Method for rapidly selecting center frequency through transient electromagnetic forward modeling in induced polarization medium

Abstract

The invention discloses a method for rapidly selecting a center frequency in transient electromagnetic forward modeling in an induced polarization medium, and belongs to the technical field of geophysical exploration, and the method comprises the following steps: firstly, determining a three-dimensional forward modeling time step iteration parameter and a center frequency range, and constructing a Cole-Cole model; then successively carrying out Pade series approximation on the discretized center frequency vector and the discretized angular frequency vector at each frequency point to obtain an expansion format of each frequency point; the time domain impulse response of the expansion format, the time domain impulse response of the Cole-Cole model and the relative errors of the time domain impulse response of the expansion format and the Cole-Cole model are calculated respectively, relative error two-dimensional distribution is obtained, the Cole-Cole model is expanded through a Pade series approximation method, and three-dimensional forward modeling containing the induced polarization effect is started. According to the invention, a rapid center frequency selection method can be provided for carrying out transient electromagnetic three-dimensional positive calculation when the underground is a polarization medium, and the method is the basis of geophysical transient electromagnetic detection.

Description

Technical Field

[0001] The present invention relates to the technical field of geophysics, and in particular to a method for quickly selecting a central frequency for transient electromagnetic forward modeling in a polarization medium. Background Art

[0002] The transient electromagnetic method is an important branch of geophysical electromagnetic detection methods. The primary field excited by the transient electromagnetic emission source induces a secondary field in the underground medium. By analyzing the received secondary field, the resistivity distribution of the underground medium is inferred to achieve the purpose of underground space detection. When the transient electromagnetic method is used to detect media with relatively high polarizability (polarization media) such as sulfide ores and metal ores, the induced polarization effect will occur. Specifically, when an external electromagnetic field is applied, the charge movement in the medium exhibits non-linear and dispersive characteristics. When using the transient electromagnetic method to detect polarization media, the electromagnetic response obtained sometimes rapidly decays into negative values at a certain moment. The appearance of this sign-changing phenomenon is due to the polarization effect. Therefore, it is very important to carry out three-dimensional simulation of transient electromagnetic in polarization media for accurate detection of underground space by transient electromagnetic.

[0003] In the three-dimensional simulation of transient electromagnetic in polarization media, it is generally considered that the polarization medium parameters satisfy the Cole-Cole model. When using this model for three-dimensional simulation of transient electromagnetic, it is necessary to transform the Cole-Cole model relationship in the frequency domain to the time domain through Fourier transform and then substitute it into the time-domain Maxwell equation for solution. This process will introduce fractional derivatives and form a Maxwell equation set with fractional derivatives. To solve the fractional Maxwell equation set, generally, the fractional Maxwell equation set is transformed into an integer order through the Padé series, and then numerically discretized and solved. When performing the Padé series expansion, it is necessary to approximate and expand with a central frequency as the base point. The selection of the central frequency at different times in the transient electromagnetic simulation affects the approximation accuracy of the Padé series for the fractional partial differential equation.

[0004] Currently, there are two ways to select the central frequency of the Padé series in the three-dimensional simulation process of transient electromagnetic in polarization media: One way is to refer to experience and use a fixed central frequency in a relatively long time period of transient electromagnetic simulation. This way will affect the stability and accuracy of the simulation, resulting in large errors in the signals near the late stage and sign change; Another way is to directly perform adaptive central frequency selection in the three-dimensional simulation process, that is, adjust the central frequency as the transient electromagnetic simulation time changes. This way can improve the accuracy compared with the first way, but since it is necessary to continuously perform three-dimensional forward modeling attempts during the adaptive process, it will consume a large amount of computing time, which is much longer than the time of three-dimensional simulation of transient electromagnetic in non-polarization media, and is completely unacceptable in complex models and inversion processes.

[0005] Based on the above, a method for quickly selecting the central frequency in the transient electromagnetic forward modeling of a polarization medium is proposed. Summary of the Invention

[0006] The purpose of the present invention is to provide a method for quickly selecting the central frequency in the transient electromagnetic forward modeling of a polarization medium to solve the problems in the background technology.

