Modeling of Weather Radar Maps and Weather Forecasting using Radial Basis Function Neural Networks

<Contributed to MeLi Wiki by Professor Dimitrios Charlampidis, Department of Electrical Engineering, University of New Orleans>

Introduction to Weather Radar Imaging

Weather radars are designed to detect precipitation in the atmosphere. NEXRAD (Next-generation Radar) is the radar used by the National Weather Service within the United States. The NEXRAD network consists of several Doppler weather radars located at various locations within the country. NEXRAD’s technical name is WSR-88D (Weather Surveillance Radar, 1988, Doppler).

The radar collects data as the radar-beam scans the surrounding area in a 3-dimensional manner. In particular, the radar collects data at different elevation angles, at different azimuth angles, and at fixed set of distances from the radar. In other words, the location of each collected data point, with respect to the radar’s position, is identified by three parameters, namely, the distance from the radar, the elevation angle of the radar beam, and the azimuth angle.

The value of each data point is equal to the reflectivity value, Z, in decibels (dB) associated to the location from which the point is collected. Reflectivity is the ratio between the energy backscattered by a target and the energy send out by the radar. Although reflectivity is unitless, the term dBZ is commonly used to indicate that the particular value corresponds to reflectivity, Z, in dB. Usually, higher reflectivity values indicate stronger precipitation. Researchers have investigated specific relationships between reflectivity, Z, and the corresponding precipitation-rate, R. Such relationships are known as Z-R relationships. Although there has not been an agreement among researchers on a single Z-R relationship, it has been found that relationships of the form <math>Z=aR^b</math>, where <math>a</math> and <math>b</math> are positive values that can be chosen depending on the particular scenario at hand, are acceptable. One such relationship is the Marshal-Palmer relationship, <math>Z = 300 R^{1.6}</math>. It should be mentioned at this point, that the radar-beam widens as it travels away from the radar. Therefore, it is more accurate to say that each collected data point is associated to the average reflectivity value backscattered from a small target rather than a single point target.

The WSR-88D can operate in two basic modes. The first mode is the clear-air mode, which is on when no precipitation is present. The clear-air mode aims in evaluating air movements. When the radar operates under this mode, the scanning speed is low. The second mode is the precipitation mode. This mode is used to measure reflectivity, and thus precipitation, in 3-dimensional volumes around the radar. The WSR-88D radar completes a multi-elevation volume scan approximately every 5 minutes. Each elevation scan includes more than 360 rays, while each ray collects about 250 data points up to a distance of 250 km away from the radar. Although two consecutive data points within the same ray are distant apart by 1km, the azimuth angle between two neighboring rays is not fixed. The data associated with each elevation is thus collected in polar coordinates, while usually the data processing is performed in Cartesian coordinates. Therefore, the data is usually rearranged from polar to the Cartesian coordinate system, using data interpolation.

An example of reflectivity images obtained from the first and second radar elevations are shown in Figure 1. Each pixel in the reflectivity images corresponds to <math>1 km^2</math>, while the radar location corresponds to the central image pixel. The intensity of each pixel is associated to a reflectivity value, such that high intensity represents strong precipitation. The images contain several types of echoes, such as small-isolated events, large-strong precipitation events, and clutter. In particular, the spurious events located relatively close to the radar are most likely resulted due to clutter. The second elevation image contains a smaller amount of clutter echoes, since the radar-beam side-lobes are less likely to hit the ground. On the other hand, the first elevation image is more likely to include precipitation echoes far away from the radar, since at those distances higher elevation beams tend to pass above echo tops. The image resolution decreases with respect to the distance from the radar for two main reasons. First, the radar-beam widens as it travels away from the radar. Second, the distance between neighboring radar rays increases with respect to the distance from the radar. Therefore, the algorithm employed during the conversion from polar to Cartesian coordinates is required to make use of more significant interpolation at distances far away from the radar.

Figure 1.

A scenario, in which the radar and a single radar-beam are pointing at a particular elevation and azimuth angle, is shown in Figure 2. It is indicated that, practically, the radar beam widens as it travels away from the radar, and also that it may bend depending on the atmospheric conditions, such as temperature and humidity. Severe bending of the radar beam is caused due to refraction and may strike the earth repeatedly for distances of hundreds of kilometers producing so called anomalous propagation (AP). Additionally, echoes resulting from other sources are usually undesired. Examples of such targets are man-made structures such as buildings and aircrafts, birds, and insects. Moreover, there may be scattering resulting from radar-antenna sidelobes striking the earth close to the radar, which is referred to as ground clutter.

