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Publication# Multimodal water age distributions and the challenge of complex hydrological landscapes

Abstract

Travel time distributions (TTDs) are concise descriptions of transport processes in catchments based on water ages, and they are particularly efficient as lumped hydrological models to simulate tracers in outflows. Past studies have approximated catchment TTDs with unimodal probability distribution functions (pdf) and have successfully simulated tracers in outflows with those. However, intricate flow paths and contrasting water velocities observed in complex hydrological systems may generate multimodal age distributions. This study explores the occurrence of multimodal age distributions in hydrological systems and investigates the consequences of multimodality for tracer transport. Lumped models based on TTDs of varying complexity (unimodal and multimodal) are used to simulate tracers in the discharge of hydrological systems under well-known conditions. Specifically, we simulate tracer data from a controlled lysimeter irrigation experiment showing a multimodal response and we provide results from a virtual catchment-scale experiment testing the ability of a unimodal age distribution to simulate a known and more complex multimodal age system. Models are based on composite StorAge Selection functions, defined as weighted sums of pdfs, which allow a straightforward implementation of uni- or multi-modal age distributions while accounting for unsteady conditions. These two experiments show that simple unimodal models provide satisfactory simulations of a given tracer, but they fail in reproducing processes occurring at different temporal scales. Multimodal distributions, instead, can better capture the detailed dynamics embedded in the observations. We conclude that experimental knowledge of flow paths and the systematic use of data from multiple, independent tracers can be used to validate the assumption of water age unimodality. Multimodal age distributions are more likely to emerge in landscapes where the distributions of flow path lengths and/or water velocities are themselves multimodal. In general, age multimodality may not be particularly pronounced and detectable, unless comparable amounts of water with contrasting ages reach a given outflow.

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Multimodal distribution

In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal. When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the antimode.

Unimodality

In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal".

Normal distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

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