scholarly journals Spatially Restricted Integrals in Gradiometric Boundary Value Problems

2009 ◽  
Vol 44 (4) ◽  
pp. 131-148 ◽  
Author(s):  
M. Eshagh
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Legendre Functions ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

scholarly journals On the joint inversion of SGG and SST data from the GOCE mission

2003 ◽  
Vol 1 ◽  
pp. 87-94 ◽  
Author(s):  
P. Ditmar ◽  
P. Visser ◽  
R. Klees
Keyword(s):  
Gravity Field ◽  
Joint Inversion ◽  
Normal Matrix ◽  
Nuisance Parameters ◽  
Design Matrix ◽  
Explicit Computation ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field

Abstract. The computation of spherical harmonic coefficients of the Earth’s gravity field from satellite-to-satellite tracking (SST) data and satellite gravity gradiometry (SGG) data is considered. As long as the functional model related to SST data contains nuisance parameters (e.g. unknown initial state vectors), assembling of the corresponding normal matrix must be supplied with the back-substitution operation, so that the nuisance parameters are excluded from consideration. The traditional back-substitution algorithm, however, may result in large round-off errors. Hence an alternative approach, back-substitution at the level of the design matrix, is implemented. Both a stand-alone inversion of either type of data and a joint inversion of both types are considered. The conclusion drawn is that the joint inversion results in a much better model of the Earth’s gravity field than a standalone inversion. Furthermore, two numerical techniques for solving the joint system of normal equations are compared: (i) the Cholesky method based on an explicit computation of the normal matrix, and (ii) the pre-conditioned conjugate gradient method (PCCG), for which an explicit computation of the entire normal matrix is not needed. The comparison shows that the PCCG method is much faster than the Cholesky method.Key words. Earth’s gravity field, GOCE, satellite-tosatellite tracking, satellite gravity gradiometry, backsubstitution

scholarly journals Expansion of a class of functions into an integral involving associated Legendre functions

1994 ◽  
Vol 17 (2) ◽  
pp. 293-300
Author(s):  
Nanigopal Mandal ◽  
B. N. Mandal
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Mathematical Physics ◽  
Legendre Functions ◽  
Expansion Formula ◽  
Associated Legendre Functions ◽  
General Integral ◽  
Class Of Functions

A theorem for expansion of a class of functions into an integral involving associated Legendre functions is obtained in this paper. This is a somewhat general integral expansion formula for a functionf(x)defined in(x1,x2)where-1<x1<x2<1, which is perhaps useful in solving certain boundary value problems of mathematical physics and of elasticity involving conical boundaries.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals Legendre functions of fractional degree: transformations and evaluations

2016 ◽  
Vol 472 (2188) ◽  
pp. 20160097 ◽  
Author(s):  
Robert S. Maier
Keyword(s):  
Boundary Value Problems ◽  
Algebraic Curves ◽  
Applied Mathematics ◽  
Classical Case ◽  
Elliptic Integrals ◽  
Legendre Functions ◽  
Coordinate Transformations ◽  
Symbolic Manipulation ◽  
Associated Legendre Functions ◽  
Complete Elliptic Integrals

Associated Legendre functions of fractional degree appear in the solution of boundary value problems on wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth or one-sixth to the classical case. These transformations or identities facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They generalize both Ramanujan's transformations of elliptic integrals and Whipple's formula, which relates Legendre functions of the first and second kinds. The proofs employ algebraic coordinate transformations, specified by algebraic curves.

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