The ionospheric F2 region around the peak electron density height hmF2 of about 250–500 km causes the most pronounced impact on transionospheric radio wave propagation. Therefore, the peak electron density of the F2 layer NmF2 is a key parameter for characterizing the ionosphere. We present an empirical model approach that allows determining global NmF2 with a limited number of model coefficients. The nonlinear approach needs 13 coefficients and a few empirically fixed parameters for describing the NmF2 dependencies on local time, geographic/geomagnetic location and solar irradiance and activity. The model approach is applied to a vast quantity of global NmF2 data derived from GNSS radio occultation measurements by CHAMP, GRACE and COSMIC satellite missions and about 60 years of processed NmF2 data from 177 worldwide ionosonde stations. The model fits to these input data with the same standard and root mean squared (RMS) deviations of 2 × 1011 m−3. The proposed Neustrelitz global NmF2 model (Neustrelitz Peak Density Model - NPDM) is climatological, i.e., the model describes the average behavior under quiet geomagnetic conditions. A preliminary comparison with the electron density NeQuick model reveals similar results for NmF2 with RMS deviations in the order of 2 × 1011 m−3 and 5 × 1011 m−3 for low and high solar activity conditions, respectively.
 The ionosphere is a weakly ionized atmospheric layer lying between about 50 km and several Earth radii. It is assumed to be composed of several regions namely D (∼50–90 km), E (∼90–140 km), F1 (∼140–210 km) and F2 (∼210–1000 km) [Davies, 1990]. The upper ionosphere namely protonosphere or plasmasphere extends from about 1000 km up to 5–6 Earth radii at the equatorial plane.
 The F2 region is the most ionized ionospheric region having the highest electron density variability. Therefore, it is responsible for most of the ionospheric impacts such as propagation delays or advances, raypath bending on transionospheric radio wave propagation. The maximum/peak electron density of the F2 layer NmF2 may range from about 1012 to 1013 m−3 [Dolukhanov, 1971] during day time and during nighttime it can go lower. The peak density height may vary approximately from 250 to 350 km at mid latitudes and from 350 to 500 km at equatorial latitudes. However, the structure and peak densities in the ionosphere vary greatly with sunspot cycle, season and day time, with geographical location and with solar disturbances [Davies, 1990]. In fact, the peak electron density limits the maximum useable frequency (MuF) for terrestrial signal propagation on Earth as well as defines the minimum required frequency for transionospheric radio wave propagation from Earth to space or vice versa. A high frequency (HF) signal transmission can be interrupted or even lost due to regular and irregular variations of the bottom side plasma density including the NmF2. Moreover, the knowledge of NmF2 is required to mitigate higher order ionospheric propagation effects such as raypath bending errors in precise positioning [see Hoque and Jakowski, 2008, 2011] using Global Navigation Satellite Systems (GNSS), e.g., the Global Positioning System (GPS), the Global'naya Navigatsionnaya Sputnikovaya Sistema (GLONASS) and the future Galileo. Therefore, the ability to model and predict the temporal and spatial variations of the peak electron density is of great use for both ionospheric research and GNSS applications.
 In this paper, we propose an empirical NmF2 model describing regular variations of the peak plasma density. The regular variation of the solar radiation with the solar zenith angle causes temporal and spatial variations of NmF2 depending on solar activity, day time, season and geographic location. In addition, there is a geomagnetic control since the movements of electrons and ions are influenced by the geomagnetic field. Depending on these complex dependencies, NmF2 can change by several orders of magnitude. However, during ionospheric disturbances the change of the peak density can happen within a few minutes over short geographic distances, e.g., in the auroral zone [e.g., Ho et al., 1996; Jakowski et al., 1999; Förster and Jakowski, 2000; Liu et al., 2006].
