A random walk-based magnetotelluric time series synthesis method

Abstract

The present invention discloses a method for synthesizing magnetotelluric time series based on random walks, comprising: performing magnetotelluric forward modeling calculations based on a given geoelectric model to obtain an initial electric field spectrum, a magnetic field spectrum, and an impedance tensor; introducing a random walk to simulate a non-stationary, non-Gaussian distributed magnetotelluric time series; normalizing the amplitudes of the random walk and random interference, multiplying them by an amplitude coefficient, and summing the two to obtain a complete time series; performing a Fourier transform on the generated electric field time series to obtain an electric field spectrum; obtaining a relationship between the electric field spectrum, the magnetic field spectrum, and the impedance tensor; performing an inverse Fourier transform on the obtained magnetic field spectrum to obtain a magnetic field time series, and finally obtaining a MT four-component time series. The present invention provides a method for stably synthesizing non-stationary, non-Gaussian distributed magnetotelluric time series, which can be used as a test standard for new MT signal processing methods.

Description

Technical Field

[0001] The present invention relates to the technical field of electromagnetic sounding, and in particular to a magnetotelluric time series synthesis method based on random walk. Background Art

[0002] Magnetotelluric (MT) is a geophysical exploration method that uses naturally occurring alternating electromagnetic fields to study the electrical structure of the Earth's interior. Natural MT signals have a wide bandwidth and weak field strength, making them susceptible to various noise influences. This can severely distort MT signals, leading to unreliable or even erroneous inversion interpretation results. Therefore, processing of measured MT data is essential. Because the signal and noise components of measured data are unknown, it is necessary to synthesize theoretical time series data as a benchmark before applying processing methods to measured data.

[0003] Currently, scholars have conducted numerous studies on the simulation of MT time series (Loddo et al., 2002; Wang et al., 2023), with similar basic concepts. Typically, starting from the frequency domain, the magnetic field spectrum is first generated, and then the electric field spectrum is calculated based on the impedance tensor obtained by forward modeling. Finally, the MT four-component time series is obtained through inverse Fourier transform or the splicing and superposition of multiple sine and cosine functions. CN114966874B proposes a method for synthesizing frequency domain electromagnetic time series data based on forward modeling. Its time series synthesis concept is similar to that of Wang et al., and it also requires first determining the spectral characteristics of the magnetic field. Then, a new electromagnetic field spectrum is obtained by scaling the electromagnetic field spectrum obtained by forward modeling. Finally, based on the new electromagnetic field spectrum information, the time series is synthesized by splicing and superposing multiple sine and cosine functions in the time domain. When applied, the simulated time series may exhibit stable and Gaussian distribution characteristics due to improperly generated magnetic field spectrum. However, in reality, the measured MT time series is often non-stationary and exhibits non-Gaussianity in statistical characteristics, and the current simulation methods cannot simulate MT data that conforms to the non-stationary and non-Gaussian characteristics.

[0004] Furthermore, the signal's inherent characteristics are crucial for validating the effectiveness of the proposed method. If the simulated signal's characteristics differ from its true characteristics, the data processing method applied to the measured data will not produce the expected results, or may even yield erroneous results. While existing MT signal synthesis methods can synthesize time series with morphologies similar to true MT signals, these synthesized time series do not necessarily possess the non-stationary and non-Gaussian properties of true MT signals, potentially rendering the resulting MT signal processing methods unreliable. Therefore, it is necessary to synthesize MT time series with non-stationary and non-Gaussian properties as a validation standard for new signal processing methods. Summary of the Invention

[0005] The present invention provides a method for stably synthesizing non-stationary, non-Gaussian distributed magnetotelluric time series, which can be used as a test standard for new MT signal processing methods.

[0006] The present invention solves the technical problem by adopting the following technical solutions:

[0007] A magnetotelluric time series synthesis method based on random walk includes the following steps:

[0008] Step S1, given a geoelectric model, performing magnetotelluric forward calculation to obtain the initial electric field spectrum, magnetic field spectrum and impedance tensor;

[0009] Step S2, introducing random walk to simulate non-stationary, non-Gaussian distributed random walk time series;

[0010] Step S3: Random interference may exist in the natural MT signal. A Gaussian distribution sequence is generated as random interference. To conveniently control the amplitude of the generated time series, the amplitudes of the random walk and random interference are normalized, and then multiplied by the amplitude coefficient and the amplitude coefficient respectively. Finally, the two are added together to obtain the complete time series.

[0011] Step S4, taking the complete time series generated in step S3 as the electric field time series, and performing Fourier transform on each of them to obtain the electric field spectrum;

[0012] Step S5, calculating the magnetic field spectrum according to the relationship between the electric field spectrum, the magnetic field spectrum and the impedance tensor;

[0013] Step S6: Perform inverse Fourier transform on the obtained magnetic field spectra to obtain magnetic field time series, and finally obtain MT four-component time series.

