Useful approximations for converted-wave AVO

Transcription

1 GEOPHYSICS VOL. 66 NO. 6 NOVEMBER-DECEMBER 001); P FIGS. 3 TABLES. Useful approximations for converted-wave AVO Antonio C. B. Ramos and John P. Castagna ABSTRACT Converted-wave amplitude versus offset AVO) behavior may be fit with a cubic relationship between reflection coefficient and ray parameter. Attributes extracted using this form can be directly related to elastic parameters with low-contrast or high-contrast approximations to the Zoeppritz equations. The high-contrast approximation has the advantage of greater accuracy; the low-contrast approximation is analytically simpler. The two coefficients of the low-contrast approximation are a function of the average ratio of compressionalto-shear-wave velocity α/β) and the fractional changes in S-wave velocity and density β/β and ρ/ρ). Because of its simplicity the low-contrast approximation is subject to errors particularly for large positive contrasts in P-wave velocity associated with negative contrasts in S-wave velocity. However for incidence angles up to 40 and models confined to β/β < 0.5 the errors in both coefficients are relatively small. Converted-wave AVO crossplotting of the coefficients of the low-contrast approximation is a useful interpretation technique. The background trend in this case has a negative slope and an intercept proportional to the α/β ratio and the fractional change in S-wave velocity. For constant α/β ratio an attribute trace formed by the weighted sum of the coefficients of the low-contrast approximation provides useful estimates of the fractional change in S-wave velocity and density. Using synthetic examples we investigate the sensitivity of these parameters to random noise. Integrated P-wave and converted-wave analysis may improve estimation of rock properties by combining extracted attributes to yield fractional contrasts in P-wave and S-wave velocities and density. Together these parameters may provide improved direct hydrocarbon indication and can potentially be used to identify anomalies caused by low gas saturations. INTRODUCTION Conventional P-wave amplitude versus offset AVO) analysis Ostrander and Gassaway 1983) is an important exploration tool that exploits offset-dependent P-wave reflectivity associated with hydrocarbons or lithology variations. Convertedwave P-SV AVO may provide useful complementary information to that obtained from P-wave AVO analysis. Here we discuss this technology under assumption of an isotropic earth. Garotta and Granger 1987) Garotta 1988) Miles and Gassaway 1989) and Zaengle and Frasier 1993) discuss the utility of combined P-P and P-SV AVO. Two advantages of P-SV AVO are 1) the slope of the converted P-SV AVO is primarily dependent on the SV-wave impedance contrast and ) shear waves are less perturbed by transmission through overlying gas sands than are compressional waves. With the development of ocean-bottom cables which include multicomponent seismometers there is now more opportunity for the use of P-SV AVO technology. A simple example that illustrates the usefulness of P-SV AVOisshown in Figure 1. For this case we use the first model shown in Table 1 from Castagna and Smith 1994). The P-P reflectivity for shale/brine sand and shale/gas sand interfaces Figure 1a) shows amplitude increase with incidence angle for both cases but the gas sand case has stronger amplitude increase and a larger reflection response throughout the entire angle range. Though detection of gas appears more robust on the P-P graph the P-SV reflectivity Figure 1b) clearly shows a distinctive difference that can help us better characterize the fluid type in this sandstone reservoir. Near-vertical incidence P-SV reflectivity is very small since little conversion occurs at these angles. As the incidence angle increases the gas sand case shows positive amplitudes which increase with angle while the Manuscript received by the Editor January ; revised manuscript received April Petrobras E&P Av. Chile 65 sala 1401-D Rio de Janeiro RJ Brazil. aramos@ep.petrobras.com.br. University of Oklahoma Institute for Exploration and Development Geosciences 100 East Boyd St. Norman Oklahoma castagna@ou.edu. c 001 Society of Exploration Geophysicists. All rights reserved. 171

