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Correlation Modeling of the Gravity Field in Classical Geodesy

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Handbook of Geomathematics

Abstract

The spatial correlation of the Earth’s gravity field is well known and widely used in applications of geophysics and physical geodesy. This chapter develops the mathematical theory of correlation functions, as well as covariance functions under a statistical interpretation of the field, for functions and processes on the sphere and plane, with formulation of the corresponding power spectral densities in the respective frequency domains, and with extensions into the third dimension for harmonic functions. The theory is applied, in particular, to the disturbing gravity potential with consistent relationships of the covariance and power spectral density to any of its spatial derivatives. An analytic model for the covariance function of the disturbing potential is developed for both spherical and planar application, which has analytic forms also for all derivatives in both the spatial and the frequency domains (including the along-track frequency domain). Finally, a method is demonstrated to determine the parameters of this model from empirical regional power spectral densities of the gravity anomaly.

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References

  • Alfeld P, Neamtu M, Schumaker LL (1996) Fitting scattered data on sphere-like surfaces using spherical splines. J Comput Appl Math 73: 5–43

    Article  MATH  MathSciNet  Google Scholar 

  • Baranov V (1957) A new method for interpretation of aeromagnetic maps: Pseudo-gravimetric anomalies. Geophysics 22:359–383

    Article  MathSciNet  Google Scholar 

  • Brown RG (1983) Introduction to random signal analysis and Kalman filtering. Wiley, New York

    Google Scholar 

  • Fengler MJ, Freeden W, Michel V (2004) The Kaiserslautern multiscale geopotential model SWITCH-03 from orbit perturbations of the satellite CHAMP and its comparison to models EGM96, UCPH2002_02_05, EIGEN-1S and EIGEN-2. Geophys J Int 157: 499–514

    Article  Google Scholar 

  • Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull GĂ©odĂ©sique 59(4):342–360

    Article  Google Scholar 

  • Forsberg R (1987) A new covariance model, for inertial gravimetry and gradiometry. J Geophys Res 92(B2):1305–1310

    Google Scholar 

  • Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere, with applications in geomathematics. Clarendon Press, Oxford

    Google Scholar 

  • Heller WG, Jordan SK (1979) Attenuated white noise statistical gravity model. J Geophys Res 84(B9):4680–4688

    Google Scholar 

  • Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy. Springer, Berlin

    Google Scholar 

  • Jeffreys H (1955) Two properties of spherical harmonics. Quart J Mech Appl Math 8(4):448–451

    Article  MATH  MathSciNet  Google Scholar 

  • Jekeli C (1991) The statistics of the Earth’s gravity field, revisited. Manuscripta Geodaetica 16(5):313–325

    Google Scholar 

  • Jekeli C (2005) Spline representations of functions on a sphere for geopotential modeling. Report no.475, Geodetic Science, Ohio State University, Columbus, Ohio, http://www.geology.osu.edu/~jekeli.1/OSUReports/reports/report_475.pdf

  • Jordan SK (1972) Self-consistent statistical models for the gravity anomaly, vertical deflections, and the undulation of the geoid. J Geophys Res 77(19):3660–3669

    Article  Google Scholar 

  • Jordan SK, Moonan PJ, Weiss JD (1981) State-space models of gravity disturbance gradients. IEEE Trans. Aerospace Electronic Syst AES-17(5):610–619

    Google Scholar 

  • Kaula WM (1966) Theory of satellite geodesy. Blaisdell Publ. Co., Waltham, MA

    Google Scholar 

  • Lauritzen SL (1973) The probabilistic background of some statistical methods in physical geodesy. Report no.48, Geodaestik Institute, Copenhagen, Denmark

    Google Scholar 

  • Lyche T, Schumaker LL (2000) A multiresolution tensor spline method for fitting functions on the sphere. SIAM J. Sci Comput 22(2):724–746

    Article  MATH  MathSciNet  Google Scholar 

  • Mandelbrot B (1983) The fractal geometry of nature. Freeman and Co., San Francisco, CA

    Google Scholar 

  • Marple SL (1987) Digital spectral analysis with applications. Prentice-Hall, Prin < $$$ > , NJ

    Google Scholar 

  • Milbert DG (1991) A family of covariance functions based on degree variance models and expressible by elliptic integrals. Manuscripta Geodaetica 16:155–167

    Google Scholar 

  • Moritz H (1976) Covariance functions in least-squares collocation. Report no.240, Department of Geodetic Science, Ohio State University, Columbus, OH

    Google Scholar 

  • Moritz H (1978) Statistical foundations of collocation. Report no.272, Department of Geodetic Science, Ohio State University, Columbus, OH

    Google Scholar 

  • Moritz H (1980) Advanced physical geodesy. Abacus Press, Tunbridge Wells, Kent

    Google Scholar 

  • Olea RA (1999) Geostatistics for Engineers and Earth Scientists. Kluwer Academic, Boston, MA

    Google Scholar 

  • Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008. Presented at the General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008

    Google Scholar 

  • Priestley MB (1981) Spectral analysis and time series analysis. Academic Press, London

    MATH  Google Scholar 

  • Schreiner M (1997) Locally supported kernels for spherical spline interpolation. J. Approx. Theory 89:172–194

    Article  MATH  MathSciNet  Google Scholar 

  • Schumaker LL, Traas C (1991) Fitting scattered data on sphere-like surfaces using tensor products of trigonometric and polynomial splines. Numer Math 60:133–144.

