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A synthetic aperture radar (SAR) system requires external absolute calibration so that radiometric measurements can be exploited in numerous scientific and commercial applications. Besides estimating a calibration factor, metrological standards also demand the derivation of a respective calibration uncertainty. This uncertainty is currently not systematically determined. Here for the first time it is proposed to use hierarchical modeling and Bayesian statistics as a consistent method for handling and analyzing the hierarchical data typically acquired during external calibration campaigns. Through the use of Markov chain Monte Carlo simulations, a joint posterior probability can be conveniently derived from measurement data despite the necessary grouping of data samples. The applicability of the method is demonstrated through a case study: The radar reflectivity of DLR’s new C-band Kalibri transponder is derived through a series of RADARSAT-2 acquisitions and a comparison with reference point targets (corner reflectors). The systematic derivation of calibration uncertainties is seen as an important step toward traceable radiometric calibration of synthetic aperture radars.

A synthetic aperture radar (SAR) system is a measurement system that acquires measurement data for earth-observation applications. Many applications like soil moisture [

The necessary radiometric calibration is a two-step process and is divided into relative and absolute calibration [

The result of a measurement is only meaningful if, besides the estimate of the value of the measurand, a statement about the

The final, combined measurement uncertainty results from several uncertainty contributions, and the calibration uncertainty is one of them. If a SAR system is absolutely calibrated, the uncertainty with which the radar reflectivity of the reference point target is known has a direct impact on the combined uncertainty. Another source of uncertainty is the estimation of the calibration factor from external calibration measurements. Specifically, this second source of uncertainty is addressed in this paper.

Currently, the derivation of measurement uncertainties in radiometric SAR system calibration is not systematically conducted. To the best of our knowledge, no operator of a previous or current space-borne SAR system claims to provide traceable radiometric calibration, and the quoted measurement uncertainties in the respective data product specifications or other publications [

One piece of the puzzle of achieving traceable radiometric SAR system calibration is the derivation of the calibration factor from external calibration acquisitions. We propose for the first time to apply Bayesian statistical data analysis and hierarchical modeling to estimate parameters like the absolute calibration factor

Bayesian data analysis has been proven advantageous in many parameter estimation problems [

The suitability of the proposed method is shown through a case study in Sections 3 (campaign setup) and 4 (data analysis). The objective of the case study is conceptually identical to the derivation of the absolute calibration factor, although here the reflectivity of a point target shall be accurately determined. The presented case study is based on an external calibration campaign which was executed in April 2013. 15 corner reflectors were deployed as reference targets, and the reflectivity of DLR’s new C-band Kalibri transponder prototype was derived from a series of eight data acquisitions from the Canadian RADARSAT-2 SAR system.

Finally, the proposed approach is further discussed in Section 5 and conclusions are given in Section 6.

This paper adopts the reasoning laid out in [

Targets whose ERCS is accurately known can be taken as a reference to calibrate SAR systems. The aim of the presented case study is to derive the ERCS of DLR’s next-generation C-band Kalibri transponder prototype so that in principle it can be taken as a calibration normal for subsequent SAR calibration campaigns.

In order to avoid confusion between the symbols for (E)RCS and standard deviation, the (E)RCS will not be denoted with the customary letter

Bayesian statistics is, like classical (frequentist) statistics, a well established field with applications in many scientific areas. It is extensively covered in the literature (see for example [

In Bayesian statistics, all unknown quantities are handled as random variables and are described by probability distributions, which in turn describe a measure of the state of knowledge of the parameter’s value. In Bayesian analysis, the probability function of a parameter (prior) is updated by incorporating new information,

If a population parameter (e.g., the case study transponder ERCS) is called ^{2}. If now some new data

Describing parameters by probability distributions is a natural fit when not only the best estimate of a parameter needs to be stated, but also a confidence interval needs to be quantified, as is the case for calibration. Deriving a confidence interval from a distribution is straight forward for any kind of distribution,

Deriving posterior distributions is in practice mostly achieved through numerical methods, which allow to consider more complex problems and arbitrary distributions. The simulation method used in the case study is the Markov chain Monte Carlo (MCMC) approach [

An external radiometric calibration campaign results in a pool of data samples (see [

What is the best estimate of the calibration factor (and its respective confidence interval) if several types of reference point targets (

Solving this problem with classical (frequentist) statistics would require to estimate the population mean of each group, and deriving the calibration factor after ERCS compensation between groups. The information on the variance within each group is lost, and a reliable statement of the final uncertainty or confidence interval on the estimated calibration factor is difficult to achieve. With hierarchical Bayesian modeling though, the variance within each group (target type) and the variance across all target types can be derived simultaneously because group and total dispersion are handled within a joint probability model.

