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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.
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Remote Sens. 2013,5, 6667-6690; doi:10.3390/rs5126667
OPEN ACCESS
remote sensing
ISSN 2072-4292
www.mdpi.com/journal/remotesensing
Article
Hierarchical Bayesian Data Analysis in Radiometric SAR
System Calibration: A Case Study on Transponder Calibration
with RADARSAT-2 Data
Bj¨
orn J. D¨
oring *, Kersten Schmidt, Matthias Jirousek, Daniel Rudolf, Jens Reimann,
Sebastian Raab, John Walter Antony and Marco Schwerdt
Microwaves and Radar Institute, German Aerospace Center (DLR), Oberpfaffenhofen,
D-82234 Weßling, Germany; E-Mails: kersten.schmidt@dlr.de (K.S.); matthias.jirousek@dlr.de (M.J.);
daniel.rudolf@dlr.de (D.R.); jens.reimann@dlr.de (J.R.); sebastian.raab@dlr.de (S.R.);
john.walterantony@dlr.de (J.W.A.); marco.schwerdt@dlr.de (M.S.)
*Author to whom correspondence should be addressed; E-Mail: bjoern.doering@dlr.de;
Tel.: +49-8153-28-2246; Fax: +49-8153-28-1449.
Received: 19 October 2013; in revised form: 20 November 2013 / Accepted: 28 November /
Published: 4 December 2013
Abstract: 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.
Keywords: synthetic aperture radar; radiometric calibration; external calibration;
Bayesian data analysis; transponder
Remote Sens. 2013,56668
1. Introduction
A synthetic aperture radar (SAR) system is a measurement system that acquires measurement data for
earth-observation applications. Many applications like soil moisture [1] or forest biomass estimation [2]
require absolutely radiometrically calibrated data products. The absolute radiometric calibration ensures
that systematic effects have been characterized and are compensated so that the derived radar ground
reflectivity appears to be independent of the measurement system. Not until then can the data be used
for physical parameter inversion modeling.
The necessary radiometric calibration is a two-step process and is divided into relative and absolute
calibration [3,4]. During relative calibration, the system transfer function, antenna pattern, instrument
drift (due to temperature and time), antenna mis-pointing and mis-polarization, SAR mode (Stripmap,
ScanSAR, etc.), atmosphere, and processor effects are characterized and compensated. After relative
calibration, the mapped reflectivities are still only given as system indications, i.e., as digital numbers
derived during processing. The digital numbers are then related to the definition of a physical unit
through absolute calibration. Absolute calibration is, for practical reasons and an expected higher
calibration quality, executed as an external calibration for space-borne SAR systems [3]. Reference
point targets (typically corner reflectors or active radar calibrators [36]) with known radar reflectivities
are distributed within an imaged scene, and their backscatters are related to the indications from the
SAR system through a proportionality constant, often called calibration factor K. Usually the external
calibration is repeated for a number of acquisitions, each using a number of reference point targets,
in order to reduce random effects through averaging.
1.1. Problem Statement
The result of a measurement is only meaningful if, besides the estimate of the value of the measurand,
a statement about the uncertainty of the estimate is given. Uncertainty is defined here as “the parameter
which characterizes the dispersion of the values that could reasonably be attributed to the measurand” [7].
Stating uncertainties is the prerequisite for measurement results which are traceable to national standards.
Metrological traceability is necessary for the comparability of measurement results, the foundation for
quantitative scientific and commercial applications.
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 [5,6,811]
remain questionable. At the moment, a comprehensive uncertainty analysis, preferably in accordance
with the ISO Guide to the expression of uncertainty in measurement (GUM) [7], is lacking. This
diminishes the scientific integrity of the data products and hinders further adoption of radiometric
Remote Sens. 2013,56669
measurement data for advanced quantitative analysis and physical parameter inversion modeling, within
and across SAR missions.
