Development of neuro-fuzzy models to account for temporal and spatial variations in a lumped rainfall–runoff model
Introduction
Mathematical models are widely used in water resources applications despite considerable difficulties arising from catchment heterogeneity, strong non-linearity in its response to precipitation and uncertainties in parameter estimation. In many practical cases, simple lumped models of either the black-box or conceptual type often perform adequately, and compare well with the more complex distributed-models, particularly for flood modelling in small catchments and for behaviour within the range of the data used to calibrate its parameters (Beven, 2000). Larger catchments can be modelled by associating different sub-lumped-models with different spatial units within the catchment (e.g. Chen and Adams, 2006, Marechal and Holman, 2005, Ajami et al., 2004). Similarly different sub-models could be used to represent the various temporal patterns in the system’s response (e.g. Shamseldin and O’Connor, 1996, Ahsan and O’Connor, 1994, Kachroo and Natale, 1992). The success of this approach is primarily because of its ability to capture some of the non-linearity in the catchment behaviour resulting from its spatial heterogeneity and time-varying character. The choice of a suitable lumped model for use in each of the sub-catchments is critical to its success and practicality. It should have a small number of parameters to reduce the total number to be estimated for the combined model thereby reducing the computational requirements. This also is likely to reduce potential problems caused by model over-parameterisation, such as ill-conditioning (Bruen and Dooge, 1992), or equifinality (Beven, 1993) in which a number of different combinations of parameter values give similar model fits and so a single optimal parameter set is difficult to determine.
One obvious symptom of non-linearity is the very different responses of the catchment to different flow regimes. The direct way of dealing with this is to build the complicated non-linear physical relationships into the model. An alternative is to have a different, but simple, sub-model for each different flow regime. For instance, Chen and Adams (2006) used a number of sub-models to simulate spatial variation in the rainfall–runoff relationship. The estimated runoffs from all sub-models were combined together using an artificial neural network to estimate the total runoff. Moreover, they investigated the suitability of using sub-models of three different conceptual models including the Xinanjiang model (Zhao and Liu, 1995), the soil moisture accounting and routing (SMAR) model (O’Connell et al., 1970) and the tank model (Sugawara, 1995). A significant improvement was obtained when using different sub-models compared to a single lumped model. Kachroo and Natale (1992) also used three sub-models using the same simple linear model (SLM) (Nash and Foley, 1982) structure with different parameter sets to represent the response during low, medium and high flow regimes. Although the total number of parameters is tripled, all of the sub-lumped-model parameters could be calibrated using the least-squares criterion. The choice of which of the sub-model to use at each time step is guided by a type of wetness index taken as the current observed discharge in this case. When no observed discharge is available at the current time step, (e.g. when either (a) simulating or (b) forecasting beyond a single time step) the discharge simulated by the lumped model is used for this index. The combined-sub-lumped models have shown significant improvement over the lumped one.
Building on these efforts to improve the performance of combined-sub-lumped-models, this paper reports the investigation of a fuzzy method proposed to combine sub-lumped-models of two types, black box model and conceptual model. The former is the simple linear model (SLM) (Nash and Foley, 1982) and the latter is the soil moisture accounting and routing model (SMAR) (O’Connell et al., 1970). Each of the two models has been included into a framework of a special type of neuro-fuzzy model (NFM), called an adaptive neuro-fuzzy inference system. The first objective is to produce a combined-lumped-model better able to represent the spatial and temporal variability of the catchment’s response to rainfall. The resulting NFM addresses the temporal variations in response by using a number of sub-models for the SLM and the SMAR models for different regimes (e.g. separate sub-models for floods and low flow situations). Each of the sub-models describes a particular feature in the temporal pattern of the catchment’s response. The NFM is assessed by applying it to eleven different catchments from around the world. In the second part of this study an NFM (for the SLM and for the SMAR model) is developed that is able to identify homogenous spatial units within a catchment on which the sub-models can be based. In this, the NFM structure of the first part is further coupled to a subtractive fuzzy clustering algorithm (Vernieuwe et al., 2005) to determine the homogeneous spatial units using a number of spatial variables specified on a catchment grid. Finally, using one of the catchments which has the required spatial database, namely, the Brosna, the NFM developed in the second part of the study is tested and its results compared with those of the corresponding model developed in the first part of the study.
