Amplitude analysis of isotropic P-wave reflections

GEOPHYSICS, VOL. 71, NO. 6 NOVEMBER-DECEMBER 2006; P. C93 C103, 7 FIGS., 2 TABLES. 10.1190/1.2335877 Amplitude analysis of isotropic P-wave reflections Mirko van der Baan 1 and Dirk Smit 2 ABSTRACT The analysis of amplitude variation with offset AVO of seismic reflections is a very popular tool for detecting gas sands. It is assumed in AVO, however, that plane-wave reflection coefficients can be used directly to analyze amplitudes measured in the time-offset domain. This is not true for near-critical angles of reflection. Plane-wave reflection coefficients incorporate the contribution of the head wave. A plane-wave decomposition such as a proper -p transform must be applied to the seismic data for accurate analysis of reflection coefficients near critical angles. Amplitudes after plane-wave decomposition are related directly to the planewave reflection coefficients; geometric-spreading corrections are no longer required, and polarization effects of P-P reflections recorded on the z-component are also removed. Conventional, linearized expressions for the isotropic P-Pwave reflection coefficient depend on contrasts in three parameters, and they require background information about average P-wave/S-wave velocity ratios. We derive a new reduced-parameter expression that depends only on two free parameters without loss of accuracy. No extra prior parameter information is needed either. The reduction in free parameters is achieved by explicitly incorporating P-wave moveout information. A new AVO strategy is developed that requires moveout analysis of three reflections: the target horizon, the reflections directly above and below the target horizon, and the amplitudes of the target horizon. The new AVO expression can be used in the time-offset domain for precritical arrivals and in the -p domain for precritical and critical reflections. INTRODUCTION In exploration seismology, the analysis of amplitude variation with offset AVO of seismic reflections is a very popular and powerful tool for detecting the presence of gas sands. Using the Zoeppritz equations, Ostrander 1984 shows that specific variations of the reflection coefficient with the angle of incidence are strongly indicative of the presence of gas sands. He thereby makes amplitude analysis a direct hydrocarbon indicator. The Zoeppritz equations are nonlinear and very cumbersome. Fortunately, linearized expressions for the isotropic P-P-wave reflection coefficient have existed for a long time Bortfeld, 1961; Richards and Frasier, 1976. The inconvenience of the linearized reflection coefficient is that it depends on perturbations in three parameters e.g., the differences in the P-wave velocity, the S-wave velocity, and the density across the interface and requires prior information about Poisson s ratio. Well control is needed to provide such prior information. We present a newavo strategy that involves measurable quantities only instead of Poisson s ratios or other prior information. Only two parameters can be extracted robustly with an AVO analysis involving moderate angles say, up to 35 while using the linearized reflection coefficients Stolt and Weglein, 1985; de Nicolao et al., 1993. As a result of improvements in acquisition quality and longer spread lengths, there is a push to analyze P-P-wave amplitude behavior up to the critical angle because it may then become possible to retrieve all three parameters with sufficient accuracy Wang, 1999. While it may be possible to retrieve all three desired parameters using a very-long-offset AVO analysis involving critical angles, two questions arise: 1 Are the Zoeppritz plane-wave reflection and transmission coefficients appropriate for an AVO analysis up to the critical angle in the time-offset domain? 2 Can we derive alternative reduced-parameter expressions that are as accurate as the conventional three-parameter ones but depend on fewer free parameters? We show first that amplitude analysis should be done in the domain of intercept time versus radial slowness -p r instead of the time-offset domain. This is because, near critical angles, there is a significant difference between the actual measured amplitudes of reflections in the time-offset domain and those predicted by assuming Manuscript received by the Editor December 14, 2004; revised manuscript received March 20, 2006; published online October 9, 2006. 1 University of Leeds, School of Earth and Environment, Department of Earth Sciences, Leeds, LS2 9JT, United Kingdom. E-mail: mvdbaan@earth.leeds. ac.uk. 2 Shell International Exploration and Production, Volmerlaan 8, Postbus 60, 2280AB Rijswijk, The Netherlands. E-mail: dirk.smit@shell.com. 2006 Society of Exploration Geophysicists. All rights reserved. C93

C94 van der Baan and Smit the plane-wave reflection coefficients apply directly in this domain. The latter is the assumption made in conventional AVO analyses. The -p r transform is a plane-wave decomposition Chapman, 1981; Brysk and McCowan, 1986. It reconstructs the amplitudes of the transient plane waves, thereby allowing us to use the plane-wave reflection coefficients for accurate AVO analysis at both precritical and critical angles. Plane-wave decomposition has the additional advantage that it corrects automatically and simultaneously for the geometric spreading of all wave modes and types i.e., all pure-mode and convertedwave reflections and multiples in laterally homogeneous media without the need for traveltime picks or velocity information Wang and McCowan, 1989; Van der Baan, 2004. As a further advantage, we prove here that a plane-wave decomposition also corrects for isotropic P-P-wave polarization effects on the vertical component. We next introduce a new linearized equation for the isotropic P-Pwave reflection coefficient that depends on two free parameters only while retaining the same accuracy as the linearized expression