Skip to main content

Effective Media for Transversely Isotropic Models Based on Homogenization Theory: With Applications to Borehole Sonic Logs


Sonic log records, including measurements of wave speeds in boreholes, provide critical input to the geological, geophysical, and petrophysical studies of a region under exploration. 1D background models are routinely built based on sonic log records for applications such as seismic imaging of hydrocarbon reservoirs and microseismic source inversions. Smoothing or ‘upscaling’ techniques are required to produce models in coarser scales than the very fine layers in the raw log data. In this paper, we follow the recently popular homogenization theory, derive its application to the special case of 1D TI models for both P-SV and SH waves, and show that it is consistent with the Backus averaging technique commonly used to upscale 1D fine-layered models. We examine a study case of sonic log data from a well in the Horn River Basin in northeastern British Columbia, a region known for its tight shale-gas deposit. We demonstrate the computational accuracy and efficiency gained by proper upscaling procedures for spectral-element simulations of seismic wave propagation, and discuss the effect of control parameters on wavefield recovery.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  • Abdulaziz, A. M. (2013). Microseismic imaging of hydraulically induced-fractures in gas reservoirs: A case study in Barnett Shale Gas Reservoir, Texas, USA. Open Journal of Geology, 3(5), 361–369.

    Article  Google Scholar 

  • Alkhalifah, T., & Tsvankin, I. (1995). Velocity analysis for transversely isotropic media. Geophysics, 60(5), 1550–1566.

    Article  Google Scholar 

  • Allaire, G. (1992). Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 23(6), 1482–1518.

    Article  Google Scholar 

  • Allaire, G., & Conca, C. (1998). Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM Journal on Mathematical Analysis, 29(2), 343–379.

    Article  Google Scholar 

  • BC Oil and Gas Commission. (2012). Investigation of observed seismicity in the Horn River Basin. BC Oil and Gas Commission, Victoria, British Columbia, Canada. Retrieved from website:

  • Backus, G. E. (1962). Long-wave elastic anisotropy produced by horizontal layering. Journal of Geophysical Research, 67(11), 4427–4440.

    Article  Google Scholar 

  • Bensoussan, A., Lions, J. L., & Papanicolaou, G. (1978). Asymptotic Analysis of Periodic Structures. North Holland.

  • Bertram, M. B., & Margrave, G. F. (2011). Recovery of low frequency data from 10Hz geophones. In Expanded Abstracts, Recovery-CSPG CSEG CWLS Convention, Calgary.

  • Beucler, É., & Montagner, J. P. (2006). Computation of large anisotropic seismic heterogeneities (CLASH). Geophysical Journal International, 165(2), 447–468.

    Article  Google Scholar 

  • Bodin, T., Capdeville, Y., Romanowicz, B., & Montagner, J. P. (2015). Interpreting radial anisotropy in global and regional tomographic models. In The Earth’s Heterogeneous Mantle (pp. 105–144). Switzerland: Springer International Publishing.

    Google Scholar 

  • Briane, M. (1994). Homogenization of a non-periodic material. Journal de mathématiques pures et appliquées, 73(1), 47–66.

    Google Scholar 

  • Brokešová, J., & Málek, J. (2010). New portable sensor system for rotational seismic motion measurements. Review of Scientific Instruments, 81(8), 084501.

    Article  Google Scholar 

  • Capdeville, Y., Guillot, L., & Marigo, J. J. (2010a). 1-D non-periodic homogenization for the seismic wave equation. Geophysical Journal International, 181(2), 897–910.

    Google Scholar 

  • Capdeville, Y., Guillot, L., & Marigo, J. J. (2010b). 2-D non-periodic homogenization to upscale elastic media for P-SV waves. Geophysical Journal International, 182(2), 903–922.

    Article  Google Scholar 

  • Capdeville, Y., & Marigo, J. J. (2007). Second order homogenization of the elastic wave equation for non-periodic layered media. Geophysical Journal International, 170(2), 823–838.

