StorAge selection (SAS) functions

../../_images/SAS_functions.png

Travel time distributions are calculated with fractional SAS functions (-; see [vanderVelde2012]):

\[\overleftarrow{p}_{Q}(T, t)=\frac{\partial}{\partial T} \Omega_Q(P_S(T, t), t)\]

with

\[P_S(T, t)=\frac{S_T(T, t)}{S(t)}\]

where \(T\) is the water age, \(t\) is the time step, is the backward travel time distribution of a specific hydrologic flux, \(Q(T,t)\) is the probability distribution of the hydrologic flux (where \(Q(T,t)\) is the cumulative probability distribution), \(S_T(T,t)\) is the cumulative age-ranked storage (mm), \(S(t)\) is the mobile storage volume (mm; i.e. storage volume below permanent wilting point is not considered) and \(P_S (T,t)\) is the cumulative probability distribution of the storage (where \(p_S (T,t)\) is the probability distribution).

Uniform

The uniform distribution function has no age preference.

\[\Omega_Q(T,t)=P_S(T,t)\]

Power

The power distribution function provides flexibility to represent a preference for younger water (\(k < 1\)) or preference for older water (\(k > 1\)).

\[\Omega_Q(T,t)=P_S(T,t)^k\]

Kumaraswamy

The Kumaraswamy distribution function (Kumaraswamy, 1980) provides flexibility to represent a preference for younger water (\(\alpha_Q = 1\) and \(\beta_Q > 1\)) or preference for older water (\(\alpha_Q > 1\) and \(\beta_Q = 1\)).

\[\Omega_Q(T,t)=1-((1-(P_S(T,t))^{\alpha_Q})^{\beta_Q})\]

Exponential

\[\Omega_Q(T,t)=1-e^{-k \cdot (P_S(T,t)}\]

Dirac

\[\begin{split}\Omega_Q(T,t)= \begin{cases}0, & T \leq T_{dirac} \\ 1, & T > T_{dirac} \end{cases}\end{split}\]

where \(T_{dirac}\) is the water age of the pulse. Please note, that a closed form of \(P_Q\) using the Dirac distribution is not available.

Gamma

\[\Omega_Q(T,t)=\frac{\gamma(\alpha, \beta \cdot P_S(T,t)}{\Gamma(\alpha)}\]

where \(\gamma\) is the regularized lower incomplete gamma function. Please note, that a closed form of \(P_Q\) using the Gamma distribution function is not available (see [Harman2015]).

Exponential

\[\Omega_Q(T,t)=1-e^{-k \cdot (P_S(T,t)}\]

Time-variant SAS function parameters

SAS function parameters can be time-variant. For example, time-variant may be described by a linear relationship of the storage volume:

\[b_Q(t)=c_1+c_2 \cdot (\frac{S(t)}{S_{sat}-S_{pwp}})\]

Short description of SAS parameterization

SAS parameters are defined in sas_params_q where _q corresponds to the flux e.g. transp.

  • 1: Uniform SAS function

  • 2: Dirac SAS function

  • 3: Kumaraswamy SAS function

  • 31: Kumaraswamy SAS function with time-variant preference for younger water

  • 32: Kumaraswamy SAS function with time-variant preference for older water

  • 35: Kumaraswamy SAS function with time-variant preference (e.g. preference for younger water while wetter conditions and preference for older water while drier conditions)

  • 36: Kumaraswamy SAS function with time-variant parameter a and constant parameter b

  • 37: Kumaraswamy SAS function with time-variant with time-variant parameter b and constant parameter a

  • 4: Gamma SAS function

  • 5: Exponential SAS function

  • 6: Power SAS function

The array of sas_params_q encompasses eight dimensions: - First array dimension of sas_params_q contains SAS function type (e.g. 1) - Second array dimension of sas_params_q contains first SAS parameter (only considered if SAS function type is Kumaraswamy, Gamma, Exponential or Power) - Third array dimension of sas_params_q contains second SAS parameter (only considered if SAS function type is Kumaraswamy or Gamma) - Fourth array dimension of sas_params_q contains lower boundary for temporal variation of SAS parameter (only considered if SAS function type is Kumaraswamy, Gamma, Exponential or Power) - Fifth array dimension of sas_params_q contains upper boundary for temporal variation of SAS parameter (only considered if SAS function type is Kumaraswamy, Gamma, Exponential or Power) - Sixth array dimension of sas_params_q contains lower boundary of storage used for temporal variation of SAS parameter (e.g. 200 mm; only considered if SAS function type is Kumaraswamy, Gamma, Exponential or Power) - Seventh array dimension of sas_params_q contains upper boundary of storage used for temporal variation of SAS parameter (e.g. 400 mm; only considered if SAS function type is Kumaraswamy, Gamma, Exponential or Power)