One of the largest expected ionospheric effects of the increasing concentrations of greenhouse gases is a decrease in the height of the F2 region peak density, hmF2, by ∼20 km. Analyses of long-term trends of hmF2 in historical data to confirm this change must rely on models of hmF2 based on a limited number of ionogram characteristics, because a full analysis of historical analog (pre–digital age) ionograms to derive the plasma frequency profiles and hmF2 was a tedious and usually rare procedure. This paper discusses the accuracy of three models/formulas of hmF2 that have been used for long-term trend studies, namely (1) the simple parabolic model, which sets hmF2 equal to the ionogram virtual height at 0.834 foF2, (2) the Dudeney (1974, 1983) model, and (3) the Bilitza et al. (1979) model. The models have been validated using ionograms calculated from plasma frequency profiles given by the International Reference Ionosphere (IRI). The best model would be the one with the smallest seasonal and solar cycle variations in the errors, even if it has a systematic error. Analysis of the errors (relative to the IRI) in the three models shows that all three are useful, but at different times of day.
 The last 15 to 20 years have seen a rise in interest in the possible effects on the upper atmosphere of global warming, and various investigators have attempted to confirm the existence of changes in ionospheric characteristics that are predicted by global atmospheric models. A good starting point for a discussion of long-term changes in the ionosphere is the paper by Rishbeth , who lists four possible causes of such variations (1) variations of solar activity, (2) the secular variation of the geomagnetic field, (3) global cooling of the upper atmosphere (vs. warming of the lower atmosphere), and (4) changes in minor constituents. Ionosonde locations affected by variations of the geomagnetic field, particularly near the South Atlantic anomaly, should be excluded if global warming/cooling is the main interest. Following a suggestion by Roble and Dickinson  that expected increases in the mixing ratios (more or less the concentrations) of mesospheric carbon dioxide and methane will cool the thermosphere by about 50 K, Rishbeth  examined the consequences for the ionosphere. He showed that the cooling and associated composition changes, as described by Roble and Dickinson, would lower the E- and F2-layer peaks by about 2 km and 20 km respectively, but that changes in the E- and F2-layer electron densities would be small, only of the order of 1%.
 The ionospheric characteristics of interest for long-term trend studies were scaled manually from “analog” film ionograms made by the ionosondes that preceded the modern digital ionosondes. These older ionosondes were susceptible to systematic height errors that would affect all height characteristics, including M(3000)F2 and hpF2. The latter parameter is the ionogram virtual height at a frequency equal to 0.834 foF2, which has been used as a substitute for the desired hmF2 [see, e.g., Chandra et al., 1997; Sharma et al., 1999]. The parameter M(3000)F2 is called the M factor, or obliquity factor, and was scaled by an overlay procedure described in section 2. All scaled frequency characteristics, such as foE, foF1, and foF2, as well as M(3000)F2, would be affected by “jitter” (random/irregular motion) in the locations of the frequency markers as the film was pulled mechanically past a CRT slit.
 It is very important to be aware of these possible errors when analyzing long time series of historical ionospheric characteristics. Most of the investigators who have analyzed scaled ionospheric characteristics have appreciated the value of having the ionograms all come from the same ionosonde, and scaled by the same (experienced) person. For example, Sharma et al.  used 30 years of Ahmedabad data that had been scaled by a single person. Much of the Australian data was scaled by station operators with 30 or more years of experience. However, multiple ionosondes were usually involved. It is not clear that investigators obtaining historical data from World Data Centers will be aware of the idiosyncrasies of that data. Unfortunately, most data that went into databases from analog ionosondes is “naked”, with no attributes describing either the ionosonde or the scaler (experience). One way to handle the uncertainties in the scaled data is to process data from ionosondes relatively close to each other (preferably in different administrations) that should yield the same long-term trends, such as done by Clilverd et al. .
