## Abstract

The critical frequency of ionospheric F2 layer (foF2) is a measure of the highest frequency of radio signal that may be reflected back by the F2 layer, and it is associated with ionospheric peak electron density in the F2 layer. Accurate long-term foF2 variations are usually derived from ionosonde observations. In this paper, we propose a new method to observe foF2 using a stand-alone global positioning system (GPS) receiver. The proposed method relies on the mathematical equation that relates foF2 to GPS observations. The equation is then implemented in the Kalman filter algorithm to estimate foF2 at every epoch of the observation (30-s rate). Unlike existing methods, the proposed method does not require any additional information from ionosonde observations and does not require any network of GPS receivers. It only requires as inputs the ionospheric scale height and the modeled plasmaspheric electron content, which practically can be derived from any existing ionospheric/plasmaspheric model. We applied the proposed method to estimate long-term variations of foF2 at three GPS stations located at the northern hemisphere (NICO, Cyprus), the southern hemisphere (STR1, Australia) and the south pole (SYOG, Antarctic). To assess the performance of the proposed method, we then compared the results against those derived by ionosonde observations and the International Reference Ionosphere (IRI) 2012 model. We found that, during the period of high solar activity (2011–2012), the values of absolute mean bias between foF2 derived by the proposed method and ionosonde observations are in the range of 0.2–0.5 MHz, while those during the period of low solar activity (2009–2010) are in the range of 0.05–0.15 MHz. Furthermore, the root-mean-square-error (RMSE) values during high and low solar activities are in the range of 0.8–0.9 MHz and of 0.6–0.7 MHz, respectively. We also noticed that the values of absolute mean bias and RMSE between foF2 derived by the proposed method and the IRI-2012 model are slightly larger than those between the proposed method and ionosonde observations. These results demonstrate that the proposed method can estimate foF2 with a comparable accuracy. Since the proposed method can estimate foF2 at every epoch of the observation, it therefore has promising applications for investigating various scales (from small to large) of foF2 irregularities.

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## Acknowledgements

DDW would like to thank the Erasmus Mundus INTACT project for fully supporting his 2-month stay at Department of Electrical Engineering, Frederick University, Cyprus. We would like to thank the Australian Space Weather Service (www.sms.bom.gov.au) and the National Institute of Information and Communications Technology (NICT) of Japan (http://wdc.nict.go.jp/) for freely providing foF2 data set used in this research. We would also like to thank the editors, Prof. Jürgen Kusche and Dr. Thomas Hobiger, and five anonymous reviewers for their valuable comments that helped to improve the quality of this paper. This research was partially supported by the “Program Penelitian, Pengabdian kepada Masyarakat, dan Inovasi (P3MI) Kelompok Keahlian ITB.”

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## Appendices

### Appendix 1: Analytical solution of VIEC

Recalling Eq. (8), the VIEC is mathematically defined as:

*Ne* is the ionospheric electron density, and *h* is the height of a point above the GPS receiver. \(h_\mathrm{{r}}\) denotes the heights of the GPS receiver, and \(h_\mathrm{{utl}}\) represents the height of the upper transition level or the height of the plasmasphere base/the topside ionosphere. d*h* represents the step size integration. For a simplicity, the term \(t_k\) is dropped here.

In order to analytically solve the integral in Eq. (28), one should formulate a model for the electron density. Following Alizadeh et al. (2015), the ionospheric electron density may be modeled as a superposition of the bottomside and topside of the ionospheric electron densities. It is mathematically expressed as:

It has been widely shown that the Chapman function offers a simple way to explain the vertical structure of *Ne* through the ionosphere (Davies 1965). The general form of the Chapman function is:

where

\(Ne^\mathrm{{b/t}}\) represents either the ionospheric electron density for the bottomside or topside ionosphere. *hmF*2 is the peak ionospheric density height, and \(H_i\) is the ionospheric scale height. Depending on the assumptions adopted for describing the electron combination process, the Chapman function has two distinct formulations: the so-called \(\alpha \)-Chapman layer (\(c=0.5\)) and \(\beta \)-Chapman layer (\(c=1\)). The \(\alpha \)-Chapman layer assumes that the electrons recombine directly with positive ions and no negative ions are present. In the \(\beta \)-Chapman layer, it is assumed that the electron loss is through attachment to neutral particles. As the height changes, the ionospheric behavior changes from \(\alpha \) to \(\beta \) or vice versa (Davies 1965; Stankov et al. 2003). In order to accommodate different behaviors between the bottomside and topside ionosphere, Ezquer et al. (1992) uses the \(\beta \)-Chapman layer for the bottomside and the \(\alpha \)-Chapman layer for the topside ionosphere:

Now, by substituting Eqs. (32) and (33) into Eq. (28), VIEC can be rewritten as:

Analytical solutions for the exponential integrals in Eq. (34) are derived in “Appendix 2”. Combining Eq. (34), Eqs. (41) and (44), the expression for VIEC can be simplified as:

The above VIEC model is identic with that derived by Ezquer et al. (1992) using the Base Point Model (BPM). Furthermore, Ezquer et al. (1992) found that their model produces the VIEC values that are close to those observed by the ground-based ionosonde technique.

