J. Cent. South Univ. (2021) 28: 1797-1812

DOI: https://doi.org/10.1007/s11771-021-4732-8

Inversion of self-potential anomalies caused by simple polarized bodies based on particle swarm optimization

LUO Yi-jian(罗议建)^{1, 2, 3}, CUI Yi-an(崔益安)^{1, 2, 3}, XIE Jing(谢静)^{1, 2, 3},LU He-shun-zi(陆河顺子)^{1, 2, 3}, LIU Jian-xin(柳建新)^{1, 2, 3}

1. School of Geosciences and Info-Physics, Central South University, Changsha 410083, China;

2. Hunan Key Laboratory of Nonferrous Resources and Geological Hazard Detection,Central South University, Changsha 410083, China;

3. Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education,Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: Prticle swarm optimization (PSO) is adopted to invert the self-potential anomalies of simple geometry. Taking the vertical semi-infinite cylinder model as an example, the model parameters are first inverted using standard particle swarm optimization (SPSO), and then the searching behavior of the particle swarm is discussed and the change of the particles’ distribution during the iteration process is studied. The existence of different particle behaviors enables the particle swarm to explore the searching space more comprehensively, thus PSO achieves remarkable results in the inversion of SP anomalies. Finally, six improved PSOs aiming at improving the inversion accuracy and the convergence speed by changing the update of particle positions, inertia weights and learning factors are introduced for the inversion of the cylinder model, and the effectiveness of these algorithms is verified by numerical experiments. The inversion results show that these improved PSOs successfully give the model parameters which are very close to the theoretical value, and simultaneously provide guidance when determining which strategy is suitable for the inversion of the regular polarized bodies and similar geophysical problems.

Key words: self-potential; inversion; particle swarm optimization

Cite this article as: LUO Yi-jian, CUI Yi-an, XIE Jing, LU He-shun-zi, LIU Jian-xin. Inversion of self-potential anomalies caused by simple polarized bodies based on particle swarm optimization [J]. Journal of Central South University, 2021, 28(6): 1797-1812. DOI: https://doi.org/10.1007/s11771-021-4732-8.

1 Introduction

The self-potential (SP) method is a classical geophysical exploration method, which detects the existence of a natural electric field in the earth through specific instruments and equipment, and processes and interprets the acquired data to achieve the purpose of geological information detection. The SP method has the advantages of simple principles and equipment, high production efficiency, low cost, secondary pollution-free, no disturbance to the geological structure. It is widely used in hydro-geological survey [1, 2], mineral exploration [3, 4], geological mapping [5], archaeological prospection [6], contaminated plume tracking, microbial activity detections and ecological restorations [7-9]. The obtain of SP data is pretty convenient and fast, while the inversion of these data to get the correct geological information is a long-term research focus and difficulty. Traditional inversion methods such as the Newton’s method and the conjugate gradient method have been widely applied in geophysics [10-12]. However, these optimization algorithms are heavily dependent on the selection of the initial solution and are very likely to fall into the local optimal solution once the initial solution is not so appropriate. Therefore, to improve this phenomenon, a large number of heuristic global optimization algorithms have been introduced into geophysical inversion in recent years. Inspired by the foraging behavior of various animal groups, a series of swarm intelligence optimization algorithms such as ant colony optimization (ACO), cockroach swarm optimization (CSO), grasshopper optimization algorithm (GOA), grey wolf optimizer (GWO), particle swarm optimization (PSO) and whale optimization algorithm (WOA), have been proposed. And some of these algorithms have been widely used in the inversion of gravity [13], magnetic [14, 15], seismic [16], and geoelectric fields [17-19]. Inspired by familiar physical phenomena, researchers have proposed a series of optimization algorithms such as black hole algorithm (BHA), simulated annealing (SA), and simulated atomic transition algorithm (SATA), which have also been proved effective in gravity [20], seismic [21] and geoelectric inversion [22]. Inspired by the process of biological reaction and evolution under the stimulus of the external environment, genetic algorithm (GA), immune algorithm (IA) and artificial neural network (ANN), etc., have been proposed and effectively applied in electric methods, especially in the inversion of SP [23-25], DC resistivity [26] and magnetotelluric data [27]. Furthermore, WU et al [28, 29] applied convolutional neural network (CNN) to fault identification in seismic prospecting, breaking the limitation that most intelligent optimization algorithms are difficult in solving high-dimensional problems and contributing new ideas to the development of intelligent optimization algorithms in geophysical exploration. Besides the algorithms mentioned above, differential evolution algorithm (DEA), teaching-learning-based optimization (TLBO), and some other new algorithms have tremendous potential in geophysics as well [30].

