Irregularities in the electron density of the ionosphere cause phase and amplitude scintillation on transionospheric VHF and UHF radar signals, particularly at lower radio frequencies. The design of radar and other transionospheric systems requires good estimates of the coherence bandwidth (CB) and coherence time (CT) imposed by a turbulent ionosphere. CB and CT measurements of the equatorial ionosphere, made using the Advanced Research Project Agency Long-range Tracking and Identification Radar 158 MHz and 422 MHz phase coherent radar located on Kwajalein (9.4°N, 167.5°E), are presented as a function of the two-way S4 scintillation index at 422 MHz The log linear regression equations are CT = 1.46 exp(−1.40 S4) s at 158 MHz and CT = 2.31 exp(−1.10 S4) s at 422 MHz. CT also varies by a factor of 2–3 depending on the effective scan velocity through the ionosphere, veff. The CT and CB, as a function of S4, have been compared to those from the Trans-Ionospheric Radio Propagation Simulator, a phase screen model. A close agreement is achieved using appropriate values of veff and midrange values of phase spectral index and outer scale. Validation of CB is, however, limited by insufficient radar chirp bandwidth. Formulating the model in terms of the two-way S4 index (an easily measurable parameter) rather than more fundamental phase screen parameters (which are difficult to obtain), improves its utility for the systems engineer. The frequency dependencies (spectral indices) of S4 and of CT are also presented to allow interpolation and some extrapolation of these results to other frequencies.
 Turbulent processes in the ionosphere create a spectrum of electron density irregularities which forward scatter VHF and UHF radio signals passing through them. The resultant phase and amplitude fluctuations (or scintillation) at the receiver limits the time and bandwidth over which the channel is coherent. The coherence time (CT) and coherence bandwidth (CB) are useful engineering parameters for the design of transionospheric radio and radar systems. For example, the ionospheric CB may limit either the data transmission rate in a satellite communications link or the range resolution of a transionospheric radar. The ionospheric CT may limit the number of pulses available for coherent pulse integration, which could, for example, limit the along-track resolution in a space-based synthetic aperture radar image [van de Kamp et al., 2009; Belcher and Rogers, 2009].
 Studies of scintillation over a two-way (radar) path are relatively uncommon due to the expensive radar facilities required; but see, for example, papers by Knepp [1983a, 1983b] and Coster et al. [1993, 2002] who have addressed the impact of the ionosphere on radar phase-coherent pulse integration. This paper extends the earlier work by Cannon et al.  and a companion paper [Rogers et al., 2009]. It uses the measurements from the Advanced Research Project Agency (ARPA) Long-range Tracking and Identification Radar (ALTAIR) radar in association with the Trans-Ionospheric Radio Propagation Simulator (TIRPS) [Rogers et al., 2009] to understand the relationships between CT and CB and parameters such as S4. In so doing we have been able to further refine the TIRPS model and we gain insight into a number of important issues.
2. Wideband Ionospheric Distortion Experiment
 The measurements in this study were collected as part of the joint UK-U.S. Wideband Ionospheric Distortion Experiment (WIDE) [Cannon et al., 2006]. WIDE uses the ALTAIR monostatic radar located on Kwajalein (9.4°N, 167.5°E) to track orbiting calibration spheres. Linear FM chirps were transmitted coherently at 158 MHz (7 MHz bandwidth) and 422 MHz (18 MHz bandwidth) with a pulse repetition frequency (PRF) in the range 285 ± 50 Hz and with PRF changes occurring no more than once every 20 s. The complex received signal was recorded for analysis as described below. For these studies we used only object 2826, which is at an orbital altitude of 767±6 km altitude and at 70° inclination.
 After proof-of-concept trials in January 2005, subsequent measurements took place during the August–October 2005 and 2006 seasonal peak in scintillation occurrence at Kwajalein [Aarons, 1993]. Late evening passes, when the likelihood of strong scintillation was greatest [e.g., Aarons, 1982], and passes with high maximum elevation were selected. Although the measurements were taken near the minimum of the solar activity cycle, strong scintillation was nonetheless observed on several occasions, so providing a wide range of conditions. For each pass the data were processed to generate a series of scattering functions (CSFs) [Cannon et al., 2006; Rogers et al., 2009] from which the CB and CT could be derived.
