Improved method for surveying groundwater using transient electromagnetic method in high resistivity surrounding rock area
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- SICHUAN HYDROPOWER ENG INVESTIGATION
- Filing Date
- 2023-07-21
- Publication Date
- 2026-05-12
AI Technical Summary
In areas with high resistivity surrounding rock, conventional transient electromagnetic survey methods cannot effectively handle negative responses, resulting in reduced inversion accuracy and survey depth.
Forward modeling was performed using the Cole-Cole model to extract zero-frequency resistivity and three excitation polarization parameters. Groundwater was then interpreted using quasi-two-dimensional regularized Newton inversion, and a joint interpretation was performed by combining zero-frequency resistivity and excitation polarization parameters.
It improves the accuracy and efficiency of groundwater inversion in areas with high resistivity surrounding rock, solves the problems of insufficient inversion accuracy and exploration depth in existing technologies, and realizes multi-parameter interpretation.
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Figure CN116953800B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of surveying technology, and in particular to an improved method for surveying groundwater in high-resistivity surrounding rock areas using transient electromagnetic methods. Background Technology
[0002] Under normal circumstances, transient electromagnetic (TEM) methods for locating groundwater distribution rely on the relatively low resistivity of aquifers compared to the surrounding rock. Therefore, conventional inversion imaging typically uses only resistivity as a single parameter for interpretation. However, in recent years, with the gradual improvement of the signal-to-noise ratio of geophysical survey instruments, the quality of late-stage data has generally improved. It has been observed that in some high-resistivity surrounding rock areas, anomalous negative responses are frequently observed in the late-stage traces of TEM data during groundwater exploration using TEM methods.
[0003] Research has revealed that this is due to the induced polarization effect caused by groundwater containing electronic conductors in high-resistivity surrounding rock areas. The total response of transient electromagnetics can be understood as consisting of an induced response and a polarization response. In the later stages, the direction of the induced polarization current is opposite to that of the induced current. When the intensity of the polarization current in the opposite direction is greater than that of the induced current, that is, when the polarization response in the opposite direction is greater than the induced response in the positive direction, the total response will show a negative response in the later stages. Especially in high-resistivity surrounding rock areas, the transient electromagnetic induced response decays rapidly, and the earlier the polarization response in the opposite direction dominates, the more obvious the negative anomaly becomes.
[0004] When processing data, conventional resistivity inversion methods cannot fit these negative responses. Previous solutions involved removing these negative responses before inversion, which greatly reduced the accuracy and depth of the inversion. Summary of the Invention
[0005] The purpose of this invention is to provide an improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods. Addressing the issue that the negative response in the late-stage traces of transient electromagnetic methods in high-resistivity surrounding rock areas is caused by the induced polarization effect of groundwater, this invention can utilize this negative response to extract three parameters: zero-frequency resistivity and the other three induced polarization parameters. Furthermore, it employs a quasi-two-dimensional regularized Newton inversion interpretation method for groundwater exploration.
[0006] The technical solution adopted by this invention to solve its technical problem is as follows:
[0007] An improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods includes the following steps:
[0008] Acquire high-resistivity surrounding rock areas and collect transient electromagnetic data in high-resistivity surrounding rock areas;
[0009] Check if there is a negative response in the late channel. If so, introduce the Cole-Cole model to replace the resistivity model for forward modeling.
[0010] Zero-frequency resistivity is extracted, along with three excitation polarization parameters: charge rate, time constant, and frequency correlation coefficient.
[0011] The zero-frequency resistivity and three excitation polarization parameters are solved simultaneously using a quasi-two-dimensional regularized Newton inversion.
[0012] The zero-frequency resistivity of the aquifer is relatively low, while the three excitation polarization parameters—charge rate, time constant, and frequency correlation coefficient—are relatively high. The exploration of the underground aquifer is completed by jointly interpreting the zero-frequency resistivity and the three excitation polarization parameters.
[0013] As a further optimization, the acquisition of high-resistivity surrounding rock areas includes the following steps:
[0014] Select the area to be tested;
[0015] Analyze the on-site hydrogeological characteristics and basic geological materials of the area to be tested, as well as the on-site electrical tests, and evaluate whether the area to be tested conforms to the characteristics of high-resistivity surrounding rock.
