Response to 'Comments on "Combining spatial transition probabilities for stochastic simulation of categorical fields" with communications on some issues related to Markov chain geostatistics'

Cao et al. (2011a) recently presented a method to implement the Tau model suggested by Journel (2002) for conditional simulation of categorical fields. In this article, a spatial Markov chain (SMC) model proposed by Li (2007a), as stated by them, was compared. Li (2007a) presented the Markov chain random field (MCRF) theory (or ideas) with a case study using a specific MCRF model and further suggested the Markov chain geostatistics (MCG) framework based on the primitive MCRF theory. A MCRF model may be called a SMC model and was also called in this way by Li (2007a), because a MCRF means a random field generated or defined by a special SMC. One reason that we always used MCRF instead of SMC (or more simply Markov chain) to name our models or algorithms in our later publications is because SMC is a very general name and may cover all of those existing Markov chain models used for spatial data. Unfortunately, Cao et al. (2011a) provided an incorrect SMC model equation with misinterpretations on the MCRF approach. Following Allard et al. (2011), Cao et al. (2011a, 2011b) also stated that the MCRF approach was a simplified form of the Bayesian maximum entropy (BME) approach proposed by Bogaert (2002); Li and Zhang (2012) commented on Allard et al. (2011), which was published in a soil science journal, to clarify some misunderstandings. Although this comment aims to clarify some misinterpretations by Cao et al. (2011a, 2011b) on the MCRF approach in geographical information science, our main purpose is to communicate with Cao and other colleagues who may have interests or misunderstandings in related issues, including those new issues raised by Allard et al. in their response letter to our comments on their paper. We think such a comment and communication is necessary to boost mutual understanding because it concerns the fate of MCG and the whole field of categorical spatial variable modeling. Therefore, we will only explain and discuss on several major issues rather than pick out specific sentences.