I'm not sure how to determine the best input values for a search radius to create a TPI for the purpose of predictive soil mapping.
Would this depend on the spatial extent of the area I wish to map?
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There really is no expected for this unitless metric. The TPI is the delta of the focal pixel from the mean of the focal window. Obviously, the metric is sensitive to outliers and the larger the focal window the more likely an outlier effect. The best way to incorporate it into an analysis is to develop explicit hypothesis around the effect of a given scale and, perhaps, even test a range of scales. The extent of your analysis would not provide any type of empirical definition of this metrics scale parameter. You could have a small area that is locally hyper-variable or a large homogeneous area.
The R spatialEco package provides a function "hsp" for deriving the Hierarchical Slope Position (Murphy et al., 2010). The HSP is a scale decomposition, across a range of defined scales, providing an integrated scale version of the TPI metric.
Murphy M.A., J.S. Evans, and A.S. Storfer (2010) Quantify Bufo boreas connectivity in Yellowstone National Park with landscape genetics. Ecology 91:252-261
I would think that there are a number of factors to consider:
It is often worth using rough cuts the value(s) you are considering with sub-sets of the data to do trial runs or if there is sufficient computing power &/time available doing multiple runs with different parameter values.