Correspondence between the low- and high-frequency limits for anisotropic parameters in a layered medium

GEOPHYSICS VOL. 74 NO. MARCH-APRIL 009 ; P. WA5 WA33 7 FIGS. 10.1190/1.3075143 Correspondence between the low- and high-frequency limits for anisotropic parameters in a layered medium Mercia Betania Costa e Silva 1 and Alexey Stovas ABSTRACT Wave propagation in a layered medium when the wavelength is much greater than each layer thickness low frequency produces a response equivalent to that of wave propagation in an equivalent single-layer medium. This equivalent medium is transversely isotropic with symmetry about a vertical axis VTI and the elastic parameters are computed with the Backus averaging technique. Conversely when the wavelength is comparable to each layer thickness high frequency the directional dependence of the phase velocity in the transmission response also can be simulated by replacing the layered medium with a single homogeneous medium with properties derived from a time average. It then can be treated approximately as a VTI medium. To compute the medium parameters a method based on fitting the traveltime parameters is used. We investigated the relationship between Thomsen s anisotropic parameters and computed for the equivalent medium in the low-frequency limit and for the homogenized medium in the high-frequency limit. In our experiments we used a medium in which layers of only two isotropic materials alternate repeatedly. For the high-frequency limit we obtained solutions for PP- and SS-wave propagation. INTRODUCTION Seismic anisotropy in a given rock is frequency dependent Mukerji and Mavko 1994. This dependence usually is caused by fluid factors Shapiro and Hubral 1996 and is known as intrinsic anisotropy. If we consider different isotropic rocks combined in one layered medium the resultant medium is layer-induced anisotropic with a pronounced frequency dependency Backus 196. The correspondence between intrinsic and layer-induced anisotropy is studied by Berryman et al. 1999 Bakulin 003 Bakulin and Grechka 003 and Stovas et al. 006. Layer-induced anisotropy usually is defined and computed for the long-wavelength or low-frequency LF limit. The equivalent medium is transversely isotropic with a vertical axis of symmetry VTI with parameters given by the Backus 196 averaging technique. The behavior of layer-induced anisotropy in the high-frequency HF case also is reported in the literature such as in Grechka and Tsvankin 00 and Bakulin and Grechka 003. In this case observing reflection or transmission effects caused by replacement of the layered medium by a homogeneous medium is the most common approach used to study the induced anisotropy. The replacement medium is a single time-average medium and its response is computed using traveltime parameters Ursin and Stovas 006. Propagation of different frequencies through a layered medium results in different apparent medium properties. Depending on frequency these properties vary from the ones in the equivalent medium LF limit to the ones in the time-average medium HF limit. Stovas and Ursin 007 show the phase velocity regimes for different propagation frequencies. They also compute the transmission response as a function of frequency and investigate the transition zone between the LF and HF limits. The behavior in the frequency range between these two limits depends on the reflection coefficient and thickness ratio of the layers. Here we consider a medium made up of repeated layers of two isotropic materials and compute the layer-induced anisotropic parameters and Thomsen 1986 for propagation near the two frequency limits. This leads to the equivalent medium in LF and the time-average medium computed for PP- and SS-wave propagation in HF. We show that the time-average medium from PP- and SS-traveltime parameters computed from this medium of isotropic layers does not yield exactly a VTI medium. Nevertheless the approximate solution can be used. We also discuss some comparisons between the LF and HF limits for and using synthetic and real data. Manuscript received by the Editor 8 January 008; revised manuscript received 4 September 008; published online 11 March 009. 1 Formerly Norwegian University of Science and Technology Department of Petroleum Engineering and Applied Geophysics Trondheim Norway; presently EMGS Trondheim Norway. E-mail: mercia.silva@emgs.com. Norwegian University of Science and Technology Department of Petroleum Engineering and Applied Geophysics Trondheim Norway. E-mail: alexey@ ipt.ntnu.no. 009 Society of Exploration Geophysicists. All rights reserved. WA5

