Abstract
Global optimization algorithm using very fast simulated annealing has been used for modeling of magnetic anomaly over various idealized geobodies for mineral/ore deposit investigation. The present technique shows that it can match the observed data very well assuming some simple geometrical bodies such as sphere, horizontal cylinder and thin dyke or sheet. Uncertainty associated with the modeling was also studied. It has been found that the obtained source parameters are strongly identical when all the parameters are optimized. The present study shows that amplitude coefficient depends on the shape factor. The relation between them shows a multimodel-type uncertainty. The shape factor modeled from several VFSA runs shows the different kind of subsurface structure. Constraining the shape factor gives reliable results and analysis of histograms and cross-plots also gives the utmost reliable results, which show a unimodel character. Inversion of synthetic noise-free and noisy data shows the worth of the present work. The present algorithm was also tested in five field case studies for mineral/ore exploration. Computation time for this present work is very minimal.
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Introduction
Elucidation of magnetic data is mostly used to identify different geological structures in various exploration studies. Such geological structures can be categorized as spheres, cylinders, thin dykes or sheets. For these types of structures, many techniques have been proposed and used for elucidation of magnetic field anomalies for simple geometrical models. These methods were tried to best estimate from magnetic observation values of different magnetic parameters, such as depth to buried body, amplitude coefficient, effective magnetization angle and location of body of magnetic anomalies produced by simple shaped structures. To model magnetic field data supposing stationary source geometrical models, many modeling procedures have been developed. For mainly parametric modeling, only geometrical shape is assumed and the different relevant parameters are modeled using various techniques. A review of magnetic anomaly modeling can be found in Nabighian et al. (2005).
Gay (1963, 1965) and McGrath (1970) have developed the curve matching techniques. Other elucidation methods that were developed include Fourier transform techniques, Hilbert transforms, monograms, characteristic points and distance approaches, least-squares residual anomalies (Bhattacharyya 1965; Grant and West 1965; Mohan et al. 1982; Prakasa Rao et al. 1986; Abdelrahman 1994; Abdelrahman and Sharafeldin 1996). Many linear and linearized inversions such as least squares, linearized least squares, normalized local wave number method, analytic signal derivatives, second-horizontal derivatives, Euler deconvolution method, simplex algorithm, fair function minimization have also been developed (McGrath and Hood 1973; Silva 1989; Salem and Ravat 2003; Salem et al. 2004; Salem 2005; Salem and Smith 2005; Tlas and Asfahani 2011a, b; Abdelrahman and Essa 2015; Tlas and Asfahani 2015). Global optimization methods such as simulated annealing, very fast simulated annealing, regularized inversion, particle swarm optimization and higher-order horizontal derivative methods (Gokturkler and Balkaya 2012; Sharma and Biswas 2013; Biswas and Sharma 2014a, b; Mehanee 2014a, b; Biswas 2015; Singh and Biswas 2015; Biswas 2016; Biswas and Acharya 2016; Ekinci 2016) have been effectively applied to solve nonlinear parametric inversion problems. A combined work of various other modeling methods can be found in Abo-Ezz and Essa (2016), Abdelrahman and Essa (2005) and Abdelrahman et al. (2003, 2009, 2012).
However, in general, the determination of certain parameters, namely depth, shape factor, amplitude coefficient and effective magnetization angle of buried geological structure, is accomplished by certain modeling or inversion methods as mentioned above. Therefore, the correctness of the results obtained by the above-mentioned methods relies on the precision by which the anomaly can be separated from the observed magnetic anomaly. In the present work, a fast and efficient modeling method using the very fast simulated annealing (VFSA) global optimization algorithm was applied for modeling magnetic field anomalies. The method was also applied to determine the correct estimate of all model parameters and uncertainty associated with the modeling. The VFSA optimization is a robust optimization technique that can examine a broad model space. The resolution for this method is very accurate, and it has the ability not to get trapped in local minima (Sharma and Biswas 2011; Sharma 2012; Sen and Stoffa 2013; Sharma and Biswas 2013). The present technique can be applied to noise-free data, noisy data and field data for single structure as well as multiple structures in deciphering subsurface structures for mineral exploration, with the slightest uncertainty in the final model.
Methodology
Forward Modeling
The general equations for magnetic anomaly m(x) for idealized geobodies in a Cartesian coordinate system are given below (Fig. 1).
