This paper presents the Spatially Explicit Hydrologic Response (SEHR) model developed at the Laboratory of Ecohydrology of the Ecole Polytechnique Fédérale de Lausanne for the simulation of hydrological processes at the catchment scale. The key concept of the model is the formulation of water transport by geomorphologic travel time distributions through gravity-driven transitions among geomorphic states: the mobilization of water (and possibly dissolved solutes) is simulated at the subcatchment scale and the resulting responses are convolved with the travel paths distribution within the river network to obtain the hydrologic response at the catchment outlet. The model thus breaks down the complexity of the hydrologic response into an explicit geomorphological combination of dominant spatial patterns of precipitation input and of hydrologic process controls. Nonstationarity and nonlinearity effects are tackled through soil moisture dynamics in the active soil layer. We present here the basic model set-up for precipitation–runoff simulation and a detailed discussion of its parameter estimation and of its performance for the Dischma River (Switzerland), a snow-dominated catchment with a small glacier cover.

Hydrological processes result from natural processes that vary strongly in space, such as
precipitation, evaporation or infiltration into the subsoil

Some models use an arbitrary (calibrated) discharge routing function to smooth the response computed
at the scale of a (sub-)catchment, e.g. with a triangular function as in the HBV model

Most existing catchment-scale model applications show an explicit parameterization of channelled flow
only for the largest rivers in the analysed system, for which typically a kinematic wave-based
routing is used, assuming that at the subcatchment level, the effect
of travel times in channels are negligible at the considered spatio-temporal resolution. This might
typically hold for hourly discharge simulation with subcatchments of a few 10–100

Traditionally, the range of aforementioned hydrological models is classified into (semi-)lumped and
(semi-)distributed

We propose the term of a spatially explicit hydrologic response (SEHR) model for any model that explicitly parameterizes both, spatial patterns of water storage evolution as well as the effect of geomorphology on the travel time of water having different spatial origins. We believe that the term spatially-explicit is more generic than the often used terms semi-lumped, semi-distributed or distributed model, which refer to specific set-ups in terms of spatial variability of state variables and of parameters.

Hereafter, we describe a simple catchment-scale hydrologic model developed at the Laboratory of
Ecohydrology (ECHO) of the Ecole Polytechnique Fédérale de Lausanne, SEHR-ECHO, that
explicitly accounts for the spatial variabilities in the runoff generation process and the
heterogeneity of the flow-paths within the catchment. The model builds on the geomorphic theory of
the hydrologic response

The general model concept is introduced in Sect.

The SEHR-ECHO model is composed of two main components (Fig.

The source areas are extracted from a digital elevation model (DEM) with the well-known Taudem
algorithm

Flow diagram of the precipitation–discharge computation. The grey boxes highlight the model output time series.

The precipitation–runoff module solves the mass balance equations at the source area
scale. This component is driven by precipitation, temperature and potential evaporation input
time series, which need to be properly provided at the source area scale. The choice of methods to
interpolate the observed input time series to this scale depends on the variable (precipitation,
temperature), on the application and on the simulation time step

The precipitation–runoff transformation module has the following key elements: interception and
re-evaporation of intercepted water, rainfall/snowfall separation, evolution of water stored in the
snowpack in solid form, evolution of the liquid water content of the snowpack and equivalent
precipitation (rain and meltwater)-runoff transformation. If a source area has partial glacier
cover, the runoff resulting from the glacier is computed separately (Fig.

Interception

Part of the intercepted water is assumed to re-evaporate during the same time step, limited by potential
evaporation

It is noteworthy that the above formulation assumes that interception is an instantaneous process, which takes place at timescales smaller than the simulation time step (i.e. sub-hourly). Only the evaporated water is subtracted from the incoming precipitation, which corresponds to a return of non-evaporated water as throughfall.

