Basic Mathematics

Abstract

The basic mathematics useful for this book is divided into discrete least squares theory, collocation , coordinate systems, Legendre’s polynomials , spherical and ellipsoidal harmonics , the fundamentals of potential theory and regularization . Most numerical applications are based on linear least squares theory, either in the spatial domain (mainly for local studies) or by spherical harmonics in regional and global applications. For example, linear regression analysis, discrete and continuous least squares collocation are described. As problems in geodesy and geophysics are frequently non-linear, the linearization of such a problem is also presented. After introducing Legendre’s polynomials and spherical harmonics, the latter type of series is used for spectral smoothing and combining sets of data. The gravitational potential on and outside the ellipsoid is also presented in ellipsoidal harmonics . One section is devoted to the basics of potential theory , including some basic concepts, Newton ’s integral for the potential, Laplace ’s and Poisson ’s equations and Gauss ’ and Green ’s formulas, as a well as basic boundary value problems, as a background for the rest of the book. Considering that most problems related with gravity inversion are inverse problems, regularization is needed to reach a practical solution. Hence, various approaches to regularization of solutions to inverse problems are shortly described and compared.

Appendix: Answers to Exercises

Exercise 2.1

From Eq. (2.37a) one obtains the Taylor series

$$ l_{P0}^{ - 1} = r_{P}^{ - 1} \left[ {1 - \frac{1}{2}\left( {s^{2} - 2st} \right) + \frac{{\left( { - \frac{1}{2}} \right)\left( { - 1 - \frac{1}{2}} \right)}}{1 \times 2}\left( {s^{2} - 2st} \right)^{2} + \cdots } \right] = r_{P}^{ - 1} \left[ {1 + st + s^{2} \frac{{3t^{2} - 1}}{2} + \cdots } \right], $$

and by comparing with Eq. (2.38a) the solution follows.

Exercise 2.2

The left member of Eq. (2.42) yields for n = 0, 1 and 2:

$$ \begin{aligned} & \int\limits_{ - 1}^{1} {1dt = 2} ,\int\limits_{ - 1}^{1} {t^{2} dt = 2/3} \quad {\text{and}}\quad \int\limits_{ - 1}^{1} {\left( {\frac{{3t^{2} - 1}}{2}} \right)^{2} dt} \\ & \quad = \int\limits_{ - 1}^{1} {\left( {\frac{{9t^{4} - 6t^{2} + 1}}{4}} \right)dt} = \frac{1}{4}\left[ {\frac{{9t^{5} }}{5} - \frac{{6t^{3} }}{3} + t} \right]_{ - 1}^{1} = \frac{2}{5}, \\ \end{aligned} $$

The second equation in Eq. (2.42) is shown in the same way.