Radio receivers capable of high-rate sampling such as GPS scintillation monitors and the ALTAIR VHF/UHF tracking radar can measure ionospheric phase fluctuations suitable for scintillation modeling using phase screen techniques. For modeling purposes, the phase variations caused by the refractive effects of electron density irregularities encountered along the propagation path are desired. The phase fluctuations measured by ground-based receivers, however, also include the unwanted effects of diffraction (phase scintillations). In this paper, we investigate the effect of phase scintillations on the accuracy of phase screen simulation when using the phase measured on the ground as a proxy for the ionospheric screen. Using stochastic and deterministic (measured) phase screens, we quantitatively assess the accuracy of this approach by cross-correlating the predicted and measured intensity fluctuations. We find that the intensity cross-correlation is less than unity even in the weak scatter limit, due to the presence of weak phase scintillations. This correlation decreases rapidly with increasing irregularity strength once rapid transitions in the phase (strong phase scintillations) develop. We demonstrate that, when using the measured phase on the ground as a proxy for the ionospheric screen, both the temporal structure of simulated fluctuations and their statistics deviate increasingly from those of the measurements as the turbulence strength increases, especially when strong phase scintillations are present. We also demonstrate that back-propagating the complex signal up to ionospheric altitudes prior to the forward propagation calculation yields improved results, but some errors still remain as a consequence of neglecting amplitude fluctuations which develop inside the random medium.
 Plasma instability mechanisms in the ionosphere generate large-scale depletions and irregularities in the ambient electron density over a wide range of spatial scales (plasma turbulence). Radio waves that propagate through these irregularities experience scattering and diffraction, causing random fluctuations in amplitude and phase referred to as scintillations. The scintillation of satellite signals can severely degrade the performance of satellite communications systems, global navigation systems such as the Global Positioning System (GPS), and space radars used to conduct cloud-free, day-and-night observations of the Earth's surface. Ionospheric irregularities and scintillations constitute one of the most important space weather threats to modern technological systems which increasingly rely on trans-ionospheric radio propagation.
 Several authors have used satellite measurements of in situ density fluctuations to predict trans-ionospheric propagation effects [Franke et al., 1984; Wernik et al., 1980; Bhattacharyya et al., 2000; Costa and Basu, 2002; Costa et al., 2011]. The principal difficulty with this approach is that in situ measurements sample the density fluctuations at the orbital altitude of the satellite, whereas trans-ionospheric propagation effects are due to the integrated effect of density variations encountered at all altitudes along the signal path, the latter of which must generally be modeled. It is not obvious how density variations sampled at one altitude are related to density variations at other altitudes (if indeed they are). For example, if density irregularities associated with an equatorial plasma bubble (EPB) do not rise to the altitude of the in situ probe, there may be no fluctuations detected even if the bubble causes significant scintillations in radio waves due to the presence of irregularities at lower altitudes.
 Ground-based observations of phase from a signal transmitted by a satellite in the topside ionosphere (or higher) have a distinct advantage over in situ density observations in that the ionospheric channel is probed at all altitudes relevant to the problem. However, it is crucial to interpret the time series of signal phase properly and to understand its limitations in providing a faithful representation of the underlying irregularity structure. In particular, it is important to quantify the contribution from phase scintillations caused by diffraction. FollowingBhattacharyya et al. , we consider phase scintillations as distinct from the phase variations imposed by refraction due to density irregularities encountered along the propagation path (TEC variations). Bhattacharyya et al.  and also Knepp  recognized that measurements of phase fluctuations on the ground are have contributions both from TEC variations as well as the diffraction effects they impart on the propagating wave (phase scintillations). Whereas Knepp was concerned primarily with the degree to which phase scintillations may corrupt ground-based measurements of TEC, here we are interested in the effect phase scintillations have on phase screen radio propagation modeling.
 More specifically, the question we address in this paper is, under what conditions can the phase fluctuations measured by a receiver on the ground serve as a proxy for the refractive phase change (i.e., TEC variations) imparted by the disturbed ionosphere in a trans-ionospheric phase screen simulation? The refractive phase change imparted by the ionosphere is required for phase screen simulation, but the measured phase on the ground also includes the unwanted effects of diffraction that develop while the wave propagates. In the context of the phase screen approximation, the distinction between screen phase and ground phase was noted earlier byBeach . The measured phase also includes contributions from geometric Doppler, system noise, and potentially also artifacts related to phase cycle repair and detrending. These effects must be removed prior to the simulation, or at least kept at manageable levels.
 There are several reasons one may want to use the measured phase fluctuations in a phase screen calculation. First, the predicted intensity fluctuations on the ground can be compared directly with the measured intensity fluctuations. This fact can be used to evaluate different radio propagation models and to test assumptions regarding the altitude of the scattering layer, the scan velocity of the line of sight through the drifting irregularities, and the anisotropy of the irregularities. For example, Caton et al. used tracking data from the VHF/UHF Advanced Research Project Agency Long-Range Tracking and Instrumentation Radar (ALTAIR) during over-flights of passive calibration spheres in low-Earth orbit to demonstrate the validity of a one-dimensional (1-D) phase screen model to represent the propagation channel through the disturbed ionosphere. The 1-D phase screen used in their analysis was constructed from radar phase-derived estimates of the total electron content, but without accounting for the effects of phase scintillations. In this paper, we examine how these neglected phase scintillations degrade the accuracy of the propagation calculation. Furthermore, we will show how theCaton et al. results could have been improved by first back-propagating the complex waveform measured at the ground up to ionospheric altitudes prior to the forward propagation calculation. Second, to some extent one can evaluate the quality of the receiver's measurements of phase fluctuations by verifying their consistency with its measurements of intensity fluctuations, since the two are closely related when the scatter is weak. This can help to confirm the receiver is measuring actual fluctuations caused by the ionosphere and not thermal noise, residual effects of platform motion, phase artifacts caused by inadequate cycle slip repair, etc.
