Introduction

Gravity exploration method has traditionally been utilized to assess and describe the variant in subsurface rock densities and offer an important understanding sight of the subsurface geology (Nabighian et al. 2005; Abdelrahman et al. 2013; Rezaie 2019; Kumar et al. 2020; Gadirov et al. 2022; Essa and Diab 2023). The assessment of model parameters of buried source bodies is a vital step in gravity anomaly elucidation (Abdelrahman and Essa 2015; Essa and Géraud 2020; Ekinci et al. 2021). It is utilized to predict the geothermal potential reservoir parameters (i.e., location, depth) and also used in several geophysical investigations such as hydrocarbon exploration, cavities and tunnels detection, ore and mineral exploration, archeological sites investigation, and geological features or formations that exist beneath the surface of the Earth (Ekinci and Yiğitbaş, 2015; Essa et al. 2020, 2022; Dilalos and Alexopoulos 2020; Rosid and Saraswati 2020; Young et al. 2020; Guglielmetti and Moscariello 2021; Kheyrollahi et al. 2021; Chavanidis et al. 2022; Saleh et al. 2022; Wang et al. 2022).

There are several methods established to infer gravity data, i.e., Asfahani and Tlas (2012) presented an approach that combined Fair function minimisation and stochastic optimisation to inferring residual gravity anomalies for spherical and cylindrical model structures. Chakravarthi et al. (2016) developed two algorithms in the space-domain to scrutinize gravity anomalies for sedimentary basins by an exponential-density function. Tlas and Asfahani (2018) mentioned an easy interpretation method for gravity anomalies depending on the quadratic curve regression. Essa et al. (2020) described a fast imaging technique called “the R-parameter imaging technique” for inferring gravity data acquired along profiles. Essa et al. (2021c) developed an inversion algorithm based on the utilization of the analytic signal to investigate gravity profiles. Mehanee (2022) developed a nonlinear optimization algorithm for inverting the gravity profile for a 2D faulted thin block and was dependent on the simultaneous algorithm optimization.

Moreover, the utilization of metaheuristic optimization algorithms (particle swarm, simulated annealing, genetic, differential evolution, ant colony, cuckoo search, bat, …, etc.) has been used too much in the geophysical data inversion and given stable results. For example, Datta et al. (2012) used an ant colony algorithm to elucidate a total gradient potential field. Toushmalani and Saibi (2015) applied a cuckoo search for inverting 3D gravity data. Ekinci et al. (2016) proposed a useful use for differential evolution (DE) to interpret residual gravity anomalies. Essa and Munschy (2019) described a method that depends on using a particle swarm optimizer (PSO) for inferring second-moving average gravity anomalies. Rao and Biswas (2021) used simulated annealing to infer gravity anomalies due to 2-D faults. Essa and Diab (2023) used the Bat algorithm to infer 2-D gravity anomaly profiles for geothermal exploration and volcanic activity.

Using a deductive awareness of either density or geometry, it is possible to identify fault structures and reduce the complexity of parameter space for gravity inversion. This approach enables a distinctive solution to be obtained, which can help in understanding the subsurface geology and potential mineral resources. For instance, if the density of the fault zone is known from geological observations, this information can be used as a constraint for gravity inversion, which can then provide a better estimate of the position and geometry of the fault. Similarly, if the geometry of the fault is well-defined from seismic or other geophysical data, this information can be used as a constraint for gravity inversion, which can then provide a more precise guess of the density distribution within the fault zone. Overall, using a deductive approach to assessing fault structure can improve the reliability and accuracy of gravity inversion techniques, and ultimately help in making more informed decisions for resource exploration and management. The process of utilizing gravity data inversion to assess basement relief continues to be employed, despite limited information about basement depth from a small number of boreholes and no prior knowledge of density contrast. When it comes to solving these problems, global optimization algorithms have a significant edge over local optimization algorithms. Global optimization techniques can address complex criteria without requiring an initial model to be preset.