Ulich et al.  drew attention to five problem areas associated with determining long-term changes in the ionosphere: (1) consistency of derived trends, (2) quality control of the data, (3) significance of trends, (4) empirical formulas for hmF2, and (5) cycles in the data. We are mostly concerned here with the fourth point, the empirical formulas for hmF2. However, the quality of the data is also addressed. The models we consider are the parabolic model (hmF2 = hpF2), and the Dudeney  and Bilitza et al.  models. For a simple parabolic layer and neglecting the Earth's magnetic field, the peak height is equal to the virtual height at 0.834 foF2 [Davies, 1965, section 220.127.116.11]. This height is called hpF2.
 This paper addresses two general issues, (1) the accuracy of historical scaled values of M(3000)F2 and hpF2, and (2) the accuracy of the three simple models of hmF2. The value of M(3000)F2 is not required by the parabolic model, for which different issues exist. Section 2 of the paper describes the limitations of the parabolic model. Section 3 provides a brief interpretation of M(3000)F2, and describes how it was scaled from the earlier analog (usually film) ionograms. Section 4 uses ionograms calculated for a two-layer Chapman model to illustrate the errors in M(3000)F2 that could be present in historical data as a result of offsets in the ionogram height grid. Section 5 provides a general overview of the different methods that can be used to validate different models of hmF2. Section 6 briefly describes the procedures to derive ionograms from the plasma frequency profiles given by the International Reference Ionosphere (IRI), and section 7 presents validation results for several locations, mostly Slough (U.K.). Section 8 then summarizes the conclusions of the paper, and makes recommendations regarding the use of the three models for long-term studies.
2. Limitations of the Parabolic Model
 For a simple parabolic layer and neglecting the Earth's magnetic field, the peak height is equal to the virtual height at 0.834 foF2 [Davies, 1965, section 18.104.22.168]. This is the basis of the parabolic model, which thus assumes that the ionosphere can be represented by a single parabolic layer, and that the Earth's magnetic field can be neglected. The former assumption is completely invalid during the day, when there are cusps in the ionogram due to the F1 layer (when it is present) or the E layer.
 The value of hpF2 is derived from an ionogram simply by first scaling foF2, and then measuring the virtual height at the frequency 0.834 foF2, which would be a fixed distance below foF2 for the logarithmic scale used with film ionograms. The scaling error would be a function of the errors in picking the wrong point on the trace and reading off the virtual height. Experience with ionograms suggests a random error of at least ±10 km. There could also be height errors if the height grid displayed on the ionogram has an offset error (discussed in the section 4).
 It is of some interest to look at an ionogram for which the 0.834 factor would give a very large error during the day. Figure 1 shows the Madimbo Digisonde ionogram for 2006 247 105959UT (13 LT). Madimbo is in South Africa, at [22.39°S, 30.88°E, dip = −58.4°].
 The continuous line in Figure 1 shows the derived plasma frequency profile, with a peak at (6.1 MHz, 278.4 km). This profile is derived by inversion of the ordinary ray trace, which is the red trace (or the one to the left of each pair of near-vertical traces at the same altitude). The virtual height at 0.834 foF2 (5.1 MHz) is 402 km, giving an hmF2 error of +123.6 km. The F2 trace at 5.1 MHz has clearly been influenced by the presence of an underlying F1 layer peak. It is not known how many of the values of hpF2 in the ionospheric databases were influenced (values of hmF2 too high) by the extra retardation produced by the underlying E or F1 layers. It seems obvious that if the parabolic model has any utility, it would be only at night, when there is no F1 layer and foE is small.
