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

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

W03022 SCHÜTZE ET AL.:

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 performed 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 transformed 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. Performance 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 effort 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 aforementioned problems. [43] Figure 10a clearly shows the differences of the computational performance 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 effort 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 performance when taking into account that the new SOM-MIO can perform 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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