# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## 4 Accuracy Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Layer No.** & \(\mathbf{v_{P}}\) & \(\mathbf{v_{S}}\) & \(\mathbf{\rho}\) & \(\mathbf{R_{1}}\) & \(\mathbf{R_{2}}\) \\ \hline \multirow{3}{*}{Model 1} & 1 & 3000 & 1414 & 2.29 & / & / \\  & 2 & 3800 & 2103 & 2.43 & 0.1468 & -0.1468 \\  & 3 & 3000 & 1414 & 2.29 & / & / \\ \hline \multirow{3}{*}{Model 2} & 1 & 3000 & 1414 & 2.29 & / & / \\  & 2 & 2400 & 897 & 2.17 & \(-\)0.1376 & 0.1376 \\  & 3 & 3000 & 1414 & 2.29 & / & / \\ \hline \multirow{3}{*}{Model 3} & 1 & 3000 & 1414 & 2.29 & / & / \\  & 2 & 3200 & 1586 & 2.33 & 0.0409 & 0.0388 \\  & 3 & 3400 & 1759 & 2.37 & / & / \\ \hline \multirow{3}{*}{Model 4} & 1 & 3400 & 1759 & 2.37 & / & / \\  & 2 & 3200 & 1586 & 2.33 & \(-\)0.0388 & -0.0409 \\ \cline{1-1}  & 3 & 3000 & 1414 & 2.29 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rock properties of thin-bed models, units of velocities and densities are m/s and g/cm\({}^{3}\), respectively. Figure 4: Comparison of Model 1’s \(R_{pp}\) calculated by true value and the second-order series approximation, and top-interface and bottom-interface \(R_{pp}\) calculated by the Zoeppritz equations, (**a**) amplitude, and (**b**) phase. Figure 5: Comparison of Model 2’s \(R_{pp}\) calculated by true value and the second-order series approximation, and top-interface and bottom-interface \(R_{pp}\) calculated by the Zoeppritz equations, (**a**) amplitude, and (**b**) phase. Figure 6: Comparison of Model 3’s \(R_{pp}\) calculated by true value and the second-order series approximation, and top-interface and bottom-interface \(R_{pp}\) calculated by the Zoeppritz equations, (**a**) amplitude, and (**b**) phase. Figure 8 shows \(R_{PP}\)'s relative errors caused by the approximate formula under the small-incidence assumption. In the numerical analysis, thin-bed thickness varies from \(\lambda/100\) to \(\lambda/8\), where \(\lambda\) is the P-wave wavelength in the middle layer. Approximate amplitude errors of Models 1-2 are less than 5% for incidence angles smaller than 20 degrees, and are less than 10% for incidence angles less than 25 degrees. When incidence angles are over 25 degrees, amplitude errors increase rapidly with an incidence-angle increase. Phase errors are less than 10% for incidence angles smaller than 30 degrees. For Models 3 and 4, amplitude accuracy by the approximate formula is higher than those of Models 1-2. Approximate errors of Model 3 are less than 5% for incidence angles smaller than 25 degrees and are less than 10% for incidence angles smaller than 29 degrees. For Model 4, amplitude errors versus incidence angles have similar variation regularity with those of Model 3, while, phase errors are smaller than 5% for incidence angles less than or equal to 30 degrees. To sum up, thin-bed \(R_{PP}\) are significantly different from the top- or bottom-interface \(R_{PP}\) calculated by the Zoeppritz equations. The second-order series approximation has high accuracy for incidence angles less than 20 degrees and deviates progressively from the true values over 20 degrees. Correspondingly, the relative errors are less than 5% for incidence smaller than 20 degrees. Figure 7: Comparison of Model 4’s \(R_{PP}\) calculated by true value and the second-order series approximation, and top-interface and bottom-interface \(R_{PP}\) calculated by the Zoeppritz equations, (**a**) amplitude, and (**b**) phase.