The design and operation of transionospheric VHF and UHF radars requires knowledge of amplitude and phase scintillation due to ionospheric scattering. Phase coherence is of particular importance where long coherent integration periods and large bandwidths are required. A thin phase screen, parabolic equation based, Trans-Ionospheric Radio Propagation Simulator (TIRPS) is described. Modeled channel scattering functions (CSFs) are compared to experimental VHF and UHF data derived from the Advanced Research Projects Agency Long-range Tracking and Instrumentation Radar on Kwajalein Island (9.4°N, 166.8°E). TIRPS quantitatively reproduces the experimental results, including the quasi-parabolic profile observed in the measured CSFs under strong turbulence conditions. Variations in the simulated CSF with ionospheric phase screen parameters are also presented. Under conditions of high integrated strength of turbulence (CkL), a low phase spectral index (p = 1), indicating relatively dense small-scale irregularities, produces pronounced range spreading. Conversely, when the spectral index is high (p = 4), indicative of strong focusing/defocusing by large-scale irregularities, there is increased Doppler spreading and, when the outer scale of irregularities is large, a greater likelihood of asymmetry of the CSF about the zero Doppler axis.
 In recent years the feasibility of space-based synthetic aperture radar (SAR) operating at frequencies below 500 MHz has been examined for both environmental monitoring [Rignot et al., 1995] and military applications when foliage- and ground-penetrating properties are required. Radars operating between 420 and 450 MHz are also used for ground-based radar space-object tracking.
 Unfortunately, at these low frequencies, the ionosphere can cause significant pulse distortion [Belcher, 2008], polarization rotation [Wright et al., 2003] and loss of phase coherence across a synthesized aperture [van de Kamp et al., 2009] and over a pulse integration period. This is particularly true in the equatorial region (±20° geomagnetic latitude) where the ionosphere can be highly structured and turbulent, forming irregularities in the electron density. These irregularities scatter the radio signals, causing amplitude and phase fluctuations known as scintillation.
 Understanding and mitigating these effects is critical in the design of transionospheric radar systems, particularly wideband systems. The effects of two-way (backscattered) propagation through a strongly turbulent atmosphere was first investigated by Yeh , who derived expressions for the backscattered intensity and the mutual coherence between forward and backscattered waves. Knepp and Houpis  extended this theory and experimentally demonstrated the predicted enhancements in backscattered signal intensity for a monostatic radar. More recently, Cannon et al.  presented the ionospheric distortion on wideband transionospheric VHF and UHF radar waveforms in terms of the channel scattering function (CSF) [Bello, 1963]. The CSF is a representation of the time-varying complex impulse response of a backscattered radar waveform which provides a simultaneous characterization of both ionospherically induced delay spread and Doppler spread.
 In this paper we describe a model of the transionospheric distortion and we compare it to sample channel scattering measurements. In a companion paper (N. C. Rogers et al., Measurements and simulation of ionospheric scattering on VHF and UHF radar signals: Coherence times, coherence bandwidths, and S4, submitted to Radio Science, 2008) we extend this analysis to a larger data set to quantify important parameters such as coherency time and coherence bandwidth.
2. Measurements of the Transionospheric Channel Scattering Function
 Our measurements were conducted as part of the joint UK/U.S. Wideband Ionospheric Distortion Experiment (WIDE) which employed the Advanced Research Projects Agency Long-range Tracking and Instrumentation Radar (ALTAIR). This is a monostatic radar located at the Ronald Reagan Ballistic Missile Defense Test Site (RTS) on Kwajalein Atoll (9.395 °N, 167.469 °E). It employs a 6 MW transmitter which feeds a 42-m-diameter parabolic dish. Upgoing linear frequency-modulated radar chirps were transmitted simultaneously at 158 MHz (7 MHz bandwidth) and 422 MHz (18 MHz bandwidth) with a pulse repetition frequency (PRF) in the range 285 ± 50 Hz with PRF changes occurring no more than once every 20 s.
 The results presented here describe signals transmitted to, and reflected back from, a calibration sphere (object 2826 in the NORAD catalog) of 0.3 m2 radar cross section (optical) in an orbit at 767 ± 6 km altitude and at 70° inclination. The sphere presented a constant aspect angle insensitive radar cross section, ensuring that any scintillation observed was due to atmospheric signal distortion rather than target tumbling. A 20° elevation mask was used to exclude measurements that might be contaminated by ground multipath. The radar measured a combination of the Doppler frequency shift due to target's range rate and a shift and spread in the Doppler frequency imposed by the irregular ionospheric medium. The Doppler frequency component due to target range rate has been subtracted from the radar measurements, as has the component due to bulk ionospheric refraction (as calculated from the relative delay of 158 and 422 MHz signals).
