On the global model for foF2 using neural networks



[1] The use of neural networks (NNs) has been employed in this work to develop a global model of the ionospheric F2 region critical frequency, foF2. The main principle behind our approach has been to utilize parameters other than simple geographic coordinates, on which foF2 is known to depend, and to exploit the ability of NNs to establish and model this nonlinear relationship for predictive purposes. The foF2 data used in the training of the NNs were obtained from 59 ionospheric stations across the globe at various times from 1964 to 1986, on the basis of availability. To test the success of this approach, one NN (NN1) was trained without data from 13 stations, selected for their geographic remoteness, which could then be used to validate the predictions of the NN for those remote coordinates. These stations were subsequently included in our final NN (NN2). The input parameters consisted of day number (day of the year), universal time, solar activity, magnetic activity, geographic latitude, angle of meridian relative to subsolar point, magnetic dip angle, magnetic declination, and solar zenith angle. Comparisons between foF2 values determined using NNs and the International Reference Ionosphere (IRI) model (from Union Radio Scientifique Internationale (URSI) and International Radio Consultative Committee (CCIR) coefficients) with observed values are given with their root-mean-square (RMS) error differences for test stations. The results from NN2 are used to produce the global behavior of hourly values of foF2 and are compared with the IRI model using URSI and CCIR coefficients. The results obtained (i.e., RMS error differences), which compare favorably with the IRI models, justify this technique for global foF2 modeling.

1. Introduction

[2] Global models of foF2 have been of major importance in various ionospheric models, providing a global distribution of F2 peak density [Rush et al., 1983, 1984; Nisbet, 1971; Jones and Obitts, 1970; International Telecommunication Union, 1982; Fox and McNamara, 1988; Bilitza, 2001]. These models have been used in different capacities such as radio wave propagation studies and telecommunication applications. Significant efforts have been made in the past at improving global models of foF2 [Fox and McNamara, 1988; Rush et al., 1989; Bilitza, 2001; Fuller-Rowell et al., 2000]. The motivation behind these efforts has been because of the significant role that the maximum electron density plays in the study of the ionosphere and its effect on radio communications.

[3] Bradley et al. [2004] illustrated the need for the existing models to be updated because of an increase in the latest data and analysis techniques. For instance, most of the existing models are only based on worldwide ionosonde data for the years 1954 and 1958 [Bradley, 1990]. Also, Rush et al. [1983, 1984] coefficients are based on data from July 1975 to June 1976 and July 1978 to July 1979. These periods were considered as representative of solar minimum and solar maximum conditions, respectively, and may not accurately represent periods of solar maximum and minimum observed over a number of years. In addition, there is evidence of long-term ionospheric changes over greater timescales than a single solar cycle as a possible indicator of the atmospheric greenhouse effect [Bremer, 1992]. One other major problem is that most of the existing global models provide a better reflection of the Northern Hemisphere than the Southern Hemisphere because of the disproportionate global distribution of ionosonde stations.

[4] In the last decade, a number of researchers have demonstrated that neural networks (NNs) can be used successfully as an alternative to classical methods to predict ionospheric parameters, particularly the peak electron density. For instance, forecasting of foF2 values up to 24 hours ahead [Wintoft, 2000; Tulunay et al., 2000], single-station modeling and regional mapping of foF2 and M(3000)F2 [Xenos, 2002], long-term trends in foF2 [Poole and Poole, 2002], the development of a new global foF2 empirical model [Oyeyemi and Poole, 2004], and short-term prediction of foF2 [Wintoft and Cander, 1999; McKinnell and Poole, 2000, 2001] have been demonstrated using NNs.

[5] In this paper we describe the results obtained using NNs to develop a global foF2 model. The results obtained compare favorably with the existing International Reference Ionosphere (IRI) model using Union Radio Scientifique Internationale (URSI) and International Radio Consultative Committee (CCIR) coefficients, which justify the use of this technique for global foF2 modeling.

