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Full-time domain matching pursuit and empirical mode decomposition based sparse fixed-point seismic inversion

Full-time domain matching pursuit and empirical mode decomposition based sparse fixed-point... Sparse seismic inversion is wildly utilized for reservoir prediction and resolution improvement. Matching pursuit (MP) is an effective algorithm for solving L0-norm and obtaining sparse inversion results. Sparse seismic inversion based on MP (MPSI) can estimate the sparse parameters of subsurface from observations by controlling the iterations and threshold. However, the low convergence and stability limit the application of MPSI. To accelerate the convergence of MP, the full-time domain matching pursuit (FTMP) algorithm is rs fi t proposed. The seismic inversion based on FTMP can realize the multi-point inversion simultaneously instead of searching the inversion results one by one, which is the process of MPSI. Also, the prior model constraint is then involved in the objective function to improve the stability and the layer-boundary fidelity of the inversion results. Furthermore, the empirical mode decomposition (EMD) algorithm is introduced into inversion framework to recover the features of the seismic signal from the noisy seismic signal. The fixed-point (FP) algorithm is adopted to solve the objective function in this study. The optimal inversion results can be estimated after nit fi e iterations by the FP algorithm. Combining the FTMP sparse seismic inversion framework, prior model constraint, EMD and FP algorithms, a complete sparse seismic inversion method named the full-time domain matching pursuit-based sparse fixed-point seismic inversion is ultimately proposed. The synthetic and field examples are utilized to demonstrate the stability and practicality of this approach. Compared with MPSI, the inversion results of the proposed method have higher resolution and fidelity of the layer-boundary. Keywords: matching pursuit, inversion, fixed-point, sparse 1. Introduction Impedance data have some advantages, such as high reso- lution since data are acquired from the seismic amplitude Geological inversion is a crucial tool in geological interpreta- information and well logs (AlBinHassan et al. 2006). Also, tion for hydrocarbon exploration to estimate the models that impedance data contain a lot information about thin layers. are consist with the observation data (Tarantola 1984;Bu- Thus, more meaningful information about the earth can be land & Omre 2003;Hamild&Pidlisecky 2016;Wang 2016). found in impedance profiles (Dai et al. 2019). Impedance inversion has been one of the most used meth- Models of the subsurface can be estimated by tfi ting seis- ods of geological inversion for reservoir detection since this mic data and inversion parameters (Yin et al. 2016;Zhang method was proposed in 1983 (Cooke & Schneider 1983). © The Author(s) 2022. Published by Oxford University Press on behalf of the Sinopec Geophysical Research Institute. This is an Open Access article distributed under the terms of 255 the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 1. The diagrammatic sketch of EMD: (a) calculation of IMF1 and (b) calculation of IMF2. The (a) to (d) labels in parts (a) and (b) are the processes of the IMF1 and IMF2, corresponding to the step 2 and step 3 of EMD algorithm. The blue, red and green lines refer to the emax(t), emin(t) and mean(t), respectively. et al. 2017). Bayesian inversion introduces the Gaussian or Cauchy distribution as the prior information to obtain inver- sion results. However, the sparse inverted reflection coe-ffi cient and blocky impedance cannot be obtained in this inver- sion framework (Dai et al. 2019). In the past few decades, the signal sparse representation (SPR) has been wildly used in the signal processing field. The purpose of the SPR is to represent a signal with a few atoms in a given dictionary so that a concise representation of the signal can be obtained. The sparse signal is useful for researchers to get the information that is contained in the signal. Matching pursuit (MP) is a classical sparse decom- position algorithm (Mallat & Zhang 1993). The MP is an iterative greedy algorithm, which is widely used in signal Figure 2. Comparison between original seismic signal (a) and de-noising reconstruction (Do et al. 2008) and time-frequency analysis seismic signal (c), and (b) is the IMF1. (Wang 2007, 2010) and so on. Sparse seismic inversion can estimate the sparse inverted reflection coefficient and blocky &Li 2016). Regarding impedance inversion, the acoustic P-wave impedance of the subsurface simultaneously (Yin impedance can be obtained from post-stack seismic data, et al. 2017;Zong et al. 2011, 2012, 2015;Dai et al. 2018). and the reservoir prediction can be realized from the spatial Two kinds of technique can be adopted to obtain the sparse inversion results: introducing the regularization term into change of impedance (Dai et al. 2016). However, impedance the objective function and using non-linear inversion frame- inversion suffers from multi-solutions. The band-limited works (Sacchi 1997;Chen&Yin 2007;Dai et al. 2019, 2021). characteristic of seismic data is a key reason for the multi- In this study, the FTMP framework and fixed point (FP) solutions, which means several models can satisfy the same seismic response (Yin et al. 2017;Grana 2020). An effective algorithm are adopted to obtain the sparse inversion results. way to reduce the effect of multi-solutions is to introduce The sparsity adaptive matching pursuit (SAMP) algorithm the priori constraint into the inversion framework, and the was proposed in 2008 by Do for practical compressed sensing prior model is also helpful for improving lateral continuity (Do et al. 2008). However, the SAMP has limitations for non- (Jin et al. 1992; Thierry et al. 1999;Yin &Zhang 2014;Yin sparse signals. The seismic signals are typically non-sparse 256 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 3. The diagrammatic sketch of the FTMP-based FP sparse seismic inversion method. MPFPSIR is the acronym for the full-time domain matching pursuit-based fixed-point seismic inversion result. The red blocks refer to the location of local maxima. Figure 5. The parameters used to generate the synthetic model: (a) P-wave velocity, (b) density and (c) P-wave impedance. and non-stationary signals. Thus, we improve this algorithm and propose the FTMP algorithm. The multi-point inver- sion results can be obtained simultaneously by FTMP. Also, both sparse and non-sparse inverted reflection coefficient can be obtained by changing the iterations. Since the seismic data usually contain the noise, the EMD algorithm (Huang 2001) is involved in the proposed inversion framework to improve the robustness of our method. The features of the seismic signal can be extracted from the noisy seismic signal by the EMD algorithm. Besides, the prior model constraint Figure 4. Flow chart of the proposed method. FPSI refers to fixed-point seismic inversion. is also introduced into the objective function to alleviate the 257 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. multi-solution and improve lateral continuity of the inver- sion results. Thus, the objective function has two terms: seis- mic inversion in the time domain and the prior model con- straint. In this study, the FP algorithm is employed to solve the objective function, the inversion results can be obtained after n fi ite iterations (Pei et al. 2021). Briefly, we propose a FTMP-based sparse FP inversion algorithm to estimate the reflection coefficient and impedance information of the subsurface. 