**2. Development of the GPS-based neural network model**

#### **2.1. Data and methods**

Three main sets of data were used in this chapter, these include: (i) GPS data, (ii) sunspot number (SSN) data, and (iii) disturbance storm time (DST) data. The next section will dwell on GPS data which is of major interest in this chapter.

#### *2.1.1. GPS data*

The GPS data used in this chapter were derived from dual-frequency receivers on the NIGNET (Nigerian Permanent GNSS Network, www.nignet.net). A brief description of how ionospheric information is usually obtained from dual-frequency GPS receivers is presented.

How are dual-frequency GPS receivers able to estimate ionospheric delays? The delays introduced on radio signals by the ionosphere are frequency-dependent; the lower frequency signals are more delayed while the higher frequency signals are less delayed. More precisely, the delay (t) is inversely proportional to the radio frequency (f) as shown in Eq. (2a) [14].

$$t = 40.3 \times \frac{\text{TEC}}{cf^2} \tag{2a}$$

The proportionality expressed in Eq. (2a) forms the underlying principle for deriving ionospheric information (precisely TEC) using dual-frequency GPS receivers. This is because two

lite will be delayed differently by the ionosphere so they arrive the same receiver at different times. The delays that will be experienced by the two radio signals are, respectively, given by

<sup>1</sup> <sup>=</sup> 40.3 <sup>×</sup> \_\_\_\_

<sup>2</sup> <sup>=</sup> 40.3 <sup>×</sup> \_\_\_\_

Subtracting Eq. (2b) from Eq. (2c), we get the time delay between arrivals of the two signals

<sup>1</sup> <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_ 40.3 *TEC c* ( \_\_1 *f* 2 <sup>2</sup> <sup>−</sup> \_\_1 *f* 1 2

Dual-frequency GPS receivers compute the TEC using Eq. (3b) which is obtained by making

40.3( \_\_1 *f* 2 <sup>2</sup> <sup>−</sup> \_\_1 *f* 1 2 )

The TECs computed in this manner using the pseudo-range measurements alone are usually noisy; differential carrier phase measurements are used to obtain precise measures of the relative TECs, and a combination with the pseudo-range measurements provide the absolute slant TEC values (STECs) along the receiver-satellite path [15–17]. The computed TECs are referred to as slant, to distinguish them from the unique TEC that will be obtained for a particular location when the satellite is exactly overhead the location (that is, satellite elevation = 90°). This unique TEC is called the vertical TEC (VTEC). VTECs are usually derived

where br and bs are, respectively, the receiver and satellite biases, S(E) is the mapping func-

*RE* <sup>×</sup> cos(*E*) \_\_\_\_\_\_\_\_\_ *RE* <sup>+</sup> *hs* )

2 ] −\_\_1 2

<sup>2</sup> − *t*

*TEC c f* 1

*TEC c f* 2

) transmitted at the same time from the same satel-

GPS Modeling of the Ionosphere Using Computer Neural Networks

<sup>2</sup> (2b)

http://dx.doi.org/10.5772/intechopen.75087

<sup>2</sup> (2c)

) (3a)

*<sup>S</sup>*(*E*) (4)

(3b)

51

(5)

and f2

radio signals (having frequencies, f1

*t*

*t*

∆*t* = *t*

TEC subject of the formula from Eq. (3a).

from the STECs using Eq. (4).

tion defined by Eq. (5) [18].

*TEC* = \_\_\_\_\_\_\_\_\_\_ *<sup>c</sup>* <sup>∆</sup>*<sup>t</sup>*

*VTEC* <sup>=</sup> *STEC* <sup>−</sup> (*br* <sup>+</sup> *bs*) \_\_\_\_\_\_\_\_\_\_\_

*<sup>S</sup>*(*E*) <sup>=</sup> \_\_\_\_\_\_ <sup>1</sup> cos(*z*) <sup>=</sup> [<sup>1</sup> <sup>−</sup> (

Eq. (2b) and Eq. (2c).

as in Eq. (3a).

c = 2.998 × 108 ms−1 is the speed of electromagnetic waves in vacuum, and TEC is the Total Electron Content. TEC is a parameter of the ionosphere that represents the total number of free electrons contained in a 1 m squared column, along the path of the signal through the ionosphere. It is this parameter of the ionosphere that is modeled in this chapter. Eq. (2a) shows that the ionospheric delay is directly proportional to the TEC, therefore the radio signals are more delayed when they travel through a route in the ionosphere with more number of free electrons.

