Definitions of * A*,

*and*η

*of the simple-geometric bodies.*q

Open access peer-reviewed chapter

Submitted: 23 June 2021 Reviewed: 28 April 2022 Published: 30 May 2022

DOI: 10.5772/intechopen.105092

Rapid imaging technique, so-called “R-parameter”, utilized for interpreting a gravity anomaly profile. The R-parameter based on calculating the correlation factor between the analytic signal of the real anomaly and the analytic signal of the forward anomaly of assumed buried source denoted by simple geometric shapes. The model parameters (amplitude, origin, depth, and shape factor) picked at the maximum value of the R-parameter. The technique has been proved on noise free and noisy numerical example, numerical example showing the impact of interfering sources. Furthermore, the introduced technique has been successfully applied to visualize a salt dome gravity anomaly profile, USA. The obtained results are in good agreement with those reported in the published studies and that with that obtained from drilling.

- anomaly
- salt dome
- R-parameter method
- depth

Gravity data interpretation has been widely used to appraise the different types of subsurface structures and their locations [1, 2, 3, 4, 5, 6, 7, 8]. Gravity methods have been widely applied to ore and mineral exploration [9, 10, 11, 12, 13], hydrocarbon exploration [14, 15, 16], cave detection [17, 18], hydrogeology [19, 20], geothermal and volcanic activity [21, 22, 23], locating of unexploded military ordnance [24], environmental and engineering application [25, 26] and archaeological investigations [27, 28].

The quantitative interpretation of gravity data using simple models (spheres and cylinders) is common in exploratory geophysics and continues to be of interest [29, 30, 31, 32, 33]. In geologic contexts with a single gravity anomaly, it can be quite appropriate [34]. A single isolated causal body can invert this recorded gravity anomaly to establish its distinctive inverted parameters and fit the recorded data.

The simple geometric models can be matched with the subsurface structures encountered during application of several approaches for inversion [35, 36, 37, 38, 39]. These methods include graphical and numerical characteristic points approaches [40, 41, 42], ratio technique [43], Fourier transform method [44], the neural network algorithms [45], Mellin transform technique [36], and Werner deconvolution technique [46]. However, the drawbacks of these methods based on tending to generate high number of invalid solution due to few numbers of points and data used, noise or window size incompatibility. As a result, these approaches are subjective, which can lead to significant inaccuracies in calculating the buried anomalous body's characteristic inverse parameters [41, 47], which is to be expected. Gupta [48] and Essa [49] developed techniques depending on successive minimization approaches, which utilize the whole measured data to assess the depth parameter and then used some of characteristic points to continue in estimating the rest parameters such as amplitude coefficient. Shaw and Agarwal [37] used the Walsh transform scheme to determine the depth of buried bodies. Mehanee [47] used the regularized conjugate gradient method to construct an effective iterative method based on the use of logarithms of the model parameters for gravity inversion. The method inverts the residual gravity data acquired along profile for evaluating a depth and amplitude coefficient of buried bodies and suitable for subsurface imaging and mineral exploration.

Here, the study proposed an application of the robust R-parameter imaging method to interpret residual gravity data along a profile over idealized geometric bodies such as semi-infinite vertical cylinder, infinitely long horizontal cylinder, and sphere models. The goal is to establish the underlying approximative model by determining the body parameters, which include its origin, depth, amplitude coefficient, and shape. The R-parameter imaging method depends on the correlation coefficient amongst the analytic signal of the collected and calculated gravity data. The optimum solution occurs at the maximum R-parameter value.

The benefit behind the use of this method is fall in estimating the depth and body location with an acceptable value compared to the true ones and used the whole gravity data points of the profile, instead of just a few characteristic points. In addition to the method does not require priori information of the subsurface and directly interpret the anomaly from the given observed data. This chapter begins with a layout of the forward modeling, which contains a theoretical gravity formula, an R-parameter imaging approach description, numerical models test without and including noise, and a field data for slat dome investigation.

