# An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock
## 1 Introduction  
The amplitude variation with offset (AVO) or amplitude variation with incidence angle (AVA) inverts from the measured amplitudes of seismic reflectivities to the rock properties [1; 2; 3]. Due to the complexity of Zoeppritz equations, most AVO/AVA inversions are based on linear approximations [4; 5; 6]. The most widely used classical approximations are proposed by [PERSON] and [PERSON] [7; 8]. These classical approximations, however, are only valid for weak elastic contrasts and relatively small incidence angles [9; 10; 11]. Actually, there are many target zones that have strong elastic contrast interfaces or long-offset seismic data used. Thus, the accurate partial derivatives of the reflection coefficients with respect to the unknown parameters should be derived for more accurate AVO/AVA inversion [12; 13; 14]. Rock physics plays a significant role in oil and gas explorations, such as hydrocarbon detection, lithology determination and fluid identification [15; 16; 17]. Biot-Gassmann equations are widely used in rock physics, which establish the relationship between elastic moduli (bulk and shear moduli) and seismic P-and S-wave velocities [18; 19; 20; 21]. Fluid substitution is a technique used to predict P-, S-wave velocities and density based on the analyses of their porosity and dry rock moduli [22; 23; 24; 25]. The elastic moduli of dry rock could be obtained from the rock physics experiments, well logging or numericalmodeling [26; 27]. [PERSON] and [PERSON] predicted dry rock bulk modulus, rigidity modulus and reflection coefficients with the application of laboratory rock samples and well logs [28]. [PERSON] and [PERSON] derived the analytical approximations of bulk and shear moduli of dry rock based on the differential effective medium theory [29]. However, the direct inversion of dry rock moduli from seismic data is always a challenging problem. During recent decades, some researchers proposed to invert the fluid or elastic moduli from seismic data. [PERSON] et al. provided an AVO approximate equation through incorporating the dry and saturated properties of rock into the P-wave velocity expression, which is named as the lambda-mu-rho (LMR) method [30]. [PERSON] et al. combined this concept with the Biot-Gassmann theory to derive an approximation equation for AVO inversion [31]. Based on [PERSON]'s fluid factor, [PERSON] et al. carried out the Bayesian-based elastic impedance inversion, which is more suitable for seismic data with large incidence angles [32]. [PERSON] et al. further proposed FMR-AVA inversion (Fluid Factor, Mu (Shear modulus), Rho (Density)-Amplitude Variation with Angle) using the approximations [33]. We propose to combine the exact Zoeppritz equations with the Biot-Gassmann equations. The best way to get an accurate elastic moduli of dry rock from seismic data is to use AVO/AVA inversion with exact Zeoppritz equations. 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]. [PERSON] et al. carried out joint PP and PS AVA seismic inversion using exact Zoeppritz equations [38]. Nevertheless, the direct inversion of bulk and shear moduli of dry rock was not taken into consideration in the previous studies. We propose to combine the exact Zoeppritz equations with the Biot-Gassmann equations and derive the accurate Jacobian matrix for inverting dry rock moduli directly without any simplifications. This derived matrix can be used further to invert bulk and shear moduli of dry rock. Due to the usage of exact Zoeppritz equations, it overcomes the weaknesses that the approximate Zoeppritz equations have.