Amplitude and AVO responses of a single thin bed

GEOPHYSICS, VOL. 68, NO. 4 (JULY-AUGUST 2003); P. 1161 1168, 14 FIGS., 1 TABLE. 10.1190/1.1598108 Amplitude and AVO responses of a single thin bed Yinbin Liu and Douglas R. Schmitt ABSTRACT The seismic reflection characterizations of a thin layer are important for reservoir geophysics. However, discussions on the reflection for a thin layer are usually restricted to precritical angle incidence. In this work, an exact analytical solution is derived to model the reflection amplitude and amplitude variation with offset (AVO) responses of a single thin bed for arbitrary incident angles. The results show that the influence of an ultra-thin bed is great for opposite-polarity reflections and is small for identical-polarity reflections. Opposite-polarity precritical reflection amplitudes first decrease in magnitude with the wavelength/thickness ratio to a local minimum, then increase to a maximum, and finally decrease gradually to zero as the layer vanishes. Opposite-polarity postcritical reflections monotonically decrease from near unity to zero, proportional to the thickness of the layer. Identicalpolarity precritical reflection amplitudes first increase in magnitude with the wavelength/thickness ratio to a local maximum, then decrease to a minimum, and finally increase to the amplitude of a single bottom reflection when the layer vanishes. Identical-polarity postcritical reflections have magnitudes near unity. The AVO responses for both opposite and identical-polarity acoustic thin beds gradually increase with angle. The influence of the Poisson s ratio of the thin bed is small for either small incidence angles or thicknesses less than 7% of the seismic wavelength, but is large for high incidence angles or thicknesses greater than 13% of the wavelength. A decrease of Poisson s ratio causes a pronounced AVO response that reaches its maximum at the quarter-wavelength tuning thickness. INTRODUCTION An ultra-thin gas-sand layer has often a detectable seismic reflection response (e.g., Schmitt, 1999). Generally, the reflections from a thin layer are concerned with seismic resolution, detection, and amplitude variation with offset (AVO). Resolution and detectability for a thin layer have been studied by Widess (1973), Neidell and Poggiagliolmi (1977), Koefoed and de Voogd (1980), Kalweit and Wood (1982), de Voogd and Rooijen (1983), Gochioco (1991), Chung and Lawton (1995, 1996), Liu and Schmitt (2001), and others. In exploration geophysics, the generally accepted threshold for vertical resolution of a layer is a quarter of the dominant wavelength (Yilmaz, 1987). In this paper, the layer is called a thin layer when 1 <λ/d 4, and an ultra-thin layer when λ/d > 4, where λ is the dominant wavelength within the layer and d is the layer thickness. Widess s classic paper (1973) studied the normal pulse reflections from the top and bottom of a thin layer with equal amplitude and opposite polarity, and noted that reflections from very thin beds are not always small. Neidell and Poggiagliolmi (1977) and Kallweit and Wood (1982) studied the reflections of a thin layer and a wedge model, and showed that the thickness information is encoded in the amplitude and shape of the reflected wavelet when the thickness is less than the tuning thickness of layer. De Voogd and Koefoed (1980) and Gochioco (1991) demonstrated that the seismic response of coal seams as thin as λ/20 to λ/50 can give rise to a distinct reflection signal. Chung and Lawton (1995, 1996) studied the reflection characterizations for different sedimentary formations and showed that the amplitude dependence on the thickness is nonlinear. Almoghrabi and Lange (1986), Lange and Almoghrabi (1988), and Juhlin and Young (1993) studied the AVO response of a thin layer by considering multiples, and showed the AVO response of a thin layer may differ significantly from the AVO response of a simple interface. The simplified methods used by the above authors gave a clear description of the interference between the top and bottom interfaces, especially for seismic resolution. However, these methods only apply to either normal incidence