Introduction

Gravity data interpretation is valuable and it is utilized in exploring archaeology, minerals, ores, and oils in different places around the world (Cella et al. 2008; Essa 2013; Mehanee 2014; Abdelrahman and Essa 2013, 2015; Mehanee and Essa 2015; Essa and Géraud 2020). Simple geometric models such as Spheres, cylinders, thin sheets or faults are considered simple geological structures because they play an important role in many exploration issues. Several interpreted approaches are often used to calculate their parameters. Furthermore, these models are not geologically exact, but they can be frequently used in gravity interpretation in evaluating the body parameters (Hinze et al. 2013; Essa et al. 2021).

The different geometrical shapes of subsurface structures confirm the inapplicability of interpreting the gravity data. This, consequently, will cause the production of a similar gravity anomaly that clearly appears on the earth's surface. Therefore, it is important to minimize these issues by increasing the completeness of the data, decreasing the measurement error of the data space, setting the appropriate parameters, and having a priori information followed by the inversion process (Abdelrahman et al. 2003; Biswas et al. 2017; Eshaghzadeh et al. 2019; Essa and Munschy 2019).

Various idealized bodies cannot be found in real subsurface geological situations. Besides, these suggested models are favored and applied in the gravity interpretation. The objective of the inversion process is to conquer the parameters of the target, i.e., shape factor, depth, and amplitude coefficient. The prior research showed and deliberated different graphical and arithmetical methods for inferring gravity data due to simple sources (Odegard and Berg 1965, Nettleton 1976, Thompson 1982, Kilty 1983, Marson and Klingele 1993, Abdelrahman et al. 2002, Abdelrahman et al. 2006, Essa 2007a, Asfahani and Tlas 2008, Abedi et al. 2009, Essa 2011). Nevertheless, the shortcomings of these methods that depend on characteristic points and curves emphasized human imprecisions in evaluating the buried structures’ parameters (Essa 2014). Thus, this confirms the neediness for an accurate inversion method that produces approximately actual geometry parameters for the buried target structures.

To facilitate this study, an inversion technique based on a fast inversion method was developed using the analytical signal of gravity anomaly to appraise the parameters of the subsurface structures, namely; depth, amplitude, and shape. The analytical signal calculation depends on the derivatives which suppressed and treated different background gravity anomaly trends especially for many anomalies in lateral directions. In consistent with this, Srivastava and Agarwal (2010) and Ekinci et al. (2017) demonstrated the application of metaheuristic algorithms such as particle swarm and differential evolution to invert the magnetic anomalies through the analytical signal approach.

The objective of this study was to improve the efficient algorithm based on the simple bodies by finding a robust and exact method for simple model structures elucidation and examining the uncertainty in explanation and conceivable method to avoid vagueness for a consistent result. In our study, the technique of creating data with and without noise showed that the obtained parameters were in line with the proposed ones. In addition, the run of the three field examples from Slovakia, Cuba, and India confirmed the accessibility of our technique to obtain accurate worthy geological or geophysical information.

The method

The gravity anomaly (g) generated by three most popular simple shapes (a sphere, a horizontal cylinder and a semi-infinite vertical cylinder) along the profile is represented as follows (Gupta 1983; Asfahani and Tlas 2015; Tlas and Asfahani 2019):

$$g\left(x,z,{s}_{n}\right)=\frac{K}{{\left[{\left(x-{x}_{o}\right)}^{2}+{z}^{2}\right]}^{{s}_{n}}} , i=1, 2, 3,\dots , N$$
(1)

where

$$K = \left\{ {\begin{array}{*{20}c} {\frac{4}{3}\pi G\Delta \sigma zR^{3} } \\ {} \\ {2\pi G\Delta \sigma zR^{2} } \\ {} \\ {\pi G\Delta \sigma zR^{2} } \\ \end{array} ,} \right.s_{n} = \left\{ {\begin{array}{*{20}c} {1.5\;{\text{for}}\;{\text{a}}\;{\text{sphere}}\;{\text{model}}} \\ {} \\ {1.0\;{\text{for}}\;{\text{a}}\;{\text{horizontal}}\;{\text{cylinder}}\;{\text{model}}} \\ {} \\ {0.5\;{\text{for}}\;{\text{a}}\;{\text{vertical}}\;{\text{cylinder}}\;{\text{model}}} \\ \end{array} } \right.,$$

z is the depth, K is the amplitude coefficient, which depends on the shape, sn is the shape index, x is the location coordinate, xo is the position of the anomaly central point, Δσ is the contrast in density between the target structure and the surrounds, G is the universal gravitational constant (= 6.67 × 10–11), and R is the radius of the buried body.