2. Inversion 2.1. Full-time domain matching pursuit algorithm Supposing that a compressible signal can be written as : y = Figure 6. Comparisons between model values (black) and inverted Φx + e,where y refers to a vector with N sample points, Φ is model values (red) of iteration = 1: (a) seismic signal, (b) reflection co- a N × M projection matrix, x represents a signal of lengthM efficient and (c) P-wave impedance. and e denotes the noise. The compressed sensing theory meansthatthe x is S sparse if the x can be approximated from the coefficients with S ≤ M (Do et al. 2008). Regarding the MP algorithm, the dictionary D should be established to ob- tain the best matched atoms using global optimization. When the residual is less than the threshold, the seismic signal can be represented as n−1 ⟨ ⟩ s t = R ,a a + R , (1) ( ) k k k n k=0 where s(t) indicates the non-stationary seismic signal, R refers to the residual after k-1 iterations and is also the tar- get signal of k iterations. a and R denote the best atom and k n residual of k iterations, respectively. In conventional MP, just one best atom can be obtained Figure 7. Comparisons between model values (black) and inverted after every iteration. We thus propose the FTMP algorithm, model values (red) of iterations = 2: (a) seismic signal, (b) reflection coef- and several best matched atoms can be found after every iter- ficient and (c) P-wave impedance. ation. The equation for FTMP can be written as s (t) = A a + R ,a ∈ D, (2) k k n k k=0 where the A denotes the amplitude of best matched atoms a . Regarding seismic inversion, the parameters of the sub- surface can be obtained if we have the best atoms (seismic wavelets) and observations. Thus, the seismic signal can be rewritten as: s(t) = W m + N ,where W , m and k k n k k k=0 N refer to the wavelets, reflection coefficients and noise after N iterations, respectively. The details of inversion and FTMP are given in the next section. 2.2. FTMP- and EMD-based FP inversion Figure 8. Comparisons between model values (black) and inverted The classical forward model is model values (red) of iterations = 3: (a) seismic signal, (b) reflection coef- ficient and (c) P-wave impedance. S t = Wm + N, (3) ( ) 258 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. where W, m and N indicate the seismic wavelet matrix, reflec- and the integral matrix is tion coefficient and noise, respectively. To improve the anti- ⎡ ⎤ noise ability of the proposed approach, the EMD algorithm is ⎢ ⎥ involved in the proposed inversion framework to recover the H = . (9) ⎢ ⎥ ⋮⋮⋱ seismic signals from the noisy seismic signals. A signal can be ⎢ ⎥ ⎣11 1 1⎦ decomposed into different intrinsic mode functions (IMFs) N×N by the EMD algorithm. Thus, the seismic signal can be writ- If the seismic trace and the prior model of P-wave ten as impedance are two N × 1 vectors. Thus, the seismic wavelet matrix and the integral matrix are two N × N matrixes. Con- sidering equations (3) and (6), the simultaneous equations S (t) = IMF + r, (4) can be obtained: i=1 S t = Wm + N ( ) where IMF and r denote the IMFs and residua, . (10) L = Hm respectively. ini The EMD algorithm includes four steps: (i) find the The objective function can be written as extreme points of a signal; (ii) calculate the envelopes ( ) of local maximum points (emax(t)) and local mini- 2 ‖ ‖ J (m) =min ‖S (t) − Wm‖ + L − Hm . ‖ ini ‖ 2 2 mum points(emin(t)); (iii) calculate the average mean(t) (11) (mean(t) = 0.5(emax(t) + emin(t))) and judge whether In this study, the objective function (equation (11)) is mean(t)isaIMF(as showninfigure 1) and (iv) calculate solved by the FP algorithm. Regarding the FP algorithm, sup- the residual signal R (R = h(t) − mean(t)), and repeat the e e pose that 𝜉 (x) is a function of the independent variables x. previous steps for residual signal R (Huang 2001). First, The FP of the function can be written as: 𝜉 (x ) = x .Using i i the noisy signal is decomposed into different IMFs. Then, the FP algorithm to estimate the solution contains two steps. the signal after de-noising can be written as Thefirststepistoderivethefunction: 𝜉 (x) = x.Next,theso- lution can be obtained by the iterative equation: 𝜉 (x ) = C t = C t − IMF1, (5) N−1 ( ) ( ) x (Istratescu 1981;Agarwal et al. 2001). This algorithm is also called Banach FP theorem in mathematics. As for post- where C (t)and C(t) refer to the noisy signal and de-noising stack seismic inversion, the independent variables m is added signal, respectively. Note that if the seismic data are accompa- to both sides of the forward model, that is d + m = Gm + nied by the noise, the EMD should be used to recover them. m. Thus, the FP solution can be obtained by simplifying However, the EMD is not necessary for the noise-free mod- the equation, that is m = (I − G)m + d ⇔ x = 𝜉 (x)(Pei els. To show the processing of EMD more clearly, gfi ure 1 et al. 2021). Because the seismic data commonly contain the exhibits the processing of IMF1 and IMF2. The de-noising noise, we can thus consider that the FP is found when the signal (as shown in figure 2c) is calculated by equation (5), |𝜉 (x) − x| < err, no matter whether in model tests or field C (t) is shown in gfi ure 2a and IMF1 is shown in gfi ure 2b. data applications, where err is the threshold. The solution of Back to the inversion, the prior model constraint is widely the objective function of FP algorithm can thus be expressed used in seismic inversion, that is: as L = Hm, (6) ini m = 𝛿 [(I − 𝛼 W) m + 𝛼 S(t)] i+1 time t i t where L , H and m refer to the prior model of inversion +𝛿 [(I − 𝛼 H) m + 𝛼 L ] , (12) ini ini ini i ini ini parameters, integral matrix and reflection coecffi ient, respec- where 𝛼 and 𝛼 refer to the step sizes of inversion param- tively.Notethatthe priormodel is notequal to thelow fre- t ini eters, respectively. 