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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An overview of increasingly indispensable radiometric technologies
Microwave radiometers have become a central part of Earth observation and radioastronomy. Most existing reference works on the subject, however, largely omit two key types of radiometers: interferometric radiometers and polarimetric radiometers. The extensive applications of these two classes of radiometer and their potential for mapping distant celestial bodies and enhancing Earth observation has made it critical for the next generation of radiometric scientists and engineers to be familiar with this technology and its principles.
Principles of Interferometric and Polarimetric Radiometry meets this crucial need with a first-in-class overview of this key subject. Beginning with an introduction to the foundational concepts of microwave radiometry, it proceeds to work through a careful revision of the field’s major theory and techniques, with a particular emphasis on interferometric and polarimetric systems. The result promises to revolutionize the use of microwave passive sensors for Earth observation and beyond.
Principles of Interferometric and Polarimetric Radiometry readers will also find:
- Broad approach that can be brought to bear in any area of microwave radiometry
- Detailed discussion of topics including stochastic processes, analytic signals, microwave networks, and many more
- Extensive appendices incorporating key mathematics and special functions
Principles of Interferometric and Polarimetric Radiometry is ideal for graduate or advanced undergraduate courses in radiometry and microwave remote sensing.
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Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
Foreword
About the Author
Preface
Acknowledgments
1 Signals, Receivers, and Antennas
1.1 Random Variables, Real and Complex
1.2 Stochastic Processes
1.3 Analytic Signals
1.4 Phasors of Random Signals
1.5 Microwave Networks
1.6 Antennas
References
Notes
2 Microwave Radiometry
2.1 Thermal Emission
2.2 Polarization
2.3 Antenna Temperature
2.4 Total Power Radiometers
References
Notes
3 Interferometry and Polarimetry
3.1 Historical Perspective
3.2 A Single Baseline
3.3 The Visibility Equation
3.4 Correlation Measurement
References
Notes
4 Aperture Synthesis
4.1 Synthetic Beam
4.2 Radiometric Sensitivity
4.3 Spatial Sampling
4.4 Imaging
References
Notes
5 Instrument Techniques
5.1 Frequency Conversion
5.2 In-phase and Quadrature (IQ) Mixer
5.3 Quarter Period Delay
5.4 Digital Techniques
References
Notes
6 Calibration and Characterization
6.1 Calibration Standards
6.2 Parameter Retrieval
6.3 Nonlinearity
6.4 Calibration Rate
References
Note
A Definitions and Concepts
A.1 Complex Vectors
A.2 Useful Complex Number Identities
A.3 Energy Conservation and Unitary Matrix
A.4 Spherical Coordinates and Solid Angle
A.5 Quadrature Equation Inversion
A.6 Special Functions
A.7 Fourier Transform
A.8 Discrete Fourier Transform
Reference
Notes
Index
End User License Agreement
List of Tables
Appendix
Table A.1 Fourier transform properties.
Table A.2 Fourier transform pairs.
List of Illustrations
Chapter 1
Figure 1.1 Qualitative examples of stochastic processes. (a) Stationary. (b)...
Figure 1.2 Impulse response of an integrator and corresponding amplitude squ...
Figure 1.3 Quadrature filter symbol (a) and argument of the frequency respon...
Figure 1.4 Analytic signal and complex envelope in the frequency domain.
Figure 1.5 A single circuit loop for illustrating the use of voltage waves.
Figure 1.6 (a) Waves definition in a two-port network. (b) Thevenin equivale...
Figure 1.7 One-port linear network to show the thermal noise representation ...
Figure 1.8 Interconnection between two one-port noisy networks at different ...
Figure 1.9 Definitions of noise in a two-port network. (a) Equivalent temper...
Figure 1.10 Setup for computing the noise equivalent temperature at output
Figure 1.11 Flux diagram to illustrate the cascade connection of networks.
Figure 1.12 Microwave receiver. (a) S-parameter representation. (b) Linear s...
Figure 1.13 (a) Waves definitions in an N-port microwave network. (b) Matche...
Figure 1.14 (a) Elevation and azimuthal unit vectors. (b) Ludwig third defin...
Figure 1.15 An antenna radiating a noise signal.
Figure 1.16 Antenna model in reception using a voltage generator with intern...
Figure 1.17 (a) Antenna located in the -axis off the center of coordinates ...
Figure 1.18 Link formed by a transmitting and a receiving antenna.
Figure 1.19 Modeling a lossy antenna with a lossless antenna and a passive n...
Chapter 2
Figure 2.1 (a) Extended source of thermal radiation observed at a distant po...
Figure 2.2 Solid angles and areas used to define the spectral brightness. (a...
Figure 2.3 Linear polarization frame rotation.
Figure 2.4 Definition of the ground reference plane. The incidence angle is
Figure 2.5 An antenna collecting energy from an extended source of radiation...
Figure 2.6 (a) Antenna surrounded by a constant source (flat target). (b) An...
Figure 2.7 Narrow beam antenna pointing to and extended source.
Figure 2.8 A scanning directive antenna.
Figure 2.9 Conceptual block diagram of a total power radiometer. The receive...
Figure 2.10 (a) Normalized frequency response (amplitude). (b) Amplitude of ...
Figure 2.11 (a) Spectrum of instantaneous power. (b) Relative standard devia...
Figure 2.12 Power spectral density of the receiver output signal (a) and of ...
Figure 2.13 A quadratic detector. (a) Block diagram. (b) Example of circuit ...
Chapter 3
Figure 3.1 Interferometer geometry.
