Compensation of Background Ionospheric Effect on L-Band Geosynchronous SAR with Fully Polarimetric Data

Abstract

The L-band geosynchronous synthetic aperture radar (GEO-SAR) has been widely praised for its advantages of short revisit time, wide coverage and stable backscattering information acquisition. However, due to the ultra-long integrated time, the echo will be affected by the time-variant background ionosphere, leading in particular to defocusing in the azimuth direction. Existing compensation methods suitable for low Earth orbit SAR (LEO-SAR) are based on the SAR image or the semi-focused image at the ionospheric phase screen, assuming that the ionosphere is time-frozen for a short integrated period; thus, accurate reconstruction of the time-variant characteristics for the ionosphere in GEO-SAR cannot be achieved. In this paper, a compensation method of background ionospheric effects on L-band GEO-SAR with fully polarimetric data is proposed. Considering the continuous variation of the ionosphere within the synthetic aperture, a decompression processing is proposed to reconstruct the echo by recovering the temporal sampling according to the imaging geometry. By virtue of the Faraday rotation angle, the time-variant total electron content (TEC) is accurately estimated with the reconstructed echo. Based on the established error model, the ionospheric effects are well compensated with the estimated TEC. Simulations with the real SAR data from ALOS-2 and the measured time-variant TEC from USTEC validate the effectiveness and performance of the proposed method. The impacts from thermal noise and polarimetric calibration error are also quantitatively analyzed. From this, the error thresholds are given to guarantee compensation accuracy, namely 18.96 dB for SNR, −15.63 dB for crosstalk and −1.02 dB to 0.31 dB for the amplitude of the channel imbalance, and the argument of the channel imbalance is suggested to be maintained as close to zero as possible.

1. Introduction

L-band fully polarimetric (FP) synthetic aperture radar (SAR) has superior advantages in penetration and temporal coherence, which has been widely used in geological surveys and biomass measurements, such as the Phased Array-type L-band SAR (PALSAR) of Advanced Land Observation Satellite (ALOS) series [1,2,3]. Geosynchronous synthetic aperture radar (GEO-SAR) could achieve much shorter revisit time (even continuous observation for a given area) and greatly wider coverage (thousands of kilometers) compared to the traditional low Earth orbit SAR (LEO-SAR) systems, thereby providing much richer information [4]. To date, several assumed L-band GEO-SAR missions have been proven to retrieve relatively stable backscattering information from the target and plan to realize measurements of water data, snow mass monitoring and mapping of terrain deformation [5,6,7,8].

Meanwhile, the L-band echo is known to be more sensitive to the ionosphere during propagation, which will lead to serious distortions for radar images [9,10]. According to the variation and distribution of the ionosphere, two kinds of typical structures are recognized, i.e., the background ionosphere and irregularity [11]. The former represents the large-scale and slow-variant structure of electrons which is always regarded as time-frozen for LEO-SAR system in a short observation. In addition, the fluctuant part known as irregularity could introduce ionospheric scintillation to the signal, but it is time- and region-restricted [12], which can be partly avoided by orbit design, such as the dawn–dusk orbit of the BIOMASS mission [13]. Thus, from the view of imaging, ionospheric effects on the LEO-SAR signal are mainly about group delay and range dispersion due to the background ionosphere [11]. The corresponding compensation methods, such as maximum contrast and multi-look autofocus, have also been proposed to solve these problems [14,15].

However, for the GEO-SAR system, the ionosphere cannot be treated as time-frozen for more than 1 min of integrated time and larger than 1000 km of swath [16]. Therefore, defocusing and positioning errors occur due to the phase advance and time delay related with the time-variant total electron content (TEC), especially for the azimuth signal [17,18]. To solve this problem, there are two main kinds of ways for compensation at present. The first kind realizes ionospheric compensation with TEC inversion through difference frequency. Typically, Hobbs et al. proposed a TEC measuring method based on dual-frequency signals, which can be used to compensate for the ionospheric effects [19]. However, the bandwidth requirement for this method is quite high (to improve compensation accuracy) [19]. The other kind of compensation uses autofocus processing, just as Long et al. have systematically researched the compensation of ionospheric effects for the GEO-SAR and put forward a compensation method based on phase grade autofocus (PGA) [20]. However, PGA relies on the strong scattering point in the image, which fails for the homogeneous scene [21,22]. Guarnieri et al. prove the effectiveness of PGA when dealing with atmospheric phase distortion but specifically for wet tropospheric interference [23]. Furthermore, Li et al. use contrast optimization autofocus (COA) to compensate for the time-variant ionospheric effect on GEO-SAR focusing, but the requirement of a strong target in the image is still indispensable [24]. Hu and Ding have applied autofocus processing based on minimum entropy to compensate for the ionospheric effects on GEO-SAR imaging, but Hu mainly concentrates on scintillation compensation [25], and Ding’s method considers the different types of imaging errors as a whole including the ionosphere [26]. In addition, Zhang et al. rely on the optimization processing of coherent character to solve background ionospheric phase distortion [27]. As the coherent character of the target is used during processing, the applicable conditions are quite strict, where the block processing and the isolate-prominent point target are necessary. Therefore, there are still some limitations to the existing methods to well compensate for the time-variant ionospheric effects on GEO-SAR systems.

