""" Amplitute versus Offset (AVO) ============================= """ # %% import numpy as np import matplotlib.pyplot as plt plt.rcParams['font.size']=14 plt.rcParams['font.family']='arial' # %% import rockphypy # import the module from rockphypy import AVO from rockphypy import Fluid # %% # # Zeoppritz Equation # ~~~~~~~~~~~~~~~~~~ # The isotropic and elastic behaviour of a P wave incident on a boundary at any angle is described by the Zoeppritz (1919) equations, Aki and Richards (1980) give the results in the following convenient matrix form # # .. math:: # \left(\begin{array}{cccc}\downarrow \uparrow & \downarrow \uparrow & \uparrow \uparrow & \uparrow \uparrow \\ \mathrm{PP} & \mathrm{SP} & \mathrm{PP} & \mathrm{SP} \\ \downarrow \uparrow & \downarrow \uparrow & \uparrow \uparrow & \uparrow \uparrow \\ \mathrm{PS} & \mathrm{SS} & \mathrm{PS} & \mathrm{SS} \\ \downarrow \downarrow & \downarrow \downarrow & \uparrow \downarrow & \uparrow \downarrow \\ \mathrm{PP} & \mathrm{SP} & \mathrm{PP} & \mathrm{SP} \\ \downarrow \downarrow & \downarrow \downarrow & \uparrow \downarrow & \uparrow \downarrow \\ \mathrm{PS} & \mathrm{SS} & \mathrm{PS} & \mathrm{SS}\end{array}\right)=\mathbf{M}^{-1} \mathbf{N} # # # where each matrix element is a reflection or transmission coefficient for displacement # amplitudes. The first letter designates the type of incident wave, and the second letter # designates the type of reflected or transmitted wave. The arrows indicate downward ↓ # and upward ↑ propagation, so that a combination ↑↓ indicates a reflection coefficient, # while a combination ↓↓ indicates a transmission coefficient. # The matrices M and N are given by # # .. math:: # M=\left(\begin{array}{cccc}-\sin \theta_{1} & -\cos \theta_{\mathrm{S} 1} & \sin \theta_{2} & \cos \theta_{\mathrm{S} 2} \\ \cos \theta_{1} & -\sin \theta_{\mathrm{S} 1} & \cos \theta_{2} & -\sin \theta_{\mathrm{S} 2} \\ 2 \rho_{1} V_{\mathrm{S} 1} \sin \theta_{\mathrm{S} 1} \cos \theta_{1} & \rho_{1} V_{\mathrm{S} 1}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 1}\right) & 2 \rho_{2} V_{\mathrm{S} 2} \sin \theta_{\mathrm{S} 2} \cos \theta_{2} & \rho_{2} V_{\mathrm{S} 2}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 2}\right) \\ -\rho_{1} V_{\mathrm{P} 1}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 1}\right) & \rho_{1} V_{\mathrm{S} 1} \sin 2 \theta_{\mathrm{S} 1} & \rho_{2} V_{\mathrm{P} 2}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 2}\right) & -\rho_{2} V_{\mathrm{S} 2} \sin 2 \theta_{\mathrm{S} 2}\end{array}\right) # # # # .. math:: # \mathrm{N}=\left(\begin{array}{cccc}\sin \theta_{1} & \cos \theta_{\mathrm{S} 1} & -\sin \theta_{2} & -\cos \theta_{\mathrm{S} 2} \\ \cos \theta_{1} & -\sin \theta_{\mathrm{S} 1} & \cos \theta_{2} & -\sin \theta_{\mathrm{S} 2} \\ 2 \rho_{1} V_{\mathrm{S} 1} \sin \theta_{\mathrm{s} 1} \cos \theta_{1} & \rho_{1} V_{\mathrm{S} 1}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 1}\right) & 2 \rho_{2} V_{\mathrm{S} 2} \sin \theta_{\mathrm{S} 2} \cos \theta_{2} & \rho_{2} V_{\mathrm{S} 2}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 2}\right) \\ \rho_{1} V_{\mathrm{P} 1}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 1}\right) & -\rho_{1} V_{\mathrm{S} 1} \sin 2 \theta_{\mathrm{S} 1} & -\rho_{2} V_{\mathrm{P} 2}\left(1-2 \sin ^{2} \theta_{\mathrm{S} 2}\right) & \rho_{2} V_{\mathrm{S} 2} \sin 2 \theta_{\mathrm{S} 2}\end{array}\right) # # # :math:`\theta` and :math:`\theta_s` are the angles of P- and S-wave propagation,respectively. # As we can see, the Zoeppritz equations are complicated and do not give an intuitive feel for how rock properties impact the change of amplitude with angle. So simpler approximations are required. # # Aki-Richards approximation to Zeoppritz Equation # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Aki and Richards (1980) also derived a simplified form by assuming small layer # contrasts. The results are conveniently expressed in terms of contrasts in VP, VS, and ρ as # follows # # .. math:: # R_{pp}(\theta)=\frac{1}{2}\left(\frac{\Delta \rho}{\bar{\rho}}+\frac{\Delta V_{p}}{\bar{V}_{p}}\right)-2\left(\frac{\bar{V}_{S}}{\bar{V}_{p}}\right)^{2}\left(\frac{\Delta \rho}{\bar{\rho}}+\frac{2 \Delta V_{S}}{\bar{V}_{S}}\right) \sin ^{2}(\theta)+\frac{\Delta V_{p}}{2 \bar{V}_{p}} \tan ^{2}(\theta) # # # .. math:: # \Delta \rho=\rho_2-\rho_1, \overline{\rho} = \frac{\rho_1+\rho_2}{2}, \Delta V_p=V_{p2}-V_{p1}, \Delta V_s=V_{s2}-V_{s1} # # # .. math:: # \overline{V_p}=\frac{V_{p1}+V_{p2}}{2}, \overline{V_s}=\frac{V_{s1}+V_{s2}}{2} # # # Intercept and gradient # ~~~~~~~~~~~~~~~~~~~~~~ # # The expression is equivelent to: # # .. math:: # R_{pp}(\theta)=\frac{1}{2}(\frac{\Delta \rho}{\bar \rho}+\frac{\Delta V_p}{\bar{V_p}})+\left [ \frac{\Delta V_p}{2\bar{V_p}}-2\left ( \frac{\bar{V_s}}{\bar{V_p}} \right )^2\ \left ( \frac{\Delta \rho}{\bar \rho}+\frac{2\Delta V_S}{\bar V_s} \right ) \right ] \sin^2\theta +\frac{1}{2}\frac{\Delta V_p}{\bar{V_p}}(\tan^2\theta-\sin^2\theta) # # # and this can be written as three-term form: # # .. math:: # R_{PP}(\theta)\approx R_{P0}+B \sin^{2} \theta+C(\tan^{2} \theta-\sin ^{2} \theta) # # # where :math:`R_{P0}` represents the zero offset P wave section, B describes the variation at intermediate offsets and is often called the AVO gradient, and C dominates at far offsets near the critical angle, it controls the turning point for the angle dependent reflectivity at medium to large angles close to critical angle if it exists. # # When assume small incident angle, the three-term approximation can be reduced to two-term approximation: # # .. math:: # R_{PP}(\theta)\approx R_{P0}+B \sin^{2} \theta # # # # Example # ~~~~~~~ # Assume constant vertical velocity in the overburden above a horizontal reflector. The reflector shown represent a clayrich overburden overlying a thick water saturated sandstone. The elastic properties of the clay are as below. Calculate the P-to-P reflectivity of the clay-wet sand interface for the angles up to 90 degree. # # %% # the clay rich layer is denoted as 1 and the wet sandstone layer is denoted as 2 vp1=2190 vs1=716 den1=2118 vp2=2760 vs2=1473 den2=2229 theta=np.linspace(0,90,100) R_pp,R_ps, Rpp0, gradient=AVO.Aki_Richards(theta, vp1,vp2,vs1,vs2,den1,den2) # plot plt.plot(theta,R_pp,'b-',lw=3,label='Rpp') plt.plot(theta,R_ps,'r-',lw=3,label='Rps') plt.ylim(-0.5,0.5) plt.xlim(0,90) plt.xlabel(' \\theta ') plt.ylabel('R') plt.xticks(np.arange(min(theta), max(theta)+1, 10)) plt.grid(ls='--') plt.legend() # %% # # **Reference**: Mavko, G., Mukerji, T. and Dvorkin, J., 2020. The rock physics handbook. Cambridge university press. #