[0007] To achieve the above object, the present invention provides a method for quickly selecting the central frequency in the transient electromagnetic forward modeling of a polarization medium, including the following steps:

[0008] S1. Determine and input the three-dimensional forward time step iteration parameters and the central frequency range according to the simulation time range and simulation accuracy requirements of the three-dimensional forward model;

[0009] S2. Discretize the central frequency range using logarithmic averaging to obtain the discretized central frequency vector and the corresponding angular frequency vector;

[0010] S3. Read the model file to obtain the polarization medium parameters and construct the Cole-Cole model;

[0011] S4. According to the model in S3, perform Padé series approximation on the central frequency vector and the angular frequency vector in S2 at each frequency point to obtain the expansion format at each frequency point;

[0012] S5. Calculate the time-domain impulse response of the expansion format in S4 and the time-domain impulse response directly calculated by the Cole-Cole model in S3 respectively;

[0013] S6. Calculate the relative error of the two time-domain impulse responses obtained in S5 to obtain the two-dimensional distribution of the relative error with respect to the acquisition time and the central frequency;

[0014] S7. According to the two-dimensional distribution, use the Padé series approximation method to expand the Cole-Cole model and start the three-dimensional forward modeling with the induced polarization effect until the requirements of the simulation time range are met.

[0015] Preferably, in S1, the three-dimensional forward time step iteration parameters include the start time point, the end time point, the initial time step, and the time step doubling factor.

[0016] Preferably, in S2, the discretized central frequency vector is expressed as:

[0017] f = (f1 f2 … f end );

[0018] The corresponding angular frequency vector is expressed as:

[0019] ω = (ω1 ω2 … ω end)。

[0020] Among them, f1, f2, …, f end represent the separated center frequencies, and ω1, ω2, …, ω end represent the corresponding separated angular frequencies.

[0021] Preferably, in the step S3, the polarization medium parameters include direct current conductivity, charging rate, frequency correlation coefficient, and time constant;

[0022] The construction process of the Cole-Cole model is as follows:

[0023] S31. Establish the relevant relationship of the Cole-Cole model, which is expressed by the formula:

[0024]

[0025] Among them, σ(ω) is the frequency-dependent conductivity, σ0 is the direct current conductivity, η is the charging rate, c is the frequency correlation coefficient, τ is the time constant, ω is the angular frequency vector, and i is the imaginary unit;

[0026] S32. Define the frequency correlation coefficient c < 1, perform Padé series expansion and Taylor series expansion on the fractional derivative, and substitute the formula of the relevant relationship of the Cole-Cole model into Ohm's law J = σ(ω)E to obtain the form of Ohm's law under the Cole-Cole polarization medium model:

[0027] σ0E + σ0τ c (iω) c E = J + (1 - η)τ c (iω) c J;

[0028] Among them, J is the current density and E is the electric field strength;

[0029] S33. Perform Padé approximation on the fractional exponential term (iω) c and express it as:

[0030]

[0031] Among them, P m and Q n are the Padé series correlation coefficients, P(iω) and Q(iω) represent the individual coefficients of each order expansion in the form of Padé series expansion, P 和 (iω) and Q 和 (iω) are respectively the sum of a group of coefficients of each order expansion in the form of Padé series expansion, and M and N are the orders of approximation;

[0032] S34. Let \(k = M = N\), and obtain the expression of the Cole-Cole model approximated by the Padé series of Ohm's law, which is expressed as:

[0033]

[0034] where \(A\) k and \(B\) k are coefficients, \(A\) k =\(\sigma_0(Q\) k +\(\tau\) c P\) k ); \(B\) k =\(Q\) k +\(\tau\) c (1 - \(\eta\))\(P\) k .

[0035] Preferably, in the said S4, the specific steps of successively performing Padé series expansion at each frequency point and obtaining the specific values of coefficients \(A\) k and \(B\) k are as follows:

[0036] 1) The expansion form at a certain frequency point is expressed as:

[0037]

[0038] where \(\omega_0\) represents the frequency point;

[0039] 2) The coefficient terms of \(p\) k and \(q\) k are obtained by using the coefficient terms of Taylor expansion, which is expressed as:

[0040]

[0041] where \(s\), \(n\), and \(l\) are the relevant coefficients of Taylor expansion.