Diagrams of cross-sectional views, geometries and parameters of (a) sphere, (b) an infinitely long horizontal cylinder, (c) thin dyke and (d) thin sheet
For sphere, the equation can be written as (Rao et al. 1977; Prakasa Rao and Subrahmanyam 1988):
For horizontal cylinder, the equation can be written as (Prakasa Rao et al. 1986):
For thin dyke, the equation can be written as (Gay 1963; Atchuta Rao et al. 1980; Abdelrahman and Sharafeldin 1996):
For thin sheet, the equation can be written as (Gay 1963):
In all of the above equations, k is the amplitude coefficient, z is the depth from the surface to the center of the body (sphere, cylinder) and depth from the surface to the top of the body (dyke and thin sheet), x i (i = 1,…, N) is the horizontal position coordinate, θ is the effective magnetization angle or the index parameter, and q is the shape factor. The shape factor for sphere is 2.5, horizontal cylinder 2, and thin dyke and thin sheet 1.
For multiple structures, the equation can be written as (Biswas and Sharma 2014a):
where m j (x i ) is the magnetic anomaly at x i location for jth body and M is the number of bodies.
Inversion
Various global optimization methods have been developed in the past based on different important principles. One important principle is the Boltzmann’s law of statistical mechanics. The important aspect of Boltzmann’s law is how to accomplish the lowest energy state. The main objective of geophysical inversion is to apply the same Boltzmann’s law and to minimize an objective function or the error function in geophysical data modeling. Various optimization methods such as simulated annealing (SA), genetic algorithms (GA), artificial neural networks (ANN), particle swarm optimization (PSO) and differential evolution (DE) (El-Kaliouby and Al-Garni 2009; Monteiro Santos 2010; Sharma and Biswas 2011; Sen and Stoffa 2013; Sharma and Biswas 2013; Biswas 2015; Ekinci 2016; Ekinci et al. 2016; Balkaya et al. 2017) were regularly used to optimize geophysical data and have been applied to derive diverse geophysical information (Rothman 1985, 1986; Dosso and Oldenburg 1991; Zhao et al. 1996; Martínez et al. 2010; Li et al. 2011; Sharma 2012; Sen and Stoffa 2013). Sen and Stoffa (2013) discussed in detail the SA. It draws analogy from an idealized physical annealing process. The VFSA is a variant of SA, the fundamental basis of both SA and VFSA is the analogy to warming of metal in a heat bath, and as it is later continuously allowed to chill, it tempers into a condition of minimum energy. The same analogy is applied for geophysical optimization. Unlike SA, the VFSA is fast, reliable, takes very less memory and has a very high resolution but needs precise fine-tuning of the parameters (Ingber and Rosen 1992). A detailed description of VFSA can be found in the above-mentioned literature. For VFSA global optimization technique, it requires a search range \( (P_{i}^{\hbox{min} } \;{\text{and}}\;P_{i}^{\hbox{max} } \)—minimum and maximum value of ith parameter or model space) for each model parameter, and during the process each parameter is optimized within the search range to find the best model that fits the observed response well. The misfit error (φ) between the observed and model response is taken as (Sharma and Biswas 2013):
where N is amount of data points, v 0 i and v c i are the ith observed and model data, v 0max and v 0min are the maximum and minimum values, respectively, of the observed data. The above objective function is taken so as to avoid the problem of data near zero crossing, which affects (increases) the objective function. The entire process of finding the misfit error has been discussed in many literature such as Sharma (2012) and Sharma and Biswas (2013).
For finding appropriate models in this study, calculations were executed at 2000 distinctive temperature levels with 50 moves (Nv) at one temperature level. The VFSA technique is iterated 10 times, and 10 appropriate results were obtained. Global mean model and uncertainty investigation have been carried out based on the different sampling techniques developed by Mosegaard and Tarantola (1995) and Sen and Stoffa (1996). Global model, probability density function (PDF), covariance and correlation matrices and uncertainty investigation can be found in previous work, namely Sharma (2012) and Sharma and Biswas (2013).
The present algorithm for VFSA has been coded in Window 8 environment by means of MS FORTRAN Developer Studio. An easy computer with core i7 processor and with 2 GB RAM has been used. The calculation procedure for a single structure is almost 35 s. For multiple structures, the whole computation process takes about 50 s. A flowchart (Fig. 2) for the whole VFSA process is shown below.