    Article  MATH  MathSciNet  Google Scholar 

  • Tscherning CC (1976) Covariance expressions for second and lower order derivatives of the ano-malous potential. Report no.225, Department of Geodetic Science, Ohio State University, Columbus, OH, http://geodeticscience.osu.edu/OSUReports.htm

  • Tscherning CC, Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations and deflections of the vertical implied by anomaly degree variance models. Report no.208, Department of Geodetic Science, Ohio State University, Columbus, OH, http://geodeticscience.osu.edu/OSUReports.htm

  • Turcotte DL (1987) A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars. J Geophys Res 92(B4):E597–E601

    Google Scholar 

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Author information

Authors and Affiliations

  1. Division of Geodetic Science, School of Earth Sciences, Ohio State University, Columbus, OH, USA

    Christopher Jekeli

Authors
  1. Christopher Jekeli
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Editor information

Editors and Affiliations

  1. Geomathematics Group, Technische Universität Kaiserslautern, 3049, 67653, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, PockelsstraĂŸe 14, 38106, Braunschweig, Germany

    Thomas Sonar

Appendices

Appendix 1

The planar reciprocal distance model, Eq. (78), for the covariance function of the disturbing potential is repeated here for convenience with certain abbreviations

$${\varphi }_{T,T}\left ({s}_{1},{s}_{2};{z}_{1},{z}_{2}\right ) = \frac{{\sigma }^{2}} {{M}^{1\left /\right . 2}},$$
(A.1)

where

$$M = {\beta }^{2}+{\alpha }^{2}{s}^{2},\quad \beta = 1+\alpha \left ({z}_{ 1} + {z}_{2}\right ),\quad {s}^{2} = {s}_{ 1}^{2}+{s}_{ 1}^{2},\quad {s}_{ 1} = {x}_{1}-{x}_{1}^{{\prime}},\quad {s}_{ 2} = {x}_{2}-{x}_{2}^{{\prime}}.$$
(A.2)

The primed coordinates refer to the first subscripted function in the covariance and the unprimed coordinates refer to the second function. The altitude levels for these functions are z 1 and z 2, respectively. Derivatives of the disturbing potential with respect to the coordinates are denoted \(\partial T\left /\right . \partial {x}_{1} = {T}_{{x}_{1}}\), \({\partial }^{2}T\left /\right .\left (\partial {x}_{1}\partial z\right ) = {T}_{{x}_{1}z}\), etc. The following expressions for the cross-covariances are derived by repeatedly using Eqs. (40) and (52). The arguments for the resulting function are omitted, but are the same as in Eq. (1).