1.2. Objective, Approach, and Paper Structure
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 including uncertainties and confidence intervals from SAR images. The proposed
method is also applicable for similar data analysis tasks in radiometric calibration.
Bayesian data analysis has been proven advantageous in many parameter estimation
problems [12,13], but to the best of our knowledge so far it has not yet been applied for absolute
radiometric SAR calibration. Bayesian analysis lends itself well to hierarchical data modeling [14], and
is therefore a very good fit for the estimation problems that one faces during an external SAR calibration
campaign. A general but short review of the methodology and an outline of its application in the domain
of absolute radiometric calibration is given in Section 2.
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 5and conclusions are given in Section 6.
1.3. Note on Point Target RCS versus ERCS
This paper adopts the reasoning laid out in [15]. According to it, a reference point target’s brightness
in a SAR image is not generally proportional to its radar cross section (RCS), which is a body property
depending on frequency and incidence angle. Rather, a target’s brightness in a SAR image depends on
the measured complex amplitudes and the weighted averaging (over the chirp pulse bandwidth and the
azimuth angles) executed by the SAR processor. This other quantity is called equivalent radar cross
section, or ERCS.
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 σbut with its alternative form ς.
Remote Sens. 2013,56670
2. Methodology for Parameter Estimation from SAR Data
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 [12,13,16]), and the
following aims to only sketch basic principles and to highlight its usefulness in the frame of absolute
radiometric SAR system calibration.
2.1. Introduction to Bayesian Statistics and Numerical Methods
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, i.e., data, to derive a posterior probability function.
If a population parameter (e.g., the case study transponder ERCS) is called θ, then it can be described
by a probability distribution function p(θ). Even before a measurement is done, a certain, subjective,
(prior) probability density can be attributed to the parameter. In case of the transponder’s ERCS, one
might claim that the parameter needs to be positive and that it certainly will not be above 100dBm2.
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 θgiven 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]:
p(θ|y) = p(θ)p(y|θ)
p(y)(1)
The likelihood function p(y|θ)describes how likely the data is, given the parameter θ.
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, i.e., the analysis is not limited to Gaussian distributions.
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 [12]. The posterior distribution
is approximated by sequentially drawing samples from approximate iterative intermediate distributions
until the simulation converges to the target distribution. Several software packages exist which simplify
computations; here the PyMC library is used [17].
2.2. Hierarchical Models
An external radiometric calibration campaign results in a pool of data samples (see [5] for an
example), one per reference point target ERCS per SAR image. Depending on the research question,
the data often needs to be grouped to estimate group-level parameters (for compensation) or to select
group-level models (for validation). Typical questions are:
Remote Sens. 2013,56671
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.
In fact, all questions previously raised can be conveniently answered by setting up a single joint
hierarchical model. Partial results contribute to the particular derivation of posterior distributions without
loss of information. On the other hand, a classical (frequentist) approach would require independent
analyses, and carrying over results from one model to the next would result in loss of information
(due to the summarizing nature of the classical (frequentist) approach).
Further details on hierarchical models can be found in [12].
3. Case Study: Measurement Campaign Goal and Setup
The following case study applies hierarchical Bayesian data analysis to a practical problem. This
Section 3describes the campaign goal and setup, whereas the following Section 4applies the
proposed method of hierarchical Bayesian data analysis for the estimation of a parameter in absolute
radiometric calibration.
3.1. Introduction and Goal
The DLR currently develops and manufactures a set of next-generation active radar calibrators
(transponders) [18], see Figure 1. These Kalibri transponders will be used in the upcoming C-band
Remote Sens. 2013,56672
Sentinel-1 SAR mission. The most critical transponder parameter is its radar reflectivity expressed in
ERCS because the transponder will be used as a calibration standard for absolute radiometric calibration.
Any error in the calibration of the calibration standards has a direct impact on the derived absolute
calibration factor for Sentinel-1. First ERCS measurements of the Kalibri prototype were performed in
DLR’s compact antenna test range.