In this article, a compensation method of background ionospheric effects on GEO-SAR with fully polarimetric data is proposed. The background ionospheric effects on the GEO-SAR signal are discussed in Section 2. Based on the transmission characteristic of electromagnetic waves in the ionosphere, the error model of time delay and phase advance is established. Considering the ultra-long integrated time of GEO-SAR, the specific ionospheric effect with time-variant TEC is specifically analyzed for both azimuth and range signals. In Section 3, the compensation method of background ionospheric effects based on FR estimation is proposed. In order to separate the time-variant TEC within the synthetic aperture, decompression processing is conducted considering the imaging geometry. Then, the corresponding phase error at each pulse is compensated for through time-variant FR angle estimation. Simulations and analyses with real SAR data from ALOS-2 and time-variant TEC from USTEC are shown in Section 4. The performance and problems of the proposed method and previous related studies are discussed in detail in Section 5. Our follow-up research is also presented in this section. Finally, conclusions are drawn in Section 6.

2. Background Ionospheric Effects on the GEO-SAR Signal

2.1. Properties of Electromagnetic Wave Transmission in the Ionosphere

As a typical kind of plasma, the group refractive index of the electromagnetic wave in the ionosphere can be expressed by the approximation of the Appleton–Hartree expression as [11]

$$n_g=\sqrt{1-\frac{f_N^2}{f^2}}\approx 1+\frac{f_N^2}{2f^2}$$

(1)

where f is the frequency of the electromagnetic waves, and f_N is the plasma frequency as [11]

$$f_N=\sqrt{\frac{N_e{e}^2}{4{\pi }^2{\epsilon }_0{m}_e}}\approx \sqrt{80.56N_e}$$

(2)

where e is the charge of an electron, m_e is the electron mass, ε₀ is the permittivity of vacuum, and N_e is the electron density.

Based on the group refractive index in (1), the time delay caused by the ionosphere compared with a vacuum is given by [28]

$$\mathsf{\Delta }t\left(f\right)={\displaystyle \underset{path}{\int }\left(n_g-1\right)ds=\frac{40.28}{c{f}^2}}{\displaystyle \underset{path}{\int }{N}_{e}ds}=\frac{K\cdot \mathrm{TEC}}{c{f}^2}$$

(3)

where c is the speed of light, K is a constant of 40.28 m³/s², and TEC is the total electron content along the propagation path with the unit of TECU (1 TECU = 10¹⁶ m⁻²).

Considering the time delay at the carrier frequency, it could represent the delay of the signal envelope, which is known as the group delay (marked with the red arrow line in Figure 1). Since the time delay Δt(f) is related to the frequency f, different frequency components of the chirp pulse are delayed differently, resulting in a change in the chirp rate, which is the root of the defocusing caused by the ionosphere. Accordingly, the affected chirp rate b′ is derived as (4) according to the relationship between the bandwidth and the changed pulse length.

$$b^{\prime }=\frac{Bw}{T_p+(\mathsf{\Delta }{t}_{high}-\mathsf{\Delta }{t}_{low})}=\frac{Bw}{T_p+\frac{2K\cdot \mathrm{TEC}}{c\left(f_c+\frac{Bw}{2}\right)}-\frac{2K\cdot \mathrm{TEC}}{c\left(f_c-\frac{Bw}{2}\right)}}$$

(4)

where Bw is the bandwidth, T_p is the pulse length, f_c is the carrier frequency, and Δt_high and Δt_low are the time delays at the highest and lowest frequencies within the bandwidth, respectively.