 For developing a global NmF2 model we consider a set of non linear equations with 13 polynomial coefficients. The polynomial coefficients are derived from a nonlinear fit with NmF2 measurements in least square senses. Two types of NmF2 measurement data have been used in the present study: space based GNSS radio occultation (RO) measurements and ground based ionosonde observations. The ionospheric electron density profiles derived from RO measurements contain the NmF2 and hmF2 and their location information. While GRACE (Gravity Recovery And Climate Experiment) and COSMIC (Constellation Observing System for Meteorology, Ionosphere and Climate, also known as FORMOSAT-3) data have been used for establishing a more recent (2006–2010) database, CHAMP (CHAllenging Minisatellite Payload) data have been used for inputs under solar maximum as well as minimum conditions (2001–2008). The ionosonde vertical sounding technique for measuring NmF2 is a classical technique developed already eight decades ago. About 60 years of processed NmF2 data from over 200 worldwide ionosonde stations are publicly available. Depending on data quality we have used NmF2 data from 177 ionosonde stations.
2. Modeling Database
 As already mentioned, we have used space based RO measurement and ground based ionosonde measurement as input database for NmF2. The establishment of the GPS and other GNSS satellites in medium Earth orbits (MEO) and different low Earth orbiting (LEO) satellites, e.g., CHAMP [Reigber et al., 2002], GRACE (http://www.csr.utexas.edu/grace/), COSMIC (http://www.cosmic.ucar.edu/) etc. together make it possible to sound the Earth's atmosphere by the limb sounding technique. A number of satellite and minisatellite missions such as GPS/MET (GPS Meteorology Instrument [Ware et al., 1996]), SAC-C (Satelite de Aplicaciones Cientificas C [Colomb et al., 2001]), CHAMP, GRACE and COSMIC satellite network carry onboard GNSS receivers for RO measurements. Even remote sensing synthetic aperture radar (SAR) satellites TerraSAR-X (http://www.infoterra.de/terrasar-x-satellite) and TanDEM-X (http://www.infoterra.de/tandem-x-satellite) carry onboard limb antennas for RO measurements. However, occultation measurements from TerraSAR-X and TanDEM-X are not yet available.
 Radio occultation observations can effectively be used to derive the vertical structure of electron density in the ionosphere. The reconstruction technique has been described by different authors [Hajj and Romans, 1998; Jakowski et al., 2002; Jakowski, 2005]. The German Aerospace Center (DLR) Neustrelitz routinely processed CHAMP RO data for retrieving vertical electron density profiles since April 2001 until August 2008. About 300,000 CHAMP reconstructed profiles covering high, medium and low solar activity periods have been used in the present study. Since April 2008 DLR Neustrelitz is using RO observations of GRACE satellite for retrieving vertical electron density profiles [Jakowski et al., 2010] (for details about GRACE mission, see http://www.gfz-potsdam.de). About 60,000 such profiles have been used in the present study. Electron density profiles derived from CHAMP and GRACE are available at http://swaciweb.dlr.de/.
 The joint United States - Taiwanese COSMIC satellite constellation (for details see http://www.cosmic.ucar.edu/) consisting of 6 microsatellites produces thousands of occultation data sets per day since April 2006. The COSMIC Data Analysis and Archival Center (CDAAC) routinely processes raw RO data for retrieving vertical ionospheric profiles in near real time. Since April 2006 about 2.5 millions ionospheric electron density profiles have been processed by the CDAAC. They are available at http://cosmic-io.cosmic.ucar.edu/cdaac/index.html. We have used these profiles for NmF2 modeling.
 It should be noted that the RO technique has some limitations. Investigation by several authors [Jakowski et al., 2004; Angling, 2008; Yue et al., 2010] shows that the retrieved NmF2 and hmF2 by RO techniques are generally in good agreement with ionosonde observations at medium latitudes. However, the reliability of the retrieved electron density degrades at low latitudes especially at the Equatorial Anomaly (EA) region. Yue et al.  found that the Abel retrieval used in RO technique overestimates electron density to the north and south of the crests of the EA and introduces artificial plasma caves below the EA crests.
 The other source of NmF2 data used in this study is the ionosonde observation. The ionosonde observations allow estimating the virtual height (the equivalent altitude of a reflection that would produce the same effect as the actual refraction) and critical frequency (e.g., foF2) of the ionosphere using a sweep-frequency (usually from 0.1 to 30 MHz) pulsed radar device. The maximum ionospheric ionization NmF2 can be obtained from the critical frequency measurements by NmF2 = 1.24 × 10−2 (foF2)2 in International System (SI) of units.
