**1. Introduction**

The Earth's gravity field reflects of the Earth's interior and is an interesting subject in Geodesy and Geophysics with various applications. Geodesy aims to determine three types of the shape and size of the Earth, the Earth's surface, geoid as the physical shape, reference ellipsoid as the mathematical one. Physical Geodesy deals with determination of the physical shape of the Earth or the geoid, which is a reference for heights, from gravimetric data. In this chapter, short descriptions of three known methods of geoid determination such as Remove-Compute-Restore (RCR) [1], Stokes-Helmert (SH) [2] and least-squares modification of the Stokes formula with addition corrections (LSMSF) [3] are presented.

In Geophysics, understanding the Earth's physics, dynamics and interior geometry is of interest using such data. Gravity measurements can be analysed over small or large area depending on the geophysical purpose. For instance, in exploration Geophysics they are used to detect or discover near surface resources and for such a goal precision and accuracy of these data should be high. Here, such applications are named small-scale Geophysics. However, understanding or studying the deep Earth's interior physics, dynamics or geometry does not require high spatial

resolutions and long wavelength portions of the gravity data are more suitable. In addition, large areas are considered for such purpose and therefore, here, such subjects are called large-scale Geophysics. Some of these large-scale phenomena are modelling the Mohorovičić discontinuity, elastic thickness of the lithosphere, sublithospheric/lithospheric stress, and thickness of ocean water, sediments, and ice; land uplift, mantle viscosity and groundwater storage; and post-seismic studies of Earthquakes, detecting the epicentre points of shallow Earthquakes, which are briefly presented in this chapter.

where

computed geoid.

final solution again.

wavelengths from gravity data.

frequencies from the RTM effect.

one-step simultaneously.

**2.2 Stokes-Helmert approach**

**3**

Δ*g*EGM *n N*EGM *n*

*DOI: http://dx.doi.org/10.5772/intechopen.97459*

*The Earth's Gravity Field Role in Geodesy and Large-Scale Geophysics*

domain, d*σ* the surface integration element, Δ*g*EGM

)

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> m*¼�*n* Δ*g*EGM *nm N*EGM *nm* )

*nm* and *N*EGM

*N* is the geoid height, *S*ð Þ *ψ* the Stokes function [4] converting gravity to geoid, Δ*g* the gravity anomaly, Δ*g*RTM the residual terrain effect (RTM) on the anomaly, <sup>Δ</sup>*<sup>g</sup>* � <sup>Δ</sup>*g*RTM � � <sup>∗</sup> means the downward continued <sup>Δ</sup>*g*- <sup>Δ</sup>*g*RTM,*σ*<sup>0</sup> the integration

spherical harmonic coefficients of Δ*g* and *N* of degree *n* and order *m*, derived from an EGM, limited to the maximum degree *L*, *Ynm*ð Þ *θ*, *λ* the spherical harmonic with arguments of co-latitude *θ* and longitude *λ*,*N*RTM the restored RTM effect on the

The first term on the right-hand side (rhs) of Eq. (1), is the Stokes integral, which converts the gravity anomalies to geoid height, and since the long and short frequencies of the anomalies are removed, the solution of this integral is the geoid height excluding these frequencies. In fact, the addition of the second and third terms of Eq. (1) is to restore these frequencies back to the computed geoid height. In the following, some issues regarding the RCR method is presented and discussed:

1.Three types of data are used in Eq. (1), terrestrial gravity data, EGM and RTM with own error properties. According to the error propagation law, the error of the reduced gravity anomalies is the square root of summation of variances of the terrestrial data, EGM and RTM, which is surely larger than the error of gravity data. If the discretisation error of the Stokes integral is assumed small, in the restoring step, the errors of the EGM and RTM will be propagated to the

2.Most portion of the geoid signal comes from its low frequencies, and by

3.After the removal step, gravity data are converted to a medium frequency geoid height, using Stokes formula, least-squares collocation (LSC) [5] or Fast Fourier Transform (FFT); see e.g. [1]. The low frequencies are restored by the

same EGM applied limited to the same maximum degree, and the high

4.Downward continuation (DWC) of the gravity data should be performed before applying the Stokes formula or FFT. However, by using LSC, the conversion of the gravity data to the geoid heights and DWC can be done in

The Stokes-Helmert (SH) method was proposed by Vanicek and Martinec [2] and developed further by Martinec [6]. Theoretically, the gravity data on a spherical surface are needed to numerically solve the Stokes integral for computing a

removing it by an EGM, this part of signal is assumed as known. By increasing the maximum degree of it, more portion of the geoid signal removed and restored. This means that the sensitivity to the terrestrial data will be reduced. The high frequencies of the geoid comes from topographic masses, and by considering it known as well, the main task will be to recover the medium

*Ynm*ð Þ *θ*, *λ* (2)

*nm* are, respectively, the
