# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## 4 Accuracy Analysis  
We tested the accuracy of the second-order series approximation on four representative thin-bed models, including (1) high-impedance thin bed, (2) low-impedance thin bed, (3) low-to-high impedance transition thin bed, and (4) high-to-low impedance transition thin bed. Table 1 lists rock properties of these models, including P-wave velocity (\(v_{P}\)), S-wave velocity (\(v_{S}\)), density (\(\rho\)), the normal-incident P-wave reflection coefficient of thin-bed top-interface (\(R_{1}\)), as well as the normal-incident P-wave reflection coefficient of thin-bed bottom-interface (\(R_{2}\)). Figure 3: Algorithmic steps of the second-order series approximation. We compare the second-order series approximation and the true value of thin-bed \(R_{PP}\) in Figures 4-7 for Models 1-4 respectively. For a better understanding of differences between thin-bed and single-interface responses, \(R_{PP}\) of thin-bed top- and bottom- interfaces based on the Zoeppritz equations are also curved in Figures 4-7. In the numerical analysis, the thin-bed thickness is set as 3 m and incidence frequencies are set as 20, 30, and 40 Hz, respectively. Meanwhile, we plot relative errors of \(R_{PP}\) induced by the approximate formula under the small-incidence assumption for Models 1-4 in Figure 8. Considering small-incidence assumption of the approximate formula, the maximum incidence angle in all cases is set as 30 degrees. Due to that thin-bed \(R_{PP}\) are complex valued, we discuss them in terms of amplitude and phase components, respectively. Figures 4 and 5 show \(R_{PP}\) of Models 1 and 2 versus incidence angles, respectively. Models 1 and 2 are high-impedance or low-impedance thin beds with strong impedance contrasts. Destructive interference effects cause that thin-bed reflected amplitudes are obviously lower than those of top- and bottom-interfaces. Meanwhile, the thin-bed reflected phases are not constant zero or \(\pi\) as those of top- and bottom-interfaces, respectively. Compared with the true \(R_{PP}\), the approximated values show a similar change regularity that a larger incidence angle and a lower incidence frequency result in a lower reflected amplitude. For incidence angles smaller than 20 degrees, approximate thin-bed amplitudes are almost the same as the true values. When incidence angles are over 20 degrees, the approximate thin-bed amplitudes are lower than the true values and show a larger deviation from the true values at a larger incidence angle. The approximated phase is almost the same as the true phase in the angle range from 0 to 30 degrees. Figures 6 and 7 show \(R_{PP}\) of Models 3 and 4 versus incidence angles, respectively. Models 3 and 4 are impedance transition thin beds with equal polarities, which differ from opposite polarities of Models 1-2. Constructive interference effects cause that thin-bed reflected amplitudes are higher than those of top- and bottom-interfaces. Compared with Models 1-2 cases, \(R_{PP}\)'s amplitudes of Models 3 and 4 are less sensitive to frequency variation. The approximate Amplitude-Versus-Angle (AVA) curves coincide with the true AVA curves for incidence angles smaller than 20 degrees and deviate progressively from the true for incidence angles over 20 degrees. Comparison of Models 3 and 4 shows that the former's phases are nearly zero while these of the latter are nearly \(\pi\).