of Richards and Frasier 1976. The popular Shuey equation Shuey, 1985 is a reformulation of the Richards and Frasier expression. The number of free parameters is reduced by explicitly incorporating traveltime information in the amplitude analysis. For completeness, we also derive a reduced-parameter expression for the linearized transmission coefficient. We therefore take the stance that it is better to estimate two out of two free parameters robustly using a reducedparameter reflection coefficient than it is to try to recover three free Figure 1. Waveforms for the reflection of a single interface predicted by two different approaches: a a primaries-only reflectivity method and b ray tracing. The reflectivity section displays the correct amplitude behavior, whereas the ray-traced section assumes that the plane-wave reflection coefficients are related directly to the amplitudes of primary reflections as observed in the time-offset domain. The amplitudes deviate most near the critical angle occurring at an offset of 2.7 km. See also Figure 2 for the extracted amplitudes along the primary reflection. parameters with large associated uncertainties using conventional linearized AVO expressions. We demonstrate first that the Zoeppritz plane-wave reflection coefficients do not describe accurately the amplitudes of reflections near critical angles. We then show how amplitudes of critical reflections can be analyzed accurately by applying a plane-wave decomposition, such as a proper -p transform, on the data. Next, we derive the new reduced-parameter expressions for the linearized reflection and transmission coefficients and outline the proposed inversion strategy. Finally, we present a synthetic example. PLANE-WAVE REFLECTION COEFFICIENTS AND TIME-OFFSET AMPLITUDES In AVO analysis, plane-wave reflection and transmission coefficients are used to analyze amplitudes in the time-offset domain after geometric-spreading, polarization, and source-and-receiver-array corrections have been applied Ostrander, 1984. However, true amplitudes in the time-offset domain and those predicted by planewave reflection and transmission coefficients may deviate considerably, especially near critical angles. Figure 1 displays a primary reflection in a simple two-layer model. The seismograms are created using two different techniques. Figure 1a displays the true waveforms of the primary reflection and associated head wave i.e., refraction along the interface. It was constructed using a partial implementation of the reflectivity method that computes primary reflections with their associated head waves only. The reflectivity method produces exact waveforms in horizontally layered media Chapman and Orcutt, 1985. Figure 1b was created by means of ray tracing by assuming the plane-wave reflection coefficients hold in the time-offset domain for spherical waves. It reproduces the waveforms as anticipated by a conventional time-offset AVO analysis. The medium parameters are identical to those of model 1 in Table 1 except that the P-wave velocity of the lower half-space is increased to 3.0 km/s to create a visible separation between the head wave and the primary. The first layer is 1 km thick. Figure 2a contains the absolute amplitudes extracted along the two synthetic gathers, and Figure 2b displays the actual absolute reflection coefficients. A polarity reversal occurs twice at incidence angles of 25 and 49. Amplitudes are extracted from the envelope of the waveforms for added accuracy. Several noticeable differences occur between the two synthetic shot gathers. The first polarity reversal causes the reflection to disappear around an offset of 900 m in both the true and the ray-traced gathers Figures 1 and 2a. On the other hand, only a weakening of the reflection occurs at the second polarity reversal in the reflectivity gather, whereas the ray-traced gather again predicts a total absence of energy. Geometric ray tracing is a high-frequency approximation. It does not take into account the effect of the Fresnel zone. The phase rotation that occurs in the reflection coefficients around the critical Table 1. Elastic parameters of the two models considered. Model 1 is also considered in Mallick (1993). Model 1 km/s 2 km/s 1 km/s 2 km/s 1 10 12 g/m/s 2 2 10 12 g/m/s 2 1 g/cm 3 2 g/cm 3 1 2.40 2.55 0.98 1.755 2.113 6.160 2.2 2.0 2 2.40 2.55 0.98 1.255 2.113 3.150 2.2 2.0

P-wave amplitude analysis C95 angle combined with the high amplitudes of postcritical reflections turns the second polarity reversal anticipated by ray theory into a zone of weak but nonzero amplitudes. Afurther discrepancy between the two shot gathers is that no head wave i.e., refraction along the interface is present in the ray-traced section. Head waves are not included in the zeroth-order ray approximation, which is the basis for all ray tracers Červený, 2001, sections 5.6 and 5.7. The final difference is that the ray-traced section shows a sudden increase in the amplitudes around the critical angle, whereas the amplitudes in the reflectivity section are much smoother along the reflected arrival. The actual maximum amplitude occurs at a much larger offset than the point of critical reflection Figure 2a. The abrupt increase in the amplitudes in Figure 1b reflects the sudden increase visible in the reflection coefficient near the critical angle Figure 2b. However, the plane-wave reflection coefficients are related to the total energy of all waves with a common incidence angle. The plane-wave reflection coefficient around the critical angle thus includes both the contribution of the primary reflection and the head wave. Therefore, amplitudes of reflected arrivals in the timeoffset domain cannot be compared directly with those predicted by the plane-wave reflection coefficients if we are close to a critical angle. This is evident in Figure 2a. What is the relevance of this discussion for AVO analysis? de Nicolao et al. 