Figure 1. Artist’s rendering of DLR’s new C-band Kalibri transponder, mounted on a
two-axis positioner.
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 including its confidence
interval. The data analysis, see Section 4, uses hierarchical Bayesian modeling as described above. The
research question answered in the case study, derivation of the transponder ERCS ςt, is conceptually
identical to the derivation of the absolute calibration factor K(linking SAR system indications Pto the
measurement quantity, (E)RCS) because only the roles of knowns and unknowns are reversed:
ςt=P
KK=P
ςt
(2)
Therefore, the approach used in the case study is equally applicable to the radiometric calibration of
SAR systems.
3.2. RADARSAT-2 Products
The RADARSAT-2 SAR system operates at a center frequency of 5.405GHz. The mode-dependent
bandwidth goes up to 100MHz. 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 Wide Ultra-Fine mode in VV polarization
and delivered as single-look complex (SLC) images. The products offer a high nominal single-look
Remote Sens. 2013,56673
geometric resolution of 1.6 m ×2.8m (range ×azimuth resolution) [19]. A high-resolution mode was
chosen as to increase the target-to-clutter ratio for the used point targets. Special care had to be taken in
choosing the dynamic range of the processed products so that the peak of the transponder pulse response
is still within the processed image’s dynamic range.
An overview of all overpass times and beams is shown in Table 1.
Table 1. Overview of approximate acquisition times (in UTC).
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
3.3. Reference Point Targets
Within this study, the RADARSAT-2 system was considered radiometrically uncalibrated because of
the mismatch between the system’s specified relative radiometric calibration (<1dB) [19] and the target
uncertainty of about 0.2dB. Therefore, reference point targets have been placed in the imaged scenes for
manual, day-to-day 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 5.405GHz 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 [15]. The peak RCS of an ideal corner reflector can be estimated
with the physical optics approximation
ς=4π
3
a4
λ2(3)
where ais the corner’s inner leg length and λis the wavelength [20]. The resulting values are listed in
Table 2.
Table 2. Corner reflectors with two different inner-leg lengths were used during
the campaign.
Size Peak RCS Number of Targets
1.5m 38.38 dBm29
3.0m 50.43 dBm26
Remote Sens. 2013,56674
3.4. Target Alignment
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 (i.e., at perfect alignment) is
taken as the reference value for the computation of the transponder ERCS. The corner reflector RCS
is comparably insusceptible to misalignments. On the other hand, the transponder’s antenna pattern falls
of more rapidly if a deviation from the main beam direction occurs, so more care needed to be taken
when a transponder was aligned.
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.
3.5. Imaged Area
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 1.5m 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
D26. The locations are shown in Figure 2. The 3.0 m corners were located within a larger rectangular
area of about 14km ×8km.
The site names for locations with a 3.0m corner are D24,D25, and D27 to D30. An overview of the
3.0m corner sites is shown in Figure 3.
4. Case Study: Data Analysis and Results
4.1. Overview
The RADARSAT-2 datatakes were processed by MDA and the final single-look complex (SLC)
images were the starting point for the data analysis.
Remote Sens. 2013,56675
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 3reproduced
with permission from Stamen Design, based on data from OpenStreetMap.)
Version November 28, 2013 submitted to Remote Sens. 9 of ??