The proportionality expressed in Eq. (2a) forms the underlying principle for deriving ionospheric information (precisely TEC) using dual-frequency GPS receivers. This is because two radio signals (having frequencies, f1 and f2 ) transmitted at the same time from the same satellite will be delayed differently by the ionosphere so they arrive the same receiver at different times. The delays that will be experienced by the two radio signals are, respectively, given by Eq. (2b) and Eq. (2c).

that is similar to the human brain; the networks are composed of simple elements operating in parallel and inspired by the biological nervous system. NNs can learn trends and patterns in particular data they are given and consequently be able to correctly predict unseen and future trends for the data. A neural network can be trained to perform a particular function by adjusting the value of connections (also called weights) between elements [7]. The true power and advantages of neural networks lies in the ability to represent both linear and nonlinear relationships directly from the data being modeled. Traditional linear models are simply inadequate when it comes for true modeling data that contains non-linear characteristics [8]. Recent explosion of ionospheric data from the GNSS is spurring interest in using computer neural networks for ionospheric modeling. A number of works have shown that neural networks (NNs) are good candidates for ionospheric modeling [6, 7, 10–13]. In this chapter, neural networks have been used to develop a regional model of the ionosphere over Nigeria. Predictions from the model have also been demonstrated to be more improved in terms of accuracy when compared to predictions from global ionospheric models like the IRI-Plas (International Reference Ionosphere—extended to the Plasmasphere) and the NeQuick.

Three main sets of data were used in this chapter, these include: (i) GPS data, (ii) sunspot number (SSN) data, and (iii) disturbance storm time (DST) data. The next section will dwell

The GPS data used in this chapter were derived from dual-frequency receivers on the NIGNET (Nigerian Permanent GNSS Network, www.nignet.net). A brief description of how ionospheric information is usually obtained from dual-frequency GPS receivers is presented. How are dual-frequency GPS receivers able to estimate ionospheric delays? The delays introduced on radio signals by the ionosphere are frequency-dependent; the lower frequency signals are more delayed while the higher frequency signals are less delayed. More precisely, the

> *TEC c f*

c = 2.998 × 108 ms−1 is the speed of electromagnetic waves in vacuum, and TEC is the Total Electron Content. TEC is a parameter of the ionosphere that represents the total number of free electrons contained in a 1 m squared column, along the path of the signal through the ionosphere. It is this parameter of the ionosphere that is modeled in this chapter. Eq. (2a) shows that the ionospheric delay is directly proportional to the TEC, therefore the radio signals are more delayed when they travel through a route in the ionosphere with more number of free electrons.

<sup>2</sup> (2a)

delay (t) is inversely proportional to the radio frequency (f) as shown in Eq. (2a) [14].

**2. Development of the GPS-based neural network model**

on GPS data which is of major interest in this chapter.

*t* = 40.3 × \_\_\_\_

**2.1. Data and methods**

50 Multifunctional Operation and Application of GPS

*2.1.1. GPS data*

$$t\_1 = 40.3 \times \frac{\text{TEC}}{cf\_1^2} \tag{2b}$$

$$t\_2 = 40.3 \times \frac{\text{TEC}}{cf\_2^2} \tag{2c}$$

Subtracting Eq. (2b) from Eq. (2c), we get the time delay between arrivals of the two signals as in Eq. (3a).

$$
\Delta t = t\_2 - t\_1 = \frac{40.3 \text{ TEC}}{\text{C}} \left( \frac{1}{f\_2^2} - \frac{1}{f\_1^2} \right) \tag{3a}
$$

Dual-frequency GPS receivers compute the TEC using Eq. (3b) which is obtained by making TEC subject of the formula from Eq. (3a).

$$\text{TEC} = \frac{c \,\Delta t}{40.3 \left(\frac{1}{f\_1^T} - \frac{1}{f\_1^T}\right)}\tag{3b}$$

The TECs computed in this manner using the pseudo-range measurements alone are usually noisy; differential carrier phase measurements are used to obtain precise measures of the relative TECs, and a combination with the pseudo-range measurements provide the absolute slant TEC values (STECs) along the receiver-satellite path [15–17]. The computed TECs are referred to as slant, to distinguish them from the unique TEC that will be obtained for a particular location when the satellite is exactly overhead the location (that is, satellite elevation = 90°). This unique TEC is called the vertical TEC (VTEC). VTECs are usually derived from the STECs using Eq. (4).

$$VTEC = \frac{STEC - (b\_r + b\_s)}{S(E)} \tag{4}$$

where br and bs are, respectively, the receiver and satellite biases, S(E) is the mapping function defined by Eq. (5) [18].

$$S(E) = \frac{1}{\cos(\mathfrak{z})} = \left[1 - \left(\frac{R\_{\underline{x}} \times \cos(E)}{R\_{\underline{x}} + h\_{\underline{s}}}\right)^2\right]^{\frac{1}{\frac{1}{2}}} \tag{5}$$

z and E are, respectively, the zenith and elevation angles in degrees; RE and hs are, respectively, the mean Earth radius and the ionosphere (effective) height above the Earth surface in km. The value of hs used for this chapter is 350 km.

GPS Data obtained from the NIGNET are in RINEX (Receiver Independent Exchange) format. The RINEX format is the standard data interchange format for raw satellite navigation system data. RINEX format data obtained from the NIGNET were processed into VTEC data using software developed by Dr. Gopi Seemala (seemala.blogspot.in). The software works basically on the principles highlighted above, and as expressed in Refs. [15, 19].