 We have used historical NmF2 data of 141 ionosonde stations from the Space Physics Interactive Data Resource (SPIDR) (available at http://spidr.ngdc.noaa.gov/spidr/) archive. The National Geophysical Data Center (NGDC) is disseminating historical ionosonde data records to the scientific research community via SPIDR. SPIDR currently includes over 60 years of processed ionospheric data from over 200 ionosondes worldwide.
 All available data from 1950 to 2010 have been used for the selected stations. We checked the solar cycle variation, annual-semiannual variation and daily variation of foF2 for each station. For some stations the median and mean variations deviate significantly from each other. For some stations the mean and median of foF2 measurements exceed the typical maximum 25 MHz. In some cases the averaged diurnal variation significantly deviates from the typical cosine pattern with the maximum around local noon time. All those stations are excluded in the analysis thus reducing the number of stations to 141 although SPIDR data are available for more than 200 stations. It should be mentioned here that we have sorted foF2 and used medians for further processing of the data.
 We have used historical foF2 data of 27 ionosonde stations from the Ionosphere Prediction Service (IPS) Australia. As before, we checked the solar cycle variation, annual-semiannual variation and daily variation of foF2 for each IPS station data to exclude erroneous measurements from the analysis. The IPS stations are well distributed over Australia. Additionally, we have used 9 stations data from the National Oceanic and Atmospheric Administration (NOAA) archive ftp://ftp.ngdc.noaa.gov/ionosonde/data/. The data are also checked before using in further processing.
 To view the ionosonde data coverage a global map of used ionosonde stations from SPIDR, IPS and NOAA is given in Figure 1. The verification stations used for validation purposes are also indicated in the map.
 Using RO and ionosonde data together, we have obtained a large database of NmF2 which includes day and night, summer, winter and equinoxes, high, medium and low solar activity, high, medium and low geomagnetic activity conditions at high, medium and low latitudes. The database has been sorted with respect to solar radio flux variation, seasonal variation, local time variation and geomagnetic latitude and longitude variations. For seasonal variation, NmF2 values are averaged for 27 days interval and the fourteenth day is taken as the reference day. The spatial resolution in the meridional direction has been restricted to 2.5°. The spatial resolution in the zonal direction is not equal; the maximum resolution 5° is taken at the equator and the resolution has been gradually decreased to 360° at the two poles. The time resolution for the local time is restricted to 1 h. Thus, the length of the input data matrix has been reduced to about 1.3 millions. We have simplified the computational complexity in the fitting procedures keeping the size of input data vectors minimum.
3. Basic Modeling Approach
 For developing the global NmF2 model we follow the basic NTCM (Neustrelitz TEC Model) approach used for the global TEC modeling as described in our recent papers Jakowski et al. [2011a, 2011b] and Feltens et al. . The different terms describing the dependencies from local time, season, geomagnetic field and solar activity are combined in a multiplicative way as
 The main dependencies of NmF2 are described by the factors Fi which contain explicitly the model functions and coefficients.
 The variation with local time (LT in hours) is split into diurnal (D), semi-diurnal (SD) and ter-diurnal (TD) harmonic components. The model function describing the local time variation is given by F1 as
where the angular phases of the diurnal, semi-diurnal and ter-diurnal variations VD, VSD and VTD are defined by
respectively. We have found about 2 h delay in the response of the ionospheric/thermospheric system to the daily solar excitation. Our investigation shows that the daily maximum of NmF2 occurs at 14 LT instead of at 12 LT for most cases. Therefore, we take the phase shift LTD = 14 h in equation (3). The solar zenith angle χ dependency is considered by the following expressions.
where is the geographic latitude and δ is the declination of the sun (all angles in radians). The dependency on the solar zenith angle has been described by three functions cosχ*, cosχ** and cosχ***. The function cosχ* is the expression of the solar zenith angle at local noon given by Davies [1990, p. 64]. Equation (7) indicates that cosχ** = 0 at geographic poles, i.e., at ϕ = 90°. This makes the contribution of diurnal, semi-diurnal and ter-diurnal terms in equation (2) equal to zero. To avoid such situation we introduce cosχ*** term additionally in equation (2). The value PF1 = 0.4 in equation (8) is chosen in such a way that the term cosχ*** has always a positive contribution. We have found that cosχ*** is best fitted with NmF2 data although was found to be best fitted with TEC data used by Jakowski et al. [2011a, 2011b].
