# An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock
## 5 Numerical Examples
### Oil-Water Interface  
Seismic waves penetrate from the oil-saturated sandstone to the water-saturated sandstone. The water, oil and gas saturations in two sandstone layers are \(S_{w1}=0~{}S_{o1}=0.6\), \(S_{g1}=0.4\), \(S_{w2}=1.0\), \(S_{o2}=S_{g2}=0\), respectively. The partial derivative curves of reflectivities \(R_{pp}\) and \(R_{ps}\) with respect to the elastic moduli \(K_{d1}\), \(K_{d2}\), \(\mu_{d1}\) and \(\mu_{d2}\) of dry rock are complex numbers.  
Figure 2 shows the real and imaginary values of partial derivatives of reflectivity \(R_{pp}\) with respect to the bulk moduli \(K_{d1}\) and \(K_{d2}\). One can observe that there is a peak (singular point) at the critical angle (\(50^{\circ}\)). It is because the partial derivative matrix \(\frac{\partial A}{\partial K_{d1}}\) contains \(\tan\alpha^{\prime}\). When the incidence angle is larger than the critical angle, the results of the partial derivatives become complex numbers rather than real numbers. Due to the existence of this discontinuous point, big errors will be observed around the critical angle when carrying out seismic inversion of bulk moduli.  
Figure 3 shows the real and imaginary curves of partial derivatives of reflectivity \(R_{PP}\) with respect to the shear moduli \(\mu_{d1}\) and \(\mu_{d2}\), respectively. Different from Figure 2, there is a zero point rather than a singular point at the critical angle. The polarities of \(R_{PP}\) with respect to \(\mu_{d1}\) and \(\mu_{d2}\) are opposite.  
Figure 2: The partial derivatives of the reflectivity \(R_{pp}\) with respect to the bulk moduli \(K_{d1}\) (**a**) and \(K_{d2}\) (**b**) of the dry rock (oil–water interface). Solid lines: the real values of partial derivatives of \(R_{pp}\) with respect to \(K_{d1}\) and \(K_{d2}\), respectively; Dashed lines: the imaginary values of partial derivatives of \(R_{pp}\) with respect to \(K_{d1}\) and \(K_{d2}\), respectively.  
Figure 4 shows the partial derivatives of reflectivity \(R_{ps}\) with respect to the bulk moduli \(K_{d1}\) and \(K_{d2}\), respectively. Figure 4a,b show similar characteristics to Figure 2a,b, but their polarities are different. Additionally, a singular point also is found at the critical angle. The absolute value of the partial derivatives of the PP-wave is almost two times larger than that of the PS-wave. One can observe that the slopes of all curves are nearly zero when the incidence angle is smaller than \(40^{\circ}\), which means the reflectivity \(R_{ps}\) is not sensitive to the bulk moduli and not suitable to be used to invert the bulk moduli.  
Figure 5 shows the partial derivatives of reflectivity \(R_{ps}\) with respect to the shear moduli \(\mu_{d1}\) and \(\mu_{d2}\) of the dry rock, respectively. Different from Figure 3, only one zero point is found here. The variation of the \"real\" part in Figure 5 is smaller than that in Figure 3, when the incidence angle is larger than the critical angle. This phenomenon could be caused by fluid substitution.  
Figure 4: The partial derivatives of reflectivity \(R_{ps}\) with respect to the bulk moduli \(K_{d1}\) (**a**) and \(K_{d2}\) (**b**) of the dry rock (oil–water interface). Solid lines: the real values of partial derivatives of \(R_{ps}\) with respect to \(K_{d1}\) and \(K_{d2}\), respectively; Dashed lines: the imaginary values of partial derivatives of \(R_{ps}\) with respect to \(K_{d1}\) and \(K_{d2}\), respectively.  
Figure 3: The partial derivatives of reflectivity \(R_{pp}\) with respect to the shear moduli \(\mu_{d1}\) (**a**) and \(\mu_{d2}\) (**b**) of the dry rock (oil–water interface). Solid lines: the real values of partial derivatives of \(R_{pp}\) with respect to \(\mu_{d1}\) and \(\mu_{d2}\), respectively; Dashed lines: the imaginary values of partial derivatives of \(R_{pp}\) with respect to \(\mu_{d1}\) and \(\mu_{d2}\), respectively.