reflections (Widess, 1973; Neidell and Poggiagliolmi, 1977; Koefoed and de Voogd, 1980; Kalweit and Wood, 1982; de Voogd and Rooijen, 1983; Gochioco, 1991; Chung and Lawton, 1995, 1996) Manuscript received by the Editor March 25, 2002; revised manuscript received February 4, 2003. Formerly University of Alberta, Institute for Geophysical Research, Department of Physics, Edmonton, Alberta T6G 2J1, Canada; presently 517-4739 Dalton Dr. NW, Calgary, Alberta T3A 2L5, Canada. E-mail: yinbin@telus.net. University of Alberta, Institute for Geophysical Research, Department of Physics, Edmonton, Alberta T6G 2J1, Canada. E-mail: doug@ phys.ualberta.ca. c 2003 Society of Exploration Geophysicists. All rights reserved. 1161

1162 Liu and Schmitt or precritical incidence angles (Almoghrabi and Lange, 1986; Lange and Almoghrabi, 1988; Juhlin and Young, 1993). Moreover, these methods are all based on ray theory and should be restricted to the case in which the wavelength is small relative to the dimension of the structure. Thin bed reflections can be accurately described only by wave theory. In this paper, the amplitude and AVO responses of thin or ultra-thin beds are quantitatively studied by an exact analytical solution. Model THEORY AND ALGORITHM Real reservoir structures and reflection seismic data are very complex. Despite this, simplified geometries (e.g., a layered model) with simplified media approximation (e.g., acoustic media) may help us to extract physical essences from complex background. Our models consist of an acoustic thin layer (models I and II) embedded between two half-spaces (Figure 1). An elastic model (model III) is also discussed in order to study the influence of Poisson s ratio. The parameters and properties of these three models are listed in Table 1. Parameters θ c1 = sin 1 (α 1 /α 2 ) and θ c2 = sin 1 (α 1 /α 3 ) are the P-wave critical angles on the top and bottom interfaces relative to the incident wave, respectively. Model I uses Widess s (1973) calculated parameters and denotes a thin high-velocity layer. Model II denotes a thin transition layer, whereas model III represents the Wabasca gas formation in the Western Canadian Sedimentary Basin (Schmitt, 1999). The densities for models I and II are uniform. Models I and III produce opposite-polarity P-wave reflections, whereas model II produces identical-polarity P-wave reflections. The opposite- (identical-) polarity reflections mean that wavelet reflections from the top and bottom of a thin layer are opposite- (identical-) polarity (Kallweit and Wood, 1982). Generally, a high- or low-impedance bed (ρ 2 α 2 >ρ 1 α 1 and ρ 3 α 3,orρ 2 α 2 <ρ 1 α 1 and ρ 3 α 3 ) produces opposite-polarity reflections, whereas increased or decreased impedance bed FIG. 1. Thin bed model, reflection rays, and the corresponding time delay δt = (2d/α 2 ) cos θ 2. (ρ 1 α 1 >ρ 2 α 2 >ρ 3 α 3 or ρ 1 α 1 <ρ 2 α 2 <ρ 3 α 3 ) produces identicalpolarity reflections. Method A plane monochromatic wave with unit amplitude illuminates the thin layer (Figure 1). For acoustic media, the reflection coefficient can be written as (see Appendix A) R(ω) = i( ) Z 1 Z 3 Z2 2 sin(kz2 d) + (Z 1 Z 2 Z 2 Z 3 ) cos(k z2 d) i ( ) Z 1 Z 3 + Z2 2 sin(kz2 d) + (Z 1 Z 2 + Z 2 Z 3 ) cos(k z2 d), (1) where Z i = (ρ i α i )/cos θ i, and k z2 = (ω/α 2 ) cos θ 2. Parameters ρ i, α i, and θ i (i = 1, 2, 3) are the densities, velocities, and incident or refracted angles, respectively; d is the thickness of the thin layer. For the postcritical angle incidence (θ 1 >θ c1 and θ c2 ), k z2 becomes imaginary (Snell s law), and the waves propagated within the layer are evanescent waves (also called inhomogeneous waves). When k z2 d = (2πd/λ 2 ) cos θ 2 = mπ, equation (1) becomes R(ω) = Z 3 Z 1. (2) Z 1 + Z 3 The reflection coefficient in this case looks as if the layer were absent. The reflection coefficient will be equal to zero if the two half-spaces have the same impedances (Z 1 = Z 3 ). For an ultra-thin layer (k z2 d 1), we take the first order of approximation for R(ω) and have cos k z2 d 1, sin k z2 d = (2πd/λ 2 ) cos θ 2. For opposite-polarity reflection, Z 1 = Z 3, and the R(ω)in equation (1) can be written