## 5 Cased Study
### GNSS Signal Correction  
GNSS signal denoised by EMD can meet the requirements of general engineering projects, while in terms of controlling projects such as the long-span bridge, the denoised data are treated with error correction to meet the requirements associated with real-time observation and final prediction of foundation settlement. In this paper, the hyperbolic model is applied to fitting the parameters of GNSS signal and the settlement data obtained by precise leveling, respectively. The fitting parameters of the two groups of data are obtained, which are expressed as \(a\), \(b\) and \(a^{\prime}\), \(b^{\prime}\), respectively. The fitting figure is shown in Figure 15.  
Figure 14: Denoising GNSS signal. Figure 13: EMD decomposition of GNSS signal. The objective of the present work is to develop ANN models which are accurate and robust for the prediction and interpolation of foundation settlement of bridge tower. They are composed of several layers of many interconnected neurons operating in parallel. Since ANN have many inputs (external stimuli) and outputs (responses) and allow nonlinearity in the transfer functions of the neurons, they can be used to solve multivariate and nonlinear modeling problems. Of course, ANNs are gross simplifications of the human brain function. In any case, they can be viewed as elaborate, nonlinear systems which, given an external input, will estimate the corresponding output, even if the inputs are noisy or incomplete. In order to correct the foundation settlement of bridge tower, two three-layer backpropagation neural network structures were chosen since three-layer fully connected ANNs are capable of forming an arbitrarily close approximation to any continuous nonlinear mapping. By using a trial-and-error approach which starts with a small number of hidden neurons and then increasing the size until the training performance is acceptable, three-layer network structures with varying numbers of hidden neurons and different training parameters have been investigated. Currently, there are no standard rules for choosing the proper neural network structure. There are, however, some guidelines to place an upper bound on the network size such as (i) using no more than three layers (excluding input layer) for the fully connected multilayer networks, and (ii) using fewer hidden layer neurons than training data. Specific structures are problem dependent and are fixed by trial-and-error, especially when given little or no prior knowledge of the problem. In order to obtain parameters \(a\) and \(b\), two three-layer backpropagation neural network structures were chosen since three-layer fully connected ANNs are capable of forming an arbitrarily close approximation to any continuous nonlinear mapping. And the Levenberg-Marquardt backpropagation training algorithm, a standard numerical optimization technique which approaches second-order training speed without having to compute the Hessian matrix, was applied in this work because of its fast convergence in training. For overcoming the \"overfitting\" (or overtraining) problem and improving the generalization, the early stopping technique (i.e., cross-validation) was used during training. The transfer functions, training parameters, and training performance of the developed ANN models are summarized in Table 2, and the corresponding training processes for obtaining the parameters \(a\) and \(b\). In this study, the 63 precis leveling datasets obtained at Measurement Points were divided into two groups, one containing 33 datasets, and the other 30, for neural network parameter training and training result evaluation. The study applied three different types of hidden node numbers and transfer functions, with the times, rate, and accuracy of the training set to be 200, 0.01, and 0.0001 respectively. Table 2 shows the computational accuracy of the rectification of GNSS signal for the BP neural network of the three types of hidden node numbers and transfer functions.