Flowchart of VFSA algorithm
Results for Hypothetical Models
The VFSA inversion was executed using noise-free and noisy synthetic data (10% uniformly random noise; i.e., multiplied by an arbitrary draw between 1 and 1.10) and 20% Gaussian noise (i.e., multiplied by a Gaussian random value with mean and standard deviation as 1 and 0.2) for a sphere, horizontal cylinder, thin-dyke and thin-sheet-type structures. Firstly, model parameters were inverted for every synthetic data. Next, q was constrained to the true value of structural feature, as mentioned earlier, and inversion was repeated. However, to know the strength of the method and to evaluate the precise model parameters, the search space or ranges for every model parameters were varied. Initially, the search space/range for every parameter was kept large depending on the true value \( (P_{i}^{\hbox{min} } \;{\text{and}}\;P_{i}^{\hbox{max} } \)—see tables) and the misfit for every parameter was computed. Next, after ensuring that the interpreted value of each parameter was within the search range, the search ranges were reduced within the probable range (minimum and maximum value for each model parameter derived from first run) so as to find out the more plausible solution with very least uncertainty. For all the different synthetic and noisy models, an appropriate search range for every model parameter was chosen and one VFSA inversion was executed. Next, convergence of every model parameters such as k, x 0, z, θ and q was studied along with the misfit of each parameter. After studying the proper convergence and φ, 10 VFSA inversions were executed. Next, histograms for each model parameter were arranged by means of accepted models whose φ is lower than 10−4. Statistical mean model was also calculated with models that have φ less than 10−4. The statistical mean models that lie within one standard deviation (1σ) from the mean were selected for further analysis.
Sphere (Noise-Free and Noisy Data: Model 1)
Synthetic data (Table 1) were generated for a spherical model using Eq. 1. The convergence and their φ are shown in Figure 3a. Figure 4a shows the histograms were set using all accepted models whose φ is less than 10−4. From Figure 4a, it can be visualized that four model parameters demonstrate broad-ranging solutions where k deviates from a wide range with its peak around 3000 nT. Likewise, z and q also vary over a range with histogram peaks close to their true values for every model parameter. Table 1 shows that k has a large uncertainty. Other parameters are fairly close to the true value but are not situated with the expected ambiguity. The correlation matrix shown in Table 2 has been calculated from the finest shaped models that are lying within 1σ of the mean. This unveils that a solid connection exists between diverse model parameters, and it demonstrates that the model parameters are mutually dependent and cannot be resolved very precisely.
Convergence pattern for various model parameters and misfit: (a) gravity data; (b) magnetic data
Histograms of all accepted models with misfit <10−4 for noise-free synthetic data and PDF (q unconstrained): (a) sphere—Model 1; (b) horizontal cylinder—Model 2; (c) thin dyke—Model 3; and (d) thin sheet—Model 4
To avoid such undependable results, at first the histograms of q in Figure 4a were examined and discovered that q shifts from 2.25 to 2.75, which indicates that the optimization algorithm shows a spherical structure with crest close to 2.5. Next, q was constrained to its real value and the optimization was repeated. Again, the convergence and φ for every model parameter were studied (Fig. 3b). The histograms of every single obtained model with φ less than 10−4 are shown in Figure 5a, which uncover that the histograms take after the Gaussian distribution. Consequently, the PDF for every model whose φ is less than 10−4 is calculated and overlaid on the corresponding histogram (Fig. 5a). At last, models with every parameter within the PDF of 1σ from the mean are chosen to figure out the statistical mean model and associated ambiguity (Table 1). Figure 6a presents an examination among the observed and the desired model response. Table 1 describes the elucidated mean model and associated ambiguity. Table 3 shows the correlation matrix registered for models within 1σ from the mean, while q is constrained. The correlation matrix demonstrates that k is distinctly linked with depth (z) and negatively correlated with location of the body (x 0). This implies that if k expands, z ought to increase and x 0 must decrease.
Histograms of all accepted models with misfit <10−4 for noise-free synthetic data and PDF (q constrained): (a) sphere—Model 1; (b) horizontal cylinder—Model 2; (c) thin dyke—Model 3; and (d) thin sheet—Model 4
Fit among observed and model data for sphere—Model 1 using: (a) noise-free synthetic data; (b) random noisy synthetic data; and (c) Gaussian noisy synthetic data
Next, VFSA inversion was performed utilizing 10% random noise (Table 4). The inversion was repeated for noise-free data. Statistical mean model was processed and is shown in Table 1. Huge ambiguity has been detected in k. The outcomes have been slandered within the sight of noise in contrast with the inversion of noise-free data. The estimated q was 2.46, which delivers an unverifiable appraisal for amplitude coefficient as φ gets higher. The estimated q acquired from noisy data demonstrates a spherical body. In this manner, q was constrained at 2.5, as the noise-free data, and inversion was performed again. The values of φ of elucidated models vary from 1–10−3 to 10−4 (10% random noisy data). Models with φ less than 0.01/0.02 were chosen for statistical investigation of occurrence of noisy data. The histograms for noisy data are not presented here, but they are like the ones in Figure 4a. The mean model shown in Table 4 for noisy data demonstrates that the true model lies outside the assessed uncertainty. Figure 6b shows an examination among the observed and the mean model for noisy data. To check the high power of noise-corrupted data, 20% Gaussian noise was added to the data to test whether the VFSA can really recover the genuine model parameters and to ponder the impact of various sorts of noises. The obtained mean model parameters are given in Table 5. Figure 6c demonstrates the observed and the desired model responses for the data with 20% Gaussian noise. Again, uncertainty exists as with noisy data as well as noise-free data.
