Zhang Qiu* , China University of Petroleum; Junhua Zhang , China University of Petroleum; Zhang ... effect; Chi Zhaohuan and Liu Cai researched the method of.
Research on spectral inversion method based on Moore-Penrose algorithm Zhang Qiu* , China University of Petroleum; Junhua Zhang , China University of Petroleum; Zhang Xiao Hui,China University of Petroleum;Hu Wen, China University of Petroleum; Liu Lei, Research Institute of Exploration and Production of Sinopec Shengli Oilfield; Summary It is a difficult problem of thin reservoir prediction for geophysicist. Because of the limitation of seismic frequency band, the resolution of existing seismic data is limited. And it is difficult to identify thin reservoirs whose thickness less than 5m. Spectral inversion uses even and odd components of the reflection coefficient sequence, it can identify the thin layer better than the conventional inversion method. Predecessors have made much substantial research, and achieved a lot of valuable conclusion. Most of them focus on the spectral inversion under sparse condition, but it is difficult to know whether there is a reflection coefficient or not in practical application. In this paper, we apply the Moore-Penrose algorithm to the field of spectral inversion, in order to improve the resolution of spectral inversion under non-sparse condition. Firstly, using the model to test the effectivity of the proposed method in this paper. Secondly, adding noise to the model to test the stability of the proposed method. Finally, applying this method to the actual seismic data, so as to prove it has advantages in broaden seismic frequency band and improve seismic resolution. And provide high resolution seismic data for the following prediction and research of thin reservoir. Introduction With the further exploration and production of oil and gas field, how to identify thin reservoir, subtle reservoirs and other unconventional oil and gas reservoirs are becoming more and more important. Because of the limitation of seismic frequency band, the resolution of existing seismic data is limited, it is difficult to identify thin reservoirs whose thickness less than 5m. Spectral inversion uses the even and odd components of the reflection coefficient sequence, it can identify the thin layer better than the conventional inversion method. Predecessors have made much substantial research, Portniaguine and Castagna researched the problem of spectral inversion, and compared the minimum L1 norm, the minimum L2 norm and the sparse spike. Puryear and Chopra make research about the spectral inversion in detail, they derive the basic formula of spectral inversion. From them research we know when the reflection coefficient sequence can be determined simultaneously the sparse reflection coefficient obtained by spectral inversion can improve the thin layer resolution effectively. Castaño K P and Ojeda G apply spectral inversion based on genetic algorithm to detect thin layer. Chen Ke and Qin Dewen researched the method of spectral inversion based on simulated annealing algorithm, and discuss the results of spectral inversion under sparse and
non-sparse conditions, The results show that the spectral inversion is more accurate in sparse condition. Cao Jianhua and Qiu Zhihai applied the spectral inversion technique to analyze post stack seismic data, and then compare the original seismic data with the processed data, they found that the processed data has higher resolution so that the thin reservoir can be identified clearly. Liu Wanjin and Zhou Hui used the reflection coefficient obtained by spectral inversion to get the high resolution data by convolution with a special Ricker wavelet, then analyzed its attribute and made a good effect; Chi Zhaohuan and Liu Cai researched the method of extracting wavelet for spectral inversion, and compared with the non-time-varying wavelet, the results show that spectral inversion has higher resolution when using time-varying wavelet; Zhu Weixing and Zhang Chunxiao improved simulated annealing algorithm to improve the speed of the spectral inversion and make the spectral inversion more practical; Fu Wei and Liu Cai discussed the problem of time window selection of spectral inversion, and proposed a method can automatic select window length, and make a good effect in practical application. Predecessor’s research mostly focused on spectral inversion under sparse conditions, but it is difficult to know whether there is a reflection coefficient or not in practical application. In this paper, in order to improve the resolution of spectral inversion under non-sparse conditions, we apply the MoorePenrose algorithm to the field of spectral inversion. Firstly, we derive the formula that reflection coefficient sequence satisfied in matrix form. Because of the limitation of known condition, solving this formula is equivalent to solving an illconditioned linear system of equations, while the MoorePenrose algorithm has the great advantage in solving linear equations. Secondly, using the model to test the effectivity of the method we propose. Then, adding noise to the model verify the stability of the method. Finally, applying this method to the actual seismic data, so as to prove it has advantages in broaden seismic frequency band and improve seismic resolution. And provide high resolution seismic data for the prediction and research of thin reservoir. Theory and Method (1)Theory of Spectral Inversion The reflection coefficient of one seismic trace can be seen as a series of pulse signal. So, it can be decomposed into even and odd components. According to the theory of spectral inversion: any reflection coefficient sequence can be decomposed into even and odd components. Odd component is not conducive to identify the thin layer, while a small
Applications of the improved spectral inversion in the thin reservoir prediction account of even component can improve the ability of identify thin layer. In the ideal conditions, spectral inversion can identify any thin layer, and obtain the reflection coefficient accurately. In actual conditions, spectral inversion also has strong ability to identify thin layer.