GPS Data used were from the 14 stations illustrated in **Figure 1** and in **Table 1**. All available data coving the periods from years 2011 to 2016 were used. To obtain instantaneous values of VTEC for a given location, VTEC values from the various satellites that are visible over the location at the time were averaged excluding those from satellites with elevation angles less than 25°. The reason for excluding data associated with low elevation angles is usually to minimize multipath errors. Multipath errors are errors associated with signals that bounce off (or reflected from) nearby buildings, trees, or other structures before they reach the receiver antenna. The problem with these signals is that the resulting range will be greater than the actual straight path range between the satellite and receiver, because the signal first has to bounce off other structures before they reach the receiver antenna. The multipath problem is typical of signals coming from low elevation satellites; the lower the satellite elevation angles (especially satellites close to the horizon), the more likely signals from them are to bouncing

off other structures before they reach the receiver antenna. This problem is mostly the reason why research-class GNSS receivers are installed such that their antennas are raised above nearby structures/buildings (or away from the structures/buildings), and the antennas are built in such a shape that the receiving surface faces the sky. In this way, radio signals that are reflected from structures beneath the antenna do not get received by the antenna even when they hit the bottom surface of the antenna. The resulting VTEC data were further averaged in

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53

The other two set of data used in this chapter are the DST and SSN data. The DST is a measure of the disturbances in the Earth's magnetic field, it is an index often used to describe the level of geomagnetic activity during storms. On the other hand, the SSN is a count of the number of sunspots present on the surface of the Sun. It is a measure of the Sun's activeness (the level of activity going on in the Sun), and is found to be cyclical, reaching its peak in about

The idea in formulating the input layer structure of a neural network is to consider parameters/factors that affect the output parameter (which is VTEC in this chapter). VTEC has been convincingly proven to be affected by both geomagnetic storm activity [20] and solar activity [21]. The practice in neural networks is to supply parameters like DST and SSN which are

1-hour intervals to reduce data and to lessen spikes on the data profiles.

**Table 1.** Description of NIGNET stations used in this work.

*2.1.2. Other data*

every 11 years.

**Figure 1.** Map of Nigeria showing locations of GPS stations used in this work.


**Table 1.** Description of NIGNET stations used in this work.

off other structures before they reach the receiver antenna. This problem is mostly the reason why research-class GNSS receivers are installed such that their antennas are raised above nearby structures/buildings (or away from the structures/buildings), and the antennas are built in such a shape that the receiving surface faces the sky. In this way, radio signals that are reflected from structures beneath the antenna do not get received by the antenna even when they hit the bottom surface of the antenna. The resulting VTEC data were further averaged in 1-hour intervals to reduce data and to lessen spikes on the data profiles.

#### *2.1.2. Other data*

z and E are, respectively, the zenith and elevation angles in degrees; RE and hs

used for this chapter is 350 km.

on the principles highlighted above, and as expressed in Refs. [15, 19].

**Figure 1.** Map of Nigeria showing locations of GPS stations used in this work.

km. The value of hs

52 Multifunctional Operation and Application of GPS

tively, the mean Earth radius and the ionosphere (effective) height above the Earth surface in

GPS Data obtained from the NIGNET are in RINEX (Receiver Independent Exchange) format. The RINEX format is the standard data interchange format for raw satellite navigation system data. RINEX format data obtained from the NIGNET were processed into VTEC data using software developed by Dr. Gopi Seemala (seemala.blogspot.in). The software works basically

GPS Data used were from the 14 stations illustrated in **Figure 1** and in **Table 1**. All available data coving the periods from years 2011 to 2016 were used. To obtain instantaneous values of VTEC for a given location, VTEC values from the various satellites that are visible over the location at the time were averaged excluding those from satellites with elevation angles less than 25°. The reason for excluding data associated with low elevation angles is usually to minimize multipath errors. Multipath errors are errors associated with signals that bounce off (or reflected from) nearby buildings, trees, or other structures before they reach the receiver antenna. The problem with these signals is that the resulting range will be greater than the actual straight path range between the satellite and receiver, because the signal first has to bounce off other structures before they reach the receiver antenna. The multipath problem is typical of signals coming from low elevation satellites; the lower the satellite elevation angles (especially satellites close to the horizon), the more likely signals from them are to bouncing

are, respec-

The other two set of data used in this chapter are the DST and SSN data. The DST is a measure of the disturbances in the Earth's magnetic field, it is an index often used to describe the level of geomagnetic activity during storms. On the other hand, the SSN is a count of the number of sunspots present on the surface of the Sun. It is a measure of the Sun's activeness (the level of activity going on in the Sun), and is found to be cyclical, reaching its peak in about every 11 years.

The idea in formulating the input layer structure of a neural network is to consider parameters/factors that affect the output parameter (which is VTEC in this chapter). VTEC has been convincingly proven to be affected by both geomagnetic storm activity [20] and solar activity [21]. The practice in neural networks is to supply parameters like DST and SSN which are well established to affect VTEC as inputs during the training of a network that will predict VTEC. For this reason, the DST and SSN parameters corresponding to instances of the VTEC data used in this chapter were used as inputs during the training of the neural networks in this chapter.

**5.** Latitude (to learn the spatial variations latitude-wise)

**7.** SSN (to learn variations of the VTEC with solar activity)

then selecting the best of them using a performance index.

