Using ionospheric sounding together with fast computational inverse processing, it is now possible to obtain good real-time ionospheric models for use in geolocation for over-the-horizon (OTH) radar. However, deflection of HF propagation paths by traveling ionospheric disturbances (TIDs) remains a troubling cause of coordinate registration errors. Bandwidth and coverage limitations in ionospheric soundings preclude the ability to model TID structure in real time in most cases. It would be useful if TID-induced path deflections could be related to radar-measurable quantities like Doppler shift. As a first step in studying this possibility, we have considered the relationship between Doppler shift and group range rate for point-to-point HF propagation paths in TID environments. The nature of group range rate–Doppler correlation is exposed in three ways: (1) simple theoretical modeling, (2) ray tracing in simulated TID environments, and (3) analysis of OTH radar measurements of a fixed beacon. It is shown that group range rate and Doppler shift for fixed-point propagation paths are usually proportional with a ratio that depends on whether ionospheric motion or density changes predominate in the TID environment.
 Traveling ionospheric disturbances (TIDs) remain a troubling cause of coordinate registration (CR, geolocation) errors for over-the-horizon (OTH) radar. They are also the principal remaining mechanism limiting the performance of SIFTER, an enhanced target detection algorithm for OTH radar [Fridman and Nickisch, 2004], because the associated range/azimuth deflections spread the target return in the SIFTER computational space. In SIFTER, a portion of the effect of TIDs is mitigated by ionospheric-induced Doppler (IDOP) correction. Here the Doppler shifts imposed by the moving ionosphere are removed by aligning the Bragg line signatures in the radar returns. (These Bragg line signatures are enhancements in radar backscatter corresponding to coherent ocean wave scatter from wave components with half the radar's operating wavelength that are moving toward and away from the radar.) IDOP correction, however, does not affect the range and azimuth domains of the data. In a series of measurements where the U.S. Navy Relocatable Over-the-Horizon Radar (ROTHR), AN/TPS-71, dwelled on a fixed beacon for long periods of time, the beacon appeared to move several tens of kilometers in times as short as a few minutes. These apparent motions are clearly due to TIDs. An example of this is shown in Figures 1 and 2, which show the measured group range (or group path, denoted P′, which is the speed of light times propagation delay) and azimuth of the fixed beacon over a period of about 3 hours. The azimuthal swing near 0450 UT, for example, corresponds to an apparent motion of 80 km over a time span of only 15 min. It is easy to see that such artificial motion will be detrimental to trackers tuned for slow-moving maritime targets. Furthermore, since OTHR CR algorithms are not currently designed to account for TID apparent motion, very large CR errors will inevitably result.
 So far it has been impossible to mitigate the effect of TID-induced motions on OTH radar return signals. Conventional OTH radar soundings have neither the spatial nor temporal resolution to resolve TIDs. In a generalized CREDO approach, we augmented ROTHR soundings with total electron content data derived from low Earth orbit satellite beacons and were thus able to “image” TID structure and account for it in the CREDO ionosphere model [Fridman and Nickisch, 2001]. But augmenting ROTHR with a suitable satellite beacon data source is impractical in the near term.
 We know that to a large extent, IDOP is caused by TID activity. We also know a lot about the physics of TIDs. They are driven by acoustic gravity waves (AGWs), which are buoyancy waves in the neutral atmosphere akin to water waves in the ocean. Because of the exponentially decreasing density of the neutral atmosphere with altitude, the energy from fairly minor disturbances in the lower atmosphere turns into large-amplitude waves at altitude, and these drive the ionosphere up and down the geomagnetic field lines; the geometry of TID undulations is constrained by the geomagnetic field. There is potential, then, to correlate IDOP with TID-induced deviations and to model this relationship. Recent studies of TIDs have been done by Kirchengast et al.  and Ma et al. .
 We report here on a study of TID-induced apparent range deflections of HF sky wave signals. This study is based on both numerical simulation and on real OTH returns of a fixed beacon (transponder). For the simulations, we apply ray tracing in modeled TID environments using an implementation of the Bottone AGW/TID model [Bottone, 1992], which includes geomagnetic constraints and temporal evolution of the effects. Using these numerical simulations, we analyze IDOP–group range rate correlations with the goal of eventually developing a mitigation strategy for TID wander. The idea is to use the indication of TID activity present in normal OTH radar IDOP measurements to estimate the associated apparent wander of the OTH radar signals and to mitigate this wander. Previous studies of the HF signatures of TIDs were done by Georges and Stephenson  and Earl .
 In section 2 we present theoretical expectations for the correlation of group range rate with Doppler shift in TID environments. Results obtained in our numerical simulations are presented in section 3, and in section 4 we present a preliminary analysis of ROTHR fixed-beacon measurements. Concluding remarks are given in section 5.
2. Theoretical Expectations for Group Range Rate–Doppler Correlations
 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
 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.
