# Robust, Model-Based External Calibration of Multi-Channel Airborne SAR Sensors Using Range Compressed Raw Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

## 3. Target Response Model and Analysis

#### 3.1. Acquisition Geometry and Response Model

#### 3.2. Response Analysis

#### 3.3. Residual Errors

## 4. Model-Based Calibration

#### 4.1. Range Impulse Response

#### 4.2. Antenna Mispointing

#### 4.3. Phase Center Positions and Troposphere

#### 4.4. Interferometric Baselines

#### 4.5. Calibration Constants

## 5. Results and Discussion

#### 5.1. Radiometry

#### 5.2. SAR Impulse Response Function

#### 5.3. Geometric Accuracy

#### 5.4. Single-Pass Interferometry and Polarimetry

#### 5.5. Calibration of Under-Sampled SAR Data

#### 5.6. Limits of Applicability

## 6. Conclusions

## 7. Patents

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A flowchart illustrating the calibration process. The columns, from right to left, correspond to the target analysis of Section 3.2, the estimation of individual calibration corrections described in Section 4 and the inputs used by estimation step (derived from the previous iteration of analysis).

**Figure 2.**An illustration of the processing steps involved in the analysis of the response of a 1.5 m trihedral reflector in SAR data acquired at L-band in VV polarisation.

**Figure 3.**The impact on various analysis steps on the target Radar Cross Section (RCS) over azimuth, for the target of Figure 2. From top to bottom and left to right: (

**i**) The backscatter intensity along the hyperbolic target range history in the original raw data. (

**ii**) The target intensity after clutter filtering. (

**iii**) The target intensity after the transmit and receive antenna patterns have been accounted for and, as a dotted line, the expected target RCS variation. (

**iv**) The residual RCS offset between the expected and the measured target response.

**Figure 4.**The results of phase and delay analysis for the target shown in Figure 2, imaged at X-band with a bandwidth of 384 MHz. The two plots in the top row show the original phase in the raw data (left) and the wrapped residual phase $\delta {\varphi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$, obtained after the expected target phase has been removed from the raw response (right). The plot in the bottom row illustrates the consistency of the residual delay $\delta {\tau}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ and the absolute residual phase $\delta {\chi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$.

**Figure 5.**The phase of the C-band VV-pol range IRF estimated from a data comprising the range histories of reference targets in multiple calibration acquisitions. The solid line indicates the estimated correction. It is used to compensate for the systematic variations observed as a function of radar frequency. A similar analysis is carried out for the IRF amplitude.

**Figure 6.**The impact of antenna pointing calibration on the residual RCS errors along azimuth, illustrated by the response of a single 1.5 m trihedral reflector imaged at C-band in VV polarization. The sub-plots show a superposition of the target backscatter intensity without antenna pattern correction and the antenna gain (left) and the residual RCS error $\delta {\mathrm{RCS}}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$.

**Figure 7.**An illustration of how baseline calibration addresses systematic errors in the polarimetric phase. The plots show the impact of adjusting the spatial separation of the H and V phase centers of the primary F-SAR X-band antenna. Each image shows the HH/VV phase difference measured, in a single acquisition, over 12 trihedral reflectors. Each column shows the phase difference within the range history of a single reflector. Reflectors (columns) are arranged in order of increasing off-nadir angle. The difference between the left and right plots corresponds to a baseline refinement of only $\left(3.4,0.8,0.1\right)$ mm in the instrument coordinate system.

**Figure 8.**A comparison of RCS analysis results before and after C-band antenna pointing calibration. The first two rows visualize the residual RCS errors $\delta {\mathrm{RCS}}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ within the range history of reference targets in the co-polar channels of four independent calibration data acquisitions. The plots in the bottom row illustrate the corresponding RCS mismatch derived from the analysis of focused SAR imagery. The estimated, optimal pointing correction, which was constrained to be identical in H and V, amounted to $\left(\Delta {\mathit{roll}}_{e},\Delta {\mathit{pitch}}_{e},\Delta {\mathit{yaw}}_{e}\right)=\left(-0.{288}^{\xb0},0.{00}^{\xb0},-0.{44}^{\xb0}\right)$.