 Each CSF was calculated from sequences of 1024 contiguous radar chirp returns which, at an average pulse repetition frequency (PRF) of 285 Hz, constitute a 3.6 s sample. The component of pulse spreading due to ionospheric dispersion was removed by calculating the total electron content (TEC) from the differential group delay of the 158 and 422 MHz chirp returns. Measurements presented in this paper were derived from channel scattering functions (CSFs) for five passes of object 2826, giving a total of 609 CSFs (see Table 1). Note that the local time at Kwajalein is 11 h 10 m ahead of UT. Data for elevations below 20° were excluded from the measurements to minimize the impact of ground multipath and any tropospheric perturbations. This also ensures that the radar tracker had established a good lock on the target. Signal-to-noise ratios ranged between 9 and 53 dB.
Table 1. Dates, Times, Maximum Elevations, and Number of CSFs for Passes of Satellite 2826
Local Time Range
Maximum Elevation (deg)
Number of CSFs (Above 20° Elevation)
18 Jan 2005
21 Oct 2005
25 Oct 2005
14 Sep 2006
15 Sep 2006
Total = 609
 Under a homogeneous ionosphere (with no irregularities), the CSF profile would be confined to a single Doppler bin at 0 Hz (the Doppler bin width is 1/3.6 s = 0.28 Hz). The delay profile would exhibit a sinc-squared profile (the linear chirp autocorrelation function) convolved with the Hann filter. This results in a 3 dB width ≈ 1.5/2B, where B is the chirp bandwidth(=107 ns at 158 MHz and 42 ns at 422 MHz).
3. Trans-Ionospheric Radio Propagation Simulator
 The Trans-Ionospheric Radio Propagation Simulator (TIRPS) has been developed to simulate WIDE measurements. The model, based on a thin phase screen, parabolic equation propagation technique [Knepp, 1983a, 1983b], is described in detail by Rogers et al. . The phase screen, which represents the effect of ionospheric irregularities, is defined to be a log-log linear spectrum (equation (1)) which flattens off for irregularities greater than an outer scale size Lo.
where κ = spatial frequency, re = classical electron radius, λ = radio wavelength, G is a geometric enhancement factor [see Rino, 1979], CkL = vertical integrated strength of turbulence at the 1 km scale, θ = propagation zenith angle at the ionospheric pierce point (IPP), p = phase spectral index, Γ = Euler's gamma function and κ0 = outer scale frequency = 2π/Lo. Equation (1) is derived from equation (16) of Rino  where we have changed the dependent variable from temporal to spatial frequency and used the definition CkL = CsL(1000/2π)p+1. In the simulations the three important variables are p, Lo and (G.CkL.sec(θ)), the latter combination having been treated as a single variable parameter.
 TIRPS uses Rino's equation to define the phase screen spectrum only. This uses the assumption that a thin phase screen adequately represents the extended scattering region, as demonstrated by Bramley . Thereafter the Parabolic Equation (PE) approximation [Levy, 2000] is used to describe the wave propagated through free space up to orbit height. This approximation to the full electromagnetic wave equation assumes propagation close to the line-of-sight axis, and assumes that refractive index fluctuations Δn/n ≪ 1 (which implies fractional electron density variations ΔNe/Ne ≪ 1). Errors in the standard PE approximation due to off-axis propagation are of the order 10−3 at 10° off-axis [Levy, 2000, p. 11] and so are negligible for the ALTAIR radar geometry which has a 3 dB beam width of approximately 3° at 158 MHz or 1° at 422 MHz. Assuming a typical postsunset F region electron density of 1011 m−3, the root-mean-squared ΔNe/Ne is less than 0.2 for CkL <= 1033 rising to 0.6 at CkL = 1034 (based on Rino's definition of Cs (equation (7) of Rino ) and assuming a scattering path length L = 200 km, Lo = 10 km, and p = 2.5). The assumption ΔNe/Ne ≪ 1 in the PE method may therefore become unrealistic above around CkL = 1033. In this paper we have not attempted to estimate the scale of the associated errors as this would require the implementation of more complex numerical techniques for solving the full electromagnetic wave equation. Nonetheless the TIRPS model has been shown by Rogers et al.  to provide reasonable approximations to the observed channel scattering functions even at high CkL values.