Finally, it is well known that \({\text {NmF2}}\) (unit \(\hbox {el/m}^3\)) can be converted into foF2 (unit MHz) via the following relation:

Now, by combining Eq. (36) with Eq. (35), the following relation holds:

Equation (37) is an important result by which one can be able to determine \({\text {foF2}}\), if \({\text {VIEC}}\) and \(H_i\) are known. It is worth mentioning here that the accuracy of \({\text {foF2}}\) will be affected by possible errors in the Chapman function, particulary in the simplification of the ionospheric scale height. In this derivation, we assume that the same scale height can be used for the topside and bottomside Chapman functions and that in both regions the scale height does not vary with height. This assumption is inaccurate since the scale height is inversely proportional to the ion mass and the dominant ions in the ionosphere change from molecular ions (O2+, N+) in the bottomside to O+ in the F-region and to light ions (H+, He+) in the topside. However, we should take this assumption in order to analytically solve the integrals in Eq. (34) and to derive a simple model that relates \({\text {foF2}}\) with \({\text {VIEC}}\).

In order to assess the effect of this inaccurate assumption on the calculated \({\text {foF2}}\) values, we use the \({\text {foF2}}\) and \({\text {VIEC}}\) values derived by the IRI-2012 model. Here, we denote \({\text {foF2}}_{\text {IRI}}\) and \({\text {VIEC}}_{\text {IRI}}\) as the values deduced by the IRI-2012 model. Furthermore, we re-arrange Eq. (37) into:

where the scale height \(H_i\) is calculated using the method described in Sect. 4.1.3. The \({\text {foF2}}\) values as determined by Eq. (38) are compared against those of \({\text {foF2}}_{\text {IRI}}\), and the deviation \(\left( {\text {foF2}}-{\text {foF2}}_{\text {IRI}}\right) \) reflects the effect of the inaccurate assumption.

We arbitrarily selected three sites with the same longitude (\(106^{\circ }\) E) but different latitudes (\(0^{\circ }\) N, \(40^{\circ }\) N and \(80^{\circ }\) N). Daily values at 14 : 00 UT for \({\text {foF2}}_{\text {IRI}}\) and \({\text {VIEC}}_{\text {IRI}}\) were determined, and the corresponding values for \({\text {foF2}}\) were then calculated using Eq. (38). Figure 8 shows the deviation between \({\text {foF2}}\) and \({\text {foF2}}_{\text {IRI}}\) during 2009 (low solar activity) and 2012 (high solar activity). According to the figure, it can be seen that the deviations for all sites during low and high solar activities are less than 1 MHz. This may indicate that the simplification of the scale height assumption does not significantly affect the accuracy of \({\text {foF2}}\) and, hence, Eq. (37) is accurate enough for this present purpose.

### Appendix 2: Analytical solutions for the exponential integrals

### Integration through the bottomside ionosphere

Recalling the first integral in Eq. (34):

From Eq. (31), one would obtain the following relations: \({\hbox {d}}h=H_i\,{\hbox {d}}z\), \(z_{R}=\frac{h_\mathrm{{r}}-hmF2}{H_i}\), \(z_{hmF2}=0\). Substituting them into Eq. (39) yields:

Let \(x={\text {exp}}(-z)\), it would be easy to verify that \({\hbox {d}}x=-x\,{\hbox {d}}z\), \(x_R={\text {exp}}(-z_R)\), and \(x_{hmF2}=1\). Substituting them into Eq. (40) yields:

For various \(h_\mathrm{{r}}\), *hmF*2 and \(H_i\), the values inside the bracketed term are close 1.

### Integration through the topside ionosphere

Recalling the second integral in Eq. (34):

From Eq. (31), one would obtain the following relations: \({\hbox {d}}h=H_i\,{\hbox {d}}z\), \(z_{hmF2}=0\), \(z_\mathrm{{utl}}=\frac{h_\mathrm{{utl}}-hmF2}{H_i}\). Substituting them into Eq. (42) yields:

It can be derived that integral in Eq. (43) has analytical solution as (see Abramowitz and Stegun 1964):

For typical values of \(h_\mathrm{{utl}}\), *hmF*2 and \(H_i\), the values inside the bracketed term do not significantly deviate from 0.68.

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Wijaya, D.D., Haralambous, H., Oikonomou, C. *et al.* Determination of the ionospheric foF2 using a stand-alone GPS receiver.
*J Geod* **91**, 1117–1133 (2017). https://doi.org/10.1007/s00190-017-1013-2

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DOI: https://doi.org/10.1007/s00190-017-1013-2