Figure 2.

Forecasting – Tracking of Storm Fronts

About the Problem

Weather forecasting is defined as the prediction of precipitation event paths and/or other precipitation characteristics in the future. In particular, nowcasting , i.e. the prediction of the rain event paths just a few minutes in the future, can be quite successful. As discussed in the literature , applications of nowcasting include point-to-point communications in poor weather conditions. For instance, excellent prediction of heavy storm motion can assist in determining the path attenuation due to precipitation. Forecasting may involve tracking of storm fronts in order to be able to analyze the behavior of storm front motion.

It is interesting to mention that tracking of precipitation events may be useful in applications other than forecasting, including clutter and AP removal. In this case, the precipitation paths do not have to be predicted in the future but simply identified in the past and present. A moving event is more likely to correspond to precipitation, as opposed to clutter which usually remains fairly static in terms of position and intensity for fairly large amounts of time. Techniques used in the literature for clutter rejection in weather imagery , often do not utilize the temporal characteristics of precipitation events.

Tracking of front paths may present two main challenges:

• The first challenge is the need for computational efficiency. As it can be inferred from discussion presented in the previous section, radar scans produce large volumes of data.
• The second challenge is the fact that there are several precipitation events within a single image with different motion characteristics. More specifically, each event may be appearing or disappearing, merging or splitting, growing or contracting, moving or remaining stationary, and intensifying or dissipating. Apparently, the process of linking corresponding precipitation events between consecutive images is not a trivial problem, even for the case of nowcasting.

Solution to the Problem

A first attempt to solving the weather forecasting problem could include tracking of pixels on an individual basis. In other words, the purpose of this approach would be to try to associate pixels present in past and current weather maps based on their intensity and past/current location. Then, the motion of the particular pixel could be examined in order to predict the position and intensity of the pixel in a future weather map. It is apparent that such an approach would be considerably difficult to implement, since it would be especially difficult to match pixels in consecutive images considering that there might be several closely located pixels having the same or similar intensity.

A second approach would move one step further and attempt to track neighborhoods of pixels. For instance, such an approach is based on the cross-correlation between pixel neighborhoods in past and current weather maps. In this case, the motion characteristics of pixel neighborhoods are be examined in order to predict the new position of the pixel neighborhood in the future weather map.

The above concept can be extended one step further. In order to solve the linking or association problem, rain events can be approximated and modeled as a collection of several well-defined 2-dimensional functions. In this case, the problem of precipitation event tracking can be reduced to the tracking of the parameters that define the 2-dimensional functions. The advantage of a modeling approach over correlation-based techniques is that by modeling the weather radar maps, each rain event can be described as a single entity, or perhaps as a mixture of a few entities. On the other hand, it is more likely that correlation approaches may define pixel neighborhoods without examining in detail if these neighborhoods are actually associated to a particular rain event or to several independent rain events. The advantage of identifying rain events as single entities is that the motion patterns, including travel direction of the event, rotation of the event, intensification or attenuation of the event, merging/splitting of the event can be more effectively studied.

A simple example that illustrates the above concept is shown in Figure 3 next. There are two consecutive images shown as First Frame and Second Frame. For simplicity, two events are included in each image. Each event is shown to have a single reflectivity value throughout its extent. Moreover, each event is depicted as a perfect ellipsis. Of course, the particular scenario is not a realistic one. However, it is used here for illustration purposes. If these events were supposed to be tracked on a pixel-by-pixel basis, a huge number of associations would have to be made between the first and second frames. Furthermore, such an association would not necessarily be possible, since it is not easy to determine which pixel within the first frame corresponds to a pixel in the second frame. On the other hand, if each event was modeled as a single ellipsis with a particular orientation and center, then it would be relatively easy to observe how much each event had moved in terms of distance and rotation. A similar approach could be followed in order to track the size and intensity of each event.

Figure 3.

