**3. Gravity field and large-scale Geophysics**

In Geophysics, the gravimetric data are used for different purposes; e.g. in exploration and prospecting for detecting near surface sources, or studying the Earth's deep interior, which are named here the large-scale Geophysics. The Earth gravity field is determined in two ways. If the temporal variations of the gravity is considered the time-variable gravity field can be determined, otherwise, the static field. In this section, some of the well-known applications of static and timevariable gravity data in large-scale Geophysics are presented and discussed.

### **3.1 Static gravity field and large-scale Geophysics**

A static gravity field reflects the physics of the Earth's interior, which is not fully known. Therefore, different assumptions are used to extract the desired information from the gravity field. Here, the use of the static gravity data and modelling of crustal structure, elastic thickness and rigidity, ice thickness, bathymetry, sediment basement, lithospheric and sub-lithospheric stresses due to mantle convection are presented briefly.

## *3.1.1 Determination of Moho depth*

One the assumptions about the Earth's interior is Isostasy, which is a state of equilibrium between the crust and upper mantle. Aity-Heiskanen, in which the mountains have roots beneath to keep them in isostatic balance, and Partt-Hayford theory, which states that the mountains loads are compensated by density variations inside the crust are two known models of Isostasy. The gravimetric isostasy mean that the isostatic gravity anomaly (Δ*g*<sup>I</sup> ) should be zero to have the crust in isostatic equilibrium. The mathematical description of the gravimetric isostasy is [4]:

$$
\Delta \mathbf{g}^{\mathrm{I}} = \Delta \mathbf{g} - \Delta \mathbf{g}^{\mathrm{TRSCI}} + \Delta \mathbf{g}^{\mathrm{CMP}} = \mathbf{0} \tag{5}
$$

where Δ*g* is gravity anomaly, Δ*g*TBSCI total effect of the topographic and bathymetric masses, sediments, crustal crystalline and ice on Δ*g* and finally, Δ*g*CMP the compensation effect on Δ*g*. Eq. (5) means that there are some compensation attraction, which is equal to the gravitational difference between the effect of loads on the crust and gravity.

In Eq. (5), when <sup>Δ</sup>*<sup>g</sup>* =0, then <sup>Δ</sup>*g*TBSCI <sup>¼</sup> <sup>Δ</sup>*g*CMP, meaning that the gravimetric isostasy becomes the Airy-Heiskanen model having a local compensation property. The presence of Δ*g* in Eq. (5), makes the compensation mechanism regional and Δ*g* acts as a smoother or regularisation factor of the compensation [7].

Two factors are important for modelling the compensation depth, so-called the Mohorovic discontinuity (Moho), a) the mean compensation depth (*D*~ 0) and b) the density contrast (Δ*ρ*) between the crust and upper-mantle. If either of Δ*ρ* or *D*~ <sup>0</sup> is known the other one can be estimated from the model. The variation of Moho depth around *D*~ <sup>0</sup> can be determined by; see [7]:

$$\Delta\tilde{D} = \frac{1}{4\pi GR\Delta\rho} \sum\_{\substack{n=0\\n\neq 1}}^{\infty} \frac{2n+1}{n-1} \beta\_n^\* \Gamma\_n \sum\_{m=-n}^{n} \left(\Delta \mathbf{g}\_{nm}^{\text{TRSCI}} - \Delta \mathbf{g}\_{nm}\right) Y\_{nm}(\theta, \lambda) \tag{6}$$

*The Earth's Gravity Field Role in Geodesy and Large-Scale Geophysics DOI: http://dx.doi.org/10.5772/intechopen.97459*

where *G* is the Newtonian gravitational constant and

$$\beta\_n^\* = \begin{cases} 1 & \text{over occurs} \\ \left(1 + (n+2)\frac{\hat{D}\_0}{2\hat{R}}\right)^{-1} & \text{over continues} \end{cases} \tag{7}$$

$$
\Gamma\_n = \left(\frac{R}{R - \tilde{D}\_0}\right)^{n+2}.\tag{8}
$$

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

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 determination of Δ*ρ* Δ*D*~ .

CRUST1.0 [8] is a global model having information about the thicknesses and densities of sediments, crustal crystalline, topographic heights and bathymetric depths, and the Moho depths with a spatial resolution of 1° � 1° . This means that Δ*g*TBSCI *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 to degree and order 180, corresponding to the resolution 1° � 1° . **Figure 1b** showed the contribution of Δ*g* ranging from �15 to 15 km to the estimated Moho depth.

