On Backus average for oblique incidence David R. Dalton∗, Michael A. Slawinski † 6 January 12, 2016 1 0 2 n a Abstract J 2 1 We postulate that validity of the Backus (1962) average, whose weights are layer thicknesses, is limited to waves whose incidence is nearly vertical. The accuracy of this average decreases with ] h the increase of the source-receiver offset. However, if the weighting is adjusted by the distance p - travelled by a signal in each layer, such a modified average results in accurate predictions of trav- o eltimesthroughtheselayers. e g . s c 1 Introduction i s y h Hookean solids, which are commonly used in seismology as mathematical analogies of physical p materials, are defined by their mechanical property relating linearly the stress tensor, σ, and the [ straintensor,ε, 1 v 3 3 (cid:88)(cid:88) 6 σ = c ε , i,j = 1,2,3, ij ijk(cid:96) k(cid:96) 6 9 k=1 (cid:96)=1 2 wherecisthe elasticitytensor. The Backus(1962) averageallows usto quantifythe responseof a 0 wavepropagatingthroughaseriesofparallelHookeanlayerswhosethicknessesaremuchsmaller . 1 thanthewavelength. 0 6 AccordingtoBackus(1962),theaverageoff(x )of“width”(cid:96)(cid:48) is 3 1 ∞ : (cid:90) v f(x ) := w(ζ −x )f(ζ)dζ, (1) i 3 3 X −∞ r a wherew(x )istheweightfunctionwiththefollowingproperties: 3 ∞ ∞ ∞ (cid:90) (cid:90) (cid:90) w(x ) (cid:62) 0, w(±∞) = 0, w(x )dx = 1, x w(x )dx = 0, x2w(x )dx = ((cid:96)(cid:48))2. 3 3 3 3 3 3 3 3 3 −∞ −∞ −∞ ∗DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected] †DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected] 1 These properties define w(x ) as a probability-density function with mean 0 and standard devia- 3 tion(cid:96)(cid:48),explainingtheuseoftheterm“width”for(cid:96)(cid:48). The long-wavelength homogeneous media equivalent to a stack of isotropic or transversely isotropiclayerswiththicknessesmuchlessthanthesignalwavelengthareshownbyBackus(1962) to be transversely isotropic. The Backus (1962) formulation is reviewed by Slawinski (2016) and Bos et al. (2016), where formulations for generally anisotropic, monoclinic, and orthotropic thin layersarealsoderived. Bosetal.(2016)examineassumptionsandapproximationsunderlyingthe Backus (1962) formulation, which is derived by expressing rapidly varying stresses and strains in terms of products of algebraic combinations of rapidly varying elasticity parameters with slowly varying stresses and strains. The only mathematical approximation in the formulation is that the average of a product of a rapidly varying function and a slowly varying function is approximately equaltotheproductoftheaveragesofthetwofunctions. 2 Formulation Let us consider a stack of ten isotropic horizontal layers, each with a thickness of 100 meters (Brisco,2014). TheirelasticityparametersarelistedinAppendixA,Table1. For vertical incidence, the Fermat traveltime through these layers is 229.47 ms. If we perform the standard Backus average—weighted by layer thickness of these ten layers, as in equation (2), below—then, the equivalent density-scaled elasticity parameters are (cid:104)c (cid:105) = 18.84, (cid:104)c (cid:105) = 1111 1212 3.99, (cid:104)c (cid:105) = 10.96, (cid:104)c (cid:105) = 3.38 and (cid:104)c (cid:105) = 18.43; their units are 106m−2s−2. With these 1133 2323 3333 parameters and for the vertical incidence, the resulting P-wave traveltime through the equivalent transversely isotropic medium is 232.92 ms, which—in comparison to the Fermat traveltime—is highby3.45ms. To examine the layer-thickness weighting, let us consider one of the equivalent-medium pa- rameters, n (cid:80) h c i 2323i cTI = i=1 , (2) 1212 n (cid:80) h j j=1 where h is the thickness of the ith layer, which herein is 100 m for each layer; thus, each layer is i weightedequallyby0.1. If we consider a P-wave signal whose takeoff angle, with respect to the vertical, is π/6, this signal reaches—in accordance with Snell’s law—the bottom of the stack at a horizontal distance of1072.89m. ItsFermattraveltimeis330.58ms. If we perform the standard Backus average, the traveltime in the equivalent medium, which corresponds to the ray angle of 47.01◦, is 343.87 ms, which is higher by 13.3 ms than its Fermat counterpart. If, however, we weight the average by the distance travelled in each layer, as in equation (3), below, the equivalent elasticity parameters become (cid:104)c (cid:105) = 20.126, (cid:104)c (cid:105) = 1111 1212 4.100, (cid:104)c (cid:105) = 12.059, (cid:104)c (cid:105) = 3.450 and (cid:104)c (cid:105) = 19.762. In such a case, the traveltime 1133 2323 3333 is 332.44 ms—which is higher by only 1.9 ms—and is an order of magnitude more accurate than 2 using the standard approach. The distances travelled in each layer and the resulting weights are giveninAppendixA,Table2. Insuchacase,expression(2)becomes n (cid:80) d c i 2323i cTI = i=1 , (3) 1212 n (cid:80) d j j=1 whered isthedistancetravelledintheithlayer,which—forverticalincidence—isequaltoh . i i AccordingtoLemma2ofBosetal. (2016),thestabilityconditionsarepreservedbytheBackus average. In other words, if the individual layers satisfy these conditions, so does their equivalent medium. ThisremainstrueforthemodifiedBackusaverage. 