SMAR Model¶

Overview¶

SMAR (Soil Moisture Accounting and Routing) is an eight-parameter daily lumped rainfall-runoff model originally developed by O'Connell, Nash, and Farrell at University College Galway, Ireland (O'Connell et al., 1970). It was designed as a general-purpose conceptual model for operational streamflow forecasting and has been widely applied in Irish, tropical, and semi-arid catchments.

The variant implemented in HOLMES is the retained version described in Annex 1 of Perrin (2000), corresponding to HOOPLA HM16. It represents the catchment as a 16-layer discretised soil column (each layer 25 mm thick, total depth \(Z = 400\) mm) feeding two parallel routing reservoirs — one linear (groundwater) and one quadratic (surface) — with a fractional-delay unit hydrograph at the outlet.

What makes SMAR distinctive among the HOLMES models is its explicit vertical soil discretisation. While most models represent soil moisture as a single bulk reservoir (GR4J, HYMOD, Bucket) or as a few discrete stores (SACRAMENTO, NAM), SMAR divides the soil into 16 physical layers and applies evapotranspiration to each layer with an exponentially decaying intensity (\(C^i\), where \(C < 1\) and \(i\) is the layer index). This means the top layers dry out first and more intensely, while deeper layers retain moisture longer — a realistic representation of how plant roots extract water preferentially from the surface. The model is a good pedagogical choice when students need to understand how vertical soil structure affects runoff generation and recession behavior.

Key Concepts¶

16-layer soil column: The soil is discretised into 16 layers of 25 mm each (total \(Z = 400\) mm). Water infiltrates from the top layer downward; each layer fills to capacity before overflowing to the next. Drainage from the bottom layer constitutes interflow.
Exponentially decaying evapotranspiration: Each soil layer \(i\) is subject to evapotranspiration at rate \(C^i \cdot E_n\), where \(C = X_3 < 1\) is a calibratable coefficient and \(E_n\) is the remaining atmospheric demand. The top layer receives the strongest ET, and each subsequent layer receives exponentially less — mimicking declining root-zone water uptake with depth.
Moisture-dependent direct runoff: A fraction \(H' = X_1 \cdot S / S_{\max}\) of net precipitation runs off directly without entering the soil, where \(S\) is the moisture content of the top five layers. As the soil wets up, direct runoff increases — a simple saturation-excess mechanism.
Exponential infiltration capacity: The maximum infiltration rate decays exponentially with soil moisture: \(F_r = Y_m \cdot \exp(-X_2 \cdot S / S_{\max})\), where \(Y_m = 200\) mm/d is a fixed maximum. Dry soils absorb rain readily; near-saturated soils severely limit infiltration.
Dual routing reservoirs: Interflow from the soil column is split between a linear groundwater reservoir (fraction \(1 - X_8\)) and a quadratic surface reservoir (fraction \(X_8\)). The linear reservoir produces smooth, sustained baseflow; the quadratic reservoir generates peakier, faster-decaying flow that dominates during storms.
Quadratic routing: The surface routing reservoir drains via \(Q_t = T^2 / (T + X_4)\), the same non-linear law used by GARDENIA. This produces fast recession from high storage and slow recession from low storage.
PET correction coefficient: The input PET is scaled by a calibratable factor \(X_7\) before use, allowing the model to compensate for systematic biases in the PET estimate (similar to GARDENIA's \(X_5\)).
Fractional delay routing: A short pure delay \(X_6\) (typically 0.5 to 5 days) shifts the hydrograph to account for channel travel time, using a two-element fractional interpolation between adjacent integer delays.

How It Works¶

The SMAR model processes precipitation and evapotranspiration through the following steps each day:

Step 1: PET correction and net inputs. The input PET is scaled by the correction coefficient: \(E_{\text{corr}} = X_7 \cdot E\). Net precipitation and net evapotranspiration are computed as complementary residuals: \(P_n = \max(0, P - E_{\text{corr}})\) and \(E_n = \max(0, E_{\text{corr}} - P)\). Only one of these is non-zero on any given day — on wet days \(P_n > 0\) and \(E_n = 0\); on dry days \(E_n > 0\) and \(P_n = 0\).

Step 2: Direct runoff. A moisture-dependent fraction of net precipitation runs off directly: \(P_{r1} = H' \cdot P_n\), where \(H' = X_1 \cdot S / S_{\max}\). Here \(S\) is the moisture sum of the top five layers (capacity \(S_{\max} = 125\) mm). When the soil is wet (\(S \approx S_{\max}\)), \(H'\) approaches \(X_1\) and substantial rain bypasses the soil column entirely; when the soil is dry, almost all rain infiltrates.

