1. Introduction

The International Reference Ionosphere (IRI) has been developed since 1978 (Rawer et al., 1978) and is established as the most standard and reliable ionospheric empirical model. Since a large amount of ionosonde data has been used, IRI derives a relatively accurate electron density (Ne) profile below the F2 peak. However, the IRI model might still have some shortcomings in the topside ionosphere, because very limited satellite data are included. Bilitza (2004) and Bilitza et al. (2006) based on Alouette/ISIS topside sounder observations reported that the IRI model overestimates Ne above the F2 peak height. Furthermore, Kakinami et al. (2008) found an in-situ Ne observation at a 600-km altitude with the Hinotori satellite which differed from the IRI Ne. This shortcoming also results in a difference between the Total Electron Content (TEC) reproduction and real observations, because the TEC is calculated using an integration of the Ne profile. Meanwhile, the IRI model overestimates the TEC in the equatorial region (Bilitza and Williamson, 2000) during high solar activity, while the IRI model overestimates and underestimates the TEC over Taiwan (24°N 120E) during low and high solar activity, respectively (Kakinami et al., 2009).

Six FORMOSAT-3/COSMIC (F3/C) micro satellites which constituted a global positioning system (GPS) occultation experiment (GOX) payload were launched on 14 April, 2006, and put into a low Earth orbit of 800-km altitude with a 72° inclination. An average of 1800 electron density profiles was obtained in a day globally. Lei et al. (2007) reported that Ne profiles measured with F3/C are consistent with NeF2 and hmF2 obtained with incoherent scatter radars at Millstone Hill and Jicamarca. An Ne profile obtained using GOX has an advantage in its coverage of observations, compared with the peak density and peak altitude of the ionosphere obtained with ground-based observations, because it was able to cover ocean and desert areas, where there are usually no receivers. Taking advantage of this feature, we have constructed an empirical model of the Ne profile measured with F3/C, which reproduces the Ne profile globally. In this paper, we describe the methodology in constructing the model and compare the empirical model based on F3/C observations with the IRI model.

2. Methodology of the Construction of an Empirical Model

The methodology of constructing an empirical model based on F3/C data is described in this section. Henceforth, the empirical model based on Ne profiles obtained by F3/C is referred to as the F3/C model. The F3/C model reproduces Ne as functions of the solar EUV flux, day of year (DOY), local time (LT), altitude and location. A similar methodology has been applied to empirical models of transition height (Marinov et al., 2004; Kutiev and Marinov, 2007), vertical scale height in the topside ionosphere (Kutiev et al., 2006) and TEC over Taiwan (Kakinami et al., 2009). Daily EUV (0.1–50 nm) measured by the Solar Heliospheric Observatory (SOHO) (Judge et al., 1998) and posted on the web site of the Space Science Center of the University of Southern California (http://www.usc.edu/dept/space_science/), is used to construct the model. The released EUV fluxes are modified using the Sun-Earth distance because the released daily EUV fluxes are adjusted at 1 AU. Data are not used in the model construction when the EUV exceeds 1011 photon/cm2 sec due to a solar flare. The EUV variation used in the F3/C model, which is excluded the fluctuated day, is shown in Fig. 1. Observed Ne profiles are accumulated under geomagnetically quiet conditions (Kp < 4) from 29 June, 2006, to 17 October, 2009. Since Ne profiles are calculated using the Abel inversion with an assumption of spherical symmetry, the errors mainly appear below 250 km around the equatorial ionization anomaly region (Liu et al., 2010), which leads to a negative value of the Ne profile in some cases. Therefore, observed Ne profiles showing a negative value are not used in the model construction. At first, the empirical model which has functions of EUV, DOY and LT is constructed in each 3-dimensional bin whose size is 30° in longitude, 18° in latitude and 20 km (40 km) altitude between 150 and 390 (390 and 590) km altitude with steps of 10° in longitude and 6° in latitude. It is assumed that the solar flux variation of log10(Ne) is proportional to the EUV flux, while the DOY and LT variation of log10(Ne) are derived using a combination of trigonometric functions of wavenumbers 1–3 and 1–4, respectively. The modeled functions are calculated for each bin. The functions for F, DOY and LT are defined as follows:

(1)
(2)
(3)

where a, b, c are coefficients for fitting. DOY is normalized by the length of the year. The function reproducing Ne is assumed to be the product of these 3 functions. Then we can obtain:

(4)

where n = (i − 1) × 63 + (j − 1) × 9 + k, i = 1, 2, j = 1,…,7, k = 1,…, 9. In order to calculate the coefficients α, more than 126 data are required. The α can be obtained by solving the following normal-equation matrix:

(5)
Fig. 1.
figure 1

EUV variation observed with SOHO from 29 June, 2006, to 17 October, 2009. Only data used in the construction of the F3/C model are shown.