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## 2. Forward modeling

g x j x o z z o η q A = A z o − z η x j − x o 2 + z o − z 2 q , j = 1 , 2 , 3 , … , n E1

A closed-form solution for the gravity anomaly caused by simple geometric structures at a measured point (* x*) along a profile (Figure 1) is given by [29, 50, 51].

where * x* and

Case | |||
---|---|---|---|

Sphere | 1 | 3/2 | |

Horizontal cylinder | 2^{2} | 1 | 1 |

Vertical cylinder | ^{2} | 0 | 1/2 |

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## 3. Methodology

ξ s x = ∂ g ∂ x − i ∂ g ∂ z , i = − 1 , E2ξ s x = ∂ g ∂ x 2 + ∂ g ∂ z 2 . E3ξ s x = A z o − z η − 1 4 q 2 x j − x o 2 z o − z 2 + η x j − x o 2 − z o − z 2 2 q − η 2 x j − x o 2 + z o − z 2 q + 1 , E4R x o z o = ∑ j n ξ so x j ξ st x j ∑ j n ξ so x j 2 ∑ j n ξ st x j 2 . E5

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

where

The amplitude of the analytic signal

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

where * j* = 1, 2, 3, ….,

The analytic signal * x* and

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## 4. Numerical examples

### 4.1 Model 1

#### 4.1.1 Noise-free data

#### 4.1.2 Noisy data

### 4.2 Model 2: neighboring effect

To verify constancy in performance of the proposed method, numerical example without noise (noise-free) and with a 20% random noise (noisy) is tested. Another numerical example to evaluate the accuracy and stability in assessing the model parameters in case of interference/neighboring influence.

The R-parameter imaging method is applied to noise-free numerical gravity anomaly due to simple model consisting of a a horizontal cylinder model (* q* = 1) with

Table 2 shows a different shape values that employed in the interpretation process. The results (Figure 3 and Table 2) reveal that at * q* = 1, the R-max = 1 and indicating that the method is stable and capable of capturing thel exact values of the model parameters.

Shape factor | Maximum R-parameter |
---|---|

(q) | (R-max ) |

0.5 | 0.6696 |

0.6 | 0.7768 |

0.7 | 0.8897 |

0.8 | 0.9620 |

0.9 | 0.9930 |

1.1 | 0.9962 |

1.2 | 0.9883 |

1.3 | 0.9791 |

1.4 | 0.9700 |

1.5 | 0.9611 |

We applied the same procedures (by utilizing Eq. (4) as the forward modelling formula in this case) to the analytic signal data presented in Figure 2c to explore the recital of the current scheme when used to the analytic signal data themselves instead of the residual gravity data. Figure 4 shows the outcomes, which are match with those derived from the above-mentioned elucidation of gravity data (Table 3).

Estimated model parameters | Analytic signal data | Gravity anomaly data |
---|---|---|

(mGal m) | 100 | 100 |

(m)_{o} | 5 | 5 |

1 | 1 | |

(m)_{o} | 51 | 51 |

Given the lack of totally noise-free gravity field data, a 20% random noise (Figure 5a) has been introduced to the data in Figure 2a. The horizontal and vertical derivatives, besides the magnitude of the analytic signal of the measured gravity anomaly, are depicted in Figure 5b and c. The R-parameter values were evaluated utilizing Eq. (5) and created a 2-D mosaic surface (Figure 5d). The maximum R-parameter value is 0.94. The imaging-derived model parameters (* z* = 6 m and

The amplitude coefficient (* A*) is substantially overstated (Figure 6a–d) when the analytic signal data (Figure 5c) is interpreted. This is unsurprising given that the technique tries to fit the real data, and the anomalous body's inferred depth (

Estimated model parameters | Analytic signal data | Gravity anomaly data |
---|---|---|

(mGal m) | 212.90 | 100 |

(m)_{o} | 7 | 5 |

1 | 1 | |

(m)_{o} | 51 | 51 |

The performance of the proposed inversion method with complicated field anomalies and the effect of interfering subsurface structures was investigated. To achieve this, we once again generate a synthetic model data from multiple source bodies as a horizontal cylinder model with _{1} = 100 mGal m, _{1} = 3 m, _{1} = 30 m, and _{1} = 1) and a sphere model with _{2} = 400 mGal m^{2}, _{2} = 4 m, _{2} = 80 m, and _{2} = 1.5) (Figure 7a).