## *3.1.2 Elastic thickness and rigidity*

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

**Figure 1.** *(a) Global Moho flexure, and (b) the contribution of the gravity data to the Moho flexure.*

flexural isostasy, by this assumption the resulted Δ*g* from the lithospheric properties will be [7]:

$$\Delta \mathbf{g} = \Delta \mathbf{g}^{\text{TBSCl}} - 4\pi G R \Delta \rho \sum\_{n=0}^{\infty} \Gamma\_n^{-1} \frac{n-1}{2n+1} \beta\_n^\* \, \mathbf{C}\_n^{-1} \sum\_{m=-n}^{n} \overline{K}\_{nm} Y\_{nm}(\theta, \lambda) \tag{9}$$

with

$$\overline{K}\_{nm} = \left(\overline{\rho}d\right)\_{nm} + \left(\rho^{\rm S}d\_{\rm S}\right)\_{nm} + \left(\rho^{\rm C}d\_{\rm C}\right)\_{nm} + \left(\rho^{\rm I}d\_{\rm I}\right)\_{nm} \tag{10}$$

surface topography, the geoid is needed. The satellite gravimetry data or gravity models can be used without any involvement with the sea surface topography, but

> *<sup>n</sup> β* <sup>∗</sup> *<sup>n</sup> C*�<sup>1</sup> *<sup>n</sup> ρ*<sup>S</sup> *d*S � �

*R*

masses. Eq. (13) means the compensated gravitational potentials of sediment and crustal crystalline by the flexure isostasy. *A* and *B*<sup>n</sup> are the contribution of the mean

The important factor in bathymetry using this method is the elastic thickness of the lithosphere over oceans, which can be independently determined with a proper

*<sup>T</sup>*<sup>e</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>7</sup> ffiffi

Determination the thickness of continental ice and its changes is important these days because of global warming. The continental ice is melted and water flow enters

*tnm* � *<sup>v</sup>*<sup>B</sup>*=*Iso *nm* � �*Ynm*ð Þ� *<sup>θ</sup>*, *<sup>λ</sup>*

� Δ*ρ* 1 *R* Γ�<sup>1</sup> <sup>0</sup> *β* <sup>∗</sup> <sup>0</sup> *C*�<sup>1</sup>

<sup>þ</sup> *<sup>R</sup>*Δ*ρ*Γ�<sup>1</sup>

*nm* are gravitational potential of the sediment and crustal crystalline

*t*

*<sup>n</sup> β* <sup>∗</sup> *<sup>n</sup> C*�<sup>1</sup> *R*2 *Aδ<sup>n</sup>*<sup>0</sup> *B*0

*nm* <sup>þ</sup> *<sup>ρ</sup>*<sup>C</sup>*d***<sup>C</sup>** � � *nm* � � (13)

<sup>p</sup> (16)

� � (12)

<sup>0</sup> *d*<sup>0</sup> (14)

*<sup>n</sup>* (15)

they have low resolutions. If the average depth of ocean *d*<sup>0</sup> is available, the

variations of the seafloor topography around it will be [7]:

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

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

X*n m*¼�*n*

<sup>1</sup> � <sup>1</sup> � *<sup>d</sup>*<sup>0</sup>

*Bn* <sup>¼</sup> *<sup>R</sup> <sup>R</sup>* � *<sup>d</sup>*<sup>0</sup>

approximation from the age of the oceanic lithosphere by [14]:

where *t* is the age of oceanic lithosphere in Ma.

� �3 !

*R* � �*<sup>n</sup>*þ<sup>2</sup>

2*n* þ 1 *Bn*

where *δ<sup>n</sup>*<sup>0</sup> stands for the Kronecker delta and

*nm* � � <sup>þ</sup> <sup>4</sup>*πGR*Δ*ρ*Γ�<sup>1</sup>

<sup>Δ</sup>*<sup>d</sup>* ¼ � <sup>1</sup>

*<sup>v</sup>*<sup>B</sup>*=*Iso *nm* ¼ � *<sup>v</sup>*<sup>S</sup>

*v*S

**Figure 2.**

*nm* and *v*<sup>C</sup>

*3.1.4 Ice thickness*

**9**

4*πGρ*

X∞ *n*¼0

*Elastic thickness determined from GOCE data over Africa [13].*

*nm* <sup>þ</sup> *<sup>v</sup>*<sup>C</sup>

*<sup>A</sup>* <sup>¼</sup> <sup>1</sup> 3*R*

depth and its flexural compensation.

