Introduction

Nowadays, gravity data was gathered in large quantities for environmental and geological applications, such as oil and mineral explorations, groundwater investigations, geothermal investigations, and volcanic activity studies. The accurate estimation of the depth to the body sources improves in budgeting and preparing drill boreholes and exploration programs. In the oil industry, for example, accurate assessments of basement depth are important for understanding the basin exploration (the thickness of the sedimentary layer) more clearly and reduce the exploration risk factor (Li and Oldenburg 1983; Florio 2020). Gravity data are needed for oil exploration mapping of sedimentary basins and salt structures (Sarsar Naouali et al. 2017; Essa et al. 2021a). In mineral and ore exploration, the estimation of the dominant orebody parameters (depth, types, shape,…etc.) reduces the risk associated with mining and plan for future investments (Snowden et al. 2002; Sultan et al. 2009; Zhang et al. 2022). In engineering and environmental studies, the gravity method has important uses in exploring, locating, and distinguished the subsurface voids, karst, buried hazardous metal containers, mapping landfills, delineating abruptly dipping geologic contacts and disruptions, visualizing regions of potential stress amplifications such as faults (Hinze et al. 2013; Eshanibli et al. 2021). In geothermal exploration, the gravity data elucidation mainly purposes to assess the location and depth of the sources, which are crucial to targeting the geothermal potential reservoirs anomalies (Athens and Caers 2021). In volcanic activities studies, the gravity method can clarify the geodynamic process based on their temporal variations, which have got up from earthquakes and volcanic activity (Lichoro et al. 2019; Casallas-Moreno et al. 2021).

Generally, the gravity method has been receiving attention in more targets as declared above because it can elucidate and visualize the variant anomaly structures. So, several methods for the complete interpretation were developed and categorized as follows: 1) Conventional and non-conventional methods, which depend on the characteristic distance and points, nomograms, matching curves, window and depth curves, various types of transformations, minimization algorithms, gradients-based, moving average, Euler deconvolution, DEXP, fair function minimization (Telford et al. 1990; Abdelrahman et al. 2003; Al-Garni 2008; Asfahani and Tlas 2012; Essa 2013; Abdelrahman and Essa 2015; Ekinci and Yiğitbaş 2015; Hiramatsu et al. 2019; Cooper 2021). 2) Methods depend on 2D and 3D imaging, modeling, and inversion (Zhang et al. 2001; Asfahani and Tlas 2015; Essa et al. 2020). 3) Methods used simple geometric shapes such as spheres, cylinders, faults, and contacts (Biswas 2015; Tlas and Asfahani 2018; Uzun et al. 2020; Essa 2021). 4) Methods rely on the application of the global optimization algorithms such as particle swarm optimization (PSO), differential evolution algorithms (DE), simulated-annealing (SA), genetic algorithms (GA), ant-colony optimization (AC), gravitational search algorithm (GSA) (Biswas 2016; Kaftan 2017; Pallero et al. 2017; Essa and Munschy 2019; Essa and Géraud 2020; Rathee and Chhillar 2020; Pace et al. 2021).

Bat algorithm represents one of the recent proposed metaheuristic algorithms simulating the echolocation attitude of bats to obtain global optimization in geophysics especially, a potential field. In recent decades, many metaheuristic inspiring algorithms have been developed to solve complex problems (Mirjalili et al. 2014). For example, the genetic algorithm (GA) (Holland 1984; Montesinos et al. 2005), the particle swarm optimization (PSO) algorithm (Kennedy and Eberhart 1995; Roshan and Singh 2017; Essa 2021; Essa et al. 2021b), the differential evolution (DE) algorithm (Storn and Price 1997; Ekinci et al. 2016), the simulated annealing (SA) algorithm (Kirkpatrick et al. 1983; Biswas 2015), the ant colony optimization (ACO) algorithm (Dorigo et al. 1996; Dorigo and Stützle 2003), bee colony algorithm (Karaboga and Basturk 2008) dolphin echolocation (Kaveh and Farhoudi 2013), and gravitational search algorithm (Rashedi 2009).

