Effective precipitation computation2

Effective precipitation computation

The simulated hydrologic processes account for the losses - evaporation, infiltration and surface storage - calculated according to the conventions and equations in the order presented below. The remaining precipitation is called effective precipitation, defined generally as:

\[ I_{eff}\left(t\right) = I\left(t\right) -I_{E}\left(t\right) -I_{I}\left(t\right) -I_{S}\left(t\right) \]

Equation: Effective precipitation accounting for evaporation, infiltration, and surface storage losses.

Where:

Table: Variables in the effective precipitation equation.

\(I(t)\)	Actual precipitation intensity at time t
\(I_{E}(t)\)	Evaporation loss at time t. It should be noted that the evaporation loss for the catchment is accounted only if the evaporation explicitly is activated. Otherwise, it is replaced by a constant decay rate
\(I_{s}(t)\)	Initial loss (wetting and surface storage) at time t
\(I_I(t)\)	Accumulated depth at time t

The individual terms in the loss equation are fundamentally different, as some terms are continuous where others are discontinuous. If the calculated loss is negative, it is set to zero. The losses have a dimension of velocity [LT-¹].

The actual precipitation, I(t), is assumed to be uniformly distributed over the individual catchments. Otherwise, it may vary as a random time function.

The surface storage, I_S(t), is the loss due to wetting and filling the depressions and holes in the terrain. The model begins with the surface storage calculation after the wetting process is completed. The surface storage is filled only if the current infiltration rate is smaller than the actual precipitation intensity reduced by evaporation. The actual surface storage loss is calculated according to the following:

\[ I_S(t) = \begin{cases} I(t) - I_E(t) - I_I(t) & \text{for } y(t) \leq y_S \\ 0 & \text{for } y(t) > y_S \end{cases} \]

Equation: Surface storage loss depending on current water storage level.

Where:

Table: Variables in the surface storage loss equation.

\(I(t)\)	Precipitation intensity at time t
\(I_{s}(t)\)	Surface storage loss at time t
\(I_I(t)\)	Infiltration loss at time t
\(I_{E}(t)\)	Evaporation loss at time t
ys	Surface storage depth
\(y(t)\)	Accumulated depth at time t

The infiltration, I_I(t), is the water loss to the lower storage caused by the porosity of the catchment surface. It is assumed that the infiltration starts when the wetting of the surface has been completed. The infiltration loss is calculated according to the following relations:

\[ I_{i}(t) = \begin{cases} I_{H}(t) & \text{for } I(t) - I_{E}(t) \geq I_{H}(t) \\ I(t) - I_{E}(t) & \text{for } I(t) - I_{E}(t) < I_{H}(t) \\ 0 & \text{for } I(t) \leq I_{E}(t) \end{cases} \]

Equation: Infiltration rate under Horton method accounting for maximum capacity and surface wetting.

Where:

Table: Variables in Horton's infiltration capacity equation.

\(I_I(t)\)	Infiltration loss at time t
\(I_H(t)\)	Horton’s infiltration at time t (see below)* *

By default the infiltration is taken into account in the calculation if the option is activated. However, the French Urban catchment will disregard any infiltration, even if it has been activated.

Valid for Global and French Rural models, only: The infiltration is a complex phenomenon, dependent on the soil porosity, moisture content, groundwater level, surface conditions, storage capacity, etc. The model calculates the infiltration loss capacity using the well-known Horton’s equation, per default in its original form:

\[ I_H(t) = I_{Imin} + (I_{Imax} - I_{Imin}) \cdot e^{-k_a \cdot t} \]

Equation: Horton infiltration capacity equation with exponential decay from maximum to minimum rate.

Where:

Table: Variables in the integrated Horton's equation.

\(I_H(t)\)	Infiltration loss calculated according to Horton
\(I_{Imax}\)	Maximum infiltration capacity (at start)
\(I_{I m i n}\)	Minimum infiltration capacity (at full saturation)
t	Time since the start of the storm
ka	Time factor (characteristic soil parameter) for wetting conditions

By default the model is able to calculate the infiltration following the standard and the integrated form of Horton’s equation. However the French rural model only offers the possibility to calculate the infiltration only according to the standard form.