2 17 Ramos and Castagna brine sand case shows negative amplitudes with only a slight amplitude increase. The incidence angles in Figures 1a and 1b are the same but the offsets are different since the reflected P-SV angle is smaller than that of the P-P wave. We demonstrate that P-SV AVO provides additional information to the traditional AVO technique reducing the degree of uncertainty associated with velocities and density estimation. Computation of P-P and P-SV reflectivity through exact Zoeppritz solvers can be easily accomplished in modern computers. However these solvers offer very little in terms of physical insight into the problem and do not let us explore the powerful features of the AVO crossplot analysis. In addition the reduction of the reflectivity problem to two or three coefficients allows the construction of attribute traces directly from curve fit through the amplitudes in common midpoint CMP) gathers. Since simple approximations for reflection coefficients were introduced e.g. Bortfeld 1961; Chapman 1976; Richards and Frasier 1976; Aki and Richards 1980; Hilterman 1983; Shuey 1985; Smith and Gidlow 1987; and Mallick 1993) AVO has become a technique that relies highly on elastic parameter estimation from curve shape. Therefore it is important to utilize reflection coefficient approximations that can be accurate yet simple enough to give insight into the physics of the problem at hand. Aki and Richards 1980) for example give useful approximations for the P-P and P-SV reflection coefficient problem. We simplify the latter to obtain a useful low-contrast approximation for P-SV AVO. From the exact P-SV reflectivity we derive an accurate highcontrast approximation for the P-SV converted-wave reflection coefficient and assess the errors related to the simpler low-contrast approximation. The differences between the coefficients of the low-contrast and the high-contrast approximations and a cubic fit to the exact P-SV reflection coefficient equation proved to be important in this analysis. A framework for crossplot analysis of extracted P-SV-wave AVO parameters is presented. We show how this approach may help identify low partial gas saturations. APPROXIMATIONS FOR CONVERTED-WAVE AVO Exact nonnormal incidence P-P and P-SV reflectivity is given by the well-known Zoeppritz equations which express reflection coefficients as a complicated function of elastic parameters. As in the case of P-P AVO analysis of P-SV data is facilitated by simpler yet acceptably accurate reflection coefficient approximations. For angles of incidence up to 30 and small contrasts in elastic properties P-P AVO can be simplified to a linearized approximation with only two coefficients Shuey 1985; Swan 1993): R PP θ) A + B sin θ 1) where A = 1 α α + ρ ) ρ 1a) and B = 1 α α β α ) β β + ρ ). 1b) ρ FIG. 1. Four possible variations of amplitude with incidence angle for an incident P-wave reflected as a) a P-wave and b) an SV-wave. Notice the distinctive separation in AVO behavior between shale over brine sand and shale over gas sand. In equations 1ab) ρ/ρ with ρ = ρ ρ 1 and ρ = ρ + ρ 1 )/ is the fractional change in density subscripts 1 and indicate the upper and lower medium respectively); α/αwith α = α α 1 and α = α + α 1 )/ is the fractional change in P-wave velocity; and β/β with β = β β 1 and β = β + β 1 )/ is the fractional change in S-wave velocity. In equation 1) θ is the average of the angles of incidence θ 1 ) and refraction θ ) across the interface. As we shall see the P-SV reflection coefficient R PS ) depends primarily on the fractional changes in density and S-wave velocity. Starting with the exact expression for R PS as given by Aki and Richards 1980) and printed for completeness in

3 Appendix A and assuming low incidence angles θ 1 ) we obtain an approximation of the form [see equation B-1)] R [1] PS θ 1) A 1 sin θ 1 + B 1 sin 3 θ 1. ) In equation ) the coefficient A 1 is a relatively simple function of µ change in shear modulus) ρ change in density) I α average P impedance) and I β average S impedance) along with P and S velocities and density. The coefficient B 1 is a complicated function of the same parameters. The complexity of B 1 is because no assumptions on small density and velocity contrasts between the two half-spaces are considered. As we demonstrate equation ) is accurate for a large number of possible exploration models. We refer to equation ) as the high-contrast approximation for the R PS reflection coefficient. Equation ) clearly shows that for near-vertical incidence sin θ = 0) converted-wave amplitudes are much smaller than those at large incidence angles. The low-contrast approximation is derived from the approximation given by Aki and Richards 1980) for R PS which is accurate for small contrasts in elastic properties and small angles of incidence. Rewriting this approximation and replacing the horizontal slowness p by sin θ/αwhere α is the average P-wave velocity between the two media and θ is the average angle of incidence and refraction for the P-wave. Using φ as the average angle of reflection and refraction for the SV-wave yields Approximations for Converted-Wave AVO 173 ) 1 ρ R PS θ) cos φ ρ + cos θ β ρ α ρ + cos θ 4β β α β sin θ αβ ) ρ + cos φ ρ + αβ β sin 3 θ. 3) cos φ β α 3 Equation 3) can be simplified further by making the following substitutions: cos θ 1 sin θ and 1 cos φ 1 + β α sin θ and collecting the powers of sin θ to obtain R PS θ) A sin θ + B sin 3 θ + C sin 5 θ 4) where [ A = β ] [ β 1 α β + β ) ] ρ 4a) α ρ [ β B = α + β ) ] [ β 3 β + α β 4 α + β ) ] ρ 4b) α ρ [ β ) ] [ 4 ) ] β 1 β 4 ρ C = +. 4c) α β α ρ Table 1. P- and S-wave velocities and densities for 5 sets of brine sands gas sands and shales. The velocities are from full-waveform sonic logs or dipole sonic logs or were measured in the laboratory. The densities are from compensated density logs or were estimated from lithology and porosity information from Castagna and Smith 1994). Brine sand Shale Gas sand α β ρ α β ρ α β ρ Model km/s) km/s) g/cm 3 ) km/s) km/s) g/cm 3 ) km/s) km/s) g/cm 3 )

