Radio Science

Forecasting total electron content maps by neural network technique

Authors


Abstract

[1] Near-Earth space processes are highly nonlinear. Since the 1990s, a small group at the Middle East Technical University in Ankara has been working on a data-driven generic model of such processes, that is, forecasting and nowcasting of a near-Earth space parameter of interest. The model developed is called the Middle East Technical University Neural Network (METU-NN) model. The METU-NN is a data-driven neural network model of one hidden layer and several neurons. In order to understand more about the complex response of the magnetosphere and ionosphere to extreme solar events, we chose this time the series of space weather events in November 2003. Total electron content (TEC) values of the ionosphere are forecast during these space weather events. In order to facilitate an easier interpretation of the forecast TEC values, maps of TEC are produced by using the Bezier surface-fitting technique.

1. Introduction

[2] Unpredictable variability of the ionospheric parameters due to space weather–borne disturbances limits the efficiency of communications, radar, and navigation systems. Forecasting the number of electrons in a column of 1 m2 cross section along a path through the ionosphere or the total electron content (TEC) values is crucial for satellite-based navigation systems, especially in the disturbed space weather conditions. For more information, see “Space Weather–Total Electron Content of the Ionosphere,” Rutherford Appleton Laboratory, http://www.chilbolton.rl.ac.uk/weather/tec.htm.

[3] It is most desirable to drive mathematical ionospheric forecasting and mapping models based on physics to incorporate them in ionospheric services and activities. However, this is a very complex and prohibitively difficult task. There has been no concrete, adoptable mathematical model agreed upon up to now to our knowledge.

[4] Neural networks are data-driven (data-based) modeling approaches inspired by simplified human brain processing. They provide parallel processing of complex nonlinear processes. Neurons in neural networks are defined as information-processing units consisting of connecting links, an adder, and activation function. The adder is for summing bias and the input signals weighted in the neuron's connecting links. An activation function follows the adder for limiting the extreme amplitudes of the output of the neuron [Haykin, 1999]. An artificial neural network is a system of interconnected computational elements, the neurons, operating in parallel, arranged in patterns similar to biological neural nets and modeled after the human brain [Tulunay, 1991; Y. Tulunay et al., 2004a].

[5] The architecture of neural networks is formed by determination of the neuron structures and their connections. In a layered neural network, the neurons are organized in the form of layers. In multilayer feed-forward neural networks, source nodes in input layer project onto hidden neurons in the hidden layer. The last hidden layer in the architecture projects onto an output layer of nodes. Those networks are strictly feed forward, and they are fully connected in the sense that every neuron in each layer of the network is connected to every other neuron in the adjacent forward layer [Haykin, 1999; Y. Tulunay et al., 2004a].

[6] Neural network–based approaches are promising in modeling of ionospheric processes [Tulunay, 1991; Williscroft and Poole, 1996; Altinay et al., 1997; Cander et al., 1998; Wintoft and Cander, 1999; Francis et al., 2000; Y. Tulunay et al., 2001, 2004b; Vernon and Cander, 2002; E. Tulunay et al., 2004; Radicella and Tulunay, 2004; Stamper et al., 2004; McKinnell and Poole, 2004]. Space weather centers provide forecasts of solar and geophysical parameters. As an example, the Lund Space Weather Center uses artificial intelligence (AI) in its methods to forecast Kp parameters [Boberg et al., 2000].

[7] In general, mapping of an ionospheric quantity such as foF2 or TEC means that a surface fitting is performed on the basis of known values of that quantity on specified points of a surface. Mapping carried out by using a certain method extrapolates the known discrete values continuously to the whole surface. There are various widely used ionospheric mapping techniques using both the ionosonde-derived TEC and the GPS-TEC [Samardjiev et al., 1993; Cander, 2003; Jakowski et al., 2004; Stamper et al., 2004; Zolesi et al., 2004]. Samardjiev et al. [1993] used contouring techniques for ionospheric mapping including Kriging technique, which performs best when compared with inverse distance squared technique and minimum curvature technique. Cander [2003] discussed the findings of the European Union Action COST 251 and plans for the COST 271 on TEC forecast and mapping. Jakowski et al. [2004] and Stamper et al. [2004] presented near-real-time and real-time TEC mappings over Europe. Zolesi et al. [2004] presented a method based on a regional model of the standard vertical incidence monthly median ionospheric characteristics, which was updated with real-time ionospheric observations for mapping of ionospheric conditions over Europe. It is suitable to be used in real time for operational applications. These studies did not report mapping based on forecasts.

