Array passive source multi-mode surface wave frequency dispersion-based artifact removal method

Abstract

The invention discloses an artifact removal method based on array passive source multi-mode surface wave frequency dispersion, which relates to the field of seismic imaging, and comprises the following steps: calculating the relative distance and orientation between stations in an array; calculating a noise cross-correlation function between the stations; constructing a causal cross-spectral density matrix according to the noise cross-correlation function between the stations; and substituting the relative distance and orientation between the stations and the causal cross-spectral density matrix into a modified weighted cross-correlation bunching analysis or modified modified bunching analysis expression, and removing artifacts in a beam pattern and a frequency dispersion pattern. According to the method, the occurrence of artifacts in the frequency dispersion diagram is effectively reduced, and the accuracy of multi-mode frequency dispersion curve extraction and mode recognition is remarkably improved, so that the result of subsequent earth velocity structure inversion is more accurate and reliable.

Description

Technical Field

[0001] The invention relates to the field of seismic imaging, and in particular to an artifact removal method based on array passive source multi-mode surface wave dispersion. Background Art

[0002] Since the 21st century, the emergence of background noise imaging methods has undoubtedly been a milestone breakthrough in the field of seismological research. Different from the traditional research path based on natural earthquake event information, this method relies on background noise data and uses seismic interference methods to effectively extract surface wave dispersion information, thereby achieving accurate inversion of the earth's velocity structure. The background noise imaging method has shown unique advantages in the study of the earth's internal structure with its excellent repeatability, high-resolution imaging effects, and convenient and efficient operation procedures. In recent years, with the rapid development of node-type seismograph technology, a large number of dense and ultra-dense seismic arrays have been deployed around the world. The establishment of these arrays has greatly enriched the channels for obtaining seismic data, provided more sufficient data support for seismological research, and strongly promoted the in-depth research and wide application of array methods.

[0003] In practical applications, the deployment of seismic arrays faces a series of constraints. Limited by the current level of technology and the actual geographical environment, geological conditions and other factors, it is very difficult to achieve a high-density deployment of seismic arrays. And in the actual operation process, in order to ensure the stability and consistency of data acquisition, seismic arrays usually adopt a regularly distributed layout pattern. Although this mode facilitates data collection and processing to a certain extent, it also brings some problems. Due to the similarity of distance and direction between stations, the array repeatedly samples at a fixed distance and direction, which makes it difficult to fully and comprehensively sample the spatial noise field.

[0004] The limitations in the above array layout and sampling process will directly affect the results of subsequent data processing and analysis. Taking the weighted cross-correlation beamforming analysis (WCBF) method and the modified beamforming analysis (MCBF) method as examples, a large number of artifacts often appear in the beam diagrams and dispersion diagrams generated by them. The existence of these artifacts seriously interferes with the accurate extraction of dispersion curves and the accurate judgment of surface wave patterns. Dispersion curves and surface wave pattern recognition are crucial for the inversion of the earth velocity model, which indirectly affects the accuracy and reliability of the inversion results of the earth velocity model. At present, CN 116400406A proposes an array-based passive source multi-mode surface wave dispersion curve extraction method. Although it can use the short-term background noise data recorded by the station array to extract a multi-mode surface wave dispersion curve with sufficient accuracy, the method proposed in the patent still has many artifacts that interfere with the beam diagram and dispersion diagram. How to overcome these problems and further improve the resolution of multi-mode dispersion maps extracted by the array method based on background noise data, and thereby improve the accuracy of Earth velocity model inversion, has become a key issue in the current field of seismology that urgently needs to be studied and resolved. Summary of the invention

[0005] In view of the above-mentioned deficiencies in the prior art, the present invention provides an artifact removal method based on array passive source multi-mode surface wave dispersion, which solves the interference problem of artifacts and power branches in the dispersion diagram.

[0006] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a method for removing artifacts based on array passive source multi-mode surface wave dispersion, comprising the following steps:

[0007] S1. Calculate the relative distance and orientation between stations in the array;

[0008] S2, calculate the noise cross-correlation function between stations;

[0009] S3, constructing the causal cross-spectral density matrix based on the noise cross-correlation function between stations;

[0010] S4. The relative distance and orientation between stations and the causal cross-spectral density matrix are introduced into the modified weighted cross-correlation beamforming analysis or the modified modified beamforming analysis expression to remove artifacts in the beam diagram and dispersion diagram.