4 174 Ramos and Castagna Disregarding the fifth power of sin θ which is small for incidence angles <30 weare left with the following lowcontrast two-coefficient approximation for a P-SV reflection coefficient: R [] PS θ) A sin θ + B sin 3 θ. 5) The superscript and subscripts and 1 in equations 5) and ) help differentiate the low-contrast and high-contrast approximations respectively. Equation 4) is a function of only three parameters: the fractional change in S-wave velocity the fractional change in density and the ratio between shear and compressional average velocities. The major difference in the methodology to obtain approximations given by equations ) and 3) is that for the first case the exact equation A-1) is linearized with respect to the horizontal slowness while in the second case the exact equation is linearized with respect to the contrasts in α β and ρ.the choice of equations with only two coefficients [equations 1) ) and 5)] is made because the third coefficient can usually be neglected for the range of recorded incidence angles. Also the third coefficient may be difficult to extract robustly Swan 1993). Therefore in practice only two parameters can be estimated effectively from P- and S-wave AVO inversion. Finally crossplot analysis and trace-attribute extraction from converted-wave AVO is facilitated if two coefficients are used instead of three. Attempts to crossplot three coefficients of a linearized reflection coefficient approximation are described by Smith 1996). Figures a and b show the variations of P-SV reflectivity with incidence angle for models 1 and 3 shown in Table 1 which are typical shale/gas sand/shale models. In these figures the high-contrast approximation B-1) and the low-contrast approximation 5) are compared with the exact equation A-1) and its two-coefficient cubic fit of the type A sin θ 1 + B sin 3 θ 1 where θ 1 is the incidence angle. The cubic equation fits the exact equation quite well in most cases and the high-contrast equation ) approximates the exact equation better than the low-contrast equation 5). The magnitudes of the contrasts in elastic properties are α =0.3 km/s β =0.09 km/s and ρ =0.15 g/cm 3 for Figure a and α =0.36 km/s and β =0.8 km/s and ρ =0.15 g/cm 3 for Figure b. The S-wave velocity contrast is large for Figure b; but because of the small contrast in P-wave velocity the high-contrast approximation tends to be close to the exact equation. The low-contrast approximation works relatively well in Figure a where S-wave velocity contrast is small but differs significantly from the exact equation when the contrast is larger Figure b). This difference is caused by the larger error in B when compared to the exact equation. Table summarizes the relative errors in the coefficients A 1 B 1 A and B for the example shown in Figures and 3. To allow proper comparison of the low-contrast approximation with the other curves equation 5) was properly scaled to replace sin θ average angle) by sin θ 1 incidence angle). Figures 3a and 3b show two other cases: a gas sand over brine sand case Figure 3a model 16 of Table 1) and a shale/brine sand/shale model Figure 3b model of Table 1). In the first case the magnitudes of the contrasts in elastic properties are α =0.66 km/s β =0.04 km/s and ρ =0.15 g/cm 3.Despite the small absolute contrast in S-wave velocity there is a significant difference in P-wave velocity and both the highand low-contrast approximations deviate from the exact equation at angles of incidence >5.However the high-contrast equation ) is clearly a better approximation as demonstrated by the small relative errors of A 1 and B 1 when compared to the cubic fit to the exact equation Table ). In the second case Figure 3b) the contrasts in elastic properties are α = 1.15 km/s β =0.91 km/s and ρ =0.36 g/cm 3.There is a significant change in both P- and S-wave velocities. However FIG.. Approximations for converted-wave reflection coefficient compared with exact equation and its cubic fit for models 1 and 3 Table 1). Magnitudes of the contrasts in elastic properties in these models are a) α =0.3 β =0.09 and ρ =0.15 and b) α =0.36 β =0.8 and ρ =0.15.

