# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## Appendix D  
\(A_{0}\), \(A_{2}\) are complex valued, their real, imaginary, amplitude and phase components are as follows. For \(A_{0}\), there are:  
\[\text{Re}(A_{0})=\left[\left(\frac{z_{P3}}{z_{P1}}-1\right)\cos\tau{n_{1}}^{ \prime}+\left(\frac{z_{P2}}{z_{P1}}-\frac{z_{P3}}{z_{P2}}\right)\sin\tau{n_{2} }^{\prime}\right]/{m_{0}}^{\prime} \tag{117}\]  
\[\text{Im}(A_{0})=\left[2\left(\frac{z_{P2}}{z_{P1}}-\frac{z_{P3}}{z_{P1}} \frac{z_{P3}}{z_{P2}}\right)\sin\tau\cos\tau\right]/{m_{0}}^{\prime} \tag{118}\]  
\[Amplitude(A_{0})=\left[\text{Re}^{2}(A_{0})+\text{Im}^{2}(A_{0})\right]^{1/2} \tag{119}\]  
\[Phase(A_{0})=\tan^{-1}[\text{Im}(A_{0})/\text{Re}(A_{0})] \tag{120}\]  
For \(A_{2}\), there are:  
\[\text{Re}(A_{2})=\left[ \begin{array}{c}{c_{3}}^{\prime}{n_{1}}^{\prime}+{c_{ 7}}^{\prime}{n_{2}}^{\prime}+{c_{13}}^{\prime}\text{Re}(A_{0})+{c_{14}}^{ \prime}\text{Im}(A_{0})+{c_{15}}^{\prime}\text{Re}(A_{1})+{c_{16}}^{\prime} \text{Im}(A_{1})\\ +{c_{11}}^{\prime}{n_{1}}^{\prime}\text{Re}(B_{0})+{c_{11}}^{\prime}{n_{2}}^{ \prime}\text{Im}(B_{0})+{c_{12}}^{\prime}{n_{1}}^{\prime}\text{Re}(B_{1})+{c_ {12}}^{\prime}{n_{2}}^{\prime}\text{Im}(B_{1})\end{array} \right]/{m_{0}}^{\prime} \tag{121}\]  
\[\text{Im}(A_{2})=\left[ \begin{array}{c}{c_{7}}^{\prime}{n_{1}}^{\prime}-{c_{ 3}}^{\prime}{n_{2}}^{\prime}-{c_{14}}^{\prime}\text{Re}(A_{0})+{c_{13}}^{ \prime}\text{Im}(A_{0})-{c_{16}}^{\prime}\text{Re}(A_{1})+{c_{15}}^{\prime} \text{Im}(A_{1})\\ -{c_{11}}^{\prime}{n_{2}}^{\prime}\text{Re}(B_{0})+{c_{11}}^{\prime}{n_{1}}^{ \prime}\text{Im}(B_{0})-{c_{12}}^{\prime}{n_{2}}^{\prime}\text{Re}(B_{1})+{c_ {12}}^{\prime}{n_{1}}^{\prime}\text{Im}(B_{1})\end{array} \right]/{m_{0}}^{\prime} \tag{122}\]  
\[Amplitude(A_{2})=\left[\text{Re}^{2}(A_{2})+\text{Im}^{2}(A_{2})\right]^{1/2} \tag{123}\]  
\[Phase(A_{2})=\tan^{-1}[\text{Im}(A_{2})/\text{Re}(A_{2})] \tag{124}\]  
with  
\[\text{Re}(B_{0})={2n_{1}}^{\prime}/{m_{0}}^{\prime},\text{Im}(B_{0})=-2{n_{2}} ^{\prime}/{m_{0}}^{\prime} \tag{125}\]  
\[\text{Re}(A_{1})=\left[ \begin{array}{c}{a_{3}}^{\prime}{n_{3}}^{\prime}+{a_{7 }}^{\prime}{n_{4}}^{\prime}+{a_{10}}^{\prime}\text{Re}(A_{0})+{a_{11}}^{ \prime}\text{Im}(A_{0})\\ +{a_{9}}^{\prime}{n_{3}}^{\prime}\text{Re}(B_{0})+{a_{9}}^{\prime}{n_{4}}^{ \prime}\text{Im}(B_{0})\end{array} \right]/{m_{1}}^{\prime} \tag{126}\]  
\[\text{Im}(A_{1})=\left[ \begin{array}{c}{a_{7}}^{\prime}{n_{3}}^{\prime}-{a_{3} }^{\prime}{n_{4}}^{\prime}-{a_{11}}^{\prime}\text{Re}(A_{0})+{a_{10}}^{\prime} \text{Im}(A_{0})\\ -{a_{9}}^{\prime}{n_{4}}^{\prime}\text{Re}(B_{0})+{a_{9}}^{\prime}{n_{3}}^{ \prime}\text{Im}(B_{0})\end{array} \right]/{m_{1}}^{\prime} \tag{127}\]\[\text{Re}(B_{1}) = \left[ \begin{array}{c}{b_{3}}^{\prime}{n_{3}}^{\prime}+{b_{7}}^{ \prime}{n_{4}}^{\prime}+{b_{11}}^{\prime}\text{Re}(A_{0})+{b_{12}}^{\prime} \text{Im}(A_{0})\\ +{b_{13}}^{\prime}\text{Re}(B_{0})+{b_{14}}^{\prime}\text{Im}(B_{0})\end{array}  \right]/{m_{1}}^{\prime}\] (114) \[\text{Im}(B_{1}) = \left[ \begin{array}{c}{b_{7}}^{\prime}{n_{3}}^{\prime}-{b_{3}}^ {\prime}{n_{4}}^{\prime}-{b_{12}}^{\prime}\text{Re}(A_{0})+{b_{11}}^{\prime} \text{Im}(A_{0})\\ -{b_{14}}\text{Re}(B_{0})+{b_{13}}^{\prime}\text{Im}(B_{0})\end{array}