Figure 2. Locations of the transponder on the DLR premises and of the 1.5 m corners on the
adjacent airport in Oberpfaffenhofen, Germany.1
Transponder
D26
D26a
D26b
D26c D26d
D26e
D26f
D26g
D26h
500 m
N48°40E11°160
N48°50E11°180
N
of about 14km×8km. The site names for locations with a 3.0 m corner are D24,D25, and D27 to D30.218
An overview of the 3.0m corner sites is shown in Fig. 3.219
4. Case Study: Data Analysis and Results220
4.1. Overview221
The RADARSAT-2 datatakes were processed by MDA and the final single-look complex (SLC)222
images were the starting point for the data analysis.223
The primary analysis goal is to derive the transponder ERCS and, equally importantly, an associated
uncertainty statement. In principal, the transponder ERCS ςtcan be derived if the ERCS of a reference
target (placed in the same scene) ςris known according to the proportionality
ςt=P
t
P
r
ςr.(4)
Here, P
tand P
rare the point target intensities of the transponder and the reference target, derived from224
the processed SAR images and possibly expressed as digital numbers.225
The analysis is split into two parts:226
1Map tiles in Fig. 2and 3reproduced with permission from Stamen Design, based on data from OpenStreetMap.
Figure 3. Map of imaged area, showing the locations of the 3.0 m corners.
Version November 28, 2013 submitted to Remote Sens. 10 of ??
Figure 3. Map of imaged area, showing the locations of the 3.0 m corners.
D24 D25
D27
D28
D29
D30
2 km
See Fig. 2
N47°580E11°110
N48°60E11°210
N
The primary analysis goal is to derive the transponder ERCS and, equally importantly, an associated
uncertainty statement. In principal, the transponder ERCS ςtcan be derived if the ERCS of a reference
target (placed in the same scene) ςris known according to the proportionality
ςt=P
t
P
r
ςr(4)
Here, P
tand P
rare the point target intensities of the transponder and the reference target, derived from
the processed SAR images and possibly expressed as digital numbers.
Remote Sens. 2013,56676
The analysis is split into two parts:
1. Point target analysis: Extract the relative point target impulse response powers for all point targets
in all scenes (see Section 4.2).
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
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 [21]. Thus the integral method was chosen for
this analysis.
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) beta naught
look-up table delivered together with the images [22]. It was confirmed that no obvious incidence-angle
dependence remained in the data.
The following steps were performed to derive a targets’ integrated impulse response intensity:
1. Define a search window around the point target in the georeferenced, processed image.
2. Find and record the brightest pixel location.
3. Define an analysis window, centered on the brightest pixel of the previous step.
4. Estimate the clutter power from four non-overlapping areas surrounding the peak.
5. Integrate the target power over a cross area (see Figure 4and Table 3), capturing all pixels with
significant point target power.
6. 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 [3].
Figure 4. Integration area for the integral method.
Version November 28, 2013 submitted to Remote Sens. 11 of ??
Figure 4. Integration area for the integral method.
Square width
Cross width
Cross length
Table 3. Cross parameters according to Fig. ?? used during analysis.
Parameter Pixels
Cross length 21
Cross width 3
Square width 5
1. Point target analysis: Extract the relative point target impulse response powers for all point targets227
in all scenes (see Sec. ??).228
2. Parameter estimation: Set up a statistical model to derive the estimated transponder ERCS and229
corresponding uncertainty from all datatakes (see Sec. ??).230
4.2. Power Estimation for Point Targets from SAR Images231
The imaged point targets appear as bright crosses in the processed image. Two methods are232
distinguished when deriving the point target power: the peak and the integral method. With the peak233
method, the target power is described by the pixel with the peak power (complex pixel amplitude234
squared) whereas with the integral method, the target power results as the sum of pixel powers over235
a relevant region around the target. It was shown that the integral method is advantageous as it leads to236
accurate results even if the image is not perfectly focused [?]. Thus the integral method was chosen for237
this analysis.238
The goal of this analysis step is to derive a table of relative point target intensities for all point targets239
in all images. Although an absolute scale is not necessary (and the RADARSAT-2 absolute calibration240
factor is ignored here), it is crucial that the point target intensities derived from different images are all241
calibrated with respect to each other. This is achieved by applying the (range-independent) beta naught242
look-up table delivered together with the images [?]. It was confirmed that no obvious incidence-angle243
dependence remained in the data.244
The following steps were performed to derive a targets’ integrated impulse response intensity:245
Remote Sens. 2013,56677
Table 3. Cross parameters according to Figure 4used during analysis.
