Introduction

In most of the geophysical exploration problems, it is assumed that a geological structure that can be characterised passably by different sheet type structures. The model is frequently used in both gravity as well as magnetic interpretation to find the depth and other parameters of geological structures. Appraisal of the depth of a buried structure from the gravity and magnetic data has drawn considerable attention in exploration of minerals (Biswas et al. 2014a, b; Mandal et al. 2015, 2013). Wide interpretation procedures have been developed to interpret the gravity and magnetic data assuming fixed source geometrical models. In almost all the cases, these methods consider the diverse parameters of the buried body being a priori assumed, and the parameters may thereafter be obtained by different interpretation methods.

Many interpretation techniques were developed in the past and many new inversion methodologies are also present in the recent times. The techniques include graphical methods (Nettleton 1962, 1976), curves matching standardized techniques (Gay 1963, 1965; McGrath 1970), Fourier transform (Odegard and Berg 1965; Bhattacharyya 1965; Sharma and Geldart 1968), Euler deconvolution (Thompson 1982), Mellin transform (Mohan et al. 1986), Hilbert transforms (Mohan et al. 1982), Monograms (Prakasa Rao et al. 1986), least squares minimization approaches (Gupta, 1983; Silva 1989; McGrath and Hood 1973; Lines and Treitel 1984; Abdelrahman 1990; Abdelrahman et al. 1991; Abdelrahman and El-Araby 1993; Abdelrahman and Sharafeldin 1995a), ratio methods (Bowin et al. 1986; Abdelrahman et al.1989), characteristic points and distance approaches (Grant and West 1965; Abdelrahman 1994), neural network (Elawadi et al. 2001), Werner deconvolution (Hartmann et al. 1971; Jain 1976; Kilty 1983); Walsh Transformation (Shaw and Agarwal 1990), Continual least-squares methods (Abdelrahman and Sharafeldin 1995b; Abdelrahman et al. 2001a, b; Essa 2012, 2013), Euler deconvolution method (Salem and Ravat 2003), Fair function minimization procedure (Tlas and Asfahani 2011a; Asfahani and Tlas 2012), DEXP method (Fedi 2007), deconvolution technique (Tlas and Asfahani 2011b); Regularised inversion (Mehanee 2014); Simplex algorithm (Tlas and Asfahani 2015). Recently simulated annealing methods (Gokturkler and Balkaya 2012), Very fast simulated annealing (Biswas 2015; Biswas and Sharma 2015; 2014a, b; Sharma and Biswas 2013) and particle swarm optimization (Singh and Biswas 2016) have been used to solve similar kind of non-linear inversion problems for different type of subsurface structures. Many other interpretation methods for gravity and magnetic data can be found in various literatures (Abdelrahman and Essa 2015, Abdelrahman and Sharafeldin 1996; Abdelrahman 1994; Asfahani and Tlas 2007, 2004; Tlas et al. 2005).

In the present work, Very fast simulated annealing (VFSA) is used to determine the various model parameters related to thin sheet type structures for gravity and magnetic anomalies. Since, VFSA optimization is able to search a enormous model space without negotiating the resolution and has the ability to avoid becoming trapped in local minima (Sen and Stoffa 2013; Sharma and Kaikkonen 1998, 1999a, b; Sharma and Biswas 2011, 2013 Sharma 2012; Biswas and Sharma 2015, 2016) and is used in interpreting the gravity and magnetic anomaly data. The applicability of the proposed technique is appraised and discussed with the help of synthetic data and two field examples. The method can be used to interpret the gravity and magnetic anomalies occurred due to a thin sheet type mineralized bodies.

Theory

Forward modeling

The general expression of a gravity anomaly g(x) for thin sheet at any point on the surface (Fig. 1) is given by the equations (after Gay 1963):

Fig. 1
figure 1

A diagram showing cross-sectional views, geometries and parameters for thin sheet type structure

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$$g\left( x \right) = k\left[ {\frac{{x_{0} \sin \theta + z\cos \theta }}{{x_{0}^{2} + z^{2} }}} \right]$$
(1)

The general expression of a magnetic anomaly m(x) for thin sheet at any point on the surface (Fig. 1) is given by the equations (after Siva Kumar Sinha and Ram Babu 1985):

$$m\left( x \right) = k\left[ {\frac{{x_{0} \cos \theta + z\sin \theta }}{{x_{0}^{2} + z^{2} }}} \right]$$
(2)

where, k is the amplitude coefficient, z is the depth from the surface to the top of the body (Thin Sheet), x 0 (i = 1,…,N) is the horizontal position coordinate, θ is the angle.