$$\begin{array}{rcl} &{\phi }_{{T}_{{x}_{ 1}},T} = \frac{{\sigma }^{2}{\alpha }^{2}{s}_{1}} {{M}^{3\left /\right . 2}} = -{\phi }_{T,{T}_{{x}_{ 1}}} \end{array}$$
(A.3)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}},T} = \frac{{\sigma }^{2}{\alpha }^{2}{s}_{2}} {{M}^{3\left /\right . 2}} = -{\phi }_{T,{T}_{{x}_{ 2}}} \end{array}$$
(A.4)
$$\begin{array}{rcl} {\phi }_{{T}_{z},T} = -\frac{{\sigma }^{2}\alpha \beta } {{M}^{3\left /\right . 2}} = {\phi }_{T,{T}_{z}} \end{array}$$
(A.5)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{1}}} = \frac{{\sigma }^{2}{\alpha }^{2}} {{M}^{5\left /\right . 2}}\left (M - 3{\alpha }^{2}{s}_{ 1}^{2}\right ) \end{array}$$
(A.6)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{2}}} = -3 \frac{{\sigma }^{2}{\alpha }^{4}} {{M}^{5\left /\right . 2}}{s}_{1}{s}_{2} = {\phi }_{{T}_{{x}_{ 2}},{T}_{{x}_{1}}} \end{array}$$
(A.7)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{z}} = -3\frac{{\sigma }^{2}{\alpha }^{3}\beta } {{M}^{5\left /\right . 2}} {s}_{1} = -{\phi }_{{T}_{z},{T}_{{x}_{ 1}}} \end{array}$$
(A.8)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}},{T}_{{x}_{2}}} = \frac{{\sigma }^{2}{\alpha }^{2}} {{M}^{5\left /\right . 2}}\left (M - 3{\alpha }^{2}{s}_{ 2}^{2}\right ) \end{array}$$
(A.9)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}},{T}_{z}} = -3\frac{{\sigma }^{2}{\alpha }^{3}\beta } {{M}^{5\left /\right . 2}} {s}_{2} = -{\phi }_{{T}_{z},{T}_{{x}_{ 2}}} \end{array}$$
(A.10)
$$\begin{array}{rcl} {\phi }_{{T}_{z},{T}_{z}} = \frac{{\sigma }^{2}{\alpha }^{2}} {{M}^{5\left /\right . 2}}\left (2M - 3{\alpha }^{2}{s}^{2}\right ) = {\phi }_{{ T}_{{x}_{1}},{T}_{{x}_{1}}} + {\phi }_{{T}_{{x}_{2}},{T}_{{x}_{2}}} \end{array}$$
(A.11)
$$\begin{array}{rcl} {\phi }_{T,{T}_{{x}_{ 1}{x}_{1}}} = -{\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{1}}} = {\phi }_{{T}_{{x}_{ 1}{x}_{1}},T} \end{array}$$
(A.12)
$$\begin{array}{rcl} {\phi }_{T,{T}_{{x}_{ 1}{x}_{2}}} = -{\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{2}}} = {\phi }_{{T}_{{x}_{ 1}{x}_{2}},T} \end{array}$$
(A.13)
$$\begin{array}{rcl} &{\phi }_{T,{T}_{{x}_{ 1}z}} = -{\phi }_{{T}_{{x}_{ 1}},{T}_{z}} = -{\phi }_{{T}_{{x}_{ 1}z},T} \end{array}$$
(A.14)
$$\begin{array}{rcl} {\phi }_{T,{T}_{{x}_{ 2}{x}_{2}}} = -{\phi }_{{T}_{{x}_{ 2}},{T}_{{x}_{2}}} = {\phi }_{{T}_{{x}_{ 2}{x}_{2}},T} \end{array}$$
(A.15)
$$\begin{array}{rcl} {\phi }_{T,{T}_{{x}_{ 2}z}} = -{\phi }_{{T}_{{x}_{ 2}},{T}_{z}} = -{\phi }_{{T}_{{x}_{ 2}z},T} \end{array}$$
(A.16)
$$\begin{array}{rcl} {\phi }_{T,{T}_{zz}} = {\phi }_{{T}_{z},{T}_{z}} = {\phi }_{{T}_{zz},T} \end{array}$$
(A.17)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{1}{x}_{1}}} = \frac{3{\sigma }^{2}{\alpha }^{4}{s}_{1}} {{M}^{7\left /\right . 2}} \left (-3M + 5{\alpha }^{2}{s}_{ 1}^{2}\right ) = -{\phi }_{{ T}_{{x}_{1}{x}_{1}},{T}_{{x}_{1}}} \end{array}$$
(A.18)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{1}{x}_{2}}} = \frac{3{\sigma }^{2}{\alpha }^{4}{s}_{2}} {{M}^{7\left /\right . 2}} \left (-M + 5{\alpha }^{2}{s}_{ 1}^{2}\right ) = -{\phi }_{{ T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}}} = {\phi }_{{T}_{{x}_{2}},{T}_{{x}_{1}{x}_{1}}} = -{\phi }_{{T}_{{x}_{1}{x}_{1}},{T}_{{x}_{2}}} \end{array}$$
