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Improving weather radar estimates of rainfall using feed-forward neural networks
Neural Networks 20 (2007) 519–527 www.elsevier.com/locate/neunet
2007 Special Issue
Improving weather radar estimates of rainfall using feed-forward neural networks Reinhard Teschl ∗ , Walter L. Randeu, Franz Teschl Department of Broadband Communications, Graz University of Technology, Inffeldgasse 12/I, 8010 Graz, Austria
Abstract In this paper an approach is described to improve weather radar estimates of rainfall based on a neural network technique. Other than rain gauges which measure the rain rate R directly on the ground, the weather radar measures the reflectivity Z aloft and the rain rate has to be determined over a Z –R relationship. Besides the fact that the rain rate has to be estimated from the reflectivity many other sources of possible errors are inherent to the radar system. In other words the radar measurements contain an amount of observation noise which makes it a demanding task to train the network properly. A feed-forward neural network with Z values as input vector was trained to predict the rain rate R on the ground. The results indicate that the model is able to generalize and the determined input–output relationship is also representative for other sites nearby with similar conditions. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Feed-forward neural network; Weather radar; Precipitation; Rainfall; Reflectivity factor; Drop-size distribution; Vertical profile of reflectivity; Z –R relationship
1. Introduction This paper shows that neural networks can be successfully used for modelling the relationship between weather radar measurements of rainfall and rain gauge data even if the radar data stem from high altitudes, because of mountainous terrain, and even if the radar data are very noisy due to several sources of error. Section 2 gives an overview of the weather radar and how it is currently used to estimate rainfall. In this section also existing neural network approaches to estimate rainfall from weather radar measurements are mentioned. In Section 3 details about the nature and possible errors of radar and rain gauge data are given. Section 4 describes the development of the neural network covering also aspects as input/output variables, details of the learning method and network design. Section 5 discusses the results of the model. The conclusions are given in Section 6. 2. Weather radar rainfall estimation The weather radar is an important tool in meteorology and hydrology and a key instrument in the field of weather ∗ Corresponding author. Tel.: +43 316 873 7431; fax: +43 316 463697.
E-mail address: [email protected] (R. Teschl). c 2007 Elsevier Ltd. All rights reserved. 0893-6080/$ - see front matter doi:10.1016/j.neunet.2007.04.005
forecasting. Although the weather radar can more likely be considered as a qualitative than a quantitative instrument it has always been used to estimate rainfall amounts. The main advantage of weather radars in the estimation of rainfall is their high spatial and temporal resolution and the gapless spatial coverage. If substituting a gauge for each radar spatial sample ten thousands of rain gauges would be needed. Weather radars also provide a high temporal resolution. Today’s weather radars make a full volume scan within about 5 min. The perhaps biggest advantage of weather radars is that they reveal a three-dimensional structure of precipitation. By displaying in what height, what intensity of precipitation can be assumed, weather radars give us an impression of the type of weather system: whether it is an advective situation with lower measured intensities and echoes up to mid-altitudes, or it is a convective situation with very high intensities and echoes up to 15 km. Because of these advantages radar measurements of precipitation have enjoyed widespread operational usage and will remain so in future (Bringi & Chandrasekar, 2001). But other than rain gauges which measure precipitation directly on the ground, the radar determines the reflectivity aloft. Due to this principle, several sources of errors originate (see Section 3.1). Estimating the rain rate on the ground from
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radar reflectivity measurements is still an important topic in radar meteorology. Bringi and Chandrasekar (2001) divide rainfall estimation techniques by radar into physical and statistical/engineering approaches. Physical approaches attempt to estimate rain rates by radar measurements together with an underlying rain model. The rain model generally describes the shape of the rain-drops and their size distribution. Such an approach is used in nearly all operational weather radars. The rain rate is estimated from the measured reflectivity factor by nonlinear relationships. Physical approaches do not include any feedback, e.g. from rain gauges as opposed to statistical/engineering approaches where a feedback is used. A neural network approach as it is presented in this paper tries to estimate the ground rainfall from the radar measurements. For the training and testing process feedback from rain gauges has to be used, while once established, this method is no longer dependent upon gauges. 2.1. Operational radar rainfall estimation Radars work by sending out an electromagnetic beam and measuring how much of the energy of that beam is reflected back. Weather radars use frequencies at which meteorological echoes like raindrops, snowflakes, hail, etc. are ideally detectable. These are in general frequencies between 3 and 10 GHz. Most of the operational weather radars work at C- or S-band (6 resp. 3 GHz). The main measuring unit of weather radars is the reflectivity factor Z . The Austrian weather radar network consists of four (soon five) C-band radars. The data they provide are described in Section 3.2. The radars in Austria and the rain rate estimation technique are comparable with those in most European countries. In Austria we distinguish between 14 intensity levels for the measured reflectivity factor. With a fixed Z –R relationship this leads to 14 levels of measurable rain rates (Fig. 1(a)). The levels reach from 0.04 mm/h up to around 150 mm/h with gradually increasing distance between the levels. The spatial resolution of the measurement is 1 km × 1 km × 1 km. Fig. 1(b) and (c) show two so called RHI-plots of radar data. RHI stands for range-height indicator. It is a type of radar display on which the reflectivity factor is displayed as a function of range and elevation angle. The RHI may be produced by scanning the radar antenna in elevation with the azimuth fixed. Fig. 1(b) shows a RHI for an advective weather situation. Due to the fact that mountains shield the radar waves, only precipitation in higher altitudes is detectable. That is why often only the upper parts of advective weather situations are visible. Fig. 1(c) shows a RHI for a convective weather event. Such clouds can reach up to more than 10 km. The plot also shows that in such events usually very high reflectivities are measured. When estimating rainfall from radar measurements therefore usually the highest value that is measured from 0 to an altitude of 15 km is taken. Fig. 1(d) shows such a weather image of Austria. All the dots in greyscale represent the predicted rainfall in a 1 km × 1 km region. Such measurements can differ by a factor 10 with rain gauge measurements on the ground.
2.2. Proposed improvements Artificial Neural Networks are a proved and efficient method to model complex input–output relationships (Aliev, Bonfig, & Aliew, 2000). In this paper therefore an approach is described to improve weather radar estimates of rainfall by using an Artificial Neural Network (ANN). In this study we use a feedforward neural network to model the relationship between weather radar data aloft and rain gauge measurements on the ground. The network accepts radar data as an input and is trained to predict the rain rate as measured by the rain gauge. The problem of estimating ground rainfall using radar measurements aloft has already been researched with neural networks. Xiao and Chandrasekar (1997) applied a Backpropagation Neural Network (BPNN) for rainfall estimation from radar data. Liu, Chandrasekar, and Xu (2001) developed a Radial Basis Function Neural Network (RBFNN) to estimate ground rainfall using the vertical profile of the radar reflectivity as input vector. Li, Chandrasekar, and Xu (2003) showed that the radar reflectivity from 1 to 4 km height above the rain gauge is the best input vector to a RBFNN for estimating the ground rainfall compared to several other choices. Xu and Chandrasekar (2005) also used this vertical profile as input vector to their RBFNN. Teschl, Randeu, and Teschl (2005) as well defined four superposed radar reflectivity measurements as input vector and added the highest level where precipitation was detected as an additional parameter. In the current study it is analysed if an ANN can lead to better estimates of ground rainfall even in an alpine environment, like in Austria, where low level radar measurements are seldomly available. In contrast to previous studies using radar reflectivity data starting from 1 km above the rain gauge, in the study area the lowest elevation where radar data aloft the rain gauge are available is 3 km (above Mean Sea Level, m. s. l.). The objective of the present study is to find a relationship that is not only valid for the special site on which the model was trained, but also can be applied to other comparable terrain where no rain gauges are available. Henceforth only radar data are necessary and the model gives a better estimate for the rain rate on the ground. In other words it improves the pristine radar data. On the site that has been used to train the network, a variety of other meteorological data like temperature, wind speed, humidity, sunshine duration, etc. were available. These have not been chosen though as input data, since the objective was to make the model applicable on sites were solely radar data are available. 3. Rain gauge vs. weather radar 3.1. Sources of error Rain gauges measure the rain rate quite exactly but the measured rate is only significant for a certain location. A few kilometres farther the situation can be quite different, especially during convective rain events and in alpine terrain. On the other hand weather radars deliver data with good spatial and temporal resolution. They typically determine the rain rate for every square kilometre. One weakness, however, is their often
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Fig. 1. Weather radar images. (a) Levels of rain rate as distinguished by the Austrian weather radar system. The levels reach from 0.04 mm/h up to 153.76 mm/h with gradually increasing distance between the levels. (b) Range–height indication for an advective weather situation. (c) Range–height indication for a convective weather situation. (d) Radar image of Austria. Composite of four Austrian weather radars. The shown values represent the highest that occurred in any measured altitude.
