**4.1 Crustal density corrections**

In order to compute the stripped refined Bouguer gravity disturbance, i.e. freeair gravity disturbance corrected for topography, bathymetry, ice thickness and sediment basins (i.e. stripping corrections), [34] developed and applied a uniform mathematical formalism of computing the gravity corrections of the density variations within the Earth's crust. This operation can be summarized as the correction

$$
\delta \mathbf{g}^{\text{TBlS}} = \delta \mathbf{g}^t + \delta \mathbf{g}^b + \delta \mathbf{g}^i + \delta \mathbf{g}^s \tag{31}
$$

where *δg<sup>t</sup>* is the topographic gravity correction, and *δg<sup>b</sup>*, *δg<sup>i</sup>* and *δg<sup>s</sup>* are the stripping gravity corrections due to the ocean (bathymetry), ice and sediment density variations, respectively.

Applying a spherical approximation of the Earth, each gravity correction on the right-hand side of Eq. (31) can be computed using the following spherical harmonic series:

$$\delta \mathbf{g}^q(P) = \frac{GM}{R^2} \sum\_{n=0}^{n\_{\text{max}}} (n+1) \sum\_{m=-n}^n c\_{nm}^q Y\_{nm}(\mathbf{P}),\tag{32}$$

with superscript *<sup>q</sup>* being one of *t, b, i* or *s,* and *GM* <sup>¼</sup> <sup>3986005</sup> � <sup>10</sup><sup>8</sup> <sup>m</sup><sup>3</sup> <sup>s</sup>�<sup>2</sup> is the geocentric gravitational constant. The coefficient *c q nm* of a particular volumetric mass density (or density contrast) layer *q* (i.e., topography, bathymetry, glacial ice and sediments) is defined by:

$$\mathcal{L}\_{nm}^{q} = \frac{2}{(2n+1)} \frac{1}{\rho\_{\varepsilon}} \left[ \frac{\left(\rho^{q} L\_{q}\right)\_{nm}}{R} + \frac{n+2}{2} \frac{\left(\rho^{q} L\_{q}^{2}\right)\_{nm}}{R^{2}} + \dots \right] \tag{33}$$

where *ρ<sup>q</sup>* is the Earth's mean mass density, and the coefficients (*ρqLi <sup>q</sup>*) are evaluated (from discrete data of density *ρ<sup>q</sup>* and thickness *Lq*) by applying a discretization to the following integral convolution

$$\left(\rho^q L\_q\right)\_{nm} = \frac{1}{4\pi} \iint\_{\sigma} \rho^q L\_q^j Y\_{nm} \, d\sigma, j = 1, 2, \dots, n \tag{34}$$

and reduced by the general NIE correction, but one may also estimate the DGIA

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>X</sup>*<sup>n</sup>*

where *γ* is normal gravity, *Ynm* and *Anm* are spherical harmonics and coefficients of the gravitational potential (see [37]). Here the limits of the series are based on the optimum correlation between the present land uplift and the gravity field in the

The main gravimetric input data to be used in the following VMM Moho model


is the global Earth gravitational field model (e.g. XGM2019e) in the harmonic window from *n*<sup>1</sup> ¼ 10 to *n*<sup>2</sup> ¼ 180. The gravity disturbance data were corrected for the gravitational signals of mass density variations due in different layers of the Earth's crust (i.e. stripping gravity corrections) and for the gravity contribution from deeper masses below the crust (i.e. non-isostatic effects). The NIEs were computed using the seismic crustal model CRUST1.0, and the stripping corrections for different crustal heterogeneous data utilized the global topographic models DTM2006 and Earth2014. The preliminary gravimetric Moho solution was combined with the CRUST1.0 model in a least-squares procedure (see Section 2.2). The

The statistics of the stripping gravity corrections and refined Bouguer gravity disturbance are presented in **Table 1**. It shows the largest corrections for bathymetry and NIE, but also ice cap corrections have some extreme values. The sum of the corrections varies roughly within �600 mGal with the STD of 178 mGal. **Figure 1** depicts the Bouguer gravity disturbances corrected for the ocean (bathymetry), ice, sediment variations and the NIEs, respectively. As one can see from the figure, these features can drastically change the Bouguer gravity disturbance from the free-air disturbance over oceans due to the application of the bathymetric stripping gravity correction. It also changes in central Greenland and Antarctica due to the applied ice density variation stripping gravity correction

