Zoeppritz equation-based prestack inversion and its application in fluid identification

Prestack seismic inversion methods adopt approximations of the Zoeppritz equations to describe the relation between reflection coefficients and P-wave velocity, S-wave velocity, and density. However, the error in these approximations increases with increasing angle of incidence and variation of the elastic parameters, which increases the number of inversion solutions and minimizes the inversion accuracy. In this study, we explore a method for solving the reflection coefficients by using the Zoeppritz equations. To increase the accuracy of prestack inversion, the simultaneous inversion of P-wave velocity, S-wave velocity, and density by using prestack large-angle seismic data is proposed based on generalized linear inversion theory. Moreover, we reduce the ill posedness and increase the convergence of prestack inversion by using the regularization constraint damping factor and the conjugate gradient algorithm. The proposed prestack inversion method uses prestack large-angle seismic data to obtain accurate seismic elastic parameters that conform to prestack seismic data and are consistent with logging data from wells.

Generalized linear joint PP-PS inversion based on two constraints

Conventional joint PP-PS inversion is based on approximations of the Zoeppritz equations and assumes constant VP/VS;therefore,the inversion precision and stability cannot satisfy current exploration requirements.We propose a joint PP-PS inversion method based on the exact Zoeppritz equations that combines Bayesian statistics and generalized linear inversion.A forward model based on the exact Zoeppritz equations is built to minimize the error of the approximations in the large-angle data,the prior distribution of the model parameters is added as a regularization item to decrease the ill-posed nature of the inversion,low-frequency constraints are introduced to stabilize the low-frequency data and improve robustness,and a fast algorithm is used to solve the objective function while minimizing the computational load.The proposed method has superior antinoising properties and well reproduces real data.

Wide-angle incidence and P-wave transmission

Polarity reversals may occur to transmitted P waves if the incidence angle is greater than the critical incidence angle. We analyze the characteristics of reflection and transmission coefficients under the condition of wide incidence angle based on Zoeppritz equations. We find that for specific conditions, as the incidence angle increases, the characteristic curve of the transmitted P-wave coefficient enters the third quadrant from the first quadrant through the origin, which produces a transition in the transmitted P wave and the corresponding coefficient experiences polarity reversal. We derive the incidence angle when the transmitted P-wave coefficient is zero and verify that it equals zero by using finite-difference forward modeling for a single-interface model. We replace the water in the model reservoir by gas and see that the reservoir P-wave velocity and density decrease dramatically. By analyzing the synthetic seismogram of the transmitted P wave in the single-interface model, we show that the gas-saturated reservoir is responsible for polarity reversal.

An approximation to the P-wave reflection coefficient for amplitude variation with long-offset seismic data

In recent years, long-offset exploration has been widely used, especially on marine seismic surveys. Conventional AVO analysis is insufficient for long-offset seismic data. To widen the application range of AVO analysis, we present a new P-wave reflection coefficient approximation applicable to long-offset data. Our result is similar to the well known Shuey formula which can be treated as an approximation to our results for short-offset seismic data.

Jacobian matrix for the inversion of P-and S-wave velocities and its accurate computation method

The optimization inversion method based on derivatives is an important inversion technique in seismic data processing,where the key problem is how to compute the Jacobian matrix.The computation precision of the Jacobian matrix directly influences the success of the optimization inversion method.Currently,all AVO(Amplitude Versus Offset) inversion techniques are based on approximate expressions of Zoeppritz equations to obtain derivatives.As a result,the computation precision and application range of these AVO inversions are restricted undesirably.In order to improve the computation precision and to extend the application range of AVO inversions,the partial derivative equation(Jacobian matrix equation(JME) for the P-and S-wave velocities inversion) is established with Zoeppritz equations,and the derivatives of each matrix entry with respect to Pand S-wave velocities are derived.By solving the JME,we obtain the partial derivatives of the seismic wave reflection coefficients(RCs) with respect to P-and S-wave velocities,respectively,which are then used to invert for P-and S-wave velocities.To better understand the behavior of the new method,we plot partial derivatives of the seismic wave reflection coefficients,analyze the characteristics of these curves,and present new understandings for the derivatives acquired from in-depth analysis.Because only a linear system of equations is solved in our method,the computation of Jacobian matrix is not only of high precision but also is fast and efficient.Finally,the theoretical foundation is established so that we can further study inversion problems involving layered structures(including those with large incident angle) and can further improve computational speed and precision.

The Goos-Hnchen shift of wide-angle seismic reflection wave

The partial derivative equations of Zoeppritz equations are established and the derivatives of each matrix entry with respect to wave vectors are derived in this paper.By solving the partial derivative equations we obtained the partial derivatives of seismic wave reflection coefficients with respect to wave vectors,and computed the Goos-Hnchen shift for reflected P-and VS-waves.By plotting the curves of Goos-Hnchen shift,we gained some new insight into the lateral shift of seismic reflection wave.The lateral shifts are very large for glancing wave or the wave of the incidence angle near the critical angle,meaning that the seismic wave propagates a long distance along the reflection interface before returning to the first medium.For the reflection waves of incidence angles away from the critical angle,the lateral shift is in the same order of magnitude as the wavelength.The lateral shift varies significantly with different reflection interfaces.For example,the reflected P-wave has a negative shift at the reflection interface between mudstone and sandstone.The reflected VS-wave has a large lateral shift at or near the critical angle.The lateral shift of the reflected VS-wave tends to be zero when the incidence angle approaches 90°.These observations suggest that Goos-Hnchen effect has a great influence on the reflection wave of wide-angles.The correction for the error caused by Goos-Hnchen effect,therefore,should be made before seismic data processing,such as the depth migration and the normal-moveout correction.With the theoretical foundation established in this paper,we can further study the correction of Goos-Hnchen effect for the reflection wave of large incidence angle.

Applying accurate gradients of seismic wave reflection coefficients (SWRC) to the inversion of seismic wave velocities

Through solving the Zoeppritz's partial derivative equations, we have obtained accurate partial derivatives of reflected coefficients of seismic wave with respect to Pand S-wave velocities.With those partial derivatives, a multi-angle inversion is developed for seismic wave velocities.Numerical examples of different formation models show that if the number of iterations goes over 10, the relative error of inversion results is less than 1%, whether or not there is interference among the reflection waves.When we only have the reflected seismograms of P-wave, and only invert for velocities of P-wave, the multi-angle inversion is able to obtain a high computation precision.When we have the reflected seismograms of both P-wave and VS-wave, and simultaneously invert for the velocities of P-wave and VS-wave, the computation precisions of VS-wave velocities improves gradually with the increase of the number of angles, but the computation precision of P-wave velocities becomes worse.No matter whether the reflected seismic waves from the different reflection interface are coherent or non-coherent, this method is able to achieve a higher computation precision.Because it is based on the accurate solution of the gradient of SWRCs without any additional restriction, the multi-angle inversion method can be applied to seismic inversion of total angles.By removing the difficulties caused by simplified Zoeppritz formulas that the conventional AVO technology struggles with, the multiangle inversion method extended the application range of AVO technology and improved the computation precision and speed of inversion of seismic wave velocities.