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Corresponding author: C. S. Carrano, Institute for Scientific Research, Boston College, Boston, MA 02467-3862, USA. (firstname.lastname@example.org)
 We develop a phase screen model called the Synthetic Aperture Radar (SAR) Scintillation Simulator (SAR-SS) for predicting the impacts of ionospheric scintillation on SAR image formation. SAR-SS consists of a phase screen generator and a propagator. The screen generator creates a 2-D random realization of spatial phase fluctuations resulting from the traversal of small-scale field-aligned irregularities in the ionosphere. It accounts for the motion of the radar platform, the drift of the ionospheric irregularities, and the oblique angle of propagation, all of which determine the scale sizes of the irregularities sampled by the radar beam. The propagator solves the 3-D parabolic wave equation using the split step technique to compute the ionospheric transfer function for two-way propagation. This ionospheric transfer function is used to modulate the SAR signal due to terrestrial features in order to assess the ionospheric impact on SAR image formation in the small target approximation. We compare simulated and observed PALSAR imagery over Brazil during disturbed ionospheric conditions. We demonstrate that SAR-SS can reproduce the field-aligned streaks in PALSAR imagery caused by irregularities in the equatorial ionosphere that have been observed by previous authors. The field-aligned streaks exhibited a dominant wavelength larger than the Fresnel break scale, which suggests that refractive scatter was dominant over diffraction as the physical mechanism responsible for the scintillation of the radar signal in this case. The spectral index of phase fluctuations in the screen was quite large (9.0), suggesting that these irregularities were possibly associated with bottomside sinudoidal irregularities rather than equatorial plasma bubbles.
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 The effects of the ionosphere on synthetic aperture radar (SAR) have been reviewed by previous authors [Ishimaru et al., 1999; Xu et al., 2004] and can be categorized into two basic types. The first type consists of effects caused by the ionospheric background such as refraction, polarization rotation, group delay, and phase advance. These effects can be largely mitigated for an L band SAR if a suitable model for the background ionosphere is employed during SAR processing [Chapin et al., 2006], or better still, if the radar can measure Faraday rotation and account for this effect during SAR processing [Nicoll et al., 2007; Roth et al., 2010]. The second type consists of effects caused by small-scale ionospheric structures (on the order of the Fresnel scale) which cause scintillation of the radar radio waves. These small-scale structures are generated by plasma instability processes which act predominantly in the polar cap and equatorial zone at night, the effects being more pronounced during periods of high solar activity [Aarons, 1982]. As radio waves traverse these small-scale structures, they scatter randomly in different directions and travel paths of different distances, resulting in spatial variations in signal phase. These phase variations cause mutual interference as the radio wave propagates through free space, causing a diffraction pattern on the ground with spatial fluctuations in both amplitude and phase. These signal fluctuations are intensified as the reflected wave traverses the ionosphere a second time during the return path to the space radar. Ionospheric effects increase dramatically as the operating frequency of the radar decreases, such that the impacts can be performance limiting for radars operating at L band and lower frequencies [Chapin et al., 2006]. Amplitude and phase fluctuations which decorrelate across the synthetic aperture of the radar reduce the effective resolution of a SAR image [Xu et al., 2004], and alter critical differential phase relationships between images collected during satellite revisits that are required by InSAR and change detection applications.
 The SAR data examined in this paper was collected by the Phased Array type L band Synthetic Aperture Radar (PALSAR) on board the Japanese Advanced Land Observation Satellite (ALOS) [Shimada et al., 2006]. Due to its operating frequency in the L band (1270 MHz) where ionospheric effects are important, a wide variety of ionospheric effects on PALSAR images have been reported in the literature [e.g., Pi et al., 2011, and references therein]. Here we are principally concerned with the impacts due to scintillation, which are manifest in ALOS/PALSAR images of low-latitude terrain as streaks that appear to be approximately aligned with the magnetic field [Shimada et al., 2008].
 In this paper we present a new phase screen model called the SAR Scintillation Simulator (SAR-SS) for predicting the impacts of ionospheric scintillation on SAR image formation and interferometry. SAR-SS consists of a phase screen generator and a propagator. The screen generator creates a 2-D random realization of spatial phase fluctuations resulting from the traversal of small-scale irregularities in the equatorial ionosphere. The irregularities are specified statistically in terms of a power spectral density that depends on (1) the vertically integrated strength of turbulence, (2) the phase spectral index, (3) the outer scale, and (4) the anisotropy ratio along and transverse to the local magnetic field direction. The screen generator accounts for the motion of the radar platform, the drift of the ionospheric irregularities, and the oblique angle of propagation, all of which determine the scale sizes of the irregularities sampled by the radar beam. The statistical parameters specifying the irregularities can be input to the simulator manually, or provided by the WideBand ionospheric scintillation Model (WBMOD), a global climatological model of scintillation constructed from an extensive database of observations [Secan et al., 1995].