Figure 7b and c illustrate the horizontal and vertical gradients of the composite gravity anomaly, as well as the amplitude of the analytic signal. The R-parameter values were determined using Eq. (5) for each source location and a 2-D mosaic surface S of 101 × 11 m in the X- and Z-directions constructed and discretized into 1-m intervals in both directions. The 2-D mosaic (Figure 7d) indicates that the R-max value for each source is 0.8 and 0.62 at * q* equal 1 and 1.5, respectively. The obtained model parameters for horizontal cylinder and sphere are

To better understand the procedure, we tainted the composite anomaly (Figure 7a) with a 20% noise level (Figure 8a). The horizontal and vertical derivatives, as well as the corresponding amplitude of the analytic signal, are shown in Figure 8b and c. The retrieved R-parameter image is shown in Figure 8d, with R-max values of 0.79 and 0.61 for a horizontal cylinder and a sphere, respectively. The drop in maximum parameter values compared to (Figure 7d) is attributable to the noise introduced into the data as well as the effect of the nearby objects. The model parameters for the first and second bodies revealed by imaging are: _{1} = 126.9 mGal m, _{1} = 3.8 m, and _{1} = 30 m and _{2} = 556.3 mGal m^{2}, _{2} = 4.7 m, and _{2} = 80 m, respectively, which are quite adjacent to the real values.

Figure 9 depicts the results of the study of the noisy analytic signal data seen in Figure 8c. The amplitude coefficients and burial depths recovered from the elucidation are exaggerated (Figure 9a–d), as shown in Table 5, which coincides with and confirms the aforementioned results.

Estimated model | Noisy contaminated interference/neighboring effect | |||
---|---|---|---|---|

Analytic signal data | Gravity anomaly data | |||

Parameters | First anomaly | Second anomaly | First anomaly | Second anomaly |

A (mGal m^{2q−η}) | 174.2 mGal m | 1554.3 mGal m^{2} | 126.9 mGal m | 556.3 mGal m^{2} |

z_{o} (m) | 4.2 | 7.6 | 3.8 | 4.7 |

q | 1 | 1.5 | 1 | 1.5 |

x_{o} (m) | 30 | 80 | 30 | 80 |

On the basis of the theoretical models presented above, it can be inferred that the technique described here is stable and robust.

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## 5. Field data

### 5.1 Humble dome anomaly, Houston, USA

### 5.2 Louisiana dome anomaly, USA

A published field example over a salt dome anomaly is examined in order to thoroughly test the applicability of the established methodology. For a variety of reasons, this case was chosen. First, the residual gravity profile was created by a simple body that may be truthfully inferred. Second, the drilling information helps in estimating the density contrast of the underlying body. Knowing the density contrast, the radius can be calculated and the depth to the top also can be inferred by using the definition of the amplitude coefficient (Table 1). Moreover, the depth of the vertical cylinder model is measured to the top but the depth of a horizontal cylinder and sphere model is measured to the center of the body (Figure 1). Third, the gravity data was taken from an area with recognized drilling information, allowing the results obtained from the technique proposed here to be cross-validated against those received via drilling.

Gravity map was acquired over the Humble Dome, Houston, Texas. At the earth's surface, the measurements of this salt dome structure reveal a negative circular contoured Bouguer anomaly ([41], Figures 8–** 16**). A Bouguer gravity profile is taken across the center of the Humble salt dome gravity map in Houston ([41], Figures 8–

The R-parameter method procedures were applied to the residual gravity anomaly profile for the available shape parameters (Table 6). Figure 10b–d express the horizontal and vertical derivative anomalies, the amplitude analytic signal, and the R-parameter 2-D mosaic. It is found that the R-max value is 0.99 corresponds to a spherical shape (* q* = 1.5). The best estimated model parameters are:

R-max | |
---|---|

0.5 | 0.999462 |

0.6 | 0.997341 |

0.7 | 0.996274 |

0.8 | 0.992509 |

0.9 | 0.986895 |

1 | 0.984110 |

1.1 | 0.987549 |

1.2 | 0.992912 |

1.3 | 0.996863 |

1.4 | 0.998792 |

The humble dome anomaly has been interpreted by several authors assuming a spherical source to decide the depth of the salt body. The obtained results agree well with those depths to the center that obtained by the published literatures of [36, 41, 47, 54, 55, 56] (Table 7).