[0011] Furthermore, the specific method of calculating the noise cross-correlation function between stations in step S2 is: obtaining the noise record of each station pair; dividing the noise record of each station pair into noise records of different time periods with a time window of fixed length and a step size; calculating the cross-correlation of the noise records of each time period of the station pair and superimposing them to obtain the noise cross-correlation function between the stations;

[0012] The noise cross-correlation function between stations is expressed as:

[0013] cor ij (ω)= <d * (x i ,ω)d(x j ,ω)>

[0014] Where i represents the i-th station in the array, j represents the j-th station in the array; cor ij (ω) represents the noise cross-correlation function between station i and station j in the frequency domain, x i represents the vector position of station i, x j represents the vector position of station j; ω represents the angular frequency; d(x i ,ω) is the Fourier spectrum of the noise record of station i, d(x j ,ω) is the Fourier spectrum of the noise record of station j, * means taking the conjugate, and <> means taking the long-term average.

[0015] Furthermore, the cross-correlation of the noise records of the station for each time period is calculated as follows: the noise records of the station for each time period are normalized in the time domain using a sliding average to obtain the result after the first normalization; the result after the first normalization is Fourier transformed to obtain the result after the Fourier transform; the result after the Fourier transform is normalized using a sliding average to obtain the result after the second normalization; the result after the second normalization is multiplied in the frequency domain to obtain the cross-correlation of the noise records of the station for each time period.

[0016] Furthermore, the specific method of step S3 is: cor ij (ω) is Fourier transformed to obtain the noise cross-correlation function of station i and station j in the time domain; the part of t≥0 in the noise cross-correlation function of station i and station j in the time domain is taken to constitute the causal response, and the part of t≤0 in the noise cross-correlation function of station i and station j in the time domain is taken to constitute the non-causal response; based on the causal response and the non-causal response, the causal cross-spectral density matrix is ​​constructed according to the station pair, and its expression is:

[0017]

[0018] in, is the causal cross-spectral density matrix, θ ij represents the direction between station i and station j; cor ij (t) represents the causal response between station i and station j, cor ij (-t) represents the non-causal response between station i and station j; t represents the time variable; N represents the number of stations in the array; FT{} represents Fourier transform.

[0019] Furthermore, the modified weighted cross-correlation clustering analysis in step S4 is a linear combination of two basis search results, and its expression is:

[0020] WCBF(k,ω,θ)=(CC+SS) / 2

[0021]

[0022] θ-π / 2≤θ ij ≤θ+π / 2

[0023] Where WCBF(k,ω,θ) represents the modified weighted cross-correlation bunching analysis result, N represents the number of stations in the array; r ij represents the distance between station i and station j; θ ij represents the azimuth between station i and station j; k represents the wave number of the surface wave, k = ω / c, c represents the phase velocity of the surface wave; Re is the real part, Im is the imaginary part; θ represents the search azimuth; N(θ) represents the condition that satisfies θ-π / 2≤θ ij The number of stations ≤θ+π / 2, π is an angle of 180 degrees; CC and SS are both intermediate parameters; is the causal cross-spectral density matrix; ω represents the angular frequency, cos is the cosine function, and sin is the sine function.

[0024] Furthermore, the modified modified cluster analysis in step S4 is a linear combination of the two basis search results, and its expression is:

[0025] MCBF(k,ω)=(XX+YY) / 2

[0026]

[0027] Wherein, MCBF(k,ω) represents the modified corrected bunching analysis result, and YY and XX are intermediate parameters.

[0028] The beneficial effects of the present invention are:

[0029] 1. By dividing the noise cross-correlation function in the time domain into causal signals and non-causal signals and constructing a causal cross-spectral density matrix, the relationship between the signals can be better analyzed, and a dispersion diagram with less artifact influence can be obtained.

