Improving seismic interpretation: a high-contrast approximation to the reflection coefficient of a plane longitudinal wave

Petroleum Science, Dec 2013

Linearized approximations of reflection and transmission coefficients set a foundation for amplitude versus offset (AVO) analysis and inversion in exploration geophysics. However, the weak properties contrast hypothesis of those linearized approximate equations leads to big errors when the two media across the interface vary dramatically. To extend the application of AVO analysis and inversion to high contrast between the properties of the two layers, we derive a novel nonlinearized high-contrast approximation of the PP-wave reflection coefficient, which establishes the direct relationship between PP-wave reflection coefficient and P-wave velocities, S-wave velocities and densities across the interface. (A PP wave is a reflected compressional wave from an incident compressional wave (P-wave).) This novel approximation is derived from the exact reflection coefficient equation with Taylor expansion for the incident angle. Model tests demonstrate that, compared with the reflection coefficients of the linearized approximations, the reflection coefficients of the novel nonlinearized approximate equation agree with those of the exact PP equation better for a high contrast interface with a moderate incident angle. Furthermore, we introduce a nonlinear direct inversion method utilizing the novel reflection coefficient equation as forward solver, to implement the direct inversion for the six parameters including P-wave velocities, S-wave velocities, and densities in the upper and lower layers across the interface. This nonlinear inversion algorithm is able to estimate the inverse of the nonlinear function in terms of model parameters directly rather than in a conventional optimization way. Three examples verified the feasibility and suitability of this novel approximation for a high contrast interface, and we still could estimate the six parameters across the interface reasonably when the parameters in both media across the interface vary about 50%.

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Improving seismic interpretation: a high-contrast approximation to the reflection coefficient of a plane longitudinal wave

Pet.Sci. Improving seismic interpretation: a high-contrast Yin Xingyao 0 Zong Zhaoyun 0 Wu Guochen 0 0 China University of Petroleum (Beijing) and Springer-Verlag Berlin Heidelberg 2013 amplitude versus offset (AVO) analysis and inversion in exploration geophysics. However, the weak properties contrast hypothesis of those linearized approximate equations leads to big errors when the two media across the interface vary dramatically. To extend the application of AVO analysis and inversion to high contrast between the properties of the two layers, we derive a novel nonlinearized high-contrast (A PP wave is a reflected compressional wave from an incident compressional wave (P-wave).) This novel approximation is derived from the exact reflection coefficient equation with Taylor expansion for the incident angle. Model tests demonstrate that, compared with the reflection coefficients of the linearized approximations, the reflection coefficients of the novel nonlinearized approximate equation agree with those of the exact PP equation better for a high contrast interface with a moderate incident angle. Furthermore, we introduce a nonlinear direct inversion method utilizing the novel reflection P-wave velocities, S-wave velocities, and densities in the upper and lower layers across the interface. This nonlinear inversion algorithm is able to estimate the inverse of the nonlinear function in terms of model and suitability of this novel approximation for a high contrast interface, and we still could estimate the six parameters across the interface reasonably when the parameters in both media across the interface vary High-contrast interface; reflection coefficient; amplitude variation with angle; multi- 1 Introduction The Zoeppritz equation (Zoeppritz and Erdbebnenwellen, 1 9 1 9 ) a n d i t s a p p r o x i m a t i o n s a s t h e f u n d a m e n t a l mathematical formulae for describing the amplitudes of PP reflected waves from P-wave incident plane waves in exploration geophysics under plane wave approximation play an important role in AVO analysis/inversion (Smith 2012b). The Zoeppritz equation gives the precise values of intrinsic nonlinearity makes it less appropriate in practical applications. Therefore, linearized approximations with different parameterization of the Zoeppritz equations are different types of linearized approximations see Russell et al (2011). The linearized approximations are derived under the hypothesis of weak property contrasts between layers or limited incident angle. However, these assumptions do not hold especially at unconformities or at interfaces between al, 2011). Therefore, in this paper, we attempt to derive an approximation of the PP reflection coefficient to adjust to high contrast situations. We utilize the forward modeling and nonlinear inversion method to test the feasibility and suitability of this novel approximation. Two forward modeling models with different degrees