 The outline of this paper is as follows. In section 2, we review the phase screen technique for 1-way and 2-way trans-ionospheric propagation scenarios. Insection 3we perform numerical experiments using stochastically generated phase screens and comment on the effects of diffraction on the propagated waves. Next we quantify the effects of phase scintillations on phase screen simulation using a two-stage simulation approach that we developed specifically for this purpose. In the first stage, a phase screen calculation is performed to simulate the phase on the ground, given a stochastically specified phase screen. In the second stage, a subsequent phase screen calculation is performed using the phase fluctuations obtained from the first stage calculation (i.e., on the ground) as the phase screen. To quantify the results, we compute the cross-correlation of phase fluctuations on the ground between the first and second stage calculations, and also the cross-correlation of intensity fluctuations on the ground between the first and second stage calculations. This two-stage simulation approach closely approximates the situation where the phase measured on the ground is used as a proxy for the screen in a phase screen calculation. Finally, insection 4, we present the results of phase screen simulations using detrended phase measurements provided by a GPS receiver and the ALTAIR VHF radar as a proxy for the screen. The altitude of the ionospheric screen is determined in each case by back-propagating the received complex signal on the ground. We show that the cross-correlations of intensity for the GPS and ALTAIR VHF observations are consistent with their predicted values determined via numerical simulation. We also show that back-propagating the complex signal up to ionospheric altitudes prior to the forward propagation step improves these correlations.
2. Phase Screen Simulation
 The phase screen technique assumes the refractive effects on the propagating wavefronts due to ionospheric irregularities can be adequately represented by a thin phase-changing screen suitably located at ionospheric altitudes. The geometry of the problem is given inFigure 1. Here d1 is the slant distance from the transmitter or reflecting object (TX) to the screen, d2 is the slant distance from the phase screen to the receiver (RX), and θ is the propagation (nadir) angle. The flat Earth representation shown in Figure 1 is used for illustration only. We use a spherical Earth model to compute the distances and angles used in the propagation calculations.
 The transmitted signal is approximated by a spherical wave U0(ρ, z) = A0exp(i2πR/λ)/R, where A0 is the amplitude, λ = c/f is the radio wavelength, and R is the distance from the transmitter (TX). The time dependence of the signal is omitted for clarity, but is implied. In the Fresnel approximation, the complex amplitude at the receiver (RX) after passage through the phase screen is given by
 In equations (1) and (2), ρ is a 2D position vector in the horizontal plane, zRX is the vertical coordinate of the receiver, US(ρ) is the complex amplitude of the wave at the phase screen, k = 2π/λ is the signal wave number, and F is the horizontal Fourier transform as a function of the horizontal wave number κ. Bernhardt et al. considered a purely phase-changing screen where the complex amplitude of the radio wave after passage through the screen is given byUS(ρ) = exp[−iφ(ρ)], where φ(ρ) is the phase in the screen. Here, we write the ionospheric transfer function in the general form (2) to emphasize that this formulation can accommodate both phase and amplitude changing screens.
 A receiver on the ground measures a scaled version of the function D(ρ). Equation (2) may be inverted to infer the complex amplitude of the wave at any altitude:
 This procedure can be interpreted as back-propagation of the complex amplitude received on the ground back up to the altitudeHs. If the scatter is sufficiently weak that only phase fluctuations develop while the radio wave travels through the random medium, then the variance of US will approach unity for some altitude Hs*. In this case, Hs* can be interpreted as the altitude of the phase changing screen, and the phase fluctuations introduced by the ionosphere can be retrieved by taking the argument of US(ρ). These phase fluctuations, i.e., φ = arg[US(ρ)], are referred to as the “equivalent” phase screen, since they produce the same diffraction pattern on the ground as the actual ionospheric irregularities do even though the latter may be extended in altitude. In practice, the variance of US generally does not attain unity for any screen altitude Hs, because amplitude fluctuations also develop while the radio wave propagates in the random medium. The tendency for amplitude fluctuations to develop inside the random medium increases with the scattering strength. In any case, we can vary the height of the screen in order to minimize the variance of US, but in this case the function US at this optimal altitude (Hs*) must now be interpreted as an equivalent phase and amplitude changing screen. If the strength of scatter is weak or moderate, we will see that quite reasonable results may be obtained by neglecting the amplitude of US(ρ) and considering only its phase, the latter of which we refer to here as an “approximate” phase changing screen. Sokolovskiy et al.  followe a by inverse diffraction methods.
 Note that, by symmetry (reciprocity), D(ρ) in equations (2) and (3)remains unchanged if we exchange the locations of the transmitter and receiver. For the case of a ground-based radar that tracks a space-based object which behaves as a point scatterer with unit reflectivity, we can account for two-way passage through the screen by squaringD [Rogers and Cannon, 2009]. In the calculations that follow, we will use one-dimensional phase screens, so thatρ and κ are scalars oriented along the path of the ionospheric penetration point (IPP) as the transmitter moves in its orbit.