 The so-called summer day time bite out (BO) effect in the mid latitude is modeled by the following functions.
 In which
 We have modeled BO function taking into account its dependencies on geographic location, season and local time. We have found that the maximum bite out effect occurs at mid latitude during summer and midday. Therefore, the parameters determining corresponding geographic latitude and day of year are set as ϕBO = 45° and doyBO = 181, respectively. Although the maximum BO effect occurs at midday, it may vary between 11:30–14:30 LT depending on the season. Therefore, the phase shift LTBO is modeled by a cosine function of doy. The half widths of Gaussian functions are best fitted as σBO1 = 14° and σBO2 = 3 h. Considering that the northern and southern hemisphere experience alternate summer and winter seasons and thus alternate BO effect over a year, the sign term ϕ/∣ϕ∣ is introduced in equations (9) and (10).
 As applied for the TEC modeling [see Jakowski et al., 2011a], the seasonal variation of NmF2 is modeled by two components: the annual (A) and the semi-annual (SA) variation by the following expression.
 In which
 The phase shifts with respect to the beginning of the year are found to be doyA = 18 days and doySA = 6 days for the annual and semi-annual variation, respectively. The F2 layer shows a marked correlation with the geomagnetic field resulting in a specific behavior [Davies, 1990; Jakowski et al., 2007]. The geomagnetic field dependency is given by the geomagnetic latitude ϕm dependency.
 The geomagnetic latitude is generally based on a simple dipole representation of the Earth's magnetic field. A better coordinate would be one that is based on a multipole representation like the International Geomagnetic Reference Field (IGRF) model [Mandea and Macmillan, 2000]. For simplicity, we have used a simple dipole representation of the Earth's magnetic field.
where ϕc1 and ϕc2 are the northward and southward crests at 16°N and −15°N, respectively. The corresponding half widths are found to be the same and equal to σc given by equation (18) in which σcLT = 12 h. For NmF2 modeling we have not fixed the half widths of the Gaussian function at certain values as we did for TEC modeling. Our investigation shows that the crests on both sides of the geomagnetic equator are prominent during day time but disappear during nighttime for most cases. Considering this, we have not fixed the half width of the Gaussian function rather modeled the half width in such a way that it is minimum at 14 LT and maximum during nighttime. We have found that the half width varies from 10–15 degrees over a day.
 The strong solar activity dependence of the NmF2 is formulated by
where F10.7 is the solar radio flux in flux units (1 flux units = 10−22 Wm−2Hz) and the parameter δF10.7 is best fitted as δF10.7 = 12. As described above, the basic modeling approach includes five major dependencies of NmF2 on the local time, season, geomagnetic latitude, equatorial anomaly and solar cycle.
 In developing the NmF2 model we had to produce representative profiles in latitude, solar zenith angle, annual, semi-annual etc. and study functional dependencies separately for them; in this way we find good functions to represent these profiles. For a better understanding of the fitting process and function selection, we have shown some example plots in Figure 2. Figure 2a confirms that we can represent F10.7 variation by a formula given by equation (19). Figure 2b shows that the seasonal variation can be successfully represented by the annual and semi-annual variation given by equation (11). Figure 2c shows the necessity of the BO function in the modeling. We see that the fitting is improved after introducing BO function in the daily variation. Figure 2d shows that the equatorial anomaly (i.e., crest on both sides of the equator) is prominent during daytime but not clearly visible during nighttime. Varying the half width of the Gaussian function we can model this feature; Figure 2d confirms this.
 In comparison with the CCIR and URSI approaches for foF2, we find that the CCIR and URSI maps use a set of 12 files each one containing over 2000 coefficients for one month of the year. The CCIR and URSI maps use a special magnetic coordinate specifically introduced for representing the magnetic field control of ionospheric parameters. In our approach, the latitudinal dependence is described by just one coefficient c9 and there are additional functions for describing the EA. For the basic modeling approach, we avoid introducing more functions for the description of the latitudinal dependence outside the EA. The ionospheric characteristics such as the mid latitude trough or small scale features will be investigated in future studies.
