**5. A global VMM solution**

where *ρ<sup>q</sup>* is the Earth's mean mass density, and the coefficients (*ρqLi*

evaluated (from discrete data of density *ρ<sup>q</sup>* and thickness *Lq*) by applying a

*σ ρq L j*

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

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

Assuming that the seismic Moho model CRUST1.0 is known and correct, the

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

CRUST1*:*<sup>0</sup> *nm* � *<sup>c</sup>*

*m*¼�*n c NIE*

*VMM*

*nm* , *c*CRUST1*:*<sup>0</sup> *nm* are the spherical harmonic coefficients of the gravity

*<sup>B</sup>* is the refined Bouguer gravity disturbance corrected for the gravi-

X*n*max *n*¼0

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

tational contributions of topography and density variations of the oceans, ice,

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

Delayed GIA (DGIA) expresses the delayed adjustment process of the Earth to

ð Þ¼ *<sup>P</sup> <sup>δ</sup>gTBISN*

discretization to the following integral convolution

*Geodetic Sciences - Theory, Applications and Recent Developments*

leads to non-isostatic effects (NIEs) (see [18, 19, 35]).

corrected on the isostatic gravity disturbance.

gravity effect of the NIEs can be determined by:

*<sup>δ</sup>gNIE* <sup>¼</sup> *GM R*2

> *c NIE nm* ¼ *c*

disturbances of the NIE, VMM and CRUST1.0, respectively.

*δgI*

*nm* <sup>¼</sup> <sup>1</sup> 4*π* ðð

*ρq Lq* � �

**4.2 Non-isostatic effects**

where

Here *cNIE*

Here *δgTBISN*

**32**

*nm* , *cVMM*

sediments and NIEs, i.e. by Eq. (31).

**4.3 Glacial isostatic adjustment (GIA)**

*<sup>q</sup>*) are

*qY*n*<sup>m</sup> dσ*, *j* ¼ 1, 2, … , *n* (34)

*nm Ynm*ð Þ P (35)

*nm* (36)

*<sup>B</sup>* ð Þþ *P AC*ð Þ¼ *P* 0*:* (37)

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

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


#### **Table 1.**

*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 XGM2019e coefficients. δg<sup>t</sup> , δgb, δg<sup>I</sup> and δg<sup>S</sup> are the topographic/bathymetric, ice and sediment stripping 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 ocean, ice and sediment density variations.*

(**Figure 1d**). In **Figure 1e** one can see large stripping corrections in sediment basins, and the NIEs are also very significant (**Figure 1f**).

**Figure 1g** shows the refined Bouguer gravity disturbance after applying the above corrections. This disturbance has a span of about 500 mGal, to be compared with the approximate span of 250 mGal of the free-air disturbance. Notable is the large positive disturbances on the oceans corresponding to the effect of filling the oceans with topographic masses. The DGIA effect, demonstrated for Fennoscandia and, depicted in **Figure 1h**, is very small compared to other corrections.

In the least-squares procedure of the combined VMM solution the weights of the two types of data were chosen as follows. The weights of the gravity disturbances were estimated from their inverse variances by Eq. (23), while the weights for CRUST1.0 data were those published in [12]. **Figures 2** and **3** depict the results of the MD and density contrast undulations and their estimated standard errors. Their extreme values for continental and oceanic crusts and mean values are reported in **Table 2**.

To validate the STE of the VMM solution for crustal depth, we determined the

*).*

global mean of it by Eq. (27) using the seismic model CRUST19. The result is 1.73 km, which is in fair agreement with the 1.20 km given in **Table 2**. Also, as one can see from **Figure 4** the test parameter in Eq. (29) for validating the VMM solution of MD from the seismic model CRUST19 is mainly in the range 1, which suggest rather close agreements of estimated MDs and their error estimates.

**Quantities Max. Mean Min. STD MD (km) Global** 70.26 23.78 7.55 13.17

**STE MD (km) Global** 8.15 1.20 0.05 0.94

*Statistics of global estimates of MD and MDC in the VMM approach for 1° 1° block data. STD is the standard deviation. STE is the standard error obtained in the least-squares adjustment. Units for MD and*

**Ocean** 43.19 14.98 7.55 **Land** 70.26 40.03 18.37

**Ocean** 7.34 2.06 0.05 **Land** 8.15 2.49 1.05

**) Global** 649.99 340.49 20.98 100.90 **Ocean** 637.36 281.01 20.98 **Land** 649.99 440.01 69.34

**) Global** 132.26 17.44 0.09 14.17 **Ocean** 99.98 35.21 0.09 **Land** 132.26 38.65 19.09

*(a) The MDC estimated by combined approach, and (b) its standard error. (unit kg/m<sup>3</sup>*

*(a) The MD estimated from combined approach, and (b) its standard error. (unit km).*

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

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

**Figure 2.**

**Figure 3.**

**MDC (kg/m<sup>3</sup>**

**STE MDC (kg/m<sup>3</sup>**

*MDC are km and kg/m3*

*, respectively.*

**Table 2.**

**35**

**Figure 1.**

*(a) The free-air gravity disturbance computed using the XGM2019e coefficients complete to degree 180 of spherical harmonics, (b) the topographic gravity correction, (c) the bathymetric stripping gravity correction, (d) the ice density variation stripping gravity correction, (e) the sediments density variation stripping gravity corrections, (f) non-isostatic effects and (g) refined Bouguer gravity disturbances after applying the above corrections. (h) the DGIA effect in Fennoscandia. Unit: mGal.*

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

(**Figure 1d**). In **Figure 1e** one can see large stripping corrections in sediment basins,