Structure assessment of dipping faults is also of significant concern in applied geophysics for adequately understanding the advancement of various geological features. Faults belong to the most valuable geological structures for regional studies and specific other explorations. Recognizing regional faulting and tectonic risks involves knowing the subsurface the structure and behavior of dipping faults, which is crucial for both scientific investigations and practical applications in the energy and mining industries. Therefore, many researchers designed optimization techniques for accurately predicting fault geometric features using global and local optimization algorithms (Roy and Kumar 2021; Nibisha et al. 2022; Roy et al. 2022).

In this paper, we explained how to infer 2D gravity anomalies for faults along profiles by determining their characteristic parameters through the usage of a particle swarm optimizer method (PSO) for the various-order of the gravity horizontal gradient. The benefit of using these gradients is fallen on eliminating the regional field effect. The efficacy of the current inversion algorithm was confirmed by different synthetic models with different scenarios. Practically, it is corroborated by two field data sets from USA and Tunisia. The results from this scheme were compared with available results from drilling, geology, and other used methods.

Methods

Forward modeling

Gravity anomaly along the profile in the Cartesian coordinate system for any geometric style represented by two-sided dipping faults (Fig. 1) is (Essa 2013; Essa et al. 2021a):

$$g\left( {x_{i} ,x_{o} ,z_{{{\text{up}}}} ,z_{{{\text{down}}}} ,M,\theta } \right) = M\left[ {1 + \frac{1}{\pi }\tan^{ - 1} \left( {\frac{{\left( {x_{i} - x_{o} } \right)}}{{z_{{{\text{up}}}} }} + \cot \theta } \right) - \frac{1}{\pi }\tan^{ - 1} \left( {\frac{{\left( {x_{i} - x_{o} } \right)}}{{z_{{{\text{down}}}} }}} \right) + \cot \theta } \right],$$
(1)

where \(x_{i}\) and xo (m) signify the locations of the collected measurement points and fault trace origin, M = 2\(\pi\)fΔσt \(\times \,10^{5}\) (mGal) and called the amplitude coefficient, t is the thickness (m) of the layer, f = 6.67 × \(10^{ - 11}\)\(\left( {\frac{{{\text{m}}^{3} }}{{{\text{kg}} \times {\text{s}}^{2} }}} \right)\), zup and zdown represent depths to upthrown downthrown parts, and θ is the dip angle of faults.

Fig. 1
figure 1

Schematic illustration shows a two-sided dipping fault source with its parameters

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Formulation of the inverse scheme

Using different orders of horizontal derivatives was considered a pioneering technique to diminish the effect of the regional gravity anomaly. For example, Essa and Munschy (2019) mentioned the first-order group. Here, we described along the profiles the third horizontal derivative anomaly (∆gxxx), which is given as

$$\Delta g_{xxx} \left( {x_{i} ,s} \right) = \frac{{g\left( {x_{i} + 3s} \right) - 3g\left( {x_{i} } \right) + 3g\left( {x_{i} } \right) - g\left( {x_{i} - 3s} \right)}}{{8s^{3} }}$$
(2)

and the fourth horizontal derivative anomaly (∆gxxxx) is

$$\Delta g_{xxxx} \left( {x_{i} ,s} \right) = \frac{{g\left( {x_{i} + 4s} \right) - 4g\left( {x_{i} + 2s} \right) + 6g\left( {x_{i} } \right) - 4g\left( {x_{i} - 2s} \right) + g\left( {x_{i} - 4s} \right)}}{{16s^{4} }}$$
(3)

where s represents the window length.

The particle swarm optimizer (PSO)

To deal with extremely nonlinear issues, recent trends employ evolutionary, stochastic heuristic and nature-inspired algorithms. Differential evolution, particle swarm optimizer, cuckoo search, and firefly algorithms are examples of nature-inspired algorithms that are simple and efficient at explaining a wide range of real-world issues.

In this paper, the applied algorithm is the particle swarm optimizer (PSO) was recognized by Eberhart and Kennedy (1995) and inspired by the bird’s search for food in nature. The birds presented particles that work in the search field to find the solution. More applications for this algorithm are found in Singh and Biswas (2016), Anderson et al. (2020), and Essa et al. (2022).

