**2. The suggested combined gravity or self-potential formula**

Firstly, the gravity anomaly formula due to simple geometric shapes is [15, 16, 18]

$$\log(\varkappa\_i, z, q) = K \frac{z^m}{\left(\left(\varkappa\_i - \varkappa\_o\right)^2 + z^2\right)^q}.\tag{1}$$

Secondly, the self-potential anomaly formula for the same simple geometric models is [14]

$$V(\mathbf{x}\_i, \mathbf{z}, \theta, q) = K \frac{\mathbf{x} \cos \theta + \mathbf{z} \sin \theta}{\left( \left( \mathbf{x}\_i - \mathbf{x}\_o \right)^2 + \mathbf{z}^2 \right)^q}. \tag{2}$$

In Refs. [1, 37], Eqs. (1) and (2) were used to join together to produce a combined gravity or self-potential formula for the simple geometric structures such as a semi-infinite vertical cylinder, a dike, a horizontal cylinder, and a sphere (**Figure 1**) as follows:

$$J(\mathbf{x}\_i) = K \frac{c \mathbf{x}\_i (\cos \theta)^n + \mathbf{z}^p (\sin \theta)^m}{\left( (\mathbf{x}\_i - \mathbf{x}\_o)^2 + \mathbf{z}^2 \right)^q},\tag{3}$$

where *K* is the amplitude coefficient, which depends on the shape of the buried model, *z* is the depth, *θ* is the polarization angle, *xi* is the horizontal coordinates, *xo* is the origin location of the buried structure, *q* is the shape (i.e., equals 1.5 for a sphere, 1.0 for a horizontal cylinder, and 0.5 for a semi-infinite vertical cylinder), *c*, *n*, *p*, and m are constants, which depend on the shape [37]. Eq. (3) is the combined formula for interpreting gravity or self-potential data. So, three suggested approaches were applied to estimate the unknown model parameters as follows:

### **2.1 The least-squares approach**

Essa [37] developed this approach, which was relied on solving the problem of finding the depth from the measured data by solving a nonlinear form *F*(*z*) = 0 by minimizing it in a least-squares sense. After that, the estimated depth was used in estimating other parameters (the polarization angle and the dipole moment for

*Combined Gravity or Self-Potential Anomaly Formula for Mineral Exploration DOI: http://dx.doi.org/10.5772/intechopen.92139*

### **Figure 1.**

*A sketch diagram for the simple geometric bodies as follows: a sphere model (top panel), a horizontal cylinder model (middle panel), and a semi-infinite vertical cylinder model (bottom panel).*

self-potential data or the amplitude coefficient for gravity data) via suggesting the shape of the buried structure (the semi-infinite vertical cylinder, the dike, the horizontal cylinder and the sphere) at the lowest root-mean-squared error. This approach is a semiautomatic because that need assuming the shape of the buried structures (a priori information needed) and applied all observed points in estimating the model parameters.

### **2.2 Werner deconvolution approach**

Werner deconvolution was proposed by Werner in 1953 [38]. This approach is used to estimate mainly the origin location and the depth of the buried structures. Werner proposed to transform the equation of unknown parameters into a rational function. Eq. (3) can be rewritten in linear form follow:

$$J(\mathbf{x}\_i)\left((\mathbf{x}\_i - \mathbf{x}\_o)^2 + \mathbf{z}^2\right)^q - \mathbf{K}c\mathbf{x}\_i(\cos\theta)^n + \mathbf{K}\mathbf{z}^p(\sin\theta)^m = \mathbf{0},\tag{4}$$

$$J(\mathfrak{x}\_i)\left(\left(\mathfrak{x}\_i - \mathfrak{e}\_1\right)^2 + \mathfrak{e}\_2\right)^q - \mathfrak{e}\_3\mathfrak{x}\_i + \mathfrak{e}\_4 = \mathbf{0},\tag{5}$$

where *<sup>e</sup>*<sup>1</sup> <sup>¼</sup> *xo*,*e*<sup>2</sup> <sup>¼</sup> *<sup>z</sup>*2,*e*<sup>3</sup> <sup>¼</sup> *Kc cos* ð Þ*<sup>θ</sup> <sup>n</sup>* ,*e*<sup>4</sup> <sup>¼</sup> *Kzp*ð Þ *sin<sup>θ</sup> <sup>m</sup>:*

Eq. (5) is linear form in the four variables *e*1, *e*2, *e*3, and *e*4, so that a mathematically unique solution can be found for them from evaluating the equation at four points by assuming the shape of the buried structure.

### **2.3 The particle swarm approach**

The particle swarm was suggested by [39] and has many various applications, for example, in geophysics [40–42]. For more detail in this approach, you find it many published literature [43, 44]. The model parameters values of the unknowns are relied upon the objective function, so that every problem can be resolved. In this approach, the particles represent the parameter which we are invert. In the beginning, each particle has a location and velocity. After that each particle changes its location (*Pbest*) at every iteration until reach the optimum location ( *Jbest*). This operation is done by using the following forms:

$$V\_i^{k+1} = c\_3 V\_i^k + c\_1 rand \left( P\_{best} - \boldsymbol{\pi}\_i^{k+1} \right) + c\_2 rand \left( f\_{best} - \boldsymbol{\pi}\_i^{k+1} \right),\tag{6}$$

$$\mathbf{x}\_{i}^{k+1} = \mathbf{x}\_{i}^{k} + \mathbf{V}\_{i}^{k+1},\tag{7}$$

where *v<sup>k</sup> <sup>i</sup>* is the velocity of the particle *i* at the *k*th cycle, *xk <sup>i</sup>* is the current *i* modeling at the *k*th cycle, rand is the random number between [0, 1], *c*<sup>1</sup> and *c*<sup>2</sup> are positive constant numbers and equal 2, *c*<sup>3</sup> is the inertial coefficient which control the velocity of the particle and usually taken less than 1, *x<sup>k</sup> <sup>i</sup>* is the positioning of the particle *i* at the *k*th cycle.

The five source parameters (*K*, *z*, *θ*, *xo*, and *q*) can be assessed by using the particle swarm approach on the subsequent objective function (*Obj*):

$$Obj = \sqrt{\frac{\sum\_{j=1}^{N} \left(f\_j^o - f\_j^c\right)^2}{N}},\tag{8}$$

where *N* is the data points number, *J o <sup>j</sup>* is the observed gravity or self-potential anomaly, and *J c <sup>j</sup>* is the estimated anomaly at the point *xj*.