3. Errors in Scaling M(3000)0F2
 foF2 and M(3000)F2 are two of the most important characteristics scaled from ionograms. Their product is equal to MUF(3000)F2, the highest ordinary-mode frequency that would be supported on a 3000 km HF communications circuit centered on the ionosonde at that time. M(3000)F2 is variously known as the M factor or the obliquity factor. Its relevance to hmF2 is that there are several simple models that relate M(3000)F2 to hmF2. The earliest and simplest such model is the Shimazaki  formula:
 Incidentally, this formula shows that an error ΔM would lead to a height error of approximately −(1490/M2)ΔM. For a typical value of M = 3, the error would be −165 ΔM. The standard uncertainty in scaled values of M(3000)F2 is ±0.05 [Piggot and Rawer, 1972], which would correspond to a height uncertainty of ∼8 km. Following Bradley and Dudeney , the Dudeney and Bilitza models include a correction term ΔM (different for the two models) to take account of underlying ionization that is ignored in the Shimazaki model:
 The Dudeney and Bilitza expressions for ΔM both rely on the availability of scaled critical frequencies for the E and F2 layers, foE and foF2. The scaled values of M(3000)F2 thus play a key role in the determination of model values of hmF2. However, the manual scaling of M(3000)F2 that was performed prior to about the mid 1980s was not a trivial exercise. The procedure for scaling M(3000)F2 is described in section 1.5, Conventions for Determining MUF Factors of the URSI Handbook of Ionogram Reduction, known as UAG-23A [Piggot and Rawer, 1972] [see also Davies, 1990, section 6.6.2]. Figure 2 is a scanned copy of the relevant figure from UAG-23A.
 The curve to the left represents the ionogram trace, with the (logarithmic) frequency scale on the bottom and the virtual height scale on the left. The other curve and the top “M-Factor” scale represent the so called MUF slider, which is drawn on a transparent sheet that is laid on top of a projected ionogram. The ionogram and MUF slider vertical scales (virtual heights) are identical, as are the logarithmic frequency scales. The distances between the 1, 2, 4, 8 and 16 MHz markers on the film will be identical if there is no jitter in the motion of the film. The MUF slider is moved manually parallel to the frequency axis until the slider curve is tangential to the F2 ionogram trace. [The application of the MUF slider to the ionogram is effectively the graphical method of determining the vertical and oblique frequencies that are reflected from the same real height for vertical propagation (the ionogram) and for oblique propagation (the MUF curve). If the MUF slider curve cuts the ionogram trace at two points, we have the situation corresponding to propagation by both the low and high rays. When the MUF curve is tangential to the ionogram trace, the low and high rays coincide, and the oblique frequency is equal to the MUF for a 3000 km circuit.] The value at which the extrapolated vertical F2 cusp cuts the slider M-factor scale is the value of M(3000)F2 for the ionogram. Almost all values of M(3000)F2 will fall in the narrow range 2.0 to 4.0, with a nominal scaling error of 0.05. The reported values of M(3000)F2 are of the form nnn (100 times the actual value), and are never qualified. If a reliable value cannot be obtained, the value is set to zero. Descriptive letters such as F can be used (nnnF), indicating that the accuracy of the value may have been affected by the presence of frequency spreading.
 Obviously, there is much room for error in this procedure. For example, the two sets of height scales must be carefully aligned vertically and horizontally, the slider must be placed so it is just tangential (no overlap) to the F2 trace, and the F2 cusp must be carefully extrapolated (by eye) until it hits the slider scale and the value of M(3000)F2 then read and saved. A common failing of new or casual scalers is to project the location of the point of tangency vertically to the M-factor scale, rather than the frequency of the F2 cusp, resulting in overestimates of M(3000)F2.
 Many of the human errors in scaling M(3000)F2 vanished with the introduction of digitizing tablets, since the value of M(3000)F2 could be derived by a program once the scaler digitized points from the relevant part of the F2 trace, following Paul . The errors are even more reduced when the ionogram and MUF slider are presented on a computer monitor screen and the operator works by mouse clicks. It seems highly likely that the errors in scaled values of M(3000)F2 changed their essential nature with each change in the scaling technology.
