# P-Wave Reflection Approximation of a Thin Bed and Its Application
## Appendix A  
In Equation (1), \(A_{0}\) and \(A_{2}\) are defined as  
\[A_{0}=\frac{1}{m_{0}}\left[\left(\frac{z_{P3}}{z_{P1}}-1\right)\cos\tau+j \left(\frac{z_{P2}}{z_{P1}}-\frac{z_{P3}}{z_{P2}}\right)\sin\tau\right], \tag{1}\]  
and  
\[\begin{array}{l}A_{2}=\frac{1}{m_{0}}\left[-\frac{\frac{2}{2}}{2}\tau\sin \tau+2\frac{4g_{1}^{2}}{r_{1}^{2}}(d_{1}-1)+\frac{2}{r_{1}}\cos\tau+b_{1}\frac{ z_{P3}}{z_{P1}}+jb_{3}\frac{z_{P3}}{z_{P3}}\right](1+A_{0})\\ +\frac{1}{m_{0}}\left(b_{6}\frac{z_{P1}}{z_{P1}}+\frac{2b_{P3}}{r_{1}^{2}}+jb_ {4}\frac{z_{P2}}{z_{P1}}\right)(1-A_{0})+\frac{b_{5}}{m_{0}}\left[l_{1}l_{2} \left(\frac{r_{3}^{2}}{4}-1\right)B_{0}+\left(\frac{\tau}{2}-1\right)B_{1} \right]\\ +\frac{1}{m_{0}}\left[\begin{array}{c}\left(b_{1}+\frac{\cos\tau}{r_{1}} \right)\frac{z_{P3}}{z_{P1}}+\frac{2}{r_{1}^{2}}\cos\tau+l_{1}a_{3}\left( \frac{2}{r_{2}}\frac{z_{P2}}{z_{P1}}+\frac{1}{r_{1}}\frac{z_{P3}}{z_{P2}} \right)\\ +\frac{2j\sin\tau}{r_{1}^{2}}\frac{z_{P3}}{z_{P2}}+j\sin\frac{\tau}{r_{1}} \frac{z_{P3}}{z_{P1}}-j\frac{b_{P3}}{r_{1}}\end{array}\right]A_{1}\end{array} \tag{2}\]  
where  
\[B_{0}=\frac{2}{m_{0}}, \tag{3}\]  
\[\begin{array}{l}A_{1}=\frac{1}{m_{1}}\left[\left(\frac{a_{3}}{d_{1}}-\cos q \right)\frac{z_{53}}{z_{51}}+j\left(a_{5}-\frac{z_{53}}{z_{51}}\sin q\right) \right](1+A_{0})+\frac{1}{m_{1}}\left(l_{1}l_{2}\frac{z_{53}}{z_{51}}-a_{4} \right)B_{0}\\ +\frac{1}{m_{1}}\left[\left(\frac{1}{r_{1}}-\frac{l_{1}}{r_{2}}\frac{z_{53}} {z_{51}}\right)a_{6}+\frac{2l_{1}}{r_{2}}\cos\tau\frac{z_{52}}{z_{51}}+jl_{1} \sin\tau\frac{z_{53}}{z_{51}}\right](1-A_{0})\end{array} \tag{4}\]  
and  
\[\begin{array}{l}B_{1}=\frac{1}{m_{1}}\left(1-\frac{a_{24}a_{1}}{d_{1}}+ja_ {1}a_{5}\right)(1+A_{0})-\frac{1}{m_{1}}[l_{1}l_{2}a_{2}+(\cos q+j\sin q)a_{ 4}]B_{0}\\ +\frac{1}{m_{1}}\left[\frac{2}{r_{1}}-ja_{2}l_{1}\sin\tau+\frac{2l_{1}}{r_{2}} \frac{z_{52}}{z_{51}}(a_{1}\cos\tau-1)\right](1-A_{0})\end{array} \tag{5}\]  
with  
\[m_{0}=(z_{P3}/z_{P1}+1)\cos\tau+j(z_{P2}/z_{P1}+z_{P3}/z_{P2})\sin\tau, \tag{6}\]  
\[m_{1}=(z_{S3}/z_{S1}+1)\cos q+j(z_{S2}/z_{S1}+z_{S3}/z_{S2})\sin q, \tag{7}\]  
\[a_{1}=\cos q+j\frac{z_{S1}}{z_{S2}}\sin q,\ a_{2}=\cos q+j\frac{z_{S2}}{z_{S1}} \sin q,\ a_{3}=\cos q-\cos\tau, \tag{8}\]  
\[a_{4}=\frac{2l_{1}l_{2}}{r_{3}}\frac{z_{S3}}{z_{S1}},\ a_{5}=\frac{r_{1}l_{1}}{ r_{2}}\left(\sin q-\frac{2}{r_{2}}\sin\tau\right),\ a_{6}=2\left(\cos q+j\sin q\frac{z_{S3}}{ z_{S2}}\right), \tag{9}\]\[b_{1}=\frac{jl_{1}}{r_{2}}\Big{(}\sin q-\frac{2}{r_{2}}\sin\tau\Big{)},\ b_{2}= \sin\tau-\frac{2}{r_{2}}\sin q, \tag{10}\]  
\[b_{3}=\frac{l_{1}^{2}}{2}(\sin\tau+\tau\cos\tau)-\frac{l_{1}^{2}}{r_{2}}\sin q+ \frac{2}{r_{1}^{2}}\sin\tau, \tag{11}\]  
\[b_{4}=4\frac{l_{1}^{2}}{r_{2}^{2}}\Big{(}\frac{1}{r_{2}}\sin q-\sin\tau\Big{)} +\frac{l_{1}^{2}}{2}(\sin\tau-\tau\cos\tau)-\frac{1}{2}\sin\tau, \tag{12}\]  
\[b_{5}=\frac{2l_{1}l_{2}}{r_{3}^{2}}\frac{z_{p3}}{z_{p1}},\ b_{6}=\frac{1}{2} \Big{(}l_{1}^{2}\tau\sin\tau-\cos\tau\Big{)}+2a_{3}\Bigg{(}\frac{l_{1}^{2}}{r_ {2}^{2}}-\frac{1}{r_{1}^{2}4l_{1}}\Bigg{)} \tag{13}\]  
\[\tau=\frac{\omega h}{v_{P2}},\ q=\frac{\omega h}{v_{S2}},\ d_{1}=\frac{\rho_{ 2}}{\rho_{1}}, \tag{14}\]  
where \(z_{P}=\rho v_{\rm P}\) are P-wave impedances, \(z_{S}=\rho v_{\rm S}\) are S-wave impedances, \(r=v_{\rm P}/v_{\rm S}\) are the ratios of P-wave velocities to S-wave velocities and their subscripts 1, 2, 3 refer to three layers of the single thin bed, respectively. \(l_{k}=v_{{\rm P}k+1}/v_{{\rm P}k}\) are the ratios of the P-wave velocities in adjacent layers with \(k=1\), 2, respectively; \(v_{P}\) and vs. are P-wave and S-wave velocities, respectively, \(\rho\) is density, \(\delta\) is S-wave transmitted or reflected angle and their subscripts 1, 2, 3 refer to three layers of the thin bed, respectively; \(\theta\) is P-wave incidence angle, \(\theta_{2}\) and \(\theta_{3}\) are P-wave transmitted or reflected angle in the target thin bed and its floor, respectively; \(h\) is the thickness of the middle layer, \(\omega\) is angular frequency and \(j\) is the imaginary symbol.