Estimation of Elastic Parameters Using Two-Term Fatti Elastic Impedance Inversion

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1 Journal of Earth Science, Vol. 6, No. 4, p , August 15 ISSN X Printed in China DOI:.7/s Estimation of Elastic Parameters Using Two-Term Fatti Elastic Impedance Inversion Jin Zhang 1, Huaishan Liu 1, Siyou Tong 1, Lei Xing* 1, Xiangpeng Chen, Chaoguang Su 1. College of Marine Geoscience, Ocean Univesity of China, Qingdao 66, China. Shengli Geophysical Research Institute of SINOPEC, Dongying 5, China ABSTRACT: Elastic impedance (EI) inversion has been widely used in industry to estimate kinds of elastic parameters to distinguish lithology or even fluid. However, it is found that conventional three-term elastic impedance formula is unstable even with slight random noise in seismic data, due to the ill-conditioned coefficient matrix of elastic parameters. We presented two-term Fatti elastic impedance inversion method, which is more robust and accurate than conventional three-term elastic impedance inversion. In our method, density is ignored to increase the robustness of inversion matrix. Besides, P-impedance and S-impedance, which are less sensitive to random noise, are inverted instead of V P and V S in conventional three-term elastic impedance. To make the inversion more stable, we defined the range of K value as a constraint. Synthetic tests claim that this method can obtain promising results with low SNR (signal noise ratio) seismic data. With the application of the method in a D line data, we achieved λρ, μρ and V P /V S sections, which matched the drilled well perfectly, indicating the potential of the method in reservoir prediction. KEY WORDS: two-term Fatti EI, random noise, K value. INTRODUCTION The concept of elastic impedance (EI) was initially introduced by Connolly (1999) and has been widely used in lithology discrimination and fluid prediction. In Connolly s paper, EI was formulated in terms of P-wave velocity, S-wave velocity, density and incident angle, based on the approximation of Zoeppritz s equations (1919) by Aki and Richards (19). The advantage of EI formulation is that reflection coefficient in terms of EI has the same form as the normal incidence reflection coefficient; therefore, a standard poststack inversion procedure can be taken to invert prestack angle gathers with EI formula. To overcome the shortcoming that EI value varies with incident angle dramatically, which makes inconvenient displaying of AI (acoustic impedance) and EI logs together, Whitcombe () normalized EI formula by scaling an appropriate factor. Compared with AVO (amplitude versus offset), EI inversion can remove the effects of the angledependent wavelet and may help identify residual NMO (normal moveout) corrections (Cambois, ). Therefore, EI inversion should be more robust than AVO. Whitcombe et al. () modified EI formula and derived extended elastic impedance (EEI), which can better optimize the elastic parameters to be fluid or lithology discriminator. Bruce et al. () and Santos et al. () developed new EI formulas respectively by introducing the ray parameter to obtain more accurate reflection coefficient, compared with Connolly s EI formula. *Corresponding author: xingleiouc@ouc.edu.cn China University of Geosciences and Springer-Verlag Berlin Heidelberg 15 Manuscript received September 1, 14. Manuscript accepted January 5, 15. However, the coefficient of density in conventional three-term EI inversion is close to zero, which makes the ill-conditioned inverse problem. Even a small amount of noise will lead to large errors in elastic parameters estimation. Mallick et al. () used a simple four-layer model to prove that with added as little as % random noise in the synthetic data, estimation of V P, V S and density from conventional EI inversion are poor, while P-impedance (I P ) and S-impedance (I S ) and Poisson s ratio are still reasonably well. Lu and McMechan (4) also pointed out with 7% random noise in the calculated EI value, even I S becomes unstable and contains unreasonable values. He used triangular smoothing to make inverted I S better. Tsuneyama and Mavko (7) illustrated that even with added 1% random noise, Connolly s EI inversion results in extremely large deviation. To reduce the negative influence of random noise, they first performed least-square linear fitting of elastic impedances in the lnei sin θ plane to obtain the general trend of elastic impedances and then applied Hashin-Shtrikman bounds to constrain the data distribution. Conventional three-term EI inversion with V P, V S and density as the object parameters is easier to be contaminated by random noise, which will make the inversion unstable and may result in unreasonable value. While P-wave impedance (I P ) and S-wave impedance (I S ) are less sensitive to random noise than V P and V S. Therefore, Fatti EI inversion was adopted to invert I P and I S, instead of V P and V S. To decrease the ill condition of the inversion matrix, we neglected density term without decreasing the inversion accuracy much. Besides, to make the whole inversion more stable and decrease the inversion errors, we treat K value as a constraint incorporated into EI inversion. To test the accuracy of two-term Fatti EI inversion method, we constructed a model from parts of well-log curves, added 5%, Zhang, J., Liu, H. S., Tong, S. Y., et al., 15. Estimation of Elastic Parameters Using Two-Term Fatti Elastic Impedance Inversion. Journal of Earth Science, 6(4): doi:.7/s