In comparison to traditional optimization techniques, these algorithms are common among researchers because of their versatility and superior ability to deal with a variety of problems. These swarm intelligence algorithms have been developed and applied to various real-world problems, but the use of the Bat algorithm in geophysical data analysis is relatively new (i.e., in seismic refraction; Poormirzaee 2017; Poormirzaee et al. 2019). Fister (2013) concluded that the Bat algorithm outperforms the PSO after conducting various experiments on the implementation of the Bat algorithm. Although methods like genetic algorithms (GA) and PSO can be quite beneficial, they still have certain limitations when it comes to multi-modal optimization issues (Yang 2013).

The proposed Bat algorithm technique have several advantages: One of the key advantages of this technique is that it provides highly quick convergence by switching from exploration to exploitation at a reasonably early stage. This makes it a good choice for applications that require a quick response, such as classifications. Bat algorithm may be used as both a global and a local optimizer and is capable of effectively handling multi-model problems. As the iteration advances, Bat algorithm uses the controlling parameter to update the parameter. Bat algorithm is committed to preserving the population's diversity of solutions. The drawbacks of the Bat algorithm technique lie in the following items: It has a lack of good exploration. It required the parameter tuning to achieve better search output. Switching between exploration and exploitation requires a better control strategy.

Finally, the present study aims to infer and elucidate the 2D gravity data profile acquired over geothermal and volcanic areas in different regions in the world to estimate the source origin (depth & location) using the Bat algorithm technique, which is discussed below.

The structure of the paper organized as follows: “Bat algorithm” section covers the fundamentals of echolocation as well as the conventional formulation of the Bat algorithm. The “Forward modeling” section describes the modeling and formulation of the proposed Bat algorithm. The “Methodology” section provides the methodology of the proposed Bat algorithm to invert gravity data. The “Synthetic datasets” section discusses the synthetic dataset examples accuracy and efficiency. The “Field datasets” section discusses the applicability of the proposed Bat algorithm to various field data examples. Finally, “Conclusions” section is drawn.

Bat algorithm

Bat algorithm (BA), which is a natural-inspired metaheuristic algorithm, was introduced by Yang (2010). The echolocation actions of micro-bats inspired this algorithm. Micro-bats use echolocation to locate their nest in the dark, avoid obstacles, and track prey. Bats release a very noisy sound pulse within 8 to 10 kHz in range and then listing to an echo backing from nearby objects. Every pulse is a few thousandths of a second long (up to about 8 to 10 ms). When bats are close to prey or an object, their pulses rate increases while their sound volume decreases (Yang 2010). As a result, the echolocation activity of micro-bats can be expressed in a way to optimize objective functions. The main rules of the Bat algorithm can shorten in three stages: first, Bats are applying the echolocation to assess distance; second, Bats are locating their sources objects by flying at a stable frequency in a range of [Qmin, Qmax] with an initial speed (Vi) at position (Xi); and third, the loudness (Li) and rate of pulse emission (ri) which depend on the distance between the bat and the source.

The wavelength spectrum [Kmin, Kmax] refers to the frequency range [Qmin, Qmax]. For an optimization problem, changing the frequency or wavelength may be used to change the movement range of bats (Eqs. 1–3). The selection of a suitable frequency or wavelength range is critical, and it should select to be equal to the scale of the domain of awareness before toning down to smaller ranges. The spectrum of [0, 0.5] was calculated as the optimal frequency range in this analysis after the code runs with various values. The pulse rate, ri, can be simply in the range [0, 1], which 0 denotes no pulse and 1 is the maximum pulse emission rate. The range is determined by the target's proximity. Moreover, the initial loudness, i.e., Li, can be normally within the range [1, 2] (Yang 2010). The loudness of the bats decreases as they approach their prey, whereas the rate of pulse emission increases. Only if the new solutions improve can the Bat algorithm update the loudness and emission rates, indicating that the bats are approaching the optimal solution (Fister 2013; Eqs. 4–5). The equations below presented the relationship between algorithm parameters (Yang 2010).