Inversion: Very Fast simulated annealing global optimization

The Global optimization methods such as simulated annealing, genetic algorithms, artificial neural networks and particle swarm optimization have been used in various geophysical data sets (e.g., Rothman 1985, 1986; Dosso and Oldenburg 1991; Sen and Stoffa 2013; Sharma and Kaikkonen 1998, 1999a, b; Zhao et al. 1996; Juan et al. 2010; Sharma and Biswas 2011, 2013; Sharma 2012; Biswas and Sharma 2014a, b, 2015; Biswas 2015; Singh and Biswas 2016). The Very Fast Simulated Annealing (VFSA) is a global optimization method is used for finding the global minimum of a function. The process comprises of heating a solid in a heat bath and then slowly allowing them to cool down and anneal into a state of minimum energy. The same principal when used to geophysical inversion aims to minimize an objective function called error function. The error function is analogous to the energy function in a way that error function is directly proportional to the degree of misfit between the observed data and the computed data.

The following misfit (φ) between the observed and model response is used for data interpretation (Sharma and Biswas 2013).

$$\varphi = \frac{1}{\text{N}}\sum\nolimits_{{\varvec{i = 1}}}^{\text{N}} \, \left( {\frac{{\varvec{V}_{\varvec{i}}^{0} - \varvec{V}_{\varvec{i}}^{\text{c}} }}{{\left| {\varvec{V}_{\varvec{i}}^{0} } \right| + \left( {\varvec{V}_{{\varvec{max}}}^{\varvec{0}} - \varvec{V}_{{\varvec{min}}}^{\varvec{0}} } \right)/2}}} \right)^{2} \,$$
(3)

where N is number of data point, \(V_{i}^{0}\) and \(V_{i}^{c}\) are the ith observed and model responses and \(V_{\rm{max}}^{0}\) and \(V_{\hbox{\rm{min}} }^{0}\) are the maximum and minimum values of the observed response, respectively.

The detailed VFSA algorithm is not discussed here and referred the work of Sen and Stoffa (2013), Sharma (2012) and Sharma and Biswas (2013), Biswas (2015). In VFSA optimization, parameters such as Initial temperature 1.0, cooling schedule 0.4, number of iterations 2000 and number of moves per temperature 50 is used in the present study. Global model, Probability Density Function (PDF) and Uncertainty analysis has been done based on the techniques developed by Mosegaard and Tarantola (1995) and Sen and Stoffa (1996).

The code was developed in Window 7 environment using MS FORTRAN Developer studio on a simple desktop PC with Intel Pentium Processor. For each step of optimization, a total of 106 forward computations (2000 iteration × 50 number of moves × 10 VFSA runs) were performed and accepted models stored in memory.

Results and discussion

Gravity data

Synthetic example

The VFSA global optimization is instigated using noise-free and noisy synthetic data (10 and 20 % Gaussian noise) for gravity anomaly over a thin sheet type model. Initially, all model parameters are optimized for each data set.

Model 1

Firstly, synthetic data are generated using Eq. (1) for a sheet model (Table 1) and 10 % Gaussian noise is added to the synthetic data. Inversion is implemented using noise-free and noisy synthetic data to retrieve the actual model parameters and study the effect of noise on the interpreted model parameters. Primarily, a suitable search range for each model parameter is selected and a single VFSA optimization is executed. After studying the proper convergence of each model parameter (k, x 0 , z, and θ) and misfit (Fig. 2) by adjusting VFSA parameters (such as initial temperature, cooling schedule, number of moved per temperature and number of iterations), 10 VFSA runs are performed. Then, histograms (Fig. 3a) are prepared using accepted models whose misfit is lower than 10−4. The histograms in Fig. 3a depict that all model parameters (k, x 0 , z, and θ) show closer to the actual solution. A statistical mean model is also computed using models that have misfit lower than 10−4 and lie within one standard deviation. Table 1 depicts that the estimated mean model and uncertainty.

Table 1 Actual model parameters, search range and interpreted mean model for noise free and 10 % Gaussian noise with uncertainty-Gravity data (Model 1)
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Fig. 2
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Convergence pattern for various model parameters and misfit for gravity data

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Fig. 3
figure 3

Gravity data: a histograms of all accepted models having misfit <10−4 for noise-free synthetic data for thin sheet-Model 1 and b histograms of all accepted models having misfit <10−2 for noisy synthetic data for Thin sheet-Model 2

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Figure 4a depicts cross-plots for noise free data between the model parameters k, z, and θ using accepted models with misfit lower than 10−4 (green) and models within the pre-defined high PDF region (red). This shows that all parameters are well resolved and pointing towards its actual value and there is no uncertainty in each model parameters. Figure 5a depicts a comparison between the observed and the mean model response.

