# P-Wave Reflection Approximation of a Thin Bed and Its Application
## 3 Numerical Analysis and Application  
### Approximation Accuracy Analysis  
In order to verify the accuracy of the linear analytic approximation of the thin-bed P-wave reflection coefficient derived in this paper, we conduct tests on four representative thin-bed models, including: (1) high-impedance thin bed, (2) low-to-high impedance transition layers, (3) low-impedance thin bed and (4) high-to-low impedance transition layers [10; 33]. Table 1 lists the elastic parameters and Figure 2 plots the reflectivity logs of these four models. Figure 2 shows that Models 1 and 3 produce opposite-polarity reflections, while Models 2 and 4 produce identical-polarity reflections. Compared with Models 2-4, Model 1 has stronger impedance contrasts, which are utilized to test the approximate formula's applicability in thin-bed models with strong impedance contrasts. In the numerical simulations, we set thin-bed thicknesses as \(\lambda\)/8, \(\lambda\)/10, \(\lambda\)/20, \(\lambda\)/30, \(\lambda\)/40, \(\lambda\)/50 and \(\lambda\)/60, where \(\lambda\) is the P-wavelength in the target thin bed. Considering the small-incidence assumption, the incidence angle in all cases is set from 0 degree to 30 degrees with a 1-degree sample interval. For a better understanding of the behavior of the linear analytic approximate formula, we compare the linear analytic approximation and the true value of thin-bed P-wave reflection coefficients [20] by illustrating \(\lambda\)/10 and \(\lambda\)/20 examples, as shown in Figure 3. In order to prove that the parameters P and G are derived accurately, the cases of thin-bed thicknesses equal to zero, i.e., single-interface cases, are also shown in Figure 3. The absolute values of relative errors are plotted in Figure 4. Thin-bed PP-wave reflection coefficients are complex valued. Therefore, we discuss thin-bed P-wave reflection coefficients in terms of their amplitude and phase sections, which are shown in Figures 3 and 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Layer No.** & \(\mathbf{v_{P}}\) & \(\mathbf{v_{S}}\) & \(\rho\) & \(\mathbf{I_{P}}\) & \(\mathbf{I_{S}}\) \\ \hline \multirow{3}{*}{Model 1} & 1 & 3050 & 1525 & 2.7 & 8235 & 4117.5 \\  & 2 & 6100 & 3050 & 2.7 & 16,470 & 8235 \\  & 3 & 2500 & 1525 & 2.7 & 6750 & 4117.5 \\ \hline \multirow{3}{*}{Model 2} & 1 & 3050 & 1600 & 2.7 & 8235 & 4320 \\  & 2 & 4200 & 2500 & 2.7 & 11,340 & 6750 \\  & 3 & 6100 & 3100 & 2.7 & 16,470 & 8370 \\ \hline \multirow{3}{*}{Model 3} & 1 & 2200 & 1200 & 2.3 & 5060 & 2760 \\  & 2 & 1800 & 800 & 2.2 & 3960 & 1760 \\  & 3 & 3050 & 1400 & 2.35 & 7167.5 & 3290 \\ \hline \multirow{3}{*}{Model 4} & 1 & 6100 & 3100 & 2.7 & 16,470 & 8370 \\  & 2 & 4200 & 2500 & 2.7 & 11,340 & 6750 \\ \cline{1-1}  & 3 & 3050 & 1600 & 2.7 & 8235 & 4320 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of thin-bed models. \(\mathrm{I_{P}}\) and \(\mathrm{I_{S}}\) are P-wave and S-wave impedances, respectively, units of velocities and densities in Table 1 are m/s and g/cm\({}^{3}\), respectively. Figure 2: Reflectivity logs of Models 1–4 for normal incidence: (**a**) Model 1, (**b**) Model 2, (**c**) Model 3 and (**d**) Model 4. Figure 3 shows the P-wave reflection coefficients as a function of incidence angle for Models 1-4 for only three of the modeled cases, \(\lambda\)/10, \(\lambda\)/20 and 0 m. As models with opposite-polarity reflections, Models 1 and 3 share similar regularity. In the angle range from 0 to 20 degrees, the linear analytic approximation demonstrates the same AVA gradient, but a lower AVA intercept, compared to the true PP-wave response. For incidence angles greater than 20 degrees, the linear analytic approximation starts to diverge further from the true value. An explanation for this behavior is that the linear analytic approximation relies on the small-incidence assumption. As models with identical-polarity reflections, Models 2 and 4 share similar regularity only for incidence angles smaller than 20 degrees.