4. Effects of Virtual Height Errors on M(3000)F2
 Investigations of long-term trends in ionospheric characteristics have concentrated on the ionograms made prior to the advent of digital ionosondes, because of the many years of continuous observations available from select ionosonde sites. With these ionosondes, it was usually possible to move the ionogram traces relative to the height grid to ensure that the heights of the leading edges of strong multi-hop Es layers were exact multiples of each other. If this was not done carefully and consistently, there is the possibility of systematic errors in the displayed heights. Height offsets that arose from the delay through the receiver could be accounted for in this way. With some ionosondes, the receiver delay increased with decreasing echo amplitude, making the required height correction amplitude dependent. Simulated ionograms are used here to illustrate the effects on scaled values of M(3000)F2 of systematic errors in the displayed virtual heights.
 Virtual heights have been calculated for a realistic F1/F2 ionospheric profile composed of two overlapping Chapman layers, using the program OVERCHAP (J. E. Titheridge, personal communication, ∼1976). The magnetic dip angle was set to 70°, and the gyrofrequency to 1.2 MHz at the ground. The layers are described in Table 1 by their critical frequencies, peak heights and scale heights.
Table 1. Definition of the Model F1 and F2 Chapman Layers
Critical Frequency (MHz)
5, 6, 8, 10
Peak Height (km)
Scale Height (km)
 The plasma frequency at a given altitude is the sum of the plasma frequencies given by each of the F1 and F2 layers. The “F1” is the layer below the model F2 layer (rather than an E layer). The values of M(3000)F2 were derived from the simulated ionograms using the procedure developed by Paul , after different height errors were added to the correct virtual heights. Figure 3 shows the derived values of M(3000)F2 as a function of the error in the virtual heights, for foF2 values of 5, 6, 8 and 10 MHz. The height errors that were applied to the correct virtual heights range from −40 to +40 km, in steps of 10 km.
 The largest errors occur for foF2 = 5 MHz, the smallest value of foF2 considered. This is the case for which the F2 trace is most affected by the underlying layer (foF1 = 4 MHz). The error in M(3000)F2 is +0.57, for a height error of −40 km, and −0.36 for a height error of +40 km. Note that the errors are not symmetric – they are larger for negative height errors. For a nighttime ionogram with foE = 0.5 MHz (the lower of the two Chapman layers) and foF2 = 5 MHz (and the same peak and scale heights), the errors in M(3000)F2 range from 0.22 to −0.18, about the same as for the 8 and 10 MHz cases shown in Figure 3.
 Ionosondes that have been run for decades by experienced operators could be expected to have height offset errors less than 10 km, which would lead to worst-case errors in M(3000)F2 of ±0.1 (and an error of ∼16.5 km in hmF2, assuming the approximate validity of the Shimazaki formula), which is about twice the scaling accuracy for this parameter. It is not possible many years after the fact to determine the errors in the reported values of M(3000)F2. They probably exceed the standard accuracy of ±0.05 and have a small level of random error. The effects of random errors could be eliminated by working with monthly median values of the model values of hmF2. There could still be glitches in the data with the arrival of a new scaler or method of scaling.
5. Possible Approaches to the Validation of hmF2 Formulas
 There are several possible approaches to investigate the accuracy of the various models of hmF2, given the scaled ionospheric characteristics foE, foF2, M(3000)F2 and hpF2. (1) Compare the model values of hmF2 with values derived by means other than ionograms. (2) Compare the model values of hmF2 with values derived by manual true-height analysis of traditional ionograms. (3) Compare the model values of hmF2 with values derived by true-height analysis of autoscaled digital ionograms. (4) Calculate ionograms using a global ionospheric model, derive the value of M(3000)F2, and thence the model values of hmF2, and compare these with the value given by the ionospheric model.
 When the simple models of hmF2 were first developed, errors of 5 to 10% were considered acceptable. However, if we are looking for changes of ∼20 km in a typical midaltitude F2 peak height of 300 km, we are looking for changes of ∼7%, which is within the range of the probable errors. We therefore have to be careful with using the various models for such studies.