In addition to the time delay, the phase characteristic of the radar signal is also changed during the two-way propagation, which is known as the phase advance. It can be calculated according to the time delay Δt(f) as [29]

$${\varphi }_{iono}\left(f\right)=2\cdot 2\pi f\cdot \mathsf{\Delta }t\left(f\right)=\frac{4\pi K\cdot \mathrm{TEC}}{cf}$$

(5)

From the view of frequency domain, the Taylor expansion of the additional phase in (5) is derived as (neglecting the higher-order terms)

$${\varphi }_{iono}\left(f\right)=\frac{4\pi K\cdot \mathrm{TEC}}{c{f}_c}-\frac{4\pi K\cdot \mathrm{TEC}}{c{f}_c^2}f+\frac{4\pi K\cdot \mathrm{TEC}}{c{f}_c^3}{f}^2$$

(6)

The first-order term in (6) represents the time delay corresponding to (3), and the second-order term will cause a mismatch during the matched filtering, leading to the defocusing of the chirp signal [15]. Thus, the impacts of ionospheric effects on the electromagnetic wave, including the time delay and the phase advance, could be unified as the additional phase error in (5), which conforms to the properties of the ionosphere as the dispersion medium.

2.2. Error Model of Time-Variant Background Ionospheric Effects on the GEO-SAR Signal

The ideal-point echo signal of the GEO-SAR system is traditionally expressed as [10]

$$S(\tau ,t)=W_r(\tau )\cdot W_a(t)\cdot \exp \left\{-j\frac{4\pi R(t)}{\lambda }\right\}\cdot \exp \left\{j\pi b{\left(\tau -\frac{2R(t)}{c}\right)}^2\right\}$$

(7)

where W_r(*) and W_a(*) are the envelope functions of the range and azimuth, respectively, t and τ are the azimuth and range time, respectively, b is the chirp rate of range signal, λ is the wavelength, and R(t) is the slant range.

After the two-way propagation through the ionosphere, the affected echo is derived as below, according to the error model shown in (5). Here, TEC(t) is related to the azimuth time t, which reflects the time-variant characteristic of the ionosphere.

$$S_{iono}(f_r,t)=W_r(f_r)\cdot W_a(t)\cdot \exp \left(-j\frac{4\pi R(t)}{\lambda }\right)\cdot \exp \left(-j\pi \frac{f_r^2}{b}\right)\cdot \exp \left(j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c\left(f_r+f_c\right)}\right)$$

(8)

where f_r is the frequency of the range signal.

In order to facilitate ascertaining the impacts of the background ionosphere on GEO-SAR imaging more clearly, the Taylor expansion of the ionospheric phase in (8) is derived as (neglecting the higher-order terms)

$$\begin{array}{l}{S}_{iono}(f_r,t)\approx W_r(f_r)\cdot W_a(t)\cdot \exp \left(-j\frac{4\pi R(t)}{\lambda }\right)\cdot \exp \left(-j\pi \frac{f_r^2}{b}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdot \exp \left(j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c{f}_c}-j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c{f}_c^2}{f}_r+j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c{f}_c^3}{f}_r^2\right)\\ \hspace{1em}\hspace{1em}\hspace{1em} =W_r(f_r)\cdot W_a(t)\cdot \exp \left(-j\frac{4\pi R(t)}{\lambda }\right)\cdot \exp \left(-j\pi \frac{f_r^2}{b}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdot \exp \left(j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c{f}_c}\right)\cdot \exp \left(-j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c{f}_c^2}{f}_r\right)\cdot \exp \left(j\frac{4\pi K\cdot \mathrm{TEC}\left(t\right)}{c{f}_c^3}{f}_r^2\right)\end{array}$$

(9)

For the range signal, the pulse width of the existing SAR system is always tens of microseconds, during which the change in TEC could be neglected. Thus, the first-order phase $-4\pi K{f}_r\cdot \mathrm{TEC}\left(t\right)/c{f}_c^2$ in (9), corresponding to the time delay $2K\cdot \mathrm{TEC}\left(t\right)/c{f}_c^2$ at each sampling time, will cause the group delay for the range signal (see Figure 1); the second-order phase $4\pi K{f}_r^2\cdot \mathrm{TEC}\left(t\right)/c{f}_c^3$ brings the error to the chirp rate leading to the defocusing of the range signal, which is called dispersion. That is, the background ionosphere TEC(t) at each azimuth sampling time introduces the frequency-dependent phase to the range signal. According to the properties of Fourier transform, the corresponding offset and defocusing should happen after matched filtering.

