Processes

Rain/snow partitioning

Precipitation is partitioned into rain and snow before entering any store. Three partitioning methods are available, selected via the snow_rain_process option. If snow_rain_process is not set, the default is 'snow_rain:linear', except when snow_melt_process='melt:cemaneige', which automatically selects 'snow_rain:cemaneige'.

Linear transition (snow_rain:linear)

The default partitioning applies a linear transition between pure snow and pure rain over a user-defined temperature range:

• Below prec_t_start (optional, °C, default: 0, [-2, 2]): all precipitation falls as snow.

• Above prec_t_end (optional, °C, default: 2, [0, 4]): all precipitation falls as rain.

• Between the two thresholds: the snow fraction decreases linearly.

This partitioning is used by default in GSM-Socont and in GR4J when snow_melt_process is not 'melt:cemaneige'.

Single-threshold (snow_rain:threshold)

A step-function partitioning with no transition zone. All precipitation is snow at or below the threshold, and rain above it.

• prec_t (optional, °C, default: 0, [-5, 5]): temperature threshold. Full name: snow_rain_transition:threshold.

CemaNeige rain/snow partitioning (snow_rain:cemaneige)

The CemaNeige partitioning (Valéry et al. [2014]) has no calibrated parameters. It is selected automatically when snow_melt_process='melt:cemaneige' in GR4J. The equation used depends on elevation.

At elevations ≥ 1500 m, the solid fraction is computed from the daily mean temperature \(T_\mathrm{mean}\) with fixed bounds of −1 °C and 3 °C:

\[f_{\mathrm{solid}} = \max\!\left(0,\; \min\!\left(1,\; \frac{3 - T_{\mathrm{mean}}}{4}\right)\right)\]

No additional forcing variables are needed beyond daily mean temperature.

At elevations < 1500 m, the solid fraction is approximated as the fraction of the day during which temperature is below 0 °C, assuming a linear diurnal cycle between the daily minimum (\(T_\mathrm{min}\)) and maximum (\(T_\mathrm{max}\)):

\[f_{\mathrm{solid}} = \max\!\left(0,\; \min\!\left(1,\; 1 - \frac{T_{\mathrm{max}}}{T_{\mathrm{max}} - T_{\mathrm{min}}}\right)\right)\]

The temperature_min and temperature_max forcing variables are required for hydro units below 1500 m.

Melt models

Four snow and glacier melt models are available in Hydrobricks. They differ in data requirements and spatial complexity, ranging from a simple temperature index to a radiation-enhanced approach. Choose based on the available input data and the catchment characteristics.

Available melt models:

• degree_day: classical degree-day model

• degree_day_aspect: aspect-based degree-day model

• temperature_index: Hock’s radiation-enhanced temperature-index model

• cemaneige: CemaNeige snowmelt model with thermal-state correction

Specify the melt model when instantiating the hydrological model:

socont = Socont(
   soil_storage_nb=2,
   surface_runoff="linear_storage",
   snow_melt_process="melt:degree_day"
)

Valid values for snow_melt_process: "melt:degree_day", "melt:degree_day_aspect", "melt:temperature_index", "melt:cemaneige".

Degree-day model (degree_day)

Degree-day or temperature-index approaches are widely used in snow- and glacier-dominated catchments because they require only air temperature data, which is often the most readily available meteorological variable. The general form (Rango and Martinec [1995]) is:

\[\begin{split}M_{\mathrm{DD}}(t) = \begin{cases} a_j(T_a(t) - T_T) & : T_a(t) > T_T \quad j \in \{\mathrm{snow,\, ice}\}\\ 0 & : T_a(t) \leq T_T \end{cases}\end{split}\]

where:

• \(M_{\mathrm{DD}}(t)\) is the melt rate at time step \(t\) [mm d⁻¹],

• \(a_j\) is the degree-day factor for snow or ice [mm d⁻¹ °C⁻¹],

• \(T_a\) is the air temperature [°C],

• \(T_T\) is the melt temperature threshold [°C].

This is the simplest option: melt is proportional to the temperature excess above the threshold, with a single degree-day factor per surface type (snow or ice). Requires only temperature and elevation band data. Use this model when computational simplicity or data availability is a priority.

Aspect-based degree-day model (degree_day_aspect)

Extends the standard degree-day model by assigning different degree-day factors to north-, south-, and east/west-facing slopes. Sun-exposed south-facing slopes receive more radiation and melt faster for the same air temperature; this model captures that effect without explicitly computing radiation. It requires the catchment to be discretized by elevation and aspect. Use this model when aspect strongly differentiates melt rates across the catchment.