$${Q}_{i}^{t}={Q}_{min}+\left({Q}_{max}-{Q}_{min}\right)\beta$$
(1)
$${V}_{i}^{(t+1)}={V}_{i}^{t}+({X}_{i}^{t}-{X}_{best}){Q}_{i}^{t}$$
(2)
$${X}_{i}^{(t+1)}={X}_{i}^{t}+{V}_{i}^{(t+1)}$$
(3)
$${L}_{i}^{(t+1)}=\alpha {L}_{i}^{t}$$
(4)
$${r}_{i}^{t}={r}_{i}^{0}[1-\mathrm{exp}\left(-\gamma \tau \right)]$$
(5)

where, Qi is the frequency of ith bat and updated in every iteration, \(\beta\) is a random vector of uniform distribution within a range [0, 1] and Xbest is currently the global best solution among all bat numbers, \(\alpha\) and \(\gamma\) are constants, 0 < \(\alpha\)  < 1 and \(\gamma\) > 0 and \(\tau\) is the scaling factor.

Bat algorithm utilizes a random path to produce new results from every chosen best solution in the local search, as follows:

$${X}_{new}={X}_{old}+\varepsilon {A}^{t}$$
(6)

where \(\varepsilon\) \(\in\) [-1, 1] is a random number, and Lt represents the average loudness of all bat numbers at the current stage.

In terms of accuracy and performance, the Bat algorithm (BA) outperforms most other algorithms. The bat algorithm basically becomes the regular PSO if the frequency variations are replaced by a random parameter and Li = 1 and ri = 1 are set.

Forward modeling

Gravity anomaly at an observation point, xj, along a profile (Fig. 1) is given by (Salem et al. 2004; Asfahani and Tlas 2008; Essa 2014):

$$\mathrm{g}\left({x}_{j},{x}_{0},z,\mu ,q,A\right)= \frac{A {z}^{\mu }}{{\left[{({x}_{j}-{x}_{0})}^{2}+{z}^{2}\right]}^{q}} , j=\mathrm{1,2},3,\dots ,n (7)$$

where \({x}_{j}\) and \({x}_{0}\) represent the horizontal locations of the measured points and the origin of the buried source, z is the depth of the source, q is the shape factor (dimensionless), and A is the amplitude coefficient, whose dimension is determined by the shape factors (q and μ), yielding g in mGal units. The definitions of A, q, and μ for the above idealized bodies are shown in Table 1.

Fig. 1
figure 1

The schematic diagrams for the different geometrical shapes

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Table 1 Definitions of A, q and μ of the idealized bodies shown in Fig. 1. σ is the density contrast (gm/cm3), γ is the universal gravitational constant (6.67384 × 10−11 m3 kg−1 s−2), and r is the radius of the idealized body (m)
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Materials and methods

In gravity data interpretation, it is essential to obtain accurate results for the subsurface model parameters. Therefore, an inversion algorithm of large capacities is needed to accomplish accepted evaluations of subsurface model parameters such as depth, location, the shape of the buried body, etc. In various case studies, metaheuristic inversion methods have achieved promising results. Most metaheuristic inversion algorithms are quicker, easier, and more efficient than traditional inversion methods.

A new inversion Bat algorithm code was developed. The most significant parameters are depth, location, shape of the body, in addition to the amplitude coefficient (z, x0, q, A). Therefore, the proposed algorithm is searched to find an appropriate subsurface model that fits the actual data, i.e., assess the best-fit parameters. In general, each bat's location in search space represents a solution. Bats hover haphazardly in search space and implement a solution to each iteration. Each bat's position is determined using the optimum locations. The best location is the one that has the lowest misfit function value, and it is chosen as the best solution (Xbest). The best solutions are then matched, and the best solution for every iteration is chosen. This process is repeated a predetermined number of times. Finally, after the final iteration, the Xbest is picked as the best solution. The Bat algorithm inversion code was first tested on various synthetic models in this research. After that, it was tested on real datasets.