By using a density contrast of −0.13 gm/cm^{3} of [41], then the depth to the top of the spherical body of the humble dome obtained from the proposed technique is 315 m, which in excellent covenant with the true depth (305 m) confirmed by drilling and seismic information [41]. Table 8 shows that several other researchers utilizing the same density contrast found some differences in the depths to the top of this spherical source. The use of simple geometrically bodies in the constrained class of spheres, horizontal cylinders, and vertical cylinders is thus suggested as a way to accurately apply the current methodology to extract depth information. As a result, if exact density contrasts are used, the related radii can be correctly computed as well.

A residual gravity map was acquired over a salt dome off the coast of Louisiana, USA ([41], Figures 8–** 20**). The residual gravity anomaly profile [57] is redrawn across the center of the map, normal to the causal anomaly’s striking. The residual gravity anomaly profile of about 13,000 m long was digitized at sampling interval of 200 m (Figure 11a).

By applying the R-parameter method procedures mentioned before to the residual gravity anomaly profile of Louisiana we get the available shape parameters corresponding to the maximum R-parameter (R-max) as shown in Table 9. Figure 11b–d shows the horizontal and vertical derivative anomalies, the amplitude analytic signal, and the R-parameter 2-D mosaic of the Louisiana anomaly. It is found that the R-max value is 0.96 corresponds to * q* = 0.9 which is approximated by horizontal cylinder shape. The best estimated model parameters are:

R-max | |
---|---|

0.5 | 0.943033 |

0.6 | 0.934839 |

0.7 | 0.926284 |

0.8 | 0.957334 |

1 | 0.957410 |

1.1 | 0.942955 |

1.2 | 0.926284 |

1.3 | 0.909983 |

1.4 | 0.894917 |

0.881254 |

The Louisiana dome anomaly has been interpreted by different authors assuming a horizontal source to determine the depth to the center of the salt body. The obtained results have a good agreement with those depths to the center that obtained by the published literatures of [5, 41] (Table 10). In addition, the proposed method has the lowest misfit compared to the other method (Table 10).

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## 6. Discussion

The proposed method of R-parameter imaging technique was deployed to visualizes the salt dome anomalies from the gravity data measured across a 2D profile. The method fitting the anomaly of the measured gravity profile by a single geometric shape body (sphere & cylinder). Such as the spherical source (* q* = 1.5) suggestion based on the maximum R-parameter (R-max = 0.99) of the Humble dome anomaly (Figure 10) and the approximated horizontal cylinder (

The obtained results by the R-parameter method of the Humble dome anomaly was compared with other results in the published literature (Table 7) and confirmed with drilling to insure the depth to the top of the buried anomaly (Table 8). For the Louisiana dome anomaly, the obtained results by R-parameter approach was compared with the pervious published literature and weighted by the misfit error between the observed and calculated anomaly for the different techniques used (Table 10) to increase the efficiency of the proposed method.

In over all the obtained results using the R-parameter method to investigate the salt dome anomalies is good and acceptable in the two given field examples.