Is there a significant systematic dependence on the chosen antenna beam (or near/far range, left/right looking geometries, or ascending/descending orbits) for radiometric measurements?

Once again the same set of data samples as before should be grouped, but this time by antenna beam (or near/far range, left/right looking acquisitions, or ascending/descending orbits). For each group, a posterior distribution for the respective calibration factor can now be derived. Comparing the different posterior distributions allows to conclude if a significant radiometric inter-beam offset exists.

For a check on plausibility: Is the ERCS of one of the reference point targets systematically different from the others? (Here repeated overpasses over the same set of targets is assumed.) In order to answer this question, the overpass-dependent effect of the SAR system and the atmosphere should be modeled out of the analysis. This can be done by grouping the samples according to overpass and target ID. All target samples of one overpass can be used to compensate for SAR system and atmospheric effects, and in a second step the group ERCS of each target can be determined.

In fact, all questions previously raised can be conveniently answered by setting up a single

Further details on hierarchical models can be found in [

The following case study applies hierarchical Bayesian data analysis to a practical problem. This Section 3 describes the campaign goal and setup, whereas the following Section 4 applies the proposed method of hierarchical Bayesian data analysis for the estimation of a parameter in absolute radiometric calibration.

The DLR currently develops and manufactures a set of next-generation active radar calibrators (transponders) [

The campaign was conducted for a second, independent derivation and verification of the Kalibri prototype ERCS. It consisted of eight repeated RADARSAT-2 acquisitions of a test site in which 15 trihedral reference corner reflectors of two different sizes (inner leg lengths of 1.5 m and 3.0 m) and the Kalibri transponder prototype were deployed and aligned for the respective acquisition. The data were acquired in April 2013.

The goal of the campaign was to derive the estimated transponder ERCS _{t}

The RADARSAT-2 SAR system operates at a center frequency of 5.405 GHz. The mode-dependent bandwidth goes up to 100 MHz. Center frequency and maximal bandwidth are therefore identical to those of the Sentinel-1 mission for which the Kalibri transponder was designed.

All eight RADARSAT-2 products were acquired in the

An overview of all overpass times and beams is shown in

Within this study, the RADARSAT-2 system was considered radiometrically uncalibrated because of the mismatch between the system’s specified relative radiometric calibration (

In total 15 trihedral corner reflectors were used as the reference point targets to derive the transponder ERCS. The comparatively large number was deemed necessary in order to profit from averaging during data analysis. Reflectors with two different inner leg lengths were deployed, resulting in two distinct values for their ERCS.

The RADARSAT-2 system operates at a center frequency of 5.405 GHz with a small relative bandwidth of 2%. Under this precondition, a corner’s ERCS is, with sufficient accuracy, equal to a corner’s RCS at the center frequency [

The corner reflectors were realigned together with the transponder for each upcoming overpass, so that all corners could be used for all overpasses during analysis. The alignment angles were computed based on the predicted RADARSAT-2 orbit for the respective overpass and the point target location (latitude, longitude).

An accurate corner alignment is necessary because the peak RCS (

The alignment of the corner reflectors (made out of aluminum) was performed manually with an inclinometer and a compass. A local magnetic declination of 2.5° was accounted for when using a compass during azimuth alignment. The alignment standard uncertainty for both axes was estimated to be not above 0.5°.

The transponder was mounted on a positioner unit, which allows a motorized two-axis alignment in azimuth and elevation with high mechanical repeatability (better 0.1°). The alignment was zeroed in elevation with a water-level. In azimuth, a compass could not be used like for the corners because the positioner’s ferromagnetic components result in a misreading. Instead, the true North direction was determined by measuring a reference azimuth direction with a GPS device. For this, two reference points where placed several dozens of meters away from the transponder, one in front, one behind, and both on the line of the current alignment. By measuring the accurate GPS positions, the azimuth orientation could be computed by determining the angle between the connecting line and true North. The standard uncertainty of this method was estimated to be 0.1° through cross-validation.

All acquired RADARSAT-2 scenes imaged an area around the DLR site at Oberpfaffenhofen, Germany. The transponder was built up right on the DLR premises, allowing easy wiring and monitoring. The 1.5 m corners were installed in the immediate vicinity on the protected grasslands surrounding the airstrip at Oberpfaffenhofen. The site names for sites on which the 1.5 m corners are installed begin with

The site names for locations with a 3.0 m corner are

The RADARSAT-2 datatakes were processed by MDA and the final single-look complex (SLC) images were the starting point for the data analysis.