5 Approximations for Converted-Wave AVO 175 only when the changes in P- and S-wave velocities are positive shale over brine sand case) is there a significant difference between the high-contrast approximation and the exact equation. This difference as demonstrated in model of Table is mostly from the error in B 1. The analysis of the coefficients of each of the approximations lends insight into the errors. Figure 4 shows two crossplots of A for the high- Figure 4a) and low-contrast Figure 4b) approximations versus A from the cubic fit to the exact equation. The computations are made for all models shown in Table 1 for a range of incidence angles from 0 to 40.Inboth cases there is good agreement between A and the coefficient derived from the cubic fit. This fact implies that A the low-contrast approximation) is well constrained and is generally very similar to A 1 the high-contrast approximation). In fact if one assumes that the contrast in S impedance is very small and that the products of P and S impedances of the media involved are all approximately the same one can algebraically demonstrate that A becomes approximately A 1.Figure 5 shows similar crossplots for B. Inthis case B 1 the high-contrast approximation) is constrained while there is a significant scatter of the values around the 45 diagonal for B the low-contrast approximation) which increases with its magnitude. The confidence in B is therefore reduced for larger magnitudes. We demonstrate the range of elastic parameters for which the estimate of B is reasonable. CONVERTED-WAVE AVO CROSSPLOT P-wave AVO crossplots are normally used to separate shales and fully brine-saturated rocks from gas-saturated rocks. In general these rocks follow a well-defined line or background trend) in the A intercept) versus B gradient) crossplot while the gas-saturated rocks deviate from this line. Equations for the background trend involving P-waves are given by Castagna et al. 1998). Generally A and B from P-wave AVO are negatively correlated for background rocks but can be positively correlated at very high ratios of compressional-to-shear-wave velocity α/β) such as those found in soft shallow sediments Castagna et al. 1998). Here we develop the general background trend equation for the P-SV case in which A and B are assumed to be A and B of the low-contrast approximation. By algebraic manipulation of B in equation 4b) and recognizing A in the B expression we obtain the relationship β B = A + α β α ) β β + 3 β 4 α β α 1 ) ρ ρ. 6) Equation 6) indicates that B and A are usually negatively correlated and the line defined in the A B plane may not pass through the origin. Assuming that density is a constant factor times the P-wave velocity raised to an arbitrary power g Gardner et al. 1974) we can write ρ ρ g α α. 7) Now assuming that the compressional- and shear-wave velocities α and β) obey a linear relationship such as α mβ + c 8) we can use equation 8) to write a relationship between the density contrast and the S-wave velocity contrast: ) ρ β β ρ gm α β. 9) Finally by substituting equations 9) into equation 6) we obtain a general equation for the background trend in convertedwave AVO: [ ) 3 β B = A + 4 gm α + gm ) β α )] β gm β α β. 10) Equation 10) can be used together with the slope of the general expression for the background trend associated with the P-wave AVO to estimate the relationship between α and β for the background rocks. Figure 6 shows the crossplot between A and B considering the models included in Table 1 and the interfaces indicated by the labels in the legend. These coefficients are determined using incidence angles from 0 to 40.The significant negative correlation between these coefficients is shown for all models. Very small deviation from a linear trend is observed for both brine and gas sands which represents a major difference from P-P AVO crossplots. A line fit through the models is made to indicate the typical relationship between A and B.Ifwe assume m = α/β [constant α/β ratio i.e. c vanishes in equation 8)] and g = 1/4 we can rewrite equation 10) as 1 α B A + [ β α ) 8 β ) 9 8 )] β β α β. 11) Table. Relative errors in the coefficients of the high- A 1 and B 1 ) and low-contrast A and B )approximations when compared to the cubic fit to the exact equation. Only selected models from Table 1 are shown and are used in Figures and 3. A1 rel. A rel. B1 rel. B rel. Rock type Model β/α β ρ α β/β ρ/ρ α/α error % error % error % error % Shale/gas ss Gas ss/shale Shale/gas ss Gas ss/shale Gas ss/brine ss Shale/brine ss Brine ss/shale

6 176 Ramos and Castagna Solving equation 11) for β/α yields β α 9 35 ± 1 560A + B ) β β. 1) 35 β β Estimation of β/α from equation 1) is dependent on how well we can approximate ρ/ρ by 1/4 β/β. Another way to estimate β/α is from the well-constrained coefficient A alone [using equation 4a)]. Equation 11) indicates that for constant α/β = i.e. β/α = 0.608) the background trend is a line that pass through the origin i.e. B = A.Forα/β > and β > 0 the increment in α/β ratio causes the vertical offset intercept) of the background trend to increase becoming more negative. The opposite situation is observed when β < 0. Figure 7 shows the variations in P-P Figure 7a) and P-SV Figure 7b) AVO background trends for a range of α/β ratios. The P-SV AVO trend is computed assuming β/β = 0.1. While an increase in α/β ratio rotates the background trend counter clockwise FIG. 3. Approximations for converted-wave reflection coefficient compared with exact equation and its cubic fit for models 16 and Table 1). Magnitudes for the contrasts in elastic properties in these models are a) α =0.66 β =0.04 and ρ =0.15 and b) α =1.15 β =0.91 and ρ =0.36. FIG. 4. Crossplots of A for the a) high- and b) low-contrast approximations versus A derived from the cubic fit to the exact equation. Incidence angles range from 0 to 40.Notice the good agreement of this coefficient in both cases. All models shown in Table 1 are used in this plot.

