||We introduce an adapted form of the Markov random field (MRF) for Bayesian spatial smoothing with small-area data due to Besag et al., 1991, Annals of the Institute of Statistical Mathematics, 43, 1-59. This new scheme allows the amount of smoothing to vary in different parts of a map by employing area-specific smoothing parameters, related to the variance of the MRF. We take an
empirical Bayes approach, using variance information from a standard MRF analysis to provide prior information for the smoothing parameters of the adapted MRF. The scheme is shown to produce proper posterior distributions
for a broad class of models by appeal to a proof in Ghosh et al., 1998, Journal of the American Statistical Association, 93, 273-282. We test our
method on both simulated and real data sets, and for the simulated data sets, the new scheme is found to improve modelling of both slowly-varying levels of smoothness
and discontinuities in the response surface.