# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## 5 Application Example  
Using the second-order series approximation, we built templates to estimate thin-bed properties, including P-wave impedance ratios and thin-bed thickness. The templates are constituted by amplitude and phase contour maps of \(R_{PP}\)'s \(A_{0}\), \(A_{2}\) of 120 thin-bed models. Figure 9 shows the \(R_{1}\) and \(R_{2}\) distributions of 120 thin-bed models, which reveal that those 120 models include most of the cases for thin-bed \(R_{1}\) and \(R_{2}\) ranging from \(-\)0.2 to 0.2. Meanwhile, those 120 models cover all types of thin-bed models, i.e., high-impedance thin beds, low-impedance thin beds, high-to-low impedance transition thin beds and low-to-high impedance transition thin beds, of which each type includes 30 models. Figure 8: Relative errors of \(R_{PP}\) induced by the second-order series approximation, (**a**) amplitude of Model 1, (**b**) phase of Model 1, (**c**) amplitude of Model 2, (**d**) phase of Model 2, (**e**) amplitude of Model 3, (**f**) phase of Model 3, (**g**) amplitude of Model 4, and (**h**) phase of Model 4. The thin-bed thickness of those 120 models is set as \(\lambda/10\), \(\lambda/20\), \(\lambda/30\), \(\lambda/40\), \(\lambda/50\), \(\lambda/60\), \(\lambda/70\), \(\lambda/80\), \(\lambda/90\), and \(\lambda/100\). Using the second-order series approximation, we plot amplitudes and phases contour maps of \(R_{pp}\)'s \(A_{0}\), \(A_{2}\) of those 120 thin-beds at different \(R_{1}\), \(R_{2}\) and thicknesses. Taking the cases of \(\lambda/10\), \(\lambda/20\), \(\lambda/40\), and \(\lambda/60\) for examples, the contour maps of \(A_{0}\) and \(A_{2}\) of those 120 thin-bed models are shown in Figures 10-13 respectively. Considering that the coefficients \(A_{0}\) and \(A_{2}\) are also complex valued, we discuss them in terms of amplitude and phase components, respectively. The expressions of amplitude and phase of \(A_{0}\) and \(A_{2}\) are shown in Appendix D in detail. Figures 10 and 11 show \(A_{0}\)'s amplitude and phase of those 120 models versus \(R_{1}\) and \(R_{2}\), respectively. For the \(\lambda/10\) case, amplitude contours are similar to ellipses with the major axis along the line \(R_{2}=-R_{1}\) and minor axis along the line \(R_{2}=R_{1}\). The amplitude increases with increasing \(R_{1}\) and \(R_{2}\). The amplitude growth rate of thin-bed models with equal polarities is larger than opposite-polarity cases, especially for a thinner bed. For the cases of thin beds with higher \(R_{1}\), \(R_{2}\) and thinner thickness, the amplitude contours in the direction of major axis are approximately parallel to the line \(R_{2}=-R_{1}\). \(A_{0}\)'s phases show that a positive \(R_{2}\) results in a negative phase, while a negative \(R_{2}\) results in a positive phase. For the same \(R_{2}\), a positive \(R_{1}\) results in a relatively smaller phase than that of a negative \(R_{1}\) case. Figure 10: \(A_{0}\)’s amplitude of 120 thin-bed models with different thickness, \(R_{1}\) or \(R_{2}\). Figure 9: \(R_{1}\) and \(R_{2  
Figures 12 and 13 show \(A_{2}\)'s amplitude and phase of those 120 models versus \(R_{1}\) and \(R_{2}\), respectively. Compared with \(A_{0}\), \(A_{2}\) has a relatively higher amplitude and a completely different phase variation. For \(A_{2}\), a positive \(R_{2}\) results in a positive phase, while a negative \(R_{2}\) results in a negative phase. Figure 11: \(A_{0}\)’s phase of 120 thin-bed models with different thickness, \(R_{1}\) or \(R_{2}\). Figure 12: \(A_{2}\)’s amplitude of 120 thin-bed models with different thickness, \(R_{1}\) or \(R_{2}\). When thin-bed \(R_{1}\) and \(R_{2}\) range from \(-0.2\) to \(0.2\), thin-bed properties can be estimated based on the templates of \(A_{0}\) and \(A_{2}\). Once the amplitude and phase values of \(A_{0}\) and \(A_{2}\) are obtained from thin-bed AVA data, we can exhibit the corresponding contours in the templates and seek the intersections of those contours.