Identification of intraterrane dislocation zones and associated mineralized bodies is of immense importance in exploration geophysics. Understanding such structures from geophysical anomalies is challenging and cumbersome. In the present study, we present a fast and competent algorithm for interpreting magnetic anomalies from such dislocation and mineralized zones. Such dislocation and mineralized zones are well explained from 2D fault and sheet-type structures. The different parameters from 2D fault and sheet-type structures such as the intensity of magnetization (k), depth to the top (z1), depth to the bottom (z2), origin location (x0), and dip angle (θ) of the fault and sheet from magnetic anomalies are interpreted. The interpretation suggests that there is uncertainty in defining the model parameters z1 and z2 for the 2D inclined fault; k, z1, and z2 for the 2D vertical fault and finite sheet-type structure; and k and z for the infinite sheet-type structure. Here, it shows a wide range of solutions depicting an equivalent model with smaller misfits. However, the final interpreted mean model is close to the actual model with the least uncertainty. Histograms and crossplots for 2D fault and sheet-type structures also reveal the same. The present algorithm is demonstrated with four theoretical models, including the effect of noises. Furthermore, the investigation of magnetic data was also applied from three field examples from intraterrane dislocation zones (Australia), deep-seated dislocation zones (India) as a 2D fault plane, and mineralized zones (Canada) as sheet-type structures. The final estimated model parameters are in good agreement with the earlier methods applied for these field examples with a priori information wherever available in the literature. However, the present method can simultaneously interpret all model parameters without a priori information.

Geophysical exploration employing magnetic data helps study the subsurface properties by measuring the variation of the geomagnetic field primarily caused due to the presence of mineral bodies and different geological structures [1]. Such variation of magnetic field intensity formed due to contrast of magnetic susceptibility between the host rocks and the target mineralized bodies or the structures and can be used to limit the depth, location, orientation, geometry, and magnetic moment of the subsurface structure [25]. Magnetic data has also been used for various applications such as oil-bearing zone detection [6], dike location [7], buried metallic objects [8], archeological investigation [911], geotechnical engineering [12, 13], cave detection [1416], geothermal exploration [1719], solid waste landfill [20], basement depth [21], and buried igneous intrusions [22]. Inversion of magnetic data for a complex subsurface structure is ill-posed and so inherently tough to clarify as they do not have a unique solution and might be erroneous [2326]. In such cases, the problem of various types of uncertainty, data density, noises, etc., is inherent and could not be well determined [27] and hence leads to misinformation about the model parameter estimation, and the uncertainty remains in the final model parameters. Thus, simple geometric or idealized structures reduce the uncertainty and give the best results [5, 28, 29].

Various interpretation techniques have been developed for magnetic data interpretation considering the idealized structures such as a sphere, cylinder, and dike. Such methods include monograms [30], theoretical curve matching techniques [31, 32], characteristic curve approaches [3337], least-squares means [38], Hilbert transforms [39], Euler deconvolution [4043] derivatives from analytic signals [44], Werner deconvolution [45, 46], and fair function minimization approaches [47]. Other derivative-based approaches, including the steepest descent, Gauss–Newton, and Levenberg–Marquardt, are commonly used to interpret magnetic data [4854]. However, this interpretation needs some required knowledge about the different variables and geological information to give a reliable estimate of the model parameters [55]. Hence, the conventional method needs some a priori knowledge, without which the final solutions often get trapped in local minima than global minima. Due to its limitations in delineating an effective solution, global optimization or metaheuristic optimization is necessary for finding an optimal solution that does not need an initial guess of the model parameters and can give the best result. Global optimization such as the genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE) algorithm, very fast simulated annealing (VFSA), genetic-price algorithm (GPO), and whale optimization algorithm (WOA) has been applied to numerous geophysical applications such as seismic data [5658], self-potential data [5973], and also in the interpretation of gravity and magnetic data [3, 7492].