[0016] Areas within the test area that meet the characteristics of high-resistivity surrounding rocks are selected as high-resistivity surrounding rock regions.
[0017] As a further optimization, when collecting transient electromagnetic data in areas with high resistivity surrounding rock, a receiving coil is used to collect transient electromagnetic data along the survey line at a certain point spacing.
[0018] As a further optimization, after the transient electromagnetic data acquisition is completed, multiple channels are used to display the data. During the data display, it is checked whether a negative response occurs in the late channel.
[0019] As a further optimization, the introduction of the Cole-Cole model to replace the resistivity model for forward modeling specifically refers to:
[0020] When groundwater containing electronic conductors exhibits induced polarization, the Cole-Cole complex resistivity model is used instead of the ordinary resistivity model to quantitatively describe its induced polarization characteristics. Its expression is as follows:
[0021]
[0022] In the formula, To account for the complex resistivity when the polarization effect is excited, Angular frequency, Zero-frequency resistivity, For charging rate, It is a time constant. This is the frequency correlation coefficient.
[0023] As a further optimization, the method of simultaneously solving for the zero-frequency resistivity and three excitation polarization parameters using a quasi-two-dimensional regularized Newton inversion specifically includes the following steps:
[0024] Quasi-2D inversion is a method that uses a one-dimensional model to invert the entire two-dimensional survey line data. The measurement lines at each measurement point are arranged along the column direction to form a one-dimensional column-directed inversion dataset:
[0025]
[0026] In the formula, T represents the transpose of the matrix;
[0027] The model parameters for each measurement point in the inversion setup include zero-frequency resistivity and three excitation polarization parameters. When the inversion model is an N-layer model, the model parameters for the i-th measurement point are:
[0028] In the quasi-two-dimensional inversion algorithm, the model parameter set is composed of the model parameters of all measurement points arranged along the column direction:
[0029]
[0030] Compare data fitting terms with model constraints. The equations for the regularization problem of inversion are formed by merging:
[0031]
[0032] In the formula, is the regularization factor. The least squares data fitting term is expressed as:
[0033] In the formula, F represents the forward modeling process. Weighted matrix of data, The model constraint term is expressed as follows:
[0034]
[0035] In the formula Let R be the model constraint weighting matrix, and R be the lateral constraint operator;
[0036] The regularization problem equation for the inversion is solved using Newton's method, and the expression for the model update equation in the (k+1)th iteration is obtained as follows:
[0037]
[0038] In the formula , Given a Jacobian matrix, the solution is obtained analytically.
[0039] The formula for calculating the fit difference is as follows:
[0040]
[0041] During the inversion process, the iteration stops when the fitting difference is less than 5% or the relative fitting difference changes by less than 1%.
[0042] As a further optimization, three improvement measures are applied during the inversion process, as detailed below:
[0043] Measure 1: Transform the parameters of the inversion model to the logarithmic domain and constrain them within a certain range of variation.
[0044]
[0045] In the formula For parameters of the logarithmic domain transformation model, and Let be the upper and lower limits of the model parameters, respectively. The inverse transform expression is as follows:
[0046]
[0047] Because the model parameters underwent a logarithmic domain transformation, the expression for the Jacobian matrix during the inversion process changes accordingly, and the expression is:
[0048]
[0049] Measure 2: Before performing multi-parameter inversion, early responses without negative values are captured from all measurement points. Conventional resistivity inversion is then performed on the captured responses, and the inversion results are used as the initial zero-frequency resistivity model for the quasi-two-dimensional regularized Newton four-parameter inversion. For the initial models of the three excitation polarization parameters, a uniform half-space model is selected, and a low charging rate of 0.1-0.3 is chosen to ensure that the forward response of the initial model does not exhibit a negative response. The time constant and frequency correlation coefficient are selected at a median value. and ;
[0050] Measure 3: In the early stage of the inversion iteration, fix the time constant and frequency correlation coefficient, and only invert the zero-frequency resistivity and charge rate. In the subsequent inversion process, substitute the time constant and frequency correlation coefficient into the model for updating and inversion, so that the inversion update is concentrated on the zero-frequency resistivity and charge rate with higher sensitivity, and at the same time obtain the abnormal distribution of the time constant and frequency correlation coefficient.
