Ionospheric Phase Scintillation Correction Based on Multi-Aperture Faraday Rotation Estimation in Spaceborne P-Band Full-Polarimetric SAR Data

Abstract

The spaceborne P-band fully polarimetric synthetic aperture radar (SAR) working system is highly susceptible to the scintillation effects induced by ionospheric irregularities due to its low carrier frequency. The scintillation phase error (SPE) is a dominant factor that leads to azimuth decorrelation. The aperture-dependent and spatial-varying characteristics of the SPE promote the complexity of the SPE estimation and compensation. In this paper, a methodology is described that compensates the SPE by estimating the Faraday rotation (FR) angle from fully polarimetric SAR data. The multi-aperture scheme is adopted, including the sub-aperture FR estimation, multi-aperture splicing, and overall compensation, to take the complicated characteristics of the aperture-dependent and spatial-varying SPE into account. The methodology is finally validated on simulated data derived from the airborne P-band SAR real data, and compared with an existing method. The new method does not need prior knowledge of the ionospheric height. Furthermore, its performance is investigated in relation to several key factors in different simulation conditions.

1. Introduction

The spaceborne P-band synthetic aperture radar (SAR) is instructive to geophysical surveys such as the monitoring of the global biomass, soil moisture, and carbon circulation [1,2], and for military applications to investigating targets hidden behind natural or artificial camouflage [3]. These benefits are derived from the low carrier frequency due to great advantages of the good temporal coherence and penetration capability [1,2,3]. Hence, the interest in developing this type of SAR mission continues to rise, such as the upcoming BIOMASS considered by the European Space Agency (ESA), and aims at retrieving biomass content and change by using a full-polarimetric (full-pol hereafter) working mode [4]. Nevertheless, the P-band SAR signals that propagate through the ionosphere are bound to be significantly interfered by the ionospheric effects [5,6,7]. Thereinto, the ionospheric scintillation is one of the intractable complications, which demands serious considerations and corresponding compensation methods when designing a spaceborne P-band SAR satellite [8,9,10,11,12].

The ionospheric scintillation is induced by particle precipitation and electron density irregularities of different scales [13,14,15]. Generally, the scintillation mainly occurs at the post-sunset sector of equatorial regions, which typically begins after 6:00 p.m. local time, peaks at about 10:00 p.m. local time, and gradually decays until the early morning; it can also frequently happen at auroral areas at any time of the day [16,17,18,19]. In the horizon, those equatorial irregularities well known as plasma bubbles tend to be rod-like structures that are extremely elongated and aligned to the geomagnetic field, while those auroral irregularities tend to be sheet-like, and both of the rod-like and sheet-like structures can be described by two anisotropic scales [18]. In addition, the global scintillation strongly depends on season and solar activities [13,14,15,16,17,18,19].

In the presence of the ionospheric scintillation, there will be two main types of scintillation artifacts in spaceborne SAR images at L-band or P-band [20,21,22,23,24,25]. One refers to the scintillation amplitude stripes, which have been frequently observed in equatorial areas, indicating the nice azimuth resolvability [12,16,17,20,21,22,23,24]. The other refers to the azimuth-imaging degradation due to the in-aperture decorrelation, which is mainly caused by the scintillation phase error (SPE) [12,18,21,22,23,24]. On this condition, all of the azimuth performances, including the resolution, peak-to-sidelobe ratio (PSLR), integrated-sidelobe ratio (ISLR), peak location, and peak power are degraded [12,18,24]. In comparison with the visible amplitude stripes, the azimuth-imaging degradation is considered as a more intractable issue in SAR systems. Thus, this paper particularly tackles with the SPE compensation for SAR images.

Compared with the amplitude error, the scintillation phase error (SPE) is a dominant factor that results in the in-aperture decorrelation; if not compensated, not only the imaging quality but also the interferometric performance of the spaceborne SAR will be negatively affected [24]. The accuracy of exterior measurements cannot be satisfied with the requirement of the SPE estimation [26], and the estimation of scintillation parameters can only provide the preliminary priori knowledge and cannot help to learn the overall knowledge of SPE [27,28]. There are two main types of strategies for the SPE estimation and compensation in spaceborne SAR images. One is involved in the autofocus algorithms based on the phase gradient autofocus (PGA) [29,30,31,32,33,34] and the minimum-entropy method (MEM) [35,36,37]. Another refers to the methodology that corrects polarimetric distortions induced by the scintillation on the basis of Faraday rotation (FR) estimation (FRE) in the full-pol SAR mode [38,39]. It takes the spatial-variable SPE into account by partially focusing the SAR data to the ionospheric height (effective height of phase screen as the phase screen theory), and thus its performance is strongly dependent on the prior knowledge of the ionospheric height.

Nevertheless, both of the autofocus and FRE-based strategies have seldom recognized that the SPE is both aperture-dependent and spatial-variant. In addition, the above autofocus algorithms are not robust in the SPE estimation because the PGA depends on the selection of strong scatterers in the scene, and the MEM may not have a convergent optimization procedure. Given that the FR angle (FRA) is very sensitive to polarimetric distortions, the FRE-based strategy will therefore be improved in this paper to tackle this complication of the aperture-dependent and spatial-variant SPE. Due to the fact that it is generally difficult to achieve the prior knowledge of the ionospheric height, we are dedicated to improving its robustness despite the lack of prior knowledge. The theoretical principle of the FRE-based strategy is anatomized, based on which this paper proposes an integral processing scheme for the overall SPE estimation and compensation in the case that SPE is the dominant factor causing aperture decorrelation.

This paper is organized as follows: in Section 2, we present the wave scintillation theory. Section 3 gives a description of scintillation impacts on full-pol SAR images. Then, in Section 4, an integral processing scheme for the overall SPE estimation and compensation is described in detail. In Section 5, the effectiveness of our proposed methodology is validated, and its performance is investigated in connection to several key factors. Finally, the main conclusions are drawn for the whole context in Section 6.

2. Wave Scintillation Theory

2.1. Propagation through Ionospheric Irregularities

The presence of the ionosphere will directly cause a change in the refractive index, which can be denoted by [6]

$$n\approx 1-\frac{{\omega }_p^2}{2{\omega }^2}\mp \frac{{\mu }_{0}He{\omega }_p^2}{2m_0{\omega }^3}cos\Theta ,$$

(1)

where $\omega =2\pi f$ is the angular frequency of the signal, f is the radio frequency, ${\omega }_p={\left[\left(e^2{N}_e\right)/\phantom{\left(e^2{N}_e\right)\left(m_0{\epsilon }_0\right)}\left(m_0{\epsilon }_0\right)\right]}^{1/2}$ denotes the angular plasma frequency, e and $m_0$ are the electron charge and mass, $N_e$ is the ionospheric electron density, ${\mu }_0$ and ${\epsilon }_0$ are the magnetic permeability and dielectric constants in the vacuum, H is the local geomagnetic field intensity, $B={\mu }_{0}H$ is the local geomagnetic flux density, and $\Theta$ is the angle of the propagation normal and geomagnetic field.

