# An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock
## 2 Zoeppritz and Biot-Gassmann Equations  
Zoeppritz equations describe the relationship between the seismic reflection coefficients and the elastic rock properties [39; 40]. The Zoeppritz equations for only P-wave incidence are written as:  
\[AR=B \tag{1}\]  
where  
\[R=\left[\begin{array}{c}R_{pp}\\ R_{ps}\\ T_{pp}\\ T_{ps}\end{array}\right],B=\left[\begin{array}{c}b_{1}\\ b_{2}\\ b_{3}\\ b_{4}\end{array}\right],\text{ and }A=\left[\begin{array}{cccc}a_{11}&a_{12}&a_{13}&a_{14} \\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\end{array}\right] \tag{2}\]  
where \(R_{pp}\), \(R_{ps}\), \(T_{pp}\) and \(T_{ps}\) are the reflection and transmission coefficients of P-and SV-wave (P-SV conversion), respectively. The elements a and b in the matrix A and B are  
\[a_{11}=\sin\alpha,\ a_{12}=\cos\beta,\ a_{13}=-\sin\alpha^{\prime},\ a_{14}=\cos \beta^{\prime};\]  
\[a_{21}=\cos\alpha,\ a_{22}=-\sin\beta,\ a_{23}=\cos\alpha^{\prime},\ a_{24}=\sin \beta^{\prime};\]\[a_{31}=\cos 2\beta,\ a_{32}=\ -\frac{v_{s1}}{v_{p1}}\sin 2\beta,\ a_{33}=\ -\frac{ \rho_{e2}}{\rho_{e1}}\frac{v_{p2}}{v_{p1}}\cos 2\beta^{\prime},\ a_{34}=\ -\frac{\rho_{e2}}{\rho_{e1}}\frac{v_{s2}}{v_{p1}}\sin 2\beta ^{\prime};\]  
\[a_{41}=\frac{v_{s1}^{2}}{v_{p1}^{2}}\sin 2\alpha,\ a_{42}=\frac{v_{s1}}{v_{p1}} \cos 2\beta,\ a_{43}=\frac{\rho_{e2}}{\rho_{e1}}\frac{v_{s2}^{2}}{v_{p2}^{2}v_{p1 }}\sin 2\alpha^{\prime},\ a_{44}=\ -\frac{\rho_{e2}}{\rho_{e1}}\frac{v_{s2}}{v_{p1}}\cos 2 \beta^{\prime};\]  
\[b_{1}=\ -\sin\alpha,\ b_{2}=\cos\alpha,\ b_{3}=\ -\cos 2\beta,\ b_{4}=\frac{v_{ s1}^{2}}{v_{p1}^{2}}\sin 2\alpha.\]  
Here, \(\alpha\) denotes the incidence and reflection angle of P-wave; \(\beta\) is the reflection angle of SV-wave; \(\alpha^{\prime}\) and \(\beta^{\prime}\) denote the transmission angles of the P-and SV-waves, respectively; \(v_{p1}\), \(v_{s1}\), \(\rho_{1}\) and \(v_{p2}\), \(v_{s2}\), \(\rho_{2}\) are the P-, S-wave velocities and density in layer 1 and 2, respectively (shown in Figure 1).  
The Biot-Gassmann equations describe the relationship between velocity, density and elastic moduli in porous media. It can be expressed as Hilterman, [41]:  
\[v_{pi}^{2}=\Bigg{[}(K_{di}+4/3\mu_{di})+\frac{(1-K_{di}/K_{si})^{2}}{(1-\phi_{ i}-K_{di}/K_{si})/K_{si}+\phi_{i}/K_{fi}}\Bigg{]}/\rho_{ei} \tag{3}\]  
\[v_{si}^{2}=\mu_{di}/\rho_{ei},\ \mu_{ei}=\mu_{di} \tag{4}\]  
\[\rho_{ei}=(1-\phi_{i})\rho_{si}+\phi_{i}\rho_{f} \tag{5}\]  
where \(\phi\) is porosity; \(K_{f}\), \(K_{s}\) and \(K_{d}\) denote the bulk moduli of effective fluid, solid mineral and dry rock, respectively; \(\rho_{e}\), \(\rho_{s}\) and \(\rho_{f}\) are the densities of saturated medium, solid mineral and pore fluid, respectively; \(\mu_{e}\) and \(\mu_{d}\) denote the shear moduli of saturated medium and dry frame, respectively.  
The approximation of bulk modulus of the effective fluid is as demonstrated by [PERSON], [41]:  
\[\frac{1}{K_{f}}=\frac{S_{w}}{K_{w}}+\frac{S_{o}}{K_{o}}+\frac{S_{g}}{K_{g}} \tag{6}\]  
where \(K_{o}\), \(K_{g}\) and \(K_{w}\) are the bulk moduli of oil, gas and water, respectively. \(S_{o}\), \(S_{g}\) and \(S_{w}\) are saturations when pores are filled with oil, gas and water, respectively.  
The effective density can be approximated as  
\[\rho_{f}=S_{w}\rho_{w}+S_{o}\rho_{o}+S_{g}\rho_{g} \tag{7}\]  
where \(\rho_{o}\), \(\rho_{g}\) and \(\rho_{w}\) are the densities of oil, gas and water, respectively.  
Figure 1: Reflection and transmission of only P-wave incidence at an interface between two elastic media [11]. \(n\) is the normal direction.