 The impact of the ionosphere on wideband radio signals has been represented by a channel scattering function (CSF), which is a measure of the channel's time-varying complex impulse response and maps the signal power distribution in group delay and Doppler frequency. The 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. Each returned chirp was pulse-compressed (cross-correlated with a replica of the transmitted chirp) using a Hann filter, and 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. The horizontal scale of the CSF represents the delay of the pulse-compressed chirp. For each delay sample (or range gate), the sequence of 1024 pulse-compressed chirps was transformed into a Doppler spectrum using a Fast Fourier Transform (FFT). Thus the temporal variation of the CIR in each range gate (or delay sample) over the 3.6 s period is represented as a Doppler spectrum (the vertical scale on the CSF).
 Examples of the CSF for the 158 MHz and 422 MHz chirps are presented in Figure 1 for four CSFs recorded on the 18 January 2005 pass. Figure 1a is an example of low Doppler spread and minimal delay spreading (limited by the system sampling rate); note the greater noise level in the 158 MHz CSF (Figure 1, left). Figure 1b exhibits moderate Doppler spreading. In contrast, in Figures 1c and 1d the Doppler spreading is considerable (tens of Hz), indicating rapid pulse-to-pulse fading. The 158 MHz CSFs in Figures 1c and 1d also exhibit strong delay spreading, where the energy is spread into quasi-parabolic “horns,” increasing in Doppler frequency offset with increasing relative delay.
3. Simulations of Channel Scattering Functions
 In order to better understand these measurements, the Trans-Ionospheric Radar Propagation Simulator (TIRPS) has been developed to simulate the effects of ionospheric scintillation on wideband waveforms. The model uses a split-step parabolic equation (PE) technique [Levy, 2000; Knepp, 1983] to provide a good approximation to the full electromagnetic wave equation under the paraxial approximation (i.e., where the signal is forward scattered over narrow angles of deflection about the axis of propagation). The ionosphere is represented by a one-dimensional thin phase screen at an altitude of 350 km (the approximate height of the F layer maximum). The phase screen is aligned in the direction of minimum ionospheric spatial coherence (i.e., aligned east-west across the geomagnetic field lines, assuming the irregularities are field-aligned). Following Knepp , the phase screen ϕ(x,f0) appropriate for a radio wave component at the carrier frequency f0, is generated by filtering a normally distributed, zero-mean random sequence by the analytical spatial spectrum proposed by Rino  (equation (1)).
vertical integrated strength of turbulence at the 1 km scale;
propagation zenith angle at ionospheric pierce point;
phase spectral index;
Euler's gamma function;
outer-scale frequency equal to 2π/Lo.
 This idealized spectrum provides a smooth transition from a flat spectrum at scales greater than Lo (the outer-scale size for turbulence) and a log-log linear spectrum (typical of turbulent energy-cascading processes) with the form Sϕ(κ) ∝ κ−p for scales below Lo. The simulator produces a single random realization of the phase screen whose spatial spectrum approximates to the ideal spectrum of equation (1). The average spectrum of an increasing number of realizations would converge to the ideal spectrum.
 Under the paraxial approximation it is convenient to define a “reduced” complex field u such that the electric field component of the radio wave is given as
where f is the radio frequency, thus removing the oscillatory component of the field along the axis of propagation, z. If the reduced field incident on the phase screen at z = 0 is u(x,0,f), then the field emerging from the phase screen is simply phase-advanced by the phase in the phase screen, i.e.,
noting that the phase screen's phase is proportional to 1/f. The second step in this split-step PE technique is to calculate the development of the emergent wave after it has propagated a distance z through free space. The technique, which is analogous to Fresnel-Kirchhoff diffraction, requires the multiplication of the wave's spatial spectrum by the “free space propagator” exp, thus:
where F[u(x,.,.)] = U(κ,.,.) and F −1[U(κ,.,.)] = u(x,.,.) represent a Fourier transform pair.
 The PE method above is used to determine the complex signal spectrum of an unmodulated planar wave propagated one-way through the phase screen. Since the wave incident on the phase screen is spherically divergent, rather than planar, a “reduced” propagation distance [Rino, 1982] from the phase screen zR is used:
where z1 is the range from radar to the phase screen and z2 is the range from phase screen to the target. The use of the planar phase screen theory with this correction is a satisfactory approximation except where z1/(z1 + z2) ≪ 1 or z2/(z1 + z2) ≪ 1 [see, e.g., Lee, 1977].