Of course, as mentioned above, this is just a simplified scenario presented for demonstration purposes. Nevertheless, this example illustrates that if a collection of basic functions was used to approximate the events shown in each image, tracking of precipitation events might be possible via tracking of the basic functions (or more accurately, via tracking of the function parameters). In the example presented here, the basic functions used are ellipsis. Tracking of precipitation events becomes equivalent to tracking of the ellipsis, or more accurately, tracking of the ellipsis parameters (namely center, size, direction, and intensity). In a more realistic situation, each precipitation event would most likely be approximated by more than one basic function. In addition, ellipses may not be the best functions to approximate/model precipitation events.

Radial Basis Function neural networks (RBFNN) provide an excellent tool that can be used to approximate weather radar maps as a collection of several functions. Therefore, RBFNNs are appropriate for modeling weather images, and thus provide a solution to the problem at hand. The RBFNN approach for modeling precipitation events is presented next.

Application: Radial Basis Functions for Modeling of Rainfall Maps

Next, it is explained how RBF neural networks (RBFNN) can be used for precipitation modeling in weather radar imagery. Additional details can be found in . Background information about RBFNN can be found in .

Generally about Neural Networks as Function Approximators

In general, RBFNNs can be utilized as classifiers or function approximators. It is the latter which is of interest in this application. As in the majority of Neural Networks, RBFNNs are associated with a set of parameters that are adjusted during a “training phase” in order to achieve a specific goal. In the case where RBFNNs are used as function approximators, the goal is to find the underlying function that associates a set of inputs (scalars or vectors) with a set of corresponding outputs.

As an example, let us assume that it is known that $N$ different vectors <math>\vec{x}^p</math>, <math>p=1,…N</math> are associated one-to-one with $N$ scalars <math>y^p</math>, <math>p=1,…N</math>. In other words, vector $\vec{x}^1$ is associated to scalar <math>y^1</math>, vector <math>\vec{x}^2</math> is associated to scalar <math>y^1</math>, etc. Although the correspondence between <math>\vec{x}^p</math> and <math>y^p</math> is known, the specific mechanism with which <math>y^p</math> can be produced from <math>\vec{x}^p</math> might be unknown. It is the RBFNN’s job to provide such mechanism. More specifically, the RBFNN works as a function that if is supplied with an input \<math>vec{x}^p</math>, it produces the output <math>y^p</math>.

Of course, the following question could be raised by someone that is introduced to this concept for the first time: “Why are we interested in finding a function that produces the correct output when an input is given, since we already know what output corresponds to what input?” There can be several answers to this question. A general answer is that in certain applications it is exactly the identification of the underlying mechanism which is of importance, and not so much the specific set of inputs and outputs. As an example, one may consider a medical imaging application where several features are extracted from patients’ X-ray images, while the disease from which these patients suffer is also known for a limited group of patients. However, it is possible that the number of features extracted from the X-ray images is significantly large. Therefore, determining which features and what specific feature values are associated to a particular disease may be a very complex problem. A Neural Network can be trained to “learn” a mapping between the inputs (X-ray features) and the outputs (patient’s particular disease), so that when a new patient’s X-ray becomes available, the patient can be properly diagnosed.

RBFNN Structure

As mentioned earlier, RBFNNs, similarly to all Neural Networks, are associated with a set of parameters that need to be adjusted in order for the Neural Network to “learn” the correct mapping between inputs and outputs. The set of parameters of a Neural Network is directly dependent on the Neural Network’s architecture.

In particular, the RBFNN is comprised of a hidden layer and an output layer, while each layer consists of a set of nodes. The input data are presented to the RBFNN in the form of vectors. Each input vector is presented to each one of the hidden layer nodes, and one response per node is obtained. The hidden layer nodes perform a non-linear operation on the input vectors. Subsequently, the responses of all hidden nodes are weighted by hidden-to-output layer weights, and are combined in order to produce an overall output. The output should ideally be equal to a desired output. The difference between the obtained and desired output is used to adjust or train the network parameters, so that the error is reduced. The network parameters consist of the hidden-to-output layer weights, and parameters associated to the hidden layer functions represented by each hidden layer node. The RBFNN architecture is presented in Figure 4.

Figure 4.

The overall input-output mapping $\phi$ resulted by applying the RBFNN on an input vector $\vec{x}$ is:

<math>\phi(\vec{x}) = \sum_{i=1}^M w^i g_i(\vec{x})</math>

assuming that the combiner function sums all the hidden-layer node outputs, or

<math>\phi(\vec{x}) = \max_{i} w^i g_i(\vec{x})</math>

assuming that the combiner function simply keeps the maximum out of all hidden-layer node outputs.