[8] In this work, a method was developed to perform neural network–based TEC forecast mapping for the first time to the best knowledge of the authors. The Middle East Technical University Neural Network (METU-NN) model [Y. Tulunay et al., 2004a] was used to produce TEC forecast maps over Europe using Bezier surfaces which are being used for surface generation in computer graphics [Rogers and Adams, 1990]. Brief information concerning mapping and Bezier surfaces is presented in section 4. In this work, up to 1 hour in advance forecast of the 10 min TEC maps over Europe during November 2003 space weather events using the METU-NN and Bezier surfaces has been introduced and the results are presented.

[9] Neural network models are designed and trained with significant inputs. In our approach, the basic inputs for the model are the past TEC values and the temporal inputs as explained in section 3. No other parameters are used in inputs. This is the first time METU-NN-based TEC forecast mappings are introduced, and the forecast results are promising for system operators. This paper leads to availability of TEC forecast mapping results for making comparisons in future studies. The neural network architecture of METU-NN is modular. Because of modularity, the model and its input parameters are open to new developments depending on future requirements. The subblocks in the METU-NN have one input layer, one hidden layer with the neurons, and one output layer. The neural network is trained and is used to forecast the TEC values for the grids located over Europe. Using these forecast TEC values of the grids, TEC maps as Bezier surfaces are presented.

[10] The main contributions of this work are organization of data for teaching complex processes, neural network–based modeling of a highly complex nonlinear process such as TEC mapping, and general demonstration of learning capability by calculating cross correlations and general demonstration of reaching a proper operating point by calculating errors. The METU-NN model can also be used in filling data gaps.

2. Preparation of Data for the METU-NN

[11] Ten minute vertical TEC data have been evaluated from the GPS measurements that took place between 1 November and 11 December 2003 over Europe centered over Italy on the basis of slant TEC data (G. Ciraolo, private communication, 2004). The geographic coverage of the TEC data is between latitudes of (35.5°N, 47.5°N) and longitudes of (5.5°E, 19.5°E). The data belong to the 104 grid locations spaced every 2° longitude by 1° latitude intervals in space. These data consisted of the training, test, and validation subsets during the development and operation modes of the modeling process.

[12] Table 1 illustrates how the data were assigned to be employed by the METU-NN model during the “training,” “test,” and “validation” modes. In particular, the period of major space weather events was chosen for the validation mode. That is, the solar active region, the sunspot group 484 (or near the sunspot group 501), was the seat of a major coronal mass ejection (CME) on 18 November 2003. This CME triggered a geomagnetic storm on 20 November 2003 at around 0800 UT. This storm was qualified by the 3 hour planetary magnetic index of Kp as 8+. For more information, see “News and information about the Sun-Earth Environment,” http://www.spaceweather.com, and National Geophysical Data Center, ftp://ftp.ngdc.noaa.gov/STP/GEOMAGNETIC_DATA/INDICES/KP_AP/. However, in principle, all the data subsets were chosen from periods of similar Zurich sunspot numbers. The models contain intrinsic information about the solar activity.

Table 1. Selection of the Time Periods for the Input Data
PhaseYearDays
Train20031–15 Nov
Test200330 Nov to 11 Dec
Validation200316–29 Nov

3. Construction of the Neural Network–Based Model

[13] The construction work of the neural network–based model is carried out in the development mode. It is composed of a “training phase or learning phase” and a “test phase” [Y. Tulunay et al., 2004a]. Training and test phases are best performed with independent but statistically similar data sets. The METU-NN model, similar to neural network modeling in general, employs a large amount of data and considerable computing time. It is natural that the nonlinear inherent processes are to be learned by the model during the learning phase as fast as possible. The Levenberg-Marquardt back-propagation algorithm is chosen to be the most convenient one during the training and development phases for this work. The Levenberg-Marquardt algorithm is an approximation to Newton's method [Hagan and Menhaj, 1994; Haykin, 1999]. Instead of the basic back-propagation algorithm, the Levenberg-Marquardt back-propagation algorithm using the approximation to Newton's method is faster in terms of computation time and is more accurate near an error minimum. The Newton's method modification to the steepest descent algorithm and random initializations of the model parameters provide the model parameters to reach near-global optimum values in the training.