 The propagator of the SAR-SS model solves the 3-D parabolic wave equation (PWE) using the split step technique [Costa and Basu, 2002; Rino, 2011] to compute the transfer function for two-way propagation through the ionosphere. This transfer function is used to modulate the SAR signal due to terrestrial features in order to assess the ionospheric impact on SAR image formation. We note that SAR-SS is similar to an existing phase screen model called SAR-TIRPS [Rogers and Cannon, 2009], except that the later was constructed using a 1-D phase screen. The SAR-SS model uses a 2-D phase screen, which is necessary for simulating SAR images in the general case where the radar propagation path intersects the magnetic field direction at arbitrary angles.
 The outline of this paper is as follows. In sections 2 and 3we present the propagator and phase screen generator of the SAR-SS model, respectively.Section 4 explains how the ionospheric transfer function is applied to a SAR image to simulate the impacts of scintillation. In section 5we use the SAR-SS model to add the effects of small-scale irregularities in the ionosphere to a PALSAR SAR image collected during quiet ionospheric conditions, and the results are compared with a PALSAR image collected during a revisit over the same terrain during disturbed ionospheric conditions. We analyze the cross-field intensity fluctuations caused by the irregularities, and we discuss the spatial structure of these irregularities. We conclude the paper insection 6.
2. Split Step Solution of the Parabolic Wave Equation
 In this section, we develop the split step solution to the parabolic wave equation for the case of two-way (radar) propagation through ionosphere in the thin phase screen approximation. We begin by considering one-way propagation of a monochromatic plane wave through the phase screen. The split step solution to the parabolic wave equation is expressed in terms of the ionospheric transfer function. The ionospheric transfer function for two-way propagation follows from two successive applications of the one-way result. Finally, we correct for the spherical curvature of the wavefronts to produce the split step solution for the case of two-way propagation of a spherical wave through the ionosphere.
 The scalar Helmholtz equation governs the propagation of radio waves through a weakly inhomogeneous medium in the absence of currents and depolarization effects [Rino, 2011]:
 In (1), E(r) is the electric field, n is the index of refraction, and k is the wave number of the radio wave. Throughout this development, we will omit the ei2πft dependence of the radio wave for clarity but it is implied, where f is the frequency. The wave number is related to the wavelength λ by k = 2π/λ. We define the reduced electric field (also called the complex amplitude) U(r) as
 The reduced electric field is employed to remove the oscillatory component of the field along the propagation direction.
 The coordinate system chosen for the propagation calculation is shown in Figure 1. The origin is located at the ionospheric penetration point (IPP) along the propagation path, at the center of the radar beam and at the center of the ionospheric scattering layer. The x, y, and z axes point toward geomagnetic north, geomagnetic east, and vertically downward, respectively. The thickness of the scattering layer, assumed to contain homogeneously distributed irregularities in the refractive index, is L. In this coordinate system, the wave vector of the transmitted radar signal is given by
 Substituting (2) and (3) into (1), and assuming the medium changes sufficiently slowly in the direction of propagation, yields a form of the PWE for the reduced field where the radio propagation angles appear explicitly:
 In (4), Δn is the fluctuating part of the index of refraction, which is assumed to be small (an additional term of order Δn2 has been neglected). In what follows, we solve (4)in the thin screen approximation using the split step approach, first for the case of space-to-ground propagation and then for the case of two-way propagation.
 We begin by considering the first penetration of the radar signal through the ionosphere. Inside the ionospheric scattering layer, we neglect the diffraction terms which involve second derivatives of U.In the high-frequency limit (corresponding to the straight line propagation assumption of geometric optics), the phase change imparted by the ionosphere is proportional to the integral of electron density fluctuations ΔNe along the line of sight [Rino and Fremouw, 1977]:
where ρ is a distance vector in the transverse (horizontal) plane, = (cos φ, sin φ) is a unit vector in the transverse plane, re is the classical electron radius, and dl is a differential length element along the propagation path. The straight line propagation assumption is appropriate for radars operating at L band frequencies or higher. To simplify the solution of (4), we assume the phase change imparted by the ionosphere can be represented using a thin phase-changing screen. In this case, the solution to(4) gives the reduced electric field after single passage through the scattering layer as U(ρ, 0+) = Ut eiϕ(ρ), where Ut is the complex amplitude of the transmitted plane wave.