Step 3: Infiltration and surface excess. The infiltration capacity \(F_r = 200 \cdot \exp(-X_2 \cdot S / S_{\max})\) limits how much of the remaining rainfall (\(P_n - P_{r1}\)) can enter the soil. Actual infiltration is \(P_s = \min(F_r, P_n - P_{r1})\), and any surplus above the infiltration capacity becomes surface excess \(P_{r2}\).

Step 4: 16-layer soil moisture accounting. Infiltrated water \(P_s\) cascades down the 16 layers sequentially: each layer absorbs water up to its 25 mm capacity, and any overflow passes to the next layer. At each layer, evapotranspiration removes water at an exponentially decreasing rate \(C^{i+1} \cdot E_n\) (where \(i\) is the zero-based layer index), reducing the remaining atmospheric demand \(E_n\) accordingly. Water that overflows from the bottom layer (layer 16) constitutes interflow.

Step 5: Linear groundwater routing. A fraction \(1 - X_8\) of the interflow enters the linear groundwater reservoir: \(L \leftarrow L + (1 - X_8) \cdot I\). The reservoir drains linearly: \(Q_l = L / X_5\), producing smooth baseflow.

Step 6: Quadratic surface routing. The remaining fraction \(X_8\) of interflow, plus any surface excess \(P_{r2}\) from Step 3, enters the quadratic routing reservoir: \(T \leftarrow T + X_8 \cdot I + P_{r2}\). The reservoir drains via the quadratic law: \(Q_t = T^2 / (T + X_4)\), producing peaked, fast-decaying flow.

Step 7: Total streamflow and delay. The total flow \(Q = Q_l + Q_t + P_{r1}\) (baseflow + surface routing + direct runoff) is injected into the fractional-delay register of size \(\lceil X_6 \rceil + 1\). The register shifts by one step each day, and the first element — clamped at zero — is returned as the simulated streamflow.

Parameters¶

The SMAR model has eight calibratable parameters:

Parameter	Description	Range	Units	Physical Interpretation
\(X_1\)	Direct flow coefficient (\(H\))	0.01–1.0	-	Maximum fraction of net precipitation that runs off directly when the top soil is fully saturated. Higher values increase the flashiness of peak flows.
\(X_2\)	Infiltration parameter (\(Y_c\))	0.01–10.0	-	Controls how fast infiltration capacity decays with increasing soil moisture. Higher values mean the soil "seals" more quickly as it wets up.
\(X_3\)	PET reduction coefficient (\(C\))	0.01–0.99	-	Base of the exponential ET decay with depth: layer \(i\) receives \(C^{i+1}\) of the remaining atmospheric demand. Lower values concentrate ET in the surface layers; values near 1 spread ET uniformly through the soil column.
\(X_4\)	Quadratic routing reservoir capacity	1–500	mm	Characteristic storage of the quadratic surface routing reservoir. Larger values slow the surface recession and spread flood peaks; smaller values produce sharper peaks.
\(X_5\)	Linear routing reservoir emptying constant	1–200	days	Residence time of the linear groundwater store (\(Q_l = L / X_5\)). Large values produce long, sustained baseflow recessions; small values give a flashier groundwater response.
\(X_6\)	Delay	0.5–5.0	days	Pure channel routing delay applied at the outlet. Shifts the hydrograph in time without changing its shape.
\(X_7\)	PET correction coefficient	0.1–2.0	-	Multiplicative factor applied to the input PET series. Values above 1 increase evapotranspiration (useful when Oudin PET underestimates for a given climate); below 1 reduces it.
\(X_8\)	Flow partitioning coefficient (\(G\))	0.01–0.99	-	Fraction of interflow routed to the quadratic surface reservoir. The complement \((1 - X_8)\) goes to the linear groundwater store. Controls the balance between fast peaked flow and slow sustained baseflow.

Understanding the parameters:

\(X_1\) is the saturation-excess runoff coefficient. At \(X_1 = 1.0\), a fully saturated soil sends all net precipitation directly to the stream; at \(X_1 = 0.01\), almost none. Expect calibrated values between 0.1 and 0.6 for most humid catchments.
\(X_2\) and \(X_3\) together control how the soil processes water. \(X_2\) governs the infiltration capacity (how quickly rain can enter the soil column), while \(X_3\) governs the ET extraction profile (how quickly the soil dries from the top down). Low \(X_3\) (0.1–0.3) creates a sharply surface-drying soil; high \(X_3\) (0.7–0.9) distributes ET evenly.
\(X_4\) and \(X_5\) are the routing parameters. \(X_4\) controls the fast (quadratic) response, while \(X_5\) controls the slow (linear) baseflow. When \(X_5 \gg X_4\), the hydrograph shows distinct fast peaks on a slow base — typical of catchments with deep aquifers.
\(X_8\) is a flow partitioning knob similar to MARTINE's \(X_5\) or GR4J's fixed 90/10 split. Values near 1 send most interflow through the fast quadratic path; values near 0 send it to groundwater. For catchments with strong baseflow, expect \(X_8 < 0.5\).