as R(ω) = πi( Z2 2 Z ) 1 2 d/λ2 cos θ 2. (3) Z 1 Z 2 Therefore, the reflection coefficient of an ultra-thin layer for opposite-polarity reflection is approximately proportional to the layer thickness. For identical-polarity reflections, or in any case where Z 1 Z 3 Z2 2 (2πd/λ 2) cos θ 2 Z 2 Z 1 Z 3, the reflection coefficient R(ω) of an ultra-thin layer (k z2 d 1) in equation (1) is almost identical to equation (2). The influence of an ultra-thin layer on the reflection coefficient can then be neglected. Figure 2 shows an example for the angular reflection coefficient spectrum for model I. The thickness of the layer was varied from d = λ 2 to d = λ 2 /100. The curves are computed in steps δθ 1 = 0.5. It can be seen that the reflection coefficients increase with increasing angle of the incidence for λ 2 /d > 2. The normal reflection coefficients are zero at λ 2 /d = 1 and 2 because the layer thickness is an integral of half-wavelength (λ 2 /2) (Brekhovskikh, 1980). There is a null at incident angle θ 1 = 25.7 for d = λ 2. This is because θ 1 = 25.7 corresponds to a refracted angle θ 2 = 60, which has an apparent thickness d = λ 2 /2[δt = (2d/α 2 ) cos θ 2 ] and so also satisfies the condition Table 1. Model parameters. α 1 (m/s) ρ 1 (g/cm 3 ) α 2 (m/s) ρ 2 (g/cm 3 ) α 3 (m/s) ρ 3 (g/cm 3 ) θ c1 θ c2 Model I 3050 2.7 6100 2.7 3050 2.7 30 Model II 3050 2.7 4575 2.7 6100 2.7 41.8 30 Model III 2200 2.3 1500 2.2 2500 2.35 61.6

Thin Bed AVO 1163 of an integral of half-wavelength. Note that the zero appears only when the layer thickness exceeds λ 2 /2. For elastic case, the analytical expressions for reflection coefficients can only be given by matrix form because of the coupling effects of P-wave and SV-wave. For multilayered mediums, the solution is described by the propagator matrix method. Interested readers may find the discussions about propagator matrices in, for example, Rokhlin et al. (1999) and Ursin and Stovas (2002). For exploration geophysics, the source waveform is not monochromatic. The reflection impulse response can be written as φ(x, z, t) = G(ω)R(ω)e i(k x x kz z ωt) dω, (4) where G(ω) is the wavelet spectrum, and R(ω) is the monochromatic reflection coefficient of the composite layer. In the following, a 50-Hz Ricker wavelet is used for all simulations. To study the influence of the SV-wave on the reflection amplitude, we compare the reflection field from both acoustic (i.e., no shear wave) and elastic thin layer reflections. Figure 3 shows the calculated reflection waveforms from model III when the top is an acoustic half-space and the thin layer and bottom half-space are elastic with Poisson s ratio of σ = 0.25 (solid) and when the thin layer and two half-spaces are all acoustic (dashed) at θ 1 = 20. It can be seen that the amplitude responses for the elastic case have a slightly smaller amplitude than those for the acoustic case. This is because in the elastic case, part of the energy converts into shear waves or Lamb waves and radiates into the bottom half-space. In the following analysis, we first deliberately ignore the influence of SV-waves, but in the final section of this paper the elastic case will be discussed. EFFECTS OF BED THICKNESS AND INCIDENT ANGLE ON REFLECTION AMPLITUDE Opposite-polarity reflection Figure 4 shows the composite reflection waveforms of model I calculated by the analytical solution for a 50-Hz Ricker wavelet at normal and 20 incidences for d = λ 2 to d = λ 2 /8 (Figure 4a) and d = λ 2 /10 to d = λ 2 /100 (Figure 4b). The waveforms for normal incidence are similar to those of Widess (1973) computed by time delay approximation. It can be seen that two reflection wavelets from the top and bottom of the thin bed overlap, and that the time delay δt for R 1 and R 2 can be approximately calculated by ray theory as δt = (2d/α 2 ) cos θ 2 (see Figure 1), where θ 2 is the refracted angle and α 2 is the P-wave velocity of the layer. Obviously, as the bed thins or the incident angle expands, the delay time δt decreases and worsens the overlap. Oblique incidence is equivalent to thinner layers, in terms of delay time. The influence of an ultra-thin bed on reflection amplitudes