RBFNN for Radar Map Modeling

In the specific case of precipitation modeling, the <math>p-th</math> input vector, <math>\vec{x}_p</math>, consists of the <math>p-th</math> pixel coordinates, i.e. <math>\vec{x}_p=(x^p_1,x^p_2), p=1,...,N</math>, where <math>N</math> represents the total number of pixels used to train the network. In particular, <math>x^p_1</math> and <math>x^p_2</math> are the horizontal and vertical pixel coordinates, respectively. The desired output associated to the <math>p-th</math> input vector should be equal to the pixel value at that location. As a reminder, pixel values in weather radar maps correspond to reflectivity values. The goal of the RBFNN is to adjust its parameters, during a training phase, in order to match the pixel locations to the appropriate reflectivity values. In other words, the RBFNN can be thought of as a function between the coordinates of the weather map and the corresponding values of the map at these coordinates. The parameters of the RBFNN are essentially the parameters of the hidden node functions <math>g_i(\vec{x})</math>. The function represented by the <math>i-th</math> hidden layer node is defined as follows:

<math>g^i(\vec{x}^p,\vec{m}^i)=exp(-d^i(\vec{x}^p,\vec{m}^i)/2)</math>

where <math>d^i(\vec{x}^p,\vec{m}^i)</math> is the square of the Euclidean distance between the <math>p-th</math> input and the prototype associated to the <math>i-th</math> node. The prototype is a vector representing the center point, <math>\vec{m}^i=(m^i_l,m^i_2)</math> of function <math>g^i(\vec{x}^p,\vec{m}^i)</math>.

A close look at the function represented by <math>g^i(\vec{x}^p,\vec{m}^i)</math> indicates that if <math>d^i(\vec{x}^p,\vec{m}^i)</math> is chosen to be the square of the Euclidean distance between <math>\vec{x}_p</math> and <math>\vec{m}_i</math>, namely <math>d^i(\vec{x}^p,\vec{m}^i)=(x^p_l-m^i_l)^2+(x^p_2-m^p_2)^2</math>, then <math>g^i(\vec{x}^p,\vec{m}^i)</math> is simply a 2D Gaussian function. In particular, if the square of the Euclidean <math>d^i(\vec{x}^p</math> is replaced by its detailed expression, the function <math>g^i(\vec{x}^p,\vec{m}^i)</math> can be written as follows:

<math>g^i(\vec{x}^p,\vec{m}^i) = exp(-[(x^p_l-m^i_l)^2+(x^p_2-m^p_2)^2]/2)</math>

which is a Gaussian function centered at <math>\vec{m}^i</math> and having a standard deviation equal to 1.

In this case, from the basic structure of the RBFNN presented in Figure 4, it can be observed that the output of a RBFNN is a mixture of Gaussian functions centered at specific locations specified by the prototype vectors <math>\vec{m}^i</math>. Moreover, the height of the <math>i-th</math> Gaussian function <math>g^i(\vec{x}^p,\vec{m}^i)</math> is equal to the hidden layer-to-output weight <math>w_i</math>.

The goal of the RBFNN during the training phase is to determine the “best” Gaussian centers and weights, so that once a specific set of coordinates is given, the reflectivity values that correspond to that location in the weather radar image can be produced as the output of the RBFNN. An example of how the RBFNN can approximate a weather radar map is shown in Figure 5.

Figure 5.

The reason why the weather events are approximated by a set of functions for the purpose of weather forecasting is because it may be more beneficial to attempt to track a whole event as opposed to individual pixels.

In general, the Mahalanobis distance can be used instead of the Euclidean distance to provide more flexibility in the design of RBFNNs. In this case, <math>g^i(\vec{x}^p,\vec{m}^i)</math> represents a directional Gaussian. The square Mahalanobis distance is defined as <math>d^i(\vec{x}^p,\vec{m}^i)=(\vec{x}^p-\vec{m}^i)^{\text{T}}K(\vec{x}^p-\vec{m}^i)</math>, where <math>K</math> represents the inverse covariance matrix. The overall output <math>\phi(\vec{x})</math> corresponding to the <math>p-th</math> input should ideally be equal to the actual pixel value located at <math>(x^p_l,x^p_2)</math>, namely <math>t^p</math>. The network output is obtained by combining the outputs of the hidden layer nodes after they are multiplied by their corresponding weights, <math>w^i</math>. In the literature ,, the output was considered to be the maximum among all weighted hidden node outputs. In other words,

<math>t^p=\underset{i}{max}(w^i g^i(\vec{x}^p))</math>

The equations used for training the network parameters are presented in detail in previous work , but they are also presented next for the purpose of completeness. The training parameters include the weights, covariance matrices, and prototypes. Training is performed in an iterative manner.