The primary analysis goal is to derive the transponder ERCS and, equally importantly, an associated uncertainty statement. In principal, the transponder ERCS _{t}_{r}_{t}_{r}

The analysis is split into two parts:

Point target analysis: Extract the relative point target impulse response powers for all point targets in all scenes (see Section 4.2).

Parameter estimation: Set up a statistical model to derive the estimated transponder ERCS and corresponding uncertainty from all datatakes (see Section 4.3).

The imaged point targets appear as bright crosses in the processed image. Two methods are distinguished when deriving the point target power: the peak and the integral method. With the peak method, the target power is described by the pixel with the peak power (complex pixel amplitude squared) whereas with the integral method, the target power results as the sum of pixel powers over a relevant region around the target. It was shown that the integral method is advantageous as it leads to accurate results even if the image is not perfectly focused [

The goal of this analysis step is to derive a table of relative point target intensities for all point targets in all images. Although an absolute scale is not necessary (and the RADARSAT-2 absolute calibration factor is ignored here), it is crucial that the point target intensities derived from different images are all calibrated with respect to each other. This is achieved by applying the (range-independent)

The following steps were performed to derive a targets’ integrated impulse response intensity:

Define a search window around the point target in the georeferenced, processed image.

Find and record the brightest pixel location.

Define an analysis window, centered on the brightest pixel of the previous step.

Estimate the clutter power from four non-overlapping areas surrounding the peak.

Integrate the target power over a cross area (see

Subtract the estimated clutter power from the integrated target power to get a clutter-compensated target power.

This procedure is in line with the one described in [

An exemplary transponder impulse response is shown in

Exemplary impulse responses for 3.0 m and 1.5 m corners are show in

The output of the previous processing step is a table which reports a relative point target power per target and scene. A graphical representation is shown in

In order to estimate the target powers from all available data, daily RADARSAT-2 and transponder drifts need to be estimated and compensated, as described in the following Section 4.3.1. Then, the transponder ERCS can be computed with the measurement model

A plot of the original data from the first processing step was show in

Also, there is one immediately apparent data outlier: The corner at site D26g on the third day was clearly not aligned and was masked out prior to further analysis.

Besides the RADARSAT-2 drift, the (mostly temperature-dependent) transponder drift can be estimated and compensated. The compensation is based on internal calibration data recorded by the transponder itself. Exemplary transponder loop gain and temperature drift data are shown in

A similar assessment has been performed for all other days. The estimated transponder drifts and their estimated error bounds are listed in

The Bayesian model requires the definition of priors,

For the eight overpass days _{d}_{d}

Besides the RADARSAT-2 drift, the transponder drift needs to be modeled. The values in column three of _{d}_{d}∼_{s,d}, σ_{s,d}_{s,d}_{s,d}

It is assumed that all measured point target powers of all targets within one target group _{g}, σ_{g}_{g}_{g}>_{g}_{g}_{g}∼^{1.5}^{7}) (_{g}∼^{6}). In a way, the _{g}_{g}

Now, for every overpass _{d,g}_{d}

Estimating parameters with the model _{t}_{30}, and _{15}) after drift compensation.

The next step is to relate the point target powers to ERCS. For this, a reference ERCS is necessary. It was chosen to be the group ERCS of the 1.5 corners. (The not more than 7 years old 1.5 corner reflectors were, mechanically speaking, in a better shape than the more than 20 years old 3.0 corners, which show apparent deformations due to damages and their continuous exposition on a field. Mechanical deformations result in a reduction of a corners monostatic ERCS because some of the incident power is reflected away from the incident beam direction. The visual observation was confirmed by looking at the ERCS dispersion within each group: _{15} = 0.15 dBm^{2} and _{30} = 0.41 dBm^{2}. These observations lead to the conclusion that the ERCS of the utilized 3.0 corners should not be used as an absolute reference, and that the 1.5 corners provide a better link to an absolute ERCS. Nevertheless, the 3.0 corners and their large ERCS helped in determining more accurately the daily RADARSAT-2 drift.)

The value of the reference ERCS is modeled as _{15}^{3.838} m^{2}^{0.02} m^{2}) (^{2}^{2})). Its location is defined by ^{2} is certainly the weakest point in the argument. It is based on previous experience gained from numerical field simulations on corners of the same size at X-band, and on plausibility. (A standard uncertainty of 0.2 dBm^{2} is plausible because with it the theoretical RCS difference of 3.0 and 1.5 corners can be (well) explained. The theoretical difference, according to ^{2}. The difference between the estimated mean target powers (in MCMC parameters: _{30} − _{15}) is 11.92 dB, ^{2}.) Nevertheless, it cannot be proofed and further work should be conducted in determining the absolute knowledge of a trihedral corner reflector’s ERCS.

