Identification and Representation of State Dependent Non-linearities in Flood Forecasting Using the DBM Methodology

  • Keith J. Beven
  • David T. Leedal
  • Paul J. Smith
  • Peter C. Young


This paper addresses the issue of identifying a state dependent input nonlinearity in a Data Based Mechanistic (DBM) flood forecasting model based on the data rather than some prior conceptualisation of nonlinearity in the system response. Four forms of nonlinear function are presented. A power law may be useful when the input non-linearity is simple. The Radial Basis Function (RBF) network method is appropriate for systems that exhibit well defined but complex input non-linearities. The Piecewise Cubic Hermite Data Interpolation (PCHIP) method also provides the flexibility to map complex input non-linearity shapes while providing the ability to maintain a natural curve. Overfit to the calibration data is a risk in both RBF and PCHIP methods when a large number of knots are used. The Takagi-Sugeno Fuzzy Inference method, together with interactive tuning, provides an alternative approach that allows human-in-the-loop interaction during the parameter estimation process but is not optimal in any statistical sense. Future work will explore the use of these methods with continuous time transfer functions and optimisation of the nonlinear function at the same time as the transfer function.


Radial Basis Function Radial Basis Function Network Flood Forecast Linear Transfer Function Data Assimilation Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was carried out as part of RPA9 and SWP1 of the Flood Risk Management Research Consortium (FRMRC) phases 1 and 2. The principal sponsors of FRMRC are: the Engineering and Physical Sciences Research Council (EPSRC) in collaboration with the Environment Agency (EA), the Northern Ireland Rivers Agency (DARDNI), the United Kingdom Water Industry Research (UKWIR) Organisation, the Scottish Government (via SNIFFER), the Welsh Assembly Government (WAG) through the auspices of the Defra/EA, and the Office of Public Works (OPW) in the Republic of Ireland. For details of the FRMRC, see


