While I am not an expert, I have worked with some people in hydrology. The traditional way to study hydrographs is to model responses as a linear time invariant system. Essentially, output signals like stream flow are related to input signals like precipitation. When precipitation increases, stream flow also increases, but there's a time delay. You can estimate this time delay deconvolving the signals. Essentially, we imagine precipitation as a signal \\(x : \mathbb{R} \to \mathbb{R}\\) and stream flow as a signal \\(y : \mathbb{R} \to \mathbb{R}\\) where stream flow is the convolution of \\(x\\) and another function (called the impulse response, I think):

\[y(t) = \int_{-\infty}^t f(\tau)x(t-\tau) d\tau \]

Essentially, we suppose \\(y\\) equals a linear combination of all past values of \\(x\\), if you estimate \\(f\\) you learn how the system responds to any input signal and you can estimate time delays and so on.

I think John was working on math for diagrams of linear time invariant systems somewhere, but I can't remember if it was in CASCADE or not.

\[y(t) = \int_{-\infty}^t f(\tau)x(t-\tau) d\tau \]

Essentially, we suppose \\(y\\) equals a linear combination of all past values of \\(x\\), if you estimate \\(f\\) you learn how the system responds to any input signal and you can estimate time delays and so on.

I think John was working on math for diagrams of linear time invariant systems somewhere, but I can't remember if it was in CASCADE or not.