[0051] The beneficial effects of this invention are as follows: Compared to the traditional method of interpreting groundwater in high-resistivity surrounding rock areas using only a single resistivity parameter, this improved method utilizes the induced polarization effect generated by transient electromagnetic waves in high-resistivity surrounding rock areas for groundwater exploration. It integrates four parameters—zero-frequency resistivity, charge rate, time constant, and frequency correlation coefficient—for joint interpretation, achieving multi-parameter interpretation. First, a complex resistivity model quantitatively describing the induced polarization characteristics of rocks and minerals is used to simulate the response containing the induced polarization effect. Then, a quasi-two-dimensional regularized Newton inversion method is used to extract the zero-frequency resistivity and the three induced polarization parameters. Simultaneously, improvements are applied during the inversion process to enhance the stability of the inversion and further improve the accuracy of the interpretation. Attached Figure Description
[0052] Figure 1 This is a flowchart of an improved method for detecting groundwater in high-resistivity surrounding rock areas using transient electromagnetic methods, as described in Embodiment 1 of the present invention.
[0053] Figure 2 This is a schematic diagram of the model established in Embodiment 2 of the present invention, wherein, Figure 2 (a) is the real model. Figure 2 (b) is the composite data profile. Figure 2 (c) is the response at the measuring point at position 0m;
[0054] Figure 3 This is a schematic diagram of the model regularized Newton inversion result in Embodiment 2 of the present invention, wherein, Figure 3 (a) is the zero-frequency resistivity. Figure 3 (b) represents the charging rate. Figure 3 (c) is the time constant. Figure 3 (d) is the frequency correlation coefficient. Figure 3 (e) represents the data fit difference RMS. Detailed Implementation
[0055] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.
[0056] Example 1
[0057] This embodiment provides an improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods. The flowchart is shown below. Figure 1 The method includes the following steps:
[0058] S1. Obtain information about high-resistivity surrounding rock areas and collect transient electromagnetic data in these areas.
[0059] S2. Check if there is a negative response in the late channel. If so, introduce the Cole-Cole model to replace the resistivity model for forward modeling.
[0060] S3. Extract the zero-frequency resistivity, and simultaneously extract the three excitation polarization parameters: charge rate, time constant, and frequency correlation coefficient.
[0061] S4. Simultaneously solve the zero-frequency resistivity and three excitation polarization parameters using quasi-two-dimensional regularized Newton inversion;
[0062] S5. The zero-frequency resistivity of the aquifer is relatively low, while the three excitation polarization parameters—charge rate, time constant, and frequency correlation coefficient—are relatively high. The exploration of the underground aquifer is completed by jointly interpreting the zero-frequency resistivity and the three excitation polarization parameters.
[0063] In high-resistivity surrounding rock areas, the induced polarization effect caused by groundwater containing electronic conductors leads to a transient electromagnetic response that can be understood as consisting of an induced response and a polarization response. In the later stages, the direction of the induced polarization current is opposite to that of the induced current. When the intensity of the reverse polarization current is greater than the induced current, meaning the reverse polarization response is greater than the positive induced response, a negative response will appear in the later stages of the overall response. This is particularly pronounced in high-resistivity surrounding rock areas, where the transient electromagnetic induced response decays rapidly, and the earlier the reverse polarization response becomes dominant, the more significant the negative anomaly. Therefore, this embodiment requires obtaining information about high-resistivity surrounding rock areas. Obtaining high-resistivity surrounding rock areas can include the following steps:
[0064] Select the area to be tested;
[0065] Analyze the on-site hydrogeological characteristics and basic geological materials of the area to be tested, as well as the on-site electrical tests, and evaluate whether the area to be tested conforms to the characteristics of high-resistivity surrounding rock.