2. Database

[6] We have used daily hourly values of foF2 accumulated by the worldwide network of ground ionosondes from 59 ionospheric stations (Table 1) across the globe. The database covers the years from 1964 to 1986 and is spread across three ionospheric regions covering, low, middle, and high latitudes. Although not all the stations have data that are equally distributed within these years, efforts have been made to ensure that we make the best use of the available data from each station within this period. For example, some stations have data from 1964 to 1976 while some have data from 1976 to 1986. In fact, some stations did not have data for a complete solar cycle. As long as a station can provide up to at least 7 years of data within a solar cycle, such a station is considered in the training process. A major problem with most, if not all, stations is that there are a lot of missing data points because of one reason or another. This problem has been overcome with the use of NNs because this technique does not require evenly distributed data points in the training procedure and there is no need to generate artificial data for the missing points.

Table 1. Ionosonde Stations Used for Training of Networks
 Station NameLatitude, deg NLongitude, E
1Resolute Bay74.7265.1
10Goose Bay53.3299.2
17St Johns47.6307.3
24Wallops Island37.9284.5
26Point Arguello34.6239.4
28Grand Bahama26.6281.8
39La Reunion−21.155.9
42Norfolk Island−29169
51Port Stanley−51.7302.2
52Campbell Island−52.5169.2
53South Georgia−54.3323.5
54Macquarie Island−54.5159
55Argentine Island−65.2295.7
57Terre Adelie−66.7140
58Halley Bay−75.5333.4
59Scott Base−77.9166.8

3. Neural Network and the Input Space

[7] A neural network (NN) can be described as a computer program that is trained to compute the relationship between its output and a set of given input parameters [Haykin, 1994; Fausett, 1994]. A NN is a network of nodes connected with directed arcs each with a numerical weight, specifying the strength of the connection. The NN is made up of an input layer, which consists of a set of inputs that feed inputs patterns to the network. The input layer is followed by at least one (sometimes more) hidden layers. The output layer, which produces the output results, then follows the hidden layer(s).

[8] For our purpose, the neural network (NN) input parameters which serve to define the location and time a prediction is required are: solar and magnetic indices, day number (day of the year), universal time, location geographic latitude, angle of meridian relative to subsolar point, solar zenith angle, and inclination and declination of the Earth's magnetic field.

[9] The relationship between the ionospheric characteristic foF2 and these input parameters is well established [Rishbeth and Garriott, 1969; Kumluca et al., 1999; Wintoft and Cander, 1999; Bilitza, 2000; McKinnell and Poole, 2000, 2001; Oyeyemi and Poole, 2004].

[10] The geographic latitude, θ, is related to the latitudinal dependence of the neutral wind and its role in blowing ionization up and down the field lines [Rishbeth and Garriott, 1969]. It has been well established by many authors [Titheridge, 1995; Jiuhou et al., 2003] that for accurate explanation and modeling of F layer parameters, and particularly the daily hourly variation of peak electron density and its height, the effects of thermospheric winds are very important and must be taken into account. On the basis of these findings both declination and inclination of the Earth's magnetic field are included in the inputs to the NN to take care of the effects of thermospheric winds. Using the well-known vertical ion drift equation

equation image

where D and I are magnetic declination and inclination, respectively, D is converted to two quadrature components according to

equation image

while I is expressed as

equation image

These three equations followed from the expansion of the drift equation above.

[11] The diurnal and seasonal variations are described by the universal time (UT) and day number (DN) in the range 0 ≤ UT ≤ 23 and 1 ≤ DN ≤ 365, respectively. Using the work of Williscroft and Poole [1996], UT and DN are converted as follows

equation image

to avoid unrealistic discontinuities between 31 December (day number 365) and 1 January (day number 1), and the midnight boundary between 2400 and 0100 UT.

[12] A number of researchers have established the existence of strong relationships between foF2 and solar and magnetic variations [Bradley, 1993; Wilkinson, 1995; Kouris et al., 1998; Bilitza, 2000; Sethi et al., 2002; Liu et al., 2003, 2004]. These two quantities play a significant role in the large variations taking place in the ionosphere, most especially the F2 region. As a measure of solar and magnetic activities, R2 (2 month running mean of the daily sunspot number) and A16 (2 day running mean of the 3 hour planetary magnetic ap index) have been used. The choice of R2 and A16 is based on the findings of Williscroft and Poole [1996], who showed that R2 and A16 are optimum parameters to produce the minimum root-mean-square error (RMSE) from a single-station NN trained to predict foF2. The meridian angle relative to the subsolar point, which provides a measure of the local time, accounts for the persistence of the ionosphere that is evidenced by the asymmetry of foF2 about noon at any station.