The 2D analytical signal (As) of gravity anomaly (Nabighian 1972; Ansari and Alamdar 2010) is:

$${A}_{s}\left(x, z\right)=\sqrt{{\left(\frac{\partial g}{\partial x}\right)}^{2}+{\left(\frac{\partial g}{\partial z}\right)}^{2}} ,$$
(2)

where \(\frac{\partial g}{\partial x}\) and \(\frac{\partial g}{\partial z}\) are the derivatives (horizontal and vertical) of gravity.

By using Eq. (1) and substituting in Eq. (2), the 2D analytical signal of the gravity anomaly is:

$${A}_{s}\left(x, z\right)=\frac{2K{s}_{n}}{{\left[{\left(x-{x}_{o}\right)}^{2}+{z}^{2}\right]}^{{s}_{n}+0.5}}.$$
(3)

The horizontal location located under the peak of analytical signal, and the value of As(xi, z) at the horizontal location xo are given as:

$${A}_{s}\left({x}_{o}, z\right)=\frac{2K{s}_{n}}{{z}^{2{s}_{n}+1}}.$$
(4)

Using the normalized equation at xi =  ± N and xi =  ± M where N and M are positive integer values (1, 2, 3, …)

$$\frac{{A}_{s}\left(N, z\right)}{{A}_{s}\left({x}_{o}, z\right)}=\frac{{z}^{2{s}_{n}+1}}{{\left[{\left(N-{x}_{o}\right)}^{2}+{z}^{2}\right]}^{{s}_{n}+0.5}} ,$$
(5)
$$\frac{{A}_{s}\left(M, z\right)}{{A}_{s}\left({x}_{o}, z\right)}=\frac{{z}^{2{s}_{n}+1}}{{\left[{\left(M-{x}_{o}\right)}^{2}+{z}^{2}\right]}^{{s}_{n}+0.5}},$$
(6)

Let \(F=\frac{{A}_{s}\left(N, z\right)}{{A}_{s}\left({x}_{o}, z\right)}\) and \(T=\frac{{A}_{s}\left(M, z\right)}{{A}_{s}\left({x}_{o}, z\right)}\) then divided Eq. (6) by Eq. (5), we get:

$$\frac{T}{F}=\frac{{\left[{\left(N-{x}_{o}\right)}^{2}+{z}^{2}\right]}^{{s}_{n}+0.5}}{{\left[{\left(M-{x}_{o}\right)}^{2}+{z}^{2}\right]}^{{s}_{n}+0.5}} ={\left\{\frac{{\left(N-{x}_{o}\right)}^{2}+{z}^{2}}{{\left(M-{x}_{o}\right)}^{2}+{z}^{2}}\right\}}^{{s}_{n}+0.5},$$
$$w = \left( {\frac{T}{F}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {s_{n} + 0.5}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${s_{n} + 0.5}$}}}} = \left\{ {\frac{{\left( {N - x_{o} } \right)^{2} + z^{2} }}{{\left( {M - x_{o} } \right)^{2} + z^{2} }}} \right\},\;{\text{and}}~{\text{then}}$$
$$z=\sqrt{\frac{w{\left(M-{x}_{o}\right)}^{2}-{\left(N-{x}_{o}\right)}^{2}}{1-w}} , M\ne N$$
(7)

Equation (7) can be deciphered for depth utilizing the simple calculation.

For all N and M values (M ≠ N), we estimated the model parameters (z, sn and K) from Eqs. (7) and (4), correspondingly. For any suggestion for initial guess for z works well for the reason that there was always one solution called a global minimum solution. Equation (7) was utilized not only to decide z but also to assess sn of the subsurface structure. Then the standard deviation (SD) of z for each sn was assessed. The minimum SD is a criterion for defining the true sn and z of the buried structure. When the true sn is utilized, the SD of z is continually less than the SD estimated for false sn. A flow chart (Fig. 1) described this method procedure.

