VOL. 84, NO. B II JOURNAL OF GEOPHYSICAL RESEARCH OCTOBER 10, 1979 The Accuracy of Geoid Undulations by Degree Implied by Mean Gravity Anomalies on a Sphere LARS SJOBERG • Department of Geodetic Science.The Ohio State University.Columbus,Ohio 43210 Formulae are derived for the resolution of the geoid undulations by degreeimplied by mean gravity anomalies. The two error sources considered are the propagation of the mean anomaly errorsand the er- ror due to lack of more detailed data than the mean anomalies. The errors are computedfor the degree variances of Tscherning and Rapp and the fi function of Meissl. Finally, the corresponding errors of point gravity anomalies are derived. 1. INTRODUCTION In a spherical approximationof the earth the geoid undula- tions can be determined from gravity anomalies (Ag) by means of Stokes' integral formula [see Heiskanenand Moritz, 1967,pp. 93-94]: Ne-- (R/¾)M{S(re,•bee)Ag} (1) where re radius of the point P from the earth'sgravity center; R radius of the mean earth sphere; ¾ = mean gravity at the sphere; M { } -- (l/4•r)fof do; o unit sphere. Also in (1), S(r•., ½•'e)= Y•2n + I R_R - P.(cos ½•'e) (1') .=2 n- I re l Formula (1) requires that the complete Ag field be known on the sphere. We now consider the following estimator of Ne: •.-- (R/¾)M {S(r•., ½•,e)Age} (2) where Ag e is the mean gravity anomaly at thepointQ. Except for the error of the zero-degreeharmonic, there are essentially two errors of the estimator •e: the error due to lack of more detailed gravity material and the error propagation of the er- roneous mean anomalies. For an earlier investigation of these effects, we refer to Rapp [1973]. The geoid undulations may be developed into a series of Laplace harmonics N,,: N=No+ Y. N. n----2 It is the purpose of this paper to studythe impactof the above error sources for each harmonic N.. In section4 the results are alsoextendedto the gravity anomaly error by degree. 2. THE ERROR DUE TO LACK OF MORE DETAILED DATA Let us assumethat the point gravity anomalies can be de- velopedinto the following series of spherical harmonics: Present address: Institutionen te6r Geodesi, Kungl, Tekniska H6g- skolan, S-100 44, Stockholm, Sweden. Copyright¸ 1979by the AmericanGeophysical Union. Paper number 9B0735. 0148-0227/79/009 B-0735501.00 n----2 where •).+2 Y.m(Q) rB radiusof the Bjerhammar sphere (rB< R); A,,., spherical harmoniccoefficients; Y--, = P-l-,l (sin qo) cos rnX, forrn _> 0; Y-.- = •,,1-,1 (sin qo) sin ImIX, form< 0. M{Y,,.,Y.,,,t} --1 M{YnmYpq } ---0 if n--p,m=q otherwise 6226 Also The mean anomalies are then defined by Agt,(Q) = Mr,{Ag(Q)} where (3) (4) (5) the local average, and Aos is area of block k. Now, following Meissl [1971, pp. 22-23], the eigenvalues X. of a kernel function K(P, Q) on a sphere are defined by X,,Y..,(P) = M{K(P, Q) Y..,(Q)} If we approximate the block mean value with that over a spherical cap of equal area (with centerQk and angularradius ½o), we obtain Mr, {Y..,(Q)} = M{B(Q•, Q) Y..,(Q)} -- fl.Y..,(QD (6) I 1 B(Q•, Q) - 2•r 1 - cos ½o for cos ½et,.e >cos ½o B(Qk, Q) = 0 otherwise where I 1 I - cos ½o 2n + I {P"' (cos ½o) - P.,(cos ½o)} and From (3), (5), and (6) we obtain r B t n+2 Agt,(Q) = • A,,m[ •-1 ,8,, Y.m(QD n.m (7)