[0042] Preferably, in the said S5, the solution method for the time-domain impulse response of the Cole-Cole model is: First, select the conductivity parameters at different frequencies required to calculate the Cole-Cole model, then calculate the frequency-domain response \(F(\omega)\) at the corresponding frequencies, and then calculate the time-domain response \(f\) cole (t) of the Cole-Cole model. The calculation formula for the time-domain impulse response of the Cole-Cole model is:

[0043] where \(F(\omega)\) is the frequency-domain response and \(Im(\cdot)\) is the imaginary extraction function.

[0044] Preferably, in the said S5, the steps for calculating the time-domain impulse response of the expansion format in S4 are: First, for the discrete \(\omega\) in step S2 i, perform Padé series expansion and calculate the relevant coefficients according to step S4, and calculate the frequency-domain response at different frequencies Then obtain the time-domain impulse response through sine-cosine transformation Specifically expressed as:

[0045]

[0046] Preferably, calculate f cole (t) The process is as follows: According to the setting of the angular frequency vector in S2, use the relevant relationship formula of the Cole-Cole model to calculate the conductivity at different angular frequencies, then calculate the electromagnetic field response at different frequencies to obtain the frequency-domain response, and use sine-cosine transformation to convert the frequency-domain response into the time pulse domain response;

[0047] The formula for calculating the electromagnetic field response at different frequencies is:

[0048]

[0049] Among them, p is the electric dipole source, p = Ids, j1 is the first-order Bessel function; λ is the wave number in the spatial domain, r = (X 2 +y 2 ) 1 / 2 , x and y are the horizontal position coordinates, r TE is the reflection coefficient;

[0050]

[0051] Calculate The process is as follows: According to the setting of the angular frequency vector in S2, use S4 to obtain the expansion form expression of each center frequency, obtain the frequency-domain response expanded at each center frequency, and use sine-cosine transformation to convert the frequency-domain response into the time pulse domain response.

[0052] Preferably, in S6, the relative error is expressed as:

[0053]

[0054] After calculating the relative error at each center frequency ω i , a two-dimensional distribution of the relative error is obtained.

[0055] Preferably, in S7, during the three-dimensional forward modeling process, when the time step changes, according to the two-dimensional distribution of the relative error obtained in S6, adopt the principle of minimum error ω best = min(RE(t)), obtain the center frequency ω best of the best Padé series expansion at the current simulation time, and continue the three-dimensional forward modeling of transient electromagnetic until the requirements of the simulation time range are met.

[0056] Therefore, the method for quickly selecting the central frequency in the transient electromagnetic forward modeling of a polarization medium according to the present invention can efficiently and dynamically select an appropriate central frequency of the Padé series according to the simulation time during the three-dimensional simulation of the transient electromagnetic in the polarization medium, while ensuring the accuracy and efficiency of the simulation, thereby better promoting the application of the transient electromagnetic in underground space detection.

[0057] This method is a central frequency selection scheme based on the quantitative distribution of relative errors, which can ensure the correctness of the central frequency selection and the accuracy of the transient electromagnetic simulation. Moreover, it is selected based on the one-dimensional simulation relative error, greatly saving the time required for central frequency selection. This method realizes the adaptive selection of the central frequency of the Padé series in the transient electromagnetic simulation with induced polarization effect.

[0058] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Description of the Drawings

[0059] Figure 1 is the flowchart of the embodiment of the present invention;

[0060] Figure 2 is the comparison diagram of the forward modeling of the Cole-Cole model and the one-dimensional forward modeling curve of the Padé series in the embodiment of the present invention;

[0061] Figure 3 is the contour map of the relative error of the central frequency in the embodiment of the present invention;

[0062] Figure 4 is the curve graph of the central frequency selection in the forward modeling simulation in the embodiment of the present invention;

[0063] Figure 5 is the one-dimensional and three-dimensional transient electromagnetic forward response diagram in the homogeneous half-space model in the embodiment of the present invention. Specific Embodiments

[0064] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0065] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention.

[0066] Embodiment

[0067] As Figure 1 shown, the present invention provides a method for quickly selecting the central frequency in the transient electromagnetic forward modeling of a polarization medium, including the following steps:

[0068] S1. Determine and input the three-dimensional forward time step iteration parameters and the center frequency range according to the simulation time range and simulation accuracy requirements of the three-dimensional forward model; including the starting time point \(t_0 = 10\) -7 s, the ending time point \(t\) end = 10 -1 s, the initial time step \(\Delta t_0 = 10\) -7 s, the time step doubling factor \(M = 2\); the proposed selection range of the center frequency is \((1\ Hz, 10\) 5 Hz).