 Next we solve (4)for the region of free-space propagation past the screen. It is convenient to seek a solution to(4) in terms of the transverse Fourier transform of U which is defined as
 In (6), κ is the transverse Fourier wave number, which is not to be confused with the wave number, k, of the transmitted radio wave. The solution to (4) for the reduced field after single passage through the phase screen can be expressed:
Equation (7) gives the split step solution to (4) subject to the thin screen approximation for the case of an incident monochromatic plane wave. A similar result was derived by Costa et al. for the case of space-to-ground radio wave propagation. The term ( ⋅ κ)z tan θ in (7) simply translates the reduced field horizontally, and can be dropped if the coordinate system is made to translate horizontally such that it follows the radio propagation path [Rino, 2011].
 We account for two-way propagation through the screen by expressing(7) in operator form and applying this operator to the downward and upward propagation paths of the radio wave. Let d1 be the slant distance from the IPP to the center of the target area (on the ground), and let d2 be the slant distance from the IPP to the radar (in orbit around the Earth). For the time being, we assume the transmitted radar signal returns to the radar without interaction with or modulation from the ground. In this case, the reduced field that returns to the horizontal plane of the radar can be expressed as the product of two operators acting on the transmitted monochromatic plane wave with initial amplitude, Ut, as follows:
where the operator D is given by:
 The operator D can be interpreted as the ionospheric transfer function corresponding to one-way propagation of a monochromatic plane wave at frequencyf after passage through the screen and traveling a slant distance s through free space. In (9), F denotes the transverse Fourier transform, which can be evaluated efficiently using the fast Fourier transform (FFT) in two dimensions.
 We can generalize this result for the case of a wideband transmitted radar pulse by decomposing the pulse into its harmonic components, applying the ionospheric transfer function operators to each of these harmonics separately, and then synthesizing the pulse [Knepp and Nickisch, 2009]. In this case, the reduced field that returns to the horizontal plane of the radar after double passage through the ionosphere when transmitting a pulse P of bandwidth B can be expressed:
 The above is the integral over all frequencies within the bandwidth of the Fourier transform of the pulse, (f), modulated by the ionospheric transfer function for two-way propagation (shown in brackets). Again, we note that the signal modulation due to the ground has been omitted at this stage.
 Finally, we correct for the curvature of the spherical wavefronts by scaling the transverse and propagation distances accordingly in (10). As discussed in Ratcliffe  (and also Ishimaru ), for a spherical wave calculation the diffraction pattern on the ground is similar to, but larger by the factor (d1 + d2)/d1 than, that which would be produced at a distance dR = d2d1/(d1 + d2) if the incident radiation came from an infinitely distant point source. In the literature, dR is often referred to as the reduced propagation distance [Rino, 1979]. The final result, accounting for the curvature of the spherical wavefronts, is then
where the ionospheric transfer function for two-way propagation is given by
3. Generating the Phase Screen
 The phase screen we use for the frequency component, f, in the wideband radar signal has contributions from the background total electron content (TEC) and the random phase fluctuations due to the small-scale structure in the turbulence:
 The frequency scaling fc/f is used to scale the screen phase from the center frequency to the desired frequency component within the radar bandwidth. The contribution from the background slant TEC is given simply by ϕTEC(ρ) = reλ ⋅ TEC(ρ). The contribution due to the turbulence, Δϕ(ρ), is generated as a random realization drawn from an ensemble with a specified power spectral density (PSD), using the algorithm described in Rino . We assume Rino's form for the power spectral density of phase after one-way passage through the phase screen:
 In (14), p is the phase spectral index, while a and b are scaling factors that elongate contours of constant correlation along and transverse to the magnetic field, respectively. The turbulence is assumed to have outer scale L0, and κ0 = 2π/L0 is the outer scale wave number. The transverse wave numbers in the geomagnetic east and geomagnetic north directions are symbolized by κx and κy, respectively. CkL is the vertically integrated strength of turbulence at the 1 km scale [Secan et al., 1995]. The coefficients A, B, and C depend on the directions of propagation and the magnetic field, and are obtained by relating the radar line of sight and the irregularity axes [Rino, 1979]:
 In (16), ψ is the magnetic inclination angle and δ is the angle at which irregularities are inclined from the xzplane (typically taken to be zero at low latitudes). SAR-SS obtains the magnetic field parameters at the IPP location from the International Geomagnetic Reference Field (IGRF) [Olsen et al., 2000].
