A novel method based on improved SFLA for IP information extraction from TEM signals

Abstract

Extracting induced polarization (IP) information from transient electromagnetic (TEM) signals is crucial for the exploration of deep mineral, oil, and gas resources.. Linear inversion technology is the preferred method for extracting IP information, but it is associated with three primary drawbacks: dependence on the initial conditions, susceptibility to falling into a local optimum, and a significant lack of uniqueness. To solve the above problems, this study presents an improved shuffle frog leaping algorithm (ISFLA) that incorporates tent chaotic distribution and an adaptive mobile factor, which is employed to extract IP information. First, a tent chaotic operator is adopted to enhance the initial population distribution, thereby improving the global search capability. Then, an adaptive mobile factor is designed to replace the random operator, balancing local and global searches. This adjustment increases solution accuracy and ensures stable convergence in the later stages. Finally, TEM inversion for a 1D layered geoelectric model with IP information is performed using the proposed ISFLA approach. The inversion results show that the ISFLA method can more effectively reconstruct the geoelectric structure, extract IP information, and exhibit greater robustness. Compared to other heuristic algorithms, the proposed method achieves superior global search ability and inversion accuracy, making it well-suited for IP information extraction.

Introduction

The transient electromagnetic method (TEM) and the induced polarization (IP) method are widely applied in geophysical exploration, including mineral exploration, hydrogeology, and engineering investigations^1,2,3,4,5. TEM studies the conduction properties, while IP focuses on the polarization effects of rocks and minerals. Both methods have well-established theoretical foundations and practical applications^6,7. The electromagnetic response of a geoelectric body reflects both electromagnetic induction and the IP effect. Due to the IP effect, the resistivity of polarizable subsurface objects is taken as a complicated function of frequency. Integrating the TEM and IP methods enhances the modeling of real geoelectric scenarios and allows the extraction of additional physical parameters, improving the inversion accuracy of the time domain electromagnetic method. The Dias model and the Cole–Cole model are usually used to simulate the IP effects observed in the Earth^8,9. Furthermore, a fundamental approximate theory has shown that the TEM response is affected by the IP effect¹⁰. Yin et al. analyzed the impact of various complex resistivity parameters on response data through forward calculations¹¹.

For the time domain electromagnetic method, one of the most immediate impacts of the IP effect is the data signal reverses in the late period¹². However, the traditional inversion imaging method based on the resistivity model cannot effectively handle the reversed data signal; therefore, it is essential to conduct time domain electromagnetic inversion that incorporates the IP effect. Over the past decade, significant advancements have been made in extracting IP information using time domain electromagnetic methods¹³. The time domain electromagnetic response with the IP effect was fitted using a basis function, and the least squares method was employed to solve the inverse problem, successfully obtaining IP information¹⁴. Subsequently, the IP information could be gradually extracted, but the regularization algorithm’s inversion results were highly sensitive to noise¹⁵. An IP information extraction method based on the thin plate model was developed¹⁶, but it was only applicable to shallow conductors. Yu et al. introduced a TEM data inversion approach based on the singular value decomposition (SVD) algorithm to extract IP parameters from field data⁵. Kang and Oldenburg performed data inversion for the 3D time domain electromagnetic method with the IP effect, successfully obtaining a polarization response¹⁷. Pelton et al. used the 1D inversion algorithm with lateral constraints to facilitate IP information extraction from time domain electromagnetic data, and their results showed that the proposed approach is highly effective for estimating IP parameters, particularly resistivity and polarizability⁹. Lin et al. further enhanced the 1D lateral constraint inversion algorithm¹⁸ by establishing an optimal initial model, setting an appropriate regularization factor, and transforming the model space. This approach successfully obtained resistivity and polarizability values that closely approximated the true values.

The studies mentioned above have demonstrated the feasibility of extracting IP information from standard electromagnetic (EM) measurements. However, utilizing IP information introduces and often exacerbates the challenges associated with traditional EM data collection and inversion⁴⁸. EM inversion is an ill-posed problem, rendering solutions susceptible to non-uniqueness and instability. Over the years, deterministic approaches have effectively addressed ill-posedness^19,20,21,22, providing unique and stable solutions by incorporating prior information via different regularization strategies^23,24. Despite these advancements, deterministic inversion approaches face limitations, including: 1) a limited capacity to formalize arbitrary and realistic prior information, and 2) a lack of reliable assessment of uncertainty in the final reconstructions. In response, statistical approaches have gained popularity due to their flexibility in defining prior distribution samples^25,26. Additionally, these methods provide a natural framework for exploring the model space and offer reliable estimates of solution uncertainty, including potential non-Gaussianity of the posterior distribution²⁷. Among statistical approaches, trans-dimensional strategies have been developed to conserve computational resources^28,29. However, the results obtained from these methods may be questionable³⁰. Recently, intermediate approaches have also been explored. These methods allow for a flexible definition of prior samples but focus on the maximum likelihood solution (the minimizer of the data misfit function), which risks overfitting the noisy measurements and may yield a data model that is not representative of reality. They are based on global optimization algorithms such as genetic algorithms³¹ or particle swarm optimization³². The present research aligns with global minimization strategies of the data misfit function, based on 1D forward modeling that incorporates IP effects. While this approach increases the complexity of forward modeling by including IP effects, it still overlooks the inherently 3D nature of the actual data.