Horizontal Cylinder (Noise-Free and Noisy Data: Model 2)
Forward model was produced for horizontal cylinder using Eq. 2 (Table 6). Inversion was executed for all types of synthetic information as it was accomplished for Model 1. Histograms of established models with φ less than 10−4 after 10 VFSA runs unmistakably demonstrate that q focuses toward 2.01, portraying a horizontal cylinder structure (Fig. 4b). Along these lines, q is constrained to its true value at 2.0 and 10 VFSA runs were executed. Histograms and PDFs for models with φ less than 10−4 are shown in Figure 5b. The mean model was processed utilizing the models as part of elevated PDF area of the model space and is illustrated in Table 6. Examinations among the observed and mean models are demonstrated in Figure 7a. The correlation matrix uncovers a comparative nature to that portrayed in Tables 2 and 3 for Model 1. VFSA inversion was executed for data with 10% random noise, and data with 20% Gaussian noise and mean models were determined. Tables 7 and 8 display the elucidated mean models and ambiguity when q was varied between 0 and 3 and when constrained at 2.0. Figure 7b and c shows the correlations among the noisy observed and mean model data.
Fit among observed and model data for horizontal cylinder—Model 2 using: (a) noise-free synthetic data; (b) random noisy synthetic data; and (c) Gaussian noisy synthetic data
Thin Dyke (Noise-Free and Noisy Data: Model 3)
Forward response was modeled for thin dyke using Eq. 3 (Table 9). Inversion was executed for all data as executed earlier. The histograms of accepted noise-free models demonstrate that q focuses toward 1.01, indicating a thin-dyke-like structure (Fig. 4c). Therefore, q was constrained at 1.0 and 10 VFSA runs were executed. Figure 5c demonstrates the histograms and PDFs. The mean model was registered utilizing the models as part of elevated PDFs of the model space and is displayed in Table 9. The genuine model was situated inside the evaluated uncertainty in the mean model. Figure 8a provides for examination among the observed and mean models. The correlation matrix uncovers a similar nature to that shown in Tables 2 and 3 for the sphere model. VFSA inversion was further executed for data with 10% random noise and synthetic data with 20% Gaussian noise. Tables 10 and 11 exhibit the mean model parameters and uncertainty. Figure 8b and c provides for an examination among the observed and mean model responses for noisy data.
Fit among observed and model data for thin dyke—Model 3 using: (a) noise-free synthetic data; (b) random noisy synthetic data; and (c) Gaussian noisy synthetic data
Thin Sheet (Noise-Free and Noisy Data: Model 4)
Forward response was modeled for thin sheet using Eq. 4 (Table 12). Inversion was executed for all data as it was accomplished for the other models. Figure 4d shows the histograms of models with φ less than 10−4 after 10 runs. After VFSA inversion, q focuses toward 1.01, indicating a thin-sheet structure. The histogram and PDFs are shown in Figure 5d. The mean model was figured utilizing the models as part of elevated PDF of the model space and is exhibited in Table 12. Figure 9a portrays an association among the observed and mean model. VFSA inversion was additionally carried out for data with 10% random noise and data with 20% Gaussian noise, and the mean model for each datasets was registered. Tables 13 and 14 display the deciphered parameters as well as uncertainty estimated when q was kept varied between 0 and 3 and fixed at 1.0. Figure 9b and c portrays an assessment between the observed and mean model responses.
Fit among observed and model data for thin sheet—Model 4 using: (a) noise-free synthetic data; (b) random noisy synthetic data; and (c) Gaussian noisy synthetic data
Multiple Thin Dykes (Noise-Free and Noisy Data: Model 5)
Forward response was modeled for thin dyke using Eq. 3 and for multiple structures using Eq. 5 (Table 15). The procedure for modeling thin dyke for a single structure was also applied here. To model the multiple structure data using three targets, first q was kept unconstrained and then it was constrained in similar way as for thin dyke. However, in this case all the structures were modeled altogether. The obtained mean model parameters are given in Table 15. The observed and mean models are illustrated in Figure 10a and b. Figure 10c shows the subsurface structure considered for this model.
Fit among observed and model data for multiple thin dykes—Model 5 using: (a) and (d) noise-free synthetic data; (b) random noisy synthetic data, (c) and (f) subsurface structure; and (e) Gaussian noisy synthetic data
Another multiple structure model was taken, and 20% Gaussian noise (Table 16) was added to check the nature of the noisy data. Inversion was executed in the same way as for multiple structures. The obtained mean model parameters are given in Table 16. The observed and mean models are illustrated in Figure 10d and e. Figure 10f shows the subsurface structure for this multiple model.