Fig. 1
figure 1

A flow chart to demonstrate the suggested method works

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Application to synthetic examples

First model

A noise-free gravity anomaly profile of a horizontal cylinder model of parameters: K = 100 mGal × m2, z = 2 m, sn = 1, xo = 0 m, and profile length = 100 m was chosen (as shown in Fig. 2). The model equation is:

Fig. 2
figure 2

Noise-free and noisy gravity anomalies of a horizontal cylinder placed at a horizontal distance of 0 m and a depth of 2 m

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$$g(x)=\frac{{100}}{{\left[(x-0)^{2}+{2}^{2}\right]}^{1}}$$
(8)

By using the Fourier transform method, the horizontal and vertical derivatives of gravity anomaly and then the analytic signal of the data were estimated (Fig. 3). The new technique was utilized to the gravity anomaly profile and inverted to retrieve the target parameters; the depth (z), the shape index (sn) and the amplitude coefficient (K), correspondingly. The solutions for all combination of N and M points were found valid.

Fig. 3
figure 3

Analytical signals of the anomaly in Fig. 2

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Table 1 displays the effect of different N and M values, i.e., for N = 1, 2, 3, 4, and 5 m and M = 1, 2, 3,…, 10 m. The computed model parameters also were explained in Table 1. It displays the model parameters (z, sn, K), in cases of using sn = 0.5, 1.0, and sn = 1.5. The correct solution was valid at the minimum SD, which occurs at sn = 1.0. So, the method was deemed to be convergent at the correct solutions. Therefore, noise-free data were inferred to authenticate the effectiveness of the method in deducing the actual model parameters.

Table 1 Numerical results of the suggested method applied to the horizontal cylinder model (K = 100 mGal × m2, z = 2 m, sn = 1, xo = 0 m, and profile length = 100 m) without and with a Gaussian random noise
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The above-mentioned synthetic data was infected with Gaussian noise (with standard deviation = 1 and mean = 0) to examine the stability of the suggested method. The estimated model parameters were presented in Table 1 for all suggested N and M values mentioned-above. Furthermore, Table 1 also indicates the model parameter solutions (z, sn, K), in all cases of using sn = 0.5, 1.0, and 1.5. The acceptable solution was seen at the lowest SD, which occurred in sn = 1.0 (Table 1). Finally, the method was convergent with the correct solutions even when the data contained a noise.

Second model

A noise-free gravity anomaly profile of a semi-infinite vertical cylinder model with K = 200 mGal × m, z = 3 m, sn = 0.5, xo = 0 m, and profile length = 100 m (Fig. 4) was adopted. The model equation is:

Fig. 4
figure 4

Noise-free and noisy gravity anomalies of a semi-infinite vertical cylinder located at a horizontal distance of 0 m and a depth of 3 m

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$$g(x)=\frac{200}{{\left[{(x-0)}^{2}+{3}^{2}\right]}^{0.5}}$$
(9)

By using the Fourier transform, the horizontal and vertical derivatives of the gravity anomaly and then the analytic signal of the data could be estimated (Fig. 5).

Fig. 5
figure 5

Analytical signals of the anomaly in Fig. 4

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The suggested technique was used to the gravity anomaly profile and inverted to recover the parameters; the depth (z), the shape index (sn) and the amplitude coefficient (K), respectively (Table 2). Table 2 indicates the solutions for all estimated parameters applying various combinations of N and M points through using different sn values (sn = 0.5, 1.0, and sn = 1.5). The true-values of the parameters are valid at the SD minima, which happens at sn = 0.5 (Table 2).

Table 2 Numerical results of the suggested method applied to the semi-vertical cylinder model (K = 200 mGal × m, z = 3 m, sn = 0.5, xo = 0 m, and profile length = 100 m) without and with a Gaussian random noise
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The above-mentioned generated synthetic data applying Eq. (9) was then contaminated with Gaussian noise (with standard deviation = 1 and mean = 0) (Fig. 4). For the same combination of N and M values, the estimated model parameters were offered in Table 2. Table 2 indicates that the model parameters estimation (z, sn, K) in the case of using sn = 0.5 were acceptable because they had the lowest SD.

Therefore, it was evident that the assessed parameters reflected the stability and efficiency of the suggested method in interpreting the gravity data.