    Article  Google Scholar 

  • Capdeville, Y., & Marigo, J. J. (2008). Shallow layer correction for spectral element like methods. Geophysical Journal International, 172(3), 1135–1150.

    Article  Google Scholar 

  • Capdeville, Y., & Marigo, J. J. (2013). A non-periodic two scale asymptotic method to take account of rough topographies for 2-D elastic wave propagation. Geophysical Journal International, 192(1), 163–189.

    Article  Google Scholar 

  • Capdeville, Y., Stutzmann, E., Wang, N., & Montagner, J. P. (2013). Residual homogenization for seismic forward and inverse problems in layered media. Geophysical Journal International, 194(1), 470–487.

    Article  Google Scholar 

  • Capdeville, Y., Zhao, M., & Cupillard, P. (2015). Fast Fourier homogenization for elastic wave propagation in complex media. Wave Motion, 54, 170–186.

    Article  Google Scholar 

  • Casarotti, E., Stupazzini, M., Lee, S. J., Komatitsch, D., Piersanti, A., & Tromp, J. (2008). CUBIT and seismic wave propagation based upon the spectral-element method: An advanced unstructured mesher for complex 3D geological media. In Proceedings of the 16th International Meshing Roundtable (pp. 579–597). Berlin, Heidelberg: Springer.

  • Chaljub, E., Komatitsch, D., Vilotte, J. P., Capdeville, Y., Valette, B., & Festa, G. (2007). Spectral-element analysis in seismology. Advances in Geophysics, 48, 365–419.

    Article  Google Scholar 

  • Chung, P. W., Tamma, K. K., & Namburu, R. R. (2001). Asymptotic expansion homogenization for heterogeneous media: Computational issues and applications. Composites Part A: Applied Science and Manufacturing, 32(9), 1291–1301.

    Article  Google Scholar 

  • Close, D., Cho, D., Horn, F., & Edmundson, H. (2009). The sound of sonic: A historical perspective and introduction to acoustic logging. CSEG Recorder, 34(05), 34–43.

    Google Scholar 

  • Dahlen, F., & Tromp, J. (1998). Theoretical global seismology. Princeton: Princeton University Press.

  • Dumontet, H. (1986). Study of a boundary layer problem in elastic composite materials. RAIRO-Modélisation mathématique et analyse numérique , 20(2), 265–286.

    Google Scholar 

  • Fichtner, A., & Igel, H. (2008). Efficient numerical surface wave propagation through the optimization of discrete crustal models—A technique based on non-linear dispersion curve matching (DCM). Geophysical Journal International, 173(2), 519–533.

    Article  Google Scholar 

  • Fichtner, A., Kennett, B. L., & Trampert, J. (2013a). Separating intrinsic and apparent anisotropy. Physics of the Earth and Planetary Interiors, 219, 11–20.

    Article  Google Scholar 

  • Fichtner, A., Saygin, E., Taymaz, T., Cupillard, P., Capdeville, Y., & Trampert, J. (2013b). The deep structure of the North Anatolian Fault Zone. Earth and Planetary Science Letters, 373, 109–117.

    Article  Google Scholar 

  • Fichtner, A., Trampert, J., Cupillard, P., Saygin, E., Taymaz, T., Capdeville, Y., et al. (2013c). Multiscale full waveform inversion. Geophysical Journal International, 194(1), 534–556.

    Article  Google Scholar 

  • Gold, N., Shapiro, S. A., Bojinski, S., & Müller, T. M. (2000). An approach to upscaling for seismic waves in statistically isotropic heterogeneous elastic media. Geophysics, 65(6), 1837–1850.

    Article  Google Scholar 

  • Goodway, B., Monk, D., Perez, M., Purdue, G., Anderson, P., Iverson, A., et al. (2012). Combined microseismic and 4D to calibrate and confirm surface 3D azimuthal AVO/LMR predictions of completions performance and well production in the Horn River gas shales of NEBC. The Leading Edge, 31(12), 1502–1511.

