Many environmental systems models, such as conceptual rainfall-runoff models,
rely on model calibration for parameter identification. For this, an observed
output time series (such as runoff) is needed, but frequently not available
(e.g., when making predictions in ungauged basins). In this study, we provide
an alternative approach for parameter identification using constraints based
on two types of restrictions derived from prior (or expert) knowledge. The
first, called

Environmental systems models, such as conceptual rainfall-runoff (CRR)
models, are abstract simplifications of real system behavior. Often, the
parameters in such models cannot be specified through direct measurements of
physical properties of the system. Further, even when a parameter is related
to measurable quantities, its value in the model typically represents an
integrated value over a much larger scale than the measurement scale. For
this reason, such models typically rely upon calibration (tuning to match
system input–output behavior for a given historical data period) to ensure
satisfactory performance when applied to specific hydrological systems of
interest

In the case of CRR, parameter values are typically specified through a
process of calibration that seeks to match the model runoff simulations to
observed hydrographs by the use of an objective function (e.g., root mean
square error, RMSE). Expert knowledge is brought to bear implicitly, by the
prior selection of the model and the specification of parameter ranges that
define the prior parameter space. Recently, several studies have tested
strategies that relate the parameters of CRR models to catchment physical
characteristics

For example,

In a complementary direction, the use of multiple objective functions or
multiple system responses for calibration

While the aforementioned studies have demonstrated that incorporation of expert and a priori knowledge can help improve the realism of models, to our knowledge, no systematic strategy has been presented in the literature for constraining the model parameters to be consistent with the patchy understanding of a modeler regarding how the real system might work. Part of the difficulty in doing this is that expert knowledge may not directly translate to quantifiable relationships (e.g., between catchment physical characteristics and model parameters) rather, it may consist of conceptual understanding about consistency relationships that must exist among modeled state variables and/or fluxes, as well as, among various model parameters. For example, the geology of a given catchment may suggest that the catchment response during intense rainfall events is characterized by a slow responding groundwater component accompanied by fast responding Hortonian overland flow. In this case, any model result that implies peak flows are composed of a strong groundwater response should be discarded or should be given low importance.

An example of such an approach toward modeling was mentioned by

In this study, we present a

The observed hydrograph and the 95 % uncertainty interval of the
modeled hydrograph derived from the complete set of

In the companion paper,

A set of parameter and process constraints is defined for each of the three
landscape entities, based on the expert knowledge. Parameter constraints are
considered to be a priori because they can be imposed without
actually running the model, while process constraints can only be imposed
after a model is run with selected parameter sets. The number of constraints
may vary from model to model, and here depends on the model complexity. For
example all of the pre-defined constraints (see the following section for
more detail) can be applied to the most complex model, FLEX

The results show that the

As mentioned earlier, two types of model constraints can be distinguished:
a priori constraints applicable to model parameters (i.e., parameter constraints)
and a posteriori constraints on
model states and fluxes

Parameter constraints provide information regarding the relationships between
parameters of the same process that correspond to different spatial
components (or units or grid cells) of a (semi-) distributed model. Such
constraints can be expressed by inequality (or equality) constraints; for example

Process constraints provide comparative information regarding the fluxes
and/or states of a model at each time step, or integrated over some specific
time period. Examples of such constraints include the following:

As an illustration, one can compare the transpiration fluxes from different spatial entities of a (semi-) distributed model. For example for two regions having similar soil type and aspect, the region with smaller normalized difference vegetation index (NDVI) is expected to transpire at a lower rate.

It is worth noting that in either case, parameter sets that satisfy the constraints are not conditional on the information provided by observations (or measurements) of the output response of the system (e.g., the runoff hydrograph), and these can therefore be determined without resorting to model calibration. Moreover, parameter sets that satisfy all of the constraints (within some acceptable range) can provide insights into how the real system can be expected to behave, assuming that it corresponds to the expert's perception of realistic (behavioral) system properties and dynamics.

Unfortunately, the use of available evolutionary algorithms to search for parameter sets that satisfy such constraints is complicated by the non-convex and potentially non-continuous parameter search space that results. In the following section, we propose a stepwise search strategy that can be used to identify behavioral parameter sets fulfilling the constraints imposed by expert knowledge.

The constraint-based search (CBS) algorithm is based on a simple parameter sampling approach to identify the parts of the feasible parameter space that satisfy the set of constraints as discussed in the previous section. At each step, the algorithm tries to generate new parameter sets that satisfy the parameter constraints, while only violating the process constraints to an acceptable level. This level could be set-up based on the desired model performance. The process of search continues until all of the parameter sets generated properly satisfy the set of imposed process constraints.

