CN105388527B - A kind of gas-oil detecting method based on complex field matching pursuit algorithm - Google Patents
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Abstract
The invention discloses a kind of gas-oil detecting method based on complex field matching pursuit algorithm, including：Step 1, based on complex field matching pursuit algorithm, the algorithm searched for using variable step great-jump-forward, search obtains the scale factor and frequency factor of best match, and then the best match that seismic channel is calculated is decomposed；Step 2, decomposed according to best match, with reference to Wigner distributions and AnMagnitude calculation obtain time-frequency spectrum；Step 3, the oil gas Monitoring Data of sampled point is obtained according to the frequency variation characteristic for obtaining the time-frequency spectrum.Gas-oil detecting method proposed by the present invention based on complex field matching pursuit algorithm, calculated by the improvement of variable step great-jump-forward searching algorithm and time-frequency spectrum, realize the Rapid matching tracing algorithm of complex field, can be while computational accuracy be ensured using the gas-oil detecting method of the present invention, carry out oil and gas detection processing rapidly and efficiently, greatly save and calculate the time, there is good actual application value.
Description
Technical Field
The invention relates to the field of seismic oil-gas detection, in particular to an oil-gas detection method based on a complex field matching tracking algorithm.
Background
Seismic oil and gas detection technology plays an increasingly important role in oil and gas exploration, and technologies related to analysis, extraction, calibration and the like of seismic oil and gas detection are rapidly developed. However, the AVO analysis and prestack multi-parameter inversion techniques for predicting oil and gas based on the change of reflection amplitude along with offset have the problems of large calculation amount, poor stability, noise interference and the like, and the method is to be perfected. Using the frequency of the seismic data to judgeOil and gas have been known for a long time, such as: M.A.Biot [1-2] (1956) Preliminarily analyzing the absorption and attenuation mechanism of seismic waves based on the oil-gas two-phase medium seismic wave propagation theory; diay [3] (1995) The influence of oil-gas containing property on the frequency of the seismic waves is analyzed based on the change of the frequency spectrum in the reservoir and above and below the reservoir. Therefore, methods for predicting hydrocarbons by analyzing time-frequency characteristics of seismic data have attracted considerable attention.
The traditional time-frequency analysis mainly utilizes window opening of mathematical transformation such as window Fourier transformation, wavelet transformation, s transformation and the like to perform time-frequency processing, so that the obtained time-frequency distribution results are all statistical results of signals in the time window, and the time-frequency analysis methods are all subjected to Heisenberg uncertainty principle [4] The time domain resolution and the frequency domain resolution are restricted with each other, and higher time frequency analysis precision cannot be obtained at the same time.
Matching pursuit algorithm by Mallat [5] The method is proposed in 1993 and is a high-precision seismic signal decomposition and reconstruction algorithm, but the application of the algorithm is restricted by the calculation efficiency. In order to improve the calculation efficiency of the algorithm, scholars at home and abroad carry out various improvements on the algorithm. Liu et al [6-7] Atomic libraries based on Ricker wavelets and Morlet wavelets are established successively in 2004 and 2005, so that the scale of the atomic libraries is greatly reduced; zhang Fanchang and the like [8] A matching tracking algorithm of the dynamic wavelet base is proposed in 2010; wang (Wang) [9] A multi-channel matching tracking algorithm is provided; to reduce the size of the atom pool, zhang Fanchang, etc [10] An atom library based on orthogonal time-frequency atoms is proposed in 2012; 2013 Zhang Fanchang [11] The seismic signals are introduced into the complex field, reducing the control parameters of the wavelets.
The basic principle of the existing complex field matching tracking algorithm is to realize the self-adaptive decomposition of signals by creating a redundant wavelet base and performing overcomplete expansion on complex signals in the wavelet base by using the matching tracking algorithm according to the characteristics of the signals.
Specifically, the existing complex field matching pursuit algorithm can be divided into the following 2 stages:
1. matching and decomposing:
1.1, a redundant wave atom library is to be established. The wavelet base is a set of matched wavelets obtained by time-shifting, modulating and phase-varying the fundamental wavelet g (t). Let g (t) be the basic wavelet that satisfies the condition, γ = (u) r ,f r ,φ r ) Representing the control parameters of the matching wavelet, the matching wavelet can be defined as:
m r (t)＝g(t-u r )exp[i(2πf r (t-u r )+φ r )]； (1-1)
wherein m is r (t) is a matching wavelet, u r Representing a time shift; f. of r Is a frequency factor; phi is a r Is the phase factor. The fundamental wavelet energy is concentrated around a few time-frequency points by time-shift and frequency factors, while the phase factor is used to control the shape change of the matching wavelet.
