*2.2.2 Homogeneity formations density distributions (model 2)*

As shown in **Figure 3**, the proposed model is consisting of a number (N) of deposited layers or formations, deposited according to Steno's Law of superposition or Depositional History, Principle of homogeneous Juxtaposition of depositions.

For simplification, assuming the model is consisting of five formations (N = 5), and densities (gm/cm<sup>3</sup> ) from top to bottom are (ρ(N), ρ(N-1), ρ(N-2), ρ(N-3), and ρ(N-4)) with thicknesses (m) are (h1, h2, h3, h4, and h5), respectively. Therefore, the average vertical densities from the top will be as follows:

$$
\rho \mathbf{v} \mathbf{2(N)} = \rho(\mathbf{N}) / (\mathbf{N} - \mathbf{4}) \tag{14}
$$

$$
\rho \mathbf{v} 2(\mathbf{N} - \mathbf{1}) = \rho(\mathbf{N}) + \rho(\mathbf{N} - \mathbf{1})/(\mathbf{N} - \mathbf{3})\tag{15}
$$

$$
\rho \mathbf{v} 2(\mathbf{N} - \mathbf{2}) = \rho(\mathbf{N}) + \rho(\mathbf{N} - \mathbf{1}) + \rho(\mathbf{N} - \mathbf{2})/(\mathbf{N} - \mathbf{2})\tag{16}
$$

$$
\rho \mathbf{v} \mathbf{2(N-3)} = \rho(\mathbf{N}) + \rho(\mathbf{N-1}) + \rho(\mathbf{N-2}) + \rho(\mathbf{N-3})/(\mathbf{N-1}) \tag{17}
$$

$$\rho \mathbf{v} \mathbf{2(N-4)} = \rho(\mathbf{N}) + \rho(\mathbf{N-1}) + \rho(\mathbf{N-2}) + \rho(\mathbf{N-3}) + \rho(\mathbf{N-4})/(\mathbf{N}) \tag{18}$$

Therefore, the average vertical densities for the above modeling is written in form of a row matrix (for the Matlab code purpose) as follows:

$$\overline{\rho \mathbf{v} \mathbf{2}} = \left[ \rho \mathbf{v} \mathbf{2} (\mathbf{N}) \ \rho \mathbf{v} \mathbf{2} (\mathbf{N} - \mathbf{1}) \ \rho \mathbf{v} \mathbf{2} (\mathbf{N} - \mathbf{2}) \ \rho \mathbf{v} \mathbf{2} (\mathbf{N} - \mathbf{3}) \ \rho \mathbf{v} \mathbf{2} (\mathbf{N} - \mathbf{4}) \right] \tag{19}$$

then the average vertical densities-contrasts are:

$$
\overline{\Delta \rho \mathbf{v} \mathbf{2}}(\mathbf{i}) = \overline{\rho \mathbf{v} \mathbf{2}}(\mathbf{i}) - \rho\_{\text{baseement}} \tag{20}
$$

And the gravity effect for model 1, is given as follows:

$$\mathbf{gB}\_{\mathsf{M2}(:,\mathsf{i})} = 2\pi \mathbf{G} \overline{\Delta p \mathbf{v} \mathbf{2}}(\mathbf{i}) \mathbf{z}(\mathbf{i}) \tag{21}$$

so that the depths can obtained by the following Eq. (22):

$$\mathbf{Z\_{M2(:,i)}} = \mathbf{abs}(\mathbf{gB\_{M2(i)}}/2\pi\mathbf{G}\overline{\Delta\rho\mathbf{v}\mathbf{2}}\tag{22}$$

where gBM2ð Þ :,i are the gravity effects of all points x(i) i.e. all vertical points (i = 1,2,3,4,5), and ZM2ð Þ :,i are the inverted depths at the same vertical points, and also the thicknesses hM2ð Þ :,i have obtained from the following equation:

**Figure 3.** *The model consists of five formations (N = 5), and densities are homogeneously distributed.* *New Semi-Inversion Method of Bouguer Gravity Anomalies Separation DOI: http://dx.doi.org/10.5772/intechopen.101593*

$$\mathbf{h}\_{\mathbf{M2}(:,)} = \sum\_{i=1}^{N} \mathbf{Z}\_{\mathbf{m2}}(\mathbf{i}) \tag{23}$$

The goal of geophysical inversion (or interpretation) is to produce models whose response matches observations with noise levels [14]. It is known that the gravity measuring tools are very sensitive only to lateral changes in the causative source, therefore there are several models, that give solutions for the observed profile (ambiguities problem). Even with, this problem the gravity anomalies often are modeled by simple geometrical shapes, (or arbitrary shapes, Talwani et al. 1960). As in all geophysical inversions, there will be ambiguities, notably between density and layer depth, and many of these were pointed out, by [15, 16]. Geophysical inversion by iterative modeling involves fitting observations by adjusting model parameters. Both seismic and potential-field model responses can be influenced by the adjustment of the parameters of rock properties [14].

The new technique in the present research is based on two synthetic models and being built first, consistent, and constrained with real data of known controlling points (or borehole), then applying the algorithm of the solved equations to determine the formations' thicknesses, the basement rocks depth, and tracing them relatively to the formations' thicknesses and depth of basement rocks at the point of a prior known real data, through the profile line of Bouguer anomaly map's covering the area of sedimentary basin.