[13] In this work we have used a feed-forward NN with back propagation to train our networks. During training, the network is presented with values of 14 inputs, which produces one output value, foF2. The back-propagation algorithm selects a training example, makes a forward and a backward pass, and then repeats until the root-mean-square (RMS) error difference does not change by more than some predetermined amount over a certain number of epochs. The backward pass is the error back propagation and the adjustment of weights. At the end of the training process, the network can be tested by interrogation data different than those used for training.

[14] In order to determine the optimum NN for predicting foF2, we trained several different NNs with different architectures. Typical examples of these architectures are (1) one hidden layer each having 35 or 55 neurons, (2) two hidden layers each having 20/20, 20/15, and 25/25 neurons, and (3) three hidden layers each having 10/10/15, 20/15/15, 20/20/15, 25/20/20, 45/30/15, and 50/30/20 neurons in the middle layers. The best NN configuration is found to be the one with three hidden layers having 45/30/15 neurons. The choice of this configuration is based on the minimization of the RMSE difference between the target and predicted values of foF2 obtained from each configuration.

4. Training

[15] In this work, we have trained two NNs (NN1 and NN2). The first network (NN1) was trained with data from 46 ionospheric stations without those stations listed in Table 2a (test stations). The test stations were chosen for their remoteness geographically from the remaining 46, to be used to test the ability of NN1 to predict foF2 spatially. The second network (NN2) was trained with data from all the available 59 stations (Table 1). Figure 1 illustrates geographical locations of training and testing stations. The reason for doing this is to compare the results from the two networks in order to estimate how well the NN can predict foF2 for locations where data are not available, for instance, within the ocean areas. Because of the large volume of the data involved (5.4 million data points), and the length of time required to train a NN with such a volume of data, only 10% of the total hourly foF2 values randomly chosen from all the available stations were used to train the NNs. Our choice of 10% of the total data set is based on the fact that the training of the NN is faster and that there is no significant difference in the errors obtained when 25% and 50% of the total data set was used to train NNs. The 10% data set was again randomly divided into training and testing data sets in the ratio 70% and 30%, respectively. The training data set is used to train the network, while the test data set is used to check whether the network has generalized.

Figure 1.

Global map of coordinates of training and testing stations. Circles and squares refer to the training and testing stations, respectively.

Table 2a. The foF2 RMSE Difference at Selected Test Stations for Different Years During Solar Minimum
 Station NameLatitude, deg NLongitude, deg ERMSE, MHzPercent Error Difference Between URSI and NN2Percent Error Difference Between CCIR and NN2Year
8Terre Adelie−66.61400.7170.7840.8450.6755.85813.9031977
11Argentine Island−65.2295.70.8460.9160.7660.70316.90323.2531977
12Scott Base−77.9166.80.7990.8370.7710.7397.50911.7081977
13Resolute Bay74.7265.10.7210.7670.6950.6914.1619.9091977
 RMSE average  0.9400.9630.9030.80714.08016.180 

5. Results and Discussions

[16] In order to estimate the performance of the networks, daily hourly values of foF2 predicted from NN1 and NN2 are compared with those determined by the existing IRI model using both the URSI and CCIR coefficients and observed values of foF2 obtained from test stations (Tables 2a and 2b) and the RMSE differences calculated. All the available observed daily hourly values of foF2 for 365 or 366 days (as the case may be) for each of the years for stations indicated in Tables 2a and 2b are used to evaluate the error differences between the observed and URSI, CCIR, NN1, and NN2 predicted values. Because it is difficult to get data for the same years for all the stations considered for testing for one reason or the other, different years for each station where data are available were used. We made efforts to ensure that we have a good representation of both periods of solar minimum and maximum activities for each station considered. As can be seen from Tables 2a and 2b we have stations representing low (1 to 5), middle (6 to 11) and high (12 and 13) latitudes. This shows that the testing covers all ionospheric regions.