The proposed algorithm that is utilized to invert the gravity data constitutes of the following procedures:

First step; The virtual bats’ initial position Xi (i = 1, 2, …, N), frequencies Qi, velocities Vi, loudness Li, and the pulse rates ri: In the search space, each bat signifies a solution. Xi represents the variables of the characteristic source parameters [i.e., depth (z), origin (x0), Amplitude coefficient (A), and shape factor (q)] randomly selected from the search space, and Vi represents the velocity for each individual, virtual bat.

Second step; Finding the Xbest.

The objective function (FObj) is defined as the normal root-mean-square (NRMSE) misfit error between the observed and calculated gravity data anomalies:

$$\mathop F\nolimits_{Obj} = \frac{100}{{Max\left( {\mathop g\nolimits_{obs} } \right) - Min\left( {\mathop g\nolimits_{obs} } \right)}}\sqrt {\frac{{\sum\limits_{j = 1}^{N} {\mathop {\left[ {\mathop g\nolimits_{obs} - \mathop g\nolimits_{cal} } \right]}\nolimits^{2} } }}{N}}$$
(8)

where, N is the total number of data points, gobs is the observed gravity and gcal is the calculated gravity. The gcal is calculated by the forward modeling algorithm. Initially, Eq. 8 is applied to compute misfits, and then the bat with the minimum misfit is selected as the Xbest.

Third step; While the program still doesn't reach the maximum number of iterations operate the following instructions:

figure a

The Bat algorithm basic steps can be outlined in the pseudo-code illustrated in Fig. 2 and the flow chart shown in Fig. 3.

Fig. 2
figure 2

Bat algorithm (BA) pseudo-code (modified after yang 2010)

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Fig. 3
figure 3

The flowchart shows the public steps of The Bat algorithm (BA)

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Results and discussion

Synthetic datasets

The introduced Bat algorithm was tested on two synthetic datasets to demonstrate its validity and accuracy in gravity data inversion. The synthetic responses are confined to the simple class of the geometrical shapes of vertical cylinder, horizontal cylinder, and spheres. Then to prove the stability of the proposed Bat algorithm inversion approach, the inversions are performed on noisy data as well as interference effect model example.

Model 1

First, a noise-free simple synthetic example of a horizontal cylinder model has been inspected. The gravity effect of a horizontal cylinder model with input parameters: A = 70 mGal.m, z = 15 m, x0 = 0 m and q = 1 is calculated using Eq. (7) with a 201-m long profile (Fig. 4a). Following the procedures of the Bat algorithm technique proposed in Sect. 2. Figure 4b shows the average loudness of all bat numbers obtained for several iterations. The number of iterations is determined based on reaching the minimum NRMSE and getting the best solution of the residual gravity anomaly (Fig. 4a). Figure 4c shows the emission rate of all bats obtained for every iteration process; the loudness of the bats decreases as they approach their prey; while the rate of pulse emission increases. Figure 4d shows the NRMSE of the global best solution (minimum objective function) vs. the iteration numbers. The NRMSE reaches the minimum after the 500 iterations process for all bat numbers. Figure 4e shows the average NRMSE of all the bats obtained for every iteration process.

Fig. 4
figure 4

Model 1: noise-free theoretical example. a The observed gravity anomaly generated by horizontal cylinder model (True model parameters), as well as the calculated gravity anomaly (Recovered model parameters) using the Bat algorithm technique, b loudness of the bats, c emission rat of the bats, d NRMSE of the global best solution (FObj) of the bats versus the iteration numbers, and e the average NRMSE of all the bats

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The global best solution of the gravity anomaly (i.e., model parameters) is obtained when the objective function (FObj) approaches the minimum of the NRMSE for the different iteration processes. Table 2 shows that the recovered model parameters of the proposed noise-free synthetic example are identical with the true-model parameters when the objective function (FObj) approaches the minimum. The result shows that the proposed Bat algorithm technique is stable and talented in recovering the true values of the model parameters of the buried model. Also, Table 2 illustrates the used search space for every model parameter, relative errors (RE), and the standard division (SD) of the given model. The search ranges are selected based on the aforementioned true-model parameters. Therefore, the search range should be approved to simulate more realistic examples where a priori information is absent.

