On effective linearity of soil process models

There are various circumstances in which it is important to know whether we can treat a model of a soil process as linear with respect to its parameters. In particular this is necessary when we decide how to apply that model so as to generate outputs at different spatial scales. Very few, if any, interesting soil models are strictly linear. However, the assumption of linearity might not be unreasonable if, for the region of interest, the variation of a particular input is fairly limited, or is limited to certain regions of the model's parameter space, or both. For this reason we propose the concept of effective linearity. We propose that the effective linearity of a model is quantified by the mean square deviation of the model output from a best linear approximation, given some distribution of inputs. This can be computed for a full set of inputs, or for one input with the other inputs assumed to have an independent non-linear effect. We computed these measures of non-linearity for a model of ammonia volatilization from the soil. We computed them for regions of different size, given a particular geostatistical model of the spatial covariation of the inputs of interest. This showed that, except for small regions, the model could not be regarded as effectively linear because of the model response to soil pH. As a result the mean square deviation of model predictions relative to a best linear approximation is large by comparison to the analytical variance of the model output.