Parameter Pixels
Cross length 21
Cross width 3
Square width 5
An exemplary transponder impulse response is shown in Figure 5. The peak power lies more
than 60dB above the clutter level. This large separation is already an indication that clutter power
compensation, in practice, is not necessary. Nevertheless, it was performed for all 126 analyzed
point targets.
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.
Exemplary impulse responses for 3.0 m and 1.5 m corners are show in Figures 6and 7, respectively.
Although the impulse response is clearly visible in the range and azimuth cuts, both corners are
comparatively hard to spot in the image patches themselves. This is due to the way the intensities in
the image patches have been clipped for visualization. All digital values above a threshold (2.7 times the
mean absolute amplitude within the patch) are set to the threshold value.
Remote Sens. 2013,56678
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.
Remote Sens. 2013,56679
4.3. Bayesian Statistics and Hierarchical Model Fitting
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 Figure 8. This is the input data to derive the
transponder ERCS and its confidence interval.
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.
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 Equation (4). The final, combined
uncertainty shall include the uncertainties which are incorporated in each step. This problem is solved
with Bayesian statistics, which directly allows to estimate the uncertainty for all derived parameters.
4.3.1. Daily RADARSAT-2 and Transponder Drifts
A plot of the original data from the first processing step was show in Figure 8. On first impression,
the data non-uniformity is dominated by a daily systematic drift. For instance, independent of the
radar target group (transponder, 3.0m, or 1.5 m corners), a radiometric drift of more than 0.5 dB can
Remote Sens. 2013,56680
be observed between the last and the previous overpass. Therefore, a daily RADARSAT-2 radiometric
drift compensation had to be included in the analysis.
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 Figure 9.
The dashed red line in the upper plot defines the point in time when the transponder RCS was corrected
to its nominal RCS. The blue markers, on the other hand, indicate times at which the transponder RCS
was merely estimated. In the case of Figure 9, the drift at the overpass time can be estimated to be
nonexistent (0.0dB). An upper bound on the uncertainty of this estimated drift can be stated from the
plot as 0.02dB, i.e., the true drift should lie, with high probability, in the range [0.02,0.02]dB.
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 0dB 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.
A similar assessment has been performed for all other days. The estimated transponder drifts and their
estimated error bounds are listed in Table 4. A higher drift and/or error is stated for overpasses when a
transponder software problem prohibited a nominal, temperature controlled operation of the prototype.
4.3.2. Hierarchical Bayesian Model
The Bayesian model requires the definition of priors, i.e., probability distributions for all model
parameters. These shall be defined in the following.
Remote Sens. 2013,56681
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 µs(dB) Maximal Error (dB) Resulting σs(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
For the eight overpass days d, eight different daily RADARSAT-2 drifts rdneed to be determined.
It is estimated from Figure 8that the daily drift, expressed as a scaling factor, will certainly be in the
range 0.4 to 1.6 (i.e.,4dB to 2 dB). The prior can thus be written as rdU(0.4,1.6), where U(a,b)
describes a uniform distribution with lower and upper bounds aand b, respectively.
Besides the RADARSAT-2 drift, the transponder drift needs to be modeled. The values in column
three of Table 4define maximum error bounds, i.e., a lower bound aand an upper bound b. According
to [7], this information can be converted to a standard uncertainty and therefore a normal distribution,
where the standard deviation σis given as
σ=1
12(ba)(5)
Hence, the daily transponder drift sdis modeled as a normal distribution N(µ,σ), where µdescribes
its mean and σits standard deviation: sdN(µs,d,σs,d). The best estimate µs,dis taken from Table 4,
where also the σs,d(resulting from Equation (5)) are listed.