[0030] 2. Using the modified corrected cross-correlation bunching analysis method and the modified weighted cross-correlation bunching analysis method, that is, using the linear combination of two basis search results, the artifacts in the dispersion diagram can be eliminated more efficiently, and the stability and reliability of the power branch signal can be greatly improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 A flow chart of the method proposed by the present invention;

[0032] Figure 2 A schematic diagram of a known earth velocity model described in an embodiment;

[0033] Figure 3 is a known multi-mode dispersion curve diagram described in the embodiment;

[0034] Figure 4 This is a schematic diagram of station distribution in the embodiment;

[0035] Figure 5 A schematic diagram of a large number of point sources described in the embodiment;

[0036] Figure 6 Schematic diagram of station distance and azimuth in the embodiment;

[0037] Figure 7 A schematic diagram of noise recording of some stations simulated by a large number of point sources in the embodiment;

[0038] Figure 8 is the noise cross-correlation function of some station pairs calculated in the embodiment;

[0039] Fig. 9 It is a schematic diagram of the causal mutual spectral density matrix;

[0040] Fig.10 : is a beam diagram obtained by using modified weighted cross-correlation beamforming analysis at 5 Hz in the embodiment;

[0041] Fig.11 : is a beam diagram obtained by using modified weighted cross-correlation beamforming analysis at 7.5 Hz in the embodiment;

[0042] Fig.12 is the beam pattern obtained using the prior art at 5 Hz;

[0043] Fig.13 is the beam pattern obtained using the prior art at 7.5 Hz;

[0044] Fig.14 It is the azimuth average dispersion diagram obtained by directly using the modified cross-correlation bunching analysis in the embodiment;

[0045] Fig.15 is the azimuthally averaged dispersion diagram obtained using the existing technology;

[0046] Fig.16 To use Fig.14 Selected multi-mode dispersion plots;

[0047] Fig.17 It is a schematic diagram of the ChinArray subarray;

[0048] Fig.18 The noise cross-correlation function obtained by using the present invention in comparative experiment 1;

[0049] Fig.19 The dispersion diagram obtained by using the present invention in comparative experiment 1;

[0050] Fig. 20 To compare the dispersion diagram obtained using the existing technology in Experiment 1;

[0051] Fig.21 This is a schematic diagram of the Tongzhou subarray;

[0052] Fig. 22 The noise cross-correlation function obtained by using the present invention in comparative experiment 2;

[0053] Fig.23 The dispersion diagram obtained by using the present invention in comparative experiment 2;

[0054] Fig.24 To compare the dispersion diagram obtained using the existing technology in Experiment 2;

[0055] Fig.25 This is a schematic diagram of the Tangshan subarray;

[0056] Fig.26 The noise cross-correlation function obtained by using the present invention in comparative experiment 3;

[0057] Fig. 27 For comparison, the dispersion diagram obtained by using the present invention in Experiment 3;

[0058] Fig.28 This is a dispersion diagram obtained by using the existing technology in comparative experiment 3. DETAILED DESCRIPTION

[0059] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0060] like Figure 1 As shown, the present invention provides an artifact removal method based on array passive source multi-mode surface wave dispersion, comprising the following steps:

[0061] S1. Calculate the relative distance and orientation between stations in the array;

[0062] In one embodiment of the present invention, the earth velocity model and the multi-mode dispersion curve are used as examples to generate simulated noise data under known conditions to introduce a specific implementation method of the present invention, wherein the earth velocity model is as follows: Figure 2 The multi-mode dispersion curve is shown in Figure 3 The simulated data uses a large number of point source excited velocity records to simulate the noise records of the array. The array contains 144 stations, and its lateral distribution is as follows Figure 4 As shown, a large number of point sources such as Figure 5 The simulated array position is a Cartesian coordinate system, and the relative position of the station pair can be directly calculated, as shown in Figure 6 As shown, the two black points give the relative distance and azimuth of station 00 and station 99. For the actual longitude and latitude data of the array, the relative distance and azimuth of the station pair need to be calculated according to the ellipsoid coordinate system.

[0063] S2. Calculate the noise cross-correlation function between stations: obtain the noise records of each station pair; divide the noise records of each station pair into noise records of different time periods with a fixed length time window and step size; calculate the cross-correlation of the noise records of each time period of the station pair and superimpose them to obtain the noise cross-correlation function between stations.