poor metering precision limiting the applicability of the radar for quantitative tasks. Several sources of error of the weather radar occur because of its working principle. Precipitation particles like rain-drops, snowflakes and hailstones but even birds and insects reflect the beam back to the radar. The measuring unit of the radar is the reflectivity factor Z . Z is used to estimate the rain rate using relationships of the form Z = a · Rb where a, b empirical parameters R rain rate in mm/h Z reflectivity factor in mm6 m−3 .
(1)
Every Z –R relationship is based on a certain drop-size distribution. The drop-size distribution in general varies for different climate zones and weather conditions. Therefore dozens of such empirical relationships have been derived. The most common Z –R relationship is that developed by Marshall and Palmer (1948) assuming an exponential dropsize distribution: Z = 200 · R 1.6 .
(2)
This relationship is used in the National Austrian Weather Radar Network from which the used radar data stem. The assumption of a fixed Z –R relationship is an inherent error source when using radar data for quantitative tasks.
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Another possible source of error is beam blockage. It is caused when objects such as trees, buildings or mountains prevent the radar beam from detecting the precipitation on the other side of them. As a result, the radar image may show no precipitation over an area where it may actually be raining. Precipitation particles also attenuate radar waves. This effect occurs twice. It affects the transmitted radar wave as well as the wave that is reflected back from the sensed volume. This effect increases with the distance to the radar since the waves to and from the sensed volume are attenuated by precipitation on the path between radar and sensed volume. Attenuation in general leads to an underestimation of precipitation by the radar since it weakens the signal that the radar receives. It is especially significant when an intense convective cell is located near the radar. Ground clutter is radar echoes from trees, buildings or other objects on the ground. Such echoes may be caused by the reflection of the beam back to the radar in the main lobe or in side lobes of the antenna pattern and they interfere with the meteorological echoes at the same range. Therefore the radar image over a certain area may wrongly show precipitation. Since the weather radar measures the reflectivity of precipitation particles aloft, they normally report precipitation earlier than rain gauges on the ground. When, however, the precipitation particles melt or evaporate on their way down to the ground, the error of the radar can be tremendous. An extreme scenario is virga. Virga is rain that falls out of the clouds but evaporates fully in a layer of dry air below the cloud, and so never reaches the ground. The radar is able to pick up the precipitation falling from the cloud but is unable to see it evaporate close to the ground. In such cases the radar would report precipitation while a rain gauge on the ground would not. Because of these and other sources of error for many – especially hydrological and quantitative – applications radar data have to be corrected. An obvious approach is to adjust radar measurements to a network of rain gauges, combining the main advantage of radar namely gapless spatial coverage with that of the rain gauge — accurate measurements on ground level. 3.2. Data available For this study rain gauge and radar data from the province of Styria, Austria, were available. The data sets extend over a two year period. The two available rain gauges, both situated below 500 m above m. s. l., are working on the tipping bucket principle with a resolution of 0.1 mm. Their updating period is 15 min. Referring to the abovementioned sources of errors of the radar system, it should be noted that rain gauges also feature measuring errors and although the rain gauge data used here are officially controlled and verified and unreliable values are ruled out, they cannot be regarded as error-free. Reflectivity measurements are gained from the Doppler weather radar station on Mt. Zirbitzkogel. The designated radar is a high-resolution C-band weather-radar situated on an altitude of 2372 m above m. s. l. The distance between the rain gauges and the weather radar is about 70 km, see Fig. 2. The radar has the following specifications:
Fig. 2. Study area in the province of Styria, Austria, showing radar and rain gauge location.
Time interval between measurements (volume scans): 5 min 3-dB-Beamwidth: 1◦ Minimum elevation angle: 0.8◦ Spatial resolution of the volume element: 1 km3 (1 km × 1 km × 1 km) ∗ Resolution in measured reflectivity: 14 levels of rain-rate, converted from reflectivity Z by using a fixed relationship (Z = 200 · R 1.6 ) ∗ Instrumented range: 220 km. ∗ ∗ ∗ ∗
Since the radar data are available every 5 min, the data of the radar and rain gauge were transferred into the same temporal resolution (15 min) by averaging the radar data. 4. Development of the neural network model 4.1. Input and output variables The first step of the development of the network was defining appropriate input and output vectors. As the network should predict the rain rate on the ground, the output vector consists of one variable namely the rain rate measured by a rain gauge on ground level. The rain gauge provides the rain rate in millimeters every 15 min with a resolution of 0.1 mm. As the study area is often affected by shower events, the measured rain rates can be quite high. As input vector for the task of estimating the rain rate for a point on the ground radar measurements from the space aloft were taken. The question was which radar measurements of the threedimensional space aloft should be chosen as input vector. Liu et al. (2001) used the reflectivity at 1 km height (nine adjacent measurements with the rain gauge at the centre of the quadratic grid) as input vector for their neural network scheme for radar rainfall estimation. Other authors (Li et al., 2003; Teschl et al., 2005; Xu & Chandrasekar, 2005) used superposed reflectivity measurements, the so called vertical profile of reflectivity as input vector. The vertical profile of reflectivity delivers also information about the nature of the precipitation event. Shower
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Fig. 3. Input data of the neural network. On the right-hand side the 14 levels of rain rate of the radar calculated from the reflectivity measurements are shown.