[0069] S2. According to the parameters defined in step S1, use the logarithmic average to discretize the center frequency range to obtain the discretized center frequency vector and the corresponding angular frequency vector; where the initial frequency is \(f_1\), and the ending frequency is \(f\) end , use the logarithmic average for discretization, and the number of discretized frequencies in the unit interval is 10, which will form the discretized center frequency vector:

[0070] \(f=(f_1\ f_2\ \cdots\ f\) end );

[0071] And the corresponding angular frequency vector:

[0072] \(\omega = (\omega_1\ \omega_2\ \cdots\ \omega\) end ), and each vector contains 51 elements. Where \(f_1\) and \(f_2\), etc. represent the separated center frequencies, and \(\omega_1\) and \(\omega_2\) represent the corresponding separated angular frequencies.

[0073] S3. Read the model file to obtain the polarization medium parameters, including the direct current conductivity \(\sigma_0 = 0.01\ S / m\), the charging rate \(\eta = 0.2\), the frequency correlation coefficient \(c = 0.8\), and the time constant \(\tau = 0.01\ s\); construct the Cole-Cole model; specifically:

[0074] S31. Construct the Cole-Cole model interrelationship, which is expressed by the formula:

[0075]

[0076] Characterize the interrelationship between the frequency-dependent conductivity \(\sigma(\omega)\) and the polarization medium parameters,

[0077] where \(\sigma(\omega)\) is the frequency-dependent conductivity, \(\sigma_0\) is the direct current conductivity, \(\eta\) is the charging rate, \(c\) is the frequency correlation coefficient, \(\tau\) is the time constant, \(\omega\) is the angular frequency vector, and \(i\) is the imaginary unit;

[0078] S32. Define the frequency correlation coefficient \(c \lt 1\), perform the Padé series expansion and Taylor series expansion on the fractional derivative, and substitute the formula of the Cole-Cole model interrelationship into Ohm's law \(J=\sigma(\omega)E\) to obtain the form of Ohm's law under the Cole-Cole polarization medium model:

[0079] σ0E + σ0τ c (iω) c E = J + (1 - η)τ c (iω) c J;

[0080] Among them, J is the current density and E is the electric field strength;

[0081] S33. For the fractional exponential term (iω) c perform Padé approximation and approximation, expressed as:

[0082]

[0083] Among them, P m and Q n are the Padé series correlation coefficients, P(iω) and Q(iω) represent the individual coefficients of each order expansion in the Padé series expansion form, P 和 (iω) and Q 和 (iω) are respectively a set of coefficient sums of each order expansion in the Padé series expansion form, and M and N are respectively the orders of approximation;

[0084] S34. In the present invention, let K = M = N, that is, the upper and lower Padé series are of the same order, and the expression of the Ohm's law for the Padé series approximation of the Cole-Cole model is obtained, expressed as:

[0085]

[0086] Among them, A k , B k are coefficients, A k = σ0(Q k + τ c P k ); B k = Q k + τ c (1 - η)P k .

[0087] S4. According to the model in S3, perform Padé series expansion on the center frequency vector and angular frequency vector in S2 at each frequency point successively and obtain the specific values of the coefficients A k , B k , and obtain the expansion format of each frequency point, specifically:

[0088] 1) The expansion form at a certain frequency point is expressed as:

[0089]

[0090] Among them, ω0 represents the frequency point,

[0091] 2) For p k and q k The coefficient terms of Taylor expansion are used for calculation, expressed as:

[0092]

[0093] where s, n, and l are the relevant coefficients of Taylor expansion.

[0094] S5. Calculate the time-domain impulse response of the expansion format in S4 and the time-domain impulse response directly calculated by the Cole-Cole model in S3 respectively, specifically:

[0095] Use the Cole-Cole model to calculate the one-dimensional transient electromagnetic response f cole (t) of the homogeneous half-space of the polarized medium;

[0096] According to the setting of the angular frequency vector ω = (ω1 ω2 … ω end ) in S2, use the relevant relationship formula of the Cole-Cole model to calculate the conductivity at different angular frequencies, and then use the following formula:

[0097]

[0098] Calculate the electromagnetic field response at different frequencies;