In exploration or crustal studies, fault and sheet-type structures are an essential feature that has been extensively interpreted for mineral, oil, and gas exploration and crustal modeling. Magnetic interpretation of a fault is usually measured as a 2D semi-infinite horizontal slab completed by either a vertical or dipping structure [34]. The magnetic anomaly over a thin sheet with finite and infinite extent is usually applied for mineral exploration studies. However, both the 2D fault and the thin sheet are very close in terms of structure and presence of mineralized bodies as the thin sheet and mineralized deposition in the fault or dislocation zones are similar. Delineation of the depth, depth extent, and the angle of the plane of a fault and sheet-type structure from magnetic data have given strong attention. However, in many cases, the parameters are not well resolved, which leads to uncertainty in final model estimation (fault or sheet), and are not well studied. Hence, the present work is focused on interpreting the different parameters from 2D fault and sheet-type structures from magnetic anomalies and the uncertainty estimation for every model parameter for finding reliable results and slightest uncertainty. As discussed in [92], we have applied the very fast simulated annealing (VFSA) inversion technique to model 2D fault and sheet-type structures. It has been theoretically proved that the algorithm must be applied in noise-free synthetic data with different degrees of noise for such studies. The uncertainty of all model parameters is also studied using the histogram analysis, 2D and 3D crossplots, and three field examples from intracontinental vis-à-vis intraterrane dislocation zones from Australia, deep-seated dislocation zones from India, and mineralized zones from Canada.

2.1. Forward Technique

The following forward equations give the measured magnetic anomalies over 2D fault and sheet-type structures.

2.1.1. 2D Fault

The magnetic effect for a 2D fault (Figure 1(a)) is given by the following equation [36, 93]:
where k is the intensity of magnetization, z1 is the depth to the top, z2 is the depth to the bottom of the fault, x0 is the origin location, and θ is the dip angle of the fault. For the vertical fault, the dip angle will be 90°.

2.1.2. 2D Sheet with Finite Depth Extent

Following [34], the magnetic effect of a 2D sheet with finite depth extent extending infinitely in the horizontal direction (Figure 1(b)) is given as
where k is the magnetic constant of the sheet, x0 is the location, z1 is the depth to the top, z2 is the depth to the bottom of the sheet, and θ is the polarization angle.

2.1.3. 2D Sheet with Infinite Depth Extent

Following [34], the vertical magnetic effect of a 2D sheet with infinite depth extent extending infinitely in the horizontal direction (Figure 1(c)) is given as
where k is the magnetic constant of the sheet, x0 is the location, z is the depth to the top, and θ is the polarization angle.

2.2. Inversion Technique

Inversion of geophysical data needs a globally best solution within several local optima. Hence, to achieve the best solution, the global search algorithm is the best to find out the optimized solutions in a controlled manner. Geophysical data inversion is ill-posed, and nonlinear models are persistent in numerous studies, and their results always require a global search process. Many global optimization methods such as the simulated annealing (SA), genetic algorithm (GA), artificial neural network (ANN), particle swarm optimization (PSO), differential evolution (DE), whale optimization algorithm (WOA), and hybrid optimization algorithm such as genetic-price algorithm (GPA) have been successfully applied in the interpretation of various geophysical data [1, 29, 89, 90]. In the present work, a modified version of the SA named the very fast simulated annealing (VFSA) global optimization algorithm has been used to interpret the magnetic anomalies over 2D fault and sheet-type structures. The fundamental idea behind this algorithm comes from the analogy of chemical thermodynamics or the heat bath algorithm [94]. VFSA has been applied in numerous geophysical interpretations, and the detailed process of the technique can be well understood after [74, 95]. Since the details are available in such literature, for brevity, it is not explained here and can be seen in detail after [95]. The significant advantage of VFSA is that it can negate the problem of linear inversion and the stability and robustness of the method. It can also help resolve the issue of nonuniqueness of different parameters and have a good resolution of the data. Moreover, it takes a minute time to complete the inversion process even if there are more than 106 models, less CPU time, and less memory and gives a high-resolution data interpretation [96].

Every inversion technique needs to find out the error in the final interpretation. Hence, for this inversion technique, the error estimation is taken after [62].
where N is the number of data, Mi0 and Mic are the ith observed and model responses for magnetic data, and Mmax0 and Mmin0 are the maximum and minimum values of the magnetic data.

The global solution is always necessary to interpret any geophysical data [1, 29, 97]. Hence, the procedure developed by [98, 99] is used for the present study. A detail of these techniques can also be found in various literature [62, 95]. Also, to see the uncertainty behind the interpretation of every model parameter, the correlation between each parameter, and the probability density function (PDF), we followed the work of [83, 100]. For brevity, it is not discussed here. These ideas are well explained in the above literature and were applied in many geophysical data such as gravity, magnetic, and self-potential data [29, 72, 91, 92]. The algorithm was developed in the Windows 10 operating system using the MS Fortran Developer Studio on a desktop PC with an i7 Intel Pentium Processor (9th Generation) and 16 GB RAM. The computation time (not CPU) for a single run is 3 seconds, and for 10 runs, it is 22 seconds. A flowchart for the entire VFSA optimization process is shown in Figure 2.