    Article  Google Scholar 

  • Grechka, V. (2003). Effective media: A forward modeling view. Geophysics, 68(6), 2055–2062.

    Article  Google Scholar 

  • Guillot, L., Capdeville, Y., & Marigo, J. J. (2010). 2-D non-periodic homogenization of the elastic wave equation: SH case. Geophysical Journal International, 182(3), 1438–1454.

    Article  Google Scholar 

  • Haboussi, M., Dumontet, H., & Billoet, J. (2001a). On the modelling of interfacial transition behaviour in composite materials. Computational Materials Science, 20(2), 251–266.

    Article  Google Scholar 

  • Haboussi, M., Dumontet, H., & Billoet, J. (2001b). Proposal of refined interface models and their application for free-edge effect. Composite Interfaces, 8(1), 93–107.

    Article  Google Scholar 

  • Haldorsen, J. B., Johnson, D. L., Plona, T., Sinha, B., Valero, H. P., & Winkler, K. (2006). Borehole acoustic waves. Oilfield Review, 18(1), 34–43.

    Google Scholar 

  • Hashin, Z., & Shtrikman, S. (1963). A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids, 11(2), 127–140.

    Article  Google Scholar 

  • Helbig, K. (1984). Transverse isotropy in exploration seismics. Geophysical Journal International, 76(1), 79–88.

    Article  Google Scholar 

  • Helbig, K. (1994). Foundations of anisotropy for exploration seismics. Oxford: Pergamon.

    Google Scholar 

  • Jones, G. A., Kendall, J. M., Bastow, I. D., & Raymer, D. G. (2014). Locating microseismic events using borehole data. Geophysical Prospecting, 62(1), 34–49.

    Article  Google Scholar 

  • Kelvin, L., & Thomson, W. (1878). Mathematical theory of elasticity (in the article, Elasticity). Encyclopedia Britannica, 7, 796–825.

    Google Scholar 

  • Kennett, B. (2009). Seismic wave propagation in stratified media. Canberra: ANU Press.

    Google Scholar 

  • Komatitsch, D., & Tromp, J. (1999). Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International, 139(3), 806–822.

    Article  Google Scholar 

  • Komatitsch, D., & Vilotte, J. P. (1998). The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures. Bulletin of the Seismological Society of America, 88(2), 368–392.

    Google Scholar 

  • Maxwell, S.C. (2005). A brief guide to passive seismic monitoring. In Proceedings of CSEG National Convention (pp. 177–178).

  • Maxwell, S. C., Urbancic, T. I., Le Calvez, J. H., Tanner, K. V., Grant, W. D., et al. (2004). Passive seismic imaging of hydraulic fracture proppant placement. In 2004 SEG Annual Meeting, Society of Exploration Geophysicists.

  • Peter, D., Komatitsch, D., Luo, Y., Martin, R., Le Goff, N., Casarotti, E., et al. (2011). Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophysical Journal International, 186(2), 721–739.

    Article  Google Scholar 

  • Sayers, C. (1998). Long-wave seismic anisotropy of heterogeneous reservoirs. Geophysical Journal International, 132(3), 667–673.

    Article  Google Scholar 

  • Snelling, P.E., de Groot, M., Hwang, K., et al. (2013). Characterizing hydraulic fracture behaviour in the Horn River Basin with microseismic data. In 2013 SEG Annual Meeting, Society of Exploration Geophysicists.

  • Stacey, R. (1988). Improved transparent boundary formulations for the elastic-wave equation. Bulletin of the Seismological Society of America, 78(6), 2089–2097.

    Google Scholar 

  • Sánchez-Palencia, E. (1980). Non-homogeneous media and vibration theory. In Lecture Notes in Physics (Vol. 127). Berlin, Heidelberg: Springer.