5.1. Obtaining hmF2 by Other Means
Bilitza et al. , for example, compared various models with incoherent scatter observations at Millstone Hill, Arecibo and Jicamarca, and with Aeros-B in situ observations. Such comparisons are severely limited by the difficulty in obtaining ground-truth observations.
5.2. Using Manual True-Height Analysis of Ionograms
 The conversion of an ionogram into the corresponding profile (of plasma frequency, which is the lingua franca of the ionosonde community, rather than electron density) is variously known as true-height analysis, real-height analysis, and ionogram inversion. This is a very tedious manual process when the traditional analog ionograms are used. Table 1 of Bilitza et al.  reports some of the heroic efforts in this area. Bradley and Dudeney  used a limited number of high quality ionograms from Argentine Islands.
Bilitza et al.  drew attention to two fundamental errors that arise in true-height analysis, which have become more obvious in the era of automatic ionogram analysis when the inversion process has become a trivial matter. The first, which is known as the “starting” problem, arises because there are usually no observations below ∼1.5 MHz, the top of the MF broadcast band. Likewise, no echoes are returned directly from within the E-F valley, giving rise to the “valley” problem. These problems have been addressed in the literature over the last few decades, notably by Titheridge [1986, 1988, and references therein, 2003], and summarized by the present author [McNamara, 2006; McNamara et al., 2007]. The practical solution to these problems is to use empirical models of the underlying ionization (i.e., below the altitude corresponding to a plasma frequency of 1.5 MHz) and of the valley.
5.3. Using Autoscaled Ionograms to Validate the hmF2 Models
 The use of profiles from actual ionograms would have the advantage that the profiles match the actual ionosphere, within the constraints of the autoscaling and true-height analysis. Modern digital ionosondes such as the Digisonde [Reinisch, 1996] offer the potential for providing extensive sets of M(3000)F2 and hmF2 that could in principle be used to validate the simple models of hmF2. In practice, however, there are still too many uncertainties in the automatic scaling (autoscaling) of the digital ionograms and the subsequent determination of the plasma frequency profile for this approach to be successful [McNamara, 2006]. There are also the uncertainties associated with the starting and valley problems. For example, different assumptions regarding the underlying ionization at night lead to quite different errors for the simple models (unpublished work by the author).
5.4. Using Ionograms From Model Ionospheres
 The use of ionograms from representative model ionospheres is a time-honored approach to deriving simple models of hmF2 [see, e.g., Bradley and Dudeney, 1973, Appendix 1]. The models were restricted to relatively simple functions that could be integrated analytically to derive the virtual heights, and the Earth's magnetic field was neglected. Other simplifying assumptions were also made [Dudeney, 1974]. Thirty years on, it is no longer necessary to represent the plasma frequency profile by convenient mathematical functions, nor to make other simplifying assumptions. The International Reference Ionosphere, IRI-2007 [Bilitza, 2007] uses empirical profile shapes based as much as possible on observations, and it is the IRI profiles that will be used for the current validations.
 In principle, the profiles from Global Assimilation of Ionospheric Measurements (GAIM) models such as the Utah State University model [Schunk et al., 2004] could also be used for validating the models of hmF2. In fact one of the methods used at AFRL for validating GAIM models [McNamara et al., 2008] matches the GAIM value of M(3000)F2 derived from the profiles with the values derived from ionosondes. However, the GAIM models require a large overhead of effort, and are still in the early stages of their development.
6. Analysis of Ionograms From the International Reference Ionosphere
 The IRI provides one of the best available (if not the best) monthly median empirical models of the subpeak ionosphere (i.e., below the peak of the F2 layer). We have determined the errors in the various simple models of hmF2 by calculating the vertical incidence ionograms for representative IRI profiles, thence the value of M(3000)F2, and evaluating the various formulas for hmF2. The starting and valley problems do not arise because the IRI specifies the plasma frequency distribution from 90 km up to the IRI value of hmF2. Some representative IRI profiles are given later, in Figure 8.
