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).
The uniform distribution function has no age preference.
\Omega_Q(T,t)=P_S(T,t)
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
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})
\Omega_Q(T,t)=1-e^{-k \cdot (P_S(T,t)}
\Omega_Q(T,t)= \begin{cases}0, & T \leq T_{dirac} \\
1, & T > T_{dirac} \end{cases}
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.
\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]_).
\Omega_Q(T,t)=1-e^{-k \cdot (P_S(T,t)}
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}})
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)