Method for fast selection of central frequency in induced polarization medium transient electromagnetic forward

Abstract

The application discloses a method for exciting polarization medium transient electromagnetic forward fast selection of central frequency, and belongs to the technical field of geophysical prospecting, and comprises the following steps: firstly, determining three-dimensional forward time step iteration parameters and a central frequency range, and constructing a Cole-Cole model; then, sequentially performing Padé series approximation on the discrete central frequency vector and the angular frequency vector at each frequency point to obtain an expansion format of each frequency point; respectively calculating time domain pulse responses of the expansion format and the Cole-Cole model and relative errors thereof, obtaining a two-dimensional distribution of the relative errors, expanding the Cole-Cole model by using the Padé series approximation method, and starting three-dimensional forward calculation of the Cole-Cole model with the exciting polarization effect. The application can provide a fast central frequency selection method for developing three-dimensional forward calculation of transient electromagnetic waves in the underground polarized medium, and is the basis of geophysical transient electromagnetic prospecting.

Description

Technical Field

[0001] This invention relates to the field of geophysics, and in particular to a method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium. Background Technology

[0002] Transient electromagnetic methods (TEM) are an important branch of geophysical electromagnetic detection methods. A primary field excited by a transient electromagnetic source induces a secondary field in the subsurface medium. By analyzing the received secondary field, the resistivity distribution of the subsurface medium can be inferred, achieving the purpose of subsurface space exploration. In highly polarizable media (polarized media) such as sulfide minerals and metallic minerals, TEM exhibits an induced polarization effect. Specifically, when an external electromagnetic field is applied, the charge movement within the medium exhibits nonlinear and dissipative characteristics. When using TEM to detect polarized media, the electromagnetic response sometimes rapidly decays into a negative value at a certain moment; this sign change phenomenon is due to the polarization effect. Therefore, conducting three-dimensional simulations of transient electromagnetic activity in polarized media is crucial for accurate TEM detection of subsurface space.

[0003] In three-dimensional transient electromagnetic simulations of polarized media, the polarized medium parameters are generally assumed to satisfy the Cole-Cole model. When using this model for transient electromagnetic simulations, the Cole-Cole model relationship in the frequency domain needs to be transformed to the time domain via Fourier transform and then substituted into the time-domain Maxwell equations for solution. This process introduces fractional derivatives, forming a system of Maxwell equations with fractional derivatives. To solve the fractional Maxwell equations, the Padé series is typically used to convert the fractional Maxwell equations to integer order, followed by numerical discretization and solution. When performing the Padé series expansion, an approximate expansion needs to be performed using a center frequency as the basis. The choice of the center frequency at different times in the transient electromagnetic simulation affects the approximate accuracy of the Padé series for the fractional partial differential equations.

[0004] Currently, there are two ways to select the center frequency of the Padé series in the three-dimensional simulation of transient electromagnetic events in polarized media: one method is to use experience as a reference and adopt a fixed center frequency over a long period of transient electromagnetic simulation. This method will affect the stability and accuracy of the simulation, leading to large errors in the signal in the later stages and near the sign change. The other method is to directly perform adaptive center frequency selection during the three-dimensional simulation, that is, to adjust the center frequency as the transient electromagnetic simulation time changes. This method can improve accuracy compared to the first method, but because it requires continuous three-dimensional forward modeling attempts during the adaptive process, it will consume a lot of computation time, far exceeding the time required for three-dimensional transient electromagnetic simulation of non-polarized media. This is completely unacceptable in complex models and inversion processes.

[0005] Based on the above, a method for rapidly selecting the center frequency in transient electromagnetic forward modeling in excited polarized media is proposed. Summary of the Invention

[0006] The purpose of this invention is to provide a method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium, so as to solve the problems in the background art.

[0007] To achieve the above objectives, the present invention provides a method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium, comprising the following steps:

[0008] S1. Based on the simulation time range and simulation accuracy requirements of the three-dimensional forward model, determine and input the three-dimensional forward modeling time step iteration parameters and center frequency range;

[0009] S2. Use logarithmic average to discretize the center frequency range to obtain the discretized center frequency vector and the corresponding angular frequency vector;

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

[0011] S4. Based on the model in S3, the center frequency vector and angular frequency vector in S2 are successively approximated by Padé series at each frequency point to obtain the expansion format of each frequency point.