 Vertical incidence ionograms (actually, only the ordinary ray trace) have been calculated for ranges of years, days, and UT for a fixed latitude and longitude. The ionograms are calculated by direct integration of the group refractive index from the base of the ionosphere up to the reflection height. The value of M(3000)F2 is then determined from the ionogram by using the method given by Paul  and now used with automatic and semi-automatic scaling systems. The value of hpF2 (the virtual height at 0.834 foF2) is derived from the calculated ionogram trace by searching and interpolating along the ionogram trace.
Figure 4 shows the IRI ionogram corresponding to the Madimbo ionogram shown in Figure 1. Comparing the IRI and Digisonde ionograms (Figure 1) shows that the IRI gives good values for foE (3.3 MHz), foF1 (4.4 MHz) and foF2, and a realistic F2 trace up to foF2, although the IRI F2 peak height is ∼50 km lower than that derived from the Digisonde ionogram. The virtual heights (ionogram traces) do not match very well on an absolute scale. For example, the virtual height at 5 MHz is ∼300 km for the IRI, and ∼400 km for the observed ionogram. The IRI cusps at foE and foF1 also bear little resemblance to the observed cusps. However, although the IRI ionograms are flawed in detail, they are sufficiently realistic to be used for the present purposes. The detailed shapes of the E and F1 cusps (the most nonphysical feature of the IRI ionograms) are not relevant.
 The important issue here is the effects of the E or F1 layers on the F2 virtual heights, which are to raise the F2 trace and change its shape in the frequency range (about the last 25% of the F2 trace) used to derive M(3000)F2. These changes can be more important in some months and years than others, as the relative sizes of foE/foF1 and foF2 changes. The F1 cusp appears only when the solar zenith angle is smaller than a threshold value.
 The values of M(3000)F2 and hmF2 are related in the IRI by the Bilitza et al.  formula. However, this relationship does not affect the present analysis. All we need out of the IRI is that it provide a reasonably realistic model of the low and midlatitude ionosphere. In fact, as discussed later, the Bilitza et al.  formula yields somewhat larger errors than does the Dudeney  formula.
 There are many ways to analyze the IRI results. Since long-term trend analyses are done station by station using many years of data, we consider the diurnal variation of the errors in hmF2 given by the three models for all cases considered, for four fixed locations.
7. Errors in Model Heights Derived From IRI Ionograms
 The height errors in the three simple models have been calculated for three long-running midlatitude ionosonde sites, Slough (opened in 1931), Canberra (Australia, opened in 1937) and Washington/Wallops Island (opened in 1934), and for an effective equatorial anomaly station (0.0°E, 30.0°N, dip latitude ∼23°N). Only the Slough results are presented in detail. The results for the three midlatitude stations are very similar, while the range of the errors at the anomaly station is smaller than for the other stations.
Figures 5 and 6are simple mass plots of the corresponding values of hmF2, Dudeney vs. IRI, and Bilitza vs. IRI. There are 7 years × 12 months × 24 hours = 2016 cases.
Figures 5 and 6 show that the spread in the Dudeney/Bilitza values of hmF2 for a given IRI value is about ±10 km. Both plots show a convex upwards curvature and a general bias with the model values of hmF2 being too high. (The red lines indicate equality.) With the curvature and bias taken into account, the errors in each model can reach 20 km, which is the size of the expected variation in hmF2 due to global warming. Thus it is not immediately obvious that the Dudeney and Bilitza models are accurate enough for use in long-term trend analyses. The remainder of the paper concentrates on determining the situations under which the errors are small enough for the three simple formulas (including the parabolic model) to be useful.