The proposed method is described in Section “Interpretation of the proposed sub-models combination method” and the NFM is reviewed in Section “Neuro-fuzzy model (NFM)”. The two rainfall–runoff models, SLM and SMAR, are briefly described in Sections “Simple linear model” and “Soil moisture accounting and routing (SMAR) model” respectively. In Section “Description of the proposed NFM”, a detailed description is given of the two NFMs applied in this study. In the final Sections “Results” and “Conclusions”, the results of the NFM applications are presented and conclusions are drawn. Suggestions for further work are added in Section “Further work”.
Section snippets
Interpretation of the proposed sub-models combination method
The method of sub-model combination used in this study is different from the flood forecast model combination methods proposed in earlier work (e.g. Shamseldin et al., 1997, See and Openshaw, 2000, See and Openshaw, 1999, Xiong et al., 2001, Abrahart and See, 2002, Coulibaly et al., 2005, Fenicia et al., 2007). In those methods, a number of models each with different internal structures were individually applied to the entire study catchment and their simulated outputs were combined. Each model
Neuro-fuzzy model (NFM)
The neuro-fuzzy model (NFM) used in this study implements the Takagi–Sugeno fuzzy approach (Takagi and Sugeno, 1985) to obtain a direct crisp value for the output variable(s) from fuzzy input variable(s). Jacquin and Shamseldin (2006) explored the application of Takagi–Sugeno fuzzy inference systems to rainfall–runoff modelling. They developed two different fuzzy models to account for the non-linearity in the catchment response due to both antecedent catchment wetness and seasonality. Vernieuwe
Simple linear model
The simple linear model (SLM) was introduced by Nash and Foley (1982) as a naı¨ve, benchmark, model against which the performance of more substantive and sophisticated rainfall–runoff models could be compared. The SLM assumes a linear time invariant relationship between rainfall and discharge, expressed by a convolution summation relation. Here, an additional term has been added in order to include, albeit crudely, losses due to evaporation in the modelling, giving the equation:
Soil moisture accounting and routing (SMAR) model
O’Connell et al. (1970) developed a quasi-physical rainfall–runoff model known as the layers model but later on renamed the soil moisture accounting and routing (SMAR) model. This model consists of two complementary components. The first implements a water balance (the soil moisture accounting procedure) between rainfall, evaporation, runoff, and simulated soil storage for each time step. The second routes the calculated runoff to the catchment outlet, taking account of attenuation and wave
Description of the proposed NFM
Ozelkan and Duckstein (2001) described any catchment model as a system composed of sub-modules to represent the sub-elements of this modelled system coupled together in order to produce a synergic effect reflected at the output of the system. The representation of the catchment model in this modal structure is equivalent to the branching structure in an algorithm flow diagram resulting from ‘IF–THEN’ fuzzy rules (Gupta and Sorooshian, 1983). In the present work, the aim is not to utilise the
Results
The key issue is to determine whether the introduction of combined sub-models to account for temporal or spatial pattern variations improves the simulation compared to that of a single lumped catchment model. First, the results corresponding to the lumped case (case 1 in Table 1 for NFM_T, and cases 1 of all combination alternatives in Table 4 for NFM_S) are calculated. These provide a baseline to be used in assessing the second set of results corresponding to the best combined case. In each
Conclusions
In this study, the NFM has been proposed to account for spatial and temporal variations in modeling the rainfall–runoff relationship. The proposed procedure was implemented with two simple lumped models, SLM and SMAR. For each model two scenarios (NFM_T and NFM_S) were used to construct sub-models to address the temporal and spatial pattern variations, respectively. In the NFM_T scenario, the two models NFM_T_SLM and NFM_T_SMAR, were applied to 11 catchments from around the world. A split
Further work
This work has shown that combinations of relative simple models can extend their ability to model a range of catchment behaviour without requiring fully distributed time-varying, physically-based models. While the combination approach has proved useful in our Brosna catchment, it should be applied to other catchments with a wider range of climatic variation and conditions to test its generality. In addition, it should be possible to extend the approach to modelling other types of data,
Acknowledgement
This study was carried out at the Centre for Water Resources Research at UCD, as part of an Irish NDP/ERTDI project funded through the Environmental Research Centre of the Environmental Protection Agency in Ireland.
References (38)
- et al.
