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

W03022 SCHÜTZE ET AL.: SOMS FOR MODELING THE RICHARDS EQUATION W03022 Figure 7. Stability analysis of the Monte Carlo subsurface simulation. training. This also enabled us to use the trained SOM-MIO **for** the subsequent inverse Monte Carlo study. In addition, this allowed the avoidance of additional simulations **with** the Hydrus-2D model. 4.3.3. Deterministic Inverse Modeling [39] The subsequent applications refer to solving the inverse problem, i.e., evaluating the water application time **for** a specified initial soil moisture (hydraulic head), application rate and either the desired horizontal extension h ini, q a, z sx ! t (Figure 8a), or the vertical extension of saturation h ini, q a, z sz ! t. Moreover, the trained SOM-MIO was applied to calculating the application rate corresponding to given depths of saturation, initial soil moisture and application time (h ini, t, z sx ! q a or h ini, t, z sz ! q a) as shown in Figure 8b. The comparison between the numerical simulation and the outcome of the SOM-MIO was based on the test data set which was ‘‘unknown’’ to the neural network, i.e., not used in the course of the training. Figure 8a shows the irrigation time necessary to obtain a certain desired horizontal or vertical saturation. The results are directly comparable to the water application times determined by the numerical simulation. Comparing the outcome of the SOM-MIO and the test data provided highly satisfactory results also **with** respect to the water application rate corresponding to a specified saturated depth (Figure 8b). Similar results were achieved **for** scenarios **with** different initial conditions, varying from soils near saturation to soils **with** very dry initial conditions (hini = 1000 cm). 4.3.4. Stochastic Inverse Modeling [40] The Monte Carlo inverse calculation was per**for**med **for** the inverse problem hini, qa, zsz ! t, i.e., **for** evaluating the water application time **for** a specified initial hydraulic head hini = 800 cm, a mean application rate qa = 3 L/h and a number of 8 desired vertical saturated depths. The Hydrus-2D model was used in a combination **with** the Nelder-Mead Simplex algorithm in order to solve the chosen inverse problem **for** each given z sz in each Monte Carlo realization. The number of vertical extensions was restricted **for** two reasons. Firstly, the Monte Carlo inverse calculation **with** the Hydrus-2D model required extensive computation time (around 14 days **for** this application), and secondly, the trained SOM-MIO did not cover the whole range of spectra data below z sz = 20 cm. [41] Figure 9a shows the Monte Carlo spectra produced by the SOM-MIO. The variability of the irrigation time increases dramatically **with** the wetted depth. Similar to the stochastic subsurface flow **modeling**, the normally distributed application rate is trans**for**med into a positively skewed distribution. A comparison of the relative variation in the two stochastic applications (Figures 6a and 9a) shows that the prediction of the irrigation time is more uncertain. Figure 8. Inverse solution as provided by the SOM-**multiple** **input**-**output** (MIO) **for** the test data set. 8of10

W03022 SCHÜTZE ET AL.: SOMS FOR MODELING THE RICHARDS EQUATION Figure 9. Variability of irrigation time t **with** wetted depth z sz obtained from Monte Carlo inverse **modeling**. Figure 9b displays five different percentiles (the 5th, 25th, median, 75th and 95th) of the predicted irrigation time **with** wetted depth from the numerical and neural framework, respectively. Again a good agreement between the results of the two inverse Monte Carlo techniques is observed. 4.3.5. Per**for**mance Analysis [42] We investigated the computational efficiency of the Hydrus-2D model and the SOM-MIO on a Dual Pentium III (800 MHz) PC. Figure 10 shows the CPU time required by the SOM-MIO **for** generating the training data, training and application **for** separately solving all the above four tasks. In addition, Figure 10 shows the computational ef**for**t which is required by the of the numerical approach as a function of an increasing number of applications. In each graph, the break-even point is given after which it becomes more and more economical to employ the SOM-MIO in order to handle the a**for**ementioned problems. [43] Figure 10a clearly shows the differences of the computational per**for**mance **with** increasing number of applications **for** both strategies. Since the computation time of the numerical model is constant in the average **for** each simulation, the neural approach has a relatively high initial ef**for**t **for** training. This becomes less important when dealing **with** the inverse problem (Figure 10c) and has low relevance in the case of a Monte Carlo simulation (Figures 10b and 10d). Both subsurface flow **modeling** and inverse calculation in a Monte Carlo setting, benefit from the very first application from the ANN based strategy. The new methodology offers yet another improvement of the per**for**mance when taking into account that the new SOM-MIO can per**for**m the different tasks **with** a unique training. 5. Summary and Conclusions [44] The speed, robustness and stability of ANN-based applications could prove to be useful when dealing **with** Monte Carlo methods or optimization problems in water resources, where a large number of realizations of a numerical model are required to obtain an appropriate solution. A 9of10 W03022 Figure 10. Computational complexity (CPU time in seconds) of the classical framework (dashed lines) and the artificial neural network-based strategy (solid lines) **with** an increasing number of applications. Black labels indicate the break-even point between both strategies.

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