WA6 Silva and Stovas FREQUENCY LIMITS FOR HOMOGENIZATION OF THE LAYERED MEDIUM It is well known that there is a dispersion of phase velocity for a wave propagating in a layered medium. Stovas and Ursin 007 illustrate the phase velocity dispersion for a wave propagating vertically in a periodically layered medium and compute its two frequency limits. The LF limit is defined by an equivalent-medium velocity computed with the Backus average whereas the HF limit is defined by time-average velocity obtained from traveltime parameters. The transition between these two limits is controlled by the layer-thicknesses ratio or volume fraction and by the reflection coefficient. V TA V EF Figure 1. Phase velocity in a medium with repeated layering Stovas and Ursin 007 : r is the reflection coefficient Mk is the number of times the layering repeats V EF is the LF limit of the velocity and V TA is the HF limit. a) Figure 1 shows the dispersion in phase velocity from a series of periodically layered models from Stovas and Ursin 007 with parameters given in Marion and Coudin 199. We observe the phase velocity curves for one specific reflection coefficient value r.48 and for various values of Mk the number of times the layers repeat within the medium. The phase velocity functions start from the LF limit V EF and as frequency increases transition to the HF limit V TA. This indicates that the limits of the transmission response from the homogenization of the layered medium are equivalent to those of a single-layer anisotropic medium. This is valid not only for the LF limit at which the model performs as an equivalent medium but also for the HF limit at which the model behaves as a time-average medium. The schematic layered medium in Figure reinforces the idea and shows the transmission responses for the last interface in HF and LF limits. These responses carry information about all the layers above and can be associated with one equivalent or homogenized anisotropic medium. Because the LF limit has well-known results for the equivalent medium with parameters specified for a VTI medium we use the same parameterization for the HF limit. We compute the traveltime parameters for a time-average medium with alternating horizontal isotropic homogeneous layers and compare those results with the traveltime parameters for a single-layer VTI model to find the anisotropic parameters and. We do not mean to imply that the layered medium converges to a VTI medium at the HF limit. By introducing the HF limit we mean that treating the layered medium in the framework of a frequency-dependent equivalent anisotropic medium requires evaluation of LF- and HF-limit anisotropic parameters. In addition because we are dealing with effective anisotropic parameters induced by layering our HF-limit anisotropic parameters are purely modeling parameters and have no physical meaning per se. To compute the two limits we treat the models according to the type of response to different wavelengths using an equivalent medium for LF and a time-average medium regime for HF. Because we are interested only in the transmission effects the traveltime parameters for the HF limit are computed after only one pair of layers for the binary medium and at the bottom of the anisotropic single-layer medium. For the LF case the Backus 196 average is performed for the two-layer medium and the traveltime parameters are computed from components of the equivalent medium. We compare the anisotropic parameters for both limits. We find a difference that agrees with the frequency dependence computed for the phase velocity by Stovas and Ursin 007. THE PP- AND SS-HIGH-FREQUENCY LIMIT Figure. Wave propagation illustration for a LF limit and b HF limit PP and SS. In a two-layer horizontal isotropic homogeneous model with seismic properties V P1 V P V S1 V S 1 and and thicknesses z 1 and z where indices 1 and define the first and second layers