 We generate a 2-D realization of phase by filtering a normally distributed, zero-mean random sequence with the power spectral density(14) using the technique described in Rino . To do this, we must choose the spatial sampling to be used in the geomagnetic east and north directions as these, together with the number of points in each direction, determine the values of the transverse wave numbers κon the spatial grid. PALSAR is a right-looking stripmap SAR system with zero squint angle on board the ALOS satellite which flies in a polar orbit. Therefore, the along-track direction is nearly aligned with geomagnetic north and the range direction is nearly aligned with geomagnetic east. For this reason we choose the number of points in the geomagnetic east and north directions to the equal to the number of points in the along-track and range directions of the SAR image, respectively. Neglecting Earth rotation during the radar propagation time, the azimuth sample spacing ΔSa and range sample spacing ΔSr in a PALSAR image are given by:
where Vp is the velocity of the satellite platform (which is assumed to be purely horizontal), PRF is the pulse repetition frequency, PSF is the pulse sampling frequency, and c is the speed of light. The ground range sampling ΔSgr in a PALSAR image is ΔSr/sin(γ), where γ is the radar incidence angle on the ground at the center of the target area. The factor sin(γ) accounts for range foreshortening. For a plane wave calculation there is no spreading of the wavefronts as the radar signal propagates. In this case, the grid spacing in the geomagnetic east and north directions can be determined from the azimuth and ground range spacing of the SAR image by rotating the radar coordinate system by the magnetic heading of the radar beam φ:
 Note that φ is the magnetic heading of the radar beam, whereas φ − π/2 is the magnetic heading of radar platform motion (since PALSAR is a right-looking stripmap SAR). If we account for the spherical curvature of the wavefronts, the dimensions of the phase screen in(18) are both reduced by the factor d1/(d1 + d2). However, it is clear that we should use the uncorrected grid spacing in (18) to construct the phase screen even for the spherical wave case, because the diffraction pattern of the plane wave calculation is to be enlarged by the factor (d1 + d2)/d1 in order to produce the spherical wave result.
 For a satellite in LEO orbit, the radar scan velocity is generally much faster than the zonal drift velocity of the ionospheric irregularities and therefore the latter can be neglected. Nevertheless, the result in (18) can be generalized to account for the irregularity drift as:
where we have assumed frozen-in motion of the plasma with geomagnetic eastward and northward components given byVDe and VDn, respectively. Once the screen is generated using the grid spacing in (18) or (19), the screen is rotated into alignment with the along-track and cross-track directions of the radar. Note that this rotation can be performed without interpolating the screen onto the SAR image grid by prerotating the wave number vectorκ by the negative of the magnetic heading γ.
4. Application of the Ionospheric Transfer Function
 A simplified model for the received SAR signal can be expressed as the sum of reflected pulses from every target (m) in the scene for every position of the radar (u) along the flight path or synthetic aperture direction [Soumekh, 1999]:
 The received SAR signal, S, is commonly referred to in the literature as the phase history data. The SAR signal is an unfocused image of the scene which must be focused to produce a single look complex (SLC) image, the magnitude of which is the SAR image. Algorithms to perform this focusing include the range Doppler algorithm (RDA) and seismic migration [see, e.g., Bamler, 1992, and references therein]. In (20), σm is the reflectivity, and Rm is the slant range to the mth target (pixel). Wr is the antenna beam pattern in the range direction, while Wa is the antenna beam pattern in the azimuth direction. Neglecting a possible dependence of σm on frequency, the summation in (20) implicitly assumes the ground clutter is produced by frequency flat point scatterers at each pixel location which contribute independently to the SAR signal. Equation (10) gives the reduced field at the radar after double passage of a single pulse through the ionosphere in the absence of interaction with the ground. Combining equations (10) and (20) gives an expression for the received SAR signal with ionospheric effects added:
 The notation in (21) indicates that the ionospheric transfer function is sampled at the azimuth location xm and slant range location ym of the mth target, which implicitly depends on the position of the radar along the flight path and also the look direction to the target. While it is possible to evaluate (21) exactly for scenarios consisting of a small number of ground targets, it becomes computationally intractable for large SAR images. In order to simulate the impacts of scintillation on large SAR images such as those from PALSAR, it was necessary to introduce a couple of simplifying assumptions. The first assumption is that if the bandwidth is sufficiently small then the ionospheric transfer function can be assumed to be independent of frequency, and equal to its value evaluated at the center frequency fc. This is a good approximation for narrow-bandwidth radars such as PALSAR (which has a bandwidth of 28 MHz). Applying this approximation to(21) gives:
 The second assumption is that the ionospheric transfer function can be pulled out of the summation over the targets, m.In this case the SAR signal with ionospheric effects added can be expressed as the product of the ionosphere-free SAR signalS(t, u) and the ionospheric transfer function:
 The consequences of this assumption are that the varying look angles from each radar position along the flight path to each target are neglected so that the same ionospheric transfer function is applied to all targets. The ionospheric transfer function which is applied in (24) is that for a spherical monochromatic wave transmitted from the central position of the radar along the flight path. The error incurred from this approximation should decrease as the size of the target area decreases, and therefore this is a small target assumption. While this approximation clearly warrants further justification (particularly for large SAR images), it does make the computation tractable and, as we will see in the next section, produces images with ionospheric impacts similar to those observed by PALSAR.
