Linear Deficit and Constant Model

Basic Concepts and Equations

The linear deficit and constant loss method (LC method) is a modification of the initial and constant method (see Schoener et al., 2021 and Schoener et al., 2023):

1)	$\begin{array}{l}f_t=\left\{\begin{array}{cl} m F_t+f_0 & \text { if } F_t< F_c \\ K_{\text {eff }} & \text { if } F_t \geq F_c \end{array}\right\} \quad\left(f_0 \geq K_{\text {eff }}\right)\end{array}$

where f_t (mm/hr or in/hr) is the potential infiltration rate at time t, F_t (mm or in) is the cumulative infiltration at time t, F_c (mm or in) is the initial deficit, m (1/hr) is the infiltration rate decay factor with respect to cumulative infiltration, and K_eff (mm/hr or in/hr) is the constant infiltration rate or effective hydraulic conductivity.

In HEC-HMS, the LC method was further modified to allow removal of soil moisture via ET in the same way as in the deficit and constant method. Between precipitation events, the soil layer will lose moisture as the canopy extracts infiltrated water. Unless a canopy method is selected, no soil water extraction will occur. This method may also be used in combination with a surface method that will hold water on the land surface. The water in surface storage can infiltrate into the soil layer and/or be removed through ET.

Infiltration

The LC model lets the potential infiltration rate f start at an initial value f₀ (mm/hr or in/hr) and decrease linearly as a function of cumulative infiltration (F_t) until reaching a constant rate K_eff when cumulative infiltration is equal to initial deficit F_c. Due to the linear relationship, only m and F_c need to be defined in addition to K_eff. Compared to other simple loss methods such as the initial and constant or curve number model, the LC method has the advantage that it does not use an initial abstraction term and will simulate runoff from the start of a rainfall event if precipitation intensity for a given time step exceeds potential infiltration rate.

Event-Based Simulation

The LC model accounts for a single, hypothetical soil layer, hereafter referred to as the active soil layer. The soil layer has a maximum capacity to hold water. Figure 1 below shows a conceptual representation of the linear deficit and constant loss method when the active soil layer is not completely saturated, i.e. the layer contains less water than the maximum storage capacity. The deficit, measured in mm or in, is the amount of water required at any point in time to bring the active layer to saturation. During event-based simulation (Figure 1, left), water will infiltrate into the soil at a rate determined by the initial deficit, decay factor, and cumulative infiltration since the onset of the storm. If at any point in time the precipitation rate exceeds the potential infiltration rate, the difference (infiltration excess) will become runoff. If the precipitation rate at a given time is equal to or less than the potential infiltration rate, all rainfall infiltrates into the soil.

Continuous Simulation

The LC method also allows for continuous simulation (see Figure 1, right) when used in combination with a canopy method that allows extraction of water from the soil due to evapotranspiration. Continuous simulation requires the specification of another loss parameter, the maximum deficit. This value can be interpreted as the porosity multiplied by the thickness of the active layer and is measured in millimeters or inches.

For continuous simulation, the modeler must select a canopy method (under subbasin elements) and specify an evapotranspiration (ET) method (under meteorologic models). ET removes water from the active soil layer between and, depending on user setting, during storm events. The potential evapotranspiration rate is taken from the meteorologic model, where a variety of methods are available for representing that process. The ET rate is used as specified by the meteorologic model without any modification. There is no further evapotranspiration after the water in the soil layer is reduced to zero. ET will start again as soon as water is present in the soil layer. Unless a canopy and ET method are selected, no soil water extraction will occur. The canopy method also allows the modeler to simulate interception, the portion of precipitation intercepted by vegetation that never reaches the ground.

Figure 1: Conceptual representation of the linear deficit and constant loss method for event-based simulation (left) and continuous simulation (right) when the active soil layer has a deficit greater than zero.

Percolation

Once the active layer has saturated (the deficit is equal to zero), the potential infiltration rate becomes equal to the constant rate. Water will percolate out of the bottom of the active soil layer at a rate equal to the actual infiltration rate (see Figure 2). Percolation water is lost from the system. Percolation will continue as long as the soil layer is at maximum storage capacity, and precipitation continues. The linear deficit and constant method should therefore not be used for systems were:

The water table is close to the surface, and the vadose zone could saturate completely during the analysis period; or
An impermeable layer is present at a depth sufficiently shallow that that a perched aquifer could form during the analysis period.

In both cases, there would be no percolation once the active layer is saturated, and all additional precipitation would become runoff.

