High-resolution Doppler ionograms taken at 5 min intervals were obtained from a KEL IPS 71 ionosonde operating over a full ionosonde sweep range. The ionograms were converted into true height profiles using the program POLAN. POLAN also produced an equivalent parabolic layer model of best fit to the true height profile. A parabolic layer model of the ionosphere is defined by three parameters, namely, peak height, maximum electron density/critical frequency, and parabolic thickness. Equations for calculating Doppler shift from a time-varying parabolic layer model have long been known but have seen remarkably little use in the absence of suitable input data and a means of verifying the results. This paper shows that these equations can provide an accurate means of calculating ionospheric Doppler shift based on standard ionospheric ionograms taken at a 5 min rate over a 24 h period when compared with actual Doppler measurements. Apart from its scientific interest, the demonstrated technique will prove particularly valuable in deriving Doppler shift and associated signal fading for the real-time control of over-the-horizon radar and HF communication links.
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 Radio waves reflected from a time varying ionosphere experience a Doppler shift. A variety of methods have been used over the years to measure Doppler shift. A common technique has involved transmitting on a single frequency and Fourier analyzing the signal returned from the ionosphere. Fourier analysis is desirable because the returned signal normally contains a spectrum of frequencies resulting from multiple reflections between the ground and the ionosphere as well as from separate layers within the ionosphere. As well, the Earth's magnetic field causes the ionosphere to be birefringent so that a single linearly polarized radio wave on entering the ionosphere separates into o and X-ray modal components which each traverse a slightly different path within the ionosphere. The resultant Doppler spectrum can range from a number of sharply defined Doppler-shifted frequencies under quiet conditions to a spread spectrum when the ionosphere is highly disturbed such as may occur during spread F.
 From a practical point of view, the multiplicity of Doppler shifted returns from the ionosphere produces signal fading and associated rapid phase shifts which can be highly deleterious in a number of HF engineering applications. Such applications may require the modeling of ionospheric propagation conditions using archival or real time measurements. An inability to measure ionospheric Doppler shift and Doppler spread simultaneously over the full range of ionospheric HF propagation has resulted in the development of statistically based mathematical models of propagation used by engineers for equipment design. Such models often fail to reflect the physical reality of ionospheric propagation and consequently the equipment so designed may not work in practice as well as theoretically expected. The Doppler measurements and calculations presented in this paper provide a physical picture of ionospheric characteristics on which improved ionospheric physical models could be based.
 A previous paper [Lynn, 2007] has described the application of a time-interleaving technique to the development of a Doppler ionosonde capable of making high-resolution Doppler measurements at every frequency of an ionosonde sweep. The resultant Doppler ionogram can be completed in less than 3 min. This technique was commercialized in the IPS KEL 71 ionosonde. Examples of the output of such an ionosonde when observing a range of ionospheric phenomena are given in that paper. Of particular interest was the discovery that in daytime, the maximum Doppler value near the critical frequency of the ionosphere could be proportional to the rate of change of critical frequency. This relationship deteriorated at night. Further investigation suggested that such a relationship could be explained theoretically in terms of a simple parabolic layer model of the F2 region.
 This paper describes in detail the derivation of Doppler shift using a time-varying parabolic layer model of the F2 region in which the parabolic layer parameters are derived from standard non-Doppler ionosonde measurements converted into true height profiles by the readily available software program POLAN. The Doppler measurements thus synthesized are then compared with the observed values obtained from a KEL IPS 71 ionosonde. Apart from the theoretical and scientific interest, a capability to derive accurate Doppler information from the many standard non-Doppler ionosondes already deployed around the world is of particular significance to both HF communications and over-the-horizon (OTH) radar.
 In its simplest theoretical form, a Doppler shift between an incident and a reflected frequency occurs when an electromagnetic wave meets an approaching or receding mirror. If the ionosphere is considered as such a mirror for reflecting radio waves then a ground based transmitter of frequency f1 will receive the reflected radio waves at a frequency f2 such that the Doppler frequency shift fd is given by
where s is twice the total free-space path between transmitter and reflector and λ is the free space wavelength. A more sophisticated derivation must take into account the fact that the radio wave on entering the ionosphere is moving through a refractive medium which is changing with height as well as with time before a reflection point is reached. In this more general case, the Doppler shift in the returned wave is given by
where the corresponding Doppler velocity V is
Here P is the total vertical phase path between the transmitter and receiver.