It is assumed that all measured point target powers of all targets within one target group g
(transponder t, 3.0 “30”, and 1.5 corners “15”) belong to the same population, which can be described
by a normal distribution N(µg,σg), having the group location (mean) µgand group scale (standard
deviation) σg>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 µgand
σgare modeled to originate from uniform distributions: µgU(101.5,107)(i.e., U(15dB,70 dB)) and
σgU(0,106). In a way, the σgare nuisance parameters which are not required to derive the transponder
ERCS, but they are nevertheless necessary in order to describe the normal distributions Ng.
Now, for every overpass d, the data yd,gis fitted depending on its group g[t,15,30]:
yd,30 N(rdµ30,σ30)
yd,15 N(rdµ15,σ15)
yd,tN(rdsdµt,σt)(6)
Remote Sens. 2013,56682
This way, the daily drift parameter rdis estimated from all available data, exploiting the fact that the drift
should be equal for data across all three groups.
Estimating parameters with the model Equation (6) results in estimates for the (relative) point target
powers for each group (µt,µ30, and µ15 ) 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: σ15 =0.15 dBm2and σ30 =0.41dBm2.
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 ς15 N(103.838 m2,100.02 m2)(i.e.,
N(38.38dBm2,0.2 dBm2)). 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 0.2dBm2is 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 0.2dBm2is 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 12.05dBm2.
The difference between the estimated mean target powers (in MCMC parameters: µ30 µ15 ) is 11.92 dB,
i.e., 0.13dB 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
0.2dBm2.) 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
ςt=µt
µ15
ς15 (7)
according to the measurement model Equation (4).
The complete Bayesian model described above is visualized in Figure 10.
4.3.3. Posterior Simulation
The hierarchical model developed in the previous section is now solved iteratively with the numerical
Markov chain Monte Carlo method (MCMC) [12,13]. The goal is to find the most probable set of
parameters (e.g., rd,µ30,etc.) which is most likely in describing the given data.
Remote Sens. 2013,56683
Figure 10. Diagram of the Bayesian model. The ellipsis symbol . . . indicates a family of
probability distributions (per group gor overpass d), whereas means that a parameter is
drawn from the respective distribution.
Version November 20, 2013 submitted to Remote Sens. 18 of 26
Figure 10. Diagram of the Bayesian model. The ellipsis symbol . . . indicates a family of
probability distributions (per group gor overpass d), whereas means that a parameter is
drawn from the respective distribution.
µg
U(15dB,70dB)
mean target group power
...
g[15,30,t]
rd
U(0.4,1.6)
RADARSAT-2 drift
...
d[1,...,8]σg
U(0,106)
std. dev. of target group power
...
g[15,30,t]
N(rd·µg,σg)
1.5 m and 3.0m corners
...
d[1,...,8]
g[15,30]
N(rd·µt·sd,σt)
transponder
...
d[1,...,8]
yg,dyt,d
ς15
N(38.38dB,0.20dB)
RCS of 1.5 m corners
sd
N(µs,d,σs,d)
daily transponder drift
ςt=µt·µ1
15 ·ς15
··· ···
···
··· ··· ···
···
···
···
∼ ∼
see Tab. 4
see Eq. 3and
Sec. 4.3.2.
see Eq. 7
see Fig. 11
for posterior
see Fig. 8
Standard
uncertainties
Result
Parameters
Model
Data
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 burn in period should be considered. To improve the (required) independence between
successive draws, only every nth simulation draw is considered, i.e.,thinning is applied.
In order to compute the parameters of the hierarchical model, which was presented in the previous
chapter, 2 ×105simulation runs were conducted, allowing for a burn in of 1 ×104and 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
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 Figure 11. The
error bars indicate the range of values which defines the 95% probability intervals. From this it can be
seen that the drift between the first and the second overpass is statistically not significant, but the drift
between the last two overpasses, for instance, is. The drift was estimated with a standard uncertainty
(according to [7]) of always less than 0.1dB.