PSO is a global optimization algorithm based on swarm intelligence, first proposed by KENNDEY et al [31]. This algorithm tries to find the optional solution by simulating the process of birds searching for food. The SP measurement can be easily interfered, which makes the measured data usually have a lot of noise. PSO can provide uncertainty analysis of the inversion results, and has the advantages of fast convergence, high precision, and high robustness [32, 33] and its effectiveness in the inversion of the synthetic and measured SP data without initial model information has been widely studied and proved. FERNANDEZ-MARTINEZ et al [34] interpreted PSO as a particular discretization of a stochastic damped mass-spring system and studied the first- and second-order stability regions of this algorithm. PEKSEN et al [35] studied the convergence process and the oscillation near the low misfit area of the model parameters. ZHU et al [36] discussed the effects of particle number, velocity proportional coefficient, velocity update strategy, and cost function on the inversion of SP data. GOKTURKLER et al [37] introduced SA, GA, and PSO into the inversion of SP data and compared their effectiveness. Under different circumstances, the emphasis of PSO’s capabilities on exploration and exploitation is often different, so different improved PSOs based on standard particle swarm optimization (SPSO) have been put forward by many scholars. SHI et al [38] discussed the linear change of the inertia weight and proposed PSO based on time decreasing inertia weights. And then they introduced a more flexible version, which can dynamically adapt the inertia weight. ANGELINE [39] proposed PSO based on natural selection (called hybrid PSO). FERNANDEZ-MARTINEZ et al [40] presented a series of improved PSOs with different update order of velocities and positions, resulting in having different exploitation and exploration capabilities. And there are also improved PSOs focusing on other parameters [41].

In this paper, the inversion of SP anomalies of simple geometry is studied using PSO. Firstly, the model parameters and the iteration process of a vertical semi-infinite cylinder (Hereinafter referred to as the cylinder) under different noise levels are obtained using SPSO. And on this basis, the dynamic change of the particle swarm in the process of convergence and the trajectories of different particles in the low misfit area are given, which is helpful to better understand the particle searching behavior. Finally, six improved PSOs aiming at improving the inversion accuracy and the convergence speed by changing the update of particle positions, and inertia weights and learning factors are introduced in the inversion of the cylinder model. And the results are compared with those obtained by SPSO. The default parameter sets of these PSOs are selected by referring to some literatures that have achieved good results, and for some continuously changing parameters in the improved PSO, the variation ranges of these parameters are also limited to the stability region proposed by FERNANDEZ-MARTINEZ et al [40]. Considering that PSOs are very sensitive to the parameter sets such as the inertia weight and the learning factor, the control variable method are adopted to ensure the fairness of the results, that is, each improved PSO only changes one parameter and can be benchmarked with SPSO and other improved PSO. Compared with previous studies, the novelty of this paper is mainly reflected in the following aspects: 1) To further reflect the effectiveness and fault tolerance of PSO, the range of each parameter’s searching space is expanded; 2) Since the parameters of different simple polarized bodies to be inverted are approximate, and considering the length of this paper, we only present the inversion results of the cylinder model; 3) The convergence speed and precision of different improved PSOs are discussed under the perspective of exploration and exploitation; 4) Our experiments can provide guidance when determining which strategy is suitable for the inversion of the regular polarized bodies and similar problems.

2 Theory

2.1 SP anomalies

Simple geometry like a sphere, cylinder and inclined sheet (Figure 1) can be used to replace regular polarized geological bodies. P and Q represent the measuring point and the polarized mass center, respectively. And their SP anomalies formulation is given by:

            (1)

where K is the electric dipole moment; α is the polarization angle; x₀ and h represent the horizontal coordinate and the depth of mass center, respectively; q is the shape factor, and 1.5, 1, and 0.5 corresponding to the sphere, horizontal infinite cylinder and vertical semi-infinite cylinder, respectively.

2.2 SPSO

In a D-dimensional space, a swarm contains n particles, and each particle contains a vector of D-dimensional positionand velocity In each iteration, the particle i will adjust its velocity and position according to its inertia and experience, and the optimal swarm experience, which can be measured according to the fitness function. The formula for updating particles’ state is given by:

            (2)

where v_i, x_i and p_i represent the current velocity, the current position and the historical best position of the particle i, respectively; p_g represents the historical best position of the particle swarm; r₁ and r₂ are the random numbers between 0 and 1; ω is the inertia weight; c₁ and c₂ represent the individual learning factor and the group learning factor, respectively. In each iteration, v_i is first updated and then used to update x_i. According to the respective characteristics of ω, c₁ and c₂, the velocity update formula in Eq. (2) is usually divided into three parts: ωv_i is the memory term, indicating the influence of the previous motion of the particle i; c₁r₁(p_i-x_i) is the self-cognition term that reflects the influence of i-th particle’s historical optimal position on its position update; c₂r₂(p_g-x_i) is the group-cognition term, which reflects the cooperation and knowledge sharing among particles.

Figure 1 Schematic diagram of simple polarized geometry:

2.3 Improved PSO

2.3.1 PSO based on the substitution method

Here we call the half part of the particle swarm with the best fitness value as “good particles”, and the other part as “bad particles”. The main idea of PSO based on the substitution method is to replace “bad particles” with “good particles” in each iteration to accelerate the iteration process. Obviously, this strategy can effectively make the particle swarm gather in some areas with a better position, but it is only suitable for the first few iterations; otherwise, the particle swarm will quickly lose its diversity.

2.3.2 PSO based on the random initialization method

PSO based on the random initialization method redistributes “bad particles” randomly in the searching space at each iteration. This improved PSO can effectively increase the diversity and searching ability of particles through such a searching strategy.

2.3.3 PSO based on adaptive weights

In the iteration process, PSO based on adaptive weights will automatically adjust the inertia weight of each particle according to their relative positions (i.e., the fitness value f). When a particle is in a relatively good position, it will get a larger inertia weight; otherwise, it will get a smaller one. The updating formula of the inertia weight is given by:

      (3)

where ω_max and ω_min represent the given maximal and minimal inertia weights, respectively; f_avg and f_min represent the average and minimal fitness value of the particle swarm, respectively.