1993 show that only two free parameters can be estimated from the linearized Richards-Frasier equation for the isotropic P-P reflection coefficient, whereas three free parameters actually govern its behavior the contrasts in density and P- and S-wave velocities. Wang 1999 argues that it is possible to retrieve all three parameters by analyzing amplitudes up to the critical angle. While this may be true, amplitudes of spherical waves in the time-offset domain near critical angles do not correspond to plane-wave amplitudes, as shown in Figures 1 and 2. An AVO analysis based on picking amplitudes of primary reflections in the time-offset domain therefore yields biased estimates for medium parameters if near-critical angles are included, although no problems occur in this case if analyzed angles are limited to approximately 35 Figures 2a and b. How then can we properly analyze amplitudes of reflected waves near critical angles? The Zoeppritz equations are derived for a plane wave impinging on an interface. A plane-wave decomposition, such as a proper -p r or f-k transform, therefore must be applied on the data first. This leads to an analysis of amplitude variation with slowness AVS. The seismic response for a given earth model is significantly simpler in the -p domain than it is in the time-offset domain. Therefore, many workers advocate that seismic inversion techniques should be applied to slant-stacked data Müller, 1971; Fryer, 1980; Treitel et al., 1982. An AVO analysis is nonetheless done conventionally in the time-offset domain. -p DOMAIN AVS ANALYSIS The seismic response of an explosive source in a laterally homogeneous isotropic earth, as recorded on the vertical component of a receiver positioned on the surface, can be expressed in the frequency domain as an integral over slowness Fuchs and Müller, 1971; Fryer, 1980: u z,r = S 2 R pp,tot,p r J 0 p r rp r dp r, 1 0 with u z the vertical displacement at the geophone, circular frequency, r offset, S the source spectrum, p r horizontal slowness, and J 0 the zeroth-order Bessel function. The remaining quantity R pp,tot is the plane-wave P-wave reflection response of the medium. This is the joint contribution of all primary reflections, multiples, and interconversions that leave the source as a P-wave and are recorded again as a P-wave. If we assume the amplitudes of the interconversions are negligible and all multiple reflections have been removed, then the reflection response R pp,tot is determined entirely by the total reflection coefficients of the primary P-P-wave reflections R pp,tot and the phase de- Figure 2. a Extracted absolute amplitudes A along the primary reflection constructed using either a reflectivity approach from Figure 1a, solid line or ray tracing from Figure 1b, dashes. babsolute plane-wave reflection coefficients R pp as a function of incidence angle. The reflection coefficients between 25 and 49 are negative, and a phase rotation occurs for postcritical reflections beyond 50. The ray-traced amplitudes are those anticipated by a time-offset AVO analysis, while the reflectivity technique reproduces the amplitudes that actually occur. The discrepancy is largest near the critical angle.

C96 van der Baan and Smit lays k p r for all interfaces k. The reflection response R pp,tot inamedium with n layers is then given by Fuchs and Müller, 1971 R pp,tot,p r = n k=1 R pp,tot p r expi k p r. The total reflection coefficient R pp,tot for interface k is the interface reflection coefficient R pp multiplied with the up- and downgoing transmission coefficients of all interfaces in the overburden: R pp,tot k p r = i=1 T` i 1 pp p r T i 1 pp p r R pp p r, with T` pp and T pp the transmission coefficients for, respectively, the down- and upgoing P-waves at interface k. By definition, T` 0 0 pp = T pp = 1. The phase delay k contains all traveltime information for primary reflection k. It is related to the sum of all vertical P-wave slownesses times the individual layer thicknesses Chapman, 1978. The integral in expression 1 is a Fourier-Bessel transform Chapman, 1981. One of the beauties of this transform is that it is its own inverse. In other words, a function gr and its transform pair Gk r are related by Figure 3. AVS analysis. a The true waveforms produced by the reflectivity technique Figure 1a are transformed to the -p r domain. No geometric spreading or polarization corrections are applied. b The amplitudes along the p r moveout curve are exactly equal to the plane-wave reflection coefficients after correction for source strength. The discrepancy at the largest slownesses is caused by the finite spread length of the time-offset section. 2 3 and gr = GJ 0 k r rk r dk r 0 Gk r = grj 0 k r rrdr, 0 with k r the horizontal wavenumber. Consequently, R pp,tot,p r = S 1 u z,rj 0 p r rrdr, 0 where the change of variables k r = p r is used explicitly. The integral in expression 6 corresponds to the mathematical formulation of a cylindrical -p r transform if an additional inverse Fourier transform over frequency is applied on the plane-wave reflection response R pp,tot,p r Chapman, 1981; Brysk and McCowan, 1986. This produces the transient i.e., time-related plane-wave response R pp,tot,p r of the medium from the measured vertical displacement field u z,r in the frequency domain. The amplitudes of the vertical displacement field u z t,r are affected by geometric spreading and the polarization of the arrivals at the receiver level. That is, there is an energy partitioning between the horizontal and vertical components that depends on the incidence angle at the surface. The plane-wave reflection response of the medium R pp,tot, on the other hand, depends only on the reflection and transmission coefficients and the phase delays equations 2 and 3. It does not contain any corrections for geometric spreading or polarization. Expression 6 states that we retrieve the plane-wave reflection response R