Figure 3.2 Example of the Fourier relationship in a 1-D interferometer imagi...
Figure 3.3 Two antennas collecting energy from an extended source of radiati...
Figure 3.4 A dual polarization antenna using an orthomode transducer (OMT). ...
Figure 3.5 Two antennas pointing to a source of thermal radiation and corres...
Figure 3.6 Simulation of the system visibility (amplitude). (a) Antenna dire...
Figure 3.7 Conceptual block diagram of cross-correlation measurement.
Figure 3.8 Implementation of a complex multiplier.
Figure 3.9 Analog multiplier of real signals. (a) Conceptual block diagram a...
Figure 3.10 Measurement of the real part of the normalized correlation using...
Figure 3.11 Numerical evaluation of (3.150) and (3.163) as a function of cen...
Chapter 4
Figure 4.1 Synthetic beam for a rectangular instrument using a rectangular a...
Figure 4.2 (a) Synthetic beam for a rectangular instrument with Blackman apo...
Figure 4.3 Synthetic beams for hexagonal sampling. (a) Rectangular window. (...
Figure 4.4 Cuts of the synthetic beams of figure 4.3. (a) Rectangular window...
Figure 4.5 Interferometric radiometer with rectangular layout. (a) Antenna p...
Figure 4.6 Reciprocal () and () grids for the instrument of figure 4.5 wit...
Figure 4.7 (a) An example of modified brightness temperature image in the wh...
Figure 4.8 Example of Y-shape instrument. (a) Antenna layout. (b) coverage...
Figure 4.9 Grid points in the () plane. (a) Rhomboidal principal period and...
Figure 4.10 Grid points in the principal hexagon of for hexagonal sampling...
Figure 4.11 Aliasing in hexagonal grids. (a) Original image. (b) Recovered i...
Figure 4.12 Example of hexagonal instrument. (a) Antenna layout. (b)
u
-
v
cov...
Figure 4.13 Cuts for of synthetic beams for hexagonal sampling comparing Y...
Figure 4.14 Cuts of the synthetic beams computed from the G-matrix of the MI...
Chapter 5
Figure 5.1 (a) Block diagram of a frequency converter. (b) Sketch of the sig...
Figure 5.2 Image rejection mixer. Setup (a) rejects the lower band while (b)...
Figure 5.3 In-phase and quadrature mixer to separate the real and imaginary ...
Figure 5.4 A quarter period delay used as quadrature filter. The period corr...
Figure 5.5 Impact of numerical averaging of the integrated output signal on ...
Figure 5.6 Instantaneous power (a) and correlation (b) sampling.
Figure 5.7 Digital implementation of a square law device (a) and a quadratic...
Figure 5.8 Digital correlator using either multibit or one bit analog-to-dig...
Figure 5.9 Noise excess due to numerical averaging with respect to analog in...
Figure 5.10 Normalized effective integration time as defined by (5.74). Plot...
Figure 5.11 Same as figure 5.9 for systems with high-frequency content on th...
Figure 5.12 Same as 5.10 for systems with high-frequency content on the inst...
Figure 5.13 Noise excess metric and effective integration time for total pow...
Figure 5.14 Digital correlator using alternate sampling to implement the qua...
Figure 5.15 Spectrum replication due to sampling.
Figure 5.16 Power spectral density of the signal before sampling. The three ...
Figure 5.17 Same as figure 5.11 for a higher value of center frequency to il...
Figure 5.18 Conceptual block diagram of an interferometer that uses a correl...
Chapter 6
Figure 6.1 Definition of antenna and calibration planes. Connection of both ...
Figure 6.2 Antenna front-end model used to describe the transfer of an inter...
Figure 6.3 A probe antenna as an external calibration standard for interfero...
Figure 6.4 Matched load as internal calibration standard. Its equivalent noi...
Figure 6.5 Setup to distribute correlated noise to two receivers.
Figure 6.6 Brightness temperature of the sky. The black line delimits the fr...
Figure 6.7 Normalized fringe washing function computed from two measured fre...
Figure 6.8 Linear input–output radiometer response to interpret calibration....
Figure 6.9 Setup for measuring the instrumental offset.
Figure 6.10 Graphical interpretation of the one-point method to retrieve the...
Figure 6.11 Setup for measuring the receiver’s nonlinearity.
Figure 6.12 Example of the linearity correction with simulated data. (a) Fir...
Figure 6.13 Two averaging strategies for calibration.
Appendix
Figure A.1 Definition of spherical system of coordinates and differential su...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Foreword
About the Author
Preface
Acknowledgments
Begin Reading
A Definitions and Concepts
Index
End User License Agreement
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IEEE Press445 Hoes LanePiscataway, NJ 08854
IEEE Press Editorial BoardSarah Spurgeon, Editor-in-Chief
Moeness AminJón Atli BenediktssonAdam DrobotJames Duncan
Ekram HossainBrian JohnsonHai LiJames LykeJoydeep Mitra
Desineni Subbaram NaiduTony Q. S. QuekBehzad RazaviThomas RobertazziDiomidis Spinellis
Principles of Interferometric and Polarimetric Radiometry
Ignasi Corbella
Universitat Politècnica de Catalunya
Barcelona
Spain
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Library of Congress Cataloging-in-Publication Data is applied for:
Hardback ISBN 9781394255108
Cover Design: WileyCover Image: © ESA – P. Carrill
To Roser, Pau and Oriol. And also to my parents Leopoldo and Isabel
Foreword
To my knowledge, this is the first book ever published dedicated to the topic of aperture synthesis (or interferometry) for the remote sensing of the Earth. Much of its contents can, however, be applied to other types of microwave radiometers and applications. The author, Prof. Ignasi Corbella of the Polytechnic University of Catalunya (UPC), Barcelona (Spain), is, in my humble opinion, the engineer, among all I know, who best understands aperture synthesis for Earth Observation, a field that started in the 1980s. Readers of this book will have the privilege to learn directly from him.