The process was initiated by randomly distributing particles for the swarm and then proceeds to search for sources through successive generations. At each step, the velocities and locations of the particles are adjusted using the following formulas:

$$v_{j}^{L + 1} = c_{3} v_{j}^{L} + c_{1} \,{\text{rand}} \,\left( {T_{{{\text{best}}}} - x_{j}^{L + 1} } \right) + c_{2} \,{\text{rand}}\,\left( {J_{{{\text{best}}}} - x_{j}^{L + 1} } \right),$$
(4)
$$x_{j}^{L + 1} = { }x_{j}^{L} + { }v_{j}^{L + 1} ,$$
(5)

where \({x}_{j}^{L}\) is the present location of the jth particle at the Lth iteration, \({v}_{j}^{L}\) signifies the velocity of the jth particle at the Lth iteration, random numbers amongst [0,1] has been used using the rand function, (\({c}_{1}\), \({c}_{2}\), and \({c}_{3}\)) are some controlling factors of the convergence. The parameter \({c}_{1}\) serves as a cognitive parameter that facilitates the individual exploration of the particles, \({c}_{2}\) is a social parameter that guides the particles to a global direction, and the inertial factor, \({c}_{3},\) adjusts the speed of particles. \({T}_{\mathrm{best}}\) is the best location, which is acquired by a model, while \({J}_{best}\) is the best global place reached by any particle in the swarm. Afterwards, the achieved best solution (\({T}_{best}\)) and the global best solution \({(J}_{\mathrm{best}})\) are stored in memory. The model’s velocity and place are updated through an iteration procedure, which terminates when the convergence.

Here, finding the global minimum for this scheme is done by applying a suitable objective function. Therefore, the following objective functional (Жobjective) is utilized to execute the dipping fault using the gravity data (M, zup, zdown, xo, and ϴ):

(6)

where u signify the number of data points, \(\Delta {g}_{j}^{\mathrm{obs}}\) represents the observed gravity field data and \(\Delta {g}_{j}^{\mathrm{calc}}\) represents the calculated gravity field data.

The flowchart of the procedures for estimating the model parameters is represented in Fig. 2 and in the incoming steps:

Fig. 2
figure 2

PSO scheme inversion process flowchart

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The gravity data of the profile is read as the first step of the procedure, then applying the different horizontal derivatives in the second step (Third, Fourth) to reduce the effect of the regional data with different orders (Eqs. 2, 3). Applying the code of the (PSO) particle swarm optimization will come as a third step to appraise the dipping fault parameters, which consider the best parameters of the fault, while the particles reach the global minimum by updating their velocity and location for each parameter. Finally, the forward model (Eq. 1) is developed using the derived parameters to investigate the discrepancy amongst the calculated and observed gravity anomalies.

The parameters of the tuning PSO scheme were selected according to various studies of several parameter groups (Fig. 3 and Table 1) and the suitable parameters are c1 = c2 = 2 and c3 = 0.9 (rapid convergence rate).

Fig. 3
figure 3

PSO-controls parameters (c1, c2, and c3) effect on rate of convergence

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Table 1 Model 1: Influence of the parameters c1, c2 and c3 on the convergence rate (Fig. 3)
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Synthetic models studies

To confirm the application of the proposed PSO approach in gravity data, several simulated data sets were examined to determine the subsurface model parameters as follows:

Model 1: effect of a regional field

The observed gravity profile (140 km) was created and consisted of a two-sided fault example (M = 390 mGal, zup = 6 km, θ = 40°, zdown = 9 km, and x0 = 70 km) with a third-order regional field background. It was examined in two manners as follows:

First, the third horizontal gradient was employed for this anomaly utilizing numerous s values (s = 2, 3, 4, 5, 6, 7, 8, 9, and 10 km) (Fig. 4a). The PSO scheme was used to evaluate the fault parameters (Table 2). The predicted parameters are M = 383.2 ± 5.67 mGal, zup = 6.77 ± 0.19 km, θ = 45.98 ± 0.94°, zdown = 8.47 ± 0.16 km, xo = 72.12 ± 0.75 km, and the root mean square of the error (RMS error) = 11.0187 mGal.