Usually, the range quadratic phase error (QPE) is used to evaluate the impact on focusing for range signal, which is derived as (10) according to (9).

$${\varphi }_{QPE}=\frac{\pi K\cdot \mathrm{TEC}\left(t\right)\cdot B{w}^2}{c{f}_c^3}$$

(10)

Considering the allowable uncompensated QPE of π/4 [30], the corresponding TEC(t) should be at least 16 TECU for 1 m resolution and 145 TECU for 3 m resolution, so that the range signal will be affected. According to the statistical analysis and release data of the ionosphere, the vertical TEC is generally less than 120 TECU [31,32]. Thus, the impact of the background ionosphere on range signal imaging is quite weak given the general state of the ionosphere for L-band SAR with a resolution more than 3 m; meanwhile, for the L-band systems with a higher resolution (such as LuTan-1 [33]), the effects on range signal should be compensated.

On the other hand, the azimuth signal is contaminated with the additional time-related phase $4\pi K\cdot \mathrm{TEC}\left(t\right)/c{f}_c$ at each sampling time according to (9). Comparing with the range signal, the change in the TEC(t) is quite large within the integrated time of azimuth signal especially for GEO-SAR. With the change rate of TEC(t), different orders of the phase error will lead to distortions of the azimuth signal, among which the first-order term causes the overall offset, and the second and higher orders cause defocusing [16]. The related error analysis and numerical simulations have been discussed by many scholars, particularly in [16,17,24].

3. Compensation of Background Ionospheric Effects Based on FR Estimation

3.1. Faraday Rotation Estimation with FP SAR Data

In magnetized plasma, the polarization plane of the electromagnetic wave is rotated by Ω for the linearly polarized wave when propagating through the ionosphere, which is known as a Faraday rotation (FR) [34]. The rotation angle is

$$\mathsf{\Omega }=\frac{K_{\mathsf{\Omega }}}{f^2}\cdot B\cos \psi \cdot \mathrm{TEC}$$

(11)

where K_Ω is a constant of 2.365 × 10⁴ A⋅m²/kg, B is the magnetic flux density, and ψ is the angle the wave makes with the Earth’s magnetic field.

Traditionally, the FR effect on the SAR signal depends on the polarization mode. For the linearly FP SAR system specifically, the signal of each polarimetric channel will be contaminated by the others according to (12) [35].

$$\left(\begin{array}{cc}{M}_{HH}& M_{HV}\\ M_{VH}& M_{VV}\end{array}\right)=\left(\begin{array}{cc}\cos \mathsf{\Omega }& \sin \mathsf{\Omega }\\ -\sin \mathsf{\Omega }& \cos \mathsf{\Omega }\end{array}\right)\left(\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right)\left(\begin{array}{cc}\cos \mathsf{\Omega }& \sin \mathsf{\Omega }\\ -\sin \mathsf{\Omega }& \cos \mathsf{\Omega }\end{array}\right)$$

(12)

where S_** and M_** are the true and measured scattering components of each polarimetric channel, respectively (the subscript “_**” represents HH, HV, VH or VV).

For now, several FR estimation methods have been proposed [36,37,38], among which Bickel and Bates’s method shows better performance [39]. The core idea of this method is to estimate the FR with the difference of cross-polarized channels in circular-polarized bases Z₁₂ and Z₂₁ as

$$\{\begin{array}{c}{Z}_{12}=M_{HV}-M_{VH}+j(M_{HH}+M_{VV})\\ Z_{21}=M_{VH}-M_{HV}+j(M_{HH}+M_{VV})\end{array}$$

(13)

Then, the FR angle is estimated as

$$\widehat{\mathsf{\Omega }}=\frac{1}{4}\arg \langle Z_{21}{Z}_{12}^{*}\rangle$$

(14)

where <*> represents the smoothing operation to reduce the influence from the thermal noise, and arg(*) gives the angle of its complex-valued argument.

3.2. Compensation Method

According to the analysis in Section 2.2, the state of the background ionosphere between adjacent radar pulses changes with time due to the quite low pulse repetition frequency (PRF) and the ultra-long integrated time of GEO-SAR. Specifically, the time-variant background ionosphere imports the errors at each pulse sampling time with TEC(t) (shown in Figure 2).