 For this study, we began with single look complex data from PALSAR and simulated the received (unfocused) SAR signal by applying the range Doppler algorithm in the inverse fashion [Khwaja et al., 2006]. We could have skipped this inversion step and used the raw unfocused SAR signal (phase history data) directly, but this data was not available at the time of this study. Next, we generated a phase screen based on the radar propagation geometry and the magnetic field orientation at the ionospheric penetration point, as described in section 3. We then calculated the ionospheric transfer function by evaluating (11) and (12). Finally, we modulated the unfocused SAR signal with the ionospheric transfer function as in (24), and then applied the RDA algorithm in the forward direction [Cumming and Bennett, 1979] to produce simulated SAR images with ionospheric effects added. We verified our use of the RDA algorithm by confirming that it properly focused the SAR signal due to a point target (not shown). We also confirmed that application of the RDA to a SAR image in the inverse fashion, followed by application of the RDA to the resulting SLC data in the forward direction with no ionospheric modulation correctly reproduced the SAR image (i.e., to confirm the unfocusing/focusing operations were invertible). We will give additional details and examples of each step in the processing in the next section as we describe our simulation results.
5. Results and Discussion
 We begin by showing examples of PALSAR imagery during two revisits of the ALOS satellite over the same terrain in Brazil. The PALSAR data was provided by Franz Meyer. User Systems extracted the SAR images and produced the SLC files which we used as input to the model. Figure 2 shows a PALSAR sublook image of a scene in Brazil centered at latitude = −4.05°, longitude = −67.91°, magnetic latitude = −6.0° on 25 December 2007 at 03:20:30 UT during quiet ionospheric conditions. Figure 3 shows a sublook of the same terrain on 26 March 2008 at 03:19:18 UT when the ionosphere was disturbed. The radar parameters corresponding to these two SAR images are summarized in Table 1. The local time for both satellite passes is 22:48 LT, which is after local sunset. Note the prominent streaks in the 26 March 2008 image oriented along the direction of the horizontal magnetic field, which is indicated by the arrow. These streaks are inclined at approximately −6.3° from the radar along-track direction, which is the magnetic heading of the radar platform motion at this location.
Table 1. Radar Parameters Corresponding to the PALSAR Images Shown in Figures 2 and 3
Pulse repetition frequency (PRF)
Pulse sampling frequency (PSF)
Radar incidence angle at ground (γ)
Slant range to target area
Altitude of radar
Velocity of satellite ground track (Vp)
Magnetic heading of satellite motion (φ − π/2)
Magnetic heading of radar beam (φ)
Number of azimuth samples
Number of slant range samples
Azimuth sample spacing (ΔSa)
Slant range sample spacing (ΔSr)
Ground range sample spacing (ΔSgr)
 In this section we demonstrate the SAR-SS model by adding simulated ionospheric effects to the image acquired by PALSAR on 25 December 2007 during quiet ionospheric conditions (Figure 2). The parameters of the phase screen were chosen to reproduce, as closely as possible, the streaks shown in the image on 26 March 2008 which was acquired by PALSAR during disturbed ionospheric conditions (Figure 3). The first step in the processing is to unfocus the SAR image in order to obtain the received SAR signal, which is the phase history measured by the radar as it travels along the synthetic aperture direction. This was accomplished by applying the range Doppler algorithm in the inverse fashion. The received SAR signal, S, is shown in Figure 4. The inverse RDA processing blurs the image because the reflected power from each point scatterer in the target area is spread in range and azimuth. Next, we assumed the height for the ionospheric phase screen to be Hp = 350 km and we evaluated IGRF model to determine the magnetic field parameters at the ionospheric penetration point along the raypath from the radar (when it was located midway along its trajectory) to the center of the target area. These magnetic field parameters are given in Table 2. Given the magnetic heading of the radar beam at the IPP location (83.5°), and the azimuth and range sample spacings for the SAR image (listed in Table 2), the grid spacings of the phase screen in the magnetic east and north directions were calculated using (18). The drift velocity of the plasma (which was not measured) should be very small relative to the scan velocity of the satellite and was therefore neglected. Following Secan et al. , an anisotropy ratio of a:b = 50:1 was assumed. The turbulent intensity (CkL), phase spectral index (p), and outer scale (L0) were selected by numerical experimentation to produce a simulated SAR image that most closely matched the image acquired by PALSAR on 26 March 2008 (Figure 3). We will make our requirement for “matching” the observations quantitative in the text that follows, but the idea was to reproduce both the dominant wavelength and amplitude of the streaks imposed on the SAR image by the ionosphere. Perhaps due to a limitation in our implementation of the phase screen generator, we found it necessary to use θ = 0° in (15) in order to generate a phase screen with structures aligned with the magnetic field direction. In all other calculations, we used the actual value of θ = 36°. The results of this numerical experimentation determined that CkL = 3.5 × 1033, p = 9, and L0 = 5 km yielded results which best matched the observations. The realization of the phase screen ϕ we generated using these parameters to have the power spectral density (14), in an ensemble averaged sense, is shown in Figure 5. For this calculation, we assumed the background TEC was zero so that ϕTEC = 0.