In the following description

The search algorithm is as follows; let

In the
case that

Note that any member of set

The final set

A conceptual illustration of possible positions of newly generated
parameter sets based on parameter sets randomly drawn from

A conceptual illustration of stepwise search for the parameter space satisfying all of the parameter and process constraints.

The models structure of FLEX

The “true” values for each of the design constraints
(corresponding to

Normalized parameter plot for FLEX

Note that the set

A synthetic case study was designed to illustrate the efficiency of the
proposed constraint-based search algorithm. First, the lumped model
FLEX

A set of (parameter and process) constraints was then designed based on the
model simulations (fluxes and states) provided by this best parameter set
(

The proposed algorithm was then applied to search for parameter sets that
satisfy the constraints mentioned in Table 1. The initial sample size
(

The parameter sets identified by the proposed search algorithm were then
compared with the best parameter set (

95 % uncertainty intervals of simulated stream flows using
parameter sets satisfying different numbers of process constraints. Darker
colors indicate the uncertainty intervals which satisfy more of the process
constraints. The hydrograph generated by

Percentages of samples generated at each stage that satisfy a given
number of process constraints. White indicates no satisfied constraints, and
progressively darker colors indicate increasing numbers of satisfied
constraints.

A similar comparison (Fig. 6) was conducted for the modeled hydrographs
associated with these parameter sets. The hydrograph for the best performing
parameter set (

Overall the search algorithm generated and evaluated 102 106 parameter sets to find 8000 feasible solutions that satisfy all of the nine constraints imposed, which corresponds to approximately 8 % efficiency. In comparison, when a conventional Monte-Carlo sampling approach was applied using 102 106 samples, none of the samples were found to be able to satisfy all of the constraints, while only two of the samples were able to satisfy at least 7 of the 9 process constraintsimposed. Clearly, the proposed search algorithm is capable of relatively rapid convergence towards the region of the parameter space where all of the constraints are satisfied.

Figure 7 illustrates how quickly the search algorithm is able to locate the
behavioral parameter sets. It depicts the percentages of generated samples
satisfying a given number of process constraints at each step of the search.
Darker colors are used to indicate the proportions of parameter sets that
satisfy progressively more of the process constraints (white indicates none
of the parameter constraints being satisfied). The initial sample of
50 000 parameter sets, see region in Fig. 7a, consists of samples
drawn uniformly from the entire parameter space, of which less than 10 %
satisfy any of the imposed constraints, and only a very few satisfy 1, 2 and
3 constraints (progressively darker shades of grey). Each progressive step
(see region in Fig. 7b) then consists of

Of course, in this illustrative case study, the constraints were specifically designed to guarantee that the observed hydrograph corresponding to the “best” performing parameter set lie within a predetermined feasible space. In principle the constraints can be specified without recourse to the information contained in the discharge time series (as discussed earlier). The main purpose of this synthetic case study was to illustrate the capability of the proposed CBS algorithm to efficiently locate behavioral parameter sets that satisfy user-specified a priori parameter and a posteriori process constraints.

One of the most challenging tasks in the development of complex conceptual
hydrological models for simulation of catchment responses to inputs is the
realistic specification of parameter values. We have presented a
constraint-based search strategy that facilitates the incorporation of expert
knowledge (i.e., the modeler's perception of catchment behavior and
characteristics) into the parameter specification process. Because the CBS
algorithm does not require observational data regarding the target system
output (e.g., runoff) it can provide a way forward for better prediction in
ungauged basins in absence of streamflow (or other system output) data for
model calibration. As constraints are much easier for understand, rather than
parameters, when discussing system behavior, they can potentially be used as
an efficient tools to bridge the gap in the dialogue between modelers and
experimentalists. Further, the approach can help to provide behaviorally more
conceptually realistic parameter sets when used in conjunction with model
calibration. Future study may apply the proposed CBS algorithm to identify
behavioral parameter sets for different kind of hydrologic models in
different regions, and compare the results with other existing parameter
specification methods and algorithms. A Matlab

Shervan Gharari is funded during his PhD program by Fonds National de la Recherche (FNR) of Luxembourg with Aides à la Formation-Recherche (AFR) project number of 1383201. Mojtaba Shafiei is partially funded by Iran's Ministry of Science, Research and Technology as exchange researcher at Delft University of Technology. Edited by: F. Tian