[0012] S5. Calculate the time-domain impulse response of the expanded 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, and obtain the two-dimensional distribution of the relative error with respect to the acquisition time and center frequency;

[0014] S7. Based on 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 until the simulation time range requirement is met.

[0015] Preferably, in S1, the three-dimensional forward modeling 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 discrete center frequency vector is represented as:

[0017] f = (f1 f2 … f) end );

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

[0019] ω=(ω1 ω2 … ω end).

[0020] Where f1 f2 … f end The center frequencies of the separation are represented by ω1 ω2 … ω end This indicates the corresponding angular frequency of the separation.

[0021] Preferably, in S3, the polarization medium parameters include DC conductivity, charge rate, frequency correlation coefficient, and time constant;

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

[0023] S31. Construct the Cole-Cole model correlation, expressed by the formula:

[0024]

[0025] Where σ(ω) is the frequency-dependent conductivity, σ0 is the DC conductivity, η is the charge 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 and Taylor series expansions on the fractional derivative, substitute the formula for the correlation relationship in the Cole-Cole model into Ohm's law J = σ(ω)E, and obtain the form of Ohm's law under the Cole-Cole polarized medium model:

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

[0028] Where J is the current density and E is the electric field strength;

[0029] S33. For the fractional exponent term (iω) c The Padé approximation is expressed as:

[0030]

[0031] Among them, P m With Q n Let P(iω) and Q(iω) be the correlation coefficients of the Padé series, respectively, and let P(iω) be the individual coefficients of each order of the Padé series expansion. 和 (iω) and Q 和 (iω) represents a set of coefficients for each order of the Padé series expansion, and M and N are the orders of the approximation, respectively.

[0032] S34. Let k = M = N, then we obtain the expression for the Padé series approximation of the Cole-Cole model using Ohm's law, expressed as:

[0033]

[0034] Among them, A k B k A is a coefficient. k =σ0(Q k +τ c P k ); B k =Q k +τ c (1-η)P k .

[0035] Preferably, in step S4, Padé series expansion is performed successively at each frequency point to obtain coefficients A. k B k The specific values ​​and steps are as follows:

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

[0037]

[0038] Where ω0 represents the frequency point;

[0039] 2) For p k q k The coefficients are obtained using Taylor expansion, and are expressed as follows:

[0040]

[0041] Where s, n, and l are the correlation coefficients of Taylor expansion.

[0042] Preferably, in step S5, the solution method for the time-domain impulse response of the Cole-Cole model is as follows: first, select the conductivity parameters at different frequencies for which the Cole-Cole model needs to be calculated; then calculate the frequency-domain response F(ω) at the corresponding frequency; and finally calculate the time-domain response f under the Cole-Cole model. cole The formula for calculating the time-domain impulse response of the Cole-Cole model is as follows:

[0043] Where F(ω) is the frequency domain response and Im(·) is the imaginary number extraction function.

[0044] Preferably, in step S5, the step of calculating the time-domain impulse response of the expanded format in step S4 is as follows: first, the discrete ω in step S2 is processed... iPerform Padé series expansion and calculate correlation coefficients according to step S4, and calculate the frequency domain response at different frequencies. The time-domain impulse response is then obtained through sine and cosine transformations. Specifically, it is expressed as follows:

[0045]

[0046] Preferably, calculate f cole The process of (t) is as follows: according to the setting of the angular frequency vector in S2, the conductivity at different angular frequencies is calculated using the Cole-Cole model correlation formula, and then the electromagnetic field response at different frequencies is calculated to obtain the frequency domain response. The frequency domain response is then converted into the time pulse domain response using sine and cosine transformation.

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

[0048]

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

[0050]

[0051] calculate The process is as follows: according to the setting of the angular frequency vector in S2, the expanded form expression of each center frequency is obtained by using S4, the expanded frequency domain response at each center frequency is obtained, and the frequency domain response is converted into the time pulse domain response by using sine and cosine transformation.

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

[0053]

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

[0055] Preferably, in step S7, during the three-dimensional forward modeling process, when the time step changes, the principle of minimizing error ω is adopted based on the two-dimensional distribution of the relative error obtained in step S6. best =min(RE(t)), which yields the center frequency ω of the optimal Padé series expansion at the current simulation time. best And continue to perform three-dimensional forward modeling of transient electromagnetics until the simulation time range requirements are met.

