Hierarchical Bayesian Data Analysis in Radiometric SAR System Calibration: A Case Study on Transponder Calibration with RADARSAT-2 Data
1.1. Problem Statement
1.2. Objective, Approach, and Paper Structure
1.3. Note on Point Target RCS versus ERCS
2. Methodology for Parameter Estimation from SAR Data
2.1. Introduction to Bayesian Statistics and Numerical Methods
2.2. Hierarchical Models
- What is the best estimate of the calibration factor (and its respective confidence interval) if several types of reference point targets (i.e., transponders and corners of different sizes) with different ERCS’ and stabilities are deployed?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.
3. Case Study: Measurement Campaign Goal and Setup
3.1. Introduction and Goal
3.2. RADARSAT-2 Products
3.3. Reference Point Targets
3.4. Target Alignment
3.5. Imaged Area
4. Case Study: Data Analysis and Results
- 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).
4.2. Power Estimation for Point Targets from SAR Images
- 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.
- Subtract the estimated clutter power from the integrated target power to get a clutter-compensated target power.
4.3. Bayesian Statistics and Hierarchical Model Fitting
4.3.1. Daily RADARSAT-2 and Transponder Drifts
4.3.2. Hierarchical Bayesian Model
4.3.3. Posterior Simulation
4.3.4. MCMC Results
4.4. Posterior Predictive Checks: Model Verification
4.5. Plausibility Check with Classical Statistics
5. Discussion of Hierarchical Bayesian Data Analysis for Radiometric Calibration
- 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.
Conflicts of Interest
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|Overpass Time||Orbit Direction||Beam Mode|
|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|
|Size||Peak RCS||Number of Targets|
|1.5 m||38.38 dBm2||9|
|3.0 m||50.43 dBm2||6|
|Overpass Date||Estimated Drift μs (dB)||Maximal Error (dB)||Resulting σs (dB)|
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Döring, B.J.; Schmidt, K.; Jirousek, M.; Rudolf, D.; Reimann, J.; Raab, S.; Antony, J.W.; Schwerdt, M. Hierarchical Bayesian Data Analysis in Radiometric SAR System Calibration: A Case Study on Transponder Calibration with RADARSAT-2 Data. Remote Sens. 2013, 5, 6667-6690. https://doi.org/10.3390/rs5126667
Döring BJ, Schmidt K, Jirousek M, Rudolf D, Reimann J, Raab S, Antony JW, Schwerdt M. Hierarchical Bayesian Data Analysis in Radiometric SAR System Calibration: A Case Study on Transponder Calibration with RADARSAT-2 Data. Remote Sensing. 2013; 5(12):6667-6690. https://doi.org/10.3390/rs5126667Chicago/Turabian Style
Döring, Björn J., Kersten Schmidt, Matthias Jirousek, Daniel Rudolf, Jens Reimann, Sebastian Raab, John Walter Antony, and Marco Schwerdt. 2013. "Hierarchical Bayesian Data Analysis in Radiometric SAR System Calibration: A Case Study on Transponder Calibration with RADARSAT-2 Data" Remote Sensing 5, no. 12: 6667-6690. https://doi.org/10.3390/rs5126667