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Radar is a useful remote sensing tool for precipitation estimation on the ground. The development of algorithms for the remote estimation of precipitation based on radar measurements has been an active research topic for many years. The problem of rainfall estimation on the ground based on radar measurements is complicated because of the space—time variability of the rainfall field.
The rainfall rate R obtained on the ground can be potentially dependent on the four-dimensional structure of precipitation aloft three spatial dimensions and time. In principle, one can obtain a functional approximation between the rainfall on the ground and the 4D radar reflectivity observations Z aloft. This function will be more complicated than a simple Z — R algorithm or a multiparameter radar rainfall algorithm. Therefore the ground rainfall estimation can be viewed as a complex function approximation problem.
Neural networks are well suited for this problem, and the theoretical basis is provided by the universal function approximation theorem Funahashi Recent research has shown that neural network techniques can be used successfully for ground rainfall estimation from radars Xiao and Chandrasekar , and other such applications Krasnopolsky et al.
This technique includes two stages, namely, 1 the training and validation stage and 2 the application stage.
In the training stage, the neural network learns the potential relationship between the rainfall rate and the radar measurements from a training dataset. When a radar measurement set is applied to the neural network, the network yields a rainfall-rate estimate as output. This output is compared with the rain gauge measurement, and their difference or the error is propagated back to adjust the parameters of the network.
This learning process is continued until the network converges. Once the training process is complete, a relationship between the rainfall rate and the radar measurements is established and the network is ready for operation.
When a radar measurement vector subsequently is applied to the network, it yields a rainfall-rate estimate. Neural networks have many advantages in the context of rainfall estimation from radar measurements. The relationship between radar measurements and rainfall rate on the ground is derived directly from a training dataset, and therefore it is not influenced by systematic variations in the radar system characteristics.
The neural network can be tuned very well for one specific kind of storm or for several storms. Once the neural network is trained, it represents a relation between radar measurements and rainfall rate. If the training dataset is large enough and representative enough, the neural network can perform very well. There is a common limitation with respect to the trade-off between generalization and accuracy of neural network—based rainfall estimation.
Therefore, developing a flexible network instead of a fixed network for rainfall estimation may be better. A neural network can learn its structure and parameters automatically from the training dataset.
One way to solve the problem is to collect new data and retrain the neural network all over again from the beginning. However, this training process is very tedious and time consuming, and to restart the training every time new data are available is not a practical solution.
The goal of this paper is to develop an adaptive neural network that is easy to train and can continuously update the structure by incorporating the latest information into an existing neural network without having to retrain from the beginning. The main feature of the adaptive neural network is that the network can adjust itself whenever new rain gauge data are available as shown in Fig. To start with, the network can be built by initial training using all the available data.
The network is in the application mode after the initial training. Once new rain gauge data are collected, the network switches into an updating mode. By using an adaptive updating algorithm, the network adjusts some of its parameters, adding or removing some neurons so as to fine-tune its structure with the new information.
The scheme not only provides a fast and efficient way to build a new neural network rainfall estimation model but also can provide a way to maintain an existing neural network rainfall estimation model and make it evolve gradually. In this paper, we have developed an adaptive neural network scheme that can be modified continuously. For this purpose, a radial basis function RBF neural network is chosen because its architecture is suited well for adaptive modification.
In the next section, the development of the adaptive neural network for rainfall estimation is presented. The performance of the adaptive neural network is evaluated in section 3. The important conclusions are summarized in section 4. Multilayer feedforward neural networks MLFNN can be used successfully for radar rainfall estimation from remotely sensed data Xiao and Chandrasekar ; Tsintikidis et al. Some of the disadvantages of BPN are that the training process is computationally demanding and the learning process is very tedious.
However, once trained, BPN can be used successfully for radar rainfall estimation. The structure and learning algorithm of a BPN make it difficult for implementing adaptive rainfall estimation algorithms. One of the alternatives suited for rainfall estimation is an RBF neural network. The RBF network has a unique structure that will make it conducive for adaptive radar rainfall estimation.