Frequency limits for anisotropic parameters WA7 respectively the anisotropic parameters are computed as functions of these model properties. Because the anisotropic parameters have nondimensional values we use the nondimensional ratios of the model parameters P V P /V P1 S V S /V S1 1 V S1 /V P1 / 1 and z /z 1 to simplify the computation. The HF limits for anisotropic parameters are computed by fitting the traveltime parameters from a two-layer isotropic model Appendix A to the traveltime parameters computed from a single-layer VTI model derived by Ursin and Stovas 006 as detailed in Appendix B. The total thickness is fixed for both models. If one considers the PP-wave propagation and equalizes traveltime parameters from the isotropic two-layer model equation A-4 and anisotropic one-layer model equation B-4 is computed easily from the first two traveltime parameters t 0 P z 1 V P1 1 P z 1 1 1 V nmo P V 1 P P1 1 0 1 P P where the P index denotes parameters computed for PP-wave propagation and is the P-wave vertical velocity for the singlelayer VTI model.after minor simplifications P becomes P 1 P P 1. To compute P and the vertical S to P velocity ratio / we use the heterogeneity coefficients S and S 3 Appendix B but the solution of equation B-1 results in physically impossible values for. This means that the resultant medium is not exactly a VTI medium which agrees with Helbig 000. Nevertheless we estimate the anisotropic parameters by applying the acoustic approximation Alkhalifah 1998 which is based on the fact that the qp-wave propagation in a VTI medium is relatively independent of the vertical S-wave velocity. This yields P 1 P 8 P 1 1 P 3 1 P 1 P 4 1 P. 4 Because the acoustic approximation performs very well for qpwaves the values for the estimated parameter P are accurate enough. For the obvious single-layer limits 0 and P 1 parameters P and P converge to zero. We also compute the anisotropic parameters for SS-wave propagation equations A-4 and B-5 : S 1 S S 1 4 S 1 1 S S 4 S 1 1 S S S 1 S S 1 S 1 S S S 1 S 4 S 1 1 S S. As in the case of PP-wave propagation computing the anisotropic parameter S does not yield exactly a single-layer VTI model because the computed vertical S to P velocity ratio S / is always larger than one which is a physically impossible solution. Nevertheless in this case the S still can be used to compute S. In addition for the single-layer limits 0 and S 1 parameters S and S converge to zero. The nonphysical values for found for PP- and SS-wave propagation were obtained because we fitted the traveltime parameters of a two-layer isotropic model to those of a single-layer VTI model. This anomaly reflects the fact that the equivalent anisotropic medium chosen to perform the fitting in the HF limit is not appropriate. Nevertheless approximations for the anisotropic parameters still can be used with reasonable accuracy. THE LOW-FREQUENCY LIMIT We compute the LF limit using the Backus 196 average. The parameters for the effective VTI medium are obtained from the original two-layer isotropic medium using the Backus average Appendix C. All parameters are a function of the same ratios of model parameters defined for the HF limits; and are given by B P B 1 1 1 S 1 S P 1 1 P 1 S P S 1 1 P S 1 1 1 S 1 P 1 1 S where the B index indicates that the anisotropic parameters were computed from the traveltime parameters defined in the Backus average. Note that the LF limit for anisotropy parameters given in equations 7 and 8 does not depend on the chosen mode of propagation. From equations 7 and 8 we also can observe that the effective model can behave as isotropic for some parameter combinations; for example if the contrast in shear modulus between the two layers is zero then S 1. 5 6 7 8

WA8 Silva and Stovas NUMERICAL AND REAL DATA EXAMPLES We consider a two-layer horizontal isotropic homogeneous model with / 1 1 P V P /V P1 1.1 S V S /V S1.9 and 1 V S1 /V P1.6. Because we have so many parameters for our model and we are interested in showing how the anisotropic parameters change with the change of some parameters whereas others are fixed we choose to fix 1 and 1.6. In Figure 3 the anisotropic parameters for the LF and HF limits from PP- and SS-wave propagation are plotted as a function of the volume fraction defined as z / z 1 z or equivalently as / 1 where z /z 1. The volume fraction equation allows us to plot the anisotropic parameters as a function of within the range 0. The limit values 0 and 1 for the volume fraction corresponding to and respectively shown in Figure 3 indicate situations in which the model is isotropic because in these limits only a single constituent is present. We notice that depending on the model parameters the LF and HF limits can have different algebraic signs for and suggesting the possibility of an equivalent medium behaving as an isotropic medium for some intermediate frequencies. In this specific example with S varying from 0.9 through 1. the parameter shows a larger variation than does. The HF-limit anisotropic parameters estimated from SS-wave propagation are larger than those estimated from PP-wave propagation for all volume fraca) Figure 3. HF and LF limits of anisotropic parameters and versus volume fraction with a S.9 and b S 1..

Frequency limits for anisotropic parameters WA9 tions and the maximum values of the anisotropic parameters occur when the volume fraction is approximately equal to 0.5 as expected. In another example we test the behavior of the anisotropic parameters with changes in the P-wave velocity contrast P Figure 4.We compare the HF limit only for PP-wave propagation because the HF limit for SS-wave propagation does not depend on the contrast in P-wave velocity. This explains why the HF limits and in Figure 4a and b curves in red are identical because we are changing only the S-wave velocity contrast S between Figure 4a and b. Note that all HF-limit curves have the value zero when there is no contrast in P-wave velocity P 1 ; otherwise for the model parameters we tested the HF-limit curves have positive values for both anisotropic parameters. The LF-limit curves are dependent on S-wave velocity contrast. When S 1 the values of and decrease with the increase of P ; when S 1 the curves have the opposite behavior. For both anisotropic parameters the HF and LF limits coincide when P S. We can observe similar behavior when plotting anisotropic parameters versus S-wave velocity contrast Figure 5. Here we display only the HF limit computed for SS-wave propagation. It also can be seen that the limits for and cross two times in all plots because S 1 and P S. In a real data experiment Figure 6 some markers were chosen in a well log from the North Sea at major P-wave velocity changes. The a) Figure 4. HF and LF limits of anisotropic parameters and versus the contrast in P-wave velocity with a S.9 and b S 1.1.