Table 2b. The foF2 RMSE Difference at Selected Stations for Different Years During Solar Maximum
 Station NameLatitude, deg NLongitude, deg ERMSE, MHzPercent Error Difference Between URSI and NN2Percent Error Difference Between CCIR and NN2Year
8Terre Adelie−66.6140.01.2031.2091.1541.0929.2279.6771979
11Argentine Island−65.2295.71.4181.3981.0191.00029.47828.4691979
12Scott Base−77.9166.81.4021.3191.1821.16317.04711.8271979
13Resolute Bay74.7265.11.3711.3741.3251.2637.8778.0791979
 RMSE average  1.3541.3151.2511.11223.42023.9261958

[17] The root-mean-square error has been used here to evaluate the performance of our networks.

equation image

where N is the number of data points and foF2obs and foF2pred are the observed and predicted foF2 values, respectively.

[18] For the solar minimum representation (Table 2a), the RMS error difference of hourly foF2 values for the years indicated is of the order of 0.6 to 1.5, 0.6 to 1.7, 0.6 to 1.1, and 0.5 to 1.1 MHz for URSI, CCIR, NN1, and NN2, respectively. From Table 2b the order is of 1.1 to 1.7, 0.9 to 1.7, 1.0 to 1.5, and 0.9 to 1.3 MHz, respectively for the solar maximum years indicated. These error differences are illustrated in Figures 2a and 2b, respectively. The average of the RMS error differences for all the years from the test stations in Tables 2a and 2b for each of URSI, CCIR, NN1, and NN2 are 0.940, 0.963, 0.903, and 0.807 MHz, respectively (for solar minimum activity) and 1.354, 1.315, 1.251, and 1.112 MHz respectively (for solar maximum activity). The RMSE average is evaluated using

equation image

where k is the total number of stations used for testing (k = 13).

Figure 2.

Bar graph illustrations of RMS error differences between measured values of foF2 and predicted values by URSI, CCIR, NN1, and NN2 for all daily hourly values of foF2 for each station for the years indicated: (a) low solar activity and (b) high solar activity.

[19] These RMSE averages are presented in Tables 2a and 2b. Also shown in Tables 2a and 2b are the percentage differences between URSI, CCIR, and NN2. The overall percentage error difference during solar minimum activity from the test stations between URSI and NN2 is 14.08% and between CCIR and NN2 is 16.18% (Table 2a). For solar maximum activity the percentage error differences are 17.86% and 15.40%, respectively (Table 2b). Figures 3a and 3b represent bar graphs illustrating the RMSE average calculated for URSI, CCIR, NN1 and NN2 during low and high solar activity, respectively.

Figure 3.

Bar graph illustrations of the RMSE average calculated for URSI, CCIR, NN1, and NN2: (a) low solar activity and (b) high solar activity, from Tables 2a and 2b, respectively.

[20] In order to test for the predictive ability of NN2 beyond the training period, 11 stations were tested as testing stations. All the available observed daily hourly values of foF2 from 1987 to 1992 were used for each of the stations in Table 3. The exception is Tortosa, where data are not available from 1987 to 1990, and instead data from 1991 to 1995 were used. Four stations (i.e., Tortosa, Camden, Leningrad, and Magadan) from these 11 stations were not part of the training stations. Similar RMSE differences, RMSE averages, and percentage error differences are as shown in Table 3. However, both the RMSE and percentage error differences from Table 3 are not as high as those obtained in Tables 2a and 2b (i.e., for years within the training period); the results obtained indicate that predictions of NN2 are not limited to the training period. Figures 4 and 5 present RMSE and RMSE average values, respectively, calculated from Table 3.

Figure 4.

Bar graph illustrations of RMS error differences between measured and URSI, CCIR, and NN2 predictions for all daily hourly values of foF2 for each station for the periods indicated (from Table 3).

Figure 5.

Bar graph illustrations of the RMSE average calculated for URSI, CCIR, and NN2 for all the years indicated in Table 3.

