Reconstruction of Synthetic Aperture Radar Raw Data under Analog-To-Digital Converter Saturation Distortion for Large Dynamic Range Scenes

: Digital storage and transmission are common processes in modern synthetic aperture radar systems; thus, analog-to-digital converters are indispensable. Such processes can lead to two types of error: quantization (or granular) error and saturation (or clipping) error, which cause sampling noise, and radiometric and harmonic distortions in ﬁnal images. Traditionally, reasonable choices of the gain and the number of quantization bits by the analog-to-digital converter based on the echo distribution can e ﬀ ectively reduce these errors. However, establishing the gain control repository of a synthetic aperture radar mission is a long process. In addition, if the dynamic range of the backscattering coe ﬃ cient is extremely large or if unexpected strong targets appear in a scene, then harmonics occur in the echo, which turns the variable gain ampliﬁer into chaos based on statistic and, inevitably, results in saturation in the raw data. Once raw data saturation occurs, the SAR system can conventionally adjust only the analog-to-digital converter in the next observation, thus reducing timeliness. Power loss compensation based on a statistical model and saturation (clipping) factor on a large-scale could compensate for the energy loss in images; however, detail interference, such as harmonic distortion, cannot be e ﬀ ectively suppressed, which will lead to false targets in the focused data. To address this particular problem, a novel anti-saturation method for large dynamic range scenes is proposed in this paper. The log-normal distribution is used in this article to describe dynamic range scenes with strong isolated targets, which mainly cause receiver saturation. Using the statistical distribution of complex scenes as a priori information, a maximum a posteriori estimation algorithm is proposed to simultaneously compensate for the saturated values in the raw data and retain the non-saturated values. Thus, the details of the weak background are well preserved, and the isolated strong targets with sparsity are reconstructed perfectly. With Monte Carlo simulation, the proposed method can improve the radiometric accuracy by 5 to 10 dB and e ﬀ ectively suppress the energy of false targets. Based on TerraSAR-X, ALOS-2, and Radarsat-1 synthetic aperture radar data, the e ﬀ ectiveness and robustness of the proposed method are also veriﬁed by simulations.


Introduction
Synthetic aperture radar (SAR) is of great importance in earth observation and geophysics research due to its all-weather and all-time observation ability. Due to the limitations of the sampling efficiency unsaturated raw data is maintained. Simulations with TerraSAR-X, ALOS-2, and Radarsat-1 data show that the proposed method can improve the relative radiometric accuracy by 5 to 10 dB and effectively suppress harmonic distortion.
The error source of ADC distortion is analyzed in Section 2. In Section 3, the a priori model is built based on a traditional backscattering distribution. The optimization equation is constructed, and the performance of the proposed method is discussed. Simulations and experiments are shown in Section 4. A discussion and conclusions are provided in Sections 5 and 6, respectively.

Analog-To-Digital Converter Saturation Distortion
The analog-to-digital converter acts on the echo sample, which leads to ADC saturation. Then, the samples (or raw data) that suffered from ADC saturation are sent to the ground, and processed by 2-D pulse compression to obtain focused data. To compress data, the focused data will be quantified again in the product, which leads to image saturation. The specific data flow of the SAR system is shown in Figure 1. It is obvious that ADC saturation and image saturation are different, as they occur during different stages of SAR data flow and have different processing intents. Then, these parts are compressed by BAQ (or another technology) and stored. For an echo, the ADC quantizes the real and imaginary signals to specific levels. As shown in Figure 3, the samples above the maximum quantization value are set to the threshold, and the others It is obvious that ADC saturation and image saturation are different, as they occur during different stages of SAR data flow and have different processing intents. Figure 2 shows a simple diagram of a radar receiver. The VGA is controlled by the AGC/MGC and amplifies the echo to an appropriate level G(s R + js I ). The real and imaginary parts of the amplified echo Gs R and Gs I are sampled separately through I/Q channels as S R [n] and S I [n]. Then, these parts are compressed by BAQ (or another technology) and stored.

ADC Error for Raw Data
Remote Sens. 2019, 11 FOR PEER REVIEW 3 unsaturated raw data is maintained. Simulations with TerraSAR-X, ALOS-2, and Radarsat-1 data show that the proposed method can improve the relative radiometric accuracy by 5 to 10 dB and effectively suppress harmonic distortion. The error source of ADC distortion is analyzed in Section 2. In Section 3, the a priori model is built based on a traditional backscattering distribution. The optimization equation is constructed, and the performance of the proposed method is discussed. Simulations and experiments are shown in Section 4. A discussion and conclusions are provided in Sections 5 and 6, respectively.