 In this paper, Doppler velocity is preferred rather than Doppler shift. Doppler velocity as defined in equation (3) is the equivalent velocity of a mirror moving with respect to the observer and as such is independent of frequency. In reflection from the ionosphere, changes in the electron density profile produce a changing phase path which is sensitive to a number of variables [Bennett, 1967]. In this paper we use a simple parabolic model of the ionosphere which restricts changes in phase path to three variables. As developed by Pickering , the phase path for a parabolic layer model of the ionosphere is given by
where x = , and f is the operating frequency, fc is the critical frequency of the parabolic layer (foF2), ym is the half width of the parabolic layer, and hm is the height of the parabolic layer peak.
 The Doppler shift for such a model when all three parameters may be varying with time is given by Boldovskaya  as
In terms of Doppler velocity, equation 3 can be rewritten as the sum of the velocity contributions associated with changes in each of the three layer parameters i.e.
 In Figure 1, the dependence on changing phase path with radio wave frequency is seen to vary from zero at the base of the layer to negative infinity at the peak of the layer. The dependence on changing phase path with varying layer thickness varies from −2 at the base of the layer to a value of −1 at the peak of the layer whereas the dependence on changing layer height is uniform throughout at a value of 2.
 Note that the equations 6–8 are time independent for a fixed value of x. For circumstances where changes in one or more of the time varying terms in equation 5 are relatively small, such terms may be ignored. This may be the explanation for the close relationship between dfc/dt and the observed Doppler during daytime on this day but not on the following night, as mentioned previously by Lynn . This matter is discussed further in section 4.
 As described by Lynn , Doppler measurements were made at 150 sounding frequencies to a Doppler resolution of 0.039 Hz over the range 2–16 MHz. The ionosonde was located at Adelaide, South Australia where local time LT = UT + 9.0 h. The Doppler ionograms were obtained at 5 min intervals for several months. The 14 November 1993 was chosen for detailed study because of the absence of sporadic E. and spread F thus allowing a continuous period of ionograms suitable for the production of accurate true height electron density profiles.
Figure 2 shows that coherent oscillations in Doppler extended over the full frequency range of the F2 layer and were always present on what is, in ionospheric terms, a very quiet day. These oscillations represent the ubiquitous presence of small-scale traveling ionospheric disturbances (TIDs) whose characteristics will be described further in a later section.
4. Synthesized Doppler Shift
 The computer program POLAN [Titheridge, 1988] was applied to the Doppler ionograms in order to obtain the three basic parameters defining a parabolic ionospheric layer. The input to POLAN requires the ionogram trace to be “skeletonized.” That is, at each ionogram frequency there must be only one virtual height value. Doppler ionograms allowed this to be done automatically by choosing the height with the highest amplitude peak in the Doppler spectrum. Such automatic processing was only practical for O-ray ionograms where the X-ray traces were removed by using suitably phased crossed-delta antennae. Occasional additional points of noise could also appear and were manually removed.
 Each skeletonized trace was processed by POLAN to produce a true height profile and the equivalent parabolic layer parameters fc, hm and subpeak thickness w (defined by Titheridge  as the total electron content up to the layer peak divided by the peak density). The ym value was calculated from the subpeak thickness using the following formula for a parabola
Figure 3 demonstrates the parabolic fit provided by POLAN to both day and night true-height profiles and the ionogram traces from which they were derived. The equivalent daytime parabolic layer includes ionization below the F2 layer. Since this varies slowly with time, its contribution to the Doppler shift should be negligible. However, any errors in the derivation of this part of the true height profile will add to those of the F2 region. It would be preferable to use a parabolic layer fit to the F2 true height profile alone. As seen in Figure 2, the equivalent nighttime parabolic layer provides a closer fit to the true height profile below the ionospheric peak.
 Values of the parabolic layer parameters derived from the observed ionograms over the course of a day are shown in Figure 4. The difference between the day and night values of ym is exacerbated by the inclusion of the sub–F2 layers in the daytime values of ym. This also results in nonrealistic values of changing layer parameters during the sunrise and sunset transitions hence the time gaps in calculated Doppler values.
