# Hierarchical Bayesian Data Analysis in Radiometric SAR System Calibration: A Case Study on Transponder Calibration with RADARSAT-2 Data

^{*}

## Abstract

**:**

## 1. Introduction

#### 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}. If now some new data y (e.g., from the case study RADARSAT-2 campaign) becomes available, one wants to update one’s believe on θ given data y. This can be written as p(θ|y) where p(·|·) describes a conditional probability. The notation follows [12]. Now, Bayes’ rule allows to compute the posterior probability p(θ|y) from a prior probability p(θ) and a likelihood function p(y|θ) [12]:

`PyMC`library is used [17].

#### 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

_{t}, is conceptually identical to the derivation of the absolute calibration factor K (linking SAR system indications P to the measurement quantity, (E)RCS) because only the roles of knowns and unknowns are reversed:

#### 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

#### 4.1. Overview

_{t}can be derived if the ERCS of a reference target (placed in the same scene) ς

_{r}is known according to the proportionality

_{t}and P

_{r}are the point target intensities of the transponder and the reference target, derived from the processed SAR images and possibly expressed as digital numbers.

- 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

_{d}need to be determined. It is estimated from Figure 8 that the daily drift, expressed as a scaling factor, will certainly be in the range 0.4 to 1.6 (i.e., −4 dB to 2 dB). The prior can thus be written as r

_{d}∼ U(0.4, 1.6), where U(a, b) describes a uniform distribution with lower and upper bounds a and b, respectively.

_{d}is modeled as a normal distribution N(μ, σ), where μ describes its mean and σ its standard deviation: s

_{d}∼ N(μ

_{s,d}, σ

_{s,d}). The best estimate μ

_{s,d}is taken from Table 4, where also the σ

_{s,d}(resulting from Equation (5)) are listed.

_{g}, σ

_{g}), having the group location (mean) μ

_{g}and group scale (standard deviation) σ

_{g}> 0. A normal distribution was chosen because it is symmetric (no plausible reason can be found for an asymmetric distribution), and because the distribution of the measured values results from several physical effects (satellite thermal drift, satellite and target alignment errors, target stability, clutter, etc.) so that the central limit theorem suggests a normal distribution as well. Each μ

_{g}and σ

_{g}are modeled to originate from uniform distributions: μ

_{g}∼ U(10

^{1.5}, 10

^{7}) (i.e., U(15 dB, 70 dB)) and σ

_{g}∼ U(0, 10

^{6}). In a way, the σ

_{g}are nuisance parameters which are not required to derive the transponder ERCS, but they are nevertheless necessary in order to describe the normal distributions N

_{g}.

_{d,g}is fitted depending on its group g ∈ [t, 15, 30]:

_{d}is estimated from all available data, exploiting the fact that the drift should be equal for data across all three groups.

_{t}, μ

_{30}, and μ

_{15}) after drift compensation.

_{15}= 0.15 dBm

^{2}and σ

_{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.)

_{15}∼ N(10

^{3.838}m

^{2}, 10

^{0.02}m

^{2}) (i.e., N(38.38 dBm

^{2},0.2 dBm

^{2})). Its location is defined by Equation (3). The standard deviation, or standard uncertainty according to [7], characterizes the state of knowledge about the reference ERCS. The statement that the ERCS of 1.5 corners can be determined with Equation (3) up to a standard uncertainty of 0.2 dBm

^{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 Table 2, is 12.05 dBm

^{2}. The difference between the estimated mean target powers (in MCMC parameters: μ

_{30}− μ

_{15}) is 11.92 dB, i.e., 0.13 dB away from the predicted value despite the already discussed deformation of the 3.0 corners. It is still possible though that the RCS of all corners is, due to deformation and the approximate nature of Equation (3), lower than assumed. Nevertheless, Equation (3) seems to characterize the absolute RCS of trihedral corners despite some mechanical deformations with a standard uncertainty of not more than 0.2 dBm

^{2}.) Nevertheless, it cannot be proofed and further work should be conducted in determining the absolute knowledge of a trihedral corner reflector’s ERCS.

