# P-Wave Reflection Approximation of a Thin Bed and Its Application
## 3 Numerical Analysis and Application
### Application Examples  
Using the linear analytic approximation of thin-bed P-wave reflection coefficients, thin-bed elastic parameters and thickness are inversed through a least-squares approach. We apply the proposed inversion flow to predict thin-bed properties from synthetic seismic data. Considering that thin-bed properties of Model 3 are close to those of some actual reservoirs, such as coal seams and unconsolidated sandstone, we take Model 3 as a thin-bed model to exhibit application of the approximate formula to synthetic data in this section. Using the true P-wave reflection coefficient, we simulate the synthetic AVA records of Model 3 using the Ricker wavelet with a 25 Hz predominant frequency. The thin-bed thickness is set as 3 m and the incidence angles range from 0 to 20 degrees. Taking the synthetic AVA records as the real seismic traces, we construct complex seismic traces through the Hilbert transform, which are shown in Figure 5. Compared with the real component, the imaginary records have the same amplitude envelope but 90-degree phase shift. Thin-bed complex reflection coefficients are derived using Equation (10). The calculation procedure of thin-bed reflection coefficients is shown in Figure 6 by taking normal responses of Model 3 as an example. Using the real and imaginary reflection coefficients, we invert for thin-bed properties by the proposed thin-bed inversion method and by the conventional AVA inversion method based on the Zoeppritz equations. As shown in Figure 7, the proposed thin-bed inversion method is superior in the characterization of thin-bed properties compared to the conventional AVA inversion method. The proposed thin-bed inversion method can derive more accurate elastic parameters of the target thin bed and its upper and lower half spaces. In particular, the inversion results can indicate the thickness of the target thin bed clearly and accurately. Figure 5: Complex seismic traces of Model 3: (**a**) real component; (**b**) imaginary component. In order to test the influence of noise on the inversion, we add 10% (mean energy ratio) and 20% white noise to the real AVA records, which are shown in Figure 8.  
Figure 6: Example of Model 3 at normal incidence for R(\(\theta\),f) = S(\(\theta\),f)/W(\(\theta\),f): (**a**) real and imaginary components of seismic records; (**b**) real and imaginary components of Ricker wavelet; (**c**) real and imaginary components of thin-bed P-wave reflection coefficients. Figure 7: Inversion results of Model 3 by the linear analytic approximation and Zoeppritz equations: (**a**) P-wave velocity; (**b**) S-wave velocity; (**c**) density. The inverted results of thin-bed properties are shown in Figure 9. For 10% white noise data, it can be seen that thin-bed thickness is basically coincident with that of the true model. The inverted P-wave and S-wave velocities deviate from the true model relatively more. Compared with the P-wave and S-wave velocities, the error in the density inversion is larger. The reason is that incidence angles of our AVA gather are from 0 to 20 degrees, which are not so large. Normally, stable density inversion requires larger incidence angle gathers. Compared to the 10% white noise case, the inverted P-wave and S-wave velocities for the 20% white noise case have higher inversion errors, while they can reveal the true size relationship of thin-bed elastic parameters. In order to test the thin-bed inversion strategy against synthetic data of multi-layer models, we set a six-layer 1D model named \"Model 5\", and synthetic P-wave AVA datasets, as shown in Figure 10. Table 2 lists the elastic parameters and layer thicknesses of Model 5. A Ricker wavelet with a 25 Hz predominant frequency is chosen as the synthetic seismic wavelet. Under the seismic resolution limit of a 25 Hz predominant frequency, Model 5 includes an impedance interface and two thin beds. The impedance interface is between layer 1 and layer 2, and is a low-to-high impedance interface.