Considering the time variation of TEC(t), the Faraday rotation angle caused by the background ionosphere changes with time, or more exactly, azimuth sampling time t (see (15)).

$$\mathsf{\Omega }\left(t\right)=\frac{K_{\mathsf{\Omega }}}{f^2}\cdot B\cos \psi \cdot \mathrm{TEC}\left(t\right)$$

(15)

Combining (8) and (15), which are both related with TEC(t), if the TEC(t) is estimated accurately through the Faraday rotation angle, the ionospheric phase error $4\pi K\cdot \mathrm{TEC}\left(t\right)/c{f}_c$ could be well compensated. Inspired by this, the fully polarimetric GEO-SAR data are firstly used to estimate the time-variant Faraday rotation angle at each azimuth sampling time. Then, the time-variant ionospheric phase can be obtained from the estimated TEC(t) according to (5), which is derived from the FR estimation in (14). The estimated time-variant ionospheric phase is written as (16). Finally, the imaging distortion is compensated with the azimuth echo at range frequency domain based on the model in (8).

$${\widehat{\varphi }}_{iono}\left(t\right)=\frac{4\pi K\cdot \mathrm{T}\hat{\mathrm{E}}\mathrm{C}\left(t\right)}{cf}=\frac{4\pi Kf}{c{K}_{\mathsf{\Omega }}B\cos \psi }\cdot \widehat{\mathsf{\Omega }}\left(t\right)$$

(16)

However, the synthesized affected SAR image cannot be used to estimate the FR directly. This is because the FR angle caused by the time-variant TEC is also compressed during focusing, which loses the information of temporal sampling. Therefore, the image should be decompressed to reconstruct the echo in azimuth firstly, so as to recover the physical distribution of the time-variant ionospheric errors.

Different from the traditional semi-focused processing of scintillation compensation for LEO-SAR [40], the GEO-SAR decompression should recover the original echo instead of the approximate echo at the altitude of the phase screen, because the phase screen model only fits for the “frozen in” ionosphere with “spatial and not temporal variations” [15]. The integrated information at each time point within the synthetic aperture is reset to its original acquisition time for the specific target, which is equivalent to a kind of reverse imaging processing. Based on the mature research studies about GEO-SAR imaging [41,42,43], this reverse imaging processing could be realized from the time or frequency domains [44].

Taking the reverse imaging processing at the frequency domain, for example, the specific processing of decompression is to obtain the frequency spectrum in azimuth with Fourier transform firstly as

$$S\left(\tau ,f_a\right)=FFT\left(S\left(\tau ,t\right)\right)$$

(17)

Then, the decompressed frequency spectrum of the azimuth echo is obtained with multiplying by the decompressed phase $\varphi (f_a)$ as shown below, which refers to the azimuth compression in [45].

$$S_{echo}\left(\tau ,f_a\right)=S\left(\tau ,f_a\right)\cdot \exp \left(j\varphi (f_a)\right)$$

(18)

The azimuth echo is reconstructed with inverse Fourier transform of (18) as

$$S_{echo}\left(\tau ,t\right)=IFFT\left(S_{echo}\left(\tau ,f_a\right)\right)$$

(19)

Based on the reconstructed azimuth echo, the time-variant FR angle is estimated with each sampling signal according to (13) and (14) as

$$\widehat{\mathsf{\Omega }}\left(t\right)=\frac{1}{4}\arg \langle Z{\left(t\right)}_{21}Z{\left(t\right)}_{12}^{*}\rangle$$

(20)

where $Z{\left(t\right)}_{12}$ and $Z{\left(t\right)}_{21}$ represents the cross-polarized signals with (13).

The corresponding time-variant ionospheric phase is obtained from (16) as

$${\widehat{\varphi }}_{iono}\left(f_r,t\right)=\frac{4\pi K{f}_r}{c{K}_{\mathsf{\Omega }}B\cos \psi }\cdot \widehat{\mathsf{\Omega }}\left(t\right)$$

(21)

Because the ionospheric phase is frequency-dependent, the compensation should be done in the frequency domain of range signal as

$$S_{corr}\left(f_r,t\right)=FFT\left(S_{echo}\left(\tau ,t\right)\right)\cdot \exp \left(j{\widehat{\varphi }}_{iono}\left(f_r,t\right)\right)$$

(22)

Then, the compensated echo is derived with inverse Fourier transform as

$$S_{corr}\left(\tau ,t\right)=IFFT\left(S_{corr}\left(f_r,t\right)\right)$$

(23)

Finally, the re-compressed processing is conducted to obtain the compensated image by removing the decompressed factor in the Doppler domain.

It should be noted that during the reconstruction of the azimuth echo, the signal-to-noise Ratio (SNR) of the radar signal will decrease, which leads to a loss of accuracy for FR estimation. In order to overcome this problem, the number of looks should be increased properly to guarantee the accuracy for FR estimation.

The complete compensating flow is summarized as follows:

• Reconstruction of the azimuth echo with decompression for each polarimetric channel;
• Estimation of the FR angle at each sampling time with FP signals;
• Calculation of the time-variant ionospheric phase with estimated $\mathrm{T}\hat{\mathrm{E}}\mathrm{C}\left(t\right)$ from the FR angle;
• Compensation for the ionospheric effects with the estimated time-variant phase;
• Re-compression of the azimuth echo.