Figure 2: Conceptual representation of the linear deficit and constant loss method for event-based simulation (left) and continuous simulation (right) when the active soil layer has a deficit equal to zero.

Required Parameters

Parameters required to utilize this method within HEC-HMS include the initial deficit (mm or in), maximum deficit (mm or in), constant rate (mm/hr or in/hr), decay factor (1/hr) and directly connected impervious area (percent).

The initial deficit is the soil moisture deficit of the active soil layer at the onset of a storm event. The potential infiltration rate decreases linearly with cumulative infiltration until the initial deficit is satisfied. Once satisfied, the potential infiltration rate becomes constant (see constant rate below).

Table 1: Proposed values for initial deficit, sand, loamy sand, and sandy loam texture classes.

Antecedent soil moisture (m³/m³)	Initial deficit (mm)
0.02	48 (36-72)
0.06	36 (23-66)
0.10	23 (10-54)
0.14	10 (0-41)
0.18	0 (0-28)
0.22 *	0 (0-16)

* or presence of physical soil crust

Table 1 provides estimates for the initial deficit parameter for different antecedent soil moisture conditions and texture classes sand, sandy loam, and loamy sand. The values are based on plot-scale rainfall simulator testing (Schoener et al., Forthcoming) carried out at different test sites in New Mexico. The table contains median values, with reasonable ranges included in parenthesis. Rainfall simulator testing also revealed that physical soil crusts can substantially increase runoff, and crusted soils and crusted soils may be modeled equivalent to wet soils.

The maximum deficit specifies the maximum soil moisture deficit of the active soil layer. This value must be equal to or larger than the initial deficit. For event-based simulations, it will not impact model results. The maximum deficit can be interpreted as the porosity multiplied by the thickness of the active layer. Table 1 suggests a reasonable maximum deficit range for sands, sandy loams and loamy sands of 36-72 mm based on plot-scale rainfall simulator testing.

The constant rate determines the potential rate of infiltration that will occur after the initial deficit is satisfied. The same rate is applied regardless of the length of the simulation. The constant rate can be interpreted as the effective hydraulic conductivity of the active soil layer. Table 2 specifies proposed values of the constant rate parameter for texture classes sand, sandy loam, and loamy sand.

Table 2: Proposed values for constant rate parameter, sand, loamy sand, and sandy loam texture classes.

Texture class	Constant rate (mm/hr)
Sand	31 (19-44)
Loamy sand	20 (14-26)
Sandy loam	15 (11-21)

The decay factor is the rate of infiltration potential decay with respect to cumulative infiltration. The default value is -3, but users can change the decay factor values within the range of 0 to -8. Sensitivity analysis has shown that the LC loss method is substantially more sensitive to changes in the initial deficit parameter compared to decay factor, and that the latter may be held constant at -3 (paper under review).

Finally, the percentage of the subbasin comprised of directly connected impervious area can be specified. Directly connected impervious areas are surfaces where runoff is conveyed directly to a waterway or stormwater collection system. These surfaces differ from disconnected impervious areas where runoff encounters permeable areas which may infiltrate some (or all) of the runoff prior to reaching a waterway or stormwater collection system. No loss calculations are carried out on the specified percentage of the subbasin; all precipitation that falls on that portion of the subbasin becomes excess precipitation and subject to direct runoff.

A tutorial describing an example application of this loss method, including parameter estimation and calibration, can be found here: Applying the Linear Deficit and Constant Loss Method.

A Note on Parameter Estimation

The parameter values presented in this section are intended as initial estimates only for the soil textures specified. Parameter guidance was developed for bare or sparsely vegetated soil plots in New Mexico under simulated rain. Evidence suggests that infiltration models can be parameterized successfully using plot-scale simulation under certain circumstances (Schoener et al., 2021). Nevertheless, initial estimates should be calibrated and validated using measured rainfall-runoff data whenever possible.

A Note on the Computational Algorithm in HEC-HMS

Potential infiltration rate over the duration of each simulation time step is calculated by averaging the potential infiltration rates at the beginning and at the end of the time step. Potential loss over the duration of the simulation time step is then calculated by assuming that the water infiltrates at the potential infiltration rate until the deficit is met. If the deficit is met before the end of the simulation time step, the water continues infiltrating at the percolation rate for the remainder of the simulation time step. The actual loss is then calculated as the minimum of the potential loss and the available precipitation. Deficit at the end of each time step is updated with the actual infiltration loss and any evaporation that happened over the time period.