# An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock
## 5 Numerical Examples
### Gradient Comparison between the Exact and the Classical Approximations of Zoeppritz Equations  
To test the accuracy of this proposed method, we compare the partial derivatives calculated using the exact Zoeppritz equations with those using two classical approximations: Aki-Richards and Shuey [6; 7]. Here, the same oil-water model is used. These two Zoeppritz approximations are used to calculate the partial derivatives of the seismic reflectivities with respect to the dry rock properties directly. Figure 8 compares the partial derivatives of reflectivity \(R_{PP}\) with respect to \(K_{d1}\) and \(K_{d2}\) using different algorithms. \(R_{kd1,Zop}\) and \(R_{kd2,Zop}\) are the results of the exact Zoeppritz equations; \(R_{kd1,Aki}\) and \(R_{kd2,Aki}\) are the results of the [PERSON] approximation; \(R_{kd1,Slua}\) and \(R_{kd2,Slua}\) are the results of the Shuey approximation. Seen in the AVO/AVA inversion, the incidence angles of seismic data are normally smaller than the critical angle. The largest incidence angle is \(50^{\circ}\) in Figure 8b,d. When the incidence angle is smaller than \(15^{\circ}\), the values calculated using these three algorithms are very similar. One can observe that differences between the exact solution and two approximations are becoming larger and larger with the increase in the incidence angle. The results prove that the approximations are only suitable for the AVO inversion with small incidence angles. Figure 8: _Cont._  
Figure 7: The partial derivatives of the reflectivity \(R_{ps}\) with respect to the elastic moduli \(K_{d1}\), \(K_{d2}\), \(\mu_{d1}\) and \(\mu_{d2}\) of the dry rock, respectively (water–oil interface). (**a**): The partial derivatives of \(R_{ps}\) with respect to \(K_{d1}\), \(K_{d2}\), \(\mu_{d1}\) and \(\mu_{d2}\), respectively; (**b**): The zoomed in curve of \(R_{ps}\) with respect to \(K_{d1}\), \(K_{d2}\) and \(\mu_{d2}\), respectively. Figure 9 shows the comparison of the absolute and relative errors of partial derivatives of reflectivity \(R_{pp}\) with respect to \(K_{d1}\) and \(K_{d2}\) using Aki-Richards and Shuey approximations. \(ReR_{kd1,Aki}\), \(ReR_{kd2,Aki}\), \(ER_{kd1,Aki}\) and \(ER_{kd2,Aki}\) are the relative and absolute errors of the Aki-Richards approximation, respectively. \(ReR_{kd1,Shu}\), \(ReR_{kd2,Shu}\), \(ER_{kd1,Shu}\) and \(ER_{kd2,Shu}\) are the relative and absolute errors of the Shuey approximation, respectively. Two zero points can be observed when the incidence angles are \(0^{\circ}\) and \(90^{\circ}\). Therefore, there are two discontinuity points on the relative errors which make other curves close to zero. Considering this circumstance, the absolute error curves are more helpful to observe the accuracy of each algorithm. Figure 9b,d show the relative and absolute errors of two classic approximations when the incidence angle is smaller than the critical angle and the errors are still very large. Figure 8: The comparison of partial derivatives of reflectivity \(R_{pp}\) with respect to \(K_{d1}\) (**a**,**b**) and \(K_{d2}\) (**c**,**d**) by using different algorithms when the incidence angles are smaller than \(90^{\circ}\) and \(50^{\circ}\), respectively. Solid lines: the real values of \(R_{pp}\) respect to \(K_{d1}\) and \(K_{d2}\) by using the exact Zoeppritz solution; Dashed lines: the real values of \(R_{pp}\) with respect to \(K_{d1}\) and \(K_{d2}\) using the Aki–Richards approximate solution; Dotted lines: the real values of \(R_{pp}\) with respect to \(K_{d1}\) and \(K_{d2}\) by using the Shuey approximate solution. Figure 9: _Cont._  
Figure 10 compares the partial derivatives of \(R_{PP}\) with respect to \(\mu_{d1}\) and \(\mu_{d2}\) using different algorithms. \(R_{\mu 1,Zapr}\) and \(R_{\mu 2,Zapr}\) are the results of the exact Zoeppritz equations; \(R_{\mu 1,Ali}\) and \(R_{\mu 2,Ali}\) are the results of the Aki-Richards approximation; \(R_{\mu 1,Shu}\) and \(R_{\mu 2,Shu}\) are the results of the Shuey approximation.