(1) Phase Scintillation: Once the signal traverses the irregular ionosphere, its phase tends to be changed by the refractive index perturbation. The signal phase perturbation is due to the second-order term of $-{\omega }_p^2/\phantom{{\omega }_p^{2}2{\omega }^2}2{\omega }^2$ in (1), which can be deduced as [6]

$$\delta \varphi ={\int }_l\frac{{\omega }_p^{2}k}{2{\omega }^2}dl=\frac{e^2\lambda }{4\pi m_0{\epsilon }_0{c}^2}{\int }_l\delta N_{e}dl=r_e\lambda ·\delta \mathrm{TEC},$$

(2)

where $k=2\pi /\phantom{2\pi \lambda }\lambda$ is the propagation wavenumber, $\lambda =c/\phantom{cf}f$ is the wavelength, $r_e=e^2/\phantom{e^2\left(4\pi m_0{\epsilon }_0{c}^2\right)}\left(4\pi m_0{\epsilon }_0{c}^2\right)$ is the classical electron radius, c is the light velocity in a vacuum, $\delta N_e$ is the irregular structure of the electron density, and $\delta \mathrm{TEC}={\int }_l\delta N_{e}dl$ is the total electron content (TEC) fluctuating part by integrating along the path l. It is denoted that the signals traversing the irregularities will experience spatially distributed phase errors, and thus the irregular ionosphere is equivalent as a thin screen that changes the signal phase and named as the phase screen (PS) [13,14,15].

(2) Polarization Scintillation: As ∓ is given in (1), a linearly polarized wave will be respectively divided into right- and left-handed circularly polarized waves, which propagates with different factors of $k\delta n_{-}$, $k\delta n_{+}$. Once passing through the ionosphere, they recombine into a new linearly polarized wave with a resultant angular deviation compared with the initial orientation of the polarization vector. This phenomenon is known as FR [5,6,7]. Therefore, in addition to the phase fluctuation, the ionospheric irregularities can lead to the polarization scintillation that is described by [6] when f outnumbers VHF (300 MHz):

$$\delta \Omega =\frac{k}{2}{\int }_l\left(\delta n_{+}-\delta n_{-}\right)dl=\frac{KBcos\Theta }{f^2}·\delta \mathrm{TEC},$$

(3)

where $K=e^3/\phantom{e^3\left(8{\pi }^2{m}_0^2{\epsilon }_{0}c\right)}\left(8{\pi }^2{m}_0^2{\epsilon }_{0}c\right)$ is a constant. Therefore, the FRA in (3) introduced by the scintillation is directly connected with the phase fluctuation in (2).

According to (3), we can explore the sensitivity of the TEC to the FRA for a system with the given frequency by evaluating $\sigma =KBcos\Theta /\phantom{\sigma =KBcos\Theta f^2}{f}^2$, well known as the TEC-to-FRA coefficient defined in [40]. For a given system, $\sigma$ is dependent on the local geomagnetic status and its relation with the wave propagation. Figure 1 shows the global distribution of $\sigma$ for the P-band (600 MHz) SAR system with a dawn/dusk orbit, which is generated by the International Geomagnetic Reference Field (IGRF) in the year 2016, and the unit is deg/TECU (1 TECU = 10${}^{16}$ electrons/m${}^2$). It is denoted that the absolute value $\left|\sigma \right|$ is larger in high-latitude areas and becomes zero near the equator where $cos\Theta =0$. Furthermore, it should be noted that the FRA value as to the P-band system will be more sensitive to the TEC than that as to the L-band system due to the lower carrier frequency.

2.2. Diffraction Process

PS theory indicates that the radio signals propagating in free space with decorrelated phase terms will mutually interfere and generate a diffraction pattern at the ground. This process can be described by means of the Kirchhoff diffraction formula, and the one-way scintillation transfer function (STF) as to the complex electric field is denoted as [13]

$$\zeta \left(\mathbf{\rho }\right)=\frac{jk}{2\pi d_R}\int \int exp\left\{j\left[\delta \varphi \left({\mathbf{\rho }}^{\prime }\right)-\frac{k}{2d_R}{\left|\mathbf{\rho }-{\mathbf{\rho }}^{\prime }\right|}^2\right]\right\}{d}^2{\mathbf{\rho }}^{\prime },$$

(4)

where $\mathbf{\rho }$ or ${\mathbf{\rho }}^{\prime }$ is the 2D spatial vector, $d_R=d_1{d}_{2}sec\theta /\phantom{d_1{d}_{2}sec\theta \left(d_1+d_2\right)}\left(d_1+d_2\right)$ means the modification factor for the spherical wave propagation [17], $d_1$ is the vertical distance from the PS to the ground, $d_2$ is the vertical distance from the PS to the radar (wave source), and $\theta$ is the incident angle at the PS. For an analytical description, a solution to the Helmholtz equation has been deduced in detail [17] and can be expressed as

$$\zeta \left(\mathbf{\rho }\right)={\mathrm{IFT}}_2\left\{{\mathrm{FT}}_2\left\{exp\left[j\delta \varphi \left(\mathbf{\rho }\right)\right]\right\}·exp\left(\frac{j{d}_{\mathrm{R}}}{2k}{\mathbf{\kappa }}^2\right)\right\},$$

(5)

where $\mathbf{\kappa }$ is the 2D spatial wavenumber vector, ${\mathrm{FT}}_2\left\{·\right\}$ refers to the Fourier transform operator, and its inverse form is written as ${\mathrm{IFT}}_2\left\{·\right\}$. It can be confirmed that (5) is actually equivalent to (4), and the detailed derivation is presented in Appendix A. Let the STF in (4) be written as [13]

$$\zeta \left(\mathbf{\rho }\right)=exp\left\{\alpha \left(\mathbf{\rho }\right)+j\delta {\varphi }_0\left(\mathbf{\rho }\right)\right\},$$

(6)

with

$$\alpha \left(\mathbf{\rho }\right)=\frac{k}{2\pi d_R}\int \int \delta \varphi \left({\mathbf{\rho }}^{\prime }\right)cos\left\{\frac{k}{2d_R}{\left|\mathbf{\rho }-{\mathbf{\rho }}^{\prime }\right|}^2\right\}{d}^2{\mathbf{\rho }}^{\prime },$$

(7)

$$\delta {\varphi }_0\left(\mathbf{\rho }\right)=\frac{k}{2\pi d_R}\int \int \delta \varphi \left({\mathbf{\rho }}^{\prime }\right)sin\left\{\frac{k}{2d_R}{\left|\mathbf{\rho }-{\mathbf{\rho }}^{\prime }\right|}^2\right\}{d}^2{\mathbf{\rho }}^{\prime },$$

(8)

where $\alpha \left(\mathbf{\rho }\right)$ refers to the log${}_e$-amplitude, and $\delta {\varphi }_0\left(\mathbf{\rho }\right)$ refers to the phase variation of the one-way STF. For distinguishing $\delta \varphi$ with $\delta {\varphi }_0$, we name $\delta \varphi$ as the PS-SPE and $\delta {\varphi }_0$ as the STF-SPE hereafter. In fact, the diffraction procedure is driven by the PS-SPE, which induces the amplitude scintillation and changes the PS-SPE structure.