<math>\Delta^p w^i = \eta (t^p - \phi(x^p)) g^i(\vec{x}^p)</math> <math>\Delta^p m^i_j = \Delta^p w^i \sum_l k^i_{j,l}(x_l^p – m_l^i)</math> <math>\Delta^p \sigma^i_j = \Delta^p w^i \sum_l k^i_{j,l}(x_j^p – m_j^i) (x_i^p – m_l^i)/\sigma_j^i</math> <math>\Delta^p h^i_{j,k} = -\Delta^p w^i \sum_l k^i_{j,l}(x_j^p – m_j^i) (x_l^p – m_l^i)/(\sigma_j^i \sigma_k^i)</math>

As mentioned earlier, the RBFNN approach essentially attempts to approximate the rain field as a mixture of Gaussians. At each location in the image, one Gaussian (specifically the Gaussian with the maximum value) is used to approximate the precipitation value.

The RBFNN is constructively trained. In other words, prototypes, and thus hidden layer nodes, may be added or deleted during training based on certain criteria. Specifically, prototypes are added if the minimum distance of an input vector from the closest prototype is larger than a user defined threshold, or if the maximum hidden layer node weighted response is lower than a user defined threshold. Prototypes are deleted in the case where they have not been activated by a minimum number of input vectors for a given number of iterations. A more detailed description of the deletion and addition steps is presented in the “Overall Process” later in the Wiki.

Speeding up Training: Pyramidal Processing

In the literature , a pyramidal approach for the modeling of the precipitation events has been used, for the purpose of speeding up the training process. More specifically, training is first performed in a lower resolution image, and then in a higher resolution image. The advantage of the pyramidal approach is that training is significantly faster in a lower resolution. On the other hand, the RBFNN parameters determined at the lower resolution are not irrelevant to the parameters in the higher resolution. In particular, the parameters determined at the lower resolution are used as the initial values for the higher resolution, after some appropriate modification.

As an example:

• If a pixel’s coordinates are <math>(23,45)</math> in a low resolution image of size <math>40 \times 40</math>, then the pixel’s coordinates in an image of size <math>80 \times 80</math>, with resolution double the one of the low resolution image, should be <math>(46, 90)</math>.
• The hidden layer-to-output weights specify the “height” of the Gaussian function in the weather radar map. Therefore, they are independent of the image resolution.
• The covariance matrix associated to a Gaussian specifies the Gaussian’s extent. As a reminder, the variance is equal to the square of the standard deviation. If the extent of the Gaussian doubles, due to a doubling of the image resolution, the standard deviation doubles as well. Therefore, if the resolution doubles, the elements of the covariance matrix become four times as large with respect to the elements of the covariance matrix associated with an image of half the resolution. Exactly the opposite is true for the inverse covariance matrix. For instance, if for a particular image the inverse covariance matrix of a Gaussian is

<math>

K=\begin{bmatrix} 0.4&-0.1 \\ -0.1 &0.2

\end{bmatrix}</math>

in an image with twice the resolution the inverse covariance matrix will be

<math>

K=\begin{bmatrix} 0.1&-0.025 \\ -0.025 &0.05

\end{bmatrix}</math>

An example of images shown at different resolutions is presented in Figure 6.

Figure 6.

Overall Process

Next, a set of steps is presented, in order to put together the overall procedure that needs to be followed in order to obtain a final weather map modeling result using RBFNNs:

1. Down-sample the original image to the lowest desirable resolution. For instance, if the original image is of size <math>400 \times 400</math>, the lowest resolution chosen could be <math>100 \times 100</math>. In this particular example, this step can be performed by finding the average pixel value in non-overlapping image blocks of size <math>40 \times 40</math>. The downsampling process is shown in the following figure:

Figure 7.

2. For each pixel <math>p</math> in the low resolution image, produce a vector <math>\vec{x}^p</math> which consists of two values, namely the horizontal and vertical pixel coordinates. Moreover, for each pixel <math>p</math>, produce a scalar, <math>y^p</math>, equal to the pixel value.

