*2.2.1 Heterogeneity formations density distributions (model 1)*

As shown in **Figure 2**, the proposed model is consisting of a number (N) of deposited layers or formations, deposited according to Walther's Law of deposition of the heterogeneous 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:

$$
\rho \mathbf{v1(N)} = \rho(\mathbf{N})/(\mathbf{N} - \mathbf{4})\tag{4}
$$

$$
\rho \mathbf{v1(N-1)} = \rho(\mathbf{N}) + \rho(\mathbf{N-1})/(\mathbf{N-3}) \tag{5}
$$

$$
\rho \mathbf{v} \mathbf{1} (\mathbf{N} - \mathbf{2}) = \rho (\mathbf{N}) + \rho (\mathbf{N} - \mathbf{1}) + \rho (\mathbf{N} - \mathbf{2}) / (\mathbf{N} - \mathbf{2}) \tag{6}
$$

$$
\rho \mathbf{v1(N-3)} = \rho(\mathbf{N}) + \rho(\mathbf{N-1}) + \rho(\mathbf{N-2}) + \rho(\mathbf{N-3})/(\mathbf{N-1})\tag{7}
$$

$$\rho \mathbf{v1(N-4)} = \rho(\mathbf{N}) + \rho(\mathbf{N-1}) + \rho(\mathbf{N-2}) + \rho(\mathbf{N-3}) + \rho(\mathbf{N-4})/(\mathbf{N}) \tag{8}$$

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

$$\overline{\rho \mathbf{v1}} = \left[ \rho \mathbf{v1}(\mathbf{N}) \ \rho \mathbf{v1}(\mathbf{N} - \mathbf{1}) \ \rho \mathbf{v1}(\mathbf{N} - \mathbf{2}) \ \rho \mathbf{v1}(\mathbf{N} - \mathbf{3}) \ \rho \mathbf{v1}(\mathbf{N} - \mathbf{4}) \right] \tag{9}$$

then the average vertical densities-contrasts are:

$$
\overline{\Delta \rho \mathbf{v} \mathbf{1}}(\mathbf{i}) = \overline{\rho \mathbf{v} \mathbf{1}}(\mathbf{i}) - \rho\_{\text{baseement}} \tag{10}
$$

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

$$\mathbf{gB\_{M1(:,i)}} = 2\pi \mathbf{G} \Delta \rho \mathbf{v1(i)} \mathbf{z(i)} \tag{11}$$

so that the depths can obtained by the following Eq. (12):

$$\mathbf{Z\_{M1(:,i)}} = \mathbf{abs}(\mathbf{gB\_{M1(i)}}/2\pi\mathbf{G}\overline{\Delta\rho\mathbf{v1}}\tag{12}$$

where gBM1ð Þ :,i are the gravity effects of all points x(i) i.e. all vertical points (i = 1,2,3,4,5), and ZM1ð Þ :,i are the inverted depths at the same vertical points, and also the thicknesses hM1ð Þ :,i have obtained from the following equation:

$$\mathbf{h\_{M1(:,)}} = \sum\_{i=1}^{N} \mathbf{Z\_{m1}(i)} \tag{13}$$

**Figure 2.** *The model consists of five formations (N = 5), and densities are heterogeneously distributed.*