of property contrast are established and we compare the reflection coefficients with the novel approximation, exact Zeoppritz equation and linearized approximation, respectively. As for the inversion method, an artificial neural network nonlinear direct inversion is introduced to estimate the six parameters with the novel approximation as a forward solver. The artificial neural network nonlinear a kind of nonlinear direct inversion approach rather than an optimization approach. It has been proved that inversion is inverse of G( ) , which is the forward solver. It can provide several solutions like the multiple realizations in stochastic inversion by Bayesian inference (Buland and Omre, 2003). (2009). In appendix A hereunder, we will give the necessary description of this method for the nonlinear inversion problem with the novel approximation equation. 2 Modeling The general theory of the P-wave reflection has been widely discussed in the literature, so we shall reproduce here only that required for an understanding of the notation and terminology that we will use in this paper. For the cases of incident longitudinal waves polarized in the plane of be expressed as (Aki and Richards, 1980) , RPP the transmission angles of the longitudinal wave and shear RPP is the reflection coefficient of the VP1, VS1 and 1 are the P-wave velocity, S-wave velocity and density in medium 1, and VP2, VS2 and 2 is the ray parameter. The linearized approximation of Eq. (1) is given by Aki and Richards (1980) as, (2) (3) (4) where V VP VP VS VS RPP 1 4VS2 p2 black), linearized approximation equation (2) (dashed blue) and our novel nonlinearized approximation equation (6) (red dots), and we can see that reflection coefficients with these three equations show good similarity in the weak contrast case. Fig. 2 displays the result of the model two, we can see show high errors compared to that from the exact equation, and the reflection coefficient from our novel approximation still shows high similarity to that from the exact equation at a moderate incident angle. 3 Nonlinear inversion To test the possibility of estimating parameters with our novel approximation, we introduce an artificial neural 2009). The inversion is formulated in a direct inversion scheme utilizing Eq. (6) as the forward solver. We restrict our computational domain to various two-layer models to test the effectiveness and potential of our novel approximation in high contrast media. Similar to Rabben et al (2008), we attempt to Taking the Taylor expansion for incident angle of Eq. (1), we can express RPP in a closed form as, VP1VP22VS1 1 2 2 VP1VP22VS2 2 3 8VS41 13 16VS21VS22 12 2 8VS42 1 2 2 A1VP1VP2 2 2VP1VP22VS1VS2 1 2 2 8VP2VS21VS2 12 2 4VP22VS1VS2 1 2 2 8VP2VS1VS22 1 2 2 2VP2 2 A2 1 (6) the wavelet estimation and the convolution in the modeling. We utilized the exact equation (1) as the synthetic model. The model vector m comprises of P-wave velocities, S-wave velocities and densities in the upper and lower layers. The observed data d incident angles of Eq. (1). The model parameters and the data parameters can be related through the nonlinear forward mapping, d G m Model One Here, we attempt to search for all possible solutions of model parameters to satisfy the observed reflection method, we suppose the inverse to G m is G d . Although the inverse mapping may not exist in entire spaces, is so smooth that the inverse inside which the mapping G of G does exist, G d m The artificial neural network direct inversion method is an inverse (not optimizing) algorithm, utilizing numerical approximation of Eq. (8) in empirically constrained subspaces. Supposing there exists an inverse mapping and its numerical approximation inside these subspaces, it works simultaneously with a population of several so-called individuals. Each individual contains a parameter vector, a data vector and the model error. The model error is used for and for their sorting from the best to the worst model. The computation records already evaluated and tested models so that these models can be reused later. Repeated usage of some models generates the possibility of efficient inversion with minimum number of forward evaluations. Besides, several solutions can be expected with this algorithm. Details of reproduce the algorithm in Appendix A but only to a level required for an understanding of the notation and terminology that we use in our examples and discussion section. 4 Examples Various two-layer models are established in the inversion test. The first one is a gas sand/shale model. The P-wave velocity (VP1), S-wave velocity (VS1) and density (Density 3, respectively, while the P-wave velocity (VP2), S-wave velocity (VS2) and density (Density 2) in the shale sand is 3,048 m/s, 1,244 m/s and 2,400 kg/m3, respectively, and we refer to this model, Eq. (1) is used to generate RPP at different incident angles to simulate observed data. The introduced nonlinear inversion method is then used to generate ten solutions for each of the six parameters. With the introduced inversion method, the results of parameters estimation are displayed in Fig. 4. Taking VP1 for example, the left figure shows the comparison between the estimated solutions and the true value. The sequence numbers from 1 to 10 indexes the ten solutions for VP1, the value at sequence number 11 gives the average value of the ten estimated values, and the value at sequence number 12 gives the true value. The right figure displays the relative error between the estimated solutions and the true value. The sequence numbers from 1 to 10 indexes the relative error between each estimated solution for VP1 and the true value, sequence number 11 gives the relative error between the average value of all ten values and the true value. 