Figure 1. Comparison of the reflection coefficients combining equations  4 and 7 (R-GEI) and the reflection coefficients combining equations  2 and 7 (R-NGEI) with the reflection coefficients from the exact Zoeppritz equation and Aki-Richards equation of the negative impedance reflector

Figure 3. Comparison of the exact Zoeppritz equation, Aki-Richards approximation, and novel equation in terms of PSR, P-wave velocity, and density of the negative impedance reflector. (a) Reflection coefficients with exact Zoeppritz equation, Aki-Richards equation, and novel equation. (b

Figure 4. (a) A comparison between the curves derived from the exact Zoeppritz equation (circle), Aki-Richard equation (point), and fluid-matrix decoupled equation (plus) in model 1, (b) a comparison between the curves derived from the exact Zoeppritz equation (circle), Aki-Richard equation

Figure 3. Comparison between the reflection coefficients as a function of the incidence angle computed with the full Zoeppritz equation and the rock-physics model (the solid black line), the Aki-Richards equation with the full/nonlinear rock-physics model (the dashed blue line), the Aki-Richards

Figure 8. Comparison of the top interface of the exact Zoeppritz formula, the Aki-Richards equation, and our proposed approximate formula. (a) Computed reflection coefficients and (b) approximation errors.

Figure 9. Comparison of the bottom interface of the exact Zoeppritz formula, the Aki-Richards equation, and our proposed approximate formula. (a) Computed reflection coefficients and (b) approximation errors.

Figure 6. A comparison of the curves derived using the exact Zoeppritz equation (the solid marked circle), the Aki-Richards equation (the solid line), the Fatti equation (the dashed line), the Russell equation (the dashed-dotted line), and the new equation (the dotted line) in (a) model 1, (b

Figure 9. A comparison between the full Zoeppritz equations (see Appendix  A ) and the two forms of the Aki-Richards equation computed over the top of a gas sand, where (a) shows the intercept-gradient-curvature approximation of equation  12 and (b) shows the Fatti et al. (1994) approximation

seismic data computed using Aki-Richards’ equation.

Figure 3. A two-layered model showing the contribution for V P (blue), V S (orange), and density (green) toward the total AVO response (red) for three-term Aki-Richards equation with (a) top layer full gas and bottom layer PRG, (b) top layer PRG and bottom layer brine, and (c) top layer

Figure 1. Comparison of exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI of the negative reflector. (a) Reflection coefficients with exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI (b) Difference of reflection coefficients

Figure 2. Comparison of exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI of the positive reflector. (a) Reflection coefficients with exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI (b) Difference of reflection coefficients

Figure 1. Comparison of the exact Zoeppritz equation, the Aki-Richards approximate equation, and the approximate equation in terms of P-wave modulus, S-wave modulus, and density. (a) Reflection coefficient with the exact Zoeppritz equation, the Aki-Richards approximate equation

Figure 11. Comparing the modeled AVO and FAVO sensitivity to gas saturation in the sand B reservoir (Figure  8 ). (a and b) Sensitivity of the AVO amplitude (following the three-term Aki-Richard equation) at angles 20° and 30° at the interface of shale ( α = 2895.60    m / s , β

Figure 13. Profiles of the inverted results of (a) PSR, (b) P-wave velocity, (c) and density with Aki-Richards reflectivity equation across the known wells A and B. The inverted PSR result is indirectly computed with P- and S-wave velocities, which are estimated from seismic data with the Aki

Figure 2. Four types of AVO reflection coefficients calculated with the exact Zoeppritz equation, Aki-Richards formula, Russell’s linear approximation, and equation  10 . (a) Calculated AVO reflection coefficients and (b) errors of AVO reflection coefficients (the dotted red lines are the AVO