Fig. 4
figure 4

a Gravity anomaly represents Model 1 in case using a third horizontal gradient method. b Gravity anomaly represents Model 1 in case using a fourth horizontal gradient method. c Gravity anomaly represents Model 1 with a 10% noise in case using a third horizontal gradient method. d Gravity anomaly represents Model 1 with a 10% noise in case using a fourth horizontal gradient method

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Table 2 Model 1: Results of the PSO-inversion algorithm applied to third horizontal derivative anomalies of the gravity profile (140 km) due to composite anomaly of two-sided dipping fault model (\(M\) = 390 mGal, zup = 6 km, \(\theta\) = 40°, zdown = 9 km and \({x}_{0}\) = 70 km) and third-order regional
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Second, Fig. 4b displays the observed and expected parameters of the two-sided fault by applying the PSO to the fourth horizontal derivatives anomalies (Table 3) and the predicted parameters are M = 390.17 ± 1.07 mGal, zup = 6 ± 0 km, θ = 40.04 ± 0.16°, zdown = 9.01 ± 0.03 km, xo = 70 ± 0.03 km, and RMS error = 0.1599 mGal.

Table 3 Model 1: Results of the PSO-inversion algorithm applied to fourth horizontal derivative anomalies of the gravity profile (140 km) due to composite anomaly of two-sided dipping fault model (\(M\) = 390 mGal, zup = 6 km, \(\theta\) = 40°, zdown = 9 km and \({x}_{0}\) = 70 km) and third-order regional
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Third, Fig. 4c displays the observed and expected parameters of the two-sided fault via using the PSO to the third horizontal derivatives anomalies in case of 10% random noises (Table 2) and the predicted parameters are M = 410.81 ± 6.91 mGal, zup = 6.99 ± 0.27 km, θ = 49.73 ± 2.66°, zdown = 9.88 ± 0.23 km, xo = 74.55 ± 1.3 km, and RMS error = 26.9787 mGal.

Fourth, Fig. 4d displays the observed and expected parameters of the two-sided fault through utilizing the PSO to the fourth horizontal derivatives anomalies with 10% random noises (Table 3) and the predicted parameters are M = 394.54 ± 6.55 mGal, zup = 6.1 ± 0.39 km, θ = 41.56 ± 4.85°, zdown = 9.17 ± 0.35 km, xo = 70.08 ± 0.54 km, and RMS error = 12.6451 mGal.

The obtained results demonstrate that the suggested method can produce accurate fault parameters and is capable of minimizing the influence of regional background and noise.

Model 2: effect of interference structures

To study the influence of structures interference, a gravity profile of 140 km was produced for combined two-sided fault and spherical structures with M = 390 mGal, zup = 6 km, θ = 40°, zdown = 9 km and x0 = 70 km (fault) and zsphere = 3 km, Msphere = 510 mGal, xo sphere = 90 km (sphere) without and with 10% random noise. The interpretation procedures are done as the following consequence:

First, the third horizontal gradient was employed for this anomaly, applying s values equal to 2, 3, 4, 5, 6, 7, 8, 9, and 10 km (Fig. 5a). Then, the PSO scheme was engaged to guess the fault parameters (Table 4) and are M = 390.49 ± 2.29 mGal, zup = 6.02 ± 0.12 km, θ = 39.96 ± 0.63°, zdown = 9.04 ± 0.12 km, xo = 70.05 ± 0.18 km and RMS error equals 0.439 mGal for a noise-free anomaly. In case of including a 10% noise, Fig. 5b shows the observed and the expected anomalies, and the parameters are M = 396.34 ± 5.34 mGal, zup = 6.38 ± 0.29 km, θ = 41.84 ± 4.23°, zdown = 9.19 ± 0.36 km, xo = 70.09 ± 0.83 km, and RMS error = 14.0570 mGal (Table 4).