The corresponding compensating flow diagram is shown in Figure 3.

4. Experiments and Analyses

4.1. Simulation Based on Real FP SAR and TEC Data

According to the geometric properties of GEO-SAR imaging, the FP SAR images are simulated using the real SAR data from ALOS-2 with 3 m resolution as the backscattering coefficients. With the parameters in

Table 1. The simulation parameters.

Parameter	Value
Orbit altitude	35,793.29 km
Orbit inclination	60 deg
Right ascension of ascending node	97 deg
Wavelength	0.24 m
Bandwidth	30 MHz
Sampling rate	33 MHz
Pulse repetition frequency	75 Hz
Pulse width	20 μs
Integrated time	208 s

, the original FP images are simulated under two kinds of scenes (heterogeneous and homogeneous, respectively), which are shown in Figure 4. The original ionospheric effects in ALOS-2 data are compensated, and the cross-polarized channels are equalized before simulation to eliminate the effects of the original ionospheric effects and polarimetric calibration errors.

The measured TEC on 1 October 2015 (36°N, 84°W) from the United States Total Electron Content (USTEC) is adopted as the time-variant TEC (shown in Figure 5) [46]. During the simulation, only part of the TEC is used according to the integrated time, which is shown in Figure 5b corresponding to the area marked with the green box in Figure 5a. Firstly, the ionospheric effect error, including phase advance and FR angle, is calculated from the TEC in Figure 5b according to Equations (5) and (11), shown in Figure 6. Then, the ionospheric effect error is added during the echo simulation. In Figure 7, the affected FP SAR images are shown after imaging processing with the affected echoes.

Comparing with the original images in Figure 4, the strong scattering points in Figure 7 become defocused in the azimuth direction, especially for the heterogeneous scene. The distortion in the range direction is not obvious, which confirms the QPE analysis in Section 2. The detailed comparison for the area marked with the red box in Figure 4 is shown in Figure 8, which reflects the time-variant ionospheric effect on the azimuth signal more distinctly. By contrast, the distortion of the homogenous scene is not that visually obvious, except for the blurring at the water–land boundary. In order to evaluate the impact from the ionosphere quantitatively, the correlation coefficient is introduced here. According to the statistical result in

Table 2. The correlation coefficient between the original and the affected as well as the compensated FP SAR images (with the proposed method) for heterogeneous and homogeneous scenes.

Channel		HH	HV	VH	VV
Scene Type
Heterogeneous scene	Affected	0.5225	0.4820	0.4764	0.5213
	Compensated	0.9917	0.9821	0.9821	0.9926
Homogeneous scene	Affected	0.5161	0.5103	0.5092	0.5129
	Compensated	0.9965	0.9601	0.9405	0.9953

, the correlation coefficient of each channel declines nearly by half, and the cross-polarized channel degenerates particularly. The profile of the artificially placed point target (area marked with the yellow box in Figure 4) is also given in Figure 9. According to the evaluation of the imaging quality in

Table 3. The evaluation result of the imaging quality for the artificially placed point target.

Target	Azimuth				Range
Index	Original	Affected	FR	PGA	Original	Affected	FR	PGA
Resolution (m)	6.1945	12.0018	6.1945	6.3881	4.5455	4.6875	4.5455	4.5455
PSLR (dB)	−13.2289	−3.2218	−13.2315	−11.0301	−13.1262	−13.2293	−13.1184	−13.0472
ISLR (dB)	−9.9549	−6.9857	−9.9591	−9.2556	−9.9207	−9.9806	−9.9196	−9.8387

, it is obvious that the azimuth signal gets defocused while the range signal is still good.

The compensation result with the proposed method is shown in Figure 8, Figure 9 and Figure 10, and the comparative result with PGA is also provided. During FR estimation, the size of the smoothing window along the azimuth direction is 64, and all the samplings of the range signal are used considering the uniformity of TEC in the range direction. The affected image is refocused after compensating with the proposed method for both scenes, while PGA is only valid for the heterogeneous scene, which conforms to its applicability with strong scattering points. The correlation coefficient and evaluation of imaging quality are also given in Table 2 and Table 3, respectively, proving the perfect performance of the proposed method and its good adaptability to different scenes. We also compare the original phase advance in the simulation and the estimated value during compensation (shown in Figure 11). The mean and standard deviation (STD) of error of the phase advance are 0.004 rad and 0.008 rad, respectively, which are quite small to guarantee the quality of imaging and polarization.