(A.19)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{1}z}} = \frac{3{\sigma }^{2}{\alpha }^{3}\beta } {{M}^{7\left /\right . 2}} \left (-M + 5{\alpha }^{2}{s}_{ 1}^{2}\right ) = {\phi }_{{ T}_{{x}_{1}z},{T}_{{x}_{1}}} = -{\phi }_{{T}_{z},{T}_{{x}_{1}{x}_{1}}} = -{\phi }_{{T}_{{x}_{1}{x}_{1}},{T}_{z}} \end{array}$$
(A.20)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{2}{x}_{2}}} = \frac{3{\sigma }^{2}{\alpha }^{4}{s}_{2}} {{M}^{7\left /\right . 2}} \left (-M + 5{\alpha }^{2}{s}_{ 2}^{2}\right ) = -{\phi }_{{ T}_{{x}_{2}{x}_{2}},{T}_{{x}_{1}}} = {\phi }_{{T}_{{x}_{2}},{T}_{{x}_{1}{x}_{2}}} = -{\phi }_{{T}_{{x}_{1}{x}_{2}},{T}_{{x}_{2}}} \end{array}$$
(A.21)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{{x}_{2}z}} = \frac{15{\sigma }^{2}{\alpha }^{3}\beta } {{M}^{7\left /\right . 2}} {s}_{1}{s}_{2} = {\phi }_{{T}_{{x}_{ 2}z},{T}_{{x}_{1}}} = -{\phi }_{{T}_{z},{T}_{{x}_{ 1}{x}_{2}}} = {\phi }_{{T}_{{x}_{ 2}},{T}_{{x}_{1}z}} = -{\phi }_{{T}_{{x}_{ 1}{x}_{2}},{T}_{z}} = {\phi }_{{T}_{{x}_{ 1}z},{T}_{{x}_{2}}}\end{array}$$
(A.22)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}},{T}_{zz}} = \frac{3{\sigma }^{2}{\alpha }^{4}{s}_{1}} {{M}^{7\left /\right . 2}} \left (4{\beta }^{2} - {\alpha }^{2}{s}^{2}\right ) = -{\phi }_{{ T}_{zz},{T}_{{x}_{1}}} = -{\phi }_{{T}_{z},{T}_{{x}_{1}z}} = {\phi }_{{T}_{{x}_{1}z},{T}_{z}} \end{array}$$
(A.23)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}},{T}_{{x}_{2}{x}_{2}}} = \frac{3{\sigma }^{2}{\alpha }^{4}{s}_{2}} {{M}^{7\left /\right . 2}} \left (-3M + 5{\alpha }^{2}{s}_{ 2}^{2}\right ) = -{\phi }_{{ T}_{{x}_{2}{x}_{2}},{T}_{{x}_{2}}} \end{array}$$
(A.24)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}},{T}_{{x}_{2}z}} = \frac{3{\sigma }^{2}{\alpha }^{3}\beta } {{M}^{7\left /\right . 2}} \left (-M + 5{\alpha }^{2}{s}_{ 2}^{2}\right ) = {\phi }_{{ T}_{{x}_{2}z},{T}_{{x}_{2}}} = -{\phi }_{{T}_{z},{T}_{{x}_{2}{x}_{2}}} = -{\phi }_{{T}_{{x}_{2}{x}_{2}},{T}_{z}} \end{array}$$
(A.25)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}},{T}_{zz}} = \frac{3{\sigma }^{2}{\alpha }^{4}{s}_{2}} {{M}^{7\left /\right . 2}} \left (4{\beta }^{2} - {\alpha }^{2}{s}^{2}\right ) = -{\phi }_{{ T}_{zz},{T}_{{x}_{2}}} = -{\phi }_{{T}_{z},{T}_{{x}_{2}z}} = {\phi }_{{T}_{{x}_{2}z},{T}_{z}} \end{array}$$
(A.26)
$$\begin{array}{rcl} {\phi }_{{T}_{z},{T}_{zz}} = \frac{3{\sigma }^{2}{\alpha }^{3}\beta } {{M}^{7\left /\right . 2}} \left (-2M + 5{\alpha }^{2}{s}^{2}\right ) = {\phi }_{{ T}_{zz},{T}_{z}} \end{array}$$
(A.27)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}{x}_{1}}} = \frac{3{\sigma }^{2}{\alpha }^{4}} {{M}^{9\left /\right . 2}} \left (3{M}^{2} - 30M{\alpha }^{2}{s}_{ 1}^{2} + 35{\alpha }^{4}{s}_{ 1}^{4}\right ) \end{array}$$
(A.28)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}{x}_{2}}} = \frac{15{\sigma }^{2}{\alpha }^{6}{s}_{1}{s}_{2}} {{M}^{9\left /\right . 2}} \left (-3M + 7{\alpha }^{2}{s}_{ 1}^{2}\right ) = {\phi }_{{ T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}{x}_{1}}} \end{array}$$
(A.29)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}z}} = \frac{15{\sigma }^{2}{\alpha }^{5}\beta {s}_{1}} {{M}^{9\left /\right . 2}} \left (-3M + 7{\alpha }^{2}{s}_{ 1}^{2}\right ) = -{\phi }_{{ T}_{{x}_{1}{x}_{3}},{T}_{{x}_{1}{x}_{1}}} \end{array}$$
