Following the discovery of the ionosphere a century ago, scientists have developed remote sensing techniques to explore its structure [e.g., Evans, 1969; Rishbeth and Williams, 1985; Reinisch, 1986], and have attempted to develop models that represent the electron density distribution in the ionosphere. Unlike the bottomside ionosphere which has been well studied and modeled, topside ionosphere electron density profile measurements and models have remained a challenge. One major challenge for topside N(h) modeling is finding a suitable mathematical representation of the profiles [e.g., Bilitza, 2009]. Representations that have been proposed include the exponential functions [e.g., Bent et al., 1972a, 1972b, 1972c; Llewellyn and Bent, 1973], Epstein functions [e.g., Rawer 1988; Radicella and Leitinger, 2001; Depuev and Pulinets, 2004], sech-squared function [e.g.,Kutiev and Marinov, 2007], Chapman function with a constant scale height [e.g., Reinisch et al., 2004; Tulasi Ram et al., 2009; McNamara et al., 2007], and a combination of Chapman functions with constant scale heights for O+ and H+ [e.g., Kutiev et al., 2006]. After comparing different analytic functions for the topside modeling, Fonda et al.  concluded that Chapman functions perform best in describing the topside ionosphere. Fonda et al. , without giving any details, suggested that a multilayer approach may be required.
 Reinisch et al.  tried using the general Chapman function that allows the scale height to vary continuously with height, based on work by Rishbeth and Garriott . This so-called ‘Vary-Chap’ technique avoids the need for an a priori selection of heights at which the scale height changes:
 This previous analysis suggested the use of Hm values derived from bottomside profiles for the construction of the topside normalized scale height function H(h)/Hm [Reinisch et al., 2007; Bilitza et al., 2011]. However, this approach led to unsatisfactory results [Kutiev et al., 2009]. The new Vary-Chap profile described in the current paper is expressed by means of a shape function S(h) which has a value of 1 at h = hm:
 Note that Y(h) can be written as where H = hmS. Therefore near the F2 peak the expression for N(h) in equation (3) becomes equal to the classical Chapman profile function. The factor multiplying the exponential function is close to 1 near the F2 peak in both equations (1) and (3). Similarly as in (2a), the shape function S(h) can be solved from (3) and expressed as (see Appendix A)
2. Topside Profile Data
 The electron density profile data used in this study were obtained from NASA's online ftp archive at http://nssdcftp.gsfc.nasa.gov. About 80,000 ISIS-2 N(h) profiles were used, 41,000 derived from manually scaled ISIS-2 ionograms using Jackson's profile inversion technique [Jackson, 1969; Jackson et al., 1980], and ∼39,000 from automatically scaled digitized ionograms [Huang et al., 2002; Bilitza et al., 2004; Benson, 2010]. These data span the period of 1971–1983, i.e., a complete solar cycle.
 Figure 1 illustrates the duality of N(h) and S(h) (equations (3) and (4)) for two midlatitude ISIS-2 profiles at low (1974) and high (1979) solar activity. The left panels show the measured topside N(h) profiles (black dots), and the right panels the calculated shape profiles S(h). Notice that S(h) varies from S = 1 at h = hmF2 to values of several hundred at altitudes above 800 km. The Vary-Chap profile functions were then re-calculated using the respective shape functions, shown in the right panels, and superposed on the measured profiles in the left panels. As expected the Vary-Chap profiles are perfect replicas of the measured profiles. Consequently we consider not only N(h) but also S(h) ‘measured quantities’.
3. Modeling the Shape Function S(h)
 The large day-to-day variability of the F2 layer peak density NmF2 and peak height hmF2 make it difficult to directly define an average topside profile for a given location and time of day. The S(h) model can be conveniently used in the IRI program [Bilitza et al., 2006] to calculate the topside profile for any location and time with NmF2 and hmF2 values specified in IRI. Similarly, the shape function model can provide real time topside profile extensions at all Digisonde stations of the GIRO network [Reinisch and Galkin, 2011], replacing the simple α-Chapman function [Huang and Reinisch, 2001] that is currently used for the Digisonde topside profiles.
 Any parameterized representation S*(h) of S(h) must be a good fit to the measured S(h) function over the entire height range from hmF2 to the altitude of the ISIS-2 spacecraft (∼1450 km), and it must be possible to analytically integrate 1/S*(h) to efficiently calculate the profile N(h) in(3). We found no simple analytical function able to accurately approximate the shape functions (Figure 1, right) with their characteristic slope changes. The function S*(h) we selected is composed of two functions, S1 and S2, where S1 controls the shape of S*(h) near hmF2, while S2 dominates the upper part of the shape profile (Figure 2):
 The intersection of S1 and S2 defines the height hT where the gradient of the slope goes through zero. This “transition height” is one of three parameters that are used in fitting S*(h) to S(h).
 S1(h) and S2(h) are specified in terms of three parameters, α, β, and hT:
 The coefficients c1(α, β, hT) and c2(α, β, hT) are given by the boundary conditions S*(hm) = 1, and S1(hT) = S2(hT), i.e.,
and c1 is then obtained from (8a).