𝛿 and 𝛿 denote the proportion of quency model. The low frequency model can be expressed time ini the previous two terms, note that 𝛿 + 𝛿 = 1. The 𝛿 as time ini time and 𝛼 should be increased to improve the resolution when [ ] seismic traces have a high signal to noise ratio (SNR). On P = Lp ,Lp ,Lp , ...,Lp . (7) low 1 2 3 n the contrary, the 𝛿 and 𝛼 should be increased to acquire ini ini the stable inversion results (Pei et al. 2021). After obtaining The prior model of P-wave impedance can be calculated thereflection coecffi ient, the‘blocky’ P-waveimpedancecan by the equation: be calculated by equation (13): [ ] ( ) L = 0.5 ln Lp ∕Lp ,Lp ∕Lp ,Lp ∕Lp , ...,Lp ∕Lp , ini 1 1 2 1 3 1 n 1 (8) P = Lp ⋅ exp 2m t dt , (13) ( ) inv 1 259 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Algorithm 1. FTMP-based sparse FP seismic inversion. Pseudo code for FTMP-based FP sparse seismic inversion Initialization: if the seismic signal is noise-free R = S(t) {Initial residue} else T T T [IMF ,IMF , ...,IMF ] = EMD(S(t)) 1 2 n R = S(t) − IMF 0 1 end Ite = 1 {Initial iteration} MaxlocF = 𝜙 {Initial location vector of local max values} m = (P − P )∕(P + P ){Initial model value of fixed-point seismic inversion} 0 low low low low i+1 i i+1 i Error = e {Threshold value} Itefp = N {Iterations of fixed-point seismic inversion} Itemp = M {Iterations of FTMP} Itefp_initial = 1{Initial iteration of fixed-point seismic inversion} Repeat: projection = ⟨W,R ⟩{Where Φ is the projection and R is the residual} 0 0 if the seismic signal is noise-free Maxloc = Max(|projection|) {Searching for the locations of all local max values} else MaxlocNoise = Sort(Max(|projection|)){Sort the local extremum from large to small} Maxloc = MaxlocNoise(1, ...,L) end MaxlocF = MaxlocF ∪ Maxloc W = W fp MaxlocF S = S fp MaxlocF I = I fp MaxlocF H = H fp MaxlocF L = L fp MaxlocF if Ite < Itefp m = 𝛿 [(I − 𝛼 W )m + 𝛼 S ] + 𝛿 [(I − 𝛼 H )m + 𝛼 L ]{FP inversion} i+1 time fp t fp i t fp ini fp ini fp i ini fp end R = S(t) − Wm {Update residue} 0 N Ite=Ite+1 if R ≤ Error break end if until halting condition true; where P denotes the inverted P-wave impedance. To show inv the process of the proposed method clearly, the pseudo code for the FTMP-based FP sparse seismic inversion is written as follows in algorithm 1: The diagrammatic sketch of the FTMP-based FP sparse seismic inversion is shown in gfi ure 3. Additionally, we also provide a workflow chart (gfi ure 4) of our method to help readers understand our method easily. The symbols and ab- breviations in gfi ure 4 are the same as those in algorithm 1. 3. Synthetic test In this section, a synthetic seismic signal is generated by the forward model. Then, the proposed approach is adopted to process the model to demonstrate the stability, practical- Figure 9. Comparisons between model values (black) and inverted ity and fidelity of our approach. P-wave velocity and den- model values (red) of iterations = 4: (a) seismic signal, (b) reflection coef- sity, which are used to generate the synthetic model, are ficient and (c) P-wave impedance. 260 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. selected from the well logs (gfi ure 5). Supposing that the th th impedance of i layer is Imp , the reflectivity of i layer can be represented as r = (Imp − Imp )∕(Imp +Imp ). The i i+1 i i+1 i equation of P-wave impedance is Imp = 𝜌 v ,where 𝜌 and i i i i th v represent the density and velocity of i layer. And the syn- thetic seismic trace is generated with a convolutional model. Besides, the seismic wavelet that we used is a 30-Hz Ricker wavelet. The sample interval is 0.002 s. To show the effect of the iterations of FTMP on the inver- sion results, inversion results from different iterations are ob- tained. Notably, the iterations refer to the iterations of FTMP rather than the FP inversion. The iterations of FP inversion are 250 fixedly, and we just change the iterations of FTMP to show the sparsity of inversion results variation with iter- Figure 10. Comparisons between model values (black) and inverted ations. Figures 6–10 show the inversion results of iterations model values (red) of iterations = 5: (a) seismic signal, (b) reflection coef- ficient and (c) P-wave impedance. of Itemp from 1 to 5, respectively. Synthetic tests show that Figure 11. Error variations with the iterations. 261 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 12. Comparisons of convergence (a) and computational efficiency (b) between MPSI and the proposed method. Figure 13. Comparisons between inversion results of different iterations of fixed-point algorithm (red) and true model values (black). (a) Iterations = 100, (b) iterations = 150, (c) iterations = 200 and (d) iterations = 250. The gray curves refer to the initial models. the sparsity of inversion parameters decreases with increas- The FTMP-based FP inversion has two loops: one is the ing Itemp.Thestableandaccuratesparseinversionresultscan FTMP and another is FP. Thus, time cost should be dis- be obtained even Itemp = 1(as showninfigure 6). Besides, cussed. Different iterations of FP are used to obtain the in- the non-sparse inversion results can be obtained in only five verted impedance (as shown in gfi ure 13)toillustratethe iterations (as shown in gfi ure 10). The error variations with computational efficiency of the proposed method. Notably, iterations are shown in gfi ure 11.Theerrorsarecalculatedby there are four iterations of FTMP. As observed in gfi ure 13, error = S(t) − Wm ,where m refers to the inversion re- the more iterations we use, the more accurate the inversion inv inv flection coefficient. Figure 11 exhibits the error is reduced to results are that can be obtained. However, the more itera- around zero when the Itemp > 4. The tests of convergence tions we use, the longer it takes. Figure 13d shows that the and the computational efficiency are shown in gfi ure 12.As inversion result of Itefp = 250 has a high similarity to model observed from figure 12, we can conclude that the conver- value. Furthermore, the FTMPs’ cost times are more <1s, gence of the proposed method is improved and the computa- even Itefp = 250, which shows the feasibility of the proposed tional efficiency of our method is similar to that of the MPSI. approach. 262 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 14. Comparisons between model values (black) and inverted model values (red). (a) SNR = 10, (b) SNR = 5, (c) SNR = 2and (d)SNR = 1. The gray lines refer to the initial models. 4. Field data example The robustness of the proposed approach should also be tested. Thus, the different SNRs Gaussian noises are added in The2Dand3DseismicdatagatheredfromeasternChinaare the synthetic model. The seismic traces with different SNRs chosen to demonstrate the feasibility of the proposed inver- and the inversion results are shown in gfi ure 14 simultane- sion framework. First, field 2D data are selected to show the ously. All the noisy synthetic seismic signals are processed by advantages and details of our method. As observed from the the proposed approach. The anti-noise ability tests show that field 2D seismic profile (as shown in gfi ure 15a), the strata with the decrease of SNR, the inversion results deviate from of this oileld fi are very fragmented, which means this pro- the model values gradually. However, due to the strong anti- filehas alow dfi elityofthe layer-boundary.Moreover, lots of noise ability of the EMD and FTMP algorithms, the inverted weak reflection areas also bring great challenges to the seis- model values still match the model values well, even when the mic inversion methods. Just the same as the synthetic exam- SNR equals 1. ples, the sparse inversion results (as shown in gfi ure 15band 263 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 15. Field data and sparse inversion results: (a) seismic profile, (b) inversion reflection coefficient profile and (c) inversion P-wave impedance profile. The white lines represent the well logs. advantages of sparse inversion results are the high-resolution c) and non-sparse inversion results (as