2.3.4 PSO based on linear weights

PSO based on linear weights can linearly reduce the inertia weight during the iteration process, which makes the inertial motion of the particle swarm decrease gradually. The update formula of the inertia weight is given by:

                 (4)

where t and maxit represent the current and the maximal number of iterations, respectively.

2.3.5 PSO based on adaptive learning factors

Similar to PSO based on adaptive weights, PSO based on adaptive learning factors can self-adjust the learning factor according to the current positions of the particle individuals. In the iteration process, the update formula of c₁ and c₂ is given by:

      (5)

       (6)

where c_max and c_min represent the maximum and minimum value of the learning factor, respectively. The advantage of using Eqs. (5) and (6) to update the learning factor is that the sum of c₁ and c₂ can be ensured to remain unchanged.

2.3.6 PSO based on linear learning factors

In the iteration process, PSO based on linear learning factors linearly changes the value of c₁ and c₂, respectively, to constantly update the parameter sets. The update formula of c₁ and c₂ is given by:

                  (7)

Similarly, the sum of c₁ and c₂ remains the same during the iteration process.

2.4 Analysis of exploration and exploitation

Discussion of exploration and exploitation of PSO can be found in many papers. TRELEA [42] pointed out that if PSO is more explorative, particles can search the entire searching space more comprehensively, but the convergence speed may be slower. However, if PSO is more exploitative, it can converge to a solution faster (although this may be a non-optimal one). FERNANDEZ-MARTINEZ et al [43] found that the parameter sets (ω, c₁, c₂) can greatly influence the capabilities of exploration and exploitation. Furthermore, they pointed out that PSO is more likely to obtain a good solution with a parameter set located in a certain region. Some parameter sets, such as (ω, c₁, c₂)=(0.729, 1.494, 1.494) used by CLERC et al [44] and (ω, c₁, c₂)=(0.9, 2.0, 2.0) used by SANTOS [45], have achieved good results on the corresponding inversion problems, and these parameter sets are all located in the region mentioned above. While attention should be paid that different parameter sets and different parameter updating strategies usually lead to various exploration and exploitation capabilities. The following is a brief explanation of the impact of parameter adjustment.

PSO based on the substitution method and PSO based on the random initialization method control the optimization process by intervening in the position update of “bad particles” in the searching space at the beginning of the iteration process. PSO based on the substitution method enables particles to converge quickly and continue searching in well-located regions in the first several iterations, which can be more advantageous for unimodal cost functions with only one optimal position. PSO based on the random initialization method enables the poorly positioned particles to be redistributed at the beginning of the iteration process, increasing the diversity of the particle swarm and enabling particles to fully search the whole searching space. Since these two improved PSOs only interfere with the distribution of particles, their parameter sets remain unchanged.

Different strategies are adopted to adjust ω in PSO based on linear weights and PSO based on adaptive weights. For a particle in PSO based on adaptive weights, the smaller the fitness value is, the larger ω it has, and thus the particle is more inclined to obey its own will and move in the direction of the last movement of the particle itself, which can promote the particle’s exploitation of the whole searching space. On the contrary, the larger the fitness value is, the smaller the value of ω is, and the particle’s movement is more affected by the historical optimal position of the particle itself and the historical optimal position of the particle swarm. PSO based on linear weights achieves the transition from exploitation to exploration by continuously decreasing the value of ω during the iteration process [38], but this improved PSO is clearly less flexible than PSO based on adaptive weights.

In addition to the update order of velocities and positions, the relative value of c₁ and c₂ is also one of the factors determining the exploitation and exploration capabilities of PSO. Keep the sum of c₁ and c₂ remain unchanged, PSO will be more explorative when c₁>c₂ and will be more exploitative when c₁2 [32]. Similar to PSO based on adaptive weights, PSO based on adaptive learning factors can adjust c₁ and c₂ according to the fitness value to promote the continuous conversion between exploration and exploitation timely. Similar to PSO based on linear weights, PSO based on linear learning factors can make c₁ and c₂ change linearly so that the algorithm can gradually change from more exploitative to more explorative in the iteration process.

For the six improved PSOs mentioned above, the first two are to adjust the distribution of different particles in the searching space, while the last four are to adjust the parameter sets during the iteration process. The performance of each improved PSO will be verified in subsequent experiments.

3 Inversion of SP anomalies based on SPSO

Here the effectiveness and robustness of SPSO in the inversion of SP anomalies are discussed by taking the cylinder model as an example. The analytical solution is obtained through Eq. (1), and then the random noise is added to simulate the observed data:

   (8)

where U_obs, U_num, U_noise and ΔU represent the observed SP anomalies, the analytical solution, the random noise, and the average SP anomalies, respectively; noise is a constant; rand is a random number between 0 and 1; M is the number of measurement points and U_i represent the SP anomalies of the measurement point i. Then the fitness function is defined by:

                     (9)

3.1 Parameters setting

Unless otherwise specified, the inversion parameters are set as follows: n=50; maxit=200; ω=0.729; c₁=c₂=2.05. The theoretical values of different inversion parameters are set as: electric dipole moment K_t=500 C·m; horizontal coordinate of mass center x_0t=2 m; depth of the mass center h_t= 5 m; polarization angle α_t=60°; shape factor q_t=0.5 (here we refer to the parameters from CUI et al [33]). And the searching spaces are set as:

3.2 Inversion results

When the absolute difference of the fitness value between two iterations is less than 0.01, the convergence condition is considered to be satisfied. In many experiments, we found that the best fitness value in each iteration does not always keep getting smaller; it may stay the same during some adjacent iterations, and then continue to decline. This is because the particle swarm can not always find a better position in every iteration. Considering this phenomenon, we proceed from the maximal iterations (i.e., the value of maxit) to the direction of the first iteration when judging whether the algorithm has converged. When the average absolute difference of the model parameters between two iterations is greater than 0.01, the algorithm is considered to have converged at this iteration.

