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Calibration

How are observations from different variables weighted?

parsac’s calibration approach maximizes the (log) likelihood. This combines all model-observations differences (after transformation, if that’s activated), each weighted by the reciprocal of its “standard deviation” (formally, squares of the differences divided by their variance). This standard deviation is currently a constant for each observed variable, it cannot yet depend on time and/or depth. It can be prescribed in your xml configuration file by adding an attribute sd="<VALUE>" to the observed variable. If it is not prescribed, parsac will instead estimate it from the model-observation differences. In that case, a variable that the model cannot capture well will automatically get a higher sd. All observations are assumed to be independent and (thus) uncorrelated.

This have several consequences:

• An increase in the number of observations will result in a greater contribution of that variable to the likelihood. Thus, parsac will place more importance on getting that variable right, potentially at the expense of other variables. This is not always desirable, in particular when using high-frequency observations. The proper solution would likely be to allow to drop the assumption of independent observations and simultaneously support prescribed or estimated autocorrelation. Observations that are closer in time could then get a lower weighting because they observe the same thing. Since this is not supported in parsac at the moment, you may want to subsample the high-frequency variables instead, e.g. with a running average.
• parsac handles observed variables with different order of magnitude well if it is allowed to estimate their sd. In that case, a change in units of a variable (i.e., applying a scale factor) would not make any difference to the outcome of calibration.
• Strong temporal fluctuations in any variable are often not that well captured by models. In particular, slight time shifts (e.g. lags) between model and data can then have disastrous effects on the “quality of fit” perceived by parsac. The consequence is that their estimated sd will typically be higher than that of a near-constant variable. But as the model-observation differences will be higher too, the variable can still contribute significantly to the likelihood.