is relatively large for oppositepolarity reflections. For example, say the reflection coefficients are about 0.2 at θ 1 = 0 and 0.22 at θ 1 = 20 for λ 2 /d = 20. A bed with λ 2 /d = 40 still reflects 11% of the amplitude of the incident wave. The critical angle at the top interface of model I is θ c1 = 30. Figure 5 is the reflection of model I for d = λ 2 to d = λ 2 /100 at two postcritical incidence angles of θ 1 = 40 (solid lines) and θ 1 = 60 (dashed lines). The reflection amplitudes decrease with decreasing thickness. The Zoeppritz equations (Aki and Richards, 1978) predict the reflection and transmission of a single interface, and show that the reflection for postcritical angles has a magnitude near unity. Therefore, the Zoeppritz equations are not suitable to study the amplitude and AVO responses for thin-layer problem. The maximum magnitude of each reflection waveform at incident angle θ 1 and layer thickness d can be calculated by taking the absolute maximum value of the waveform. Figure 6 shows the maximum reflection magnitude as a function of the wavelength/thickness ratio (λ 2 /d) at several incident angles. The curves are computed in steps δd of 0.005λ 2. The precritical amplitude response for opposite-polarity reflections first decreases to a local minimum (at λ 2 /d 2 for normal incidence) indicating destructive interference, then increases to a maximum (at λ 2 /d 4 for normal incidence) indicating FIG. 2. Reflection coefficient spectrum (single frequency) for different wavelength/thickness (λ 2 /d) in model I. FIG. 3. Comparison between elastic (Poisson s ratio σ = 0.25) and acoustic thin layer reflections in model III for the layer thickness changes from d = λ 2 to d = λ 2 /100. Solid and dashed lines denote the elastic and acoustic layers, respectively.

1164 Liu and Schmitt constructive interference at the tuning thickness, and finally gradually decreases to zero as the layer vanishes. The greater the angle of the incidence or the thinner the layer thickness, the smaller the reflection amplitudes become. The minima and maxima for high incidence angles appear at smaller thickness than those for low incidence angles because layers appear to be thinner at oblique angles. The maximum absolute amplitudes for the postcritical incidence angles (θ 1 30 ) monotonically decrease from near unity (total reflection) to zero. This is because the wave within the thin layer is evanescent, resulting in amplitude attenuation. Figure 7 shows the AVO response of model I where the thickness of the layer was varied from d = λ 2 to d = λ 2 /100. The maximum absolute amplitudes increase with increasing angle of incidence or offset. The amplitude changes are large for d = λ 2 and d = λ 2 /2(θ 1 <θ c1 ) and for d = λ 2 /4tod=λ 2 /10, but are slight for λ 2 /d > 20 when θ 1 is less than about 40. The influence of the critical angle of the top interface on the reflection coefficients and AVO is obvious for λ 2 /d < 2 (total reflection), but cannot be observed for λ 2 /d > 4. The AVO response smoothly passes the critical angle (θ c1 = 30 ) and only at near grazing incidence (θ 1 90 θ c1 ) does the effective reflectivity approach unity. Identical-polarity reflection Figure 8 shows the calculated reflection waveforms of model II at normal and 20 incidences for various λ 2 /d ratios. Figure 9 shows the maximum absolute amplitudes as a function of the wavelength/thickness ratio for several incident angles. The precritical amplitude responses for identical-polarity reflections first increase to a local maximum (at λ 2 /d 2 for normal incidence) indicating constructive interference, then decrease to a FIG. 5. Opposite-polarity reflections in model I for d = λ 2 to d = λ 2 /100 at two postcritical incidence angles. Solid and dashed lines denote θ = 40 and 60, respectively. FIG. 4. Opposite-polarity reflections in model I. The layer thickness changes from d = λ 2 to d = λ 2 /8 for (a) and from d = λ 2 /10 to d = λ 2 /100 for (b). Solid and dashed lines denote the normal and 20 incidences, respectively. FIG. 6. Maximum absolute amplitudes of opposite-polarity reflections in model I as a function of λ 2 /d for several incident angles.