pp,tot,p r of the medium by applying a cylindrical -p r transform on the vertical component data. This transform therefore removes all geometric spreading and polarization effects of isotropic P-P-wave reflections. The amplitudes measured along the primary reflections in the -p r domain after correction for the source spectrum S are identical to the total reflection coefficients given by expression 3. Van der Baan 2004 provides more background on why a plane-wave decomposition such as a proper -p transform removes geometric spreading in layered media. Figure 3a shows the shot gather displayed in Figure 1a after applying a cylindrical -p r transform, and Figure 3b dashes displays the amplitudes extracted along the primary reflection after correction for the source strength. No corrections for geometric spreading or polarization are applied at any stage. The extracted amplitudes are exactly identical to the Zoeppritz plane-wave reflection coefficients except at the largest horizontal slownesses. The discrepancy here is caused by the finite spread length of the original time-offset shot gather Figure 1a. Several assumptions go into the derivation of expression 1. The medium is laterally homogeneous, isotropic, and elastic. No converted-wave energy is recorded on the vertical component, and freesurface effects are negligible. The source has an isotropic radiation pattern that does not depend on horizontal slowness. Similarly, the receiver-array response has been removed. Furthermore, the source and receivers should be in the same layer; otherwise, polarization effects prevail. Note that the polarization effects are corrected only for 4 5 6

P-wave amplitude analysis C97 P-P reflections on the vertical component. This does not happen after applying a cylindrical -p r transform on the radial component or for converted waves in general. Marine data consist predominantly of pressure measurements. Appendix A describes how to convert pressure data to vertical displacement. Equation A-4 performs the conversion. Therefore, an AVS analysis is a viable alternative to AVO. First, source- and receiver-array corrections are applied and the source strength is normalized. Next, the 2D data are transformed to the -p r domain and the amplitudes of the target horizons are extracted. These amplitudes can be compared directly with total reflection coefficients R pp,tot from equation 3, without further corrections. An AVS analysis is applicable to all incidence angles, including nearcritical ones. AnAVS strategy is thus indispensable if amplitudes up to the critical angles are to be analyzed using the Zoeppritz equations to determine all three contrasts in medium parameters. However, we need very long offsets to record critical arrivals because critical angles only occur at relatively short offset/depth ratios for large contrasts in the P-wave velocity. For instance, the P-wave velocity increased here from 2.4 3.0 km/s, yielding a critical angle of 53 occurring at an offset/depth ratio of 2.7. Head waves also affect the total amplitude, and they cannot be muted out if near-critical angles are included in theavs analysis. In view of the large offset/depth ratios required, it is unlikely that all three free parameters can be determined accurately using conventional acquisition geometries. In the next section, we derive alternative reduced-parameter expressions that are as accurate as the conventional three-parameter ones but depend on fewer free parameters. REDUCED-PARAMETER EXPRESSIONS Conventional AVO expression The exact expression for the P-P-wave reflection coefficient in an isotropic medium is rather cumbersome e.g., Aki and Richards, 2002. It can be written in the form Mallick, 1993; Wang, 1999 R pp p r = A + Bp 2 r + Cp 4 6 r Dp r E + Fp 2 r + Gp 4 6 r + Dp, r with p r the horizontal slowness. Most coefficients A, B, C, etc., depend on the vertical P- and S-wave slownesses q,i and q,i in layers i and i + 1. Linearization of this expression in terms of the differences in P- and S-wave velocities and and density across the interface leads to Richards and Frasier, 1976 R pp p r 1 1 4 2 p 2 2 r + 1 2 1 p 2 r 2 1 4 2 p r 2. See also Bortfeld, 1961. For notational convenience, we assume a two-layer model for the moment; the interface of interest is not mentioned explicitly. The abbreviations f = f 2 f 1 for the contrast and f = f 2 + f 1 /2 for the average are used in this expression, with f 2 a parameter in the lower medium e.g., density and f 1 the same parameter in the upper medium. 7 8 This expression also can be written in terms of the difference in shear modulus = 2 22 1 12 across the interface. Given that 2 +2, the linearized reflection coefficient 8 can also be expressed as R pp p r 1 2 + + p 2 r 2 21 p 2 r 2 2 pr 2 9 10 after slightly reordering terms. The first term in this expression is related to the relative change in acoustic impedance Z because 1 2 + 1 Z 2 Z = 2 2 1 1 = R 0. 11 2 2 + 1 1 Therefore, the first term in the right-hand side of equation 11 represents the zero-slowness i.e., vertical-incidence reflection coefficient R 0. Two-parameter approximations Shuey 1985 shows that the linearized expression of Richards and Frasier 1976 for the P-P-wave isotropic reflection coefficient R pp can be written R pp p A + B sin 2 p + C tan 2 p sin 2 p, 12 with p the average of the incident and transmitted P-wave angles in, respectively, the upper and lower mediums. He originally expressed it in terms of contrasts in the P-wave velocity, density, and Poisson s ratio. The same form can be obtained if contrasts in P-wave velocity, density, and shear modulus are used as perturbation parameters. This leads to e.g., Thomsen, 1993 R pp p 2 1 + + 1 2 2 2 sin 2 p + 1 2 tan2 p sin 2 p. 13 The significance of Shuey s formulation 12 is that the three separate A, B, and C terms are related to amplitude behavior at vertical, intermediate, and near-critical angles of