3 Discussion The Backus (1962) average with weighting by the thickness of layer assumes vertical or near- vertical incidence. Consequently, such an average does not result in accurate traveltimes for the far-offsetor,inparticular,cross-welldata,whichnowadaysarecommonseismicexperiments,and werenothalf-a-centuryago,whentheBackus(1962)averagewasformulated. Ifwemodifytheweightingtobebythedistancetravelledineachlayer,thentheresultingtrav- eltimes are significantly more accurate. Such weighting, however, entails further considerations. Since the distance travelled in each layer is a function of Snell’s law, there is a need to modify the weights with the source-receiver offset. However, given information about layers, it is achievable algorithmicallybyaccountingfordistancetravelledineachlayerasafunctionofoffset. Thereisalsoaninterestingissuetoconsider. Themodifiedequivalentmediumisdefinedbyits elasticityparameters,whicharefunctionsoftheobliquenessofrayswithineachlayer. Thismeans that the equivalent-medium parameters are different for the qP waves, for the qSV waves and for theSH waves. However,sincea Hookean solidexistsinthemathematicalrealm,notthephysical world, such a consideration is not paradoxical. It is common to invoke even different constitutive equations for the same physical material depending on empirical considerations. Furthermore, it mightbepossibletoderiveelasticityparametersofasingleHookeansolid—possiblyofamaterial symmetrylowerthantransverseisotropy—whosebehaviouraccountsforbothnearandfaroffsets inthecaseofthreewaves. It is interesting to note that—in each examined case—the traveltime in the equivalent medium is greater than its Fermat counterpart through the sequence of layers. It might be a consequence of optimization, which—in the case of layers—benefits from a model with a larger number of parameters. There remains a fundamental question: Is the Fermat traveltime an appropriate criterion to consider the accuracy of the Backus average? An objection to such a criterion is provided by the following Gedankenexperiment. Consider a stack of thin layers, where—in one of these layers— waves propagate much faster than in all others. In accordance with Fermat’s principle, distance travelled by a signal within this layer is much larger than in any other layer, which might be expressed by the ratio of a distance travelled in a given layer divided by its thickness. This effect 3 is not accommodated by the standard Backus average, since this effect is offset-dependent and the average is not, but it is accommodated by the modified average discussed herein. However, a property of such a single layer might be negligible on long-wavelength signal. To address such issues, it might be necessary to consider a full-waveform forward model, and even a laboratory experimentalset-up. Asanaside,letusrecognizethat—ifwekeeptheFermattraveltimeasacriterion—makingthe propagationspeedafunctionofthewavelengthwouldnotaccommodatethetraveltimediscrepancy duetooffset. Be that as it may, it must be recognized that the discrepancy between the traveltimes in the layered and equivalent media increases with the source-receiver offset. In the limit—for a wave propagating horizontally through a stack of horizontal layers—the Backus average, even in its modifiedform,isnotvalid,duetoitsunderlyingassumptionofaloadonthetopandbottomonly. Acknowledgments We wish to acknowledge discussions with Theodore Stanoev. This research was performed in the context of The Geomechanics Project supported by Husky Energy. Also, this research was partially supported by the Natural Sciences and Engineering Research Council of Canada, grant 238416-2013. References Backus,G.E.,Long-waveelasticanisotropyproducedbyhorizontallayering,J.Geophys.Res.,67, 11,4427–4440,1962. Brisco,C.,Anisotropyvs.inhomogeneity:Algorithmformulation,codingandmodelling,Honours Thesis,MemorialUniversity,2014. Bos,L,D.R.Dalton,M.A.SlawinskiandT.Stanoev,OnBackusaverageforgenerallyanisotropic layers,arXiv,2016. Slawinski, M.A. Wavefronts and rays in seismology: Answers to unasked questions, World Scien- tific,2016. Slawinski,M.A.,Wavesandraysinelasticcontinua,WorldScientific,2015. 4 Appendix A layer c v 1111 P 1 10.56 3.25 2 20.52 4.53 3 31.14 5.58 4 14.82 3.85 5 32.15 5.67 6 16.00 4.00 7 16.40 4.05 8 18.06 4.25 9 31.47 5.61 10 17.31 4.16 Table 1: Density-scaledelasticityparameters,whoseunitsare106m−2s−2,forastackofisotropiclayers, andthecorrespondingP-wavespeedsinkms−1. layer d w i i 1 115.47 0.0773 2 139.45 0.0934 3 195.07 0.1306 4 124.12 0.0831 5 204.61 0.1370 6 126.88 0.0849 7 127.85 0.0855 8 132.17 0.0885 9 198.04 0.1326 10 130.17 0.0871 Table 2: Distances, d , in meters, travelled by the P wave in each layer, and the corresponding i averagingweights,w = d /((cid:80)10 d ). i i j=1 j 5