 Parameter α relates to the steepness of S*(h) for h > hT, and parameter β relates to the topside thickness of the F2 layer. An Sj*(h) function is fitted to each measured shape function Sj(h) to derive the parameters αj, βj, and hTj. The quality of the fitting is assessed by comparing each measured profile Nj(h) with the calculated profile Nj*(h) where Nj*(h) is given by
 It is easy to verify that the integral in (9) can be solved analytically:
 The example in Figure 3 illustrates the construction of Nj*(h): starting with the measured profile Nj(h) (dots in right panel) the Sj(h) function is calculated (dots in left panel), and the parameterized function Sj*(h) (solid line) is fitted to S(h). The fitting parameters αj, βj, and hTj for profile j are shown in the left panel. The profile Nj*(h) (solid line in the right panel) is then calculated using equation (9) and superposed on the measured profile (dots) in the right panel for comparison. To assess the performance of the least squares fitting process we compare the parameterized profile N*(h) with N(h) by calculating the average percentage error . This error is typically small, like in the example shown in Figure 3 where ε = 4.3%, confirming the good fit of S*(h) to S(h). In the statistical analysis of the shape parameters we excluded 13% of the profiles for which ε > 20%. These were usually cases where the autoscaling of the ISIS ionogram was incorrect resulting in profiles with atypical Sj functions for which the S* function was an ill fit. The judicious choice of the functions S1 and S2 generally resulted in a good fit to S(h) over the entire height range from hm to 1400 km. Comparing ε(h < hT) with ε(h > hT) showed slightly larger fitting errors for the lower heights as expected since more data points are used for hT < h < 1400 km than for hm < h < hT biasing the S* fitting in favor of the larger heights. Nevertheless, for 60% of the fitted profiles ε(h < hT) is smaller than 10%.
4. Model Parameters
 The task of specifying the topside profile model is now reduced to determining the mean parameters α, β, and hTfor each local time, location, season, and solar and magnetic activity. Our preliminary analysis binned the parameters from the ∼80,000 profiles in 10° latitude and 3-h local time bins, assuming no dependence on longitude and solar or magnetic activity. We also disregarded any dependence on hmF2 since we found only a weak dependence as illustrated for the thickness parameterβ (Figure 4) whose mean value decreases with increasing hmF2 from ∼170 km to 125 km.
5. Latitudinal Variation of the Shape Parameters at Different Seasons
 The exponent α, which is a measure of the profile gradient for h > hT, varies between 2 and 3 (Figure 5) with the smaller values at high latitudes. Smaller values of α imply smaller gradients above the transition height.
 The thickness parameter β has a minimum of ∼120 km at the equator, and maxima of ∼200 km at high latitudes for all seasons (Figure 6). The standard deviations are fairly large for latitudes beyond ∼ ± 50°, likely caused by a smaller data sample.
 The transition height parameter hT varies with latitude between ∼600 km and 800 km. Winter and autumn show three maxima (Figures 7a and 7d), one at the equator with ∼720 km, and two at high latitudes with hT ≈ 800 km. A similar variation of the transition height with latitude has been reported by Kutiev and Marinov  using a very different modeling approach.
6. Diurnal Variation of the Shape Parameters
 Figures 8–10 show the typical diurnal variations of the shape parameters, presented here for the summer months. The mean value of α (Figure 8) has a maximum after local noon at middle and low latitudes, and remains fairly constant around 2.0 at high northern latitudes. The different behavior in the southern polar region where the mean value varies from ∼2.0 to 2.7 is surprising; however the plot in Figure 8a is likely influenced by the scarcity of data for this region. The mean values for the thickness parameter β (Figure 9) show a shallow minimum of ∼100 km at local noon at middle and low latitudes, and little diurnal variation in the polar regions. The transition height parameter hT for summer (Figure 10) shows a minimum around 500 km at midday, and maximum values around 800 km at night for middle and low latitudes. In the northern polar region the mean hT values are between ∼700 km and ∼800 km. In contrast, the mean hT values for the same high latitudes in the south are only between ∼600 km to 700 km (Figure 10d), but these values are less trustworthy because of the small data sample they are based on.
7. Solar Flux Dependence
 A preliminary test was performed to determine the dependence of the model on solar activity. We selected profile data for which PF10.7 < 90 or PF10.7 > 160. Here PF10.7 equals (F10.7 + F10.7A)/2 where F10.7 is the daily solar radio flux at 10.7 cm wavelength, and F10.7A is the 81-day average of F10.7 centered on the given day. PF10.7 is widely used as a EUV proxy because it correlates well with observed ionospheric changes [Richards et al., 2006; Bilitza et al., 2007]. The “transition height” hT is in general higher for larger flux numbers (Figure 11), and the same is true for the exponent α (Figure 12). The thickness parameter β appears to be independent of the flux (Figure 13). Because of the rather small database we consider these results preliminary. A more comprehensive test has to wait until a larger topside profile database becomes available. Similar tests can then also be conducted on the magnetic activity dependence.
8. Summary and Conclusions
 The new Vary-Chap profile model for the topside ionosphere describes the profile from hmF2 to 1400 km as function of local time, latitude, and season for any foF2/hmF2 layer peak specification. The IRI model provides monthly median maps of foF2 and hmF2 that are based on CCIR or URSI coefficients, as well as the bottomside and topside N(h) profiles. The Vary-Chap topside profile can be implemented as an option into the IRI electron density model. Also, the Vary-Chap profile can replace the Chapman topside profile with constant scale height currently produced by the GIRO Digisondes.
 The Vary-Chap N(h) representation is defined by a shape function S(h). The shape function is represented by two functions S1(h) and S2(h) where S1 is the dominant function below hT, and S2 is dominant above hT. The “transition height” hT marks the height of maximum change of S(h), presumably indicating the switch from an O+ to an H+ dominated height profile.
 The mean shape function parameters α, β, and hTwere determined from ∼80,000 ISI-2 profiles by binning the data according to local time, latitude, and season thus establishing a preliminary model for S(h) that is totally derived from measured electron density profiles that are expressed as generalized Chapman functions. Additional topside profiles from ISIS and Alouette ionograms may become available soon which will allow refining the model presented here, especially with regard to solar and magnetic activity. There also are plans for a series of new topside sounders with digital data and more complete data downloads than what was afforded by the ISIS/Alouette satellites which will significantly enhance the data resource.