shown in gfi ure 16a and clear layer-boundary, and the non-sparse inversion has and b) are obtained by changing the Itemp.Comparedwith better lateral continuity. non-sparse inversion results, the sparse inversion results have A eld fi 3D seismic volume (as shown in gfi ure 18)isex- a higher del fi ity of layer-boundary and resolution. Regard- tracted from the same oileld. fi The 3D seismic volume is also ing the inversion impedance profile, the layer-boundary of processed by the proposed method. Two profiles (as shown sparse P-wave impedance profile (as shown in gfi ure 15c) in gfi ure 19a) of the seismic volume are selected to show the is clearer than that of the non-sparse P-wave impedance inversion results. The sparse and non-sparse inversion results profile (as shown in gfi ure 16b). Specifically, from the ar- are shown in gfi ure 19. The same as for 2D test, the sparse eas that are indicated with the white arrows (as shown in inversion results have higher resolution and a clearer layer- figures 15cand 16b), we can conclude that the layer- boundary than those of non-sparse inversion results. boundary of non-sparse inverted P-wave impedance (as showninfigure 16b) aremisty.And theboundariesbetween layersshowninfigure 15careclear.However,comparedwith 5. Discussion sparse inversion results, the layers of non-sparse inversion results are more continuous. The comparison between well In this section, the eld fi data are processed by both our method and MPSI, respectively. The seismic profile and in- log and inverted P-wave impedance (as shown in gfi ure 17) version results are shown in gfi ure 20 simultaneously. As ob- demonstrate the practicality and effectivity of the proposed served from inversion results, the inverted P-wave impedance method. Eeven in the weak reflection areas, the stable inver- from our method has higher lateral continuity and the sion results can also be obtained by our method. Briefly, the 264 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure16. Non-sparse inversion results: (a) inversion reflection coefficient profile and (b) inversion P-wave impedance profile. The white lines represent the well logs. layer-boundary fidelity than those of MPSI, especially in the areas that are indicated by the arrows and circles. The bound- ariesoflayersofinversion resultsfromour method areclear. Even in the weak reflection areas and fragmentized strata, the accurate inversion results can also be obtained. Since the pro- posed method is simple and effective, this principle can be generalized to the other parameters inversion easily. Because the methods have their pitfalls. Thus, the dis- Figure 17. Comparison between well logs and inversion results. The advantages of our method should be discussed. To obtain black, blue and red curves denote the well logs, non-sparse inverted P-wave impedance and inverted blocky P-wave impedance, respectively. the inversion results with high accuracy and layer-boundary fidelity, our method has two loops. The tests show that the Figure 18. Field 3D seismic volume. 265 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 19. 3D inversion results: (a) seismic profile, (b) sparse reflection coefficient inversion result, (c) sparse P-wave impedance inversion result, (d) non-sparse reflection coefficient inversion result and (e) non-sparse P-wave impedance inversion result. The white lines refer to the well logs. (f )The comparison between well log and the inversion results. The black, red and blue curves in (f) refer to the well log, non-sparse inverted P-wave impedance and sparse inverted P-wave impedance, respectively. computational efficiency of our approach is similar to that of is then involved to the objective function to alleviate the the MPSI. Thus, the parallel computation may be an effective multi-solution and improve the stability of the inversion. way to improve the computational efficiency. However, if we Since the seismic data are usually accompanied by the noise, apply this method to a section or a small 3D seismic data, the the EMD algorithm is used to recover the seismic signal computational ecffi iency is acceptable. from the noisy seismic signal. Finally, the FP algorithm is adoptedtosolve theobjective function.Regarding thein- version algorithm, the objective function has two terms: the inversion in the time domain and the prior model con- 6. Conclusions straint. If the seismic data have a high SNR, the Itefp, 𝛼 and A blocky P-wave impedance inversion method named 𝛿 should be increased to obtain the well inversion results. time FTMP- and EMD-based sparse FP seismic inversion is pro- On the contrary, the 𝛿 and 𝛼 should be increased and ini ini posed. We first propose the FTMP inversion framework. the Itefp should be decreased simultaneously to obtain the The multi-point inversion results can be obtained by the stable inversion results. According to the synthetic model FTMP simultaneously. Thus, the convergence of FTMP is examples, the sparse inversion results can be obtained even higher than that of the MPSI. The prior model constraint the Itemp = 1. That means the proposed method has high 266 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 20. (a) Seismic profile, (b) inversion P-wave impedance profile of MPSI and (c) inversion P-wave impedance profile of the proposed method. stability. As observed from field data examples, we can con- References clude that the sparse P-wave impedance inversion results Agarwal, R.P., Meehan, M. & O’Regan, D., 2001. Fixed Point Theory and have higher layer-boundary d fi elity than that of non-sparse Applications, Cambridge University Press. P-wave impedance inversion results. And the non-sparse in- AlBinHassan, N.M., Luo, Y. & Al-Faraj, M.N., 2006. 3D edge-preserving smoothing and applications, Geophysics, 71, P5–P11. version results have higher lateral continuity than that of Buland, A. & Omre, H., 2003. Bayesian linearized AVO inversion, sparse inversion results. Note that our method has two loops, Geophysics, 68, 185–198. thus the computational efficiency is similar to that of the Cooke, D.A. & Schneider, W.A., 1983. Generalized linear inversion of re- MPSI although the convergence is improved. 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Inversion of seismic reflection data in the acoustic ap- porating poroelasticity and seismic reflectivity inversion, Surveys in proximation, Geophysics, 49, 1259–1266. Geophysics, 36, 659–681. Thierry, P., Operto, S. & Lambaré, G., 1999. Fast 2-D ray+Born migra- Zhang, F. & Li, X.Y., 2016. Exact elastic impedance matrices for trans- tion/inversion in complex media, Geophysics, 64, 162–181. versely isotropic medium, Geophysics, 81, C23–C37. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Geophysics and Engineering Oxford University Press

Full-time domain matching pursuit and empirical mode decomposition based sparse fixed-point seismic inversion

Journal of Geophysics and Engineering , Volume 19 (2): 14 – Apr 30, 2022

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Oxford University Press
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© The Author(s) 2022. Published by Oxford University Press on behalf of the Sinopec Geophysical Research Institute.
