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Self-organizing maps with multiple input-output option for modeling ...

Self-organizing maps with multiple input-output option for modeling ...

Self-organizing maps with multiple input-output option for modeling

Self-organizing maps with multiple input-output option for modeling the Richards equation and its inverse solution N. Schütze and G. H. Schmitz Institute of Hydrology and Meteorology, Dresden University of Technology, Dresden, Germany U. Petersohn WATER RESOURCES RESEARCH, VOL. 41, W03022, doi:10.1029/2004WR003630, 2005 Institute of Artificial Intelligence, Dresden University of Technology, Dresden, Germany Received 8 September 2004; revised 18 October 2004; accepted 10 November 2004; published 26 March 2005. [1] Inverse solutions of the Richards equation, either for evaluating soil hydraulic parameters from experimental data or for optimizing irrigation parameters, require considerable numerical effort. We present an alternative methodology based on self-organizing maps (SOM) which was further developed in order to include multiple input-output (MIO) relationships. The resulting SOM-MIO network approximates the Richards equation and its inverse solution with an outstanding accuracy, and both tasks can be performed by the same network. No additional training is required for solving the different tasks, which represents a significant advantage over conventional networks. An application of the SOM-MIO simulating a laboratory irrigation experiment in a Monte Carlo–based framework shows a much improved computational efficiency compared to the used numerical simulation model. The high consistency of the results predicted by the artificial neural network and by the numerical model demonstrates the excellent suitability of the SOM-MIO for dealing with such kinds of stochastic simulation or for solving inverse problems. Citation: Schütze, N., G. H. Schmitz, and U. Petersohn (2005), Self-organizing maps with multiple input-output option for modeling the Richards equation and its inverse solution, Water Resour. Res., 41, W03022, doi:10.1029/2004WR003630. 1. Introduction [2] The numerical solution of the Richards equation continues to be the most popular method for modeling transient soil moisture transport, mainly, due to the various limitations of analytical and semianalytical infiltration models [Philip, 1957; Haverkamp, 1983; Schmitz and Liedl, 1998]. User friendly commercial (e.g., Hydrus-2D [Simúnek et al., 1996]) and public domain simulation programs (e.g., VS2DT [Healy and Ronan, 2000]) are routinely applied, proving themselves to be versatile and flexible tools. However, all these numerical models require the support of iterative algorithms. Together with the ambiguity of the parameters of the van Genuchten soil model [Felgenhauer et al., 1999] this may jeopardize the stability of the computation, especially in the presence of strong pressure gradients or flow near saturation [Vogel et al., 2000]. Obviously, things become even worse if the inverse solution of the Richards equation is required [Pan and Wu, 1998] which, moreover, incorporates the risk that the optimization may end up in a local minimum and, hence, provide an incorrect result. This contribution investigates a specific type of artificial neural network (ANN) to serve as an alternative strategy when dealing routinely with a multitude of repeated flow simulations as e.g., Monte Carlo–based studies and/or inverse solutions of the Richards equation. Copyright 2005 by the American Geophysical Union. 0043-1397/05/2004WR003630$09.00 W03022 [3] Applications of ANN in water resources, which mostly employ the Multilayer Perceptron architecture (MLP), are well documented in several review papers [Atiya and Shaheen, 1999; Maier and Dandy, 2000; American Society of Civil Engineers (ASCE), 2000a, 2000b; Dawson and Wilby, 2001]. A number of applications focus on the development of robust tools in rainfall-runoff modeling [Shamseldin, 1997; Sezin and Johnson, 1999] as well as stream flow forecasting [Shamseldin and O’Connor, 2001; Jayawardena and Fernando, 2001; Atiya and Shaheen, 1999]. All these ANN based models perform well in comparison with conventional methods [Hsu et al., 1995; Dawson and Wilby, 1999]. However, they yield poor results if the training data do not cover all essential scenarios [Kumar and Minocha, 2001; Minns and Hall, 1996]. In a first attempt to overcome this limitation, Smith and Eli [1995] use model generated data rather than field data. They portray an entire hydrograph by a Fourier series and obtain accurate predictions for multiple storm events. Schmitz and Schütze [2002] proposed another solution to the dilemma of insufficient training data by employing physically based flow models in order to train a specific type of ANN, namely self-organizing maps (SOM) [Kohonen, 2001]. They achieved promising results when evaluating a reduced inverse solution of the Richards equation. [4] Self-organizing maps are originally designed for objective classification, pattern recognition and image compression, i.e., they lack the ability to generate output data. Cai et al. [1994] successfully employed a standard SOM for 1of10

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