incidence, respectively. Term A in formulation 12 thus corresponds to the zero-offset reflection coefficient, i.e., A = R 0. The three-term Richards-Frasier equation is seldom used in AVO analysis. It is reduced in practice to a two-parameter expression. This can be done by assuming that only arrivals with short to intermediate incidence angles are recorded, thereby allowing us to neglect C. We then use the form of the Shuey formula only and invert for A intercept and B gradient without relating them directly to the actual medium parameters. This leads to the intercept-gradient crossplot technique Castagna et al., 1998. Alternatively, prior assumptions about coupling in changes between relative variations in medium parameters can be incorporated Smith and Gidlow, 1987. We could assume, for instance, that the relative variations in P-wave velocity and density are related linearly Gardner et al., 1974:

C98 van der Baan and Smit = k c, 14 with k c a prior known parameter. Inserting this relation into expression 13 also produces a two-parameter linearized reflection coefficient. The accuracy of the resulting expression depends in practice on the precision of our prior knowledge of the coupling parameter k c near the target horizon and our estimate of the average velocity to convert offsets to angles. Alternative formulation It is possible to derive an alternative two-parameter expression for the linearized reflection coefficient by analyzing the p r moveout curves of the target horizon and both directly overlying and underlying reflections. The -p r transform is given by = t p r r, 15 with the intercept time, t the two-way traveltime, p r the horizontal slowness, and r the offset. The total intercept time is related in laterally homogeneous media to the sum of the individual layer thicknesses z i times the vertical P-wave slownesses q,i in each layer i Diebold and Stoffa, 1981; Schultz, 1982: = i i =2 i z i q,i =2z i 2 i p 2 r 1/2. i 16 It follows from expression 16 that the normalized interval intercept time d i is directly related to the vertical P-wave slowness q,i by d i = i = 2z iq,i 0,i 2z i = i q,i, 17 i with i the two-way interval intercept time in layer i and 0,i the associated value at zero slowness Figure 4. Measuring the ratios d i therefore directly gives us information about the vertical P-wave Figure 4. Analysis of the p moveout curves yields information on the vertical P-wave slownesses in each layer. This is obtained by calculating the ratios d i = i / 0,i, where i represents the interval intercept times between two reflections and 0,i is the corresponding value at zero slowness. If the first reflection is the target horizon, then the AVO expression depends on the ratios d 1 and d 2. The ratios d 2 and d 3 are involved if the second reflection is of interest. The transmission effect through the overburden also needs to be taken into account in the latter case. slowness above and below the interface and thereby the incidence and transmission angles of the P-waves. Thus, it is unnecessary to linearize the vertical P-wave slownesses q,i in the exact Zoeppritz equation 7. Inserting the normalized interval intercept times d i in the exact equation 7 for the reflection coefficient and linearizing the resulting expression in terms of perturbations,, and yields 2 d 1 d 2 R pp p r 2 d 1 + d 2 + + d 1 d 2 d 1 + d 2 d 1 d 2 8 2 d 1 + d 2 2. 18 pr The third term in the new linearized reflection coefficient 18 is the only part that depends explicitly on the horizontal slowness p r. The factors d i depend implicitly on the horizontal slowness but are measured and thus are assumed to be known. No small-slowness i.e., small-angle approximation is made in the derivation. Matching the coefficients in this linearized expression with those of the reformulated linearized reflection coefficients 10 shows that equation 18 can be simplified further to R pp p r 1 Z 2 Z + d 1 d 2 2 d 1 + d 2 pr 2. 19 Writing out the second term in equation 19 and neglecting higherorder terms in shows that it is identical to the second term in the linearized reflection coefficient 10. The accuracy of all linearized expressions 10, 18, and 19 is highly similar except possibly for large P-wave velocity contrasts across the interface. The inclusion of moveout information thus allows us to remove explicitly the term related to the relative P-wave velocity contrast / in the linearized reflection coefficient 10, thereby leading to a new AVO expression that depends on two free parameters only. The first and second terms in the new linearized AVO equation 19 are determined predominantly by the acoustic properties of the medium. The only free parameter in these terms is the relative change in acoustic impedance Z/Z equation 11. The second term is determined from a moveout analysis of the target horizon and the reflections directly above and below it. Thus, the second term is understood to be known. The third term is the only part solely related to the elastic properties of the medium and depends explicitly on the horizontal slowness. The free parameter in this term is /. This parameter could be expressed in terms of changes in the density and S-wave velocity using expression 9, but this would yet again lead to a three-parameter expression, while only two parameters can be recovered from the linearized reflection coefficient Stolt and Weglein, 1985; de Nicolao et al., 1993. The accuracy of the reduced-parameter expression 19 depends in practice on the precision of the moveout analysis and the derived normalized ratios d 1 and d 2. A comparison of expression 19 with Shuey s formulation 12 and 13 shows that the second term in the newavo equation is related to terms B and C terms in the Shuey formulation. All other terms are identical. Therefore, it accounts for both intermediate and near-critical angles of propagation. This confirms that no short-offset approximation is made in its derivation.