where *ρ* is the density of the topographic masses when the computation point is in continents and the density contrast between the water and topographic masses when it is over ocean, *d* stands for the topography height or bathymetric depth based on the position of the computation point. *ρ*<sup>S</sup> and *d*<sup>S</sup> are, respectively the density and the thickness of sediment layers, *ρ*<sup>C</sup> and *d***<sup>C</sup>** the corresponding one for crustal crystalline, and *<sup>ρ</sup>*<sup>I</sup> and *<sup>d</sup>*<sup>I</sup> those of the ice. ð Þ• *nm* means the spherical harmonic coefficients. *Cn* is the compensation degree, which is derived from the flexure isostasy model:

$$C\_n = \frac{n^2(n+1)^2}{R^4g}\ddot{\Theta} + \Delta\rho \text{ and } \ddot{\Theta} = \begin{cases} D^{\text{Rig}} & \text{if } \text{flexural rigidity is desired} \\\\ \frac{ET\_\epsilon^3}{12(1-\nu^2)} & \text{if elastic thickness is desired} \end{cases} \tag{11}$$

*g* is the gravity attraction, *E* stands for the Young modulus and *v* the Poisson ratio. In fact, *Cn* carries the mechanical information of lithosphere including the elastic thickness.

The gravity anomaly on the left-hand side of Eq. (9), is generated from the lithosphere's mass and density structures excluding the signal from sub-lithosphere. By comparison of this gravity anomaly and the observed ones excluding the lower degrees, coming from sub-lithosphere say to degree 15 [12] elastic thickness is determined in a trial and error process.

**Figure 2** is the map of elastic thickness determined from GOCE gradiometric data over Africa in [13] the same procedure as explained for Eq. (9). The large elastic thickness over the tectonic border in the ocean is not realistic.

#### *3.1.3 Bathymetry*

Determining the ocean depths using gravity data is an old subject. Over offshore areas, hydrographic surveying methods are applicable by boats and Echo-sounders, known as traditional methods, modernised today by being equipped by GNSS technologies. However, they are costly and not practicable over oceans. Satellite altimetry data cover oceans sufficiently well and bathymetry can be done with acceptable precision, but the shortcoming is the low quality of them over shallow water. In this section, the focus will be on application of gravimetry over oceans for bathymetry purpose, based on isostasy. The theory and mathematical developments are available in [7], but they are not applied so far. Then the strengths and weaknesses of the method is still unknown.

Satellite altimetry data are the distance between the satellite and the sea surface, which is not fully-coincidence to the geoid. The departure of the sea surface from geoid is called sea surface topography. For determining the geoid from satellite altimetry, the sea surface topography should be known; and for determining the sea *The Earth's Gravity Field Role in Geodesy and Large-Scale Geophysics DOI: http://dx.doi.org/10.5772/intechopen.97459*

**Figure 2.** *Elastic thickness determined from GOCE data over Africa [13].*

surface topography, the geoid is needed. The satellite gravimetry data or gravity models can be used without any involvement with the sea surface topography, but they have low resolutions. If the average depth of ocean *d*<sup>0</sup> is available, the variations of the seafloor topography around it will be [7]:

$$\Delta d = -\frac{1}{4\pi G\rho} \sum\_{n=0}^{\infty} \frac{2n+1}{B\_n} \sum\_{m=-n}^{n} \left( t\_{nm} - v\_{nm}^{\text{B/Iso}} \right) Y\_{nm}(\theta, \lambda) - \left( \frac{R^2 A \delta\_{n0}}{B\_0} \right) \tag{12}$$

where *δ<sup>n</sup>*<sup>0</sup> stands for the Kronecker delta and

$$\boldsymbol{\upsilon}\_{nm}^{\rm B/Iso} = - \left( \boldsymbol{\upsilon}\_{nm}^{\rm S} + \boldsymbol{\upsilon}\_{nm}^{\rm C} \right) + 4 \pi G R \Delta \rho \boldsymbol{\Gamma}\_n^{-1} \boldsymbol{\beta}\_n^\* \, \mathbf{C}\_n^{-1} \left( \left( \boldsymbol{\rho}^{\rm S} \boldsymbol{d}\_{\rm S} \right)\_{nm} + \left( \boldsymbol{\rho}^{\rm C} \boldsymbol{d}\_{\rm C} \right)\_{nm} \right) \tag{13}$$

$$A = \frac{1}{3R} \left( 1 - \left( 1 - \frac{d\_0}{R} \right)^3 \right) - \Delta \rho \frac{1}{R} \Gamma\_0^{-1} \beta\_0^\* C\_0^{-1} d\_0 \tag{14}$$

$$B\_n = R\left(\frac{R - d\_0}{R}\right)^{n+2} + R\Delta\rho\Gamma\_n^{-1}\beta\_n^\*\,\mathrm{C}\_n^{-1} \tag{15}$$