[0066] Areas within the test area that meet the characteristics of high-resistivity surrounding rocks are selected as high-resistivity surrounding rock regions.
[0067] In this embodiment, the transient electromagnetic device may include a transmitting system and a receiving system. The transmitting system may be an ungrounded return source or a grounded conductor source. When collecting transient electromagnetic data in a high-resistivity surrounding rock area, the receiving coil is used to collect transient electromagnetic data along the survey line at a certain point spacing.
[0068] Furthermore, after the transient electromagnetic data acquisition is completed, multi-channel data display can be used. During data display, it's possible to check for negative responses in later channels. If a negative response is present, a Cole-Cole model can be introduced to replace the resistivity model for forward modeling. Specifically:
[0069] When groundwater containing electronic conductors exhibits induced polarization, this embodiment uses the Cole-Cole complex resistivity model instead of the ordinary resistivity model to quantitatively describe its induced polarization characteristics. The expression is as follows:
[0070]
[0071] In the formula, To account for the complex resistivity when the polarization effect is excited, Angular frequency, Zero-frequency resistivity, For charging rate, It is a time constant. This is the frequency correlation coefficient.
[0072] It should be noted that the aforementioned method of simultaneously solving for the zero-frequency resistivity and the three excitation polarization parameters using a quasi-two-dimensional regularized Newton inversion may specifically include the following steps:
[0073] Quasi-2D inversion is a method that uses a one-dimensional model to invert the entire two-dimensional survey line data. The measurement lines at each measurement point are arranged along the column direction to form a one-dimensional column-directed inversion dataset:
[0074]
[0075] In the formula, T represents the transpose of the matrix;
[0076] The model parameters for each measurement point in the inversion setup include zero-frequency resistivity and three excitation polarization parameters. When the inversion model is an N-layer model, the model parameters for the i-th measurement point are:
[0077] In the quasi-two-dimensional inversion algorithm, the model parameter set is composed of the model parameters of all measurement points arranged along the column direction:
[0078]
[0079] Compare data fitting terms with model constraints. The equations for the regularization problem of inversion are formed by merging:
[0080]
[0081] In the formula, is the regularization factor. The least squares data fitting term is expressed as:
[0082] In the formula, F represents the forward modeling process. Weighted matrix of data, The model constraint term is expressed as follows:
[0083]
[0084] In the formula Let R be the model constraint weighting matrix, and R be the lateral constraint operator;
[0085] The regularization problem equation of the inversion is solved using Newton's method, which has a high efficiency in inversion iteration. The expression of the model update equation at the (k+1)th iteration is as follows:
[0086]
[0087] In the formula , Given a Jacobian matrix, the solution is obtained analytically.
[0088] The formula for calculating the fit difference is as follows:
[0089]
[0090] In this embodiment, the iteration stops when the fitting difference is less than 5% or the relative fitting difference changes by less than 1% during the inversion process.
[0091] During the inversion process, three improvement measures can be applied to improve the accuracy of the inversion calculation. In this embodiment, the three improvement measures are as follows:
[0092] Measure 1: Imposing reasonable constraints on the model parameters can effectively reduce the problem of multiple solutions in the inversion process and avoid the appearance of model solutions without physical meaning. In this embodiment, the inversion model parameters are transformed to the logarithmic domain and constrained within a certain range of variation:
[0093]
[0094] In the formula For parameters of the logarithmic domain transformation model, and Let be the upper and lower limits of the model parameters, respectively. The inverse transform expression is as follows:
[0095]
[0096] Because the model parameters underwent a logarithmic domain transformation, the expression for the Jacobian matrix during the inversion process changes accordingly, and the expression is:
[0097]
[0098] Measure 2: Before performing multi-parameter inversion, early responses without negative values are captured from all measurement points. Conventional resistivity inversion is then performed on the captured responses, and the inversion results are used as the initial zero-frequency resistivity model for the quasi-two-dimensional regularized Newton four-parameter inversion. For the initial models of the three excitation polarization parameters, a uniform half-space model is selected, and a low charging rate (0.1-0.3) is chosen to ensure that the forward response of the initial model does not exhibit a negative response. The time constant and frequency correlation coefficient are selected at a median value. and ;
[0099] Measure 3: In the early stage of the inversion iteration, fix the time constant and frequency correlation coefficient, and only invert the zero-frequency resistivity and charge rate. In the subsequent inversion process, substitute the time constant and frequency correlation coefficient into the model for updating the inversion. This makes the inversion update mainly focus on the zero-frequency resistivity and charge rate, which have higher sensitivity. At the same time, it can also obtain the abnormal distribution of the time constant and frequency correlation coefficient to better fit the measured data.