2.3. Spectrum Description

The spectrum mechanism is generally introduced to describe the phase fluctuations induced by the ionospheric irregularities. For a simplified and more compact formulation compared with the mechanism adopted in [14], Rino’s spectrum that relieves its dependence on the inner scale [13,15,17] will be introduced in this part. For a description of the PS, the 2D spectrum density function (SDF) and its Fourier formulation of the spatial autocorrelation function (ACF) can be respectively presented as [13]

$$\Phi \left({\kappa }_x,{\kappa }_y\right)=\frac{C_{s}L·r_e^2{\lambda }^{2}ab{sec}^2\theta }{{\left[{\kappa }_0^2+\left(A_1{\kappa }_x^2+A_2{\kappa }_x{\kappa }_y+A_3{\kappa }_y^2\right)\right]}^{\left(p+1\right)/\phantom{\left(p+1\right)2}2}},$$

(9)

$$\Re \left({\rho }^{*}\right)=r_e^2{\lambda }^{2}G{C}_{s}Lsec\theta {\left|\frac{{\rho }^{*}}{2{\kappa }_0}\right|}^{\left(p-1\right)/2}·\frac{K_{\left(p-1\right)/2}\left({\kappa }_0{\rho }^{*}\right)}{2\pi \Gamma \left[\left(p+1\right)/2\right]},$$

(10)

and

$${\rho }^{*}=\sqrt{\frac{A_3{{\rho }_x}^2-A_2{\rho }_x{\rho }_y+A_1{\rho }_y^2}{A_1{A}_3-A_2^2/4}},$$

(11)

where $C_{s}L=C_{k}L{\left(2\pi /\phantom{2\pi 1000}1000\right)}^{p+1}$ is the strength of turbulence, $C_{k}L$ is the vertical integrated turbulence strength at $1$ km scale, L is the PS thickness, a,b represent the two anisotropic elongation scales, ${\kappa }_0=2\pi /L_o$ is the wavenumber with regard to the outer scale $L_o$, p is the spectrum index, $\Gamma \left(·\right)$ is the Gamma function, $K_{\xi }\left(·\right)$ is the modified Bessel function of second kind, G refers to the enhancement factor defined in [13], x and y refer to the orientations along and across the ionospheric penetration point (IPP) track, ${\kappa }_x$, ${\kappa }_y$ represent the 2D spatial wavenumbers with respect to the separations ${\rho }_x$, ${\rho }_y$, respectively. In addition, coefficients of $A_1$, $A_2$, $A_3$ are involved in the incident angle $\theta$, squint angle $\phi$, two anisotropic elongation scales a, b, geomagnetic heading angle ${\delta }_B$, geomagnetic inclination angle ${\theta }_B$ and third rotation angle ${\gamma }_B$; interested readers can refer to Appendix B for more details. In addition, the phase variance of the PS can be learned by integrating the SDF in (9) over all frequencies or calculating the limitation of the ACF in (10) as ${\rho }^{*}\to 0$, and can be denoted as [13]

$$〈\delta {\varphi }^2〉=\Re \left(0\right)=r_e^2{\lambda }^{2}G{C}_{s}Lsec\theta \frac{{\kappa }_0^{-p+1}\Gamma \left[\left(p-1\right)/2\right]}{4\pi \Gamma \left[\left(p+1\right)/2\right]}.$$

(12)

The diffraction process changes the PS-SPE morphology and leads to amplitude scintillation. Thus, the 2D ACFs of amplitude and phase scintillations after diffraction can be respectively given by [15]

$${\Re }_{\alpha }\left(\mathbf{\rho }\right)=\int \int \Phi \left(\mathbf{\kappa }\right)·{sin}^2\left(\frac{{\mathbf{\kappa }}^2{d}_R}{2k}\right)cos\left(\mathbf{\kappa }·\mathbf{\rho }\right)d^2\mathbf{\kappa },$$

(13)

$${\Re }_0\left(\mathbf{\rho }\right)=\int \int \Phi \left(\mathbf{\kappa }\right)·{cos}^2\left(\frac{{\mathbf{\kappa }}^2{d}_R}{2k}\right)cos\left(\mathbf{\kappa }·\mathbf{\rho }\right)d^2\mathbf{\kappa }.$$

(14)

In addition, the corresponding SDFs after the diffraction can be given by [15]

$${\Phi }_{\alpha }\left(\mathbf{\kappa }\right)=\Phi \left(\mathbf{\kappa }\right)·{sin}^2\left(\frac{{\mathbf{\kappa }}^2{d}_R}{2k}\right),$$

(15)

$${\Phi }_0\left(\mathbf{\kappa }\right)=\Phi \left(\mathbf{\kappa }\right)·{cos}^2\left(\frac{{\mathbf{\kappa }}^2{d}_R}{2k}\right).$$

(16)

3. The Scintillation Effects on Spaceborne P-Band Full-Pol SAR Images

The scintillation effects above will be investigated in this section and comprehensively incorporated in the full-pol SAR focused impulse response function (IRF). By using the spectrum mechanism, the above effects introduced by the irregular ionosphere can be discussed and simulated for the spaceborne P-band full-pol SAR azimuth focusing.

3.1. The Modeling of Full-Pol SAR Focused IRF

The full-pol SAR images can be modeled by the convolution between the scattering coefficients of the ground scene and the acquisition of the focused IRF, which can be presented by the consideration of the system noise and written as [41,42,43,44,45]

$$\mathbf{X}={\int }_{\mathit{r}}\left[\sum _{pq}{\mathbf{H}}_{pq}{S}_{pq}\left(\mathit{r}\right)+\mathbf{N}\right]d\mathit{r},pq=hh,hv,vh,vv,$$

(17)

where $\mathbf{X}$ refers to the SAR images of four polarized channels, $d\mathit{r}$ indicates the integration across the whole scene, ${\mathbf{H}}_{pq}$ is the focused IRF for a full-pol mode, $S_{pq}\left(\mathit{r}\right)$ is the scattering term in the $pq$ polarized channel for the scatterer at $\mathit{r}$, and $\mathbf{N}$ is the noise matrix. As a result, ${\mathbf{H}}_{pq}$ is presented as

$${\mathbf{H}}_{pq}=\sum _{n=1}^N\mathbf{R}\tilde{\mathbf{F}}{\mathbf{I}}_{pq}\tilde{\mathbf{F}}\mathbf{T}{\zeta }_{}^2\left({\rho }_{t_n;\mathit{r}}\right)·s_{rc}\left(t_n\right)p^{-1}\left(t_n-t_{n^{\prime }}\right),$$

(18)

$$s_{rc}\left(t_n\right)=\mathrm{sinc}\left\{B_r\left[\tau -\frac{2R_{t_n;\mathit{r}}}{c}\right]\right\}exp\left\{\frac{-j4\pi f_0{R}_{t_n;\mathit{r}}}{c}\right\},$$