In fact, I started learning microwave theory having Ignasi as teacher at UPC back in 1983, and he was also the supervisor of my master’s thesis in 1986, and of my PhD in 1996. Therefore, it is an honor to write this foreword to his book some 41 years after I attended his lessons at UPC. From 1993 till 1999, he was involved in several studies and experiments to develop the Microwave Imaging Radiometer with Aperture Synthesis (MIRAS) within contracts with the European Space Agency (ESA) I was responsible for. Since 2000 till today, Prof. Corbella has been part of the team supporting, in many aspects, ESA’s Soil Moisture and Ocean Salinity (SMOS) mission, which carries MIRAS as its only payload. It is the SMOS mission, in which I am involved from ESA side, which has allowed us to enjoy this very long professional link and friendship.
The development of MIRAS is a story truly worth telling. It all started with the work of David LeVine (NASA-GSFC) and Carl Swift (University of Massachusetts at Amherst) in the late 1980s on the Electronically Scanning Thinned-Array Radiometer (ESTAR). ESTAR was a one-dimensional interferometer presenting a new technological solution to deploy a very large L-band antenna in space, to make observations in the 1400–1427 MHz protected band, to provide global maps of soil moisture and ocean salinity. Inspired by this research, and following some work carried out by the French Space Agency (CNES), in 1992, ESA decided to initiate the development of a two-dimensional interferometer: MIRAS.
During the first 10 years of MIRAS, Prof. Corbella was involved in the theoretical aspects, the calibration, and the image processing of the instrument. In principle, although aperture synthesis had never been applied to remote sensing, it was a matter of translating the knowledge accumulated in radio-astronomy to the field of Earth Observation. Given this, we were all quite confident that the new type of radiometer should work very finely.
Radio-astronomy was (and is) based on a theorem that was first formulated by the Dutch physicist Pieter Hendrik van Cittert in 1934, and four years later proved in a simpler way by another Dutch physicist, Frits Zernike. Zernike was a prominent scientist who was awarded the Nobel Prize in 1953 for having invented the Phase-Contrast Microscope. The so-called Van Cittert-Zernike theorem, or in short, the VCZ theorem, was the theoretical basis of radio-astronomy, and in turn, of ESA’s initiative to embark on two-dimensional aperture synthesis for remote sensing.
Nobody would have anticipated that such solid theoretical grounds were going to be shaken by the development of MIRAS, to the point of proving them completely inadequate for the application of interferometry to Earth Observation.
Based on the so-far successful breadboarding of MIRAS subsystems and the scientific need to monitor soil moisture and ocean salinity, ESA’s Earth Science Advisory Committee recommended the implementation of the SMOS mission on April 27, 1999, as second Earth Explorer Opportunity Mission.
By 2002, the breadboarding activities (MIRAS Demonstrator Pilot Project – MDPP) were running in parallel with the Phase B of the SMOS project. The prototype built within the MDPP consisted of three linear arrays of four receivers each which could be mechanically moved along three horizontal tracks 120° equi-spaced from each other. The MDPP also included the optical harness, the correlator, and the internal calibration subsystems.
In December 2002, the MDPP demonstrator was brought inside the Electro Magnetic Compatibility (EMC) chamber of INTA (Instituto Nacional de Técnica Aeroespacial, Madrid) for the very first end-to-end test. The imaging algorithm, based on the VCZ theorem, was predicting high correlation values due to the high emissivity of the microwave absorber material (at room temperature) of the walls of the EMC chamber. To our surprise, the correlations were two orders of magnitude (!) smaller than those predicted by the VCZ theorem. After carefully assessing the breadboard, and realizing that the hardware was working well, we had to accept that the value of the correlations was real. As mentioned, SMOS was already in Phase B …
The UPC team, led by Prof. Corbella, was responsible for processing the data from the INTA test. Therefore, we all packed the hardware and went on Christmas Holidays with the heavy weight of the insignificant correlation values on our shoulders…
It was during the Christmas of 2022 that Ignasi set himself into resolving the mystery. He found a paper on noise waves and passive linear multiports of 1991 which gave the expression of the correlation of the outgoing noise waves of a passive microwave circuit. Such problem resembled the configuration of the MDPP breadboard inside the EMC chamber. The article referred to a theorem stated by a Dutch engineer, Hendrik Bosma, in his PhD thesis of 1967. According to Bosma’s theorem, the correlation of the noise waves coming out of a passive microwave circuit in thermal equilibrium with its terminations, as the MDPP breadboard inside the EMC chamber, had to null.
Based on these findings, Prof. Corbella carefully reformulated the basic measurement of interferometry, the correlation between the output signals of two receivers forming a baseline, considering the presence of the neighboring receivers as well as the physical temperature of the receivers and the target. By February 2003, he had come with a new equation, more general than the CVZ theorem, which could explain the correlations in the INTA chamber as well as those measured by radio-telescopes (for large antennas and spacings, in wavelength units, the new formulation reduced to the CVZ theorem). Later the same year, Prof. Corbella had derived the polarimetric version of his equation. The new equation, which I very deservedly coined as the Corbella Equation, was first published in TGARS, August 2004.