Fig. 5
figure 5

a Gravity anomaly represents Model 2 without noise in case using a third horizontal gradient method. b Gravity anomaly represents Model 2 with a 10% noise in case using a third horizontal gradient method. c Gravity anomaly represents Model 2 without noise in case using a fourth horizontal gradient method. d Gravity anomaly represents Model 2 with a 10% noise in case using a fourth horizontal gradient method

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Table 4 Model 2: Results of the PSO-inversion algorithm applied to third horizontal derivative anomalies of the gravity profile (140 km) due to composite anomaly of two-sided dipping fault model (\(M\) = 390 mGal, zup = 6 km, \(\theta\) = 40°, zdown = 9 km and \({x}_{0}\) = 70 km) and sphere structure (zsphere = 3 km, Msphere = 510 mGal, xo sphere = 90 km)
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Second, the fourth horizontal gradient was used for the same anomaly with the equivalent s values (Fig. 5c), and the deduced fault parameters are M = 390.43 ± 1.24 mGal, zup = 6.02 ± 0.14 km, θ = 40.65 ± 1°, zdown = 9.01 ± 0.12 km, xo = 70.06 ± 0.14 km, and RMS error = 0.79 mGal (in a case of noise-free) (Table 5). In addition, in case of including a 10% noise, Fig. 5d shows the observed and the derived anomalies, and the deduced parameters are M = 393.8 ± 7.16 mGal, zup = 6.2 ± 0.36 km, θ = 42.2 ± 3.6°, zdown = 9.19 ± 0.33 km, xo = 70.38 ± 0.59 km, and RMS equals 13.6824 mGal.

Table 5 Model 2: Results of the PSO-inversion algorithm applied to fourth horizontal derivative anomalies of the gravity profile (140 km) due to composite anomaly of two-sided dipping fault model (\(M\) = 390 mGal, zup = 6 km, \(\theta\) = 40°, zdown = 9 km and \({x}_{0}\) = 70 km) and sphere structure (zsphere = 3 km, Msphere = 510 mGal, xo sphere = 90 km)
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Model 3: effect of multi-faults (horst model)

A 140-km gravity anomaly was created through two neighboring two-sided faults consisting of a horst model with M1 = 410 mGal, zsh = 3 km, θ1 = 55°, z1 = 7 km, xo1 = 50 km and M2 = 460 mGal, zsh = 3 km, θ2 = 30°, z2 = 5 km, xo2 = 80 km (Fig. 6a). The interpretation procedures as declared above are:

Fig. 6
figure 6figure 6

a Gravity anomaly represents Model 3 without noise in case using a third horizontal gradient method. b Gravity anomaly represents Model 3 with a 10% noise in case using a third horizontal gradient method. c Gravity anomaly represents Model 3 without noise in case using a fourth horizontal gradient method. d Gravity anomaly represents Model 3 with a 10% noise in case using a fourth horizontal gradient method. e Error percentage in each calculated parameter through Model 1 without and with 10% random noise. f Error percentage in each calculated parameter through Model 2 without and with 10% random noise. g Error percentage in each calculated parameter through Model 3 without and with 10% random noise

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First (using the third horizontal gradient method), Fig. 6a displays the observed and predicted anomalies of the two neighboring faults with M1 = 410.21 ± 1.4 mGal, zsh = 2.99 ± 0.06 km, θ1 = 55 ± 0.15°, z1 = 7.06 ± 0.05 km, xo1 = 49.99 ± 0.01 km, M2 = 462.45 ± 1.91 mGal, z2 = 4.99 ± 0.06 km, θ2 = 29.98 ± 0.09°, xo2 = 80 ± 0.01 km (in case of noise-free) and RMS error is 2.6953 mGal. Moreover, in the case of a 10% noise included, Fig. 6b shows the observed and predicted anomalies with an RMS error of 27.36 mGal, and the deduced parameters are M1 = 414.55 ± 6.58 mGal, zsh = 3.11 ± 0.47 km, θ1 = 55.86 ± 3.34°, z1 = 7.06 ± 0.53 km, xo1 = 49.67 ± 0.81 km, M2 = 464.94 ± 6.24 mGal, z2 = 5.14 ± 0.4 km, θ2 = 31.93 ± 3.59°, xo2 = 79.94 ± 0.67 km (Table 6).