Analog-To-Digital Converter Saturation Distortion
The analog-to-digital converter acts on the echo sample, which leads to ADC saturation. Then, the samples (or raw data) that suffered from ADC saturation are sent to the ground, and processed by 2-D pulse compression to obtain focused data. To compress data, the focused data will be quantified again in the product, which leads to image saturation. The specific data flow of the SAR system is shown in Figure 1. It is obvious that ADC saturation and image saturation are different, as they occur during different stages of SAR data flow and have different processing intents. Then, these parts are compressed by BAQ (or another technology) and stored. For an echo, the ADC quantizes the real and imaginary signals to specific levels. As shown in Figure 3, the samples above the maximum quantization value are set to the threshold, and the others For an echo, the ADC quantizes the real and imaginary signals to specific levels. As shown in Figure 3, the samples above the maximum quantization value are set to the threshold, and the others are set to the most recent quantized values. Thus, the continuous probability distribution changes to a discrete distribution, which leads to ADC errors. Specifically, the saturated error is related to the values Remote Sens. 2019, 11, 1043 4 of 21 exceeding the threshold, and the quantization error takes values between -0.5 and 0.5, conforming to a uniform distribution.

ADC Error for Raw Data
Remote Sens. 2019, 11 FOR PEER REVIEW 4 are set to the most recent quantized values. Thus, the continuous probability distribution changes to a discrete distribution, which leads to ADC errors. Specifically, the saturated error is related to the values exceeding the threshold, and the quantization error takes values between -0.5 and 0.5, conforming to a uniform distribution. The echo is amplified G times by the VGA, and the ADC performs M -bit quantization. The signal s obeys a zero-mean Gaussian distribution. 1 1 e x p 2 2 . For specific quantization bits M , there is an optimal choice of k that minimizes Q S E E + . For a typical SAR system, M is typically selected as 8 [2]. It is obvious that quantization error changes much more gradually than the saturation level at various k values in Figure 4. Hence, the ADC error of strong targets mainly comes from saturation noise, because the threshold set by the variance of the weak background is much smaller than the optimized value of the strong scatterers. The echo is amplified G times by the VGA, and the ADC performs M-bit quantization. The signal s obeys a zero-mean Gaussian distribution.
Here, s is s R or s I , and σ is the standard deviation. The saturated threshold of the signal is ± 2 M−1 − 0.5 /G, which is expressed as a multiple of the signal standard deviation ±kσ. The quantization noise ranges from ±0.5/G, which can also be expressed as ±kσ/ 2 M − 1 . The relative errors of quantization E Q and saturation E S are where C(k) is the confidence level of the standard normal distribution in [−k, k]. For specific quantization bits M, there is an optimal choice of k that minimizes E Q + E S . For a typical SAR system, M is typically selected as 8 [2]. It is obvious that quantization error changes much more gradually than the saturation level at various k values in Figure 4. Hence, the ADC error of strong targets mainly comes from saturation noise, because the threshold set by the variance of the weak background is much smaller than the optimized value of the strong scatterers. . The relative noise of quantization and saturation at 8-bits. The ADC error is dominated by saturation or quantization noise when variable k is lower or higher than the optimum value, respectively.

Effects of Saturation on Imaging
Because of ADC saturation in I/Q channels (real and imaginary parts), there are phase and amplitude errors in the raw data samples, as shown in Figure 5. The phase errors and amplitude errors of a zero-mean complex Gaussian echo with a clipping factor of 30% and normalized mean power.
As shown above, the irregular phase errors of the samples do not exceed 4 π and have little effect on the focused results of the main lobes via coherent integration [22,23]. It is demonstrated that the effect of pulse phase errors on range compression with the chirp scaling algorithm is negligible [24]. Thus, the phase errors caused by saturation have no impact on the resolution or position of real targets, even if the system uses 1-bit quantization (i.e., only sign information remains) [25]. The amplitude errors of raw data lead to power loss for the main lobes in the pulse compressed result [23,25]. However, the relative energy intensity between different targets in the same data . The relative noise of quantization and saturation at 8-bits. The ADC error is dominated by saturation or quantization noise when variable k is lower or higher than the optimum value, respectively.

Effects of Saturation on Imaging
Because of ADC saturation in I/Q channels (real and imaginary parts), there are phase and amplitude errors in the raw data samples, as shown in Figure 5.
Remote Sens. 2019, 11 FOR PEER REVIEW 5 Figure 4. The relative noise of quantization and saturation at 8-bits. The ADC error is dominated by saturation or quantization noise when variable k is lower or higher than the optimum value, respectively.