The fact that the VCZ theorem could not explain all interferometric experiments, like the test of the MDPP breadboard in the EMC chamber (by two orders of magnitude), was further evidenced when the opposite test was carried out: the imaging of the Cold Sky at the Dwingeloo radio-observatory facility in The Netherlands. In the Dwingeloo experiment, four receivers were manually moved along three arms to get an image of the Cold Sky. While the VCZ theorem was predicting now very low correlations based on the low brightness temperature of the Cosmic Microwave Background Radiation (CMBR), around 2.7 K, the expected values according to the Corbella equation were two orders of magnitude larger in this case. The measurements were perfectly in accordance with Corbella’s predictions and, once again, about two orders of magnitude away from the VCZ theorem-predicted values.
The Corbella Equation was then adopted to process the data of the SMOS mission instead of the VCZ theorem. The calibration approach had to be adapted, and, fortunately, the Flat Target Response of the instrument could be obtained by using the CMBR and the subsequent Flat Target Transformation could be devised to remove the −Tr term Corbella had introduced into the VCZ equation. With the data processing and calibration following his equation, Ignasi had brought the SMOS project back on track half way through Phase B.
It is worth noting that it took some time for some engineers to accept and understand the profound implications of the “Corbella Equation.” After all, the VCZ theorem, established by renowned (Nobel Prize awarded) scientists, had been working perfectly well in radio-astronomy for decades. It was only a matter of making new experiments in new conditions (as taking the MDDP breadboard inside the INTA EMC chamber), realizing of other scientists’ research (as Bosma’s PhD thesis), and having a clear and humble mind to put the pieces of the puzzle together. This ultimately culminated in a scenario where David beat Goliath with a new more general formulation of aperture synthesis.
This book starts with a tedious Chapter 1 on probability concepts and stochastic processes, Hilbert transform, and analytic signals. However, the usefulness of its contents will be much appreciated throughout the rest of the book, as they will be frequently called upon to make progress in the formulation of the different flavors of interferometers.
Chapter 2 introduces the basic concepts of radiometry. Prof. Corbella has made an effort to explain interferometry and all its equations in a way that collapses to the case of a conventional total power radiometer. In this respect, it makes interferometry easier to digest for engineers knowledgeable in conventional radiometry.
The Corbella Equation is derived in Chapter 3. This chapter can be considered central and including the nucleus of the aperture synthesis theory applied to remote sensing. An engineer confronted with the design of this type of radiometers should absorb these contents thoroughly. This chapter introduces also the tools necessary to derive the sensitivity of an interferometer, and the subtleties which occurr when digitizing the output signals.
The discussion on sensitivity continues more deeply in Chapter 4, which focuses on the image processing. The basic theory on how to retrieve the brightness temperature from the visibility function are explained in great detail. Key concepts such as the reciprocal grids, the apodization window, the G-matrix, the aliasing, and the noise floor are found here.
The implementation of the interferometry theory into practical instruments is covered by Chapter 5. The nomenclature of this chapter is dense, but precise and consistent with the previous chapters. The reader is invited to understand and carefully realize the subtle differences between the different instrument architectures, including the effects of digitalization of the signals and the different ways complex correlations can be obtained (whether using two or four multipliers, with or without digitalization).
Finally, Chapter 6 gives the reader an overview on the characterization and calibration of an interferometric radiometer, based on the Corbella Equation. The key parameters which have to be found to correct the raw visibilities into a calibrated set are described, together with the methods to retrieve their value.
For every new topic in engineering and science, there is always a “first reference book,” a “bible.” I believe this book is the “bible” on polarimetric aperture synthesis for remote sensing. This book will be tremendously useful for engineers working with this type of instruments, as well as a valuable reference for a university course on interferometry for Earth Observation.
I can see a future for aperture synthesis in remote sensing, as demonstrated by the SMOS mission and, in great part, thanks to Prof. Corbella’s contributions. Interferometry combines well with formation flying as a way to deploy extremely large apertures. When the techniques to master formation flying will be sufficiently developed, interferometry will spread further. I do hope graduated students as well as practicing engineers will find in this book a great support for decades to come.
Oegstgeest 26 June, 2024(14 years, 7 months, and 24 days fromSMOS launch, still up and running)
Manuel Martín-Neira
Senior Microwave Radiometer EngineerEuropean Space Agency
About the Author
Ignasi Corbella was born in Barcelona, Spain, in 1955. He received the telecommunication engineering degree and doctoral degree in telecommunication engineering from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1977 and 1983, respectively.
In 1976, he was with the Microwave Laboratory, School of Telecommunication Engineering, UPC, as a research assistant, where he worked on passive microwave integrated-circuit design and characterization. In 1979, he was with Thomson-CSF, Paris, France, working on microwave oscillators’ design. In 1982, he was Assistant Professor, in 1986, Associate Professor, and, in 1993, Full Professor with UPC, where he is currently teaching basic microwaves and antennas at the undergraduate level and graduate courses on nonlinear microwave circuits. Since 1993, he has been actively participating as a researcher with the European Space Agency (ESA) SMOS mission in the frame of several contracts, directly with ESA, with the payload prime contractor EADS-Casa Espacio (Spain) or with the operational processor prime contractor Deimos Engenharia (Portugal). His expertise includes, among others, fundamentals of interferometric aperture synthesis radiometry, radiometer calibration, image reconstruction, radiometer hardware specification, and payload characterization.