[14] As the training advances, the training error starts to decrease, and it eventually goes to zero. Zero error corresponds to a memorization state. Memorization means the loss of the generalization capability of the neural network. The METU-NN model parameters are recorded when the gradient of the error in the validation phase of the development procedure becomes near zero. To prevent memorization, the training is halted immediately. Also, independent validation data are used. Errors are then calculated again. The decrease in the validation error is noted. Training is restarted, and the training cycle is repeated. When the gradient of the error in the validation phase becomes near zero and if the error is then an increasing sequence, a “stop training” signal is produced, and with this, training is terminated. Thus getting stuck in local minima is avoided. The model is then ready for its actual use in the operation mode for forecasting of the TEC. Otherwise the training is not terminated until the gradient of the error becomes near zero again. Thus the model parameters are optimized and are fixed at the end of the construction procedure. In the operation mode, Bezier surface maps of the validation data are used for calculating the errors, point by point, to measure the performance of the model.

[15] The value of the TEC at the time instant k is designated by f(k). The output is f(k + 10) in 10 min in advance forecasts and f(k + 60) in 60 min in advance forecasts. It is the value of the TEC to be observed up to 1 hour later than the present time for this work. There are 419 inputs fed into the METU-NN model. Three of the inputs are the temporal inputs, i.e., dnd, Cm, and Sm. The rest of the input parameters are the inputs related to the history of the TEC values for the grids over Europe, i.e., 104 f(k) values, 104 Δ1(k) values, 104 Δ2(k) values, and 104 RΔ(k) values. Table 2 presents the input parameters employed for present values of the TEC for the grids,

equation image

first differences,

equation image

second differences,

equation image

relative differences,

equation image

serial date number difference, dnd, where dnd is present date number minus the first date number of the data of interest,

equation image

cosine component of the minute, m, of the day,

equation image

and sine component of the minute of the day,

equation image

When 10 min in advance forecasting is required, h is set to 10 min, and when 1 hour in advance forecasting is required, h is set to 60 min.

Table 2. Input Parameters of the METU-NN Blocks
Input ParametersNotation
Present values of the TEC for the gridsTEC(k)
First differences for the gridsΔ1(k)
Second difference for the gridΔ2(k)
Relative difference for the gridRΔ(k)
Serial date number differencednd(k)
Cosine component of the minute of the dayCm
Sine component of the minute of the daySm

[16] Date numbers start with 1 January of year 0 as date number 1. By calculating the serial date number difference, the start value is shifted to the first date of the data of interest. In this study, the first date of the data of interest is 1 November 2003, 0005 UT.

[17] Figure 1 shows the architecture of the neural network modules. The modular structure of the METU-NN provides the development and operation modes to be fast and robust. In this work, the METU-NN model has 104 modules of neural networks. The number of modules corresponds to the number of grids for the region of interest. The inputs of the modules are the present TEC values (number 104) and the first differences (number 104) for each grid; first, second, and relative differences (number 3) for the grid of interest; the present date number difference (number 1); and the present trigonometric components (number 2) of minute of the day. Thus, for each module, there are 214 inputs. When common inputs of the modules are not counted, the overall number of inputs for the METU-NN model is 419. Hidden neurons correspond to state-like variables of the system of interest. During training, the state-like variables of all of the METU-NN modules are determined for each grid. The METU-NN model has 104 outputs corresponding to 104 modules. The output of each module is the forecast value of TEC for the grid of interest. For the modules, among the various neural network structures, the best configurations are found to be the ones with one hidden layer. Six neurons are used in the hidden layer of the modules.

Figure 1.

Architecture of the neural network modules. (METU-NN model has 104 modules, each having 214 inputs, six hidden neurons in one hidden layer, and one output. METU-NN has a total of 419 distinct inputs and 104 outputs.)

4. Brief Information Concerning Mapping and Bezier Surfaces

[18] Mapping covers a portion of land. As an example, consider the European area, which is bounded by the latitudes (35.5°N, 47.5°N) and longitudes (5.5°E, 19.5°E). This area is partitioned by using a grid structure. Grid points or local control points are thus defined.

[19] In practice, the number of control points can be increased by increasing the number of defining polygon vertices. Local control provides the capability of including possible variations around a local control point without interfering with other distant localities of the mapping area.

[20] Bezier surfaces, which are used in such mapping for the first time, have some advantages [Rogers and Adams, 1990]. The availability of the GPS data to be used for TEC evaluation provides a larger number of polygon vertices for Bezier surfaces. Thus better surface fit is achieved.