3. Ray-Tracing Results
 Our purpose in studying TIDs is to look for possible ways of mitigating their effects in OTH radar processing algorithms for detection, tracking, and coordinate registration (geolocation). In this section we show results obtained using ray tracing in model ionospheres with superimposed TIDs. With ray tracing, we can compute all quantities of interest, particularly group path, phase path, and Doppler shift for rays that connect a receiver location and a fixed ground location (using ray homing). By computing ray paths for a wide variety of variables (such as mean ionosphere model, TID strength, ground range, signal frequency, etc.), we hope to discern correlations between TID-affected quantities that are easily measured by an actual OTH radar and interesting quantities that cannot be measured directly. In particular, we believe that existing ionospheric Doppler (IDOP) correction algorithms can be used to estimate Doppler shifts due to TIDs, so a relationship between Doppler shift and TID-caused modifications of group path would be very helpful.
 For our initial investigations, we used an undisturbed ionosphere model that was stratified (i.e., the electron density varied with altitude but not with latitude or longitude), derived from a moderately strong vertical sounding. On top of this we superimposed various realizations of the Bottone model for TIDs [Bottone, 1992]. This allowed us to simulate numerically what an OTH radar might observe when staring at a fixed beacon in a TID environment. Figure 3 displays the ratio of plasma frequencies of a Bottone model TID-disturbed ionosphere to the undisturbed background. The time shown is 120 min following initiation by an impulsive source at 29°N latitude, corresponding to a 5 Mt detonation. Maximum plasma frequency deviations are ±15% and may be somewhat large compared to typical naturally occurring TID variations.
 Our computations provide the deviations of group path, phase path, launch azimuth, Doppler shift, etc., for HF rays shot from a hypothetical radar location to a fixed ground point (which, in our computations, we generally located some 2000 km distant). We computed these quantities over an extended period of time while TIDs (caused by sources away from the propagation path) moved across the propagation region. These computations reveal a persistent correlation between the time rate of change of the group path, P′, and the TID-induced Doppler shift. Figures 4 and 5show examples of this correlation. In Figures 4 and 5, the solid curve is dP′/dt, the rate of change of the group path (obtained by numerically differencing the group paths output by our ray-tracing code). Each dashed curve is the (one-way) Doppler shift fD directly reported by the ray-tracing code (converted to phase path rate), multiplied by a constant factor. The best proportionality between the two curves is determined by minimizing the root-mean-square difference between the two.
 Recall equation (1), which shows that fD is proportional to phase path rate dP/dt. Consequently, a correlation between group path rate and Doppler shift implies a correlation between group path rate and phase path rate. Our algorithm multiplied fD by (c/f) before fitting; hence the derived proportionality factor is a dimensionless number, relating group path rate dP′/dt to phase path rate dP/dt. For a fairly wide set of cases, the proportionality constant varies from about 3 to slightly more than 5, similar to the proportionalities implied by the analysis of section 2.
4. OTH Beacon Results
 We have acquired a data set of ROTHR returns from a fixed beacon. The data set contains measurements from both the Virginia and Texas ROTHRs for 2 days in each of May and September 1999. The beacon was in Jamaica. Many examples of obvious TID activity are present in the data set. These data show evidence of the kinds of correlation exhibited by our ray-tracing calculations shown in section 3. Figure 6 shows examples that exhibit varying levels of correlation. Note that the proportionality factors (ratio of group path rate to phase path rate) are above unity in all examples, but none are as high as in most of our ray-tracing examples of section 3. In fact, of all the cases we have looked at so far in this data set, the largest proportionality factor is only 2.55, whereas in our ray-tracing analysis, the factors were always larger than 3.
 Further fixed-beacon data collections will be required to understand the extent to which these correlations manifest and what range of proportionality factors are typical. If, as is suggested by this data set, the proportionality factors from real data are smaller than those provided by our numerical simulations, it will be an interesting exercise to understand what modifications of the Bottone AGW/TID model are required to restore agreement.
 The nature of the correlation between ionospheric-induced Doppler shift (IDOP, proportional to phase path rate) and group range rate (group path rate) for HF propagation between fixed ground stations in TID environments has been studied theoretically, numerically, and by direct measurement. The simple theoretical analysis of section 2 exposes two mechanisms causing group range rate–Doppler correlations. Vertical motion of the ionosphere causes a direct proportionality between group path rate and phase path rate, where dP′/dt ≈ dP/dt. However, compression (or rarefaction) of the ionosphere, while continuing to yield a proportionality, strengthens the effect to dP′/dt ≥ 3 dP/dt. Our numerical simulations described in section 3 used ray tracing in model TID environments with results consistent with the theory of section 2, but dominated by electron density compression/rarefaction effects, as the ratio of group path rate to phase path rate always exceeded 3. The ROTHR fixed beacon measurements presented in section 4 also show strong group range rate–Doppler correlations, but here the ratio of group path rate to phase path rate was always below 3, indicating significant contributions from vertical ionospheric motion in addition to compression/rarefaction effects caused by passing TIDs.
 This work was funded by the DOD Counter Narcoterrorism Technology Program Office at Dahlgren, Virginia.