Numerical results for a gravity model due to horizontal cylinder without and with two levels of 5% and 10% of random noise (A = 400 mGal m^{2}, z = 5 m, q = 1, and profile length = 120 m) using the gradient method.

## Abstract

Gravity data interpretation is useful in exploring regions that have different geological structures, which contain minerals, ores and oil deposits. There are different numerical methods for the model parameters (depth (z), origin location (xo), shape parameter (q) and amplitude coefficient (A)) evaluation of a covered structure such as gradient method, particle swarm optimization technique and Werner deconvolution method. In this study, application of these methods is utilized to appraise the model parametric quantity of the covered structures. The application of these methods was demonstrated by different engineered data without and with various range of noise (5%, 10%) and applied for a real example from Egypt. The result values of each method were compared together and with those published and drilling information.

### Keywords

- gravity anomaly
- depth
- werner deconvolution
- PSO
- gradient method

## 1. Introduction

Gravity method is a non-ruinous geophysical procedure that measures contrasts in the gravitational field of the earth at many various areas. It has much beneficial utilization in hydrocarbon exploration, mineral prospecting, archeological investigations, environmental applications and crustal imaging [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The main objective of the gravity interpretation is evaluating the model parameters (depth, amplitude coefficient, origin location, and shape parameter) of gravity oddities delivered by basic geometrical formed structures (spheres, cylinders). Clarification of gravity data is constantly connected with the ill-posed and non-unique problems. To overcome these issues, we find a preferred geometry to subsurface structures with a known density followed by the inversion processes [12, 13]. Understanding of gravity data can be performed utilizing basic geometrical models, forward modeling and inversion. Analytical formula for basic geometrical shapes and many approaches have been produced to translate the gravity anomaly expecting the body of basic geometry (sphere, horizontal cylinder and vertical cylinder). These techniques have varying complexity in the interpretation.

All different simple models may not be found in real subsurface geological situations, they usually are preferred in practical inversion of many isolated sources. The target of an inversion process is to recover the converse parameters of the model (depth, amplitude coefficient, origin location and shape factor). Many scientists showed and discussed several graphical and numerical approaches developed in past and significantly in the present time [10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. However, the disadvantages of these methods that depend on characteristic points and curves subject to person errors in calculating the inverted parameters of the subsurface structures which can prompt significant errors in assessing the inverse parameters of the covered structure [10, 11]. Thus, the outcomes from these techniques need the accessibility of density information as a noteworthy aspect of the commitment, alongside similar depth information got from geology and/or geophysics. Consequently, the resultant model can shift comprehensively relying upon these factors since the inverse problems are not well-postured and are along these lines unsteady and non-unique [33].

The interpretation of the gravity data is attempted here using three methods: the gradient method [34], the particle swarm optimization and Werner deconvolution method [21]. Analysis of the gravity anomalies can allow obtaining more detailed information on the geological structures that partially outcrops or covered totally in depth. In overall, these different methods are utilized in this work to searching the sources nature of gravity anomalies. The results of applied three different methods are compared together. A synthetic example without and with various level of noise (5% and 10%) used to show the stability of these methods. The proposed techniques are additionally tested on a gravity data from Egypt. To judge satisfaction and fulfillment of these approaches is finished by contrasting the acquired results with other accessible geological or geophysical information in the published literatures.

## 2. The methods

Different three algorithms used to interpret the gravity anomaly (mGal) produced by most common three shapes (spheres, horizontal cylinders and vertical cylinders) (Figure 1) represented by:

where

In the above equation, z is the depth (m), A is the amplitude coefficient (mGal × m^{2q}) that depends on the shape parameter, q is the parameter related to the shape of the body (dimensionless), x_{i} is the position coordinate (m), x_{o} is the origin location (m), σ is the density contrast between the target and the surroundings, G is the gravitational constant parameter which equal 6.67 × 10^{−11} SI units, and R is the radius of the covered body (m), as follow:

### 2.1. The gradient method

The gradient algorithm [34] depends on the utilizing the numerical fourth horizontal gradient registered from the measured gravity anomaly utilizing filter of successive window lengths to evaluate the depth and shape of covered structures. The numerical fourth gradient gravity value at point x_{i} is figured from measured gravity data g(x_{i}) by:

where s is a window length or graticule spacing.