 A receiver on the ground measures temporal fluctuations, rather than spatial ones, as the diffraction pattern implied by (1) and (2)sweeps past the receiver. Assuming the random medium is invariant over the measurement interval (this is the Taylor hypothesis of frozen-in flow), spatial fluctuations and temporal fluctuations can be related by a model-dependent effective scan velocity [Rino et al., 2011]. Here we use Rino's power law spectral model, for which the effective scan velocity, Veff, is calculated as described by Rino . The effective scan velocity depends on the locations and velocities of the transmitter and receiver, as well as the anisotropy, orientation, and drift velocity of field-aligned ionospheric irregularities. Following the approach implemented by the Wideband Scintillation Model [Secan et al., 1995] and also Rogers et al. , we remove the component of IPP velocity parallel to the line of sight in order to minimize spectral smearing.
3. Simulation Using Stochastic Screens
 In this section, we perform simulations using stochastically generated phase screens, which are realizations of the following one-dimensional power law spectrum with an outer-scale,L0 [Rogers et al., 2009]:
 We note that the spectral form in (4) appears in Rino as the result of a 1-D scan through a fully 3-D structure and propagation model. Here we use this result to define a 1-D phase screen and perform a 2-D propagation calculation which is not mathematically equivalent to the fully 3-D treatment given by Rino. Nevertheless, if the same effective scan velocity is used to relate spatial fluctuations and temporal fluctuations, the 2-D propagation model captures the correct temporal frequencies in the signal fluctuations subject to the assumed anisotropic structure of the irregularities. Following this reasoning, the spatial sampling, Δρ, is determined from the time between successive samples of phase, Δt, as:
 In (5), the transverse scaling factor, (d1 + d2)/d1, accounts for the spherical spreading of the wavefronts. The sampling requirements for Δρ and the number of samples in the screen, N, are motivated by need to resolve all the scale sizes that influence the propagation calculation and are spelled out in detail by Knepp . All of the phase screen simulations in this paper satisfy the sampling requirements given by Knepp . Properly sampled realizations of the screen with spectrum (4) will have the following total phase variance [Rino, 1979]:
 We note that the phase variance in (6) includes the contribution from all scale sizes to the phase fluctuations; it is not the detrended measurement of phase fluctuations that is provided by GPS scintillation monitors such as the GSV 4004B and discussed at length by Beach .
 We consider two specific propagation scenarios in this paper. Scenario 1 corresponds to 1-way propagation from GPS satellite PRN08 to an Ashtech Z-XII GPS receiver located at Ascension Island (7.96°S, 14.41°W) on 13 Mar 2002 at 22:24 UT. Scenario 2 corresponds to 2-way propagation between the ALTAIR VHF radar (9.4°N, 167.5°E) and RIGIDSPHERE-2 on 5 Oct 2005 at 10:20 UT. RIGIDSPHERE-2 (LCS-4, NORAD ID 05398), is smooth radar calibration sphere with a diameter of 112.9 cm and a radar cross section of 1 m. It orbits Earth with perigee/apogee of 743/834 km at 87.6° inclination. Additional details of this ALTAIR observation were published previously byCaton et al. . The parameters defining the spectrum of phase fluctuations and the propagation geometry for these two scenarios are summarized in Table 1. The zonal irregularity drift velocity, VD, was determined using a collocated pair of VHF links to geostationary satellites provided by the AFRL-SCINDA network using the spaced-receiver technique. The phase enhancement factor,G, and the effective scan velocity, Veff, were computed from the propagation geometry, drift velocity, and local magnetic field direction according to the procedure outlined by Rino . We assumed the irregularities to be elongated along the magnetic field direction with anisotropy ratio 50:1. The height of the screen, Hs, was estimated via back-propagation. We conducted simulations using several values of the phase spectral index, ranging from 2 to 4, in addition to the values listed inTable 1 (p = 4.0 for GPS and p = 3.5 for ALTAIR) which were inferred directly from the observations.
Table 1. Parameters Defining the Spectrum of Phase Used in the Phase Screen Simulations for Scenario 1 (One-Way Propagation) and Scenario 2 (Two-Way Propagation)
 We carried out two-stage phase screen simulations in order to investigate the effect of phase scintillations on the accuracy of phase screen simulation when using the phase measured on the ground as a proxy for the screen. For the first stage calculation, we generated realizations of the screen phase,φS(ρ), according to the spectrum (4) and the parameters listed in Table 1. The radio wave was then propagated through the screen using equations (1)–(2) with φ(ρ) = φS(ρ) to determine the Stage 1 complex amplitude on the ground, U1(ρ, zRX) = U(ρ, zRX). From this, we computed the Stage 1 phase on the ground as φ1(ρ) = arg[U1(ρ, zRX)]. For the second stage calculation, we propagated the wave through the screen again by evaluating equations (1)–(2) but this time with φ(ρ) = φ1(ρ) for Scenario 1 and φ(ρ) = φ1(ρ)/2 for Scenario 2 (to account for two-way propagation in the later case). This yielded the Stage 2 complex amplitude on the ground,U2(ρ, zRX) = U(ρ, zRX), from which we computed the Stage 2 intensity I2(ρ) = |U(ρ, zRX)|2 and phase φ2(ρ) = arg[U(ρ, zRX)] on the ground. To quantify the differences between the Stage 1 and Stage 2 intensity and phase fluctuations, we computed the intensity and phase cross-correlations:
where “cov” signifies the covariance and “var” signifies the variance. In addition to these metrics, we also calculated, for both the Stage 1 and Stage 2 results, the RMS of normalized intensity fluctuations (S4), the time to 50% intensity decorrelation (τ), the phase spectral index (p), and the RMS of phase fluctuations (σφ). The phase spectral index was measured by fitting the power-spectral density of phase with a linear function in log-log space between frequencies 0.5 Hz and 10 Hz.