Table 2. Geophysical and Propagation Parameters Used to Generate the Phase Screen Shown in Figure 5a
Note that the grid spacings listed in this table have been scaled to account for the spreading of the spherical wavefronts. The physical sizes of grid spacings at the screen are smaller than those listed by a factor d1/(d1 + d2).
Integrated turbulence strength (CkL)
3.5 × 1033
Phase spectral index (p)
Outer scale (L0)
Anisotropy ratio (a:b)
Screen altitude (Hp)
Propagation angle (θ)
Slant distance from target to IPP (d1)
Slant distance from IPP to radar (d2)
Reduced propagation distance (dR)
Magnetic declination at IPP
Magnetic inclination at IPP (ψ)
Number of azimuth samples
Number of slant range samples
Magnetic east screen spacing (ΔSe)
Magnetic north screen spacing (ΔSn)
Azimuth sample spacing (ΔSa)
Ground range sample spacing (ΔSgr)
 We note that the phase screen was generated with grid spacings in magnetic east and north given by ΔSe = 7.08 m and ΔSn = 4.03 m, respectively. Once the screen is rotated into the coordinates of the SAR image (by an angle equal to minus the magnetic heading of radar beam), these grid spacing translate to the same spacings in the azimuth and ground range directions as the SAR image itself, which are ΔSa = 3.20 m and ΔSgr = 7.49 m, respectively. Given the number of samples in the azimuth direction (6144) and slant range direction (4496), the size of the SAR image on the ground is 19,661 m in azimuth and 33,675 m in ground range. These are also the spatial dimensions of the phase screen used to calculate the ionospheric transfer function (using the reduced propagation distance), since the intention is to scale (i.e., spread out) the diffraction pattern from the plane wave calculation in order to produce the spherical wave result. The physical dimensions of the screen are smaller than the scale of the SAR image by the factor, d1/(d1 + d2), and this is why the scale used in Figure 5 is smaller than the scale used in the other figures.
 After the phase screen was generated, we calculated the ionospheric transfer function DITF using (9) and (12). The result is shown in Figure 6. The received SAR signal with ionospheric effects added, Siono, was calculated using the approximate formulation given in (24) and is shown in Figure 7. Finally, we focused this SAR data using the RDA in the forward sense to produce the SAR image with ionospheric effects added shown in Figure 8. The qualitative similarity between the observed PALSAR image (Figure 3) and the simulated image (Figure 8) suggests the SAR-SS model can reproduce the field-aligned streaks in PALSAR images caused by small-scale structure in the equatorial ionosphere. The streaks in both images have a dominant wavelength of approximately 2 km. It should be reiterated, however, that the approximate formulation given in(24) neglects the varying radar look angles to the individual ground targets. The variation in radar look angle to each ground scatterer changes significantly as the radar moves along the synthetic aperture direction, which in this case happens to be nearly along the magnetic field lines such that the correlation length of the ionospheric irregularities is maximal. Therefore, the ionospheric perturbation lies mostly along the range direction for which the radar look angles to the ground scatterers varies less. Therefore, it is possible that the good agreement we have obtained between the simulated and observed images, in spite of the approximation made in (24), may be partly fortuitous and due to the fact that ALOS/PALSAR flew in a near-polar orbit. It would be highly desirable to test the approximation(24) for a satellite which flies in an equatorial orbit, but data from such a system was not available at the time of this study.
 Using the weak scatter analytic theory of Rino  and Fremouw and Ishimaru , summarized in Appendix A, it can be shown that these phase screen parameters and propagation conditions correspond to a one-way scintillation intensity index (Sm) equal to 0.07 and a two-way scintillation intensity index (S4) equal to 0.15, assuming weak scatter and perfect correlation along the downward and upward propagation paths.
