I am new to signal processing and have come into the subject from the study of rivers and basic geophysics. I am trying to test an idea previously put forward that the sediment transport response of rivers can be modeled as a linear impulse-response system. Transport along a river reach changes as the upstream supply of water or sediment changes. Working from an impulse response convolution, I derived an algebraic equation for the sediment transport (flux) of a river reach as measured at the downstream end of the reach. Here is the derivation of the algebraic equation (Equation 27 below), and I apologize in advance because I prepared the write-up snippet as an image from the master LaTex file:
When I use Equation 27 to model the measured sediment transport at the downstream end of a river reach I observe that when the system is perturbed by an increase in the supply of water and sediment at the upstream end (elapsed time ~2100-2400 and 4050-4310 minutes in the Figure that follows), Equation 27 predicts the mirror image in behavior of what I measure. This is depicted in the following figure:
In the Figure the measured transport is the heavy dark line, the modeled response with Equation 27 is the light blue dotted line, and the inverse of Equation 27 for periods of supply perturbation is the dark gray dashed line. Note that to calculate the responses shown in the Figure I reset the elapsed time when the upstream supply changes - but here I am plotting the results against the running elapsed time. It can't be a coincidence that the piece-wise modeled-inverse curve (dark gray dashed line) does a pretty good job of matching the measured transport. However, I have no idea why nor do I know if I can arrive there mathematically from the starting point of Equation 20. Any assistance or guidance on this would be greatly appreciated.