Although the above nonlinear inversion methods have been widely used in EM data inversion, there are some well-known shortcomings for TEM data with IP information. First, the resistivity of underground polarizable bodies is modeled as a complicated function that varies with frequency, increasing the computational burden during forward iterations. This, in turn, reduces the computational efficiency of Monte Carlo search inversion algorithms (e.g., SA, PSO, and GA). Second, the inclusion of IP parameters in the inversion model complicates the nonlinear inversion process, making convergence more challenging. Therefore, it is a key problem to find an efficient nonlinear inversion method that effectively balances both global and local searche strategies for extracting IP information from TEM signals.

The shuffle frog leaping algorithm (SFLA) is a heuristic swarm intelligence search algorithm that effectively intergrates global information interaction with local search through a grouping operator and meme search³³. During the evolutionary process, only the position of the worst individual in the population is modified. In contrast to other swarm intelligence search algorithms that require adjustments to all individuals, the SFLA demonstrates higher execution efficiency. Consequently, this study proposes an improved SFLA (ISFLA) for TEM signals containing IP information, based on tent chaotic distribution and an adaptive mobile factor for extracting IP information. The proposed approach offers several advantages: (1) The tent chaotic distribution operator is utilized to adjust the initial population distribution to improve the population diversity within the meme group and enhance the global search capabilities. (2) An adaptive mobile factor is designed to replace the random operator to balance the early search and late convergence capabilities. Simulation results demonstrate that the proposed ISFLA approach is more effective than the existing methods in reconstructing the geoelectric structures and extracting IP information. Thus, it exhibits strong robustness in environments with interference noise. In comparison other nonlinear inversion algorithms, the proposed algorithm offers superior convergence performance and inversion accuracy, and can effectively extracting IP information from TEM signals.

TEM forward modeling theory with IP information

Dias model

Numerous mathematical models have been developed to delineate rocks and minerals with IP information, and the Dias model, proposed by Dias in 1968, is one of the most used³⁴. The expression for calculating complex resistivity in the Dias model as follows:

$$\rho \left( \omega \right){ = }\rho \left( {1 - m\left( {1 - \frac{1}{{1 + i\omega \tau^{\prime}\left( {1 + \frac{1}{\mu }} \right)}}} \right)} \right)$$

(1)

where $\mu = i\omega \tau + \left( {i\omega \tau^{\prime\prime}} \right)^{1/2}$, $\tau^{\prime} = \frac{{\tau \left( {1 - \delta } \right)}}{{\delta \left( {1 - m} \right)}}$, $\tau^{\prime\prime} = \tau^{2} \eta^{2}$, and $\rho \left( \omega \right)$ represents the complex resistivity related to frequency, including the polarization efficiency; ρ is taken as the resistivity without polarization efficiency; m signifies the polarizability; τ refers to the time constant; $\eta$ is the electrochemical parameter; and $\delta$ is the polarization resistivity coefficient.

Cole–cole model

The Cole–Cole model, initially proposed by the Cole brothers and later refined by Pelton et al.⁹, is a widely used model for interpreting IP responses. The formula for calculating complex resistivity in the Cole–Cole model is as follows:

$$\rho \left( \omega \right) = \rho \left( {1 - m\left( {1 - \frac{1}{{1 + \left( {i\omega \tau } \right)^{c} }}} \right)} \right)$$

(2)

where c is considered to be the correlation coefficient of the frequency.

TEM forward modeling

The x- and y-axes in the Cartesian coordinate system are located on the ground, with the vertical z-axis oriented downward. A circular transmitting coil with a horizontal radius is placed on the ground, and the transmitted current is designated as I. Kaufman’s derivation indicates that the z-component of the vertical magnetic field (H_z) generated by the TEM central loop source in the frequency domain is given by³⁵:

$$H_{z} = Ia\int_{0}^{\infty } {\frac{{\lambda^{2} }}{{\lambda + {{\lambda_{1} } \mathord{\left/ {\vphantom {{\lambda_{1} } {R_{1}^{ * } }}} \right. \kern-0pt} {R_{1}^{ * } }}}}J_{1} \left( {\lambda a} \right){\text{d}}\lambda }$$

(3)

where J₁ denotes the first-order Bessel function, λ represents the integral variable for the Hankel transformation, and R₁^* is defined as the input impedance in the first layer, which can be determined using the recurrence formula³⁵. The frequency domain electromagnetic response, as specified in Eq. (3), can be calculated using the Hankel integral with 47-point filter coefficients. Subsequently, the time domain electromagnetic response (TEM data) can be obtained through the Gaver-Stepfest transformation and convolution with the transmitted current.

Additionally, by substituting the actual resistivity from Eq. (3) into the complex resistivity expressed in Eq. (1) or Eq. (2), the TEM response data with the IP effect can be obtained.