4. Coherence Time and Coherence Bandwidth and S4 Measurement and Simulation
where ΔB is the Doppler spread and ΔT is the delay spread, both of which are determined from the CSFs. Doppler spreads are calculated for the central 68.3% of the total power in the CSF (integrated across all delay samples) as this represents the root-mean-squared spread for a Normal power distribution, equivalent to −2.2 dB from the peak. The spread is calculated as the difference between the 50 + (68.3/2) percentiles of two cumulative distributions, one summed in the positive delay direction and the other in the negative delay direction. To avoid integrating noise in the measured CSFs, only those points within 6 dB of the peak were used in the power summation. Delay spreads were calculated in a similar manner. The S4 index (the ratio of the standard deviation of signal intensity to its mean) was calculated over 10s periods (i.e., the central time of the CSF ± 5s). This period is somewhat shorter than is conventional for GPS satellite measurements and represents a compromise between averaging over many Fresnel zones and minimizing the spatial blurring which is inevitable for a fast moving LEO satellite.
Figure 1 presents a log linear scatterplot of CT for 158 MHz CSFs against the 422 MHz S4 index. Taking all measurements together, there is a close to exponential decline in CT with S4. The log linear regressions (gray dashed line in Figure 1 for 158 MHz) are represented by the equations:
Each point in Figure 1 has also been categorized into one of four ranges of the effective scan velocity, veff [Rino, 1979]. (Note that veff was generally much lower than the ionospheric penetration point (IPP) horizontal scan speed since each pass was aligned close to the magnetic meridian (i.e., along the irregularities).) It is clear that the data are well ordered by veff with a very useful reduction in spread. The corresponding log linear regression equations are shown in Table 2 for two veff ranges in the two frequency bands. Table 2 indicates, very broadly, that at both frequencies, a threefold to fourfold increase in veff results in a twofold increase in CT at S4 = 1.0.
Table 2. Regression Equations for Various Ranges of veff for CT at 158 and 422 MHza
veff Range (m s−1)
Regression Equation for CT (s)
CT Regression Value at S4 = 1.0 (s)
S4 is the two-way S4 at 422 MHz. The regression equations are evaluated for S4 = 1.
1.70 exp(−1.46 S4)
0.74 exp(−1.19 S4)
2.19 exp(−0.83 S4)
1.55 exp(−1.17 S4)
 The solid lines in Figure 1 represent mean values obtained from 10 random realizations of the TIRPS model, each line representing a different fixed value of spectral index, p. Each point on the line is connected in order of increasing log10(G.CkLsec(θ)) from 32 to 36 in increments of 0.5, resulting in generally increasing S4. A veff of 730 m s−1 (the mean of all veff values), an outer scale of 10 km and a midrange elevation of 30° are assumed. Choosing p = 3 provides a good match to the composite exponential fit, with a similarly good fit observed in the 422 MHz data (dashed line in Figure 3).
Figure 2 reproduces the same ALTAIR measurements of CT but the superposed lines describe TIRPS simulations with fixed p = 3 and varying veff. As previously described, measurements with greater veff have lower CT values and this trend is reproduced in the simulations. For example, the lower red line (veff = 1500 m s−1) in Figure 2 provides a good fit to measurements in the category veff = 1400–1630 m s−1 (red squares) and similarly the other simulated TIRPS predictions pass through the appropriate range of ALTAIR measurements (though perhaps less well for the lowest veffrange). It is apparent that the inclusion of veff is a necessary prerequisite of any modeling procedure of the coherency time; without it, CT predictions may be in error by factors of between 2 and 3.
 Similar good agreements are observed between the 422 MHz simulations and measurements (Figure 3), although the 422 MHz coherence times were upper limited by the CSF measurement period (∼3.6 s) for S4 < 0.5. (A similar limitation is evident in the 158 MHz graphs but only for the lowest S4 < ∼0.2.)