events or convective precipitation are characterized by high reaching clouds up to 10 km and more, whereas advective rainfall events show low level precipitation. This information is regarded as important as the rain rate is also related to the type of rainfall. During advective events the typical rain rate is several mm per hour whereas the same rainfall amount can be observed within 10 min during an average shower event. The accuracy of the radar in this part of Austria is dependent upon the type of rainfall — low level stratiform events or high reaching convective shower cells (Teschl & Randeu, 2005). The highest level where precipitation was detected is an additional parameter, because there is a significant correlation between measured rainfall on the ground and the highest level of precipitation as measured by the radar. In this paper also the vertical profile of reflectivity is taken as input vector. Because of the mountainous terrain in the study area and the occurring effect of beam blockage, reflectivity measurements directly above the rain gauge are not available. Because of the orography the lowest elevation where radar data above the rain gauge are available is 3 km above m. s. l. Four superposed radar reflectivity measurements from 3 to 7 km above m. s. l. were taken as inputs. Fig. 3 shows the setup. Weather radar and rain gauges do not measure the same dimension. In the following, two considerations are presented. Because of its elevation, precipitation is detected by the radar earlier than by the rain gauge. The weather radar measures the reflectivity at 3 km above m. s. l. (minimum) whereas the rain gauge detects the precipitation later when it reaches the ground. An average time lag of 5 min was detected between radar and rain gauge time series. This seems consistent with the mean falling velocity of raindrops. Therefore the network has been set up to predict the ground rainfall 5 min later. The input and output vectors are shifted accordingly. The other consideration addresses the different character of the measurements between weather radar and rain gauge. Other than the rain gauges which deliver point measurements, the weather radar provides volume measurements. This fact makes it inconsistent in the strict sense to simply compare weather radar and rain gauge measurements quantitatively. It would be
desirable to have several rain gauges within the small area of the weather radar grid and to calculate with mathematical methods the areal precipitation out of the point measurements of the rain gauges to compare it with the radar measurements. But the rain gauge density in the study area does not allow this. For modelling the radar rain gauge relationship only one rain gauge site was available. For the development process of the neural network, several sources of data are necessary: Data for training the network, and data for testing the trained network. The dataset was divided into training, validation and test datasets. The test dataset, however, stems from another location, 10 km apart. The training dataset was used to train the network — to determine the weights of the model. The validation dataset was used to determine the number of the hidden neurons. The error on the validation dataset was monitored during the training process and the training was stopped when the validation error increased. In order to examine if the relationship found between radar reflectivity and rain gauge is representative for other sites, data from the other rain gauge and the associated radar data were used as test dataset. This dataset was used to determine the performance of the trained network. The partitioning between training and validation datasets stemming from the first rain gauge site was done randomly. Each of these datasets comprises 449 pairs of radar and corresponding rain gauge measurements. The test dataset originating from the other rain gauge site contains 1310 pairs of measurements. 4.2. Network architecture The types of neural networks that have been used for weather radar rainfall estimation are the back-propagation feed-forward neural network (BPNN) and the radial basis function neural network (RBFNN). Both network types can solve the desired complex function approximation task. The BPNN is one of the most common types of neural networks. It consists of three types of layers: input, hidden and output layer. A BPNN is a feed-forward neural network. It is called “feed-forward” because all of the data information flows in one direction. The
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neurons of one layer are connected with the neurons of the following layer, there is no feedback. Xiao and Chandrasekar (1997) successfully developed a neural network of this type for radar rain-rate estimation. According to Liu et al. (2001) some of the disadvantages of BPNNs are that the training process is computationally demanding. They have chosen a RBFNN for their weather radar rainfall estimation task. A RBFNN consists of two layers, a hidden radial basis layer and an output linear layer. According to Demuth and Beale (1998) RBFNNs require more neurons than BPNNs, and they work best when many training vectors are available. As the number of training vectors was limited the BPNN was chosen for the radar rainfall estimation problem. Here a fully connected BPNN was used where all neurons of two successive layers are connected with each other. The network function of a BPNN is determined largely by the number of neurons in the different layers and the weighted connections between them. An area of conflict is that a small network may have insufficient degrees of freedom (weights and biases) to represent the relationship between radar reflectivity and ground measured rainfall, and a large network with many weights to be adapted may memorize fluctuations in the training data and is therefore not able to generalize. The aspect of generalization capability of the network is very important for this special task, because it is assumed that weather radar measurements contain an amount of observation noise that should not be memorized by the network. Thus the method used to determine the architecture of the BPNN was to start with a small network, to increase the number of hidden nodes and to choose the network with the best performance. When the mean square error on the validation dataset increased, the training was stopped and the minimum of the validation error was taken as an indicator for the performance. The training was performed several times starting with randomly chosen initial weights. Thus a network with two hidden layers each with five nodes was determined. Accordingly the network consists of five nodes in the input layer representing the four superposed radar reflectivity measurements a3 , a4 , a5 , a6 starting at 3 km above m. s. l. The fifth input parameter is the highest level h (in km above m. s. l.) where precipitation was detected. The output layer consists of one node. Except in the output layer where an unbounded linear activation function was used, nonlinear sigmoid functions where utilized (see Fig. 4. for the scheme of the network). An important point concerning the training process of a neural network is the question of the training algorithm. During the training process the weights and biases of the network are modified according to the training algorithm to minimize the error function. Here the mean square error function was used. The error was calculated as the difference between target and network output. The standard training algorithm is the gradient descent algorithm. This method works by moving the weights and biases in the direction of the negative gradient of the performance function. For practical problems this method is often too slow (Demuth & Beale, 1998). In this study two faster training methods were utilized and their influence on the performance of the network was analysed.
Fig. 4. Scheme of the architecture of the used backpropagation feedforward network. Input parameters are the four superposed radar reflectivity measurements from 3 to 7 km above m. s. l. (a3 , a4 , a5 , a6 ) and the highest level h (in km above m. s. l.) where precipitation was detected. The output parameter is the rain rate on the ground. Below the scheme the activation functions utilized in each layer are displayed.
One method was the Levenberg–Marquardt (LM) algorithm (Hagan & Menhaj, 1994). The LM-algorithm is a numerical optimization technique for neural network training. It usually converges much faster than the standard gradient descent training algorithm but it has high memory requirements. The alternatively applied training algorithm is a modification of the LM algorithm. It uses Bayesian regularization in combination with LM training (Foresee & Hagan, 1997) and is known to produce networks which generalize well. The artificial neural network models in this paper are developed using the Matlab Neural Network Toolbox. 4.3. Performance evaluation The performance of the network was evaluated by comparing the rain rate from the rain gauge and that from the radar measurement with statistical measures. The used statistical measures are the correlation coefficient (CORREL), the bias and the root mean square error (RMSE). n P
CORREL = s
(ri − r¯ )(ai − a) ¯ s n n P P ¯ 2 (ri − r¯ )2 · (ai − a) i=1
i=1 n X
1 (ri − ai ) n i=1 v u n u1 X (ri − ai )2 . RMSE = t n i=1
BIAS =
(3)
i=1
(4)
(5)
In Eqs. (3)–(5) n represents the pairs of variates, r rain rate from the rain gauge, and a that from the radar measurements. The values r and a are given in mm per 15 min.
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Fig. 5. Regression analysis between the estimated rain rate from the pristine radar data and the corresponding rain rate measured by the rain gauge. The analysis was done separately with the datasets that were later used for training (a) and for validation (b) of the neural network. The solid straight line is the regression line, showing a significant underestimation of the rain rate by the radar. The dashed straight line is the ideal regression line if there would be on the average neither an under- nor an overestimation by the radar. (Rain rate is in mm/15 min.) Table 1 Performance evaluation of the network
Correlation coefficient RMSE (mm/15 min) BIAS (mm/15 min) Correctly classified (%)
Training dataset
Validation dataset
Test dataset (from new location, 10 km apart)
0.6820 (0.5777) 0.6696 (0.8693) −0.0004 (0.3745) 22.5 (14.5)
0.6307 (0.5249) 0.7508 (0.9206) 0.0043 (0.3929) 20.3 (18.3)
0.3742 (0.2876) 0.9674 (1.0909) 0.0753 (0.4359) 19.5 (13.9)
(Values in parenthesis are for pristine radar data and do not involve any training with neural networks).
