Multiwave Amplitude Compensation and Its Sensitivity Analysis to AVO Inversion in Viscoelastic Media

The precision of the amplitude-coefficientversus-offset(AVO)inversion result depends on the accuracy of reflection amplitude input. However, for field data there are several other important factors affecting AV0, in addition to the variation of reflection coefficient. These factors include the effect of the free surface, geometric spreading, thin-bed effect, medium absorption, multiples, anisotropy and so on (Strudley, 1991; Sheriff, 1975). Geometric spreading and absorption are major factors. Hence, a series of processes of amplitude compensation needs to be performed before AV0 inversion. Huhral and Kev (1 980)deduced an analytic expression for the spreading factor of a compressional wave in plane layered medium and indicated that the expression is not suitable to multiwave.Samec et al.(1991)studied the influence of viscoelasticity and anisotropy on AVO interpretation and pointed out that one cannot ignore these factors when accurate AVO analysis and inversion are desired.Dong(1999)discussed quantitatively the influence of migration stretch and thin-bed tuning on AVo analysis and interpretation, and derived two conditions of determining AVO detectability. Nicolao et al.(1993)and Martinez(1993)studied the reliability of inversion results at a plane reflector in homogenous elastic media. Joakim(1 998)investigated qualitatively and semi-quantitatively the error of inversion results caused by amplitude attenuation in 1D viscoelastic media. These analyses above were limited to the analyses of P-wave. Also the missing from these studies is the expressions of wavefront spreading and absorption for multi-waves in a layered viscoelastic medium. In this paper, we extend the model given by Nicolao et al.(1993) and Joakim (1998)to viscoelastic media and derive expressions of wavefront spreading and absorption for P-P and P-S converted waves in a layered viscoelastic medium. The imaging matrix at a plane reflector between vls coelastic media can be determined in the frequency domain using lineariaed reflection coefficients through Born approximation. We quantitatively analyze the sensitivity by studying eigenvalues and eigenvectors of the imaging matrix. The results show that two linear combinations of petrophysieal parameters can be determined from the multi-wave AVO inversion in the case of amplitude compensation. Muhi-wave AV0 contains the information of attenuation in the media. However, the sensitivity of multi-wave AV0 inversion to attenuation is small. Especially, if we have difficulty in determining accurately quality factor(Q).the average value of Q within layers is the best approximation when the absorption to stratiform media Is compensated in the attenuation of model.

ALGORITHM OF MULTIWAVE AV0 INVERSION WITH AMPLITUDE COMPENSATION

We consider a layered viscoelastic medium with down-going P-wave as the incident wave. The reflected waves shall be up-going P-wave and mode-converted S-wave. Let α, β, ρ and Q respectively be P-wave velocity, S-wave velocity, density and quality factor of the homogenous layers. The reflection amplitude of the p-wave and S-wave are related to source amplitude factor, A_s; propagation factor, A_p, and reflection response, A_r. It can be expressed in potential function

where A_s is the complex amplitude of the source, which can he both frequency dependent and anisotropic; k_x and k_z are the horizontal and vertlcal wavenumbers of a plane wave, respectively; . Let k_P be the total P-wave wavenumber; k_S, the S-wave wavenumber; k_Px and k_Pz, ,the horizontal and vertical wavenumbers of a P-wave, and k_Sx.and k_Sx, ,the horizontal and vertlcal wavenumbers of S-wave. Using the relation among wavenumher, velocity and Irequency, and the relation of medium wavenumber k_m and the vertical wavenumher of P-wave k_Pz, we have

Here θ, Ψare the P-wave incidence angle and the P_S wave reflection angle, respectively.

Caleulation of A_p

A_p onsists of wavefront spreading factor A_r, medium absorption A_s and transmission coefficient A_t. Since A_t can be directly calculated from the Zoeppritz equation, we only discuss the calculation of A_r and A_a in a viscoelastic medium.

Calculation of A_r

Based on the three laws of refraction, reflection and transmission (Hubral and Key, 1980), we can derive wavefront spreading factors in a layered or slanted layered medium for both P-P and P-S waves in terms of the radius of curvature matrix. After some algebra, we obtain

where α_k^d, β_k^d, respectively, are the incident and transmission angles of downgoing P-wave or S-wave at the kth reflector; α_k^u, β_k^u are the incident and transmission angle of upgoing P-wave or S-wave at the kth reflector; R_E is the radius of curvature matrix for the P-P and P-S wavefronts. v₁^u, v₁^d, oare the velocities ofupgoing P or S wave in the first layer. Using the law of wavefront curvature, we have

where R_x, is the radius of wavefront in the plane perpendicular to the plane of incidence; R_y, is the radius of wavefront in the plane of incidence. Using the three laws of refraction, reflection and transmission, we obtain the radius of curvature of the P-P and P-S waves through recursion. We have

where v_i^u, v_i^d, orespectively are the velocities of upgoing and downgoing waves in the ith layer. _si^u, s_i^d are the ray lengths of upgoing and downgoing waves in the ith layer.

In horizontal layered medium, there exist the following relationships

where h_i is thickness of the ith layer. After simplifying equation (3), the multiwave wavefront spreadingfactor in nth layer is

Calculation of Medium Absorption Factor A_a

In general, wavenumber in a viscoelastic medium is complex. Let

where k' is the real part of multiwave wavenumber. The imaginary part is the absorption coefficient. Then the exponent in equation (1) becomes

where k'_x is the real part of k_x; k'_z is the real part of k_z. For convenience, from now on we will denote k'_x by k_x, k'_z by k_z, , and k' by k.