In the complex field matching pursuit algorithm, the phase and amplitude of the matched wavelets are solved as a whole, so that the matched wavelets with zero phase are generally selected in the establishment of the wavelet base:
m r (t)＝g(t-u r )exp[i2πf r (t-u r )]； (1-2)
experimental analysis shows that Morlet wavelets are more suitable for quantitative analysis of energy and frequency spectrum of seismic records, so that Morlet is generally selected as basic wavelets, and formula (1-2) can be written as follows:
wherein a parameter k is newly introduced r Representing a scale factor and controlling the time window width of the wavelet;
at this time, γ = (k) r ,u r ,f r )。
1.2 calculating the complex seismic channel S of the original seismic channel S c And calculate S c And the instantaneous frequency to which the maximum point corresponds. The real seismic trace is generally used as the real part of the complex seismic trace to actually carry outThe Hilbert transform of the seismic traces constructs complex seismic traces as imaginary parts of the complex seismic traces:
S c (t)＝s(t)+iΗ[s(t)]； (1-4)
wherein the content of the first and second substances,representing the Hilbert transform of the seismic traces s.
S is obtained by calculation according to the formula (1-4) c The position of the envelope maximum is u r And calculating the instantaneous frequency f corresponding to the position s 。
1.3, preferred Scale factor k r Frequency factor f r So that the wave atoms best match the seismic traces. The criteria chosen here are:
wherein the content of the first and second substances,S c (t) is the complex seismic channel, m r (t) is the matching wavelet, k r-best Scale factor for best match, f r-best Frequency factor for best match, D k Is a preferred interval of scale factors, D f Is a preferred interval of the frequency factor.
According to the formulae (1-5), in the given preferred range D k Inner, ensure f r ＝f s Unchanged, for k r Go through traversal and optimization to obtain k r-best (ii) a In the given preferred range D f In, ensure k r ＝k r-best To f for r Go through traversal and optimization to obtain f r-best (ii) a Computing matching wavelets m according to preferred parameter combination formula (1-3) r (t)。
1.4 solving equation S c (t)＝Am r Obtaining A by (t) + R (t) to enable | | | R (t) | | to be minimum, wherein A is a complex number, representing amplitude and phase, and the result of the iterative matching is real [ Am |) r (t)]Calculating the matched result and seismic traceThe residual error of s. And judging the error according to the residual error, and if the error does not meet the requirement, taking the residual error of the iteration as a new seismic channel to continue the iteration until the precision meets the requirement.
Finally, the seismic trace s may be represented as:
wherein the content of the first and second substances,for best matching wavelets, A n To characterize the terms amplitude and phase, M is the number of iterations.
2. And a time-frequency analysis stage:
after complex field matching pursuit decomposition, the seismic trace s is finally decomposed into a series of best matching waveletsn =1,2. The time-frequency distribution is also determined by the following formula:
the wavelet decomposition reconstruction is carried out by using a complex field matching tracking algorithm, although a time frequency spectrum with higher resolution can be obtained, the method still has certain limitation. For example, in the matching decomposition stage, in the process of optimizing a scale factor and a frequency factor in respectively given intervals, the method is due to F (k) r ,f r ) The method comprises the steps that actual seismic channel data are contained, and an analytic expression of the actual seismic channel data cannot be obtained, so that a traversing method is adopted for searching, and a large amount of time is wasted; each calculation needs to combine the formula (1-3) and the formula (1-5) to calculate, which includes a large number of complex multiplication operations. The calculation is complicated and time consuming. In the time-frequency analysis stage, a large amount of complex multiplication operations (see formula (2-1)) occur due to the fact that the complex multiplication operations are carried out in a complex number field, and the calculation efficiency is greatly reduced.
Disclosure of Invention
Although the time-frequency analysis precision of the existing complex field matching tracking algorithm meets the requirement, the calculation efficiency is greatly limited. If the existing method is used for oil gas detection processing, the requirements of actual production work cannot be met, and serious influence is caused to the actual work.
