Effective precipitation computation1

Effective precipitation computation

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

\[ I_{eff}(t) = I(t) - I_E(t) - I_W(t) - I_I(t) - I_S(t), \quad I_{eff} \geq 0 \]

Equation: Effective precipitation, defined as the actual precipitation reduced by the evaporation, wetting, infiltration and surface storage losses.

Where:

Table: Variables in the effective precipitation equation.

\(I(t)\)	Actual precipitation 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 an evaporation time series in provided
\(I_W(t)\)	Wetting loss at time t
\(I_I(t)\)	Infiltration loss at time t
\(I_S(t)\)	Surface Storage loss 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 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:

\[ I_E(t) = \begin{cases} I_E(t) & \text{for } \left(I(t) \geq I_{PE}(t)\right) \text{ OR } \left(y(t) > 0\right) \\[1ex] I(t) & \text{for } \left(I(t) < I_{PE}(t)\right) \text{ AND } \left(y(t) = 0\right) \end{cases} \]

Equation: Evaporation loss, limited by the potential evaporation and the availability of ponded water.

Where:

Table: Variables in the evaporation loss equation.

\(I(t)\)	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 actual evaporation is not explicitly specified in the simulation (i.e. Evaporation process activated – see INI file parameters – and evaporation time series specified), only a decay rate will be applied during dry periods.

The wetting, I_W(t), is a discontinuous loss. When the precipitation starts, a part of the precipitation is used for wetting of the surface if the surface is initially dry. The model assumes that the precipitation remaining after subtraction of the evaporation loss is used for wetting of the catchment surface. When the surface is wet, the wetting loss, I_W, is set to zero. This is summarised in the following expression:

\[ I_W(t) = \begin{cases} I(t) - I_E(t) & \text{for } y(t) < y_w \\[1ex] 0 & \text{for } \left(I(t) \leq I_E(t)\right) \text{ OR } \left(y(t) \geq y_w\right) \end{cases} \]

Equation: Wetting loss, applied until the wetting depth is reached.

Where:

Table: Variables in the wetting loss equation.

\(I(t)\)	Actual precipitation at time t
\(I_E(t)\)	Evaporation loss at time t
\(I_W(t)\)	Wetting loss at time t
\(y_w\)	Wetting 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 relation:

\[ I_{I}(t) = \begin{cases} I_{H}(t) & \text{for } \left(I(t) - I_{E}(t) - I_{W}(t)\right) \geq I_{H}(t) \text{ \& } \left(y(t) \geq yw\right) \\[1.5ex] I(t) - I_{E}(t) - I_{W}(t) & \text{for } \left(I(t) - I_{E}(t) - I_{W}(t)\right) < I_{H}(t) \\[1.5ex] 0 & \text{for } y(t) < yw \end{cases} \]

Equation: Infiltration loss, capped by the maximum infiltration capacity and zero once wetting depth is exceeded.

Where:

Table: Variables in the infiltration loss equation.

\(I_I(t)\)	Infiltration loss at time t
\(I_H(t)\)	Horton’s infiltration at time t
\(y_w\)	Wetting depth
\(y(t)\)	Accumulated depth at time t

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:

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

Equation: Horton's equation for the infiltration capacity, original form.

Where:

Table: Variables in Horton's infiltration capacity equation.

\(I_H(t)\)	Infiltration loss calculated according to Horton
\(I_{Imax}\)	Maximum infiltration capacity (after a long dry period)
\(I_{Imin}\)	Minimum infiltration capacity (at full saturation)
t	Time since the start of the storm
ka	Time factor (characteristic soil parameter) for wetting conditions

Urban (MIKE+)River (MIKE 11 / MIKE HYDRO)

The Horton's equation in its original form the horton's equation for the infiltration capacity, original form 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's equation for the infiltration capacity, original form 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's equation for the infiltration capacity, original form:

\[ I_{ICUM}(t_P) = \int_{0}^{t_P} I_H \, dt = I_{Imin} \cdot t_P + \frac{I_{Imax} - I_{Imin}}{k_a} \cdot \left(1 - e^{-k_a t_P}\right) \]

Equation: Integrated form of Horton's equation, giving the cumulative infiltration up to the equivalent time.

Where:

Table: Variables in the integrated Horton's equation.

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:

\[ I_{CUM}(t) = \int_{0}^{t} I_I \, dt \]

Equation: Cumulative infiltration at the elapsed time, from the actually occurred infiltration.

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 form of Horton's equation, giving the cumulative infiltration up to and the cumulative infiltration at the elapsed time, from the actually occurred can be used to determine the “equivalent” time tp. This is done iteratively, since equation the integrated form of Horton's equation, giving the cumulative infiltration up to 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's equation for the infiltration capacity, original form. By these means, the infiltration capacity has been made a function of actual water infiltrated and not just a function of time.

Per default, the program is set to “standard” Horton's equation. Application of a “standard” Horton's equation the horton's equation for the infiltration capacity, original form or its integrated form (Equations the integrated form of Horton's equation, giving the cumulative infiltration up to and the cumulative infiltration at the elapsed time, from the actually occurred) is controlled par the parameter Integrated Horton.

MIKE 11 note: the integrated Horton form of the equation is not available from MIKE 11 interface, so only its “normal” form can be used.

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:

\[ I_H(t) = I_{IT} + \left(I_{Imax} - Q_{IT}\right) \cdot e^{-\frac{1}{k_h} t} \]

Equation: Recovery of the infiltration capacity during the dry period following rainfall, inverse form of Horton's equation.

Where:

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

\(I_H(t)\)	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	Dry Horton Exponent, the time factor (characteristic soil parameter) for drying conditions

The surface storage, I_S(t), is the loss due to 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:

\[ I_S(t) = \begin{cases} I(t) - I_E(t) - I_W(t) - I_I(t) & \text{for } y(t) \leq \left(y_w + y_s\right) \\[1ex] 0 & \text{for } y(t) > \left(y_w + y_s\right) \end{cases} \]

Equation: Surface storage loss, filling the terrain depressions once the wetting and infiltration losses are satisfied.

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
\(I_W(t)\)	Wetting loss at time t
\(I_E(t)\)	Evaporation loss at time t
\(y_w\)	Wetting depth
\(y_s\)	Surface storage depth
\(y(t)\)	Accumulated depth at time t