WA30 Silva and Stovas anisotropic parameters were computed for the HF limit PP- and SSwave propagation and for the LF limit using these same markers. It can be seen that has only positive values for this data set and can range from negative LF limits to positive HF limits. These results show the small fluctuation between the HF PP-wave propagation and the LF limits which is in agreement with the results found by Bakulin and Grechka 003. In Figure 7 we use the same well-log data as used in Figure 6 and show the behavior of the discrepancy of anisotropic parameter limits with respect to a function of the HF and LF vertical velocity limits 1 V P LF /V P HF. This function was chosen according to Stovas and Ursin 007 who prove it to be controlled by the reflection coefficient. We can easily observe in Figure 7 the clear relation between the reflection coefficient and the discrepancy of anisotropic parameters with frequency. The first plot in Figure 7 shows the results for PP-wave propagation. For this data the LF limit for is always greater than its HF limit. The parameter has the opposite algebraic sign and the dispersion for both anisotropic parameters increases with an increasing reflection coefficient. For SS-wave propagation second plot in Figure 7 the HF limits for and are greater than their LF limits computed for this data and the differences in the HF and LF values for and also increase with an increasing reflection coefficient. Figure 5. HF and LF limits of anisotropic parameters and versus the contrast in S-wave velocity with a P.9 and b P 1.1. a)

Frequency limits for anisotropic parameters WA31 a) Figure 6. HF and LF limits calculated from a well log. Velocities a V P and b V S and anisotropic parameters c and d. c) d) a) Figure 7. Behavior of the anisotropic parameters for HF and LF limits as a function of the reflection coefficient for a qpand b qsv approximations.

WA3 Silva and Stovas CONCLUSIONS We computed the anisotropic parameters for the homogenization of a layered medium considering the upper and lower limits in the frequency of propagation and using the equivalent medium for the LF limit and time-average medium for the HF limit. The properties computed for the HF limit suggest that the time-average medium is not an exact VTI because the vertical V S /V P ratio cannot be computed from the traveltime parameters for either PP- or SS-wave propagation. We used an approximate solution to solve this problem and to compute the anisotropic parameters with reasonable accuracy. With numerical and real data examples we verified the behavior of the anisotropic parameter limits resulting from changes in P- and S-wave velocity contrasts and in reflection coefficient. The numerical tests show that the LF and HF limits for are more sensitive to changes in S than are the limits for. Another interesting observation is that when S 1 the HF limits for and decrease with the increase of P and when S 1 these curves have the opposite behavior. We also observed that the curves for the LF and HF limits for both anisotropic parameters cross two i.e. when S 1 and P S. From the real data example we verified that the deviation of and from LF to HF limits clearly is related to the reflection coefficient. Although equations for the anisotropic parameters at the two limits are algebraically complicated all plots presented here show strong correlation between the anisotropic parameters in LF and HF limits PP- and SS-wave propagation and velocity heterogeneity. ACKNOWLEDGMENTS The authors acknowledge the Norwegian Research Council for financial support via the ROSE project. APPENDIX A TRAVELTIME PARAMETERS FROM A TWO-LAYER MODEL The traveltime parameters in a vertically heterogeneous isotropic medium are given in terms of velocity moments defined as timeweighted moments of velocity distribution where n I n 1 I 1 n 1... A-1 z I m 0 v p m d A- where is the depth. Then the traveltime parameters two-way vertical traveltime normal moveout velocity and heterogeneity coefficients are defined by t 0 I 1 V nmo S n n n n 3.... A-3 For a two-layer model or for a model in which the two layers repeat the first four traveltime parameters are given by t 0 z 1 V 1 1 1 V nmo V 1 1 S 1 1 1 S 3 1 1 1 1 3 A-4 where V /V 1 and z /z 1 are velocity and thickness ratios and V 1 z 1 and V z are velocity and thickness of layer 1 and layer respectively. APPENDIX B TRAVELTIME PARAMETERS FROM A SINGLE VTI MODEL For a single VTI layer with thickness z P-wave vertical velocity S-wave vertical velocity and anisotropic parameters and Thomsen 1986 the traveltime parameters for qp-wave propagation are defined by Ursin and Stovas 006 where and a 0 a 1 t 0P z V nmop 0 1 a 0 S P 1 4a 1 1 a 0 S 3P 1 4a 1 1 a 0 8a 1 a 0 3 1 1 a 4 1 1. B-1 B- B-3 In the quasi-acoustic approximation Alkhalifah 1998 equation B-1 reduces to