(A.30)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}{x}_{2}}} = \frac{3{\sigma }^{2}{\alpha }^{4}} {{M}^{9\left /\right . 2}} \left ({M}^{2} - 5M{\alpha }^{2}{s}^{2} + 35{s}_{ 1}^{2}{s}_{ 2}^{2}\right ) = {\phi }_{{ T}_{{x}_{2}{x}_{2}},{T}_{{x}_{1}{x}_{1}}} = {\phi }_{{T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}{x}_{2}}} & \end{array}$$
(A.31)
$$\begin{array}{rcl} &{\phi }_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}z}} = \frac{15{\sigma }^{2}{\alpha }^{5}\beta {s}_{2}} {{M}^{9\left /\right . 2}} \left (-M + 7{\alpha }^{2}{s}_{ 1}^{2}\right ) = -{\phi }_{{ T}_{{x}_{2}z},{T}_{{x}_{1}{x}_{1}}} = -{\phi }_{{T}_{{x}_{1}z},{T}_{{x}_{1}{x}_{2}}} = {\phi }_{{T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}z}} \end{array}$$
(A.32)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{1}},{T}_{zz}} = \frac{3{\sigma }^{2}{\alpha }^{4}} {{M}^{9\left /\right . 2}} \left (-4{M}^{2} + 5M{\alpha }^{2}{s}_{ 2}^{2} + 35{\beta }^{2}{\alpha }^{2}{s}_{ 1}^{2}\right ) = {\phi }_{{ T}_{zz},{T}_{{x}_{1}{x}_{1}}} = -{\phi }_{{T}_{{x}_{1}z},{T}_{{x}_{1}z}} \end{array}$$
(A.33)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{2}},{T}_{{x}_{2}{x}_{2}}} = \frac{15{\sigma }^{2}{\alpha }^{6}{s}_{1}{s}_{2}} {{M}^{9\left /\right . 2}} \left (-3M + 7{\alpha }^{2}{s}_{ 2}^{2}\right ) = {\phi }_{{ T}_{{x}_{2}{x}_{2}},{T}_{{x}_{1}{x}_{2}}} \end{array}$$
(A.34)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}{x}_{2}},{T}_{{x}_{2}z}} = \frac{15{\sigma }^{2}{\alpha }^{5}\beta {s}_{1}} {{M}^{9\left /\right . 2}} \left (-M + 7{\alpha }^{2}{s}_{ 2}^{2}\right ) = -{\phi }_{{ T}_{{x}_{2}z},{T}_{{x}_{1}{x}_{2}}} = -{\phi }_{{T}_{{x}_{1}z},{T}_{{x}_{2}{x}_{2}}} = {\phi }_{{T}_{{x}_{2}{x}_{2}},{T}_{{x}_{1}z}}\end{array}$$
(A.35)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 1}z},{T}_{zz}} = \frac{15{\sigma }^{2}{\alpha }^{5}\beta {s}_{1}} {{M}^{9\left /\right . 2}} \left (3M - 7{\beta }^{2}\right ) = -{\phi }_{{ T}_{zz},{T}_{{x}_{1}z}} \end{array}$$
(A.36)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}{x}_{2}}} = \frac{3{\sigma }^{2}{\alpha }^{4}} {{M}^{9\left /\right . 2}} \left (3{M}^{2} - 30M{\alpha }^{2}{s}_{ 2}^{2} + 35{\alpha }^{4}{s}_{ 2}^{4}\right ) \end{array}$$
(A.37)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}z}} = \frac{15{\sigma }^{2}{\alpha }^{5}\beta {s}_{2}} {{M}^{9\left /\right . 2}} \left (-3M + 7{\alpha }^{2}{s}_{ 2}^{2}\right ) = -{\phi }_{{ T}_{{x}_{2}z},{T}_{{x}_{2}{x}_{2}}} \end{array}$$
(A.38)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}{x}_{2}},{T}_{zz}} = \frac{3{\sigma }^{2}{\alpha }^{4}} {{M}^{9\left /\right . 2}} \left (-4{M}^{2} + 5M{\alpha }^{2}{s}_{ 1}^{2} + 35{\beta }^{2}{\alpha }^{2}{s}_{ 2}^{2}\right ) = {\phi }_{{ T}_{zz},{T}_{{x}_{2}{x}_{2}}} = -{\phi }_{{T}_{{x}_{2}z},{T}_{{x}_{2}z}} \end{array}$$
(A.39)
$$\begin{array}{rcl} {\phi }_{{T}_{{x}_{ 2}z},{T}_{zz}} = \frac{15{\sigma }^{2}{\alpha }^{5}\beta {s}_{2}} {{M}^{9\left /\right . 2}} \left (3M - 7{\beta }^{2}\right ) = -{\phi }_{{ T}_{zz},{T}_{{x}_{2}z}} \end{array}$$
(A.40)
$$\begin{array}{rcl} {\phi }_{{T}_{zz},{T}_{zz}} = \frac{3{\sigma }^{2}{\alpha }^{4}} {{M}^{9\left /\right . 2}} \left (8{\beta }^{4} - 24{\beta }^{2}{\alpha }^{2}{s}^{2} + 3{\alpha }^{4}{s}^{4}\right ) = {\phi }_{{ T}_{{x}_{1}z},{T}_{{x}_{1}z}} + {\phi }_{{T}_{{x}_{2}z},{T}_{{x}_{2}z}}\end{array}$$
(A.41)