Thin Bed AVO 1165 minimum (at λ 2 /d 4 for normal incidence) indicating destructive interference, and finally increase to the amplitude of the single bottom-reflection wavelet without the thin layer. The maxima and minima shift to smaller values of λ 2 /d for the larger incident angles. Figure 9 shows that thinner layers result in larger reflection amplitudes in the precritical case because of constructive interference between identical-polarity reflections. The maximum absolute amplitudes for λ 2 /d greater than about 20 are basically invariant to the wavelength/thickness ratio. This means that the amplitude differences with and without thin layers are small. The amplitude responses of identicalpolarity reflections are not sensitive to an ultra-thin layer. A single ultra-thin layer appear to be a single interface. The postcritical reflection amplitude for identical-polarity reflections is near unity, which is similar to the total reflection of a single interface. Figure 10 shows the AVO response for identical-polarity reflections for d = λ 2 to d = λ 2 /100. The reflection amplitudes increase with the increasing angle of incidence for θ 1 <θ c2 and have a near unity value for θ 1 θ c2 (total reflection). The influence of the critical angle on reflection amplitude for identicalpolarity reflections is similar to the case of a single interface. EFFECTS OF POISSON S RATIO ON AVO RESPONSES Now let us turn our attention toward the reflection from an elastic layer and consider the influence of shear waves. We assume an identical Poisson s ratio of 0.25 for both the thin sand and the bottom half-space. The critical angle on the bottom interface is θ c2 = 61.6. Figures 11 and 12 show the amplitude and AVO responses for various λ 2 /d. It can be seen that the amplitude and AVO responses appear as a composite effect of opposite and identical-polarity reflections because of the different impedances (ρ 1 α 1 ρ 3 α 3 ) but have a little bit more complex structures. The elastic AVO responses exhibit opposite-polarity behavior for precritical angles and identical-polarity behavior FIG. 7. Maximum absolute AVO for opposite-polarity reflections for different wavelength/thickness (λ 2 /d) in model I. FIG. 9. Maximum absolute amplitudes of identical-polarity reflections in model II as a function of λ 2 /d for several incident angles. FIG. 8. Identical-polarity reflections in model II for the layer thickness changes from d = λ 2 to d = λ 2 /100. Solid and dashed lines denote the normal and 20 incidences, respectively. FIG. 10. Maximum absolute AVO for identical-polarity reflections for different wavelength/thickness (λ 2 /d) in model II.