2 Estimation of Elastic Parameters Using Two-term Fatti Elastic Impedance Inversion 557 %, % of Gaussian noise to the synthetic angle traces, then inverted I P, I S and density. We found that the inverted results are still promising even with low SNR (signal noise ratio) seismic data (S/N=1). We applied two-term Fatti EI inversion to a D line in our study area, and obtained several attribute sections, predicting our target reservoir successfully. Therefore, this method has great potential in reservoirs prediction. 1 METHOD EI inversion mainly contains two steps: first invert the EI value of angle-limited stacks at different incident angle, and then extract the desired elastic parameters. The first step can be implemented with conventional poststack seismic inversion, such as constrained sparse spike inversion (CSSI) or modelbased inversion algorithm, and the second step can be regarded as the same course of AVO inversion. From the previous study of EI inversion, we know that direct inversion of V P, V S, and density using Connolly s three-term EI formula is easily affected by random noise, while I P and I S are relatively less sensitive to the random noise. So it is more favorable that we invert I P and I S first and then derive other elastic parameters from I P and I S. Besides, reliable rock physics constraints are also important to make the inversion stable. Fatti et al. (1994) proposed the following AVO approximation with I P, I S, and density as the variables based on Aki and Richards s work (19) R 1 1 tan I P IP VS IS 4 sin VP IS 1 tan V S sin VP Wang et al. (8) followed the procedure of Connolly s EI formula derivation and the normalization method by Whitcombe () to derive normalized Fatti EI equation as where EI I a P S I P a b c I I, 1 tan, b 8Ksin, P IP IS c4ksin tan, V Sn V Pn1 VPn VPn1 K I P, I S, ρ are constants for normalization, and can be taken as the P-wave impedance, S-wave impedance and density of the upper layer, respectively. K is the average value of V S /V P of the nth layer and (n+1)th layer, but for multi-layer (1) () strata, K is the average value of V S /V P of each layer. For most rocks, V S /V P is about.5, so in industry, K is often taken as a constant around.5. As soon as we obtain the EI values at different angles using AI inversion procedure, we can invert I P, I S, and density from the following equation EI t, 1 ln I t EI t, n ln I P P IP a( 1) b( 1) c( ln( ) EI, 1) t I ln P a( ) b( ) c( ) I P IS t ln( ) IS a( i) b( i) c( i) EI t, i t ln a( n) b( n) c( n) ln( ) I P () It is worth noting that for near offset data, tan θ sin θ, and if K=.5, then c= in Eq. (), which results in the ill-conditioned coefficient matrix in Eq. (). Considering the ill condition problem resulted from the density term and density varies much less than other elastic parameter, we didn t invert density directly. Instead, we used some empirical formulas, like Gardner equation (Gardner et al., 1974) or well-log information to achieve it. Therefore, we revised Eqs. () and () into two-term form. For the range of small to moderate angles, c in Eq. (). Therefore, we got two-term Fatti EI as following EI I I P S I P IP IS a b Accordingly, I P and I S can be obtained from the following equation with several angle gathers 1 I P a1 b1 IP t ln EI t, 1 a b I P ln I P IS t ln an bn I S EI t, 1 ln I P EI t, ln (4) (5) Figure 1 shows the reflectivity coefficients of gas sand/ shale model (Ostrander, 1984) calculated with three-term and two-term Fatti EI equation, respectively. The elastic parameters of the model are listed in Table 1. From Fig. 1, we can see that the reflectivity calculated with two-term equation is very close to three-term at near offset (incident angle<º), so we can ignore the density term in EI inversion without lowing down the accuracy of reflectivity for near offset data. Besides, decreasing the number of parameters will also increase the robustness of inversion. For far offset data, the reflection coefficient calculated with two-term equation will deviate from three-term equation and density cannot be ignored. That is why accurate estimation of density needs large angle and high SNR seismic data (Downton, 5).