[0056] Therefore, the present invention provides a method for rapidly selecting the center frequency in transient electromagnetic forward modeling in polarized media. This method can efficiently and dynamically select the appropriate Padé series center frequency as the simulation time changes in three-dimensional transient electromagnetic simulation in polarized media, while ensuring the accuracy and efficiency of the simulation, thereby better promoting the application of transient electromagnetics in underground space exploration.

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

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

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

[0060] Figure 2 This is a comparison chart of the forward modeling curves of the Cole-Cole model and the one-dimensional forward modeling curves of the Padé series in an embodiment of the present invention;

[0061] Figure 3 This is a contour map of the relative error of the center frequency in an embodiment of the present invention;

[0062] Figure 4 This is a graph showing the selection of the center frequency in the forward modeling of an embodiment of the present invention.

[0063] Figure 5 The figures show the one-dimensional and three-dimensional transient electromagnetic forward response diagrams of the uniform semi-space model in an embodiment of the present invention. Detailed Implementation

[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 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 some embodiments of the present invention, but not all embodiments.

[0066] Example

[0067] like Figure 1 As shown, this invention provides a method for rapidly selecting the center frequency in transient electromagnetic forward modeling of an excited polarized medium, comprising the following steps:

[0068] S1. Based on the simulation time range and accuracy requirements of the 3D forward model, determine and input the 3D forward modeling time step iteration parameters and center frequency range; including the starting time point t0 = 10. -7 s, end time t end =10 -1 s, initial time step Δt0 = 10 -7 s, time step doubling factor M = 2; proposed selection range of center frequency (1Hz, 10 Hz). 5 Hz).

[0069] S2. Based on the parameters defined in step S1, the center frequency range is discretized using a logarithmic average to obtain the discretized center frequency vector and the corresponding angular frequency vector; where the initial frequency is f1 and the ending frequency is f end Using logarithmic mean discretization, with 10 frequencies discrete within a unit interval, a discrete center frequency vector will be formed:

[0070] f = (f1 f2 … f) end );

[0071] And the corresponding angular frequency vector:

[0072] ω=(ω1 ω2 … ω end Each vector contains 51 elements. Here, f1 and f2 represent the center frequencies of the separation, and ω1 and ω2 represent the corresponding angular frequencies of the separation.

[0073] S3. Read the model file to obtain the polarization medium parameters, including DC conductivity σ0 = 0.01 S / m, charge rate η = 0.2, frequency correlation coefficient c = 0.8, and time constant τ = 0.01 s; construct the Cole-Cole model; specifically:

[0074] S31. Construct the Cole-Cole model of mutual relationships, expressed by the formula:

[0075]

[0076] Characterizing the relationship between frequency-dependent conductivity σ(ω) and polarization medium parameters,

[0077] Where σ(ω) is the frequency-dependent conductivity, σ0 is the DC conductivity, η is the charge rate, c is the frequency correlation coefficient, τ is the time constant, ω is the angular frequency vector, and i is the imaginary unit;

[0078] S32. Define the frequency correlation coefficient c < 1, perform Padé series and Taylor series expansions on the fractional derivative, substitute the formula for the correlation relationship in the Cole-Cole model into Ohm's law J = σ(ω)E, and obtain the form of Ohm's law under the Cole-Cole polarized medium model:

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

[0080] Where J is the current density and E is the electric field strength;

[0081] S33. For the fractional exponent term (iω) c The Padé approximation is expressed as:

[0082]

[0083] Among them, P m With Q n Let P(iω) and Q(iω) be the correlation coefficients of the Padé series, respectively, and let P(iω) be the individual coefficients of each order of the Padé series expansion. 和 (iω) and Q 和 (iω) represents a set of coefficients for each order of the Padé series expansion, and M and N are the orders of the approximation, respectively.

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

[0085]

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

[0087] S4. Based on the model in S3, perform Padé series expansion on the center frequency vector and angular frequency vector in S2 at each frequency point to obtain the coefficients A. k B k The specific values ​​are used to obtain the expanded format of each frequency point, as follows:

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

[0089]

[0090] Where ω0 represents the frequency point,

[0091] 2) For p k q k The coefficients are obtained using Taylor expansion, and are expressed as follows:

[0092]

[0093] Where s, n, and l are the correlation coefficients of Taylor expansion.

