SMAP

The Soil Moisture Accounting Procedure (SMAP) is a rainfall-runoff model that simulates the transformation of precipitation into runoff based on conceptual reservoirs representing different states. The formulation follows Lopes (1982).

The MOGESTpy implementation of SMAP includes two versions: a daily timestep version and a monthly timestep version. Both versions are designed to be flexible and can be calibrated to fit observed data for specific catchments.

Daily timestep

The Daily version of SMAP consists of 3 conceptual reservoirs: surface, soil and subsurface. This version can be accessed through the mogestpy.quantity.hydrological.smap.SmapD class.

SMAP Daily Schematic — Daily SMAP Schematic

Mathematical formulation

Mathematical Reservoirs

The daily SMAP model uses three conceptual reservoirs to simulate the hydrological cycle. The water balance equations for each reservoir are:

\[\begin{split}R_{solo}(i+1) &= R_{solo}(i) + P - Es - Er - Rec \\ R_{sup}(i+1) &= R_{sup}(i) + Es - Ed \\ R_{sub}(i+1) &= R_{sub}(i) + Rec - Eb\end{split}\]

Where:

\(R_{solo}\) = soil reservoir (aerated zone) (mm)
\(R_{sup}\) = surface reservoir (mm)
\(R_{sub}\) = subsurface reservoir (saturated zone) (mm)
\(P\) = precipitation (mm)
\(Es\) = surface runoff (mm)
\(Ed\) = direct runoff (mm)
\(Er\) = actual evapotranspiration (mm)
\(Rec\) = groundwater recharge (mm)
\(Eb\) = baseflow (mm)

Initialization

The initial conditions for the three reservoirs are set as follows:

\[\begin{split}R_{solo}(1) &= Tuin \cdot Str \\ R_{sup}(1) &= 0 \\ R_{sub}(1) &= \frac{Ebin}{(1-Kk) \cdot Ad} \cdot 86.4\end{split}\]

Where:

\(Tuin\) = initial soil moisture content (-)
\(Str\) = soil saturation capacity (mm)
\(Ebin\) = initial baseflow discharge (m³/s)
\(Ad\) = drainage area (km²)
\(Kk = 0.5^{(1/kkt)}\) = baseflow recession factor

Transfer Functions

The model uses six transfer functions to compute the fluxes between reservoirs. The surface runoff separation follows the SCS (Soil Conservation Service) method.

1. Surface runoff (Es):

\[\begin{split}Es = \begin{cases} \frac{(P - Ai)^2}{P - Ai + Str - R_{solo}}, & \text{if } P > Ai \\ 0, & \text{otherwise} \end{cases}\end{split}\]

2. Actual evapotranspiration (Er):

\[\begin{split}Er = \begin{cases} Ep, & \text{if } (P - Es) > Ep \\ (P - Es) + (Ep - (P - Es)) \cdot Tu, & \text{otherwise} \end{cases}\end{split}\]

3. Groundwater recharge (Rec):

\[\begin{split}Rec = \begin{cases} \frac{Crec}{100} \cdot Tu \cdot \left(R_{solo} - \frac{Capc}{100} \cdot Str\right), & \text{if } R_{solo} > \frac{Capc}{100} \cdot Str \\ 0, & \text{otherwise} \end{cases}\end{split}\]

4. Direct runoff (Ed):

\[Ed = R_{sup} \cdot (1 - K2)\]

5. Baseflow (Eb):

\[Eb = R_{sub} \cdot (1 - Kk)\]

6. Soil moisture (Tu):

\[Tu = \frac{R_{solo}}{Str}\]

Where:

\(P\) = precipitation (mm)
\(Ai\) = initial abstraction (mm)
\(Ep\) = potential evapotranspiration (mm)
\(Crec\) = recharge coefficient (%)
\(Capc\) = field capacity (%)
\(K2 = 0.5^{(1/k2t)}\) = surface runoff recession factor
\(Kk = 0.5^{(1/kkt)}\) = baseflow recession factor

Discharge Calculation

Any overflow from the soil reservoir is converted to surface runoff. The total discharge at the basin outlet is calculated as:

\[Q = (Ed + Eb) \cdot \frac{Ad}{86.4}\]

Where:

\(Q\) = discharge (m³/s)
\(Ed\) = direct runoff (mm)
\(Eb\) = baseflow (mm)
\(Ad\) = drainage area (km²)
86.4 is the unit conversion factor (mm·km²/s to m³/s)

The SmapD parameters are described in the table below.