4. Modeling Results
 In the previous section we have formulated five main dependencies of the basic model approach of NmF2. Now the basic model approach has been applied to observation data sets for determining the 13 model coefficients by nonlinear least square methods. The coefficients and data are related through the nonlinear system of equations (1)–(19). Our goal is to find values of the coefficients that best fit the data in the sense of minimizing the sum of squares of residual errors. In this way we have obtained a set of 13 model coefficients. To assess the quality of the adjustment procedure we have computed the approximate covariance matrix for the model coefficients. The square roots of the diagonal elements of the covariance matrix are standard deviations for the individual model coefficient. The standard deviations are used to compute 95% confidence intervals for model coefficients. The solution coefficients with 95% confidence intervals are given in Table 1. The estimated 95% confidence intervals show a large degree of certainty for model coefficients.
Table 1. Optimal Set of Model Coefficients With 95% Confidence Interval
95% Confidence Interval
 The coefficients are related to the input data set described in section 2, i.e., they may change if another or more extended data set is used. Also the parameters fixed at certain values in the previous section might slightly change when other data sets are used. The model residuals, i.e., the differences between input data and NPDM values are calculated and the corresponding histogram is plotted in Figure 3. The mean deviation, standard deviation (std) and root mean squared (RMS) values of model residuals are also determined (see Figure 3).
 The model residuals are normally distributed without any bias, equal RMS and standard deviation of 2 × 1011 m−3 and show no obvious asymmetric pattern. In order to assess the quality and adequacy of the model, model results and input data are displayed in Figures 4–6 as a function of local time, day of year and geomagnetic latitude.
Figures 4–6 show that different nonlinear behaviors of the input data are represented by the NPDM. The model follows the data trend in most of the cases.
 In this section we provide preliminary test computations of the proposed NmF2 model. Besides comparison with observational data we include also a comparison with NmF2 estimations from the three dimensional electron density NeQuick model [Radicella and Leitinger, 2001; Hochegger et al., 2000; Leitinger et al., 2005; Coisson et al., 2006; Nava et al., 2008]. As reference data sets we have used both radio occultation and ionosonde data from selected time periods. It should be mentioned that using about 1.3 millions input data sets we have derived only 13 coefficients. Therefore, we assume that it may not be unjustified if we use some data sets for validation purposes that are already used as inputs to derive the model coefficients.
 In the test against radio occultation data two one-month periods May 2002 and December 2006 have been chosen as reference data sets. CHAMP data has been used for the period May 2002 and COSMIC data has been used for December 2006. The reference data includes day and night, summer and winter, high and low solar activity, high, medium and low geomagnetic activity conditions at high, medium and low latitudes. The NmF2 values are calculated by our model and NeQuick model at the same location and time window for comparisons. Subsequently, their differences from the radio occultation observation, i.e., NmF2obs − NmF2NPDM and NmF2obs − NmF2NeQuick are computed. The histograms of differences are shown in Figure 7.
 The corresponding RMS, mean and standard deviation of NmF2 differences are also given in Figure 7. The NPDM residuals are normally distributed with positive biases of 5 × 1010 m−3 and 2 × 1010 m−3 during May 2002 and December 2006, respectively. The NeQuick residuals have biases of −8 × 1010 m−3 and −5 × 1010 m−3 for the two time periods, respectively. The model residuals have equal RMS and standard deviation of about 4.5 × 1011 m−3 for NPDM and about 4.2 × 1011 m−3 for NeQuick during May 2002. During low solar activity period December 2006 both model residuals give the same RMS and standard deviations of about 2 × 1011 m−3. Comparing the statistical estimates, we see that both models perform very similarly in the test against radio occultation data.
 To validate against ionosonde data 11 ionosonde stations have been chosen as verification stations depending on the data availability and coverage over high, medium and low latitudes in the northern and southern hemisphere (see Figure 1). The name and geographic location of verification stations are given in Table 2. Two one-month periods of high and low solar activity conditions have been chosen. As already mentioned, two such potential periods of one month each are May 2002 and December 2006. Each month contains a mix of high and low geomagnetic activities. Figures 8 and 9 show comparisons among the NPDM results, ionosonde measurements and NeQuick results as a function of Universal Time (UT) at selected northern and southern hemisphere stations, respectively. The NmF2 values are averaged at each UT hour for all days in May 2002 and December 2006.