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## 7. Conclusions

In this study, we have introduced and investigated the applicability and the performance of the R-parameter imaging method in elucidating distinctive physical parameters (* A*,

- 1.
Paterson NR, Reeves CV. Applications of gravity and magnetic surveys: The state of-the-art in 1985. Geophysics. 1985; 50 :2558-2594 - 2.
Ateya IL, Takemoto S. Gravity inversion modeling across a 2-D dike like structure—A case study. Earth Planets Space. 2002; 54 :791-796 - 3.
Essa KS, Di Risio M, Celli D, Pasquali D, editors. Geophysics and Ocean Waves Studies. Rijeka: InTech; 2021. 188 p. DOI: 10.5772/intechopen.87807 - 4.
Hector B, Séguis L, Hinderer J, Descloitres M, Vouillamoz J, Wubda M, et al. Gravity effect of water storage changes in a weathered hard-rock aquifer in West Africa: Results from joint absolute gravity, hydrological monitoring and geophysical prospection. Geophysical Journal International. 2013; 194 :737-750 - 5.
Biswas A. Interpretation of residual gravity anomaly caused by a simple shaped body using very fast simulated annealing global optimization. Geoscience Frontiers. 2015; 6 :875-893 - 6.
Essa KS. A fast interpretation method for inverse modeling of residual gravity anomalies caused by simple geometry. Journal of Geological Research. 2012; 2012 :327037. DOI: 10.1155/2012/327037 - 7.
Nishijma J, Naritomib J. Interpretation of gravity data to delineate underground structure in the Beppu geothermal field, central Kyushu, Japan, Regional Studies. Journal of Hydrology. 2017; 11 :84-95 - 8.
Anderson NL, Essa KS, Elhussein M. A comparison study using particle swarm optimization inversion algorithm for gravity anomaly interpretation due to a 2D vertical fault structure. Journal of Applied Geophysics. 2020; 179 :104120 - 9.
Mehanee S, Essa KS. 2.5D regularized inversion for the interpretation of residual gravity data by a dipping thin sheet: Numerical examples and case studies with an insight on sensitivity and non-uniqueness. Earth, Planets and Space. 2015; 67 :130 - 10.
Chen YQ, Zhang LN, Zhao. Application of bi-dimensional empirical mode decomposition (BEMD) modeling for extracting gravity anomaly indicating the ore controlling geological architectures and granites in the Gejiu tin-copper polymetallic ore field, southwestern China. Ore Geology Reviews. 2017; 88 :832-840 - 11.
Essa KS, Munschy M. Gravity data interpretation using the particle swarm optimization method with application to mineral exploration. Journal of Earth System Science. 2019; 128 :123 - 12.
Essa KS, editor. Minerals. Vol. 142. Rijeka: InTech; 2019. DOI: 10.5772/intechopen.74902 - 13.
Mehanee SA. Simultaneous joint inversion of gravity and self-potential data measured along profile: Theory, numerical examples, and a case study from mineral exploration with cross validation from electromagnetic data. IEEE Transactions on Geoscience and Remote Sensing. 2022; 60 :1-20. Art no. 4701620. DOI: 10.1109/TGRS.2021.3071973 - 14.
Yuan B, Song L, Han L, An S, Zhang C. Gravity and magnetic field characteristics and hydrocarbon prospects of the Tobago Basin. Geophysical Prospecting. 2018; 66 :1586-1601 - 15.
Essa KS, Gèraud Y. Parameters estimation from the gravity anomaly caused by the two-dimensional horizontal thin sheet applying the global particle swarm algorithm. Journal Petroleum Science and Engineering. 2020; 193 :107421 - 16.
Coelho ACQM, Menezes PTL, Mane MA. Gravity data as a faulting assessment tool for unconventional reservoirs regional exploration: The Sergipe–Alagoas Basin example. 2021; 94 :104077 - 17.
Martínez-Moreno FJ, Galindo-Zaldívar J, Pedrera A, González-Castillo L, Ruano P, Calaforra JM, et al. Detecting gypsum caves with microgravity and ERT under soil water content variations (Sorbas, SE Spain). Engineering Geology. 2015; 193 :38-48 - 18.