 For both propagation scenarios, we generated sequences of realizations of the screen phase for increasing values of the RMS screen phase ranging from 0.1 to 50 radians. The same seed was used to generate the random screens in each sequence so they differ only by a scaling factor. Next, we generated animations showing how the Stage 1 and Stage 2 phase screen results are affected by diffraction as the RMS of phase fluctuations in the screen (or equivalently, the turbulent intensity) increases. These animations have been included as AGU auxiliary materials for this paper. Figure 2shows the simulation results for Scenario 1 (one-way propagation of the GPS signal) whileFigure 3shows the simulation results for Scenario 2 (two-way propagation of the ALTAIR radar signal). InFigure 2 the results are shown for 0.5, 10.15, and 50 radians of RMS phase fluctuations in the screen, whereas for Figure 3 the results are shown for Scenario 2 corresponding to 0.25, 5, and 21 radians of RMS phase fluctuations in the screen. These specific values were chosen as representative of weak scatter conditions, the start of rapid transitions in the phase due to diffraction effects, and strong scatter conditions. Only the central data portion corresponding to one outer scale of the turbulence is shown in Figures 2 and 3, for clarity. In these plots the intensity has been normalized by the mean signal intensity, while the phase has been normalized by the RMS phase in the screen so that the same phase scaling can be used for all frames of the animation. Labeled on these plots are statistics computed from the phase screen simulation results. For the intensity, these statistics are S4, τI, and Ic and for the phase these are p, σφ, and Φc. The curves and labels colored gray correspond to the Stage 1 results, while the curves and labels colored red correspond to the Stage 2 results. Differences between the Stage 1 and Stage 2 results are the direct consequence of unwanted effects of diffraction on the ground phase fluctuations (i.e., phase scintillations) that are used as a proxy for the ionospheric screen in these calculations. In these simulations, the intensity correlation, Ic, decreased monotonically as the RMS screen phase increased. This suggests that the temporal structure of intensity fluctuations simulated by the phase screen technique (Stage 2 results) deviates increasingly from the truth data (Stage 1 results) as the scattering strength increases.
 The two-stage simulation described above was then performed 25 times using different values of the seed used by the random number generator to construct the stochastic phase screen. Ensemble averages of the statistics (S4, τ, Ic, p, σφ, and Φc) for the Stage 1 and Stage 2 calculations were then calculated. Figure 4compares the ensemble-averaged statistics for the Stage 1 and Stage 2 calculations corresponding to Scenario 1 (one-way propagation).Figure 5 shows the result of an identical simulation but with p = 4 instead of p = 3. Figure 6compares the ensemble-averaged statistics for the Stage 1 and Stage 2 calculations for Scenario 2 (two-way propagation), whileFigure 7 shows the result of an identical simulation but with p = 4. In Figures 4–7 the curves corresponding to Stage 1 statistics are shown in gray, while curves corresponding to Stage 2 statistics are shown in red. Several additional simulations were conducted (results not shown) for other specified values of p, ranging from 2 to 4, to examine the changing effect of phase scintillations on the phase screen simulations as the phase spectral index changes. The upper horizontal axes in these plots show the RMS of phase fluctuations in the screen while the lower axes show the turbulent intensity, CkL. The vertical dashed lines indicate the value of RMS screen phase and CkLfor which the phase cross-correlation (Φc) drops to 95% of its maximum value. These values are summarized in Table 2.
Table 2. Values of Screen σφ and CkLfor Which the Phase Cross-Correlation Drops to 95% for Scenario 1 (One-Way Propagation) and Scenario 2 (Two-Way Propagation)a
These values can be considered as (empirically) defining the scattering strengths at which strong phase scintillations begin.
8.4 × 1034
2.3 × 1035
4.1 × 1035
6.6 × 1035
1.0 × 1036
2.8 × 1032
3.8 × 1032
3.2 × 1032
2.6 × 1032
2.0 × 1032
 In Figures 4a, 5a, 6a, and 7athe dotted line indicates where the RMS of ground phase fluctuations equals the RMS of screen phase fluctuations for the case of one-way propagation, or twice the RMS of screen phase fluctuations for the case of two-way propagation. InFigures 4b, 5b, 6b, and 7b the dotted line indicates the specified value of the phase spectral index for the screen. The dotted line in Figures 4e, 5e, 6e, and 7e indicates the value τweak= (λd2/4π)1/2/Veff, which is the Fresnel scale divided by the effective scan velocity.
