# A New Stochastic Process of Prestack Inversion for Rock Property Estimation
## 3 Stochastic Process of Inversion  
Due to its limited bandwidth of seismic data and the absence of low frequency information, it is not possible to directly recover absolute impedance from a seismic trace. Since the inversion problem is a non-unique and ill-posed issue, it is difficult to derive the best fitness in a huge model space. Hence, a preconditioned model space is initialized at the start of inversion. Each particle should be defined in a finite space, not only to help solving the non-unique problem but also for fast converge. The low-frequency background model (LFBM) contains prior information in the reservoir. Information from the well-log data and seismic interpretation can be adopted in preconditioning by the Kriging method, where faults and sediment information can be added to make LFBM more reasonable. Kriging is an estimation conception commonly used for the interpolation of spatial data. The Kriging interpolation method for LFBM is  
\[L=\sum_{i=1}^{m}\lambda_{i}L_{i}+\xi \tag{16}\]  
Here, \(\xi\) is the perturbation term induced from interpretation or prior information. \(L_{i}\) is the low-frequency model from well-log and \(\lambda_{i}\) is the distance between traces. [PERSON] theorem is stated mathematically as the following equation:  
\[P(A|B)=\frac{P(B|A)P(A)}{P(B)} \tag{17}\]  
In probability theory and statistics, given two jointly distributed random variables, \(A\) and \(B\), the conditional probability distribution of \(B\) given \(A\) is the probability distribution of \(B\) when \(A\) is known to be a particular value. Here, in issues of elastic impedance inversion, \(A\) and \(B\) are elastic parameters like P-wave velocity, S-wave velocity, density and Poisson ratio. The conditional probability distribution can refine inverted parameters reasonably and not jump-out from model space. For a set of independent and identically distributed random variables (IID), normally distributed data points X of size n, where each individual point follows normal distribution with known variance. Assuming that the data collected from the well-log is independent, then the probability of variation is expressed as:  
\[L(\theta|x)=p_{\theta}(x)=P_{\theta}(X=x) \tag{18}\]  
\[\begin{array}{l}p(X|\mu,\sigma){=}\prod\limits_{i=1}^{n}\sqrt{\frac{\sigma}{ \pi\pi}}\exp(-\frac{1}{2}\sigma(x_{i}-\mu)^{2})\\ \quad\quad=(\frac{\sigma}{2\pi})^{\frac{\theta}{2}}\exp(-\frac{1}{2}\sigma \sum\limits_{i=1}^{n}{(x_{i}-\mu)^{2}})\\ \quad\quad=(\frac{\sigma}{2\pi})^{\frac{\theta}{2}}\exp[-\frac{1}{2}\sigma (\sum\limits_{i=1}^{n}{(x_{i}-\pi)}^{2}+n(\pi-\mu)^{2})]\end{array} \tag{19}\]  
With sufficient sample points from the well-log data, the likelihood function can be accurate at a specific expectation \(\mu\) and mean square \(\sigma\) given the parameter \(\theta\).  
The inversion procedure is summarized as follows:  
Step 1: Define the preconditioned model space based on rock physics and stratigraphy.  
Step 2: Wavelet estimation at a certain angle.  
Step 3: Generate a swarm and set the stopping criterion.  
Step 4: Create objective function and build low-frequency impedance model via Kriging method.  
Step 5: Apply APSO and choose the global best and several local optimums from a generation.  
Step 6: For each time of iteration, compare several local optimums and the global best, and add movement on global best.  
Step 7: Choose the global optimum satisfying the stopping criterion.  
Step 8: Apply smooth filter on the inversion result and then output.  
The flow chart is shown in Figure 2. Before inversion, many preparations need to be settled, including: wavelet estimation, well-log data rectification, time-to-depth conversion, calibration. In this paper, we mainly focus on objective function building.