Table 3. The foF2 RMSE Difference for Some Selected Test Stations Outside the Training Period
 Station NameLatitude, deg NLongitude, deg ERMSE, MHzPercent Error Difference Between URSI and NN2Percent Error Difference Between CCIR and NN2Period
1Point Arguello34.6239.41.1931.1311.1116.8731.7681987–1992
 RMSE average  1.1731.1301.0649.2485.839 

[21] Figures 6 and 7 show examples of the diurnal variation of foF2 predicted by the NN2 model compared with URSI and CCIR and the observed values starting at 0000 UT on the first of the days indicated in the month of the year in each case. Figures 8a and 8b illustrate a similar comparison of the diurnal variation of foF2 for some stations outside the training period. These graphs serve to illustrate that all the three models successfully predict the general diurnal shape of foF2 behavior. Such differences that do exist are short-term (<∼3 hrs) variations in foF2 for which neither NNs nor the IRI models are designed to predict.

Figure 6.

Comparisons of the diurnal behavior of foF2 during solar minimum period predicted by NN2, URSI, and CCIR with observed values for 2 days starting at 0000 UT on the first of the days indicated.

Figure 7.

Comparisons of the diurnal behavior of foF2 during solar maximum period predicted by NN2, URSI, and CCIR with observed values for 2 days starting at 0000 UT on the first of the days indicated.

Figure 8a.

Comparisons of the diurnal behavior of foF2 during solar maximum periods predicted by NN2, URSI, and CCIR with observed values for 2 days starting at 0000 UT on the first of the days as indicated above. This tests for the predictive ability of NN beyond the training period and for four stations that were not included in the training (i.e., Tortosa, Magadan, Leningrad, and Camden).

Figure 8b.

Same as for Figure 8a but for solar minimum periods.

[22] As can be observed from Tables 2a, 2b, and 3, there are cases where each of the methods tends to perform better than the others. However, a comparison of NN2 results with URSI and CCIR show an improvement for Talara, Concepcion, Argentine Island, and Scott Base (Tables 2a and 2b). An exception is Djibouti from Table 2b, for which CCIR performed better than NN prediction. It is possible that this exception could be due to a measurement problem in the Djibouti database, and we plan on investigating this in the future.

[23] Figures 9a, 9b, and 9c show examples of the global distribution of daily hourly values of foF2 predicted by the NN2 model, URSI, and CCIR coefficients, respectively, for 12 October 1991 at 1200 UT. Values of foF2 are obtained using a scale size interval of 10° for both geographic longitude and latitude. Figures 10a, 10b, and 10c illustrate similar global variation of daily hourly values of foF2 for 21 June 1996 at 1200 UT. The contour maps of Figures 9 and 10 show examples of global distribution of foF2 for the periods of solar maximum and solar minimum activities, respectively. These maps are similar because the differences are not large. As can be observed from Figures 3 and 5, the RMSE averages obtained from NN2 and from NN1 (Figure 3) are less than those obtained from URSI and CCIR. On the basis of the results obtained in this work in comparison with the URSI and CCIR, one may say that these results justify consideration of this technique for a global foF2 model.

Figure 9.

Contour map of the global representation of foF2 values for 12 October 1991 at 1200 UT derived from (a) CCIR, (b) NN2, and (c) URSI models.

Figure 10.

Contour map of the global representation of foF2 values for 21 June 1996 at 1200 UT derived from (a) CCIR, (b) NN2, and (c) URSI models.

6. Conclusions

[24] This work has presented results obtained from a global foF2 model using NNs. These results have shown the potential of a NN-based global F2 layer critical frequency empirical model. The results show that on the basis of the RMSE values of Tables 2a, 2b, and 3, the CCIR model is better than the URSI model. The NN2 model is better than the CCIR model on average by a margin in the order of 15–16%. The authors feel that this is sufficient improvement to warrant the consideration of the NN2 model as an alternative to the CCIR model. The results obtained with some selected stations (Table 3) outside the training period are an indication that the NN2 model can also be used to produce daily hourly values of foF2 at any point across the globe, at any time, with minimal error. A further advantage of the NN2 model is that its use is not limited to the training period alone. One important point that needs to be made clear is that the end users of this model will only need to provide year (to assess sunspot and magnetic activity from a lookup table), geographic longitude, geographic latitude, day of the year, and time of the day as input parameters for the model. All other input parameters are calculated internally with a written program.


[25] Elijah Oyeyemi acknowledges the financial support received from the Hartebeesthoek Radio Astronomy Observatory (HartRAO), a national facility operated by the National Research Foundation (NRF) of South Africa, during 2004, and GRINTEK EWATION, a member of the MRCM group, during 2005.