# Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium
## 3 Numerical Analysis  
Synthetic seismic data were created by using interpretation results of well log data and vertical seismic profiling (VSP) data, which provided input for the AVOA inversion. It was assumed that the fracture medium had monoclinic symmetry, the azimuths of the two fracture sets were \(10^{\circ}\) and \(70^{\circ}\), respectively, the average fracture width was 1 mm, the average fracture radius was 10 m, and the average area of the fracture was 0.02 m\({}^{2}\). Figure 2 shows two sets of fracture densities from the well log interpretation results as model parameters for inversion. Figure 3 shows the well log and VSP interpretation results, among which the isotropic elastic parameters were used as the known parameters, to calculate the reflection coefficient of isotropic background and sensitivities. Figure 2: Interpretation results of well log data (fracture density). In order to test the robustness of the proposed inversion algorithm, synthetic seismic data were created by the model parameters shown in Figures 2 and 3, as well as a 30 Hz Ricker wavelet, based on the derived PP-wave reflection coefficient, Equation (13), and the convolution model shown in Figure 4a. The incident angle had 10 angles between \(0^{\circ}\) and \(45^{\circ}\), and the azimuths were \(10^{\circ}\), \(55^{\circ}\), \(100^{\circ}\), and \(145^{\circ}\). Gaussian random noises (the signal-to-noise ratios (SNRs) were 5 and 2) were added into the synthetic data, as shown in Figure 4b,c. Comparisons between the inversion results (black) of the WA parameters, the fracture density, and the true values (red) are displayed in Figure 5. We can see that, in the case of the SNR being 5, the proposed inversion method could make a stable estimation of WA parameters and fracture density; however, when the SNR was 2, the inversion results of fracture density exhibited more instabilities. Random noise had some influence on the WA parameters and fracture density, but the overall agreement was good. In general, it can be concluded that the proposed inversion method is stable and valid. Figure 4: _Cont._  
Figure 3: Interpretation results of well log and vertical seismic profiling (VSP) data (P- and S-wave velocity, density, and WA parameters). Figure 4: Synthetic seismic profiles. (**a**) noise-free, (**b**) S/N of 5, (**c**) S/N of 2. Figure 5: _Cont._  
In the approach proposed, the fracture width, radius, and strike are required to be assumed. The dimensions of fractures are a priori functions of many factors in the medium and cannot be readily assumed in practice. In order to test the impact of this assumption (the dimensions assumed for the fractures) on the predicted results, the results of inversion, using the same synthetic data as used in Figure 4a, but with the forward operators, that fracture radius is 0.25 m and fracture width is 5 mm (wrong assumptions), are shown in Figure 6a,b, respectively. Comparing the inversion results with Figure 5a, we observed that, in the case of the fracture radius being 0.25 m, the incorrect assumption of fracture radius introduced larger errors into the prediction of WA parameters and fracture density. In general, the inverted model parameters captured the amplitudes and variability of the true models. When the fracture width was 5 mm, the incorrect assumption of fracture width almost had no impact on WA parameters; however, the accuracy and stability of the fracture density inversion was seriously influenced. More accurate information on fracture radius and width may help to improve the accuracy of WA parameters and fracture density inversion. Figure 5: Comparisons between the estimated results (black curves) and the true values (red curves). The blue curves represent initial models that were constructed by smoothing the true values. (**a**) noise-free, (**b**) S/N of 5, (**c**) S/N of 2.