Radiation-enhanced temperature-index model (temperature_index)

Based on Hock [1999], this model replaces the fixed degree-day factor with one that scales with potential clear-sky solar radiation:

\[\begin{split}M_{\mathrm{HTI}}(t) = \begin{cases} (m + r_j\, I_{\mathrm{pot}})(T_a(t) - T_T) & : T_a(t) > T_T \quad j \in \{\mathrm{snow,\, ice}\}\\ 0 & : T_a(t) \leq T_T \end{cases}\end{split}\]

where:

• \(m\) is a base melt factor common to both ice and snow [mm d⁻¹ °C⁻¹],

• \(r_j\) is a radiation factor for snow or ice [mm d⁻¹ °C⁻¹ m² W⁻¹],

• \(I_{\mathrm{pot}}\) is the potential clear-sky direct solar radiation [W m⁻²].

Radiation is computed as:

\[I_{\mathrm{pot}} = I_0 \left( \frac{R_m}{R} \right)^2 \Psi_a^{\left( \frac{P}{P_0 \cos Z} \right)} \cos\theta\]

where \(I_0\) is the solar constant (1368 W m⁻²), \((R_m/R)^2\) is the orbital eccentricity correction, \(\Psi_a\) is the mean clear-sky transmissivity, \(P/P_0\) is the ratio of local to sea-level atmospheric pressure, \(Z\) is the zenith angle, and \(\theta\) is the angle between the surface normal and the solar beam. Radiation is computed at 15-minute intervals and aggregated to daily values to capture diurnal and shading effects.

This model requires the catchment to be discretized by elevation and radiation. It is the most physically realistic of the three temperature-based variants and is recommended when snow and glacier melt dominate runoff. The main trade-off is that computing the radiation field adds some preprocessing time. See Argentin et al. [2025] for a comparative evaluation of all three models.

CemaNeige snowmelt model (cemaneige)

The CemaNeige model (Valéry et al. [2014]) addresses a known weakness of plain degree-day models: they tend to overestimate melt immediately after cold spells because they respond instantly to temperature without any memory of prior cold. CemaNeige adds a thermal state variable \(e_{TG}\) that tracks how cold the snowpack is. Melt can only occur when the snowpack has fully warmed (i.e., \(e_{TG} = 0\)):

\[e_{TG}(t) = \min\!\left(0,\; CTG \cdot e_{TG}(t-1) + (1-CTG) \cdot (T_a - T_{melt})\right)\]

\[\begin{split}M_{\mathrm{pot}}(t) = \begin{cases} K_f \cdot (T_a - T_{melt}) & : e_{TG} = 0 \text{ and } T_a > T_{melt} \\ 0 & : \text{otherwise} \end{cases}\end{split}\]

An additional scaling factor accounts for partial snow cover: when the snow water equivalent (SWE) is well below the mean annual accumulation \(C_n\), only a fraction of the area is snow-covered and effective melt is reduced:

\[M(t) = \left(0.9 \cdot \min\!\left(1,\; \frac{SWE}{C_n}\right) + 0.1\right) \cdot M_{\mathrm{pot}}(t)\]

where \(C_n = 0.9 \times \overline{P_{\mathrm{snow}}}\) and \(\overline{P_{\mathrm{snow}}}\) is the mean annual solid precipitation. The factor 0.1 ensures a minimum melt rate even at very low SWE.

Parameters:

• Kf: degree-day melt factor [mm d⁻¹ °C⁻¹]

• CTG: cold content weighting factor [−], range [0, 1]. Values close to 1 give the snowpack a long thermal memory; values near 0 make it respond almost instantly to temperature.

• Tmelt: melt temperature threshold [°C]

• Cn (mean_annual_snow): mean annual solid precipitation [mm]

CemaNeige is the recommended snow model for the GR4J model:

gr4j = models.GR4J(snow_melt_process='melt:cemaneige')

Interception

GR4J interception (interception:gr4j)

Implements the GR4J zero-capacity interception store (Perrin et al. [2003]), which neutralises precipitation against PET before either reaches a store. With total precipitation \(P\) (rain plus snowmelt) and potential evapotranspiration \(E_P\):

• if \(P > E_P\), the amount \(E_P\) is sent to the atmosphere, the net precipitation \(P_n = P - E_P\) is passed on, and the net evaporative demand is \(E_n = 0\);

• if \(P \le E_P\), the amount \(P\) is sent to the atmosphere and the net evaporative demand \(E_n = E_P - P\) is published for the downstream production-store ET.

The remaining net precipitation \(P_n\) is then routed to the production store by an outflow:rest (throughfall) process. Requires the pet forcing; no calibrated parameters.

Evapotranspiration

Two ET formulations are available.