Multiple Structures (Noise-Free and Noisy Data: Model 6)
Forward response was modeled for multiple structures (sphere, dyke, cylinder and sheet) using Eqs. 1–4 and for multiple structures using Eq. 5 (Table 17), and then, 20% Gaussian noise (Table 18) was added to the response. The model was chosen to check whether the multiple models can be achieved with the present method. Inversion was executed for both types of data as discussed earlier. To elucidate these multiple structure data using four geobodies, first the shape factor was kept unconstrained and then it was constrained to its actual value in the same way as for multiple thin dykes, and the structures were modeled together. Table 17 shows the obtained mean model parameters. The observed and mean model responses are illustrated in Figure 11a and b. Figure 11c shows the subsurface structure considered for this model.
Fit among observed and model data for multiple structures—Model 6 using: (a) noise-free synthetic data; (b) Gaussian noisy synthetic data; and (c) subsurface structure for multiple bodies
Cross-Plot Analysis
Sphere (Noise-Free and 10% Random Noisy Data: Model 1)
Cross-plots among the parameters k, z and θ (Fig. 12a) were created for all established models whose φ is less than 10−4 (gray). Models within 1σ of the mean (i.e., models with elevated PDF) (black) are shown in Figure 12a when q is unconstrained. The cross-plots show that for a particular z and θ, k yields a wide range of results when q was unconstrained. Models that are within elevated PDF show extremely minute range, which suggests that the mean model parameters are extremely close to the true model parameter when q was constrained (Fig. 12b). From the cross-plots (Fig. 12a and b) and Table 1, it can be seen that the uncertainty estimated for z and θ is negligible; however, uncertainty still remains in k, which shows a large variation. The cross-plots among k, z and θ for noisy data (Fig. 12c) show a similar nature as the noise-free model. However, for noisy data, Figure 12c shows the models with φ less than 0.01 (10% random noise) (gray) and models with high PDF (black) when q was unconstrained. Cross-plots for noise-corrupted data show that the models in elevated PDF’s are constrained close to the true value when q was constrained (Fig. 12d).
Sphere Model 1 cross-plots among: (a) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (b) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for noise-free data; (c) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (d) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for random noisy data
Horizontal Cylinder (Noise-Free and 20% Gaussian Noisy Data: Model 2)
The cross-plots for horizontal cylinder show similar situation as Model 1 and are shown in Figure 13. The cross-plots among k, z and θ reveal the same patterns as for noise-free data for sphere when q was unconstrained and constrained (Fig. 13a and b) and Gaussian noisy data (Fig. 13c and d). Figure 13b and d reveals that the models within elevated PDF region (black) are constrained close to the true value of the model parameters when q was constrained.
Horizontal cylinder Model 2 cross-plots among: (a) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (b) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for noise-free data; (c) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (d) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for random noisy data
Thin Dyke (Noise-Free and 10% Random Noisy Data: Model 3)
Figure 14a shows cross-plots among the model parameters k, z and θ when q was unconstrained. The cross-plots among k, z and θ show the same nature as when q was unconstrained and constrained (Fig. 14a and b) and random noisy data (Fig. 14c and d). Figure 14b and d reveals that the models within elevated PDF region (black) are constrained close to the true value of the model parameters when q was constrained.
Thin-dyke Model 3 cross-plots among: (a) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (b) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for noise-free data; (c) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (d) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for random noisy data
Thin Sheet (Noise-Free and 20% Gaussian Noisy Data: Model 4)
The cross-plots, which depict similar nature as for sphere, cylinder and dyke, are shown in Figure 15. The cross-plots among k, z and θ show the same nature as when q was unconstrained and constrained (Fig. 15a and b) and Gaussian noisy data (Fig. 15c and d). Figure 15b and d reveals that the models within elevated PDF region (black) are restricted close to the true value of the model parameters when q was constrained.
Thin-sheet Model 4 cross-plots among: (a) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (b) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for noise-free data; (c) amplitude coefficient (k), depth (z), magnetization angle (θ) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is unconstrained; (d) amplitude coefficient (k), depth (z), shape factor (q) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for random noisy data
Field Examples for Mineral Exploration
To display the efficiency of the VFSA, six field examples of magnetic anomaly data were modeled. Four field examples represent single structures, and two examples represent multiple structures associated with ore bodies.