*v*S *nm* and *v*<sup>C</sup> *nm* are gravitational potential of the sediment and crustal crystalline masses. Eq. (13) means the compensated gravitational potentials of sediment and crustal crystalline by the flexure isostasy. *A* and *B*<sup>n</sup> are the contribution of the mean depth and its flexural compensation.

The important factor in bathymetry using this method is the elastic thickness of the lithosphere over oceans, which can be independently determined with a proper approximation from the age of the oceanic lithosphere by [14]:

$$T\_{\mathbf{e}} = 2.7\sqrt{t} \tag{16}$$

where *t* is the age of oceanic lithosphere in Ma.

#### *3.1.4 Ice thickness*

Determination the thickness of continental ice and its changes is important these days because of global warming. The continental ice is melted and water flow enters oceans and causes the sea level to rise. This is an issue which affect the Earth climate and may have risk of entering water to low land areas. Some satellite missions have been designed and developed for measuring the ice thickness using radar signals directly. This thickness can also be determined indirectly using gravimetry data. By assuming that the ice mass covers the surface of the Earth and it is part of the Earth's solid topography, its thickness can be determined using the following spherical harmonic expansion [7]:

the compensation degree. Over ocean there is a known relation between the lithospheric plate age and its elastic thickness see Eq. (16), but not over continents.

Mantle convection can also be studied by the long wavelength structure of the Earth gravity field. The Navier–Stokes equations of convection can be solved and simplified it in such a way that simple formula for shear stress at the base of

> *R <sup>R</sup>* � *<sup>D</sup>*Lith � �*<sup>n</sup>*þ<sup>1</sup>

where *τzx* and *τzy* are the shear-stresses at the base of the lithosphere toward north and east, respectively. *D*Lith is the depth of boundary between lithosphere and mantle. Eq. (21) is known as Runcorn's formulae. He assumed that the mantle convection creates only the shear stresses at the base of the lithosphere. Most importantly,

Only by these assumptions the simple formula having a direct relation with the gravity data can be obtained. Many believe that the Runcorn simple solution is not realistic and successful, in spite of different efforts for justifying the applicability of

In Eq. (21), the maximum degree of expansion should not be infinity as the mantle convection contributes mainly in low frequencies of the gravity field. In [16] degrees between 13 to 25 are suggested to reduce the contributions from the core and lithosphere. However, in [12] the degrees below 15 are considered as contribu-

**Figure 3a** and **b** show the map of the sub-lithospheric shear stresses *τzx* and *τzy*,

In [20] a better theory was developed for modelling the mantle convection using the displacement vectors of and tectonic movement. They also use the long wavelength portion of a geoid model in their solution, but the contribution of geoid is not very significant. This could be the reason that Runcorn has simplified the same mathematical models by ignoring the significant parameters and emphasising on the

By assuming that the lithosphere is an elastic shell, solution of the spherical boundary-value problem of elasticity can be applied for presenting the stress status inside the lithosphere. The stresses at base and top of the lithosphere is considered

respectively, using Eq. (21) at the lithospheric depths of Conrad and Lithgow-Bertelloni model [19] over Iran. One issue in applying Eq. (21) is the choice of the maximum degree of expansion based on the lithospheric depth. When the base of

the lithosphere is deeper, this degree should be lower and vice versa.

*3.1.7 Stress propagation through the lithosphere from its base*

X*n m*¼�*n*

*tnm*

0

BB@

*<sup>∂</sup>Ynm*ð Þ *<sup>θ</sup>*, *<sup>λ</sup> ∂θ <sup>∂</sup>Ynm*ð Þ *<sup>θ</sup>*, *<sup>λ</sup>* sin *θ∂λ*

1

CCA (21)

*3.1.6 Runcorn's theory and sub-lithospheric shear stresses*

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

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

X∞ *n*¼2

2*n* þ 1 *n* þ 1

lithosphere is obtained see [15]:

4*πG R*ð Þ � *D*Lith

a. the viscosity of mantle is constant.

c. the mantle is Newtonian.

b. the toroidal flow in the mantle is negligible.