 When the RMS phase in the screen is small (σφ < <1), weak scatter theory is applicable. Under these conditions, the S4 index increases as the square root of CkL [Yeh and Liu, 1982]. In this regime, the S4 index from the Stage 2 simulation remained fairly close to, but consistently underestimated, its value from the Stage 1 result (truth). In weak scatter, τ is independent of CkL and depends instead on the effective scan velocity and the diameter of the first Fresnel zone, which in turn depends on the signal frequency and propagation distance [Rino and Owen, 1980]. As noted by one of the reviewers, τfrom the Stage 2 calculation over-predictedτ from the Stage 1 calculation (truth) by the factor √2 in the weak scatter regime because the propagation distance is effectively twice as large. Thus, even in the weak scatter regime, the technique of using measured phase fluctuations as a proxy for the screen cannot reproduce the full set of observables. In weak scatter the normalized phase also changes little with increasing CkL. When the scatter is weak, the phase on the ground is equal to the phase in the screen modulated in the frequency domain by a sequence of nulls whose locations also depend on the diameter of the first Fresnel zone [Yeh and Liu, 1982]. These Fresnel nulls are a consequence of diffraction in weak scatter; they are the manifestation of weak phase scintillations. As is evident from Figures 4–7, the phase cross-correlation (Φc) is close to unity in the weak scatter regime, suggesting that the presence of weak phase scintillations does not cause significant decorrelation with the phase in the screen. However, the intensity cross-correlation (Ic) is significantly less than unity even in the weak scatter limit. For example, when the spectral index is p = 3, we find Ic≈ 0.7 for both Scenario 1 (Figure 4) and Scenario 2 (Figure 6) throughout the weak scatter regime. When the spectral index is p = 4, we find Ic≈ 0.85 for both Scenario 1 (Figure 5) and Scenario 2 (Figure 7) throughout the weak scatter regime. These results suggest an inherent limitation on the accuracy one can expect to achieve using the ground phase as a proxy for the screen in a phase screen simulation, even when the scatter is weak. They also suggest that weak phase scintillations may cause larger simulation errors using this approach when the irregularity spectrum is shallower. We note that while it appears from Figures 4–7 that the measured spectral slope (p) varies with CkL in the weak scatter regime, this is actually not the case. Instead, these deviations are errors in measuring the slope of the phase power spectral density (PSD) when the Fresnel nulls are present.
 As the RMS phase in the screen exceeds 1 rad, the complex amplitude of the signal on the ground makes further excursions from the unit circle, some of which begin to approach the origin in the complex plane (see Figures 2b and 3b). When the complex amplitude crosses from one quadrant to another near the origin, a rapid phase transition occurs. These rapid phase transitions coincide with the deep signal fades. This is the action of diffraction on the phase, and has been observed previously in beacon scintillation measurements [Humphreys et al., 2010; Carrano and Groves, 2010; Gherm et al., 2011]. We will refer to these rapid phase transitions as strong phase scintillations. When strong phase scintillations are present, the phase on the ground begins to deviate significantly from the screen phase at times coincident with the deepest signal fades. The RMS of phase fluctuations on the ground tend to exceed the RMS of screen fluctuations in both the Stage 1 and Stage 2 results (see Figures 4a, 5a, 6a, and 7a). Furthermore, both the intensity and phase cross-correlation decrease rapidly. We note that these correlations are independent of scattering strength for weak phase scintillations but decrease with increasing scattering strength once strong phase scintillations begin. The values of screenσφ and CkLfor which the phase cross-correlation drops to 95% for these scenarios are summarized inTable 2. These values can be considered the scattering strengths for which strong phase scintillations begin to occur, such that further increases in scattering strength will dramatically degrade the accuracy of phase screen simulations using the measured phase fluctuations as a proxy for the screen. We find that for both scenarios, when the spectrum of the irregularities is shallower, the RMS of screen fluctuations can be larger before strong phase scintillations begin to occur.
 As CkL and the RMS screen phase is increased further into the strong scatter regime, the S4 index saturates to unity while log(τ) begins to vary linearly with CkL (with a negative slope). These effects can be seen clearly in Figures 4–7, and can also be inferred from theoretical considerations [Rino and Owen, 1980]. The phase PSD becomes shallower with increasing CkLin the strong scatter regime, as the rapid phase changes due to diffraction become more prominent, approaching the phase spectral index for a discontinuous process, which is 2.0. As a result of these strong phase scintillations, the temporal structure of phase fluctuations on the ground no longer resembles that of the screen phase. Under these conditions, it can be challenging to un-wrap the phase measurements correctly since the phase change between successive samples can approach or even exceedπ radians. Hence, quantifying the RMS of phase fluctuations can be subject to error when strong phase scintillations are present. In any case, when strong phase scintillations are present the RMS of phase fluctuations on the ground (σφ) tends to exceed the RMS of phase fluctuations in the screen, often quite significantly. Comparing Figures 4–7suggest that these observations apply to the case of two-way propagation as well as one-way propagation, except that in the former case the onset of strong phase scintillations occurs at a lower value of the RMS screen phase. ComparingFigures 4a and 5a, and also comparing Figures 6a and 7a, this effect is greater when the spectrum of the irregularities is shallower. This would suggest that strong phase scintillations may have a greater impact on the accuracy of phase screen simulations when the spectrum of the irregularities is shallower. The results of our simulations suggest this is indeed the case when only weak phase scintillations are present. On the other hand, when strong phase scintillations are present we found the S4values from the Stage 2 simulation to under-predict the truth data (Stage 1 simulation) in all of our simulations. This under-prediction was most dramatic for Scenario 2 when the spectral slope was steeper (e.g.,Figure 7 with p = 4) even though the RMS of phase fluctuations on the ground was smaller (e.g., Figure 6 with p = 3).