From 1993 to 1997, he was Academic Director of the School of Telecommunications Engineering. From 2001 to 2003, he was Director of the Department of Signal Theory and Communications, UPC. From 1998 to 1999, he was Guest Researcher at the NOAA/Environmental Technology Laboratory (Boulder-Colorado), developing methods for total-power radiometer calibration and data analysis. In 2015, he was a visiting scholar at the University of Colorado at Boulder, working on emission models from layered media. From 2004 to 2010, he was a member of the SMOS Science Advisory Group (SAG) and since 2010 of the SMOS Quality Working Group (QWG).
Dr. Corbella was the General Chairman of the IEEE 2007 International Geoscience and Remote Sensing Symposium (IGARSS), Barcelona.
He also was the Scientific Coordinator of a Dictionary of Telecommunication terms in Catalan language, with more than 4000 entries, published in March 2007.
Preface
Invention, it must be humbly admitted, does not consist in creating out of void, but out of chaos. (Mary Shelley, 1831)
This book compiles the expertise the author has gathered during his 30-year-long work on Microwave Interferometric and Polarimetric Radiometry, particularly in relation to the “Soil Moisture and Ocean Salinity” (SMOS) mission of the European Space Agency. The SMOS satellite was put into orbit in November 2009 with the objective of acquiring global measurements of the two important geophysical parameters mentioned in its full name. The satellite includes a single payload named MIRAS (Microwave Imaging Radiometer with Aperture Synthesis) that is the first ever microwave interferometric radiometer designed for observing the Earth surface from space.
Except in some very few cases where text previously written by the author is reused, practically all writing is completely new, not extracted from any existing source. Many concepts and methods were indeed published in open literature or internal reports, often in collaboration with other authors (see acknowledgments) but, for the present work, the author has made an important effort to rewrite all the material from scratch in order to present the information with consistent terminology and complemented with detailed procedures and concepts not sufficiently and/or rigorously explained in the original papers. In consequence, even for the more established themes, for example in Chapters 1 and 2, the reader may find some explanations or points of view different from those found in published treatises. This feature will be probably useful to complement different visions of similar concepts.
The book starts with a review of the fundamental concepts that are used throughout the rest of the volume. They range from random variables and stochastic processes to microwave circuits and antennas. This first chapter is essential to set the terminology and symbols and is frequently referenced in all others. Even though the reader may be familiar with these themes in general, the author recommends to start exploring this chapter, at least in a “fast read” mode, before going to others more specific.
The second chapter is dedicated to the quite established topic of Microwave Radiometry, focused especially on instrumental aspects. It is included as a kind of introduction of the main subject of the book, interferometric and polarimetric radiometry, in order to present the fundamental concepts, formulate the needed hypotheses, and derive the theorems that will be extended later to more complex systems. It should be helpful in facilitating the readers the study of the rest of the book, even if they have a sound knowledge of microwave radiometric systems in general.
Interferometric and polarimetric radiometry is presented in the third chapter. This is perhaps the core of the book as it presents the fundamentals of the interferometric technique in the microwave region of the spectrum. Optical systems are simply not covered, although they may have important similitudes with what is here presented. Unavoidably, the formulation is biased to its application in Earth observation, as this is the background of the author. Nevertheless, most of the results are perfectly applicable to Radioastronomy since the fundamental concept is exactly the same. It may happen that experts in this field find some material in this book interesting for their own research.
Combination of multiple baselines to form an image is the subject of the fourth chapter. Here, the content is even more specific than that of the previous one in terms of application to large instruments dedicated to Earth observation from space or aircraft. As a matter of fact, much of the material relies on algorithms or procedures actually implemented in the MIRAS instrument. This chapter should be useful to people designing similar systems for related applications.
Chapter 5 deals with specific instrument techniques. Consistent with a top-down vision, the general theory developed in Chapters 2–4 is here materialized into dedicated technologies. Many of the items included are already implemented in existing instruments or intended for their use in future ones. Digital processing, essential in today’s systems, has been intentionally relegated to this chapter as a convenient way to treat measured data. Up to this point the world in this book is analog.
Finally, calibration and characterization are covered in Chapter 6. These are essential aspects of any radiometer, either total power or interferometric, maybe less conceptual, more oriented to techniques and procedures, and mathematically simpler. Commonly used methods are here described and particularly those utilized in the MIRAS instrument. This chapter has been written with the aim of helping engineers designing an instrument of the same or similar kind.
An appendix is also included with some specific and supposedly well-known concepts and definitions, such as Fourier transform (analog and discrete), special functions, spherical coordinates, solid angles, and others. Surely, in many cases the advanced reader will not need to consult it, although it can serve as a quick reference guide.
The author hopes that this book will help engineers and scientists facing the design of any kind of Microwave Interferometric and/or Polarimetric Radiometer, especially for application in Earth observation.
BarcelonaSeptember 12, 2024
Ignasi Corbella
Acknowledgments
Above all, I would like to express my most sincere gratitude to Dr. Manuel Martin-Neira, Senior engineer at the European Space Agency, for his careful revision of the manuscript. His opportune comments always hit in some weakness of the original writing and, once endorsed, enhanced the quality of the final version. I am also thankful to him for having written the excellent Foreword and for the generous words he expresses in it about my work and person.
A substantial amount of information in this book can be found scattered in articles and conference proceedings published by myself, always in collaboration with other colleagues. Three persons deserve to be mentioned in this regard: the already mentioned Dr. Manuel Martin-Neira, Dr. Francesc Torres and Dr. Adriano Camps, both professors at the “Universitat Politècnica de Catalunya.” This book would not have been possible without the exceptional job they have carried out during many years of fruitful cooperation. I am indebted to all three.