7 Castagna et al. 1998) in the P-P AVO it has the effect of changing the intercept of the background trend in the P-SV AVO. Another important observation from equation 11) is that if we consider a fixed α/β ratio the simple addition of seismically derived traces A and B is a scaled measure of β/β.thesame conclusion can be obtained if we solve equations 4a) and 4b) as a system of two equations and two unknowns β/β and ρ/ρ). Assuming that A and B can be measured from the P-SV AVO and α/β ratio can be roughly estimated from logs and P-P-wave AVO we obtain Approximations for Converted-Wave AVO 177 FIG. 6. Crossplot of A and B of the low-contrast approximation for data from Table 1. Notice the strong negative correlation between these coefficients. Only the fit for the shale/brine sand model is shown. FIG. 5. Crossplots of B for the a) high- and b) low-contrast approximations versus B derived from the cubic fit to the exact equation. Incidence angles range from 0 to 40. b) Notice the significant scatter of the values around the ideal 45 line as the magnitude of this coefficient increases. All models shown in Table 1 are used in this plot. FIG. 7. Background trends for a) P-P and b) P-SV AVO for a different α/β ratios. The P-SV AVO background trend is computed assuming β/β = 0.1.

8 178 Ramos and Castagna [ βα ] )A B ρ ρ = β β = [ β α + 3 β ) 13) α + 1 β ) A α + βα ] )B + 1 β α β ). 14) α + 1 The estimated parameters β/β and ρ/ρ should be affected when gas replaces the liquid phase in a porous sandstone reservoir encased in shales. In Figure 8 we model these parameters using the shale and the brine sand properties of model 9 Table 1); then we change the water saturation to 90% and 0% using equations given in Gassmann 1951). In this typical class III reservoir example Castagna et al. 1998) β/β becomes more positive and ρ/ρ becomes more negative as gas replaces water in the sand. This is because gas reduces the density of the sandstone which increases its S-wave velocity. Conversely the presence of gas in the sand initially causes a large drop and change in sign of α/α. Aswater saturation is further decreased the magnitude of α/α does not change significantly. The behavior of a class IV reservoir model 16 Table 1) is slightly different since β/β is negative for 100% water saturation and decreases its magnitude becoming less negative) as gas replaces water in the reservoir. However similar to the class III sand ρ/ρ is negative for the brine case and becomes more negative as water saturation decreases. Therefore the fractional change in density decreases in magnitude as gas saturation increases in both class III and IV reservoirs. The parameter β/β becomes important when compared to the P-wave AVO intercept [equation 1a)] which essentially retains the changes in fluid properties in α/α and ρ/ρ. An estimate of α/α can be made by replacing ρ/ρ found in equation 13) into equation 1a). The analysis of sections FIG. 8. Attributes α/α β/β and ρ/ρ for three different water saturations S w ) indicated in the figure. Brine reservoir properties are from model 9 Table 1). Notice the opposite behavior of α/α and β/β as we move from a water sand top) to a gas sand bottom). In this case low gas saturations can be distinguished from high gas saturations from the magnitudes of β/β and ρ/ρ with respect to α/α. Fluid substitution is performed using Gassmann s equations Gassmann 1951). FIG. 9. True values of α/α β/β and ρ/ρ a) compared with their corresponding estimated values b) from equations 1a) 14) and 13) for different water saturations. Notice the overall similarity in the shape of the curves shown in a) and b). In this case the magnitudes of β/β and ρ/ρ and the difference between α/α and ρ/ρ increase as gas saturation increases. Fluid substitution is performed using Gassmann s equations.

9 Approximations for Converted-Wave AVO 179 formed by these three parameters α/α β/β and ρ/ρ) may help distinguish full from partial gas saturation as seen in Figure 8. A comparison between the true values of α/α β/β and ρ/ρ with their corresponding estimated values [from equations 1a) 14) and 13)] for a range of water saturations is shown in Figure 9 where we use the same water-saturated reservoir properties as in Figure 8. True and estimated parameters compare well for the entire range of water saturations. Also as gas saturation increases in this class III reservoir the magnitude of β/β and ρ/ρ always increases while α/α initially drops changes its polarity and then decreases in magnitude. DISCUSSION The correctness of the B coefficient of the low-contrast approximation is addressed in Figure 5 where it is compared with the cubic fit to the exact equation. The scattering of the B values around the ideal diagonal line increases with its magnitude. Figure 10 shows the errors B B from cubic fit to the exact equation minus B from low-contrast approximation) for different FIG. 10. Errors in the coefficient B of the low-contrast approximation as a function of various combinations of elastic parameter changes across the interface. The horizontal axis represents the difference in B between cubic fit to the exact equation and the low-contrast approximation. All models from Table 1 are considered. Units of α and β are in km/s while ρ are in g/cm 3.