3 558 Jin Zhang, Huaishan Liu, Siyou Tong, Lei Xing, Xiangpeng Chen and Chaoguang Su Reflectivity Table 1 Gas sand/shale model (Ostrander, 1984) Layer V P (m/s) V S (m/s) ρ (g/cm ) Gas sand Shale Three-term Two-term.1 Incident angle (º) Figure 1. Comparison of reflectivities calculated with threeterm Fatti EI equation (red solid curve) and two-term equation (blue dotted curve) of Ostrander model. Actually, Eq. (5) is completely equivalent to Eq. (6), which is obtained from Eq. (7), that is, two-term Fatti AVO approximation (Fatti et al., 1994) a b I R 1 1 P 1 a b I P R 1 I S an bn IS Rn IP 1 I R 1 tan 4 sin I I P Once we achieved EI or reflectivity at different angle with the first step of EI inversion, we can also extract elastic parameters with Eq. (6). Without consideration of inverting density, two-term AVO/EI has the following advantages: (1) AVO/EI assumes the amplitude can be formulated and predicted with a simple convolutional model, while this assumption is only valid for the plane wave propagation in isotropic layered medium with no geometry spread, no scattering and transmission loss, no mode converted wave and no multiple (Mallick, 1). Convolutional model doesn t work for the seismic data of large offsets. Therefore, two-term AVO/EI for small to moderate angle data can avoid the contamination of interference from other wave modes at large angle. () Most of AVO/EI equations assume that incident angle and K keep constant. Constant incident angle implies the impedance contrasts of each layer are very small and constant K means, the average value of V S /V P, keeps constant in each layer. These two assumptions can not meet the realistic situation and sometimes will introduce big errors to EI inversion (Mallick, 1). However, errors due to constant incident and K usually increase with angle increase, which can be easily learnt from AVO/EI equation. Two-term AVO/EI for S S (6) (7) small to moderate angle gathers will decrease the errors due to constant incident angle and K. () Three-term AVO/EI equations are more accurate only for wide angle data than two-term AVO/EI equations, while for the data angle less than º, they have no obvious advantage. Two-term AVO/EI equations are more robust than three-term ones, because three-term AVO/EI equations are ill-conditioned and easily damaged with random noise (Mallick, 7, 1). It is well known that good initial model and constraints can improve the stability of inversion. In practical EI inversion, well log data and interpreted horizons are adopted to provide the initial model and inversion constraints. However, in AVO/EI equation, K=(V S /V P ) =(I S /I P ) constructs the direct relationship between I S and I P, so it can be used as a rock physics constraint to be involved into inversion, not a constant value any more. For example, we can give a range of value to K according to the well-log data and rock physical knowledge, like K min <K<K max, and if I P can be estimated enough accurately, then we get K min I P I S K max I P. We can also use some other empirical equations as the constraints for I S and I P, same as the relationship between V S and V P, such as mudrock line (Castagna et al., 1985) for water-saturated clastics; the velocity-porosity-clay relation (Han et al., 1986); the Greenberg-Castagna relation for mixed lithologies (Greenberg and Castagna, 199) and the velocity relation for clay-sand mixtures (Xu and White, 1996). The inversion procedures of our approach are shown as following: (1) Invert EI at small to moderate angle gathers with their angle wavelets using the similar poststack AI inversion method. () Because I P and I S are relatively less sensitive to the random noise, we use the two-term Fatti EI formula or the two-term AVO equation to get I P (t)/i P and I S (t)/i S, or I P /I P and I S /I S. During the course of inverting I S /I S, empirical formulas or well-log data are involved to provide K variation range as the constraints. () Get ρ/ρ, V P /V P, V S /V S and other reflectivities of elastic parameters related to I P /I P and I S /I S inverted in step (). (4) Given the value of I P and I S from the well-log data, I P and I S can be achieved and then the other elastic parameters, such as V P, V S, ρ, V P /V S, σ, λρ, μρ, λ/μ can be obtained. SYNTHETIC TESTS We test our approach using parts of real well logs of length of ms in our study area. We obtain the synthetic angle traces at 5º, 15º, 5º using Connolly s EI formula with average K value of each layer and then add different level of random noise with Gauss distribution to the synthetic angle traces to test EI inversion method. Figure shows the comparison of the true well logs (blue solid curves, left three panels are V P, V S, density and right two panels are pseudo I P and I S ) and the inverse results using Connolly s three-term EI inversion for the angle gathers with 5% random noise (green dashed curves). It is obvious that the inverted V P, V S, and density deviate much from the true model (the errors can exceed %), while I P and I S are reliable (the average errors are about %), which agrees with the results from Mallick et al. () and Lu and McMechan (4). Relative percentage errors diagram of each inverted parameter are shown in Fig. to illustrate the errors more clearly.

