**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:

$$N = \frac{R}{4\pi\gamma} \left[ \int\_{\sigma\_0}^{L} (\boldsymbol{\nu}) \Delta \mathbf{g} \, \mathrm{d}\sigma + \frac{R}{2\gamma} \sum\_{n=2}^{L} b\_n \Delta \mathbf{g}\_n^{\mathrm{EGM}} + N^{\mathrm{T}} + N^{\mathrm{A}} + N^{\mathrm{e}} + N^{\mathrm{DWC}} \right. \tag{4}$$

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 effect on the geoid.

The properties of this method are:


5.The effect of ignoring the ellipticity of the Earth will be considered as an extra correction to the geoid directly.

where *G* is the Newtonian gravitational constant and

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

series in Eq. (6) should be truncated at lower degrees [7].

depths, and the Moho depths with a spatial resolution of 1° � 1°

to degree and order 180, corresponding to the resolution 1° � 1°

*(a) Global Moho flexure, and (b) the contribution of the gravity data to the Moho flexure.*

<sup>1</sup> <sup>þ</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *<sup>D</sup>*<sup>~</sup> <sup>0</sup>

� ��<sup>1</sup>

<sup>Γ</sup>*<sup>n</sup>* <sup>¼</sup> *<sup>R</sup>*

1 over oceans

over continents (

(7)

*:* (8)

. This means that

. **Figure 1b** showed

2*R*

*<sup>R</sup>* � *<sup>D</sup>*<sup>~</sup> <sup>0</sup> � �*<sup>n</sup>*þ<sup>2</sup>

The factor Γ*<sup>n</sup>* is a signal amplifier, and when *n* increases this factor grows. For large values of *D*~ 0, this amplification starts from lower frequencies, therefore, the

Δ*ρ* can also be determined from Δ*D*~ , if its available or even the product Δ*ρ* Δ*D*~ ; e.g. see [7] in which the GOCE data are constrained to seismic data for determina-

CRUST1.0 [8] is a global model having information about the thicknesses and densities of sediments, crustal crystalline, topographic heights and bathymetric

*nm* can be generated from CRUST1.0. In addition, numerous EGMs have been provided, which applicable for computing Δ*gnm*. **Figure 1a** shows the Moho flexure/ variation computed based on Eq. (6) the CRUST1.0 model, and EGM08 [9] limited

the contribution of Δ*g* ranging from �15 to 15 km to the estimated Moho depth.

In flexural isostasy [10] the lithospheric is considered as an elastic shell, being flexes under loads. This shell bends based on its own mechanical properties and pressure of the loads. Elastic thickness (Te) is one of the properties of this shell. Admittance and coherence analyses between the topography and gravity anomalies; see [11] are known methods for estimating this elastic thickness. By combining the gravimetric and flexure isostasy models the elastic thickness or rigidity of the lithosphere can be estimated as well [7]. The main assumption of this approach is that the Moho variations derived from the gravimetric and flexural isostasy theories are equal. Therefore, the elastic thickness is estimated such a way that the Moho variation estimated from the gravimetric isostasy becomes closer to that from the

*β* ∗ *<sup>n</sup>* ¼

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

tion of Δ*ρ* Δ*D*~ .

*3.1.2 Elastic thickness and rigidity*

Δ*g*TBSCI

**Figure 1.**

**7**
