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

Abstract

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. Jacobian matrix, Zoeppritz equations, inversion of velocities, derivatives of RCs with respect to P-and S-wave velocities, large angle Citation:Liu F P, Meng X J, Wang Y M, et al. Jacobian matrix for the inversion of P-and S-wave velocities and its accurate computation method.Determination of rock lithology and identification of fluid in reservoirs are the ultimate objectives of oil and gas exploration [1][2][3][4][5][6][7][8]. Inversion of seismic data is an important approach to improve the precision of lithology interpretation. Therefore, the application of the seismic wave reflection coefficients to the inversion of P-and S-wave velocities or to the inversion of petrophysical characteristics remains a hot topic for geophysical research [9][10][11][12][13][14][15][16][17][18][19][20].Since the middle 1980s, the AVO (Amplitude Versus Offset) [9][10][11][12][13][14][15][16][17][18][19][20] or AVA (Amplitude Versus incident Angle) techniques [9-20] have been stuided extensively. Their theoretical foundation is Zoeppritz equations [8,9,14], which accurately describe the relations of reflected wave amplitudes to incidence angle of waves, velocity of waves, and density of media. Zoeppritz equations indicate that the prediction of rock lithology is possible with seismic data. But ...

An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock

“…The most difficult part of AVO/AVA inversion based on the exact Zoeppritz equations, however, is to obtain the accurate Jacobian matrix (partial derivatives of the reflection coefficients with respect to dry rock moduli) [34,35]. An accurate algorithm was proposed to compute the gradients of the seismic reflection coefficients with respect to P-, S-wave velocities and density by solving the partial derivative equations of the exact Zoeppritz equations [36,37]. Lu et al carried out joint PP and PS AVA seismic inversion using exact Zoeppritz equations [38].…”

Zoeppritz-based AVO inversion using an improved Markov chain Monte Carlo method

“…In terms of the reservoirs with strong-contrast interfaces (salt domes, heavy oil fields, basalts, etc.) and long-offset ranges, however, the conventional AVO inversion based on the approximations would be not appropriate or suitable (Larsen 1999 ; Chen et al 2009 ; Liu et al 2010 , 2012a , b , 2014 ; Wang et al 2011 ; Zhu et al 2012 ; Zhang et al 2013 ; Huang et al 2013 ; Zhi et al 2013 , 2015 ; Zong et al 2013a , b ; Lu et al 2015 ; Lehochi et al 2015 ).…”