Table 6 Model 3: Results of the PSO-inversion algorithm applied to third horizontal derivative anomalies of the gravity profile (140 km) due to composite anomaly of two two-sided dipping fault models (M1 = 410 mGal, zsh = 3 km, z1 = 7 km, \(\theta\)1 = 55°, xo1 = 50 km) and (M2 = 460 mGal, zsh = 3 km, z1 = 7 km, \(\theta\)1 = 55°, xo2 = 80 km)
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Second (using the fourth horizontal gradient method), Fig. 6c shows the observed and predicted anomalies and their deduced parameters (M1 = 410.81 ± 1.11 mGal, zsh = 2.99 ± 0.06 km, θ1 = 54.98 ± 0.16°, z1 = 7.03 ± 0.07 km, xo1 = 50 ± 0.01 km, M2 = 461.23 ± 1.15 mGal, z2 = 4.99 ± 0.08 km, θ2 = 30.03 ± 0.19°, xo2 = 80 ± 0.04 km) and RMS error is 2.0266 mGal (Table 7). In addition, after adding a 10% random noise, Fig. 6d displays the observed and predicted anomalies with M1 = 412.49 ± 3.9 mGal, zsh = 3.08 ± 0.41 km, θ1 = 54.2 ± 3.92°, z1 = 6.93 ± 0.41 km, xo1 = 50.24 ± 0.84 km, M2 = 462.42 ± 6.72 mGal, z2 = 4.87 ± 0.43 km, θ2 = 31.44 ± 3.78°, xo2 = 79.28 ± 0.25 km) and RMS error equals 24.3930 mGal.

Table 7 Model 3: Results of the PSO-inversion algorithm applied to fourth horizontal derivative anomalies of the gravity profile (140 km) due to composite anomaly of two two-sided dipping fault models (M1 = 410 mGal, zsh = 3 km, z1 = 7 km, \(\theta\)1 = 55°, xo1 = 50 km) and (M2 = 460 mGal, zsh = 3 km, z1 = 7 km, \(\theta\)1 = 55°, xo2 = 80 km)
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The combination of input model parameters and outputs are the tactical factors. Sensitivity evaluation is an approach for evaluating the reliability of a model or a judgment. Numerous methods can be used to envision the global model and diminish outcome uncertainty (Feng 2021; Essa et al. 2022). In deciding on the finest control systems, the significance of each control factor (c1, c2, and c3) was examined (Fig. 3).

The metaheuristic optimization algorithm routine finally means selecting the population values based on the scale and complexity of the optimization task. Whereas the particle swarm algorithm works successfully and effectively with higher population density, the processing complexity grows, since more search spaces are found. The doubt and integrity of the suggested scheme are also explored using the previously mentioned three synthetic models with a 10% random noise and two case studies, as discussed in the following section. To avoid the doubt of the offered method utilizing different values of s windows and each s window through considerable numbers of iterations, then taking the average of the finest parameter values from the whole s windows solutions to get more reliable results.

Field examples

To investigate the rationality and the stability of the offered scheme for the buried fault parameters estimation, two published real data sets of case studies from the United States and Tunisia were used. The particle swarm optimization scheme was implemented to these data to accomplish the optimal fit for the buried fault parameters (M, zup, θ, zdown, xo). The results were then matched with existing geologic information and any additional geophysical outcomes.

Case 1: Seattle fault system, USA

The Puget Lowland is located inside a forearc region of a Cascade subduction region. It is confined by a Cascade Volcanic Arc, the old Mesozoic terrain, and the uplifted and quarried Olympic Mountains accretion complex. The depth of the Puget Lowland approaches 25–30 km and superimposes Siletz terrain, basalts, and invasive rocks, including island-arc formations. The bedrock beneath the tailing basin reached depths between 2.195 and 2.637 km. These bedrocks are basalt embedded in mafic rock interspersed with siltstone, tuff, conglomerate, and sandstone (Symons and Crosson 1997; Rau and Johnson 1999; Brink et al. 2002) (Fig. 7a). Figure 7a indicates the position of the examined area and observed gravity profile. It is crossed over a Seattle fault region, which comprises two steep strata deformed primarily through two or more faults (Johnson et al. 1994). The Seattle fault region has a placement from east–west, depth to subsurface bedrock, and the danger of metropolitan inhabitants’ concentration. It is a complex thrust, reversing fault with nearly 7 km width and 70 km length delimiting the northern margin of the Seattle uplift.