 Coherence bandwidths (CB) are presented in Figure 4 for the 158 MHz measurements (points) and for the TIRPS simulations parameterized as in Figure 2. These measurements are upper limited due to the insufficient bandwidth of the radar waveform, and only in the 158 MHz data and above S4 = 1.3 did the CB drop significantly below this system-imposed limit. The TIRPS model (solid lines in Figure 4) reproduces this behavior reasonably well, at least for higher values of the veff parameter. The CB for the 422 MHz signals (not shown) fell below the system bandwidth on only three occasions.
 It should be noted from observations and simulations of the CSF presented by Rogers et al.  that the delay spread of the waveform under strong turbulence conditions is most evident at intensities below the −6 dB noise threshold used in this analysis. While these components would contribute little to the overall coherence bandwidth, in a radar system their presence may raise the clutter background, reducing the probability of detecting smaller secondary targets in the vicinity of the primary target.
5. Radio Frequency Dependencies of S4 and Coherence Time
5.1. S4 Index
 The results above, relate CT (and to some extent CB) to S4 at just two spot frequencies. To generalize the results to other frequencies it is necessary to examine the variation of CT and S4 with frequency. Published observations show that, in the case of low scintillation (S4(one-way) < 0.5) the S4 index follows an approximately log-log linear relation to the radio frequency f, i.e., S4 ∝ , where mS4 denotes the spectral index for S4 which varies between 1.6 for low S4 and 0.8 for high S4 on one-way paths [Davies, 1990, p. 290, and references therein].
 In Figure 5 the S4 values for 158 MHz signals are plotted against S4 on the simultaneously recorded 422 MHz signals. Diagonals on this plot indicate the value of −mS4. The same data have been replotted in Figure 6 which presents mS4 as a function of the two-way 422 MHz S4 index. At low S4 the points follow the diagonal for mS4 = 1.5 ± 0.5, while at higher S4 the spectral index reduces. A linear regression to the measurements (dashed gray line) in Figure 6 suggests a two-way S4 spectral index of = 1.53 − 0.80 S4 (2-way, 422 MHz) although the points are widely distributed about this line and the linear fit is poor above S4 (2-way, 422 MHz) = 2 where the 422 MHz waveform exhibits greater S4 than the 158 MHz waveform (i.e., mS4 < 0).
 The TIRPS model was also used to determine the spectral index of S4 as a function of S4 at the band center frequency. The solid line in Figure 6 represents these TIRPS simulated values (mean ± one standard deviation of 10 simulations) at each value of CkL with p = 3 and Lo = 10 km and veff = 730 m s−1. The simulated mS4 closely follow the linear trend observed in the measurements, reducing from approximately 1.5 at S4 = 0 to zero at S4 = 2.0. The slope of this line is slightly less negative above S4 = 2, as is also observed in the measurements.
 The solid lines in Figure 5 describe the TIRPS S4 values, where each line represents a different value of p. The simulated S4 index increased with increasing CkL but saturated close to 2.2 for a spectral index p < = 3, while higher values of S4 are realizable for p > 3. Fremouw and Ishimaru  showed that a limit of S4 = 2.2 (= √5) would be expected for a Rayleigh fading signal on perfectly correlated up and down paths. S4 greater than 2.2 is possible under conditions of strong focusing which can only be reproduced in highly anisotropic media using spectra with p > 3 [Rino, 1980]. The theoretical maximum S4 is 3.3 (= ) for Nakagami-m fading [Nakagami, 1960] on perfectly correlated up and down paths [Fremouw and Ishimaru, 1992] and in the ALTAIR data set only one point exceeded this value (maximum S4 (422 MHz) = 3.4).
5.2. Coherence Time
 Coherence times for the 158 MHz signals are plotted against those recorded simultaneously at 422 MHz in Figure 7. Diagonals in Figure 7 indicate the spectral index assuming a power law relationship with frequency. The same data are presented in Figure 8 which shows the spectral index of CT as a function of the S4 index at 422 MHz. Below S4 ∼ 0.5 the results are artificially influenced by the finite sample duration (as discussed in section 4) and may be ignored. At higher S4 there is a wide distribution in the spectral index from 0 to −2, centered about ∼−0.9 (indicated by the 30-point moving average (gray line in Figure 8)). The spread of the distribution narrows toward higher S4 (see Figure 8) and lower CT (see Figure 7). These results compare well with a coherence time spectral index of −1 which was predicted from the analytical model of Knepp [1983b], who assumed a Gaussian mutual coherence function and a κ−4 spectrum of three-dimensional electron density (equivalent to a phase spectral index, p of 3).