linear system of equations Am b . In this paper, we use the Moore-Penrose algorithm to solve this problem, and it can be seen that the solution procedure is carried out under the non-sparse condition, in other words, assuming the reflection coefficient exists at every time point, but the size is different.
First, derive the objective function of spectral inversion of two layer model, and then extend to N layer model. As shown in Figure 1, when the analysis points located in the middle of the thin layer, the reflection coefficient of the thin layer can be expressed as follows: g (t ) r1 (t T / 2) r2 (t T / 2) (1)
(2) Moore-Penrose algorithm
Figure 1 Two layer reflection coefficient model
Processing g(t) by DFT and get its amplitude spectrum G(f), using Euler's formula and assuming t=0, even component re (r1 r2 ) / 2 , odd component ro (r1 r2 ) / 2 , finally get the derivative of G(f) , the objective function of two layer model can be established as follows: S( f ) ) 2re cos( fT ) W( f ) S( f ) Im( ) 2ro sin( fT ) W( f ) Re(
(2)
Where S(f) is the Fourier transform of seismic data, W(f) is the Fourier transform of seismic wavelet.And then extend to N layer model, The objective function of N layer model is Re( Im(
N /2 S( f ) ) 2re (i , N i 1) cos( fTi ) W( f ) i 1
(3)
N /2 S( f ) ) 2ro(i , N i 1) sin( fTi ) W( f ) i 1
The Moore-Penrose generalized inverse matrix is defined as: assuming A is m n order matrix, if exists n m order matrix G to satisfy one or more of the following four conditions (Commonly referred to as Moore-Penrose conditions ) (a) AGA A (b) GAG G (c) (GA)H GA (d) ( AG)H AG We called matrix G as the Moore-Penrose generalized inverse matrix of A. In this paper, we mainly use two kinds of generalized inverse matrix G, which satisfy the first condition or satisfy the first and third conditions, recorded as Am , called minimum-norm inverse matrix. For any matrix A, its minimum-norm inverse matrix Am is always exists. When the matrix A is row nonsingular, namely R( A) m , its minimum-norm inverse matrix is as shown in equation (5): (5) Am AT ( AAT )1 As Am b is an ill-conditioned linear system of equations, there will be infinite solutions. Minimum-norm inverse matrix Am can be used to get the minimum norm solution of linear equations Am b . When the 2-norm
x2
of the
solution vector obtains the minimum value, it is called minimum norm solution, recorded as equation (6): (6) xm Amb It is difficult to know whether there is a reflection coefficient or not in practical application. In order to improve the resolution of spectral inversion under non-sparse condition, we apply the Moore-Penrose algorithm to the field of spectral inversion.
Write it in matrix form as equation (4): S ( f1 ) Re(W ( f ) ) 1 cos( f1T1 ) cos( f1T2 ) S ( f2 ) cos( f T ) cos( f T ) Re( ) 2 1 2 2 W ( f2 ) 2 cos( f H T1 ) cos( f H T2 ) Re( S ( f H ) ) W ( f H ) S ( f1 ) Im(W ( f ) ) 1 sin( f1T1 ) sin( f1T2 ) S ( f2 ) Im(W ( f ) ) 2 sin( f 2T1 ) sin( f 2T2 ) 2 sin( f H T1 ) sin( f H T2 ) S ( f ) H Im( ) W ( f H )
Examples cos( f1TN /2 ) re (1, N ) cos( f 2TN /2 ) re (2, N 1) cos( f H TN /2 ) re ( N /2, N /21)
(1)Model inversion
(4) sin( f1TN /2 ) ro(1, N ) sin( f 2TN /2 ) ro(2, N 1) sin( f H TN /2 ) ro( N /2, N /21)
Because of the limitation of known condition, solving the above matrix is equivalent to solving two ill-conditioned
To test the effectivity of the proposed method, firstly we use the theoretical single trace seismic data. The format of the model is: 239 sampling points, the sampling interval is 0.002s. The results is as shown in Figure 2. From the data in the red box in Figure 2(b), The weak seismic events is concealed in the initial seismic data, while after the processing of spectral inversion, as shown in the red box in Figure 2(c), the weak seismic events is displayed. From Figure 2(d), it can be seen that the spectral inversion can improve the main frequency and broaden frequency band.