Third model

A gravity anomaly profile of a sphere model with K = 150 mGal × m3, z = 5 m, sn = 1.5, xo = 0 m, and length = 100 m (Fig. 6) was adopted. The model equation is:

Fig. 6
figure 6

Noise-free and noisy gravity anomalies of a sphere located at a horizontal distance of 0 m and a depth of 5 m

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$$g\left(x\right)=\frac{150}{{\left[{\left(x-0\right)}^{2}+{5}^{2}\right]}^{1.5}} .$$
(10)

By using the Fourier transform, the horizontal and vertical derivatives of the gravity anomaly and then the analytic signal of the data were estimated (Fig. 7).

Fig. 7
figure 7

Analytical signals of the anomaly in Fig. 6

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The suggested technique was applied to this anomaly profile without and including Gaussian noise (with standard deviation = 1 and mean = 0) (Fig. 6) and inverted to recover the model parameters (Table 3). Table 3 explains the percentage of errors in z and K for all available combination N and M values using sn = 1.5.

Table 3 Numerical results of the suggested method applied to the sphere model (K = 150 mGal × m3, z = 5 m, sn = 1.5, xo = 0 m, and profile length = 100 m) without and with a Gaussian random noise
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So, the percentage of error in z and K was zero in the noise-free case. However, in the noisy case, the maximum error in z and K were 9.6% and 10.1% respectively.

Fourth model

A forward gravity response of three different anomalies consisted of a sphere model whose model parameters were K = 10,000 mGal × m3, z = 7 m, sn = 1.5 and xo = 15 m, a horizontal cylinder whose model parameters is K = 100 mGal × m2, z = 3 m, sn = 1 and xo = 50 m, and a horizontal cylinder whose model parameter is K = 50 mGal × m2, z = 1 m, sn = 1, and xo = 70 m (Fig. 8). Adapting the same procedures mentioned-above, the Analytical signal was created as shown in Fig. 9.

Fig. 8
figure 8

A gravity anomaly of three different models which are a sphere situated at a horizontal distance of 15 m and a depth of 7 m, a horizontal cylinder located at a horizontal distance of 50 m and a depth of 3 m, and a horizontal cylinder located at a horizontal distance of 70 m and a depth of 1 m

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Fig. 9
figure 9

Analytical signal of the anomalies in Fig. 8

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The new method was applied to the gravity response profile and inverted to retrieve the model parameters; z, sn and K, individually, to validate the outcomes for all combinations of N and M points. The estimated parameters are presented in Fig. 10 which demonstrates that the method converges at the correct solutions (z, sn) for every model. This is evident of the fact that suitable inversion outcomes were achieved by employing the new current algorithm specifically for depth and shape index, which is a targeted concern in gravity exploration for minerals which have an economic significance.

Fig. 10
figure 10

Numerical results by using the present method for interpreting anomalies in Fig. 9

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Effect of choosing xo

It was well known that incorrect estimation of the origin of buried structures causes an error in model parameters estimation while inferring the real data. Since we were attempting to study the effect of choosing the wrong origin (xo) as stated in Eq. (9), the origin of the horizontal cylinder model (K = 100 mGal × m2, z = 2 m, sn = 1, xo = 0 m, and profile length = 100 m) was anticipated to be picked incorrectly by introducing errors of 0, ± 0.2, ± 4,…, ± 1.0 m in the horizontal coordinate xi.

After the similar elucidation method applied, the results were revealed in Fig. 11which shows that the error in model parameters appraised increases with increasing error in xo. This demonstrates that the determination of xo is very significant in our method.

Fig. 11
figure 11

Error in model parameters estimated after imposing error in xo in case of noise-free and noisy model mentioned in Fig. 2

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Application to field examples

The proposed method explained in this study was ready and adjusted to investigate the residual gravity anomalies to simple bodies of various structures, e.g., vertical cylinders, spheres, and horizontal cylinders. Three field examples from Slovakia, Cuba and India were re-interpreted to examine the robustness and constancy of the suggested method. The pertinent parameters (K, z, and sn) were inferred in an integrated manner with the existing geological and geophysical outcomes.

Catholic Church Crypt, Slovakia

As it is argued for not using destructive tools during the exploring process for the archaeological buildings, the proper tool of using the microgravity technique appears on the ground. A good example of using such a technique is the Roman-Catholic Church crypt of St. Nicolas in Pukanec town. This crypt was detected and delineated for its local density variations which had been caused by a near-surface cavity. Microgravity measurements were taken on a 10 m × 20 m grid with a 1 m spacing “somewhere the spacing was less than 1 m due to restrictions on the study area” (Fig. 12). Microgravity data measuring and processing have been examined in detail in past studies (Panisova and Pasteka, 2009).