**Figure 9.**The impact of antenna pointing estimation and correction on the residual RCS mismatch. The plots illustrate variations observed within the range histories of trihedral reflectors imaged by DBFSAR at X-band. Large RCS errors (

**left**) are much reduced when target analysis takes into account the estimated antenna pointing correction (

**right**). Calibration yields independent pointing corrections for the transmit and receive antennas of $\Delta {A}_{T}=\left(1.{70}^{\xb0},-1.{58}^{\xb0},0.{03}^{\xb0}\right)$ and $\Delta {A}_{R}=\left(-0.{03}^{\xb0},-0.{84}^{\xb0},-0.{09}^{\xb0}\right)$, respectively.

**Figure 10.**The impact of the pointing estimate of Figure 9 on the backscatter amplitude in an independent SAR acquisition over the city of Landsberg am Lech.

**Figure 11.**A comparison of the 2D SAR impulse response function of a reference target imaged in VV polarisation at C-band. The top row represents the nominal performance. The IRF in the bottom row. meanwhile, is the one obtained when the estimated calibration corrections are taken into account. The plots in the second and third columns show the normalized power spectrum in range and in azimuth. The red dashed lines indicate the signal bandwidth.

**Figure 12.**The impact of antenna element phase center position calibration on phase and position errors. The measurements shown are derived from the co-polar channels of five independent F-SAR calibration data acquisitions at L-band. The first and second row visualize the absolute residual phase $\delta {\chi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ derived from range compressed raw data in different polarizations, while the last row shows the reference target range position error as measured in focused SAR imagery. In the latter plots, each target is represented by a point and lines connect targets in the same channel (see the legend in Figure 13).

**Figure 13.**Similar to the analysis of Figure 12, but with an emphasis on the azimuth dimension. The top row shows the absolute residual phase $\delta {\chi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ variation as a function of the squint angle, after the mean phase has been subtracted from each response. The second row shows the reference target azimuth position error as measured in focused SAR imagery.

**Figure 14.**Residual interferometric phase errors along the range histories of reference targets in two independent polarimetric, interferometric X-band acquisitions. The top and the bottom halves of the figure show interferograms before and after baseline calibration, respectively. Each half comprises six plots that correspond to all possible combinations of four channels. The channels in this example are the four co-polar channels that are acquired, simultaneously, by the two polarimetric antennas in the F-SAR X-band across track interferometer.

**Figure 15.**A comparison of the consistency of digital surface models (DSMs) extracted from F-SAR single pass interferometry over the Rhone glacier in Switzerland. (

**Top**): Polarimetric RGB color composites showing the backscatter amplitude in the lexicographic basis (HH, HV, VV) for two consecutive acquisitions with opposite look directions onto the glacier. (

**Middle**): The single-pass interferometric phase in the HH polarization, relative to the SRTM elevation model used for motion compensation, and the DSM derived therefrom. (

**Bottom**): The difference of the DSMs obtained, one for each look direction, assuming the nominal baseline (left) and using the refined baseline after calibration (right).

**Figure 16.**The impact of azimuth under-sampling on the residual errors obtained in the analysis of a trihedral reflector response at X-band. The left column summarizes the analysis of the original target response at the full azimuth sampling frequency. The right column shows the results obtained when only every third pulse of the raw data is used (the other pulses are discarded in pre-processing). The squint angle on the x-axis is defined in the antenna reference frame as in (6) and, for this particular response, the zero Doppler frequency corresponds to a squint of $-{4}^{\xb0}$.

**Figure 17.**A comparison of the residual errors measured in two independent F-SAR X-band acquisitions. The plots juxtapose results after calibration with (red) and without (blue) sub-sampling the input data. The comparison is based on the raw data used for calibration and illustrates the impact of the baseline and pointing changes given in Table 2. Left: interferometric phase errors obtained in the six co-polar channel combinations of Figure 14. Each data point is obtained as an average over the illumination time of a reference target. Lines connect targets in the same interferometric channel combination. Right: the residual RCS mismatch in the four co-polar channels of the interferometer. As before, data points are obtained as an average over the target illumination time.

