Calibration strategies for cross section models

[MattBrst]

Referring to the Combined calibration notebook (ZIP file here) that @dbrakenhoff presented at the TIM workshop in Delft on 29th June 2026.

In this notebook, the importance of the following parameters was emphasized:

1. Aquifer diffusivity
2. Ratio of the leakage factors $\lambda = \sqrt{kHc}$ between the river and the polder

Just some thoughts from my side:

1. Would it make sense to use the diffusivity as calibration parameter (e.g.: cal.set_parameter_by_reference(name="D", ...)) instead of fitting on Saq and keeping kaq fixed?
2. Would it make sense to use the lambda values (or the ratio of lambdas) as calibration parameters?

The advantage might be that we have fewer parameters to fit on. As a postprocessing step we then should calculate reasonable kaq, Saq, c values from the derived diffusivity and lambda values.

[mbakker7]

In the model you describe, there are four parameters: T, S (same everywhere), c_0 below the river and c_1 below the land. See figure below with initial values. For a problem where the dynamics are governed entirely by a moving river stage and the measurements are varying heads, there are only 3 independent parameters. Hence, you can pick one of these 4 variables arbitrarily and the rest follows from calibration. Parameters can be combined in groups, for example: T / S, c_0S, and c_1S. Alternatively, you could pick T / S, lambda_0 and lambda_1, or yet differently T / S, lambda_1, c0 / c1. But whatever you do, you can only pick 3.

Now the big question is probably if we could modify the Calibrate class to take this into account. For this simple case, it is clear that there are only 3 parameters but you have to physically define 4 parameters. For more layers, this may be more difficult to define.

[mbakker7]

Short follow up. The practical solution right now is to fix 1 of the 4 parameters to a reasonable number. Then calibrate the model. And after that you can vary as much as you like, as long as a set of 3 numbers (as defined above) remains the same.