Estimation of Velocity Field Using Artificial Neural Networks and Kriging Interpolation (Case Study: Iran Geodynamic GPS Network)

In this paper, two methods have been used: multi-layer perceptron artificial neural network (ANN-MLP) and universal kriging to estimate of velocity field. Neural network is an information processing system which is formed by a large number of simple processing elements, known as artificial nerves. It is formed by a number of nodes and weights connecting the nodes. The input data are multiplied by the corresponding weight and the summation are entered into neurons. Each neuron has an activation function. Inputs pass to the activation function and determine the output of neurons. The number of neurons and layers could be obtained through trial and error according to a specific problem.
One of the simplest and effective methods to use in modeling of real neurons is multi-layer perceptron neural network. This model has been established of one input layer, one or more hidden layers and one output layer. In this structure, all the neurons in one layer are connected to all neurons of the next layer. This arrangement is commonly called a network with full connectivity. Neuron numbers in each layer is determined independently. The neurons of input and output layers are determined according to the number of input and output parameters. The number of neurons in the hidden layer can be determined by trial and error through minimizing total error of the ANN. For this minimization, each ANN parameter’s share in the total error should be computed which can be achieved by a back-propagating algorithm.
One of the most famous and simplest methods is back-propagation algorithm which trains network in two stages: feed-forward and feed-backward. In feed-forward process, input parameters move to output layer. In this stage, output parameters are compared with known parameters and the errors is identified. The next stage is done feed-backward. In this stage, the errors move from output layer to input layer. Again, the input weights are calculated. These two stages are repeated until the errors reaches a threshold expected for output parameters.
Kriging is probably the most widely used technique in geostatistics to interpolate data. Kriging interpolation is a two-step process: first a regression function f(x) is constructed based on the data and a gaussian process Z is constructed through the residuals:
Y(x) =f(x) + Z(x)
where f(x) is a regression function and Z is a gaussian process with mean 0, variance σ2 and a correlation matrix ψ .Depending on the form of the regression function, kriging has been prefixed with different names. Simple kriging assumes the regression function to be a known constant, f(x) = 0. A more popular version is ordinary kriging, which assumes a constant but unknown regression function f(x) = α0. In universal kriging, more complex trend functions such as linear or quadratic polynomials are used.
In two methods, for testing and validation of results, 7 GPS station have been used. The velocity field of these stations is known with respect to Eurasia. The average relative error in test stations is obtained 13.48% for ANN-MLP and 25.38% universal kriging in northern component (VN). Also in eastern component (VE) the average relative error is obtained 18.12% for ANN-MLP and 28.61% for universal kriging. The results show the capability and efficiency of artificial neural networks approach for estimation of velocity field in this region. Another important result obtained from this research indicates that distribution and number of input points are very effective in training stage and coefficients determine.