# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## 3 Approximate Formula of PP-Wave Reflections  
For small incidence, \(\sin^{n}\theta\) decreases rapidly with increasing \(n\) in Equation (3). \(R_{PP}\) can be simplified into a second-order series approximation of sine incidence angle via a series truncation procedure as follows,  
\[R_{PP}\approx A_{0}+A_{2}\sin^{2}\theta \tag{4}\]  
Figure 2 shows the simplified procedure of the approximate formula from the exact R/T coefficient equation, i.e., Equation (1), in detail. Taking \(\sin\theta\) as an independent variable, we can obtain \(A_{0}\), \(B_{0}\), \(A_{1}\), \(B_{1}\), \(A_{2}\) in turn from the constant term, linear term and quadratic term of \(\sin\theta\) in Equation (1), respectively. Considering the constant term of sine incidence angle in Equation (1), i.e., normal incidence, then \(R_{PP}=A_{0}\), \(T_{PP}=B_{0}\), \(R_{PS}=0\), \(T_{PS}=0\). Equation (1) is simplified as  
\[\left\{ \begin{array}{l}\left(\cos\tau+j\frac{z_{p1}}{z_{p2}}\sin\tau\right)A _{0}+B_{0}=\cos\tau-j\frac{z_{p1}}{z_{p2}}\sin\tau\\ \left(\cos\tau+j\frac{z_{p2}}{z_{p1}}\sin\tau\right)A_{0}-\frac{z_{p3}}{z_{p1} }B_{0}=j\frac{z_{p2}}{z_{p1}}\sin\tau-\cos\tau\end{array} \right. \tag{5}\]  
where \(\tau=\omega h/v_{P2}\), \(z_{P}=\rho v_{P}\) are P-wave impedances and their subscripts 1,2,3 indicate three layers of the single thin bed, respectively; \(j\) is an imaginary symbol. Figure 2: Flowchart of the second-order series approximation simplified from the thin-bed exact R/T coefficient equations. Solving Equation (5), there are  
\[A_{0}=\left[\left(\frac{z_{P3}}{z_{P1}}-1\right)\cos\tau+j\left(\frac{z_{P2}}{z_{ P1}}-\frac{z_{P3}}{z_{P2}}\right)\sin\tau\right]/m_{0} \tag{6a}\] \[B_{0}=2/m_{0} \tag{6b}\]  
with \(m_{0}=(z_{P3}/z_{P1}+1){\cos\tau}+j(z_{P2}/z_{P1}+z_{P3}/z_{P2}){\sin\tau}\). When the thickness is equal to zero, the thin-bed model becomes a single interface constituted by the roof and floor of the original thin bed. Equation (6) is simplified as  
\[A_{0}=\frac{z_{P3}-z_{P1}}{z_{P3}+z_{P1}} \tag{7a}\] \[B_{0}=\frac{2z_{P1}}{z_{P3}+z_{P1}} \tag{7b}\]  
which are consistent with the normal reflection and transmission coefficients given by [PERSON] [35], respectively. Considering the linear term of sine incidence angle in Equation (1), then \(R_{PP}=A_{0}\), \(T_{PP}=B_{0}\), \(R_{PS}=A_{1}{\sin\theta}\), \(T_{PS}=B_{1}{\sin\theta}\). Equation (1) is simplified as  
\[\left\{ \begin{array}{ll}a_{1}A_{1}-B_{1}&=\left(\frac{a_{3}}{d_{1}}-\cos q \right)(1+A_{0})+l_{1}l_{2}B_{0}\\ &+j\left(l_{1}\sin\tau+\frac{2}{r_{1}}\frac{z_{51}}{z_{52}}\sin q-\frac{2l_{1} }{r_{2}}\sin q\right)(1-A_{0})\\ a_{2}A_{1}+\frac{z_{51}}{z_{51}}B_{1}&=j\left(a_{5}-\frac{z_{52}}{z_{51}}\sin q \right)(1+A_{0})\\ &+\left(\frac{2}{r_{1}}\cos q-\frac{2a_{3}l_{2}}{r_{2}}\frac{z_{52}}{z_{51}} \right)(1-A_{0})-a_{4}B_{0}\end{array} \right. \tag{8}\]  
where \(q=\omega h/v_{S2}\), \(d_{1}=\rho_{2}/\rho_{1}\); \(z_{S}=\rho v_{S}\) are S-wave impedances, \(r=v_{P}/v_{S}\) are the ratios of P-wave velocities to S-wave velocities, their subscripts 1, 2, 3 refer to three layers of the single thin bed respectively. \(l_{k}=v_{pk}+1/v_{pk}\) are the ratios of the P-wave velocities in adjacent layers with \(k=1\), 2, \(a_{1}\)-\(a_{5}\) are presented in Appendix C.  
Solving Equation (8), there are  
\[ \begin{array}{ll}A_{1}=\frac{1}{m_{1}}\left[\left(\frac{a_{3}}{d_{1}}-\cos q \right)\frac{z_{53}}{z_{51}}+j\left(a_{5}-\frac{z_{52}}{z_{51}}\sin q\right) \right](1+A_{0})+\frac{1}{m_{1}}\left(l_{1}l_{2}\frac{z_{53}}{z_{51}}-a_{4} \right)B_{0}\\ +\frac{1}{m_{1}}\left[\left(\frac{1}{r_{1}}-\frac{l_{1}}{r_{2}}\frac{z_{52}}{ z_{51}}\right)a_{6}+\frac{2l_{1}}{r_{2}}\cos\tau\frac{z_{52}}{z_{51}}+jl_{1}\sin\tau \frac{z_{53}}{z_{51}}\right](1-A_{0})\end{array} \] (9a) and \[