RMSE = √

and NNVTECi

hidden layer neurons were used on the networks.

where GPSVTECi

**2.2. Results and discussions**

*2.2.1. Sample simulations*

*2.2.1.1. Diurnal variations*

**6.** DST index (to learn variations of the VTEC with geomagnetic storm activity)

squared-errors (RMSEs). RMSEs were computed using the formula in Eq. (6).

NNVTECi

NN-predicted VTEC values, n is the number of samples predicted.

∑ i=1 n

The output layer is clearly known to have one neuron which is the GPS-VTEC to be modeled, but deciding the number of neurons in the hidden layer is an intricate aspect of neural network trainings. This is an aspect that conspicuously affects the performance of the trained networks. The most credible practice to deciding an appropriate number of hidden layer neurons has been to train several networks that vary in the number of hidden layer neurons, and

In this chapter, 20 neural networks were simulated, varying the number of hidden layer neurons in integer steps from 1 to 20. The main performance index used is the root-mean-

The criteria for deciding the best network is to choose the one that gives the least RMSE on the test dataset. Testing of the networks was done using 15% dataset that was randomly selected from the entire data and which were not used for the training. Another randomly selected 15% of the data was used for validation during the training, and the remaining 70% was used for the actual training. **Figure 2** illustrates outcomes of the RMSEs when different number of

**Figure 2** shows that the network that gave the least RMSE is the network that has 6 hidden layer neurons. The RMSE for this network is 5.03 TECU. It is this network that has been adopted as the optimal network in this study. A detailed and elementary treatment on how to

Using the Neural Network model developed in this chapter, sample simulations were made to assess predictions from the model in terms of known ionospheric variation patterns.

Diurnal variations of the ionosphere are variations in the ionosphere that are observed as the Earth makes a complete rotation about its axis. That is, the changes that are observed within an entire day as we go from morning to night. **Figure 3(a)**–**(d)** are constructed to visualize

train neural networks using MATLAB is contained in a more elementary book [28].

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

(GPSVTECi − NNV TECi

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are, respectively, the GPS-VTEC values and the

)2 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ n (6)

DST indices were obtained from the World Data Center (WDC) for Geomagnetism (http:// wdc.kugi.kyoto-u.ac.jp/dstdir/index.html), while data on SSN were obtained from the WDC-SILSO (Sunspot Index and Long-term Solar Observations, http://www.sidc.be/silso/datafiles), Royal Observatory of Belgium, Brussels.

## *2.1.3. Neural network training and testing*

The Levenberg-Marquardt back-propagation algorithm [22] as implemented in MATLAB was used in this chapter. A couple of other algorithms exist [23] but the Levenberg-Marquardt algorithm is admired for its speed and efficiency in learning [24, 25]. NNs typically have input layers, output layers, and intermediary hidden layers. Each layer could consist of one or more units or nodes (also called neurons).

As explained in the previous section, the idea in formulating the input layer structure of a neural network is to consider parameters/factors that affect the output parameter. In the previous section, the inclusion of DST and SSN as inputs was justified. Other factors that have been established to affect VTEC are time and space; VTEC is known to vary with time and space.

Particularly, VTEC changes with time in the forms of diurnal, seasonal, and long-term yearly variations [7]. For the neural networks to learn long-term yearly variations, the year for each of the GPS VTEC data was included as input for the training. To learn seasonal variations, the day of the year for each of the data was included, and to learn diurnal variations, the hour of day for each data was included.

Spatially, VTEC changes with longitude and latitude of the GPS receiver location, and so the longitudes and latitudes of the GPS receivers were included for each of the GPS VTEC data so that the networks will learn spatial variations of the VTEC. Geomagnetic longitudes and latitudes (rather than geographic longitudes and latitudes) were used since the ionospheric properties are based mainly on the interactions between the solar radiation and the Earth's geomagnetic field [6, 26]. Conversion of geographic to geomagnetic coordinates was done using the Apex Coordinate Conversion Utility Software [27].

In summary, a total of the following seven input nodes were used for the neural network training:


**5.** Latitude (to learn the spatial variations latitude-wise)

well established to affect VTEC as inputs during the training of a network that will predict VTEC. For this reason, the DST and SSN parameters corresponding to instances of the VTEC data used in this chapter were used as inputs during the training of the neural networks in

DST indices were obtained from the World Data Center (WDC) for Geomagnetism (http:// wdc.kugi.kyoto-u.ac.jp/dstdir/index.html), while data on SSN were obtained from the WDC-SILSO (Sunspot Index and Long-term Solar Observations, http://www.sidc.be/silso/datafiles),

The Levenberg-Marquardt back-propagation algorithm [22] as implemented in MATLAB was used in this chapter. A couple of other algorithms exist [23] but the Levenberg-Marquardt algorithm is admired for its speed and efficiency in learning [24, 25]. NNs typically have input layers, output layers, and intermediary hidden layers. Each layer could consist of one or more

As explained in the previous section, the idea in formulating the input layer structure of a neural network is to consider parameters/factors that affect the output parameter. In the previous section, the inclusion of DST and SSN as inputs was justified. Other factors that have been established to affect VTEC are time and space; VTEC is known to vary with time and