[0064] The noise records of some stations simulated using a large number of point sources are as follows: Figure 7 As shown in the figure, the noise records in some station noise records are divided into different time periods using a 20s time window and a 10s step size. For each station pair, the cross-correlation of the noise records in each time period is calculated: the noise records of the station pair in each time period are normalized in the time domain using a sliding average to obtain the first normalized result; the first normalized result is Fourier transformed to obtain the Fourier transformed result; the Fourier transformed result is normalized by sliding average to obtain the second normalized result; the second normalized result is multiplied in the frequency domain to obtain the cross-correlation of the noise records in each time period of the station pair. The cross-correlation of the noise records in each time period is superimposed to obtain the noise cross-correlation function of the station pair. Figure 8 The figure shows the noise cross-correlation function of some station pairs calculated, arranged vertically according to the distance between the station pairs. The black solid line is the noise cross-correlation function between station 00 and station 99. According to the distance between the station pairs, the separation of wave packets of different modes can be seen, and its trend is marked by the black dotted line. For actual data, the corresponding parameters need to be set according to the pre-extracted dispersion curve.

[0065] The noise cross-correlation function between stations is expressed as:

[0066] cor ij (ω)= <d * (x i ,ω)d(xj ,ω)>

[0067] Where i represents the i-th station in the array, j represents the j-th station in the array; cor ij (ω) represents the noise cross-correlation function between station i and station j in the frequency domain, x i represents the vector position of station i, x j represents the vector position of station j; ω represents the angular frequency; d(x i ,ω) is the Fourier spectrum of the noise record of station i, d(x j ,ω) is the Fourier spectrum of the noise record of station j, * means taking the conjugate, and <> means taking the long-term average.

[0068] S3, constructing the causal cross-spectral density matrix based on the noise cross-correlation function between stations;

[0069] Cor ij (ω) is Fourier transformed to obtain the noise cross-correlation function of station i and station j in the time domain; the part of t≥0 in the noise cross-correlation function of station i and station j in the time domain is taken to constitute the causal response, and the part of t≤0 in the noise cross-correlation function of station i and station j in the time domain is taken to constitute the non-causal response; based on the causal response and the non-causal response, the causal cross-spectral density matrix is ​​constructed according to the station pair, as Fig. 9 As shown, its expression is:

[0070]

[0071] in, is the causal cross-spectral density matrix, θ ij represents the direction between station i and station j; cor ij (t) represents the causal response between station i and station j, cor ij (-t) represents the non-causal response between station i and station j; t represents the time variable; N represents the number of stations in the array; FT{} represents Fourier transform.

[0072] S4. The relative distance and orientation between stations and the causal cross-spectral density matrix are introduced into the modified weighted cross-correlation beamforming analysis or the modified modified beamforming analysis expression to remove artifacts in the beam diagram and dispersion diagram.

[0073] The modified weighted cross-correlation clustering analysis in step S4 is a linear combination of two basis search results, and its expression is:

[0074] WCBF(k,ω,θ)=(CC+SS) / 2

[0075]

[0076] θ-π / 2≤θ ij ≤θ+π / 2

[0077] Where WCBF(k,ω,θ) represents the modified weighted cross-correlation bunching analysis result, N represents the number of stations in the array; r ij represents the distance between station i and station j; θ ij represents the azimuth between station i and station j; k represents the wave number of the surface wave, k = ω / c, c represents the phase velocity of the surface wave; Re is the real part, Im is the imaginary part; θ represents the search azimuth; N(θ) represents the condition that satisfies θ-π / 2≤θ ij The number of stations ≤θ+π / 2, π is an angle of 180 degrees; CC and SS are both intermediate parameters; is the causal cross-spectral density matrix; cos is the cosine function, and sin is the sine function.

[0078] The modified modified cluster analysis is a linear combination of two basis search results, and its expression is:

[0079] MCBF(k,ω)=(XX+YY) / 2

[0080]

[0081] Wherein, MCBF(k,ω) represents the modified corrected bunching analysis result, and YY and XX are intermediate parameters.

