# A New Stochastic Process of Prestack Inversion for Rock Property Estimation
## 2 Methods
### The Evolutionary Swarm Based on Markov Chains  
A Markov chain is a type of Markov process that has either a discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies, where we transition from one state to another state using a simple sequential procedure [29]. In a finite model space, an important property of Markov chains is that the starting state of the chain no longer affects the state of the chain after a sufficiently long sequence of transitions, which means Markov chains converge to a stationary distribution regardless of where they started. For the inversion problem, the iterative process can be regarded as a discrete-time Markov chain, a time-homogeneous sequence, as each time of iteration is dependent, and the probability of moving to the next state depends only on the present state and not to the previous state. The general Markov chains can be delineated as:  
\[\Pr(X_{M+1}=x|X_{1}=x_{1},X_{2}=x_{2},...,X_{M}=x_{M})=\Pr(X_{M+1}=x|X_{M}=x_{M}) \tag{10}\]  
where \(\Pr\) is an abbreviation for probability, \(X_{M}\) is the outcome function at time \(M\) and \(x_{M}\) is the state in the chain.  
The information on the state from \(M\) to \(M+1\) is described by the state-transition matrix, which is independent of n and not presented as sequences. Particles will be sorted in order from the best fitting to the least fitting. When initializing each Markov chain, the chain will wander in state space around the starting state. Several local optimums will then be chosen from the swarm in a generation. In the inverse problem, the state-transition matrix is a matrix whose product with the model at a certain iteration time, \(k\), gives \(x\) at a later time, \(k+1\). The outcome at any particular time k in the chain is a state \(x^{k}\). For a particular particle \(x_{i}\), the generation from \(k\) to \(k+1\) can be represented as:  
\[X_{i}^{k+1}=PX_{i}^{k} \tag{11}\]  
where \(P\) is the state-transition matrix.  
\[P=\left[\begin{array}{cccc}p_{11}&p_{12}&...&p_{1d}\\ p_{21}&p_{22}&...&p_{2d}\\...&...&...&...\\ p_{d1}&p_{d2}&...&p_{dd}\end{array}\right] \tag{12}\]  
For each element \(p_{ir}\) in matrix \(P\), an evolutionary factor, \(p_{ir}\) can be defined as  
\[p_{ir}=\left\{\begin{array}{cccc}1+4 cx(1-cx)(x_{\max}^{k}-x_{\min}^{k})/x_{i }^{k}&,&if&i< Optimum\\ \left[M+(x_{\max}^{k}-x_{\min}^{k})*Gauss(\mu,\sigma^{2})\right]/x_{i}^{k}&,&if &i> Optimum\end{array}\right. \tag{13}\]  
\[cx=\frac{(x_{i}^{k}-x_{\min}^{k})}{(x_{i}^{k}-x_{\max}^{k})} \tag{14}\]  
If \(x_{i}^{k}\) is chosen as one of local optimums, denoted as \(i<optimum\). Then, \(x_{i}^{k}\) will transform to new state if it has a better fitness of objective function. If \(x_{i}^{k}\) is out of local optimums, it means that particle \(X_{i}\) will start at a low-frequency model with Gaussian distribution information and then join to the next iteration.  
For the strategy of sorting local optimums, the number depends on the number of particles in a swarm and can be expressed as  
\[N_{l}=\gamma*N_{s} \tag{15}\]  
where \(N_{l}\) is the number of local optimums and \(N_{s}\) is particle quantity. \(\gamma\) is empirical parameter to adjust between the generation and local optimums. Usually, when the \(N_{s}\) is small, \(\gamma\) will be a little larger and vice versa.