TIDs induce apparent range and azimuth motion on OTH radar signals. That is, since the TIDs induce real variations in measured delay and received azimuth of even stationary targets but since the TIDs are not included in the radar's current ionosphere model, when converted to ground coordinates, these variations appear as variations in ground range and azimuth. Our initial investigations thus far have concentrated on TID-induced apparent range variation. For a stationary target, one might at first expect that group range rate and IDOP should be proportional since all apparent range motion for a stationary target must be due to ionospheric motion. The situation is, however, more complicated than this.
2.1. Group Range Rate–Doppler Discrepancies
 It is an experimental fact that very often the group range rates of targets tracked by OTH radar do not agree with the Doppler shifts of those same targets. This is particularly evident in the case of ship tracks, where at times a ship may appear to be receding (positive group range rate) while its Doppler indicates an approaching target (positive Doppler shift). Normally, one would expect a positive Doppler to indicate an approaching target. Here we show that group range rate and Doppler can in fact differ significantly from each other in HF propagation when the changing propagation path is taken into account.
 IDOP is closely related to the rate of change of phase path, P,
where f is the signal frequency, c is the speed of light, and the phase path is over the ray connecting the radar to the target (for simplicity, we will only consider one-way propagation here, though the results we present carry over to the two-way case as well). This ray is obtained mathematically by Fermat's principle, which states that the path of predominant energy flow is one of “stationary phase.” This is written as
where n is the refractive index, ds is the geometric distance element along the connecting path, and δ symbolically represents spatial variations of the path connecting the (fixed) end points (radar and target). Equation (2) states the fact that the path of energy flow must be one for which small deviations from the path do not give a significant phase difference. If significant phase changes occurred, destructive interference would, on the whole, cancel any net propagation of energy. The energy must flow through regions where the phase remains constant for small transverse variations of the path.
 For oblique HF propagation (and ignoring magnetoionic effects), the following expression for the ionospheric index of refraction holds quite accurately:
Here Ne is electron density, κ = 8.061 × 10−5 MHz2 cm3 is a constant, and f is the operating frequency of the radar. We have used the fact that for highly oblique HF propagation, the electron density term remains small compared to unity.
 The expression for phase path can then be written as
where Geo represents the geometric distance along the ray path. Doppler shift is the time derivative of phase, so it is proportional to the time derivative of phase path,
 As in Fermat's principle, the δ in equation (5) represents a change in the propagation path. It is the first two terms in equation (5) that can cause the discrepancy between group range rate and Doppler. The third term provides a Doppler shift due to the changing electron density along the path, often called IDOP (ionospheric-induced Doppler). The first term contains the usual “target” Doppler shift due to target motion. It also contains a contribution due to a change in the ray path, the change required for the ray to maintain contact with the target as it moves or as the ionosphere changes. The second term represents the ionospheric contribution to this “path-change” effect. Let us formally distinguish the target motion contribution from the path-change contribution in the first term, rewriting equation (5) as
Now, by Fermat's principle, the second and third terms cancel. That is,
since this just defines the ray path to the target. Thus we arrive at a simpler expression for the phase path rate (which, by equation (1), is proportional to Doppler shift) [Bennett, 1968],
Equation (8) states that the Doppler shift measured by the radar is simply composed of the contribution due to target motion augmented by the ionospherically induced Doppler shift (IDOP).
 Now let us examine group range rate. Group range rate is measured as the rate of change of delay, or group path (P′), to the target. For ionospheric propagation, it is given by
We have used equation (3) and note that because of the appearance of the index of refraction in the denominator, the sign of the electron density term is opposite that in the corresponding phase path equation, equation (3). Paralleling the development of equation (6), we arrive at
 Now we can see important differences between group range rate and Doppler. Note that in equation (10), the fourth (IDOP) term has the opposite sign relative to the target motion term as compared to equation (6). This fact is normally accounted for in OTH radar processing algorithms. However, there is a subtle but important further difference. The second and third terms in equation (10) no longer cancel by Fermat's principle because the ionospheric contribution has the opposite sign relative to equation (6). There is a definite ray path change contribution in the calculation of group range rate that does not appear in the calculation of Doppler. This contribution is responsible for the group range rate–Doppler discrepancy.
 To summarize, we have shown that group range rate and Doppler are not exactly the same thing, the difference being that group range rate is sensitive to changes in the ray path connecting the radar and target. This phenomenon is not apparent for line-of-sight radars because, with their fixed straight line ray geometry, the path-change contribution is negligible. For HF OTH radar, however, the effect can be quite large, especially in the case of slow-moving surface targets. Ionospheric motions, especially those related to large TID events, can cause quite significant changes in ray path geometry, manifesting large group range rate–Doppler discrepancies [Croft, 1972].
