**Appendix A**

(Note that *<sup>E</sup>* <sup>T</sup><sup>2</sup> <sup>¼</sup> 1, implies that assumed variance components are correct and

*Validation of the VMM MD solution by Eq. (29) and CRUST19 model. (the scale is unitless).*

The study of the Moho discontinuity has been a crucial topic in inferring the dynamics of the Earth's interior for a long time. In general, the Moho can be studied with profitable results through seismic data. However, due to the sparsity of seismic data in parts of the world, it has not been well determined. With the advent of satellite missions, it has been possible to recover the Moho constituents via satellite

So far, various isostatic models have been presented for recovering the Moho constituents, but it was not clarified which one is most appropriate to employ for geophysical and geodynamical purposes. The preliminary and simplest isostatic models proposed are the classical ones with local or regional compensation. However, those models cannot realistically image the actual Moho undulation. This is because they assume a uniform crustal density, disregarding the density irregularities distributed within the crust and sub-crust. Understanding this important role of Moho recovery has been in the center of the discussions by many geoscientists

Here we have determined the Moho constituents and their uncertainties based on the VMM technique using both gravimetric and seismic data on a global scale to a resolution of 1° � 1°. The combination of the gravimetric and seismic data in one approach as well as the joint adjustment of MD and density contrast are expected to

The basic VMM method is based on the hypothesis that the isostatic gravity disturbance vanish. However, this is the case only if the gravity component is reduced such that there are no signals from the Earth's interior below the crust. The major problem in this reduction is therefore to distinguish and remove those signals, which we utilize by estimating and removing the NIEs with the help from

The second step is to combine the gravimetric data, propagated in the VMM technique to a linear equation (with MD and MDC as the unknowns), with a seismic model, CRUST1.0. This is performed by a weighted least-squares adjustment, block by block, which has the advantage that the standard error of the unknowns can also be estimated block-wise. The weights of the gravity disturbances were based on the error estimates by Eq. (23), while the weights for CRUST1.0 data were those

the expected MDs of the two models are the same).

*Geodetic Sciences - Theory, Applications and Recent Developments*

gravity observations based on an isostatic model.

**6. Discussion and final remarks**

**Figure 4.**

during the last decades.

CRUST1.0 seismic model.

published in [12].

**36**

significantly improve the total result.

Let us assume that the compensation attraction in Eq. (2) is generated by a density contrast Δ*ρ* between the constant reference depth *D*<sup>0</sup> and the actual depth *D*. Assuming that the density contrast may change only laterally, it follows from the Newton integral in 3D, that the compensation potential becomes:

$$(Vc)\_P = G \iint\_{\sigma} \Delta \rho \int\_{R-D}^{R-D\_0} \frac{r^2 dr}{l\_P} d\sigma = G \left[ \int\_{\sigma} \Delta \rho \int\_{R-D}^{R} \frac{r^2 dr}{l\_P} d\sigma + G \right] \left[ \Delta \rho \int\_{\sigma}^{R-D\_0} \int\_{R}^{r} \frac{r^2 dr}{l\_P} d\sigma,\tag{A.1} \right]$$

where the last integral term is a constant, global mean value. Disregarding this term (which does not contribute to the Moho undulation) the integral can be written in the spectral domain after integration with respect to *r* and setting *rP* ¼ *R* (sea level radius):

$$(Vc)\_P = GR^2 \sum\_{n=0}^{\infty} \frac{1}{n+3} \iiint\_{\sigma} \frac{\Delta \rho}{n+3} \left[1 - \left(1 - \frac{D}{R}\right)^{n+3}\right] P\_n(\cos \varphi) d\sigma \tag{A.2}$$

Considering the addition theorem of fully normalized spherical harmonics (Heiskanen and Moritz 1967, p. 33):

$$P\_n\left(\cos\psi\_{PQ}\right) = \frac{1}{2n+1} \sum\_{m=-n}^{n} Y\_{nm}(P)Y\_{nm}(Q),$$

one obtains

$$\delta T\_P = \bar{\mathbf{G}} \mathbf{R}^2 \sum\_{n=0}^{\infty} \frac{\mathbf{1}}{(2n+1)(n+3)} \sum\_{m=-n}^{n} Y\_{nm}(P) \left[ \left| \Delta \rho \left[ \mathbf{1} - \left( \mathbf{1} - \frac{D}{R} \right)^{n+3} \right] Y\_{nm} d\sigma. \tag{A.3} \right|$$

As *D* is small vs. *R,* one may expand the last bracket in this equation a la Taylor as

$$\frac{1}{n+3}\left[1-\left(1-\frac{D}{R}\right)^{n+3}\right] = \frac{D}{R} - \frac{n+2}{2}\left(\frac{D}{R}\right)^2 + \frac{(n+2)(n+1)}{6}\left(\frac{D}{R}\right)^3 + \dots,\tag{A.4}$$

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

$$\begin{split} (Vc)\_P &= 4\pi GR \sum\_{n=0}^{\infty} \frac{1}{2n+1} \times \\ &\sum\_{m=-n}^{n} \left[ (D\Delta\rho)\_{nm} - \frac{n+2}{2R} \left( D^2 \Delta\rho \right)\_{nm} + \frac{(n+2)(n+1)}{6R^2} \left( D^3 \Delta\rho \right)\_{nm} - \dots \right] Y\_{nm}(P) \end{split} \tag{A.5}$$

where ðÞ*nm* are spherical harmonic coefficients. As the compensation potential coefficients are related to those of the compensation attraction *Ac* by

$$(Vc)\_{nm} = \frac{R}{n+1}(Ac)\_{nm},\tag{A.6}$$

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

$$\frac{R}{n+1}(Ac)\_{nm} = \frac{4\pi GR}{2n+1} \left[ (D\Delta\rho)\_{nm} - \frac{n+2}{2R} \left( D^2 \Delta\rho \right)\_{nm} + \frac{(n+2)(n+1)}{6R^2} \left( D^3 \Delta\rho \right)\_{nm} - \dots \right] \tag{A.7}$$

By comparing the spectra of both sides and summing up all harmonics and considering Eq. (2),

$$D\Delta\rho = \frac{1}{4\pi} \sum\_{n=0}^{\infty} \frac{2n+1}{n+1} \sum\_{m=-n}^{n} f\_{nm} Y\_{nm} + \Delta,\tag{A.8}$$

**Author details**

**39**

(KTH), Stockholm, Sweden

West (HV), Trollhättan, Sweden

Lars Erik Sjöberg1,2\* and Majid Abrehdary2

*On Moho Determination by the Vening Meinesz-Moritz Technique*

*DOI: http://dx.doi.org/10.5772/intechopen.97449*

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

provided the original work is properly cited.

1 Division of Geodesy and Satellite Positioning, Royal Institute of Technology

2 Division of Mathematics, Computer and Surveying Engineering, University

© 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,

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

$$\Delta = \sum\_{n=0}^{\infty} \sum\_{m=-n}^{\infty} \left[ \frac{n+2}{2R} \left( D^2 \Delta \rho \right)\_{nm} - \frac{(n+2)(n+1)}{6R^2} \left( D^3 \Delta \rho \right)\_{nm} + \dots \right] Y\_{nm} \tag{A.9}$$

accounts for higher order terms in the series.

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.

*On Moho Determination by the Vening Meinesz-Moritz Technique DOI: http://dx.doi.org/10.5772/intechopen.97449*