Bankura Anomaly, India
This field example deals with magnetic anomaly associated with spherical mass of gabbroic anomaly from Bankura, West Bengal, India (Verma and Bandopadhyaya 1975). Interpretation of these field data has been done by numerous researchers (Verma and Bandopadhyaya 1975; Prakasa Rao and Subrahmanyam 1988; Abdelrahman et al. 2007; Asfahani and Tlas 2007; Abdelrahman et al. 2012; Abdelrahman and Essa 2015) assuming a spherical body. The field data were obtained by digitizing at 62.5-m interval (Fig. 16) and modeled using the present inversion method.
Fit among the field and model data for the Bankura Anomaly, India
An appropriate search range (Table 19) was selected for these field data. Next, 1 and 10 solutions were derived using VFSA inversion. Modeling of the field data suggests a shape factor of 2.52. This advocates a 3-D spherical model. Afterward, q was constrained to the actual value for sphere as 2.5 and inversion was repeated as well as 10 VFSA inversion runs were executed. Model parameters whose φ is less than 0.01 were chosen to calculate PDF. Models inside elevated PDF area were used to calculate the mean model and associated ambiguity. The obtained results and assessment with other recently available results can be seen from Table 19.
The estimated depth in the current study is 1.44 km, which is quite excellent as compared with the previous results (Table 19). Comparison of various modeling methods reveals that the present approach is as reliable as the other methods. The estimated misfit was quite low for these field data. However, in most of the other modeling results, some parameters were not determined. The fits in the present study also illustrate that the elucidated model in this work is strongly matched to those in previous studies (Fig. 16). Assessment of elucidated outcome by diverse methods reveals that the current method is in excellent agreement with other modeling methods; however, none of the other modeling methods show the error of their estimated parameters. The amplitude coefficient k and the location of the body x 0 are imperative parameters that should be determined together with other model parameters. The cross-plots among k, z and θ (Fig. 17a) show similar nature as illustrated by the synthetic example for sphere.
Cross-plots among amplitude coefficient (k), depth (z), magnetization angle (θ) for all models with misfit < threshold (10−4 for noise-free data) (gray) and models with PDF > 60.65% (black) when q is constrained for the (a) Bankura Anomaly, India, (b) Parnaiba Anomaly, Brazil, and (c) Pima Copper deposit, Arizona, USA
Parnaiba Anomaly, Brazil
This field example is from Parnaiba basin, Brazil (Silva 1989). The data were acquired above a Mesozoic dyke that intruded into Paleozoic sedimentary rocks (Fig. 18). The data were digitized at 0.4-m interval. Several researchers have modeled the same anomaly using different techniques (Silva 1989; Abdelrahman and Sharafeldin 1996; Asfahani and Tlas 2004, 2007; Abdelrahman and Essa 2015; Tlas and Asfahani 2015) assuming a horizontal cylinder as well as thin-sheet structure. It was modeled by means of the present inversion.
Fit among the field and model data for the Parnaiba Anomaly, Brazil
An appropriate search range for every parameter was selected (Table 20). The inversion procedure was repeated for these field data. The q derived was 2.02, suggesting a horizontal cylindrical. Next, q was constrained at 2.0 and inversion was repeated.
The estimated depth of the body was 3.5 m, which is quite excellent as compared with past results (Table 20). However, the values of q interpreted by Silva (1989), Abdelrahman and Sharafeldin (1996), Asfahani and Tlas (2004, 2007) and Abdelrahman and Essa (2015) were considered as thin sheet. In the current results, which are in well agreement with those of Tlas and Asfahani (2015), the φ is low. Evaluation among the field data and modeled data is illustrated in Figure 18. The cross-plots among k, z and θ (Fig. 17b) show a similar nature as the synthetic data for horizontal cylinder.
Pima Copper Deposit, Arizona, USA
This field magnetic anomaly example, taken from a 750-m-long profile above the Pima Copper mine, Arizona, USA (Gay 1963), is caused by a thin dyke. The data were digitized at 6.25-m interval. Gay (1963), Abdelrahman and Sharafeldin (1996), Asfahani and Tlas (2004, 2007), Abdelrahman and Essa (2015) and Tlas and Asfahani (2015) modeled these data using different techniques assuming a thin-dyke model. The same data were re-modeled using the discussed method.
The q value obtained was 1.01, which can be fitted with either dyke or sheet. However, after studying the estimated uncertainty and the fit among the observed and modeled data, it is finally concluded that the structure was a thin dyke. Next, q was constrained at 1.0 and inversion was repeated. Table 21 represents the final results.
The depth estimated was 68 m, which is quite good as compared with previous results (Table 21). Figure 19 shows the fit among the observed and mean model, which is quite excellent compared to those using other modeling methods. The results from the present study are good and comparable with the other results. The estimated φ is quite low. Figure 17c shows the cross-plots among k, z and θ.
