Transit time distributions are not L-shaped

Abstract

A probability density function f(t) with origin at t = 0 is defined here as being ‘L-shaped’if f ʹ(t) ≤ 0 for t ≥ 0. L-shaped probability density functions, such as exponential distributions, are often employed as transit time distributions in hydrological modelling. However, the use of L-shaped transit time distributions implies including tracer particles that were already present at the observation point at time t = 0. These passive particles have no transit history by definition because ‘transit’ implies some history of movement of a particle through a hydrological system, however small that movement may be. That is, the particle must have reached the observation point by moving to it for some non-zero time and over some non-zero distance through a hydrological system such as a catchment or aquifer. By this argument, particles that happen to be already at the observation point at time t = 0 represent background noise. The distinction between the inclusion or not of particles already at the observation point might seem pedantic but does have an important implication: if transit time distributions in the hydrological environment are defined to not include non-transiting particles, they must have the property f ( 0) = 0. This negates the possibility of L-shaped transit time distributions. The exclusion of non-transiting particles also has the implication that transit time distributions must have at least one mode at t > 0, which may or may not be identifiable from field measurements.