Mathematical Formulation¶

Initialization¶

Fixed reservoir states at \(t = 0\):

\[W_i = 2 \text{ mm for } i = 1, \ldots, 16\]

\[S_0 = 100 \text{ mm}, \quad L_0 = 50 \text{ mm}, \quad T_0 = 20 \text{ mm}\]

where \(W_i\) are the individual soil layer storages, \(S\) is the top-five-layer moisture sum (recalculated each step), \(L\) is the linear routing reservoir, and \(T\) is the quadratic routing reservoir.

The fractional-delay register \(\{H_k\}\) has \(n = \lceil X_6 \rceil + 1\) elements, initialized to zero, with weights:

\[d_{n-2} = \frac{1}{X_6 - n + 3}, \quad d_{n-1} = 1 - d_{n-2}\]

PET Correction and Net Inputs¶

\[E_{\text{corr}} = X_7 \cdot E\]

\[P_n = \max(0, P - E_{\text{corr}})\]

\[E_n = \max(0, E_{\text{corr}} - P)\]

Direct Runoff¶

The soil moisture sum of the top five layers is:

\[S = \sum_{i=1}^{5} W_i, \quad S_{\max} = 125 \text{ mm}\]

The moisture-dependent direct runoff fraction:

\[H' = X_1 \cdot \frac{S}{S_{\max}}\]

\[P_{r1} = H' \cdot P_n\]

Infiltration¶

The infiltration capacity decreases exponentially with soil saturation:

\[F_r = Y_m \cdot \exp\!\left(-X_2 \cdot \frac{S}{S_{\max}}\right), \quad Y_m = 200 \text{ mm/d}\]

Actual infiltration and surface excess:

\[P_s = \min\!\bigl(F_r,\; \max(0, P_n - P_{r1})\bigr)\]

\[P_{r2} = \max\!\bigl(0,\; P_n - P_{r1} - P_s\bigr)\]

16-Layer Soil Moisture Accounting¶

For each layer \(i = 1, 2, \ldots, 16\):

\[W_i \leftarrow W_i + P_s\]

\[P_s \leftarrow \max(0, W_i - 25)\]

\[W_i \leftarrow W_i - P_s\]

\[E_i = \min\!\bigl(W_i,\; X_3^{\,i} \cdot E_n\bigr)\]

\[W_i \leftarrow W_i - E_i\]

\[E_n \leftarrow E_n - E_i\]

The remaining \(P_s\) after the bottom layer is the interflow:

\[I = P_s \big|_{i=16}\]

The top-five-layer moisture sum \(S\) is then recomputed for the next time step:

\[S \leftarrow \sum_{i=1}^{5} W_i\]

Linear Groundwater Routing¶

The linear reservoir receives a fraction \((1 - X_8)\) of the interflow:

\[L \leftarrow L + (1 - X_8) \cdot I\]

\[Q_l = \frac{L}{X_5}\]

\[L \leftarrow L - Q_l\]

Quadratic Surface Routing¶

The quadratic reservoir receives the remaining interflow plus surface excess:

\[T \leftarrow T + X_8 \cdot I + P_{r2}\]

\[Q_t = \frac{T^2}{T + X_4}\]

\[T \leftarrow T - Q_t\]

Delay Routing¶

The total flow before routing:

\[Q = Q_l + Q_t + P_{r1}\]

The register \(\{H_k\}\) is updated with the shift-and-add rule:

\[H_k \leftarrow H_{k+1} + d_k \cdot Q \quad \text{for } k = 0, 1, \ldots, n - 2\]

\[H_{n-1} \leftarrow d_{n-1} \cdot Q\]

The simulated streamflow is:

\[Q_{\text{sim}} = \max(0,\; H_0)\]

References¶

O'Connell, P. E., Nash, J. E., & Farrell, J. P. (1970). River flow forecasting through conceptual models, Part II: The Brosna catchment at Ferbane. Journal of Hydrology, 10(4), 317–329.

Kachroo, R. K. (1992). River flow forecasting. Part 5. Applications of a conceptual model. Journal of Hydrology, 133(1-2), 141–178.

Perrin, C. (2000). Vers une amélioration d'un modèle global pluie-débit au travers d'une approche comparative. PhD dissertation, Institut National Polytechnique de Grenoble, France. Annex 1, Fiche n°30 (SMAR), pp. 438–441.

Thiboult, A., Seiller, G., Poncelet, C., & Anctil, F. (2020). The HOOPLA toolbox: a HydrOlOgical Prediction LAboratory. Hydrology and Earth System Sciences Discussions.