This estimated RADARSAT-2 drift can now be applied to the measured data. The original data in
Figure 8appears now much more uniform, see Figure 12. No apparent systematic drift remains.
Remote Sens. 2013,56684
Figure 11. Estimated daily drifts rdof the RADARSAT-2 system. The error bars indicate
95% highest-probability density intervals.
Figure 12. Measured impulse response powers with RADARSAT-2 drift compensation (see
Figure 11) applied. The dispersion within one target group is now greatly reduced with
respect to Figure 8.
Remote Sens. 2013,56685
After RADARSAT-2 drift compensation, the estimated transponder drift can be taken out from the
upper plot in Figure 12. The plot in Figure 13 results. Already visually it becomes clear that the
transponder drift is small in comparison to the remaining dispersion of the measurement data. This
can also explain why at times the drift compensation results in updated values which lie further away
from the population mean.
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.
Now the most important result of the MCMC simulation is the derivation of the transponder ERCS
ςtand its standard uncertainty. The transponder ERCS was estimated to be 60.80 dBm2with a standard
uncertainty according to [7] of 0.206dBm2. The 95% highest probability density interval is given as
[60.38,61.17]dBm2. Note that the standard uncertainty is clearly dominated by the assumed ERCS
knowledge uncertainty of the 1.5 corners of 0.2dBm2.
4.4. Posterior Predictive Checks: Model Verification
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 yt,d, test statistics T(yt,d)of the observed data can be compared
to the test statistics of replicated data samples T(yrep
t,d),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 pvalue (relative number of samples above observed test statistic) can be used. At
a confidence level of 95%, the pvalue 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 pvalue close to ½.
Remote Sens. 2013,56686
Figure 14. Posterior predictive checking for predicted (modeled) transponder data yrep
t,dand
four different test statistics. Especially the most important aspect of the predicted data,
its mean, is well modeled.
Version November 28, 2013 submitted to Remote Sens. 23 of ??
Figure 14. Posterior predictive checking for predicted (modeled) transponder data yrep
t,dand
four different test statistics. Especially the most important aspect of the predicted data, its
mean, is well modeled.
55.4 55.6 55.8 56 56.2
pvalue =0.48
T(yt,d) = mean(yt,d)[dB]
40 42 44 46
pvalue =0.96
T(yt,d) = std(yt,d)[dB]
54.8 55 55.2 55.4 55.6 55.8
pvalue =0.03
T(yt,d) = min(yt,d)[dB]
55.8 56 56.2 56.4 56.6 56.8 57
pvalue =0.73
T(yt,d) = max(yt,d)[dB]
the 1.5 m corners of 0.2 dB through the method of “root-sum-square”. Lastly, the expanded uncertainty [?382
] with a coverage factor of k=2 is derived.383
This approach results in an estimated transponder ERCS of 60.80 dBm2with a standard uncertainty384
of 0.20 dB or an expanded standard uncertainty of 0.41 dB at a confidence level of 95 % (k=2). The385
result confirms the findings of the previous section.386
Note that once again the combined uncertainty is dominated by the assumed ERCS knowledge387
uncertainty of the 1.5 m corner reflectors. The transponder’s combined backscatter uncertainty is388
sufficiently low to permit the calibration of SAR instruments like Sentinel-1 to an absolute radiometric389
uncertainty of 1.00 dB at a confidence level of 99%, provided that the SAR instrument is otherwise390
sufficiently precise [?].391
5. Discussion of Hierarchical Bayesian Data Analysis for Radiometric Calibration392
The advantages of using hierarchical Bayesian data anlysis for radiometric calibration were laid out393
before: The approach can jointly answer typical analysis questions in radiometric calibration while fully394
exploiting the hierarchical nature of external calibration data and fulfilling the requirement on reporting395
measurement uncertainties and confidence intervals. This section shall add a critical discussion of the396
proposed method.397
First, the most recognized approach in metrology for deriving measurement uncertainties is the ISO398
Guide to the expression of uncertainty in measurements (GUM) [?]. The GUM, which was introduced399
in 1993, fundamentally applies classical (frequentist) statistics, and is not directly compatible with a400
Bayesian approach (which was proposed in this paper). Nevertheless, the current GUM approach has401
been repeatedly criticized, and Bayesian methods have been proposed as a consistent and universally402
applicable replacement [? ? ? ]. In practice, uncertainties derived by Bayesian statistics are often equal403
4.5. Plausibility Check with Classical Statistics
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 [7]. The third step
is then to derive a combined standard uncertainty by incorporating the ERCS knowledge uncertainty of
the 1.5 corners of 0.2dB through the method of “root-sum-square”. Lastly, the expanded uncertainty [7]
with a coverage factor of k=2 is derived.
