CN108241171B - A Method for Filtering Seismic Data and Extracting Three-Transient Attributes Using Complex-valued Gauss Integral Filters - Google Patents

Abstract

The invention proposes a complex-valued Gauss integral filter and uses it to extract the three-transient property of seismic signals. The method constructs a complex-valued filter based on the Gauss window. This filter is a zero-phase complex-valued filter, and its frequency function can approximate the ideal gate function, which effectively suppresses the Gibbs phenomenon of the Fourier frequency domain ideal door and window filter. The complex-valued Gauss integral filter is used to filter the seismic signal, the real part of the output signal is the filtering result of the original signal, and the imaginary part is the Hilbert transform of the real part of the signal. Using this method to calculate the three-transient properties of seismic signals overcomes the defect of being sensitive to noise when using the Hilbert transform method.

Description

A complex-valued Gauss integral filter for filtering and extracting seismic data Three-instant attribute method

technical field

The invention belongs to the field of oil and gas seismic exploration, and in particular relates to a structure of a complex digital filter and a method for filtering and dividing frequency filtering processing of seismic exploration signals to obtain stable and accurate instantaneous attributes of seismic data, including: a complex-valued Gauss integral filter Structure; setting and numerical realization of high and low cutoff frequencies of complex Gauss integral filter; filtering processing of seismic data; extracting instantaneous seismic attributes from filtering results.

Background technique

In oil and gas seismic exploration, instantaneous attributes are widely used post-stack seismic attributes. Instantaneous phase is often used to detect formation unconformities, faults, and lateral changes in formation; in the journal paper "Seismic attributes----Ahistorical perspective" (Geophysics, 2005), Chorpra et al. used instantaneous frequency to identify anomalies in seismic data Attenuation and thin layer tuning. Robertson et al. used transient attributes for thin layer analysis and conventional seismic interpretation in "Complex seismic trace analysis of thin beds" (Geophysics, 1984) and "Seismic interpretation 9-Complex seismic traceattributes" (The Leading Edge, 1988), respectively; Hart Channel detection by instantaneous amplitude in Channel detection in 3-D seismic data using sweetness (AAPG Bulletin, 2008); identification of thin layers by Zeng in Geologic significance of anomalousinstantaneous frequency (Geophysics, 2010) and lithofacies boundaries, etc.

Complex seismic trace analysis technology is the basis for extracting instantaneous seismic attributes, wherein instantaneous amplitude and instantaneous phase are the basic complex seismic trace attributes, and other instantaneous attributes can be generated by their differentiation, averaging, combination or transformation. The complex seismic trace is an analytical signal, which consists of two parts, the real part and the imaginary part, the real part is the original signal, and the imaginary part is the Hilbert transform of the real part. It is found that the Hilbert transform is sensitive to noise, and the error is large when calculating instantaneous attributes from seismic data with low signal-to-noise ratio, which brings difficulties to seismic attribute analysis. In order to overcome this shortcoming, Barnes pointed out in the document "A tutorial on complexseismic trace analysis" (Geophysics, 2007) that filtering and weighted average processing can effectively overcome the influence of noise when extracting instantaneous attributes. Luo et al. introduced generalized Hilbert transform to calculate instantaneous properties in the document "Generalized Hilbert transform and its applications in geophysics" (The Leading Edge, 2003), which overcomes the defect of Hilbert transform being sensitive to noise; Lu and Zhang in the document "Robust transform" In estimation of instantaneous phase using a time-frequency adaptive filter" (Geophysics, 2013), on the premise that the high-amplitude frequency components in the seismic data have a higher signal-to-noise ratio, a zero-phase adaptive filter is constructed using the time spectrum of STFT , this filter can enhance the high-amplitude frequency components and suppress the low-amplitude frequency components, so as to achieve a robust estimation of the instantaneous phase properties of the seismic signal. Lu and Zhang proposed an adaptive filter algorithm for estimating instantaneous frequency in the literature "A robust instantaneous frequency estimation method" (EAGE, ExtendedAbstracts, 2011).

The invention constructs a complex-valued zero-phase filter, which is a new method for replacing Hilbert transform when complex-track analysis is performed on seismic data. The method integrates the frequency-modulated Gauss window function in a given frequency interval to generate a time-domain zero-phase filter. Using this filter to filter the seismic signal, its response is a complex signal, and the imaginary part of the response signal is the Hilbert transform of the real part. Using this filter response can easily extract the instantaneous properties of the signal.