Fig. 4
figure 4

Gravity data: a scatter-plots between amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit <threshold (10−4 for noise-free data) (green), and models with PDF >60.65 % (red) for noise free data; b scatter-plots between amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit <threshold (10−2 for noisy data) (green), and models with PDF >60.65 % (red) for noisy data

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Fig. 5
figure 5

Gravity data: fittings between the observed and model data for Thin sheet: Model 1- a noise-free synthetic data and b 10 % Gaussian noisy synthetic data, and Model 2- c noise-free synthetic data and d 20 % Gaussian noisy synthetic data

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Next, VFSA optimization is performed using 10 % Gaussian noise added data for Model 1 (Table 1). The convergence of each model parameter and reduction of misfit is studied for a single solution. After observing the reduction of misfit systematically and stabilization of each model parameter during later iteration, ten VFSA runs are performed. The histograms in Fig. 3b also depict that all model parameters (k, x 0 , z, and θ) show closer to the actual solution. A statistical mean model is also computed using models that have misfit lower than 10−2 and lie within one standard deviation. Table 1 depicts that the estimated mean model and uncertainty for noisy model.

Figure 4b depicts cross-plots for noisy data between the model parameters k, z, and θ using accepted models with misfit lower than 10−4 (green) and models within the pre-defined high PDF region (red). However, it reveals that scatter is large for noisy data but models in high PDF region are restricted near the actual value. Figure 5b depicts a comparison between the observed and the mean model response for noisy data.

Model 2

Another synthetic data are generated using Eq. (1) for a sheet model (Table 2) and 20 % Gaussian noise is added to the synthetic data to check the effect of more noise. Inversion is implemented using noise-free and noisy synthetic data to retrieve the actual model parameters and study the effect of higher noise on the interpreted model parameters. The procedure was repeated again as discussed in Model 1. The histogram and cross plots were also studied and found the similar in nature like Model 1. For brevity, the figures are not presented here. Figure 5c, d depicts a comparison between the observed and the mean model response for noise free and noisy data.

Table 2 Actual model parameters, search range and interpreted mean model for noise free and 20 % Gaussian noise with uncertainty-Gravity data (Model 2)
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Field example

Mobrun Anomaly, Noranda, Quebec, Canada

Residual gravity anomaly map over Noranda Mining District, Quebec, Canada was taken (Grant and West 1965; Roy et al. 2000) over a massive sulphide ore body (Fig. 6). The interpretation procedure mentioned in synthetic example is again carried out for this field data. The interpreted results are shown in Table 3. Figure 6 depicts the fitting between the observed and interpreted mean model response. The depth of the body estimated in the present study is 47.9 m and is in excellent agreement with the depth obtained by Biswas 2015. Also, the misfit in the present approach is slightly less than the other method. However, it should be mentioned that Biswas 2015 interpreted this field data using horizontal cylinder, however, in the present case, it is interpreted as thin sheet type structure.

Fig. 6
figure 6

Fittings between the observed and model data for Mobrun Anomaly, Noranda, Quebec, Canada

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Table 3 Search range and interpreted mean model for Mobrun Anomaly, Noranda, Quebec, Canada
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Magnetic data

Synthetic example

The VFSA global optimization is also applied using noise-free and noisy synthetic data (10 and 20 % Gaussian noise) for magnetic anomaly over a thin sheet type model. Initially, all model parameters are optimized for each data set.

Model 1

Firstly, synthetic data are generated using Eq. (2) for a sheet model (Table 4) and 10 % Gaussian noise is added to the synthetic data. Like in gravity data, inversion is implemented using noise-free and noisy synthetic data to retrieve the actual model parameters and study the effect of noise on the interpreted model parameters. Figure 7 shows the convergence pattern for all model parameters. The inversion procedure mentioned for gravity data is also applied here and is not repeated here for brevity. Figure 8a shows the histogram for all model parameters (k, x 0 , z, and θ) is closer to the actual solution. A statistical mean model is also computed for magnetic anomaly using models that have misfit lower than 10−4 and lie within one standard deviation. Table 4 depicts that the estimated mean model and uncertainty.