Several inversion experiments are carried out, and a group of randomly selected inversion results is shown in Table 1 and Figure 2. These results show the effectiveness of SPSO, which can get rather accurate results when the noise level is low. However, it is worth noting that the accuracy of the inversion results decreases with the increase of noise levels, and the main reason for this phenomenon is that the presence of noise may produce additional local minima or increase the size of the low misfit area [32]. Therefore, efforts should be made to reduce noise in practical work. When the random noise is zero, the convergence process of the model parameters is analyzed separately, and the results are shown in Figure 3. It can be seen that before the final convergence, all the model parameters are constantly changing rather than continuously approaching the optimal solution, which is caused by the common constraints of personal experience, group experience, and inertial motion.

3.3 Behavior of particle swarm

In this part, the cylinder model is still adopted as the research object. For the convenience of discussion, we only study the particle behavior of SPSO when inverting K and h, and the other three parameters are set to the theoretical value. Firstly, the change of the particles’ optimal positions in the iteration process is studied. As shown in Figure 4, the contour maps of the low misfit area and the trajectories of the individual particles are drawn. The red point represents the theoretical value of the model parameters, and the broken line represents the trajectories of individual particles. In the process of inversion, p_g is always moving towards the theoretical solution, while individual particles have different states of motion. Some particles stay near the low misfit area and find the right direction at the beginning (such as particle No. 50), so they converge to the place nearby the theoretical solution soon. Some particles stay a little bit further to the theoretical solution (such as particles No. 20 and No. 40), but they can still converge at a relatively fast speed. However, for some particles (such as particles No. 10 and No. 30), their initial positions are far away from the theoretical solution, and their searching behaviors are restricted by many factors, which make the motion trajectories fluctuate greatly. It is conceivable that the existence of these distinct individuals enables the particle swarm to explore the searching space more comprehensively, thus making PSO can achieve remarkable results in the inversion of SP anomalies.

Secondly, the position change of the particle swarm in the iteration process is studied. As shown in Figure 5, the randomly initialized particles are distributed evenly in the whole space. The red point represents the theoretical value of the model parameters, and the white points represent the particle swarm. At the first several iterations, the particles quickly converge toward the low misfit area and the fitness value decreases gradually. Then their searches become slower. In the end, most of the particles converge to the theoretical solution and only a few particles remain in the periphery of the low misfit area. The particle swarm can quickly identify the low misfit area but need to pay a higher time cost to converge to the theoretical solution because of the flatness of the low misfit area.

Table 1 Inversion results of cylinder model based on SPSO

Figure 2 Inversion results of cylinder model with different noise levels:

4 Inversion results of improved PSO

Six different improved PSOs are introduced and discussed in this part by taking the cylinder model as an example. To make the results more convincing, 1000 experiments for each algorithm are conducted and the average value of each parameter is taken as the final results. Table 2 shows the statistical results based on SPSO, which are used to compare with the results of improved PSO.

Figure 3 Convergence process of different parameters when noise=0:

4.1 Inversion results of improved PSO

In order to make PSO based on the substitution method and PSO based on the random initialization method converge effectively, “bad particles” are only replaced in the first 10 iterations, and the subsequent iteration process still adopts the same strategy as SPSO. As shown in Tables 3 and 4, PSO based on the substitution method and PSO based on the random initialization method have obtained good results in the inversion of the cylinder model.

In PSO based on adaptive weights and PSO based on linear weights, ω_max and ω_min are set as 0.8 and 0.6, respectively, and the remaining parameter setting is the same with SPSO. The inversion results are shown in Tables 5 and 6. Compared with SPSO, PSO based on adaptive weights has a great improvement in accelerating convergence, but the accuracy of inversion is obviously reduced, and the mean error of model parameters fluctuates significantly under different noise levels. PSO based on linear weights has obtained good results in inverting the model parameters, but the convergence speed and the accuracy are not improved.

Figure 4 Contour maps of low misfit area of dipole moment K and depth h and trajectories of individual particles:

In PSO based on adaptive learning factors and PSO based on linear learning factors, c_max and c_min are set as 2.05 and 1.5, respectively and the remaining parameters are the same with SPSO. As shown in Tables 7 and 8, two improved PSOs have obtained good results. Compared with SPSO, PSO based on adaptive learning factors has a great improvement in accelerating convergence, while its inversion accuracy is severely reduced. PSO based on linear learning factors has improved both in convergence speed and accuracy.