(19)

where ${\mathbf{I}}_{pq}$ is the scattering impulse unit matrix, and $\tilde{\mathbf{F}}$ is the polarimetric distortion matrix imposed by the FR effect, which also depends upon the IPP location (the tilde denotation indicates this dependence), $\mathbf{R}$ and $\mathbf{T}$ represent the polarimetric distortion matrixes of the receiving and transmission systems, $\tau$ is the fast time, $t_n$ is the slow time within the synthetic aperture of each pixel, N is the number of real apertures, n is the sampling index, $t_1=t_c-T_a/2$, $t_n=t_1+(n-1)·T_a/(N-1)$, $t_N=t_c+T_a/2$, $T_a$ represents the synthetic aperture duration time, $t_c$ is the time when the beam center crosses the scatterer, $R_{t_n;\mathit{r}}$ indicates the slant range history as to the scatterer locating at $\mathit{r}$, $B_r$ is the system bandwidth, $f_0$ is the carrier frequency, $s_{rc}\left(t_n\right)$ is the azimuth signal after the range compression, and $p^{-1}$ means the azimuth matched filter. We consider the two-way propagation for SAR by the square STF [17], denoted as ${\zeta }_{}^2\left({\rho }_{t_n;\mathit{r}}\right)$ in (18) regarding the IPP location, which depends on both azimuth and scatterer positions.

The relevant polarimetric matrices in (18) can be written as

$${\mathbf{I}}_{hh}=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],{\mathbf{I}}_{hv}=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],{\mathbf{I}}_{vh}=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],{\mathbf{I}}_{vv}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],$$

(20)

$$\tilde{\mathbf{F}}=\left[\begin{array}{cc}cos\delta \Omega \left({\rho }_{t_n;\mathit{r}}\right)& sin\delta \Omega \left({\rho }_{t_n;\mathit{r}}\right)\\ -sin\delta \Omega \left({\rho }_{t_n;\mathit{r}}\right)& cos\delta \Omega \left({\rho }_{t_n;\mathit{r}}\right)\end{array}\right],$$

(21)

$$\mathbf{R}=\left[\begin{array}{cc}1& {\delta }_1\\ {\delta }_2& f_1\end{array}\right],\mathbf{T}=\left[\begin{array}{cc}1& {\delta }_3\\ {\delta }_4& f_2\end{array}\right],\mathbf{N}=\left[\begin{array}{cc}{N}_{hh}& N_{vh}\\ N_{hv}& N_{vv}\end{array}\right],$$

(22)

where ${\delta }_i,i=1,2,3,4$ refer to the four crosstalk terms of polarization channels, and $f_1,f_2$ denote the channel imbalance elements of receiving and transmission systems, and $N_{hh}$, $N_{hv}$, $N_{vh}$, $N_{vv}$ represent the channel noise terms.

3.2. The Scintillation Effects on Azimuth Focusing

The focused IRF in (18) is affected by both STF and FRA. The amplitude and phase scintillations of the square STF will introduce azimuth decorrelation, and thus further result in azimuth-image degradation. In fact, the FRA also fluctuates within the synthetic aperture, but it is generally such a small factor that the azimuth decorrelation and polarimetric distortions imposed by it can be neglected in most cases [40]. Thus, we explore the azimuth defocusing caused by the scintillation amplitude error (SAE) and SPE of the two-way STF, which is shown in Figure 2. The simulation is operated for a spaceborne P-band (600 MHz) system with the designed azimuth resolution of 5 m. A typical severe turbulence intensity in the ionosphere is considered as $C_{k}L={10}^{34}$, $p=3$, $L_o$ = 10 km, Ref. [17], and the anisotropic characteristic will not be involved in the point-target experiment.

Figure 2a,b show the square SAE $exp\left\{2\alpha \left({\rho }_{t_n;\mathit{r}}\right)\right\}$ and the two-way STF-SPE $2\delta {\varphi }_0\left({\rho }_{t_n;\mathit{r}}\right)$ together with the two-way PS-SPE $2\delta \varphi \left({\rho }_{t_n;\mathit{r}}\right)$, respectively. As described in Figure 2b, the STF-SPE has a close uniform tendency with the PS-SPE; the standard deviation (STD) of the two-way STF-SPE is 255.0 deg, and that of the two-way PS-SPE is 257.7 deg. It proves the diffraction procedure will not significantly change the SPE trend-like structure in the PS, which is consistent with the conclusion drawn in [31,32,33]. By employing this two-way STF $exp\left\{2\alpha \left({\rho }_{t_n;\mathit{r}}\right)+j2\delta {\varphi }_0\left({\rho }_{t_n;\mathit{r}}\right)\right\}$ to modulate the azimuth signal, the deteriorated azimuth profile of the focused IRF in (18) is shown as Figure 2c in comparison with the ideal profile. The azimuth-image defocusing mainly exhibits as the sidelobe enhancements (the integrated sidelobe ratio is 6.55 dB) so that it is meaningless to analyze the mainlobe resolvability. In Figure 2d, the azimuth profile affected by the only SAE is well preserved on both mainlobe and sidelobe qualities, whereas the azimuth profile affected by the only STF-SPE is similar to that affected by the STF. Therefore, compared with the SAE effect, the SPE is a predominant factor that leads to the azimuth decorrelation, so that it should be effective to mitigate the scintillation effects by only estimating and correcting the STF-SPE. Note that the one-way TEC variation denoted as $\delta \varphi \left({\rho }_{t_n;\mathit{r}}\right)/\left(r_e{\lambda }_0\right)$ is below $0.4$ TECU, and the FRA must be below 5${}^{\circ }$, which verifies that the polarimetric errors are far less than the considerable extent.

We further consider the impact of the diffraction procedure on the SDF, which is simulated in Figure 3. Figure 3a presents the SDFs of the simulated and theoretical PS-SPEs, which have a good consistency. In comparison with Figure 3b, the diffraction can only change the high-frequency components with a slight degree, which conforms to the function of the ${sin}^2$ operation in (15). It also presents the reason why the STF-SPE is closely consistent with the PS-SPE, as depicted in Figure 2b. By the ${cos}^2$ interaction in (16), the high-frequency components become more notable for the amplitude scintillation; therefore, it describes in Figure 3c that the spatial structures from the first Fresnel cutting scale to the outer scale have more contributions than those for the PS- or STF-SPE. As a whole, the diffraction procedure is greatly driven by the high-frequency components of the PS, both for STF-SPE and SAE. Therefore, it has $\delta \varphi \simeq \delta {\varphi }_0$, and we can directly use the overall PS-SPE estimate derived from the FRA estimate to mitigate the scintillation effects (mainly due to STF-SPE) on full-pol SAR images, while neglecting the effect of the SAE due to its negligible contributions to the azimuth decorrelation:

4. The Principle and Integral Processing Scheme of the SPE Estimation and Compensation

As is discussed in Section 3.2, the polarimetric distortions caused by the FRA that fluctuates within the synthetic aperture are negligible for polarimetric applications. However, these should be measurable based on the FRE [38] and beneficial to further estimate the PS-SPE according to (2) and (3). In this section, the principle of the FRE-based strategy for the SPE estimation will be anatomized, according to which, an integral processing scheme is proposed to estimate and compensate the SPE, and to mitigate the scintillation impacts on P-band PolSAR images.