  • Tarantola, A., Coll, B., Mosegaard, K., & Xu, P. (2000). The null information probability density for anisotropic tensors.

  • Tiwary, D. K., Bayuk, I. O., Vikhorev, A. A., Chesnokov, E. M. (2009). Comparison of seismic upscaling methods: From sonic to seismic. Geophysics, 74(2), WA3–WA14.

  • van der Baan, M., Eaton, D., Dusseault, M., et al. (2013). Microseismic monitoring developments in hydraulic fracture stimulation. In ISRM International Conference for Effective and Sustainable Hydraulic Fracturing. International Society for Rock Mechanics.

  • Wang, Z. (2002). Seismic anisotropy in sedimentary rocks, part 2: Laboratory data. Geophysics, 67(5), 1423–1440.

    Article  Google Scholar 

  • Wang, N., Montagner, J. P., Fichtner, A., & Capdeville, Y. (2013). Intrinsic versus extrinsic seismic anisotropy: The radial anisotropy in reference Earth models. Geophysical Research Letters, 40(16), 4284–4288.

    Article  Google Scholar 

  • Wong, J. (2009). Microseismic hypocenter location using nonlinear optimization. CREWES Research Report (Vol. 21).

  • Yang, D., Malcolm, A., Fehler, M., & Huang, L. (2014b). Time-lapse walkaway vertical seismic profile monitoring for CO2 injection at the SACROC enhanced oil recovery field: A case study. Geophysics, 79(2), B51–B61.

    Article  Google Scholar 

  • Yang, D., Malcolm, A., & Fehler, M. (2014a). Using image warping for time-lapse image domain wavefield tomography. Geophysics, 79(3), WA141–WA151.

  • Yilmaz, Ö. (2001). Seismic data analysis (Vol. 1). Tulsa: Society of exploration geophysicists.

    Book  Google Scholar 

  • Zhu, L., & Rivera, L. A. (2002). A note on the dynamic and static displacements from a point source in multilayered media. Geophysical Journal International, 148(3), 619–627.

    Article  Google Scholar 

Download references


We would like to thank Yann Capdeville for his 1D homogenization program available online at, which is used in this paper to obtain the effective media. We are grateful to Yann Capdeville for pointing out the derivation mistakes that we made in an earlier draft, and we thank him and another anonymous reviewer for their generous advice that helped us further reshape this paper. Our thanks also goes to the developers of SPECFEM2D package for their continued community support. We would like to thank Dan Walker from BC Oil and Gas Commission for providing the bore-well sonic log data in the Horn River Basin region. This research was supported by the Discovery Grant No. 487237 of the Natural Sciences and Engineering Research Council of Canada (NSERC). Computations for this study were performed on hardware purchased through the combined funding of Canada Foundation for Innovation (CFI), Ontario Research Fund (ORF), and University of Toronto Startup Fund, and partly hosted by the SciNet HPC Consortium. SciNet is funded by CFI under the auspices of Compute Canada, the Government of Ontario, ORF-Research Excellence, and the University of Toronto. R.S. is supported by the SUMIT project of Ontario Research Fund.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Chuangxin Lin.


Appendix A: Kelvin notation for second-order and fourth-order tensors

Helbig (1994) demonstrated one of the canonical ways of transforming a second-order symmetric 3D tensor (e.g., stress and strain) to a 6-D vector, and a fourth-order symmetric 3D tensor (e.g., elastic tensor) to a \(6\times 6\) matrix is the Kelvin notation (Kelvin and Thomson 1878; Tarantola et al. 2000), where indices (ij) (\(i=1,2,3\), \(j=1,2,3\)) are mapped into an index \(\alpha\), (\(\alpha =1,\ldots ,6\)), as

$$\begin{aligned} \begin{array}{cccccccc} ij &{} = &{} 11 &{} 22 &{} 33 &{} 23,32 &{} 13,31 &{}12,21. \\ \Downarrow &{} {} &{} \Downarrow &{} \Downarrow &{} \Downarrow &{} \Downarrow &{} \Downarrow &{} \Downarrow \\ \alpha &{} = &{} 1 &{} 2 &{} 3 &{} 4 &{} 5 &{} 6. \\ \end{array} \end{aligned}$$

