Data for yield maps can be obtained from modern combine harvesters equipped with a differential global positioning system and a yield monitoring system. Due to delay and smoothing effects in the combine harvester the recorded yield data for a location represents a shifted weighted average of yield previously harvested along the swath. The unobserved yield is assumed to be a Gaussian random field and the yield monitoring system data is modelled as a convolution of the yield and an impulse response function. This results in an unusual spatial covariance structure (depending on the driving pattern of the combine harvester) for the yield monitoring system data. Parameters of the impulse response function and the spatial covariance function of the yield are estimated using maximum likelihood. The fitted model is assessed using certain empirical directional covariograms and the yield is finally predicted using the inferred statistical model.