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1742-2132
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1742-2140
DOI
10.1093/jge/gxac014
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Abstract

Sparse seismic inversion is wildly utilized for reservoir prediction and resolution improvement. Matching pursuit (MP) is an effective algorithm for solving L0-norm and obtaining sparse inversion results. Sparse seismic inversion based on MP (MPSI) can estimate the sparse parameters of subsurface from observations by controlling the iterations and threshold. However, the low convergence and stability limit the application of MPSI. To accelerate the convergence of MP, the full-time domain matching pursuit (FTMP) algorithm is rs fi t proposed. The seismic inversion based on FTMP can realize the multi-point inversion simultaneously instead of searching the inversion results one by one, which is the process of MPSI. Also, the prior model constraint is then involved in the objective function to improve the stability and the layer-boundary fidelity of the inversion results. Furthermore, the empirical mode decomposition (EMD) algorithm is introduced into inversion framework to recover the features of the seismic signal from the noisy seismic signal. The fixed-point (FP) algorithm is adopted to solve the objective function in this study. The optimal inversion results can be estimated after nit fi e iterations by the FP algorithm. Combining the FTMP sparse seismic inversion framework, prior model constraint, EMD and FP algorithms, a complete sparse seismic inversion method named the full-time domain matching pursuit-based sparse fixed-point seismic inversion is ultimately proposed. The synthetic and field examples are utilized to demonstrate the stability and practicality of this approach. Compared with MPSI, the inversion results of the proposed method have higher resolution and fidelity of the layer-boundary. Keywords: matching pursuit, inversion, fixed-point, sparse 1. Introduction Impedance data have some advantages, such as high reso- lution since data are acquired from the seismic amplitude Geological inversion is a crucial tool in geological interpreta- information and well logs (AlBinHassan et al. 2006). Also, tion for hydrocarbon exploration to estimate the models that impedance data contain a lot information about thin layers. are consist with the observation data (Tarantola 1984;Bu- Thus, more meaningful information about the earth can be land & Omre 2003;Hamild&Pidlisecky 2016;Wang 2016). found in impedance profiles (Dai et al. 2019). Impedance inversion has been one of the most used meth- Models of the subsurface can be estimated by tfi ting seis- ods of geological inversion for reservoir detection since this mic data and inversion parameters (Yin et al. 2016;Zhang method was proposed in 1983 (Cooke & Schneider 1983). © The Author(s) 2022. Published by Oxford University Press on behalf of the Sinopec Geophysical Research Institute. This is an Open Access article distributed under the terms of 255 the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 1. The diagrammatic sketch of EMD: (a) calculation of IMF1 and (b) calculation of IMF2. The (a) to (d) labels in parts (a) and (b) are the processes of the IMF1 and IMF2, corresponding to the step 2 and step 3 of EMD algorithm. The blue, red and green lines refer to the emax(t), emin(t) and mean(t), respectively. et al. 2017). Bayesian inversion introduces the Gaussian or Cauchy distribution as the prior information to obtain inver- sion results. However, the sparse inverted reflection coe-ffi cient and blocky impedance cannot be obtained in this inver- sion framework (Dai et al. 2019). In the past few decades, the signal sparse representation (SPR) has been wildly used in the signal processing field. The purpose of the SPR is to represent a signal with a few atoms in a given dictionary so that a concise representation of the signal can be obtained. The sparse signal is useful for researchers to get the information that is contained in the signal. Matching pursuit (MP) is a classical sparse decom- position algorithm (Mallat & Zhang 1993). The MP is an iterative greedy algorithm, which is widely used in signal Figure 2. Comparison between original seismic signal (a) and de-noising reconstruction (Do et al. 2008) and time-frequency analysis seismic signal (c), and (b) is the IMF1. (Wang 2007, 2010) and so on. Sparse seismic inversion can estimate the sparse inverted reflection coefficient and blocky &Li 2016). Regarding impedance inversion, the acoustic P-wave impedance of the subsurface simultaneously (Yin impedance can be obtained from post-stack seismic data, et al. 2017;Zong et al. 2011, 2012, 2015;Dai et al. 2018). and the reservoir prediction can be realized from the spatial Two kinds of technique can be adopted to obtain the sparse inversion results: introducing the regularization term into change of impedance (Dai et al. 2016). However, impedance the objective function and using non-linear inversion frame- inversion suffers from multi-solutions. The band-limited works (Sacchi 1997;Chen&Yin 2007;Dai et al. 2019, 2021). characteristic of seismic data is a key reason for the multi- In this study, the FTMP framework and fixed point (FP) solutions, which means several models can satisfy the same seismic response (Yin et al. 2017;Grana 2020). An effective algorithm are adopted to obtain the sparse inversion results. way to reduce the effect of multi-solutions is to introduce The sparsity adaptive matching pursuit (SAMP) algorithm the priori constraint into the inversion framework, and the was proposed in 2008 by Do for practical compressed sensing prior model is also helpful for improving lateral continuity (Do et al. 2008). However, the SAMP has limitations for non- (Jin et al. 1992; Thierry et al. 1999;Yin &Zhang 2014;Yin sparse signals. The seismic signals are typically non-sparse 256 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 3. The diagrammatic sketch of the FTMP-based FP sparse seismic inversion method. MPFPSIR is the acronym for the full-time domain matching pursuit-based fixed-point seismic inversion result. The red blocks refer to the location of local maxima. Figure 5. The parameters used to generate the synthetic model: (a) P-wave velocity, (b) density and (c) P-wave impedance. and non-stationary signals. Thus, we improve this algorithm and propose the FTMP algorithm. The multi-point inver- sion results can be obtained simultaneously by FTMP. Also, both sparse and non-sparse inverted reflection coefficient can be obtained by changing the iterations. Since the seismic data usually contain the noise, the EMD algorithm (Huang 2001) is involved in the proposed inversion framework to improve the robustness of our method. The features of the seismic signal can be extracted from the noisy seismic signal by the EMD algorithm. Besides, the prior model constraint Figure 4. Flow chart of the proposed method. FPSI refers to fixed-point seismic inversion. is also introduced into the objective function to alleviate the 257 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. multi-solution and improve lateral continuity of the inver- sion results. Thus, the objective function has two terms: seis- mic inversion in the time domain and the prior model con- straint. In this study, the FP algorithm is employed to solve the objective function, the inversion results can be obtained after n fi ite iterations (Pei et al. 2021). Briefly, we propose a FTMP-based sparse FP inversion algorithm to estimate the reflection coefficient and impedance information of the subsurface. 