Parameter co-variance with data not normally considered in a non-linear, physically-based model

Hi all,
I’m using a pretty standard non-linear model in soil physics that relates pressure head (|\psi|) to volumetric water content (\theta):

\theta (\psi) = \theta_r + \frac{\theta_s - \theta_r}{[1 + (\alpha |\psi|) ^ n] ^ {1 - 1/n}}

However, I’d like to extend this model to include some secondary information I have about organic matter (values are continuous, varying from 0 to 1, or 0 to 100%; O_m). That is, I’m curious if it is possible to include O_m in the model, to understand how it might be influencing the model parameters above (i.e., \theta_r, \theta_s, \alpha, n).

This seems conceptually similar to a hierarchical model, except that O_m is continuous, and order matters. On that basis, this kind of extension seems logical, but I’ve not seen anyone apply this kind of model to a non-linear model, using a continuous covariate, so it is unclear to me how to operationalize this kind of modeling framework.

If my question is not well conceived and you think I need some remedial improvements to my understanding, I’d be happy to take any book or article suggestions that you might think are useful for understanding similar issues.

Also, is their a name for the kind of framework I’m trying to operationalize? I was thinking it might be a joint distribution model, since O_m and the other parameters should co-vary, but perhaps my grasp of the nomenclature is not accurate.


It is still a hierarchical model. Loosely, you are going to need to make \theta_r, \theta_s, \alpha and n be the same size as the number of observations and their prior expectation should be some function of O_m. If those four parameters are required to be positive, then you might do something like exp(a + b * O_m) and you can even estimate a and b as parameters.

Huh. That’s super fascinating. Thanks for the insights.

Do you, by chance, have any references you’d recommend for better understanding these concepts? I think I can operationalize what you’ve suggested, but I think I’d need a bit more of a deep dive to help explain it to others, and better understand it myself.


I don’t, but maybe some else does. I don’t even know much of anything about soil or physics. I can just tell from the equation you wrote that one could write down a hierarchical but non-linear data-generating process for whatever the left-hand side of that equation in. Closest / easiest thing I know of is