**3.2 The uncertainties in VMM Moho depth and density contrast**

Assuming that all observation errors are stochastic with expectation zero, an error propagation of the least squares solution in Eq. (21) yields that the covariance matrix of **X**^ becomes

$$\mathbf{Q}\_{\mathbf{X}\mathbf{X}} = \sigma\_0^2 \left(\mathbf{A}^T \mathbf{Q}^{-1} \mathbf{A}\right)^{-1} \tag{25}$$

approximate t-test to judge whether the estimates *x* and *y* from the two models are

Nowadays, the Earth's gravity field has been recognized as an important source of information about the Earth's structure. Such data contain both short- and longwavelength features, i.e., signals from the topographic and bathymetry geometries and density heterogeneities in the topography, ice caps, sediment basins and also in

The long-wavelength contribution to the gravity field, say to spherical harmonic degree and order 10, may be assumed to be related to the mantle and below located

To isolate the gravity data caused only by the geometry and density contrast of the Moho interface, all aforementioned signal contributors to the gravity data must be removed by applying the so-called stripping corrections and NIEs [34] and NIEs (see section 4.2). Another gravity correction corresponds to the gravimetric effect of filling-up all oceans with masses to a standard density of 2670 kg/m<sup>3</sup>

Finally, by removing also normal gravity from the resulting stripped free-air gravity observation, one obtains the refined Bouguer gravity disturbance. As a result, the ideal stripped Bouguer gravity disturbance can be explained as caused by a spherical

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

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

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

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

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

mass density (or density contrast) layer *q* (i.e., topography, bathymetry, glacial ice

*m*¼�*n c q*

*n* þ 2 2

*q*

*ρqL*<sup>2</sup> *q* � �

*nm <sup>R</sup>*<sup>2</sup> <sup>þ</sup> …

*<sup>δ</sup>gTBIS* <sup>¼</sup> *<sup>δ</sup>g<sup>t</sup>* <sup>þ</sup> *<sup>δ</sup>g<sup>b</sup>* <sup>þ</sup> *<sup>δ</sup>g<sup>i</sup>* <sup>þ</sup> *<sup>δ</sup>g<sup>s</sup>* (31)

*nmYnm*ð Þ P , (32)

*nm* of a particular volumetric

3

5 (33)

Earth without solid Earth topography and mass anomalies below the crust.

.

statistically equal or not, if they are (weighted) mean values.

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

**4. Corrections to gravimetric data**

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

heterogeneities.

**4.1 Crustal density corrections**

density variations, respectively.

and sediments) is defined by:

*c q nm* <sup>¼</sup> <sup>2</sup>

*δg<sup>q</sup>*

geocentric gravitational constant. The coefficient *c*

ð Þ 2*n* þ 1

1 *ρe* 2 4

ð Þ¼ *<sup>P</sup> GM R*2

X*n*max *n*¼0

*ρqLq* � � *nm R* þ

series:

**31**

the mantle and core/mantle topography variations.

where *σ*<sup>2</sup> <sup>0</sup> is the variance of unit weight, which can be unbiasedly estimated by

$$
\sigma\_0^2 = \mathbf{L}^T \mathbf{Q}^{-1} (\mathbf{L} - \mathbf{A}\hat{\mathbf{X}}).\tag{26}
$$

Note that there is no denominator in Eq. (26), because in the present adjustment example with 3 observations and 2 unknowns per pixel there is only 1 degree of freedom.

### **3.3 Verification of the solutions**

First, we will find an estimate of the variance *σ*<sup>2</sup> *<sup>x</sup>* of the solution *x* for the MD or MDC by assuming that we know another solution *y* with variance *σ*<sup>2</sup> *<sup>y</sup>* . If both solutions have vanishing expected errors, the solution becomes

$$
\sigma\_\mathbf{x}^2 = \sigma\_\mathbf{y}^2 + E\{\mathbf{x}^2 - \mathbf{y}^2\}. \tag{27}
$$

The correlation coefficient between *x* and *y* follows from

$$k = \left[\sigma\_\mathbf{x}^2 + \sigma\_\mathbf{y}^2 - E\left\{\left(\mathbf{x} - \mathbf{y}\right)^2\right\}\right] / \left(2\sigma\_\mathbf{x}\sigma\_\mathbf{y}\right) \tag{28}$$

One can also plot the t-test parameter of the normalized (and unitless) difference between *x* and *y*:

$$T = \frac{\varkappa - \wp}{\sqrt{\sigma\_{\varkappa}^{2} + \sigma\_{\jmath}^{2} - 2k\sigma\_{\varkappa}\sigma\_{\jmath}}} \tag{29}$$

to study the expected difference.

To verify Eqs. (27)–(29), one may start from the substitutions that the true value for *x* and *y* is given by

$$
\overline{\mathfrak{X}} = \mathfrak{x} - \mathfrak{e}\_{\mathfrak{x}} = \mathfrak{y} - \mathfrak{e}\_{\mathfrak{y}},
\tag{30}
$$

where *ex* and *ey* are random errors with zero-expectations.

In practice, *x* and *y* are the Moho quantities at a pixel estimated from two models, and the expectation operator should be replaced by the (weighted) mean value over the central and surrounding pixels. Note that the solution in Eq. (27) is independent on whether *x* and *y* are correlated or not. Eq. (29) can be used in an

approximate t-test to judge whether the estimates *x* and *y* from the two models are statistically equal or not, if they are (weighted) mean values.