 The signal of the chirp-modulated wave is obtained by filtering the unmodulated field u with the spectrum of the transmitted waveform. The baseband spectrum of the upgoing linear frequency-modulated chirps transmitted by ALTAIR may be expressed as [Klauder et al., 1960]:
where Δ is the frequency sweep range (or “chirp bandwidth”), Tch is the chirp duration, and fd is the f − f0 (the Doppler frequency).
and Z() is the complex Fresnel integral given by
The modulated signal phase and amplitude uup is thus determined at each point in the orbit for which a pulse is reflected by the target. The time separation between upward and downward ionospheric transitions is just 3 ms at zenith or 6 ms at 20° elevation, so given that ionospheric drift speeds are typically around 100 ms−1 in the equatorial region, the ionospheric motion (up to 0.6 m) is well below the inner-scale size for field-aligned irregularities (approximately 2–5 m [Wheelon, 2001, p. 97]) and so is negligible. The received field pattern following two-way transition u2-way(x,f) is, therefore, calculated as the square of that on a one-way path uup(x, f), noting that zR is the same on up and down paths.
 The pulse-compressed complex impulse response (CIR) is then determined for each transverse position x:
where H(fd) = F[h(τ)] is the temporal FFT of the Hann function which is used to pulse shape the waveform and its matched filter. A spatial FFT from x to κ (using a Hann filter h(x)) is then used to produce a spatial CSF
The satellite velocity vector does not generally align with the phase screen which is perpendicular to the principal axis of the irregularities. Therefore, to convert from a spatial to a temporal CSF (for comparison with those recorded by ALTAIR) the model uses an effective scan velocity veff of the ionospheric pierce point (IPP) along the phase screen [Rino, 1982]. This is based on the satellite and ionospheric drift velocities; the alignment of the irregularities relative to both the geomagnetic field and the radio propagation vector; and the axial ratios of the irregularities. Details of its calculation are given in Appendix A. Given that the diffraction pattern at orbital height is sampled at the radar PRF, the required spatial sampling interval is
where the factor (z1 + z2)/z1 corrects for the spherical divergence between phase screen and target range.
 Ionospheric drift velocity forms a small component of the IPP scan velocity. In the following simulations we have assumed an approximate ionospheric drift velocity of 100 ms−1 in the geomagnetic east direction. While no drift measurements were available for 18 January 2005, this approximation is based on typical drift velocities measured at similar local times using a correlation analysis of VHF geosynchronous satellite beacon signals received on east-west spaced antennas on Kwajalein atoll (part of the global monitoring network for SCINDA (Scintillation Network Decision Aid)). An axial ratio of 30:1:1 has been assumed for irregularities aligned to a geomagnetic field determined by the U.S./UK World Magnetic Model [McLean et al., 2004].
 To help prevent aliasing, absorbing layers (half Hann windows) are applied at the extreme eighths of the screen (equation (13)).
where Lx is the length of the screen.
 In the simulations described below, a phase screen of Nx = 8192 points has been implemented. The 40-μs duration 158 MHz chirp waveform was sampled with 1024 points at the ALTAIR radar sampling interval of 50 ns, while the 150 μs 422 MHz chirp was sampled with 8192 points at the radar sampling interval of 25 ns.
 Using an effective velocity veff of 1514 ms−1 an elevation of 24° and PRF = 262 Hz (chosen for direct comparison with observations presented in Figures 1c and 1d and assuming a 100 ms−1 eastward ionospheric drift), the screen spacing dx = 11.6 m, giving a total screen length of Nxdx = 94.8 km. Only the central 1024 points of the field pattern at the target range are used in the calculation of the CSF. For comparison, the first Fresnel zone radius, is 857 m at 158 MHz and 524 m at 422 MHz.
 The results of the simulations of CSFs, presented below are mainly dependent on three variables used in the phase screen spectrum (equation (1)): p, Lo and CkL. Empirical climatological models such as WBMOD [Secan et al., 1995] predict CkL values ranging over several orders of magnitude, peaking at around 1036 in the equatorial, postsunset ionosphere. In equation (1), CkL is augmented by geometric factors G [Rino, 1979] and sec(θ) which are both of order 1; so the simulations presented below treat G CkL sec(θ) as a single combined variable. Measurements of phase spectral index, p, reported in the literature generally range between 1 and 4, with a mode between 2.5 and 2.8. [e.g., Livingston et al., 1981]. Outer-scale lengths, Lo, also vary widely over an approximate range of 5 to 20 km [Wheelon, 2001, p. 96].