[0100] Example 2
[0101] This embodiment is based on Embodiment 1, and establishes the following... Figure 2 The model shown is as follows: Figure 2 As shown in (a), the surface layer is a uniform overburden layer with a thickness of 30m, while the deeper part is a high-resistivity basement. A water-bearing stratum with induced polarization exists in the middle, exhibiting a trapezoidal distribution. The model parameters for each part are shown in Table 1. This profile is 1000m long and has 101 measuring points with a spacing of 10m. One-dimensional forward modeling is performed on each measuring point to synthesize pseudo-two-dimensional profile data. The grounding conductor source is arranged parallel to the measuring line. The parameters of the transmitting and receiving devices are shown in Table 2. A model is generated for each measuring point from 10m... -5 s to 10 -2 The 50-channel response of the measurement point is calculated, and corresponding random noise and background noise are applied to the response of each measurement point.
[0102] The generated noisy data response profile is as follows Figure 2 As shown in (b), the vertical axis is a logarithmic domain coordinate. It can be seen from the figure that a large number of negative responses appear in the late-stage data above the aquifer. The response curve of the measuring line at the 0m position is shown in the figure. Figure 2 As shown in (c), the image coordinates are double logarithmic coordinates, where the solid line is the response without noise, and it can be seen that there is a significant negative response in the late channel (dashed line part), and the dashed line is the response curve after applying two kinds of noise, which shows that it is affected by background noise (dotted line) and random noise, and exhibits jumping and lifting phenomena.
[0103]
[0104]
[0105] The improved regularized Newton inversion algorithm proposed in this embodiment is used to perform trial calculations on this profile data. The first step is to apply corresponding range constraints to the four inversion parameters: zero-frequency resistivity, charge rate, time constant, and frequency correlation coefficient, with range constraints of [1, 5000]. [0,0.98], [0,0.1]s, [0,0.6], Step 2: Extract the first 34 early response data (before 1ms) from all measurement points, use conventional resistivity inversion, and use this inversion result as the initial model for zero-frequency resistivity in regularized Newton inversion. Step 3: The initial model for the three induced polarization parameters is also set the same as that for single-point inversion. In the first 3 iterations of inversion, fix the time constant and frequency correlation coefficient, and only invert zero-frequency resistivity and charge rate. In subsequent iterations, update all four parameters simultaneously. The inversion results are as follows: Figure 3 As shown, from Figure 3 (a)- Figure 3 (e) It can be seen that the abnormal distributions of the four parameters match the actual model distributions well.