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The α are calculated in each 3-dimensional bin. Finally, the modeled Ne is obtained by linearly interpolating between the bins.

3. Results and Discussion

In order to estimate the accuracy of the F3/C model, the root-mean-square error (RMS) is calculated as follows,

(6)

where n, o i , m i , denote the data number, the Ne observed with F3/C, and the Ne derived by the F3/C model, respectively. RMSs at each altitude level are displayed in Fig. 2. The RMS is over 50% below 200 km while the RMS decreases with increasing altitude, and shows a minimum value at 300 km. The RMS increases with increasing altitude over 300 km and reaches about 45% around 500 km. Below 250 km, the estimation error of the Ne profile is very large due to the fundamental issue of the Abel inversion (Liu et al., 2010), and the data used in the construction have inherent errors. Such errors might produce a high dispersion and show high RMS. On the other hand, since a similar high variability is seen in the Ne model using the HINOTORI data at 600 km (Kakinami et al., 2008), the dispersion of Ne might be high above 400 km.

Fig. 2.
figure 2

Variation of root mean square error of the F3/C model with altitude.

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The peak Ne calculated with the F3/C model are compared with those measured with ionospondes (Millstone Hill at 71.5°W, 42.6°N and Darwin at 131.0°E, 12.7°S) and IRI2007 with standard options (henceforth, we use the same IRI version) from 29 June, 2006, to 17 October, 2009, under geomagnetically quiet conditions, Kp < 4 (Fig. 3). The results with IRI match the observations at Millstone Hill, while those with the F3/C model overestimate the observations. Meanwhile, both models overestimate at Darwin in many cases. This result indicates that the reproduction of peak density by the IRI also has shortcomings at some locations.

Fig. 3.
figure 3

Comparison of peak electron density derived by the F3/C model and the IRI model with ionosonde observations at 1200 LT at Millstone Hill (left) and Darwin (right) from 29 June, 2006, to 17 October, 2009. Red and blue dots indicate the F3/C and the IRI model results.

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In order to compare the observed Ne using F3/C with the F3/C and IRI model, residual ratios, (o i − m i )/m i , where o i and m i denote the observed Ne at 410 km and the modeled Ne at 410 km, are calculated. Figure 4 displays the local time variation of the residual ratios for the F3/C and IRI models. Though the F3/C model slightly overestimates the observations by 10–20% during night time in all latitudes, the F3/C model agrees with the observations on the whole. However, the IRI model always overestimates observations by 40–60% at all local times and latitudes. Figure 5 shows the solar flux variation of the residual ratios for the F3/C and IRI models. The F3/C model shows a good agreement with the observations except for EUV > 24 × 1010 photon/cm2 sec. As shown in Fig. 1, since the data number for EUV > 24 × 1010 photon/cm2 sec is very small, the reliability of the F3/C model is less than that for EUV < 24 × 1010 photon/cm2 sec. On the other hand, the IRI model overestimates the observations when EUV is low. The accuracy of the IRI model improves with an increase of EUV. This tendency of the IRI model to solar flux is similar to that of TEC over Taiwan (Kakinami et al., 2009).

Fig. 4.
figure 4

Local time variation of the residual ratios for the F3/C (blue) and IRI (red) models in (a) dip latitude = −30 ∼ −10, (b) dip latitude = −10 ∼ 10 and (c) dip latitude = 10 ∼ 30. Dots and error bars denote median and quartiles.

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Fig. 5.
figure 5

Solar flux variation of the residual ratios for the F3/C (blue) and IRI models (red) in dip latitude = −10 ∼ 10. Dots and error bars denote median and quartiles.

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The altitude profile of Ne derived by the F3/C and the IRI models above Inamori Hall, Kagoshima University (42.5°N 288.5°E), where the IRI 2009 workshop was held, are shown in Fig. 6. The Ne derived by the IRI model is higher than that of the F3/C model at, and above the peak height in all seasons as shown in Figs. 4 and 5. In addition, the peak altitude hmF2 derived by the IRI model is higher than that of the F3/C model in all seasons. The vertical scale height (VSH) of the IRI model, which is defined as −dh/d(ln Ne) (Kutiev et al., 2006), is slightly lower than that of the F3/C model in June. The Ne derived by IRI is lower than that of the F3/C model in the lower ionosphere in all seasons except December.

Fig. 6.
figure 6

Electron density profiles derived by the F3/C (solid line) and IRI models (dashed line) at 42.5°N 288.5°E at 1200 LT in March (a), June (b), September (c) and December (d). Intensities of EUV applied to calculations are 1.99, 1.99, 1.84 and 2.20 × 1011 photon/cm2 sec, which are the actual conditions of solar flux in each month of 2007.

