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We address the problem of the prediction of a spatial categorical variable by revisiting the maximum entropy approach. We first argue that, for predicting category probabilities, a maximum entropy approach is more natural than a least-squares approach, such as (co-)kriging of indicator functions. We then show that, knowing the categories observed at surrounding locations, the conditional probability of observing a category at a location obtained with a particular maximum entropy principle is a simple combination of sums and products of univariate and bivariate probabilities. This prediction equation can be used for categorical estimation or categorical simulation. We make connections to earlier work on prediction of categorical variables. On simulated data sets we show that our equation is a very good approximation to Bayesian maximum entropy (BME), while being orders of magnitude faster to compute. Our approach is then illustrated by using the celebrated Swiss Jura data set.

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[1] We propose a new cosimulation algorithm for simulating a primary attribute using one or several secondary attributes known exhaustively on the domain. This problem is frequently encountered in surface and groundwater hydrology when a variable of interest is measured only at a discrete number of locations and when the secondary variable is mapped by indirect techniques such as geophysics or remote sensing. In the proposed approach, the correlation between the two variables is modeled by a joint probability distribution function. A technique to construct such relation using underlying variables and physical laws is proposed when field data are insufficient. The simulation algorithm proceeds sequentially. At each location of the domain, two conditional probability distribution functions (cpdf) are inferred. The cpdf of the main attribute is inferred in a classical way from the neighboring data and a model of spatial variability. The second cpdf is inferred directly from the joint probability distribution function of the two attributes and the value of the secondary attribute at the location to be simulated. The two distribution functions are combined by probability aggregation to obtain the local cpdf from which a value for the primary attribute is randomly drawn. Various examples using synthetic and remote sensing data demonstrate that the method is more accurate than the classical collocated cosimulation technique when a complex relation relates the two attributes.Citation: Mariethoz, G., P. Renard, and R. Froidevaux (2009), Integrating collocated auxiliary parameters in geostatistical simulations using joint probability distributions and probability aggregation, Water Resour. Res., 45, W08421,

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