SUMMARY OF THE INVENTION

technical problem to be solved

Extracting instantaneous attributes using complex-track analysis technology of seismic data requires Hilbert transform on seismic data, and Hilbert transform is sensitive to noise, which makes the extracted instantaneous attributes have large errors, which is not conducive to seismic data interpretation and analysis. In order to overcome the deficiency of Hilbert transform, the present invention proposes a complex-valued Gauss integral filter, which replaces Hilbert transform to realize complex-track analysis of seismic data, and the method can extract instantaneous seismic attributes conveniently and robustly.

Technical solutions

A complex-valued Gauss integral filter is a time-domain filter generated by integrating a Gauss window in a specified frequency interval after frequency modulation, and is characterized in that the expression is as follows:

Among them, f ₁ and f ₂ are the high and low cutoff frequencies of the filter respectively, and 0≤f ₁ <f ₂ ; σ is the resolution factor of the complex Gauss window; j is the imaginary unit, namely t is time.

A discretization method of a complex-valued Gauss integral filter, characterized in that the steps are as follows:

Step 1: Set the filter high and low cutoff frequencies f ₁ and f ₂ , and f ₂ >f ₁ is 0;

Step 2: Set the resolution factor σ of the complex Gauss window, take σ≥2;

Step 3: Obtain the time sampling interval Δt of the digital signal to be filtered, and set the time sampling interval of the complex-valued Gauss integral filter sequence to be equal to the time sampling interval of the digital signal to be filtered;

Step 4: Set the frequency sampling interval Δf of the numerical integration of the integral filter, take Δf≤0.001Hz;

Step 5: Take the filter time interval as [-T, T], where T>0, for each fixed time t, use numerical integration to calculate the value F _σ (t) of the filter sequence at this time.

A method for filtering seismic data and extracting three-instantaneous attributes using a complex-valued Gauss integral filter is characterized in that the steps are as follows:

Step 1: For the seismic data to be filtered, extract several typical sections, do FFT for each channel, and obtain the average frequency spectrum of the seismic section to determine the frequency range of the seismic data;

Step 2: Generate a complex-valued Gauss integral digital filter:

Step 2a: According to the frequency range of seismic data and practical application needs, set the low cutoff frequency f ₁ and high cutoff frequency f ₂ of the complex-valued Gauss integral filter, set the frequency domain sampling interval Δf of the numerical integration, and set the resolution of the FM Gauss window. rate factor σ, usually σ≥2;

Step 2b: For any given time t, calculate the value of the complex-valued Gauss integral filter at time t:

in, The operator "[ ]" means to round the value, and is an imaginary unit;

Step 2c: Perform complex trace analysis on the seismic data, the filter time sampling interval Δt is equal to the time domain sampling interval of the seismic data; the filter time interval is [-T, T], where T>0, the complex-valued Gauss integral The filter numerical sequence has 2M+1 complex-valued points, where, The operator "[ ]" means rounding the value;

The value of T is as follows: if at time t, there is

||F _σ (t)||≤1.0e-6

Then, if there are 6 consecutive positive integers such that

||F _σ (t+kΔt)||≤1.0e-6, k=1, 2, …, 5

method to take

T=t+5Δt

Obtain the numerical sequence of Gauss integral complex filter, denoted as

{F _σ (k)}, k=-M,-M+1,...,-1,0,1,M-1,...,M

F _σ (k) represents the value of the filter at time kΔt;

Step 3: Use the complex-valued Gauss integral digital filter F _σ (k) generated in Step 2 to filter the seismic data Tr(n), n=0, 1, ..., N track by track:

where Re(G _Tr (n)) is the real part of G _Tr (n), which is the result of filtering the seismic trace Tr(n) by the complex-valued Gauss integral filter; Im(G _Tr (n)) is the G _Tr ( The imaginary part of n) is the Hilbert transform of Re(G _Tr (n));

Step 4: Calculate the instantaneous amplitude, instantaneous phase and instantaneous frequency instantaneous properties of the seismic data using Re(G _Tr (n)) and Im(G _Tr (n)) obtained in Step 3:

The instantaneous amplitude InAm(n) is

The instantaneous phase InPh(n) is

InPh(n)=arctan(Im(G _Tr (n))/Re(G _Tr (n))), n=0,1,...,N

The instantaneous frequency InFr(n) is

beneficial effect

The present invention proposes a complex-valued Gauss integral filter and a method for filtering seismic data and extracting three-transient attributes, using the complex-valued Gauss integral filter to replace the Hilbert transform to realize seismic data complex trace analysis. The method has stable performance, overcomes the insufficiency of the Hilbert transform to be sensitive to noise, and can stably and accurately extract the three-transient properties of earthquakes. By modifying the filter high and low cutoff frequencies, low-pass or band-pass filters of different frequency bands can be generated to suit different seismic data analysis needs. The filtering result is a complex-valued analytical signal, which can easily extract the three-transient properties. The principle of the method is clear, the implementation is simple, and there are few user operations.

