# P-Wave Reflection Approximation of a Thin Bed and Its Application
## 3 Numerical Analysis and Application
### Approximation Accuracy Analysis
For Model 2 with a 30-degree critical angle, the AVA curves of the linear analytic approximation diverge further from the true value with increasing angle over 20 degrees. Especially near the critical angle, the true P-wave reflection coefficient increases sharply, while the linear analytic approximation becomes invalid. For Model 4 with no critical angles, the linear analytic approximation is nearly the same as the true values. Comparison of the \(\lambda\)/10 and \(\lambda\)/20 cases shows that a thinner bed results in a higher  
Figure 3: Comparison of the linear analytic approximation and the true value of P-wave reflection coefficients: (**a**) amplitude of Model 1; (**b**) phase of Model 1; (**c**) amplitude of Model 2; (**d**) phase of Model 2; (**e**) amplitude of Model 3; (**f**) phase of Model 3; (**g**) amplitude of Model 4; (**h**) phase of Model 4.  
accuracy for amplitude and phase responses of the linear analytic approximation. For single-interface cases, the true equations of thin-bed R/T coefficients are simplified into Zoeppritz equations [20]. Approximate amplitudes are coincident with the true amplitudes for incidence angles less than 20 degrees. Meanwhile, approximate phases are zero or pi, which are equal to the true phases. The high accuracy of approximate values for single-interface cases confirms that derivations of the parameters P and G are accurate. Figure 4 shows errors in amplitude (left) and phase (right) caused by the linear analytic approximation for Models 1-4 with incidence angles for the full range of \(\uplambda\)h cases tested. Comparison between different \(\uplambda\)h cases for small incidence shows that approximation errors decline with a thinner bed and drop below 5% rapidly with decreasing thickness of the thin bed. For models with critical angles, i.e., Models 1-3, the relative errors versus incidence angle and thickness share a similar phenomenon. For incidence angles less than 20 degrees and thicknesses less than \(\uplambda\)10, the amplitude  
Figure 4: Absolute values of relative errors of thin-bed P-wave reflection coefficients caused by the linear analytic approximation: (**a**) amplitude of Model 1; (**b**) phase of Model 1; (**c**) amplitude of Model 2; (**d**) phase of Model 2; (**e**) amplitude of Model 3; (**f**) phase of Model 3; (**g**) amplitude of Model 4; (**h**) phase of Model 4.  
errors are less than 10%. For incidence angles greater than 20 degrees, the amplitude errors are larger than 10%, and increase with increasing incidence angle and thin-bed thickness. The amplitude errors rise quickly for incidence angles near the critical angles, and the linear analytic approximation becomes invalid. Applicability of the approximate formula for Model 1 shows that the linear analytic approximation is valid for thin-bed models with strong impedance contrasts. For model with no critical angles, i.e., Model 4, approximation errors smaller than 5% for a thickness smaller than \(\lambda\)/8. Compared with the Model 1-3 cases, the linear analytic approximation for Model 4 has a relatively higher accuracy, which exhibits a better applicability of the linear analytic approximation for low-to-high impedance transition layers. From numerical tests, we can conclude that: when incidence angles are less than 20 degrees and thin-bed thicknesses are less than one-tenth of the P-wave wavelength, the approximation amplitude errors are smaller than 10%, which verifies the applicability of the linear analytic approximation.