**3. Methodology**

The gravitational anomaly's analytic signal is written as follows [52, 53]:

$$\mathfrak{L}\_i(\mathfrak{x}) = \frac{\partial \mathfrak{g}}{\partial \mathfrak{x}} - i \frac{\partial \mathfrak{g}}{\partial \mathfrak{z}}, \qquad i = \sqrt{-1}, \tag{2}$$

where *<sup>∂</sup><sup>g</sup> <sup>∂</sup><sup>x</sup>* and *<sup>∂</sup><sup>g</sup> <sup>∂</sup><sup>z</sup>* are the horizontal and vertical gradients of the gravity anomaly. *Gravity Anomaly Interpretation Using the R-Parameter Imaging Technique over a Salt Dome DOI: http://dx.doi.org/10.5772/intechopen.105092*

#### **Figure 1.**

*Geometry and parameters of sphere (top), semi-infinite vertical cylinder (middle and infinitely long horizontal cylinder (bottom) (re-drawn from [39]).*


#### **Table 1.**

*Definitions of* A*,* η *and* q *of the simple-geometric bodies.*

The amplitude of the analytic signal j j *ξs*ð Þ *x* can be calculated as follows [44]:

$$|\xi\_x(\mathbf{x})| = \sqrt{\left(\frac{\partial \mathbf{g}}{\partial \mathbf{x}}\right)^2 + \left(\frac{\partial \mathbf{g}}{\partial \mathbf{z}}\right)^2}. \tag{3}$$

By an adapting the horizontal and vertical derivatives to Eq. (1), and putting the obtained outcomes into Eq. (3), we get the following:

$$|\xi\_s(\mathbf{x})| = A(\mathbf{z}\_o - \mathbf{z})^{(\eta - 1)} \frac{\sqrt{4q^2 \left(\mathbf{x}\_j - \mathbf{x}\_o\right)^2 \left(\mathbf{z}\_o - \mathbf{z}\right)^2 + \left(\eta \left(\mathbf{x}\_j - \mathbf{x}\_o\right)^2 - \left(\mathbf{z}\_o - \mathbf{z}\right)^2 \left(2\mathbf{q} - \eta\right)\right)^2}}{\left(\left(\mathbf{x}\_j - \mathbf{x}\_o\right)^2 + \left(\mathbf{z}\_o - \mathbf{z}\right)^2\right)^{(q+1)}},\tag{4}$$

where *j* = 1, 2, 3, ...., *n*. To calculate the horizontal location (*xo*) and depth (*zo*) of the buried target (**Figure 1**), the 2-D X-Z mosaic of the correlation coefficient (*R*) is constructed from the analytic signal amplitudes j j *ξso*ð Þ *x* of the measured data and j j *ξst*ð Þ *x* calculated from the theoretical generated data by a supposed simplegeometric source S(*xo*, *zo*), and is expressed as:

$$R(\mathbf{x}\_o, \mathbf{z}\_o) = \frac{\sum\_j^n |\xi\_{so}(\mathbf{x})|\_j |\xi\_{st}(\mathbf{x})|\_j}{\sqrt{\sum\_j^n |\xi\_{so}(\mathbf{x})|\_j^2 \sum\_j^n |\xi\_{st}(\mathbf{x})|\_j^2}}. \tag{5}$$

The analytic signal j j *ξso*ð Þ *x* is assessed numerically along the profile using Eq. (3). To map the relevant discrepancy of the R-parameter from which *xo* and *zo* are appraised, discretization in the X- and Z-directions is done around the anticipated spatial location of the supposed source. The R-parameters value (R-max) reaches the maximum when the depth and location of the assumed source match the true ones.