7.1. Parabolic Model Height Errors for Slough
 The Slough ionospheric observatory started operation in 1931 [Rishbeth and Davis, 2001]. We approximate its location (51.5°N, 359.4°E) by (50°N, 0°E). Note that UT = LT. Figure 7 presents the diurnal variation of the errors in the parabolic model value of hmF2 for seven years 2000(1)2006, days 15(30)345, and UT 0(1)23, for Slough. [The abbreviation 0(1)23 means, for example, hours from 0 to 23, in steps of 1.] There are 7 years × 12 months = 84 curves.
 The strange curves in the middle of the day are due to a failure in the search procedure when there is a strong F1 cusp in the IRI ionograms, but such occasional failures are not important here. Most of the high values of the error occur for solar minimum, early afternoon, in the summer. In spite of the large daytime errors, the figure suggests that the parabolic model could be used for long-term studies for midnight ionograms, for which the spread in the errors is small (∼5 km). The bias of ∼25 km should not be a problem. Figure 8 shows the 84 IRI midnight profiles for Slough, illustrating the seasonal and solar cycle variability.
 The IRI value of foE, which is used in the Dudeney  and Bilitza et al.  formulas, is set to ∼0.4 MHz during the night for Slough. The peak height of the E layer is ∼110 km. The profiles below the point (1.5 MHz, ∼230 ± 40 km) correspond to the “underlying ionization” that must be taken into account in real-height ionogram analysis. [The ionogram calculations start at 90 km.]
7.2. Dudeney Model Height Errors for Slough
Figure 9 presents the diurnal variation of the errors in the Dudeney model values of hmF2 for years 2000(1)2006, days 15(30)345, and UT 0(1)23, for Slough. At midnight, the error is about 5 ± 13 km. The errors for a solar maximum year (2000) all lie above zero, while for a solar minimum year (2006), the errors range from about −25 to +15 km. Again at midnight, the percentage errors lie in the range −2 to 7%, but they reach ±10% during the day (06–18 UT).
 The size of the daytime errors conflicts with the use of the model for long-term studies over multiple years and solar cycles using daytime ionograms. The seasonal and solar cycle variations of the errors are better illustrated in Figure 10, which shows the errors at noon versus the elapsed hours since the start of the year 2000.
 There are seven cycles, corresponding to the seven years. The high values in each cycle correspond to day 345 (November) and the lower values to day 135 (May). It can be seen that the amplitude of the seasonal variation increases with decreasing solar activity (from ∼25 to 40 km), while the bias in the errors changes from about +15 to −5 km. The solar cycle variation is similar to that for Argentine Islands shown in Figure 2 of Jarvis et al. , and its effect on long-term trends is illustrated in Figure 3 of that paper.
7.3. Bilitza Model Height Errors for Slough
Figure 11 presents the diurnal variation of the errors in the Bilitza et al.  model values of hmF2 for years 2000(1)2006, days 15(30)345, and UT 0(1)23, for Slough. Figure 11 shows that the Bilitza errors do not have a reduced spread near midnight, or at any other time. The spread at 0 UT (midnight) is 40 km, in contrast to the ∼4 km and ∼26 km spreads for the parabolic and Dudeney models. The Bilitza errors for midnight also have a larger range than the Dudeney errors for Canberra and Wallops Island. However, the solar cycle variation of the errors at midday is less extreme than for the Dudeney model (Figure 10), as illustrated in Figure 12.
 The 15 km change in the error bias found for the Dudeney model (Figure 10) does not exist for the Bilitza model, which includes a solar cycle term in the formula for ΔM. The seasonal variation of the errors has a range similar to that of the Dudeney model, and has the maxima and minima in the same months (November and May).