Braitenberg C, Sampietro D, Pivetta T, Zuliani D, Barbagallo A, Fabris P, et al. Gravity for detecting caves: Airborne and terrestrial simulations based on a comprehensive karstic cave benchmark. Pure and Applied Geophysics. 2016; 173 :1243-1264 - 19.
Rodell M, Chen J, Kato H, Famiglietti JS, Nigro J, Wilson CR. Estimating groundwater storage changes in the Mississippi River basin (USA) using GRACE. Hydrogeology Journal. 2007; 15 :1 - 20.
Epuh EE, Okolie CJ, Daramola OE, Ogunlade FS, Oyatayo FJ, Akinnusi SA, et al. An integrated lineament extraction from satellite imagery and gravity anomaly maps for groundwater exploration in the Gongola Basin. Remote Sensing Applications: Society and Environment. 2020; 20 :100346 - 21.
Lichoro CM, Árnason K, Cumming W. Joint interpretation of gravity and resistivity data from the Northern Kenya volcanic rift zone: Structural and geothermal significance. Geothermics. 2019; 77 :139-150 - 22.
Njeudjang K, Essi JMA, Kana JD, Teikeu WA, Nouck PN, Djongyang N, et al. Gravity investigation of the Cameroon Volcanic Line in Adamawa region: Geothermal features and structural control. Journal of African Earth Sciences. 2020; 165 :103809 - 23.
Mulugeta BD, Fujimitsu Y, Nishijima J, Saibi H. Interpretation of gravity data to delineate the subsurface structures and reservoir geometry of the Aluto-Langano geothermal field, Ethiopia. Geothermics. 2021; 94 :102093 - 24.
Butler DK, Wolfe PJ, Hansen RO. Analytical modeling of magnetic and gravity signatures of unexploded ordnance. Journal of Environmental and Engineering Geophysics. 2001; 6 (1):33-46 - 25.
Tinivella U, Giustiniani M, Cassiani G. Geophysical methods for environmental studies. International Journal of Geophysics. 2013; 2 :950353. DOI: 10.1155/2013/95035 - 26.
Setyawan A, Fukuda Y, Nishijima J, Kazama T. Detecting land subsidence using gravity method in Jakarta and Bandung Area, Indonesia. Procedia Environmental Sciences. 2015; 23 :17-26 - 27.
Padín J, Martín A, Anquela AB. Archaeological microgravimetric prospection inside don church (Valencia, Spain). Journal of Archaeological Science. 2012; 39 :2 - 28.
Klokočník J, Cílek V, Kostelecký J, Bezděk A. Gravity aspects from recent Earth gravity model EIGEN 6C4 for geoscience and archaeology in Sahara, Egypt. Journal of African Earth Sciences. 2020; 168 :103867 - 29.
Salem A, Ravat D, Mushayandebvu MF, Ushijima K. Linearized least-squares method for interpretation of potential-field data from sources of simple geometry. Geophysics. 2004; 69 (3):783-788 - 30.
Su Y, Cheng LZ, Chouteau M, Wang XB, Zhao GX. New improved formulas for calculating gravity anomalies based on a cylinder model. Journal of Applied Geophysics. 2012; 86 :36-43 - 31.
Essa KS. Gravity interpretation of dipping faults using the variance analysis method. Journal of Geophysics and Engineering. 2013; 10 :015003 - 32.
Singh A, Biswas A. Application of global particle swarm optimization for inversion of residual gravity anomalies over geological bodies with idealized geometries. Natural Resources Research. 2016; 25 :297-314 - 33.
Essa KS, Elhussein M. Gravity data interpretation using new algorithms: A comparative study. In: Zouaghi T, editor. Gravity-Geoscience Applications, Industrial Technology and Quantum Aspect. Rijeka: InTech; 2018. pp. 3-17. DOI: 10.5772/intechopen.68576 - 34.
Abdelrahman EM, El-Araby TM, Essa KS. Shape and depth solutions from third moving average residual gravity anomalies using the window curves method. Kuwait Journal of Science and Engineering. 2003; 30 :95-108 - 35.
Nettleton LL. Gravity and magnetics for geologists and seismologists. AAPG Bulletin. 1962; 46 :1815-1838 - 36.