3.1. Synthetic Example

3.1.1. Search Space and Fine-Tuning of Parameters

Finding a globally best-fit model or solution requires an initial search space for all model parameters. In the present study, initially, the search space is defined, and a single run for the inversion process is completed. After studying the interpreted model parameters, whether it has been computed within the search space, the search space is again reduced within a probable range to get the best-fit models with less uncertainty in model parameters. Next, the convergence pattern for all parameters is studied to see whether all the model parameters are close to the actual value and the error minimization process in the final interpretation (Figure 3). Finally, ten runs are performed to delineate the mean model. Next, the interpreted data for all parameters whose errors are below 10-4 are taken for statistical investigation, which falls inside one standard deviation. Different models and actual values were selected, and the inversion process was repeated with the help of noise-free data and different degrees of Gaussian noise (with mean 1 and standard deviation 0.2).

3.1.2. Model 1 (Inclined Fault)

The initial model for a 2D inclined fault-type structure (k=100 nT, x0=75 m, z1=10 m, z2=30 m, and θ=60°) was formed using Equation (1) (Figure 4(a)). The inversion process discussed above was executed, and the parameters for such model (k, x0, z1, z2, and θ) were interpreted (Table 1). Next, histograms were prepared from the analyzed data to understand that every parameter is interpreted precisely (Figure 5(a)). Subsequently, 10% noise was corrupted in the noise-free data, and the inversion technique was repeated. Histograms were also set for noisy data (Figure 5(b)). From the interpretation of this data, it is visualized that the VFSA inversion can precisely delineate all the model parameters. Figures 4(a) and 4(b) illustrate the observed and model responses for both the noise-free and noisy data.

3.1.3. Model 2 (Vertical Fault)

A different model for a vertical fault-type structure (k=50 nT, x0=80 m, z1=15 m, z2=40 m, and θ=90°) was taken, and the inversion process is repeated considering noise-free and noisy data (10%) (Figures 6(a) and 6(b)). It can be understood from the histogram analysis that the inversion process can precisely delineate every model parameter (Figures 7(a) and 7(b)). Figures 6(a) and 6(b) demonstrate the observed and model responses for both the noise-free and noisy data. The final interpreted parameters are given in Table 2.

3.1.4. Model 3 (Finite Sheet)

This sheet-type model (k=100 nT, x0=250 m, z1=10 m, z2=20 m, and θ=60°) was taken where both the top and the bottom of the sheet can be interpreted, and the model was prepared using Equation (2). The inversion method was implemented for these sheet-type model parameters (k, x0, z1, z2, and θ), and the model responses for noise-free and noisy data (10%) are demonstrated in Figures 8(a) and 8(b). The histogram for this type of structure was also achieved, and the interpreted data show that it can delineate the model parameters correctly (Figures 9(a) and 9(b)). The final elucidated model parameters are shown in Table 3, and the observed and model responses are illustrated in Figures 8(a) and 8(b).

3.1.5. Model 4 (Infinite Sheet)

The alternate sheet-type model (k=10 nT, x0=250 m, z=10 m, and θ=60°) was taken where the depth to the end of the sheet is at infinity using Equation (3). This infinite-depth extent of the sheet-type model is also interpreted using the inversion process considering both the noise-free and noisy data (20%) (Figures 10(a) and 10(b)). Additionally, the histogram of all model parameters was also interpreted to see that the model parameters are close to the initial value (Figures 11(a) and 11(b)). The final fittings between the observed and model responses for noise-free and noisy synthetic data are shown in Figures 10(a) and 10(b), and the elucidated model parameters are shown in Table 4.

3.1.6. Uncertainty Analysis

To comprehend the relation between the model parameters, its uncertainty in explaining the precise parameters, and how various models come near to the initial (actual) value, a 2D crossplot investigation was designed for all models. For the 2D inclined fault, parameters such as k, z1, z2, and θ were taken for such analysis. Figure 12(a) shows the various plots between k, z1, z2, and θ, and the parameters are near their actual value for noise-free data (Model 1). However, from the crossplots, the parameters such as z1 and z2 show a small wide range in the plots. From these crossplot data, the parameters for this model are close to the actual data (blue), and the final mean model parameters are within the uncertainty value and lie within the high PDF (red). Figure 12(b) shows the same for noisy data as discussed above for noise-free synthetic data, suggesting that the parameters z1 and z2 show some uncertainty in the final interpretation.

