**1. Introduction**

Traditionally, the structure of the Earth's interior is divided according to its chemical and physical properties into crust, mantle, outer core and inner core. The oceanic crust ranges from 5 to 10 km depth, while the continental crust ranges from 35 to 70 km depth. The layer below the crust is the mantle, which is the thickest layer of the Earth. It can be divided into the upper (extending down to 660 km from the Earth's surface) and lower mantle (down to 2900 km beneath the surface). The innermost layer of the Earth is the core, which can be decomposed into the outer and inner core. (A modern decomposition of the Earth's interior is based on its main mechanical properties: the lithosphere and asthenosphere, of main interest in global geodynamics, plate tectonics and motion, but not for this study).

The geoscientist typically uses three sources of information to figure out the interior of the Earth's structure:

The first source is understood by direct evidence from rock samples by drilling projects. In this way, the scientist attempts to drill holes in the Earth's surface, to a maximum depth of about 12 km, and explode rocks for inferring the conditions within the Earth's interior. The drilling method is severely limited, because it is difficult to drill a deep hole due to the high pressure and temperature, and it is also a very time-consuming and expensive technology (see [1]).

The second source includes the records of seismic waves, which are generated, for example, by earthquakes, explosions, volcanoes and other natural sources. Accordingly, specialists can detect information about the Earth's interior, e.g. depth to density discontinuities, through detailed analysis of seismic data. Also, by studying the velocity of the wave, it can to some extent be used for estimating the density of the medium. At this point it deserves to be mentioned that the seismic data are also expensive to collect and therefore sparse and in-homogeneously distributed around the Earth (see [2]).

waves. For global studies the most frequently used crustal models are the CRUST2.0 [11] and CRUST1.0 models [12], compiled with 2° � 2° and 1° � 1° resolutions, respectively. More recently, [13] developed a global crustal thickness model and velocity structure from geostatistical analysis of seismic data, and we hereafter call

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

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

Over large areas of the world with a sparse coverage of seismic data, in particular

methods for estimating the MDC [6] and for reducing the Bouguer gravity anomaly for non-isostatic effects [18, 19]. [20] demonstrated that the MD estimated from the isostatic gravity disturbance based on solving the VMM model has a better agreement with the CRUST2.0 seismic model than those computed by the isostatic gravity anomaly. Their argument was also theoretically explained by [21]. [22] estimated the MD and MDC using a combination of the CRUST2.0 and a GOCE global gravity models. [23] showed that the application of the Bouguer gravity disturbance and the no-topography correction in the VMM model to determine the MD provides very similar results, suggesting the preference of the gravity disturbance to the traditional Bouguer gravity anomaly for gravity inversion. [4, 5] computed combined Moho constituent model according to the VMM method. [24] estimated a new MDC model named MDC2018, using the marine gravity field from satellite altimetry in combination with a seismic-based crustal model and Earth's topographic/bathymetric data. Finally, [25] estimated a combined Moho model for marine areas via satellite altimetric - gravity and seismic crustal

Isostasy is an important concept in Earth sciences describing the state of equilibrium (or mass balance) to which the mantle tends to balance the mass of the crust in the absence of external disturbing forces. "*When a certain area of the crust reaches the state of isostasy, it is said to be in isostatic equilibrium (or balance), and the depth at which isostatic equilibrium prevails is called the depth of compensation"* [26]. However, the transport of material over the Earth's surface, such as glaciers, volcanism, and sedimentation, etc., are factors that disturb isostasy, yielding

Four principle models of isostasy related with the crustal depth and/or density can briefly be listed as those of (a) Airy/Heiskanen (A/H; [27–29]), (b) Pratt/ Hayford (P/H; [30, 31]), (c) Vening Meinesz (VM; [14]), and (d) the Vening Meinesz-Moritz (VMM; [7, 17]). Common for the isostatic models is that the Bouguer gravity anomaly Δ*g<sup>b</sup>* (or disturbance *δg<sup>b</sup>*) is fully compensated by a compensation attraction below the crust such that the isostatic gravity anomaly and

A/H and P/H are local models, implying that the compensation attraction operates along the vertical of the observation point, implying that the sum of the

<sup>Δ</sup>*g<sup>I</sup>* <sup>¼</sup> *<sup>δ</sup>g<sup>I</sup>* <sup>¼</sup> <sup>0</sup>*:* (1)

at sea, a gravimetric-isostatic or combined gravimetric/seismic method can be prosperous. For example, [14] modified the Airy/Heiskanen theory ([15], Section 3.4) by introducing a regional isostatic compensation model based on a thin plate lithospheric flexure model [16, p. 114]. [17, Section 8] generalized the Vening Meinesz hypothesis from a regional to global compensation. [7] expressed the Vening Meinesz-Moritz (VMM) problem as that of solving a non-linear Fredholm integral equation, and presented some solutions for recovering the MD. The VMM method was also followed up by some additional theoretical studies, such as

this model CRUST19.

models.