Ignasi Corbella
1Signals, Receivers, and Antennas
A radiometer, as a measuring system, processes input information in the form of signals and converts them to usable magnitudes proportional to observational parameters. Since all signals encountered in radiometry are of random nature, it is of paramount importance to establish a solid background of knowledge on stochastic processes. With the objective in mind of establishing notation and in order to have concepts and definitions at hand, the most relevant of them are collected in this chapter. Section 1.1 is an introduction to random variables, a fundamental tool on which random signals, developed in the section 1.2, are based. The reader is assumed to be familiar with basic random variables and processes, so only the definitions and methods needed for this book are included. More complete treatments can be found in classical books on statistics and its application to signal theory [1–5].
Receivers and antennas are the two basic building blocks of any radiometer, so mastering the main concepts about these two components is fundamental to perceive the details of radiometric operation. Both of them are also used in Telecommunications or Radar and are subjects of basic university courses. Consequently, there is a considerable amount of literature about them. This chapter presents their usage in microwave radiometry, aimed at gathering all important definitions and relations used later in this book. To know more, the interested reader may wish to complement the information with basic textbooks, as for example [6–10] among others. In any case, some familiarity with microwave circuits, including noise, and antenna theory is assumed.
1.1 Random Variables, Real and Complex
A real random variable is rigorously defined in, for example, [1]. It is fundamentally described by the “probability density function,” which is the probability that the variable takes a given value within a differential interval. For two random variables, the “joint density function” is the same concept applied to a differential area in a cartesian plane defined by all possible values of both. Two real random variables are said to be independent if they describe completely unrelated events. In this case, their joint density function is equal to the product of both individual density functions. Not surprisingly, if two variables are independent, arbitrary functions of each one of them are also independent.
A complex random variable is defined by the joint density function of its real and imaginary parts. Most of the definitions and equations provided in this section apply to complex variables, although they are also valid for real ones just by setting the imaginary part equal to zero. To avoid confusion, real variables are represented by upright roman typeface () while complex variables use italic shape (). Complex conjugation is denoted by a superscript asterisk.
1.1.1 Definitions
The “expected value” or “mean” of a real random variable is
where is the probability density function. In [1], it is demonstrated that this operation is linear so, for any two random variables and , where and are arbitrary constants.
Given a complex random variable , the following definitions apply:
The standard deviation is always real and positive and it has the same units as the random variable itself. Conjugating a random variable does not change the standard deviation. The following property is immediate from the definition:
where and are complex constants, and is a complex random variable, not to be confused with , the real part of . The additive constant does not change the standard deviation.
The generalization of variance and autocorrelation for two arbitrary complex random variables and is
The covariance is a complex number satisfying the Hermitian property: . By convention, if and represent both the same variable, the covariance becomes the variance: .
1.1.1.1 Correlated, Uncorrelated, and Independent Random Variables
Variables and are said to be “uncorrelated” if their covariance vanishes, which is equivalent to
This definition does not necessarily imply that the real and imaginary parts of one of the variables are uncorrelated with those of the other, as it is readily deduced by expanding the product of two complex numbers. But the opposite is true: if the real and imaginary parts are uncorrelated, then the complex variables also. In this case, the two following identities hold
Complex variables are independent only if the corresponding real and imaginary parts are independent. As expected, independent variables are always uncorrelated (see [1] for a proof) so the above equations hold also for them. The converse is only true in some cases, as for example in real variables with Gaussian statistics. In this particular situation uncorrelated and independent are equivalent concepts. In general, it cannot be assumed that individual functions of two uncorrelated variables are also uncorrelated.
A property similar to (1.3) is:
where all Greek letters are complex constants and and complex random variables. Furthermore, the covariance of the two linearly related variables and above is
where (1.3) has been used to write the last equality. These variables are said to be “correlated,” which in many textbooks is quantified by the so-called “correlation coefficient”
with an absolute value equal to one in this case. In [1] it is demonstrated the inverse statement: if the absolute value of the correlation coefficient is 1, then the variables are linearly related. For uncorrelated variables this coefficient is null and in general its amplitude is bounded between zero and one. The correlation coefficient is no longer used in this book.
As expected, if two variables are correlated with respect to a third one they are correlated to each other. Making equal to in (1.7)
and therefore the correlation coefficient has unit amplitude.
1.1.2 Operations
The sum of complex random variables
has variance
The double sum can be split into terms with equal indices () and the rest, in turn grouped by mutually conjugated terms. Then
If variables are uncorrelated (or, more specifically, if the real part of the covariance vanishes), then the second term vanishes and the variance of the sum becomes equal to the sum of variances.
If furthermore all variables have equal standard deviation (), then . On the other hand, if all variables are replicas of the same one (), then and the standard deviation is . In this case, the variables are correlated. These two important results are highlighted here below for clarity
The average of N random variables is their sum (1.11) divided by the number of them . Using (1.3) makes it apparent that the standard deviation is also that of the sum divided by
where is the square root of the variance (1.13). If the real part of the covariance between any two random variables is null and all variables have the same standard deviation , then
which means that averaging uncorrelated random variables reduces the standard deviation of a single one (assumed the same for all) by a factor of . If they are correlated, the standard deviation is unchanged, as is clearly deduced from the second line of (1.15). Obviously, due to the linearity of the mean operator, the mean of is, in any case, equal to the average of the mean values
which, in case of equal mean variables reduces to . In other words, averaging preserves the mean.