[0082] In this embodiment, Fig.10 The beam pattern obtained using the modified weighted cross-correlation beamforming (WCBF) at 5 Hz, with the power expressed as the intensity of the incident surface wave at the X and Y velocities. Fig.11 The beam pattern is obtained using the modified weighted cross-correlation beamforming (WCBF) at 7.5 Hz. Fig.12 : is the beam pattern obtained using the prior art at 5 Hz. Fig.13 The beam pattern obtained at 7.5 Hz using the prior art. Compared with the prior art, the beam pattern obtained using the present invention has a more continuous power change along the azimuth, a smaller artifact effect, and a clearer mode branch. The beam patterns of all frequencies are superimposed along the azimuth and arranged according to frequency to obtain an azimuth average dispersion diagram without artifact effects.

[0083] Fig.14 To directly use the azimuthally averaged dispersion map obtained by the modified cross-correlation bunching analysis (MCBF), Fig.15 It is the azimuth average dispersion map obtained by using the prior art. It can be seen from the results in the figure that compared with the prior art, the dispersion map obtained by the method proposed in the present invention has less artifacts, more continuous mode branches and higher resolution.

[0084] After obtaining the dispersion diagram, the frequency-speed values ​​can be selected from the dispersion diagram along the power extreme value branch to form a multi-mode dispersion curve. Fig.16 To use Fig.14 Selected multi-mode dispersion curves. The obtained multi-mode dispersion curves are intended to be used for subsequent earth velocity structure inversion.

[0085] In order to verify the beneficial effects of the present invention, the following comparative experiments were conducted for different sub-arrays.

[0086] Comparative Experiment 1: The dispersion diagrams corresponding to the ChinArray subarrays are obtained using the method proposed by the present invention and the prior art. Fig.17 As shown in Figure 1, the ChinArray subarray consists of 104 broadband seismometers, with a station spacing of about 70 km and a noise record length of about 30 days. The time window length used to calculate the noise cross-correlation function is 1000 s and the step length is 500 s, as shown in Figure 1. Fig.18 As shown in FIG. 1 , the noise cross-correlation function can be observed according to the arrangement of the station distances, and an obvious surface wave waveform can be observed. The dispersion diagram obtained by the present invention is shown in FIG. Fig.19 As shown, the dispersion diagram obtained using the existing technology is as follows Fig. 20 shown.

[0087] Comparative experiment 2: The dispersion diagram corresponding to the Tongzhou subarray is obtained by using the method proposed by the present invention and the prior art. Fig.21 As shown in Figure 1, the Tongzhou subarray consists of 56 short-period seismometers, with a station spacing of about 1 km and a noise record length of about 45 days. The time window length used to calculate the noise cross-correlation function is 300 seconds and the step length is 150 seconds, as shown in Figure 1. Fig. 22 As shown in FIG. 1 , the noise cross-correlation function can be observed according to the arrangement of the station distances to observe two clearly separated mode surface wave waveforms. The dispersion diagram obtained by the present invention is shown in FIG. Fig.23 As shown, the dispersion diagram obtained using the existing technology is as follows Fig.24 shown.

[0088] Comparative experiment 3: The dispersion diagram corresponding to the Tangshan subarray is obtained by using the method proposed by the present invention and the prior art. Fig.25 As shown in Figure 1, the Tangshan subarray consists of 30 node-type seismometers, with a station spacing of about 30m and a noise record length of about 8 days. The time window length used to calculate the noise cross-correlation function is 50s and the step length is 25s, as shown in Figure 1. Fig.26 As shown in FIG. 1 , the noise cross-correlation function can be arranged according to the distance between stations to observe the surface wave waveform, but the wave packets are not separated. The dispersion diagram obtained by the present invention is shown in FIG. Fig. 27 As shown, the dispersion diagram obtained using the existing technology is as follows Fig.28 shown.

[0089] Through comparative analysis between the present invention and the prior art, it can be seen that the present invention can significantly and efficiently eliminate artifacts in the dispersion diagram, greatly reducing the possibility of result deviation and pattern misjudgment caused by the presence of artifacts. At the same time, the present invention can also effectively reduce the disturbance phenomenon on the power branch, greatly improving the stability and reliability of the power branch signal. After actual verification and data comparison, the present invention is effective in improving the resolution of the power branch, and can provide more accurate and reliable data support for subsequent data analysis and processing, thereby fundamentally improving the performance and application value of the entire method.