Figure 5 Contour maps of low misfit area of dipole moment K and depth h and distribution of the particle swarm during iteration process:

Table 2 Inversion results of SPSO

Table 3 Inversion results of PSO based on the substitution method

Table 4 Inversion results of PSO based on the random initialization method

Table 5 Inversion results of PSO based on adaptive weights

Table 6 Inversion results of PSO based on linear weights

Table 7 Inversion results of PSO based on adaptive learning factors

Table 8 Inversion results of PSO based on linear learning factors

4.2 Performance evaluation of improved PSO

To show the performance characteristics of each improved PSO more intuitively, the iteration times and average error of each algorithm in different noise levels are given in Figure 6. From the discussion of exploration and exploitation given in Section 2.4, we can conclude that SPSO has achieved a good balance between exploration and exploitation (c₁=c₂), while those improved PSOs have a different emphasis on exploration and exploitation in different situations. As shown in Figure 6, PSO based on linear learning factors has made considerable improvements in both convergence speed and inversion accuracy. PSO based on adaptive weights and PSO based on adaptive learning factors have also made a great improvement in accelerating convergence. On the contrary, they are far behind SPSO in keeping accuracy. PSO based on the substitution method and PSO based on the random initialization method have higher precision than SPSO, but their convergence speeds are slower. PSO based on linear weights has no obvious improvements in accelerating convergence and even the inversion accuracy is reduced, therefore, it is considered to have no improvements than SPSO.

In addition to the convergence speed and accuracy, the computation complexity of SPSO and different improved PSOs are analyzed by adopting big oh notation. Compared with SPSO, each improved PSO only changes the update strategy of particle positions or partial parameters rather than the structure, so the results are found the same, all of which are O(Dnmaxit), where D is the number of parameters to be inverted, n is the number of particles, and maxit is the maximal iterations.

Furthermore, the average computation time of each algorithm for 200 iterations is counted on a computer with Inter(R) Core(TM) i5-5200U CPU @ 2.20 GHz processor and 4.00 GB of RAM, and the results are shown in Figure 6(c). All these algorithms have completed the optimization process in a very short time. Compared with SPSO, PSO based on adaptive weights and PSO based on adaptive learning factors have a greater runtime because of the addition of judging the adaptive weights and the learning factors, while the other improved PSOs do not change much.

5 Discussion

Experiments show that all versions of PSO have achieved good results in the inversion of SP anomalies, at the same time, the shortcomings of these algorithms have been exposed. It can be seen from previous studies that the selection of the parameter sets has a great impact on the exploration and exploitation capabilities of SPSO. Although some improved PSOs introduced here reduce such impact through certain strategies, from the inversion results shown in Figure 6, however, we may find except for PSO based on linear learning factors, other improved PSOs fail to make breakthroughs in inversion accuracy and convergence speed simultaneously. As pointed out in the no-free-lunch theorem [46], it is usually difficult to improve both exploration and exploitation capabilities of an algorithm by simply adjusting one parameter. That means when we try to improve the exploration capability of an algorithm, its exploitation capability will more likely to be weakened, and vice versa. Simultaneously attention should be made that there are no “one-size-fits-all” parameter sets that can be applied to all the inversion problems. When the parameter sets have achieved great success, we can predict that good results can also be obtained when using this parameter sets in other geophysical problems of the same type (i.e., the same demand for exploration and exploitation capabilities of an algorithm), but these parameter sets and corresponding algorithms may not necessarily achieve similar results when in the face of different types of problems [47]. Therefore, our experiments only provide guidance when determining which strategy is suitable for the inversion of the regular polarized bodies and similar problems. Considering that SPSO and PSO based on linear learning factors have achieved relatively good results in the experiments, these two algorithms are obviously more suitable for the geophysical problems represented by the inversion of the SP anomalies caused by simple polarized bodies, which require relatively balanced exploitation and exploration capabilities of an algorithm. While other improved PSOs may be more suitable for the cases with explorative or exploitative preferences.

Figure 6 (a) Average iteration times, (b) mean error, and (c) computation time of each improved PSO in different noise levels

Limited by the paper length, we do not compare our inversion results with those of other global optimization algorithms. Besides, the lack of measured data prevents us from proving the effectiveness of these algorithms in practical applications. Therefore, the following work will be carried out in the future: 1) Field data will be inverted to further verify the validity of these versions of PSOs; 2) Some other global optimization algorithms will be introduced in the inversion of SP anomalies; 3) Parallel processing and sub-population combination [48] are both optional schemes that will be introduced to make these algorithms more practical and efficient.

6 Conclusions

PSO is adopted to invert SP anomalies of simple geometry (here refer to the cylinder model). Firstly the model parameters are successfully inverted using SPSO. Then the searching behavior of the particle swarm in the inversion of the cylinder model and the change of particle distribution during the iteration process are discussed. The existence of different particle behaviors enables the particle swarm to explore the searching space more comprehensively, thus making PSO can achieve remarkable results in the inversion of SP anomalies. Finally, six improved PSOs aiming at improving the inversion accuracy and the convergence speed by changing the update of particle positions, inertia weights and learning factors are introduced in the inversion of the cylinder model. These improved PSOs have the same computation complexity, and have successfully given the model parameters that are very close to the theoretical value, which indicates their effectiveness in the inversion of SP anomalies. SPSO and PSO based on linear learning factors have been considered to be more suitable for the geophysical problems represented by the inversion of the SP anomalies caused by simple polarized bodies, which require relatively balanced exploitation and exploration capabilities of an algorithm. While other improved PSOs may be more suitable for the cases with explorative or exploitative preferences.