Now, the final transponder ERCS is deterministically related to the already derived model parameters through

The complete Bayesian model described above is visualized in

The hierarchical model developed in the previous section is now solved iteratively with the numerical Markov chain Monte Carlo method (MCMC) [_{d}_{30},

If the model is well posed, then the iterative MCMC will converge to the true distribution of the parameters. This also means that the first draws/simulation runs need to be discarded, and only values after this

In order to compute the parameters of the hierarchical model, which was presented in the previous chapter, 2 ^{5} simulation runs were conducted, allowing for a burn in of 1 ^{4} and a thinning of 20. These parameters were determined empirically by observing the traces and autocorrelations of the model parameter draws.

The solution of the hierarchical model describes all parameters at the same time. Nevertheless, the results can be visualized step by step.

The first result is the estimated RADARSAT-2 drift. The estimated drift is shown in

This estimated RADARSAT-2 drift can now be applied to the measured data. The original data in

After RADARSAT-2 drift compensation, the estimated transponder drift can be taken out from the upper plot in

Now the most important result of the MCMC simulation is the derivation of the transponder ERCS _{t}^{2} with a standard uncertainty according to [^{2}. The 95% highest probability density interval is given as [60.38^{2}. Note that the standard uncertainty is clearly dominated by the assumed ERCS knowledge uncertainty of the 1.5 corners of 0.2 dBm^{2}.

In Section 4.3.2, a normal distribution was chosen in order to model the observed integrated pixel intensities. Here it shall be demonstrated that this model is indeed plausible and adequate.

Focusing on the transponder data _{t,d}_{t,d}

As a means of verification, the result of the previous section can be reproduced approximately with classical (frequentist) statistics. One way of handling the hierarchical data structure is to derive one transponder ERCS per overpass, and then to combine the resulting eight ERCS values through averaging into an overall transponder ERCS. The disadvantage of this simplified approach (in comparison to the approach using hierarchical Bayesian data analysis as shown before) is that information about the uncertainty of each of the eight transponder ERCS values is lost and does not contribute to the combined uncertainty.

After averaging, an estimated standard deviation of the mean can be derived from the eight estimated ERCS values, resulting in a standard uncertainty for the estimated transponder ERCS [

This approach results in an estimated transponder ERCS of 60.80 dBm^{2} with a standard uncertainty of 0.20 dB or an expanded standard uncertainty of 0.41 dB at a confidence level of 95% (

Note that once again the combined uncertainty is dominated by the assumed ERCS knowledge uncertainty of the 1.5 corner reflectors. The transponder’s combined backscatter uncertainty is sufficiently low to permit the calibration of SAR instruments like Sentinel-1 to an absolute radiometric uncertainty of 1.00 dB at a confidence level of 99%, provided that the SAR instrument is otherwise sufficiently precise [

The advantages of using hierarchical Bayesian data anlysis for radiometric calibration were laid out before: The approach can jointly answer typical analysis questions in radiometric calibration while fully exploiting the hierarchical nature of external calibration data and fulfilling the requirement on reporting measurement uncertainties and confidence intervals. This section shall add a critical discussion of the proposed method.

First, the most recognized approach in metrology for deriving measurement uncertainties is the ISO

From the outset, using Markov chain Monte Carlo (MCMC) simulations to infer model parameters appears to be more complicated than employing classical statistics. The added initial work is offset though by a joint probability model, which allows to derive model parameters on arbitrary hierarchical levels without loss of information. Hence the improved analysis justifies the initial extra work.

If numerical methods like MCMC are used, problems of non-convergence can occur and must be addressed during analysis. Care must be taken in the assessment of simulation results, and plausibility checks should be added.

The presented work proposed for the first time in the field of radiometric SAR system calibration to exploit hierarchical Bayesian data analysis. It was claimed that Bayesian statistics is well suited to the analysis of calibration data because of the following key factors:

Within Bayesian statistics, probability distributions are used in describing model parameters. The distributions convey a meaning of uncertainty. Bayesian statistics is therefore an appropriate choice for calibration, where an estimated parameter is meaningless without a statement of its uncertainty.

Hierarchical joint probability models are well suited to describe data that is typically acquired during an external radiometric SAR calibration campaign. During data analysis, depending on the research question, parameters often need to be estimated on different levels or for different groups. Hierarchical Bayesian modeling is well suited to derive model parameters for different interdependent parameters, especially when numerical methods like Markov chain Monte Carlo simulations are used.

