# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## 2 R/T Coefficients Expressed by Series Functions  
For a thin bed in elastic isotropic media, as shown in Figure 1, the exact displacement R/T coefficient equations for the P-wave incident are as follows [23],  
\[\mathbf{M}\Big{[}\begin{array}{cccc}R_{PP}&R_{PS}&T_{PP}&T_{PS}\end{array} \Big{]}^{T}=\mathbf{n} \tag{1}\]  
where \(R_{PP}\), \(R_{PS}\), \(T_{PP}\), \(T_{PS}\) are displacement R/T coefficients of P-waves and converted S-waves, respectively; \(\mathbf{M}\) is a \(4\times 4\) matrix, \(\mathbf{n}\) is a \(4\times 1\) vector, which is presented in Appendix A in detail.  
According to [PERSON]'s rule, the displacement R/T coefficients \(R_{PP}\), \(R_{PS}\), \(T_{PP}\), \(T_{PS}\) can be calculated. When the thin-bed thickness is equal to zero, Equation (1) will be reduced to that for R/T coefficients equations of a single interface between the thin-bed roof and floor, which is consistent with the Zoeppritz equations [35].  
From Equation (1), seismic theoretical characteristics analysis of a thin bed is easily achieved. However, complicated matrix calculations prevent its application in thin-bed AVO analysis and inversion. To solve Equation (1) approximately, the trigonometric function parameters are rewritten as a series expansion of sine incidence angle (see Appendix B).  
Substituting Equations (A18)-(A34) into Equation (1), the elements of \(\mathbf{M}\) and \(\mathbf{n}\) are all expressed by series functions of sine incidence angle (sin\(\theta\)). If the elements are even or odd series functions of sin\(\theta\), we mark them by 'even' or 'odd' in Equation (1) respectively as follows,  
\[\left[\begin{array}{cccc}odd&even&odd&even\\ even&odd&even&odd\\ even&odd&even&odd\\ odd&even&odd&even\end{array}\right]\left[\begin{array}{c}R_{PP}\\ R_{PS}\\ T_{PP}\\ T_{PS}\end{array}\right]=\left[\begin{array}{c}odd\\ even\\ even\\ odd\end{array}\right] \tag{2}\]  
From Equation (2), it can be seen that \(R_{PP}\), \(T_{PP}\) are even power series functions of sin\(\theta\), while \(R_{PS}\), \(T_{PS}\) are odd power series functions of sin\(\theta\). Therefore, \(R_{PP}\), \(R_{PS}\), \(T_{PP}\), \(T_{PS}\) can be expressed by series functions of sin\(\theta\) as follows,  
\[\left\{\begin{array}{c}R_{PP}=\sum\limits_{n=0}^{\infty}A_{2n}\sin^{2n} \theta\\ R_{PS}=\sum\limits_{n=0}^{\infty}A_{2n+1}\sin^{2n+1}\theta\\ T_{PP}=\sum\limits_{n=0}^{\infty}B_{2n}\sin^{2n}\theta\\ T_{PS}=\sum\limits_{n=0}^{\infty}B_{2n+1}\sin^{2n+1}\theta\end{array}\right. \tag{3}\]  
where \(A\) and \(B\) are coefficients of series functions.  
Figure 1: A thin-bed model with two horizontal interfaces. \(P\) is incident P-wave, \(PS\) is reflected or transmitted S-wave, \(PP\) is reflected or transmitted P-wave; \(h\) is thin-bed thickness; \(\theta\) is incidence or reflected angle of P-wave, \(\theta_{3}\) is P-wave transmitted angle, \(\delta_{1}\) and \(\delta_{3}\) are S-wave reflected and transmitted angles respectively; \(v_{P}\) and \(v_{S}\) are P-wave and S-wave velocities respectively, \(\rho\) is density, and their subscripts 1, 2, 3 refer to three layers of the thin bed respectively.