## 2 Method
### Markov Chain Co-Simulation
#### 2.2.2 Transiogram  
Traditionally, variograms are commonly used to describe the correlation between variables. However, variograms cannot describe directional asymmetry when land cover categories appear sequentially, thus they cannot effectively express the parallel relationship between categories. Markov chain conditional simulation provides a general framework for a non-linear non-kriging geostatistical approach. It needs a powerful index of spatial heterogeneity to realize multi-type simulation. [PERSON] [31] proposed the concept of transiogram, which is a one-dimensional transition probability function (conditional probability between two points) model at distance \(h\):  
\[p_{\bar{\mathrm{ij}}}(h)=Pr[z(x+h)=j|z(x)=i] \tag{1}\]  
where \(x\) is a specific location, and \(P_{\bar{\mathrm{ij}}}(h)\) is the transition probability of a random variable \(z\) from class \(i\) to class \(j\). When \(h\) increases gradually, \(P_{\bar{\mathrm{ij}}}(h)\) forms a graph, i.e., a transiogram, \(P_{\bar{\mathrm{ii}}}(h)\) denotes an auto-transiogram, which describes the dependency of a type itself; a cross-transiogram describes the dependency between types (including cross correlation, parallel relationship, and directional asymmetry). Transiograms can estimate multistep transition probabilities from sparse point samples for two-dimensional Markov chain simulations, which involves the rich spatial heterogeneity characteristics of land cover types. The transition probabilities of different spatial steps (or lag) can form a one-dimensional continuous transition probability graph. The role of transiograms on Markov chain geostatistics is similar to that of variograms for Kriging geostatistics. Transiograms can only rely on a large number of reliable and soundly distributed sample points, which need an amount of visual interpretation. When the number of sample points is insufficient, transiograms may show false fluctuations and cannot convey reliable information [31]. In recent years, some organizations participating in land cover mapping around the world have released their validation data to the public, thereby providing a reference for subsequent research. Geoweb-based tagging system also enables users to tag geographical information for land cover data acquisition [32]. We collected these reference sample data, and they were mainly from the following websites: (1) GOFC-GOLD Land Cover Project Office in coordination with reference data producers ([[http://www.gofcgold.wurnl/sites/gofcgold.refataportal.php](http://www.gofcgold.wurnl/sites/gofcgold.refataportal.php)]([http://www.gofcgold.wurnl/sites/gofcgold.refataportal.php](http://www.gofcgold.wurnl/sites/gofcgold.refataportal.php))). The GOFC-GOLD includes the consolidated GLC 2000 reference (GLC200 ref) [33], the consolidated GlobCover 2005 reference (GlobCover2005 ref) [34], the System for Terrestrial Ecosystem Parameterization (STEP) reference [35], the Visible Infrared Imaging Radiometer Suite Surface-Type reference [36] and the GLCNMO 2008 datasets [37]; (2) Geo-Wiki crowdsourced data ([[https://www.geo-wiki.org/](https://www.geo-wiki.org/)]([https://www.geo-wiki.org/](https://www.geo-wiki.org/))); (3) DCP volunteers ([[http://www.confluence.org](http://www.confluence.org)]([http://www.confluence.org](http://www.confluence.org))); (4) other research institutions, such as the global validation sample set developed by Tsinghua University [38]; [[http://data.ess.tsinghua.edu.cn/data/temp/GlobalLandCoverValidationSampleSet_v1.xlsx](http://data.ess.tsinghua.edu.cn/data/temp/GlobalLandCoverValidationSampleSet_v1.xlsx)]([http://data.ess.tsinghua.edu.cn/data/temp/GlobalLandCoverValidationSampleSet_v1.xlsx](http://data.ess.tsinghua.edu.cn/data/temp/GlobalLandCoverValidationSampleSet_v1.xlsx)). ); (5) Flickr photo-sharing website (www.flickr.com); and (6) LACO-Wiki open access online portal for land cover validation ([[http://www.laco-wiki.net](http://www.laco-wiki.net)]([http://www.laco-wiki.net](http://www.laco-wiki.net))).