**2.2 A least-squares solution for both the Moho depth and the Moho density contrast**

The Moho component *χ*, the product of *D* and Δ*ρ*, can be estimated from Eq. (11) or (12) and applied as an observation together with seismic data for solving both the MD and the MDC in a least-squares adjustment. Then a linear set of equations including gravimetric data (*l*1), and seismic data for MD (*l*2) and MDC (*l*3) can be written for each pixel (*P*):

$$
\Delta\rho\_P dD + D\_P d\Delta\rho = l\_1 - \varepsilon\_1 \tag{16}
$$

$$dD = l\_2 - \varepsilon\_2 \tag{17}$$

$$d\Delta\rho = l\_3 - \varepsilon\_3,\tag{18}$$

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

Where *dD* and *d*Δ*ρ* are the (unknown) corrections to the initial values *DP* and Δ*ρP*, *l*<sup>1</sup> ¼ *χ* � *χ<sup>P</sup>* � *I k <sup>P</sup>*, where *I* 0 *<sup>P</sup>* ¼ 0, and *ε<sup>i</sup>* are the errors of the observations. In matrix form the adjustment system can be written

$$\mathbf{A}\mathbf{X} = \mathbf{L} - \mathbf{e},\tag{19}$$

where

one obtains from (8) the iterative formula

2*n* þ 1 *n* þ 1

*<sup>P</sup>* is the first-order solution:

*f n* <sup>4</sup>*<sup>π</sup>* � <sup>X</sup>*<sup>n</sup> m*¼�*n*

*Geodetic Sciences - Theory, Applications and Recent Developments*

*χk*þ<sup>1</sup> *<sup>P</sup>* <sup>¼</sup> *<sup>χ</sup>*<sup>0</sup>

*I k <sup>P</sup>* <sup>¼</sup> <sup>1</sup> *R*Δ*ρ<sup>P</sup>*

*<sup>χ</sup><sup>P</sup>* <sup>≈</sup> <sup>1</sup> 4*π*

Newton integral. See Appendix A.

**contrast**

**28**

written for each pixel (*P*):

X*n*2 *n*¼*n*<sup>1</sup>

*n* þ 2 2*R*Δ*ρ<sup>P</sup>*

X*n*2 *n*¼*n*<sup>1</sup>

" #

*<sup>χ</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> 4*π*

Alternatively, we may present Eq. (11) by the iterative formula:

*<sup>P</sup>* þ *I k*

ðð

*χk P* � �<sup>2</sup> � *<sup>χ</sup><sup>k</sup>*

*σ*

2*n* þ 1 *n* þ 1

Note that the solution *χ<sup>P</sup>* is the product of the MD and MDC. If one of the parameters is known, the other can be determined by the equation. Hence, gravity data alone cannot be used to distinguish between the two Moho constituents. Hence, additional information, e.g., from seismic and/or geological data, is needed to separate the two. However, as we shall see later, usually such data is not taken for granted in the VMM technique, but the gravity data used in Eq. (8) is typically applied to improve a priori Moho constituents in a least-squares procedure.

The solution (8) can be derived from Eq. (1), and from the inversion of a 3-D

The Moho component *χ*, the product of *D* and Δ*ρ*, can be estimated from Eq. (11) or (12) and applied as an observation together with seismic data for solving both the MD and the MDC in a least-squares adjustment. Then a linear set of equations including gravimetric data (*l*1), and seismic data for MD (*l*2) and MDC (*l*3) can be

Δ*ρPdD* þ *DPd*Δ*ρ* ¼ *l*<sup>1</sup> � *ε*<sup>1</sup> (16)

*dD* ¼ *l*<sup>2</sup> � *ε*<sup>2</sup> (17)

*d*Δ*ρ* ¼ *l*<sup>3</sup> � *ε*3, (18)

**2.2 A least-squares solution for both the Moho depth and the Moho density**

*χ*<sup>2</sup> � �*<sup>k</sup> n*

2*n* þ 1

*Q* � �<sup>2</sup>

*f n*

sin <sup>3</sup> *ψ*

Again, this integral is very local, which suggests the use of a flat-Earth approximation. Also, assuming that ð Þ *n*<sup>2</sup> þ 2 *D*0*=*ð Þ 2*R* <1, Eq. (7) leads to the approximate

, ; *k* ¼ 0, 1, 2, … (11)

*dσ<sup>Q</sup> :* (14)

*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>f</sup> <sup>n</sup>:* (12)

*<sup>P</sup>*, ; *k* ¼ 0, 1, 2, … , (13)

<sup>1</sup> <sup>þ</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *<sup>D</sup>*0*=*ð Þ <sup>2</sup>*<sup>R</sup> :* (15)

*χk*þ<sup>1</sup> *<sup>P</sup>* <sup>¼</sup> <sup>X</sup>*n*<sup>2</sup>

where *χ*<sup>0</sup>

where

solution:

*n*¼*n*<sup>1</sup>

$$\mathbf{A} = \begin{bmatrix} \Delta \rho\_p & D\_P \\ \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{bmatrix}, \qquad \mathbf{X} = \begin{bmatrix} dD \\ d\Delta \rho \end{bmatrix} \qquad \text{and} \qquad \mathbf{L} = \begin{bmatrix} l\_1 \\ l\_2 \\ l\_3 \end{bmatrix}. \tag{20}$$

Assuming that the observation errors are random with expectation zero and covariance matrix **Q**, the weighted least squares solution of this system becomes:

$$
\hat{\mathbf{X}} = \left(\mathbf{A}^T \mathbf{Q}^{-1} \mathbf{A}\right)^{-1} \mathbf{A}^T \mathbf{Q}^{-1} \mathbf{L}.\tag{21}
$$

From this result, the adjusted MD and MDC for point *P* are obtained by:

$$
\hat{D} = D\_P + d\hat{D} \text{ and } \Delta\hat{\rho} = \Delta\rho\_P + d\Delta\hat{\rho}. \tag{22}
$$

As the first equation ð Þ *l*<sup>1</sup> is a linearization, it could make sense to iterate the adjustment procedure by replacing the previous initial values *DP* and Δ*ρ<sup>P</sup>* in Eq. (16) by their adjusted values *D*^,Δ^*ρ* and repeat the above computation procedure until sufficient convergence.