P-wave amplitude analysis C99 Appendix B contains the reduced-parameter equations for the transmission coefficients. The derived expressions B-4 and B-5 depend on a single parameter only; the relative change in acoustic impedance Z/Z. PRACTICAL IMPLEMENTATION How are the new reduced-parameter transmission and reflection coefficients 19, B-4, and B-5 best used to invert for the actual medium parameters in AVO analysis? Either a time-offset approach or a -p r -domain approach can be chosen. A time-offset implementation is appropriate as long as near-critical angles are not included. Time-offset domain For a time-offset-based approach, we pick the amplitudes and traveltimes of the reflections after geometric spreading, polarization, and source- and receiver-array corrections. The required horizontal slowness p r is given by the slope of the moveout curves, i.e., p r = t/r. It is recommended that first an analytic curve be fitted through the traveltime picks. Otherwise, small picking errors will yield large uncertainties in the estimated slope and thus the horizontal slowness p r. The p r moveout curves are then computed using expression 15. Layer stripping of the p r moveout curves yields the interval intercept times i p r. The required ratios d i are obtained after normalization with the zero-slowness value 0,i Figure 4 and equations 16 and 17. At this point we have two options: either we ignore the transmission effect through the overburden and directly analyze the local interface reflection coefficients R pp equation 19, or we include the transmission coefficients in our analysis and invert for the total reflection coefficients R pp,tot equation 3. If we include the transmission effect through the overburden, then the total reflection coefficient R pp,tot for interface k is computed using expression 3, where the transmission coefficients at each preceding layer are given by equations B-4 and B-5. Expressions B-4 and B-5 depend on previously established relative impedance contrasts Z/Z only, and no extra inversion parameters are required to account for the transmission effect through the overburden. On the other hand, if we ignore the transmission effect, then the total reflection coefficient R pp,tot is equated to the interface reflection coefficient R pp equation 19. In a top-down fashion, we then invert for the free parameters Z/Z and / at each interface. This approach is similar to the one adopted by Buland et al. 1996. Alternatively, a crossplot technique can be devised. The amplitudes of the target horizon are extracted and then corrected for the moveout of the target horizon and the reflections directly above and below by subtracting the second term in equation 19. The moveoutcorrected amplitudes are plotted versus horizontal slowness squared p r2. The resulting line should have intercept Z/2Z and gradient 2/. The extracted intercept and gradient can then be scrutinized as in a conventional crossplot analysis. The advantage of this approach over conventional crossplot analyses is that prior information about average Poisson s ratios or P-wave/S-wave velocity ratios is not required, and no offset-to-angle conversion is needed. The resulting gradient and intercept value relate directly to the medium properties. In addition, no small-offset approximation has been made, i.e., the C term in Shuey equation 12 has not been neglected. -p r domain In a -p r -based approach, we proceed in a similar way. The z-component common-midpoint CMP gathers are transformed to the -p r domain after source- and receiver-array corrections have been applied. No geometric spreading or polarization corrections are required. The cylindrical -p r transform automatically corrects for these effects. Next, the amplitudes and the p r moveout curves of the reflections are picked. The amplitudes correspond directly to the total reflection coefficients see Figure 3. We can then proceed as before by inverting for the medium parameters using a layer-stripping approach or by analyzing horizons of interest using crossplots of intercept value versus gradient. NUMERICAL EXAMPLE We consider two numerical examples to study the accuracy of the new AVO expression 19 and to demonstrate the developed novel AVO strategy. Accuracy of reduced-parameter expressions Two synthetic examples illustrate the accuracy of the new AVO approximation 19 compared with the exact expression 7 and approximation 10 of Richards and Frasier 1976. All Shuey-type approximations Shuey, 1985 are based on the original Richards-Frasier approximation and have the same accuracy unless additional approximations have been made e.g., short-offset assumptions. Table 1 displays the P- and S-wave velocities and the density above and below the interface. The only difference between the two models is that the jump in S-wave velocity is considerably smaller in the second case. Figure 5a and b displays the reflection coefficients for models 1 and 2, respectively, and Figure 5c and d displays the corresponding transmission coefficients. All linearized approximations perform equally well, as anticipated. They lead to accurate predictions if the relative contrasts in all elastic parameters are small to moderate model 2, even though the jump in shear modulus is large. The approximations are less accurate for the large-contrast model model 1, but they do predict the overall behavior of the reflection coefficient up to the point of critical incidence. Multilayer example Finally, we present a synthetic example to illustrate the whole procedure. The exact waveforms of the P-wave primaries are computed for a 14-layer model based on a North Sea well log. Table 2 displays the model, which is adapted from Ursin and Dahl 1992. Figure 6a shows the resulting shot gather in the time-offset domain. Random noise is imposed to render the example more realistic. A total spread length of 5 km is considered, with a receiver spacing of 25 m. The target horizon is the weak twelfth reflection at 2 s zero-offset traveltime. It has a maximum offset-depth ratio of 2.35. This is a challenging target because it disappears in the noise for offsets larger than 3km. The shot gather is first transformed to the -p r domain by means of a cylindrical -p r transform Figure 6b. The target horizon is better visible in this domain because the transform attenuates some of the random noise. Next, the amplitudes of the target horizon are extracted along with the p r moveout curves of the target and the re-

C100 van der Baan and Smit flections directly above and underneath. Moveout analysis using the layer-stripping approach of Schultz 1982 leads to estimated interval velocities of 2.83 and 2.74 km/s above and below the target horizon, respectively. True values are 2.75 and 2.64 km/s. The second term in reduced-parameter expression 19 is then subtracted from the extracted amplitudes, and the resulting values are plotted versus horizontal slowness squared Figure 7, dashes. No Figure 5. Comparison of the different approximations for the P-P-wave isotropic reflection a, b and transmission c, d coefficient as a function of horizontal slowness for two synthetic examples. Model 1 a, c has a large S-wave contrast, whereas the contrast is much smaller for model 2 b, d. All linearized approximations have nearly the same accuracy. Solid line is exact reflection and transmission coefficients. Dashed line is Richards-Frasier approximations 10 and B-2. Dot-dashed line is intermediate reduced-parameter expressions 18 and B-3. Dotted line is final reduced-parameter reflection and transmission coefficients expressions 19 and B-4. geometric spreading or polarization corrections are applied. The corrected amplitudes will lie on a straight line if the second moveout-related term in expression 19 has been removed correctly. Figure 7 also displays the exact interface reflection coefficient after correction for the second term in equation 19 while ignoring the