For the 2D vertical fault model (Model 2), all the parameters (k, z1, z2, and θ) were selected to perceive the connection between them. Figure 13(a) demonstrates the relationship between parameters k, z1, z2, and θ for noise-free data, and Figure 13(b) reflects the relationship between k, z1, z2, and θ for noisy data. In both cases, it can be well understood that the parameters k, z1, and z2 show a broad distribution (blue), but the final mean model is in the high PDF zone (red) with minimum uncertainty.

Hence, from the study of crossplot analysis for 2D inclined and vertical faults, it can be understood that the depth to the top of the fault and bottom end of the fault plane provides some uncertainty for the inclined fault plane. The intensity of magnetization and the depth to the top and bottom show some uncertainty for the 2D inclined fault.

As discussed above for the 2D fault plane, crossplots for the 2D sheet-type structure with finite depth (k, z1, z2, and θ) are also studied for noise-free data and noisy synthetic data (Model 3). Figure 14(a) shows the crossplots for noise-free data, and Figure 14(b) shows those for noisy data. Here also, it can be seen from the crossplots that the parameters k, z1, and z2, show a widespread solution, but the final estimated parameters are within the expected uncertainty.

As discussed above that for the 2D vertical fault plane, three parameters show some uncertainty; hence, we have carried out a study to see a 3D crossplot for the parameters k, z1, and z2. Figures 15(a) and 15(b) show the 3D crossplots for noise-free and noisy data for the 2D vertical fault plane. It can be understood from the crossplot that all three parameters show some uncertainty (yellow) which are equivalent models with lower misfits. However, the final mean model is near the actual value and within the high PDF (red) but with a slightly wide range of solutions. This advocates that a small uncertainty prevails between these parameters, although the final mean model is near the actual value.

Finally, to understand any uncertainty related to the 2D sheet-type structure with infinite depth extent (k, z, and θ) (Model 4), all parameters (k, z, and θ) were selected for this study. Figure 16(a) demonstrates the 3D crossplot for noise-free data, and it can be seen that the parameters (k and z) show some widespread solutions (yellow) and are equivalent models with lower misfits. The final mean model parameters are within expected uncertainty and in the area of the high PDF (red). Figure 16(b) displays the noisy data where the results are the same.

3.2. Field Example

For theoretical modeling of geophysical data, it must be confirmed using field data for such types of structures to show the efficacy of the inversion method to delineate the model parameter estimation and further the uncertainty in such parameters. Field examples are always linked with different types of noises [24, 25]. To validate the theoretical models, three field data were taken from various literature, and the inversion procedure was carried out and compared with the previous results and drilling data.

It must be mentioned that without prior geological knowledge of the subsurface structure, interpretation of the subsurface structure from geophysical data would lead to erroneous results. For example, the presence of mineralized zones within the subsurface structure such as fault/dislocation/fracture zones or sheet-type structures cannot be well delineated if the geological understanding is poor. In such a case, mineralized bodies present in the fault or simple sheet-type mineralized bodies cannot be well determined and can lead to wrong information of the subsurface. Hence, it is recommended that with a priori knowledge of the local geology, it will clarify the subsurface structure and help determine the structure precisely. Though assessment and possibility in magnetic data interpretation are generally acknowledged, the idea behind the interpretation of the field data is to visualize the subsurface structure related to 2D inclined and vertical and 2D finite and infinite sheet-type structures and their geological implications.

3.2.1. Perth Basin, Australia

The field case was taken from the total field magnetic anomaly data from the Perth basin in Australia (Figure 17(a)). It is an elongated northwestern basin which is a rift structure formed during intracontinental setting lying in the southwestern margin part of the Carnarvon Basin [101]. This basin is well known for hydrocarbon exploration, where more than twenty oil fields are found. Based on the geological information and borehole data, the magnetic anomaly was observed as a deep-seated N-S striking fault-like structure [101]. The length of the profile is 40 km, and it was digitized with an interval of 1 km. The magnetic anomaly was interpreted using the inversion procedure, and the parameters such as k, x0, z1, z2, and θ were interpreted. The earlier investigation advises that the fault plane is inclined; however, to confirm the same, the data was analyzed considering both the inclined and vertical faults (using Equation (1)). The results from the inversion suggest that the fault plane is an inclined fault plane with the parameters estimated as k=184.3 nT, x0=20.7 km, z1=7.8 km, z2=11.8 km, and θ=36.2°. Uncertainty analysis was also carried out in this field example (Figure 18), and it also shows that there is a small, varied solution for the parameters k, z1, and z2; however, the estimated mean model is within the limits of uncertainty. This field anomaly was also interpreted by different methods [29, 47, 70, 89, 101104], and the results are summarized in Table 5. The observed and model responses and the subsurface structure are shown in Figure 17(a).