TEM forward modeling procedure with IP information

The following section describes the TEM forward modeling procedure with the IP effect:

• (1) The geoelectric model parameters (including resistivity, thickness, and polarizability) are defined;

• (2) The IP response model, known as the Dias model or Cole–Cole model, is presented;

• (3) The real resistivity ρ(w) in the geoelectric model is replaced with the frequency-dependent complex resistivity ρ;

• (4) The impedance transformation between adjacent layers are determined based on the complex resistivity of each layer;

• (5) The TEM response data containing IP information are obtained by calculating the theoretical impedance of the ground using the recurrence formula.

Improved shuffled frog leaping algorithm inversion

SFLA basic theory

The SFLA mathematical model is described as follows: P represents the frog population, X(0) = {X₁, X₂, $\cdots$, X_P} symbolizes the initial frog population, and d identifies the dimension of the feasible solution space. Then, X_i = [x_i1, x_i2, $\cdots$, x_id] is set as the i-th frog in the solution space. The initial population of frogs is generated randomly and distributed uniformly throughout the solution space. Frogs are then ranked by their fitness levels in descending order , and the population is divided into k meme groups. Each group contains n frogs, with the relationship P = k $\times n$. The grouping is as follows: the first frog is placed in the first meme group, and the second frog in the second meme group, and so on, until the k-th frog is assigned to the k-th meme group. Then, the (k + 1)-th frog is placed the first meme group, and the (k + 2)-th frog is placed in the second meme group, and this process repeats until all frogs are assigned to a group. The specific expression for meme grouping is as follows:

$${\mathbf{M}}_{i} = \left\{ {{\mathbf{X}}_{i + m(l - 1)} \in {\mathbf{P}}\left| {1 \le l \le k} \right.} \right\}, \, 1 \le i \le k$$

(4)

where M_i represents the i-th meme group. The frogs with the best and worst fitness levels are designated as X_b and X_w, respectively, and the frog with the highest fitness in the entire population is designated as X_g.

Then, the worst frog in each meme group is identified through a local search operation (the local search iterations are denoted as iter_l), and the update formula is defined as follows:

$$\Delta_{w} \left( {t + 1} \right) = rand \times \left( {{\mathbf{X}}_{b} \left( t \right) - {\mathbf{X}}_{w} \left( t \right)} \right)$$

(5)

$${\mathbf{X}}_{w} \left( {t + 1} \right) = {\mathbf{X}}_{w} \left( t \right) + \Delta_{w} \left( {t + 1} \right),S_{min} \le \Delta_{w} \left( t \right) \le S_{max}$$

(6)

where rand represents a random value between 0 and 1, t is regarded as the number of iterations, and S_min and S_max are taken as the allowable movement ranges for the frogs.

During the local update within the meme group, if frog X_w(t + 1) exhibits a superior fitness level compared to X_w(t), it replaces the latter. Otherwise, Eq. (7) replace Eq. (5), and the local search is repeated:

$$\Delta_{w} \left( {t + 1} \right) = rand \times \left( {{\mathbf{X}}_{g} \left( t \right) - {\mathbf{X}}_{w} \left( t \right)} \right)$$

(7)

If the frog’s fitness still not improved, X_w(t) is replaced by a new frog, X_w(t + 1), that is randomly generated. Then, all the frogs in the meme group are globally shuffled (i.e., remixed and sorted), and the meme group is divided again for the local search. The process continues until the defined global iteration value iter_g is reached.

Improved SFLA algorithm (ISFLA)

The traditional SFLA model achieves a better balance between local and global search performance by extracting IP information from TEM signals, which enhances both the global optimal solution and execution efficiency. Nevertheless, the SFLA has several limitations, including an uneven initial population distribution, a random moving step size, and slow convergence speed. To solve the above problems, this paper proposes an improved SFLA (ISFLA) model that utilizes tent chaotic distribution and an adaptive mobile factor to extract IP information.

Chaotic distribution is a uniform distribution function characterized by randomness, ergodicity, and regularity, which helps maintains population diversity. It is widely applied in swarm intelligence algorithms^36,37,38. Compared with other chaotic maps, the tent map has piecewise linearity, making it advantageous for constructing a uniform initial population. Therefore, the tent map is selected to construct the initial population, and its specific formulation is expressed as follows:

$$x_{n + 1} = \left\{ {\begin{array}{*{20}c} {ax_{n} ,{ 0} \le x_{n} \le 0.5 \, } \\ {a\left( {1 - x_{n} } \right),{ 0}{\text{.5}} \le x_{n} \le 1 \, } \\ \end{array} } \right.$$

(8)

The distribution range of chaotic sequence values is denoted by $x_{n} \in \left[ {0,1} \right]$, with a representing the control parameter (typically, a = 2).