Acknowledgment

The anthors would like to thank YANG gang-qiang for his advice on algorithm coding.

Contributors

The overarching research goals were developed by CUI Yi-an and LUO Yi-jian. LUO Yi-jian conducted the literature review and wrote the first draft of the manuscript. XIE Jing and LU He-shun-zi edited the draft of manuscript. LIU Jian-xin oversaw the orderly conduct of the research.

Conflict of interest

LUO Yi-jian, CUI Yi-an, XIE Jing, LU He-shun-zi, and LIU Jian-xin declare that they have no conflict of interest.

References

[1] JARDANI A, DUPONT J P, REVIL A. Self-potential signals associated with preferential groundwater flow pathways in sinkholes [J]. Journal of Geophysical Research: Solid Earth, 2006, 111(B9): B09204. DOI: 10.1029/2005JB004231.

[2] XIE Jing, CUI Yi-an, ZHANG Li-juan, MA Chang-ying, YANG Bing, CHEN Xiao-le, LIU Jian-xin. 3D forward modeling of seepage self-potential using finite-infinite element coupling method [J]. Journal of Environmental and Engineering Geophysics, 2020, 25(3): 381-390. DOI: 10.32389/jeeg19-038.

[3] KAWADA Y, KASAYA T. Marine self-potential survey for exploring seafloor hydrothermal ore deposits [J]. Scientific Reports, 2017, 7(1): 13552. DOI: 10.1038/s41598-017-13920-0.

[4] HERITIANA A R, RIVA R, RALAY R, BONI R. Evaluation of flake graphite ore using self-potential (SP), electrical resistivity tomography (ERT) and induced polarization (IP) methods in east Coast of Madagascar [J]. Journal of Applied Geophysics, 2019, 169: 134-141. DOI: 10.1016/j.jappgeo. 2019.07.001.

[5] MEHANEE S A. Tracing of paleo-shear zones by self-potential data inversion: Case studies from the KTB, Rittsteig, and Grossensees graphite-bearing fault planes [J]. Earth, Planets and Space, 2015, 67(1): 1-33. DOI: 10.1186/s40623-014-0174-y.

[6] EPPELBAUM L V. Quantitative analysis of self-potential anomalies in archaeological sites of Israel: An overview [J]. Environmental Earth Sciences, 2020, 79(15): 1-15. DOI: 10.1007/s12665-020-09117-w.

[7] ABBAS M, JARDANI A, SOUEID AHMED A, REVIL A, BRIGAUD L, BEGASSAT P, DUPONT J P. Redox potential distribution of an organic-rich contaminated site obtained by the inversion of self-potential data [J]. Journal of Hydrology, 2017, 554: 111-127. DOI: 10.1016/j.jhydrol.2017.08.053.

[8] CUI Yi-an, ZHU Xiao-xiong, WEI Wen-sheng, LIU Jian-xin, TONG Tie-gang. Dynamic imaging of metallic contamination plume based on self-potential data [J]. Transactions of Nonferrous Metals Society of China, 2017, 27(8): 1822-1830. DOI: 10.1016/S1003-6326(17)60205-X.

[9] XIE Jing, CUI Yi-an, ZHANG Li-juan, GUO You-jun, WANG Jia-xin, FANIDI M, LIU Jian-xin. Numerical modeling of biogeobattery system from microbial degradation of underground organic contaminant [J]. SN Applied Sciences, 2020, 2(2): 1-11. DOI: 10.1007/s42452-020-2008-9.

[10] MEHANEE S A. An efficient regularized inversion approach for self-potential data interpretation of ore exploration using a mix of logarithmic and non-logarithmic model parameters [J]. Ore Geology Reviews, 2014, 57: 87-115. DOI: 10.1016/ j.oregeorev.2013.09.002.

[11] SINDIRGI P, OZYALIN S. Estimating the location of a causative body from a self-potential anomaly using 2D and 3D normalized full gradient and Euler deconvolution [J]. Turkish Journal of Earth Sciences, 2019, 28(4): 640-659. DOI: 10.3906/yer-1811-14.

[12] OLIVETI I, CARDARELLI E. Self-potential data inversion for environmental and hydrogeological investigations [J]. Pure and Applied Geophysics, 2019, 176(8): 3607-3628. DOI: 10.1007/s00024-019-02155-x.

[13] LIU Shuang, HU Xiang-yun, LIU Tian-you. A stochastic inversion method for potential field data: Ant colony optimization [J]. Pure and Applied Geophysics, 2014, 171(7): 1531-1555. DOI: 10.1007/s00024-013-0712-8.

[14] AGARWAL A, CHANDRA A, SHALIVAHAN S, SINGH R K. Grey wolf optimizer: A new strategy to invert geophysical data sets [J]. Geophysical Prospecting, 2018, 66(6): 1215-1226. DOI: 10.1111/1365-2478.12640.

[15] LIU Shuang, LIANG Miao, HU Xiang-yun. Particle swarm optimization inversion of magnetic data: Field examples from iron ore deposits in China [J]. Geophysics, 2018, 83(4): J43-J59. DOI: 10.1190/geo2017-0456.1.