Using equations (2)-(5) and the relations among , Q and wavenumber, we can express the ab-sorption coefficient of P-wave and the absorption coefficient of S-wave into

Consequently, the absorption coefficients of Pwave and S-wave can be written in a single form in horizontally layered medium. That is

Equations (7) and (11) are the formulae of am-plitude compensation suitable for AVO inversion inlayered viscoelastic media.

Calculation of A_r

In a homogenous model, the relative perturbations of the model parameters can be written as

Using Born approximation (Nicolao et al., 1993), we obtain the relationship among reflection coefficient, incident angle (θ) and depth(z). It is

where R(θ) is deterrnined by the equation given by Aki and Richards (1989), which consists of the reflection coefficients of Pwave and S-wave for multi-wave prospecting, i. e.,

Fourier transforming equation (13) with respect tovariable z yields

Inversion Algorithm

Let the frequency spectrum of the source be 1. The spreading factor and absorption of multiwave in homogenous background model can be obtained by simplifying equations (7) and (11). Further, using equation (15), we can obtain the following matrix equation for the reflection amplitudes and incident angles,

where d is the vector of reflection amplitudes defined by equation (1), and

G is the sensitivity matrix, which will form the base of the inversion algorithm and

where are the factors affecting multiwave amplitude compensation; q is the amount of attenuation in the medium, which is half of the reciprocal of Q. A_pp(θ, , A_ps(θ, ) is the factors of amplitude compensation for P-P and P-S waves, and can be expressed in vector form

SENSITIVITY ANALYSIS OFAVO INVERSION WITH AMPLITUDE COMPENSATION

The matrix G in equation (16) is the imagingr matrix from model space to data space. The magnitude of the eigenvalues of G determines the stability of model parameters estimated (Gu, 2000), and reveals whether or not the best solution or the best linear combination of solutions can be obtained. The condition number of matrix G can be determined by singular value decomposition (SVD). This enables us to perform quantitative analysis of the sensitivity of imaging matrix in the case of inversion with amplitude compensation.

Assuming that z is equal to 2 000 m, and , medium wavenumber k_m=0. 06 m^-1, This corresponds to a wave with frequency around 10 to 12 Hz at vertical incidence and 19 to 24 Hz at 60 degree incidence with a velocity between 2 000 and 2 500 m/s.

Figure 1a and b respectively shows the condition number and the ratio of the two largest eigenvalues for different values of maximum incidence angle and attenuation. The condition number and the ratio in crease with the increasing of incidence angle. This shows that in the case of amplitude compensation the sensitivity of AVO to elastic parameters at large incidence angles is stronger than that at small incidence angles. That is to say, the elastic parameters are more likely to be determined at large incidence angles. Meanwhile, these figures show that the condition number and the eigenvalue ratio increase slightly with the increasing attenuation. Although multiwave AVO contains information of media attenuation, their sensitivity to attenuation is small. In other words, the effect of Q value on elastic parameters is small, which implies that an average Q value for all layers over the reservoir is probably sufficient when it is difficult to determine accurate Q value for each layer in an attenuation model(Clayton and Stolt, 1981).

Figure  1.  Condition number and eigenvalue (λ) of imaging matrix with various maximum incident angles(θ_max) and attenuation parameters (q).(a). condition number of multi-wave imaging matrix; (b).ratio of the 2nd largest to the largest eigenvalues for imaging matrix.

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Figure 2 and 3 show the isochron for each component of the eigenvectors in the model space corresponding to the largest and the 2nd largest eigenvalues, respectively. From these figures, we observe that the eigenvector for the largest eigenvalue points mainly at Δα/α and Δρ/ρ, corresponding to relative P-wave impedance variation. While the eigenvector for the 2nd largest eigenvalue mainly points at Δβ/β, Δρ/ρ, i. e. relative S-wave impedance variation. Therefore, in the case of amplitude compensation, two linear combinations of elastic parameters can be determined from multi-wave AVO inversion using equation (16). Furthermore, since the gradient direction of eigenvectors as a function attenuation is invariable (Fig. 1), variation in attenuation does not change the combination of elastic parameters.

Figure  2.  Eigenvectors for maximum eigenvalue with various maximum incident angles(θ_max) and attenuation parameters (q) in model space. (a).Δρ/ρ, (b). Δα/α; (e). Δβ/β.

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Figure  3.  Eigenvectors for the secondary maximum eigenvalue with various maximum incident angles(θ_max) and attenuation parameters (q) in model space.(a).Δρ/ρ; (b). Δα/α; (c).Δβ/β.

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CONCLUSIONS

(1) Two linear combinations of elastic parameters can be determined from multi-wave AVO in the case of amplitude compensation. The reliability of the elastic parameters estimated by AVO inversion is higher in large incidence angle.

(2) Although multi-wave AVO contains the information of media attenuation, the sensitivity of multi-wave AVO inversion to attenuation is small. The average Q value of all layers above reservoir is probably the best approximate value when it is difficult to determine an accurate Q value for each layers in the model.

(3) Absorption in viscoelastic media does not change the combination relation of elastic parameters in AVO inversion.