Is there a way to determine the quality of an impulse-response relationship? I'm new to system ID so be gentle.

I have a modeled signal of streamflow that I would like to map to an observed signal of stream height. This lends itself nicely to a simple impulse-response kind of relationship.

Problem is that the streamflow is modeled. So sometimes the magnitude of the impulse does not really match the magnitude of the response (e.g. big impulse, small response followed by small impulse, big response later in the time series). Even worse, sometimes I have false positives (modeled impulse, no response) and false negatives (no impulse, but there is a response). I have hundreds of sites I'm working with, and it's tedious to go through each one by hand and eyeball whether there's a good input-output relationship.

So my question is: Is there a way to compare the impulse signal to the response signal to determine the "quality" of the relationship? That way I can start working with the "good" sites, and also try to characterize what makes a site good vs bad based on other factors. Is cross-correlation the way to look at it? Or is there a better way to look at this?
I don't yet see the connection of your input with "impulses" in that if I understand correctly you are looking at the settled state from prior changes rather than dynamic responses. What you are asking for I believe is a statistical correlation between "Stream Flow" and "Stream Height"; given a stream flow (in the steady state), what is the "settled" stream height that would be sustained by that flow. I would argue that to be a correlation problem to the degree you are looking for a linear relationship between the two. We have other posts that detail how to compute the correlation coefficient, if you need further info on that you could look here:

Noise detection