece of log-wavelegth errors i the gravity aomaly data, [11] proposed the low-degree GOCE-oly method, which implies that the modificatio parameters s k are chose so that the gravity aomaly cotributio is exactly zero up to the maximum degree of the GOCE-derived EG. It follows from Eqs. (4) ad (6), that this situatio is achieved whe the followig low-degree GOCE-oly coditios are fulfilled: k + 1 s e s = Q ; =,3,, (8) k k k = 1 I case =, Eq. (8) provides us with a system of equatios with the same umber of equatios as ukows. However, due to the fact that the ubiased estimator i Eq. (1) yields umerically the same geoid height for all choices

5 of s k for which the modified kerels coicide iside the spherical cap σ (o matter how the kerels behaves outside the cap), stadard trucated sigular value decompositio, or some other umerical techique, eeds to be applied to fid a solutio. As Eq. (8) cotais the oly coditios that are posed o the parameters, the method i questio will be called the pure low-degree GOCE-oly modificatio i this paper. A hybrid versio of the method for the case > is also suggested i [11]. I this case, the modificatio parameters are chose so that the expected global mea square error i Eq. (6) is miimised, at the same time as the coditios i Eq. (8) are fulfilled. This method is called the optimum low-degree GOCE-oly modificatio ad requires apriori kowledge of the sigal ad gravity aomaly error degree variaces i Eq. (6). The optimum solutio ca easily be foud usig well-kow methods from the differetial calculus, utilisig agrage multipliers to esure the fulfilmet of the low-degree GOCE-oly coditio. The solutio is obtaied by solvig the resultig system of equatios by trucated sigular value decompositio. Now, usig the same iput data ad type of table as before, but limitig us to the correlated oise model for the gravity aomalies, the propagated RS-values for the pure ad optimum low-degree GOCE-oly methods are preseted i Table. The same sigal ad oise degree variaces as applied i the error propagatio were used as apriori values for the weightig. Table. Expected global RS-errors for the pure ad optimum low-degree GOCE-oly modificatio methods. Correlated oise with σ = 1 mgal used as error model for g. Global RS [mm] ethod, Parameters Pure, = = Optimum, =, = 7 Degrees All All Total g EG (GOCE) Bias As ca be see i Table, the gravity aomaly errors do ot ifluece the degrees up to, which meas that the GOCE-model is left completely utouched exactly as required. O the egative side, we see that the basic problem is the huge trucatio error, which does ot reduce sigificatly as additioal parameters are added i the optimum versio of the method. It seems impossible to get rid of the trucatio error as log as the GOCE-oly coditio i Eq. (8) must be strictly fulfilled. Hece, we coclude that the pure ad optimum low-degree GOCE-oly methods are ot good ideas. 6 THE EAST SQUARES ODIFICATION ETHOD Cosiderig the results i the last sectio, it seems best to allow for some low-degree ifluece of gravity aomaly errors, as log as the expected global RS-error i Eq. (6) is small. Thus, we are led to the least squares modificatio of Stokes formula [8], i which the modificatio parameters are chose so that the total expected global mea square error (Eq. 6) of the ubiased geoid estimator is miimised. As i Sect. 5, the solutio is provided by the methods of the differetial calculus. This gives us a system of equatios [8], which ca be solved by trucated sigular value decompositio. It is also possible to utilise other umerical methods; see []. The method aturally requires apriori values for the sigal ad oise degree variaces. Assumig that the correct apriori models are kow, the propagated RS-errors for the correlated error model are preseted i the left part of Table 3. Table 3. Expected global RS-errors for the least squares modificatio method usig true ad pessimistic apriori oise model for g. = =. Global RS [mm] True oise model g Correlated, σ = 1 mgal Correlated, σ = 1 mgal Apriori oise model g Correlated, σ = 1 mgal Ucorrelated, σ = mgal Degrees All All Total g EG (GOCE) Bias As ca be, the improvemet is remarkable compared to the earlier methods. It might be objected that it is a serious limitatio of the least squares method that apriori values of the sigal ad oise degree variaces have to be kow. As the error degree variaces for GOCE will be estimated quite accurately, the most crucial problem here is that the gravity