For example, the stress and strain tensors are transformed to vectors

$$\begin{aligned} \begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix}=\begin{bmatrix} \sigma _{11} \\ \sigma _{22} \\ \sigma _{33} \\ \sqrt{2}\,\sigma _{23} \\ \sqrt{2}\,\sigma _{13} \\ \sqrt{2}\,\sigma _{12} \end{bmatrix} \quad \text { and }\quad \begin{bmatrix} E_1 \\ E_2 \\ E_3 \\ E_4 \\ E_5 \\ E_6 \end{bmatrix}=\begin{bmatrix} e_{11} \\ e_{22} \\ e_{33} \\ \sqrt{2}\,e_{23} \\ \sqrt{2}\,e_{13} \\ \sqrt{2}\,e_{12} \end{bmatrix}, \end{aligned}$$

while the symmetric fourth-order tensor B becomes (\(B_{ijkl}=B_{jikl}=B_{ijlk}\))

$$\begin{aligned} B_{\alpha \beta }={\left\{ \begin{array}{ll} B_{iikk} \quad{ \text {when} }\quad i=j,\, k=l\\ \sqrt{2} B_{iikl} \quad {\text {when }}\quad i=j,\,k\ne l\\ \sqrt{2} B_{ijkk} \quad {\text {when }}\quad i\ne j, \,k=l\\ 2B_{ijkl} \quad {\text {when} }\quad i\ne j,\, k\ne l. \end{array}\right. } \end{aligned}$$

Note here, we do not use different symbols for the tensor \(B_{ijkl}\) and the matrix representation \(B_{\alpha \beta }\), as the distinction should be clear based on the number of indices. Under this notation, the norm of the fourth-order tensor is preserved in the matrix representation:

$$\begin{aligned} B_{ijkl}B_{ijkl}=B_{\alpha \beta }B_{\alpha \beta }, \end{aligned}$$

and double-dot product of two fourth-order tensors B and D can be computed by matrix multiplications, and converted back based on Eq. (42):

$$\begin{aligned} E_{ijmn}=B_{ijkl}D_{klmn} \quad \Leftarrow \quad E_{\alpha \gamma }=B_{\alpha \beta }D_{\beta \gamma }. \end{aligned}$$

For example, elastic tensor can be represented by the matrix

$$\begin{aligned} \mathbf {C}=\begin{bmatrix} C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16} \\ C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26} \\ C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36} \\ C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46} \\ C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56} \\ C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66} \end{bmatrix}= \begin{bmatrix} C_{1111}&C_{1122}&C_{1133}&\sqrt{2}\, C_{1123}&\sqrt{2}\,C_{1113}&\sqrt{2}\,C_{1112} \\ C_{2211}&C_{2222}&C_{2233}&\sqrt{2}\,C_{2223}&\sqrt{2}\,C_{2213}&\sqrt{2}\,C_{2212} \\ C_{3311}&C_{3322}&C_{3333}&\sqrt{2}\,C_{3323}&\sqrt{2}\,C_{3313}&\sqrt{2}\,C_{3312} \\ \sqrt{2}\,C_{2311}&\sqrt{2}\, C_{2322}&\sqrt{2}\,C_{2333}&2C_{2323}&2C_{2313}&2C_{2312} \\ \sqrt{2}\,C_{1311}&\sqrt{2}\,C_{1322}&\sqrt{2}\,C_{1333}&2C_{1323}&2C_{1313}&2C_{1312} \\ \sqrt{2}\,C_{1211}&\sqrt{2}\,C_{1222}&\sqrt{2}\,C_{1233}&2C_{1223}&2C_{1213}&2C_{1212}. \end{bmatrix}. \end{aligned}$$