[0014] Furthermore, in step S1, the electric field spectrum , the magnetic field spectrum , the impedance tensor ;

[0015] is the electric field spectrum in the x direction, is the electric field spectrum in the y direction, is the magnetic field spectrum in the x direction, is the magnetic field spectrum in the y direction, is the impedance of the xx component, is the impedance of the xy component, is the impedance of the yx component, is the impedance of the yy component.

[0016] Furthermore, in step S1, the given geoelectric model is not limited in the application, and a one-dimensional, two-dimensional or three-dimensional model is used; the magnetotelluric forward modeling method is not limited, and a numerical simulation method or ModEM calculation is used, as long as it can output the electric field spectrum. E , magnetic field spectrum H and the impedance tensor Z.

[0017] Furthermore, in step S2, random walk It is expressed as follows:

[0018] ,

[0019] Where, is a Gaussian distributed random number with mean 0 and variance 1. represents time; in addition, the random walk Also expressed as , so we can see that the random walk The variance of grows over time, Naturally maintains non-stationary and non-Gaussian properties.

[0020] Furthermore, the complete time series generated in step S3 can be used as the electric field or magnetic field time series in the magnetotelluric signal; the complete time series generated can be used as the electric field time series and :

[0021] ,

[0022] Where, and Respectively represent the minimum and maximum values ​​of the time series; and Respectively represent and The random walk in the corresponding amplitude coefficients are and ; and Respectively represent and The random interference in the corresponding amplitude coefficients are and .

[0023] Furthermore, in step S4, the electric field spectrum is expressed as:

[0024] ,

[0025] and is the electric field time series in the x-direction and y-direction, is the Fourier transform of the electric field time series in the x direction, is the Fourier transform of the electric field time series in the y direction.

[0026] Furthermore, in step S5, the relationship between the electric field spectrum, the magnetic field spectrum, and the impedance tensor is as follows:

[0027] ,

[0028] in, E is the electric field spectrum, H is the magnetic field spectrum, Z is the impedance tensor.

[0029] Furthermore, in step S5, the magnetic field spectrum H Expressed as:

[0030] ,

[0031] The magnetic field spectrum can be obtained based on the above formula, or it can be obtained by scaling the proportional relationship between the four-component spectra of the electromagnetic field.

[0032] Furthermore, in step S6, the magnetic field time series and It is expressed as follows:

[0033] .

[0034] Beneficial effects of the present invention:

[0035] This method can stably synthesize non-stationary and non-Gaussian MT time series. Furthermore, the synthesized MT time series conforms to the frequency domain characteristics of the natural magnetotelluric field. The resulting impedance response is essentially consistent with the theoretical response of a given model and is similar to existing measured data, making it a useful benchmark for new signal processing methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 Flow chart of the method of the present invention.

[0037] Figure 2 It is a one-dimensional anisotropic geoelectric model diagram;

[0038] Figure 3 The four-component time series of magnetotelluric elements simulated by the present invention;

[0039] Figure 4 The frequency distribution histogram corresponding to the time series of the magnetotelluric four components simulated by the present invention;

[0040] Figure 5 The distribution diagram of each frequency domain parameter corresponding to the magnetotelluric four-component time series simulated by the present invention;

[0041] Figure 6 The apparent resistivity and phase curves corresponding to the magnetotelluric four-component time series simulated by the present invention. DETAILED DESCRIPTION

[0042] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0043] Reference Attachment Figure 1 The present invention provides a magnetotelluric time series synthesis method based on random walk, comprising the following steps:

[0044] Step S1: Given a geoelectric model, perform magnetotelluric forward modeling to obtain the initial electric field spectrum, magnetic field spectrum, and impedance tensor. The present invention is not limited to the given geoelectric model when applied, and can use a one-dimensional, two-dimensional, or three-dimensional model. The magnetotelluric forward modeling method is not limited, and can use numerical simulation methods such as the finite element method, or can also use existing open source software such as ModEM, as long as it can output the electric field spectrum. E , magnetic field spectrum H and the impedance tensor Z That's it.

[0045] Electric field spectrum , the magnetic field spectrum , the impedance tensor ;

[0046] is the electric field spectrum in the x direction, is the electric field spectrum in the y direction, is the magnetic field spectrum in the x direction, is the magnetic field spectrum in the y direction, is the impedance of the xx component, is the impedance of the xy component, is the impedance of the yx component, is the impedance of the yy component.