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

[0095] The one-dimensional transient electromagnetic response f in a homogeneous half-space of a polarized medium is calculated using the Cole-Cole model. cole (t);

[0096] According to the angular frequency vector ω=(ω1 ω2 … ω in S2) end The conductivity at different angular frequencies is calculated using the Cole-Cole model correlation formula, and then the following formula is used:

[0097]

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

[0099] Where p is an electric dipole source, p = Ids, J1 is a first-order Bessel function; λ is the wavenumber in the spatial domain, r = (x 2 +y 2 ) 1 / 2 x and y are the horizontal position coordinates, r TE The reflection coefficient is a function related to the electrical conductivity of the underground medium and the thickness of the formation; in this invention, the model is a uniform half-space model, and the reflection coefficient r is... TE The following formula can be used for calculation:

[0100]

[0101] in,

[0102] H at each frequency point is obtained by calculating using the above method. z After (r), the sine and cosine transforms shown in the following formula can be used to convert to the time domain and obtain the time domain response f under the Cole-Cole model. cole (t), represented as:

[0103]

[0104] Where F(ω) is the frequency domain response and Im(·) is the imaginary number extraction function.

[0105] Using the Padé series expansion, the one-dimensional transient electromagnetic response f in a uniform half-space is calculated. pade (t):

[0106] For the discretized center frequency position ω=(ω1 ω2 … ω end The conductivity expression for each center frequency is obtained using S4, and then expressed by the following formula:

[0107]

[0108] Find all frequency domain responses expanded at each center frequency.

[0109] Then, the sine and cosine transforms are applied to the time domain as shown in the following formula, and the time domain response is calculated. Represented as:

[0110]

[0111] Indicates at ω i The time-domain response obtained by performing Padé series expansion at the frequency location. For example... Figure 2 As shown, the response f of a one-dimensional polarized medium model calculated using the Cole-Cole model under the current polarization model parameters is illustrated. cole (t), such as Figure 2 The black dashed line in the middle shows the model response compared to that calculated using Padé series expansion. like Figure 2 The solid gray line represents the frequency, which is 1000 Hz, or ω. i =2000π.

[0112] S6. Calculate the time-domain response f directly obtained from the Cole-Cole model. cole (t) obtained by Padé series expansion The relative error between them

[0113]

[0114] Then at each center frequency ω i After calculating the relative error using the above formula at each location, a two-dimensional distribution of the relative error with respect to the acquisition time and center frequency, RE(ω,t), is obtained. Its distribution is as follows: Figure 3 As shown, lighter-colored positions indicate smaller relative errors.

[0115] S7. Based on the two-dimensional distribution, the Cole-Cole model is expanded using the Padé series approximation method, and the three-dimensional forward modeling involving the induced polarization effect begins. During the three-dimensional forward modeling, when the time step changes, based on the two-dimensional distribution of relative error obtained in S6, the average relative error of the center frequency vector in S2 within the same time step is calculated iteratively, and the minimum error principle ω is adopted. best =min(RE(t)), which yields the center frequency ω of the optimal Padé series expansion at the current simulation time. best And continue to perform three-dimensional forward modeling of transient electromagnetics until the simulation time range requirements are met.

[0116] like Figure 4 The figure shows the final case of adaptive selection of the center frequency during the forward modeling stage. Figure 5 The black curve represents the result calculated using the analytical method, while the gray curve represents the result calculated using the center frequency selection method of this invention. The two results show a very good correspondence.