Also, the depth computed using the following form derived from the above equation:

### 2.2. The particle swarm optimization (PSO)

PSO-algorithm was created by [35]. It’s relying upon the reenactment of the apparent conducts of birds, fishes and insects in food searching. PSO-algorithm is applied in many issues, like model construction [36], biomedical images [37], electromagnetic optimizations [38] and hydrological problems [39]. In this calculation, the birds representing the particles or models, every molecule has a location vector which speak to the parameters esteem and a velocity vector. So, for a four-dimensional improvement issue, each molecule or individual will have a location in four-dimensional spaces which speak to a solution [40]. Each molecule changes its location at every movement of the operation of the algorithm, this location refreshed amid the iteration procedure considering the best location reached by the molecule which is called the T_{best} model and the best location obtained by any particle in the community called the J_{best} model, this refreshment is clarified in Eqs. (4) and (5) [41]

where *i* at the *k*th cycle, *k*th cycle, rand() is an arbitrary number in the vicinity of 0 and 1, c_{1} and c_{2} are positive constant numbers which ascendency the person and the sociable behavior, they are typically taken as 2 [41] yet some recent researches give that picking c1 more prominent than c_{2} however c_{1} + c_{2} ≤ 4 may give better outcomes [42], c_{3} is the inertial coefficient which control the velocity of the molecule, since the substantial esteems may shuffle the molecules to miss up the great arrangements and the small esteems may bring about insufficient place for exploration [41], it’s usually taken less than 1, *k*th cycle.

The four model parameters (z, A, x_{o}, and q) can be evaluated by using the PSO-algorithm to reach the misfit by using the following objective function:

where N is the number of data points,

### 2.3. Werner deconvolution method

Werner deconvolution method [21, 43] was also originally developed for magnetic interpretation. Also, Werner deconvolution has been used for gravity interpretation. The method is particularly useful when the profile anomaly of interest can be expressed as a rational function of the form of Eq. (1). As identified by [43], Eq. (1) can be rewritten in linear form as follow:

where

Eq. (7) is linear form in the four variables e_{1}, e_{2}, e_{3} and e_{4}, so that a numerically remarkable arrangement can be found for them from evaluating the equation at four points.

The Root Mean Square error (RMS) between the data and model responses is evaluated as follows

This is considered as a rule in evaluating the best-fitted model parameters (z, A, x_{o}, q) of the covered structure.

## 3. Synthetic example

Noisy-free gravity anomaly for a horizontal cylinder with A = 400 mGal m^{2}, z = 5 m, q = 1, m = 1 and profile length of 120 m. Our analysis begins by applying the fourth horizontal gradient separation technique (Eq. (2)) to the gravity anomaly utilizing distinctive s-values (s = 2, 3, 4 and 5 m) (Figure 2). By applying this inversion technique, we evaluated z and A values at different q for every s-value and after that ascertained the average depth and RMS (Table 1). Table 1 exhibits the estimation consequences of the interpretation of noise free data. The assessed parameters from the proposed technique are in a decent concurrence with the model of the horizontal cylinder where z = 5 m, A = 400 mGal m^{2} and q = 1. At long last, we can watch that the minimum RMS (RMS = 0 m) occurs at the true model parameters.

s (m) | Vertical cylinder model, q = 0.5 | Horizontal cylinder model, q = 1 | Sphere model, q = 1.5 | |||
---|---|---|---|---|---|---|