Description of drawings

Figure 1. Low-pass filter with low cut-off frequency f ₁ =0Hz and high cut-off frequency f ₂ =40Hz. Comparison of the influence of the change of resolution factor on the filter: Figure 1 (a), (b), (c) and ( d) When the resolution factor is 1, the theoretical spectrogram of the low-pass filter, the FFT spectrum of the time-domain numerical filter, and the real and imaginary parts of the time-domain numerical filter; Figure 1 (A), (B) ), (C) and (D) are the theoretical spectrogram of the low-pass filter, the FFT spectrum of the time-domain numerical filter, and the real and imaginary parts of the time-domain numerical filter when the resolution factor is 4; 1 It can be seen from the comparison that as the resolution factor increases, the filter spectral function approaches the ideal gate function.

Figure 2: Band-pass filter with low cut-off frequency f ₁ = 10Hz and high cut-off frequency f ₂ = 20 Hz, the comparison of the effect of the change of resolution factor on the filter: Figure 2 (a), (b), (c) and ( d) are the theoretical spectrogram of the band-pass filter, the FFT spectrum of the time-domain numerical filter, and the real and imaginary parts of the time-domain numerical filter when the resolution factor is 1; Figure 2 (A), (B) ), (C) and (D) are the theoretical spectrogram of the bandpass filter, the FFT spectrum of the time-domain numerical filter, and the real and imaginary parts of the time-domain numerical filter when the resolution factor is 4; 2 It can be seen from the comparison that as the resolution factor increases, the filter spectral function approaches the ideal gate function.

Figure 3. Comparison of the effect of Hilbert transform and complex-valued Gauss integrator filter to achieve complex-track analysis: Figure 3 shows the real part, imaginary part and instantaneous phase of the noise-free 10Hz sinusoidal analytical signal in subgraphs (a), (b) and (c) Figure; sub-figures (A1), (B1) and (C1) are the noise-containing signal with a signal-to-noise ratio of 23dB generated by adding noise to the real signal shown in figure (a), the quadrature signal generated by Hilbert transform, and The instantaneous phase properties calculated by it; subfigures (A2), (B2) and (C2) are the noise-containing signals with a signal-to-noise ratio of 23dB generated by adding noise to the real signal shown in subfigure (a). The real and imaginary parts of the generated analytical signal and the instantaneous phase properties of the signal computed therefrom by the Gauss integrator filter. It can be seen that the instantaneous phase properties calculated by complex-valued Gauss integral filter for complex-track analysis are more accurate.

Fig. 4 Processes the original seismic data with a low-pass complex-valued Gauss integrator filter with low cut-off frequency f ₁ =0 Hz and high cut-off frequency f ₂ =125 Hz, and calculates instantaneous phase, instantaneous frequency and instantaneous amplitude properties: (a) Original seismic trace profile; (b) the imaginary part profile of the original seismic profile filtered by the complex-valued Gauss integral filter; (c) the instantaneous phase attribute profile of the original seismic profile extracted from the filtering result of the complex-valued Gauss integral filter; (d) the complex-valued The instantaneous frequency attribute profile of the original seismic section extracted by the filtering result of the Gauss integral filter; (e) the instantaneous amplitude attribute profile of the original seismic section extracted from the filtering result of the complex-valued Gauss integral filter; (f) the frequency spectrum of the original seismic section (red The curve is the average spectrum).

Fig. 5 Using a low-pass complex-valued Gauss integrator filter with a low cutoff frequency f ₁ =0Hz and a high cutoff frequency f ₂ =30Hz, the original seismic section is subjected to frequency division processing and complex trace analysis, and three instantaneous seismic sections are extracted: (a) The frequency-divided real seismic section is obtained by filtering the original seismic section with a low-pass complex Gauss integral filter; (b) the frequency-divided virtual seismic section is obtained by filtering the original seismic section with a low-pass complex Gauss integral filter; (c) The original seismic section is filtered by a complex valued Gauss integral filter and its instantaneous phase property profile is extracted; (d) the instantaneous frequency property profile extracted by frequency division by a low-pass complex valued Gauss integral filter; (e) a low-pass complex valued Gauss integral filter The instantaneous amplitude attribute profile extracted by the Gauss integrator filter; (f) the frequency spectrum of each channel of the real seismic profile filtered by the complex-valued Gauss integrator filter.