In order to overcome the defects of the existing time-frequency analysis method and overcome the defect that the existing algorithm is slow to improve urgently, a generally applicable fast algorithm is selected for fast matching, the time-frequency analysis precision is ensured, and the oil-gas prediction accuracy is ensured. Aiming at the algorithm of complex field matching pursuit based on the Morlet wavelet atomic library, the invention provides the improvement of the variable step jump type search optimization method, and greatly improves the efficiency of matching pursuit decomposition while ensuring the precision; in the time-frequency analysis stage, wigner distribution and A are adopted n The module value of the method improves the time frequency spectrum calculation, does not influence the precision of the time frequency spectrum, avoids complex multiplication operation and further improves the calculation efficiency; and finally, carrying out oil gas detection according to the high-resolution time frequency spectrum obtained by calculation.
In order to achieve the purpose, the invention provides an oil-gas detection method based on a complex field matching pursuit algorithm, which comprises the following steps: step 1, searching for a scale factor and a frequency factor which are optimally matched by using a variable step size jump type searching algorithm based on a complex field matching tracking algorithm, and further calculating to obtain the optimal matching decomposition of a seismic channel; step 2, decomposing according to the best matching, and combining Wigner distribution with A n Calculating the module value to obtain a time frequency spectrum; and 3, acquiring oil gas monitoring data of sampling points according to the obtained frequency variation characteristics of the time frequency spectrum.
The oil gas detection method based on the complex field matching pursuit algorithm realizes the complex field fast matching pursuit algorithm by the improved calculation of the variable step jump type search algorithm and the timely frequency spectrum.
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The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principles of the invention. In the drawings:
fig. 1 is a flow chart of an oil-gas detection method based on a complex field matching tracking algorithm according to an embodiment of the invention.
FIG. 2 shows F (k) according to an embodiment of the present invention r ) The change curve is shown schematically.
FIG. 3 is a diagram illustrating results obtained after processing a model using a conventional method in an embodiment.
FIG. 4 is a diagram illustrating results obtained after processing a model using the method of the present invention in one embodiment.
Fig. 5 is a diagram illustrating a result obtained by processing actual data according to a conventional method in an embodiment.
FIG. 6 is a diagram illustrating results obtained after actual data is processed by the method of the present invention in one embodiment.
FIG. 7 is a schematic diagram illustrating the results of hydrocarbon testing of an actual seismic section using the method of the present invention in one embodiment.
Detailed Description
The technical means adopted by the invention to achieve the predetermined purpose are further described in the following figures and preferred embodiments of the invention.
Fig. 1 is a flow chart of an oil-gas detection method based on a complex field matching tracking algorithm according to an embodiment of the invention. As shown in fig. 1, the method includes:
step 1, searching for a scale factor and a frequency factor which obtain the best matching by using a variable step size jump type searching algorithm based on a complex field matching tracking algorithm, and further calculating to obtain the best matching decomposition of a seismic channel;
step 2, decomposing according to the optimal matching, and combining Wigner distribution with A n Calculating the module value to obtain a time frequency spectrum;
and 3, acquiring oil gas monitoring data of sampling points according to the frequency of the obtained time frequency spectrum.
Specifically, step 1 is a matching decomposition stage, and a step-variable jump search algorithm is adopted to optimize matching parameters.
Through experimental tests, the function F (k) can be found when one parameter is fixed r ,f r ) The curve of the change with the other parameter has a parabolic trend. In the following, preferred k r Are specifically described as examples. In a given interval D ([ 0.3,3)]) Internal, fixed f r Then F (k) r ) The variation curve of (c) is shown in fig. 2.
According to the characteristics, the best matching scale factor is searched by adopting a variable step jump type search algorithm, and the method comprises the following basic steps:
step 11, initializing calculation;
according to the formula (1-5), f is fixed r ＝f s Will k is r ＝k min Substituting, calculating F (k), and recording F best ＝F(k r )，k r-best ＝k r Skip step sl = (k) max -k min )/δ 1 ；
Wherein f is s Is the instantaneous frequency, k, of the maximum envelope point of the seismic trace min Is a preferred interval D k Minimum value of (a), k max Is a preferred interval D k Maximum value of, δ 1 According to the preferred interval D k Determines the adjustment factor for the initial jump step.