Table 4 Actual model parameters, search range and interpreted mean model for noise free and 10 % Gaussian noise with uncertainty-Magnetic data (Model 1)
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Fig. 7
figure 7

Convergence Pattern for various model parameters and misfit for magnetic data

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Fig. 8
figure 8

Magnetic data: a histograms of all accepted models having misfit <10−4 for noise-free synthetic data for thin sheet-Model 1 and b histograms of all accepted models having misfit <10−2 for noisy synthetic data for Thin sheet-Model 2

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Figure 9a depicts cross-plots for noise free data between the model parameters k, z, and θ using accepted models with misfit lower than 10−4 (green) and models within the pre-defined high PDF region (red). This also shows that all parameters are well resolved and pointing towards its actual value and there is no uncertainty in each model parameters. Figure 10a depicts a comparison between the observed and the mean model response.

Fig. 9
figure 9

Magnetic data: a scatter-plots between amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit <threshold (10−4 for noise-free data) (green), and models with PDF >60.65 % (red) for noise free data; b scatter-plots between amplitude coefficient (k), depth (z), magnetization angle (θ) for all models having misfit <threshold (10−2 for noisy data) (green), and models with PDF >60.65 % (red) for noisy data

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Fig. 10
figure 10

Magnetic data: fittings between the observed and model data for Thin sheet: Model 1- a noise-free synthetic data and b 10 % Gaussian noisy synthetic data, and Model 2- c noise-free synthetic data and d 20 % Gaussian noisy synthetic data

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VFSA optimization is performed using 10 % Gaussian noise added data for Model 1 (Table 4). The histograms in Fig. 8b also depict that all model parameters (k, x 0 , z, and θ) show closer to the actual solution. A statistical mean model is also computed using models that have misfit lower than 10−4 and lie within one standard deviation. Table 4 depicts that the estimated mean model and uncertainty for noisy model.

Figure 4b depicts cross-plots for noisy data between the model parameters k, z, and θ using accepted models with misfit lower than 10−4 (green) and models within the pre-defined high PDF region (red). As, it is a noisy data the scatter is large but models in high PDF region are restricted near the actual value. Figure 10b depicts a comparison between the observed and the mean model response for noisy data.

Model 2

Alternative synthetic data are also generated using Eq. (2) for a sheet model (Table 5) and 20 % Gaussian noise is added to the synthetic data to check the effect of more noise. Inversion is executed using noise-free and noisy synthetic data to retrieve the actual model parameters and study the effect of higher noise on the interpreted model parameters. The procedure was repeated again as discussed in Model 1 for magnetic data. The histogram and cross plots were also studied and found the similar in nature like Model 1. For brevity, the figures are not presented here. Figure 10c, d depicts a comparison between the observed and the mean model response for noise free and noisy data.

Table 5 Actual model parameters, search range and interpreted mean model for noise free and 20 % Gaussian noise with uncertainty magnetic data (Model 2)
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Field example

Pishabo Lake anomaly, Canada

Total magnetic anomaly from Pishabo Lake, Ontario (McGrath 1970) was taken from an olivine diabase dike (Fig. 11). The interpretation process mentioned in synthetic example is again applied for this field data. The interpreted results are shown in Table 6. Figure 11 depicts the fitting between the observed and interpreted mean model response. The depth of the body estimated in the present study is 324 m and is in excellent agreement with the depth obtained by Abdelrahman et al. (2012). Moreover, the depth and shape of the concealed structure obtained by the present method approve very sound with the surface geologic records shown by McGrath (1970).

Fig. 11
figure 11

Fittings between the observed and model data for Pishabo Lake anomaly, Canada

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Table 6 Search range and interpreted mean model for Pishabo Lake anomaly, Canada
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Conclusions

A proficient and reliable method is employed for the interpretation of gravity and magnetic anomaly over thin sheet type structure using a VFSA global optimization method for exploration studies. The problematic determination of the appropriate shape, depth, index parameter and amplitude coefficient of a buried structure from a residual gravity and magnetic anomaly profile can be well resolved using the present method. The present study discloses that, while optimizing all model parameters (amplitude coefficient, location, depth, angle) together, the VFSA approach yields a very good results without any uncertainty in the final model parameters. The efficacy of this approach has been successfully proved, established and validated using noise-free and noisy synthetic data. The metier of this method for practical application in mineral exploration has also been efficaciously exemplified on some field examples with many complex geological structures and depths of burial. The estimated gravity and magnetic inverse parameters for the field data are found to be in excellent agreement with the other methods as well as from the geological and drilling results. The actual (not CPU) time for the whole computation process is nearly 35 s.