This approach results in an estimated transponder ERCS of 60.80dBm2with a standard uncertainty of
0.20dB 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.
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 1.00dB at a confidence level of 99%, provided that the SAR instrument is otherwise
sufficiently precise [23].
Remote Sens. 2013,56687
5. Discussion of Hierarchical Bayesian Data Analysis for Radiometric Calibration
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
Guide to the expression of uncertainty in measurements (GUM) [7]. The GUM, which was introduced
in 1993, fundamentally applies classical (frequentist) statistics, and is not directly compatible with a
Bayesian approach (which was proposed in this paper). Nevertheless, the current GUM approach has
been repeatedly criticized, and Bayesian methods have been proposed as a consistent and universally
applicable replacement [2426]. In practice, uncertainties derived by Bayesian statistics are often equal
or approximately equal to uncertainties derived by classical statistics [25] so that both approaches lead
to desired results. Yet thanks to the availability of numerical tools, Bayesian computations are now often
simpler than classical approaches if hierarchical data is analyzed [14]. Therefore, taking a Bayesian
approach for deriving the measurement uncertainty of the absolute calibration factor seems justified.
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.
6. Conclusions
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
Remote Sens. 2013,56688
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
0.21dBm2.
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 traceable radiometric SAR system calibration. Uncertainties and confidence intervals can now
be derived with the same diligence that is presently applied to the estimation of calibration parameters.
Acknowledgments
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 D¨
ohring 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.
Conflicts of Interest
The authors declare no conflict of interest.
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© 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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... Different methods were analytically compared [8] and conducted in order to increase calibration confidence through cross-comparisons. Thus, a standard uncertainty of 0.2 dB has been achieved [9][10][11]. • DLR's remote controlled trihedral corner reflector is shown in Figure 2b. This corner reflector is turned up-side down, allowing the opening facing downwards to be rotated when the corner reflector is not operated. ...
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... To repeat, the major source for measurement uncertainty stems from the multipath effect which was attributed an uncertainty of 0.375 dB. Could this be suppressed due to a better measurement setup, the remaining standard uncertainty would have been only 0.08 dB, which is a significant improvement of the currently claimed transponder RCS uncertainty of 0.2 dB [3]. ...
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Preface; Acknowledgements; 1. Role of probability theory in science; 2. Probability theory as extended logic; 3. The how-to of Bayesian inference; 4. Assigning probabilities; 5. Frequentist statistical inference; 6. What is a statistic?; 7. Frequentist hypothesis testing; 8. Maximum entropy probabilities; 9. Bayesian inference (Gaussian errors); 10. Linear model fitting (Gaussian errors); 11. Nonlinear model fitting; 12. Markov Chain Monte Carlo; 13. Bayesian spectral analysis; 14. Bayesian inference (Poisson sampling); Appendix A. Singular value decomposition; Appendix B. Discrete Fourier transforms; Appendix C. Difference in two samples; Appendix D. Poisson ON/OFF details; Appendix E. Multivariate Gaussian from maximum entropy; References; Index.