Frequency limits for anisotropic parameters WA33 t 0P z V nmop 0 1 S P 1 8 S 3P 1 8 3 B-4 where the anisotropic parameter / 1 is that introduced by Alkhalifah 1998. For qsv-wave propagation we have where t 0S z V nmos 0 1 b 0 S S 1 4b 1 1 b 0 S 3S 1 4b 1 1 b 0 8b 1 b 0 3 b 0 B-5 b j a j j j 1... B-6 with a j and defined in equations B- and B-3. APPENDIX C BACKUS AVERAGING FOR TWO CONSTITUENTS The stiffness coefficients for the equivalent medium are given by Backus 196 averaging. For a two-layer isotropic model these coefficients are given by where c 11 A 1 A 3 A c 13 A 3 A c 33 1 A c 44 1 A 4 4 A 1 1 V 1 P1 1 1 1 S 1 1 S P C-1 A 1 1 1 V P1 1 1 P A 3 1 1 1 1 1 1 S A 4 1 1 1 V P1 1 1 1 S P C- where P V P /V P1 S V S /V S1 1 V S1 /V P1 / 1 and z /z 1 are the ratios of properties of the two layers and V P1 1 z 1 V P and z are the P-wave velocity density and thickness for layer 1 and layer respectively. From definitions of and the layer-induced anisotropic parameters we can use equations C-1 and C- to obtain c 11 c 33 A 1A A 3 1 c 33 c 13 c 44 c 33 c 44 c 33 c 33 c 44 A 3 1 A 4 A 3 1 A A 4 A. C-3 REFERENCES Alkhalifah T. 1998 Acoustic approximations for processing in transversely isotropic media: Geophysics 63 63 631. Backus G. E. 196 Long-wave elastic anisotropy produced by horizontal layering: Journal of Geophysical Research 67 447 4440. Bakulin A. 003 Intrinsic and layer-induced vertical transverse isotropy: Geophysics 68 1708 1713. Bakulin A. and V. Grechka 003 Effective anisotropy of layered media: Geophysics 68 1817 181. Berryman J. G. V. Grechka and P. A. Berge 1999 Analysis of Thomsen parameters for finely layered VTI media: Geophysical Prospecting 47 959 978. Grechka V. and I. Tsvankin 00 Processing-induced anisotropy: Geophysics 67 190 198. Helbig K. 000 Layer-induced elastic anisotropy: Part Inversion of compound parameters to constituent parameters: Brazilian Journal of Geophysics 1 173 185. Marion D. and P. Coudin 199 From ray to effective medium theories in stratified media: An experimental study: 6nd Annual International meeting SEG ExpandedAbstracts 1341 1343. Mukerji T. and G. Mavko 1994 Pore fluid effects on seismic velocity in anisotropic rocks: Geophysics 59 33 44. Shapiro S. A. and P. Hubral 1996 Elastic waves in finely layered sediments: The equivalent medium and generalized O Doherty-Anstey formulas: Geophysics 61 18 1300. Stovas A. M. Landrø and P. Avseth 006 AVO attribute inversion for finely layered reservoirs: Geophysics 71 no.3 C5-C36. Stovas A. and B. Ursin 007 Equivalent time-average and effective medium for periodic layers: Geophysical Prospecting 55 871 88. Thomsen L. 1986 Weak elastic anisotropy: Geophysics 51 1954 1966. Ursin B. and A. Stovas 006 Travel-time approximations for a layered transversely isotropic medium: Geophysics 71 no. D3-D33.