These measures were calculated for the pristine radar data (the radar reflectivity measurements of the lowest elevation transformed into rain rate using the standard Z –R-relationship Z = 200 · R 1.6 ) and the rainfall estimates of the neural network after the training process. In addition to the statistical measures a measure was used to take the quantization intervals of the radar into consideration. The weather radar on Mt. Zirbitzkogel delivers data with 14 levels of rain rate. So in the strict sense the radar does not deliver a value but an interval. For processing, all values lying within the quantization interval are assigned to the mean value. Thus the quantization error can be minimized to the half width of the quantization interval. But the quantization intervals increase with the rainfall intensity and so there can be a high bias and root mean square error between rain gauge and radar measurements even if the radar measurements are error-free. Therefore the fourth measure is the percentage of correctly classified radar measurements, saying the rain gauge measurement falls in the interval the radar delivers. This measure relates to rain rates >0 mm, because periods without rainfall would decrease its significance. 5. Results and discussion First the radar estimated rain rate was compared with the rain rate measured by the rain gauge. This radar estimated
rain rate does not involve any training with neural networks. It uses the radar reflectivity measurements of the lowest elevation transformed into rain rate using the standard Z –Rrelationship Z = 200 · R 1.6 . This pristine radar data exhibit a significant underestimation of the rain rate. This can be seen in Fig. 5 which shows a regression analysis between pristine radar and corresponding rain gauge data, separately for the datasets that were later used for training and for validation of the neural network. On the average the radar underestimates the rain rate by 0.3745 mm/15 min in the later training and by 0.3929 mm/15 min in the later validation dataset. The comparison of both regression analyses shows the same characteristic which indicates a proper division of the whole data into training and validation data. The general underestimation is ascribed mainly to attenuation (the site is about 70 km apart from the radar) to beam blockage (because of the mountainous terrain) and to the fact that low level precipitation below 3 km above m. s. l. cannot be detected by the radar. Then the neural network was trained and validated and its estimation for the rain rate was also compared with the rain rate measured by the rain gauge. Table 1 summarizes the performance evaluation of the network for the training, validation and the test dataset separately. The values in parenthesis are for pristine radar data and do not involve any
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Table 2 Performance evaluation of the network when using different training configurations
Correlation coefficient RMSE (mm/15 min) BIAS (mm/15 min) Correctly classified (%)
Pristine data
Neural network model
With data transformation
With modified training alg.
0.2876 1.0909 0.4359 13.9
0.3742 0.9674 0.0753 19.5
0.3329 0.9980 0.0730 21.1
0.3923 0.9683 0.0909 20.7
training with neural networks. The performance enhancement compared to the pristine radar data enhancement is similar for training and validation data. It should be noted though that the figures of the training dataset are on no account representative for the performance of the network when validated on new data. The comparison should emphasize that the error of the training dataset was not driven to a very small value without stopping when the error on the validation dataset increased. It is important that the network does not memorize training data but learns to generalize to new data. The performance evaluation of the neural network with the validation dataset shows an enhancement compared with the pristine radar data. The correlation coefficient increased from 0.5249 to 0.6307, while root mean square error and bias decreased to 0.7508 and 0.0043 respectively. When precipitation occurred the network delivers in 20.3% of the cases the correct quantization interval. This is a slight improvement compared to 18.3% of the pristine radar data. The performance of the neural network can be seen when simulating it with new data (data which was not involved in the training process nor was it used to determine when to stop training). In our case the new data derive from another rain gauge site. The data were formatted in the same way as the training and validation data. Again the radar data was shifted five minutes with respect to the rain gauge measurements. The elevation of this rain gauge is approximately the same as the first one. The sites are about 10 km apart. Though the measures of the test dataset are inherently poorer than the validation dataset, the relative enhancement of the performance figures is quite as high. The correlation coefficient increased to 0.3742, RMSE and bias decreased. Such as the validation dataset the neural network model puts more values in the correct quantization interval (19.5% against 13.9%). It was also examined if certain preprocessing steps like data transformation or changes in the training algorithm can lead to better results. A data transformation method and another training algorithm were applied. Such methods often improve the performance of a neural network. A survey of data transformation methods is given by Shi (2000). Here the applied data transformation method normalizes the mean and standard deviation of the training dataset. The inputs and targets have been normalized so that they have zero mean and unity standard deviation, (see Demuth and Beale (1998)). After the training process the outputs were converted back into the same units which were used for the original targets. The sotrained network, which has the same architecture as the first one, showed improved performance in bias and classification while the other figures are poorer than training without data transformation (see Table 2).
The alternatively applied training algorithm is a modification of the LM algorithm using Bayesian regularization in combination with LM. The results of training our network with the use of Bayesian regularization are: best measure concerning correlation, poorest bias, average RMSE and classification measure. Table 2 compares the performance of the three training configurations. All exhibit better measures than the pristine data. A general enhancement of data transformation and modified training algorithm over the first neural network model cannot be recognized. Altogether the results show that the neural network model has the ability to generalize. The relationship between the vertical reflectivity profile and the rain gauge measurements was learned without memorizing the peculiarity of the training data. The relationship found by the neural network delivers better results than the Z –R relationship used by the Austrian weather radars. 6. Summary and conclusion By means of an artificial neural network the radar reflectivity profile aloft a rain gauge was mapped to the measurements on the ground. Many sources of error are inherent to the weather radar and it is important to understand that not all of them can be corrected or enhanced by the means of data driven models. But what could be done was to find a better general relation between reflectivity profile and rain gauge than the exponential Z –R-relationship. Because of the errors of the radar data, the training process is fundamental. The errors of the training data set can be driven to a very low value with sufficiently big networks, but what is crucial is the performance on new data. The test dataset used here comes from another rain gauge site. This was thought to be important to test the ability to generalize. Although no low-level radar measurements were available the ANN led to better estimates of ground rainfall. In this paper a new performance measure was presented. The radar data exhibit 14 levels of reflectivity and therefore there will always be a quite high quantization error and thus a bias and a root mean square error between radar and rain gauge time series. The new measure gives the percentage of precipitation measurements classified correctly by the radar or the neural network model. This measure forms together with bias, RMSE and correlation coefficient the performance criteria. According to these measures an enhancement of the neural network model over the pristine radar data was determined.