[21] TEC values are forecast by using the METU-NN model. Mapping is performed over the area of interest by using a Bezier surface. The Bezier surface is advantageous since it can provide more control points to increase the quality of fit as compared with other surface patches such as bilinear, ruled, linear Coons, and Coons bicubic surface patches. Coons bicubic surface needs the specification of precise, nonintuitive mathematical information such as position, tangent, and twist vectors as in the cubic spline curves [Rogers and Adams, 1990]. Therefore there are difficulties limiting its use in practice. These difficulties are overcome by using Bezier surfaces.

[22] In this work, 104 grid locations corresponding to 104 defining polygon vertices are used to obtain sufficient control in mapping. The TEC forecast value at any location on the Bezier surface can be calculated as follows:

equation image

where

equation image
equation image

where

B

matrix values correspond to the METU-NN outputs for the grids;

n + 1 = 8

is the number of longitude grids for each latitude;

m + 1 = 13

is the number of latitude grids for each longitude;

u

is the normalized longitude variable in the region of interest;

w

is the normalized latitude variable in the region of interest.

5. Results

[23] The TEC-trained METU-NN model was used for forecasting TEC values up to 1 hour in advance during 16–29 November 2003. The time period includes the major November 2003 space weather event. Then maps of TEC are constructed by using the Bezier surface mapping technique. Observed TEC values are used only for the grid locations. METU-NN is trained with the observed TEC data to give the outputs, the forecast TEC values, for the grid locations. The TEC mapping is not performed during training because the observed TEC values for the whole region are not a priori except the grid locations. After the forecast operation, TEC mapping is performed. Figure 2a illustrates the variations of both the 10 min in advance forecast and observed TEC values for the grid location (13.5°E, 41.5°N) during the whole period of interest, 16–29 November 2003. Figure 2b, covering the period 19–21 November 2003, is a subset of Figure 2a. Similarly, Figure 3a illustrates the variations of both the 1 hour in advance forecast and observed TEC values for the grid location (13.5°E, 41.5°N) during 16–29 November 2003. Figure 3b, covering the period 19–21 November 2003, is a subset of Figure 3a. The diurnal minute-long variation of the TEC values is shown in the vertical axis, and the horizontal axis is the days of the November 2003 in minute intervals. To a first approximation, the agreement between the forecast and observed GPS TEC values are in very good agreement on visual inspection.

Figure 2a.

Observed (dots) and 10 min ahead forecast (solid line) TEC during 16–29 November 2003 for the single grid point (13.5°E, 41.5°N).

Figure 2b.

Observed (dots) and 10 min ahead forecast (solid line) TEC during 19–21 November 2003 for the single grid point (13.5°E, 41.5°N).

Figure 3a.

Observed (dots) and 1 hour ahead forecast (solid line) TEC during 16–29 November 2003 for the single grid point (13.5°E, 41.5°N).

Figure 3b.

Observed (dots) and 1 hour ahead forecast (solid line) TEC during 19–21 November 2003 for the single grid point (13.5°E, 41.5°N).

[24] Figure 4 illustrates the variations of the forecast and observed TEC values for the 1 hour in advance TEC maps for the big geomagnetic storms of 20 November 2003, at 0930, 1340, 1530, and 1720 UT, respectively.

Figure 4.

Observed and 1 hour ahead forecast TEC map examples during 20 November 2003.

[25] Figure 5a presents the scatter diagram of the 10 min in advance forecast and observed TEC data for whole of the 104 grid locations during 16–29 November 2003. In order to appreciate the performance of the METU-NN, individual reference can be made to Figures 5b, 5c, and 5d, which present the scatter diagrams of the forecast and the observed TEC data at the grid locations (11.5°E, 38.5°N), (13.5°E, 41.5°N), and (15.5°E, 44.5°N), respectively, during 16–29 November 2003.

Figure 5a.

Scatter diagram (dots) with best fit line (solid line) for the 10 min ahead forecast mapping and observed TEC values for all grid points for the validation time 16–29 November 2003.

Figure 5b.

Scatter diagram (dots) with best fit line (solid line) for the 10 min ahead forecast mapping and observed TEC values for the single grid point (11.5°E, 38.5°N) for the validation time 16–29 November 2003.

Figure 5c.

Scatter diagram (dots) with best fit line (solid line) for the 10 min ahead forecast mapping and observed TEC values for the single grid point (13.5°E, 41.5°N) for the validation time 16–29 November 2003.

Figure 5d.