Socont ET (et:socont)

Used in GSM-Socont. Actual ET is proportional to PET, scaled by the filling ratio of the reservoir with a square-root exponent (Schaefli et al. [2005]):

\[E(t) = E_P(t) \left(\frac{S(t)}{S_{\max}}\right)^{0.5}\]

where:

• \(E_P(t)\) is the potential evapotranspiration [mm d⁻¹],

• \(S(t)\) is the current water content of the reservoir [mm],

• \(S_{\max}\) is the maximum storage capacity [mm].

ET decreases as the reservoir empties; it equals PET when the reservoir is full. No calibrated parameters. Requires the pet forcing.

GR4J ET (et:gr4j)

Used internally by GR4J. Withdraws actual evapotranspiration from the production store to the atmosphere, using the net evaporative demand \(E_n\) published by the interception step (Perrin et al. [2003]):

\[E_s = S \cdot \frac{2 - S/X_1}{1 + (1 - S/X_1) \tanh(E_n/X_1)} \tanh(E_n/X_1)\]

where \(X_1\) is the production store capacity and \(S\) is the store content after this step’s infiltration. No calibrated parameters beyond \(X_1\).

Infiltration

Socont infiltration (infiltration:socont)

Used in GSM-Socont for the ground land cover. Water infiltrates into the slow reservoir at a rate that decreases quadratically as the reservoir fills (Schaefli et al. [2005]):

\[\mathrm{Inf}(t) = S(t) \left(1 - \left(\frac{S(t)}{S_{\max}}\right)^2\right)\]

where \(S(t)\) is the current water content and \(S_{\max}\) is the maximum capacity. When the reservoir is empty, all incoming water infiltrates; when full, infiltration is zero and the remainder becomes surface runoff. No calibrated parameters.

GR4J infiltration (infiltration:gr4j)

Computes the portion \(P_s\) of net precipitation \(P_n\) that enters the GR4J production store (Perrin et al. [2003]). In the GR4J model this infiltration term is not applied on its own but as part of the combined production:gr4j step:

\[P_s = X_1 \left(1 - \left(\frac{S}{X_1}\right)^{\!2}\right) \frac{\tanh\!\left(\dfrac{P_n}{X_1}\right)} {1 + \dfrac{S}{X_1}\,\tanh\!\left(\dfrac{P_n}{X_1}\right)}\]

where \(X_1\) is the production store capacity and \(S\) is its current water content. The rate increases with \(P_n\) and decreases as the store fills, going to zero when \(S = X_1\). No calibrated parameters.

Percolation

Constant percolation (percolation:constant)

Drains water from one reservoir to another at a fixed rate, independent of storage:

\[\mathrm{Perc}(t) = r\]

where \(r\) is the percolation rate [mm d⁻¹]. Used in GSM-Socont when soil_storage_nb=2 to connect the first slow reservoir to the baseflow reservoir.

Parameters:

• percol (alias for percolation_rate): constant drainage rate [mm d⁻¹], range [0, 10]. Full name: slow_reservoir:percolation_rate.

GR4J percolation (percolation:gr4j)

Drains water from the GR4J production store (Perrin et al. [2003]):

\[\mathrm{Perc} = S \left[1 - \left(1 + \left(\frac{4S}{9\,X_1}\right)^{\!4}\right)^{\!-1/4}\right]\]

where \(S\) is the current production-store water content and \(X_1\) is its capacity. Percolation increases non-linearly with the filling ratio and goes to zero when the store is empty. In the GR4J model it is applied as part of the combined production:gr4j step. No calibrated parameters.

Production

GR4J production (production:gr4j)

Used internally by GR4J. Applies the complete production-store update in a single discrete step and routes its output to the unit-hydrograph input (Perrin et al. [2003]). For a start-of-step store level \(S\) and net precipitation \(P_n\):

1. Infiltration \(P_s\) fills the store (see infiltration:gr4j), raising the level to \(S + P_s\).

2. Evapotranspiration \(E_s\) is withdrawn separately by the et:gr4j process and sent to the atmosphere.

3. Percolation \(\mathrm{Perc}\) drains the store (see percolation:gr4j).

The water routed to the unit-hydrograph input is

\[P_R = (P_n - P_s) + \mathrm{Perc}\]

i.e. the net precipitation that did not infiltrate plus the percolation leaving the store. Because GR4J is solved as a discrete model, the production store is computed directly (no ODE solver), so the per-step update is exact and solver-independent. No calibrated parameters beyond \(X_1\).

Surface runoff / Quick flow

Two formulations control how water leaves the surface runoff storage in GSM-Socont. The choice is made via the surface_runoff option at model instantiation.