Effects of Saturation on Imaging
Because of ADC saturation in I/Q channels (real and imaginary parts), there are phase and amplitude errors in the raw data samples, as shown in Figure 5. As shown above, the irregular phase errors of the samples do not exceed 4 π and have little effect on the focused results of the main lobes via coherent integration [22,23]. It is demonstrated that the effect of pulse phase errors on range compression with the chirp scaling algorithm is negligible [24]. Thus, the phase errors caused by saturation have no impact on the resolution or position of real targets, even if the system uses 1-bit quantization (i.e., only sign information remains) [25]. The amplitude errors of raw data lead to power loss for the main lobes in the pulse compressed result [23,25]. However, the relative energy intensity between different targets in the same data As shown above, the irregular phase errors of the samples do not exceed π/4 and have little effect on the focused results of the main lobes via coherent integration [22,23]. It is demonstrated that the effect of pulse phase errors on range compression with the chirp scaling algorithm is negligible [24]. Thus, the phase errors caused by saturation have no impact on the resolution or position of real targets, even if the system uses 1-bit quantization (i.e., only sign information remains) [25].
The amplitude errors of raw data lead to power loss for the main lobes in the pulse compressed result [23,25]. However, the relative energy intensity between different targets in the same data block is maintained well. Based on numerical simulations, at large-scales, the power loss of focused data Remote Sens. 2019, 11, 1043 6 of 21 caused by saturated raw data can be estimated and compensated according to the clipping factor of raw data [12][13][14] (as shown in Figure 6), and the radiometric accuracy can be improved.
Remote Sens. 2019, 11 FOR PEER REVIEW 6 block is maintained well. Based on numerical simulations, at large-scales, the power loss of focused data caused by saturated raw data can be estimated and compensated according to the clipping factor of raw data [12][13][14] (as shown in Figure 6), and the radiometric accuracy can be improved. Figure 6. Compensation factor for the power loss of focused data, which is caused by the saturation of raw data. The clipping factor is calculated in saturated raw data, the real and imaginary parts of which follow a zero-mean Gaussian distribution.
Therefore, the errors of sampling can be mitigated by coherent integration of pulse compression and compensation of power loss during imaging process, which slightly impact the images of the targets themselves. However, saturation noise remains in the product, which reduces the radiometric resolution and may cause false objects. Thus, even low bits are used in the ADC or BAQ, and we can still obtain effective images (although they are contaminated by ADC saturation).

Harmonic Distortion Due to Saturation
If there are a few extremely strong targets in a scene, in addition to saturation noise, the raw data of the isolated strong targets can lead to harmonic distortion, which causes false targets and parasitic sidelobes in focused data (as shown in Figure 7) [16].  Compensation factor for the power loss of focused data, which is caused by the saturation of raw data. The clipping factor is calculated in saturated raw data, the real and imaginary parts of which follow a zero-mean Gaussian distribution. Therefore, the errors of sampling can be mitigated by coherent integration of pulse compression and compensation of power loss during imaging process, which slightly impact the images of the targets themselves. However, saturation noise remains in the product, which reduces the radiometric resolution and may cause false objects. Thus, even low bits are used in the ADC or BAQ, and we can still obtain effective images (although they are contaminated by ADC saturation).

Harmonic Distortion Due to Saturation
If there are a few extremely strong targets in a scene, in addition to saturation noise, the raw data of the isolated strong targets can lead to harmonic distortion, which causes false targets and parasitic sidelobes in focused data (as shown in Figure 7) [16].
Remote Sens. 2019, 11 FOR PEER REVIEW 6 block is maintained well. Based on numerical simulations, at large-scales, the power loss of focused data caused by saturated raw data can be estimated and compensated according to the clipping factor of raw data [12][13][14] (as shown in Figure 6), and the radiometric accuracy can be improved.

Figure 6.
Compensation factor for the power loss of focused data, which is caused by the saturation of raw data. The clipping factor is calculated in saturated raw data, the real and imaginary parts of which follow a zero-mean Gaussian distribution.
Therefore, the errors of sampling can be mitigated by coherent integration of pulse compression and compensation of power loss during imaging process, which slightly impact the images of the targets themselves. However, saturation noise remains in the product, which reduces the radiometric resolution and may cause false objects. Thus, even low bits are used in the ADC or BAQ, and we can still obtain effective images (although they are contaminated by ADC saturation).