1166 Liu and Schmitt for postcritical angles. AVO responses basically increase with increasing angle of the incident angle for the precritical angles and are complex for the postcritical angles (bottom interface) because of the effects of the SV-wave and the different material properties between the top and bottom half-spaces. Note the very small sharp peaks and valleys on the curves are due to numerical noise in the calculation. Wet sands have relatively high Poisson s ratios compared with gas sands (Rutherford and Williams, 1989; Hilterman, 2001). We choose different Poisson s ratios to study the AVO responses of thin gas sands. The Poisson s ratio of the bottom half-space was fixed at σ 2 = 0.25, while the thin layer was given two Poisson s ratios: σ 2 = 0.1(gas sand) or σ 2 = 0.4(wet sand). Figures 13 and 14 show the influence of Poisson s ratio on the AVO responses for d = λ 2 to d = λ 2 /8 (Figure 13) and d = λ 2 /15 to d = λ 2 /100 (Figure 14). This influence is large for d >λ 2 /8 at moderate or large angle of incidence (θ 1 > about 15 for model III), but is small for either small angle of incidence or d <λ 2 /15. The increase of amplitude with incident angle or offset is stronger for small Poisson s ratio (gas sand) than that for large Poisson s ratio (wet sand), which is similar to the AVO responses of a single interface for class 3 gas sand (e.g., Rutherford and Williams, 1989; Hilterman, 2001). The influence of Poisson s ratio on the AVO responses is also dependent on the thickness of the layer and reaches its maximum at the tuning thickness (d = λ 2 /4). The combination of two increasing AVO effects (interference and small Poisson s ratio) may results in a sizeable increase for AVO response in thin gas sand. The influence of Poisson s ratio on AVO responses is both large and complex for the postcritical angle incidence (bottom interface) because shear waves play a dominant role for large incidence angles (θ 1 >θ c2 ). FIG. 11. Maximum absolute amplitudes of the thin-sand reflections as a function of λ 2 /d for several incident angles. The sand layer and bottom half-space are elastic with Poisson s ratio σ = 0.25. FIG. 13. Maximum absolute AVO for the thin-sand reflections (σ = 0.1 for solid and 0.4 for dashed lines) for d = λ 2 to d = λ 2 /8 in model III. FIG. 12. Maximum absolute AVO for the thin-sand reflections for different wavelength/thickness (λ 2 /d) in model III. The sand layer and bottom half-space are elastic with Poisson s ratio σ = 0.25. FIG. 14. Maximum absolute AVO for the thin-sand reflections (σ = 0.1 for solid and 0.4 for dashed lines) for d = λ 2 /15 to d = λ 2 /100 in model III.

Thin Bed AVO 1167 CONCLUSIONS AND DISCUSSIONS For exploration geophysics, seismic AVO now is one of the major criteria for recognizing potential hydrocarbon reserves. However, traditional AVO analysis is based on the Zoeppritz equations and only contains single-interface information. Many observed seismic attributes cannot be explained by this kind of oversimplified model [for example, the low-frequency shadow that appeared at the bottom reflection from a gas sand (e.g., Taner et al., 1979)]. On the other hand, seismic stratigraphy (e.g., Payton, 1977; Anstey, 1982) studies the seismic reflection patterns to identify the conditions under which the rocks were deposited. These mainly contain the volume scattering information within depositional sequences. The volume information is more useful than just a single interface reflection for reservoir characterization because it carries stratigraphic structure, lithological change, and pore fluid information within depositional sequences. However, the multiple scattering of interfering