H:\CCTWP\sphcov.wpd



Prediction of spherical harmonic coefficients

using Least-Squares Collocation - basic theory.



by


C.C.Tscherning

Department of Geophysics

Juliane Maries Vej 30,

Dk-2100 Copenhagen Ø, Denmark


20 May 1999.



Abstract: In this note the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first and second derivatives of T. These equations forms the basis for the prediction of spherical harmonic coefficients, a feature now implemented in the GRAVSOFT program GEOCOL.

1. Introduction.

When using least-squares collocation (LSC) available information about the spherical harmonic coefficients of the anomalous gravity potential, T, can be used directly as observations, or used in a remove-restore procedure, see Forsberg & Tscherning (1981). However, if coefficients are to be predicted, we need the explicit covariances (or values of functionals applied on a reproducing kernel, see Tscherning (1993)). As shown in section 2 the covariances are simply the observation functional applied on the solid spherical harmonic function of a specific degree and order, multiplied by a constant depending on the degree.

The equations have been implemented in the form of new versions of the subroutines COVAX, COVBX and COVCX (Tscherning, 1976) in the program COVFIT (Knudsen, 1987). This program has been used for the calculation of covariances, and used to illustrate the order of magnitude of typical covariances.

The question of which method is the most efficient to be used for the calculation of the predicted coefficients and their error-estimates is discussed in section 3.





2. Covariances between spherical harmonic coefficients and point related quantities.

Let P, Q be two points with coordinates (latitude, longitude, r), , respectively, and having the spherical distance . R is the mean radius of the Earth, Yij are the surface spherical harmonics, Pi the Legendre polynomials and the degree-variances. Then the covariance between the values of the anomalous potential T in P, Q is


The covariance between the coefficient GM Cij /R and the anomalous potential is obtained by applying the functional Lij on the covariance function, where






Then we have




When applying the functional twice we get its norm, squared,




or


The covariance with an arbitrary functional Lk is then


Example:


The correlation can also be calculated


3. Prediction of spherical harmonic coefficients.

An approximation to T determined using LSC will have the form


where bn are the solutions to the normal equations, Ln are the observation functionals and N is the number of observations. The value of a predicted quantity is obtained by applying the associated functional on this expression. For the spherical harmonic coefficients we then have


The calculation of the observation functional applied on the solid spherical harmonic function is generally done by recursion, starting with the (0,0) term. This means that a better computational strategy would be to calculate all coefficients simultaneously up to and inclusive degree i. However, we get a problem when we want to calculate the error estimates


because we then must store all the quantities cov(Ln,Lij) in the expression eq. (10) simultaneously, and subsequently evaluate the expressions for all degrees up to and inclusive degree i. This will obviously be possible when the degree i and the number of observations are small.

The strategy adopted is for the moment to predict all quantities up to degree i, but only compute the error estimate for the ij'th coefficient.









4. Conclusion.

The procedures described above have been implemented in a new version of COVFIT, denoted covfit12.f and in a new version of GEOCOL, geoco115.f .

Both programs are now in the testing phase. They can be found at ftp.gfy.ku.dk, directory cct.

References.



Forsberg, R. and C.C.Tscherning: The use of Height Data in Gravity Field Approximation by Collocation. J.Geophys.Res., Vol. 86, No. B9, pp. 7843-7854, 1981.

Knudsen, P.: Estimation and Modelling of the Local Empirical covariance Function using gravity and satellite altimeter data. Bulletin Geodesique, Vol. 61, pp. 145-160, 1987a.

Tscherning, C.C.: A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation. Reports of the Department of Geodetic Science No. 212, The Ohio State University, Columbus, Ohio, 1974.

Tscherning, C.C.: Covariance Expressions for Second and Lower Order Derivatives of the Anomalous Potential. Reports of the Department of Geodetic Science No. 225, The Ohio State University, Columbus, Ohio, 1976b.

Tscherning, C.C.: Computation of covariances of derivatives of the anomalous gravity potential in a rotated reference frame. Manuscripta Geodaetica, Vol. 18, no. 3, pp. 115-123, 1993.

This is a contribution to the ESA sponsored project E2M. Last update 1999.05.20, comments to: cct@gfy.ku.dk