Harmonic Distortion Due to Saturation
If there are a few extremely strong targets in a scene, in addition to saturation noise, the raw data of the isolated strong targets can lead to harmonic distortion, which causes false targets and parasitic sidelobes in focused data (as shown in Figure 7) [16].  Suppose that there are two close point targets with slant ranges R 1 (η) and R 2 (η), respectively, and the echo data associated with these targets is where τ is the fast time and η is the slow time, T r is the pulse width, K r is the chirp rate, λ is the wavelength, σ 1 e jϕ 1 and σ 2 e jϕ 2 are the complex scattering coefficients of the two targets, and A 1 (η) and A 2 (η) are antenna pattern weights. At a specific time η 0 , if the two targets have the same scattering intensity and are close to each other in the range direction, namely, σ 1 ≈ σ 2 and A 1 (η 0 ) ≈ A 2 (η 0 ), then the echo within a pulse width can be simplified to The raw data can be considered a chirp signal with a low-frequency cosine envelope (as shown in Figure 8). With dechirp processing, the pulse compressed result is where F [•] is the Fourier transform. From Equation (7), it can be seen that the pulse compressed result depends on the low-frequency component of the echo.
Remote Sens. 2019, 11 FOR PEER REVIEW 7 Suppose that there are two close point targets with slant ranges ( ) respectively, and the echo data associated with these targets is where τ is the fast time and η is the slow time, r T is the pulse width, r K is the chirp rate, λ is the wavelength, The raw data can be considered a chirp signal with a low-frequency cosine envelope (as shown in Figure 8). With dechirp processing, the pulse compressed result is is the Fourier transform. From Equation (7), it can be seen that the pulse compressed result depends on the low-frequency component of the echo.  With a saturation threshold S a , the envelope has harmonic distortion which leads to several additional frequency components [26]. The pulse compression result in Equation (7) is rewritten as The first part with phase information in Equation (8) has not changed with saturation, which keeps the normal focusing along both the range and the azimuth, and the additional frequency components in the second part result in a series of false targets after pulse compression (as shown in Figure 9a). For several strong isolated targets, the harmonic envelope is complicated due to saturation, which leads to parasitic sidelobes in the given range (as shown in Figure 9b). The same phenomenon also occurs along the azimuth, which can be explained similarly.
Remote Sens. 2019, 11 FOR PEER REVIEW 8 With a saturation threshold a S , the envelope has harmonic distortion which leads to several additional frequency components [26]. The pulse compression result in Equation (7) is rewritten as The first part with phase information in Equation (8) has not changed with saturation, which keeps the normal focusing along both the range and the azimuth, and the additional frequency components in the second part result in a series of false targets after pulse compression (as shown in Figure 9a). For several strong isolated targets, the harmonic envelope is complicated due to saturation, which leads to parasitic sidelobes in the given range (as shown in Figure 9b). The same phenomenon also occurs along the azimuth, which can be explained similarly.

Maximum A Posteriori (MAP) Log-Normal Distribution
The log-normal distribution is used in radar detection and has been widely accepted [20]. As discussed above, the ADC error for scenes with isolated strong targets is mainly caused by saturation, which leads to false objects and parasitic sidelobes in a given range. For special scenes consisting of large isolated dynamic range objects (such as buildings) [21], the probability density function of backscattering coefficient γ is where V and β are the mean value and variance of ln γ , respectively. As shown in Figure 10, the container slice of Figure 7 is fitted to a log-normal distribution.

Maximum A Posteriori (MAP) Log-Normal Distribution
The log-normal distribution is used in radar detection and has been widely accepted [20]. As discussed above, the ADC error for scenes with isolated strong targets is mainly caused by saturation, which leads to false objects and parasitic sidelobes in a given range. For special scenes consisting of large isolated dynamic range objects (such as buildings) [21], the probability density function of backscattering coefficient γ is where V and β are the mean value and variance of ln γ, respectively. As shown in Figure 10, the container slice of Figure 7 is fitted to a log-normal distribution.
For scatterers in a given range ( ) where r T is the pulse width. Based on a discretized representation, = + S Ax n (12) where the vector x expresses the scatterers along the range, the amplitudes of which obey ( ) X p x , S is the echo caused by x , matrix A is the relationship between S and x , which can be seen as a cyclic convolution matrix with a chirp signal, and n is complex Gaussian white noise with variance 2 n σ [27]. The relationship between echo S and the scatterers along range x is shown in Figure 11. The MAP version of (12) is as follows. The amplitude of the scatterers |x| = √ γ obeys the following relation.
For scatterers in a given range x(t), the echo is where T r is the pulse width. Based on a discretized representation, where the vector x expresses the scatterers along the range, the amplitudes of which obey p |X| (|x|), S is the echo caused by x, matrix A is the relationship between S and x, which can be seen as a cyclic convolution matrix with a chirp signal, and n is complex Gaussian white noise with variance σ 2 n [27]. The relationship between echo S and the scatterers along range x is shown in Figure 11.
For scatterers in a given range ( ) where r T is the pulse width. Based on a discretized representation, = + S Ax n (12) where the vector x expresses the scatterers along the range, the amplitudes of which obey ( ) X p x , S is the echo caused by x , matrix A is the relationship between S and x , which can be seen as a cyclic convolution matrix with a chirp signal, and n is complex Gaussian white noise with variance 2 n σ [27]. The relationship between echo S and the scatterers along range x is shown in Figure 11. The MAP version of (12) is as follows. The MAP version of (12) is as follows.
The estimated result is thenx where There is no explicit solution for Equation (14); however, using an iterative method, we could update |x i |, V and β in Equation (15) and obtain an approximate solution.