z (m) | A (mGal m) | z (m) | A (mGal m^{2}) | z (m) | A (mGal m^{3}) | |

2 | 3.9 | 465.8 | 5 | 400 | 4.0 | 2753.37 |

3 | 3.8 | 461.2 | 5 | 400 | 3.9 | 3447.5 |

4 | 4.1 | 447.2 | 5 | 400 | 3.9 | 3916.2 |

5 | 4.2 | 433.1 | 5 | 400 | 4.2 | 4230.3 |

Average | 4.0 | 451.9 | 5 | 400 | 4.0 | 3586.8 |

RMS (mGal) | 17.57 | 0 | 25.71 | |||

With 5% random noise | ||||||

2 | 4.2 | 465.8 | 5.1 | 400.0 | 4.1 | 2753.3 |

3 | 3.7 | 461.2 | 4.9 | 410.5 | 3.8 | 3447.5 |

4 | 3.8 | 447.2 | 5.2 | 417.7 | 3.7 | 3916.2 |

5 | 3.9 | 433.1 | 4.9 | 424.6 | 4.3 | 4230.2 |

Average | 3.9 | 451.8 | 5.0 | 413.2 | 3.9 | 3586.8 |

RMS (mGal) | 17.87 | 0.65 | 27.23 | |||

With 10% random noise | ||||||

2 | 4.0 | 351.7 | 5.0 | 479.4 | 3.9 | 2079.0 |

3 | 4.0 | 410.5 | 4.8 | 481.1 | 4.0 | 3068.7 |

4 | 4.0 | 406.3 | 4.7 | 432.9 | 3.9 | 3557.9 |

5 | 3.9 | 404.2 | 5.1 | 422.83 | 4.0 | 3947.8 |

Average | 3.9 | 393.2 | 4.9 | 479.4 | 3.9 | 3163.4 |

RMS (mGal) | 13.65 | 4.4 | 22.29 |

Because of the real data are tainted with random noise, random noise of 5 and 10% imposed on the gravity anomaly to see the effect of these noises on the inversion method. The fourth horizontal gradients were evaluated using the same s-values mentioned above (Figures 3 and 4). Table 1 also demonstrates the computational outcomes of the interpretation of noisy gravity data. The average depth of 5 m and the solution with minimum RMS (0.65 mGal) gives in case of 5% noise and depth 4.9 m and RMS of 4.4 mGal in case of 10% noise. This shows that this method is useful when applied to noisy gravity data. In addition, we use Werner deconvolution method to the same gravity anomaly utilizing the same window size every 2 m. we used 11 clustered solutions to calculate the average estimated depth is 5 m, A = 400 mGal m^{2}, and q = 1 with RMS = 0 mGal. Also, as mentioned above we use the same Werner deconvolution method for the noisy gravity anomalies. The average estimated depth of the cluster solutions is 5.3 m, A = 410.1 mGal m^{2} and q = 1 with RMS = 0.82 mGal in case of adding 5% random noise. Also, the average estimated depth of the cluster solutions is 5.6 m, A = 425.3 mGal m^{2} and q = 1 with RMS = 1.20 mGal in case of adding 10% random noise (Table 2).

Vertical cylinder model, q = 0.5 | Horizontal cylinder model, q = 1 | Sphere model, q = 1.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

z (m) | A (mGal m) | x_{o} (m) | z (m) | A (mGal m^{2}) | x_{o} (m) | z (m) | A (mGal m^{3}) | x_{o} (m) | |

Average | 4 | 458.37 | 0 | 5 | 400 | 0 | 3 | 3875.3 | 0 |

RMS (mGal) | 18.04 | 0 | 55.96 | ||||||

With 5% random noise | |||||||||

Average | 4.3 | 465.51 | 0 | 5.3 | 410.1 | 0 | 3.4 | 3884.4 | 0 |

RMS (mGal) | 17.60 | 0.82 | 42.82 | ||||||

With 10% random noise | |||||||||

Average | 4.5 | 468.26 | 0 | 5.6 | 425.3 | 0 | 3.6 | 3910.1 | 0 |

RMS (mGal) | 17.28 | 1.20 | 38.25 |

The PSO-algorithm was connected to the same synthetic gravity anomaly. In this circumstance, it is noise free data, so we start testing our technique using 100 models. The best model came after 700 cycles, the used extent of the parameters are showed up in Table 3. The assessed model parameters which control the body measurements are in good correlation with the proposed values (Table 3) corresponding to zero RMS. Since, the uproarious data considered as a basic part in geophysics, thusly, we applied our method to 5% arbitrary random noise gravity data caused by horizontal cylinder model appear with a particular true objective to inquire about the effect of noise corrupted data. The assessed indicate parameters (z, A, x_{o}, q) are presented in Table 3. Table 3 exhibits that the RMS error is 0.32 mGal. Plus, we forced 10% of subjective random noise on the comparable synthetic anomaly. Also, Table 3 demonstrates the inverted parameters and shows that the RMS error is 0.64 mGal.

Type of body | Parameters | Used ranges | Result | RMS (mGal) |
---|---|---|---|---|

