Views
5 years ago

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 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 before performing 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 form 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). Before 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 perform 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 outputs 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 without further

Hydro-Morphodynamic modeling and flood risk mapping ... - Simhydro
Multiplicative Noise Removal Using Self-Organizing Maps - INC
Self-Organizing Maps for Time Series 1 Introduction
Multiplicative Noise Removal Using Self-Organizing Maps - INC
The modeling of turbulent reactive flows based on multiple mapping ...
Local Dynamic Modeling With Self-organizing Maps And Applications
Multiple-inputs Dual-outputs Process Characterization and ... - JSTS
Dynamic Cluster Recognition with Multiple Self-Organising Maps
Using Psycho-Acoustic Models and Self-Organizing Maps to Create ...
Error modeling and kernel modification in a multiple input - multiple ...
A HYBRID DEVICE OF SELF ORGANIZING MAPS (SOM) AND ...
Experimental comparison of recursive self-organizing maps for ...
comparison of qtl mapping models with multiple traits and multiple ...
The impact of network topology on self-organizing maps
Mixture Generalized Linear Models for Multiple Interval Mapping of ...
Evolving Self-Organizing Maps for On-line Learning, Data Analysis ...
Multiple Input Multiple Output Iterative Water-Filling Algorithm for ...
Modeling Semantic Similarities in Multiple Maps - Pattern ...
Considering topology in the clustering of the Self-Organizing Maps
Nonlinear Spring Model of Self-Organizing Map Arranged in Two ...
Data exploration using self-organizing maps - CS Course Webpages
Editorial: Advances in Self-Organizing Maps The Self ... - Hal-SHS
Self-Organizing Maps Applied in Visualising Large ... - CiteSeerX
Combining Self-organizing Maps with Mixtures of Experts
On Modeling of Self-organizing Systems
Application of ART2 Networks and Self-Organizing Maps
A self-organizing model of disparity maps in the primary visual cortex