**1.2 Gravimetric-isostatic Moho models**

so-called non-isostatic effects (NIEs).

disturbance vanish:

**25**

The third set of information in modeling the Earth's interior is the recent gravity field models, generated through modern satellite gravity missions such as Challenging Mini-satellite Payload (CHAMP), Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady state Ocean Circulation Explorer (GOCE), which can provide a global and homogeneous coverage of data. An improvement can also be obtained in the accuracy and spatial resolution of these models by combining them with airborne and ground-based gravity data as well as satellite altimetry data over the oceans. Other important sources for studying Earth's interior are its magnetic field and meteorites.

### **1.1 Background of Moho modeling**

The primary interface of the Earth's interior is the boundary between the Earth's crust and mantle, which is called the Mohorovičić discontinuity (or Moho). This discontinuity was first discovered in 1909 by the Croatian seismologist Andrija Mohorovičić, when analyzing seismograph records of an earthquake in the Kapula valley, namely *P*-waves (compressional waves) and *S*-waves (shear waves). He noticed that the *P*-waves, which travel deeper into the Earth, moved faster than those that travel nearer the surface. Accordingly, he concluded that the Earth is not homogeneous, and at a specific depth there must be a boundary surface, which distinguishes two media with different compositions, and by which the seismic waves propagate with different velocities (see [3]).

Currently the Moho interface can be studied using two main methods: the gravimetric and seismic ones. These methods cannot provide exactly the same results, as they are based on different hypotheses, different types, qualities and spatial distributions of data (see, e.g. [4, 5]).

The seismic methods are the major traditional techniques in modeling the thickness of the Earth's crust (the Moho depth, MD), where the base of the crust is defined as the Moho. Another Moho constituent is the Moho Density Contrast (MDC), which can be estimated from the change of velocity of a seismic wave passing through the Moho boundary. Models based on seismic data can be locally very accurate but useless in areas without adequate seismic observations, particularly over large portions of the oceans. In addition, the seismic data acquisition is costly with lack of global coverage [6].

In contrast, while using satellite gravity data, information on the Moho can be inferred from a uniform and global data set. However, Moho models based on gravity data are in general characterized by simplified hypotheses to guarantee the uniqueness of the solution of the inverse gravitational problem (see, e.g. [7]). As we will show in Section 2.1, gravity data alone cannot separate the MD from the MDC, but additional information is needed to solve this problem. In any case, due to the complementary information described above, a combined gravimetricseismic method could be fruitful in modeling the Moho.

Much research using seismic surveys for recovering the Moho interface has been performed in the last decades. For instance, [8, 9] compiled global Moho models based on seismic data analysis, and [10] estimated the MD using seismic surface

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

The second source includes the records of seismic waves, which are generated, for example, by earthquakes, explosions, volcanoes and other natural sources. Accordingly, specialists can detect information about the Earth's interior, e.g. depth to density discontinuities, through detailed analysis of seismic data. Also, by studying the velocity of the wave, it can to some extent be used for estimating the density of the medium. At this point it deserves to be mentioned that the seismic data are also expensive to collect and therefore sparse and in-homogeneously distributed

The third set of information in modeling the Earth's interior is the recent gravity field models, generated through modern satellite gravity missions such as Challenging Mini-satellite Payload (CHAMP), Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady state Ocean Circulation Explorer (GOCE), which can provide a global and homogeneous coverage of data. An improvement can also be obtained in the accuracy and spatial resolution of these models by combining them with airborne and ground-based gravity data as well as satellite altimetry data over the oceans. Other important sources for studying

The primary interface of the Earth's interior is the boundary between the Earth's crust and mantle, which is called the Mohorovičić discontinuity (or Moho). This discontinuity was first discovered in 1909 by the Croatian seismologist Andrija Mohorovičić, when analyzing seismograph records of an earthquake in the Kapula valley, namely *P*-waves (compressional waves) and *S*-waves (shear waves). He noticed that the *P*-waves, which travel deeper into the Earth, moved faster than those that travel nearer the surface. Accordingly, he concluded that the Earth is not homogeneous, and at a specific depth there must be a boundary surface, which distinguishes two media with different compositions, and by which the seismic

Currently the Moho interface can be studied using two main methods: the gravimetric and seismic ones. These methods cannot provide exactly the same results, as they are based on different hypotheses, different types, qualities and

ness of the Earth's crust (the Moho depth, MD), where the base of the crust is defined as the Moho. Another Moho constituent is the Moho Density Contrast (MDC), which can be estimated from the change of velocity of a seismic wave passing through the Moho boundary. Models based on seismic data can be locally very accurate but useless in areas without adequate seismic observations, particularly over large portions of the oceans. In addition, the seismic data acquisition is