Fig. 7
figure 7figure 7

a Case 1: Geological map for the area including the Seattle Fault System, USA (after Brink et al. 2002). b Observed and predicted gravity anomalies deduced from using the third horizontal gradient method. c Third horizontal derivative anomalies using different s values of b. d Convergence rate of the PSO scheme after applying the third horizontal gradient method. e Observed and predicted gravity anomalies deduced from using the fourth horizontal gradient method. f Fourth horizontal derivative anomalies using the s values of b. g Convergence rate of the PSO scheme after applying the fourth horizontal gradient method

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Figure 7b shows the observed and estimated anomalies with M = − 51.49 ± 6.73 mGal, zup = 1.51 ± 0.1 km, θ = 38.55 ± 3.62°, zdown = 3.88 ± 0.37 km, xo = 27.26 ± 0.87 km) and RMS error equals 1.47 mGal (Table 8). Also, the third horizontal derivative anomalies using different values of s (s = 1 km, s = 1.5 km, s = 2 km, s = 2.5 km, s = 3 km, s = 3.5 km, s = 4 km, s = 4.5 km, s = 5 km) are shown in Fig. 7c. Moreover, the convergence rate of the PSO algorithm demonstrated in Fig. 7d.

Table 8 Results of the PSO-inversion algorithm applied to the third horizontal derivative anomalies of the Seattle fault, USA
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In case of utilizing the fourth horizontal gradient, Fig. 7e explains the observed and estimated anomalies with M = − 51.56 ± 3.24 mGal, zup = 1.52 ± 0.17 km, θ = 38.93 ± 2.97°, zdown = 3.86 ± 0.33 km, xo = 27.29 ± 0.97 km) and RMS error is 0.72 mGal (Table 9). In addition, the fourth derivative anomalies are displayed in Fig. 7f for the same s values declared above. Finally, the convergence rate is mentioned in Fig. 7g. A comparison results are demonstrated in Table 10.

Table 9 Results of the PSO-inversion algorithm applied to the fourth horizontal derivative anomalies of the Seattle fault, USA
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Table 10 Case 1: achieved results of the Seattle fault, USA
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Case 2: Bizerte Molassic Basin, Tunisia

Figure 8a, the study area includes a 1:50 km scale geological map of Metline and Porto-Farina, the oldest series of exposures are Triassic evaporates positioned close to the town of El Alia, along the El Alia fault. In the southern district of Metline city, the Upper Cretaceous is signified by “Cenomanian marly limestone, Senonian marls, and Middle Campanian–Lower Maastrichtian marl-limestone alternations and outcrops.” Green clays from the El Haria formation are a representative fossil of the Late Maastrichtian–Paleocene. At El Alia, Metline, and Menzel Jemil, huge outcroppings of Globigerina limestone correlate to the lower Eocene deposits. The Middle-Upper Miocene succession was signified through “marls, clays, sandstones, conglomerates, and gypsum layers” (El Ghali and Ben Ayed 2000). In addition, more geologic setting details are found in several published literature (Ben Ayed et al. 1979; Bejaoui et al. 2016; Zaghdoudi et al. 2021).

Fig. 8
figure 8figure 8

a Case 2: Geological map of the Bizerte Molassic Basin, South of Bizerte City, Northeastern Tunisia (after El Ghali and Ben Ayed 2000; Zaghdoudi et al. 2021). b Observed and predicted gravity anomalies deduced from using the third horizontal gradient method. c Third horizontal derivative anomalies using different s values of b. d Convergence rate of the PSO scheme after applying the third horizontal gradient method. e Observed and predicted gravity anomalies deduced from using the fourth horizontal gradient method. f Fourth horizontal derivative anomalies using the s values of b. g Convergence rate of the PSO scheme after applying the fourth horizontal gradient method

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Figure 8b explains the original and estimated anomalies with M = − 4 ± 0.79 mGal, zup = 0.31 ± 0.11 km, θ = 78.07 ± 3.59o, zdown = 0.94 ± 0.13 km, xo = 2.01 ± 0.13 km, and RMS error equals 1.00 mGal (Table 11). In addition, Fig. 8c shows the third derivative anomalies using different values of s (s = 0.028, 0.042, 0.056, 0.070, 0.084, 0.098, 0.112, 0.126, and 0.140 km). Figure 8d shows the convergence rate of the PSO algorithm engaged to third horizontal anomalies.