4.1. Review of FRE

To introduce the FRE, it is generally supposed that the FRA is invariant within the integrated aperture and the FR effect is the only polarimetric distortion source. Thus, it has $\mathbf{M}={\mathbf{F}}_0{\mathbf{SF}}_0$, which can be expanded as

$$\left[\begin{array}{cc}{M}_{hh}& M_{vh}\\ M_{hv}& M_{vv}\end{array}\right]=\left[\begin{array}{cc}cos\Omega & sin\Omega \\ -sin\Omega & cos\Omega \end{array}\right]\left[\begin{array}{cc}{S}_{hh}& S_{vh}\\ S_{hv}& S_{vv}\end{array}\right]\left[\begin{array}{cc}cos\Omega & sin\Omega \\ -sin\Omega & cos\Omega \end{array}\right],$$

(23)

where $\mathbf{S}$ represents the true scattering matrix, and $M_{mn},mn=hh,hv,vh,vv$ denote the four terms of the measured scattering matrix $\mathbf{M}$.

Based on (23) and its linear or circular polarimetric covariance matrix (PCM) [40], various FR estimators were proposed to estimate the FRA from the full-pol SAR image [45,46,47,48,49,50]. Among these proposed estimators, the Bickel and Bates’ ($B\&B$) estimator has been proved to be the most robust estimator [38,40,50]. Hence, this estimator will be adopted to be a typical implement for FRE in this paper. Derived from the PCM of circular bases, the $B\&B$ estimator can be denoted as [45]

$$\hat{\Omega }=\frac{1}{4}arg〈Z_{21}{Z}_{12}^{*}〉,$$

(24)

where $〈·〉$ is the mathematical expectation operation, and the two circularly polarized bases of $Z_{21}$, $Z_{12}$ are given by

$$\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]=\left[\begin{array}{cc}1& j\\ j& 1\end{array}\right]\left[\begin{array}{cc}{M}_{hh}& M_{vh}\\ M_{hv}& M_{vv}\end{array}\right]\left[\begin{array}{cc}1& j\\ j& 1\end{array}\right].$$

(25)

In fact, the performance of FRE is affected by other polarimetric distortions, including the system noise, channel imbalance, and crosstalk [38,40,46,47,48,49,50]. Among these factors, the system noise most significantly influences the FRE performance of the $B\&B$ estimator [38,40], thus selected for further discussions. In the FRE application, to improve the FRE accuracy, the number of looks will be increased for the signal enhancement and the noise depressing, which is equivalent to perform an averaging window [38]. The STD of the FRA estimates is presented as a function of the overall SNR [38]

$$\Delta \Omega =\frac{1}{4\gamma }\sqrt{\frac{1-{\gamma }^2}{2Q}},\gamma =\frac{\mathrm{SNR}}{1+\mathrm{SNR}},$$

(26)

where $Q=Q_1{Q}_2$ indicates the number of looks ($Q_1$, $Q_2$ denote the azimuth and range looks number).

In addition to using the FRA estimate to correct polarimetric distortions, it can also be applied to estimating the background ionosphere by $\overline{\mathrm{TEC}}=\Omega /\sigma$, where $\overline{\mathrm{TEC}}$ is the homogeneous component of the TEC [49]. The accuracy of the retrieved TEC can be given by

$$\Delta TEC=\frac{\sigma \Delta \Omega -\hat{\Omega }\Delta \sigma }{{\sigma }^2}.$$

(27)

It shows that the error source originates from two aspects. One is the residual FRA, and the other is the inaccurate calculation of the TEC-to-FRA coefficient that is mainly derived from the inaccuracy of the projected geomagnetic flux density $Bcos\Theta$. According to (2), the ionospheric phase error can be further presented as $\overline{\varphi }=r_e\lambda ·\widehat{\Omega }/\sigma$. Given a precise $\sigma$, which means the local geomagnetic field is accurately known, it has $\Delta \overline{\varphi }=r_e\lambda ·\Delta \Omega /\sigma$. Because $\sigma$ is close to zero in the equatorial regions, it is infeasible to derive the background TEC or phase error according to the FRA estimate for a small $\left|\sigma \right|$ [38,50]. Therefore, the methods to retrieve the TEC or phase error from the estimated FRA are only suitable for the medium- or high-latitude areas, but not for the equatorial areas.

4.2. An Ideal Condition

Due to the FRA that fluctuates within the synthetic aperture, it is difficult to derive an analytical expression of the measured scattering matrix like (23) based on (17). The direct application of FRE to the SPE estimation is bound to be invalid, and hence demands the proper modification.

According to (18), the scintillation factors of the STF and FRA are dependent on different apertures for a scatterer and spatial-variant across the holistic scene. Therefore, to describe this characteristic, a schematic diagram is shown in Figure 4. It is reasonably supposed for a side-look radar that the scatterers in the whole scene with the same azimuth position have a collective real aperture. As shown in Figure 4, the first to the last real apertures as to a scene are respectively with regard to different covered PS regions with the similar size. Like the real apertures’ tracks, the PS region also moves along the IPP track direction. Note that these covered PS regions as to vicinal real apertures are staggered and overlapped. As to the holistic working time, it is thus corresponding to an overall PS region that covers full apertures for all scatterers in the SAR scene.

As a result, Ref. (17) can be rewritten as the sum of real aperture images ${\mathbf{X}}_1,{\mathbf{X}}_2,\cdots ,{\mathbf{X}}_N$ and given by

$$\mathbf{X}=\sum _{n=1}^N{\mathbf{X}}_n=\sum _{n=1}^N\left\{{\int }_{\mathit{r}}\left[\sum _{pq}{\mathbf{H}}_{pq,n}{S}_{pq}\left(\mathit{r}\right)+{\mathbf{N}}_n\right]d\mathit{r}\right\},$$

(28)

$${\mathbf{H}}_{pq,n}=\mathbf{R}{\mathbf{F}}_n{\mathbf{I}}_{pq}{\mathbf{F}}_n\mathbf{T}·{\zeta }^2\left({\rho }_{n;\mathit{r}}\right)·s_{rc}\left(t_n\right)p^{-1}\left(t_n-t_{n^{\prime }}\right),$$

(29)

in which ${\mathbf{H}}_{pq,n}$ is the full-pol focused IRF of the nth aperture, corresponding to the nth PS region ${\rho }_{n;\mathit{r}}$, shown in Figure 4. The matrix ${\mathbf{F}}_n$ with respect to $\delta \Omega \left({\rho }_{n;\mathit{r}}\right)$ and the STF $\zeta \left({\rho }_{n;\mathit{r}}\right)$ lose their dependence on t, and are spatially distributed across ${\rho }_{n;\mathit{r}}$. Therefore, the measured scattering matrix ${\mathbf{M}}_n$ for the nth real aperture full-pol SAR image can be presented as

$${\mathbf{M}}_n={\zeta }^2\left({\rho }_{n;\mathit{r}}\right)·\mathbf{R}{\mathbf{F}}_n\mathbf{S}{\mathbf{F}}_n\mathbf{T}+{\mathbf{N}}_n$$

(30)

For the ideal condition without the channel noise, imbalance and crosstalk, it can be simplified as ${\mathbf{M}}_n={\zeta }^2\left({\rho }_{n;\mathit{r}}\right)·{\mathbf{F}}_n\mathbf{S}{\mathbf{F}}_n$. Then, it is feasible to perform the FRE in the use of the $B\&B$ estimator for retrieving the FRA distribution $\widehat{\delta \Omega }\left({\rho }_{n;\mathit{r}}\right)$. Taking advantage of the reciprocal overlapping of $\widehat{\delta \Omega }\left({\rho }_{1;\mathit{r}}\right)$, $\widehat{\delta \Omega }\left({\rho }_{2;\mathit{r}}\right)$, ⋯, $\widehat{\delta \Omega }\left({\rho }_{N;\mathit{r}}\right)$, a full-aperture splicing operation is demanded for estimating the overall FRA. On this basis of this, we can obtain the overall PS-SPE and further derive STF by using (4) or (5).

