Improvement of consistency in model performance and parameter estimates

To improve consistency in hydrologic models performance several approaches have been proposed in the literature: increasing parameters temporal transferability, ensemble model weighting, time variable parameterisations and enhancement of model structure.

Model calibration improvement

Hartmann and Bardossy (2005) proposed a linear combination of Nash-Sutcliffe efficiency coefficients (NSE) calculated not only for daily flows, but also for flows averaged over longer periods (e.g. weeks, months, seasons, years) and for the transformed flows (e.g. square root transformation). They carried out DSST with calibration over wet period and evaluation in dry one to appraise several calibration strategies – combinations of flow series according to which NSE was calculated. Parameter estimates obtained with NSE with daily and annual flows resulted in high model performance in terms of smaller decrease in NSE and flow bias in the evaluation period.

Gharari et al. (2013) advocated multi-objective calibration over several sub-periods of equal length, resulting in several Pareto fronts. They assumed that minimisation of the Euclidian distance to all sub-period Pareto fronts would result in the Pareto sets (so-called Minimum Distance Pareto Front – MDPF) that would perform consistently. The performance of MDPF over the sub-periods of the short testing period was almost as good as the performance of the Pareto fronts obtained over each sub-period. The MDPF performance over the long testing period was consistent, although suboptimal in some years compared to the Pareto front obtained over the full calibration period.

Conditional parameterisations

To obtain more consistent model performance some researchers applied time variable parameterisations (e.g. on monthly or seasonal basis). To obtain these conditional parameterisations, the model parameters are optimised in various climatic conditions (e.g.

wet or dry periods).

Fenicia et al. (2009) assumed that changes in catchment properties would reflect in changes in the model parameters. They tried to explain a rainfall-runoff anomaly in the Meuse catchment behaviour (i.e. decrease in runoff from 1930 to 1965) by varying the model parameters in time. They calibrated the conceptual FLEX model with 10 free parameters using the GLUE method over consecutive 4-year long periods by employing

indicated that two time variable parameters could explain the anomaly: namely, time to peak and the parameter relating changes in forest transpiration to the forest age. They attributed the decrease in the former parameter to the catchment urbanisation and river engineering works. Variability in the latter parameter was attributed to forest rotation i.e.

changing age of the forests and consequently ET.

Muleta (2012) carried out a sensitivity analysis (SA) of the SWAT model parameters over wet and dry seasons and in entire calibration period. The wet and dry seasons were selected according to mean monthly runoff. The SA revealed that sensitivity of some parameters related to soil conductivity, evaporation and interception capacity changes between wet and dry seasons. He optimised the principal model parameters and obtained two version of the model. The first version comprised temporally invariant parameters, while the parameters of the second one varied over the seasons. Two versions of the model were evaluated by conducting SST. The model with varying parameters outperformed its counterpart in most of the evaluation periods.

Choi and Beven (2007) calibrated the TOPMODEL using the GLUE framework and various objective functions. Behavioural parameter sets were updated according to model performance over the years after the calibration period (globally conditioned models).

There were numerous behavioural parameter sets in individual years, but only a few sets were behavioural over the full record period. To account for seasonal shifts in runoff generations mechanism, they calibrated the model in a dynamic manner (multi-period conditioned models) over 15 fuzzy clusters of time. The clusters were sampled according to precipitation, precipitation variance, maximum daily precipitation and PET.

Behavioural parameters’ posterior pdfs varied considerably over the clusters and none of the parameter sets was behavioural over all clusters. Minimum number of the behavioural sets was obtained over dry clusters due to poor model performance in dry periods, which was attributed to the model structural deficiencies. In the evaluation period, the multi-period conditioned model resulted in significantly higher percentage of flow observations within the prediction band than the globally conditioned one.

Zhang et al. (2011) calculated six aridity indices for each water year of the hydrologic record. They performed the principal component analysis (PCA) of the indices to reduce redundancy in data since all aridity indices are based on daily temperatures. The fuzzy

C-means clustering method was applied to the principal components resulting in five clusters. Every year was assigned to a particular cluster and split into the warm and cold seasons. Distributed SWAT model was calibrated in every season over all clusters (i.e.

ten model calibrations) by employing the SCE calibration algorithm. The number of free parameters was reduced after the sensitivity analysis prior to the model calibration. The results in the calibration and evaluation periods were compared to the results of the model calibrated in the full record period. The “multi-period” model outperformed the “single-period” model in both periods in terms of NSE and flow bias. In addition, “multi-“single-period”

model resulted in narrower prediction intervals and in larger percentage of observation encompassed by the prediction band.

Model ensemble and model averaging

Oudin et al. (2006) applied dynamic weighting of two model parameterisations obtained with NSE calculated with flows and log-transformed flows. . They examined four different weighting strategies: (1) equal weights, (2) sinusoidal weights, (3) weight that is equal to normalised soil moisture (form 0 to 1) and its complement, and (4) weights calculated using the nonlinear functions of simulated soil moisture. The fourth weighting strategy resulted in the highest model performance.

Weighting of the outputs from different hydrologic models within Hierarchical Mixtures of Experts framework (HME) is employed by Marshall et al. (2007). HME is based on their individual models and gating functions that control weighting, i.e. probability of using the individual models. The gating function relates probability of using a model with the predictor variables, such as antecedent precipitation. HME allows that model with the same structure but different parameters have different weights – probabilities. Marshall et al. (2007) used HME with parsimonious models (3 free parameters) and simple gating functions. The results obtained by employing HME with three models outperformed those of the single model.

over previous computational time step. Model ensembles obtained in this way outperformed individual models in calibration and evaluation periods.

Model structure improvement

Time variability of model parameters is assumed by de Vos et al. (2010) to be due to model structural inadequacy. These authors calibrated the lumped conceptual model HyMod in (1) single- (traditional calibration) and multi-objective manner over entire calibration period, and (2) over 12 clusters of time (dynamic calibration). The clusters are selected according to daily precipitation, 10-day moving average of precipitation and soil moisture simulated by the GR4J model. They successively improved the model structure by introducing a parameter for correcting the observed precipitation rates, upgrading linear reservoirs to the nonlinear ones, and introducing the routing function to the model.

The corrections to the model are made so that traditionally calibrated model performs as well as the dynamically calibrated one.

Efstratiadis et al. (2014) enhanced the lumped hydrologic DM0 model to account for catchment urbanisation. They proposed two alternatives: (1) the liner reservoir coefficient for direct runoff simulation which was proportional to the share of urbanised areas (model DM1), and (2) application of a distributed version of the model DM2 which involved Hydrologic Response Units (HRUs). In the distributed model the catchment is delineated in two HRUs. One HRU included urbanised and the other non-urbanised areas in the catchment. Different parameter sets are assigned to the HRUs. They tested the models following the protocol presented by Thirel et al. (2014). The performance of the models DM0 and DM1 was similar, while the distributed model DM2 performed considerably better.

1.5.3. Model transferability in time and assessment of the climate change impact