Valid for Global model, only (not valid for the French model): The Horton's equation in its original form, Equation the horton infiltration capacity equation with exponential decay from maximum to, yields in realistic results only if applied to events for which the rainfall intensity always exceeds the infiltration capacity. However, typical values for infiltration capacity parameters I_Imax and I_Imin are often greater than typical rainfall intensities. Thus, when Equation the horton infiltration capacity equation with exponential decay from maximum to is used, with infiltration capacity I_H being the function of time only, I_H will decrease even if rainfall intensities are very small. In other words, this results in a reduction in infiltration capacity regardless of the actual amount of water entering the soil. This problem is solved by implementing the integrated form of the Equation the horton infiltration capacity equation with exponential decay from maximum to:

\[ I_{ICUM}(t_P) = \int_0^{t_P} I_H dt = I_{Imin} \cdot t_P + \frac{I_{Imax} - I_{Imin}}{k_a} \cdot (1 - e^{-k_a t_P}) \]

Equation: Integrated cumulative infiltration capacity recovery with exponential decay in Horton model.

Where:

Table: Variables in the inverse Horton's equation for infiltration recovery.

ICUM(tp)	A cumulative infiltration (m) at time tp, i.e. the area under the Horton's curve

This assumes that the actual infiltration has been equal to the infiltration capacity at any time within the period 0->t. Since this is only the case for rainfall intensities higher than the infiltration capacity, this needs to be corrected as:

\[ I_{C U M}(t)=\int_{0}^{t}I_{I}d t \]

Equation: Cumulative infiltration as the integral of infiltration rate over elapsed time.

i.e. the cumulative infiltration at the elapsed time t is calculated using the actually occurred infiltration.

At any elapsed time t, the two equations the integrated cumulative infiltration capacity recovery with exponential decay in and the cumulative infiltration as the integral of infiltration rate over elapsed time can be used to determine the “equivalent” time tp. This is done iteratively, since equation the integrated cumulative infiltration capacity recovery with exponential decay in cannot be solved explicitly for t_p.

The time t_p on cumulative Horton's curve will always be less or equal than the actually elapsed time, implying that the available infiltration capacity at time t_p will be greater than or equal to that given by the equation the horton infiltration capacity equation with exponential decay from maximum to. By these means, the infiltration capacity has been made a function of actual water infiltrated and not just a function of time.

Application of a “standard” Horton's equation the horton infiltration capacity equation with exponential decay from maximum to or its integrated form (equations the integrated cumulative infiltration capacity recovery with exponential decay in and the cumulative infiltration as the integral of infiltration rate over elapsed time) is controlled by the switch IntegratedHorton, found in INI file (see relevant documentation for reference). Per default, the program is set to “standard” Horton's equation.

In the dry period following the rainfall, the infiltration capacity is gradually recovered to the initial value, using an inverse form of the Horton's equation:

\[ I_{H}(t)~=~I_{I T}+(I_{I m a x}-Q_{I T})\cdot e^{-1/k_{h}t} \]

Equation: Horton infiltration rate recovery during dry periods following rainfall.

Where:

Table: Variables in the Horton infiltration-capacity recovery equation.

\(I_H(t)\)	Infiltration loss capacity calculated according to Horton
\(I_{Imax}\)	Maximum infiltration capacity (after a long dry period)
\(I_{IT}\)	Infiltration capacity at the threshold between the wetting and drying period
t	Time since the start of the recovery process
kh	Time factor (characteristic soil parameter) for drying conditions

The evaporation, I_E(t), is a continuous loss that is normally of less significance for single event simulations. However, on a long-term basis, evaporation accounts for a significant part of hydrological losses. If included in the computation, the evaporation is the first part subtracted from the actual precipitation, according to the following:

\[ I_E(t) = \begin{cases} I_E(t) & \text{for } (I(t) \geq I_{PE}(t)) \text{ OR } (y(t) > 0) \\ I(t) & \text{for } (I(t) < I_{PE}(t)) \text{ AND } (y(t) = 0) \end{cases} \]

Equation: Evaporation loss computed as the minimum of potential evaporation and rainfall.

Where:

Table: Variables in the evaporation loss equation.

\(I_a(t)\)	Actual precipitation at time t
\(I_{E}(t)\)	Evaporation loss at time t
\(I_{PE}(t)\)	Potential evaporation at time t
\(y(t)\)	Accumulated depth at time t

If the evaporation has not been explicitly included in the simulation (i.e. evaporation process is activated and a time series is included), a constant decay rate is applied instead (see Section 7.5.4).