2.2. Group Range Rate and Doppler Correlation
 Very often, group range rate and Doppler are correlated or even proportional (excluding times where discrete changes of propagation mode occur on the link connecting radar to target). For now, just consider a static target (like a beacon) so that all Doppler shifts are induced by the ionosphere alone (IDOP). Correlation of group range rate with IDOP can be justified by considering a simplified flat Earth ionosphere model in which the electron density increases linearly with altitude above a minimum height h0, but with h0 or its strength changing with time,
 For a planar ionosphere with the simple relationship (11), any ray path will be a straight line from the ground to altitude h0 and will trace a parabola above h0. The equation of this parabola is [Budden, 1966, p. 179]
where z > h0, the horizontal distance x is measured from the launch point, and γ0 is the ray's launch elevation angle.
 We can construct analytic formulae for phase path P(t) and group path P′(t) using equations (4) and (9), noting that ds = , and can differentiate them. The task is simplified by the Breit-Tuve theorem, which states that in a planar, stratified ionosphere, the group path connecting two ground points with ground distance D is equal to the two legs of the isosceles triangle whose base connects the points and whose other two legs are at launch angle γ0 from the horizontal. That is,
 For our problem, the ground distance D is fixed, and γ0 varies with time. Moreover, we can use equation (8) for phase path P with the first term set to zero since it refers to target motion, which is zero for the fixed-beacon case of interest here. Our simple, stratified model of the time-varying ionosphere can be simplified further by setting either α or h0 in equation (11) to a constant.
2.3. Case 1: Rising/Falling Ionosphere
 The constant α, varying h0 case is equivalent to the entire ionosphere rising or falling with time, and the rate change of Ne is
where Vz ≡ dh0/dt. After much calculus and algebra, we can solve equation (8) for this model to find
where the expansion is made in sin γ0, assuming low-elevation rays. Meanwhile, the group path rate can be had by differentiating equation (13),
where γ0 is determined by the need to connect ground points D apart by reflection from the current ionosphere. This angle can be determined by noting that x(zmax), the downrange distance of the highest point of the parabola in equation (12), must equal half the total distance,
This can be approximated in the low-angle case (in which h0 ≪ D and f2/ακ ≪ D) by
All this implies that to leading order in sin γ0,
 In fact, the path rates of change for this simple height-changing case are dominated by the fact that the bottom of the ionosphere is rising (or falling, if Vz is negative), and the straight legs of the ray below h0 increase at the same rate 2Vz sin γ0. This model yields a proportionality ratio between group path and phase path rates near unity.
2.4. Case 2: Ionosphere With Changing Strength
 A different result occurs if we assume constant layer height h0 but time-varying gradient strength α in equation (11). We can compute the difference between phase and group paths by subtracting equation (9) from (4),
The first two equalities of equation (21) are general, while the last merely assumes that the ionosphere is stratified and planar and begins at altitude h0. Here zc is the maximum altitude reached by the ray. In general, the ray's tangent at any altitude is found from Snell's law,
(Note that Snell's law, equation (23), looks different from usual because we are expressing it not in terms of the angle from vertical but its complement, the elevation angle γ.) Substituting this into equation (21) and performing trigonometric substitutions yields
Using equation (3) for the index of refraction n and (11) for our model of Ne, we find
The rate change of the group and phase paths can be had by differentiating equations (13) and (26),
We can relate dγ0/dt to dα/dt by differentiating equation (16) and noting that D is a constant,
For this model, we assume dh0/dt = 0 and use equation (29) to express dα/dt in terms of dγ0/dt. After much algebra and trigonometric substitution, equation (28) becomes
This can be divided into equation (27) to give the relation
 We see that in this model, the group path rate dP′/dt will be at least 3 times the phase path rate dP/dt. The lower the launch angle γ0, the closer this ratio will be to 3. The rates will be nearly proportional with time, as long as the launch angle does not vary too dramatically.
 In general, one might expect some combination of these two limiting case 1 and case 2 effects, so that the proportionality might range from 1 to quite large values. Note, however, that the “flat ionosphere” limitation of the simple analytical model used in the above two cases does not capture other geometrical effects that can occur in a real undulating TID environment, such as effects associated with the tilt of the ray reflection region or relatively discrete changes in the propagation mode structure as different phases of the TID pass through the reflection region. These can cause the apparent recession of an approaching ship, for instance, that was referred to at the beginning of this section. The two analytical cases above only capture the effects of height changes and density changes caused by the TID and show that density changes are sufficient to cause apparent group range rate–Doppler discrepancies.