Claims

1. A method for removing artifacts based on array passive source multi-mode surface wave dispersion, characterized in that: The following steps are involved: S1. Calculate the relative distance and orientation between stations in the array; S2, calculate the noise cross-correlation function between stations; S3, constructing the causal cross-spectral density matrix based on the noise cross-correlation function between stations; S4. The relative distance and orientation between stations and the causal cross-spectral density matrix are introduced into the modified weighted cross-correlation beamforming analysis or the modified modified beamforming analysis expression to remove artifacts in the beam diagram and dispersion diagram.

2. The method for removing artifacts based on array passive source multi-mode surface wave dispersion according to claim 1, characterized in that: The specific method of calculating the noise cross-correlation function between stations in step S2 is: obtaining the noise record of each station pair; dividing the noise record of each station pair into noise records of different time periods with a fixed length time window and step size; calculating the cross-correlation of the noise records of each time period of the station pair and superimposing them to obtain the noise cross-correlation function between the stations; The noise cross-correlation function between stations is expressed as: heart ij (ω)= <d * (x i ,ω)d(x j ,oh)> Where i represents the i-th station in the array, j represents the j-th station in the array; cor ij (ω) represents the noise cross-correlation function between station i and station j in the frequency domain, x i represents the vector position of station i, x j represents the vector position of station j; ω represents the angular frequency; d(x i ,ω) is the Fourier spectrum of the noise record of station i, d(x j ,ω) is the Fourier spectrum of the noise record of station j, * means taking the conjugate, and < > means taking the long-term average.

3. The method for removing artifacts based on array passive source multi-mode surface wave dispersion according to claim 2, characterized in that: The cross-correlation of the noise records of the station for each time period is calculated as follows: the noise records of the station for each time period are normalized in the time domain using a sliding average to obtain the result after the first normalization; Perform Fourier transform on the result after the first normalization to obtain the result after Fourier transform; Perform sliding average normalization on the result after Fourier transformation to obtain the result after the second normalization; The results after the second normalization are multiplied in the frequency domain to obtain the cross-correlation of the noise records of the station for each time period.

4. The method for removing artifacts based on array passive source multi-mode surface wave dispersion according to claim 2, characterized in that: The specific method of step S3 is: ij (ω) is Fourier transformed to obtain the noise cross-correlation function of station i and station j in the time domain; the part of t≥0 in the noise cross-correlation function of station i and station j in the time domain is taken to constitute the causal response, and the part of t≤0 in the noise cross-correlation function of station i and station j in the time domain is taken to constitute the non-causal response; based on the causal response and the non-causal response, the causal cross-spectral density matrix is ​​constructed according to the station pair, and its expression is: in, is the causal cross-spectral density matrix, θ ij represents the direction between station i and station j; cor ij (t) represents the causal response between station i and station j, cor ij (-t) represents the non-causal response between station i and station j; t represents the time variable; N represents the number of stations in the array; FT{} represents Fourier transform.

5. The method for removing artifacts based on array passive source multi-mode surface wave dispersion according to claim 1, characterized in that: The modified weighted cross-correlation clustering analysis in step S4 is a linear combination of two basis search results, and its expression is: WCBF(k,ω,θ)=(CC+SS) / 2 θ-π / 2≤θ ij ≤θ+π / 2 Where WCBF(k,ω,θ) represents the modified weighted cross-correlation bunching analysis result, N represents the number of stations in the array; r ij represents the distance between station i and station j; θ ij represents the azimuth between station i and station j; k represents the wave number of the surface wave, k = ω / c, c represents the phase velocity of the surface wave; Re is the real part, Im is the imaginary part; θ represents the search azimuth; N(θ) represents the condition that satisfies θ-π / 2≤θ ij The number of stations ≤θ+π / 2, π is an angle of 180 degrees; CC and SS are both intermediate parameters; is the causal cross-spectral density matrix; ω represents the angular frequency; cos is the cosine function, and sin is the sine function.

6. The method for removing artifacts based on array passive source multi-mode surface wave dispersion according to claim 5, characterized in that: The modified modified cluster analysis in step S4 is a linear combination of the two basis search results, and its expression is: MCBF(k,ω)=(XX+YY) / 2 Wherein, MCBF(k,ω) represents the modified corrected bunching analysis result, and YY and XX are intermediate parameters.

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