2. Inversion 2.1. Full-time domain matching pursuit algorithm Supposing that a compressible signal can be written as : y = Figure 6. Comparisons between model values (black) and inverted Φx + e,where y refers to a vector with N sample points, Φ is model values (red) of iteration = 1: (a) seismic signal, (b) reflection co- a N × M projection matrix, x represents a signal of lengthM efficient and (c) P-wave impedance. and e denotes the noise. The compressed sensing theory meansthatthe x is S sparse if the x can be approximated from the coefficients with S ≤ M (Do et al. 2008). Regarding the MP algorithm, the dictionary D should be established to ob- tain the best matched atoms using global optimization. When the residual is less than the threshold, the seismic signal can be represented as n−1 ⟨ ⟩ s t = R ,a a + R , (1) ( ) k k k n k=0 where s(t) indicates the non-stationary seismic signal, R refers to the residual after k-1 iterations and is also the tar- get signal of k iterations. a and R denote the best atom and k n residual of k iterations, respectively. In conventional MP, just one best atom can be obtained Figure 7. Comparisons between model values (black) and inverted after every iteration. We thus propose the FTMP algorithm, model values (red) of iterations = 2: (a) seismic signal, (b) reflection coef- and several best matched atoms can be found after every iter- ficient and (c) P-wave impedance. ation. The equation for FTMP can be written as s (t) = A a + R ,a ∈ D, (2) k k n k k=0 where the A denotes the amplitude of best matched atoms a . Regarding seismic inversion, the parameters of the sub- surface can be obtained if we have the best atoms (seismic wavelets) and observations. Thus, the seismic signal can be rewritten as: s(t) = W m + N ,where W , m and k k n k k k=0 N refer to the wavelets, reflection coefficients and noise after N iterations, respectively. The details of inversion and FTMP are given in the next section. 2.2. FTMP- and EMD-based FP inversion Figure 8. Comparisons between model values (black) and inverted The classical forward model is model values (red) of iterations = 3: (a) seismic signal, (b) reflection coef- ficient and (c) P-wave impedance. S t = Wm + N, (3) ( ) 258 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. where W, m and N indicate the seismic wavelet matrix, reflec- and the integral matrix is tion coefficient and noise, respectively. To improve the anti- ⎡ ⎤ noise ability of the proposed approach, the EMD algorithm is ⎢ ⎥ involved in the proposed inversion framework to recover the H = . (9) ⎢ ⎥ ⋮⋮⋱ seismic signals from the noisy seismic signals. A signal can be ⎢ ⎥ ⎣11 1 1⎦ decomposed into different intrinsic mode functions (IMFs) N×N by the EMD algorithm. Thus, the seismic signal can be writ- If the seismic trace and the prior model of P-wave ten as impedance are two N × 1 vectors. Thus, the seismic wavelet matrix and the integral matrix are two N × N matrixes. Con- sidering equations (3) and (6), the simultaneous equations S (t) = IMF + r, (4) can be obtained: i=1 S t = Wm + N ( ) where IMF and r denote the IMFs and residua, . (10) L = Hm respectively. ini The EMD algorithm includes four steps: (i) find the The objective function can be written as extreme points of a signal; (ii) calculate the envelopes ( ) of local maximum points (emax(t)) and local mini- 2 ‖ ‖ J (m) =min ‖S (t) − Wm‖ + L − Hm . ‖ ini ‖ 2 2 mum points(emin(t)); (iii) calculate the average mean(t) (11) (mean(t) = 0.5(emax(t) + emin(t))) and judge whether In this study, the objective function (equation (11)) is mean(t)isaIMF(as showninfigure 1) and (iv) calculate solved by the FP algorithm. Regarding the FP algorithm, sup- the residual signal R (R = h(t) − mean(t)), and repeat the e e pose that 𝜉 (x) is a function of the independent variables x. previous steps for residual signal R (Huang 2001). First, The FP of the function can be written as: 𝜉 (x ) = x .Using i i the noisy signal is decomposed into different IMFs. Then, the FP algorithm to estimate the solution contains two steps. the signal after de-noising can be written as Thefirststepistoderivethefunction: 𝜉 (x) = x.Next,theso- lution can be obtained by the iterative equation: 𝜉 (x ) = C t = C t − IMF1, (5) N−1 ( ) ( ) x (Istratescu 1981;Agarwal et al. 2001). This algorithm is also called Banach FP theorem in mathematics. As for post- where C (t)and C(t) refer to the noisy signal and de-noising stack seismic inversion, the independent variables m is added signal, respectively. Note that if the seismic data are accompa- to both sides of the forward model, that is d + m = Gm + nied by the noise, the EMD should be used to recover them. m. Thus, the FP solution can be obtained by simplifying However, the EMD is not necessary for the noise-free mod- the equation, that is m = (I − G)m + d ⇔ x = 𝜉 (x)(Pei els. To show the processing of EMD more clearly, gfi ure 1 et al. 2021). Because the seismic data commonly contain the exhibits the processing of IMF1 and IMF2. The de-noising noise, we can thus consider that the FP is found when the signal (as shown in figure 2c) is calculated by equation (5), |𝜉 (x) − x| < err, no matter whether in model tests or field C (t) is shown in gfi ure 2a and IMF1 is shown in gfi ure 2b. data applications, where err is the threshold. The solution of Back to the inversion, the prior model constraint is widely the objective function of FP algorithm can thus be expressed used in seismic inversion, that is: as L = Hm, (6) ini m = 𝛿 [(I − 𝛼 W) m + 𝛼 S(t)] i+1 time t i t where L , H and m refer to the prior model of inversion +𝛿 [(I − 𝛼 H) m + 𝛼 L ] , (12) ini ini ini i ini ini parameters, integral matrix and reflection coecffi ient, respec- where 𝛼 and 𝛼 refer to the step sizes of inversion param- tively.Notethatthe priormodel is notequal to thelow fre- t ini eters, respectively. 