4 Estimation of Elastic Parameters Using Two-term Fatti Elastic Impedance Inversion 559 VP (m/s) VS (m/s) Figure. Results of conventional three-term EI inversion for the angle gathers with 5% random noise (blue solid curves denote the true model, sand green dashed curves denote the inversion results). Error percentage (%) Figure. Relative errors of Fig. (horizontal axis denotes the relative error).

5 5 Jin Zhang, Huaishan Liu, Siyou Tong, Lei Xing, Xiangpeng Chen and Chaoguang Su Unstability due to the ill-conditioned matrix can be decreased with damping least-squares method or SVD. We solved Eq. () with damping least-squares method and choose damping factor λ=.1 after many tests. Figure 4 shows that the inversion results have been much improved with damping least-squares method. The inverted elastic parameters have similar trends with the true model. However, the errors of inverted I P and I S get higher due to the existence of damping factor. Figure 5 shows the results obtained from the two-term Fatti EI inversion and we can see that all of them have a good match with the true model. During the inversion, we use Gardner equation to obtain density. Figure 6 shows the relative errors of Fig. 5. It is obvious that two-term Fatti EI inversion gets better results than the conventional EI inversion and damping least-squares solving. During the two-term Fatti EI inversion, we defined the range of K according to the well-log data as a constraint, which will improve the inversion results. This is because if we take constant K, such as K=.5 or average K of well-log data, which means each layer has the same value of V S /V P, when some layers deviate this assumption, big errors will be introduced into inversion. Figure 7 shows the results from two-term Fatti EI inversion with constant K as the average value of V S /V P of each layer. We can see that compared with Fig. 5, inverted I S and V S have big errors in some layers due to constant K. The biggest deviation can reach %, which is shown in Fig. 8. Figures 9 and show the inversion results and relative errors of our approach for the angle gathers with % random noise (S/N=/1). Figures 11 and 1 show the inversion results and relative errors of our approach for the angle gathers with % random noise (S/N=1/1). We can see that even with a low SNR, our approach can obtain stable and reliable estimation of elastic parameters. INVERSION OF FIELD DATA We applied our approach to a cross-well seismic line of Guojuzi sag in China. Prior to the EI inversion, a series of amplitude-preserved processes were taken, such as spherical compensation, surface consistent deconvolution, random noise attenuation, inverse Q filter, prestack time migration, to make sure the final prestack amplitudes represent the reflection strength of the subsurface interfaces. Though the SNR of seismic data has been much improved after process, the residual random noise will still do much bad influence to the elastic parameters extraction if we use conventional three-term EI inversion. The seismic line is across two wells, among which gs1 is treated as a known well and da1 is treated as a test well. Our known reservoir was drilled at depth of 8 m (around ms) in Well gs1. Through the analysis of well-log data of Well gs1, we found that the oil sand has relatively lower V P /V S, lower λρ, higher μρ, compared with the surrounding mudstone Figure 4. Conventional three-term EI inversion results with damping least-squares method for the angle gathers with 5% random noise (blue solid curves denote the true model, sand green dashed curves denote the inversion results).