 Detailed examination of the data in Figure 4 shows short period (<2 h) variations in hm and ym to be essentially in phase. This is to be expected from the downward phase motion of TIDs. A decrease in the height occurs first at the critical frequency of a layer and takes a finite time to work its way down to the base of the layer. Thus the slab width of the subpeak ionosphere will decrease as the height at the peak decreases. Since the phase path partial differentials for hm and ym have opposite sign (Figure 1), in-phase changes in hm and ym will tend to cancel. In contrast, the variations in both these parameters are roughly 90° out of phase with the variations in critical frequency fc, such that peak deviations of hm, fc relative to the undisturbed background occur during the associated periods of falling height. A subsequent paper will show these relationships to be true in greater detail and over a greater range of magnitude and variational period than considered here.
Figure 5 compares the summed Doppler velocity components of equation (9) with the measured Doppler velocity at 0.98 fc. The overall agreement between the synthesized and observed values of Doppler velocity is remarkable considering the expected accumulation of experimental error both in the measurement of Doppler shift and in the derivation of true height profiles from successive ionograms whose quality necessarily varied.
Figure 6 separates out the three velocity components of equation (9) for comparison with the observed Doppler velocity to determine their relative importance. Examination of Figure 6 shows that V1(fc) follows the observed Doppler variations during the day but differs significantly from the observations during the night. V3(hm) follows the observed variations both day and night whereas V2(ym) shows little relationship with observation apart from the tendency to be in antiphase with the height- dependent velocity V3(hm) in daytime.
 A previous paper [Lynn, 2007] noted that the rate of change of fc correlated closely with the measured values of Doppler shift in daytime for the observations made on the day examined here. This relationship ceased at night indicating a change in the relative importance of time variations in the three ionospheric parameters fc, ym, and hm in determining Doppler shift. In Figure 7, the Doppler velocity measured in daytime is compared with values of dfc/dt normalized to the scale of the other parameters as well as the complete synthesized Doppler velocity. The relation between dfc/dt and measured Doppler in daytime is comparable in accuracy with the Doppler velocity calculated from the full parabolic model indicating that the number of additional parameters in equation 5 contributed little except noise with contributions from hm and ym largely cancelling out. The relationship deteriorated at night because the fluctuations in fc were smaller and noisier while the changes in height became much greater than during the day.
 If Figure 4 is representative, it would seem that major background changes in parabolic layer thickness rarely occur at middle latitudes outside the periods of sunrise and sunset in a quiet ionosphere. Some consideration of the parabolic model (Figure 1) shows that the Doppler velocity component due to changes in fc will be greatest near the peak of the layer. In contrast, the effect of a uniform change in height will be constant at all frequencies.
 The observations discussed so far refer to Doppler measurements made in the vicinity of fc. The ability of the model to predict Doppler shift at lower frequencies within the F2 layer is of greater practical interest since no man-made HF propagation system is intentionally operated at the ionospheric critical frequency because of its instability and diurnal time variation. Practical systems usually operate at radio frequencies which can be expected to maintain propagation for long periods of time without the requirement for constant frequency changes. The behavior of the parabolic model was thus investigated at a fraction of fc.
 In Figure 8a, measured values of fc (open circles), foF1 (pluses), and frequency f = 0.8 fc (solid line) are shown. Figure 8b compares the synthesized (crosses) and observed (solid line) values of Doppler velocity at frequency f. In daytime, the synthesized value of Doppler velocity was still in reasonable agreement but showed signs of deterioration in the early part of the day. This presumably arises from the fact that both hm and ym in daytime are not derived from an accurate fit to the F2 layer but represents a parabolic layer which contains the total electron density below the ionospheric peak. The measured Doppler velocity diminishes as the value of foF1 is approached which should represent the base of the equivalent F2 parabolic layer but remains significantly higher than the base of the daytime subpeak equivalent layer given by POLAN.
 The agreement between observed and calculated Doppler velocity at night as seen in Figure 8b remains very good throughout and, although not shown, continues to be good right down to the minimum frequency of the ionosonde. Here the model parabolic layer and the actual ionospheric profile fit closely and the relatively small changes in fc appeared to have no effect on the total Doppler velocity in comparison with changes in hm and ym.
5. TID Descent Velocities
Figure 9 is a plot of daytime reflection frequency at a number of fixed true heights in the ionosphere (isoheight plot). As expected for atmospheric gravity waves, the disturbances in frequency and thus electron density are seen to descend through the profile. Measurements were taken at selected peaks and troughs as marked in Figure 9 and plotted as a function of true height versus time delay with results shown in Figure 10. Such measurements are necessarily crude because the electron density waveforms often change in shape over the observed height range but they do give a reasonably consistent descent velocity of 57 ± 20 m/s.