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Figure 7 shows the seasonal variation of peak Ne (NmF2) at 1200 LT derived by the F3/C (top panel) and the IRI (bottom panel) models. The maximum of NmF2 is located beside the magnetic equator for both models. NmF2 derived by IRI is higher than that of the F3/C model in all seasons. A longitudinal structure of NmF2 exists in both models. Small patch-like maxima of NmF2 appear in the F3/C model, while wide-longitude-range maxima of NmF2 appear in IRI in March (Fig. 7(a)). A four-peak (2-peak) longitudinal structure is detected in the F3/C (IRI) model in June. The IRI model shows a clear 2-peak longitudinal structure in September. On the other hand, the F3/C model displays a 2-peak or more small-scale longitudinal structure in September. A four or three-peak structure appears in both models in December.

Fig. 7.
figure 7

Peak electron density maps derived by F3/C (top) and IRI model (bottom) at 1200 LT in March (a), June (b), September (c) and December (d). White curves indicate magnetic equator. Intensities of EUV applied to calculations are the same as Fig. 6.

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The seasonal variation of Ne at 450 km derived by the F3/C and IRI models are shown in Fig. 8. The Ne derived by the IRI model is generally higher than that of the F3/C model around the magnetic equator. In contrast to the NmF2 shown in Fig. 7, a large-scale longitudinal structure, which is similarly reported by many scientists (e.g. Sagawa et al., 2005; Immel et al., 2006; Lin et al., 2007; Kakinami et al., 2011), is clearly seen around the magnetic equator in the F3/C model. According to Kakinami et al. (2011), the 4-peak structure of Ne at 660 km observed with the DEMETER satellite is the most pronounced in September while a 3-peak structure appears in March and December. Though the IRI model also shows a longitudinal structure, the longitudinal structure retrieved with the F3/C model is more pronounced than that of the IRI and its seasonal variation agrees with previous studies.

Fig. 8.
figure 8

Electro density maps at 450 km derived by F3/C (top) and IRI model (bottom) at 1200 LT in March (a), June (b), September (c) and December (d). White curves indicate magnetic equator. Intensities of EUV applied to calculations are the same as Fig. 6.

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Seasonal variations of VSH at 450 km derived by the F3/C and IRI models are shown in Fig. 9. The IRI model shows a maxima of VSH along the magnetic equator with a 10° latitude width while the F3/C model only displays a small maxima of VSH around 140–240°E and 300°E near the magnetic equator. In the F3/C model, maxima of VSH around 140–240°E near the magnetic equator are always significant in all seasons, which show a maximum value in March. However, a maximum of VSH around 300°E is only significant in March. The longitudinal structure seen in the VSH derived by the F3/C model differs from those of Ne at 450 km (Fig. 8). Meanwhile, 3 or 4 maxima of VSH derived by the IRI model appear around the magnetic equator. The location of each peak roughly corresponds to the locations of the maximum of Ne shown in Fig. 8. A significant difference in the VSH between the F3/C and the IRI models appears in the middle latitude. The VSH derived by the IRI model only shows a maximum around the magnetic equator, a sharp drop away from the magnetic equator and is almost constant in low and middle latitudes. Meanwhile, VSH derived by the F3/C model shows maxima (minima) around magnetic equator (beside geomagnetic equator) and increases with increasing latitude in low and middle latitudes. Similar VSH results are observed not only in F3/C data (Liu et al., 2008) but also by the topside sounder onboard the Alouette and ISIS satellites (Kutiev and Marinov, 2007). Especially, VHS derived by the F3/C model higher than 50°S is higher than the magnetic equator in September and December.

Fig. 9.
figure 9

Vertical scale height at 450 km derived by the F3/C model (top) and IRI model (bottom) at 1200 LT in March (a), June (b), September (c) and December (d). White curves indicate magnetic equator. Intensities of EUV applied in the calculations are the same as Fig. 6.

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4. Summary

We have constructed an empirical model based on Ne profiles observed with F3/C under geomagnetically quiet conditions (Kp < 4). The F3/C model derives global Ne profiles between altitudes of 150 and 590 km as functions of the solar EUV flux, day of year, local time and location. The Ne above the F2 peak, and the F2 peak altitude, derived by the F3/C model is lower than those derived by the IRI. The F3/C model reproduces a longitudinal structure better than the IRI model. The F3/C model also derives VSH which show good agreement with measurements obtained from topside sounders onboard the Alouette and ISIS satellites. Since the F3/C model has a big advantage of greater data coverage, it helps us to understand a variety of upper ionospheric phenomenon. As a result, the F3/C model contributes an improvement over the IRI model.