Table 2 Model-1: True and recovered model parameters of the noise-free synthetic example with corresponding RE and SD for each parameter using the Bat algorithm
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A 20% random Gaussian noise has been added to the free-noise observed gravity anomaly (Fig. 5a) to test the proposed Bat algorithm stability. By applying the procedures of the Bat algorithm declared-above to the noisy data, the best results of the recovered model parameters will be corresponding to the minimum objective functions FObj (NRMSE). The loudness and emission rate of the bats are shown in Figs. 5b and c. Figures 5d and e depict the NRMSE of the global best solution (min objective function) and the average NRMSE of all the bats. Table 3 shows that the recovered model parameters of the proposed noisy synthetic example (i.e., corresponding to min objective function FObj) are not significant influenced by the contaminated 20% random Gaussian noise, and recovered parameters are close to the true ones. As a result, it can be inferred that the BA technique suggested here is stable concerning for noise. The RE and SD of the recovered model parameters are illustrated in Table 3.

Fig. 5
figure 5

Model 1: Noisy theoretical example. a The observed gravity anomaly generated by horizontal cylinder model (True model parameters) after added 20% random Gaussian noise, as well as the calculated gravity anomaly (Recovered model parameters) using Bat algorithm technique, b loudness of the bats, c emission rat of the bats, d NRMSE of the global best solution (FObj) of the bats versus the iteration numbers, and e the average NRMSE of all the bats

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Table 3 Model-1: True and recovered model parameters of the noisy synthetic example (20% random Gaussian noise) with corresponding RE and SD for each parameters using the Bat algorithm
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Model 2

The gravity data of an anomalous buried object can be impacted by an interfering effect (that is, the action of surrounding structures) in specific geologic contexts (Mehanee and Essa 2015). We computed the composite gravity response (using formula (7)) of two surrounding geologic objects, namely a horizontal cylinder model with true model values [A1 = 50 mGal.m, z1 = 17 m, x01 = -50 m and q1 = 1] and a sphere model with true model values [A2 = 250 mGal.m2, z2 = 17 m, x02 = 50 m, and q2 = 1.5], along a profile length of 201 m (Fig. 6a) to test this influence on the truthfulness of the characteristic parameters estimated from the Bat algorithm methodology presented here. Applying the procedures of the Bat algorithm technique described-above, the calculated gravity response of the two models is obtained in Fig. 6a. The obtained average loudness of the composite response is shown in Fig. 6b, while the emission rat of bat for the composite anomaly is shown in Fig. 6c. The NRMSE of the global best solution (min objective function, FObj) is given in Fig. 6d, and the average NRMSE of all the bats is shown in Fig. 6e. Figure 6 and Table 4 show that the Recovered model parameters of the two introduced models are identical to the true ones. This result supports that the Bat algorithm technique is stable in the cases of multi-structures relying on the extent of the neighboring effect.

Fig. 6
figure 6

Model-2: Interference effect. a The composite gravity anomaly generated by horizontal cylinder and sphere model (True model parameters), as well as the calculated gravity response of them (Recovered model parameters) using the Bat algorithm technique, b loudness of the bats, c emission rat of the bats, d NRMSE of the global best solution (FObj) of the bats versus the iteration numbers, and (e) the average NRMSE of all the bats

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Table 4 Model-2: True and recovered model parameters of the composite interference gravity anomaly of a horizontal cylinder and a sphere model with corresponding RE and SD for each model parameter using the Bat algorithm
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To further investigate the procedure of BA on Multi-structure and surrounding effect, we contaminated the composite gravity response (Fig. 6a) with a noise level of 20% of random Gaussian noise (Fig. 7a). Figures 7b and c present the average loudness and emission rate of the bat to the overall composite gravity anomaly, respectively. Figures 7d and e show the NRMSE of the global best solution (min objective function, FObj), and the average NRMSE of all the bats, respectively. Figure 7a and Table 4 illustrate that the calculated composite anomaly and the recovered model parameters of the two introduced models even after being contaminated with noise. Also, they reveal that the recovered model parameters differ slightly from the true ones due to the effect of both the surrounded objects and noise. In general, the model parameters recovered using BA from the noisy interference sources for the first and second models are in good matching with the exact values (Table 5). These results support and confirm the Bat algorithm method find.