Figure 11 is a plot of night reflection frequency at a number of fixed true heights in the ionosphere (isoheight plot). Some measured night descent velocities derived at the points shown in Figure 11 are displayed in Figure 12. These show a larger range of velocities (−50 to −113 m/s than in daytime. These results suggest a need to take a statistical number of such measurements over many days to see if there any definite relationships can be established between descent velocities and time of day, TID characteristics and background wind velocity and direction.
 Note that if the whole ionosphere moves in the vertical plane then hm will vary accordingly but there will be no change in ym. Such conditions can be seen at night in Figure 4 (19–27 LT). Here small-scale short-period changes in hm continue to be associated with similar changes in ym. However, the large-scale changes in the height occurring over a period of several hours are not reflected in changes in ym. At this larger scale, the generic relationship between an increased electron density deviation from the background level continues to be seen in the middle of an extended period of falling height (23–26 LT).
6. Second Hop Doppler
Lynn  gave some examples of individual Doppler ionograms showing the second hop Doppler shift to be twice that of the first hop. However, Figures 2, 9, and 11 show descending TID wavefronts to be the chief source of Doppler variation within the F2 layer. Since tilted TID wavefronts can be expected and indeed are well known to produce off-vertical angle reflections, it is not immediately obvious that the second hop Doppler variations should consistently be double that of the first hop. To test this hypothesis in detail, Doppler measurements by the KEL ionosonde were taken from the second hop, converted to equivalent vertical velocity by dividing by two and compared with the values deduced from the first hop. The results shown in Figure 13 demonstrate the close agreement thus obtained.
 The results described so far demonstrate that ionospheric Doppler shift, derived from ionograms taken at 5 min intervals, is determined by changes in the total subpeak ionospheric profile which are well represented by changes in the equivalent parabolic layer. These changes in turn are usually caused by the spatial and temporal properties of traveling ionospheric disturbances. The three parabolic layer parameters fc, hm and ym do not vary independently when responding to the presence of TIDs but show a consistent phase relationship. Such a relationship appears to be generic among ionospheric disturbances which move ionization up-and-down magnetic field lines irrespective of the driver, as previously commented upon [Lynn, 2007; Lynn et al., 2008].
 As Bennett et al.  has pointed out, the X-ray can be seen to a first approximation to have the same properties as the O-ray displaced in frequency by the O-X separation. The Doppler shift on the X mode is then approximately that of the corresponding component of the O mode, as was shown from the Doppler ionograms presented by Lynn . Thus at any given frequency, there will usually be a slight Doppler frequency difference between the O-rays and the X-rays at most frequencies below fc unless the ionogram is being traversed by a strongly Doppler-shifted TID (see examples in the work of Lynn et al. ).
 Theory tells us that propagating gravity wave TIDs have a sloping wavefront. The descent velocities measured here could be converted into wavefront tilt if the horizontal propagation velocities were known. However, horizontal velocity measurements were unavailable. The approximately linear descent velocities however suggest that TID wavefronts are themselves essentially linear through the F2 region. For TIDs of similar period, different descent velocities will result from varying degrees of wave tilt, horizontal velocity and background wind as well as the possible presence of more than one wavefront component.
 TIDs are not the only possible source of large scale ionospheric movement. Other sources include large-scale changes in electric field such as occur at the magnetic equator or during ionospheric storms [Lynn et al., 2006, 2008].
 Finally, according to Boldovskaya , the Doppler results for vertical incidence can be easily extended to oblique paths by multiplying equation 5 by sin β where β is the launch angle. Thus as path length increases, the Doppler shift from the same time-varying equivalent parabolic layer will diminish.
 This paper shows that vertical ionospheric Doppler frequency shift can be calculated with high accuracy from standard ionograms converted to true height ionospheric profiles which are then represented by a simple equivalent parabolic layer model. The time variation of such a parabolic layer model allows exact analytic calculations of Doppler shift. The parabolic layer model is a function of three parameters, hm, ym and fc, representing the height, thickness and maximum electron density of the F2 layer. These three parameters are shown to be sufficient to characterize the ionospheric Doppler effect of ubiquitous traveling ionospheric disturbances, the major source of ionospheric variation in the geomagnetically quiet ionosphere.