10 1730 Ramos and Castagna combinations of elastic parameter changes across the interface. Computations are made for angles of incidence up to 30 for all graphs. In this case we consider indiscriminately all 150 models that can be computed from Table 1. As the changes in elastic parameters increase the error also increases becoming especially significant for large positive variations in P- and S-wave velocities. Some large errors in B associated with small density contrasts are in fact related to large changes in P- and S-wave velocities in such models. Analysis of Figure 10d reveals that for models confined to β/β < 0.5 the errors in the B coefficient are relatively small even considering that α and ρ could vary freely on both sides of the interface. While the expression of B low-contrast approximation) can be described by a simple function of three parameters β/β ρ/ρ and β/α [equation 4b)] the expression of B 1 highcontrast approximation) is a complicated equation involving different combinations of α β and ρ for each medium. Clearly B does not depend on α/α; however the exact equation and the high-contrast approximation show dependency on this parameter since these expressions are explicit functions of α 1 and α Table Figure 5). Therefore part of the error associated with B can be explained by its complete independence of α/α. Figure 11 shows the magnitude of the exact SV-wave reflection coefficient for an incident P-wave [equation A-1)] considering different contrasts in P-wave and S-wave velocity and fixed density contrast. In this case the α/β ratio in both media is kept fixed at 1.73 for all computations so that the contrasts in S-wave velocity are the same as those for P-wave velocity. When the P-wave velocity of the lower medium is less than that of the upper medium i.e. α < 0 the curves shown in gray) are smooth and there is no change in phase of the SV-wave. This explains why both approximations yield reasonable results for α < 0 class III and IV reservoirs). When the lower medium has higher P-wave velocity than the upper medium i.e. α > 0 the incident P-wave may reach a critical angle. In this case the reflection coefficient of the SV-wave will undergo abrupt changes like a sudden fall followed by a rise or change in phase. The displacement of these notches in the curves toward zero offset can be observed as the P-wave velocity of the lower medium is increased. This fact explains why both the high- and low-contrast approximations fail for α 0Figures 3 10b and 10e). Figure 1 shows the P-SV reflectivity behavior when only the P-wave velocity contrast always positive) is changed and both S-wave velocity and density contrasts are kept fixed. As the P-wave velocity contrast increases the approximations deviate from the exact equation at smaller incident angles. Also there is a clear influence of P-wave velocity contrast on the P-SV reflectivity causing a poorer performance of the lowcontrast approximation when compared to the high-contrast approximation both computed for α 1 /α = 0.7). Despite the differences in the linearization process from the exact equation the low-contrast approximation is linearized with respect to the contrasts in α β and ρ while the highcontrast approximation is linearized in terms of horizontal slowness) there is excellent agreement for the A coefficient of both approximations. Figure 4 has shown that these coefficients are also very accurate when compared to the cubic fit to the exact equation. There is however a difference in the B coefficients between the approximations. The comparison between the B coefficients of the low-contrast and high-contrast FIG. 11. Magnitude of an exact SV-wave reflection coefficient for an incident P-wave for variable P- and S-wave velocity contrasts and fixed density contrast. Notice the displacement of the notch in the curves toward zero offset as the P-wave velocity of the lower medium increases. Also notice there is no change in phase of the SV-wave when P-wave velocity of the lower medium is less than that of the upper medium. The α/β ratio in both media is kept fixed as 1.73 for all cases. FIG. 1. Magnitude of the exact SV-wave reflection coefficient for an incident P-wave considering variable contrast in P-wave velocity and fixed S-wave velocity and density contrasts. Notice the displacement of the notch in the curves toward zero offset as the P-wave velocity of the lower medium increases. Highand low-contrast approximations are compared with the exact curve for α 1 /α = 0.7. Both approximations fail for large angles of incidence.