𝛿 and 𝛿 denote the proportion of quency model. The low frequency model can be expressed time ini the previous two terms, note that 𝛿 + 𝛿 = 1. The 𝛿 as time ini time and 𝛼 should be increased to improve the resolution when [ ] seismic traces have a high signal to noise ratio (SNR). On P = Lp ,Lp ,Lp , ...,Lp . (7) low 1 2 3 n the contrary, the 𝛿 and 𝛼 should be increased to acquire ini ini the stable inversion results (Pei et al. 2021). After obtaining The prior model of P-wave impedance can be calculated thereflection coecffi ient, the‘blocky’ P-waveimpedancecan by the equation: be calculated by equation (13): [ ] ( ) L = 0.5 ln Lp ∕Lp ,Lp ∕Lp ,Lp ∕Lp , ...,Lp ∕Lp , ini 1 1 2 1 3 1 n 1 (8) P = Lp ⋅ exp 2m t dt , (13) ( ) inv 1 259 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Algorithm 1. FTMP-based sparse FP seismic inversion. Pseudo code for FTMP-based FP sparse seismic inversion Initialization: if the seismic signal is noise-free R = S(t) {Initial residue} else T T T [IMF ,IMF , ...,IMF ] = EMD(S(t)) 1 2 n R = S(t) − IMF 0 1 end Ite = 1 {Initial iteration} MaxlocF = 𝜙 {Initial location vector of local max values} m = (P − P )∕(P + P ){Initial model value of fixed-point seismic inversion} 0 low low low low i+1 i i+1 i Error = e {Threshold value} Itefp = N {Iterations of fixed-point seismic inversion} Itemp = M {Iterations of FTMP} Itefp_initial = 1{Initial iteration of fixed-point seismic inversion} Repeat: projection = ⟨W,R ⟩{Where Φ is the projection and R is the residual} 0 0 if the seismic signal is noise-free Maxloc = Max(|projection|) {Searching for the locations of all local max values} else MaxlocNoise = Sort(Max(|projection|)){Sort the local extremum from large to small} Maxloc = MaxlocNoise(1, ...,L) end MaxlocF = MaxlocF ∪ Maxloc W = W fp MaxlocF S = S fp MaxlocF I = I fp MaxlocF H = H fp MaxlocF L = L fp MaxlocF if Ite < Itefp m = 𝛿 [(I − 𝛼 W )m + 𝛼 S ] + 𝛿 [(I − 𝛼 H )m + 𝛼 L ]{FP inversion} i+1 time fp t fp i t fp ini fp ini fp i ini fp end R = S(t) − Wm {Update residue} 0 N Ite=Ite+1 if R ≤ Error break end if until halting condition true; where P denotes the inverted P-wave impedance. To show inv the process of the proposed method clearly, the pseudo code for the FTMP-based FP sparse seismic inversion is written as follows in algorithm 1: The diagrammatic sketch of the FTMP-based FP sparse seismic inversion is shown in gfi ure 3. Additionally, we also provide a workflow chart (gfi ure 4) of our method to help readers understand our method easily. The symbols and ab- breviations in gfi ure 4 are the same as those in algorithm 1. 3. Synthetic test In this section, a synthetic seismic signal is generated by the forward model. Then, the proposed approach is adopted to process the model to demonstrate the stability, practical- Figure 9. Comparisons between model values (black) and inverted ity and fidelity of our approach. P-wave velocity and den- model values (red) of iterations = 4: (a) seismic signal, (b) reflection coef- sity, which are used to generate the synthetic model, are ficient and (c) P-wave impedance. 260 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. selected from the well logs (gfi ure 5). Supposing that the th th impedance of i layer is Imp , the reflectivity of i layer can be represented as r = (Imp − Imp )∕(Imp +Imp ). The i i+1 i i+1 i equation of P-wave impedance is Imp = 𝜌 v ,where 𝜌 and i i i i th v represent the density and velocity of i layer. And the syn- thetic seismic trace is generated with a convolutional model. Besides, the seismic wavelet that we used is a 30-Hz Ricker wavelet. The sample interval is 0.002 s. To show the effect of the iterations of FTMP on the inver- sion results, inversion results from different iterations are ob- tained. Notably, the iterations refer to the iterations of FTMP rather than the FP inversion. The iterations of FP inversion are 250 fixedly, and we just change the iterations of FTMP to show the sparsity of inversion results variation with iter- Figure 10. Comparisons between model values (black) and inverted ations. Figures 6–10 show the inversion results of iterations model values (red) of iterations = 5: (a) seismic signal, (b) reflection coef- ficient and (c) P-wave impedance. of Itemp from 1 to 5, respectively. Synthetic tests show that Figure 11. Error variations with the iterations. 261 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 12. Comparisons of convergence (a) and computational efficiency (b) between MPSI and the proposed method. Figure 13. Comparisons between inversion results of different iterations of fixed-point algorithm (red) and true model values (black). (a) Iterations = 100, (b) iterations = 150, (c) iterations = 200 and (d) iterations = 250. The gray curves refer to the initial models. the sparsity of inversion parameters decreases with increas- The FTMP-based FP inversion has two loops: one is the ing Itemp.Thestableandaccuratesparseinversionresultscan FTMP and another is FP. Thus, time cost should be dis- be obtained even Itemp = 1(as showninfigure 6). Besides, cussed. Different iterations of FP are used to obtain the in- the non-sparse inversion results can be obtained in only five verted impedance (as shown in gfi ure 13)toillustratethe iterations (as shown in gfi ure 10). The error variations with computational efficiency of the proposed method. Notably, iterations are shown in gfi ure 11.Theerrorsarecalculatedby there are four iterations of FTMP. As observed in gfi ure 13, error = S(t) − Wm ,where m refers to the inversion re- the more iterations we use, the more accurate the inversion inv inv flection coefficient. Figure 11 exhibits the error is reduced to results are that can be obtained. However, the more itera- around zero when the Itemp > 4. The tests of convergence tions we use, the longer it takes. Figure 13d shows that the and the computational efficiency are shown in gfi ure 12.As inversion result of Itefp = 250 has a high similarity to model observed from figure 12, we can conclude that the conver- value. Furthermore, the FTMPs’ cost times are more <1s, gence of the proposed method is improved and the computa- even Itefp = 250, which shows the feasibility of the proposed tional efficiency of our method is similar to that of the MPSI. approach. 262 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 14. Comparisons between model values (black) and inverted model values (red). (a) SNR = 10, (b) SNR = 5, (c) SNR = 2and (d)SNR = 1. The gray lines refer to the initial models. 