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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 2. Determination of the enclosing triangle using the Delaunay triangulation. After the triangulation we can determine the enclosing triangle (or tetrahedron **for** higher dimensions) established by the units of the trained SOM **for** a presented **input** vector (see Figure 2). During the application of the SOM-MIO, firstly, the enclosing triangle tj(m1, m2, m3) is calculated **for** a masked sample vector DxxSOM (see Figure 2). Assuming a 3D sample vector, each **output** value can be determined using the three corner units m1, m2, m3 of the enclosing triangle: y 0 ¼ um1Dy þ vm2Dy þ wm3Dy: ð10Þ The calculated **output** y’ is derived from the barycentric coefficients u, v, w of the point xSOMDx **with** respect to the triangle vertices m1, m2, m3 (see Mallet [2002] **for** details). [25] For **input** values located outside the convex hull of the SOMs units, the triangle centroid nearest to the point is determined. The nearest triangle plane then defines a gradient which can be used to calculate the new **output** value based on the actual barycentric coefficients. Thus the employed interpolation scheme is a robust calculation method, as it additionally allows **for** an extrapolation, if the **output** values always lie outside the range given by the training data. 3. Model Building With the SOM-MIO [26] When using the SOM-MIO **for** **modeling** the first approach is the approximation of a subsurface flow model to a required accuracy **for** executing simulation tasks. After successful training this can be per**for**med much faster than by any numerical model, while the robustness of the purely algebraic algorithm guarantees an easy and straight**for**ward operation. The alternative method of using ANNs in flow **modeling** refers to the inverse problem, e.g., the determination of adequate boundary conditions according to the final state of a subsurface flow process. Such inverse problems can also be per**for**med by standard ANNs. However, the standard ANNs generally require a new training process, which finally leads to a second, independent network. Over and above the achievements of the standard networks the SOM-MIO can solve both the 4of10 simulation task as well as the inverse solution **with** the same network. [27] The operation of an ANN as a mapping function requires a set of data (x, y), which represents the **input** x 2 X, X < n **with** x =(x1, ..., xn) and the **output** y 2 Y, Y < m **with** y = (y1, ..., ym) of the considered problem. After training, the neural network should be able to calculate the mapping f: f : X ! Y: ð11Þ The accuracy of the mapping is normally limited by the availability of measured training data. For overcoming these limitations, the vectors x and y are generated synthetically using a numerical flow model which creates a scenario database **for** training neural networks over an adequate range (see Figure 3). The learning algorithm then recalls **input**-**output** data in a certain strategy, e.g., randomly. During an unsupervised training, the SOM develops its own continuous topological mapping by self-organization on the samples (x, y) from (X, Y) **with** respect to the underlying probability density function r(x, y). In other words, regions in the space (X, Y) from which sample vectors (x, y) are drawn **with** a high probability of occurrence are mapped **with** a superior resolution than regions from which sample vectors are drawn **with** a low probability of occurrence. After training, the SOM-MIO (**with** the extensions described in section 2.2) can be applied in different mapping directions. For simulating the subsurface flow the SOM-MIO then provides the mapping f: f : ðxini; xbound; tÞ ! y; ð12Þ **with** initial conditions xini, the simulation time t and boundary conditions xbound. The solution of the inverse problem by a neural network is defined by the following mappings f1 0 , f2 0 and f3 0 : f 0 1 : y; xini; ð tÞ ! xbound; f 0 2 : y; xbound; ð tÞ ! xini; f 0 3 : y; xini; ð xboundÞ ! t: ð13Þ Depending on the selected mapping task, the external parameters, namely the simulation time t, the initial conditions x ini, and the boundary conditions x bound, serve as **input** or **output** parameters during the application of the trained ANN. When applying either of the four different mapping functions these parameters and the simulation results y can be varied **with**in the bounds of the generated database. While the mapping function f can be used as a simulation model, the defined inverse functions are applicable to control problems (e.g., f 1 0 and f3 0 in irrigation see section 4), to monitoring tasks (e.g., f 1 0 , f2 0 ) or likewise to water resources planning (e.g., f 1 0 and f3 0 ). 4. Application of the SOM-MIO to an Irrigation Experiment [28] The new methodology was applied to model the subsurface flow as well as to solve different inverse problems

W03022 SCHÜTZE ET AL.: SOMS FOR MODELING THE RICHARDS EQUATION Figure 3. Unsupervised training of a self-**organizing** map (SOM) using flow models. of an irrigation experiment per**for**med by Meshkat et al. [1999]. They investigated the efficiency of the sand tube irrigation (STI) method and validated their numerical model using a laboratory experiment (Figure 4). An axisymmetric version of the SWMS-2D model [Simúnek et al., 1996], a **for**mer freeware version of the Hydrus-2D model, was applied **for** simulating a quasi three-dimensional flow pattern. 4.1. Numerical Simulation of the Irrigation Experiment [29] The irrigation experiment was conducted by Meshkat et al. [1999] on a cylindrical lysimeter (diameter = 1 m, height = 0.7 m, see Figure 4) filled **with** a maury silt loam. A sand tube, located in the upper part of the domain, had a 9 cm 5of10 W03022 radius and a 28 cm depth. The sand tube diameter was chosen based on the extent of water spread over the soil surface during the applied drip irrigation treatment. The sand tube was dimensioned such that the capillary rise of water in the soil matrix would be less than the height of the sand column. [30] The database **for** training the SOM-MIO was generated using a similar numerical model and identical model parameters as described by Meshkat et al. [1999]. Thus the van Genuchten parameters **for** the sand used as **input**s to the model are qr = 0.02, qs = 0.42, aVGN = 0.024 cm 1 , nVGN = 4.13, Ks = 180.0 cm/h and **for** the maury silt loam qr = 0.14, qs = 0.525, aVGN = 0.024 cm 1 , nVGN = 1.393, Ks = 1.8 cm/h. We also followed Meshkat et al. [1999] in implementing an artificial impervious barrier (assumed Ks = 0.000018 cm/h)

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