4.3. The Basic Principle

However, the above scheme does not take other interference factors into account, especially the system noise. As to the real aperture image, the lack of azimuth coherent accumulation will cause low SNR and bring about poor FRA estimates. Therefore, the sub-aperture separation technology comes into being as a robust scheme that can well accommodate the existence of the system noise. For one aspect, the coherent accumulation is intensified for sub-aperture images so as to improve the SNR compared with the real aperture images. For another aspect, the SPE or FRA could be closely constant within the sub-aperture, as long as the separated sub-aperture length is suitably selected to maintain a strong ionospheric correlation.

It is assumed that the integrated aperture can be divided into M sub-apertures ($N=\upsilon M$, $\upsilon$ is the number of real apertures in each sub-aperture). The sub-aperture length projected in the PS height is given by

$$L_{sub}\approx \frac{T_a}{M}·\frac{d_1}{d_1+d_2}·V_s$$

(31)

where $V_s$ is the satellite velocity. Then, the strong ionospheric correlation should be confirmed, and thus $L_{sub}$ is confined by an upper limit

$$L_{sub}\le \left\{\underset{{\rho }_x}{arg}\left[\Re \left({\rho }_x,{\rho }_y=0\right)={\beta }_0\right]\right\},$$

(32)

where the correlation degree ${\beta }_0$ (usually larger than 0.707) is set according to different demands for the processing accuracy. Then, the ACF of the PS-SPE in (10) is applied to calculating the ionospheric correlated distance in the along-track direction. If (32) is established, which is similar to (28), (17) can be rewritten as the sum of sub-aperture images ${\mathbf{X}}^{\prime }{}_1$, ${\mathbf{X}}^{\prime }{}_2$, ⋯, ${\mathbf{X}}^{\prime }{}_M$

$$\mathbf{X}=\sum _{m=1}^M{{\mathbf{X}}^{\prime }}_m=\sum _{m=1}^M\left\{{\int }_{\mathit{r}}\left[\sum _{pq}{{\mathbf{H}}^{\prime }}_{pq,m}{S}_{pq}\left(\mathit{r}\right)+{{\mathbf{N}}^{\prime }}_m\right]d\mathit{r}\right\},$$

(33)

$${{\mathbf{H}}^{\prime }}_{pq,m}\approx \mathbf{R}{\mathbf{F}}_{m_c}{\mathbf{I}}_{pq}{\mathbf{F}}_{m_c}\mathbf{T}·{\zeta }^2\left({\rho }_{m_c;\mathit{r}}\right)·\sum _{n=m_1}^{m_2}{s}_{rc}\left(t_n\right)p^{-1}\left(t_n-t_{n^{\prime }}\right),$$

(34)

$$m_1=\left(m-1\right)\upsilon +1,m_2=m\upsilon ,m_c=\mathrm{int}\left\{\left(m_1+m_2\right)/\phantom{\left(m_1+m_2\right)2}2\right\},$$

(35)

where $\mathrm{int}\left\{·\right\}$ takes the nearest integer, $m_c$ is the center of the sub-aperture. As a result, the measured scattering matrix ${\mathbf{M}}^{\prime }{}_m$ for the mth sub-aperture image is presented as

$${\mathbf{M}}^{\prime }{}_m\approx {\zeta }^2\left({\rho }_{m_c;\mathit{r}}\right)·\mathbf{R}{\mathbf{F}}_{m_c}\mathbf{S}{\mathbf{F}}_{m_c}\mathbf{T}+{\mathbf{N}}^{\prime }{}_m,$$

(36)

In this paper, impacts of the channel imbalance and crosstalk will not be involved because the estimation errors imposed by them are far less than those induced by the system noise. Thus, the accuracy of the FRE as to each sub-aperture image can be theoretically given by

$$\Delta \delta {\Omega }_{sub}=\frac{1}{4{\gamma }_{sub}}\sqrt{\frac{1-{\gamma }_{sub}^2}{2Q}},{\gamma }_{sub}\approx \frac{\mathrm{SNR}/M}{1+\mathrm{SNR}/M},$$

(37)

which will be affected by both $\upsilon$ and L ($L_1$, $L_2$). Even though the larger $\upsilon$ and L can improve the FRE accuracy, they should be appropriately selected since $\upsilon$ is confined by the upper limit in (32), and the increase of the looks number is bound to hinder the description of irregularity structures.

To sum up, in existence of the system noise, the FRE should be operated as to each sub-aperture image to retrieve the local FRA estimation $\widehat{\delta \Omega }\left({\rho }_{m_c;\mathit{r}}\right)$. Similar to the scheme of the ideal condition, the full-aperture splicing will be operated by taking advantage of the information redundancy of local estimates in order to obtain the overall FRA and PS-SPE. Nevertheless, the feasibility to further retrieve both STF-SPE and SAE deserves deliberation, which will be analyzed in the next subsection.

4.4. The Integral Processing Scheme for the Overall SPE Estimation and Compensation

Based on the basic principle discussed above, the flow chart of the integral processing scheme is illustrated in Figure 5, which is mainly divided into two parts of the overall SPE estimation and compensation. The detail steps are as follows.