The estimation of the aggregation state of precipitation is based on a simple temperature threshold

The evolution of the water equivalent of the snowpack height

The snowmelt

Refreezing

The snowpack is assumed to have a certain water retention capacity

It is noteworthy that rainfall

The water fluxes

The partitioning of equivalent precipitation into surface runoff, fast and slow subsurface runoff
and transpiration resulting from water infiltration and percolation in the subsoil is performed via
a minimalist description of the soil moisture dynamics at the source area scale

It is noteworthy that this soil moisture dynamics equation, if forced with Poisson infiltration, can
be solved exactly for a number of cases and forms the basis of substantial analytic work on the
probabilistic properties of stream flow

The leakage

The transpiration rate is a linear function of relative soil moisture between the wilting point,

The infiltrated water corresponds to the equivalent precipitation from which direct surface runoff
is subtracted:

The water mobilized from the active layer

The linear reservoir equations for the fast and slow subsurface runoff thus read as

Note that

If a source area has a partial glacier coverage, ice is assumed to start melting if

The transport of the runoff components through the river network uses
a linear approach, assuming that most relevant nonlinear processes are
captured through the source area-scale precipitation–runoff transformation.
This assumption only holds for systems where flow velocity can be assumed to
be relatively constant in time (independent of discharge) and space. The
total discharge at the catchment outlet is obtained by convolution of each of
the fluxes

Sketch of the flow paths from a catchment with five source areas

The probability density functions of travel times,

The travel time distributions within channelled states

The simulated discharge at the catchment outlet becomes

The model requires temperature, precipitation and potential evaporation time series for each subcatchment and, for model calibration, at least one concomitant discharge time series observed at the catchment outlet. The numerical implementation uses a fixed time step and a fourth-order Runge–Kutta scheme to compute the soil moisture evolution. The other stores (fast and slow subsurface flux stores, solid and liquid snow stores) are solved with explicit time stepping, which is justified given that these stores have only one outflux, linearly dependent on the storage.

The provided code (see Sect. “Code availability”) is developed in Matlab R2010b. The parameterization
of each of the presented hydrological processes can easily be modified. The basic model structure
(passing of variables and parameters among functions) has been designed for an easy combination with
the now widely used optimization algorithms developed by

The physical parameters of SEHR-ECHO, which describe the physiographic catchment characteristics and
can be extracted from topographic data, are listed in Table

Model parameters describing the physiographic catchment characteristics. The units are given in generic terms (L: length).

Model parameters that have to be calibrated or estimated otherwise with indication of a priori values, a reference value for parameters that are not calibrated here and key references for further details. A total of 12 parameters are calibrated here. The maximum value of the reservoir coefficient is the numerical time step.

Location of the Dischmabach catchment within Switzerland

It is, in particular, recommended to relate the mean residence time

It might be tempting to derive the spatial variability of the active soil depth from soil production
theory

The saturated hydraulic conductivity can easily be distributed in space according to observed land
use and soil types; an example is discussed in Sect.

The Dischmabach catchment, located in the southeast of Switzerland near Davos
(Fig.

The subcatchments as well as the river network characteristics required to run the model (network
topology, river reach lengths) are identified with TauDEM Version 5

The temperature time series for each subcatchment is obtained through a linear interpolation of the
temperature observed at the Davos weather station (1594

Given the important heterogeneity of land use in this catchment, we distribute the saturated
hydraulic conductivity according to land use types, which are available from the Swiss land use
database at a resolution of 100

For each subcatchment, the saturated hydraulic conductivity is then obtained as

Impervious subcatchment areas are accounted for by setting the soil depth of the corresponding portions to 0 (in total 1.2 km

For the purpose of this paper, the model is calibrated on daily and hourly discharge with simple
Monte Carlo simulation: we draw a high number of random parameter sets in the a priori parameter
ranges and retain the best simulations with respect to the well-known Nash–Sutcliffe performance
criterion

For hydrological regimes with a strong annual discharge cycle, the above NSE value is not very
meaningful since any model that reproduces the annual cycle more or less will have a high NSE value

Given the insignificant role of Horton direct runoff in this environment, this runoff mechanism is
deactivated here (assuming infinite infiltration capacity). For all other processes, the a priori
parameter ranges are obtained based on existing literature (see Table

A similar scaling approach is also used to ensure that the degree-day factor for ice

For the Dischma catchment, a total of 12 model parameters have to be
calibrated, seven for the water input–runoff transform and five for the
glacier and snowmelt simulation. Here, these calibration parameters have been
estimated through simple Monte Carlo simulation to illustrate the main
features of the SEHR-ECHO model. Figure

Observed and simulated hydro-meteorological time series of model state variables for the
parameter set of Table

Time series of the streamflow components (direct surface flow, fast subsurface flow and
slow (deep) subsurface flow) corresponding to the parameter set of Table

The calibrated parameter values corresponding to the parameter set with the best NSE value at the daily time step for the calibration period. The performance criteria values of this set are:

The splitting between the three hillslope scale runoff generation processes
corresponds to the expected pattern: Fig.