 In the interest of a quantitative comparison, we extracted the average normalized intensity fluctuations for the observed and simulated SAR images in the cross-field direction (i.e., perpendicular to the horizontal component of the magnetic field). This was accomplished as follows. First, the quiet and disturbed PALSAR images (from 25 December 2007 and 26 March 2008, respectively) were cross correlated to determine the spatial offset between them. The quiet image was shifted by this spatial offset and then subtracted from the disturbed image to suppress the contribution from the ground. The resulting image was then rotated by 6.3° (the negative of the magnetic heading of the radar beam), and then averaged down in the vertical direction. This yielded the average amplitude in the cross-field direction, which was then squared to obtain the intensity, and finally normalized (by the mean intensity). The results are shown inFigure 9. There are 4496 samples in the cross-field direction and the spatial sampling is 7.08 m. The black curve was extracted from the PALSAR observation on 26 March 2008, while the red curve was extracted from our simulated PALSAR image. For comparison, the blue curve shows the squared magnitude of the ionospheric transfer function in the cross-field direction. Note the similarity in both the magnitude and frequency content of the observed (black) and simulated (red) cross-field intensity fluctuations. TheS4index calculated from the cross-field intensity fluctuations was 0.11 for the observations and 0.10 for the simulation. These values compare reasonably well with the theoretical value ofS4for two-way propagation (0.15). Note that the variations in the ionospheric transfer function (blue) are somewhat larger than variations in the cross-field intensity extracted from the simulated image. This may suggest that focusing the SAR image alters to some extent the structure of the ionospheric perturbation imposed on the received SAR signal.
Figure 6shows the power spectral densities (PSD) of the cross-field intensity fluctuations for the observations (black), simulation (red), and the squared magnitude of the ionospheric transfer function (blue). We usedWelch's method with a Hanning window and 50% overlap to compute these PSDs. To accomplish this, we subdivided the 4496 samples in the cross-field direction into 6 segments of samples 749 each, with 50% overlap between segments, and then calculated the spectral coefficients for each segment via FFT. The spectral coefficients for each segment were weighted by the coefficients of a Hanning window and then averaged together. As discussed byWelch , the averaging helps to minimize the effects of spectral leakage. The spatial frequencies shown in the figure are given in terms of inverse scale and not spatial wave number. For example, the spatial frequency corresponding to the outer scale is 1/L0 and this is shown in the figure as a black dashed line at spatial frequency 0.2 km−1.
 For weak scatter due to a thin layer of irregularities, the PSD of intensity fluctuations is simply related to the PSD of the phase in the screen [Yeh and Liu, 1982]:
 In (25), K is the transverse spatial wave number, and dR is the slant propagation range past the scattering region to the receiver. The factor modulating the phase PSD, Φϕ(K), is referred to as the Fresnel filter function. The Fresnel filter function in (25) has a maximum at the Fresnel break frequency fb = 1/(2λdR)1/2, and a sequence of local minima at the frequencies fn = (n/λdR)1/2, where n = 1, 2, … In Figure 10, the Fresnel break frequency fb is shown with a green dashed line and the first three Fresnel minima fn are shown with green dotted lines.
 As a consequence of (25), the spectral index of intensity fluctuations is the same as the phase spectral index provided the scatter is weak. Unfortunately, it is not possible to measure the spectral index of intensity fluctuations directly from the PSD shown in Figure 10. The reason for this is that the spectral index must be measured where the PSD of intensity fluctuations exhibits a log-log linear regime. According to weak scatter theory, this regime occurs for scales smaller than the Fresnel break scale. Due to speckle noise in the SAR images, however, the intensity PSD are corrupted by noise for frequencies higher than about 2 km−1. While it is possible to bin down the SAR image pixels to suppress the noise, this procedure alters the shape of the spectrum (since a filtering operation is implicitly involved) and therefore binning has been avoided. Since the Fresnel break frequency is 3.1 km−1, the log-log linear regime is not evident in our intensity PSD (due to the speckle noise) and therefore we cannot measure the spectral index directly. However, it is possible to infer the phase spectral index indirectly by determining the value for which the PSDs of observed and simulated cross-field intensity fluctuations agree most closely.
 In this study, the parameters that specify the phase screen (CkL, p, and L0) were selected to obtain the best agreement possible between these two PSDs, and also their integral over frequency which gives the S4 index. The similarity between the observed (black) and simulated (red) PSDs shown in Figure 10 suggests that we have indeed determined screen parameters which are consistent with the observations. It is interesting to note that a phase spectral index of 9 was required to obtain good agreement with the observations. This suggests the irregularities responsible for the scintillation of the radar signal were perhaps better characterized by a Gaussian spectrum than a power law spectrum. We note that bottomside sinusoidal irregularities in the ionosphere are well characterized by a Gaussian spectrum [Cragin et al., 1985; Basu et al., 1986]. We also note that in Figure 10 the dominant contribution to the intensity fluctuations comes from spatial frequencies significantly smaller than the Fresnel break frequency. When the phase spectral index of the irregularities is close to 3 (which is typical of ionospheric turbulence associated with equatorial plasma bubbles), these spatial frequencies are filtered out by the action of diffraction and do not contribute appreciably to the scintillation. When the phase spectral index is larger than 5, as in the case studied here, spatial scales larger than the Fresnel scale (and, in fact, spatial scales up to the outer scale) can contribute appreciably to the scintillation due to refractive focusing effects [Booker and MajidiAhi, 1981]. In this case, refractive scatter due to focusing dominates relative to diffractive scatter as the physical mechanism responsible for producing the scintillations.