3.2.2. Dehri Aeromagnetic Anomaly, Bihar, India

The field example was taken from a deep-seated fault from the southwest part of Dehri, Bihar, India (Figure 17(b)). The area is enveloped by the Vindhyan group of rocks and is connected to the Bijawar group of rocks. The anomaly was digitized, and 17 data points were taken for the inversion. Earlier literature suggests that the fault plane is an example of a vertical fault [32, 47, 101, 104]. The present inversion technique was applied to interpret the magnetic anomaly data, and the parameters are estimated as k=283.0 nT, x0=32.0 km, z1=10.9 km, z2=27.2 km, and θ=40.3°. The interpretations from other techniques are summarized in Table 6, and the observed and model responses are illustrated in Figure 17(b).

3.2.3. Pishabo Lake Anomaly, Ontario, Canada

The total magnetic anomaly data as a field example (Figure 19) was taken over an outcropping olivine diabase dike of gabbroic composition from Pishabo lake, Ontario, Canada [105]. The field anomaly was digitized at an interval of 22 m. The estimated parameters from this field data are k=140563.2 nT, x0=1.6 m, z=324.2 m, and θ=37.9°. The field data was also interpreted by different forward equations [3] and different interpretation and inversion techniques [4, 105, 106]. The estimated parameters are in good agreement with the other results and are summarized in Table 7. Uncertainty analysis was also carried out for this field example, and it also shows that the parameters k and z show some uncertainty, but the estimated final model is within the limits of uncertainty (Figure 20). The field data and the model responses are shown in Figure 19.

Interpretation of magnetic anomalies has a vast range of utility, from crustal studies such as intraterrane dislocation zones to mineral exploration. The choice of specific methods in the interpretation of magnetic anomalies is vital for such studies. However, the understanding of different parameters for magnetic data is also crucial to delineate accurate subsurface information. In the present study, magnetic anomalies from 2D inclined and vertical faults, as well as the 2D thin sheet with finite and infinite depth extent, have been studied. The parameters such as the depth to the top and bottom of the fault and sheet (z1 and z2), location (x0), magnetic intensity/constant (k), and dip angle (θ) have been studied, and the uncertainty estimation has also been studied for each parameter. The results show that the magnetic intensity (k) and depth from the top and bottom (z1andz2) show some uncertainty for the 2D fault-type structure, and the magnetic constant (k) and depth to the top (z) offer a broad solution for the sheet-type structure with infinite extent. Although these parameters (z1 and z2) are vital for the delineation of subsurface structures, the final model is close to the actual values. The inversion results are shown with noise-free data and noisy Gaussian data with validation from three field examples from intracontinental/intraterrane dislocation zones (Australia), deep-seated dislocation planes (India), and mineralized zones (Canada). The results are in good agreement with other interpretation methods. The efficiency of this method is that it can be used to interpret all parameters of subsurface structures and can be well applied in a multifaceted geological environment. Furthermore, the present inversion does not require prerequisite information such as geology and structures within the subsurface. However, it would be better to resolve the subsurface structure if a priori information is available for all inversion/optimization methods.

The data will be available upon request.

Highlights. The global optimization algorithm examines magnetic anomalies. Model parameter estimation of synthetic and noisy data for 2D fault and sheet-type structures is demonstrated. 2D magnetic fault and sheet-type structures are well demarcated by inversion of field data. Uncertainty estimation for model parameters is well demonstrated. Interpretation of magnetic anomalies from intraterrane zones and exploration is presented.

This work forms a part of the Ph.D. thesis of KR.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The entire work has been carried out by both authors.

KR would like to thank the Council of Scientific and Industrial Research (CSIR), New Delhi, for the research fellowship. AB would like to thank the Institution of Eminence (IoE), Banaras Hindu University, Varanasi, for the seed grant to pursue this work.

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