**2. Geoid determination as a purpose in Geodesy**

The geoid is a reference surface for heights and if this reference is not enough precise and accurate, all determined heights from it will be unreliable. Having a precise geoid model simplifies the lengthy and costly work of levelling by simply using a global navigation satellite systems (GNSS) receiver, the height above the geoid can be determined. However, to reach to this goal, deep knowledge about the Earth's gravity field and the Physical Geodesy theories, skills in numerical modelling and precise data in all frequency bands are required.

The main task of Physical Geodesy is to develop theories and methods to model a precise geoid. Different approaches have been developed toward this goal. As known, the surface terrestrial gravity data are sensitive to high frequencies and near surface mass variations, but their low frequencies of the signal are weak unlike satellite-only Earth gravity models (EGMs) having better qualities only in the low frequency band. In geoid modelling approaches the terrestrial gravity data are used for recovering the high frequencies of the geoid and the satellite EGMs for the lower. Generally, in geoid modelling the following issues should also be considered:


The differences between the geoid modelling methods are related to how mathematically these issues are handled. In the following the three methods of RCR, SH and LSMSF are shortly presented.

#### **2.1 Remove-compute-restore approach**

In the RCR scheme, the low and high frequencies of terrestrial gravity data are removed by an EGM and topographic heights. Because the low frequencies are global and for converting of the gravity data with a regional coverage to the geoid, the low frequencies of geoid cannot be recovered well. In addition, removing the effect of topographic terrain makes the gravity data smoother, and simplifies the computations. After computing the geoid excluding the low and high frequencies, the removed frequencies are restored to it to complete all frequencies of the geoid. Mathematically, the idea is presented by:

$$N = \frac{R}{4\pi\gamma} \left[ \left| \mathbf{S}(\boldsymbol{\nu}) \left( \left( \Delta \mathbf{g} - \Delta \mathbf{g}^{\rm RTM} \right)^{\*} - \sum\_{n=2}^{L} \Delta \mathbf{g}\_{n}^{\rm EGM} \right) \right| \mathbf{d}\boldsymbol{\sigma} + \sum\_{n=2}^{L} N\_{n}^{\rm EGM} + N^{\rm RTM} \tag{1}$$

*The Earth's Gravity Field Role in Geodesy and Large-Scale Geophysics DOI: http://dx.doi.org/10.5772/intechopen.97459*

where

$$
\begin{pmatrix}
\Delta \mathbf{g}\_n^{\mathrm{EGM}} \\
N\_n^{\mathrm{EGM}}
\end{pmatrix} = \sum\_{m=-n}^{n} \Delta \mathbf{g}\_{nm}^{\mathrm{EGM}}
\begin{Bmatrix} \\ \\ \end{Bmatrix} Y\_{nm}(\theta, \lambda) \tag{2}
$$

*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 domain, d*σ* the surface integration element, Δ*g*EGM *nm* and *N*EGM *nm* are, respectively, the 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 computed geoid.

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:


#### **2.2 Stokes-Helmert approach**

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

geoid height. In addition, this integral is the solution of the gravimetric boundaryproblem, the Laplace equation, with gravity anomalies at the boundary, the geoid. This means that this solution is theoretically valid where there is no mass outside the boundary surface. However, in practice, the gravity data are collected at the Earth's surface, on topographic and under atmospheric masses. The presence of such masses violates the theory. Therefore, the gravitational effects on the gravity data should be removed to fulfil the Laplace equation. The result will be a notopography and no-atmosphere computational space, or the Helmert space. After removing these effects the gravity data still remain above the boundary and need to be continued downward. By solving the Stokes integral numerically, these continued data are converted to a surface similar to geoid, known as co-geoid. The next step will be to convert this co-geoid by restoring the effects of topographic and atmospheric masses. The principle of SH method is:

4.The TE and AE have their own error properties, in addition by removing the long wavelengths of the anomalies by an EGM, Therefore, the reduced anomalies contaminate larger stochastic error than the measured ones. The errors of the gravity anomalies and EGMs are not considered in the solution.