**Figure 1g** shows the refined Bouguer gravity disturbance after applying the

In the least-squares procedure of the combined VMM solution the weights of the two types of data were chosen as follows. The weights of the gravity disturbances were estimated from their inverse variances by Eq. (23), while the weights for CRUST1.0 data were those published in [12]. **Figures 2** and **3** depict the results of the MD and density contrast undulations and their estimated standard errors. Their extreme values for continental and oceanic crusts and mean values are

*(a) The free-air gravity disturbance computed using the XGM2019e coefficients complete to degree 180 of spherical harmonics, (b) the topographic gravity correction, (c) the bathymetric stripping gravity correction, (d) the ice density variation stripping gravity correction, (e) the sediments density variation stripping gravity corrections, (f) non-isostatic effects and (g) refined Bouguer gravity disturbances after applying the above*

*corrections. (h) the DGIA effect in Fennoscandia. Unit: mGal.*

above corrections. This disturbance has a span of about 500 mGal, to be compared with the approximate span of 250 mGal of the free-air disturbance. Notable is the large positive disturbances on the oceans corresponding to the effect of filling the oceans with topographic masses. The DGIA effect, demonstrated for Fennoscandia and, depicted in **Figure 1h**, is very small compared to other

and the NIEs are also very significant (**Figure 1f**).

*Geodetic Sciences - Theory, Applications and Recent Developments*

corrections.

**Figure 1.**

**34**

reported in **Table 2**.

**Figure 2.** *(a) The MD estimated from combined approach, and (b) its standard error. (unit km).*

**Figure 3.** *(a) The MDC estimated by combined approach, and (b) its standard error. (unit kg/m<sup>3</sup> ).*

To validate the STE of the VMM solution for crustal depth, we determined the global mean of it by Eq. (27) using the seismic model CRUST19. The result is 1.73 km, which is in fair agreement with the 1.20 km given in **Table 2**. Also, as one can see from **Figure 4** the test parameter in Eq. (29) for validating the VMM solution of MD from the seismic model CRUST19 is mainly in the range 1, which suggest rather close agreements of estimated MDs and their error estimates.


#### **Table 2.**

*Statistics of global estimates of MD and MDC in the VMM approach for 1° 1° block data. STD is the standard deviation. STE is the standard error obtained in the least-squares adjustment. Units for MD and MDC are km and kg/m3 , respectively.*

Our estimated results can be summarized as follows. The global means of MD

recent CRUST19 seismic model, showing that the differences between the models vary within the extremes �23.4 and 32.9 km, with a global average of 0.91 km and an RMS fit of 4 km. The normalized differences were generally within the limits �1,

This study was supported by project no. 187/18 of the Swedish National Space

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

*R*�*D*

where the last integral term is a constant, global mean value. Disregarding this

<sup>1</sup> � <sup>1</sup> � *<sup>D</sup>*

X*n m*¼�*n*

ðð

*σ*

Considering the addition theorem of fully normalized spherical harmonics

2*n* þ 1

*Ynm*ð Þ *P*

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

*D R* � �<sup>2</sup> *R*

*r*<sup>2</sup>*dr lP*

*dσ* þ *G*

ðð

*σ* Δ*ρ R*�ð *D*<sup>0</sup>

� �*<sup>n</sup>*þ<sup>3</sup> " #*Pn*ð Þ cos *<sup>ψ</sup> <sup>d</sup><sup>σ</sup>* (A.2)

*Ynm*ð Þ *P Ynm*ð Þ *Q* ,

<sup>Δ</sup>*<sup>ρ</sup>* <sup>1</sup> � <sup>1</sup> � *<sup>D</sup>*

<sup>þ</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 6

*R*

� �*<sup>n</sup>*þ<sup>3</sup> " #*Ynmdσ:* (A.3)

*D R* � �<sup>3</sup>

þ …, (A.4)

*R*

*r*<sup>2</sup>*dr lP*

*dσ*, (A.1)

Newton integral in 3D, that the compensation potential becomes:

ðð

*σ* Δ*ρ* ð *R*

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*

*dσ* ¼ *G*

, respectively, ranging

. The MD results were validated by the

and MDC are 23.8 � 0.05 km and 340.5 � 0.37 kg/m<sup>3</sup>

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

between 7.6–70.3 km and 21.0–650.0 kg/m<sup>3</sup>

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

which should be regarded as acceptable.

**Acknowledgements**

Agency (SNSA).

**Appendix A**

ð Þ *Vc <sup>P</sup>* ¼ *G*

(sea level radius):

one obtains

*<sup>δ</sup>TP* <sup>¼</sup> *GR*<sup>2</sup>X<sup>∞</sup>

Taylor as

1 *n* þ 3

**37**

*n*¼0

<sup>1</sup> � <sup>1</sup> � *<sup>D</sup>*

*R* � �*<sup>n</sup>*þ<sup>3</sup> " #

ðð

*σ* Δ*ρ R*�ð *D*<sup>0</sup>

ð Þ *Vc <sup>P</sup>* <sup>¼</sup> *GR*<sup>2</sup>X<sup>∞</sup>

(Heiskanen and Moritz 1967, p. 33):

*R*�*D*

*n*¼0

1 ð Þ 2*n* þ 1 ð Þ *n* þ 3

1 *n* þ 3 ðð

Δ*ρ n* þ 3

*σ*

*Pn* cos *<sup>ψ</sup>PQ* � � <sup>¼</sup> <sup>1</sup>

¼ *D*

X*n m*¼�*n*

*<sup>R</sup>* � *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> 2

*r*<sup>2</sup>*dr lP*

**Figure 4.** *Validation of the VMM MD solution by Eq. (29) and CRUST19 model. (the scale is unitless).*

(Note that *<sup>E</sup>* <sup>T</sup><sup>2</sup> <sup>¼</sup> 1, implies that assumed variance components are correct and the expected MDs of the two models are the same).