4. Simulation Using Deterministic Screens
 In this section, we show the results of phase screen simulations using actual phase measurements on the ground as a proxy for the phase screen. We compare these results with the stochastic screen simulations described in the previous section. Finally, we compare these results with simulations where the measured complex signal is back-propagated up to ionospheric altitudes prior to the forward propagation step. We begin by describing the experimental observations.
4.1. GPS Results
 In Scenario 1, we monitored the intensity and phase fluctuations of GPS satellite PRN08 using an Ashtech Z-XII receiver located at Ascension Island on 13 Mar 2002 at 22:24 UT.Table 1 lists the screen and propagation parameters for this scenario. No cycle slips were detected in the measured phase during this period, so cycle slip repair techniques were not applied. To remove the geometric (Doppler) contribution to the phase due to satellite motion, the measured GPS L1 phase was detrended using a 6th order Butterworth filter with 3dB cutoff at 1/60 s = 0.017 Hz. We use this detrended phase as the screen in expressions (1)–(2) to compute the intensity and phase on the ground. We note that so long as the filter cutoff frequency is much smaller than the Fresnel break frequency (0.3 Hz in this case), the specific value chosen for the cutoff has little effect on the predicted intensity on the ground using the phase screen.
 In order to evaluate (2), the altitude of the phase screen must be specified. We estimate the altitude of the phase screen via back-propagation as follows. From the measured intensityI and phase φ on the ground we first calculate the complex amplitude, which according to (1) is a scaled version of the ionospheric transfer function for one-way propagation and for two-way propagation. Here we have neglected the amplitude modulation imposed byU0, which is negligible for the short time series and long trans-ionospheric propagation distances considered here. In this case, the correct spatial sampling to use is
which is the same as the sample spacing in (5)but without the transverse scaling factor. The transverse scaling factor is not needed when using ground measurements to specify the phase screen since the wavefronts have already spread by the time they reach the ground. The back-propagation is performed by usingequation (3) to evaluate the complex amplitude US(ρ) of the wave at 40 discrete altitudes ranging from approximately 100 km to 460 km. The scintillation index of the back-propagated wave is plotted as a function of screen altitude inFigure 8. Note that a clear minimum is evident at the altitude Hs* = 302 km. We use this as the altitude of the screen in our subsequent calculations. The fact that the minimum value of S4achieved during the back-propagation step is not zero suggests that an amplitude and phase changing screen is required to identically reproduce the diffraction pattern sampled on the ground by the receiver. This suggests that amplitude fluctuations had developed while the radio wave traveled in the extended random medium. In any case, we discard the amplitude fluctuations and take arg[US(ρ)] as the “approximate” phase screen for this scenario. Since the observed value of S4 = 0.43 corresponds to a scattering strength (CkL = 5.0 × 1035) which is smaller than the threshold for strong scintillations given in Table 2, we expect that strong phase scintillations are absent from the measurements for this Scenario.
 Despite the apparent simplicity of this approach, we note that the effective scan velocity Veffis a strong function of altitude for this scenario and therefore it must be recalculated at each step during the back-propagation.Veff is also quite sensitive to the zonal irregularity drift, VD. In fact, using the zonal irregularity drift of 131 m/s which was measured using spaced VHF receivers, the altitude of the screen was originally inferred to be 236 km, which is unphysical. Given that measurements of the zonal drift at VHF and GPS from the same location can differ for a number of geophysical reasons [de Paula et al., 2010], we felt justified in artificially increasing the measured VDfrom 131 m/s to 152 m/s so that the screen height inferred by back-propagating the wave would occur at the more plausible altitude of 302 km. We acknowledge that this inferred screen height is therefore quite likely in error, but in spite of this we can still demonstrate the benefits of back-propagating the wave back to ionospheric altitudes prior to the forward propagation step.
Figure 9acompares the measured phase on the ground (black) with the approximate phase screen determined via back-propagation (red), whileFigure 9b compares their corresponding power spectral densities (PSD). Labeled on this plot are the σφ values for the measured phase (black) and the approximate phase screen (red). The spectral index pof the measured phase was determined to be 4.1 by performing a least squares fit to the screen phase PSD in log-log space between 0.25 Hz and 1.5 Hz. The approximate phase screen and the measured phase exhibit some minor differences but are quite similar overall (the correlation between them is 0.98), presumably because only weak phase scintillations are present in this scenario.Figure 9b compares the intensity fluctuations following a forward propagation calculation through the approximate phase screen (red) with the measured intensity fluctuations (black), while Figure 9d compares their corresponding spectra. Last, Figure 9e compares the intensity fluctuations following a forward propagation calculation through the screen consisting of the measured phase fluctuations (red) with the measured intensity fluctuations (black), while Figure 9f compares their corresponding spectra. While a 10 min segment of GPS measurements was used in these simulations, only a 60 s segment is shown in Figures 9a–9f for clarity. The quantities S4 and τ, are labeled on these plots in red or black to indicate the simulation data or measured data, respectively.