In our study, we propose the use of a tent chaotic distribution to improve the initial population structure, mitigating the effects of uneven distribution in the traditional SFLA model. Furthermore, the traditional SFLA model employs a single random operator, denoted as rand (as shown in Eq. (5) and Eq. (7)), which restricts the local search range within each meme group to either X_b—X_w or X_g—X_w. This limitation can cause the algorithm to become trapped in local optima during later stages. To address this , we introduce an adaptive mobile factor designed to improve the rand operator, as outlined below:

$$\Delta_{w} \left( {t + 1} \right) = \tan \left( {\frac{\pi }{4} \times \frac{i}{{iter_{g} }}} \right) \times \left( {{\mathbf{X}}_{b} \left( t \right) - {\mathbf{X}}_{w} \left( t \right)} \right)$$

(9)

$$\Delta_{w} \left( {t + 1} \right) = \tan \left( {\frac{\pi }{4} \times \frac{i}{{iter_{g} }}} \right) \times \left( {{\mathbf{X}}_{g} \left( t \right) - {\mathbf{X}}_{w} \left( t \right)} \right)$$

(10)

where i represents the current iteration.

As illustrated in Eqs. (9) and (10), the moving step size gradually increases with the number of iterations. The adaptive mobile factor is used to adjust the moving step, enhancing the global search capability during the initial stages of the local search. This adjustment broadens the optimization range for individual frogs, enabling better distribution across the feasible region. In the later stages, the moving step size increases significantly to maximize the global search potential. Furthermore, the adaptive adjustment of the moving step mitigates unnecessary detailed searches, thereby speeding up convergence.

As shown in Eqs. (7)-(10), an improved SFLA model based on tent chaotic distribution and an adaptive mobile factor is presented. The primary advancements of the presented approach are as follows: (1) A tent chaotic distribution is used to improve the initial population, which ameliorates the population uniformity within the meme group and enhances the global search ability. (2) The adaptive mobile factor replaces the random operator (i.e., rand). In the early stage of evolution, a smaller moving step is chosen to appropriately restrict the global search ability of the individual frogs, which helps maintain population diversity and prevents premature convergence. In the late stages, a larger moving step is used to ensure the global search ability and prevent falling into local optima, thereby accelerating the convergence and improving accuracy.

Inversion framework

The inverse problem for IP information extraction is regarded as an optimization problem that involves exploring a set of geoelectric models to best explain the observed data. Suppose that the mapping F between model space m and data space d be defined as follows (Kratzer and Macnae, 2012)¹⁴:

$${\mathbf{d}} = F\left( {\mathbf{m}} \right)$$

(11)

The data space, designated by d, is represented by N vertical magnetic field (i.e., H_z = [H_z1, H_z2, $\cdots$, H_zN]), and model space m is expressed as l layer model parameters (i.e., m = [ρ₁, ρ₂, $\cdots$, ρ_l; h₁, h₂, $\cdots$, h_l1; m_c1, m_c2, $\cdots$, m_cl]), which consist of (3l-1) model parameters. Additionally, ρ_i, h_i, and m_ci represent the resistivity, thickness, and polarizability of the i-th layer, respectively.

Assuming that the real earth m_true is in the model space and that the observation data d_obs is noise-free, the equation is defined as follows:

$${\mathbf{d}}_{obs} = F\left( {{\mathbf{m}}_{true} } \right)$$

(12)

Accordingly, the inversion process depends on three elements: 1) Calculate the forward model F for transfer impedance. 2) Define an objective function that aligns with model fitting criteria. 3) Determine the search algorithm for the optimal model m_true. In our study, the forward model F is adopted, as detailed in Sect. “TEM forward modeling”. The ISFLA model is employed as the optimization algorithm, with the objective function specified as follows:

$$Fitness = \sqrt {\frac{1}{N}\sum\limits_{{{\text{i}} = 1}}^{N} {[(E_{i}^{obs} - E_{i}^{pre} )/E_{i}^{pre} ]^{2} } } {\kern 1pt}$$

(13)

where $E_{i}^{obs}$ is regarded as observation data, and $E_{i}^{pre}$ represents the inversion data. During TEM inversion, the fitness function is described as the fitting error for the vertical magnetic field (H_z), aiming to minimize the this fitness function. Furthermore, in the synthetic example presented in this paper, the geoelectric model with the lowest fitness value for the TEM data is selected as the optimal inversion model, which alleviates the ill-posed problem associated with TEM inversion.

Inversion steps

In this study, we propose an ISFLA model based on tent chaotic distribution and an adaptive mobile factor for IP information extraction. The specific implementation are as follows:

• (1) Initialize the population: using the iterations iter, the population size P, the quantity of meme groups k, and the control parameter a, generate the initial population within the optimization range of geoelectric model parameters;

• (2) Calculate the fitness function: according to the frog population, the TEM forward algorithm with IP information is used to calculate the fitness function.

• (3) Grouping operator: the frogs are ranked in decreasing order of fitness values and assigned to k meme groups, and the globally optimal frogs are updated simultaneously;

• (4) Local position update operator: the frog with the lowest fitness value in each meme group is updated using the local position operator;

• (5) Global shuffling: Frogs within each meme group are shuffled to create a new population and update the global optimal frog X_g;

• (6) Termination condition: If the termination condition is met, the current global optimal solution is saved. If not, the process returns to step 2;

• (7) Result output: the inversion results are evaluated based on the global optimal output solution.

The flowchart of the inversion algorithm is presented in Fig. 1.

Fig. 1

Flow chart of the inversion algorithm.

