By Zdenek Martinec
This e-book bargains a simultaneous remedy of the idea and numerical program of boundary-value difficulties relating to the selection of an exact geoid from gravimetric info. the subsequent matters are mentioned: topographical results and their computations in targeted gravimetric geoid choice, the downward continuation of a harmonic functionality, Stokes' challenge formulated on an ellipsoid of revolution, round Stokes' challenge with ellipsoidal corrections considering boundary stipulations for an anomalous strength, and the altimetry-gravimetry boundary-value challenge. the reply to a couple of clinical difficulties, raised and mentioned in geodetic literature during the last years, are available right here. The publication is meant for scientists and complicated graduate students.
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Additional info for Boundary-Value Problems for Gravimetric Determination of a Precise Geoid
6. act as regularization factors removing the ill-posedness of matrix A. , Cruz, 1985). 3, but now, in contrast with preceeding Stokes's two-boundary-value problem 23 example, we will not multiplied height H(t~) by 10. In this case, it is not possible to carry out the eigenvatue analysis of matrix A starting from degree j~,] = 20 and going up to degrees j m ~ ~ 104 - 10s due to a huge consummation of computer time and memory. We have to confine ourselves to a smaller range of sought spherical harmonics.
Let us discuss the differences. 15). The difference, -r:GpoH2(~), is always negative and may reach up to -5 metres in high mountains. Therefore, eqn. 15) derived from the planar approximation of the geoid is biased. 11) in the terrain roughness term. We cannot neglect the term Z~/2R with respect to H(~) in the logarithmic function because for distances over 1° the magnitude of the term t2o/2Rmay be comparable with H(~) or H(f~'). Moreover, the logarithmic function is completely neglected in planar approximation 5Ap(fl).
We have to confine ourselves to a smaller range of sought spherical harmonics. That is why we choose j~,] = 10000 and j , ~ in the range between 10300 and 11600. 6. We can again observe that eigenvalue spectra intersect the zero-level starting at degree j~,~o - 10500. It means that whenever jm~,: >_ j~,~o, the spectrum of matrix A contains an eigenvalue which is very close or equal to zero. Consequently, matrix A becomes ill-conditioned or even singular. 5. To carry out the eigenvalue analysis of matrix A needs a lot of computer time.