At first, the sub-aperture separation strategy will be adopted to perform the FRE by using the $B\&B$ estimator, in order to obtain the local FRA estimate for each sub-aperture full-pol SAR image. Secondly, the full-aperture splicing is performed to retrieve the overall FRA and PS-SPE. The schematic diagram is illustrated in Figure 6, where ${\widehat{\delta \Omega }}_{1c}$, ${\widehat{\delta \Omega }}_{2c}$, $\cdots$, ${\widehat{\delta \Omega }}_{Mc}$ are the local estimates of the FRA with the same coverage size in the PS plane. They are mutually overlapped to constitute a full-aperture PS region, and $\Delta D\approx L_{sub}$ is the staggered distance of two neighboring sub-regions. The splicing operation as to ${\widehat{\delta \Omega }}_{1c}$, ${\widehat{\delta \Omega }}_{2c}$, taken for example, is given by an discrete expression, which is presented as

$${\widehat{\delta \Omega }}_{12}={\widehat{\delta \Omega }}_{1c}\left(:,1:\Delta d\right)\oplus \left\{{\beta }_1{\widehat{\delta \Omega }}_{1c}\left(:,\Delta d+1:\mathrm{end}\right)+{\beta }_2{\widehat{\delta \Omega }}_{2c}\left(:,1:\mathrm{end}-\Delta d\right)\right\}\oplus {\widehat{\delta \Omega }}_{2c}\left(:,\mathrm{end}-\Delta d+1:\mathrm{end}\right),$$

(38)

where ‘:’ represents the indexing range, ‘$\mathrm{end}$’ is the sampled number of one of the matrix dimension, ⊕ indicates the ‘end-to-end’ splicing along the azimuth orientation, $\Delta d$ is the sampled number with regard to $\Delta D$, and ${\beta }_1$, ${\beta }_2$ are the weighing factors for the overlapped region in order to give a buffer to avoid discontinuity around the splice place, and to confirm the smooth splicing (they have a tendency depicted as Figure 7 for the splicing continuity). In reality, the PS altitude is unknown and therefore $\Delta D$ cannot be straightly acquired by (31). Instead, $\Delta d$ can be estimated by maximizing a correlation function and presented as

$$\Delta \widehat{d}=\frac{1}{N_r}{\displaystyle \sum _{i=1}^{N_r}}arg\underset{d}{max}\left\{\frac{{\displaystyle \sum _{j=1}^{\mathrm{end}-d}}{\widehat{\delta \Omega }}_{1c}\left(i,j+d\right)·{\widehat{\delta \Omega }}_{2c}\left(i,j\right)}{\sqrt{{\displaystyle \sum _{j=1}^{\mathrm{end}-d}}{\left[{\widehat{\delta \Omega }}_{1c}\left(i,j+d\right)\right]}^2}·\sqrt{{\displaystyle \sum _{j=1}^{\mathrm{end}-d}}{\left[{\widehat{\delta \Omega }}_{2c}\left(i,j\right)\right]}^2}}\right\},$$

(39)

where $N_r$ is the across-track samplings of the estimated FRA. Followed by (38), the splicing should be further operated as to ${\widehat{\delta \Omega }}_{12}$, ${\widehat{\delta \Omega }}_{3c}$ for obtaining ${\widehat{\delta \Omega }}_{123}$. In addition, we can then retrieve the overall FRA distribution, denoted as $\widehat{\delta \Omega }\left({\rho }_{n;\mathit{r}}\right)$, and, based on which, the overall PS-SPE $\widehat{\delta \varphi }\left({\rho }_{n;\mathit{r}}\right)$ can be estimated by using (2) and (3).

The last is to mitigate the scintillation impacts by the overall SPE compensation. The strategy of the real aperture separation will be employed to ensure the phase continuity of the image. The compensation should be operated according to

$${\mathbf{X}}_{\mathrm{com}}=\sum _{n=1}^N{\widehat{\mathbf{F}}}_n^{-1}{\mathbf{X}}_n{\widehat{\mathbf{F}}}_n^{-1}·exp\left\{-2j\widehat{\delta \varphi }\left({\rho }_{n;\mathit{r}}\right)\right\}.$$

(40)

5. Simulation Validation

5.1. Simulation of the Affected P-Band Full-Pol SAR Image

Due to the lack of a spaceborne P-band full-pol SAR system at present, the airborne P-band full-pol SAR images that were observed in the center of ${35.5}^{\circ }$N, ${110.5}^{\circ }$E, in December 2015, would be used to perform the simulation validation for our proposed methodology. The involved system and ionospheric parameters are summarized in

Table 1. System parameters of a spaceborne P-band full-Pol SAR system.

Parameters	Specification	Unit
Radar altitude	700	km
Carrier frequency	600	MHz
System bandwidth	56	MHz
Pulse sampling frequency	60	MHz
Doppler bandwidth	1223.72	Hz
Pulse repetition frequency	1740	Hz
Azimuth sample spacing	3.9267	m
Slant range sample spacing	2.5	m
Inclination angle at ground	30	deg
Radar Squint angle at ground	90	deg

and

Table 2. Ionospheric Parameters.

Parameters	Specification	Unit
Ionospheric height	350	km
Outer scale $L_o$	10	km
Spectrum index p	3	-
Integrated turbulence strength $C_{k}L$	$1\times {10}^{34}$	-
Anisotropic scale ratio $a:b$	$5:1$	-
Geographic heading angle	12	deg
Geomagnetic declination angle	$1.70$	deg
Geomagnetic heading angle ${\delta }_B$	$10.30$	deg
Geomagnetic inclination angle ${\varphi }_B$	$49.99$	deg
TEC-to-FRA coefficient $\sigma$	$1.12$	deg/TECU

, respectively. Here, we take a side-look P-band system with a designed azimuth resolution (≈5 m) into account. The ionospheric altitude is the barycenter altitude of the electron density profile because the electrons are primarily concentrated in a narrow layer with an altitude range from 250 to 400 km [12]. Because the method proposed in the paper is not very sensitive to the exact ionospheric height, the empirical height of 350 km is chosen. A highly turbulent case $C_{k}L={10}^{34}$, $p=3$, $L_o$ = 10 km with an intermediate configuration $a:b=5:1$ between the rod-like and sheet-like configurations discussed in Appendix B is simulated for the mid-latitude areas [17]. A sun-synchronous orbit is employed for the simulated system, and hence the geographic heading angle is set as 12 deg, which is similar to the Phased Array Type L-band Synthetic Aperture Radar (PALSAR) [17]. According to the local time (December 2015) and position ($35.5^{\circ }$N, ${110.5}^{\circ }$E), and the geomagnetic declination and inclination angles, the TEC-to-FRA coefficient can be obtained by using the IGRF.

The PS-SPE and STF of the ionospheric scintillation is then generated according to the detailed steps described in [17]. The simulation of scintillation-influenced P-band full-pol SAR images is according to the real aperture principle presented in (28). As shown in Figure 8, it presents a comparison of the original and influenced images by taking the HH polarized channel for example. It shows that the image resolvability in azimuth is greatly distorted by the highly turbulent ionosphere, and the right side in Figure 8 is highlighted for a comparison of the local areas with many strong scatterers.

5.2. Realization of the Proposed Methodology

By using the simulated and influenced P-band full-pol SAR images shown in Figure 8b, the proposed methodology is realized, and the schematic diagram of the overall SPE estimation is shown in Figure 9. The original simulated and injected FRA or PS-SPE distribution is denoted (bottom) for a comparison. Meanwhile, $\mathrm{SNR}=20$ dB, $M=16$, and $Q={64}^2$ is considered, for example, in the realization of the integral processing scheme. As described in Figure 9, the FRA estimate for each sub-aperture full-pol SAR image is firstly retrieved by using the $B\&B$ estimator denoted in (24). Taking the first and second estimates, for example, the splicing operation is performed by means of (38). And for another estimate, it can also be observed that the similarity of the overlapped region is very high so that $\Delta d$ is accurately estimated. In addition, for the following splicing operations, the continuity of the overall FRA estimate is well preserved after the full-aperture splicing. It finally illustrates a perfect match between the simulated and estimated FRA distributions. Because of the smoothing impact by the averaging window for FRE, it is impracticable to further calculate the STF-SPE and SAE, which has been discussed in Section 4.4. Moreover, the STD of the simulated PS-SPE is $163.2$ deg; and the STD of the difference between the simulated and estimated PS-SPE maps is $23.1$ deg, which verifies the effectiveness of the overall SPE estimation.

