Impulse Response Functions as Probability Distributions

Abstract

A multitude of environmental and hydrologic processes embody both the elements of chance and the descriptive laws of physics. A finer process description at one scale is lost through the processes of integration in time and space and through averaging. This justifies simplification in representation of the processes. It is hypothesized that if an environmental process is described by a linear or linearized governing equation, then the solution of this equation for a unit impulse (or Dirac delta) function can be interpreted as a probability density function for describing the probabilistic properties of the process. This hypothesis is tantamount to mapping from the unit impulse response (UIR) function, h(t), to the probability density function (PDF), f(x), where h is the UIR as a function of time or space variable denoted by t and f is the PDF as a function of the random variable of the process. For example, the impulse response of a diffusion equation for pollutant transport described by the space-time variation of concentration can be used as a probability distribution for pollutant concentration in a medium, such as a river, a lake, conduit storm water, soil, or a saturated geologic formation. Likewise, the impulse response of a linearized diffusion model of channel flow can be interpreted as a probability distribution for frequency analysis of extreme values (such as floods, droughts, hurricanes, earthquakes, and so on). Similarly, the impulse response of a linear reservoir can be used as an exponential probability distribution model. The impulse response of a cascade of equal linear reservoirs is the gamma distribution, which has a number of applications in environmental and water resources data analysis. In this vein, a number of impulse responses of physically based equations that apply to environmental and hydrologic processes and data are discussed and illustrated by using field or laboratory data.