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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 4. Soil monolith **with** sand tube [from Meshkat et al., 1999] (reproduced by permission of ASCE). Dimensions are in centimeters. along a part of the vertical sand column in the used subsurface flow model. 4.1.1. Initial and Boundary Conditions [31] A constant hydraulic head was used to initialize the computation. The irrigation process was characterized by a constant flux boundary **with** a constant application rate (variations between 0.6 L/h and 6 L/h) into the sand tube (between grid points O and H, Figure 4). In the domain of the silt loam (between grid points H and D), the upper boundary condition was described by a potential evaporation rate of 0.051 cm/h. All other boundaries had a zero-flux condition. 4.1.2. Finite Element Grid [32] The computational grid used **for** the numerical simulations differs from the one chosen by Meshkat et al. [1999]. Whilst they applied a quadrilateral grid (Figure 4) **with** 1376 nodes and 1302 elements, we generated a triangular grid of 1282 nodes and 2443 elements **with** Meshgen, a supplementary tool of the Hydrus-2D software. However, the number of grid points used in the actual simulation were comparable to the amount used by Meshkat et al. [1999] since all quadrilateral grids are subdivided into triangles by the numerical flow model be**for**e per**for**ming the computation. Meshkat et al. [1999] reported that the simulated values of the wetting front modeled **with** SWMS-2D closely matched the experimental measurements. Our simulations, shown in Figure 5, likewise resulted in soil moisture patterns close to those simulated by Meshkat et al. [1999] and this implies a similar match of our simulation results provided by Hydrus-2D. 4.2. Setup of the SOM-MIO 4.2.1. Generation of the Database [33] For this application, the SOM-MIO was designed in a three-dimensional structure. The computation of the subsurface flow was carried out **for** a simulation time of 12 hours. The varying initial pressure head h ini at the upper boundary was the first variable in the permutation scheme **for** generating the set of feature vectors (h ini, t, q a, z sx, z sz) which **for**m the training database. The other components of the feature vectors are the application time t, the application rate **for** irrigation q a, and the wetted depths z sx and z sz, Figure 5. Simulation snapshots of trickle irrigation in a soil monolith **with** sand tube: the white line is the boundary of the nearly saturated zone, and black dashed and solid lines are the 300 cm equipotential lines **for** simulations by Meshkat et al. [1999] and Hydrus-2D, respectively. 6of10

W03022 SCHÜTZE ET AL.: SOMS FOR MODELING THE RICHARDS EQUATION Figure 6. Variability of saturated depth z sz **with** time obtained from Monte Carlo subsurface simulation. defined by 0.95 * qs (see Figure 5). The scenarios were created using initial conditions hini = { 1000 + i * 45 20 cm}i=0 and an upper boundary condition qa ={0+j * 10 0.606 L/h}j=1. This resulted in a total of 460 simulations. The decision variable was defined as the depth of the wetting front in the z direction, zsz, and in the x direction, 50 zsx, at a given time t ={0+k * 864 s}k=0. This procedure created a database consisting of 460 * 51 = 23,460 feature vectors (hini, t, qa, zsx, zsz). Be**for**e training the SOM-MIO, the calculated feature vectors were separated into a training data set and a test data set, each consisting of 11,730 feature vectors. The test data were used solely **for** evaluating the accuracy of the SOM-MIO in solving the inverse problem. All components were equally weighted by normalizing the features x =(hini, t, qa, zsx, zsz) in the range of [0, 1] applying xnorm =(xi min(X))/(max(X) min(X)). 4.2.2. Training the SOM-MIO [34] The three-dimensional SOM-MIO, consisting of 2000 neurons, was trained using learning parameters proposed by Kohonen [2001]. The initial learning rate was set at as(0) = 1. The neighborhood radius around the best matching unit (BMU) was set to decrease from initially s(0) = 3 to 1 during subsequent training steps. The training was initiated by presenting all the 11,730 feature vectors (hini, t, qa, zsx, zsz) of the training data set to the SOM-MIO. With an increasing number of ten repeated presentations of this complete training data, the approximation of the training data achieved a satisfactory result after only 987 s on a Dual Pentium III (800 Mhz) PC. 4.3. Application of the Trained SOM-MIO [35] The trained SOM-MIO was employed to per**for**m four different tasks, i.e., (1) deterministic subsurface flow **modeling**, (2) stochastic subsurface flow **modeling**, (3) deterministic inverse **modeling**, and (4) stochastic inverse **modeling**. For the stochastic **modeling** tasks a Monte Carlo technique was used to investigate the effects arising from uncertainty and variation regarding the application rate q a of the applied drip irrigation equipment. For each Monte Carlo realization a normally distributed random number generator f N **with** a mean of zero and standard deviation s q = 0.3 provided 7of10 W03022 the stochastic component of the fluctuating application rate qa*: qa* ¼ qa þ fN xjm; sq : ð14Þ 4.3.1. Deterministic Subsurface Flow Modeling [36] The first application of the trained SOM-MIO consisted of a subsurface flow simulation **for** evaluating the vertical and horizontal saturated depth **for** a given initial hydraulic condition, the water application time and a certain water application rate (hini, t, qa ! zsx, zsz). The water flow in the soil monolith (Figure 4) predicted by the SOM-MIO was consistent **with** the simulated water flow of the Hydrus-2D model, and thus also **with** the transition of the wetted depth as simulated and observed by Meshkat et al. [1999]. 4.3.2. Stochastic Subsurface Flow Modeling [37] Monte Carlo subsurface flow simulations were carried out **with** the numerical model and the SOM-MIO **for** an exemplary application rate qa = 3 L/h and an initial pressure head hini = 800 cm. The number of simulations, nmc, which are required statistically in order to get representative first and second moments of the **output** zsz, was determined by comparing the results of the deterministic scenario and the mean of the **output**s from an increasing number of Monte Carlo runs at 50 different times t. The necessary number of Monte Carlo runs was assigned to nmc = 750 when the mean value and second moments of both was essentially identical (see Figure 7). [38] Figure 6a shows the spectra of the scattered values of the wetted depth in z direction **for** different irrigation times t. The variability of z sz due to deviations in the application rate increases significantly **with** irrigation time. Furthermore, it is worth mentioning that the type of the distribution changes **with** time and becomes an asymmetrical probability distribution. Figure 6b shows the 5th to 95th percentile of the distribution of z sz **with** time. In this example, the agreement between the two Monte Carlo simulations is very good. The similarity of the results from the Hydrus- 2D model and the SOM-MIO indicates that the SOM-MIO can be employed in Monte Carlo studies **with**out further

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