Reconstruction Method for Saturation
Because saturation independently occurs in the real and imaginary parts of the raw data, Equation (12) could be converted to real values [27] R(S) where R(•) and I(•) keep the real and imaginary parts of the variables. We still used S, A, and x to express the real values in (16). According to the values, the raw data S = [s 1 , s 2 , . . . s N ] T can be divided into three types: non-saturated S 0 , positive saturated S + , and negative saturated S − . The corresponding indexes are U 0 , U + , and U − .
The matrix A = a T 1 , a T 2 , · · · , a T N T in which a n is row vector could even be divided as shown in Equation (18), and Figure 12 shows the schematic diagram. To achieve anti-saturation, the penalty function of the elements in S is constructed as follows [17].
This equation can be given in matrix form as [18] ( ) ( ) ( ) 0, 0 n a n n a n a n n a S S diag g S S S S diag g S S and where α is the regularization parameter. The estimated result is as follows.
To maintain the unsaturated part of the raw data and reconstruct the saturated values, the reconstruction of the raw data Ŝ was given by (21). To achieve anti-saturation, the penalty function of the elements in S is constructed as follows [17].
This equation can be given in matrix form as [18] where G + = diag g + , g +n = 1, (S +n − S a ) ≤ 0 0, (S +n − S a ) > 0 and S a = [S a , S a , . . . S a ] T . Thus, the MAP (13) estimate was revised aŝ where α is the regularization parameter. The estimated result is as follows.
To maintain the unsaturated part of the raw data and reconstruct the saturated values, the reconstruction of the raw dataŜ was given by (21). The workflow of the proposed method is shown in Figure 13. With the reconstructed raw data, traditional imaging algorithms can be used in the follow-up processing.
The workflow of the proposed method is shown in Figure 13. With the reconstructed raw data, traditional imaging algorithms can be used in the follow-up processing.

Accuracy Assessment
We used radiometric accuracy improvement (RAI) to evaluate the reconstruction performance. The RAI value could be calculated as follows.
The relative value of reduced saturation (RRS) was used to evaluate the suppression effect of the false targets and saturated noise within a weak background. Specifically, the RRS was calculated as Because of the nonlinear processing in the proposed method, we used Monte-Carlo simulation to estimate the performance. In this simulation, several strong targets were placed at the center of distributed weak targets, and the ratio of the intensity of strong targets to that of the background was 30 dB. As shown in Figure 14(a), with a 30% saturation factor, the reconstructed results had fewer parasitic sidelobes and higher radiometric accuracy than the original image. With different saturation factors and S/W values (i.e., the ratio of the strong target quantity to the weak target quantity), as shown in Figure 14(b), the proposed method could improve the RAI by 5 to 10 dB.

Accuracy Assessment
We used radiometric accuracy improvement (RAI) to evaluate the reconstruction performance. The RAI value could be calculated as follows.
where I s is the complex value of the saturated image, I r is the complex value of the reconstructed image, and I 0 is the complex value of the original image. The larger the RAI and RRS values, the better the reconstruction and the suppression performances. Because of the nonlinear processing in the proposed method, we used Monte-Carlo simulation to estimate the performance. In this simulation, several strong targets were placed at the center of distributed weak targets, and the ratio of the intensity of strong targets to that of the background was 30 dB. As shown in Figure 14a, with a 30% saturation factor, the reconstructed results had fewer parasitic sidelobes and higher radiometric accuracy than the original image. With different saturation factors and S/W values (i.e., the ratio of the strong target quantity to the weak target quantity), as shown in Figure 14b

One-Dimensional Pulse Compression
The parameters of the one-dimensional experiment are shown in Table 1, and the pulse compression result was weighted with a Hamming window to exclude traditional sidelobes. 50% There are two strong point targets and one weak target at the center of the scene. As shown in Figure 15, with a 50% saturation factor, the raw data was severely distorted, and the false target caused by the saturated targets covered the real weak target. Using the proposed method, the reconstructed raw data perfectly fitted the unsaturated raw data, and the weak target reappeared at the correct position.

One-Dimensional Pulse Compression
The parameters of the one-dimensional experiment are shown in Table 1, and the pulse compression result was weighted with a Hamming window to exclude traditional sidelobes. There are two strong point targets and one weak target at the center of the scene. As shown in Figure 15, with a 50% saturation factor, the raw data was severely distorted, and the false target caused by the saturated targets covered the real weak target. Using the proposed method, the reconstructed raw data perfectly fitted the unsaturated raw data, and the weak target reappeared at the correct position.