Table 11 Results of the PSO-inversion algorithm applied to the third horizontal derivative anomalies of the Bizerte Molassic Basin, Tunisia
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Finally, Fig. 8 shows the original and estimated anomalies with M = − 3.8 ± 0.78 mGal, zup = 0.32 ± 0.09 km, θ = 77.97 ± 4.33°, zdown = 0.95 ± 0.2 km, xo = 2.017 ± 0.15 km, and RMS error is 0.97 mGal (Table 12). Figure 8f displays the fourth gradient anomalies for similar s values. Moreover, Fig. 8g displays the rate of convergence of the PSO scheme used for fourth horizontal anomalies.

Table 12 Results of the PSO-inversion algorithm applied to the fourth horizontal derivative anomalies of the Bizerte Molassic Basin, Tunisia
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Discussion

Our results of synthetic investigations, which included the weight of the regional field, interference of neighboring structures, and complex fault system, explain that the utilization of a particle swarm optimizer incorporated with the horizontal gradient method (with different orders) can avoid the noise and regional anomaly in the collected gravity data and gives a respected view for deducing the subsurface two-sided fault parameters (the amplitude coefficient, depths of up- and downthrown, dip angle, and fault trace location). Furthermore, the sensitivity error in each parameter (M, zup, θ, zdown, xo) in all synthetic cases is demonstrated in Fig. 6e, f, and g including the effect of noise. Also, attained results for two field cases reflected the accuracy of the existing method.

Gravity anomaly in the Seattle Fault Region (Case 1: USA) for deducing its parameters (Fig. 7b, e) has been investigated. The attained results for this fault structure are matched with available geologic and geophysical information (Table 10). For example, the depths for up- and down-thrown deduced by Brink et al. (2002) equal 1.5 km and 3.5 km, respectively. In addition, depths deduced by Essa et al. (2021b) are 1.46 km and 3.73 km, respectively. Moreover, Anderson et al. (2020) postulated this fault as a one-side vertical fault and deduced the depth as 2.2 km. Therefore, our achieved results were compared with others and found acceptable uncertainties (Table 10). Besides, the gravity anomaly in the Bizerte Molassic Basin (Case 2: Tunisia) for deducing fault parameters (Fig. 8b, e) has been inspected. The accomplished results for this fault structure are matched with available geologic and geophysical information from Zaghdoudi et al. (2021). The patterns of the calculated gravity anomalies in the two cases matched well with the observed anomalies.

Finally, the motivation for exploiting the particle swarm scheme is to capitalize on its benefits in overwhelming the instability and non-uniqueness of the gravity anomalies inversion. In addition, the current algorithm can get a stable and acceptable accuracy for the parameters of fault and can be confirmed by available data from other published literature.

Conclusions

The use of the particle swarm optimizer scheme for identifying and deducing the third and fourth horizontal derivative anomalies is likely to be valuable in the geophysical investigation, because it has numerous advantages, including (1) eliminating the influence of deep-structure (regional anomaly), (2) eradicating the weight of neighboring structures and noise responses, and (3) accurately deducing the two-sided fault source parameters.

The proposed method is simple, automatic, and does not involve any graphical support. The synthetic examples investigated demonstrated that the suggested approach is stable in terms of neighboring effect and noise. It has additionally been successively exploited to two real-world data sets from the United States and Tunisia over faults structures and produced positive results. From the validation of the results, this method is robust and firm. Finally, the implementation of PSO to estimate buried fault parameters using real data sets from the United States and Tunisia provides evidence for the rationality and stability of the proposed scheme. The accuracy of the results obtained using PSO can be used to support informed decision making in any investment projects.