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Acknowledgement We gratefully acknowledge the provision of weather radar data from Austro Control, Aviation Weather Service (Vienna, Austria), and of rain gauge data from the Office of Styrian Regional Government, Hydrographic Department 19A (Graz, Austria). References Aliev, R., Bonfig, K. W., & Aliew, F. (2000). Soft computing: Eine grundlegende Einf¨uhrung. Berlin: Verlag Technik. Bringi, V. N., & Chandrasekar, V. (2001). Polarimetric Doppler weather radar — Principles and applications. Cambridge, MA: Cambridge University Press. Demuth, H., & Beale, M. (1998). Neural network toolbox user’s guide — For use with MATLAB, Version 3. Natick, MA: The MathWorks, Inc. Foresee, F. D., & Hagan, M. T. (1997). Gauss–Newton approximation to Bayesian learning: In Proc. of the 1997 international conference on neural networks (Vol. 3), pp. 1930–1935. Hagan, M. T., & Menhaj, M. (1994). Training feed-forward networks with
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the Marquardt algorithm. IEEE Transactions on Neural Networks, 5(6), 989–993. Li, W., Chandrasekar, V., & Xu, G. (2003). Investigations in radar rainfall estimation using neural networks. In 2003 Proc. IGARSS. Liu, H., Chandrasekar, V., & Xu, G. (2001). An adaptive neural network scheme for rainfall estimation from WSR-88D observations. Journal of Applied Meteorology, 40(11), 2038–2050. Marshall, J. S., & Palmer, W. McK. (1948). The distribution of raindrops with size. Journal of Meteorology, 5, 165–166. Shi, J. J. (2000). Reducing prediction error by transforming input data for neural networks. Journal of Computing in Civil Engineering, 14(2), 109–116. Teschl, R., & Randeu, W. L. (2005). A method to enhance 3D radar rain rate data by interpolating over adjacent grid data. In Proc. WSN05. Teschl, R., Randeu, W. L., & Teschl, F. (2005). A neural network based method to adjust weather radar estimates of rainfall to rain gauge measurements using the vertical reflectivity profile. In Proc. 32nd conference on radar meteorology. Xiao, R., & Chandrasekar, V. (1997). Development of a neural network based algorithm for rainfall estimation from radar observations. IEEE Transactions on Geoscience and Remote Sensing, 35, 160–171. Xu, G., & Chandrasekar, V. (2005). Operational feasibility of neural-networkbased radar rainfall estimation. IEEE Geoscience and Remote Sensing Letters, 2(1), 13–17.
2007 Special Issue
Improving weather radar estimates of rainfall using feed-forward neural networks Reinhard Teschl ∗ , Walter L. Randeu, Franz Teschl Department of Broadband Communications, Graz University of Technology, Inffeldgasse 12/I, 8010 Graz, Austria
Abstract In this paper an approach is described to improve weather radar estimates of rainfall based on a neural network technique. Other than rain gauges which measure the rain rate R directly on the ground, the weather radar measures the reflectivity Z aloft and the rain rate has to be determined over a Z –R relationship. Besides the fact that the rain rate has to be estimated from the reflectivity many other sources of possible errors are inherent to the radar system. In other words the radar measurements contain an amount of observation noise which makes it a demanding task to train the network properly. A feed-forward neural network with Z values as input vector was trained to predict the rain rate R on the ground. The results indicate that the model is able to generalize and the determined input–output relationship is also representative for other sites nearby with similar conditions. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Feed-forward neural network; Weather radar; Precipitation; Rainfall; Reflectivity factor; Drop-size distribution; Vertical profile of reflectivity; Z –R relationship
1. Introduction This paper shows that neural networks can be successfully used for modelling the relationship between weather radar measurements of rainfall and rain gauge data even if the radar data stem from high altitudes, because of mountainous terrain, and even if the radar data are very noisy due to several sources of error. Section 2 gives an overview of the weather radar and how it is currently used to estimate rainfall. In this section also existing neural network approaches to estimate rainfall from weather radar measurements are mentioned. In Section 3 details about the nature and possible errors of radar and rain gauge data are given. Section 4 describes the development of the neural network covering also aspects as input/output variables, details of the learning method and network design. Section 5 discusses the results of the model. The conclusions are given in Section 6. 2. Weather radar rainfall estimation The weather radar is an important tool in meteorology and hydrology and a key instrument in the field of weather ∗ Corresponding author. Tel.: +43 316 873 7431; fax: +43 316 463697.
E-mail address: [email protected] (R. Teschl). c 2007 Elsevier Ltd. All rights reserved. 0893-6080/$ - see front matter doi:10.1016/j.neunet.2007.04.005
forecasting. Although the weather radar can more likely be considered as a qualitative than a quantitative instrument it has always been used to estimate rainfall amounts. The main advantage of weather radars in the estimation of rainfall is their high spatial and temporal resolution and the gapless spatial coverage. If substituting a gauge for each radar spatial sample ten thousands of rain gauges would be needed. Weather radars also provide a high temporal resolution. Today’s weather radars make a full volume scan within about 5 min. The perhaps biggest advantage of weather radars is that they reveal a three-dimensional structure of precipitation. By displaying in what height, what intensity of precipitation can be assumed, weather radars give us an impression of the type of weather system: whether it is an advective situation with lower measured intensities and echoes up to mid-altitudes, or it is a convective situation with very high intensities and echoes up to 15 km. Because of these advantages radar measurements of precipitation have enjoyed widespread operational usage and will remain so in future (Bringi & Chandrasekar, 2001). But other than rain gauges which measure precipitation directly on the ground, the radar determines the reflectivity aloft. Due to this principle, several sources of errors originate (see Section 3.1). Estimating the rain rate on the ground from
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radar reflectivity measurements is still an important topic in radar meteorology. Bringi and Chandrasekar (2001) divide rainfall estimation techniques by radar into physical and statistical/engineering approaches. Physical approaches attempt to estimate rain rates by radar measurements together with an underlying rain model. The rain model generally describes the shape of the rain-drops and their size distribution. Such an approach is used in nearly all operational weather radars. The rain rate is estimated from the measured reflectivity factor by nonlinear relationships. Physical approaches do not include any feedback, e.g. from rain gauges as opposed to statistical/engineering approaches where a feedback is used. A neural network approach as it is presented in this paper tries to estimate the ground rainfall from the radar measurements. For the training and testing process feedback from rain gauges has to be used, while once established, this method is no longer dependent upon gauges. 2.1. Operational radar rainfall estimation Radars work by sending out an electromagnetic beam and measuring how much of the energy of that beam is reflected back. Weather radars use frequencies at which meteorological echoes like raindrops, snowflakes, hail, etc. are ideally detectable. These are in general frequencies between 3 and 10 GHz. Most of the operational weather radars work at C- or S-band (6 resp. 3 GHz). The main measuring unit of weather radars is the reflectivity factor Z . The Austrian weather radar network consists of four (soon five) C-band radars. The data they provide are described in Section 3.2. The radars in Austria and the rain rate estimation technique are comparable with those in most European countries. In Austria we distinguish between 14 intensity levels for the measured reflectivity factor. With a fixed Z –R relationship this leads to 14 levels of measurable rain rates (Fig. 1(a)). The levels reach from 0.04 mm/h up to around 150 mm/h with gradually increasing distance between the levels. The spatial resolution of the measurement is 1 km × 1 km × 1 km. Fig. 1(b) and (c) show two so called RHI-plots of radar data. RHI stands for range-height indicator. It is a type of radar display on which the reflectivity factor is displayed as a function of range and elevation angle. The RHI may be produced by scanning the radar antenna in elevation with the azimuth fixed. Fig. 1(b) shows a RHI for an advective weather situation. Due to the fact that mountains shield the radar waves, only precipitation in higher altitudes is detectable. That is why often only the upper parts of advective weather situations are visible. Fig. 1(c) shows a RHI for a convective weather event. Such clouds can reach up to more than 10 km. The plot also shows that in such events usually very high reflectivities are measured. When estimating rainfall from radar measurements therefore usually the highest value that is measured from 0 to an altitude of 15 km is taken. Fig. 1(d) shows such a weather image of Austria. All the dots in greyscale represent the predicted rainfall in a 1 km × 1 km region. Such measurements can differ by a factor 10 with rain gauge measurements on the ground.