Since the elastic tensor is symmetric \(C_{ijkl}=C_{klij}\), the corresponding \(\mathbf {C}\) matrix is also symmetric. The Kelvin matrix notation of the elastic tensor for a TI medium becomes

$$\begin{aligned} C_{ijkl} \Rightarrow {C}_{\alpha \beta } = \left[ \begin{array}{cccccc} A &{} B &{} F &{} 0 &{} 0 &{} 0 \\ B &{} A &{} F &{} 0 &{} 0 &{} 0 \\ F &{} F &{} C &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2L &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 2L &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2N \\ \end{array}\right] , \end{aligned}$$

where A, C, L, N, and F are the Love parameters discussed in Sect. 2.2, and \(B=A-2N\).

Appendix B: Detailed expressions for H and G matrices

With the expressions for \(\partial _m{\chi _n^{kl}}\) computed in Sect. 2.2, if we define auxiliary variables

$$\begin{aligned}&p=\left\langle \frac{C_{13}}{C_{33}} \right\rangle , \quad q=\left\langle C^{-1}_{33} \right\rangle ,\quad r=\left\langle \frac{C_{23}}{C_{33}} \right\rangle , \quad s=\left\langle C^{-1}_{44} \right\rangle ,\quad t=\left\langle C^{-1}_{55} \right\rangle \nonumber \\&\eta =\frac{C_{13}}{C_{33}},\quad \theta =C^{-1}_{33},\quad \xi =\frac{C_{23}}{C_{33}},\quad \phi =C^{-1}_{44},\quad \psi =C^{-1}_{55},\quad \nu =C_{66} \nonumber \\&\alpha =C_{11}-\frac{C_{13}^2}{C_{33}},\quad \beta =C_{12}-\frac{C_{13}C_{23}}{C_{33}},\quad \gamma =C_{22}-\frac{C_{23}^2}{C_{33}} \end{aligned}$$

then the Kelvin notation of the fourth-order tensor \(\mathbf {G}\) in Sect. 2.2 becomes

$$\begin{aligned} {\mathbf G} = \begin{bmatrix} 1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ \frac{p}{q} \theta - \eta&\frac{r}{q} \theta - \xi&\frac{\theta }{q}&0&0&0 \\ 0&0&0&\frac{\phi }{s}&0&0 \\ 0&0&0&0&\frac{\psi }{t}&0 \\ 0&0&0&0&0&1 \end{bmatrix}. \end{aligned}$$

When the velocity model does not have any fine-scale variations, clearly, \(G_{31}\) and \(G_{32}\) vanish and \({\mathbf G}\) becomes an identity matrix.

The product of \({\mathbf C}\) matrix (Eq. 46) and \({\mathbf G}\) matrix (Eq. 34) gives

$$\begin{aligned} {\mathbf H} = \begin{bmatrix} \alpha +\frac{p}{q}\eta&\beta +\frac{r}{q}\eta&\frac{\eta }{q}&0&0&0 \\ \beta +\frac{p}{q}\xi&\gamma +\frac{r}{q}\xi&\frac{\xi }{q}&0&0&0 \\ \frac{p}{q}&\frac{r}{q}&\frac{1}{q}&0&0&0 \\ 0&0&0&\frac{1}{s}&0&0 \\ 0&0&0&0&\frac{1}{t}&0 \\ 0&0&0&0&0&\nu \\ \end{bmatrix}, \end{aligned}$$

and it can be shown that in the absence of fine-scale structures, \({\mathbf H}\) matrix becomes the \({\mathbf C}\) matrix.