11 approximations shows similar results to that of Figure 5 i.e. an increasing scatter of the difference for large magnitudes of B. This scatter is in turn associated with larger contrasts in elastic properties between the media involved and the effect of the P-wave velocity contrast. Estimating α/α β/β and ρ/ρ may fail if the true values of these contrasts exceed certain limits Table 3). Figure 13 comparies true values of β/β and α/α and the corresponding estimated values computed using equations 13) 14) and 1a) for all models in Table 1. Correct values for β/α were used to estimate these parameters assuming β/α can be obtained elsewhere). Estimation of β/β becomes less reliable when the magnitude of the true value of β/β is >0.5. A similar behavior can be observed in α/α. Figure 14 shows the sensitivity of the estimated parameters ρ/ρ and β/β to random noise. Noise-free and noisy P-SV synthetic seismograms are shown in Figures 14a and 14b. These synthetics are computed for a 100-m-thick sand embedded in shale with elastic properties given in model 9 Table 1). Water saturation in the sand is assumed as 0%. The effect of random noise on the estimation of ρ/ρ and β/β from P-SV AVO for the top of the sand is shown in Figures 14c and 14d respectively. The exact values of these parameters are for β/β and for ρ/ρ.ass/n ratio drops to very small values the estimation of these parameters is greatly affected. Another natural source of problems for the P-SV AVO is the background velocity model used for angle estimation. A wrong velocity model can introduce inconsistencies in the incidence angles amplifying the error in the estimated parameters. CONCLUSIONS Approximations for Converted-Wave AVO 1731 The simple two-coefficient low-contrast approximation to the converted-wave reflection coefficient exhibits larger errors than the two-coefficient high-contrast approximation for large contrasts in S-wave and P-wave velocities across the interface. The errors in both approximations when compared to the cubic fit to the exact reflection coefficient equation are mainly associated with the coefficient Bwhich is better defined in the high-contrast approximation. The errors in the low-contrast approximation are greatest for large positive contrasts in P-wave velocity associated with negative contrasts in S-wave velocity. On the other hand because of the smoothness of the P-SV reflectivity for negative P-wave velocity contrast both approximations perform better under such conditions even when large contrasts are considered. Consequently estimation of β/β and ρ/ρ using the low-contrast approximation is improved at the top of low-impedance reservoirs. FIG. 13. Comparison between true values of β/β and α/α and the corresponding estimated values computed using equations 13) 14) and 1a). Correct values for β/α were used to estimate these parameters assuming that β/α can be obtained from P-wave AVO). Notice the problems in the estimation of β/β when the magnitude of its true value is >0.5. Table 3. Typical relative errors in the estimation of α/α β/β and ρ/ρ as a function of errors in A and B for petrophysical models randomly picked from Table 1. Model Set A error % B error % α/α error % β/β error % ρ/ρ error % Shale/brine sand Brine sand/shale Shale/gas sand Gas sand/shale Gas sand/brine sand

12 173 Ramos and Castagna FIG. 14. a) P-SV synthetic seismogram computed for 100-m-thick sand embedded in shale with elastic properties from model 9 Table 1). Water saturation in the sand is assumed to be 0%. b) Noisy synthetic seismogram. cd) The effect of random noise on estimating ρ/ρ and β/β from P-SV AVO for the top of the sand arrow) respectively. Notice the problems in the estimation of these parameters as the S/N ratio drops to very small values. Converted-wave AVO crossplots can provide a powerful calibration tool for P-waveAVO. Considering the two coefficients A and B) of the P-SV reflection coefficient approximations it is possible to establish a similar framework to that already in use for P-waves Castagna et al. 1998). The background trend in the P-SV case has a negative slope and an intercept proportional to the α/β ratio and the fractional change in S-wave velocity. A combination of the P-SV AVO crossplot intercept with the P-P AVO crossplot slope can improve the estimation of a general α/β ratio for the background rocks. For constant α/β ratio weighted summations of A and B attributes from converted-wave AVO yield valuable estimates of fractional changes in density and S-wave velocity. The combined interpretation of this attribute and the P-wave intercept can be useful in separating pay zones from nonpay zones. ACKNOWLEDGMENTS The authors thank the sponsors of the O.U. Reservoir Characterization Consortium. A. R. thanks Petrobras for permission to publish this work. Thanks also to Martin Tygel for useful discussions. REFERENCES Aki K. and Richards P. G Quantitative seismology: W.H. Freeman Co. Bortfeld R Approximation to the reflection and transmission coefficients of plane longitudinal and transverse waves: Geophys. Prosp