[16] ZHANG Jin, SHEN Peng, ZHAO Wei-na, GUO Xu-bing, WANG Xing, CHEN Song, XU Xiu-gang. AVA simultaneous inversion of prestack seismic data using particle swarm optimization [J]. Journal of Earth Science, 2018, 29(6): 1390-1397. DOI: 10.1007/s12583-017-0809-6.

[17] LI Si-yu, WANG Shu-ming, WANG Peng-fei, SU Xiao-lu, ZHANG Xin-song, DONG Zhi-hui. An improved grey wolf optimizer algorithm for the inversion of geoelectrical data [J]. Acta Geophysica, 2018, 66(4): 607-621. DOI: 10.1007/ s11600-018-0148-8.

[18] ABDELAZEEM M, GOBASHY M, KHALIL M H, ABDRABOU M. A complete model parameter optimization from self-potential data using Whale algorithm [J]. Journal of Applied Geophysics, 2019, 170: 103825. DOI: 10.1016/ j.jappgeo.2019.103825.

[19] GOBASHY M, ABDELAZEEM M, ABDRABOU M, KHALIL M H. Estimating model parameters from self-potential anomaly of 2D inclined sheet using whale optimization algorithm: Applications to mineral exploration and tracing shear zones [J]. Natural Resources Research, 2020, 29(1): 499-519. DOI: 10.1007/s11053-019-09526-0.

[20] SRIGUTOMO W, HERIYANTO M, AUFA M H. Gravity inversion of talwani model using very fast simulated annealing [J]. Journal of Mathematical and Fundamental Sciences, 2019, 51(2): 177-190. DOI: 10.5614/j.math. fund.sci.2019.51.2.7.

[21] YU Peng, WANG Jia-lin, WU Jian-sheng, WANG Da-wei. Constrained joint inversion of gravity and seismic data using the simulated annealing algorithm [J]. Chinese Journal of Geophysics, 2007, 50(2): 465-475. DOI: 10.1002/cjg2.1056. (in Chinese)

[22] SUNGKONO, WARNANA D D. Black hole algorithm for determining model parameter in self-potential data [J]. Journal of Applied Geophysics, 2018, 148: 189-200. DOI: 10.1016/ j.jappgeo.2017.11.015.

[23] EL-KALIOUBY H M, AL-GARNI M A. Inversion of self-potential anomalies caused by 2D inclined sheets using neural networks [J]. Journal of Geophysics and Engineering, 2009, 6(1): 29-34. DOI: 10.1088/1742-2132/6/1/003.

[24] DAS M B, SUNDARARAJAN N. Analysis of self-potential anomalies due to 2D horizontal cylindrical structures—An artificial neural network approach [J]. Arabian Journal of Geosciences, 2016, 9(7): 1-10. DOI: 10.1007/s12517-016-2492-9.

[25] DI MAIO R, PIEGARI E, RANI P, CARBONARI R, VITAGLIANO E, MILANO L. Quantitative interpretation of multiple self-potential anomaly sources by a global optimization approach [J]. Journal of Applied Geophysics, 2019, 162: 152-163. DOI: 10.1016/j.jappgeo.2019.02.004.

[26] SCHWARZBACH C, BORNER R U, SPITZER K. Two-dimensional inversion of direct current resistivity data using a parallel, multi-objective genetic algorithm [J]. Geophysical Journal International, 2005, 162(3): 685-695. DOI: 10.1111/ j.1365-246X.2005.02702.x.

[27] WANG He, LIU Meng-lin, XI Zhen-zhu, PENG Xing-liang, HE Hang. Magnetotelluric inversion based on BP neural network optimized by genetic algorithm [J]. Chinese Journal of Geophysics, 2018, 61(4): 1563-1575. (in Chinese)

[28] WU Xin-ming, LIANG Lu-ming, SHI Yun-zhi, FOMEL S. FaultSeg3D: Using synthetic data sets to train an end-to-end convolutional neural network for 3D seismic fault segmentation [J]. Geophysics, 2019, 84(3): IM35-IM45. DOI: 10.1190/geo2018-0646.1.

[29] WU Xin-ming, LIANG Lu-ming, SHI Yun-zhi, GENG Zhi-cheng, FOMEL S. Multitask learning for local seismic image processing: Fault detection, structure-oriented smoothing with edge-preserving, and seismic normal estimation by using a single convolutional neural network [J]. Geophysical Journal International, 2019, 219(3): 2097-2109. DOI: 10.1093/gji/ ggz418.

[30] SUNGKONO. An efficient global optimization method for self-potential data inversion using micro-differential evolution [J]. Journal of Earth System Science, 2020, 129(1): 1-22. DOI: 10.1007/s12040-020-01430-z.

[31] KENNEDY J, EBERHART R. Particle swarm optimization [C]// IEEE International Conference on Neural Networks-Conference Proceedings. New York: IEEE, 1995: 1942-1948.

[32] FERNANDEZ-MARTINEZ J L, GARCIA-GONZALO E, NAUDET V. Particle swarm optimization applied to solving and appraising the streaming-potential inverse problem [J]. Geophysics, 2010, 75(4): WA3-WA15. DOI: 10.1190/ 1.3460842.

[33] CUI Yi-an, ZHU Xiao-xiong, CHEN Zhi-xue, LIU Jia-wen, LIU Jian-xin. Performance evaluation for intelligent optimization algorithms in self-potential data inversion [J]. Journal of Central South University, 2016, 23(10): 2659-2668. DOI: 10.1007/s11771-016-3327-2.

