Concepts

This page introduces the StorAge Selection (SAS) framework at a level appropriate for users of the package. For the full mathematical treatment, see Harman (2015) and Benettin et al. (2022) and Harman and Xu Fei (2024).

What is StorAge Selection?

StorAge Selection (SAS) is a framework for modeling transport through a control volume — a catchment, a soil column, a lake, or any system where fluid enters, is stored, and eventually leaves.

The key idea is simple: at each moment in time, the water leaving a system is a mixture of water that entered at different times in the past. The SAS function describes which ages of water are preferentially selected for discharge.

Age-ranked storage

The foundation of SAS is the concept of age-ranked storage. At any time \(t\), the water in storage can be sorted by how long it has been in the system. We define:

\[S_T(T, t) = \int_0^T s_T(\tau, t)\, d\tau\]

where \(s_T(\tau, t)\) is the density of water of age \(\tau\) in storage at time \(t\), and \(S_T(T, t)\) is the cumulative storage up to age \(T\). The total storage is \(S(t) = S_T(\infty, t)\).

The SAS function

The SAS function \(\Omega_Q(S_T, t)\) is a cumulative distribution function that maps age-ranked storage to cumulative probability:

\[\Omega_Q(S_T, t) = P(\text{discharged water has age-ranked storage} \leq S_T)\]

When \(\Omega_Q\) is a straight line from 0 to \(S(t)\), all ages of water are equally likely to be discharged — this is the well-mixed case. When \(\Omega_Q\) rises steeply at small \(S_T\), young water is preferentially discharged. When it rises steeply near \(S(t)\), old water is preferred.

The backward transit time distribution for outflow \(Q\) is:

\[p_Q(T, t) = \frac{\partial \Omega_Q}{\partial S_T} \cdot s_T(T, t)\]

Types of SAS functions

MESAS supports several types of SAS functions:

Piecewise-linear: Defined by breakpoints in \((S_T, P)\) space. The simplest is a uniform distribution: "ST": [0, S_max]. Adding more breakpoints creates more complex shapes.
Gamma distribution: A flexible family with shape, scale, and location parameters (a, scale, loc). The shape parameter \(a\) controls whether young water (\(a < 1\)) or intermediate-aged water (\(a > 1\)) is preferentially discharged. Specified with "func": "gamma".
Beta distribution: Defined over a finite interval, useful when total storage volume is known. Has two shape parameters \(a\) and \(b\). Specified with "func": "beta".
Kumaraswamy distribution: Similar to the beta but with a closed-form CDF, making it computationally efficient. Specified with "func": "kumaraswamy".

All distribution-based SAS functions accept loc and scale parameters that shift and stretch the distribution along the \(S_T\) axis.

Solute transport

Once the water fluxes are solved, solute transport follows directly. The concentration of water of a given age in storage determines the concentration of the outflow:

\[C_Q(t) = \int_0^\infty \frac{m_T(T,t)}{s_T(T,t)} \, p_Q(T,t) \, dT\]

where \(m_T(T,t)\) is the age-ranked solute mass in storage.

MESAS also supports:

First-order reactions via k1 and C_eq parameters
Evapoconcentration via the alpha parameter
Multiple solutes tracked simultaneously

Time-varying parameters

Any SAS function or solute parameter can be made time-varying by setting its value to a string that names a column in the input DataFrame. For example, setting "scale": "S_0" in a gamma specification causes the scale parameter to be read from the "S_0" column at each timestep.