4, we can see that all six parameters can be inverted well and the relative error is around 3% for each parameter. from medium 1 to medium 2 in the second model. It shows respect to the parameters in medium 1. Fig. 6 displays the result of estimating six parameters with the second model 30 25 % i,o 20 t a r e ng 15 a h c itve 10 a l e R 5 0 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number 10 % ,ro 6 r r e e 5 v it a le 4 R 9 8 7 3 2 1 0 10 9 8 7 % ,ro 6 r r e e 5 v it lea 4 R 3 2 1 0 3500 3000 2500 2000 s / m ,1 VP1500 1000 500 0 3500 3000 2500 /s 2000 m ,2P V1500 1000 500 0 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number (Continued) 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number when the maximum incident angle is 31 degree. We can see that even in high contrast media, the inversion with the exact reflection coefficient equation still estimates the six parameters reasonably. Fig. 7 to Fig. 12 display the inversion results with the 2-D surface model. Taking the P-wave velocity in upper medium for example, Fig. 7(a) - 7(c) display P-wave velocity in upper medium of the true model, inverted result and the relative error between the true model and inverted result, respectively. in Fig. 7 to Fig. 12. From the inverted results, we can see that, with the high-contrast approximation and the nonlinear inversion algorithm, we can obtain reasonable inversion results, and the relative error is around 4% for each parameter. 3500 3000 2500 3 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number 5500 5000 4500 4000 3500 /s 3000 m ,2P2500 V 2000 1500 1000 500 0 5500 5000 4500 4000 3500 /s 3000 m ,1S 2500 V 2000 1500 1000 500 0 5500 5000 4500 4000 3500 /s 3000 m2S2500 V 2000 1500 1000 500 0 3 2 1 0 10 9 8 7 ,r%6 o r re 5 e v it la 4 e R 3 2 1 0 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number 10 9 8 7 % ,r 6 o r r ee 5 v it la 4 e R 3 2 1 0 10 9 8 7 3 2 1 0 ,r%6 o r r e 5 e v it la 4 e R (Continued) 5500 5000 4500 4000 3 /m3500 g ,k 3000 1 y its 2500 n e D2000 1500 1000 500 0 5500 5000 4500 4000 3 m3500 / g ,k 3000 2 itsy 2500 n eD2000 1500 1000 500 0 1 2 3 4 5 6 7 8 9 10 11 12 Sequence number 1 2 3 4 5 6 7 8 9 10 11 Sequence number (a) (b) (c) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 20 40 60 80 100 20 40 60 80 100 In line In line Fig. 8 P-wave velocities in the lower medium (a) True model, (b) Inverted result, and (c) Relative error 20 40 60 80 100 Inline 20 40 60 80 100 20 40 60 80 100 Inline Inline Fig. 9 S-wave velocities in the upper medium (a) True model, (b) Inverted result, and (c) Relative error 20 40 60 80 100 In line 20 40 60 80 100 In line (c) 20 40 60 80 100 In line (a) 20 40 60 80 100 In line 20 40 60 80 100 20 40 60 80 100 In line In line Fig. 11 Densities in the upper medium (a) True model, (b) Inverted result, and (c) Relative error 2180 2160 2140 2120 2100 10 20 30 ine 40 l ss 50 roC 60 70 80 90 2220 2200 5 Conclusions In this paper, we derived a high-contrast approximation of the exact PP reflection coefficient in terms of six parameters including P-wave velocities, S-wave velocities and densities in upper and lower layers around a reflector. We utilized the forward modeling and inversion method to test the validity and feasibility of this novel approximation. Forward modeling tests demonstrated the priority of the novel approximation to the linearized approximation in reflection coefficient modeling. A nonlinear direct inversion method was introduced to estimate the six layer parameters around multi-parameters with our novel nonlinearized approximation of exact reflection coefficient equation could still get reasonable inversion result even when the parameters in both Acknowledgements We would like to acknowledge the sponsorship of the National 973 Program of China (2013CB228604) and the National Grand Project for Science and Technology acknowledge the support of the Australian and Western Ve n t u r e P a r t n e r s , a s w e l l a s t h e We s t e r n A u s t r a l i a n Energy Research Alliance (WA:ERA), and Foundation WX2013-04-01). Buland A and Omre H. Bayesian linearized AVO inversion. Geophysics. prediction from seismic prestack data. Geophysics. 2008 . 73(3): C13-C21 Technical Program Expanded Abstracts. 2004 continental sandstone interbeds and deep volcanic rocks. Applied Karimi O, Omre H and Mohammadzadeh M. Bayesian closed-skew Gaussian inversion of seismic AVO data for elastic material amplitude polynomial methods for characterisation of hydrocarbon seismic data from unconsolidated sediments containing gas hydrate and free gas. Geophysics. 2004. 69(1): 164-179 Rabben T E, Tjelmeland H and Ursin B. Nonlinear Bayesian joint Rimstad K, Avseth P and Omre H. Hierarchical Bayesian lithology/ poroelasticity. Geophysics. 