*τzx*

*<sup>τ</sup>zy* � � <sup>¼</sup> *<sup>g</sup>*

he assumed that:

this theory [16–18].

weakest one.

**11**

tions from sub-lithosphere.

$$d\_{\rm I} = \frac{1}{4\pi GR} \sum\_{n=0}^{\infty} \left( -\frac{\Delta \rho^{\rm I}}{2n+1} + \Delta \rho \rho^{\rm I} \Gamma\_n^{-1} \beta\_n^\* \, ^\*\mathrm{C}\_n^{-1} \right)^{-1} \sum\_{m=-n}^{n} \left( t\_{nm} - v\_{nm}^{\rm TSC/Iso} \right) Y\_{nm}(\theta, \lambda) \tag{17}$$

where Δ*ρ*<sup>I</sup> is the density contrast between the upper crust and ice, *ρ*<sup>I</sup> stands for the density of ice, and

$$\begin{split} \boldsymbol{\nu}\_{nm}^{\text{TSC/Iso}} &= -\left(\boldsymbol{\nu}\_{nm}^{\text{T}} + \boldsymbol{\nu}\_{nm}^{\text{S}} + \boldsymbol{\nu}\_{nm}^{\text{C}}\right) \\ &+ 4\pi G r \Delta \rho \boldsymbol{\Gamma}\_{n}^{-1} \boldsymbol{\beta}\_{n}^{\*} \, \mathrm{C}\_{n}^{-1} \Big( \left(\boldsymbol{\rho}^{\text{T}} \boldsymbol{H}\right)\_{nm} + \left(\boldsymbol{\rho}^{\text{S}} \boldsymbol{d}\_{\text{S}}\right)\_{nm} + \left(\boldsymbol{\rho}^{\text{C}} \boldsymbol{d}\_{\text{C}}\right)\_{nm} \Big). \end{split} \tag{18}$$

Note that Eq. (17) is based on the linear approximation of the involved binomial terms related to the topographic heights. Such an approximation is good as long as the heights are not large and the maximum degree of the expansion is not high. For example, for a height of 10 km and maximum degree 360, the relative error of this approximation will be 11%, for degree 180 is 4% and when the eight is 5 km for the maximum degree of 360 it will be 4% and less than 1% for 180. Since we have applied isostasy principle to obtain this equation, higher resolution than 180 is not needed, then approximation should be rather fine. One issue is the elastic thickness of lithosphere which is needed to determine the compensation degrees, which should be known from independent sources.

### *3.1.5 Sediment basement determination*

Sediments are located at the surface of the upper-crust resulted from erosion during a long period of time. They are compacted by time meaning that their density will be high at their bottom and low at the surface. Therefore, the process of determining their thickness is not simple because the sediment density is an exponential function increasing by depth. In [7] some of the density contrast models have been presented and the gravitational potential of sediments have been modelled in spherical harmonics series. If we assume an average density for sediments the following approximate formula can be used to determine its thickness

$$d\_{\rm S} = \frac{\rho^{\rm S}}{-4\pi \text{GR}} \sum\_{n=0}^{\infty} \left( -\frac{1}{2n+1} + \Delta\rho \Gamma\_n^{-1} \beta\_n^\* \, ^\*\mathcal{C}\_n^{-1} \right)^{-1} \sum\_{m=-n}^{n} \left( t\_{nm} - v\_{nm}^{\rm BC, \rm Iso} \right) Y\_{nm}(\theta, \lambda) \tag{19}$$

where Δ*ρ*<sup>I</sup> is the density contrast between the upper crust and ice, *ρ*<sup>I</sup> stands for the density of ice, and

$$\boldsymbol{\nu}\_{nm}^{\rm BC/Iso} = -\left(\boldsymbol{\nu}\_{nm}^{\rm B} + \boldsymbol{\nu}\_{nm}^{\rm C}\right) + 4\pi GR \Delta \rho \boldsymbol{\Gamma}\_n^{-1} \boldsymbol{\beta}\_n^\* \, \mathrm{C}\_n^{-1} \left(\left(\boldsymbol{\rho}^{\rm B} H\right)\_{nm} + \left(\boldsymbol{\rho}^{\rm C} d\_{\rm C}\right)\_{nm}\right). \tag{20}$$

One important parameter which should be known for sediment thickness determination using Eq. (19) is the elastic thickness of lithosphere, needed for computing the compensation degree. Over ocean there is a known relation between the lithospheric plate age and its elastic thickness see Eq. (16), but not over continents.
