# A New Stochastic Process of Prestack Inversion for Rock Property Estimation
## 4 Results
### Example Analysis  
In the inversion process, the sideline inversion is performed first to verify the validity of the inversion results. The perturbation distribution of p-wave velocity, s-wave velocity, and density was established by extracting the track set of the sidetrack and combining the low-frequency information in the logging curve. After the initial particle swarm was generated, the inversion of the three parameters was carried out. The prior model is established using the Gaussian distribution. Then the distribution is sampled randomly to generate some initial disturbance models and form an initial particle swarm.  
Through the inversion of the whole particle swarm, the evolution factors are estimated, and the convergence state is judged. We take some optimal solutions from all the particles, and the single-channel multi-particle inversion results are shown in Figure 4, One is the result of compressional wave inversion, the other is the result of compressional wave impedance inversion. Although the inversion results are inconsistent in the particles, the particles of these optimal solutions are close to each other in the search space, and the inversion results are consistent with the rock physics model.  
Figure 4: _Cont_.  
Figure 3: Well placement and Vp/Vs ratio map of working area.  
In addition, as shown in Figure 5, by extracting the objective function value of a particle in the particle swarm, it can be found that, with the increase in the number of iterations, the particle is constantly converging, that is, all the particles in the final particle swarm will move in the same direction in the search space, which further confirms the effectiveness of the algorithm.  
Rock physical analysis was performed on the well data to analyze the relationship between gas and non-gas layer parameters in order to preprocess the model space before inversion. Additionally, the difference in rock attribute parameters between the gas layer and non-gas layer could be obtained. The analysis results are shown in Figure 6:  
Figure 4: Single channel multiple particle inversion results.  
Figure 5: The value of the objective function changes with the number of iterations.  
Figure 6: (a) Gas and non-gas porosity histogram, (b) Gas and non-gas P-impedance histogram, (c) Cross-plot of Porosity and impedance of gas and non-gas layer.  
Furthermore, the aspect ratio \(\mathrm{Vp}/\mathrm{Vs}\) of the gas layer and non-gas layer was analyzed, as shown in Figure 7. It can be seen that the gas layer can be distinguished from the non-gas layer in the \(\mathrm{Vp}/\mathrm{Vs}\) attribute, and its numerical range can be obtained.  
After verifying the effectiveness of the bypass inversion, inversion is carried out for the whole section of the working area. Figure 8a-c respectively shows the low-frequency model, P-impedance and Poisson's ratio in the section obtained by inversion. It can be seen from the section that the inversion results maintain the low-frequency information while adding a lot of detail, and are consistent with the seismic waveform variation characteristics. Figure 8c shows the Poisson's ratio profile calculated by three-parameter inversion results. Poisson's ratio is generally used as a gas indicator and, combined with Poisson's Ratio characteristics of the gas reservoir in the well, we can further use Poisson's ratio to indicate the presence of a gas reservoir in the target layer.  
Figure 8: (**a**) Background model (**b**) P-impedance of inverted results (**c**) Poisson’s ratio of inverted results.  
Figure 7: \(\mathrm{Vp}/\mathrm{Vs}\) histogram among gas-layer (red), non-gas layer (green), and target layer (yellow).