7.4. Model Errors at Midnight
 The parabolic model has been much maligned, basically because it gives such large and biased height errors during the day. However, if we concentrate on midnight, when the parabolic model errors have a small spread, we find that the spread in the errors is significantly less than that for the Dudeney and Bilitza models. In fact, the spread in the errors in the midnight values of hmF2 for the 84 year/month cases for Slough is a factor of ∼5 smaller for the parabolic model than for the Dudeney model.
 The parabolic model errors for the anomaly station are smallest at midnight, in both average and spread, at ∼15 ± 2 km. The Dudeney errors at midnight for this station are the smallest for the day, at ∼0 ± 5 km. The largest errors occur at ∼06 and ∼18 UT/LT.
 The errors in the Dudeney and Bilitza models at midnight both have an upward trend with decreasing solar activity (from 2000 to 2006). This trend is smaller for the Dudeney model (∼10 km) than for the Bilitza model (∼25 km), which is why the midnight values are less spread for the Dudeney model than for the Bilitza model.
7.5. Model Errors at Midday
 There would obviously be a desire to use daytime ionograms as well as the nighttime ionograms that have the smaller seasonal and solar cycle variability of the errors in hmF2. The logical model to use would be the Bilitza model, since the errors have no solar-cycle dependence in the middle of the day, in contrast to the Dudeney model. However, as shown in Figure 11, there is still a large seasonal variability in the errors, from about −20 to +10 km, which would impose noise on the long-term trend. This formula has a bias of about −5 km. The daytime errors in the parabolic model are very large and very spread, so that model is no use for daytime ionograms.
8. Discussion of Results
 For long-term trend analyses, we would like to know how the accuracy of the model values of hmF2 changes, if at all, with season and solar activity. The best model would be the one with the smallest seasonal and solar cycle variations in the errors, even if it has a systematic error. In what might be considered a surprising turn of events, it has been shown that the best model of hmF2 to use for long-term studies would be the parabolic model, but that such studies should be restricted to midnight ionograms. The systematic positive bias in the midnight parabolic values of hmF2 should cause no problems. A random scaling error of ±10 km should be expected, and largely overcome by using monthly median values of hpF2. Note that UAG-23A [Piggot and Rawer, 1972, section 1.41] recommends against the scaling of this parameter for general use because of the large errors in the model for daytime ionograms. Thus hpF2 is not available for most stations. The parabolic model had to be implemented at the time of the original scaling, since it is the virtual height at 0.834 foF2 that is saved as the parameter hpF2. The papers by Chandra et al.  and Sharma et al.  indicate that this parameter was in fact scaled for Ahmedabad.
 For those stations at which the value of hpF2 was not scaled, the Dudeney  model is recommended over the Bilitza et al.  model for midnight ionograms. The scatter in the model errors is smallest at midnight, and is smaller for the Dudeney model (because the errors have a smaller solar-cycle variation). During the day, the Bilitza et al.  formula gives the smaller range of errors because it includes a successful solar cycle variation term. The uncertainty in the scaled values of M(3000)F2 should be expected to be at least as large as the standard scaling accuracy of ±0.05, with a superimposed random component. Assuming the validity of the Shimazaki formula, an uncertainty of 0.1 would correspond to an uncertainty of ∼15 km in hmF2. If the errors are indeed random, they could be overcome by using monthly median values of hmF2 derived using the Dudeney and Bilitza formulas. There is little point in using the original Shimazaki formula for hmF2, since the Dudeney and Bilitza formulas give smaller errors, especially during the day. The Shimazaki errors are very similar to those of the parabolic model shown in Figure 7.
 Note that the validity of the conclusions presented here is predicated on the assumption that the IRI is a better representation of the subpeak ionosphere than the convenient analytical models used to establish the simple models of hmF2. The errors inherent in the simple formulas may be the cause of the situation reported by Rishbeth and Davis  “Studies of decades of ionosonde data from Europe and the Antarctic give a rather conflicting picture…”. Restriction of long-term analyses to midday and midnight ionograms with the appropriate model of hmF2 might help to clarify this conflicting picture.