 As mentioned earlier, the screen parameters in the SAR-SS model can be specified manually, or given by the WBMOD climatological model. Since WBMOD uses the average value ofp = 2.5, using WBMOD to specify the screen parameters in the simulation would not correctly reproduce the wavelength of the streaks in the PALSAR images we considered. One may expect better performance in an average sense using WBMOD to simulate streaks in SAR images when they are caused by the more common equatorial plasma bubbles (EPBs), for which the spectral index is generally close to this assumed value of 2.5. For the case of scintillation due to EPBs, the dominant streak wavelength in the SAR images is expected to be close to the Fresnel break scale.
 To test whether this is true, we examined the effect of changing the spectral index. To do this we repeated the simulation with all parameters held constant except that we changed p from 9 to 3 and CkL from 3.5 × 1033 to 2.25 × 1034. The turbulent intensity was changed to match the S4indices calculated from the observed and simulated intensity fluctuations in the cross-field direction. For these parameters, the SAR image with ionospheric effects added is shown inFigure 11, while the PSDs of the intensity fluctuations (and also the squared magnitude of the ionospheric transfer function) in the cross-field direction are shown inFigure 12. When the phase spectral index is 3 the streaks have a dominant spatial frequency of approximately 0.2 km−1, which is close to the Fresnel break frequency fb = 3.1 km−1.
 We note that since this paper was drafted, User Systems has carefully examined a large number of PALSAR images exhibiting streaks over equatorial South America and Africa. They found that these streaks are consistently aligned with the projection of the magnetic field vector at ionospheric heights onto the ground plane, provided that one properly accounts for the apparent “magnification” of the cross-track direction that occurs because the ionospheric altitude is a significant fraction of the radar platform altitude (K. Chotoo, personal communication, 2011). Their findings are consistent with the results we present here. We intend to apply the SAR-SS model to simulate other affected PALSAR images with different angles of propagation with respect to the magnetic field, both to validate the algorithm and also to study the characteristics of the ionospheric irregularities responsible for degrading these SAR images.
 We developed a phase screen model called the SAR Scintillation Simulator (SAR-SS) for predicting the impact of ionospheric scintillation on SAR image formation. We compare simulated and observed PALSAR imagery over Brazil during disturbed ionospheric conditions. We demonstrate that SAR-SS can reproduce the field-aligned streaks in PALSAR imagery caused by small-scale structure in the equatorial ionosphere that have been observed by previous authors. The streaks in the images we examined had a dominant wavelength larger than the Fresnel break scale which suggests that refractive scatter due to focusing was dominant over diffraction as the physical mechanism responsible for the scintillation of the radar signal. The spectral index of phase fluctuations in the screen was quite large (9.0), suggesting that these irregularities were possibly associated with bottomside sinudoidal irregularities rather than equatorial plasma bubbles. We show via simulation that if the phase spectral index were 3.0, which is more typical of propagation through equatorial plasma bubbles [Carrano et al., 2011], the dominant wavelength of the streaks would be close to the Fresnel break scale as suggested by weak scatter theory.
Rino and Fremouw  and Rino showed that for one-way propagation through a thin layer of ionospheric irregularities in the Born approximation (single scatter), the scintillation intensity index can be expressed as:
re = classical electron radius (2.818 × 10−15 m),
λ = wavelength,
CkL = vertically integrated strength of turbulence at the 1 km scale,
dR = reduced slant propagation range,
θ = zenith angle of propagation,
Γ = Euler's gamma function,
and ℑ = a combined geometry and propagation factor.
 The spectral slope, ν, is defined such that p = 2ν is the phase spectral index and p(3) = 2ν+ 1 is the spectral slope of three-dimensional in situ density fluctuations. The reduced slant propagation rangedR corrects for the spherical curvature of the wavefronts and is given by dR = d1d2/(d1 + d2) where d1 and d2 are the slant distances from the target area and radar to the phase screen, respectively. For highly elongated irregularities traversed in a direction well away from along their principal axis, the factor ℑ is approximately given by
 Assuming Nakagami statistics, Fremouw and Ishimaru derived a relationship between the scintillation intensity index for a two-way propagation path (S4) and the scintillation intensity index for a one-way propagation path (Sm). For the case of perfect correlation between the downward and upward paths, their result is
 The authors thank David Belcher and Neil Rogers for their helpful comments on SAR modeling, Franz Meyer for providing the PALSAR data, and User Systems who extracted the imagery and provided valuable feedback on our analysis. This work was sponsored by Air Force contract FA8718-09-C-0041.