  1. 1.
    Beven, K.: Rainfall-Runoff Modelling: The Primer. Wiley, New York (2001) Google Scholar
  2. 2.
    Colgan, L., Spence, R., Rankin, P.: The cockpit metaphor. Behav. Inf. Technol. 14(4), 251–263 (1995) CrossRefGoogle Scholar
  3. 3.
    Cunge, J.A.: On the subject of a flood propagation computation method (Muskingum method). J. Hydraul. Res. 7(2), 205–230 (1969) CrossRefGoogle Scholar
  4. 4.
    Dooge, J.C., Strupczewski, W.G., Napiorkowski, J.J.: Hydrodynamic derivation of storage parameters of the Muskingum model. J. Hydrol. 54(4), 371–387 (1982) CrossRefGoogle Scholar
  5. 5.
    Fritsch, F.N., Carlson, R.E.: Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17, 238–246 (1980) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Leedal, D., Beven, K.J., Young, P.C., Romanowicz, R.J.: Data assimilation and adaptive real-time forecasting of water levels in the Eden catchment, UK. In: Samuels, P., Huntington, S., Allsop, W., Harrop, J. (eds.) Flood Risk Management Research and Practice. Taylor and Francis, London (2008) Google Scholar
  7. 7.
    Lees, M., Young, P.C., Beven, K.J., Ferguson, S., Burns, J.: An adaptive flood warning system for the river Nith at Dumfries. In: White, W.R., Watts, J. (eds.) River Flood Hydraulics. Institute of Hydrology, Wallingford (1994) Google Scholar
  8. 8.
    Nash, J.E.: A note on the Muskingham flood routing method. J. Geophys. Res. 64, 1053–1056 (1959) CrossRefGoogle Scholar
  9. 9.
    Pappenberger, F., Beven, K.J., Hunter, N., Gouweleeuw, B., Bates, P., de Roo, A.: Cascading model uncertainty from medium range weather forecasts (10 days) through a rainfall-runoff model to flood inundation predictions within the European flood forecasting system (EFFS). Hydrol. Earth Syst. Sci. 9(4), 1430–1449 (2005) CrossRefGoogle Scholar
  10. 10.
    Park, J., Swandberg, I.W.: Universal approximation using radial-basis-function networks. Neural Comput. 3(2), 246–257 (1991) CrossRefGoogle Scholar
  11. 11.
    Pielke, R.A. Jr., Pielke, R.A. Sr.: Hurricanes: Their Nature and Impacts on Society. Wiley, New York (1997) Google Scholar
  12. 12.
    Ratto, M., Young, P.C., Romanowicz, R., Pappenberger, F., Saltelli, Pagano A.: Uncertainty, sensitivity analysis and the role of data based mechanistic modeling in hydrology. Hydrol. Earth Syst. Sci. 11, 1249–1266 (2007) CrossRefGoogle Scholar
  13. 13.
    Romanowicz, R.J., Young, P.C., Beven, K.J.: Data assimilation and adaptive forecasting of water levels in the river Severn catchment, United Kingdom. Water Resour. Res. 42, W06407 (2006) CrossRefGoogle Scholar
  14. 14.
    Romanowicz, R.J., Young, P.C., Beven, K.J., Pappenberger, F.: A data based mechanistic approach to nonlinear flood routing and adaptive flood level forecasting. Adv. Water Resour. 31(8), 1048–1056 (2008) CrossRefGoogle Scholar
  15. 15.
    Sherman, L.K.: Streamflow from rainfall by the unit-hydrograph method. Eng. News-Rec. 108, 501–505 (1932) Google Scholar
  16. 16.
    Smith, P., Beven, K.J., Tych, W., Hughes, D., Coulson, G., Blair, G.: The provision of site specific flood warnings using wireless sensor networks. In: Samuels, P., Huntington, S., Allsop, W., Harrop, J. (eds.) Flood Risk Management Research and Practice. Taylor and Francis, London (2008) Google Scholar
  17. 17.
    Young, P.C.: Recursive approaches to time-series analysis. Bull. Inst. Math. Appl. 10, 209–224 (1974) Google Scholar
  18. 18.
    Young, P.C.: Recursive Estimation and Time-Series Analysis. Springer, Berlin (1984) MATHGoogle Scholar
  19. 19.
    Young, P.C.: Time variable and state dependent modelling of nonstationary and nonlinear time series. In: Subba Rao, T. (ed.) Developments in Time Series Analysis, pp. 374–413. Chapman and Hall, London (1993) Google Scholar
  20. 20.
    Young, P.C.: Data-based mechanistic modelling and validation of rainfall-flow processes. In: Anderson, M.G., Bates, P.D. (eds.) Model Validation: Perspectives in Hydrological Science, pp. 117–161. Wiley, Chichester (2001) Google Scholar
  21. 21.
    Young, P.C.: The identification and estimation of nonlinear stochastic systems. In: Mees, A.I. (ed.) Nonlinear Dynamics and Statistics, pp. 127–166. Birkhäuser, Boston (2001) Google Scholar
  22. 22.
    Young, P.C.: Advances in real-time flood forecasting. Philos. Trans. R. Soc. Lond. A 360(1796), 1433–1450 (2002) CrossRefGoogle Scholar
  23. 23.
    Young, P.C., Beven, K.J.: Computation of the instantaneous unit hydrograph and identifiable component flows with application to two small upland catchments comment. J. Hydrol. 129(1–4), 389–396 (1991) CrossRefGoogle Scholar
  24. 24.
    Young, P.C., Beven, K.J.: Data-based mechanistic (DBM) modelling and the rainfall-flow nonlinearity. Environmetrics 5, 335–363 (1994) CrossRefGoogle Scholar
  25. 25.
    Young, P.C.: Top-down and data-based mechanistic modelling of rainfall-flow dynamics at the catchment scale. Hydrol. Process. 17, 2195–2217 (2003) CrossRefGoogle Scholar
  26. 26.
    Young, P.C., Garnier, H.: Identification and estimation of continuous-time data-based mechanistic (DBM) models for environmental systems. Environ. Model. Softw. 21(8), 1055–1072 (2006) CrossRefGoogle Scholar
  27. 27.
    Young, P.C., Castelletti, A., Pianosi, F.: The data-based mechanistic approach in hydrological modelling. In: Castelletti, A., Sessa, R.S. (eds.) Topics on System Analysis and Integrated Water Resource Management, pp. 27–48. Elsevier, Amsterdam (2007) Google Scholar
  28. 28.
    Young, P.C.: Real-time updating in flood forecasting and warning. In: Pender, G.J., Faulkner, H. (eds.) Flood Risk Science and Management, Oxford, UK, pp. 163–195. Wiley-Blackwell, Oxford (2010) CrossRefGoogle Scholar
  29. 29.
    Young, P.C.: Gauss, Kalman and advances in recursive parameter estimation. J. Forecast. 30, 104–146 (2010) (special issue celebrating 50 years of the Kalman Filter) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  • Keith J. Beven
    • 1
  • David T. Leedal
    • 1
  • Paul J. Smith
    • 1
  • Peter C. Young
    • 1
  1. 1.Lancaster Environment CentreLancaster UniversityLancasterUK

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