And step 12, carrying out jump type search according to the current step length, wherein the formula (1-5) is substituted into K = K r + sl, calculating F (K);
when F (K)>F best When, remember k r-best ＝K，F best ＝F(k r )，k r K, continue to step 12;
when F (K)<F best Step 13 is executed.
In the step 13, the step of the method is that,by reducing the jump step size appropriately, updating the step size to the current step sizeContinuing the search of step 12 and recording the number N of times of reducing the step length;
wherein, delta 2 A decay factor that is the step size of the jump.
Step 14, when k is r >k max Or when N is larger than a set value, ending the search.
Step 15, recording the k obtained by searching r-best At interval k with the current jump step sl r-best -sl,k r-best ]Traversing search is carried out in the range to obtain the best matching scale factor k r-best 。
By the step-variable jump-type search algorithm, the number of the searched sampling points can be greatly reduced, and the calculation efficiency is improved. For frequency factor f r The same applies to the optimization algorithm of (1). That is, the frequency factor f for obtaining the best match is searched according to the principle of the step-by-step jump search of steps 11-15 r-best 。
According to the scale factor and frequency factor of the best matching, under the guidance of the principle of minimum residual error, calculating to obtain the best matching decomposition of the seismic channel s, namely a series of best matching wavelets of the seismic channel sAnd corresponding A n ，n＝1,2,...,M。
Step 2 is a time-frequency analysis stage, combining Wigner distribution and A n The modulus of (c) is calculated as a time spectrum.
The existing complex domain time spectrum calculation formula is shown in the formula (1-6), wherein a large number of complex numbers are generated for product operation, and the efficiency is extremely low. The invention adopts the method of combining the Wigner distribution of matched Morlet sub-wave atoms with the module value of A to improve the calculation of time spectrum, thereby avoiding the product operation of complex number and improving the operation efficiency.
The specific derivation process is as follows:
matching wavelets m r The Wigner distribution of (t) is expressed as:
wherein the content of the first and second substances,is m r Conjugation of (1);
for ease of calculation, equations (1-3) are converted to:
m r (t)＝exp[a(t-u r ) 2 +ib(t-u r )]； (2)
wherein m is r (t) for the matching wavelets,b＝2πf r-best ；
according to formula (2), we obtain:
substituting formula (3) into formula (1) yields:
according to the document [9 ]]Relation in (1)Converting formula (4) to:
computing m using a Hilbert transform r Envelope of (t):
env[m r (t)]＝|m r (t)+iH[m r (t)]|； (6)
according to the formula (5), the formula (6) and the modulus | A | of A, the matched wavelet m is obtained r (t) the improved time-frequency spectrum calculation formula is as follows:
spp＝WVDenv[m r (t)]|A|； (7)
where spp is the matching wavelet m r (t) time-frequency spectrum, WVD is matching wavelet m r (t) Wigner distribution, | A | is the modulus of A, A is the characteristic wavelet m r The terms of amplitude and phase of (t).
According to equation (7), the time-frequency spectrum SPP of the seismic trace s is calculated as follows:
wherein, the first and the second end of the pipe are connected with each other,is a series of best matching wavelets, WVD, for seismic traces s n For matching waveletsWigner distribution, | A n L is A n Modulus of (A) n Is to characterize the waveletN =1,2, M.
The calculation effect of the formula (8) is consistent with the precision of a time-frequency spectrum calculation formula of the existing complex number field matching tracking algorithm, but the calculation process shows that the improved formula avoids the complex number product operation, reduces the calculation amount and saves the time.
For a clearer explanation of the hydrocarbon detection method based on the complex field matching pursuit algorithm, a specific embodiment is described below, however, it should be noted that the embodiment is only for better explaining the present invention, and should not be construed as an undue limitation to the present invention.
Step 101, reading seismic channel data s and storing the seismic channel data s in columns.
Step 102, setting parameters according to actual conditions: preferred interval D of scale factor k The frequency factor preferably has a radius R f 。
Step 103, calculating the envelope S of the complex seismic channel of the original seismic channel S c And the position corresponding to the maximum envelope point, i.e. the instantaneous shift factor u r And calculating the instantaneous frequency f at the maximum envelope point s 。
Step 104, in the given preferred range D k In the interior, let f r ＝f s Keeping the scale factor k unchanged, and obtaining the best matching scale factor k by using a step-length-variable jump type search algorithm r-best 。
Let k r ＝k r-best Remain unchanged at f s -R f ,f s +R f ]Within the range, the best matching frequency factor f is obtained by utilizing a step-variable jump type search algorithm r-best 。
Step 105, using the best matching parameter γ = (u) obtained above r ,k r-best ,f r-best ) Computing the best matching Morlet wavelet m r (t)。
Solving A by using a damping least square principle, and calculating the residual error R = s-real [ Am ] after matching r ]. Judging the error according to R, and stopping iteration if the error is less than 0.1 s; otherwise let s = R, go to step 103 to continue the iteration.