Fig. 6 Using a band-pass complex-valued Gauss integrator filter with a low cutoff frequency f ₁ =20Hz and a high cutoff frequency f ₂ =35Hz, the original seismic section is subjected to frequency division processing and complex trace analysis, and three instantaneous seismic sections are extracted: (a) The frequency-divided real seismic section is obtained by filtering the original seismic section by the band-pass complex Gauss integral filter; (b) the frequency-divided virtual seismic section is obtained by filtering the original seismic section by the band-pass complex Gauss integral filter; (c) The instantaneous phase property profile extracted by the frequency division of the pass complex Gauss integrator filter; (d) the instantaneous frequency property profile extracted by the frequency division of the bandpass complex Gauss integrator filter; The instantaneous amplitude attribute profile extracted by frequency; (f) the frequency spectrum of each channel of the real seismic profile filtered by the band-pass complex Gauss integral filter.

Detailed ways

Seismic data contains rich stratigraphic structure information. However, noisy seismic data reduces the resolution of seismic data, which is not conducive to describing stratigraphic structure with seismic data; in addition, instantaneous seismic attributes are often used in stratigraphic structural analysis, but the Hilbert transform used to calculate instantaneous attributes is sensitive to noise, making the extracted instantaneous attributes Distortion can not reflect the real stratigraphic structure information, which is not conducive to describing the stratigraphic structure. To this end, a complex-valued Gauss integral filter is proposed.

Principle of the present invention:

1) Complex-valued Gauss integral filter

The complex-valued Gauss integral filter is a time-domain filter generated by integrating a Gauss window in a specified frequency range after frequency modulation. Its composition is as follows

Among them, f ₁ and f ₂ are the high and low cutoff frequencies of the filter respectively, and 0≤f ₁ <f ₂ ; σ is the resolution factor of the complex Gauss window; j is the imaginary unit, namely It can be seen from formula (1) that the complex-valued Gauss integral filter is a complex-valued filter, and its real part is an even function in the time domain, and its imaginary part is an odd function. Therefore, this filter is a zero-phase filter, and the filtering does not change the phase information of the original signal. its frequency function for

The complex-valued Gauss integrating filter has the following characteristics:

a) The complex-valued Gauss integral filter is a time-domain zero-phase complex filter, which is an integral function whose integral variable is the dominant frequency of the complex Gauss window;

b) Set the appropriate integration interval, that is, given the high and low cutoff frequencies of the filter, different low-pass or band-pass complex-valued Gauss integral filters can be constructed, which can be used for signal filtering or frequency division processing;

c) By adjusting the resolution factor of the complex Gauss window, the frequency domain response of the filter can be approximated to an ideal rectangular gate function, the filter sidelobes can be suppressed, and the frequency domain resolution of the filter can be effectively controlled;

d) Filter the seismic trace with a complex-valued Gauss integral filter to generate a complex seismic trace, wherein the imaginary part of the complex seismic trace is orthogonal to the real part, that is, the imaginary part is the Hilbert transform of the real part.

e) The complex-valued Gauss integral filter has good noise filtering performance. It uses the filtering results of the seismic traces to calculate the instantaneous seismic attributes, which overcomes the defect that the instantaneous attributes calculated by the Hilbert transform method are sensitive to noise. Four seismic data attributes of virtual seismic trace, instantaneous amplitude, instantaneous phase and instantaneous frequency can be easily extracted from the filtering results.

2) Seismic data filtering and instantaneous attribute extraction

Seismic data is composed of seismic traces as the basic unit, so the filtering of seismic data and the extraction of instantaneous seismic attributes are also implemented trace by trace. Assume that any seismic trace is taken, denoted as Tr(t), and the complex-valued Gauss integral filter F _σ (t) is used to filter Tr(t), and the result is denoted as G _Tr (t), that is, we have

G _Tr (t)=F _σ (t)*Tr(t)

=Re(Fσ(t))*Tr(t)+ _jIm ( _Fσ (t))*Tr(t)

=Re(G _Tr (t))+jIm(G _Tr (t)) (3)

In formula (3), the operator "*" represents the time domain convolution operation. The high and low cutoff frequencies of the complex-valued Gauss integral filter F _σ (t) are different, and different low-pass or band-pass filters can be generated, which can be used to filter the original seismic trace or perform frequency division processing on the seismic trace. The instantaneous seismic attributes can be easily extracted from the filtering results, and the instantaneous amplitude InAm(t) is

The instantaneous phase InPh(t) is

InPh(t)=arctan(Im(G _Tr (t))/Re(G _Tr (t))) (5) The instantaneous frequency InFr(t) is

Other instantaneous properties can be derived from the above properties, and will not be described here.