Simulation experiment

Evaluation indicator

This study presents an example of extracting IP information from TEM data using the ISFLA inversion approach. Six theoretical geoelectric models are constructed to verify the efficacy of the proposed method. In the synthetic tests, the parameterization of the real model and the discretization used for the inversion are identical. While this approach artificially enhances the algorithm’s reconstruction performance, it is considered suboptimal. Although the number and positions of the model parameters are unknown in practical applications, limiting its practical advantages, the proposed algorithm demonstrates the ability to avoid human interference and can reconstruct the target geoelectric model with reasonable accuracy. The algorithm simulation is performed in MATLAB R2018a on an Intel (R) Core (TM) i5-7500 processor with 3.40 GHz speed and 8.0 G memory.

To objectively evaluate the inversion efficiency of the ISFLA algorithm for extracting IP information, the mean square error (MSE), absolute percentage error (APE) and total relative error (TRE) are adopted in our study³⁶. The relevant definitions are as follows:

MSE:

$${\text{MSE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {Y_{i} - y_{i} } \right)^{2} }$$

(14)

APE:

$${\text{APE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\frac{{\left| {Y_{i} - y_{i} } \right|}}{{\left| {y_{i} } \right|}} \times 100\% }$$

(15)

TRE:

$${\text{TRE}} = \sum\limits_{j = 1}^{D} {\left| {\frac{{Z_{j} - z_{j} }}{{z_{j} }}} \right|} \times 100\%$$

(16)

where ${\text{y}}_{\text{i}}$ is taken as the TEM data with IP information, ${\text{Y}}_{\text{i}}$ is viewed as the observed data, and N represents the number of observation data channels. Additionally, $Z_{j}$ and $z_{j}$ denote the inversion values and actual values for the layered geoelectric model, respectively, in which D indicates the parameter dimension for the layered geoelectric model. The indicators mentioned above demonstrate the following: MSE denotes the prediction error, APE presents the relative error of the response data, and TRE indicates the relative error for the layered geoelectric model. The smallerthese values, the better the performance of the inversion.

Three-layer (K-type) geoelectric model

Taking the three-layer (K-type) geoelectric model as an example, the inversion performance of the proposed approach is evaluated by studying the polarization layers located at different positions, and the geoelectric model parameters (Dias model) are set as follows: resistivity values of ρ₁ = 50 Ω·m, ρ₂ = 200 Ω·m, and ρ₃ = 20 Ω·m; thickness of h₁ = 100 m and h₂ = 200 m; and a polarizability parameter of m = 0.3. The basic parameters for the ISFLA are as follows: the population size P = 20, the meme group number k = 4, the number of frogs in each meme group n = 5, the local iteration number iter_l = 10, and the global iteration number iter_g = 20.

The inversion results for the polarization layers located in the first layer, middle layer and bottom layer are shown in Table 1. The results indicate that the fitting performance for the bottom layer polarization is the best (the lowest MSE, APE, and TRE values), and that the fitting performance (the highest MSE, APE and TRE values) for the middle layer polarization is the worst. Overall, the inversion results at the different polarization layers show lower MSE, APE and TRE values for the three-layer K-type geoelectric model. Moreover, the inversion results of the geoelectric model with the polarization layer at different positions are illustrated in Fig. 2, which includes the data-fitting curve of the vertical magnetic field (H_z) and the SFLA iteration number.

Table 1 IP information extraction and inversion results for the three-layer (K-type) geoelectric model.

Fig. 2

Inversion results of the three-layer (K-type) geoelectric model with polarization information: (a) resistivity; (b) polarization parameters; (c) H_Z data-fitting curve; (d) iterative curve of the fitness value.

As shown in Fig. 2, the three-layer (K-type) geoelectric model demonstrates a high degree of fit between the H_z inversion data curve and the theoretical model, achieving effective iterative convergence. The proposed method allows for the accurate inversion and extraction of resistivity, thickness, and polarizability parameters. The results indicate that the improved SFLA algorithm, which incorporates a tent chaotic distribution and an adaptive mobile factor, effectively avoids local optima and enhances global search capabilities.

Three-layer (A-type) geoelectric model

This study uses the three-layer (A-type) geoelectric model as an example to evaluate the inversion performance of the proposed method with different polarization models.The geoelectric model parameters are set as follows: resistivity values of ρ₁ = 50 Ω·m, ρ₂ = 100 Ω·m, and ρ₃ = 200 Ω·m; thickness of h₁ = 100 m and h₂ = 200 m; and a polarizability parameter of m₂ = 0.3 (applicable only to middle layer). All other parameters are consistent with those outlined in Sect. “Three-layer (K-type) geoelectric model”.

Table 2 presents the inversion results for both the Dias and Cole–Cole models using the proposed approach, with the polarization layer located in the middle. The evaluation indicators indicate that the inversion result for the Dias model outperforms that of the Cole–Cole model. Furthermore, the inversion results for both models are depicted in Fig. 3, which shows the fitness degree between the H_z data model and the theoretical model, as well as the iterations of the proposed algorithm.

Table 2 IP information extraction and inversion results for the three-layer (A-type) geoelectric model.

Fig. 3

Inversion results of the three-layer (A-type) geoelectric model with an intermediate polarization layer. (a) resistivity; (b) polarizability parameters; (c) H_Z data fitting curve; (d) iterative curve of fitness value.