2.2. Proposed improvements Artificial Neural Networks are a proved and efficient method to model complex input–output relationships (Aliev, Bonfig, & Aliew, 2000). In this paper therefore an approach is described to improve weather radar estimates of rainfall by using an Artificial Neural Network (ANN). In this study we use a feedforward neural network to model the relationship between weather radar data aloft and rain gauge measurements on the ground. The network accepts radar data as an input and is trained to predict the rain rate as measured by the rain gauge. The problem of estimating ground rainfall using radar measurements aloft has already been researched with neural networks. Xiao and Chandrasekar (1997) applied a Backpropagation Neural Network (BPNN) for rainfall estimation from radar data. Liu, Chandrasekar, and Xu (2001) developed a Radial Basis Function Neural Network (RBFNN) to estimate ground rainfall using the vertical profile of the radar reflectivity as input vector. Li, Chandrasekar, and Xu (2003) showed that the radar reflectivity from 1 to 4 km height above the rain gauge is the best input vector to a RBFNN for estimating the ground rainfall compared to several other choices. Xu and Chandrasekar (2005) also used this vertical profile as input vector to their RBFNN. Teschl, Randeu, and Teschl (2005) as well defined four superposed radar reflectivity measurements as input vector and added the highest level where precipitation was detected as an additional parameter. In the current study it is analysed if an ANN can lead to better estimates of ground rainfall even in an alpine environment, like in Austria, where low level radar measurements are seldomly available. In contrast to previous studies using radar reflectivity data starting from 1 km above the rain gauge, in the study area the lowest elevation where radar data aloft the rain gauge are available is 3 km (above Mean Sea Level, m. s. l.). The objective of the present study is to find a relationship that is not only valid for the special site on which the model was trained, but also can be applied to other comparable terrain where no rain gauges are available. Henceforth only radar data are necessary and the model gives a better estimate for the rain rate on the ground. In other words it improves the pristine radar data. On the site that has been used to train the network, a variety of other meteorological data like temperature, wind speed, humidity, sunshine duration, etc. were available. These have not been chosen though as input data, since the objective was to make the model applicable on sites were solely radar data are available. 3. Rain gauge vs. weather radar 3.1. Sources of error Rain gauges measure the rain rate quite exactly but the measured rate is only significant for a certain location. A few kilometres farther the situation can be quite different, especially during convective rain events and in alpine terrain. On the other hand weather radars deliver data with good spatial and temporal resolution. They typically determine the rain rate for every square kilometre. One weakness, however, is their often
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Fig. 1. Weather radar images. (a) Levels of rain rate as distinguished by the Austrian weather radar system. The levels reach from 0.04 mm/h up to 153.76 mm/h with gradually increasing distance between the levels. (b) Range–height indication for an advective weather situation. (c) Range–height indication for a convective weather situation. (d) Radar image of Austria. Composite of four Austrian weather radars. The shown values represent the highest that occurred in any measured altitude.
poor metering precision limiting the applicability of the radar for quantitative tasks. Several sources of error of the weather radar occur because of its working principle. Precipitation particles like rain-drops, snowflakes and hailstones but even birds and insects reflect the beam back to the radar. The measuring unit of the radar is the reflectivity factor Z . Z is used to estimate the rain rate using relationships of the form Z = a · Rb where a, b empirical parameters R rain rate in mm/h Z reflectivity factor in mm6 m−3 .
(1)
Every Z –R relationship is based on a certain drop-size distribution. The drop-size distribution in general varies for different climate zones and weather conditions. Therefore dozens of such empirical relationships have been derived. The most common Z –R relationship is that developed by Marshall and Palmer (1948) assuming an exponential dropsize distribution: Z = 200 · R 1.6 .
(2)
This relationship is used in the National Austrian Weather Radar Network from which the used radar data stem. The assumption of a fixed Z –R relationship is an inherent error source when using radar data for quantitative tasks.
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Another possible source of error is beam blockage. It is caused when objects such as trees, buildings or mountains prevent the radar beam from detecting the precipitation on the other side of them. As a result, the radar image may show no precipitation over an area where it may actually be raining. Precipitation particles also attenuate radar waves. This effect occurs twice. It affects the transmitted radar wave as well as the wave that is reflected back from the sensed volume. This effect increases with the distance to the radar since the waves to and from the sensed volume are attenuated by precipitation on the path between radar and sensed volume. Attenuation in general leads to an underestimation of precipitation by the radar since it weakens the signal that the radar receives. It is especially significant when an intense convective cell is located near the radar. Ground clutter is radar echoes from trees, buildings or other objects on the ground. Such echoes may be caused by the reflection of the beam back to the radar in the main lobe or in side lobes of the antenna pattern and they interfere with the meteorological echoes at the same range. Therefore the radar image over a certain area may wrongly show precipitation. Since the weather radar measures the reflectivity of precipitation particles aloft, they normally report precipitation earlier than rain gauges on the ground. When, however, the precipitation particles melt or evaporate on their way down to the ground, the error of the radar can be tremendous. An extreme scenario is virga. Virga is rain that falls out of the clouds but evaporates fully in a layer of dry air below the cloud, and so never reaches the ground. The radar is able to pick up the precipitation falling from the cloud but is unable to see it evaporate close to the ground. In such cases the radar would report precipitation while a rain gauge on the ground would not. Because of these and other sources of error for many – especially hydrological and quantitative – applications radar data have to be corrected. An obvious approach is to adjust radar measurements to a network of rain gauges, combining the main advantage of radar namely gapless spatial coverage with that of the rain gauge — accurate measurements on ground level. 3.2. Data available For this study rain gauge and radar data from the province of Styria, Austria, were available. The data sets extend over a two year period. The two available rain gauges, both situated below 500 m above m. s. l., are working on the tipping bucket principle with a resolution of 0.1 mm. Their updating period is 15 min. Referring to the abovementioned sources of errors of the radar system, it should be noted that rain gauges also feature measuring errors and although the rain gauge data used here are officially controlled and verified and unreliable values are ruled out, they cannot be regarded as error-free. Reflectivity measurements are gained from the Doppler weather radar station on Mt. Zirbitzkogel. The designated radar is a high-resolution C-band weather-radar situated on an altitude of 2372 m above m. s. l. The distance between the rain gauges and the weather radar is about 70 km, see Fig. 2. The radar has the following specifications:
Fig. 2. Study area in the province of Styria, Austria, showing radar and rain gauge location.
Time interval between measurements (volume scans): 5 min 3-dB-Beamwidth: 1◦ Minimum elevation angle: 0.8◦ Spatial resolution of the volume element: 1 km3 (1 km × 1 km × 1 km) ∗ Resolution in measured reflectivity: 14 levels of rain-rate, converted from reflectivity Z by using a fixed relationship (Z = 200 · R 1.6 ) ∗ Instrumented range: 220 km. ∗ ∗ ∗ ∗
Since the radar data are available every 5 min, the data of the radar and rain gauge were transferred into the same temporal resolution (15 min) by averaging the radar data. 4. Development of the neural network model 4.1. Input and output variables The first step of the development of the network was defining appropriate input and output vectors. As the network should predict the rain rate on the ground, the output vector consists of one variable namely the rain rate measured by a rain gauge on ground level. The rain gauge provides the rain rate in millimeters every 15 min with a resolution of 0.1 mm. As the study area is often affected by shower events, the measured rain rates can be quite high. As input vector for the task of estimating the rain rate for a point on the ground radar measurements from the space aloft were taken. The question was which radar measurements of the threedimensional space aloft should be chosen as input vector. Liu et al. (2001) used the reflectivity at 1 km height (nine adjacent measurements with the rain gauge at the centre of the quadratic grid) as input vector for their neural network scheme for radar rainfall estimation. Other authors (Li et al., 2003; Teschl et al., 2005; Xu & Chandrasekar, 2005) used superposed reflectivity measurements, the so called vertical profile of reflectivity as input vector. The vertical profile of reflectivity delivers also information about the nature of the precipitation event. Shower
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Fig. 3. Input data of the neural network. On the right-hand side the 14 levels of rain rate of the radar calculated from the reflectivity measurements are shown.