6 aomaly degree variaces are ot kow. However, the least squares method is rather isesitive to the choice of weights. This is well kow from other areas of geodesy. et us illustrate this feature by meas of the results from a error propagatio, i which the ucorrelated error model with a very high stadard deviatio (σ = mgal) was used i the estimatio of the least squares parameters, while the correlated model with the same stadard deviatio as before (σ = 1 mgal) was utilised i the error propagatio. The results are preseted i the right part of Table 3. The isesitivity to the weightig is clearly illustrated. It ca further be observed that the resultig estimator is isesitive to log-wavelegth gravity aomaly errors at the same time as the trucatio error is low. Thus, it seems suitable i the GOCE case to use the least squares method with pessimistic weights for the gravity aomalies. 8 CONCUSIONS Whe a GOCE satellite-oly EG, which is expected to be highly accurate up to a rather high maximum degree, will be combied with terrestrial gravity aomalies, it is importat to apply a modificatio of Stokes formula that is isesitive to low-degree errors i the gravity aomalies. It is also importat that the trucatio error is small. I the error propagatios preseted i this paper, it was foud that oe of these requiremets are fulfilled by the stadard remove-compute-restore method, while the low-degree GOCE-oly method fails because of its huge trucatio error. The method most suitable for GOCE was foud to be the (ubiased) least squares modificatio of Stokes formula. It was also demostrated that the lack of a good apriori error model for the terrestrial gravity aomalies is ot a big problem for a successful applicatio of the method. I the GOCE case, a reasoably pessimistic error model of the terrestrial data seems like a good choice i matchig various error sources i a least squares sese. REFERENCES 1. Ditmar P., Kusche J., ad Klees R., Computatio of spherical harmoic coefficiets from gravity gradiometry data to be acquired by the GOCE satellite: regularisatio issues, Joural of Geodesy 77: , 3.. Ellma, A., O the umerical solutio of parameters of the least squares modificatio of Stokes formula, IAG Symp. Series 16, IUGG, Spriger Verlag Forsberg R., A Study of Terrai Reductios, Desity Aomalies ad Geophysical Iversio ethods i Gravity Field odellig, Rep 355, Dept Geod Sci, Ohio State Uiv, Columbus, Forsberg R., Spectral properties of the gravity field i the Nordic coutries, Iteratioal Symposium o the Defiitio of the Geoid, Florece, 6-3 may, ESA (1999), Gravity Field ad Steady-State Ocea Circulatio issio, ESA SP-133(1), Report for missio selectio of the four cadidate earth explorer missios, ESA Publicatios Divisio, pp. 17, July Rummel, R., Spherical spectral properties of the Earth s gravitatioal potetial ad its first ad secod derivatives, i: Geodetic Boudary Value Problems i View of the Oe Cetimeter Geoid, F Sasò ad R Rummel (eds.), ecture Notes i Earth Scieces 65, pp , Sideris. G., Regioal geoid determiatio, i: Vaicek P., Christou N. T. (Eds.): The geoid ad its geophysical iterpretatio, CRC Press, Boca Rato, A Arbor odo, Tokyo, Sjöberg. E. Refied least squares modificatio of Stokes formula, auscripta Geodetica 16: , Sjöberg. E., O the topographic effects by the Stokes-Helmert method of geoid ad quasi-geoid determiatios, Joural of Geodesy, 74: 55-68,. 1. Sjöberg. E., Topographic ad atmospheric correctios of gravimetric geoid determiatio with special emphasis o the effects of degree zero ad oe, Joural of Geodesy 75: 83-9,. 11. Sjöberg. E., Improvig modified Stokes formula by GOCE data, to appear i Boll. Geod. Szi. Aff. 61, Sjöberg. E. ad Huegaw A., Some modificatios of Stokes formula that accout for trucatio ad potetial coefficiet errors, Joural of Geodesy 74: 3-38, Seeuw, N., va Geldere., The polar gap, i: Geodetic Boudary Value Problems i View of the Oe Cetimeter Geoid, F Sasò ad R Rummel (eds.), ecture Notes i Earth Scieces 65, pp , Tscherig C. C., Rapp R. H., Closed covariace expressios for gravity aomalies, geoid udulatios, ad deflectios of the vertical implied by aomaly degree variace models, Rep. 8, Dept Geod Sci, Ohio State Uiv, Columbus, Wog. ad Gore R., Accuracy of geoid heights from modified Stokes kerels, Geophysical Joural of the Royal Astroomical Society 18, 81 91, 1969.

Stokes s Kernel Modifications

Stokes s Kernel Modifications Stokes s Kerel Modificatios Algeria Key words: stochastic modificatio, determiistic modificatio, modified kerel, trucatio error. SUMMARY The icomplete global coverage of accurate gravity measuremets excludes