Fig. 7
figure 7

Model-2: Noisy interference effect. a The noisy composite gravity anomaly generated by data set shown in Fig. 6a after adding 20% random Gaussian noise (True model parameters), as well as the calculated gravity response of them (Recovered model parameters) using the Bat algorithm technique, b loudness of the bats, c emission rat of the bats, d NRMSE of the global best solution (FObj) of the bats versus the iteration numbers, and e the average NRMSE of all the bats

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Table 5 Model-2: True and recovered model parameters of the noisy composite interference gravity anomaly of a horizontal cylinder and a sphere model with corresponding RE and SD for each model parameters using the Bat algorithm
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Based on the numerical models presented above, the method described here is stable and appropriate for field gravity data interpretation, as explained in the following section.

Field datasets

Two field cases for geothermal investigation and volcanic activity study are analyzed to examine the applicability of the established methodology.

The Phenaimata gravity anomaly, Gujarat, India

The Phenaimata igneous complex lies northern the Narmada River and is part of the Narmada-Tapti tectonic zone (NTTZ) (Fig. 8). The Narmada rift appears to regulate the positioning of this plug-like entity. This intrusive is located 23 km southwest of Chhota Udaipur, on the left bank of the Heran River (Fig. 9). It's also a bimodal igneous complex, with plutonic and volcanic rock types representing tholeiitic and alkaline magmatism. Basalt, layered gabbro, diorite, nepheline-syenite, lamprophyres, and granophyres make up the Phenaimata plug, which is a differentiated igneous complex (Sukeshwala and Sethna 1969; Hari et al. 2011 and 2014). The Phenaimata complex is made up of two-thirds basalts and one-third alkaline plutonic series. The Phenaimata igneous complex produced orthopyroxene gabbro, according to Hari et al. (2011). According to the petrological modeling, the origin of gabbroic rocks was caused by magma accumulation in the crust, contemporaneous assimilation, and fractional crystallization. Also, reverse magnetization has been found in olivine gabbro and syenite rocks and implying emplacement toward the end of the major Deccan volcanism period in the Phenaimata igneous complex (Basu et al. 1993; Poornachandra Rao et al. 2004).

Fig. 8
figure 8

Simplified geological and tectonic map of the Deccan traps (green) and adjacent area showing locations of some important igneous complexes in the NW and central India. Grey color rectangles indicate the present study area covering Phenaimata igneous complex (after Krishnamurthy et al., 2000; Singh et al. 2014)

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Fig. 9
figure 9

Detailed geology and physiography of the area around Phenaimata igneous intrusive. BB' shows profile adopted for gravity interpretation using the Bat algorithm technique (after Hari et al. 2011; Singh et al. 2014)

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The Bouguer gravity anomaly map in the Phenaimata igneous exposure displays an elliptical-shaped anomaly closure to the North of the complex, with the principal axis of the anomalies aligned roughly in the ENE–WSW direction, which aligns with the Narmada-Tapti tectonic zone's orientation (Fig. 10). A north–south BB' 30 km long profile was taken perpendicular to the closure anomaly of the Bouguer gravity map using a 0.5 km sample interval. The BB' residual gravity anomaly profile (Fig. 11a) was obtained after separating the regional anomaly from the BB' Bouguer gravity anomaly profile using a suitable separation technique.