Table 2. Geographic Coordinates of the Verification Stations
Geographic Latitude (°N)
Geographic Longitude (°E)
 We see that the NPDM performance is comparable to the NeQuick performance for both test periods except at high latitude station Tromsø (69.7°N, 19°E) and low latitude station Vanimo (−2.7°N, 141°E) during May 2002. The NPDM overestimates the ionosonde observation by about 0.2 × 1012 m−3 at Tromsø but underestimates the observation by about 1–2 units of 1012 m−3 at Vanimo during May 2002.
 To compare again both models performance, we have computed the differences between ionosonde observation and model results for all hourly values of NmF2 and estimated the RMS of differences at each verification station for the selected two one-month periods. The bar plots of RMS estimates are given in Figure 10. We see that the NeQuick performs better than the NPDM for most of the stations except at Chongqing and Kwajalein during May 2002 and December 2006, respectively, and at Townsville during May 2002. The NPDM shows the worst performance at Vanimo during May 2002.
 We have found that the NeQuick model shows better performance than the NPDM model on average. However, their performances are comparable and it should be remembered that to compute foF2/NmF2 the NeQuick model uses the ITU recommended CCIR  coefficients [Jones and Gallet, 1962, 1965] which are derived from worldwide ionosonde observations. Thus the NeQuick model uses a set of 12 files each one containing over 2000 coefficients. The proposed NPDM uses only 13 coefficients and a few empirically fixed parameters for global NmF2 distribution. In terms of number of coefficients the NPDM model is much simpler than the NeQuick model.
 In operational ionospheric parameters reconstruction using real time GNSS observations, the use of any background model is very helpful (see use of the background model for TEC reconstruction [Jakowski et al., 2011a, 2011b]). The proposed NPDM can be very effective as a background model for near real time NmF2 reconstruction and prediction in operational services and radio systems (see examples of reconstructed NmF2 map at http://swaciweb.dlr.de/daten-und-produkte/public/nmf2/).
 The proposed NPDM is climatological, i.e., the model describes the average behavior under quiet geomagnetic conditions. During disturbed geomagnetic conditions, e.g., during storms the ionosphere does not follow the average behavior. Therefore, an empirical model based on climatology cannot predict the variability and dynamics of the ionosphere. In fact, we need separate correction models for ionospheric parameters for storm times. As an example, Fuller-Rowell et al.  developed a storm-time correction model for the F2-peak electron density to be used with IRI.
 This paper presents the global maximum electron density NmF2 model NPDM covering all temporal and spatial variations of global NmF2 under varying solar activity conditions with a reasonable performance. The approach limits number of coefficients to only 13. The model approach is applied to a vast quantity of global NmF2 data derived from radio occultation and ionosonde measurements. The model fits to these input data with a standard deviation of 2 × 1011 m−3. Comparing NPDM and NeQuick models with selected radio occultation and ionosonde data, similar results have been obtained with RMS deviations in the order of 5 × 1011 m−3 and 2 × 1011 m−3 for high and low solar activity conditions, respectively. The performance of the model may be further improved by extending the database used. The NmF2 database is expected to increase rapidly in the upcoming years enabling us to further improve the set of model coefficients and parameters.
 The authors are grateful to the sponsors and operators of the CHAMP and GRACE missions; Deutsches GeoForschungsZentrum (GFZ) Potsdam, German Aerospace Center (DLR), and the U.S. National Aeronautics and Space Administration (NASA). The authors are also grateful to the sponsors and operators of the FORMOSAT-3/COSMIC mission; Taiwan's National Science Council and National Space Organization (NSPO), the U.S. National Science Foundation (NSF), National Aeronautics and Space Administration (NASA), National Oceanic and Atmospheric Administration (NOAA), and the University Corporation for Atmospheric Research (UCAR). The authors would like to thank the Australian Space Weather Agency for publishing ionosonde data via IPS Radio and Space Services. The authors would like to thank NGDC for disseminating historical ionosonde data via SPIDR. This research is essentially supported by the Ministry of Education and Science of Mecklenburg-Vorpommern under grant AU 07 008.