Estimation of linear models of coregionalization by residual maximum likelihood

Observations of ancillary soil properties spatially correlated to a soil property of interest may be used to increase the precision and reduce the sampling costs of a geostatistical survey. The relationship between such coregionalized properties must be expressed as a linear model of coregionalization but the conventional estimator of the linear model of coregionalization is biased unless the mean value of each property is constant across the study region. However, the mean value of a soil property may vary according to a spatial trend or a deterministic relationship with other factors which vary within the study region. We therefore propose that a linear mixed model should be fitted to coregionalized soil properties by residual maximum likelihood. This approach simultaneously fits spatial trends or deterministic relationships and random effects to the observations with minimum bias. We implement a residual maximum likelihood estimator for coregionalized properties and suggest a criterion to decide what order of spatial trend and which deterministic relationships should be included in the model. The effectiveness of the estimator is proved upon simulated data and upon observations of zinc and cadmium concentrations from the Swiss Jura.