6 Estimation of Elastic Parameters Using Two-term Fatti Elastic Impedance Inversion Figure 5. Results of two-term Fatti EI inversion for angle gathers containing 5% random noise (blue solid curves denote the true models and green dashed curves denote the inversion results). 4 Error percentage (%) Figure 6. Relative errors of the inversion results in Fig. 5 (horizontal axis denotes the relative error).

7 56 Jin Zhang, Huaishan Liu, Siyou Tong, Lei Xing, Xiangpeng Chen and Chaoguang Su Figure 7. Two-term Fatti EI inversion results with contant K for angle gathers containing 5% random noise (blue solid curves denote the true models and green dashed curves denote the inversion results). Error percentage (%) Figure 8. Relative errors of the inversion results in Fig. 7 (horizontal axis denotes the relative error).

8 Estimation of Elastic Parameters Using Two-term Fatti Elastic Impedance Inversion Figure 9. Results of two-term Fatti EI inversion for angle gathers containing % random noise (blue solid curves denote the true models and green dashed curves denote the inversion results). 4 Error percentage (%) Figure. Relative errors of the inversion results in Fig. 9 (horizontal axis denotes the relative error).

9 564 Jin Zhang, Huaishan Liu, Siyou Tong, Lei Xing, Xiangpeng Chen and Chaoguang Su Figure 11. Results of two-term Fatti EI inversion for angle gathers containing % random noise (blue solid curves denote the true models and green dashed curves denote the inversion results). 4 Error percentage (%) Figure 1. Relative errors of the inversion results in Fig. 11 (horizontal axis denotes the relative error).

10 Estimation of Elastic Parameters Using Two-term Fatti Elastic Impedance Inversion 565 We chose 5, 15 and 5 angle gathers, used the two-term Fatti EI equation to do the EI inversion and got kinds of attribute sections. During the inversion, we defined the range of K as a constraint obtained from Well gs1 well-log. After we got I P and I S from two-term Fatti EI inversion, λρ, μρ and V P /V S sections can be achieved easily, because λρ=i P I S, μρ=i S, and V P /V S =I P /I S. The inverted results of λρ, μρ and V P /V S are displayed in Figs respectively. All these sections agree with the drilling results of Well da1 (located at CDP 157). Our target reservoir is encountered at 914 m (around ms) of Well da1 as white ellipse denotes, which has relatively lower λρ, higher μρ and lower V P /V S. 4 CONCLUSION Due to the ill-conditioned coefficient matrix of the three-term EI inversion, the course of conventional EI inversion becomes easily contaminated with tiny random noise in seismic data, which will lead to unreasonable estimation of elastic parameters. In this paper, we present a two-term Fatti EI formula through ignoring the density term to decrease the ill condition, which is more robust and keeps almost the same accuracy with conventional three-term EI formula in near offset data. We inverted I P and I S first, which are less sensitive to random noise, then got other elastic parameters of rocks with empirical equations or well logs information. During the inversion, we defined the range of K as a constraint to make the inversion more stable. Synthetic tests confirmed that our approach can make promising result even with high level of random noise (S/N=1) in seismic data. Our approach successfully delineated target reservoir and had good agreements with the drilled well, so two-term Fatti EI inversion has promised great potential in reservoir prediction, especially for near offset seismic data. Figure 1. λρ section with two wells (gs1 located at CDP 18, da1 located at CDP 157). White ellipse denotes the predicted reservoir. Figure 14. μρ section with two wells (gs1 located at CDP 18, da1 located at CDP 157). White ellipse denotes the predicted reservoir.