Fig. 10
figure 10

Bouguer gravity anomaly map of Phenaimata igneous complex, Gujarat, India. + indicates gravity stations. BB' shows profile location adopted for gravity interpretation using the Bat algorithm technique (after Singh et al. 2014)

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Fig. 11
figure 11

The Phenaimata anomaly, Gujarat, India. a The observed gravity anomaly profile (blue squares), and the calculated best-fitting gravity response (red circles) using the Bat algorithm technique, b loudness of the bats, c emission rat of the bats, d NRMSE of the global best solution (FObj) of the bats versus the iteration numbers, and e the average NRMSE of all the bats

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The Bat algorithm methodology was applied to the residual gravity anomaly using the probable search range of the characteristic parameters (Table 6). Figures 11b and c illustrate the average bat loudness and emission rate for the residual gravity profile, respectively. Figures 11d and e illustrate the NRMSE of the global best solution (min FObj) and the average NRMSE of all bats, respectively. The best optimum model parameters correspond to the min FObj. The minimum FObj is 0.0002 mGal, and the best model parameters are [A = 960 mGal.km, z = 5.3 km, x0 = 1 km, q = 1.5, and μ = 1], which recommend that the intrusive body of the Phenaimata igneous complex anomaly is approximated by sphere like model (Table 6).

Table 6 The Phenaimata anomaly, Gujarat, India. Recovered model parameters with the corresponding minimum FObj and the SD for each parameter using the Bat algorithm
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The measured and computed gravity anomalies of the Phenaimata igneous complex anomaly are in good agreement, as shown in Fig. 11a. Using a density of the volcanic intrusive rock (Gabbro) of 2.86 gm/cm3 and average density of the surrounding rocks of 2.65 gm/cm3, then the density contrast (Δρ) is 0.21 gm/cm3 according to Singh et al. (2014), the depth to the top will be 2.76 km, which is in good agreement with the results obtained by joint modeling of Singh et al. (2014) (Fig. 12).

Fig. 12
figure 12

Results of 2.5 D joint modeling of Gravity–Magnetic anomalies along N-S profile BB' over the Phenaimata intrusive complex Bougeur gravity map. Where ρ is density, S is susceptibility and M is magnetization are given in CGS units (after Singh et al. 2014)

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The Hohi volcanic gravity anomaly, Kyushu, Japan

The Hohi volcanic zone in central Kyushu, southwest Japan, is a 70 km long, 45 km wide volcano-tectonic depression with Plio-Pleistocene volcanic materials widely dispersed in an E-W orientation. With a limited amount of volcanic-clastic material, these volcanic rocks are mostly made up of andesitic lava flows and dacitic pyroclastic-flow deposits. From around 5 Ma to the present, the Hohi volcanic zone provides a strong indication that volcanic activity and subsidence alternated under a regional extensional stress field, with the depression mainly remunerated by filling of an equal amount of volcanic material (Kamata 1989a, b).

Based on drill-hole, geochronologic, and gravity data, the buried Shishimuta caldera was identified beneath post-caldera lava domes and lacustrine deposits in the center of the Hohi volcanic zone. The Yabakei pyroclastic flow, which erupted 1.0 Ma ago with a bulk volume of 110 km3, is the source of the Shishimuta caldera in the center of the Hohi volcanic zone (Fig. 13). The Shishimuta caldera is an 8-km wide, 3-km deep breccia-filled funnel depression with a V-shaped negative Bouguer gravity anomaly of up to 36 mGal (Fig. 14). The andesitic breccia fills the caldera and has a relatively low density; it was likely produced by fragmentation of disturbed ceiling rock during the intense Yabakei eruption and subsequent collapse (Aramaki 1984). Komazawa and Kamata (1985) and Kamata (1989a, b) introduced a gravity model that defined the V-shaped depression with depth to the Pre-Tertiary basement up to 3.8 km below sea level.

Fig. 13
figure 13

Geologic map of the Kusu basin and Shishimuta area of the Hohi volcanic zone, Kyushu, Japan (modified after Kamata 1989b). Dotted circles represent the locations of drill holes and opened circles refer to the location of 3000- m deep drill holes with spot coring

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Fig. 14
figure 14

Bouguer gravity map of the central part of Hohi volcanic zone, Kyushu, Japan. Contour interval is 2 mGals. Assumed density is 2.3 g/cm3. The dotted line refers to the location of the Shishimuta caldera (modified after Kornazawa and Kamata 1985). The gravity anomaly profile (Fig. 15a) subjected to interpretation using the Bat algorithm is denoted by AB

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A NE-SW AB profile was taken normal to the trend of the Shishimuta caldera anomaly of the previous Bouguer gravity map to be subjected to the BA interpretation procedures (Fig. 15). The AB Bouguer gravity anomaly profile with a length of 13.5 km was digitized at 0.1 km sampling intervals (Fig. 15a).