The applicability of the method for radiometric calibration was demonstrated through a case study. The data of an external calibration campaign was analyzed. The campaign goal was to derive the ERCS of DLR’s C-band Kalibri transponder prototype. Due to hierarchical Bayesian data analysis, it was possible to estimate and compensate the overpass-dependent drift of the RADARSAT-2 system and to derive the transponder ERCS with a remaining standard uncertainty (according to GUM) of only 0.21 dBm^{2}.

In order to convert the case study approach to an operational uncertainty analysis procedure for SAR missions, a database of point-target measurements should be set up. Filling the database incrementally with measurements of permanently installed reference point targets over the mission lifetime would allow one to continually derive radiometric uncertainty estimates based on Bayesian statistics.

The authors are convinced that Bayesian statistics and hierarchical modeling are an important step toward

The campaign was only feasible due to the commitment and collaboration of several partners and many DLR colleagues. Ron Caves from MDA Systems Ltd. rendered the campaign possible by organizing the acquisitions and by providing the processed RADARSAT-2 datatakes. The EDMO airport in Oberpfaffenhofen and namely Werner Döhring greatly simplified the campaign work by permitting the installation of all utilized 1.5 corners right on the grassland surrounding the airstrip, directly next to the DLR premises.

On the side of DLR, the campaign was made possible by Manfred Zink and the additional campaign team members Christo Grigorov, Thomas Kraus, and Sebastian Ruess.

The authors declare no conflict of interest.

Artist’s rendering of DLR’s new C-band Kalibri transponder, mounted on a two-axis positioner.

Locations of the transponder on the DLR premises and of the 1.5 m corners on the adjacent airport in Oberpfaffenhofen, Germany. (Map tiles in this and

Map of imaged area, showing the locations of the 3.0 m corners.

Integration area for the

Transponder impulse response for the first overpass on 7 April 2013. A large target-to-clutter ratio is apparent. The four red squares indicate the areas from which the clutter power was estimated.

3.0 corner impulse response at site D24 for the first overpass on 7 April 2013. Also see caption of

1.5 corner impulse response at site D26 for the first overpass on 7 April 2013. Also see caption of

Uncompensated and unmasked data, the immediate result of the processing described in Section 4.2. The data points lie on a common ordinate; for better visibility, one region per target group is plotted.

Estimated transponder drift and temperature stability (main influence for gain drift) for the overpass on 21 April 2013. The transponder operates with its nominal loop gain if a relative drift of 0 dB is detected. The loop gain was adjusted at 16:30 (red dotted line), after which the drift was monitored (blue circles) until the overpass (dashed green line). After drift correction, the drift is estimated to be with high probability within the range [−0.02,0.02] dB at 17:03 UTC.

Diagram of the Bayesian model. The ellipsis symbol . . . indicates a family of probability distributions (per group

Estimated daily drifts _{d}

Measured impulse response powers with RADARSAT-2 drift compensation (see

Visualization of the Kalibri transponder drift compensation with data from

Posterior predictive checking for predicted (modeled) transponder data

Overview of approximate acquisition times (in UTC).

7 April 2013 17:11:09 | ascending | U17W2 |

8 April 2013 05:20:16 | descending | U16W2 |

14 April 2013 17:06:59 | ascending | U11W2 |

15 April 2013 05:16:06 | descending | U22W2 |

18 April 2013 05:28:36 | descending | U5W2 |

21 April 2013 17:02:49 | ascending | U5W2 |

24 April 2013 17:15:19 | ascending | U22W2 |

25 April 2013 05:24:26 | descending | U10W2 |

Corner reflectors with two different inner-leg lengths were used during the campaign.

1.5 m | 38.38 dBm^{2} |
9 |

3.0 m | 50.43 dBm^{2} |
6 |

Cross parameters according to

Cross length | 21 |

Cross width | 3 |

Square width | 5 |

Daily transponder RCS drift, estimated maximal error bounds on the estimated drift, and resulting standard uncertainties according to

_{s} |
_{s} | ||
---|---|---|---|

2013-04-07 | 0.00 | 0.05 | 0.03 |

2013-04-08 | 0.00 | 0.02 | 0.01 |

2013-04-14 | 0.02 | 0.03 | 0.02 |

2013-04-15 | 0.03 | 0.02 | |

2013-04-18 | 0.00 | 0.07 | 0.04 |

2013-04-21 | 0.00 | 0.02 | 0.01 |

2013-04-24 | 0.05 | 0.05 | 0.03 |

2013-04-25 | 0.02 | 0.03 | 0.02 |