The four nearest neighbors in four cardinal directions can be regarded as conditionally neverless independent given the state of the surrounded central location in a sparse data space [17]. Consequently, the neighborhood choice for the Co-MCRF model needs only to use the four nearest neighbors in four cardinal directions, allowing (8) to be further simplified =bi0r0pi1i0(h10)��g=24pi0ig(h0g)��f0=1n[bf0r0pi1f0(h10)��g=24pf0ig(h0g)].(9)Here,??top[i0(u0)?�O?i1(u1),��,i4(u4);r0(u0)] we assume that the last visited location of the spatial Markov chain is always within the four nearest neighbors; if it is not so, we assume that the spatial Markov chain comes through one of them (Figure 2). Such a simplified Co-MCRF model provides the MCRF approach the capability of dealing with large data sets.

Figure 2Illustration of the Markov chain random field colocated cosimulation model with quadrant search and one auxiliary variable for random-path sequential simulation. Double arrows represent the moving directions of the spatial Markov chain. Dashed arrows …A tolerance angle is required because nearest neighbors in a neighborhood may not be located exactly along cardinal directions. To cover the whole space of a search area, sectors can be substituted for cardinal directions, and we can seek one nearest neighbor from each sector to represent the neighborhood (Figure 2). If we consider four cardinal directions, the sectors representing cardinal directions are quadrants. There may be no data to occur in some quadrants within a search range at the boundary strips or at the beginning of a simulation when sample data are very sparse.

Consequently, the size of a neighborhood may be less than four. Equation (9) can always be adapted to the situation. In case no data can be found in the whole search area, we assume the spatial Markov chain comes from a location outside the search range. By choosing a suitable search radius based on the density of sample data, this situation rarely occurs.The MCSS algorithm was developed based on the above quadrant search method and was effective in simulating multinomial classes in two horizontal dimensions [20]. The colocated Co-MCSS algorithm used in this paper is an extension Brefeldin_A of the random-path MCSS algorithm; therefore, their computation processes are similar. 2.4. Transiogram Modeling and Cross-Field Transition Probability MatrixTo perform simulations using Co-MCSS, transiogram models are needed to provide transition probability values at any needed lag distances. The transiogram was formally established in recent years to meet the needs of related Markov chain models [18].