For the three-layer (A-type) geoelectric model, the Dias model demonstrates superior accuracy in the inversion and extraction of the resistivity, thickness, and polarizability parameters compared to the Cole–Cole model. It also shows better fitting between the Hz data curves and the theoretical data, along with improved iteration convergence. The inversion results prove that TEM nonlinear inversion using the Dias model better represents the actual geoelectric model with IP information.

Three-layer (H-type) geoelectric model

In TEM forward modeling, different response components exhibit varying sensitivities to changes in the geoelectric model parameters. Taking the three-layer (H-type) geoelectric model as an example, we investigate the impact of using the magnetic field component ${H}_{z}$ or apparent resistivity ${\rho }_{a}$ as the fitness function on the inversion results. The geoelectric model parameters are defined as follows: resistivity values of ρ₁ = 100 Ω·m, ρ₂ = 25 Ω·m, and ρ₃ = 200 Ω·m; thickness of h₁ = 100 m and h₂ = 200 m; and a polarizability parameter of m₂ = 0.3 (applicable only to the middle layer). All other parameters are consistent with those specified above.

In our study, the inversion results for the middle layer polarization, with either the magnetic field component ${H}_{z}$ or apparent resistivity ${\rho }_{a}$ taken as the fitting error in the fitness function, are shown in Table 3. The results indicate that using the magnetic field component as the fitting error yields superior inversion outcomes (smaller MSE, APE, and TRE values). Moreover, the data in Table 3 indicate that using the magnetic field component as the fitness function can significantly reduce the computational load during the TEM forward solution, resulting in enhanced calculation efficiency and reduced processing time.

Table 3 IP information extraction and inversion results for the three-layer (H-type) geoelectric model.

The inversion results for the geoelectric model and the fitting of the magnetic field component or apparent resistivity for the Dias model are provided in Fig. 4. The magnetic field component serves as the fitting error in the fitness function, enabling a more accurate inversion and extraction of the resistivity, thickness, and polarizability parameters. The calculated data exhibit a high level of agreement with the observed data. Furthermore, the inversion results reveal that the magnetic field component ${H}_{z}$ is more sensitive to TEM inversion with the IP effect than the apparent resistivity ${\rho }_{a}$.

Fig. 4

Inversion results of the three-layer (H-type) geoelectric model with an intermediate polarization layer: (a) resistivity; (b)polarizability parameters; (c) H_Z data fitting curve; (d) ${\uprho }_{\text{a}}$ data-fitting curve.

Three-layer (Q-type) geoelectric model

Taking the three-layer (Q-type) geoelectric model as an example, this study compares the inversion performance of the proposed tent-SFLA, Logistic SFLA, Chebyshev-SFLA, Henon-SFLA and Kent-SFLA models. The geoelectric model parameters are: resistivity values of ρ₁ = 100 Ω·m, ρ₂ = 25 Ω·m, and ρ₃ = 5 Ω·m; thickness of h₁ = 100 m and h₂ = 200 m; and a polarizability parameter of m₂ = 0.3 (applicable only to the middle layer). All models use the same initial parameters for fair comparison, detailed in Table 4.

Table 4 Parameter settings of SFLA.

For the selected three-layer (Q-type) geoelectric model (Dias model), the inversion results of different chaotic-SFLA models are presented in Table 5.. The model inversion results, fitting degree of the ${H}_{z}$ curve, and the iteration numbers are depicted in Fig. 5. As displayed in Fig. 5 (a) and Fig. 5 (b), all methods effectively perform resistivity and thickness parameter inversion and extract polarizability information. The chaotic tent-SFLA model outperforms other chaotic-SFLA models, especially in extracting precise polarizability parameters. Figure 5 (c) indicates that all chaotic-SFLA methods provide a better fit for the observed data, attributed to their independence from initial values and strong search ability. To further illustrate the inversion performance, the fitness convergence curves for all chaotic-SFLA models are presented in Fig. 5 (d).

Table 5 Inversion performance of various chaotic-SFLA inversion methods.

Fig. 5

Inversion results of the three-layer (Q-type) geoelectric model with an intermediate polarization layer: (a) resistivity; (b) polarizability parameters; (c) H_Z data fitting curve; (d) iterative curve of fitness value.

As illustrated in Fig. 5, all the chaotic-SFLA inversion methods can gradually search for better model parameters, and the chaos operation facilitates a continuous decline in fitness and convergence. According to Table 4 and Fig. 5, the proposed tent-SFLA model achieves a lower fitness value compared with the other chaotic SFLA models, which indicates that the initial population acquisition mechanism of the tent chaotic distribution algorithm possesses a stronger global search capability. Consequently, the polarization information can be effectively extracted in the later stages. The proposed algorithm adopts the initial population acquisition strategy based on a uniform statistical distribution, which enhance its global search ability relative to the other chaotic models, resulting in the attainment of the minimum fitness value and the optimal IP information. Furthermore, the model performance evaluation indicators and fitness convergence curve reveal that the proposed algorithm achieves a faster convergence and greater efficiency.