3. Initialize the RBFNN by assuming a specific number of hidden-nodes, and thus hidden-node functions, <math>g^i(\vec{x}^p)</math>. Randomly initialize the centers <math>m^i</math> and the weights <math>w^i</math> associated to each function.

4. Present the <math>p-th</math> input/output pair to the RBFNN, where the pixel coordinate vectors, <math>\vec{x}^p</math>, are the input, and the pixel values, <math>y^p</math>, are the desired outputs. If the RBFNN combination function is considered to be the maximum out of all node responses, then find the node, <math>i_{max}</math>, that produces the maximum response (i.e. find which node results in the maximum value of <math>g^i(\vec{x}^p)</math>).

5.

(a) If the maximum response out of all node responses is lower than a user specified threshold for this <math>p-th</math> pixel, it is implied that there is no Gaussian that can approximate the value of the pixel well. In this case, a new node is added to the hidden layer of the RBFNN, and the particular pixel coordinates become the center of Gaussian associated to the new node. The weight and inverse covariance matrix associated to the node can be randomly initialized.

(b) Otherwise, if the <math>p-th</math> pixel is far away from any Gaussian function, again, a new node is added to the hidden layer of the RBFNN, and the particular pixel coordinates become the center of Gaussian associated to the new node. The weight and inverse covariance matrix associated to the node can be randomly initialized. The distance of the <math>p-th</math> point from a Gaussian function can be defined to be the Euclidean distance between the <math>p-th</math> point and the center of the Gaussian function.

(c) If none of the two cases (a) or (b) is true, then apply the RBFNN training equations to update only the parameters of the node <math>i_{max}</math> that provided the maximum response.

6. If a node is never chosen by more than <math>S</math> pixels, after all pixels have been presented to the RBFNN for a user-specified number of times, then delete the node.

7. Stop the training process when the change in centers and weights from one iteration to the other drops below a user-defined threshold.

8. Go back to step 2 and repeat steps 2 to 5 for the next resolution image. However, step 3 should be replaced by a step in which the parameters obtained by training the RBFNN for the lower resolution image are used as the initialization step for the next resolution. In particular, assuming that the next resolution corresponds to an image which is twice as large, the initial value of the Gaussian centers in the next resolution should be twice as much as in the previous resolution. The elements in the inverse covariance matrix should be four times smaller than in the previous resolution. The weight values should stay the same.

Experimental Results

Figure 8 depicts an example in precipitation modeling using RBFNNs. More specifically, the image to the left is the original precipitation map, while the image to the right is the modeled precipitation map. Some differences between the original and modeled image are apparent. Nevertheless, these may not be important for the purpose of forecasting, since the main precipitation components/echoes have been identified and represented by Gaussian envelopes.

The ellipses included within the modeled precipitation map correspond to the Gaussian envelopes produced by the RBF neural network. Of course, Gaussian envelopes are of infinite spatial extent. However, the sizes of the major and minor axes of the ellipses have been chosen to be proportional to the variance of the Gaussian envelopes. Essentially, these ellipses simply provide an idea of how the Gaussian envelopes are oriented and their relative size, for illustration purposes.

Figure 8.

Figures 9 (a) and (b) present a sequence of two weather radar images obtained at two different points in time. The image sizes are <math>400 \times 400</math>. The two images are very similar since they are obtained at only two minutes apart. Figures 8 (c) and (d) present the approximated weather radar maps obtained using the RBFNN process, for a lowest resolution image of size <math>40 \times 40</math>. The weather radar maps have been modeled using 6 Gaussians each. The difference with respect to the Gaussian parameters between the two images can be used to predict the parameters of the Gaussian functions in the future image. Of course, if the parameters of all Gaussians associated to a particular image are known, the image can be reconstructed.

Figure 9.

The parameter prediction can be performed as follows:

<math>r_{t+1} = r_t + \alpha (r_t - r_{t-1})</math>

where <math>\alpha</math> is a user defined positive scalar. In the above equation, <math>r</math> can represent any of the Gaussian paramenters, such as a Gaussian center coordinate, <math>m^p_1</math> or <math>m^p_2</math>, a hidden weight, or an element of the inverse covariance matrix, $K$. The subscripts <math>t-1</math>, <math>t</math>, and <math>t+1</math> correspond to the past, present, and future weather radar maps, respectively.