2011. 76(3): C19-C29 nonlinear equations. Technical Computing Prague. ISBN 978-807080-733-0. Praha. 2009. p.90 Shuey R T. A simplification of the Zoeppritz equations. Geophysics. inversion of PP reflections from plane interfaces using effective Smith G C and Gidlow P M. Weighted stacking for rock property estimation and detection of gas. Geophysical Prospecting. 1987. Ulvmoen M and Omre H. Improved resolution in Bayesian lithology/ fluid inversion from prestack seismic data and well observations: Ayzenberg M, Tsvankin I, Aizenberg A, et al. Effective reflection coefficients for curved interfaces in transversely isotropic media. Ulvmoen M, Omre H and Buland A. Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well Bortfeld R. Approximations to the reflection and transmission B73-B82 Ursin B, Bauer C, Zhao H S, et al. Combined seismic inversion and gravity modeling of a shallow anomaly in the southern Barents Sea. The artificial neural network inversion was initially direct inversion approach rather than optimization approach. It mainly contains the following steps. Problem initialization starting model population, even if doing so is easy. We just mimin < mi < mimax (A-1) The starting population of models is generated from the defined range with uniform probability. The number of models in the starting population q is not very important because it will change during iterations. A suitable choice can be 30 q 60 . At each iteration, the current population of models M B mB , dB , err B is sorted according to the individual errors between the models and the candidate solution. The diameter of the population R defines the size of a subspace, inside which the next population of models will be generated, and the index of the prediction function ip specifies the prediction method used for predicting the candidate solution, including linear regression (ip=1), radial basis function network (RBFN) (ip=2), and Kriging prediction (ip=3). Both of these parameters can be tuned during the inversion, but in the beginning they are both set to 1. Prediction of population and candidate solution There is a geometrical criterion in distinct iteration cycles before population predicting, one model is selected as the center of the population (mC), and the other surrounding models are located randomly in the distance R measured from the center of the population. The prediction population is generated in such a way that the center is located close to the expected solution, and surrounding models are located randomly along the surface of a hypersphere with diameter R and center mC. Provided both the diameter R and the center mC are known, new models of the predicting population can be obtained as follows. Firstly, The matrix tensor Cm whose value on the principal diagonal line is defined as the square of the difference between the maximum and minimum of each model parameter can be decomposed using a Choleski decomposition as Cm=L LT. Secondly, a random six dimensional unit vector g is generated. Then, the proposed candidate model can be expressed as, mC RL g (A-2) In any case when the candidate model mC is outside the parametric hypercube, it is projected along the direction (mg mC) to the closest face of the parametric hypercube. The archive of already evaluated models is checked and the model {mk, dk, errk} is selected, and this model is the closest archive model to mg and still not connected to the predicting population. We need to compute the distance sg between this model and the candidate model. If the distance sg is smaller than the distance between neighboring surrounding models, the model {mk, dk, errk} is connected to the population, otherwise, the candidate model mg is evaluated and is connected to the predicting population and copied to the archive for future use. Finally, the second step made 1 times to obtain a population of total size q. The prediction population above can be used for estimating the solution (8) with the prediction algorithm. The prediction algorithm is implemented for the local approximation of the inverse mapping. Different prediction algorithms with different index ip can be selected to estimate the solutions, however the best solution is often to use different prediction algorithms even inside each individual inverse problem. Therefore, in our case, we use the prediction algorithm in a cyclic manner according to the variable ip. Freeman and Co. 1980 Alemie W and Sacchi M D. High-resolution three-term AVO inversion by means of a Trivariate Cauchy probability distribution . Geophysics. Geophysics . 2003 . 68 ( 4 ): 1140 -1149 Vedanti N and Sen M K. Seismic inversion tracks in situ combustion: A case study from Balol oil field , India. Geophysics . 2009 . 74 ( 4 ): B103 -112 Wang H Y , Sun Z D , Wang D , et al. Frequency-dependent velocity prediction theory with implication for better reservoir fluid its application . 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Xingyao Yin, Zhaoyun Zong, Guochen Wu. Improving seismic interpretation: a high-contrast approximation to the reflection coefficient of a plane longitudinal wave, Petroleum Science, 2013, 466-476, DOI: 10.1007/s12182-013-0297-y