Do a Bit More with Convolution

Abstract

Convolution is a form of superposition that efficiently deals with input varying arbitrarily in time or space. It works whenever superposition is applicable, that is, for linear systems. Even though convolution is well-known since the 19th century, this valuable method is still missing in most textbooks on ground water hydrology. This limits widespread application in this field. Perhaps most papers are too complex mathematically as they tend to focus on the derivation of analytical expressions rather than solving practical problems. However, convolution is straightforward with standard mathematical software or even a spreadsheet, as is demonstrated in the paper. The necessary system responses are not limited to analytic solutions; they may also be obtained by running an already existing ground water model for a single stress period until equilibrium is reached. With these responses, high-resolution time series of head or discharge may then be computed by convolution for arbitrary points and arbitrarily varying input, without further use of the model. There are probably thousands of applications in the field of ground water hydrology that may benefit from convolution. Therefore, its inclusion in ground water textbooks and courses is strongly needed.

Residence times of groundwater and nitrate transport in coastal aquifer systems: Daweijia area, northeastern China

“…Lumped parameter models (LPMs) have been effectively used for estimating groundwater MRT in many natural systems, particularly in karst aquifers (Mangin, 1994;Zuber et al, 2004;Olsthoorn, 2008;Long and Putnam, 2009). Although these models are simple and represent ideal systems, they require less data than more complex methods.…”

Estimation of heterogeneous aquifer parameters using centralized and decentralized fusion of hydraulic tomography data

“…Second, the resulting spatiotemporal cross-covariance matrix is typically large and difficult to manipulate. In this study, we propose to assimilate temporal moments of the impulse response function of transient drawdown data (Harvey and Gorelick, 1995;Von Asmuth and Maas, 2001;Li et al, 2005;Bakker et al, 2008;Olsthoorn, 2008;Von Asmuth et al, 2008), rather than the hydraulic head data themselves.…”

A time-series analysis framework for the flood-wave method to estimate groundwater model parameters

“…Note that the dependence of the response function on spatial coordinates is omitted in this notation. The response function can also be interpreted as the weighting function in a moving average process (Olsthoorn 2008). As a comparison, in runoff hydrology, the familiar unit hydrograph is the response function relating precipitation (the input signal) to stream discharge (the output signal).…”

“…Groundwater heads are calculated with MicroFEM as suggested by Olsthoorn (2008) by first evaluating the step responses at the place of the piezometers, after which head fluctuations are obtained by convolution of the block response functions with their corresponding stress time series. Finally, the simulated groundwater heads are obtained by adding the drainage base (which is the mean river stage here) as given in Eq.…”