Particularly, VTEC changes with time in the forms of diurnal, seasonal, and long-term yearly variations [7]. For the neural networks to learn long-term yearly variations, the year for each of the GPS VTEC data was included as input for the training. To learn seasonal variations, the day of the year for each of the data was included, and to learn diurnal variations, the hour of

Spatially, VTEC changes with longitude and latitude of the GPS receiver location, and so the longitudes and latitudes of the GPS receivers were included for each of the GPS VTEC data so that the networks will learn spatial variations of the VTEC. Geomagnetic longitudes and latitudes (rather than geographic longitudes and latitudes) were used since the ionospheric properties are based mainly on the interactions between the solar radiation and the Earth's geomagnetic field [6, 26]. Conversion of geographic to geomagnetic coordinates was done

In summary, a total of the following seven input nodes were used for the neural network

this chapter.

space.

training:

Royal Observatory of Belgium, Brussels.

54 Multifunctional Operation and Application of GPS

*2.1.3. Neural network training and testing*

units or nodes (also called neurons).

day for each data was included.

using the Apex Coordinate Conversion Utility Software [27].

**1.** Hour of Day (to learn diurnal variations of the VTEC)

**4.** Longitude (to learn the spatial variations longitude-wise)

**2.** Day of Year (to learn the seasonal variations) **3.** Year (to learn the long-term yearly variations)


The output layer is clearly known to have one neuron which is the GPS-VTEC to be modeled, but deciding the number of neurons in the hidden layer is an intricate aspect of neural network trainings. This is an aspect that conspicuously affects the performance of the trained networks. The most credible practice to deciding an appropriate number of hidden layer neurons has been to train several networks that vary in the number of hidden layer neurons, and then selecting the best of them using a performance index.

In this chapter, 20 neural networks were simulated, varying the number of hidden layer neurons in integer steps from 1 to 20. The main performance index used is the root-meansquared-errors (RMSEs). RMSEs were computed using the formula in Eq. (6).

$$\text{RMSE} = \sqrt{\frac{\sum\_{i=1}^{k} (\text{GPSVTE}\_i - \text{NNVTE}\_i)^2}{n}} \tag{6}$$

where GPSVTECi and NNVTECi NNVTECi are, respectively, the GPS-VTEC values and the NN-predicted VTEC values, n is the number of samples predicted.

The criteria for deciding the best network is to choose the one that gives the least RMSE on the test dataset. Testing of the networks was done using 15% dataset that was randomly selected from the entire data and which were not used for the training. Another randomly selected 15% of the data was used for validation during the training, and the remaining 70% was used for the actual training. **Figure 2** illustrates outcomes of the RMSEs when different number of hidden layer neurons were used on the networks.

**Figure 2** shows that the network that gave the least RMSE is the network that has 6 hidden layer neurons. The RMSE for this network is 5.03 TECU. It is this network that has been adopted as the optimal network in this study. A detailed and elementary treatment on how to train neural networks using MATLAB is contained in a more elementary book [28].

#### **2.2. Results and discussions**

#### *2.2.1. Sample simulations*

Using the Neural Network model developed in this chapter, sample simulations were made to assess predictions from the model in terms of known ionospheric variation patterns.

#### *2.2.1.1. Diurnal variations*

Diurnal variations of the ionosphere are variations in the ionosphere that are observed as the Earth makes a complete rotation about its axis. That is, the changes that are observed within an entire day as we go from morning to night. **Figure 3(a)**–**(d)** are constructed to visualize

**Figure 2.** Plot of the RMSEs for varied number of hidden layer neurons.

diurnal variations in the ionosphere over the Nigerian region. The figures are, respectively, images of the VTEC over Nigeria for 05:00 UT (06:00 local Nigerian time, which is around local sunrise), 11:00 UT (12:00 local Nigerian time, which is around local midday), 17:00 UT (18:00 local Nigerian time, which is around local sunset), and 23:00 UT (24:00 local Nigerian time, which is around local midnight) of 1st July 2014. The day was arbitrarily chosen for this illustration. Local time in Nigeria is UT + 1. In the color scheme used for the figure (and for all other figures in this chapter), the blue colors indicate lower VTECs, the red colors indicate higher VTECs, and the green-yellow colors indicate moderate VTECs (see the associated color bars for exact VTEC values in each case).

This is because the Sun rises from the east. At sunset (**Figure 3(c)**), the VTECs are higher west-

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**Figure 3.** VTEC maps over Nigeria for (a) 06:00 LT, (b) 12:00 LT, (c) 18:00 LT, and (d) 24:00 LT, of 1st July 2014.

Seasonal variations have to with variations that are observed as the Earth makes a complete revolution about the Sun. That is, the changes that are observed within an entire year as we go through the seasons. **Figure 4(a)**–**(d)** are constructed to illustrate seasonal variations over Nigeria during year 2012. The figures are, respectively, VTEC maps of the Nigerian region for 11:00 UT (local midday) of 20th March 2012 (the March equinox day), 21st June 2012 (the June solstice day), 22nd September 2012 (the September equinox day), and 21st December 2012 (the December solstice day). The year 2012 was arbitrarily chosen for the

**Figure 4(a)** and **(c)** illustrates that the VTECs are relatively high during the equinoxes. This is because Nigeria is located close to the equator, and as such receives much sunlight during the

wards than eastwards. This is because the Sun sets to the west.