Appendix 2

The along-track PSD of the disturbing potential, given by Eq. (83), can be shown to be

$${S}_{T,T}\left ({f}_{1};{s}_{2};{z}_{1},{z}_{2}\right ) = \frac{2{\sigma }^{2}} {\alpha } {K}_{0}\left (2\pi {f}_{1}\,d\right ),$$
(B.1)

where K 0 is the modified Bessel function of the second kind and zero order, and

$$d = \sqrt{\frac{{\beta }^{2 } } {{\alpha }^{2}} + {s}_{2}^{2}}.$$
(B.2)

In the following along-track PSDs of the derivatives of T, also the modified Bessel function of the second kind and first order, K 1, appears. Both Bessel function always have the argument, 2πf 1 d; and, the arguments of the along-track PSDs are the same as in Eq. (1).

$$\begin{array}{rcl} &{S}_{T,{T}_{{x}_{ 1}}} = i2\pi {f}_{1}{S}_{TT} = -{S}_{{T}_{{x}_{ 1}},T} \end{array}$$
(B.3)
$$\begin{array}{rcl} {S}_{T,{T}_{{x}_{ 2}}} = \frac{2{\sigma }^{2}\left (2\pi {f}_{1}\right ){s}_{2}} {\alpha d} {K}_{1} = -{S}_{{T}_{{x}_{ 2}},T} \end{array}$$
(B.4)
$$\begin{array}{rcl} {S}_{T,{T}_{z}} = -\frac{2{\sigma }^{2}\left (2\pi {f}_{1}\right )\beta } {{\alpha }^{2}d} {K}_{1} = {S}_{{T}_{z},T} \end{array}$$
(B.5)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}}} ={ \left (2\pi {f}_{1}\right )}^{2}{S}_{ TT} \end{array}$$
(B.6)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{2}}} = i2\pi {f}_{1}{S}_{{T}_{{x}_{ 2}},T} = {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{1}}} \end{array}$$
(B.7)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{z}} = -i2\pi {f}_{1}{S}_{T,{T}_{z}} = -{S}_{{T}_{z},{T}_{{x}_{ 1}}} \end{array}$$
(B.8)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}}} = \frac{2{\sigma }^{2}\left (2\pi {f}_{1}\right )} {\alpha d} \left (\left (1 -\frac{2{s}_{2}^{2}} {{d}^{2}} \right ){K}_{1} - 2\pi {f}_{1}\frac{{s}_{2}^{2}} {d} {K}_{0}\right ) \end{array}$$
(B.9)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}},{T}_{z}} = -\frac{2{\sigma }^{2}\left (2\pi {f}_{1}\right )\beta {s}_{2}} {{\alpha }^{2}{d}^{3}} \left (2{K}_{1} + 2\pi {f}_{1}d\,{K}_{0}\right ) = -{S}_{{T}_{z},{T}_{{x}_{ 2}}} \end{array}$$
(B.10)
$$\begin{array}{rcl} {S}_{{T}_{z},{T}_{z}} = {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}}} + {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}}} \end{array}$$
(B.11)
$$\begin{array}{rcl} {S}_{T,{T}_{{x}_{ 1}{x}_{1}}} = -{S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}}} = {S}_{{T}_{{x}_{ 1}{x}_{1}},T} \end{array}$$
(B.12)
$$\begin{array}{rcl} {S}_{T,{T}_{{x}_{ 1}{x}_{2}}} = -{S}_{{T}_{{x}_{ 1}},{T}_{{x}_{2}}} = {S}_{{T}_{{x}_{ 1}{x}_{2}},T} \end{array}$$
(B.13)
$$\begin{array}{rcl} {S}_{T,{T}_{{x}_{ 1}z}} = -{S}_{{T}_{{x}_{ 1}},{T}_{z}} = -{S}_{{T}_{{x}_{ 1}z},T} \end{array}$$
(B.14)
$$\begin{array}{rcl} {S}_{T,{T}_{{x}_{ 2}{x}_{2}}} = -{S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}}} = {S}_{{T}_{{x}_{ 2}{x}_{2}},T} \end{array}$$
(B.15)
$$\begin{array}{rcl} {S}_{T,{T}_{{x}_{ 2}z}} = -{S}_{{T}_{{x}_{ 2}},{T}_{z}} = -{S}_{{T}_{{x}_{ 2}z},T} \end{array}$$
(B.16)
$$\begin{array}{rcl} {S}_{T,{T}_{zz}} = {S}_{{T}_{z},{T}_{z}} = {S}_{{T}_{zz},T} \end{array}$$
(B.17)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}{x}_{1}}} = i{\left (2\pi {f}_{1}\right )}^{3}{S}_{ T,T} = -{S}_{{T}_{{x}_{1}{x}_{1}},{T}_{{x}_{1}}} \end{array}$$
(B.18)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}{x}_{2}}} = \left (2\pi {f}_{1}\right ){S}_{T,{T}_{{x}_{ 2}}} = -{S}_{{T}_{{x}_{ 1}{x}_{2}},{T}_{{x}_{1}}} = {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{1}{x}_{1}}} = -{S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}}} \end{array}$$