Fig.  12
figure 12

The St. Nicolas church view and a location of the study area in Pukanec-city, southern central Slovakia

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For further interpretation, the method suggested in our study can be used on the final residual Bouguer anomaly profile (Fig. 13) with negative anomaly amplitude exceeding –30 up to –40μGal. To detect and interpret the existing negative gravity anomaly and estimate the model parameters of the buried anomaly source. The negative residual Bouguer anomaly profile was digitized with an interval of 0.5 m. These digitized residual anomaly profile data were subjected to our new algorithm to achieve the prospected depth, amplitude coefficient, and shape of the buried target anomaly as shown in Table 4.

Fig.  13
figure 13

Upper panel: The misfit between the observed and calculated data. Middle panel: The measured gravity anomaly (black circles) over a crypt, St. Nicolas Church, Slovakia and the fitted response (open circles) computed from the present method. Bottom panel: Geologic sketch for the crypt from the present method and Panisova and Pasteka method (2009)

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Table  4 Numerical results of the present method applied to a crypt, St. Nicolas Church, Slovakia
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Table 4 shows the optimum parameters, least SD, z = 2.08 m, and k = -7.66 mGal × m2. The acceptable fit between the observed gravity anomaly over a crypt, St. Nicolas Church, Slovakia (black circles) and calculated gravity anomaly from the present method (open circles) was achieved with a least SD (Fig. 13) and the analytical signal of these gravity data is shown in Fig. 14. The discrepancy between the observed and calculated fields in some places is shown in Fig. 13. Explanations for this can refer to the inadequacy in identifying the corrections of the building influence or the deformation of the residual anomaly field. In addition, Fig. 13 depicts that the crypt could be situated approximately 1 m to the crypt top and 2.08 m depth to the center of the crypt.

Fig.  14
figure 14

Analytical signal of the anomaly in Fig. 13

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Chromite deposit body

Davis et al. (1957) declared that the chromite deposits in the Camagüey Area, Cuba are initiated in complex geological environs including serpentinized peridotite and dunite with little quantities of gabbro, troctolite, and anothosite. The geologic complex environs are seen inhibited into metamorphic rocks and superimposed by upper Cretaceous volcanic rocks with limestone and radiolarian cherts. Chromite is a black, hard moderately heavy mineral that varies widely in composition. The chromites of this area have an average composition and occur as anhedral or occasionally subhedral grains, commonly in a region millimeter or less. They are disseminated principally in dunite and less commonly in peridotite, troctolite, or anorthosite, and as larger masses but all within the peridotite zone although surrounded by a dunite envelope. The texture ranges from that of a normal rock through increasing proportions of chromite and an increase in grain size to a massive, moderately coarse-grained aggregate consisting entirely of chromite (Flint et al. 1948; Ulloa Santana et al. 2011) (Fig. 15).

Fig.  15
figure 15

Geologic map of the Camagüey area, Cuba (after Ulloa Santana et al. 2011)

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Davis et al. (1957) and Roy (2001) displayed the gravity map over a chromite ore in the Camagüey Area, Cuba (Fig. 16). From this figure, the marked black line with 180 m length was taken to represent the residual gravity profile (Fig. 17). This profile was digitized at an equal interval of 18 m. For the purpose of guessing z, sn and K via the analytical signal of the gravity data, the suggested algorithm was utilized (Fig. 18) as shown in Table 5.

Fig.  16
figure 16

Gravity anomaly map over a chromite deposit in the Camagüey area, Cuba (after Davis et al. 1957), the black line (Profile) represents the profile used in this study

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Fig.  17
figure 17

Upper panel: The misfit between the observed and calculated data. Lower panel: A residual gravity anomaly (black circles) over a chromite body in Camagüey district, Cuba, and the predicted response (open circles) calculated from the present method

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Fig.  18
figure 18

Analytical signal of the anomaly in Fig. 17

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Table  5 Numerical results of the present method applied to a chromite body in Camagüey district, Cuba
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The optimum inverse parameters at minimum SD (SD = 4.18 m and SD = 1.38 m in cases of z and K, respectively) are given at sn = 1, z = 49.49 m and K = 16.4 mGal × m2. The best-fit-model is shown in Fig. 14 (open circle). This recommends that the suppressed structure shape is a cylinder model with 49.5 m depth. The estimated parameters (shape and depth) of the ore body were matched with those attained by different published interpreted methods (Table 6). Table 6 explains that the application of the suggested method gives an optimal result (λ = 0.083 mGal) for the buried mineral target, especially the depth, which opens the field for more investigations in the future using different methods.