seismic waves within a depositional sequence or fractured reservoir remains poorly understood (Liu and Schmitt, 2002). The results of a single thin bed given above have significant implications on seismic stratigraphic interpretation using amplitude. We believe that the integration for well log analysis, seismic AVO, and seismic stratigraphy, as well as giving us more physical insights into wave scattering within reservoir and depositional sequences, will help us to perform more accurate stratigraphic and lithological interpretations. ACKNOWLEDGEMENTS The authors thank Guoping Li, Encana, for his suggestions regarding thin-layer seismic attributes. The comments and suggestions of the associate editor (H. W. Swan), the assistant editor (J. M. Carcione), and three reviewers (D. C. Lawton, C. Ribordy, and an anonymous reviewer) improved the communication of this paper. This work was sponsored by the Seismic Heavy Oil Consortium in the Department of Physics, University of Alberta, and by the contributors of the Petroleum Research Fund, administered by the American Chemical Society for partial support of this research. REFERENCES Aki, K., and Richards, P. G., 1978, Quantitative seismology: Theory and methods, vol. I: W. H. Freeman. Anstey, N. A., 1982, Simple seismics: International Human Resources Development Corp. Almoghrabi, H., and Lange, J., 1986, Layers and bright spots: Geophysics, 51, 699 709. Brekhovskikh, L. M., 1980, Waves in layered media: Academic Press. Chung, H. M., and Lawton, D. C., 1995, Amplitude responses of thin beds: Sinusoidal approximation versus Ricker approximation: Geophysics, 60, 223 230. Chung, H. M., and Lawton, D. C., 1996, Frequency characteristics of seismic reflections from thin beds: Can. J. Expl. Geophys., 31, 32 37. de Voogd, N., and den Rooijen, H., 1983, Thin-layer response and spectral bandwidth: Geophysics, 48, 12 18. Gochioco, L. M., 1991, Tuning effect and interference reflections from thin beds and coal seams: Geophysics, 56, 1288 1295. Hilterman, F. J., 2001, Seismic amplitude interpretation: 2001 Soc. Expl. Geophys./Eur. Assn. Geosci. Eng. Distinguished Instructor Short Course. Juhlin, C., and Young, R., 1993, Implication of thin layers for amplitude variation with offset (AVO) studies: Geophysics, 58, 1200 1204. Kallweit, R. S., and Wood, L. C., 1982, The limits of resolution of zerophase wavelets: Geophysics, 47, 1035 1046. Koefoed, O., and de Voogd, N., 1980, The linear properties of thin layers, with an application to synthetic seismograms over coal seams: Geophysics, 45, 1254 1268. Lange, J. N., and Almoghrabi, H. A., 1988, Lithology discrimination for thin layers using wavelet signal parameters: Geophysics, 53, 1512 1519. Liu, Y., and Schmitt, D. R., 2001, Amplitude and AVO responses of a single thin bed: Can. Soc. Expl. Geophys. Conv., Expanded Abstracts, 5 8. 2002, Seismic scale effects: dispersion, attenuation, and attenuation by multiple scattering of waves, Can. Soc. Expl. Geophys. Conv., Expanded Abstracts, 1 3. Neidell, N. S., and Poggiagliolmi, E. 1977, Stratigraphic modeling and interpretation Geophysical principles, in Payton, C. E, Eds., Seismic stratigraphy Application to hydrocarbon exploration: Am. Assn. Petr. Geol. Memoir 26, 389 416. Payton, C. E., Ed., Seismic Stratigraphy Application to hydrocarbon exploration: AAPG Memoir 26. Rokhlin, S. I., Xie, Q., Liu, Y., and Wang, L., 1999, Ultrasonic study of quasi-isotropic composites, in Thompson, D. O., and Chimeti, D. E., Eds., Review of progress in QNDE: Plenum Press, 18, 1249 1256. Rutherford, S. R., and Williams, R. H., 1989, Amplitudes-versus-offset variations in gas sands: Geophysics, 54, 680 688. Schmitt, D. R., 