Fig. 1. Possible neighborhoods within a search radius in a Markov chain random field if only considering nearest known neighbors in four cardinal directions: (a) four known neighbors; (b) three known neighbors; (c) two known neighbors; (d) one known neighbor; and (e) no known neighbors. The white cell represents the unknown one to be estimated and black cells represent known data locations in cardinal directions.
Fig. 2. The search circle, search sectors, and the considered neighborhoods of nearest known neighbors. The search circle is divided into four equal search sectors. Left: four nearest known neighbors are found, one from each sector. Right: two nearest known neighbors are found because the other two sectors have no known data locations.
Fig. 3. The reference soil map of seven soil classes and randomly distributed sample data sets (dense: 646 points, 2.9% of total pixels; medium: 179 points, 0.8% of total pixels; sparse: 50 points, 0.2% of total pixels). The seven soil classes (soil series) include: 1, loamy alluvium; 2, Sparta; 3, Plainfield; 4, Dakota; 5, Richwood; 6, others (e.g., water); and 7, peat and muck.
Fig. 4. Experimental transiograms headed by Soil Class 1, estimated from each of the three sample data sets: (a) from the dense data set; (b) from the medium data set; and (c) from the sparse data set. Note that estimated points for each experimental transiogram are connected as a line so that different experimental transiograms can be readily differentiated.
Fig. 6. Simulated results by Markov chain sequential simulation, conditioned on the dense data set: (a) optimal prediction map; (b) and (c) two simulated realizations; (d) maximum occurrence probability map; (e) occurrence probability map of Class 1; (f) occurrence probability map of Class 4.
Fig. 7. Simulated results by Markov chain sequential simulation, conditioned on the medium data set: (a) optimal prediction map; (b) and (c) two simulated realizations; (d) maximum occurrence probability map; (e) occurrence probability map of Class 1; (f) occurrence probability map of Class 4.
Fig. 8. Simulated results by Markov chain sequential simulation, conditioned on the sparse data set: (a) optimal prediction map; (b) and (c) two simulated realizations; (d) maximum occurrence probability map; (e) occurrence probability map of Class 1; (f) occurrence probability map of Class 4.
Fig. 9. Proportions of classes in sample data sets and simulated realizations (averaged from 100 realizations) generated by Markov chain sequential simulation.
Fig. 13. Simulated results by sequential indicator simulation with ordinary kriging, conditioned on the dense dataset: (a) optimal prediction map; (b) and (c) two simulated realizations; (d) maximum occurrence probability map; (e) occurrence probability map of Class 1; (f) occurrence probability map of Class 4.
Fig. 14. Simulated results by sequential indicator simulation with ordinary kriging, conditioned on the medium dataset: (a) optimal prediction map; (b) and (c) two simulated realizations; (d) maximum occurrence probability map; (e) occurrence probability map of Class 1; (f) occurrence probability map of Class 4.
Fig. 15. Simulated results by sequential indicator simulation with ordinary kriging, conditioned on the sparse dataset: (a) optimal prediction map; (b) and (c) two simulated realizations; (d) maximum occurrence probability map; (e) occurrence probability map of Class 1; (f) occurrence probability map of Class 4.
Fig. 16. Average maximum probabilities estimated from maximum occurrence probability maps generated by Markov chain sequential simulation (MCSS) and sequential indicator simulation with ordinary kriging (SISoik).