13 Approximations for Converted-Wave AVO 1733 Castagna J. P. and Smith S. W A comparison of AVO indicators: Amodeling study: Geophysics Castagna J. P. Swan H. W. and Foster D Framework for AVO gradient and intercept interpretation: Geophysics Chapman C. H Exact and approximate generalized ray theory in vertically inhomogeneous media: Geophys. J. Roy. Astr. Soc Gardner G. H. F. Gardner L. W. and Gregory A. R Formation velocity and density The diagnostic basis for stratigraphic traps: Geophysics Garotta R Amplitude-versus-offset measurements involving converted waves: 58th Ann. Internat. Mtg. Soc. Expl. Geophys. Expanded Abstracts Garotta R. and Granger P. Y Comparison of responses of compressional and converted waves on a gas sand: 57th Ann. Internat. Mtg. Soc. Expl. Geophys. Expanded Abstracts Gassmann F Elastic waves through a packing of spheres: Geophysics Hilterman F Unpublished course notes in Castagna J. P. and Backus M. M. Eds. Offset dependent reflectivity Theory and practise of AVO analysis: Soc. Expl. Geophys Mallick S A simple approximation to the P-wave reflection coefficient and its implication in the inversion of amplitude variation with offset: Geophysics Miles D. R. and Gassaway G. S Three-component AVO analysis: 59th Ann. Internat. Mtg. Soc. Expl. Geophys. Expanded Abstracts Ostrander W. J. and Gassaway G The use of offset dependent reflectivity in exploration: 53rd Ann. Internat. Mtg. Soc. Expl. Geophys. Expanded Abstracts 637. Richards P. G. and Frasier C. W Scattering of elastic waves from depth dependent inhomogeneities: Geophysics Shuey R. T A simplification of the Zoeppritz equations: Geophysics Smith G. G parameter Geostack: 66th Ann. Internat. Mtg. Soc. Expl. Geophys. Expanded Abstracts Smith G. G. and Gidlow P. M Weighted stacking for rock property estimation and detection of gas: Geophys. Prosp Swan H. W Properties of direct AVO hydrocarbon indicators in Castagna J. P. and Backus M. M. Eds. Offset dependent reflectivity Theory and practise of AVO analysis: Soc. Expl. Geophys Zaengle J. F. and Frasier C. W Correlation and interpretation of P-P and P-SV data Zamora gas field Yolo County California in Castagna J. P. and Backus M. M. Eds. Offset dependent reflectivity Theory and practise of AVO analysis: Soc. Expl. Geophys APPENDIX A EXACT EQUATION FOR P-SV REFLECTION COEFFICIENT Aki and Richards 1980) give the following expression for the exact P-SV reflection coefficient as a function of the ray parameter p: R PS p) = where cos θ 1 α 1 ab + cd cos θ cos φ α β β 1 D) a = ρ 1 β p ) ρ 1 1 β 1 p ) b = ρ 1 β p ) + ρ 1 β1 p c = ρ 1 1 β 1 p ) + ρ β p d = ρ β p ρ 1 β1) D = EF + GHp ) α 1 p A-1) E = b cos θ 1 + c cos θ α 1 α F = b cos φ 1 + c cos φ β 1 β G = a d cos θ 1 cos φ α 1 β H = a d cos θ cos φ 1. α β 1 In these equations θ 1 and θ represent the incident and transmission angle for the P-waves; φ 1 and φ represent the reflection and transmission angles for an SV-wave: α 1 β 1 and ρ 1 represent the P and S velocities and density of the incident medium; and α β and ρ represent the same elastic parameters in the transmission medium. APPENDIX B AHIGH-CONTRAST APPROXIMATION FOR P-SV REFLECTION COEFFICIENT The first step in deriving this approximation is substituting the cosine functions in equation A-1 with their corresponding expressions in p: cos θ 1 = 1 p α1 cos θ = 1 p α cos φ 1 = 1 p β1 cos φ = 1 p β. After substituting the cosine functions and doing some algebraic manipulations we expand the entire expression in terms of p in a Taylor series up to the third order. We then collect the terms in the new expression in terms of powers of p and substitute p with sin θ 1 /α 1 where θ 1 is the incidence angle) to obtain the following nearly exact approximation for low incidence angles: where and B 1 = with R [1] PS p) A 1 sin θ 1 + B 1 sin 3 θ 1 A 1 = 1 ) α β ρ ρ + ρ 1 µ I α I β B-1) B-) µ α 3 1 R α R β [M 1 + M + M 3 + M 4 M 5 + M 6 + M 7 )] M 1 = 4 ρ + ρ) β 1 M = α α β β + β ) ρ1 4 µ ) 1 M 3 = α 1 ρ ρ + ρ ) 1 µ β 1 µ α β

14 1734 Ramos and Castagna ) M 4 = ρ ) 1 ρ + ρ β 1 R α R β α β µ M 5 = ρ µ ) ρ µ ) α β 1 α 1 β ) 4 µ α M 6 = R ρ 1 β 4 µ + α 1 ρ α α 1 ) M 7 = R α 4 µ β ρ 1 β 4 µ + β 1 ρ β 1 and R α = I α α 1 α R β = I β β 1 β I α = α 1ρ 1 + α ρ I β = β 1ρ 1 + β ρ µ = β ρ β 1 ρ 1.