The estimated overall PS-SPE distribution is then adopted to mitigate the ionospheric scintillation impacts in accordance with [43], and the processing results are described in Figure 10. It is indicated in Figure 10a that the azimuth resolvability of SAR image is greatly improved; for example, the strong scatterers in the red rectangle are more distinguishable, in spite of a slightly worse performance than the original image in Figure 8a because of the slightly residual SPE. In order to further present a quantitative description for this correction result, we introduce the correlation coefficient between two images, similar to the zero-baseline interferometric coherence. As shown in Figure 11, the mean amplitude value of the correlation coefficient is 0.3276 before the correction (left), while it is 0.6285 after the correction (right), which validates the effectiveness of the proposed methodology.

5.3. Comparison with an Existing Method

According to the principle of the proposed methodology, it does not need the prior knowledge of the ionospheric height. By using the same simulated data in Figure 8b, an existing method proposed in [38,39] is reproduced here for comparison, which estimates the FRA and SPE by projecting the full-pol SAR images onto the known or estimated ionospheric height. As shown in Figure 12, the results of the SPE estimation are achieved by using different values of the ionospheric height.

As listed in

Table 3. Comparison with an Existed Method.

	Proposed Method		Existed Method
Ionospheric height/km	/	350	300	250
STD difference of the PS-SPE/deg	23.1	16.2	34.1	59.9

, the proposed method obtained a not bad estimation of 23.1 deg in the lack of prior knowledge of the ionospheric height. When an accurate value (350 km) of the ionospheric height is adopted, the first result of existing method presents a perfect match with the original PS-SPE map. The STD of the difference between the simulated and estimated PS-SPE maps is 16.2 deg, which validates a more effective performance than the result of the proposed methodology shown in Figure 9.

Nevertheless, when the inaccurate values of the ionospheric height are adopted with the errors of 50 km and 100 km, the STDs of the difference between the simulated and estimated PS-SPE maps are 34.1 deg and 59.9 deg, respectively. It indicates that the performance of the existing method is degraded with the increasing error of the adopted ionospheric height, and this conclusion has also been acknowledged in [38,39]. Hence, compared with the method proposed in [38,39], the methodology proposed in this paper shows a more robust performance when the ionospheric height is very rarely known or estimated.

5.4. Performance Analysis

Given a fixed scintillation condition, the estimation and correction performance of our proposed methodology strongly depend on the SNR, the looks number (windowing size), and the sub-apertures number. Then, the performance is presented as

Table 4. The STD of the difference between the simulated and estimated PS-SPEs (Unit: deg).

Sub-Apertures	Looks	$\mathbf{SNR}=30$ dB	$\mathbf{SNR}=25$ dB	$\mathbf{SNR}=20$ dB	$\mathbf{SNR}=15$ dB	$\mathbf{SNR}=10$ dB
$M=4$	$Q={32}^2$	38.0615	39.5625	43.8549	57.0672	89.3160
	$Q={64}^2$	37.9445	38.3507	39.3897	41.1356	50.9886
	$Q={128}^2$	39.6925	39.8376	40.0809	39.6572	42.0098
	$Q={256}^2$	45.4937	45.4930	45.7315	45.4746	45.8095
$M=8$	$Q={32}^2$	23.6533	27.1690	36.4672	60.0620	109.9155
	$Q={64}^2$	23.0196	23.9478	26.5459	32.5227	50.7443
	$Q={128}^2$	26.1762	26.4209	27.4353	28.8149	33.1549
	$Q={256}^2$	37.9136	37.9606	38.0416	38.9005	39.1953
$M=16$	$Q={32}^2$	18.3778	25.3219	40.0312	73.5612	141.4067
	$Q={64}^2$	16.2273	18.1108	23.1145	35.1844	63.0175
	$Q={128}^2$	21.6689	21.9303	23.3200	25.6889	34.1432
	$Q={256}^2$	37.3379	37.4811	37.6860	38.3050	38.6001
$M=32$	$Q={32}^2$	19.4000	30.0065	50.4975	92.4399	182.8159
	$Q={64}^2$	15.0709	18.6617	27.0715	44.7555	85.1934
	$Q={128}^2$	21.1801	21.7107	23.6001	28.0775	42.9435
	$Q={256}^2$	38.0003	38.3761	38.4514	39.1747	40.4093

and the four plots in Figure 13. As shown in Table 4, the STD of the difference between the simulated and estimated PS-SPEs is presented as to different SNR levels (10∼30 dB), looks number (Q = 32²∼256²), and sub-aperture number (M = 4∼32). Each red-marked item refers to the best processing result for a typical SNR level. In attempt to further evaluate the correction performance, the correlation coefficient between the original and compensated images is calculated by the mean and STD of its amplitude (the mean amplitude of the correlation coefficient before the compensation is about 0.21). As shown in Figure 13, the x-axis is against the SNR level, and the four colors in each plot denote different looks number, and the four subplots indicate different sub-aperture numbers.

The information from Table 4 and Figure 13 is summarized as: (1) the correction performance is basically consistent with the estimation performance. (2) The estimation and correction performance generally improves with the increased SNR level. (3) A smaller M (larger sub-aperture size) or larger Q indicates less dependence of the estimation and correction performance upon the SNR level. (4) The selections of M and Q should be suitable to optimize the method performance in different SNR levels. For instance, in case of the best noise status ($\mathrm{SNR}=30$ dB), $M=32$, $Q={64}^2$ is the optimal processing choice; in case of the worst noise status ($\mathrm{SNR}=10$ dB), $M=8$, $Q={128}^2$ is the optimal processing choice.

6. Conclusions

In this paper, we investigate the approach to the ionospheric scintillation compensation in future spaceborne P-band full-pol SAR systems. It has been firstly confirmed that the SPE is a far more dominant factor that results in the azimuth decorrelation, compared with the SAE. In addition, the PS-SPE proves to be approximate to the STF-SPE, so that it is practicable to merely estimate and compensate the PS-SPE, while neglect distortions caused by the diffraction. Thus, an integral processing scheme is proposed to estimate and compensate the overall PS-SPE by taking advantage of the FRA’s high sensitivity to polarimetric distortions. First of all, the sub-aperture separation is employed to address the aperture-dependent and spatial-variant SPE and estimate the FRA for each sub-aperture image, which has been theoretically derived in the context. Secondly, the full-aperture splicing is performed for retrieving the overall FRA and PS-SPE, which is finally further applied to the scintillation mitigation. Experiments on the simulated data validate the effectiveness of the proposed methodology, and its performance is dependent on the SNR level, looks number, and sub-apertures number. Last but not the least, compared with the existing method, it presents more robustness despite the lack of the accurate knowledge of the ionospheric height.