Four-layer (KH-type) geoelectric model inversion

To verify the ISFLA algorithm’s effectiveness, a four-layer (KH-type) geoelectric model is constructed and compared with the PSO algorithm⁴⁴, the differential evolution (DE) algorithm⁴⁵, the artificial bee colony (ABC) algorithm⁴⁶, and the standard SFLA model⁴⁷. The parameters of the geoelectric model are defined as follows: resistivity values of ρ₁ = 100 Ω·m, ρ₂ = 300 Ω·m, ρ₃ = 50 Ω·m, and ρ₄ = 100 Ω·m; thickness of h₁ = 100 m, h₂ = 200 m, and h₃ = 300 m; and polarizability parameters of m₂ = m₃ = 0.3 (representing only intermediate polarization and underlying polarization). All algorithms have the same initial parameters, detailed in Table 6.

Table 6 Simulation parameter settings for the four-layer (KH-type) geoelectric model.

Considering the Dias model with intermediate polarization, Table 7 compares inversion performance and computation time for nonlinear algorithms.. The SFLA approaches (both the standard SFLA and the proposed ISFLA methods) outperform the PSO, DE, and ABC algorithms. This superiority is due to the higher demands on global search capability when extracting IP information. The SFLA model based on meme group evolution outperforms the PSO, DE, and ABC algorithms, which rely on individual evolution during the global search. The inversion performance of the ISFLA is notably better than that of the standard SFLA model due to the use of the tent chaotic distribution operator in the ISFLA, which generates the initial population and randomly expands the search range of the frogs, thereby enhancing optimization ability. Furthermore, the adaptive mobile factor is employed to balance the relationship between local development and global exploration. These two strategies in the ISFLA improve global search capability, enhance convergence efficiency, and significantly reduce the risk of premature convergence in IP information extraction. However, these update strategies also increases the computing time for the proposed approach.

Table 7 Inversion performance for different optimization algorithms.

Moreover, Fig. 6 presents the model inversion results, the fitting degree of H_z and the fitness iterative curve for all algorithms. As shown in Fig. 6, for the four-layer (KH-type) geoelectric model, the proposed algorithm effectively performs the inversion of resistivity, thickness, and polarizability parameters. The ${H}_{z}$ inversion data align closely with the theoretical values, and the ISFLA achieves a higher convergence speed and a lower fitness value. These results above indicate that the nonlinear ISFLA algorithm, based on tent chaotic distribution and an adaptive mobile factor, achieves superior inversion accuracy and enhance global search capability, making it well-suited for IP information extraction.

Fig. 6

Inversion results of four-layer (KH-type) geoelectric model with intermediate polarization layer: (a) resistivity; (b) polarizability parameters; (c) H_Z data fitting curve; (d) iterative curve of fitness value.

Five-layer (HKH-type) geoelectric model (Robustness analysis)

Due to the presence of noise in the field data, this paper establishes a five-layer (HKH-type) geoelectric model (Dias model) to investigate the inversion robustness of the proposed method on TEM data with different levels of noise. The model parameters are as follows: resistivity values of ρ₁ = 300 Ω·m, ρ₂ = 50 Ω·m, ρ₃ = 500 Ω·m, ρ₄ = 50 Ω·m, and ρ₅ = 1000 Ω·m; thickness of h₁ = 100 m, h₂ = 200 m, h₃ = 300 m, and h₄ = 500 m; and polarization parameters of m₃₌m₄ = 0.3 (indicating polarization for the third and fourth layer only). All other parameters remain as previously mentioned.

With the Dias model and intermediate polarization, the inversion performance of the ISFLA approach on the TEM data with noise is presented in Table 8. As shown in Figs. 7 (a) and (b), the proposed algorithm can precisely reconstruct the resistivity, thickness and polarizability parameters for the HKH-type geoelectric model without noise. For the data with 10% and 20% random noise, the proposed algorithm also effectively inverts the model’s parameters, although the accuracy decreases slightly. Specifically, parameters for the high-resistivity layers and the middle-layer polarizability show greater deviation. Due to the inherent non-uniqueness of inverse problem, numerous potential models can be derived from data with equivalent noise levels. Nevertheless, the inversion results in this study correspond to the best-fitting model that most accurately aligns with the noise data, which effectively represent the real geoelectric model. As shown in Figs. 7 (c) and (d), the inversion results remain relatively stable despite increasing noise levels, demonstrating the robustness of the ISFLA method. Thus, geoelectric model inversion using the ISFLA algorithm can yield consistent results even with noisy data, showing minimal impact from noise. As shown in Table 8, as noise levels rise, the inversion error (MSE, APE, and TRE) gradually increases, but the proposed algorithm consistently maintains a low inversion error due to the tent chaotic distribution and the adaptive mobile factor, which enhance the robustness of the SFLA. Additionally, the computing time for five-layer model inversion is greater than that for the three-layer and four-layer models. This is primarily because the population dimension of the ISFLA algorithm and the computational requirements of the TEM forward model increase as the complexity of the geoelectric model grows.

Table 8 IP information extraction and inversion results of the five-layer geoelectric model with noise data.