5.The reduced gravity anomalies in the Helmert space need to be continued downward to see level prior to integrating them. To do so, inverse solution of the Poisson integral is applied; see [6], which is an ill-posed problem and

**2.3 Least-squares modification of stokes formula with additive corrections**

Unlike the RCR and SH methods, neither the TE and AE nor the long wavelength portion of the anomalies are removed from the gravity anomalies. Instead, the terrestrial anomalies and EGM are spectrally weighted which means that the Stokes integral is modified in such a way their errors and the truncation error of the Stokes integral outside the integration cap are minimised in a least-squares sense. This method is called least-squares modification of Stokes formula [3]. In this method, the terrestrial anomalies are integrated directly by the modified Stokes formula to estimate a geoid model. Later the total TE and AE, DWC and ellipsoidal corrections will be added to the modelled geoid to make it precise. This method can be mathematically presented by:

complicated when the resolution of the anomalies is high.

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

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

*<sup>N</sup>* <sup>¼</sup> *<sup>R</sup>* <sup>4</sup>*πγ* ðð *σ*0

effect on the geoid.

**5**

*<sup>S</sup><sup>L</sup>*ð Þ *<sup>ψ</sup>* <sup>Δ</sup>*g*d*<sup>σ</sup>* <sup>þ</sup>

The properties of this method are:

applying the stokes formula.

*R* 2*γ* X *L*

contributes to solution according to their precision.

equation needs to be solved numerically.

*n*¼2

*bn*Δ*g*EGM

where *bn* is a parameter depending on type of modification, *N*<sup>T</sup> and *N*<sup>A</sup> are, respectively, the total TE and AE. *N<sup>e</sup>* is the ellipsoidal correction, *N*DWC the DWC

1.The measured gravity anomalies are used in the modified Stokes integral. However, gridded anomalies are not at the boundary surface, which is not theoretically corrected, also no mass should exist outside the geoid when

2.This method considers the errors of the terrestrial data, EGM and truncation of the integral formula and modify the integral in an optimal way, meaning that the quality of the data play important role in geoid modelling. The data

modified Stokes integral will contain biases. However, the total TE and AE will be removed. In fact, the gravitational potential of these masses are computed for points at the surface of the Earth. Later they are continued downward to the boundary, and subtracted from the indirect gravitational potential of the points under the masses at the boundary. Such a potential will be converted

4.The DWC process is done directly on the potential, the gravity data converted to the potential and continued downward analytically. Therefore, no integral

3.Because of neglecting the TE and AE on the gravity data, results of the

simply to correction to geoid using the Bruns formula. Note that no compensation or condensation mechanism is required in this method

*<sup>n</sup>* <sup>þ</sup> *<sup>N</sup>*<sup>T</sup> <sup>þ</sup> *<sup>N</sup>*<sup>A</sup> <sup>þ</sup> *<sup>N</sup><sup>e</sup>* <sup>þ</sup> *<sup>N</sup>*DWC (4)

$$N = \frac{R}{4\pi\gamma} \iiint\_{\sigma\_0} \mathbf{S}^L(\boldsymbol{\nu}) \left(\Delta \mathbf{g}^L - \Delta \mathbf{g}\_{\rm dir}^{\rm TAe}\right) \mathbf{d}\boldsymbol{\sigma} + \frac{R}{2\gamma} \sum\_{n=2}^L \frac{2}{n-1} \Delta \mathbf{g}\_n^{\rm EGM} + N\_{\rm Ind}^{\rm TAe} \tag{3}$$

where *<sup>S</sup><sup>L</sup>*ð Þ *<sup>ψ</sup>* is the modified Stokes function, <sup>Δ</sup>*g*TAe dir the joint direct effect of topographic and atmospheric masses as well as the ellipticity of the Earth, which should be removed from the gravity data. Δ*g<sup>L</sup>* means the gravity anomalies excluding the frequencies to degree *L*. *N*TAe Ind is the joint indirect effect of the removed masses and ellipticty.

The SH method has the following properties:


*The Earth's Gravity Field Role in Geodesy and Large-Scale Geophysics DOI: http://dx.doi.org/10.5772/intechopen.97459*