Socont runoff (runoff:socont)

The default (surface_runoff="socont_runoff"). Models overland flow on an inclined plane with a linearly varying water depth, following Manning’s equation (Schaefli et al. [2005]):

\[q = \beta \sqrt{\tan J} \; h^{5/3}\]

where:

• \(\beta\) is a runoff coefficient [m^4/3 s^-1] that absorbs Manning’s roughness and the plane width,

• \(J\) is the mean terrain slope [°]; internally converted to m m^-1 via \(\tan J\),

• \(h\) is the water depth at the downslope end of the plane [m].

Parameters:

• beta: runoff coefficient [m^4/3 s^-1], range [100, 30 000]. Full name: surface_runoff:runoff_coefficient.

• J: mean terrain slope [°], range [0, 90]. Derived from terrain data; not a calibration parameter. Full name: surface_runoff:slope.

It is the original Socont parameterisation.

Linear storage runoff (outflow:linear)

Alternative (surface_runoff="linear_storage"). Replaces the Manning-based formulation with a simple linear reservoir:

\[Q_{\mathrm{quick}}(t) = k_{\mathrm{quick}} \cdot S(t)\]

Parameters:

• k_quick: response factor [d^-1], range [0.05, 1]. Full name: surface_runoff:response_factor.

Use this formulation when a simpler quick-flow representation is preferred.

Transfer

GR4J routing (routing:gr4j)

Used internally by GR4J. Transforms the production-store output \(P_R\) into discharge through two unit hydrographs, a non-linear routing store, and a groundwater-exchange term (Perrin et al. [2003]).

The inflow is split 90 % / 10 % between two unit hydrographs:

• UH1 (S-curve \(S\!H_1\), time base \(X_4\)) feeds the routing store and carries \(0.9\,P_R\);

• UH2 (S-curve \(S\!H_2\), time base \(2X_4\)) feeds the direct branch and carries \(0.1\,P_R\).

The S-curves are

\[S\!H_1(t) = \left(\frac{t}{X_4}\right)^{5/2}, \quad 0 \le t \le X_4\]

\[\begin{split}S\!H_2(t) = \begin{cases} \tfrac{1}{2}\left(\dfrac{t}{X_4}\right)^{5/2} & 0 \le t \le X_4 \\[2ex] 1 - \tfrac{1}{2}\left(2 - \dfrac{t}{X_4}\right)^{5/2} & X_4 < t \le 2X_4 \end{cases}\end{split}\]

with \(S\!H = 0\) below the range and \(S\!H = 1\) above it; the unit-hydrograph ordinates are the successive differences of the S-curve.

The groundwater exchange depends on the routing-store filling ratio:

\[F = X_2 \left(\frac{R}{X_3}\right)^{7/2}\]

where \(R\) is the routing-store content. A positive \(X_2\) imports water into the catchment; a negative one exports it (a loss). The routing store is updated with the UH1 output and \(F\), then releases

\[Q_R = R \left[1 - \left(1 + \left(\frac{R}{X_3}\right)^{\!4}\right)^{\!-1/4}\right]\]

The direct branch yields \(Q_D = \max(0,\; \text{UH2 output} + F)\), and the total discharge is \(Q = Q_R + Q_D\).

Parameters:

• X2 (exchange_factor): groundwater exchange coefficient [mm d⁻¹], default 0. Full name: uh_input:exchange_factor.

• X3 (routing_capacity): routing-store capacity [mm], default 90. Full name: uh_input:routing_capacity.

• X4 (uh_base_time): unit-hydrograph time base [d], default 1.7, must be > 0.5. Full name: uh_input:uh_base_time.

Reservoir outflow

The following outflow mechanisms are used by storage bricks across all models.

Linear outflow (outflow:linear)

The most common outflow type. Outflow is proportional to the current storage:

\[Q(t) = k \cdot S(t)\]

Parameter: response_factor [d^-1]. Used for the slow reservoirs, glacier area storages, and the linear quick-flow option.

Overflow (outflow:overflow)

Releases water instantaneously when storage exceeds its maximum capacity. No parameters. Used to prevent the slow reservoir from overfilling.

Direct outflow (outflow:direct)

Routes the entire storage content to the target in a single time step. No parameters. Used internally in GSM-Socont to pass glacier meltwater directly to the basin-level lumped storages. Because it withdraws the full content, it cannot run in parallel with any other process drawing from the same storage; configuration validation rejects such a combination (use outflow:rest instead when several outflows must share a store).

Rest outflow (outflow:rest)

Routes whatever water remains in the store after all the other (sibling) processes have taken their share. It works on both directly-computed and solver-handled bricks: on a direct brick the siblings have already withdrawn their share, so the remaining content is exactly “the rest”; on a solver brick the siblings’ current outflow rates are subtracted so the demands sum to the available content. It must therefore be declared after the other withdrawing processes. No parameters. Used for surface runoff in the Socont ground land cover (water not infiltrated becomes runoff) and in GR4J for throughfall (the net precipitation \(P_n\) passed on to the production store).