 From the plots in Figure 9, it is clear that back-propagating the measured complex amplitude up to ionospheric altitudes prior to the forward propagation step improves the simulation results. InFigure 9b, both the S4index and decorrelation time of the simulated intensity fluctuations match the measurements almost perfectly. In fact, the simulated and measured intensity fluctuations generally agree on a fade-by-fade basis, their correlation is 0.98, and their corresponding PSDs agree very well for frequencies less than ∼0.6 Hz. Artifacts in the phase spectrum above 1 Hz degrade this agreement at high frequencies. When using the measured phase as a proxy for the screen, the simulatedS4index (0.50) over-predicts the measured value (S4 = 0.43) by 16%. The decorrelation time, however, agrees with the measurements about equally well regardless of whether back-propagation was used. In any case, it is clear that when using the measured phase as a proxy for the screen (Figure 9c), the temporal structure of the simulated intensity fluctuations deviates from that of the measurements in multiple places, presumably because errors in the screen phase are causing slight differences in how the power is focused and defocused at the ground. The intensity cross-correlation in this case is 0.69, which is somewhat higher than the value 0.44 we obtained via numerical experimentation in the previous section. The intensity correlation 0.44 was determined by looking up the value ofIcor in Figure 5f at the scattering strength which corresponds to the value S4 = 0.43 in Figure 5d. These results suggest that a modest loss of accuracy is incurred using the measured phase as a proxy for the screen due to the presence of weak phase scintillations. The back-propagation step is largely able to mitigate the unwanted effects of diffraction on the phase for this scenario, thereby improving the propagation results significantly (the intensity correlation increases from 0.69 without back-propagation to 0.98 with back-propagation).
4.2. ALTAIR VHF Results
 In Scenario 2, the ALTAIR VHF radar recorded the intensity and phase of the reflected signal from RIGIDSPHERE-2 on 5 Oct 2005 starting at 10:20 UT.Table 1 lists the screen and propagation parameters for this scenario. The radar periodically adjusted the pulse repetition frequency (PRF) to maintain the object within the tracking range gate. We selected a 20 s data segment during which the PRF remained constant. The geometric (Doppler) contribution to the phase due to motion of the calibration sphere was removed by subtracting from the measured phase, the reference phase provided by MIT Lincoln Laboratory which was generated using a precise orbit determination technique. Due to the scintillations of the radio wave, the measured phase and reference phase were not always within the same cycle, and hence slips in the phase occurred. These slips were corrected by using a polynomial fit to the measured minus reference phase within a 30 sample moving window to predict the next value, adjusting by 2π radians as needed. A linear fit to the phase was then used to detrend the data. The effective cutoff frequency in this case is 0.1 Hz, which is close to the Fresnel break frequency. It is possible, therefore, that this simulation may incur some error due to the relatively short data interval length.
 As in the previous example (Scenario 1), the altitude of the phase screen must be specified in order to proceed with the propagation calculations. Once again, we estimate this altitude via back-propagation. Using the sample spacing given in(8)the back-propagation is performed usingequation (3) to evaluate the complex amplitude US(ρ) of the wave at 40 discrete altitudes ranging from approximately 100 km to 460 km. The scintillation index of the back-propagated wave is plotted as a function of screen altitude inFigure 10. In this case a minimum is evident at the altitude of Hs* = 388 km, and we will take as the altitude of the screen in our subsequent calculations. Note that in this example the minimum S4achieved during the back-propagation step is still relatively large (>1), which suggests that significant amplitude fluctuations had developed while the radio wave traveled in the extended random medium. This is not surprising since the scattering conditions are much stronger in this case than the previous (Scenario 1). In any case, we discard the amplitude fluctuations as before and take arg[US(ρ)] as the “approximate” phase screen for this scenario. As might be expected, however, larger errors are incurred by neglecting the amplitude fluctuations in this case since they are larger than in Scenario 1. We note that the values of S4during the back-propagation shown inFigure 10 become less erratic once Hsincreases past approximately 300 km. This occurs because the back-propagation step is able to remove many (but not all) of the rapid phase transitions in the screen that are caused by diffraction and which alter the focusing and defocusing of the propagating wavefronts. Since the observed value ofS4 = 1.72 corresponds to a scattering strength (CkL = 7.5 × 1032) which exceeds the threshold for strong scintillations given in Table 2, we expect that strong phase scintillations are present in the measurements for this Scenario.
Figure 11acompares the measured phase on the ground (black) with the approximate phase screen determined via back-propagation (red), whileFigure 11b compares their corresponding power spectral densities (PSD). Labeled on this plot are the σφ values for the measured phase (black) and the approximate phase screen (red). The spectral index pof the measured phase was determined to be 3.5 by performing a least squares fit to the screen phase PSD in log-log space between 0.4 Hz and 4 Hz. The approximate phase screen and the measured phase differ significantly (the correlation between them is only 0.45) for this scenario, presumably because strong phase scintillations are present. Note that some, but not all, of the rapid transitions in the ground phase caused by diffraction have been removed during the back-propagation step. For example, there are two large phase transitions evident in the ground phase shown inFigure 11a during the interval 9–12 s which are not present in the approximate phase screen. Figure 11b compares the intensity fluctuations following a forward propagation calculation through the approximate phase screen (red) with the measured intensity fluctuations (black), while Figure 9d compares their corresponding spectra. Last, Figure 11e compares the intensity fluctuations following a forward propagation calculation through the screen consisting of the measured phase fluctuations (red) with the measured intensity fluctuations (black), while Figure 11f compares their corresponding spectra. The quantities S4 and τ, are labeled on these plots in red or black to indicate the simulation data or measured data, respectively.