 Results of TIRPS simulations of the CSFs recorded in Figure 1 are presented in Figure 2 for 158 MHz (Figure 2, left) and 422 MHz (Figure 2, right). These have used system and orbit parameters appropriate to the examples in Figures 1c and 1d with midrange values of p = 2.5, Lo = 10 km and a high value of G CkL sec(θ) = 1035. The standard deviation of phase in the phase screen, σϕ, was 12.6 and 4.6 radians for 158 MHz and 422 MHz, respectively. The simulation provides a good reproduction of the main features of the ALTAIR CSFs at both frequencies, including the characteristic quasi-parabolic increases in delay with increasing Doppler offset.
Figure 3 presents the variation of the simulated CSF for the 158 MHz waveform for CkL increasing over 4 orders of magnitude, while keeping other phase screen parameters unchanged. In these simulations σϕ was 0.5, 1.3, 4.7, 12.6 radians for G CkL sec(θ) = 32, 33, 34, 35, respectively. This sequence of CSFs illustrates the increase in Doppler spreading and the spread of energy into the horns with increasing turbulence strength.
Figure 4 presents 158 MHz CSF simulations for which the CkL is kept constant at 1034 while p and Lo are varied through most of their natural range. Evidently the proportion of energy in the horns is greatest when the phase spectral index p is small (see Figure 4a). Under these conditions small-scale irregularities dominate and scatter the wave through large angles and consequently induce large delay values, τ. Scatterers before and after the moving satellite IPP induce positive and negative Doppler shifts, respectively. As p is increased from 3 to 4 (Figure 4d) there is a slight enhancement in Doppler spreading and for larger outer scales a strong asymmetry often develops about the 0 Hz axis. With larger Lo there is an increased likelihood of a large linear component of signal phase being present over the sample period (1024 pulse returns) and this gives rise to the Doppler offsets observed.
 The model has also been applied to satellites in higher, lower-velocity orbits. In these cases the Doppler separation of the CSF horns increased, and they spread to more positive delays.
 Understanding the transionospheric propagation channel is important in the design of high-performance radar systems operating below 1 GHz. Measurements which provide the full complex impulse response (described here through the channel scattering function) are necessarily limited through time and cost, and a robust and validated modeling approach is needed.
 By intercomparison between model and measurement we have shown how a thin phase screen, PE technique can indeed simulate the two-way channel scattering functions (CSF) (delay and Doppler power profiles) for an equatorial transionospheric radar. The CSF simulations quantitatively reproduce the delay and Doppler measured using the 158/422 MHz ALTAIR radar and also faithfully reproduce the characteristic features such as the quasi-parabolic “horns.”
 The application of the model under extreme conditions of spectral slope also serve to elucidate the reasons why nonuniform scattering is sometimes seen (high spectral index) and why at other times broad spreading of the signal is seen in both Doppler and delay (low spectral index). These results show that with an appropriate choice of phase screen parameters, the TIRPS simulator can be used to assist the design and operation of transionospheric radars performing coherent pulse integration.
Appendix A:: Calculation of G and veff
 Ionospheric electron density irregularities are aligned to the geomagnetic field, B. Following Rino , we describe the shape of an irregularity as the ratio of coherence lengths along and across the field. An illustration of an irregularity of axial ratio a:b:1 is shown in Figure A1. Here x represents geomagnetic north, y is geomagnetic east and z is down. Hence B lies in the plane y = 0 and ψ represents the geomagnetic dip (angle from horizontal). a is the largest ratio of coherence length along the geomagnetic field to that perpendicular to it, while b relates to a secondary axis of elongation perpendicular to a and rotated an angle δ from the y axis. Normally δ ≈ 0, implying elongation in the magnetic east-west direction (i.e., along the L shell). In this Cartesian frame, the propagation vector, k has polar coordinates defined as
 The orientation and elongation of the irregularities is important when modeling the ionospheric medium as a single phase screen. In this case the apparent power spectral density of the irregularities is modified by the geometric enhancement factor G, as defined in equation (A2) to equation (A11) below [from Rino, 1982].
 1. Calculate satellite velocity in a Cartesian frame aligned to the transmitted wave propagation direction , up (increasing radar elevation) and right (increasing radar azimuth);
 2. Scale velocity down to that at the ionospheric pierce point (IPP), vIPP, by multiplying by the factor z1/(z1 + z2) where z1 is the range from radar to the IPP and z2 is the range from IPP to the target. The component of vIPP in the direction of propagation is set to zero.
 3. Rotate coordinate frame to determine vIPP in geomagnetic coordinates local to the IPP. i.e.,
Then, using Rino's  derivation of the spatial correlation structure of the phase, we determine the effective velocity as:
The ionospheric drift velocity, vd, is subtracted from vIPP prior to calculation of veff.
 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.