I am having difficulty in visualising both unit step response function and impulse response function of groundwater heads resulting from flood wave in nearby stream. The functions have been calculated by Venetis 1968 using Laplace transform.
http://hydrologie.org/hsj/133/133013.pdf
A unit step response will usually have a wide width defined over an interval, whilst an impulse response has infinitesimal or no width.
The response of groundwater head (output) to a change in head at a river boundary (input) can be described, mathematically, as a linear system. Linear systems can be characterised in terms of the output response to an impulse or a step input. The impulse input occurs over an infinitesimally short interval of time, with a total value of one head unit multiplied by one time unit. It is a useful mathematical construction for integration into more complex input functions. The step input represents an instantaneous change of one head unit to a steady input value . The step input response is often of more practical value because complex input series can be described in terms of a series of step changes of various magnitudes. The total response to a complex input is then obtained by combining the individual step responses.
The unit step response is the impulse response convolved with a unit step, and corresponds to the running integral of the impulse response. The impulse response represents the effect of a flood wave on the groundwater head for a system where the increase in groundwater head leads to lateral dispersal through the aquifer (dispersal time is not small compared to the duration of the flood peak), and/or increased loss from the groundwater system (e.g. baseflow input to stream). The unit step response captures the situation where the dispersal time is small compared to the duration of the flood (i.e. groundwater level approaches equilibrium across the aquifer within the flood event - e.g. for a localised fluvial aquifer), and there is no significant loss from the aquifer.
The unit step repsents the integration of impulse and its value is constat for a certain wide of time
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