The seismic methods are the major traditional techniques in modeling the thick-

In contrast, while using satellite gravity data, information on the Moho can be inferred from a uniform and global data set. However, Moho models based on gravity data are in general characterized by simplified hypotheses to guarantee the uniqueness of the solution of the inverse gravitational problem (see, e.g. [7]). As we will show in Section 2.1, gravity data alone cannot separate the MD from the MDC, but additional information is needed to solve this problem. In any case, due to the complementary information described above, a combined gravimetric-

Much research using seismic surveys for recovering the Moho interface has been performed in the last decades. For instance, [8, 9] compiled global Moho models based on seismic data analysis, and [10] estimated the MD using seismic surface

around the Earth (see [2]).

Earth's interior are its magnetic field and meteorites.

*Geodetic Sciences - Theory, Applications and Recent Developments*

waves propagate with different velocities (see [3]).

spatial distributions of data (see, e.g. [4, 5]).

costly with lack of global coverage [6].

**24**

seismic method could be fruitful in modeling the Moho.

**1.1 Background of Moho modeling**

waves. For global studies the most frequently used crustal models are the CRUST2.0 [11] and CRUST1.0 models [12], compiled with 2° � 2° and 1° � 1° resolutions, respectively. More recently, [13] developed a global crustal thickness model and velocity structure from geostatistical analysis of seismic data, and we hereafter call this model CRUST19.

Over large areas of the world with a sparse coverage of seismic data, in particular at sea, a gravimetric-isostatic or combined gravimetric/seismic method can be prosperous. For example, [14] modified the Airy/Heiskanen theory ([15], Section 3.4) by introducing a regional isostatic compensation model based on a thin plate lithospheric flexure model [16, p. 114]. [17, Section 8] generalized the Vening Meinesz hypothesis from a regional to global compensation. [7] expressed the Vening Meinesz-Moritz (VMM) problem as that of solving a non-linear Fredholm integral equation, and presented some solutions for recovering the MD. The VMM method was also followed up by some additional theoretical studies, such as methods for estimating the MDC [6] and for reducing the Bouguer gravity anomaly for non-isostatic effects [18, 19]. [20] demonstrated that the MD estimated from the isostatic gravity disturbance based on solving the VMM model has a better agreement with the CRUST2.0 seismic model than those computed by the isostatic gravity anomaly. Their argument was also theoretically explained by [21]. [22] estimated the MD and MDC using a combination of the CRUST2.0 and a GOCE global gravity models. [23] showed that the application of the Bouguer gravity disturbance and the no-topography correction in the VMM model to determine the MD provides very similar results, suggesting the preference of the gravity disturbance to the traditional Bouguer gravity anomaly for gravity inversion. [4, 5] computed combined Moho constituent model according to the VMM method. [24] estimated a new MDC model named MDC2018, using the marine gravity field from satellite altimetry in combination with a seismic-based crustal model and Earth's topographic/bathymetric data. Finally, [25] estimated a combined Moho model for marine areas via satellite altimetric - gravity and seismic crustal models.

#### **1.2 Gravimetric-isostatic Moho models**

Isostasy is an important concept in Earth sciences describing the state of equilibrium (or mass balance) to which the mantle tends to balance the mass of the crust in the absence of external disturbing forces. "*When a certain area of the crust reaches the state of isostasy, it is said to be in isostatic equilibrium (or balance), and the depth at which isostatic equilibrium prevails is called the depth of compensation"* [26].

However, the transport of material over the Earth's surface, such as glaciers, volcanism, and sedimentation, etc., are factors that disturb isostasy, yielding so-called non-isostatic effects (NIEs).

Four principle models of isostasy related with the crustal depth and/or density can briefly be listed as those of (a) Airy/Heiskanen (A/H; [27–29]), (b) Pratt/ Hayford (P/H; [30, 31]), (c) Vening Meinesz (VM; [14]), and (d) the Vening Meinesz-Moritz (VMM; [7, 17]). Common for the isostatic models is that the Bouguer gravity anomaly Δ*g<sup>b</sup>* (or disturbance *δg<sup>b</sup>*) is fully compensated by a compensation attraction below the crust such that the isostatic gravity anomaly and disturbance vanish:

$$
\Delta \mathbf{g}^I = \delta \mathbf{g}^I = \mathbf{0}.\tag{1}
$$

A/H and P/H are local models, implying that the compensation attraction operates along the vertical of the observation point, implying that the sum of the masses of the crust and its compensation along each vertical is assumed to be constant from place to place.

