# Research Status of and Trends in 3D Geological Property Modeling Methods: A Review
## 3 Interpolation Methods for 3D Property Modeling
### Markov Model  
The Markov random field (MRF) model has been used as a spatial continuity model for subsurface sedimentary facies [86]. By including higher-order interaction conditions, models with realistic physical properties can be obtained theoretically. These methods explicitly produce a model that relies on hundreds of parameters for the multivariate distribution of random variables. Initially, [PERSON] [87] proposed a simple Metropolis-Hastings algorithm [88]. In each iteration of the model with K classifications, first, a possible new data template is obtained by replacing the current facies at node u with a randomly selected one of the K-1 other facies. Second, the new facies is accepted according to the probability defined by the Metropolis-Hastings algorithm. This Markov chain should converge to the specified MRF after a sufficient number of iterations. In addition, [PERSON] [89] selected a one-way path of Markov chain Monte Carlo (McMC) in an MRF-related model, called the Markov mesh model (MMM). MMM is an approximate simulation that is improved by multiple iterations of the McMC algorithm. However, the algorithm can produce good results only for models that appear to be asymmetric. [PERSON] [90] also proposed a method based on the Metropolis-Hastings (MH) algorithm to simulate a Markov chain that can converge to posterior distributions. A simple sampling mechanism is used to propose a new value for the random variable and then, with s being a certain probability, accept/reject the new variable. However, the author changed the directional MH algorithm, in which the block of spatial variables was updated at each iteration. In other words, instead of calculating a new random variable at a specific location, the author first generated a directional auxiliary random variable and then gave the value on the line defined by the current node and the auxiliary direction. The proposed method still assumes a Gaussian random field prior model but uses a nonlinear likelihood function to explore the original nonlinear posterior distribution. The modeling results of all these methods suffer from the slow convergence of Markov chain simulations. Moreover, these methods cannot control the reproduction of large-scale geological structures in their implementations. It is necessary to find a suitable MRF model for each specific sedimentary environment. Estimating the parameters of the MRF model is difficult and CPU-demanding. Moreover, the specification of the model, i.e., how to select the higher-order statistics (or cliques in MRF terminology) used for the MRF model, is still an open question. Consequently, modeling new geological environments is a difficult task, especially in 3D environments, and there has been no report on the successful application of this technique in this regard. Due to the slow convergence of these algorithms, an optimal number of iterations is required before performing the simulation. For example, stopping prematurely can lead to images that are not representative of a particular model. In addition, the model parameters in these methods defy a clear geometric interpretation,thereby making the parameter estimation rely on the computationally costly maximum likelihood estimation technique, and it is likely difficult to derive this analytical expression. Gibbs sampling was originally proposed as a statistical method by [PERSON] [91]. This method is also a special case of the Metropolis algorithm for sampling from complex joint/marginal distributions. [PERSON] [92] applied the Gibbs sampling McMC method to the multiple-point simulation of geological phenomena. This process is similar to that of the opportunistic Markov chain model, where an initial random image is iteratively refined by resampling the node values from the multiple-point statistics of the previous training image. The main challenge is to find the conditional probability of the facies at each point. In the method proposed by [PERSON], a technique based on the linear combination of observed data is used.