events or convective precipitation are characterized by high reaching clouds up to 10 km and more, whereas advective rainfall events show low level precipitation. This information is regarded as important as the rain rate is also related to the type of rainfall. During advective events the typical rain rate is several mm per hour whereas the same rainfall amount can be observed within 10 min during an average shower event. The accuracy of the radar in this part of Austria is dependent upon the type of rainfall — low level stratiform events or high reaching convective shower cells (Teschl & Randeu, 2005). The highest level where precipitation was detected is an additional parameter, because there is a significant correlation between measured rainfall on the ground and the highest level of precipitation as measured by the radar. In this paper also the vertical profile of reflectivity is taken as input vector. Because of the mountainous terrain in the study area and the occurring effect of beam blockage, reflectivity measurements directly above the rain gauge are not available. Because of the orography the lowest elevation where radar data above the rain gauge are available is 3 km above m. s. l. Four superposed radar reflectivity measurements from 3 to 7 km above m. s. l. were taken as inputs. Fig. 3 shows the setup. Weather radar and rain gauges do not measure the same dimension. In the following, two considerations are presented. Because of its elevation, precipitation is detected by the radar earlier than by the rain gauge. The weather radar measures the reflectivity at 3 km above m. s. l. (minimum) whereas the rain gauge detects the precipitation later when it reaches the ground. An average time lag of 5 min was detected between radar and rain gauge time series. This seems consistent with the mean falling velocity of raindrops. Therefore the network has been set up to predict the ground rainfall 5 min later. The input and output vectors are shifted accordingly. The other consideration addresses the different character of the measurements between weather radar and rain gauge. Other than the rain gauges which deliver point measurements, the weather radar provides volume measurements. This fact makes it inconsistent in the strict sense to simply compare weather radar and rain gauge measurements quantitatively. It would be
desirable to have several rain gauges within the small area of the weather radar grid and to calculate with mathematical methods the areal precipitation out of the point measurements of the rain gauges to compare it with the radar measurements. But the rain gauge density in the study area does not allow this. For modelling the radar rain gauge relationship only one rain gauge site was available. For the development process of the neural network, several sources of data are necessary: Data for training the network, and data for testing the trained network. The dataset was divided into training, validation and test datasets. The test dataset, however, stems from another location, 10 km apart. The training dataset was used to train the network — to determine the weights of the model. The validation dataset was used to determine the number of the hidden neurons. The error on the validation dataset was monitored during the training process and the training was stopped when the validation error increased. In order to examine if the relationship found between radar reflectivity and rain gauge is representative for other sites, data from the other rain gauge and the associated radar data were used as test dataset. This dataset was used to determine the performance of the trained network. The partitioning between training and validation datasets stemming from the first rain gauge site was done randomly. Each of these datasets comprises 449 pairs of radar and corresponding rain gauge measurements. The test dataset originating from the other rain gauge site contains 1310 pairs of measurements. 4.2. Network architecture The types of neural networks that have been used for weather radar rainfall estimation are the back-propagation feed-forward neural network (BPNN) and the radial basis function neural network (RBFNN). Both network types can solve the desired complex function approximation task. The BPNN is one of the most common types of neural networks. It consists of three types of layers: input, hidden and output layer. A BPNN is a feed-forward neural network. It is called “feed-forward” because all of the data information flows in one direction. The
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neurons of one layer are connected with the neurons of the following layer, there is no feedback. Xiao and Chandrasekar (1997) successfully developed a neural network of this type for radar rain-rate estimation. According to Liu et al. (2001) some of the disadvantages of BPNNs are that the training process is computationally demanding. They have chosen a RBFNN for their weather radar rainfall estimation task. A RBFNN consists of two layers, a hidden radial basis layer and an output linear layer. According to Demuth and Beale (1998) RBFNNs require more neurons than BPNNs, and they work best when many training vectors are available. As the number of training vectors was limited the BPNN was chosen for the radar rainfall estimation problem. Here a fully connected BPNN was used where all neurons of two successive layers are connected with each other. The network function of a BPNN is determined largely by the number of neurons in the different layers and the weighted connections between them. An area of conflict is that a small network may have insufficient degrees of freedom (weights and biases) to represent the relationship between radar reflectivity and ground measured rainfall, and a large network with many weights to be adapted may memorize fluctuations in the training data and is therefore not able to generalize. The aspect of generalization capability of the network is very important for this special task, because it is assumed that weather radar measurements contain an amount of observation noise that should not be memorized by the network. Thus the method used to determine the architecture of the BPNN was to start with a small network, to increase the number of hidden nodes and to choose the network with the best performance. When the mean square error on the validation dataset increased, the training was stopped and the minimum of the validation error was taken as an indicator for the performance. The training was performed several times starting with randomly chosen initial weights. Thus a network with two hidden layers each with five nodes was determined. Accordingly the network consists of five nodes in the input layer representing the four superposed radar reflectivity measurements a3 , a4 , a5 , a6 starting at 3 km above m. s. l. The fifth input parameter is the highest level h (in km above m. s. l.) where precipitation was detected. The output layer consists of one node. Except in the output layer where an unbounded linear activation function was used, nonlinear sigmoid functions where utilized (see Fig. 4. for the scheme of the network). An important point concerning the training process of a neural network is the question of the training algorithm. During the training process the weights and biases of the network are modified according to the training algorithm to minimize the error function. Here the mean square error function was used. The error was calculated as the difference between target and network output. The standard training algorithm is the gradient descent algorithm. This method works by moving the weights and biases in the direction of the negative gradient of the performance function. For practical problems this method is often too slow (Demuth & Beale, 1998). In this study two faster training methods were utilized and their influence on the performance of the network was analysed.
Fig. 4. Scheme of the architecture of the used backpropagation feedforward network. Input parameters are the four superposed radar reflectivity measurements from 3 to 7 km above m. s. l. (a3 , a4 , a5 , a6 ) and the highest level h (in km above m. s. l.) where precipitation was detected. The output parameter is the rain rate on the ground. Below the scheme the activation functions utilized in each layer are displayed.
One method was the Levenberg–Marquardt (LM) algorithm (Hagan & Menhaj, 1994). The LM-algorithm is a numerical optimization technique for neural network training. It usually converges much faster than the standard gradient descent training algorithm but it has high memory requirements. The alternatively applied training algorithm is a modification of the LM algorithm. It uses Bayesian regularization in combination with LM training (Foresee & Hagan, 1997) and is known to produce networks which generalize well. The artificial neural network models in this paper are developed using the Matlab Neural Network Toolbox. 4.3. Performance evaluation The performance of the network was evaluated by comparing the rain rate from the rain gauge and that from the radar measurement with statistical measures. The used statistical measures are the correlation coefficient (CORREL), the bias and the root mean square error (RMSE). n P
CORREL = s
(ri − r¯ )(ai − a) ¯ s n n P P ¯ 2 (ri − r¯ )2 · (ai − a) i=1
i=1 n X
1 (ri − ai ) n i=1 v u n u1 X (ri − ai )2 . RMSE = t n i=1
BIAS =
(3)
i=1
(4)
(5)
In Eqs. (3)–(5) n represents the pairs of variates, r rain rate from the rain gauge, and a that from the radar measurements. The values r and a are given in mm per 15 min.