Integrate the FM Gauss window over the specified frequency interval to generate a zero-phase time-domain complex-valued filter. As the resolution parameter of the complex Gauss window increases, the frequency function of the filter will approach the ideal gate function, so that the filter has a very high resolution. The present invention realizes complex-valued Gauss integral filter by numerical method. The implementation method is as follows:

1) Set the filter high and low cutoff frequencies f ₁ and f ₂ , and f ₂ >f ₁ ≥0. Wherein, when f ₁ =0, the filter is a low-pass filter; when f ₁ >0, the filter The filter is a band-pass filter;

2) Set the resolution factor σ of the complex Gauss window, and take σ≥2. As the resolution factor σ increases, the frequency function of the filter approximates the ideal gate function;

3) Obtain the time sampling interval Δt of the digital signal to be filtered, and set the time sampling interval of the complex-valued Gauss integral filter sequence to be equal to the time sampling interval of the digital signal to be filtered;

4) Set the frequency sampling interval Δf of the numerical integration of the integral filter, take Δf≤0.001Hz;

5) Take the filter time interval as [-T, T], where T>0, take the duration Δt as the time interval, and obtain the filter numerical sequence F _σ (k) (k=-N,-N+1,..., -1, 0, 1, ..., N-1, N), F _σ (k) represents the filter value at time kΔt, The operator [·] means to round up the data. For each fixed time t, numerical integration is used to calculate the value F _σ (t) of the filter sequence at that time.

The complex-valued Gauss integral filter sequence F _σ (k) generated by the above solution is used to filter the seismic signal. Its implementation method is as follows:

1) In the seismic data to be filtered, take out one or several seismic sections, use FFT to calculate the frequency spectrum of each seismic trace one by one, and calculate the average frequency spectrum to determine the frequency range of the seismic data to be processed;

2) Determine the high and low cutoff frequencies of the Gauss integral complex filter according to actual needs, and set the filter time domain sampling interval. filter F _σ (t);

3) If Tr(t) is a seismic trace in the seismic data, the filtering result is

G _Tr (t)=F _σ (t)*Tr(t) (7)

The result of formula (7) is a complex-valued sequence;

4) The complex-valued sequence generated by the filter, according to formulas (4), (5) and (6), the instantaneous amplitude, instantaneous phase and instantaneous frequency of the seismic trace can be obtained.

Specific examples:

The first step is data analysis.

For the seismic data to be processed, extract several seismic profiles, perform FFT on each trace, and obtain the average frequency spectrum of the seismic traces in the profile statistically to determine the frequency range and dominant frequency band of the seismic data;

The second step is to implement a complex-valued Gauss integral numerical filter

1) According to the actual application requirements, set the low cutoff frequency f ₁ and high cutoff frequency f ₂ of the complex-valued Gauss integral filter, set the frequency domain sampling interval Δf of the numerical integration, and set the resolution factor σ of the FM Gauss window, which is often taken as σ ≥2.

2) For any given time t, use numerical integration to calculate the value of the complex-valued Gauss integral filter at time t according to equation (1), and the result is shown in equation (8).

In formula (8), The operator "[ ]" means to round the value, and is an imaginary unit;

3) Do complex-track analysis on the seismic data, the filter time sampling interval Δt is equal to the time domain sampling interval of the seismic data. The filter time interval is [-T, T] (T>0), and the complex-valued Gauss integral filter numerical sequence has 2M+1 complex-valued points, among which, The operator "[ ]" means to round a value.

The value of T is as follows: if at time t, there is

||F _σ (t)||≤1.0e-6 (9) Then, if there are 6 consecutive positive integers, such that

||F _σ (t+kΔt)||≤1.0e-6, k=1, 2, …, 5 (10)

method to take

T=t+5Δt (11)

According to formula (8), the numerical sequence of Gauss integral complex filter can be obtained, which is denoted as

{ _Fσ (k)}, k=-M,-M+1,...,-1,0,1,M-1,...,M (12)

In formula (12), F _σ (k) represents the value of the filter at time kΔt.