Moreover, in order to make the impact of the ionospheric effects and the performance of compensation for different polarimetric channels clearer, a strong scattering area within one sampling unit (area marked with the green box in Figure 4) is picked out to provide the profiles in both the azimuth and range directions in Figure 12. Comparing the azimuth and range profiles of the affected area in Figure 12b,e, it is obvious that the ionospheric effects on the azimuth signal is more severe than that of the range signal, which is consistent with the analyses and simulations above. In addition, the compensated profiles in Figure 12c,f also indicate the effectiveness of the proposed method, which could well solve the defocusing problems and recover the distributions of intensity for different polarimetric channels simultaneously.

4.2. Performance Analysis

The compensating performance of the proposed method is determined by the estimation of the ionospheric phase in (21). As the geomagnetic field can be obtained accurately based on the International Geomagnetic Reference Field (IGRF), the only factor determining the compensation is the accuracy of the FR angle. According to the polarimetric scattering model of the linearly FP SAR as shown below, two kinds of errors will lead to the misestimation of the FR angle: one is the polarimetric calibration error including the crosstalk and channel imbalance; the other is the thermal noise [47].

$$\begin{array}{l}\left(\begin{array}{cc}{M}_{HH}& M_{HV}\\ M_{VH}& M_{VV}\end{array}\right)=\left(\begin{array}{cc}1& {\delta }_2\\ {\delta }_1& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& f_1\end{array}\right)\left(\begin{array}{cc}\cos \mathsf{\Omega }& \sin \mathsf{\Omega }\\ -\sin \mathsf{\Omega }& \cos \mathsf{\Omega }\end{array}\right)\left(\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdot \left(\begin{array}{cc}\cos \mathsf{\Omega }& \sin \mathsf{\Omega }\\ -\sin \mathsf{\Omega }& \cos \mathsf{\Omega }\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& f_2\end{array}\right)\left(\begin{array}{cc}1& {\delta }_3\\ {\delta }_4& 1\end{array}\right)+\left(\begin{array}{cc}{N}_1& N_2\\ N_3& N_4\end{array}\right)\end{array}$$

(24)

where δ_i, i = 1–4, represents the crosstalk, f_i, i = 1–2, represents the channel imbalance, and N_i, i = 1–4, represents the thermal noise in each polarimetric channel.

Based on the simulated FR SAR data in Section 4.1, the impact of the polarimetric calibration error and thermal noise is analyzed quantitatively. The simulation parameters are given in

Table 4. The simulation parameters of thermal noise and polarimetric calibration error.

Parameter	Value
SNR	15~40 dB
Amplitude of crosstalk	−10~−40 dB
Argument of crosstalk	−60~60 deg
Amplitude of channel imbalance	−2~2 dB
Argument of channel imbalance	−45~45 deg

. Due to the error of the thermal noise and polarimetric calibration, the FR estimation becomes biased [47]. Thus, both the mean and STD of the estimation error are calculated to evaluate the performance. The mean of the phase error can be treated as a constant for each sampling time, which will not lead to the defocusing in azimuth. However, the corresponding constant TEC at each sampling time will deteriorate the range signal according to (9). The STD of the phase error, on the other hand, represents the randomness of the ionospheric phase within the integrated time, which will lead to the distortion in azimuth.

For the thermal noise, the error of the estimated phase is calculated in Figure 13, indicating that the error becomes large as the SNR decreases. If we take π/4 as the error threshold for imaging, the SNR should be better than 18.96 dB to guarantee the compensation accuracy for both the azimuth and range signals. For the amplitude of crosstalk, the mean of the estimated phase error is much more severe than that of STD (in Figure 14a), which means the interference from the quadrature channel will cause a larger overall bias than a fluctuation. In comparison, the argument of crosstalk is secondarily determining, or even the partial introduction of argument is conducive to the elimination of interference. Similarly, an error threshold of −15.63 dB for crosstalk is drawn according to the simulation result. By contrast, the influence from the channel imbalance is a little different. According to the result in Figure 15, if the channel imbalance can be kept within −1.02 dB to 0.31 dB, the residual error after ionospheric compensation will not affect imaging. Meanwhile, for the argument part, the restriction is more severe. According to the results in Figure 15b,c, the trend of error caused by the channel imbalance remains almost unchanged with different amplitudes of channel imbalance. It means the argument matters more than that of the amplitude for channel imbalance. Thus, the argument of channel imbalance is suggested to be maintained at zero as far as possible so as to guarantee the compensation accuracy.