One-Dimensional Pulse Compression
The parameters of the one-dimensional experiment are shown in Table 1, and the pulse compression result was weighted with a Hamming window to exclude traditional sidelobes. 50% There are two strong point targets and one weak target at the center of the scene. As shown in Figure 15, with a 50% saturation factor, the raw data was severely distorted, and the false target caused by the saturated targets covered the real weak target. Using the proposed method, the reconstructed raw data perfectly fitted the unsaturated raw data, and the weak target reappeared at the correct position.

Simulation Based on the Real SAR Data
The real SAR data were used here to verify the effectiveness and performance of the proposed method. Because this method is robust under different circumstances, the data from varieties of SAR platforms (with different carrier frequencies and resolutions) should be covered in simulations. However, most users seldom have access to the SAR raw data. Therefore, both the raw data and the focused data were used in the simulations. The only difference between the raw data and the focused data is that the former were directly implemented with saturation error, while the latter should be converted into raw data through radar echo simulation first and then undergo saturation.
During the implementation of saturation, the real part and imaginary part of the echo were separated at first. Any data of these two parts beyond the saturated threshold were set to the threshold value and then recombined into the saturated raw data. Then, the reconstructed raw data could be obtained with the proposed method. Finally, the reconstructed data were processed with the traditional chirp scaling imaging algorithm.
In comparative experiments, power loss compensation (PLC) factors refer to the theoretical curve in Figure 6 for traditional power loss compensation processing.

Simulations Based on the Real Focused Data
With the parameters listed in Table 2, we simulated the original image (in Figure 16a) with backscattering coefficients from TerraSAR-X data with 1 m resolution. The image contained isolated strong targets (marked with red boxes) and was free from saturation issues. Figure 16, Table 3 and Table 4 give the comparison results.

Simulation Based on the Real SAR Data
The real SAR data were used here to verify the effectiveness and performance of the proposed method. Because this method is robust under different circumstances, the data from varieties of SAR platforms (with different carrier frequencies and resolutions) should be covered in simulations. However, most users seldom have access to the SAR raw data. Therefore, both the raw data and the focused data were used in the simulations. The only difference between the raw data and the focused data is that the former were directly implemented with saturation error, while the latter should be converted into raw data through radar echo simulation first and then undergo saturation.
During the implementation of saturation, the real part and imaginary part of the echo were separated at first. Any data of these two parts beyond the saturated threshold were set to the threshold value and then recombined into the saturated raw data. Then, the reconstructed raw data could be obtained with the proposed method. Finally, the reconstructed data were processed with the traditional chirp scaling imaging algorithm.
In comparative experiments, power loss compensation (PLC) factors refer to the theoretical curve in Figure 6 for traditional power loss compensation processing.

Simulations Based on the Real Focused Data
With the parameters listed in Table 2, we simulated the original image (in Figure 16a) with backscattering coefficients from TerraSAR-X data with 1 m resolution. The image contained isolated strong targets (marked with red boxes) and was free from saturation issues. Figure 16, Tables 3 and 4 give the comparison results.       With the parameters listed in Table 5, another simulation result with ALOS-2 focused data is shown in Figure 17. And Tables 6 and 7 give the comparison results. PLC (dB) -0.43 -0.43 -0.43 -0.43 -0.43 dB With the parameters listed in Table 5, another simulation result with ALOS-2 focused data is shown in Figure 17. And Table 6 and Table 7 give the comparison results.  Compared with Figures 16a and 17a, the saturated images in Figures 16b and 17b became noisy, and obvious false targets (marked with green boxes) were caused by harmonics. In addition, isolated strong targets, which are generally areas of focus, were distorted, especially at the image edges (shown in the enlarged part of Figure 16b), resulting in difficulties in detection and recognition. After reconstructed processing with the proposed method, the false targets caused by saturation were effectively suppressed, and the real targets were recovered, as shown in the enlarged portions of Figures 16c and 17c. Meanwhile, the reconstructed results with power loss Compared with Figures 16a and 17a, the saturated images in Figures 16b and 17b became noisy, and obvious false targets (marked with green boxes) were caused by harmonics. In addition, isolated strong targets, which are generally areas of focus, were distorted, especially at the image edges (shown in the enlarged part of Figure 16b), resulting in difficulties in detection and recognition. After reconstructed processing with the proposed method, the false targets caused by saturation were effectively suppressed, and the real targets were recovered, as shown in the enlarged portions of Figures 16c and 17c. Meanwhile, the reconstructed results with power loss compensation, which is regarded as a classical anti-saturation method [13,14], are presented in Figures 16d and 17d for comparison. It is obvious that, although PLC could recover the energy loss of the image, false targets still existed, as shown in the enlarged portions of Figures 16d and 17d. We calculated the RAIs of the saturated and compensated images for the isolated strong targets listed in Tables 3 and 6. The results indicate that the RAI was improved by 9.47 and 12.96 dB and 10.69 and 10.43 dB on average for the proposed method and PLC, respectively, which confirms the better performance of the proposed method. Moreover, the improvement of RAI coincides with the theoretical compensation precision discussed above. Tables 4 and 7 suggest that the mean energy of false targets was effectively suppressed by 8.16 and 0.61 dB with the proposed method (compared with an increase of 0.43 and 2.22 dB for PLC), which significantly improves image quality.