A simple non-linear rainfall–runoff model with a variable gain factor
Journal of Hydrology
(1994) Prophecy, reality and uncertainty in distributed hydrological modelling
Advances in Water Resources
(1993)- et al.
Integration of artificial neural networks with conceptual models in rainfall–runoff modelling
Journal of Hydrology
(2006) - et al.
Development of rainfall–runoff models using Takagi–Sugeno fuzzy inference systems
Journal of Hydrology
(2006) - et al.
Non-linear modeling of the rainfall–runoff transformation
Journal of Hydrology
(1992) - et al.
Development and application of a soil classification-based conceptual catchment-scale hydrological model
Journal of Hydrology
(2005) - et al.
River flow forecasting through conceptual models. Part 1. A discussion of principles
Journal Hydrology
(1970) - et al.
River flow forecasting through conceptual models. Part II: The Brosna catchment at Ferbane
Journal of Hydrology
(1970) - et al.
Fuzzy conceptual rainfall–runoff model
Journal of Hydrology
(2001) - et al.
A nearest neighbour linear perturbation model for river flow forecasting
Journal of Hydrology
(1996)
Methods for combining the outputs of different rainfall–runoff models
Journal of Hydrology
Application of an empirical infiltration equation in the SMAR conceptual model
Journal of Hydrology
Comparison of data-driven Takagi–Sugeno models of rainfall-discharge dynamics
Journal of Hydrology
A non-linear combination of the forecasts of rainfall–runoff models by the first-order Takagi–Sugeno fuzzy system
Journal of Hydrology
Multi-model data fusion for river flow forecasting; an evaluation of six alternative methods based on two contrasting catchments
Hydrology and Earth System Sciences
Calibration of a semi-distributed hydrologic model for streamflow estimation along a river system
Journal of Hydrology
Nonlinear flood routing with multilinear models
Water Resources Research
Uniqueness of place and process representations in hydrological modeling
Hydrology and Earth System Sciences
Time Series Analysis: Forecasting and Control
Cited by (17)
A novel approach for the prediction of the incipient motion of sediments under smooth, transitional and rough flow conditions using Geno-Fuzzy Inference System model
2019, Journal of HydrologyCitation Excerpt :Obviously, a need to extend this approach for all hydrodynamic flow conditions to contribute to the current knowledge is present. Knowing that Neuro-Fuzzy approach which is a powerful combination of Artificial Neural Networks (ANN) and Adaptive Neural Fuzzy Inference System (ANFIS) has been successfully used in hydrology, hydraulics and other water resources fields such as rainfall-runoff modeling (Şen and Altunkaynak, 2006; Nasr and Bruen, 2008; Remesan et al., 2009), flood hazard risk mapping ( Sonmez and Bizimana, 2018), modeling of lake water level fluctuations (Altunkaynak and Şen, 2007; Bizimana et al., 2016), stream flow prediction (Özger, 2009), model for equilibrium scour in sediment transport (Uyumaz et al., 2006). Moreover, Tütmez et al. (2007) who compared the spatial interpolation of mechanical properties of rocks by using Mamdani and Sugeno fuzzy models, concluded that Sugeno fuzzy model was better than Mamdani fuzzy model in investigating the randomness presented in mechanical properties of rocks under nonlinear distribution.
A geomorphology-based ANFIS model for multi-station modeling of rainfall-runoff process
2013, Journal of HydrologyCitation Excerpt :The efficiency of spatiotemporal ANN was compared with two hybrid neural-geostatistics and multivariate time series-geostatistics models and it was found that the ANNs could provide accurate predictions (Nourani and Ejlali, 2012). Hence, few attempts have been made in hydrology to use ANN and ANFIS for spatiotemporal modeling (see e.g., Nasr and Bruen, 2008). Furthermore, because of the fluctuation, periodic pattern and paucity, involved in the rainfall and runoff time series, the data may not be employed without an appropriate data preprocessing.
Derivation of a fuzzy national phosphorus export model using 84 Irish catchments
2013, Science of the Total EnvironmentA novel application of a neuro-fuzzy computational technique in event-based rainfall-runoff modeling
2010, Expert Systems with ApplicationsImplementation on the evolutionary machine learning approaches for streamflow forecasting: case study in the Seybous River, Algeria
2020, Journal of Korea Water Resources Association