5. Discussion

According to the simulation in Section 4.1, the performance of the PGA and the proposed method is contrasting, especially for the homogenous scene. As for the PGA itself, the estimated error within the synthetic aperture should be the same for each scattering point, which is quite a strict assumption. In order to meet this condition, the shifting and windowing are necessary to acquire the echo of the isolated scattering point approximatively. However, the interference from the background clutter and adjacent scatters cannot be completely eliminated in real SAR images. Thus, the PGA will predictably lose its efficiency for the homogenous image (such as the rainforest and desert) and the heterogeneous image with low SCR/SNR. Moreover, the PGA processing does not depend on the analytic error model and removes the error directly from the image to increase the visual effect. Thus, the true error at each sampling, i.e., the time-variant ionospheric error in the echo instead of the image, could not be obtained. This is why the estimated phase advance of the proposed method is shown in Figure 11, while the PGA cannot be shown.

The proposed method uses the FR angle to estimate the time-variant TEC, which is born with two-dimensional resolution capability corresponding to the SAR data. Thus, the time-variant characteristic of TEC can be accurately reflected based on the pulse sampling. Moreover, the Bickel and Bates’s FR estimation is based on the argument of the circular basis, which has less restriction on the target. Moreover, the variance of the estimated FR is only related with the SNR level according to the theoretical error model [48]. Thus, the performance of applicability with different scenes for the proposed method is quite good, as we have shown in Section 4. As for the polarimetric calibration error, the amplitude and phase of the distortions could give rise to a noticeable bias in the FR estimation, which will affect the ionospheric phase correction. The amplitude of crosstalk and the argument of channel imbalance are proved to have a greater impact on the FR bias, which is consistent with Quegan’s analysis in [47]. Thus, the polarimetric calibration of the FP SAR system is quite necessary, which could greatly weaken the impact from polarimetric distortions.

Of course, there are still some problems that need to be further considered for the proposed method:

(1)

The first problem concerns about the “zero Faraday rotation area”. According to the FR model in (11), the FR is closely related to the angle between the beam and geomagnetic field. If the angle is close to 90°, the FR angle will be almost zero. Combining the distribution of the geomagnetic field and the typical side-looking model of SAR, the zero FR area usually appears near the equator [39]. In this particular area, the TEC cannot be accurately estimated with the FR angle, so the corresponding method becomes invalid. Aiming to solve this problem, we are now trying to use the spatial variation of the geomagnetic field under a multi-dimensional model of the ionosphere to avoid the zero FR area problem.

(2)

The other problem concerns the limitation of the observing swath with FP mode. Due to the double of PRF, the observing swath is reduced by half compared to the single-polarimetric mode. Thus, for the wide swath modes, such as ScanSAR and TOPSAR, the FP observation is traditionally abandoned. The GEO-SAR system is proposed and recommended owing to its almost global coverage capability. Thus, the FP data may not be provided for ionospheric effect compensation. To solve this contradiction, we are working on the transmission of orthogonal polarimetric signals with frequency division multiplexing (FDM) technology, which can extend the swath and well preserve the FP information.

6. Conclusions

In this paper, a compensation method of background ionospheric effects on L-band GEO-SAR with fully polarimetric data is proposed. Based on the Faraday rotation estimation, the time-variant TEC can be acquired accurately for ionospheric effect compensation.

Considering the ultra-long synthetic aperture time of the GEO-SAR system, the time-variant characteristic of the background ionosphere is mainly concerned, and the corresponding error model is established. Combining the coherent integration of the SAR signal, decompression processing is firstly used to reconstruct the echo for each polarimetric channel. Then, the time-variant TEC is obtained accurately based on FR angle estimation. Finally, the ionospheric effect is compensated for according to the error model we established previously.

With the real FP SAR data from ALOS-2 and the measured time-variant TEC from USTEC, the simulation is conducted to prove the effectiveness and performance of the proposed compensation method. For different scenes, the proposed method could recover the data to more than 90% of the correlation coefficient, and the residual error of phase is less than 0.01 rad. Two kinds of typical error sources are also discussed to evaluate the performance of the method, including the thermal noise and the polarimetric calibration errors, from which the thresholds of error are given to guarantee compensation accuracy, in particular 18.96 dB for SNR, −15.63 dB for crosstalk and −1.02 dB to 0.31 dB for the amplitude of the channel imbalance, and the argument of the channel imbalance is suggested to be maintained as to zero as possible. Furthermore, the comparison between the proposed and existing methods is discussed in detail and the preliminary work of our follow-up research is also presented.