1999, Seismic attributes for monitoring of a shallow heated heavy oil reservoir: A case study: Geophysics, 64, 368 377. Taner, M. T., Koehler, F. and Sheriff, R. E., 1979, Complex seismic trace analysis: Geophysics, 44, 1041 1063. Ursin, B., and Stovas, A., 2002, Reflection and transmission response of a layered isotropic viscoelastic media: Geophysics, 67, 324 325. Widess, M. B., 1973, How thin is a thin bed?: Geophysics, 38, 1176 1180. Yilmaz, O., 1987, Seismic data processing: Soc. Expl. Geophysics. APPENDIX A COEFFICIENTS OF REFLECTION AND TRANSMISSION FOR A SINGLE LAYER A plane harmonic wave with unit amplitude illuminates the thin layer (Figure 1). For acoustic media, the displacement potentials in the top half-space, thin layer, and the bottom halfspace can be written respectively (e.g., Aki and Richards, 1978) as φ 1 = e i(k x1 x+k z1 z ωt) + R(ω)e i(k x1 x k z1 z ωt), (A-1) φ 2 = A(ω)e i(k x2 x+k z2 z ωt) + R(ω)e i(k x2 x k z2 z ωt), (A-2) φ 3 = T (ω)e i(k x3 x+k z3 z ωt). (A-3) where k xi =ω/c = (ω/α i ) sin θ i ; k zi = (ω/α i ) cos θ i ; i = 1, 2, 3; α i and θ i are the velocities and incident or refracted angles, respectively; and c and ω are phase velocity and angle frequency, respectively. R(ω) and T (ω) are the generalized reflection and transmission coefficients, and A(ω) and B(ω) are the amplitudes of the potential within thin layer. The solution can be written as the form of the propagator matrix by the continuous boundary conditions for displacements and pressures on the top and bottom interfaces (e.g., Rokhlin et al., 1999): S 1 = BS 3, (A-4) B = XD 1 X 1, (A-5) where S 1 = (u z1, p 1 ) T and S 3 = (u z3, p 3 ) T are the vertical displacement and pressure vectors at the top and bottom interfaces, respectively. X and D are 2 2 matrixes within the layer, which can be written as ( ) α 2 α 2 X =, (A-6) ρ 2 c 2 ρ 2 c 2 ( ) e ik z2 d 0 D =. (A-7) 0 e ik z2 d

1168 Liu and Schmitt Substituting equations (A-6) and (A-7) into equation (A-5), we have iη 2 sin(k z2 d) cos(k z2 d) B = ρ 2 c 2 iρ 2 c 2 sin(k z2 d), (A-8) cos(k z2 d) η 2 where η 2 = c 2 /α2 2 1. The displacements and pressures in the top and bottom interfaces can be written as S 1 = [ [η 1 (1 R(ω)],ρ 1 c 2 [1 + R(ω)] ] T, (A-9) S 3 = [ η 3 T (ω),ρ 3 c 2 T(ω) ] T, (A-10) where η i = c 2 /αi 2 1(i = 1 and 3). Substituting equations (A-8), (A-9), and (A-10) into equation (A-4) and solving equation (A-4) for R(ω)and T (ω), we have T (ω) = 2Z 1 Z 2 i ( Z 2 2 + Z 1Z 3 ) sin(kz2 d) + (Z 1 Z 2 + Z 2 Z 3 ) cos(k z2 d). (A-14) If the material properties for the top and bottom half-spaces are identical, we have i ( Z2 2 R(ω) = Z 1) 2 sin(kz2 d) i ( Z1 2 + Z ) 2 2 sin(kz2 d) + 2Z 1 Z 2 cos(k z2 d), (A-15) R(ω) = i( ρ2 2η ) 1η 3 ρ 1 ρ 3 η2 2 sin(kz2 d) + (ρ 2 ρ 3 η 1 η 2 ρ 1 ρ 2 η 2 η 3 ) cos(k z2 d) i ( ρ2 2η ) 1η 3 + ρ 1 ρ 3 η2 2 sin(kz2 d) + (ρ 2 ρ 3 η 1 η 2 + ρ 1 ρ 2 η 2 η 3 ) cos(k z2 d), (A-11) T (ω) = 2ρ 1 ρ 2 η 1 η 2 i ( ρ2 2η ) 1η 3 + ρ 1 ρ 3 η2 2 sin(kz2 d) + (ρ 2 ρ 3 η 1 η 2 + ρ 1 ρ 2 η 2 η 3 ) cos(k z2 d). (A-12) Assuming Z i = ρ i α i / cos θ i (i = 1, 2, 3) and applying sin θ 1 / sin θ 2 = α 1 /α 2, sin θ 1 / sin θ 3 = α 1 /α 3, and sin θ 2 / sin θ 3 = α 2 /α 3, we have R(ω) = i( Z 2 2 Z 1Z 3 ) sin(kz2 d) + (Z 2 Z 3 Z 1 Z 2 ) cos(k z2 d) i ( Z 2 2 + Z 1Z 3 ) sin(kz2 d) + (Z 1 Z 2 + Z 2 Z 3 ) cos(k z2 d), (A-13) 2Z 1 Z 2 T (ω) = i ( Z1 2 + Z ) 2 2 sin(kz2 d) + 2Z 1 Z 2 cos(k z2 d). (A-16) Equations (A-11 A-16) are the different forms for the generalized reflection and transmission coefficients for a single layer in acoustic case. Brekhovskikh (1980) also derived similar forms by two kinds of different methods (i.e., input impedance and multiple superposition).