Scatter diagram (dots) with best fit line (solid line) for the 10 min ahead forecast mapping and observed TEC values for the single grid point (15.5°E, 44.5°N) for the validation time 16–29 November 2003.

[26] Figure 6a presents the scatter diagram of the 1 hour in advance forecast and observed TEC data for whole of the 104 grid locations during 16–29 November 2003. Figures 6b, 6c, and 6d present the scatter diagrams of the forecast and the observed TEC data at the grid locations (11.5°E, 38.5°N), (13.5°E, 41.5°N), and (15.5°E, 44.5°N), respectively, during 16–29 November 2003. Best fit lines of near to 45° slopes, almost passing through the origins in Figures 5a–6d, indicate small forecasting errors.

Figure 6a.

Scatter diagram (dots) with best fit line (solid line) for the 1 hour ahead forecast mapping and observed TEC values for all grid points for the validation time 16–29 November 2003.

Figure 6b.

Scatter diagram (dots) with best fit line (solid line) for the 1 hour ahead forecast mapping and observed TEC values for the single grid point (11.5°E, 38.5°N) for the validation time 16–29 November 2003.

Figure 6c.

Scatter diagram (dots) with best fit line (solid line) for the 1 hour ahead forecast mapping and observed TEC values for the single grid point (13.5°E, 41.5°N) for the validation time 16–29 November 2003.

Figure 6d.

Scatter diagram (dots) with best fit line (solid line) for the 1 hour ahead forecast mapping and observed TEC values for the single grid point (15.5°E, 44.5°N) for the validation time 16–29 November 2003.

[27] In order to examine the performance of the METU-NN during the geomagnetic storm on 20 November 2003, reference can be made to Figures 7a, 7b, 8a, and 8b. Figures 7a and 8a illustrate the scatter diagrams of the 10 min in advance and 1 hour in advance forecasts and observed TEC data for whole of the 104 grid locations during 20 November 2003. Figures 7b and 8b present the scatter diagrams of the forecast and the observed TEC data at the grid location (13.5°E, 41.5°N) during 20 November 2003.

Figure 7a.

Scatter diagram (dots) with best fit line (solid line) for the 10 min ahead forecast mapping and observed TEC values for all grid points for the day 20 November 2003.

Figure 7b.

Scatter diagram (dots) with best fit line (solid line) for the 10 min ahead forecast mapping and observed TEC values for the single grid point (13.5°E, 41.5°N) for the day 20 November 2003.

Figure 8a.

Scatter diagram (dots) with best fit line (solid line) for the 1 hour ahead forecast mapping and observed TEC values for all grid points for the day 20 November 2003.

Figure 8b.

Scatter diagram (dots) with best fit line (solid line) for the 1 hour ahead forecast mapping and observed TEC values for the single grid point (13.5°E, 41.5°N) for the day 20 November 2003.

[28] Summarizing the results, the METU-NN model with Bezier surface TEC mapping learned the shape of the inherent nonlinearities during the severe space weather conditions of the November 2003 period. In other words, the METU-NN system reaches the global error minimum by reaching the correct operating point.

[29] The overall absolute TEC error map for 10 min in advance forecasts is plotted in Figure 9. Similarly, the TEC error map for 1 hour in advance forecasts is plotted in Figure 10. It is interesting to note that forecasts inside the region of interest exhibit a better match with the observed data, leading to smaller error values in the inner grids when compared with the corner grids. The reason is that the presence of the neighbor grids increases the learning performance of the model for forecasting. The authors propose selecting a wider area in training than the area in operation. This may be achieved by discarding the outermost grids of the area of interest during operation and performance analysis. In the current study none of the grids are discarded, and the overall performance of the model is presented for discussion.

Figure 9.

Absolute error map of observed and 10 min ahead forecast TEC between 16 November 2003, 1000 UT, and 30 November 2003, 0000 UT.

Figure 10.

Absolute error map of observed and 1 hour ahead forecast TEC between 16 November 2003, 1000 UT, and 30 November 2003, 0000 UT.