#### 4.3.3. Posterior Simulation

_{d}, μ

_{30}, etc.) which is most likely in describing the given data.

^{5}simulation runs were conducted, allowing for a burn in of 1 × 10

^{4}and a thinning of 20. These parameters were determined empirically by observing the traces and autocorrelations of the model parameter draws.

#### 4.3.4. MCMC Results

_{t}and its standard uncertainty. The transponder ERCS was estimated to be 60.80 dBm

^{2}with a standard uncertainty according to [7] of 0.206 dBm

^{2}. The 95% highest probability density interval is given as [60.38,61.17] dBm

^{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}.

#### 4.4. Posterior Predictive Checks: Model Verification

_{t,d}, test statistics T (y

_{t,d}) of the observed data can be compared to the test statistics of replicated data samples T ( ${y}_{t,d}^{\text{rep}}$), i.e., samples which were generated numerically through the model [12]. If this analysis is conducted for the four test statistics mean, standard deviation, minimum, and maximum value across all eight overpasses, Figure 14 results. A good model fit is found if the test statistic of the observed data (vertical line) lies close to the center of mass of the histogram. As a criterion, the p value (relative number of samples above observed test statistic) can be used. At a confidence level of 95%, the p value should therefore be within the range of 2.5% to 97.5%. This is observed for all four test statistics, and especially the most important aspect of the model, its mean, is well reproduced by the model with a p value close to ½.

#### 4.5. Plausibility Check with Classical Statistics

^{2}with a standard uncertainty of 0.20 dB or an expanded standard uncertainty of 0.41 dB at a confidence level of 95% (k = 2). The result confirms the findings of the previous section.

## 5. Discussion of Hierarchical Bayesian Data Analysis for Radiometric Calibration

## 6. Conclusions

- 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.

^{2}.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Artist’s rendering of DLR’s new C-band Kalibri transponder, mounted on a two-axis positioner.

**Figure 2.**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 Figure 3 reproduced with permission from Stamen Design, based on data from OpenStreetMap.)

**Figure 5.**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.

**Figure 6.**3.0 corner impulse response at site D24 for the first overpass on 7 April 2013. Also see caption of Figure 5.

**Figure 7.**1.5 corner impulse response at site D26 for the first overpass on 7 April 2013. Also see caption of Figure 5.

**Figure 8.**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.

**Figure 9.**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.

**Figure 10.**Diagram of the Bayesian model. The ellipsis symbol . . . indicates a family of probability distributions (per group g or overpass d), whereas ∼ means that a parameter is drawn from the respective distribution.

**Figure 11.**Estimated daily drifts r

_{d}of the RADARSAT-2 system. The error bars indicate 95% highest-probability density intervals.

**Figure 13.**Visualization of the Kalibri transponder drift compensation with data from Table 4. For visual guidance, the gray line marks the sample mean of the data before transponder drift compensation.

**Figure 14.**Posterior predictive checking for predicted (modeled) transponder data ${y}_{t,d}^{\text{rep}}$ and four different test statistics. Especially the most important aspect of the predicted data, its mean, is well modeled.

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 dBm^{2} | 9 |

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

**Table 3.**Cross parameters according to Figure 4 used during analysis.

Parameter | Pixels |
---|---|

Cross length | 21 |

Cross width | 3 |

Square width | 5 |

**Table 4.**Daily transponder RCS drift, estimated maximal error bounds on the estimated drift, and resulting standard uncertainties according to Equation (5).

Overpass Date | Estimated Drift μ_{s} (dB) | Maximal Error (dB) | Resulting σ_{s} (dB) |
---|---|---|---|

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 | −0.01 | 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 |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

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

**AMA Style**

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/rs5126667

**Chicago/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