 From the plots shown in Figure 11, it is clear that back-propagating the measured complex amplitude back to ionospheric altitudes prior to the forward propagation step significantly improves the simulation results. With back-propagation (Figure 11b), the S4 index of the simulated intensity fluctuations is 1.69, which is within 2% of the measured result (S4 = 1.72). When using the measured phase as the screen (Figure 11c), the S4index of the simulated intensity fluctuations is under-predicted by 26%. The decorrelation time of the intensity fluctuations is 0.27 sec with back-propagation, which is 23% lower than the measured result (τ = 0.35 sec). Without back-propagation, however,τis underestimated by 66%. The correlation between the simulated and measured intensity fluctuations with back-propagation (Figure 11c) is 0.52 and the PSDs of simulated and measured intensity fluctuations agree fairly very well for frequencies less than ∼1.5 Hz, beyond which the measured phase has apparently descended below the noise floor. When using the measured phase as a proxy for the screen (Figure 11e), the temporal structure of the simulated intensity fluctuations deviates from that of the measurements in multiple places, e.g., Figure 11ebetween 7 and 14 s, presumably because large phase transitions (e.g., in the central area of the screen) are changing how the power is focused and defocused at the ground. The intensity cross-correlation in this case is only 0.15, which is somewhat less than the value 0.35 we obtained via numerical experimentation. These results suggest that a significant loss of accuracy is incurred using the measured phase as a proxy for the screen due to the presence of strong phase scintillations. The back-propagation step is able to partially mitigate the unwanted effects of diffraction on the phase, thereby improving the propagation results significantly (the intensity correlation increases from 0.15 without back-propagation to 0.52 with back-propagation). Not all rapid transitions in the phase were removed by back-propagation, possibly because these rapid transitions and (possibly also system noise) caused errors in repairing the cycle slips present in the measured phase data. Even with back-propagation, significant errors remain in the simulation as a consequence of neglecting amplitude fluctuations that develop within the random medium.
 The question addressed by this paper is, when can the measured phase on the ground serve as a proxy for the refractive phase change imparted by the ionosphere in a phase screen simulation? The brief answer suggested by our investigation is that this technique provides acceptable simulation results when only weak phase scintillations are present, but that accuracy degrades rapidly as the strength of scatter increases once strong phase scintillations develop. Here, we define strong phase scintillations to be the rapid transitions in the phase that occur when the complex amplitude of the signal on the ground approaches the origin in the complex plane, crossing rapidly from one quadrant to another. To answer this question in more a quantitative fashion, we conducted a number of experiments using a two-stage phase screen technique. In the first stage, a stochastic screen was generated as a realization of a power law spectral model with an outer-scale cutoff. A spherical wave was propagated through the screen and the phase on the ground was calculated. In the second stage, the phase on the ground was used as the screen and a new spherical wave was propagated through it to obtain the intensity and phase on the ground. We calculated the cross-correlation of phase and intensity between the Stage 1 and Stage 2 phase screen simulations, and also the statisticsS4, τ, p, and σφas metrics of accuracy. We considered two propagation scenarios, one-way propagation from a GPS satellite to a ground-based receiver located at Ascension Island and two-way propagation between the ALTAIR VHF radar and a calibration sphere in LEO orbit.
 We found that the cross-correlation of phase starts at unity and decreases rapidly as the irregularity strength increases once strong phase scintillations have developed. The cross-correlation of intensity is less than unity even in the weak scatter limit, due to the presence weak phase scintillations caused by diffraction. The level of RMS of phase fluctuations (or equivalently, the turbulent intensity) for which the phase cross-correlation drops to 95%, can be considered the point at which strong phase scintillations develop and diffraction effects begin to significantly degrade the accuracy of phase screen simulations using the measured phase as a proxy for the screen. We find that the scattering strength required for strong phase scintillations to develop decreases as the spectral index of fluctuations in the screen decreases. The intensity cross-correlation decreases rapidly with increasing irregularity strength when the phase cross-correlation starts to decrease, which shows that the temporal structure deviates increasingly from those of the measurements as the turbulence strength increases due to the increased influence of phase scintillations. Statistical metrics of the simulated results includingS4, τ, p, and σφ, also deviate from the measurements in a similar fashion. Finally, these deviations tend to be larger when the irregularity index is shallower.
 Using measurements of the complex signal along a one-way propagation path to GPS receiver at Ascension Island and along a two-way propagation path to the ALTAIR VHF radar at Kwajalein Atoll, we demonstrate that back-propagating the complex signal up to ionospheric altitudes prior to the forward propagation step yields improved phase screen simulation results, but some propagation errors remain due to the neglect of amplitude fluctuations that develop inside the random medium. We also observed that receiver phase noise and artifacts associated with cycle slip repair can limit the accuracy of phase screen simulations, particularly when strong phase scintillations are present. We obtained more accurate phase screen simulation results using the GPS measurements than using the ALTAIR measurements, primarily because only weak phase scintillations were present in the former while strong phase scintillations were also present in the latter.
 The authors would like to thank Charles Rino and the two anonymous reviewers for their comments and suggestions which have significantly improved this paper. This work was sponsored by Air Force contract FA8718-09-C-0041.