[30] The quantified performance of the model can be studied in terms of the values of errors presented in Tables 3, 4, 5, and 6. Tables 3 and 4 illustrate the average error values for 10 min in advance forecasts during 16–29 November 2003 and during 20 November 2003, respectively. Tables 5 and 6 present the average error values for 1 hour in advance forecasts during 16–29 November 2003 and during 20 November 2003, respectively. The first three columns of the tables present the error values for the grid locations: (11.5°E, 38.5°N), (13.5°E, 41.5°N), and (15.5°E, 44.5°N), respectively. For the overall TEC forecast mapping, error values in the fourth columns of Tables 3–6 are small. The average absolute error, for example, in Table 5 for the 1 hour in advance forecast, is less than 2 TEC units (1 TECU = 1016 el m−2), which is important for practical applications. In practice, at the confidence interval of 96%, to reach a conclusion with a wrong estimate is less than 4% when the cross-correlation coefficients between the computed and observed TEC values as noted in Table 5.

Table 3. Error Table for 10 m in Advance Forecasts for the Validation Time Period (16–29 November 2003)
Location11.5°E, 38.5°N13.5°E, 41.5°N15.5°E, 44.5°NOverall TEC Map
Root-mean-square error, TECU0.84970.74990.76161.0147
Absolute error, TECU0.60500.53860.55820.7087
Normalized error0.06020.06020.06760.0740
Cross-correlation coefficient0.99570.99630.99590.9924
Table 4. Error Table for 10 m in Advance Forecasts for the Day 20 November 2003
Location11.5°E, 38.5°N13.5°E, 41.5°N15.5°E, 44.5°NOverall TEC Map
Root-mean-square error, TECU1.47441.21331.22971.9570
Absolute error, TECU1.04260.84600.90741.3510
Normalized error0.09110.07280.09020.1543
Cross-correlation coefficient0.99580.99750.99670.9904
Table 5. Error Table for 1 Hour in Advance Forecasts for the Validation Time Period 16–29 November 2003
Location11.5°E, 38.5°N13.5°E, 41.5°N15.5°E, 44.5°NOverall TEC Map
Root-mean-square error, TECU2.15562.05012.14442.2969
Absolute error, TECU1.57851.48461.51681.6537
Normalized error0.14290.15220.16300.1563
Cross-correlation coefficient0.97750.97460.96960.9699
Table 6. Error Table for 1 Hour in Advance Forecasts for the Day 20 November 2003
Location11.5°E, 38.5°N13.5°E, 41.5°N15.5°E, 44.5°NOverall TEC Map
Root-mean-square error, TECU4.06634.39124.33974.4966
Absolute error, TECU2.89483.28883.15373.2613
Normalized error0.19450.27320.26840.2998
Cross-correlation coefficient0.97640.96940.96450.9640

6. Conclusions

[31] Characteristics of near-Earth space play vital roles in the ionospheric and transionospheric propagation of radio waves. These parameters are subject to drastic variations depending on the space weather conditions. Thus the reliable operations of radio communication as well as navigation systems and spacecraft control systems largely depend on the reliable information concerning the ionospheric parameters such as TEC. Forecasts of TEC values in advance are especially essential in high-frequency and other types of telecommunication system planning.

[32] Space weather conditions also affect earthbound systems, such as pipelines and electric power networks. By receiving alerts and warnings, pipeline managers can provide efficient systems, decreasing the resultant corrosion rate on the pipes, and power companies can minimize resultant power outages and damages.

[33] Mapping is required in telecommunication planning as it involves the whole land, such as Europe in this case. In this paper, a data-driven, neural network model forecasting the TEC values on the grids is offered, and then Bezier surfaces are used in obtaining the forecast TEC maps over Europe, which is very important for telecommunication and navigation, especially during disturbed ionospheric conditions.

[34] Forecasts of an ionospheric process, the TEC variation, using the neural network–based METU-NN model, were employed in order to forecast the TEC values up to 1 hour in advance. The model learned the shape of the inherent nonlinearities, and the system reached the correct operating point in the operation time period of 16–29 November 2003. Forecasting errors are small. This fact is the indication of the system reaching the correct operating point within training. In other terminology, the system is prevented from reaching local minima, and it is successful in reaching the global minimum of the error cost function. The correlation coefficients are very close to unity, which means that the METU-NN model learned the shape of the inherent nonlinearities. Therefore the deviations from straight line are small in the scatter diagrams. In other words, it is shown that properly constructed neural network–based systems, trained and tested with properly organized data, are promising in modeling the complex nonlinear processes, such as the unpredictable variability of the ionospheric TEC values. To the best knowledge of the authors, this is the first time that a neural network–based mapping technique has been presented for forecast of the TEC values over Europe.

Acknowledgments

[35] This work is partially supported by Middle East Technical University (METU)–State Planning Organization, Turkey (DPT) project AFP-03-01-DPT 98K122690 and European Union project COST271.

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