(B.19)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}z}} ={ \left (2\pi {f}_{1}\right )}^{2}{S}_{ T,{T}_{z}} = {S}_{{T}_{{x}_{1}z},{T}_{{x}_{1}}} = -{S}_{{T}_{z},{T}_{{x}_{1}{x}_{1}}} = -{S}_{{T}_{{x}_{1}{x}_{1}},{T}_{z}}\end{array}$$
(B.20)
$$\begin{array}{rcl} &{S}_{{T}_{{x}_{ 1}},{T}_{{x}_{2}{x}_{2}}} = i2\pi {f}_{1}\,{S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}}} = -{S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{1}}} = {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{1}{x}_{2}}} = -{S}_{{T}_{{x}_{ 1}{x}_{2}},{T}_{{x}_{2}}} \end{array}$$
(B.21)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{2}z}} = i2\pi {f}_{1}\,{S}_{{T}_{{x}_{ 2}},{T}_{z}} = {S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{1}}} = -{S}_{{T}_{z},{T}_{{x}_{ 1}{x}_{2}}} = {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{1}z}} = -{S}_{{T}_{{x}_{ 1}{x}_{2}},{T}_{z}} = {S}_{{T}_{{x}_{ 1}z},{T}_{{x}_{2}}}\end{array}$$
(B.22)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}},{T}_{zz}} = -i2\pi {f}_{1}\,{S}_{{T}_{z},{T}_{z}} = -{S}_{{T}_{zz},{T}_{{x}_{ 1}}} = -{S}_{{T}_{z},{T}_{{x}_{ 1}z}} = {S}_{{T}_{{x}_{ 1}z},{T}_{z}} \end{array}$$
(B.23)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{2}},{T}_{{x}_{2}{x}_{2}}} & = -\frac{2{\sigma }^{2}\left (2\pi {f}_{ 1}\right ){s}_{2}} {\alpha {d}^{3}} \left (\left (6 -\frac{8{s}_{2}^{2}} {{d}^{2}} -{\left (2\pi {f}_{1}\,{s}_{2}\right )}^{2}\right ){K}_{1} + 2\pi {f}_{1}\,d\left (3 -\frac{4{s}_{2}^{2}} {{d}^{2}} \right ){K}_{0}\right ) \\ & = -{S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}}} \end{array}$$
(B.24)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{2}},{T}_{{x}_{2}z}} & = -\frac{2{\sigma }^{2}\left (2\pi {f}_{ 1}\right )\beta } {{\alpha }^{2}{d}^{3}} \left (\left (2 -\frac{8{s}_{2}^{2}} {{d}^{2}} -{\left (2\pi {f}_{1}\,{s}_{2}\right )}^{2}\right ){K}_{1} + 2\pi {f}_{1}\,d\left (1 -\frac{4{s}_{2}^{2}} {{d}^{2}} \right ){K}_{0}\right ) \\ & = {S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{2}}} = -{S}_{{T}_{z},{T}_{{x}_{ 2}{x}_{2}}} = -{S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{z}} \end{array}$$
(B.25)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}},{T}_{zz}} = -{S}_{{T}_{{x}_{ 2}},{T}_{{x}_{1}{x}_{1}}} - {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}{x}_{2}}} = -{S}_{{T}_{zz},{T}_{{x}_{ 2}}} = -{S}_{{T}_{z},{T}_{{x}_{ 2}z}} = {S}_{{T}_{{x}_{ 2}z},{T}_{z}} \end{array}$$
(B.26)
$$\begin{array}{rcl} {S}_{{T}_{z},{T}_{zz}} = {S}_{{T}_{{x}_{ 1}},{T}_{{x}_{1}z}} + {S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}z}} = {S}_{{T}_{zz},{T}_{z}} \end{array}$$
(B.27)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}{x}_{1}}} ={ \left (2\pi {f}_{1}\right )}^{4}{S}_{ T,T} \end{array}$$
(B.28)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}{x}_{2}}} = i{\left (2\pi {f}_{1}\right )}^{3}{S}_{{ T}_{{x}_{2}},T} = {S}_{{T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}{x}_{1}}} \end{array}$$
(B.29)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}z}} = -i{\left (2\pi {f}_{1}\right )}^{3}{S}_{ T,{T}_{z}} = -{S}_{{T}_{{x}_{1}z},{T}_{{x}_{1}{x}_{1}}} \end{array}$$
(B.30)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}{x}_{2}}} ={ \left (2\pi {f}_{1}\right )}^{2}{S}_{{ T}_{{x}_{2}},{T}_{{x}_{2}}} = {S}_{{T}_{{x}_{2}{x}_{2}},{T}_{{x}_{1}{x}_{1}}} = {S}_{{T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}{x}_{2}}} \end{array}$$