Table  6 A Comparative model parameters results for the chromite body in Camagüey district, Cuba
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Manganese ore body

Sarkar et al. (1967) demonstrated in detail the geology of the Nagpur area,

India is covered by soil with very scanty rock exposures. Rocks have suffered three phases of major folding deformation as revealed by surface and sub-surface data. These rock units denote series of anti-formal and syn-formal structures with steep plunge sub-parallel to strike Manganese oxide orebodies and manganese silicate rocks characterizing the metasedimentary sequence of the Precambrian rocks in the Nagpur area.

Reddi et al. (1995) show the gravity map over a manganese deposit near Nagpur, India (Fig. 19). A residual gravity profile was taken for this study area (Fig. 20). To show the residual gravity profile of the study area, a 333 m profile was digitized with an interim of 27 m (Fig. 20) and subjected to the inversion utilizing the present method. The new approach was utilized to the gravity data to appraise z, sn and A utilizing the analytical signal of the gravity data (Fig. 21). The results are illustrated in Table 7.

Fig.  19
figure 19

Gravity anomaly map over a manganese deposit near Nagpur, India (after Reddi et al. 1995), the black line (Profile) represents the profile used in this study

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Fig.  20
figure 20

Upper panel: The misfit between the observed and calculated data. Lower panel: The measured gravity anomaly (black circles) over a manganese deposit near Nagpur, India and the fitted response (open circles) computed from the present method

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Fig.  21
figure 21

Analytical signal of the anomaly in Fig. 20

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Table  7 Numerical results of the present method applied to a manganese deposit near Nagpur, India
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The minimum SD was found at sn = 1, z = 59.77 m and K = 23.22 mGal × m. The synthetic anomaly of the optimum parameters is shown in Fig. 20 (open circle). The results suggest that the shape of the buried target is a horizontal cylinder model buried at a depth of 59.77 m. The shape and the depth assessed by the present approach are in good agreement with those attained from Roy (2001), Essa (2012 and 2014), and Ekinci et al. (2016) (Table 8). Table 8 explains that our method gives the least root mean squared error (λ = 0.048 mGal). In other words, the estimated result, especially the depth of the target structure by the application of the proposed method is giving a powerful insight into the geologic subsurface.

Table  8 A Comparative model parameters results for the manganese deposit near Nagpur, India
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The findings also emphasize that the real structures may not have the classic shapes (e.g. spheres or cylinders) or structures in nature. Therefore, the modeling and inversion of the real data mentioned-above with regular geometric-structures does not revenue the real subsurface targets. Insignificant deviation of the real-structure from the modeled structure (e.g. spheres, cylinders) can be presumed to be a superposition of various types of noises on the responses represented by simple and standard geometric-structures. Hence we can catch a respectable estimation for the place and the depth of the subsurface structure of a mineralized target. Finally, it is important to mention that the suggested method can elucidate the gravity data for two mineralized structures from Cuba and India that have given a respectable result.

Conclusion

The aim of our study was to assess the target buried structure parameters depth, shape, and the amplitude coefficient from the residual gravity data. The method used in this study relies on evaluating the analytical signal of the original gravity data and it utilizes the combination of all available points on the analytic signal profile to appraise the buried target parameters. The findings of this study revealed that our method was less affected by noise than other methods. In this regard, synthetic data, noisy data, and raw data, crypt example from Slovakia, and two mineralized examples from Cuba and India, were used to display and demonstrate the stability and efficiency of the suggested method. Thus, the inverted parameters (z, sn) should be engaged by the available geological information and other provided geophysical outcomes to help in rectifying any encountered uncommon solution in geophysical exploration. While this proposed method was utilized for the simplified models, it provides an appropriate explanation for the new geological insight into the subsurface. However, this study is subject to a limitation which is the suggested method falls in the region of more complex and complicated structures which may cause more ambiguity in the results. Future line of research may overcome the above-declared facts by extending the use of this method in deciphering several problems.