[0047] Step S2, introducing random walk to simulate non-stationary, non-Gaussian distributed magnetotelluric time series;

[0048] Random Walk It is expressed as follows:

[0049] ,

[0050] Where, is a Gaussian distributed random number with mean 0 and variance 1. Indicates time; express In addition, the random walk Also expressed as , so we can see that the random walk The variance of grows over time, Naturally maintains non-stationary and non-Gaussian properties.

[0051] Step S3: There may be random interference in the natural MT signal, and a Gaussian distribution sequence is generated as random interference; in order to conveniently control the amplitude of the generated time series, the random walk is and random interference The amplitude is normalized to [-1,1] and then multiplied by the amplitude coefficient A and amplitude coefficient B , and finally add the two together to get the complete time series;

[0052] Generally speaking, the magnetic field shows more slow changes, stronger low-frequency components, and more obvious non-stationary characteristics. Under this condition, it is more appropriate to prioritize the non-stationary nature of the electric field time series, so the generated complete time series is used as the electric field time series. and :

[0053] ,

[0054] Where, and Respectively represent the minimum and maximum values ​​of the time series; and Respectively represent and The random walk in the corresponding amplitude coefficients are and ; and Respectively represent and The random interference in the corresponding amplitude coefficients are and .

[0055] Step S4, the generated electric field time series and Perform Fourier transform respectively to obtain the electric field spectrum, which is expressed as:

[0056] ,

[0057] and is the electric field time series in the x-direction and y-direction, is the Fourier transform of the electric field time series in the x direction, is the Fourier transform of the electric field time series in the y direction.

[0058] Step S5, according to the electric field spectrum E , magnetic field spectrum H and the impedance tensor Z The relationship between the three is used to obtain the magnetic field spectrum, as shown below:

[0059] ,

[0060] in, E is the electric field spectrum, H is the magnetic field spectrum, Z is the impedance tensor.

[0061] Magnetic field spectrum H Expressed as:

[0062] ,

[0063] The magnetic field spectrum can be obtained based on the above formula or by scaling the four-component spectrum of the electromagnetic field. Regardless of the method, the essence is to make the spectrum corresponding to the simulated MT time series satisfy , to ensure the accuracy and stability of the impedance response obtained after processing.

[0064] Step S6, the obtained magnetic field spectrum and Perform inverse Fourier transform respectively to obtain the magnetic field time series and , and finally we get the MT four-component time series 、 、 、 .

[0065] Magnetic field time series and It is expressed as follows:

[0066] .

[0067] This method can stably synthesize non-stationary and non-Gaussian MT time series. Furthermore, the synthesized MT time series conforms to the frequency domain characteristics of the natural magnetotelluric field. The resulting impedance response is essentially consistent with the theoretical response of a given model and is similar to existing measured data, making it a useful benchmark for new signal processing methods.

[0068] When simulating the magnetotelluric time series, the resistivity data provided by the CSDP-2 well in the South Yellow Sea were used to establish Figure 2 The one-dimensional horizontal anisotropic geoelectric model is shown and the continuous field values ​​at each frequency point are obtained through magnetotelluric forward modeling. Figure 3 The simulated 100-minute four-component MT time series is shown. The simulated MT time series shows a clear trend, which is consistent with the non-stationary characteristics of the natural MT signal.

[0069] In order to study the statistical characteristics of the synthetic MT time series, the frequency distribution histogram of the synthetic time series is drawn as follows Figure 4 As shown, for comparison, the Gaussian distribution fitting curve is also drawn. Figure 4 It can be seen that the frequency distribution of the synthesized MT four-component time series is irregular, which is quite different from the Gaussian distribution fitting curve, and is consistent with the non-Gaussian characteristics of the natural MT signal.

[0070] To investigate the frequency domain characteristics of the synthetic time series, the time series was segmented into 256 data points, with a 50% overlap between adjacent segments. Each segment was then Fourier transformed, and a series of frequency domain parameters, including the electromagnetic field power spectral density, polarization orientation, coherence, and impedance tensor, were calculated. Prior to calculation, the data were first-order differencing to eliminate the influence of low-frequency trends during the Fourier transform. Figure 5 The frequency domain parameters of each MT are shown in the figure. Figure 5 As can be seen, the power spectral density distribution of the electromagnetic field exhibits random variations similar to those of a natural MT field. The polarization directions of the electric and magnetic fields exhibit disorder over time. The coherence of the orthogonal electromagnetic fields is concentrated near 1. The estimated transfer function values ​​are concentrated near the theoretical value. These frequency-domain parameter distribution characteristics are similar to those of natural magnetotelluric signals, demonstrating that the time series synthesized by the present invention accurately simulates the frequency-domain characteristics of the natural magnetotelluric field.