Fit among the field and model data for the Pima Copper deposit, Arizona, USA
Chromite Ore, Tangarparha, Odisha, India
This field example was taken from a chromite ore body as a single structure (Mandal et al. 2015a). The mineralized body is situated at 205 m below the surface (Mandal et al. 2015a). The data for the main anomaly were digitized at 23-m interval. The data were again modeled using the VFSA technique. The procedure for the data modeling was again carried out.
The optimization results point toward a spherical-shaped body, and hence, it was modeled to be a spherical-shaped body. From the earlier literature, it is known that individual chromite ore bodies at this locality have a pod-type structure, which is closer to a spherical body. Figure 20 shows the fit among the observed and mean model, and the modeled results are given in Table 22. To further understand the robustness of the method and to model multiple structures, one more profile was taken from the same study area (Mandal et al. 2015a), as there are three main anomalous zones. The first zone was a negative anomaly, the second a high peak positive anomaly and the third again a negative anomaly. Interestingly, the inversion results point toward a single spherical-shaped body for all the structures and hence the data were modeled to represent a single spherical body as before. However, the data were also modeled with the other structures and failed to give good results considering cylinder, dyke and sheet. The results were erroneous, and the values of φ among the observed and modeled data were too high and very poor. Finally, the anomaly was modeled as three spherical bodies and results were excellent (Table 23). Figure 21a shows the fit among the observed and mean model, and Figure 21b shows the subsurface structure derived from the obtained results.
Fit among the field and model data for the chromite ore, Tangarparha, Odisha, India (profile 1—single body)
Chromite ore, Tangarparha, Odisha, India (profile 2—multiple bodies): (a) fit among the field and model data; and (b) subsurface structure
Uranium Ore, Beldih Mine, Purulia, West Bengal, India
This field example is from a uranium ore body from Beldih mine, Purulia, West Bengal (Mandal et al. 2013), representing a multiple structure. The main anomaly data were digitized at 3.5-m interval, and the data were modeled using the VFSA technique. The discussed procedure was also followed for the modeling of these data. Initially, the data were modeled considering multiple structures; however, in all the cases, the inversion results point toward a dyke-like body. Inversion results also suggest low φ for dyke compared to other structures. Finally, the data were modeled assuming thin-dyke-like bodies. It is also interesting to note that the uranium ore bodies in the area correspond with a vertical thick-sheet-like structure as modeled from self-potential anomaly (Biswas and Sharma 2014a; Biswas et al. 2014a, b; Biswas and Sharma 2016). This was also confirmed from drilling results (Katti et al. 2010), which show that the mineralization starts from near surface and extends to the subsurface at a depth of around 10–20 m and which are nearly vertical and dipping northerly to southerly. The anomaly was also well characterized from other geophysical investigations using gravity and resistivity data (Mandal et al. 2013; Sharma et al. 2014; Mandal et al. 2015b). However, since a vertical thick-sheet-type structure appears as a dyke-like structure but not a thin-sheet-type structure, the data were finally modeled considering a dyke-like structure as equivalent to a thick-sheet-type structure. Since the data 100 m further away from the anomaly are flat, those parts were excluded for modeling and the remaining three-peak anomaly zone was modeled. The observed and obtained mean model shows an excellent match for a multiple structure (Fig. 22a), and the subsurface structure was derived from the obtained model (Fig. 22b). Table 24 shows the modeling results. For comparison, the drilling result from Katti et al. (2010) is shown in Figure 23.
Uranium ore, Beldih mine, Purulia, West Bengal, India: (a) fit among the field and model data; and (b) subsurface structure
Borehole drilled at the profile shows the presence of uranium associated with quartz–magnetite–apatite rocks (from Katti et al. 2010)
Discussion
From the experiments carried out in this study for magnetic anomaly using all synthetic data for different structures, it has been established that, to get consistent results, the q has to be optimized first. By performing several VFSA runs, one can obtain the near probable shape factor. Next, from earlier runs, an approximate value of q can be obtained and the near probable q must be constrained to its true value and the other four parameters must be inverted. When q is also modeled jointly with other parameters, the ambiguity of the results is very high. However, x 0 and z can be determined well (Tables 1, 6, and 9) with little uncertainties. When q is constrained, the other parameters are resolved very precisely. This has been shown from the analysis of cross-plots. However, in principle, when q is constrained, the search range can be reduced to find out the best mean model for each parameter. This has been shown in the literature for other geophysical datasets as well (Biswas 2015; Biswas and Sharma 2015). Moreover, correlation matrices disclose strong correlation among model parameters when q is constrained. This suggests that all model parameters are dependent on each other and cannot be resolved precisely. However, correlations between all parameters turn out to be very small when q is unconstrained. Thus, optimization with constrained q is stressed here for modeling of magnetic data as well. It is noteworthy that the shape can be known from either local geology or the anomaly map, and the probable depth can be guessed as well from the half width of an anomaly. Even if such conditions are known or can be guessed, modeling must be executed in dual steps as described here in order to achieve consistent appraisal of model parameters as well as actual authentication of q of the structures. It is also important to mention that even if only slight noise is added to noise-free data the actual model is not identified exactly. In such condition, the expected ambiguity shows the precision of the result. It has been demonstrated that for all types of synthetic data, the true model is positioned inside the estimated ambiguity. However, the type of ambiguity remains the same for all synthetic data as well.