[0117] Therefore, the present invention provides a method for rapidly selecting the center frequency in transient electromagnetic forward modeling in a polarized medium. This method can provide a fast center frequency selection method for conducting three-dimensional forward modeling of transient electromagnetics when the underground medium is polarized, providing more efficient and accurate simulation results, and is the foundation 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 not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium, characterized in that, Includes the following steps: S1. Based on the simulation time range and simulation accuracy requirements of the three-dimensional forward model, determine and input the three-dimensional forward modeling time step iteration parameters and center frequency range; S2. Use logarithmic average 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 polarization medium parameters and construct the Cole-Cole model; S4. Based on the model in S3, the center frequency vector and angular frequency vector in S2 are successively approximated by Padé series at each frequency point to obtain the expansion format of each frequency point. S5. Calculate the time-domain impulse response of the expanded format in S4 and the time-domain impulse response directly calculated by the Cole-Cole model in S3, respectively. Specifically: First, select the conductivity parameters for different frequencies under the Cole-Cole model to be calculated. Then, calculate the frequency domain response F(ω) at the corresponding frequencies. Finally, calculate the time domain response f under the Cole-Cole model. cole The formula for calculating the time-domain impulse response of the Cole-Cole model is as follows: Where F(ω) is the frequency domain response, Im(·) is the imaginary number extraction function, and ω is the angular frequency vector; The steps for calculating the time-domain impulse response of the expanded scheme in S4 are as follows: First, the discrete ω in step S2... i Perform Padé series expansion and calculate correlation coefficients according to step S4, and calculate the frequency domain response at different frequencies. The time-domain impulse response is then obtained through sine and cosine transformations. Specifically, it is expressed as follows: S6. Calculate the relative error of the two time-domain impulse responses obtained in S5, and obtain the two-dimensional distribution of the relative error with respect to the acquisition time and center frequency; the relative error is expressed as: At each center frequency ω i After calculating the relative error at each location, the two-dimensional distribution of the relative error is obtained; S7. Based on the two-dimensional distribution, the Cole-Cole model is expanded using the Padé series approximation method. At different simulation times, corresponding center frequencies are selected to begin three-dimensional forward modeling with induced polarization effects, until the simulation time range requirements are met. Specifically: Based on the two-dimensional distribution, the Cole-Cole model is expanded using the Padé series approximation method, and a three-dimensional forward modeling incorporating the induced polarization effect is initiated. During the three-dimensional forward modeling, when the time step changes, the average relative error of the center frequency vector in S2 within the same time step is calculated iteratively based on the two-dimensional distribution of the relative error obtained in S6, and the minimum error principle ω is adopted. best =min(RE(t)), which yields the center frequency ω of the optimal Padé series expansion at the current simulation time. best And continue to perform three-dimensional forward modeling of transient electromagnetics until the simulation time range requirements are met.

2. The method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium according to claim 1, characterized in that: In S1, the three-dimensional forward modeling time step iteration parameters include the start time point, end time point, initial time step, and time step doubling factor.

3. The method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium according to claim 1, characterized in that: In S2, the discrete center frequency vector is represented as: f=(f1 f2 … f end ); The corresponding angular frequency vector is represented as: ω=(ω1 ω2 … ω end )。 4. The method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium according to claim 1, characterized in that: In S3, the polarization medium parameters include DC conductivity, charge rate, frequency correlation coefficient, and time constant; The construction process of the Cole-Cole model is as follows: S31. Construct the Cole-Cole model correlation, expressed by the formula: Where σ(ω) is the frequency-dependent conductivity, σ0 is the DC conductivity, η is the charge rate, c is the frequency correlation coefficient, τ is the time constant, ω is the angular frequency vector, and i is the imaginary unit; S32. Define the frequency correlation coefficient c < 1, perform Padé series and Taylor series expansions on the fractional derivative, substitute the formula for the correlation relationship in the Cole-Cole model into Ohm's law J = σ(ω)E, and obtain the form of Ohm's law under the Cole-Cole polarized 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. For the fractional exponent term (iω) c The Padé approximation is expressed as: Among them, P m With Q n Let P(iω) and Q(iω) be the correlation coefficients of the Padé series, respectively, and let P(iω) be the individual coefficients of each order of the Padé series expansion. 和 (iω) and Q 和 (iω) represents a set of coefficients for each order of the Padé series expansion, and M and N are the orders of the approximation, respectively. S34. Let K = M = N, then we obtain the expression for the Padé series approximation of the Cole-Cole model using Ohm's law, expressed as: Among them, A k 、B k A k =σ0(Q k +t c P k );B k =Q k +t c (1-h)P k 。 5. The method for rapidly selecting the center frequency in transient electromagnetic forward modeling in an excited polarized medium according to claim 4, characterized in that: In step S4, Padé series expansion is performed successively at each frequency point to obtain coefficients A. k B k The specific values ​​and steps are as follows: 1) The expanded form at a certain frequency point is expressed as: Where ω0 represents the frequency point, 2) For p k q k The coefficients are obtained using Taylor expansion, and are expressed as follows: Where s, n, and l are the correlation coefficients of Taylor expansion.

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