(B.31)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}z}} ={ \left (2\pi {f}_{1}\right )}^{2}{S}_{{ T}_{{x}_{2}},{T}_{z}} = -{S}_{{T}_{{x}_{2}z},{T}_{{x}_{1}{x}_{1}}} = -{S}_{{T}_{{x}_{1}z},{T}_{{x}_{1}{x}_{2}}} = {S}_{{T}_{{x}_{1}{x}_{2}},{T}_{{x}_{1}z}} \end{array}$$
(B.32)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{zz}} = -{\left (2\pi {f}_{1}\right )}^{2}{S}_{{ T}_{z},{T}_{z}} = {S}_{{T}_{zz},{T}_{{x}_{1}{x}_{1}}} = -{S}_{{T}_{{x}_{1}z},{T}_{{x}_{1}z}} \end{array}$$
(B.33)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{2}},{T}_{{x}_{2}{x}_{2}}} = i2\pi {f}_{1}\,{S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}}} = {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{1}{x}_{2}}} \end{array}$$
(B.34)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{2}},{T}_{{x}_{2}z}} = -i2\pi {f}_{1}\,{S}_{{T}_{{x}_{ 2}},{T}_{{x}_{2}z}} = -{S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{1}{x}_{2}}} = -{S}_{{T}_{{x}_{ 1}z},{T}_{{x}_{2}{x}_{2}}} = {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{1}z}} \end{array}$$
(B.35)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 1}{x}_{2}},{T}_{zz}} = -i2\pi {f}_{1}\,{S}_{{T}_{{x}_{ 2}z},{T}_{z}} = {S}_{{T}_{zz},{T}_{{x}_{ 1}{x}_{2}}} = -{S}_{{T}_{{x}_{ 1}z},{T}_{{x}_{2}z}} = -{S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{1}z}}\end{array}$$
(B.36)
$$\begin{array}{rcl} &{S}_{{T}_{{x}_{ 1}z},{T}_{zz}} = {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{1}z}} + {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{1}z}} = -{S}_{{T}_{zz},{T}_{{x}_{ 1}z}} \end{array}$$
(B.37)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}{x}_{2}}}& = \frac{2{\sigma }^{2}\left (2\pi {f}_{1}\right )} {\alpha {d}^{3}} \left (2\pi {f}_{1}\,d\left (3 -\frac{24{s}_{2}^{2}} {{d}^{2}} + \frac{24{s}_{2}^{4}} {{d}^{4}} +{ \left (2\pi {f}_{1}\,{s}_{2}\right )}^{2}\frac{{s}_{2}^{2}} {{d}^{2}} \right ){K}_{0}\right . \\ &\quad \left .+2\left (3 -\frac{24{s}_{2}^{2}} {{d}^{2}} - 3{\left (2\pi {f}_{1}{s}_{2}\right )}^{2} + \frac{24{s}_{2}^{4}} {{d}^{4}} + 4{\left (2\pi {f}_{1}\,{s}_{2}\right )}^{2}\frac{{s}_{2}^{2}} {{d}^{2}} \right ){K}_{1}\right ) \end{array}$$
(B.38)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}z}} & = -\frac{2{\sigma }^{2}\left (2\pi {f}_{1}\right )\beta {s}_{2}} {{\alpha }^{2}{d}^{5}} \left (2\pi {f}_{1}\,d\left (12 -\frac{24{s}_{2}^{2}} {{d}^{2}} -{\left (2\pi {f}_{1}\,{s}_{2}\right )}^{2}\right ){K}_{ 0}\right . \\ &\quad \left .+\left (24 -\frac{48{s}_{2}^{2}} {{d}^{2}} + 3{\left (2\pi {f}_{1}d\right )}^{2} - 8{\left (2\pi {f}_{ 1}\,{s}_{2}\right )}^{2}\right ){K}_{ 1}\right ) \\ & = -{S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{2}{x}_{2}}} \end{array}$$
(B.39)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{zz}} = -{S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}{x}_{2}}} - {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}{x}_{2}}} = {S}_{{T}_{zz},{T}_{{x}_{ 2}{x}_{2}}} = -{S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{2}z}}\end{array}$$
(B.40)
$$\begin{array}{rcl} {S}_{{T}_{{x}_{ 2}z},{T}_{zz}} = {S}_{{T}_{{x}_{ 1}{x}_{1}},{T}_{{x}_{2}z}} + {S}_{{T}_{{x}_{ 2}{x}_{2}},{T}_{{x}_{2}z}} = -{S}_{{T}_{zz},{T}_{{x}_{ 2}z}} \end{array}$$
(B.41)
$$\begin{array}{rcl} {S}_{{T}_{zz},{T}_{zz}} = {S}_{{T}_{{x}_{ 1}z},{T}_{{x}_{1}z}} + {S}_{{T}_{{x}_{ 2}z},{T}_{{x}_{2}z}}\end{array}$$
(B.42)

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Jekeli, C. (2010). Correlation Modeling of the Gravity Field in Classical Geodesy. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01546-5_28

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