[0106] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. An improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods, characterized in that, Includes the following steps: Acquire high-resistivity surrounding rock areas and collect transient electromagnetic data in high-resistivity surrounding rock areas; Check if there is a negative response in the late channel. If so, introduce the Cole-Cole model to replace the resistivity model for forward modeling. Zero-frequency resistivity is extracted, along with three excitation polarization parameters: charge rate, time constant, and frequency correlation coefficient. The zero-frequency resistivity and three excitation polarization parameters are solved simultaneously using a quasi-two-dimensional regularized Newton inversion. The zero-frequency resistivity of the aquifer is relatively low, while the three excitation polarization parameters—charge rate, time constant, and frequency correlation coefficient—are relatively high. The exploration of the underground aquifer is completed through joint interpretation of the zero-frequency resistivity and the three excitation polarization parameters. The method of simultaneously solving for the zero-frequency resistivity and three excitation polarization parameters using a quasi-two-dimensional regularized Newton inversion specifically includes the following steps: Quasi-2D inversion is a method that uses a one-dimensional model to invert the entire two-dimensional survey line data. The measurement lines at each measurement point are arranged along the column direction to form a one-dimensional column-directed inversion dataset: In the formula, T represents the transpose of the matrix; The model parameters for each measurement point in the inversion setup include zero-frequency resistivity and three excitation polarization parameters. When the inversion model is an N-layer model, the model parameters for the i-th measurement point are: In the quasi-two-dimensional inversion algorithm, the model parameter set is composed of the model parameters of all measurement points arranged along the column direction: Compare data fitting terms with model constraints. The equations for the regularization problem of inversion are formed by merging: In the formula, is the regularization factor. The least squares data fitting term is expressed as: In the formula, F represents the forward modeling process. Weighted matrix of data, The model constraint term is expressed as follows: In the formula Let R be the model constraint weighting matrix, and R be the lateral constraint operator; The regularization problem equation for the inversion is solved using Newton's method, and the expression for the model update equation in the (k+1)th iteration is obtained as follows: In the formula , Given a Jacobian matrix, the solution is obtained analytically. The formula for calculating the fit difference is as follows: During the inversion process, the iteration stops when the fit difference is less than 5% or the relative fit difference changes by less than 1%. During the inversion process, improvement measures are applied to perform the inversion. The specific improvement measures are as follows: Step 1: Transform the parameters of the inversion model to the logarithmic domain and constrain them within a certain range of variation: In the formula For parameters of the logarithmic domain transformation model, and Let be the upper and lower limits of the model parameters, respectively. The inverse transform expression is as follows: Because the model parameters underwent a logarithmic domain transformation, the expression for the Jacobian matrix during the inversion process changes accordingly, and the expression is: Step Two: Before performing multi-parameter inversion, early responses without negative values are extracted from all measurement points. Conventional resistivity inversion is performed on the extracted responses, and the inversion results are used as the initial zero-frequency resistivity model for the quasi-two-dimensional regularized Newton four-parameter inversion. For the three excitation polarization parameters, a uniform half-space model is selected as the initial model, with the charging rate ranging from 0.1 to 0.3 to ensure that the forward response of the initial model does not show a negative response. The time constant and frequency correlation coefficient are chosen to be at a median value. and ; Step 3: In the early stage of the inversion iteration, fix the time constant and frequency correlation coefficient, and only invert the zero-frequency resistivity and charge rate. In the subsequent inversion process, substitute the time constant and frequency correlation coefficient into the model for updating and inversion, so that the inversion update is concentrated on the zero-frequency resistivity and charge rate with higher sensitivity, and at the same time obtain the abnormal distribution of the time constant and frequency correlation coefficient.
2. The improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods according to claim 1, characterized in that, The process of obtaining the high-resistivity surrounding rock area includes the following steps: Select the area to be tested; Analyze the on-site hydrogeological characteristics and basic geological materials of the area to be tested, as well as the on-site electrical tests, and evaluate whether the area to be tested conforms to the characteristics of high-resistivity surrounding rock. Areas within the test area that meet the characteristics of high-resistivity surrounding rocks are selected as high-resistivity surrounding rock regions.
3. An improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods according to claim 1, characterized in that, When collecting transient electromagnetic data in areas with high resistivity surrounding rock, a receiving coil is used to collect transient electromagnetic data along the survey line at a certain point spacing.
4. An improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods according to claim 1, characterized in that, After the transient electromagnetic data acquisition is completed, multiple channels are used to display the data. During the data display, it is checked whether a negative response occurs in the late channel.
5. An improved method for groundwater exploration in high-resistivity surrounding rock areas using transient electromagnetic methods according to claim 1, characterized in that, The introduction of the Cole-Cole model to replace the resistivity model for forward modeling specifically refers to: When groundwater containing electronic conductors exhibits induced polarization, the Cole-Cole complex resistivity model is used instead of the ordinary resistivity model to quantitatively describe its induced polarization characteristics. Its expression is as follows: In the formula, To account for the complex resistivity when the polarization effect is excited, Angular frequency, Zero-frequency resistivity, For charging rate, It is a time constant. This is the frequency correlation coefficient.