**Author details**

and by inserting this series in Eq. (A.3) one obtains after integration

*nm* <sup>þ</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

� �

where ðÞ*nm* are spherical harmonic coefficients. As the compensation potential

*n* þ 1

*D*2 Δ*ρ* � �

By comparing the spectra of both sides and summing up all harmonics and

2*n* þ 1 *n* þ 1

*nm* � ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

� �

In practical application for Moho feature determination the lower limit for the summation is found to be about 10 (as the lower harmonics are related with deep Earth gravity anomalies) and the upper limit should not exceed about 180.

X*n m*¼�*n*

<sup>6</sup>*R*<sup>2</sup> *<sup>D</sup>*<sup>3</sup>

Δ*ρ* � �

<sup>6</sup>*R*<sup>2</sup> *<sup>D</sup>*<sup>3</sup>

Δ*ρ* � �

*nm* <sup>þ</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

� �

*nm* � …

ð Þ *Ac nm*, (A.6)

<sup>6</sup>*R*<sup>2</sup> *<sup>D</sup>*<sup>3</sup>

*f nmYnm* þ Δ, (A.8)

*nm* þ *::*

Δ*ρ* � �

*Ynm* (A.9)

*Ynm*ð Þ *P*

(A.5)

*nm* � …

(A.7)

ð Þ *Vc <sup>P</sup>* <sup>¼</sup> <sup>4</sup>*πGR*X<sup>∞</sup>

ð Þ *Ac nm* <sup>¼</sup> <sup>4</sup>*πGR*

considering Eq. (2),

where *fnm* <sup>=</sup> �*<sup>δ</sup> <sup>g</sup><sup>B</sup>*

X∞ *m*¼�*n*

<sup>Δ</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

**38**

X*n m*¼�*n*

*R n* þ 1 *n*¼0

ð Þ *<sup>D</sup>*Δ*<sup>ρ</sup> nm* � *<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

1 2*n* þ 1 �

*Geodetic Sciences - Theory, Applications and Recent Developments*

*D*2 Δ*ρ* � �

coefficients are related to those of the compensation attraction *Ac* by

ð Þ *Vc nm* <sup>¼</sup> *<sup>R</sup>*

2*R*

2*R*

one obtains the spectral equation from (A.5):

<sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>D</sup>*Δ*<sup>ρ</sup> nm* � *<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

*<sup>D</sup>*Δ*<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup> 4*π* X∞ *n*¼0

> *D*2 Δ*ρ* � �

accounts for higher order terms in the series.

*nm*/*G* and

*n* þ 2 2*R*

Lars Erik Sjöberg1,2\* and Majid Abrehdary2

1 Division of Geodesy and Satellite Positioning, Royal Institute of Technology (KTH), Stockholm, Sweden

2 Division of Mathematics, Computer and Surveying Engineering, University West (HV), Trollhättan, Sweden

\*Address all correspondence to: lsjo@kth.se

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