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Fig. 5. Regression analysis between the estimated rain rate from the pristine radar data and the corresponding rain rate measured by the rain gauge. The analysis was done separately with the datasets that were later used for training (a) and for validation (b) of the neural network. The solid straight line is the regression line, showing a significant underestimation of the rain rate by the radar. The dashed straight line is the ideal regression line if there would be on the average neither an under- nor an overestimation by the radar. (Rain rate is in mm/15 min.) Table 1 Performance evaluation of the network
Correlation coefficient RMSE (mm/15 min) BIAS (mm/15 min) Correctly classified (%)
Training dataset
Validation dataset
Test dataset (from new location, 10 km apart)
0.6820 (0.5777) 0.6696 (0.8693) −0.0004 (0.3745) 22.5 (14.5)
0.6307 (0.5249) 0.7508 (0.9206) 0.0043 (0.3929) 20.3 (18.3)
0.3742 (0.2876) 0.9674 (1.0909) 0.0753 (0.4359) 19.5 (13.9)
(Values in parenthesis are for pristine radar data and do not involve any training with neural networks).
These measures were calculated for the pristine radar data (the radar reflectivity measurements of the lowest elevation transformed into rain rate using the standard Z –R-relationship Z = 200 · R 1.6 ) and the rainfall estimates of the neural network after the training process. In addition to the statistical measures a measure was used to take the quantization intervals of the radar into consideration. The weather radar on Mt. Zirbitzkogel delivers data with 14 levels of rain rate. So in the strict sense the radar does not deliver a value but an interval. For processing, all values lying within the quantization interval are assigned to the mean value. Thus the quantization error can be minimized to the half width of the quantization interval. But the quantization intervals increase with the rainfall intensity and so there can be a high bias and root mean square error between rain gauge and radar measurements even if the radar measurements are error-free. Therefore the fourth measure is the percentage of correctly classified radar measurements, saying the rain gauge measurement falls in the interval the radar delivers. This measure relates to rain rates >0 mm, because periods without rainfall would decrease its significance. 5. Results and discussion First the radar estimated rain rate was compared with the rain rate measured by the rain gauge. This radar estimated
rain rate does not involve any training with neural networks. It uses the radar reflectivity measurements of the lowest elevation transformed into rain rate using the standard Z –Rrelationship Z = 200 · R 1.6 . This pristine radar data exhibit a significant underestimation of the rain rate. This can be seen in Fig. 5 which shows a regression analysis between pristine radar and corresponding rain gauge data, separately for the datasets that were later used for training and for validation of the neural network. On the average the radar underestimates the rain rate by 0.3745 mm/15 min in the later training and by 0.3929 mm/15 min in the later validation dataset. The comparison of both regression analyses shows the same characteristic which indicates a proper division of the whole data into training and validation data. The general underestimation is ascribed mainly to attenuation (the site is about 70 km apart from the radar) to beam blockage (because of the mountainous terrain) and to the fact that low level precipitation below 3 km above m. s. l. cannot be detected by the radar. Then the neural network was trained and validated and its estimation for the rain rate was also compared with the rain rate measured by the rain gauge. Table 1 summarizes the performance evaluation of the network for the training, validation and the test dataset separately. The values in parenthesis are for pristine radar data and do not involve any
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Table 2 Performance evaluation of the network when using different training configurations
Correlation coefficient RMSE (mm/15 min) BIAS (mm/15 min) Correctly classified (%)
Pristine data
Neural network model
With data transformation
With modified training alg.
0.2876 1.0909 0.4359 13.9
0.3742 0.9674 0.0753 19.5
0.3329 0.9980 0.0730 21.1
0.3923 0.9683 0.0909 20.7
training with neural networks. The performance enhancement compared to the pristine radar data enhancement is similar for training and validation data. It should be noted though that the figures of the training dataset are on no account representative for the performance of the network when validated on new data. The comparison should emphasize that the error of the training dataset was not driven to a very small value without stopping when the error on the validation dataset increased. It is important that the network does not memorize training data but learns to generalize to new data. The performance evaluation of the neural network with the validation dataset shows an enhancement compared with the pristine radar data. The correlation coefficient increased from 0.5249 to 0.6307, while root mean square error and bias decreased to 0.7508 and 0.0043 respectively. When precipitation occurred the network delivers in 20.3% of the cases the correct quantization interval. This is a slight improvement compared to 18.3% of the pristine radar data. The performance of the neural network can be seen when simulating it with new data (data which was not involved in the training process nor was it used to determine when to stop training). In our case the new data derive from another rain gauge site. The data were formatted in the same way as the training and validation data. Again the radar data was shifted five minutes with respect to the rain gauge measurements. The elevation of this rain gauge is approximately the same as the first one. The sites are about 10 km apart. Though the measures of the test dataset are inherently poorer than the validation dataset, the relative enhancement of the performance figures is quite as high. The correlation coefficient increased to 0.3742, RMSE and bias decreased. Such as the validation dataset the neural network model puts more values in the correct quantization interval (19.5% against 13.9%). It was also examined if certain preprocessing steps like data transformation or changes in the training algorithm can lead to better results. A data transformation method and another training algorithm were applied. Such methods often improve the performance of a neural network. A survey of data transformation methods is given by Shi (2000). Here the applied data transformation method normalizes the mean and standard deviation of the training dataset. The inputs and targets have been normalized so that they have zero mean and unity standard deviation, (see Demuth and Beale (1998)). After the training process the outputs were converted back into the same units which were used for the original targets. The sotrained network, which has the same architecture as the first one, showed improved performance in bias and classification while the other figures are poorer than training without data transformation (see Table 2).
The alternatively applied training algorithm is a modification of the LM algorithm using Bayesian regularization in combination with LM. The results of training our network with the use of Bayesian regularization are: best measure concerning correlation, poorest bias, average RMSE and classification measure. Table 2 compares the performance of the three training configurations. All exhibit better measures than the pristine data. A general enhancement of data transformation and modified training algorithm over the first neural network model cannot be recognized. Altogether the results show that the neural network model has the ability to generalize. The relationship between the vertical reflectivity profile and the rain gauge measurements was learned without memorizing the peculiarity of the training data. The relationship found by the neural network delivers better results than the Z –R relationship used by the Austrian weather radars. 6. Summary and conclusion By means of an artificial neural network the radar reflectivity profile aloft a rain gauge was mapped to the measurements on the ground. Many sources of error are inherent to the weather radar and it is important to understand that not all of them can be corrected or enhanced by the means of data driven models. But what could be done was to find a better general relation between reflectivity profile and rain gauge than the exponential Z –R-relationship. Because of the errors of the radar data, the training process is fundamental. The errors of the training data set can be driven to a very low value with sufficiently big networks, but what is crucial is the performance on new data. The test dataset used here comes from another rain gauge site. This was thought to be important to test the ability to generalize. Although no low-level radar measurements were available the ANN led to better estimates of ground rainfall. In this paper a new performance measure was presented. The radar data exhibit 14 levels of reflectivity and therefore there will always be a quite high quantization error and thus a bias and a root mean square error between radar and rain gauge time series. The new measure gives the percentage of precipitation measurements classified correctly by the radar or the neural network model. This measure forms together with bias, RMSE and correlation coefficient the performance criteria. According to these measures an enhancement of the neural network model over the pristine radar data was determined.
R. Teschl et al. / Neural Networks 20 (2007) 519–527
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