The following section describes the structure of an RBF network for radar rainfall estimation. The block diagram of the RBF neural network is shown in Fig. For the problem of rainfall estimation at a point on the ground, the input to the network can be chosen from available radar measurements over the three-dimensional space aloft.
The input vector size depends on the input data used nine for the structure shown in Fig. The rainfall rate is chosen as the output of the network. The hidden layer is an important part of the network, which determines the accuracy of the network. From Eq. Once all these parameters are determined, the network can be used for applications.
If an input vector x is applied to the RBF network, the distance of the vector to every center vector of the neurons in the hidden layer is calculated. The output of the neuron is a function of the distance [as shown in Eq.
A linear combination of the outputs from all the hidden units is the final output. It was shown in the previous section that three sets of parameters need to be determined when constructing an RBF neural network. The orthogonal least squares method is used to determine the center vectors for the hidden neurons. Once the parameters in the hidden layer are determined, the weight vector from the hidden layer to the output layer can be obtained by the linear least squares method.
This combined learning algorithm is fast, because no back propagation is involved in the process. The size vectors r j must be determined in conjunction with the center vectors c j. The generalization capability of the RBF neural network is sensitive to the size vector. If the size vector is small, the network will function very well with the training set but will have poor generalization capability. On the other hand, if it is too large, then the network will be overgeneralized.
Therefore, an appropriate size vector should be determined by several trials. We can use the procedure introduced above to develop an RBF neural network for rainfall estimation. As days go by, more data become available. Some of the data are completely new to the network, and some are similar to what the network has seen before but with slightly different output.
To incorporate the information from the new data, it is necessary to refine the network by adding or removing some neurons in the hidden layer. For data similar to what the network has seen but with different desired output as compared with what the network saw before, it is necessary to adjust the weights from the hidden layer to the output layer with the latest input—output pairs, maintaining the structure of the network.
One way to incorporate the new information from the new data into the network is simply by combining the new data with the old training dataset to form a new larger training dataset and to retrain from the beginning. The most important part in the retraining process is searching for the optimum center set from the new training dataset, and this process is tedious. This solution is neither convenient nor practical.
Another disadvantage is that a simple retraining process may not give preference to the latest data in the training process, which is required if the most current relation between reflectivity structure and rainfall is to be maintained by the neural network. Based on these reasons, it may be better to use an adaptive RBF neural network for rainfall estimation. In the adaptive learning scheme illustrated in Fig.
One of the simplest ways to modify the network for new data is to add or replace neurons as well as change the center vector. The schematic of altering the center vectors is shown in Fig. The procedure used to modify the RBF network is as follows. First, the standard orthogonal least squares method that was used to build the network from the beginning can be applied to the new dataset to come up with new center vector C 2.
The existing model has a center vector set called center vector set 1, C 1. A new center vector set can be constructed from these two center vector sets C 1 and C 2. Radar data used in this study were collected by the Melbourne WSRD in the summers of and over central Florida. Two consecutive months August and September of radar data and the corresponding rain gauge measurement records were used for this study. The WSRD volume scans were done every 6 min.
Data from rain gauges within a km radius of the radar were used. We construct radar data constant-altitude plan position indicators, and, based on the time of the radar data, 5 min. The locations of the rain gauges are shown in Figs. Figure 6 shows the location of the 19 rain gauges used in the dataset, and Fig. The analysis performed in this section is done using data collected during The rainfall estimation calculated using an adaptive neural network is compared with a fixed neural network for the period between 21 and 30 August The fixed neural network is set up based on the radar data and rain gauge measurements during the period of 1—20 August.
The RBF network is constructed using the algorithm described in section 2. The adaptive network is based on the initial model RRN F  , and is adaptively updated everyday when new rain gauge data become available. The subscript A with RRN indicates the adaptive network. Figure 8 shows a schematic that describes the test scheme. The mean hourly rain-rate estimation from the fixed network and the adaptive network over the gauge are shown in Fig.
For the convenience of evaluation, the gauge observation is shown by a solid line in all figures. The mean daily rainfall accumulation results based on the two rainfall algorithms are shown in Fig.
The statistical analysis of the rainfall estimation results is listed in table 1.
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