It must also be highlighted that a comparison of Tables 19, 20, and 21 reveals that the horizontal location ‘x 0’ of a body is an important model parameter, which many other researchers failed to model (Abdelrahman et al. 2007; Asfahani and Tlas 2007; Abdelrahman et al. 2012; Abdelrahman and Essa 2015). This suggests that these previous studies interpreted the central peak of an anomaly as the location of the body, which may not be accurate always. A minor inappropriate approximation of any one of the model parameters can influence other model parameters as well, which perhaps could be the main reason for different assessments of model parameters made by different authors (Tables 19, 20, and 21). Moreover, the amplitude coefficient (k) is one of the most important parameters that was not determined by Abdelrahman and Essa (2015). However, there is strong correlation between the amplitude coefficient (k) and depth (z) as indicated by the cross-plots in the present study. This is another reason why the depth modeled by Abdelrahman and Essa (2015) does not match with the present results and with those determined by other methods as mentioned in the field examples. However, in the present work using VFSA, φ is less compared to other studies. Although it has not been compared to results by different objective functions used in other studies, it is noteworthy that the cross-plots (Figs. 12–15 and 17) show that the number of models needed to compute the mean model is very small as they are centered on a very small region (black) of the model space.
Conclusions
A proficient technique is proposed for the elucidation of magnetic anomaly caused by idealized subsurface geobodies utilizing very fast simulated annealing (VFSA) global optimization strategy in mineral or ore deposit investigation. The VFSA inversion technique is sufficiently able to accurately derive various appropriate models in a highly multidimensional space. Moreover, the type of uncertainty in the elucidation processes has also been studied and the results show that, while inverting every single model parameter, the VFSA method produces precise results. It has been seen from the inversion that the shape factor q is very vital in deriving consistent solutions of various alternative parameters. The investigation of uncertainty demonstrates that a little alteration in q delivers a huge change in the assessed amplitude coefficient (k). In such condition, incorrect appraisals of different model parameters are likewise obtained. It has been demonstrated that the inversion strategy can choose all the model parameters precisely when q is constrained to its real value. Subsequently, understanding of magnetic anomaly information is completed by following a dual-step inversion methodology as discussed here. Firstly, all the model parameters were streamlined and the parameters were modeled by keeping the shape factor unconstrained. Next, the inversion is achieved after the initial step coordinates the estimation of shape factor around 2.5, 2.0 and 1.0 for sphere, cylinder and dyke or sheet. Then, the shape factor is constrained to 2.5, 2.0 or 1.0 depending on the shape of the anomaly and other model parameters are also inverted. Hence, the most consistent results have been achieved and uncertainty in the elucidation has likewise ended up being nonsignificant. In this way, the mean model processed from the models within elevated PDF gives mainly predictable results with minimum ambiguity. The viability of this methodology has been effectively confirmed, tried and exhibited utilizing noise-free and noise-corrupted data. The quality of this strategy for viable application in mineral investigation has likewise been effectively demonstrated in five field case studies with difficult geological structures and depth. These field case studies have demonstrated the efficiency of the proposed technique, especially if obtained inverse parameters are interpreted in an integrated manner using known geologic information, or even when a priori information is lacking. The estimated magnetic inverse parameters for irregular ore bodies are likewise modeled paying little consideration to the depth of a body. The evaluated parameters in the field case studies are strongly consistent with various inversion and modeling methods and log data information from previous studies. The obtained inversion results are consistent as well with various and numerous subsurface structures, which demonstrate the vitality of the proposed method for utilization in mineral or ore deposit investigation.
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Acknowledgments
The author thanks the Editor-in-Chief, Prof. John Carranza and two anonymous reviewers for their comments and suggestions that helped to improve the quality of the manuscript. The author also acknowledges the necessary facilities and support from the Director of Wadia Institute of Himalayan Geology, Dehradun, to complete this work.
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Biswas, A. Inversion of Source Parameters from Magnetic Anomalies for Mineral/Ore Deposits Exploration Using Global Optimization Technique and Analysis of Uncertainty. Nat Resour Res 27, 77–107 (2018). https://doi.org/10.1007/s11053-017-9339-2
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DOI: https://doi.org/10.1007/s11053-017-9339-2