106, a series of matched wavelets are obtained by complex field fast matching, tracing and decomposingAnd corresponding A n Reconstructing to obtain a new seismic channel; and (5) calculating by using the equation (8) to obtain the time frequency spectrum of the seismic channel s.
And 107, identifying oil gas display at the sample point according to the frequency change characteristics of the acquired time frequency spectrum, and completing oil gas detection.
The model data is processed by using the existing method and the present invention, the results are shown in fig. 3 and fig. 4, and the specific information of the processing results is shown in table 1.
TABLE 1 comparison of model Signal processing results
The single-channel real seismic data are processed by using the existing method and the invention, the results are shown in fig. 5 and fig. 6, and the specific information of the processing results is shown in table 2.
TABLE 2-2 comparison of actual signal processing results
From a comparison of fig. 3 and 4, and fig. 5 and 6, it can be seen that the results of the matching decomposition reconstruction of the two inverse methods are approximately consistent; the time frequency spectrum obtained by the method has the same transformation trend as the time frequency spectrum obtained by the existing method, but the time frequency spectrum obtained by the method has more obvious contrast and better effect. As can be seen from tables 1 and 2, the quality of the two reconstructed signals is almost consistent, but the calculation efficiency of the method is obviously better than that of the existing method, and the speed is obviously increased as the number of sampling points is larger.
The method is used for processing the actual seismic section and performing oil-gas detection processing by using the obtained time-frequency spectrum. The results of the hydrocarbon testing were well matched to the well as shown in figure 7.
The complex field matching pursuit algorithm-based oil-gas detection method provided by the invention realizes the complex field rapid matching pursuit algorithm by the improved calculation of the variable step jump type search algorithm and the timely frequency spectrum.
The above-mentioned embodiments are intended to illustrate the objects, technical solutions and advantages of the present invention in further detail, and it should be understood that the above-mentioned embodiments are only exemplary embodiments of the present invention, and are not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
References mentioned in the background of the invention:
[1]Biot M A.Theory of propagation of elastic waves in a fluid‐saturated porous solid.I.Low‐frequency range[J].the Journal of the Acoustical Society of America,1956,28(2):168-178.
[2]Biot M A.Theory of propagation of elastic waves in a fluid‐saturated porous solid.II.Higher frequency range[J].the Journal of the Acoustical Society of America,1956,28(2):179-191.
[3]Dilay A,Eastwood J.Spectral analysis applied to seismic monitoring of thermal recovery[J].The Leading Edge,1995,14(11):1117-1122.
[4] wangchun, shang Guoan, li Fayuan, zhu Xuejian, gu Yini extraction of uncertainty in slope spectral information based on DEM [ J ]. Earth information science, 2008,10 (4): 539-545.
[5]Mallat S G,Zhang Z.Matching pursuits with time-frequency dictionaries[J].Signal Processing,IEEE Transactions on,1993,41(12):3397-3415.
[6]Liu J,Wu Y,Han D,et al.Time-Frequency Decomposition Based On Ricker Wavel[C].2004SEG Annual Meeting.Society of Exploration Geophysicists,2004.
[7]Liu J,Marfurt K J.Matching pursuit decomposition using Morlet wavelets[C].2005SEG Annual Meeting.Society of Exploration Geophysicists,2005.
[8] Zhang Fanchang, li Chuanhui, yin Xing blazing seismic data based on a dynamic matched wavelet base rapid matched pursuit [ J ] oil geophysical exploration 2010,45 (5): 667-673.
[9]Wang Y.Multichannel matching pursuit for seismic trace decomposition[J].Geophysics,2010,75(4):V61-V66.
[10] Zhang Fanchang, li Chuanhui seismic signal fast match trace based on orthogonal time-frequency atoms [ J ]. Geophysical reports, 2012,55 (1): 277-283.