[0071] The magnetotelluric method mainly analyzes the underground electrical distribution based on the apparent resistivity and phase obtained by impedance estimation. Figure 6 The results of the least squares method for estimating apparent resistivity and phase of the synthetic time series are presented. Figure 6 The obtained apparent resistivity and phase are essentially consistent with the theoretical response, further verifying the accuracy of the time series synthesized by this method. Due to the limited duration of the simulated time series, the apparent resistivity and phase at a few low-frequency points differ slightly from the theoretical values. The accuracy of the impedance estimation in this low-frequency region can be improved by increasing the length of the simulated time series.

[0072] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A magnetotelluric time series synthesis method based on random walk, characterized in that: The steps include: Step S1, given a geoelectric model, performing magnetotelluric forward calculation to obtain the initial electric field spectrum, magnetic field spectrum and impedance tensor; Step S2, introducing random walk to simulate non-stationary, non-Gaussian distributed random walk time series; Step S3: Generate a Gaussian distribution sequence as random interference, normalize the amplitudes of the random walk and random interference, multiply them by the amplitude coefficient respectively, and finally add the two to obtain the complete time series; Step S4, taking the complete time series generated in step S3 as the electric field time series, and performing Fourier transform on each of them to obtain the electric field spectrum; Step S5, calculating the magnetic field spectrum according to the relationship between the electric field spectrum obtained in step S4, the magnetic field spectrum in step S1, and the impedance tensor in step S1; Step S6: Perform inverse Fourier transform on the obtained magnetic field spectra to obtain magnetic field time series, and finally obtain MT four-component time series.

2. The random walk-based magnetotelluric time series synthesis method according to claim 1, characterized in that: In step S1, the electric field spectrum E=[E x (ω),E y (ω)] T , the magnetic field spectrum H=[H x (ω),H y (ω)] T , the impedance tensor E x (ω) is the electric field spectrum in the x direction, E y (ω) is the electric field spectrum in the y direction, H x (ω) is the magnetic field spectrum in the x direction, H y (ω) is the magnetic field spectrum in the y direction, Z xx (ω) is the impedance of the xx component, Z xy (ω) is the impedance of the xy component, Z yx (ω) is the impedance of the yx component, Z yy (ω) is the impedance of the yy component.

3. The random walk-based magnetotelluric time series synthesis method according to claim 2, characterized in that: In step S1, the geoelectric model that needs to be given in the application is not limited, and a one-dimensional, two-dimensional, or three-dimensional model is used; the magnetotelluric forward modeling method is not limited, and a numerical simulation method or ModEM calculation is used, and it only needs to be able to output the electric field spectrum E, the magnetic field spectrum H, and the impedance tensor Z.

4. The random walk-based magnetotelluric time series synthesis method according to claim 3, characterized in that: In step S2, the random walk RW(t) is expressed as follows: RW(t)=RW(t-1)+ε t Where, ε t is a Gaussian distributed random number with mean 0 and variance 1, and t represents time; In addition, the random walk RW(t) is also expressed as From this we can see that the variance of the random walk RW(t) increases with time, and RW(t) naturally maintains non-stationary and non-Gaussian properties.

5. The random walk-based magnetotelluric time series synthesis method according to claim 4, characterized in that: The complete time series generated in step S3 is used as the electric field or magnetic field time series in the magnetotelluric signal; the complete time series generated is used as the electric field time series e x (t) and e y (t): Where min(·) and max(·) represent the minimum and maximum values ​​of the time series, respectively; RW x and RW y Represents e x (t) and e y The random walk in (t) has the corresponding amplitude coefficients A x and A y ;g x (t) and g y (t) represents e x (t) and e y The random interference in (t) has the corresponding amplitude coefficients B x and B y .

6. The random walk-based magnetotelluric time series synthesis method according to claim 5, characterized in that: In step S4, the electric field spectrum is expressed as: e x (t) and e y (t) is the time series of the electric field in the x and y directions, FFT (e x (t)) is the Fourier transform of the electric field time series in the x direction, FFT(e y (t)) is the Fourier transform of the electric field time series in the y direction.

7. The random walk-based magnetotelluric time series synthesis method according to claim 6, characterized in that: In step S5, the relationship between the electric field spectrum, the magnetic field spectrum, and the impedance tensor is as follows: E=ZH Where E is the electric field spectrum, H is the magnetic field spectrum, and Z is the impedance tensor.

8. The random walk-based magnetotelluric time series synthesis method according to claim 7, characterized in that: In step S5, the magnetic field spectrum H is expressed as: The magnetic field spectrum can be obtained based on the above formula, or it can be obtained by scaling the proportional relationship between the four-component spectra of the electromagnetic field.

9. The random walk-based magnetotelluric time series synthesis method according to claim 8, characterized in that: In step S6, the magnetic field time series h x (t) and h y (t) is expressed as follows:

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