1.1.2.1 Product
Consider now the product of two random variables . The variance of , according to the definition (1.2) is
In general this expression cannot be easily evaluated. However, if and are independent, then (1.6) allows separating the second term into the product of two individual means squared. Furthermore, in this case, the two real products and are independent so, again using (1.6), the first term is also separable,
Using now (1.2), this variance can be written in terms of the corresponding individual ones,
Note that, if one of the two variables is a constant so it has zero standard deviation, this equation is consistent with (1.3). In general, for zero mean variables
which means that the standard deviation of the product is equal to the product of standard deviations. Otherwise, it is convenient to rewrite (1.21) as
which, if the two variables involved have small standard deviation with respect to their means, can be approximated to
so, in this case, the variance of the product is equal to the sum of variances of the individual variables weighted by the mean of the other variable.
1.1.3 Normal Random Variables
A complex random variable is said to be normal if its real and imaginary parts are jointly normal, meaning that their joint density function is Gaussian:
where its determinant of the “covariance matrix”
the column vector
and its transpose. The definition (1.25) is extended to N jointly normal complex random variables with only changing the factor by and extending correspondingly the matrices. In this case, they are conveniently partitioned as
where the subscripts and refer to the real and imaginary parts of the variables, respectively.
Having four zero-mean jointly normal variables to , it can be shown that they satisfy the following important property
Its proof is out of the scope of this book, but the interested reader may consult for example [2], in which it is termed as “moment theorem for normal random variables.” It is demonstrated for real random variables using the so-called “characteristic function,” defined as the Fourier transform of the probability density function. Besides the formal proof of [11], it is quite easy to infer that it also holds for complex variables: linear combinations of jointly normal variables are known to be also jointly normal (see example 7–12 of [1]). Consequently, if and ( to 4) are jointly normal variables, the four random variables defined by their sum are also jointly normal, so they satisfy (1.29). On the other hand, both and individually satisfy the same equation, so it can be concluded that, in general, if(1.29)holds forand, it holds also for their sum. Now, since multiplying by has no effect on the four-term product, it follows that (1.29) also applies for , so it must hold for .
1.1.3.1 Circularity
Of particular interest is the case of “circular” complex jointly normal variables. They are defined such that they have zero-mean and covariance matrix (1.28) satisfying
For example, a single complex zero-mean normal random variable is circular if
and (1.25) reduces to
Since this function depends only on the amplitude of , constant probability contours are circles in the complex plane, hence the name of circular. It is interesting to highlight that (1.31) implies that the variable is uncorrelated with its conjugate by virtue of the definition (1.5). The reader is invited to check this nonintuitive result.
Similarly, two zero-mean jointly normal complex random variables and are circular if they satisfy the following conditions derived from (1.30):
The corresponding joint density function is not provided here but it is opportune to mention that these conditions imply that the variances and cross-correlation of the complex variables are
Furthermore, the application of (1.29) to the product of four variables with two of them conjugated results in
where the first term vanishes due to the circularity condition (1.33). It may be worth to remind that all four variables must have zero mean.
1.1.4 The Arc Sine Law
The so-called “arc sine law” is an elegant relation often used to simplify digital correlation measurements. Given two real, zero-mean, jointly normal variables, this law relates their cross correlation with that of the following associated variables
where the signum function is defined in (A.30). These new variables contain only the sign information so they both have zero mean and unit variance. Their cross-correlation is related to that of the original variables through this law [see (1.42) below].
An easy method to derive it is by using the classical “Price’s theorem.” If is an arbitrary function of and
this theorem states (see for example [1]) that
with arbitrary, the joint density function (1.25) and the covariance of and . This theorem does not require that these variables have zero mean.
Application of this theorem to compute the cross correlation of and is straightforward. Equation (1.37) is now written as
and (1.38) is used with . Taking into account that the derivative of the signum function is equal to twice the Dirac delta function,
and substituting (1.25) knowing that ,
which is easily integrated to yield
in which has been substituted by and the definitions of covariance (1.4) and standard deviation (1.2) have been used. Remind that and are two real, zero-mean, jointly normal variables and and are related to them through (1.36).
The above equation is the arc sine law. A generalization for transitions at a nonzero threshold (i.e. ) can be found in [12].
1.1.4.1 A Useful Approximation
Related to the usage of this law in retrieving correlation measurements, an approximation is needed in section 3.4.5 to handle non infinite integration times. The demonstration is provided here for clarity. Consider two real random variables related through a sinus function
In general the mean of differs from the sinus of the mean of (). Putting , it can be written as
which for small , using Taylor linear and quadratic expansions for the sinus and cosinus, respectively, is approximated as
where is the variance of .
1.2 Stochastic Processes
A stochastic process is a mathematical model to describe the random fluctuations of a physical quantity and/or its measurement with time. The concept is fundamental for understanding many physical phenomena, particularly in this book to interpret radiometric measurements, as for example the output voltage of a total power radiometer or the correlation between output signals in an interferometric radiometer. In what follows the terms “stochastic process,” “random process” or “random signal” will be used indistinguishably. And often the adjective “stochastic” or “random” will be omitted to simplify writing.
Signals in the real world are real. However, in order to produce more elegant results and simplify formulations, it is often useful to use a complex representation of real signals. This is a common practice in engineering and is profusely used in this book. For this reason all properties and definitions provided all along this book assume complex processes unless mentioned otherwise. Following the same convention as for random variables in section 1.1, real signals are represented by upright roman typeface [] and complex signals use italic shape [] except, exceptionally, if denoted by Greek letters. Variable denotes time (in seconds) and frequency (in Hertz).
