I have a spatial salinity field s and a model g(s) ~ Xb where the X comes from slightly modified GAM basis functions.

I am trying to deal with the following set of requirements:
1. The underlying physics are linear, and plain salinity (the identity link) is the correct response to my covariates.
2. Dispersion (variance or sd) is almost certainly proportional to the mean.
3. The data s(x,y) respect min (freshwater) and max (ocean) values. If I knew how to bound constrain an ordinary GAM it would work for my case, because my "slight modification" satisfies a min-max principle.

The toolset, as far as I know, would include some combo of:
1. changing the link function, which I think would be very hard without disturbing some physical relationships, so I only mention it to rule it out.
2. rescaling the data and rows of the model matrix using a priori information.
3. enforcing the constraints (which I now think tools like mgcv can now, perhaps through penalties - but I don't know how).
4. deviating from a canonical choice by modeling the link as the identity but changing the dispersion function.

Is there a best choice, or an alternative? If (3) is indicated as part of a good solution, how would I arrange for it, say in the context of mgcv?

Thanks,
Eli

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