[0099] where p is an electric dipole source, p = Ids, J1 is the first-order Bessel function; λ is the wave number in the spatial domain, r = (x 2 + y 2 ) 1 / 2 , x and y are the horizontal position coordinates, r TE is the reflection coefficient, which is a function related to the conductivity of the underground medium and the formation thickness; in the present invention, the model is a homogeneous half-space model, and the reflection coefficient r TE can be calculated by the following formula:

[0100]

[0101] where

[0102] After calculating H z (r) at each frequency point through the above method, then by using the sine-cosine transform method shown in the following formula, it can be transformed into the time domain to obtain the time-domain response f cole (t) under the Cole-Cole model, expressed as:

[0103]

[0104] Among them, F(ω) is the frequency-domain response, and Im(·) is the imaginary extraction function.

[0105] Using the Padé series expansion form, calculate the one-dimensional transient electromagnetic response f pade (t) of a homogeneous half-space:

[0106] For the discretized center frequency positions ω = (ω1 ω2 … ω end ), use S4 to obtain the conductivity expression form expanded at each center frequency, and then through the following formula:

[0107]

[0108] Obtain all the frequency-domain responses expanded at each center frequency

[0109] Then use the sine-cosine transform shown in the following formula to transform to the time domain and calculate the time-domain response Expressed as:

[0110]

[0111] Represents the time-domain response obtained after the Padé series expansion at the ω i frequency position. As Figure 2 shown, it shows the one-dimensional polarized medium model response f cole (t) calculated using the Cole-Cole model for the current polarization model parameters. As Figure 2 shown by the black dashed line in, and the model response calculated using the Padé series expansion As Figure 2 shown by the gray solid line in, where the expansion frequency is 1000 Hz, that is, ω i = 2000π.

[0112] S6. Calculate the relative error between the time-domain response f cole (t) directly obtained from the Cole-Cole model and that obtained from the Padé series expansion Then, after calculating the relative error using the above formula at each center frequency ω

[0113]

[0114] , obtain the two-dimensional distribution RE(ω, t) of the relative error with respect to the acquisition time and the center frequency. Its distribution is as i shown, and the light-colored positions indicate smaller relative errors. Figure 3 As shown, the light-colored positions indicate smaller relative errors.

[0115] S7. According to the two-dimensional distribution, the Cole-Cole model is expanded using the Padé series approximation method, and the three-dimensional forward modeling with induced polarization effect is started. During the three-dimensional forward modeling process, when the time step changes, according to the two-dimensional distribution of the relative error obtained in S6, the average relative error of the center frequency vector in S2 within equal time steps is calculated by traversing, and the error minimum principle ω best = min(RE(t)) is adopted to obtain the center frequency ω of the best Padé series expansion at the current simulation time best , and the three-dimensional forward modeling of transient electromagnetic is continued until the requirement of the simulation time range is reached.

[0116] As Figure 4 shown, it is the final situation of the adaptive selection of the center frequency in the forward modeling stage Figure 5 is the final result of the transient electromagnetic forward modeling. The black curve is the result calculated by the analytical method, and the gray curve is the result calculated by the center frequency selection method of the present invention. The correspondence between the two results is very good.

[0117] Therefore, the method for quickly selecting the center frequency in the transient electromagnetic forward modeling in a polarization medium of the present invention can provide a fast center frequency selection method for carrying out the three-dimensional forward calculation of transient electromagnetic when the underground is a polarization medium, provide more efficient and accurate simulation results, and is the basis for geophysical transient electromagnetic detection.

[0118] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for quickly selecting the central frequency in transient electromagnetic forward modeling in an induced polarization medium, characterized in that, It includes the following steps: S1. Determine and input the three-dimensional forward time step iteration parameters and the center frequency range according to the simulation time range and simulation accuracy requirements of the three-dimensional forward model; S2. Adopt logarithmic mean to discretize the center frequency range to obtain the discretized center frequency vector and the corresponding angular frequency vector; S3. Read the model file to obtain the polarization medium parameters and construct the Cole-Cole model; S4. According to the model in S3, perform Padé series approximation on the center frequency vector and angular frequency vector in S2 at each frequency point successively to obtain the expansion format at each frequency point; S5. Calculate the time-domain impulse response of the expansion format in S4 and the time-domain impulse response directly calculated by the Cole-Cole model in S3 respectively; S6. Calculate the relative error of the two time-domain impulse responses obtained in S5 to obtain the two-dimensional distribution of the relative error with respect to the acquisition time and the center frequency; S7. According to the two-dimensional distribution, use the Padé series approximation method to expand the Cole-Cole model, select the corresponding center frequency at different simulation times, and start the three-dimensional forward with induced polarization effect until the requirement of the simulation time range is met.