**Table 1.**The position refinements estimated for the four phase centers of the X-band polarimetric across track interferometer of F-SAR using the baseline optimization of Section 4.4. The changes are given in cm and correspond to those in the phase evaluation of Figure 14.

Baselines [cm] | |||
---|---|---|---|

$\mathit{\delta}\mathit{x}$ | $\mathit{\delta}\mathit{y}$ | $\mathit{\delta}\mathit{z}$ | |

X1h | 0.121 | 0.065 | 0.051 |

X1v | 0.014 | −0.023 | 0.000 |

X2h | −0.041 | −0.042 | −0.050 |

X2v | −0.093 | 0.000 | −0.001 |

**Table 2.**The quantitative impact of severe azimuth under-sampling (see Figure 16) on the external calibration of the four phase centers comprising the polarimetric X-band across track interferometer of F-SAR. The columns of the table are as follows: Positions denote the change in phase center coordinates, Baselines are the same changes after the overall mean offset is subtracted, while Antenna Pointing denotes the changes in the antenna pointing angles.

Positions [cm] | Baselines [cm] | Antenna Pointing [deg] | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}\mathit{x}$ | $\mathit{\delta}\mathit{y}$ | $\mathit{\delta}\mathit{z}$ | $\mathit{\delta}\mathit{x}$ | $\mathit{\delta}\mathit{y}$ | $\mathit{\delta}\mathit{z}$ | $\mathit{\delta}\mathit{roll}$ | $\mathit{\delta}\mathit{pitch}$ | $\mathit{\delta}\mathit{yaw}$ | |

X1h | 0.471 | −0.108 | 0.952 | 0.000 | 0.061 | 0.065 | 0.007 | 0.026 | −0.016 |

X1v | 0.481 | −0.214 | 0.840 | 0.010 | −0.040 | −0.047 | |||

X2h | 0.473 | −0.173 | 0.884 | 0.001 | −0.003 | −0.003 | 0.002 | 0.045 | −0.038 |

X2v | 0.472 | −0.164 | 0.894 | 0.001 | 0.005 | 0.006 |

**Table 3.**A summary of the limits of applicability of the proposed calibration procedure. Limits are quantified by the magnitude of the calibration corrections that can be reliably estimated. The left and right columns indicate the sensor parameter correction and the maximum permissible correction magnitude, respectively. The middle column indicates which of the residual errors in the target response are affected.

Parameter | Residuals affected | Max. Parameter Error |
---|---|---|

$\Delta {\overrightarrow{\mathit{A}}}_{e}$ | $\delta {\mathrm{RCS}}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ | Antenna beam width |

$\Delta {\overrightarrow{\mathit{L}}}_{e}$ | $\delta {\chi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$, $\delta {\varphi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ | $>10$ m at X-band |

$\Delta c$ | $\delta {\chi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$, $\delta {\varphi}_{pq}^{s}\phantom{\rule{-0.166667em}{0ex}}\left(\mathit{az}\right)$ | $>2000$ ppm at X-band |

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**MDPI and ACS Style**

Jäger, M.; Scheiber, R.; Reigber, A. Robust, Model-Based External Calibration of Multi-Channel Airborne SAR Sensors Using Range Compressed Raw Data. *Remote Sens.* **2019**, *11*, 2674.
https://doi.org/10.3390/rs11222674

**AMA Style**

Jäger M, Scheiber R, Reigber A. Robust, Model-Based External Calibration of Multi-Channel Airborne SAR Sensors Using Range Compressed Raw Data. *Remote Sensing*. 2019; 11(22):2674.
https://doi.org/10.3390/rs11222674

**Chicago/Turabian Style**

Jäger, Marc, Rolf Scheiber, and Andreas Reigber. 2019. "Robust, Model-Based External Calibration of Multi-Channel Airborne SAR Sensors Using Range Compressed Raw Data" *Remote Sensing* 11, no. 22: 2674.
https://doi.org/10.3390/rs11222674