Applications of the improved spectral inversion in the thin reservoir prediction The results proves the effectiveness of the proposed method, it also proves that the spectral inversion method has the ability to broaden seismic frequency band and improve seismic resolution.
(a)
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(b)
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(d) Figure 2 Comparison of Spectral inversion results (a) reflection coefficient of the model; (b) the seismic data integrated by reflection coefficient of (a) with 30HZ Ricker Wavelet by convolution; (c) the data obtained by spectral inversion; (d) the spectrum of the original data and the data processed by spectral inversion
In order to test the stability of the proposed method, adding random noise to the initial seismic data, then processing the seismic data that include noise by spectral inversion, the results is as shown in Figure 3. From the data in the red box in Figure 3(e), it can be seen that after adding noise the resolution of the seismic data obtained by spectral inversion has reduced, but the weak seismic events still can be displayed. The results show that the method has certain stability. (2)Inversion of real seismic data Select a line seismic data of Xiaohaotu block, Xiaohaotu block is located in the northern of Yi-shan slope of Ordos Basin, this gas field had obtained industrial gas flow in Tai 1, Tai 2, Shan 1, Shan 2, He 1, He 2, He 3 and Ma 5 section of the upper Carboniferous-Lower Permian series. In this area, it containing weathering crust T9 layer above the reservoir of Ma 5 section, and present a strong amplitude
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Figure 3 Comparison of Spectral inversion results after adding noise (a) reflection coefficient of the model; (b) the seismic data integrated by reflection coefficient of (a) with 30HZ Ricker Wavelet by convolution; (c) random noise; (d) the data obtained by adding random noise to the initial seismic record; (e) the data obtained by spectral inversion
characteristics in the seismic record, as shown by red line in Figure 4(a) and Figure 4 (b). The weathering crust has a strong effect on the underlying reservoir, and cover the reflection information of the underlying reservoir. The results of spectral inversion analysis is as shown in figure 5, from the data in the red box in Figure 4(a), it can be seen that the reflection information of the underlying reservoir is concealed, and present a weak signal characteristics. From figure 4(b), we can find that after the processing of spectral inversion the reservoir events of Ma 5 section is displayed, the reservoir information is enhanced. From figure 4(c), it can be seen that the oil and gas characteristics of D93 and D92 wells are not obvious, which is consistent with the results of linked-well section. After the processing of spectral inversion, from figure 4(d), we can find that the oil and gas characteristics of D93 and D92 wells are obvious,
Applications of the improved spectral inversion in the thin reservoir prediction which is consistent with the well logging data. From above results, it can be seen that spectral inversion can improve the resolution of seismic data, and provide high resolution seismic data for the prediction and research of thin reservoir.
where there is a reflection coefficient, where there is no reflection coefficient in practical application. In this paper, in order to improve the resolution of spectral inversion under non-sparse conditions, we apply the Moore-Penrose algorithm to the field of spectral inversion. From above results, it can be seen that the proposed method is effective and practical. (3) It can be seen from the derivation of the formula, the quality of the extracted wavelet has a great influence on the accuracy of the results of the spectral inversion. So we should choose an appropriate method to extract wavelet in practical application. References
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Figure 4 Comparison of Spectral inversion results (a) Through D93, D92, D62 linked-well section; (b) the seismic section obtained by spectral inversion; (c) RMS attribute of the location between 25 and 30ms under T9 layer of the original seismic data; (d) RMS attribute of the location between 25 and 30ms under T9 layer of the data processed by spectral inversion
Conclusions In this paper, based on forerunner's study, we apply the Moore-Penrose algorithm to the field of spectral inversion, then using the proposed method to process the theoretical seismic data and the real seismic data, we can come to conclusions as below (1) Spectral inversion make full use of the feature that even component is conducive to identify the thin layer to highlight the even component in the reflection coefficient sequence, and weaken the influence of the odd component to the spectral inversion, so as to achieve the purpose of identifying the thin layer and improving the resolution. (2) Predecessor’s research mostly focused on spectral inversion under sparse conditions, but it is difficult to know
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