The third step is to filter the seismic data with a complex-valued Gauss integral filter to generate complex seismic traces.

Seismic data is based on seismic traces as the basic storage unit, and seismic data retrace analysis is carried out in the sequence of sections, trace by trace. Take a seismic trace, denoted as _Tr (n), n = 0, 1, .

In equation (13), Re(G _Tr (n)) is the real part of G _Tr (n), which is the result of filtering the seismic trace Tr(n) by the complex-valued Gauss integral filter; Im(G _Tr (n)) is the imaginary part of G _Tr (n) and is the Hilbert transform of Re(G _Tr (n)).

In the fourth step, the instantaneous seismic attributes are calculated using the complex traces generated by the filtering.

Calculated using formulas (4), (5) and (6), the instantaneous amplitude InAm(n) is

The instantaneous phase InPh(n) is

InPh(n)=arctan(Im(G _Tr (n))/Re(G _Tr (n))), n=0, 1, ..., N (15)

The instantaneous frequency InFr(n) is

Other transient properties can be generated from the above three transient properties, or consult the related literature.

Claims (1)

1. A method for filtering seismic data and extracting three-transient attributes by adopting a complex value Gauss integral filter is characterized by comprising the following steps:

step 1: extracting a plurality of typical sections of seismic data to be filtered, performing FFT (fast Fourier transform) channel by channel, solving the average frequency spectrum of the seismic sections, and determining the frequency range of the seismic data;

step 2: generating a complex-valued Gauss integral digital filter: the complex value Gauss integral digital filter is a time domain filter generated by integrating a Gauss window in a specified frequency interval after frequency modulation, and is characterized in that the expression is as follows:

wherein f is₁And f₂Respectively the high and low cut-off frequencies of the filter, and f is more than or equal to 0₁＜f₂(ii) a σ is the resolution factor of the complex Gauss window; j is an imaginary unit, i.e.t is time;

step 2 a: setting the low cut-off frequency f of the complex value Gauss integral filter according to the frequency range of the seismic data and the practical application requirement₁And high cut-off frequency f₂Setting a frequency domain sampling interval delta f of numerical integration, and taking the delta f to be less than or equal to 0.001 Hz; setting a resolution factor sigma of a frequency modulation Gauss window, wherein the sigma is usually more than or equal to 2;

and step 2 b: for any given time t, the value of the complex-valued Gauss integrating filter at time t is calculated:

wherein,operator "[ ·]"denotes the rounding of a logarithmic value, andis an imaginary unit;

and step 2 c: performing multi-channel analysis on the seismic data, wherein the value of the time sampling interval delta t of the filter is equal to the time domain sampling interval of the seismic data; the filter time interval is [ -T, T [ -T]Where T > 0, the complex-valued Gauss integrator filter has 2M +1 complex-valued points in its numerical sequence, where,operationsSymbol []"means the rounding of the logarithm;

the value of T is as follows: if at time t, there is

||F_σ(t)||≤1.0e-6

Then, if there are 6 consecutive positive integers, such that

||F_σ(t+kΔt)||≤1.0e-6，k＝1，2，…，5

Method for extracting

T＝t+5Δt

Obtain the Gauss integral complex filter numerical sequence, and record as

{F_σ(k)}，k＝-M，-M+1，…，-1，0，1，M-1，…，M

F_σ(k) Represents the value of the filter at time t + k Δ t;

and step 3: using the complex value Gauss integral digital filter F generated in step 2_σ(k) Filtering the seismic data Tr (N), wherein N is 0, 1, …, and N is channel by channel:

wherein, Re (G)_Tr(n)) is G_Tr(n) the real part of the trace, Tr (n), is the result of filtering by a complex-valued Gauss integrator filter; im (G)_Tr(n)) is G_TrThe imaginary part of (n) is Re (G)_Tr(n)) a Hilbert transform;

and 4, step 4: using Re (G) obtained in step 3_Tr(n)) and Im (G)_Tr(n)) computing instantaneous amplitude, instantaneous phase and instantaneous frequency transient properties of the seismic data:

the instantaneous amplitude Inam (n) is

The instantaneous phase InPh (n) is

InPh(n)＝arctan(Im(G_Tr(n))/Re(G_Tr(n)))，n＝0，1，…，N

The instantaneous frequency InFr (n) is