Simulation Based on Real Raw Data
When reasonable a priori information was used, the proposed method overcame the limitation of sparse expression [17,18] and was applied to complex scenes, such as in Figure 18a (with various natural and artificial features). This experimental area was chosen from the RADARSAT-1 raw data in [28] of English Bay, because it encompasses a large dynamic range where saturation issues are likely to occur. To simulate saturation, the real and imaginary parts of the signal were limited by a threshold of four times the standard deviation of the one from the sea. The saturation factor of the land was 25%.
In Figure 18b, the saturated image became brighter (especially in the sea area marked with yellow boxes), and false targets appeared (marked with green boxes) that were not in the original image in Figure 18a. Similarly, the edges of the isolated strong targets (marked with red boxes) became fuzzy, as can be clearly seen in the enlarged view of Figure 18b. After reconstruction with the proposed method, the image (in Figure 18c) recovered perfectly, the luminance decreased, and false targets vanished. Moreover, the isolated strong targets were well preserved compared with those in Figure 18a, and results compensated for with PLC are also shown in Figure 18d. The energy loss of strong targets could be compensated for by using PLC, but the energy of the false targets and the saturated background were improperly strong. To obtain a quantitative reconstruction result, we calculated the RAIs of the saturated and compensated images for strong isolated targets (red boxes), as listed in Table 8. The results indicate that the RAI was improved by 9.18 dB on average for the proposed method and by 6.98 dB for PLC, which supports the results in the performance curve in Figure 14b. In addition, the mean energies of false targets (green boxes) and the scene (yellow box) decreased by 0.56 dB (in Table 9) and 0.20 dB, respectively, verifying the effectiveness of the proposed method for complex scenes. In contrast, the energies of false targets and the scene were increased by 3.08 dB with PLC due to the overall power compensation.  -3.08 -3.08 -3.08 -3.08 dB

Discussion
The false targets and radiometric losses caused by ADC saturation in SAR images can lead to serious mistakes about classification and recognition of targets in SAR applications. Figure 19 shows a polarimetric decomposition mistake, in which the containers on the top leads to a larger red area (marked with a green box), which represents incorrect backscattering characteristics. Because the echo from the HV channel, which mainly carries volume scattering information, is much weaker than that from the HH channel, which mainly carries the double bounce scattering information, ADC saturation happens in the latter. Thus, parasitic sidelobes beside the containers caused by harmonic distortion are improperly identified as artificial objects in the decomposition results, and traditional PLC processing cannot suppress them.