transmission effect through the overburden solid line. Finally, Figure 7 shows the amplitudes extracted along the target horizon after perfect correction of the moveout term using the exact interval velocities when no noise is superimposed on the shot gather dots. The latter curve overlaps the theoretical interface reflection coefficient solid line if the transmission effect through the overburden is removed. The scatter in the amplitudes of the noise-contaminated case dashes can be attributed to the presence of the superimposed noise in the shot gather. The difference in gradients between the noise-free case dotted line and the noise-contaminated case dashed line is the result of the imperfect correction for the moveout term in equation 19. Correct estimation of the interval velocities centers the noise-contaminated results around those for the noise-free scenario. The interface reflection coefficients solid line differs from the noise-free extracted amplitudes dotted line after correction because the latter includes the transmission effect through the overburden. The transmission effect is often ignored because it has only a small effect on the recorded amplitudes Ursin and Dahl, 1992 and the difference between the two curves is visibly small. After removing the second moveout-related term in equation 19, the intercept Z/2Z and gradient 2/ can be extracted from Figure 7 and scrutinized as in a conventional crossplot analysis. Table 2. Synthetic model adapted from Ursin and Dahl (1992) based on a North Sea well log. Depth km km/s km/s g/cm 3 0.2500 1.7400 0.3900 1.7000 0.4000 1.8500 0.6200 1.8000 0.5000 1.9400 0.7800 1.9000 0.6600 2.1400 0.8600 1.9500 0.7500 2.2200 0.8900 2.0000 0.7900 2.0000 1.0000 1.8000 0.8900 1.9900 0.9900 1.8500 1.0800 1.9000 0.9500 2.0000 1.3500 2.2000 1.1500 2.1500 1.5200 2.0500 1.1300 2.3000 1.8300 2.6500 1.5000 2.3500 2.1300 2.7500 1.5300 2.4000 2.2400 2.6400 1.4900 2.3000 3.0000 1.6000 2.5500 DISCUSSION The standard linearized reflection coefficient of Richards and Frasier 1976 depends on contrasts in three parameters; de Nicolao et al. 1993 show that only two free parameters can be extracted robustly in an AVO analysis involving moderate angles. Wang 1999 argues that all three parameters can be recovered if amplitudes are analyzed up to the critical angle. The plane-wave reflection coefficient can be used only to analyze amplitudes of reflected waves with small to moderate reflection angles in the time-offset domain. An AVO analysis in the time-offset domain involving the plane-wave reflection coefficients and nearcritical angles of incidence yields erroneous results, as clearly seen in Figure 2a. We first must apply a plane-wave decomposition such as a proper -p transform on the data to analyze amplitudes of near-critical arrivals correctly. This leads to an AVS analysis. The appropriate -p transform depends on acquisition geometry and source type van der Baan, 2004. A cylindrical -p r transform is to be used for 2D data resulting from a point source. Furthermore, any forward modeling approach that assumes plane-wave reflection coefficients can be used directly to compute

P-wave amplitude analysis C101 amplitudes of spherical waves in the time-offset domain yields unrealistic amplitudes around critical angles. Care therefore should be used when closely examining amplitudes predicted by, for instance, ray-tracing codes Figures 1 and 2. What are the advantages ofavs overavo analysis? InAVS analysis, we measure horizontal slowness; no offset-to-angle conversion is required. Furthermore, no geometric spreading nor polarization corrections are needed. The cylindrical -p r transform automatically corrects for these effects. A 1D assumption naturally underlies these corrections, but the most commonly applied geometricspreading corrections used in AVO analysis also assume a 1D earth Newman, 1973; Ursin, 1990. The most important inconvenience of -p r domain-based amplitude analysis is that the reconstruction quality of the plane-wave amplitudes is influenced strongly by total spread length and trace spacing. The number of aliasing artifacts introduced is linked directly to the interval and regularity of the trace spacing. Moreover, the head waves must be measured over a sufficiently long interval to reconstruct the reflection coefficients of the near-critical angles accurately. In view of the generally small relative contrasts in P-wave velocities encountered in most AVO analyses, it seems unlikely that the near-critical reflection coefficients can be reconstructed using current conventional acquisition geometries. This casts doubt on whether all three free parameters can be retrieved by AVO analysis. Therefore, it is common practice to reduce the uncertainty in AVO parameter estimation either by making a short-offset approximation and inverting solely for the A and B terms in Shuey formulations 12 and 13 or by explicitly including prior information, e.g., by assuming that relative changes in P-wave velocities and density are correlated linearly equation 14. The accuracy of the resulting equations strongly depends on the precision of our prior knowledge of the supposedly linked medium parameters. Such knowledge must be derived from local well control. The derived reduced-parameter reflection coefficient 19 depends on only two free parameters, but it is as accurate as the conventional linearized expression of Richards and Frasier 1976. It has the additional advantage that no prior information about P-wave/S-wave velocity ratios is required. The reduction of free parameters is achieved by explicitly incorporating moveout information of the target horizon and the reflections directly above and underneath. The C term in the Shuey formulations 12 and 13 depends on the relative P-wave velocity contrast. Our strategy is unusual, in the sense that the normalized ratios d i measure the interval velocities in the layers directly above and beneath the target horizon to constrain the intermediate-to-long-offset AVO response. The -p domain is the natural domain for estimating interval velocities because it circumvents the short-offset approximation implicit in the Dix equation Schultz, 1982; Stoffa et al., 1982. Thus, it produces more accurate interval velocity estimates, especially for long-offset data. The accuracy of the reduced-parameter expression 19 depends on the precision of our moveout picks. It can be used in both AVO or AVS analysis. Therefore, a reduction of free parameters in the AVO problem can be achieved by including detailed prior information on coupling between different medium parameters at the target level or by analyzing moveout of adjacent reflections as favored in this paper. Well control is vital for acquiring reliable prior information on coupling parameters and Poisson s ratios, whereas moveout information can be measured directly from the data, thereby advocating in future AVO studies the use of the reduced-parameter reflection coefficients derived here. Figure 6. Waveforms of the primaries of the 14-layer model shown in Table 2. The target horizon is the weak reflection at 2 s zero-offset time. Shown are envelopes of the waveforms in a the time-offset domain and b the -p r domain. Random noise is added in the time-offset domain to render the example more realistic. Figure 7. Extracted amplitudes of the target horizon in Figure 6, after correction for the second moveout-related term in reduced-parameter expression 19, versus horizontal slowness squared. Solid line is corrected theoretical reflection coefficients for the target horizon, ignoring the transmision effect through the overburden. Dashed line is measured amplitudes along the target horizon for the noise-contaminated case shown in Figure 6 after correction with picked normalized ratios d 12 and d 13. Dotted line is the same as the dashed curve but without superimposed noise and while using the correct interval velocities.