*2.2.1.2. Seasonal variations*

illustration.

**Figure 3** shows that within a day the VTEC values are greatest around local midday. Since the Sun is the major source of ionospheric ionization, the level of ionospheric ionization (and hence VTEC value) is usually higher during the daytime (when the solar-zenith angle is low) than at nights (when the solar-zenith angle is high). The VTEC values are also relatively high around sunset because the ionizations produced by the Sunlight do not instantly disappear (it takes about 2 hours for the ionized particles to substantially recombine when the Sun goes below horizon).

**Figure 3(a)** and **(c)** also reveals the interplay between the Sun and the ionosphere during sunrise and sunset. At sunrise (**Figure 3(a)**), the VTECs are higher eastwards than westwards.

GPS Modeling of the Ionosphere Using Computer Neural Networks http://dx.doi.org/10.5772/intechopen.75087 57

**Figure 3.** VTEC maps over Nigeria for (a) 06:00 LT, (b) 12:00 LT, (c) 18:00 LT, and (d) 24:00 LT, of 1st July 2014.

This is because the Sun rises from the east. At sunset (**Figure 3(c)**), the VTECs are higher westwards than eastwards. This is because the Sun sets to the west.

#### *2.2.1.2. Seasonal variations*

diurnal variations in the ionosphere over the Nigerian region. The figures are, respectively, images of the VTEC over Nigeria for 05:00 UT (06:00 local Nigerian time, which is around local sunrise), 11:00 UT (12:00 local Nigerian time, which is around local midday), 17:00 UT (18:00 local Nigerian time, which is around local sunset), and 23:00 UT (24:00 local Nigerian time, which is around local midnight) of 1st July 2014. The day was arbitrarily chosen for this illustration. Local time in Nigeria is UT + 1. In the color scheme used for the figure (and for all other figures in this chapter), the blue colors indicate lower VTECs, the red colors indicate higher VTECs, and the green-yellow colors indicate moderate VTECs (see the associated color

**Figure 3** shows that within a day the VTEC values are greatest around local midday. Since the Sun is the major source of ionospheric ionization, the level of ionospheric ionization (and hence VTEC value) is usually higher during the daytime (when the solar-zenith angle is low) than at nights (when the solar-zenith angle is high). The VTEC values are also relatively high around sunset because the ionizations produced by the Sunlight do not instantly disappear (it takes about 2 hours for the ionized particles to substantially recombine when the Sun goes

**Figure 3(a)** and **(c)** also reveals the interplay between the Sun and the ionosphere during sunrise and sunset. At sunrise (**Figure 3(a)**), the VTECs are higher eastwards than westwards.

bars for exact VTEC values in each case).

56 Multifunctional Operation and Application of GPS

**Figure 2.** Plot of the RMSEs for varied number of hidden layer neurons.

below horizon).

Seasonal variations have to with variations that are observed as the Earth makes a complete revolution about the Sun. That is, the changes that are observed within an entire year as we go through the seasons. **Figure 4(a)**–**(d)** are constructed to illustrate seasonal variations over Nigeria during year 2012. The figures are, respectively, VTEC maps of the Nigerian region for 11:00 UT (local midday) of 20th March 2012 (the March equinox day), 21st June 2012 (the June solstice day), 22nd September 2012 (the September equinox day), and 21st December 2012 (the December solstice day). The year 2012 was arbitrarily chosen for the illustration.

**Figure 4(a)** and **(c)** illustrates that the VTECs are relatively high during the equinoxes. This is because Nigeria is located close to the equator, and as such receives much sunlight during the equinoxes. During the equinoxes, the solar-zenith angle is lower at the equator as compared to the solstices. It is also conspicuous that the VTEC values are high during the December solstice (**Figure 4(d)**), even higher than at the September equinox (**Figure 4(c)**). This is because Nigeria is mostly located on the geomagnetic southern hemisphere. The December solstice is the summer solstice in the southern hemisphere, and so the solar-zenith angle is lower in the southern hemisphere during this season than at other seasons.

**Figure 5(a)**–**(d)** was constructed to illustrate how the ionosphere over Nigeria varies with the solar activity. The figures are, respectively, the local midday VTECs over Nigeria for the same

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A look at the solar activity cycle shows that the solar activity level was on the rise from year 2011 to year 2014. Comparing with the VTEC values for those years (**Figure 5(a)**–**(d)**), it is observed that the VTEC values are also on the increase; the VTECs are least during year 2011 (**Figure 5(a)**) which also has the least solar activity level, and greatest during year 2014 (**Figure 5(d)**) which also has the greatest solar activity level. **Figure 5(a)**–**(d)** clearly indicates that the neural network

Two of the most popular global ionospheric models (the IRI-Plas and the NeQuick) have been selected to make a comparative assessment of the neural network model developed in this

was able to learn/capture the long-term variations associated with the solar activity.

