# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## Appendix B  
Through Snell's Law and power series expansions of sine incidence angle, the parameters of the trigonometric function in Equation (1) can be written as follows:  
\[\sin\theta_{2}=\frac{v_{P2}}{v_{P1}}\sin\theta,\sin\theta_{3}=\frac{v_{P3}}{v_ {P1}}\sin\theta \tag{24}\]\[\sin\delta_{1}=\frac{v_{S1}}{v_{P1}}\sin\theta,\sin\delta_{2}=\frac{v_{S2}}{v_{P1}} \sin\theta,\sin\delta_{3}=\frac{v_{S3}}{v_{P1}}\sin\theta \tag{119}\]  
\[\cos\theta=1-\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)}\sin^{2n}\theta, \cos\theta_{2}=1-\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(} \frac{v_{P2}}{v_{P1}}\sin\theta\bigg{)}^{2n} \tag{120}\]  
\[\cos\theta_{3}=1-\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(} \frac{v_{P3}}{v_{P1}}\sin\theta\bigg{)}^{2n},\cos\delta_{1}=1-\sum_{n=1}^{ \infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(}\frac{v_{S1}}{v_{P1}}\sin\theta \bigg{)}^{2n} \tag{121}\]  
\[\cos\delta_{2}=1-\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(} \frac{v_{S2}}{v_{P1}}\sin\theta\bigg{)}^{2n},\cos\delta_{3}=1-\sum_{n=1}^{ \infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(}\frac{v_{S3}}{v_{P1}}\sin\theta \bigg{)}^{2n} \tag{122}\]  
\[\sin 2\theta=2\sin\theta\Bigg{(}1-\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2 n-1)}\sin^{2n}\theta\Bigg{)} \tag{123}\]  
\[\sin 2\theta_{3}=2\frac{v_{S3}}{v_{P1}}\sin\theta\Bigg{[}1-\sum_{n=1}^{\infty} \frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(}\frac{v_{S1}}{v_{P1}}\sin\theta\bigg{)} ^{2n}\Bigg{]} \tag{124}\]  
\[\sin 2\delta_{2}=2\frac{v_{S2}}{v_{P1}}\sin\theta\Bigg{[}1-\sum_{n=1}^{\infty} \frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(}\frac{v_{S2}}{v_{P1}}\sin\theta\bigg{)} ^{2n}\Bigg{]} \tag{125}\]  
\[\sin 2\delta_{3}=2\frac{v_{S3}}{v_{P1}}\sin\theta\Bigg{[}1-\sum_{n=1}^{\infty} \frac{C_{2n}^{n}}{2^{2n}(2n-1)}\bigg{(}\frac{v_{S3}}{v_{P1}}\sin\theta\bigg{)} ^{2n}\Bigg{]} \tag{126}\]  
\[\cos 2\delta_{1}=1-2\bigg{(}\frac{v_{S1}}{v_{P1}}\sin\theta\bigg{)}^{2},\cos 2 \delta_{2}=1-2\bigg{(}\frac{v_{S2}}{v_{P1}}\sin\theta\bigg{)}^{2} \tag{127}\]  
\[\cos 2\delta_{3}=1-2\bigg{(}\frac{v_{S3}}{v_{P1}}\sin\theta\bigg{)}^{2},\tan \theta_{2}=\frac{v_{P2}}{v_{P1}}\sin\theta\bigg{[}1+\sum_{n=1}^{\infty}\frac{ C_{2n}^{n}}{2^{2n}}\bigg{(}\frac{v_{P2}}{v_{P1}}\sin\theta\bigg{)}^{2n} \Bigg{]} \tag{128}\]  
\[\tan\delta_{2}=\frac{v_{S2}}{v_{P1}}\sin\theta\Bigg{[}1+\sum_{n=1}^{\infty} \frac{C_{2n}^{n}}{2^{2n}}\bigg{(}\frac{v_{S2}}{v_{P1}}\sin\theta\bigg{)}^{2n} \Bigg{]} \tag{129}\]  
\[\sin P=\sum_{m=0}^{\infty}\Bigg{\{}\frac{(-1)^{m}\tau^{2m+1}}{(2m+1)!}\sum_{k= 0}^{2m+1}(-1)^{k}C_{2m+1}^{k}\Bigg{[}\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2 n}(2n-1)}\bigg{(}\frac{v_{P2}}{v_{P1}}\sin\theta\bigg{)}^{2n}\Bigg{]}^{k}\Bigg{\}} \tag{130}\]  
\[\cos P=\sum_{m=0}^{\infty}\Bigg{\{}\frac{(-1)^{m}\tau^{2m}}{(2m)!}\sum_{k=0}^{2 m}(-1)^{k}C_{2m}^{k}\Bigg{[}\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)} \bigg{(}\frac{v_{P2}}{v_{P1}}\sin\theta\bigg{)}^{2n}\Bigg{]}^{k}\Bigg{\}} \tag{131}\]  
\[\sin Q=\sum_{m=0}^{\infty}\Bigg{\{}\frac{(-1)^{m}q^{2m+1}}{(2m+1)!}\sum_{k=0}^{2 m+1}(-1)^{k}C_{2m+1}^{k}\Bigg{[}\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)} \bigg{(}\frac{v_{S2}}{v_{P1}}\sin\theta\bigg{)}^{2n}\Bigg{]}^{k}\Bigg{\}} \tag{132}\]  
\[\cos Q=\sum_{m=0}^{\infty}\Bigg{\{}\frac{(-1)^{m}q^{2m}}{(2m)!}\sum_{k=0}^{2m} (-1)^{k}C_{2m}^{k}\Bigg{[}\sum_{n=1}^{\infty}\frac{C_{2n}^{n}}{2^{2n}(2n-1)} \bigg{(}\frac{v_{S2}}{v_{P1}}\sin\theta\bigg{)}^{2n}\Bigg{]}^{k}\Bigg{\}} \tag{133}\]  
with \(\tau=\omega h/v_{P2}\), \(q=\omega h/v_{S2}\).