The table next presents Mean Square Error (MSE) results that compare the RBFNN approach, with the trivial persistence method. The MSE is defined as the average squared difference between the actual future weather radar map, and the predicted weather radar map, considering all pixels in the images. In other words,

<math>MSE = \sum_p{(im_{actual}(\vec{x}^p)- im_{predicted}(\vec{x}^p))^2}</math>

where <math>im_{actual}(\vec{x}^p)</math> is the actual future image, and <math>im_{predicted}(\vec{x}^p)</math> is the predicted image using the RBFNN approach.

The persistance method does nothing more, than just assuming that the next image frame will be identical to the current image frame. The MSE results indicate that the RBFNN method performs better than the persistence method. It may appear obvious that any prediction method would work better than the persistance method. However, this is surprisingly not true. It is possible that a bad prediction could produce higher MSE results than even the trivial approach were presence and future are assumed to be identical.

MSE Error Sequence No	RBFNN Method	Persistence Method
1	6.4386	7.3891
2	7.482	7.3338
3	12.2844	14.2472
4	11.4507	14.4324

The experimental results have helped to draw some positive conclusions about the RBFNN-based algorithm, but also that there is still room for improvement:

• The pyramidal approach helps to significantly speed up the training process.
• The RBFNN method performs better, in general, than the persistence method in terms of the MSE.
• Modeling of the rain events provides good intuition about the motion patterns of these rain events.
• There could be a better approach for initializing the number of hidden nodes and the parameters of the Gaussian functions.
• There could be a better selection process about which pixels are used for training the RBFNN (not all pixels have to be used) in order to speed up the training process further.

Project

1. Implement the equations for updating the RBF model.
2. Apply the RBF model on two consecutive radar images and find the RBF parameters.
3. For each one of the Gaussian envelope centers found in the first of the two images, find the closest Gaussian center found in the second of the two images.
4. Compare the vector of parameters that correspond to each Gaussian envelope within the first image, to the closest one in the second image. The comparison can be performed using the Euclidean distance.
5. For each Gaussian envelope pair (a Gaussian envelope in the first image and the closest one in the second image), if the distance between the corresponding parameters is found to be larger than a given threshold T, the two Gaussian envelopes are not linked. For each Gaussian envelope within the first image for which the above threshold is not satisfied, steps 3 and 4 are repeated.
6. If no Gaussian envelope in the second image satisfies the threshold for a Gaussian envelope in the first image after several repetitions of steps 3 through 5, the particular Gaussian envelope in the first image is marked as DMSL. The acronym DMSL stands for Disappeared-Merged-Split-Lost. Essentially, if a Gaussian envelope within the first image is not found to be linked to another Gaussian in the second image, it may have either disappeared (exited the radar range, or dissipated as precipitation), merged with another event, split into several smaller echoes (several Gaussians), or lost (implying that the RBF network failed to produce similar Gaussian envelopes in the second image, although the echo changes between two consecutive images could allow precipitation to be modeled using a set of similarly parameterized Gaussians).
7. Those Gaussian envelopes in the second image that are not linked to any Gaussian envelope in the first image after several repetitions of steps 3 through 5, are marked as AMSL. The acronym AMSL stands for appeared-Merged-Split-Lost. Essentially, if a Gaussian envelope within the second image is not found to be linked to another Gaussian in the first image, it may have either just appeared (entered the radar range, or appeared as a new precipitation event), been the result of merging with another event, be the result of a split from a bigger event, or lost.
8. For all non-DMSL pairs, find the velocity vector that identifies the motion of the particular Gaussian envelopes.

Assessment

1. What is the purpose of weather radars, and what applications are they used for?
2. How do weather radars collect data, what type of data, and in what coordinate form (Polar, Cartesian)?
3. Provide a brief description of RBFNNs in terms of their structure, and flow/processing of data, from input to output.
4. How can RBFNNs be used in order to model 2-D data (such as image data)? More specifically, explain how are the inputs and output of the RBFNN associated to the image coordinates and image pixel values. How can RBFNNs approximate a 2D radar map as a mixture of 2D Gaussians?
5. Why is it important to model weather radar maps prior to forecasting (tracking of weather radar events)?
6. Why is pyramidal processing important?
7. What is the difference between Euclidean and Mahalanobis distance?