With the expressions for \({\mathbf H}\) and \({\mathbf G}\), we can compute the effective moduli based on the matrix representation of Eq. (8):

$$\begin{aligned} {\mathbf C}^* =\mathscr {F}\big( {\mathbf H}\big) : \mathscr {F}\big( {\mathbf G}\big) ^{-1}. \end{aligned}$$

Clearly, the filter operator \(\mathscr {F}\left( {\cdot }\right)\) only needs to be applied to \(\alpha\), \(\beta\), \(\gamma\), \(\eta\), \(\theta\), \(\xi\), \(\phi\), \(\psi\), and \(\nu\) (i.e., the Greek letters), as p, q, r, s, and t already have fine-scale structures averaged out. To reduce symbol clutter, we write the filtering of the Greek letters \(\mathscr {F}\left( {\cdot }\right)\) as \((\cdot )^*\), and the effective elastic tensor becomes

$$\begin{aligned} {\mathbf C}^* = \begin{bmatrix} \alpha ^*+\frac{(\eta ^*)^2}{\theta ^*}&\beta ^*+\frac{\eta ^*\xi ^*}{\theta ^*}&\frac{\eta ^*}{\theta ^*}&0&0&0 \\ \beta ^*+\frac{\eta ^*\xi ^*}{\theta ^*}&\gamma ^*+\frac{(\xi ^*)^2}{\theta ^*}&\frac{\xi ^*}{\theta ^*}&0&0&0 \\ \frac{\eta ^*}{\theta ^*}&\frac{\xi ^*}{\theta ^*}&\frac{1}{\theta ^*}&0&0&0 \\ 0&0&0&\frac{1}{\phi ^*}&0&0 \\ 0&0&0&0&\frac{1}{\psi ^*}&0 \\ 0&0&0&0&0&\gamma ^* \end{bmatrix}. \end{aligned}$$

Note interestingly that all the cell averaged variables p, q, r, s, and t naturally cancel out from this matrix product, which is a result of the special structure of \({\mathbf G}\) as in Eq. (34).

If we plug in the expressions for TI media, Eq. (46), to the auxiliary variables (47), we have

$$\begin{aligned}&\alpha =\gamma =A-F^2/C,\quad \beta =B-F^2/C, \nonumber \\&\eta =\xi =F/C,\quad \theta =1/C, \quad \phi =\psi =1/(2L), \quad \nu =2N, \end{aligned}$$

then we have the effective elastic tensor

$$\begin{aligned} \mathbf {C}^*= \begin{bmatrix} A^*&A^*-2N^*&F^*&0&0&0 \\ A^*-2N^*&A^*&F^*&0&0&0 \\ F^*&F^*&C^*&0&0&0 \\ 0&0&0&2L^*&0&0 \\ 0&0&0&0&2L^*&0 \\ 0&0&0&0&0&2N^* \\ \end{bmatrix} \end{aligned}$$


$$\begin{aligned}&A^* = \mathscr {F}\left( {A-F^2/C}\right) + \left[ \mathscr {F}\left( {1/C}\right) \right] ^{-1} \left[ \mathscr {F}\left( {F/C}\right) \right] ^{2}\nonumber \\&F^*=\left[ \mathscr {F}\left( {1/C}\right) \right] ^{-1}\mathscr {F}\left( {F/C}\right) \nonumber \\&C^*=\left[ \mathscr {F}\left( {1/C}\right) \right] ^{-1}\nonumber \\&L^*=\left[ \mathscr {F}\left( {1/L}\right) \right] ^{-1}\nonumber \\&N^*=\mathscr {F}\left( {N}\right) . \end{aligned}$$

Therefore, the homogenization of a TI model also produces a TI media with the effective moduli given by the expressions (54).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, C., Saleh, R., Milkereit, B. et al. Effective Media for Transversely Isotropic Models Based on Homogenization Theory: With Applications to Borehole Sonic Logs. Pure Appl. Geophys. 174, 2631–2647 (2017).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Sonic logging
  • homogenization
  • upscaling technique
  • effective media
  • 1D transverse isotropy
  • seismic wave propagation