Fig. 7

Inversion results of the five-layer (HKH-type) geoelectric model with an intermediate polarization layer: (a) resistivity; (b) polarizability parameters; (c) H_Z data fitting curve; (d) iterative curve of fitness value.

Field data inversion

To verify the practical performance of the proposed algorithm, field data from an exploration area in Sichuan Province, China, were used to evaluate the real application of the ISFLA. Additionally, the proposed inversion method was compared with the conventional SFLA and PSO algorithms. In this survey area, four 520m-long survey lines were deployed, each containing 14 measurement points spaced at 40m intervals. TEM data from all the measurement points on survey line 4# were analyzed. We found that there was a obvious signal reversals in the mid and late phase of some data points, indicating significant impact from the IP effect.

The observed data inversions were conducted using the ISFLA, SFLA and PSO methods, which extracted information on the resistivity and polarizability parameters. The inversion process utilized an initial geoelectric model with 30 layers and a maximum of 20 iterations. Specifically, 1D inversion was performed at the 400m measurement point. The resistivity curves, data fitting curves, and iterative curves generated by these three inversion algorithms are presented in Figs. 8(a)-(c), respectively. The 30 layers of resistivity data are roughly divided into 10 distinct layers, serving as a reference model. Comparatively, the proposed ISFLA method outperformed traditional PSO and SFLA methods, with better fitting to real data and convergence to lower fitness values. The geoelectric model that exhibited the best fit (lowest fitness value) for the TEM data was selected as the optimal inversion model in the field study, which alleviates the inherent ill-posed problem associated with TEM inversion. Furthermore, the proposed method was employed to invert all field data from survey line 4# to infer the geological structure of the exploration area. The resulting resistivity profile and polarizability parameter profile are presented in Fig. 9(a)-(b), indicating a three-layer model distribution: The first layer is a high-resistance region (resistivity greater than 800 Ω·m) with a thickness of approximately 50 m. The second layer is a polarized layer with a highly variable thickness of 30–80 m, a resistivity value of about 200–400 Ω·m, and a polarization parameter of approximately 0.3. The third layer is another high-resistance layer (resistivity greater than 800 Ω·m), which extends to 200 m below the surface. Therefore, it can be inferred that the shallow layer is a high-resistivity layer composed of pebbly soils and Quaternary flood deposits. The intermediate layer consists of gravelly soils, which create a low-resistivity zone due to the significant presence of groundwater. Finally, the deeper layer represents the bedrock of the exploration area. These findings clearly demonstrate the feasibility and effectiveness of the proposed algorithm for field data analysis, particularly in TEM inversion considering the impact of the IP effect, showcasing significant practical value.

Fig. 8

1D inversion results at 400m measurement point: (a) resistivity inversion; (b) inversion data-fitting curve; (c) iterative curve of fitness value.

Fig.9

Inversion results for field data: (a) resistivity profile; (b) polarizability profile.

Conclusion

This study investigates a nonlinear inversion algorithm for extracting IP information from TEM signals. IP information extraction is a complex and ill-posed problem with multiple extrema, making it essential to adopt a nonlinear algorithm to improve the global search ability and achieve accurate fitting of the weak observation data caused by IP anomalies. To address this, an improved shuffling frog leaping algorithm (ISFLA) that incorporates tent chaotic distribution and an adaptive mobile factor was proposed. This model facilitates the effective extraction of IP information from TEM signals.

The inversion results of ISFLA for the different layered geoelectric models demonstrate the following: (1) the proposed algorithm effectively extracts polarization parameters and reconstructs the real geoelectric structure for the layered model with polarization layers at different positions; (2) different polarization models and fitting function indicators influence the inversion outcome, with the Dias model and the fitting function indicator utilizing vertical magnetic field (H_z) yielding superior results; (3) compared to other chaotic optimization operators and nonlinear inversion algorithms, such as SFLA, PSO, DE, and ABC, the presented algorithm offers superior iterative fitness, inversion accuracy, and computational speed; and (4) even with 10% and 20% random interference noise, the proposed algorithm accurately reveals the actual geoelectric structure and IP information, which proves that the ISFLA approach’s strong anti-interference performance. Additionally, the ISFLA algorithm was validated with field data to assess its practical effectiveness. The superior inversion results can be attributed to several factors. The initial population generated based on the tent chaotic distribution operator allows for a broader search range with a uniform distribution in the early stages, which helps maintain the population diversity and enhances global search capability. Furthermore, the adaptive mobile factor replaces the random operator, ensuring a higher convergence rate and stability in the later stages, while improving the accuracy and robustness of IP information extraction.

Due to the complexity of actual geoelectric structures and the noise composition, the field data inversion using the ISFLA algorithm presents several challenges: (1) This study focused on inverting only the resistivity and polarizability parameters, while other parameters, such as the frequency index and time constant in the IP model, also affect the inversion results. Future work will address these parameters to improve the applicability of the approach. (2) Our study tested the ISFLA model using simple field data, which verified its feasibility in extracting IP information from TEM signals. Future research will focus on applying the method to more complex field data. (3) The synthetic tests assumed prior knowledge of layer interfaces, which may lead to an overestimation of the algorithm’s performance in field applications. Future studies will implement adaptive parameterization strategies to mitigate this limitation.