**Quantities** *δg* **(mGal) Max Mean Min STD** *δg* 285.85 �0.44 �281.40 23.84 *<sup>δ</sup>g<sup>t</sup>* 255.13 �71.06 �647.61 105.98 *δg<sup>b</sup>* 721.60 332.91 110.28 165.02 *δg<sup>I</sup>* 325.78 21.84 �2.61 56.57 *δg<sup>S</sup>* 185.31 45.48 �0.02 32.47 *δgNIE* 248.70 �134.65 �497.98 69.98 *δgTBISN* 562.82 128.87 �620.54 178.10

*Statistics of global estimates of the gravity disturbances, stripping gravity corrections and NIEs. STD is the standard deviation of the estimated quantity over the blocks. δg is the gravity disturbance computed by the*

*gravity corrections derived from the CRUST1.0, respectively. δgNIE is the non-isostatic effect. δgTBISN is the refined Bouguer gravity disturbance after applying the topographic and stripping gravity corrections due to the*

*, δgb, δg<sup>I</sup> and δg<sup>S</sup> are the topographic/bathymetric, ice and sediment stripping*

*m*¼�*n*

*AnmYnm*, (38)

effect on gravity as a separate correction by the harmonic window:

X 23

*n*¼10

*<sup>δ</sup>gDGIA* <sup>¼</sup> *<sup>γ</sup>*

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

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

adjustment was performed globally for each 1<sup>∘</sup> � <sup>1</sup><sup>∘</sup>

region.

**Table 1.**

**33**

*XGM2019e coefficients. δg<sup>t</sup>*

*ocean, ice and sediment density variations.*

**5. A global VMM solution**

#### **4.2 Non-isostatic effects**

It is important to remind the reader that in general the crust is not in complete isostatic equilibrium, and the observed gravity data are not only generated by the topographic/isostatic masses, but also from those in the deep Earth interior, that leads to non-isostatic effects (NIEs) (see [18, 19, 35]).

According to [7], the major part of the long-wavelengths of the geopotential undulation is caused by density variations in the Earth's mantle and core/mantle topography variations. Such NIEs could be the contribution of different factors, such as crustal thickening/thinning, thermal expansion of mass of the mantle [36], Glacial Isostatic Adjustment (GIA), plate flexure ([16], p. 114), and effect of other phenomena. This implies that this contribution to gravity will lead to systematic errors/NIEs of the computed Moho topography. Hence the NIEs should also be corrected on the isostatic gravity disturbance.

Assuming that the seismic Moho model CRUST1.0 is known and correct, the gravity effect of the NIEs can be determined by:

$$\delta \mathbf{g}^{N\text{IE}} = \frac{\mathbf{G}\mathbf{M}}{R^2} \sum\_{n=0}^{n\_{\text{max}}} (n+1) \sum\_{m=-n}^{n} c\_{nm}^{N\text{IE}} Y\_{nm}(\mathbf{P}) \tag{35}$$

where

$$
\sigma\_{nm}^{\text{NIE}} = \sigma\_{nm}^{\text{CRUST1.0}} - \sigma\_{nm}^{\text{VMM}} \tag{36}
$$

Here *cNIE nm* , *cVMM nm* , *c*CRUST1*:*<sup>0</sup> *nm* are the spherical harmonic coefficients of the gravity disturbances of the NIE, VMM and CRUST1.0, respectively.

The isostatic equilibrium equation in Eq. (2) is then rewritten as:

$$
\delta \mathbf{g}^I(P) = \delta \mathbf{g}\_B^{TBISN}(P) + A\_C(P) = \mathbf{0}.\tag{37}
$$

Here *δgTBISN <sup>B</sup>* is the refined Bouguer gravity disturbance corrected for the gravitational contributions of topography and density variations of the oceans, ice, sediments and NIEs, i.e. by Eq. (31).

#### **4.3 Glacial isostatic adjustment (GIA)**

Delayed GIA (DGIA) expresses the delayed adjustment process of the Earth to an equilibrium state when former ice sheet loads have vanished. The ongoing adjustment of the Earth's body to the redistribution of ice and water masses is evident in various phenomena, which have been studied to infer the extent and amount of the former ice masses, to reconstruct the sea level during a glacial cycle and to constrain rheological properties of the Earth's interior. Here we aim at answering the question whether the effect of the gravimetric DGIA correction is significant for Moho determination in Fennoscandia. Usually, this effect is part of

and reduced by the general NIE correction, but one may also estimate the DGIA effect on gravity as a separate correction by the harmonic window:

$$\delta \mathbf{g}^{DGA} = \gamma \sum\_{n=10}^{23} (n+1) \sum\_{m=-n}^{n} A\_{nm} Y\_{nm},\tag{38}$$

where *γ* is normal gravity, *Ynm* and *Anm* are spherical harmonics and coefficients of the gravitational potential (see [37]). Here the limits of the series are based on the optimum correlation between the present land uplift and the gravity field in the region.