Claims (3)
1. An oil-gas detection method based on a complex field matching pursuit algorithm is characterized by comprising the following steps:
step 1, introducing a scale factor search based on a complex field matching tracking algorithm, searching for a scale factor and a frequency factor which obtain the best matching by using a variable step size jump search algorithm, and further calculating to obtain the best matching decomposition of a seismic channel;
step 2, combining Wigner distribution and envelope of matched wavelets and A according to the optimal matching decomposition n Calculating the module value to obtain a time frequency spectrum; wherein A is n Terms characterizing the amplitude and phase of the matched wavelet;
step 3, obtaining oil gas monitoring data of sampling points according to the frequency change rate characteristics of the obtained time frequency spectrum;
in step 1, based on a complex field matching and tracking algorithm, a variable step jump type searching algorithm is used to search and obtain a scale factor and a frequency factor which are the best matched, and the method comprises the following steps:
step 11, initializing calculation;
selecting formula according to optimal matching factorFixed f r ＝f s And will k r ＝k min Substituting, calculate F (k) r ) Remember k r-best ＝k r ，F best ＝F(k r ) Skip step sl = (k) max -k min )/δ 1 ；
Wherein k is r-best Scale factor for best match, F best Is the maximum projection, f r-best Frequency factor for best match, D k A preferred interval for the scale factor, D f Is a preferred interval of frequency factors, k r Is a scale factor, f r Is a frequency factor, f s Is the instantaneous frequency, k, of the maximum envelope point of the seismic trace min Is a preferred interval D k Minimum value of (a), k max Is a preferred interval D k Maximum value of, δ 1 According to the preferred interval D k Determining an adjustment factor for the initial jump step size;
step 12, jump-type search is carried out according to the current step length, and a formula is selected and substituted into K = K according to the optimal matching factor r + sl, calculating F (K);
when F (K) > F best Time, remember k r-best ＝K，F best ＝F(k r )，k r K, continue to step 12;
when F (K) < F best Executing step 13;
step 13, reducing the step size of the jump and updating the step size to the current step sizeContinuing the search of step 12 and recording the number N of times of reducing the step length;
wherein, delta 2 A decay factor that is a step size of the jump;
step 14, when k is r ＞k max Or when N is larger than a set value, finishing the search;
step 15, recording the k obtained by searching r-best At interval k with the current jump step sl r-best -sl,k r-best ]Traversing search is carried out in the range to obtain the best matching scale factor k r-best ；
Searching for the frequency factor f that obtains the best match according to the principle of step-by-step jump search of steps 11-15 r-best ；
And calculating to obtain the best matching decomposition of the seismic channel s under the guidance of the principle of minimum residual error according to the scale factor and the frequency factor of the best matching.
2. The method of claim 1, wherein in step 2, the Wigner distribution is combined with A according to the best match decomposition n The time frequency spectrum is obtained by calculating the modulus value of (A), and the formula is as follows:
wherein SPP is the time-frequency spectrum of seismic channel s, M is the number of matching decompositions,decomposing wavelets for best match of seismic traces, WVD n Is composed ofThe Wigner distribution of (a) is,is composed ofEnvelope, | A n L is A n Modulus of (A) n Is to characterizeThe amplitude and phase of (c).
3. The method of claim 2, wherein in step 2, the formula for calculating the time-frequency spectrum is derived as follows:
matching wavelets m r The Wigner distribution of (t) is expressed as:
wherein the content of the first and second substances,is m r T is time, f is frequency, τ is time delay;
matching wavelets m r (t) is expressed as:
m r (t)＝exp[a(t-u r ) 2 +ib(t-u r )]； (2)
wherein u is r Is time shifting; a and b are parameters introduced in a specific formb＝2πf r-best ；
According to formula (2), we obtain:
substituting formula (3) into formula (1) yields:
further finishing formula (4) to obtain:
computing m using a Hilbert transform r Envelope of (t):
env[m r (t)]＝|m r (t)+iH[m r (t)]|； (6)
according to the formula (5), the formula (6) and the modulus | A | of A, the matched wavelet m is obtained r (t) the improved time-frequency spectrum calculation formula is as follows:
spp＝WVDenv[m r (t)]|A|； (7)
wherein spp is matched wavelet m r (t) time-frequency spectrum, WVD is matching wavelet m r (t) Wigner distribution, | A | is the modulus of A;
from equation (7), the time-spectrum SPP of the seismic traces s is calculated as follows:
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