Fig. 15
figure 15

The Hohi volcanic anomaly, Kyushu, Japan. a The observed gravity anomaly profile (blue squares), and the calculated best-fitting gravity response (red circles) using the Bat algorithm technique, b loudness of the bats, c emission rat of the bats, d NRMSE of the global best solution (FObj) of the bats versus the iteration numbers, and e the average NRMSE of all the bats

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The Bat algorithm interpretation procedures have been applied to the AB anomaly profile of the Shishimuta caldera of the Hohi volcanic zone using the suitable search range of the characteristic parameters. Figures 15b and c demonstrate the average bat loudness and emission rate for the Bouguer gravity profile, respectively. Figures 15d and e demonstrate the NRMSE of the global best solution (min FObj) and the average NRMSE of all bats, respectively. Note, the increase in the NRMSE of the objective function (FObj) is due to work directly on the Bouguer gravity anomaly profile instead of the residual gravity anomaly profile in the other field example mentioned-above. The best interpretive model parameters correspond to the min FObj. The minimum FObj is 0.0043 mGal, and the best-recovered model parameters are [A = -145 mGal.km, z = 3.9 km, x0 = 0.15 km, q = 0.5, and μ = 0], which recommend that the source of the Shishimuta caldera anomaly is approximated by vertical cylinder-like model (Table 7). Figure 15a shows great matching between the observed and calculated anomaly.

Table 7 The Hohi volcanic anomaly, Kyushu, Japan. Recovered model parameters with the corresponding minimum FObj and the SD for each parameter using the Bat algorithm
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Drill holes outside Shishimuta caldera (boreholes with dotted circles; a – k in Fig. 13) encounter Pre-Tertiary rocks at depths of 2.0—2.5 km (Sasada 1984; Tamanyu 1985). While drill holes up to 3 km deep within the caldera (boreholes with opened circles; y and z in Fig. 13), on the other hand, penetrate Plio-Pleistocene volcanic rocks instead of Pre-Tertiary materials (MITI 1986). According to depth modeling to gravity basement of the caldera, Komazawa and Kamata (1985) and Kamata (1989a, b) suggest the Pre-Tertiary rocks lie about 3.8 km underneath the surface (Fig. 16). The proposed technique of BA interpreted the depth to Pre-Tertiary basement rocks of the caldera about 3.9 km, which agree very well with the drilling and modeling information of Kamata (1989a, b) and Komazawa and Kamata (1985).

Fig. 16
figure 16

Interpretive N-S profile of the Shishimuta caldera (modified after Kamata 1989b)

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Conclusion

Estimation of an appropriate and precise model for representing the subsurface structures is critical in gravity data interpretation. The inversion process, which involved model parameter estimation, represents the main stage in the geophysical data analysis and interpretation and produces an appropriate model relying on the physical characteristic of the deliberated environment. A new metaheuristic algorithm called the Bat algorithm, based on the echolocation behavior of bats, was applied to gravity data to obtain the best convenient characteristic parameters and discover a satisfactory model. The best-interpreted model parameters (depth, origin location, amplitude coefficient, and geometric shape factor) are obtained corresponding to the minimum objective function after reaching the global best solution. The BA technique does not require a priori information; The developed inversion technique of the BA is simple, fast, accurate, and easy to be applied to gravity datasets. Furthermore, the appropriate efficiency and accuracy of the proposed BA technique have been confirmed on synthetic datasets with noise-free & noisy examples, as well as on multiple structures to test the interference effect. Finally, the BA technique is successfully applied to two field data examples from India and Japan for geothermal exploration and volcanic activity study. The inverted outcomes of the BA technique are found in good agreement with the available drilling & geological information and the published literatures. We concluded that the proposed technique is helpful in geothermal investigations and the study of volcanic activity and we recommend that the proposed technique of BA can extend to mineral & ore exploration in future studies.