This parameter set comes along with a number of sets that lead to equally good discharge simulations
for the reference performance criteria for the calibration period. The plots of NSE vs. NSE-log and
of NSE vs. the relative bias (Fig.

NSE vs. NSE-log for all simulated parameter sets during model calibration with daily time step (period 1981–1994), values for 100 parameter sets with best NSE values, values for the same parameter sets simulated over validation period (1993–2003) and values for the same parameter sets simulated at hourly time step over the validation period. The lines indicate the benchmark values for daily and hourly (broken line) time steps.

A notable aspect of this SEHR-Echo application is that a large number of the best-performing
parameter sets at the daily time step perform equally well during the calibration period
(1981–1992) and the validation period (1993–2000) and at the hourly time step
(Fig.

Predictions resulting from the 100 best parameter sets obtained under NSE for a daily time step (of the total 35 000 Monte Carlo simulations); top: simulation at the daily time step, bottom: simulation at the hourly time step (same parameter sets). The first half of the plot shows that the prediction limits are reasonably narrow, the reserved plotting order in the second half shows that the observations are well spanned – 79 % (daily) respectively 75 % (hourly) of the observed time steps fall into the range spanned by the simulations.

A comparable assessment of model performance at different timescales
without re-calibration is rarely reported in the literature. For an example,
see the work of

The above evidence, timescale independence and splitting between the runoff generation processes, are important hints that the model works well for the right reasons: the parameters play the role they have been designed for, rather than trying to mimic omitted runoff generation processes or to compensate for the lack of spatial differentiation of travel times, which might typically occur for lumped models.

This conclusion is also supported by the good identifiability of some of the model key parameters,
illustrated by the relatively peaky distributions in Fig.

Parameter distributions obtained for the best 100 parameter sets under NSE at a daily time
step. The

The question arises as to how far the geomorphology-based set-up and the routing scheme influence the
results. The model simulations are insensitive to hydrodynamic dispersion since its effect is
overruled by advection. Given the relatively short distances in this catchment (the longest travel
paths from a subcatchment to the outlet is 11

Sensitivity of the discharge simulation with respect to the in-stream flow velocity plotted
against the reference simulation with parameters of Table

Comparable precipitation–runoff models often either use a grid-based spatial discretization

This hypothesis is supported by the fact that if we use subcatchments divided into elevation bands,
we obtain a consistent improvement of the discharge simulation performance at the hourly time step
for the calibration and the validation period (Fig.

This paper presents a precipitation–runoff model that computes spatially explicit water fluxes at the ecosystem level and that can, thus, be used as a simulation tool for ecohydrologic applications requiring distributed discharge information. The model formulates the hydrologic response of a catchment as a convolution of the subcatchment-scale hydrologic flow processes with the river network, where the kernels account for the spatial arrangements of the subcatchments linked by the river network. The hydrologic response accommodates directly any direct information on observable physiographic catchment characteristics such as in-stream flow path lengths or subcatchment area as a proxy for subsurface residence time scaling. Remaining model parameters are calibrated on observed discharge. This spatially explicit parameterization confers the model transferability across timescales, as has been demonstrated in this paper based on a cryosphere-dominated catchment from the Swiss Alps where, due to the steep topography, travel times in unchannelled areas are dominating in-stream travel times. The main focus of this paper was on discharge simulation. Including appropriate formulations of subcatchment-scale mass transformation processes, the general modelling framework can be extended to transport processes.

Comparison of model performance of the subcatchment-based model set-up (23 subcatchments)
to elevation band set-ups. Top row: comparison to 10 elevation bands (see
Fig.

A fully annotated Matlab version of the model
is available at

The research of the first author of this paper has been funded through an Ambizione grant of the Swiss National Science Foundation, SNSF, number PZ00P2_147366. The last author acknowledges funding from the ERC Advanced Grant RINEC 22761 and the SNSF grant 200021_135241. The meteorological data were obtained from MeteoSwiss and the discharge data from the Swiss Federal Office for the Environment. Edited by: R. Marsh