 The red solid lines in Figures 7 and 8 represent the ensemble average predictions from the TIRPS model using a spectral index p = 3 and a mean effective velocity veff = 730 m s−1. Once again the TIRPS model faithfully reproduces the experimental results.
6. Summary and Conclusions
 The Wideband Ionospheric Distortion Experiment (WIDE) using the ALTAIR radar sought to measure the equatorial transionospheric propagation channel under a wide variety of ionospheric conditions and to relate those measurements to the design of wideband radio systems, especially space-based wideband synthetic aperture radars. However, from an early stage it was recognized that a dual approach of measurement and modeling was required to best achieve these aims. This led to the implementation of the Trans-Ionospheric Radio Propagation Simulator (TIRPS). By making measurements and refining this model we have been able to better understand the impact that the ionosphere has on the operation of such sophisticated systems. So far all of our data collects have occurred during the years of sunspot minimum and our experiment lacks the bandwidth to fully elucidate the issue of coherency bandwidth. As we move toward sunspot maximum, the coherency bandwidths and times are likely to reduce and this should mitigate this limitation.
 In the work of Cannon et al.  we described the WIDE experiment, our first results and provided examples of channel scattering functions and estimates of the coherency bandwidths and times. In the work of Rogers et al. , a companion to this paper, we describe the TIRPS model in some detail and undertake some detailed cross comparisons between the measurements and the model, thereby providing credence to that model.
 In this paper we have extended that cross comparison to look in more detail at the issue of coherency bandwidth and times, for a wide range of conditions from weak scintillation to strong scintillation, under conditions of strong focusing. We have demonstrated a high degree of correlation between the model and the measurements and have demonstrated that the model is able to characterize the CT of the equatorial ionosphere on wideband systems, operating in the frequency range 150 to 450 MHz. While our CB intercomparison is broadly in agreement, the lack of measurement bandwidth means that we are unable to fully validate this aspect of our model.
 We have also recognized the need to have a practical strategy for running such a model and we have demonstrated that rather than using the integrated strength of turbulence parameter CkL, we are able to use the common S4 parameter and produce meaningful results. We note however, that we have demonstrated good correlation between our model and measurements when using the 422 MHz S4 derived from our two-way measurements. It remains to be seen if this can be extended to using S4 from other sensors, particularly from GPS receivers.
 Our results may be summarized as follows:
 1. The measured CT decreased exponentially with S4 (422 MHz) up to the Rayleigh limit of S4 = 2.2. The CT decreased below 100 ms at the highest S4 (>1.5) and highest veff.
 2. The TIRPS simulations best matched this trend using a phase screen with p = 3. A higher spectral gradient (p = 4) was required to simulate the CT for S4 > 2.2, (strong focusing conditions).
 3. There was a decrease in CT for higher values of effective scan velocity (veff) and the simulations provided a good fit to the CT trend with S4 when an appropriate value of veff was used in the simulation.
 4. The coherence bandwidth (CB) was generally limited by the radar chirp bandwidth except for a small number of 158 MHz CSFs above an S4(422 MHz) of 1.3, where the CB reduced to as little as 0.7 MHz. This behavior is again broadly reproduced in the TIRPS simulations (but see comments above).
 5. To extend these results to other radio frequencies we presented the frequency dependences of CT and S4 and their variation with S4 at the higher frequency.
 6. The spectral indices of the CT measurements vary widely about a mean of about −0.9, and this is reproduced by the TIRPS model.
 7. The spectral index of the S4 measurements decreases linearly with increasing S4(422 MHz), ranging between around 1.5 at low S4 to −0.3 at the highest S4. The TIRPS simulations closely follow this trend, for p = 3 and with an appropriate veff and elevation.
 The UK contribution to this program has been conducted with funding from the United Kingdom Ministry of Defence Science and Technology program. The MIT Lincoln Laboratory contribution was sponsored by the U.S. Department of the Army under Air Force contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States government. The authors would like to thank Paul Kossey for his unfailing support in the early stages of this UK-U.S. collaborative program.