Daily SMAP parameters
Parameter	Description	Range	Default	Unit
Str	Soil saturation.	100 - 2000	100	\(\text{mm}\)
Crec	Recession coefficient.	0 - 20	0.0	\(\%\)
Capc	Field capacity.	30 - 50	40	\(\%\)
kkt	Base flow recession coefficient.	30 - 180	30	\({\text{d}^{-1}}\)
k2t	Surface runoff recession coefficient.	0.2 - 10	0.2	\({\text{d}^{-1}}\)
Ad	Drainage area.	\(-\)	1.0	\(\text{km}{^2}\)
Tuin	Initial soil moisture content.	0 - 1	0.0	\(-\)
Ebin	Initial base flow.	\(-\)	0.0	\(\text{m}^3\text{/s}\)
Ai	Initial abstraction.	2 - 5	2.5	\(\text{mm}\)

Examples

To use the daily version of SMAP, you can create an instance of the SmapD class and provide the necessary parameters. Here is an example of how to set up and run a simulation using the daily version of SMAP.

from mogestpy.quantity.hydrological.smap import SmapD

model = SmapD(
 Str=100,
 Crec=19.6125,
 Capc=30,
 kkt=47.53,
 k2t=1.430,
 Ai=2,
 Tuin=.05,
 Ebin=0.1,
 Ad=70.2
 )

precipitations = [10, 20, 15, 5, 0]  # Example precipitation data
evapotranspirations = [2, 3, 1, 0.5, 0]  # Example evapotranspiration data

discharges = model.run_to_list(precipitations, evapotranspirations)

The daily model can be calibrated against observed discharge data with the calibrate method.

from mogestpy.quantity.hydrological.smap import SmapD

model = SmapD() # Initialize with default parameters

result = model.calibrate(
    prec_arr=precipitations, # as list/np.array/pd.Series
    etp_arr=evapotranspirations, # as list/np.array/pd.Series
    eval_arr=observed_discharge, # as list/np.array/pd.Series
    variables=["Str", "Crec", "Capc", "kkt", "k2t", "Ai"],
    disp=True,
)

print(result.x)
print(result.fun)

By default the calibration uses the Kling-Gupta Efficiency (KGE) as the objective function, but other metrics can be used by specifying the obj_func argument in the calibrate method.

Monthly timestep

The Monthly version of SMAP consists of 2 conceptual reservoirs: soil and subsurface and can be access through the mogestpy.quantity.hydrological.smap.SmapM class. This version is designed for applications where monthly data is more appropriate or available, and it simplifies the model structure while still capturing the essential hydrological processes.

SMAP Monthly Schematic — Monthly SMAP Schematic

The monthly SMAP parameters are described in the table below.

Monthly SMAP parameters
Parameter	Description	Range	Default	Unit
Str	Soil saturation.	400 to 5000	1000	mm
Pes	Exponent for soil evaporation.	1 to 10	1	Dimensionless
Crec	Recharge coefficient.	0 to 1	0.5	Dimensionless
kkt	Base flow recession coefficient.	1 to 6	1.5	months
Tuin	Initial soil moisture content.	0 to 1	0.5	Fraction
Ebin	Initial base flow.	No explicit bound	0.1	m^3/s
Ad	Drainage area.	No explicit bound	1.0	km^2

Examples

Similarly, to use the monthly version of SMAP, you can create an instance of the SmapM class and provide the necessary parameters. Here is an example of how to set up and run a simulation using the monthly version of SMAP.

from mogestpy.quantity.hydrological.smap import SmapM

model = SmapM(
 Str=1000,
 Pes=1,
 Crec=0.5,
 kkt=1.5,
 Tuin=0.5,
 Ebin=0.1,
 Ad=1,
 )

precipitations = [100, 200, 150, 50, 0]  # Example precipitation data
evapotranspirations = [20, 30, 10, 5, 0]  # Example evapotranspiration data

discharges = model.run_to_list(precipitations, evapotranspirations)

The monthly model uses the same calibrate method.

from mogestpy.quantity.hydrological.smap import SmapM

model = SmapM()

result = model.calibrate(
    prec_arr=precipitations, # as list/np.array/pd.Series
    etp_arr=evapotranspirations, # as list/np.array/pd.Series
    eval_arr=observed_discharge, # as list/np.array/pd.Series
    variables=["Str", "Pes", "Crec", "kkt"],
    disp=True,
)

print(result.x)
print(result.fun)