4. Field data example The robustness of the proposed approach should also be tested. Thus, the different SNRs Gaussian noises are added in The2Dand3DseismicdatagatheredfromeasternChinaare the synthetic model. The seismic traces with different SNRs chosen to demonstrate the feasibility of the proposed inver- and the inversion results are shown in gfi ure 14 simultane- sion framework. First, field 2D data are selected to show the ously. All the noisy synthetic seismic signals are processed by advantages and details of our method. As observed from the the proposed approach. The anti-noise ability tests show that field 2D seismic profile (as shown in gfi ure 15a), the strata with the decrease of SNR, the inversion results deviate from of this oileld fi are very fragmented, which means this pro- the model values gradually. However, due to the strong anti- filehas alow dfi elityofthe layer-boundary.Moreover, lots of noise ability of the EMD and FTMP algorithms, the inverted weak reflection areas also bring great challenges to the seis- model values still match the model values well, even when the mic inversion methods. Just the same as the synthetic exam- SNR equals 1. ples, the sparse inversion results (as shown in gfi ure 15band 263 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 15. Field data and sparse inversion results: (a) seismic profile, (b) inversion reflection coefficient profile and (c) inversion P-wave impedance profile. The white lines represent the well logs. advantages of sparse inversion results are the high-resolution c) and non-sparse inversion results (as shown in gfi ure 16a and clear layer-boundary, and the non-sparse inversion has and b) are obtained by changing the Itemp.Comparedwith better lateral continuity. non-sparse inversion results, the sparse inversion results have A eld fi 3D seismic volume (as shown in gfi ure 18)isex- a higher del fi ity of layer-boundary and resolution. Regard- tracted from the same oileld. fi The 3D seismic volume is also ing the inversion impedance profile, the layer-boundary of processed by the proposed method. Two profiles (as shown sparse P-wave impedance profile (as shown in gfi ure 15c) in gfi ure 19a) of the seismic volume are selected to show the is clearer than that of the non-sparse P-wave impedance inversion results. The sparse and non-sparse inversion results profile (as shown in gfi ure 16b). Specifically, from the ar- are shown in gfi ure 19. The same as for 2D test, the sparse eas that are indicated with the white arrows (as shown in inversion results have higher resolution and a clearer layer- figures 15cand 16b), we can conclude that the layer- boundary than those of non-sparse inversion results. boundary of non-sparse inverted P-wave impedance (as showninfigure 16b) aremisty.And theboundariesbetween layersshowninfigure 15careclear.However,comparedwith 5. Discussion sparse inversion results, the layers of non-sparse inversion results are more continuous. The comparison between well In this section, the eld fi data are processed by both our method and MPSI, respectively. The seismic profile and in- log and inverted P-wave impedance (as shown in gfi ure 17) version results are shown in gfi ure 20 simultaneously. As ob- demonstrate the practicality and effectivity of the proposed served from inversion results, the inverted P-wave impedance method. Eeven in the weak reflection areas, the stable inver- from our method has higher lateral continuity and the sion results can also be obtained by our method. Briefly, the 264 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure16. Non-sparse inversion results: (a) inversion reflection coefficient profile and (b) inversion P-wave impedance profile. The white lines represent the well logs. layer-boundary fidelity than those of MPSI, especially in the areas that are indicated by the arrows and circles. The bound- ariesoflayersofinversion resultsfromour method areclear. Even in the weak reflection areas and fragmentized strata, the accurate inversion results can also be obtained. Since the pro- posed method is simple and effective, this principle can be generalized to the other parameters inversion easily. Because the methods have their pitfalls. Thus, the dis- Figure 17. Comparison between well logs and inversion results. The advantages of our method should be discussed. To obtain black, blue and red curves denote the well logs, non-sparse inverted P-wave impedance and inverted blocky P-wave impedance, respectively. the inversion results with high accuracy and layer-boundary fidelity, our method has two loops. The tests show that the Figure 18. Field 3D seismic volume. 265 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 19. 3D inversion results: (a) seismic profile, (b) sparse reflection coefficient inversion result, (c) sparse P-wave impedance inversion result, (d) non-sparse reflection coefficient inversion result and (e) non-sparse P-wave impedance inversion result. The white lines refer to the well logs. (f )The comparison between well log and the inversion results. The black, red and blue curves in (f) refer to the well log, non-sparse inverted P-wave impedance and sparse inverted P-wave impedance, respectively. computational efficiency of our approach is similar to that of is then involved to the objective function to alleviate the the MPSI. Thus, the parallel computation may be an effective multi-solution and improve the stability of the inversion. way to improve the computational efficiency. However, if we Since the seismic data are usually accompanied by the noise, apply this method to a section or a small 3D seismic data, the the EMD algorithm is used to recover the seismic signal computational ecffi iency is acceptable. from the noisy seismic signal. Finally, the FP algorithm is adoptedtosolve theobjective function.Regarding thein- version algorithm, the objective function has two terms: the inversion in the time domain and the prior model con- 6. Conclusions straint. If the seismic data have a high SNR, the Itefp, 𝛼 and A blocky P-wave impedance inversion method named 𝛿 should be increased to obtain the well inversion results. time FTMP- and EMD-based sparse FP seismic inversion is pro- On the contrary, the 𝛿 and 𝛼 should be increased and ini ini posed. We first propose the FTMP inversion framework. the Itefp should be decreased simultaneously to obtain the The multi-point inversion results can be obtained by the stable inversion results. According to the synthetic model FTMP simultaneously. Thus, the convergence of FTMP is examples, the sparse inversion results can be obtained even higher than that of the MPSI. The prior model constraint the Itemp = 1. That means the proposed method has high 266 JournalofGeophysicsand Engineering (2022) 19, 255–268 Pei et al. Figure 20. (a) Seismic profile, (b) inversion P-wave impedance profile of MPSI and (c) inversion P-wave impedance profile of the proposed method. stability. 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Journal

Journal of Geophysics and EngineeringOxford University Press

Published: Apr 30, 2022

Keywords: matching pursuit; inversion; fixed-point; sparse

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