*2.2.2. Comparison of neural network predictions with IRI-Plas and NeQuick predictions*

**Figure 5.** VTEC maps over Nigeria for local midday of 1st July, year (a) 2011, (b) 2012, (c) 2013, and (d) 2014.

day (1st July) of years 2011, 2012, 2013, and 2014.

chapter.

#### *2.2.1.3. Long-term solar cycle variations*

The ionosphere has also been established to vary with the level of solar activity. As explained earlier, the sunspot number is a good measure of the level of solar activity. The solar activity is known to have a time series cycle of about 11-years during which the activity level goes from peak to peak or trough to trough.

**Figure 4.** VTEC maps over Nigeria for local midday of (a) 20th March, (b) 21st June, (c) 22nd September, and (d) 21st December, of year 2012.

**Figure 5(a)**–**(d)** was constructed to illustrate how the ionosphere over Nigeria varies with the solar activity. The figures are, respectively, the local midday VTECs over Nigeria for the same day (1st July) of years 2011, 2012, 2013, and 2014.

A look at the solar activity cycle shows that the solar activity level was on the rise from year 2011 to year 2014. Comparing with the VTEC values for those years (**Figure 5(a)**–**(d)**), it is observed that the VTEC values are also on the increase; the VTECs are least during year 2011 (**Figure 5(a)**) which also has the least solar activity level, and greatest during year 2014 (**Figure 5(d)**) which also has the greatest solar activity level. **Figure 5(a)**–**(d)** clearly indicates that the neural network was able to learn/capture the long-term variations associated with the solar activity.

#### *2.2.2. Comparison of neural network predictions with IRI-Plas and NeQuick predictions*

equinoxes. During the equinoxes, the solar-zenith angle is lower at the equator as compared to the solstices. It is also conspicuous that the VTEC values are high during the December solstice (**Figure 4(d)**), even higher than at the September equinox (**Figure 4(c)**). This is because Nigeria is mostly located on the geomagnetic southern hemisphere. The December solstice is the summer solstice in the southern hemisphere, and so the solar-zenith angle is lower in the

The ionosphere has also been established to vary with the level of solar activity. As explained earlier, the sunspot number is a good measure of the level of solar activity. The solar activity is known to have a time series cycle of about 11-years during which the activity level goes from

**Figure 4.** VTEC maps over Nigeria for local midday of (a) 20th March, (b) 21st June, (c) 22nd September, and

southern hemisphere during this season than at other seasons.

*2.2.1.3. Long-term solar cycle variations*

58 Multifunctional Operation and Application of GPS

peak to peak or trough to trough.

(d) 21st December, of year 2012.

Two of the most popular global ionospheric models (the IRI-Plas and the NeQuick) have been selected to make a comparative assessment of the neural network model developed in this chapter.

**Figure 5.** VTEC maps over Nigeria for local midday of 1st July, year (a) 2011, (b) 2012, (c) 2013, and (d) 2014.

The IRI-Plas is the IRI (International Reference Ionosphere) extended to the plasmasphere [29]. The IRI model [3] has been widely accepted as a defector standard for specifying ionospheric parameters across the globe. The IRI-Plas model (rather than the IRI model) is selected for use in this chapter because TEC computed by the IRI-Plas model involves electron density integrations up to the GPS satellite altitudes of about 20,200 km, whereas for the IRI model, it only gets up to a maximum of 2000 km. Since, this chapter concentrates on TEC derived from the GPS, a more comprehensive comparison is therefore obtained using the IRI-Plas model rather than the IRI model. The IRI-Plas model has also been proposed for extension of the IRI model to the plasmasphere [30]. The most recent version of the IRI-Plas (the IRI-Plas 2017) was used for this comparison. The windows executable program of the IRI-Plas used was obtained from the website of the IZMIRAN Institute (http://ftp.izmiran.ru/pub/izmiran/SPIM/).

The NeQuick [31–33] is another popular global ionospheric model which has been severally compared with GNSS TEC measurements and shown to be a good representation of the

ionosphere. The NeQuick is admired because of its improved performance in predicting the topside ionosphere, and consequently versions of the IRI model from 2007 and later have included the topside formulation of the NeQuick, and has adopted it as the most mature of the different proposals to compute the topside part of the IRI electron density profile [33, 34]. The NeQuick includes routines that compute the electron density along any ray-path from ground to GPS satellite altitudes of about 20,200 km, and so also makes for a comprehensive comparison with observations from the GPS. The latest version of the NeQuick (the NeQuick-2, which is currently recommended by the ITU [35] is the one used for this comparison. The NeQuick-2 used in this chapter is the windows executable program created from the FORTRAN source code, and was obtained from the Ionosphere Radio propagation Unit of the