Mohan NL, Anandababu L, Roa S. Gravity interpretation using the Melin transform. Geophysics. 1986; 51 :14-22 - 37.
Shaw RK, Agarwal NP. The application of Walsh transform to interpret gravity anomalies due to some simple geometrically shaped causative sources: A feasibility study. Geophysics. 1990; 55 :843-850 - 38.
Abdelrahman EM, Abo-Ezz ER, Essa KS, El-Araby TM, Soliman KS. A least-squares variance analysis method for shape and depth estimation from gravity data. Journal of Geophysical Engineering. 2006; 3 :143-153 - 39.
Asfahani J, Tlas M. Estimation of gravity parameters related to simple geometrical structures by developing an approach based on deconvolution and linear optimization techniques. Pure and Applied Geophysics. 2015; 172 :2891-2899 - 40.
Siegel HO, Winkler HA, Boniwell JB. Discovery of the Mobrun Copper Ltd. sulphide deposit, Noranda Mining District, Quebec. In: Methods and Case Histories in Mining Geophysics: Commonwealth Mining Metallurgical Congress. 6th ed. Vancouver; 1957. pp. 237-245 - 41.
Nettleton LL. Gravity and Magnetics in Oil Prospecting. New York: McGraw-Hill Book Co.; 1976 - 42.
Reynolds JM. An Introduction to Applied and Environmental Geophysics. New York: John Wiley and Sons; 1997. 796 p - 43.
Bowin C, Scheer E, Smith W. Depth estimates from ratios of gravity, geoid and gravity gradient anomalies. Geophysics. 1986; 51 :123-136 - 44.
Odegard ME, Berg JW. Gravity interpretation using the Fourier integral. Geophysics. 1965; 30 :424-438 - 45.
Abedi M, Afshar A, Ardestani VE, Norouzi GH, Lucas C. Application of various methods for 2D inverse modeling of residual gravity anomalies. Acta Geophysica. 2009; 58 :317-336 - 46.
Kilty KT. Short Note: Werner deconvolution of profile potential field data. Geophysics. 1983; 48 :234-237 - 47.
Mehanee SA. Accurate and efficient regularised inversion approach for the interpretation of isolated gravity anomalies. Pure and Applied Geophysics. 2014; 171 :1897-1937 - 48.
Gupta OP. A least-squares approach to depth determination from gravity data. Geophysics. 1983; 48 :360-375 - 49.
Essa KS. A new algorithm for gravity or self-potential data interpretation. Journal of Geophysics and Engineering. 2011; 8 :434-446 - 50.
Essa KS. New fast least-squares algorithm for estimating the best-fitting parameters of some geometric-structures to measured gravity anomalies. Journal of Advanced Research. 2014; 5 :57-65 - 51.
Asfahani J, Tlas M. An automatic method of direct interpretation of residual gravity anomaly profiles due to spheres and cylinders. Pure and Applied Geophysics. 2008; 165 :981-994 - 52.
Nabighian MN. The analytic signal of two-dimensional magnetic bodies with polygonal cross-section: Its properties and use for automated anomaly interpretation. Geophysics. 1972; 37 :507-517 - 53.
Klingele EE, Marson I, Kahle HG. Automatic interpretation of gravity gradiometric data in two dimensions: Vertical gradients. Geophysical Prospecting. 1991; 39 :407-434 - 54.
Essa KS. Gravity data interpretation using the s-curves method. Journal of Geophysics and Engineering. 2007; 4 :204-213 - 55.
Asfahani J, Tlas M. Fair function minimization for direct interpretation of residual gravity anomaly pro_les due to spheres and cylinders. Pure and Applied Geophysics. 2012; 169 :157-165 - 56.
Tlas M, Asfahani J, Karmeh H. A versatile nonlinear inversion to interpret gravity anomaly caused by a simple geometrical structure. Pure and Applied Geophysics. 2005; 162 :2557-2571 - 57.
Roy L, Agarwal BNP, Shaw RK. A new concept in Euler deconvolution of isolated gravity anomalies. Geophysical Prospecting. 2000; 48 :559-575

Submitted: 23 June 2021 Reviewed: 28 April 2022 Published: 30 May 2022

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