Discussion
The false targets and radiometric losses caused by ADC saturation in SAR images can lead to serious mistakes about classification and recognition of targets in SAR applications. Figure 19 shows a polarimetric decomposition mistake, in which the containers on the top leads to a larger red area (marked with a green box), which represents incorrect backscattering characteristics. Because the echo from the HV channel, which mainly carries volume scattering information, is much weaker than that from the HH channel, which mainly carries the double bounce scattering information, ADC saturation happens in the latter. Thus, parasitic sidelobes beside the containers caused by harmonic distortion are improperly identified as artificial objects in the decomposition results, and traditional PLC processing cannot suppress them. As shown in Section 4, the proposed method performs better than the traditional PLC. Because the false targets are treated as real targets when counting the energy loss caused by saturation, indiscriminate power loss compensation is conducted in traditional processing on both the real and false targets. Thus, the energy is falsely reinforced rather than eliminated. In contrast, the proposed method reconstructs the saturated raw data but not for image processing; thus, the false targets could be clearly distinguished from the real ones and then suppressed in the focused data.
Using a log-normal distribution to describe a large dynamic range scene is key in this article. The echo from a small quantity of strong targets disrupts the receiver, and also opens up the possibility to rebuild them. We consider two reasons that the algorithm can reconstruct the raw data accurately, even if some information is lost in saturation: 1) For a scene obeying a log-normal distribution, there are two kinds of scatterers: a. distributed weak background, and b. isolated strong targets. With the appropriate amplification factor, the echo from the background should be among the quantization thresholds, and the echo from the strong targets periodically exceeds the thresholds which causes harmonic distortion. The algorithm reconstructs only the saturated part of the raw data and retains the non-saturated data. Thus, the weak background without harmonic distortion is not obviously affected by reconstructed processing. Moreover, as analyzed in Section 2.2, the phase error of the saturated data does not exceed 4 π , and at the last step of the algorithm, the phase is still controlled in the effective focus range; thus, the focusing result does not result in a loss of resolution.
2) The small number of strong but sparse targets leads to information redundancy; thus, strong targets can be obtained from the raw data with information loss. The addition of a priori information (log-normal distribution) and a penalty function reinforce the advantage of the algorithm.
However, if the distribution of the scene is not log-normal but rather chi-squared (which is a uniform distributed target), the saturation cannot be very serious, as the VGA works well.
With the rapid development of electronic devices and the continuous optimization of control strategies, saturation errors are not very common in modern spaceborne SAR systems. However, if these errors occur, lost information cannot be retrieved by relying on a single observation, and control systems adjust only at the next observation. Therefore, we find the saturation phenomenon in some large dynamic range scenes. To ensure the timeliness of observation, the saturation reconstruction algorithm is still worthy of study. In the future, remote sensing data will surge rapidly, driven by a high resolution and a wide swath, and it will be accompanied by sampling and transmitting pressures. Therefore, ADC error with a low bit rate will come into focus. Moreover, interplanetary probes cannot achieve high-speed sampling because of the limitations of power As shown in Section 4, the proposed method performs better than the traditional PLC. Because the false targets are treated as real targets when counting the energy loss caused by saturation, indiscriminate power loss compensation is conducted in traditional processing on both the real and false targets. Thus, the energy is falsely reinforced rather than eliminated. In contrast, the proposed method reconstructs the saturated raw data but not for image processing; thus, the false targets could be clearly distinguished from the real ones and then suppressed in the focused data.
Using a log-normal distribution to describe a large dynamic range scene is key in this article. The echo from a small quantity of strong targets disrupts the receiver, and also opens up the possibility to rebuild them. We consider two reasons that the algorithm can reconstruct the raw data accurately, even if some information is lost in saturation: (1) For a scene obeying a log-normal distribution, there are two kinds of scatterers: a. distributed weak background, and b. isolated strong targets. With the appropriate amplification factor, the echo from the background should be among the quantization thresholds, and the echo from the strong targets periodically exceeds the thresholds which causes harmonic distortion. The algorithm reconstructs only the saturated part of the raw data and retains the non-saturated data. Thus, the weak background without harmonic distortion is not obviously affected by reconstructed processing. Moreover, as analyzed in Section 2.2, the phase error of the saturated data does not exceed π/4, and at the last step of the algorithm, the phase is still controlled in the effective focus range; thus, the focusing result does not result in a loss of resolution.
(2) The small number of strong but sparse targets leads to information redundancy; thus, strong targets can be obtained from the raw data with information loss. The addition of a priori information (log-normal distribution) and a penalty function reinforce the advantage of the algorithm.
However, if the distribution of the scene is not log-normal but rather chi-squared (which is a uniform distributed target), the saturation cannot be very serious, as the VGA works well.
With the rapid development of electronic devices and the continuous optimization of control strategies, saturation errors are not very common in modern spaceborne SAR systems. However, if these errors occur, lost information cannot be retrieved by relying on a single observation, and control systems adjust only at the next observation. Therefore, we find the saturation phenomenon in some large dynamic range scenes. To ensure the timeliness of observation, the saturation reconstruction algorithm is still worthy of study. In the future, remote sensing data will surge rapidly, driven by a high resolution and a wide swath, and it will be accompanied by sampling and transmitting pressures. Therefore, ADC error with a low bit rate will come into focus. Moreover, interplanetary probes cannot achieve high-speed sampling because of the limitations of power consumption and transmission. The proposed method can solve the saturation problem based on data and simplify the hardware platform, which will help address these challenges.

Conclusions
In this paper, for large dynamic range scenes, we propose an anti-saturation algorithm that can reconstruct saturated raw data. This method can be used to improve the quality of images that have been polluted by saturation and to reduce the complexity of the radar receiver.
The echo from isolated strong targets results in a chaotic receiver and leads to saturated raw data. Traditional PLC methods compensate for only the energy loss in the focused data but cannot deal with false targets and saturation noise. We use a log-normal distribution to describe particular scenes consisting of weak backgrounds and strong isolated targets, such as buildings on vegetation and ships in bays, and find that the harmonic distortion caused by the isolated scatterers and ADC saturation is the main cause of false targets.
Using the statistical distribution of complex scenes as a priori information, a regularization algorithm is proposed to simultaneously compensate for the saturated values in the raw data and retain the non-saturated values. Thus, the details of the weak background are kept well, and the isolated strong targets with sparsity are reconstructed. Compared to PLC, the proposed method can improve the radiometric accuracy of real targets with better performance and effectively suppress the false targets and saturation noise. Through Monte Carlo simulation, this method can improve the radiometric accuracy by 5 to 10 dB and effectively suppress the energy of false targets caused by harmonic distortion. Based on TerraSAR-X, ALOS-2, and Radarsat-1 data, the effectiveness and robustness of the proposed method are verified via simulation.