The A/H model assumes a constant crustal density, and variations in topographic height is compensated by variations in the depth of the crust. That is, the mass excess of topography is compensated by the mass deficit of mountain roots in the upper mantle. In ocean areas anti-roots of mantle material compensates for the light mass of the ocean.

non-linear Fredholm integral equation of the first kind, which has the following

2*n* þ 1 *n* þ 1

where *Ynm* is a fully-normalized spherical harmonic, *f nm* is the corresponding

Here subscripts *P* and *Q* denote computation and integration points, respectively, *f nm* is the spherical harmonic coefficient of *f*. Note that the integral contributes significantly only locally around the computation point. The formula can be

ðð

*D*2 1 � �

*<sup>Q</sup>* � *<sup>D</sup>*<sup>2</sup> 1 � � *P*

sin <sup>3</sup>

3 7 5

*σ*

� *<sup>D</sup><sup>k</sup> P* � �<sup>2</sup>

X*n m*¼�*n*

*f nmYnm* (4)

*<sup>ψ</sup>PQ* " #*dσ<sup>Q</sup> :* (5)

*dσ<sup>Q</sup>* ; *k* ¼ 0, 1, 2, … , (6)

X∞ *n*¼0

*<sup>D</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> 4*πΔp*

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

coefficient given by the Bouguer gravity disturbance *f*, and

*D*2 1 � � *P <sup>R</sup>* � <sup>1</sup> 32*Rπ*

ðð

2 6 4

*Dk Q* � �<sup>2</sup>

> sin <sup>3</sup> *ψPQ*

As the isostatic balance of the crust is hardly valid for crustal blocks of diameter smaller than, say, 100 km ([32], p.195), the upper limit of the series in Eq. (4) of should not exceed *n*<sup>2</sup> =180. Also, as we shall see later, the low-degree harmonics in *D*1, say, below *n*<sup>1</sup> =10, are not contributing to the isostatic balance but are due to

The integrals in Eqs. (5) and (6) are local, as the integrand quickly vanishes with distance away from the computation point. Hence a flat earth approximation may

If the MDC varies laterally, the following 2nd-order approximation of Eq. (3)

*n* þ 2

<sup>2</sup>*<sup>R</sup>* ð Þ *<sup>χ</sup><sup>D</sup> <sup>n</sup>*

*f nm* ð Þ *χD nm*

� �, (8)

<sup>2</sup>*<sup>R</sup>* ð Þ *<sup>χ</sup><sup>D</sup> nm* � � (7)

� �*Ynm:* (9)

*=*Δ*ρP*, (10)

can be found in the spectral domain (cf. [6]) when introducing the notation

<sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>χ</sup>nm* <sup>þ</sup>

*f n* <sup>4</sup>*<sup>π</sup>* � *<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

2*n* þ 1 *n* þ 1

> <sup>¼</sup> <sup>X</sup>*<sup>n</sup> m*¼�*n*

*χD* ≈*χ*<sup>2</sup>

*σ*

*<sup>P</sup>* ¼ *D*<sup>1</sup> at point *P* determined by Eq. (4).

ð Þ *D*<sup>2</sup> *<sup>P</sup>* ¼ ð Þ *D*<sup>1</sup> *<sup>P</sup>* þ

improved by a few steps of iteration:

*Dk P* � �<sup>2</sup>

*<sup>R</sup>* � <sup>1</sup> 32*Rπ*

mass anomalies in the Earth's interior below the crust.

*<sup>f</sup> nm* <sup>¼</sup> <sup>4</sup>*<sup>π</sup> <sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>χ</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*<sup>2</sup> *n*¼*n*<sup>1</sup>

where *f <sup>n</sup>* and ð Þ *χD <sup>n</sup>* are the Laplace harmonics

*f n* ð Þ *χD <sup>n</sup>* � �

and, after summing up, one obtains:

Using the approximation

*<sup>P</sup>* þ

*D<sup>k</sup>*þ<sup>1</sup> *<sup>P</sup>* <sup>¼</sup> *Dk*

where *D*<sup>0</sup>

be relevant (See [6]).

*χ* ¼ *D*Δ*ρ*

**27**

first- and second-order solutions:

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

The P/H model assumes a constant depth of compensation of the solid Earth topography (including negative topography over oceans), while the density of the topography varies with topographic height.

Due to the elasticity of the Earth's crust these local models are not very realistic. Hence, [14] modified the A/H model by introducing a model with a regional compensation in which mass loads and unloads are balanced by a gentle bending or flexure of the crust over a regional area. [17] generalized the VM model from a regional to a global compensation with a spherical sea level approximation. [7] and, finally, [6] generalized the VMM model to allow for variations both in crustal density and depth. In this way the VMM can be seen as a generalization of both the A/H and P/H models with global isostatic compensations by variations of both mountain root and crustal density.

Below we will present the least-squares theory for determining a combined VMM-seismic model for both MD and MDC. The theory is finally applied in a new global model.