C102 van der Baan and Smit CONCLUSIONS Amplitudes of primaries in the time-offset domain with near-critical angles of reflection do not correspond to those predicted by plane-wave reflection coefficients even after geometric spreading and polarization effects are taken into account. A plane-wave decomposition such as a proper -p transform must be applied on the data first for accurate analysis of reflection coefficients near critical angles. This leads to an analysis of AVS. Amplitudes after planewave decomposition are related directly to the plane-wave reflection coefficients; geometric-spreading corrections are no longer required, and polarization effects of P-P reflections recorded on the z-component are also removed. In addition, no offset-to-angle conversion is needed. Only two parameters can be estimated robustly in an AVO analysis involving moderate reflection angles, while the linearized reflection coefficients depend on contrasts in three parameters. The novel reduced-parameter expressions for the isotropic P-P-wave reflection and transmission coefficients are as accurate as conventional linearized coefficients but depend on fewer free parameters. No prior information is required about Poisson s ratios or coupling between the different relative contrasts. Thus, the new AVO expressions reduce the need for extensive well control inavo studies. The newavo expressions can be used in the time-offset domain for precritical arrivals and in the -p r domain for precritical and critical reflections. ACKNOWLEDGMENTS M. van der Baan thanks Shell Expro UK for funding and Steve Hall for the many discussions on the implications of the derived reduced-parameter expressions and the advantages of amplitude analysis in the -p r domain in general. Comments and suggestions of José Carcione, Rob Ferguson, Liam Suilleabhain, Herbert Swan, Yangua Wang, and two anonymous reviewers also proved very useful. APPENDIX A CONVERTING PRESSURE DATA TO VERTICAL DISPLACEMENT Marine data consist predominantly of pressure measurements. How can one convert pressure data to vertical displacement such that the described AVS strategy applies? The general acoustic-wave equation in water can be written as Červený, 2001, his equation 2.1.22 P x i = w 2 u i t 2, A-1 where P is pressure, w is water density, and u i is displacement in the x i -direction. Transforming equation A-1 to the frequency domain while assuming time-harmonic solutions for u i and P yields u z,p r = iq wp r w P,p r, A-2 where q w p r = c w 2 p r2 1/2 is the vertical P-wave slowness in the water layer with speed c w. The constant factor i/ w can be absorbed in the source correction term S 1 in equation 6. Hence, or, equivalently, u z,p r q w p r P,p r u z,p r q w p r P,p r. A-3 A-4 In other words, pressure data P,p r that are transformed to the -p r domain need to be scaled with the vertical P-wave slowness q w in the water layer for each horizontal slowness p r to convert pressure amplitudes to vertical displacement. The latter can then be used in the AVS strategy. APPENDIX B TRANSMISSION COEFFICIENTS If the first reflection does not constitute the target horizon but a deeper reflection is of interest, then transmission effects through the overburden also need to be taken into account. The expression of the isotropic transmission coefficient T` pp for the downgoing P-P-wave is only slightly less complex than the exact formula for the associated reflection coefficient Aki and Richards, 2002. It can be expressed in the form Wang, 1999 T` ppp r = H + Ip r 2 E + Fp r 2 + Gp r 4 + Dp r 6. B-1 The coefficients in the denominator are identical to those for the exact reflection coefficient 7. Both H and I depend on the vertical P-wave slowness q,i in the upper layer. Linearization of the exact transmission coefficient B-1 leads to Richards and Frasier, 1976 T` ppp r 1+ 1 p 2 r 2 2 1 p 2 r 2 2 1 +. B-2 It is therefore first-order independent of changes in the S-wave velocity or the shear modulus. Expressing the vertical P-wave slownesses in the exact transmission coefficient B-1 in terms of the ratios d i through equation 17 and linearizing the resulting expression leads to T` ppp r 2d 1d 2 d 1 + d 2 21 + d 1 d 21 + 2d 1 d 2 2. d 2 pr B-3 The common premultiplication factor is identical to the one in expression 18 for the reduced-parameter reflection coefficient. It can be shown to equal the constant 1/4 plus higher-order terms. Coefficient matching reveals that the third term involves second-order variations only, so it can be neglected. Expression B-3 then simplifies to T` ppp r 1 2 + 1 d 1 2 d 2 1 Z Z. B-4 This new expression for the linearized downgoing P-P-wave transmission coefficient depends only on a single free parameter the relative change in the acoustic impedance Z/Z equation 11 and thereby on the vertical-incidence acoustic reflection coefficient R 0 only.