[34] FERNANDEZ-MARTINEZ J L, GARCIA-GONZALO E, FERNANDEZ-ALVAREZ J P. Theoretical analysis of particle swarm trajectories through a mechanical analogy [J]. International Journal of Computational Intelligence Research, 2008, 4(2): 93-104. DOI: 10.5019/j.ijcir.2008.129.

[35] PEKSEN E, YAS T, KAYMAN A Y, OZKAN C. Application of particle swarm optimization on self-potential data [J]. Journal of Applied Geophysics, 2011, 75(2): 305-318. DOI: 10.1016/j.jappgeo.2011.07.013.

[36] ZHU Xiao-xiong, CUI Yian, LI Xi-yang, TONG Tie-gang, JI Tong-xin. Inversion of self-potential anomalies based on particle swarm optimization [J]. Journal of Central South University (Science and Technology), 2015, 46(2): 579-585. (in Chinese)

[37] GOKTURKLER G, BALKAYA C. Inversion of self-potential anomalies caused by simple-geometry bodies using global optimization algorithms [J]. Journal of Geophysics and Engineering, 2012, 9(5): 498-507. DOI: 10.1088/1742-2132/9/5/498.

[38] SHI Y, EBERHART R. A modified particle swarm optimizer [C]// 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360). Anchorage, AK, USA: IEEE, 1998: 69-73. DOI: 10.1109/ICEC.1998. 699146.

[39] ANGELINE P J. Using selection to improve particle swarm optimization [C]//1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360). Anchorage, AK, USA: IEEE, 1998: 84-89. DOI: 10.1109/ICEC.1998.699327.

[40] FERNANDEZ MARTINEZ J L, GARCIA GONZALO E, FERNANDEZ MUNIZ Z, MUKERJI T. How to design a powerful family of particle swarm optimizers for inverse modelling [J]. Transactions of the Institute of Measurement and Control, 2012, 34(6): 705-719. DOI: 10.1177/ 0142331211402900.

[41] RATNAWEERA A, HALGAMUGE S K, WATSON H C. Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients [C]// IEEE Transactions on Evolutionary Computation. New Jersey: IEEE, 2004: 240-255. DOI: 10.1109/TEVC.2004.826071.

[42] TRELEA I C. The particle swarm optimization algorithm: Convergence analysis and parameter selection [J]. Information Processing Letters, 2003, 85(6): 317-325. DOI: 10.1016/S0020-0190(02)00447-7.

[43] FERNANDEZ MARTINEZ J L, GARCIA GONZALO E. The PSO family: Deduction, stochastic analysis and comparison [J]. Swarm Intelligence, 2009, 3(4): 245-273. DOI: 10.1007/s11721-009-0034-8.

[44] CLERC M, KENNEDY J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space [C]// IEEE Transactions on Evolutionary Computation. New Jersey: IEEE, 2004: 58-73. DOI: 10.1109/4235.985692.

[45] MONTEIRO SANTOS F A. Inversion of self-potential of idealized bodies’ anomalies using particle swarm optimization [J]. Computers & Geosciences, 2010, 36(9): 1185-1190. DOI: 10.1016/j.cageo.2010.01.011.

[46] WOLPERT D H, MACREADY W G. No free lunch theorems for optimization [J]. IEEE Transactions on Evolutionary Computation, 1997, 1(1): 67-82. DOI: 10.1109/4235.585893.

[47] ALEARDI M, PIERINI S, SAJEVA A. Assessing the performances of recent global search algorithms using analytic objective functions and seismic optimization problems [J]. Geophysics, 2019, 84(5): R767-R781. DOI: 10.1190/geo2019-0111.1.

[48] ALEARDI M, MAZZOTTI A. 1D elastic full-waveform inversion and uncertainty estimation by means of a hybrid genetic algorithm-Gibbs sampler approach [J]. Geophysical Prospecting, 2017, 65(1): 64-85. DOI: 10.1111/1365-2478.12397.

(Edited by ZHENG Yu-tong)

中文导读

基于粒子群算法的简单极化体自然电场反演

摘要：本文采用粒子群算法反演了简单极化体引起的自然电场异常。首先，引入标准粒子群算法对垂直半无限延伸柱状体的模型参数进行了反演。然后在此基础上讨论了粒子群的搜索行为，研究了粒子群在迭代过程中的位置变化情况。不同粒子行为的存在使粒子群能够更全面地探索整个解空间，保证了粒子群算法可在反演自然电位异常方面取得显著效果。最后，在标准粒子群算法的基础上引入了6种通过改变粒子位置、惯性权值和学习因子等因素的更新策略来提高反演精度或收敛速度的改进粒子群算法，并通过模型反演验证了各算法的有效性。本文通过数值实验对标准粒子群算法及各改进粒子群算法的勘探性与开发性做了讨论，对于在规则极化体自然电场反演中如何确定有效反演策略具有一定的指导意义。

关键词：自然电场；反演；粒子群算法

Foundation item: Projects(41874145, 72088101) supported by the National Natural Science Foundation of China

Received date: 2020-05-10; Accepted date: 2021-01-13

Corresponding author: CUI Yi-an, PhD, Professor; Tel: +86-13975894892; E-mail: cuiyian@csu.edu.cn; ORCID: https://orcid.org/0000-0002-4811- 642X