2. A method for quickly selecting the central frequency in transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In S1, the three-dimensional forward time step iteration parameters include the starting time point, the ending time point, the initial time step, and the time step doubling factor.

3. A method for quickly selecting a central frequency in transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In S2, the discretized center frequency vector is expressed as: f = (f1 f2 … f end ); The corresponding angular frequency vector is expressed as: ω = (ω1 ω2 … ω end ).

4. A method for quickly selecting the central frequency of transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In S3, the polarization medium parameters include the direct current conductivity, the chargeability, the frequency-dependent coefficient, and the time constant; The construction process of the Cole-Cole model is as follows: S31. Construct the relevant relationship of the Cole-Cole model, and the formula is expressed as: where σ(ω) is the frequency-dependent conductivity, σ0 is the direct current conductivity, η is the chargeability, c is the frequency-dependent coefficient, τ is the time constant, ω is the angular frequency vector, and i is the imaginary unit; S32. Define c < 1 for the frequency-dependent coefficient, perform Padé series expansion and Taylor series expansion on the fractional derivative, and substitute the formula of the Cole-Cole model relevant relationship into Ohm's law J = σ(ω)E to obtain the form of Ohm's law under the Cole-Cole polarization medium model: σ0E + σ0τ c (iω) c E = J + (1 - η)τ c (iω) c J; where J is the current density and E is the electric field strength; S33. Padé approximation is performed on the fractional-order exponential term (iω) c and is expressed as: where P m and Q n are the Padé series correlation coefficients, P(iω) and Q(iω) represent the individual coefficients of each order expansion in the Padé series expansion form, P 和 (iω) and Q 和 (iω) are respectively the sum of a group of coefficients of each order expansion in the Padé series expansion form, and M and N are respectively the orders of approximation; S34. Let K = M = N to obtain the expression of Ohm's law for the Padé series approximation of the Cole-Cole model, which is expressed as: Among them, A k and B k are coefficients, A k = σ0(Q k + τ c P k ); B k = Q k + τ c (1 - η)P k .

5. A method for quickly selecting the central frequency in transient electromagnetic forward modeling in an induced polarization medium according to claim 4, characterized in that: In S4, Padé series expansion is successively performed at each frequency point to obtain the specific values of coefficients A k and B k . The specific steps are as follows: 1) The expansion form at a certain frequency point is expressed as: where ω0 represents the frequency point, 2) For p k and q k Obtain using the coefficient terms of the Taylor expansion, expressed as: where s, n, and l are the relevant coefficients of the Taylor expansion.

6. A method for quickly selecting the central frequency in transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In the step S5, first, select the conductivity parameters at different frequencies under the Cole-Cole model that need to be calculated. Then, calculate the frequency-domain response F(ω) at the corresponding frequencies. Next, calculate the time-domain response f cole (t) of the Cole-Cole model. The calculation formula for the time-domain impulse response of the Cole-Cole model is: where F(ω) is the frequency-domain response and Im(·) is the imaginary extraction function.

7. A method for quickly selecting the center frequency in transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In S5, the steps to calculate the time-domain impulse response in the expanded format in S4 are as follows: First, for the discrete ω in step S2 i , perform Padé series expansion and calculate the correlation coefficients according to step S4, and calculate the frequency-domain responses at different frequencies Then obtain the time-domain impulse response through sine-cosine transformation Specifically expressed as:

8. A method for quickly selecting the center frequency of transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In S6, the relative error is expressed as: After calculating the relative error at each center frequency ω i a two-dimensional distribution of the relative error is obtained.

9. A method for quickly selecting the center frequency of transient electromagnetic forward modeling in an induced polarization medium according to claim 1, characterized in that: In S7, during the three-dimensional forward process, when the time step changes, according to the two-dimensional distribution of the relative error obtained in S6, adopt the principle of minimum error to obtain the center frequency of the best Padé series expansion at the current simulation time, and continue the three-dimensional forward of transient electromagnetic until the requirement of the simulation time range is met.

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