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For the purpose of visual illustration, the diurnal VTEC profiles from GPS observations for four selected days, over the OSGF station, are illustrated in **Figure 6(a)**–**(d)** alongside corresponding VTEC predictions from the NeQuick, the IRI-Plas model, and the neural network (NN) model developed in this chapter. **Figure 6(a)**–**(d)**, respectively, represents diurnal VTEC profiles over the OSGF station for 20th March 2012 (the March equinox day), 21st June 2012 (the June solstice day), 22nd September 2012 (the September equinox day), and 21st December

**Figure 6** clearly indicates that the VTEC predictions from the NN model developed in this chapter were closer to the GPS VTEC observations in most of the times than the VTEC predictions of the IRI-Plas and NeQuick. **Table 2** summarizes the RMSEs (computed using Eq. (6)) for each of the days and models illustrated in **Figure 6**. The RMSEs for each of the models were computed with reference to the GPS observations. **Table 2** shows that the prediction errors for the NN model were predominantly lower than for the other two models, except for

Asides the demonstrated capability of neural networks to very accurately learn and predict variations in the ionosphere, the better performance of the NN model could also be linked to the fact that more volume of regional GPS data (GPS data from the Nigerian region) were used in the NN model than the volume used in either of the NeQuick or IRI-

T/ICT4D Laboratory (https://t-ict4d.ictp.it/nequick2/source-code).

**Table 2.** Diurnal RMSEs of the 3 models for the days illustrated in **Figure 6**.

the December solstice day when the NeQuick prediction error was lower.

2012 (the December solstice day).

Plas models.

**Figure 6.** Diurnal VTEC plots over the OSGF stations for days of (a) March Equinox, (b) June Solstice, (c) September Equinox, and (d) December Solstice, in year 2012.


**Table 2.** Diurnal RMSEs of the 3 models for the days illustrated in **Figure 6**.

ionosphere. The NeQuick is admired because of its improved performance in predicting the topside ionosphere, and consequently versions of the IRI model from 2007 and later have included the topside formulation of the NeQuick, and has adopted it as the most mature of the different proposals to compute the topside part of the IRI electron density profile [33, 34]. The NeQuick includes routines that compute the electron density along any ray-path from ground to GPS satellite altitudes of about 20,200 km, and so also makes for a comprehensive comparison with observations from the GPS. The latest version of the NeQuick (the NeQuick-2, which is currently recommended by the ITU [35] is the one used for this comparison. The NeQuick-2 used in this chapter is the windows executable program created from the FORTRAN source code, and was obtained from the Ionosphere Radio propagation Unit of the T/ICT4D Laboratory (https://t-ict4d.ictp.it/nequick2/source-code).

For the purpose of visual illustration, the diurnal VTEC profiles from GPS observations for four selected days, over the OSGF station, are illustrated in **Figure 6(a)**–**(d)** alongside corresponding VTEC predictions from the NeQuick, the IRI-Plas model, and the neural network (NN) model developed in this chapter. **Figure 6(a)**–**(d)**, respectively, represents diurnal VTEC profiles over the OSGF station for 20th March 2012 (the March equinox day), 21st June 2012 (the June solstice day), 22nd September 2012 (the September equinox day), and 21st December 2012 (the December solstice day).

**Figure 6** clearly indicates that the VTEC predictions from the NN model developed in this chapter were closer to the GPS VTEC observations in most of the times than the VTEC predictions of the IRI-Plas and NeQuick. **Table 2** summarizes the RMSEs (computed using Eq. (6)) for each of the days and models illustrated in **Figure 6**. The RMSEs for each of the models were computed with reference to the GPS observations. **Table 2** shows that the prediction errors for the NN model were predominantly lower than for the other two models, except for the December solstice day when the NeQuick prediction error was lower.

Asides the demonstrated capability of neural networks to very accurately learn and predict variations in the ionosphere, the better performance of the NN model could also be linked to the fact that more volume of regional GPS data (GPS data from the Nigerian region) were used in the NN model than the volume used in either of the NeQuick or IRI-Plas models.

**Figure 6.** Diurnal VTEC plots over the OSGF stations for days of (a) March Equinox, (b) June Solstice, (c) September

The IRI-Plas is the IRI (International Reference Ionosphere) extended to the plasmasphere [29]. The IRI model [3] has been widely accepted as a defector standard for specifying ionospheric parameters across the globe. The IRI-Plas model (rather than the IRI model) is selected for use in this chapter because TEC computed by the IRI-Plas model involves electron density integrations up to the GPS satellite altitudes of about 20,200 km, whereas for the IRI model, it only gets up to a maximum of 2000 km. Since, this chapter concentrates on TEC derived from the GPS, a more comprehensive comparison is therefore obtained using the IRI-Plas model rather than the IRI model. The IRI-Plas model has also been proposed for extension of the IRI model to the plasmasphere [30]. The most recent version of the IRI-Plas (the IRI-Plas 2017) was used for this comparison. The windows executable program of the IRI-Plas used was obtained from

the website of the IZMIRAN Institute (http://ftp.izmiran.ru/pub/izmiran/SPIM/).

The NeQuick [31–33] is another popular global ionospheric model which has been severally compared with GNSS TEC measurements and shown to be a good representation of the

Equinox, and (d) December Solstice, in year 2012.

60 Multifunctional Operation and Application of GPS