11 566 Jin Zhang, Huaishan Liu, Siyou Tong, Lei Xing, Xiangpeng Chen and Chaoguang Su Figure 15. V P /V S section with two wells (gs1 located at CDP 18, da1 located at CDP 157). White ellipse denotes the predicted reservoir. ACKNOWLEDGMENTS We would like to acknowledge the sponsorship of the National Natural Science Foundation of China (Nos. 496 and 4118) for funding this research. We also thank Shengli Oil Field for providing the field data. Finally we thank for the anonymous reviewers for their constructive suggestion. REFERENCES CITED Aki, K., Richards, P. G., 19. Quantitative Seismology, nd Ed.. W. H. Freeman and Company, San Francisco. 557 Bruce, V., VerWest, B., Masters, R., et al.,. Elastic Impedance Inversion. Expanded Abstracts of th SEG Mtg., Calgary Cambois, G.,. AVO Inversion and Elastic Impedance. SEG Technical Program Expanded Abstracts, doi:.11/ Castagna, J. P., Batzle, M. L., Eastwood, R. L., Relationships between Compressional-Wave and Shear-Wave Velocities in Clastic Silicate Rocks. Geophysics, (4): doi:.11/ Connolly, P., Elastic Impedance. The Leading Edge, 18(4): doi:.11/ Downton, J. E., 5. Seismic Parameter Estimation from AVO Inversion: [Dissertation]. University of Calgary, Calgary Fatti, J. L., Smith, G. C., Vail, P. J., et al., Detection of Gas in Sandstone Reservoirs Using AVO Analysis: A -D Seismic Case History Using the Geostack Technique. Geophysics, 59(9): doi:.11/ Gardner, G. H. F., Gardner, L. W., Gregory, A. R., Formation Velocity and Density The Diagnostic Basics for Stratigraphic Traps. Geophysics, 9(6): 7 7. doi:.11/ Greenberg, M. L., Castagna, J. P., 199. Shear-Wave Velocity Estimation in Porous Rocks: Theoretical Formulation, Preliminary Verification and Applications. Geophysical Prospecting, (): doi:.1111/j tb71.x Han, D., Nur, A., Morgan, D., Effects of Porosity and Clay Content on Wave Velocities in Sandstones. Geophysics, 51(11): 9 7. doi:.11/ Lu, S. M., McMechan, G. A., 4. Elastic Impedance Inversion of Multichannel Seismic Data from Unconsolidated Sediments Containing Gas Hydrate and Free Gas. Geophysics, 69(1): doi:.11/ Mallick, S., 1. AVO and Elastic Impedance. The Leading Edge, (): doi:.11/ Mallick, S., 7. Amplitude-Variation-With-Offset, Elastic- Impedence, and Wave-Equation Synthetics A Modeling Study. Geophysics, 7(1): C1 C7. doi:.11/1.878 Mallick, S., Huang, X. R., Lauve, J., et al.,. Hybrid Seismic Inversion: A Reconnaissance Tool for Deepwater Exploration. The Leading Edge, 19(11): doi:.11/ Ostrander, W. J., Plane-Wave Reflection Coefficients for Gas Sands at Nonnormal Angles of Incidence. Geophysics, 49(): doi:.11/ Santos, L. T., Tygel, M., Ramos, A. C. B.,, Reflection Impedance. 64th Conference European Association of Geoscientists and Engineers, Expanded Abstracts, Florence. 18 Tsuneyama, F., Mavko, G., 7. Elastic-Impedance Analysis Constrained by Rock-Physics Bounds. Geophysical Prospecting, 55(): doi:.1111/j x Wang, B., Yin, X., Zhang, F., et al., 8. Elastic Impedance Equation Based on Fatti Approximation and Inversion. Progress in Geophysics, : (in Chinese with English Abstract) Whitcombe, D. N.,. Elastic Impedance Normalization. Geophysics, 67(1): 6. doi:.11/ Whitcombe, D. N., Connolly, P. A., Reagan, R. L., et al.,. Extended Elastic Impedance for Fluid and Lithology Prediction. Geophysics, 67(1): doi:.11/ Xu, S. Y., White, R. E., A Physical Model for Shear-Wave Velocity Prediction. Geophysical Prospecting, 44(4): doi:.1111/j tb1.x