Horizontal cylinder model | Without random Gaussian noise | |||

A (mGal m^{2}) | 100–700 | 400 | 0 | |

z (m) | 2–12 | 5 | ||

q | 0–3 | 1 | ||

x_{o} (m) | −20 to 50 | 0 | ||

With 5% random Gaussian noise | ||||

A (mGal m^{2}) | 100–700 | 395 | 0.32 | |

z (m) | 2–12 | 5 | ||

q | 0–3 | 1 | ||

x_{o} (m) | −20 to 50 | −0.01 | ||

With 10% random Gaussian noise | ||||

A (mGal m^{2}) | 100–700 | 411 | 0.64 | |

z (m) | 2–12 | 4.9 | ||

q | 0–3 | 1 | ||

x_{o} (m) | −20 to 50 | 0.02 |

## 4. Field example

So as to inspect the pertinence and effectiveness of the three showed methods on the real data, we have connected the three techniques to a gravity anomaly profile of Abu Roash dome area, the Northern Western Desert, Egypt (Figure 5). The Bouguer gravity map is situated in the West of Cairo ([44]; his Figure 11) and was mapped in 1980 by the Egyptian General Petroleum Corporation (EGPC) utilizing a density of 2.3 g cm^{−3}. The structure information is accessible from the surface geology and drilled hole data [45]. From the geology information of the area, we observe that the basement rocks (with greater prominent thickness than the above sedimentary layers) are elevated because of the high pressure in the SW direction [45]. At the Abu Roash dome, there are exposures of Cenomanian clastics at its core took after by Turonian and Senonian strata. This Cretaceous succession separated from the above Eocene sediments by an angular unconformity [45, 46, 47]. Figure 5 shows the Bouguer anomaly profile which are opposite to the heading of compression striking NW–SE, this profile was digitized at an interim of 300 m. The Bouguer anomaly accordingly acquired has been subjected to the three various methods (the fourth horizontal gradient method, Werner deconvolution method, and the PSO-technique).

Firstly, we used the fourth horizontal gradient method to four progressive windows (s = 600, 900, 1200 and 1500 m) to obtain the inverted model parameters. The four fourth horizontal gradient anomaly profiles were gotten (Figure 6). Table 4 summarized the results obtained from this method. Secondly, by applying Werner deconvolution method to the same observed gravity data, the outcomes are summarized in Table 5. Thirdly, a PSO-algorithm utilized to assess the interpretive model parameters of gravity anomaly profile. Table 6 displays the ranges and results of the evaluated parameters.

s (m) | Vertical cylinder model, q = 0.5 | Horizontal cylinder model, q = 1.0 | Sphere model, q = 1.5 | |||
---|---|---|---|---|---|---|

z (m) | A (mGal m) | z (m) | A (mGal m^{2}) | z (m) | A (mGal m^{3}) | |

600 | 1870 | −5236 | 2200 | −6160 | 2520 | −13,552,000 |

900 | 1890 | −5292 | 2250 | −6300 | 2600 | −14,175,000 |

1200 | 1920 | −5376 | 2510 | −7028 | 2830 | −17,640,280 |

1500 | 2050 | −5740 | 2630 | −7364 | 3290 | −19,367,320 |

Average | 1932.5 | −5411 | 2397.5 | −6713 | 2810 | −16,183,650 |

RMS (mGal) | 10.04 | 10.38 | 224 |

Vertical cylinder model, q = 0.5 | Horizontal cylinder model, q = 1.0 | Sphere model, q = 1.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

z (m) | A (mGal m) | x_{o} (m) | z (m) | A (mGal m^{2}) | x_{o} (m) | z (m) | A (mGal m^{3}) | x_{o} (m) | |

Average | 1870 | −5230 | 0 | 2320 | −6503 | 0 | 2750 | −17,253,450 | 0 |

RMS (mGal) | 10.06 | 10.39 | 246.99 |

Parameters | Used ranges | Result | RMS (mGal) |
---|---|---|---|

A (mGal m) | −2000 to −6000 | −5200 | 0.17 |

z (m) | 500–2500 | 1860 | |

q | 0–2.5 | 0.45 | |

x_{o} (m) | −100 to 100 | 0 |

Finally, the three inversion techniques give a full picture of the model parameters instead of various techniques which did not give a totally elucidation. The results are outlined in Table 7.

## 5. Conclusions

In this chapter, three various methods were used for modeling gravity anomaly due to simple geometrical shaped. The viability of the proposed methods (the gradient method, particle swarm optimization method and Werner deconvolution method) is used on a synthetic example including noisy-free data, contaminated data with various level of noise (5 and 10%), and a real field data from Egypt. The three approaches can enhance the quality solution and convergence traits and computational adequacy. The examination of the results with drilling information and published information detailed in the literature demonstrated the prevalence of the three methods and its potential for dealing gravity issue. Later on work, we will attempt to suggest some enhanced variant of these methods to deal with issue.

## Acknowledgments

We would like to thank Prof. Taher Zouaghi, the Editor, for his keen interest, imperative comments on the Article, and changes to this work. Also, we might want to express gratitude to Ms. Maja Bozicevic, Publishing Process Manager, for her help and collaboration in this issue.