1.2.1 Stationarity
To systematize and simplify the study of physical observations, it is fundamental to understand the concept of “stationarity” applied to a stochastic process. Later a mathematical definition will be given, but for now just an intuitive one is needed. Figure 1.1 shows two sketches of stochastic processes, both representing for example the output voltage of a radiometer. The plot on figure 1.1(a) is stationary and the one on figure 1.1(b) is not. Without any rigorous definition, the reader can understand why this adjective is adequate to describe each of them.
Figure 1.1 Qualitative examples of stochastic processes. (a) Stationary. (b) Non stationary.
Since nothing remains forever stable in time, all physical phenomena are described by nonstationary processes. For example, a radiometric measurement evolves as the target is moved in front of the antenna and experiences changes as a function of the receiver’s physical temperature and other effects. Needless to say, nonstationarity is difficult to handle mathematically so, in practice, a convenient time scale is defined for which the process is assumed stationary; and then, after characterized, its (stable) parameters are allowed to vary (“slowly”) with time. For example, the signal on figure 1.1(b) can be interpreted as a locally stationary process with slowly varying average and amplitude evolving with time. Moreover, a nonstationary process can be viewed as the result of applying a function of time to a stationary one, so fundamental studies can be safely limited to this last case. With this approach in mind, stationarity will always be implicitly assumed, understanding that the results apply only to a finite time scale and may change by long-term time evolution.
Given a stationary process characterized by a real signal , its time average is a constant defined as
The “instantaneous power” is in turn a stochastic process defined by the square of the signal . The term “power” is just a convenient name, but the physical meaning depends on that of the original signal . In this context, power does not necessarily have units of Watt. In radiometry only “finite mean power” signals are encountered, meaning that the time average of the instantaneous power is a constant
A set of samples of any process at different epochs is a random variable. This is an axiom based on the fundamental definition of a random variable, always linked to the outcome of a certain experiment (see for example section 4.1 of [1]). The two above definitions are thus interpreted as the corresponding mean(1.1) and autocorrelation(1.2) of the random variable formed by all possible values of . Therefore, all definitions and equations defined for random variables in section 1.1 are directly applicable to samples of stochastic processes. To be precise though, there is a mathematical subtlety called “ergodicity,” which is the property that a process must satisfy for the above statement to be true. In practice one needs not care about it since stationary processes describing physical experiments are ergodic.
1.2.2 Correlation and Power
Given a complex stationary random process , the following definitions apply:
The absolute value of the mean squared is sometimes called “DC power” and the variance “AC power”1 so the mean power is the sum of AC plus DC powers. A zero mean process has no DC power. For most processes used all along this book the mean power, and hence the autocorrelation function, has units of Watt (see section 1.5.1 for details). The standard deviation has always the same units as the signal itself.
Subtracting the mean to a process implies subtracting the DC power to the autocorrelation function:
which, particularized for becomes the variance of : .
For stationary processes, both mean and variance are constants and the autocorrelation function depends only on time difference . This statement is in fact the mathematical definition of (wide sense) stationarity [1]. It implies that a random variable defined as the average of samples of has a mean equal to that of the process
Similarly, the mean of a random variable defined by averaging the amplitude squared of a process is equal to the mean power
Given a second complex process , the following additional definitions apply
The two processes and are uncorrelated if samples of them at any time delay are uncorrelated. Using (1.5), the cross-correlation function becomes then a constant,
which vanishes if one of the two variables has zero-mean.
As a generalization of (1.49), the cross-correlation function of the incremental signals that result of subtracting the mean and is
from which it follows that the covariance is . In some texts, is named “covariance function” with the symbol , but this definition is not used in this book. The cross-correlation function of the “primed” variables will be used instead. Note that for uncorrelated processes .
If both and represent the same signal, the cross-correlation function becomes the autocorrelation function , so no double subscript is used, in this case. Similarly, the covariance reduces to the variance .
Sometimes, it is useful to normalize the correlation functions with respect to their value at the origin. The following naming convention is used
As a direct consequence of the definition (1.52), swapping the subscripts of the cross-correlation function is equivalent to changing the sign of the time delay and conjugate,
which implies that is always real, consistent with the definition of mean power in (1.48). Moreover, the amplitude of the autocorrelation function is an even function of and the phase an odd function
For real signals both auto- and cross-correlation functions are real, so (1.56) is valid without the conjugate operation. The autocorrelation function of a real process is an even function of .
The following identity, deduced directly from the definition (1.52), relates the cross-correlation function of two complex signals to those of their real and imaginary parts: if and , then
which, for (single signal) reduces to
1.2.2.1 Sum and Product of Signals
Having two signals and , the autocorrelation function of a linear combination of them , with and complex constants, is readily deduced from the definition (1.48)
valid for either complex or real processes.
The autocorrelation function of the complex product is
which, in general, cannot be easily expanded. However, if and are independent processes, so they are the products and . Since independent variables are also uncorrelated, from (1.5)
applicable also to real signals. Other situations are analyzed here below.
1.2.3 Jointly Normal Processes
In the particular case where the signals at a given time and their delayed versions are circular jointly normal complex random variables, property (1.35) applies. Identifying , , and , (1.61) reduces to
where . The reader may easily check that this result is compatible with (1.62). In general, particularizing the above expression for results in
which is a very elegant result. Unlike (1.22), the above result does not require the signals to be uncorrelated. Note that the standard deviation of the complex product of jointly normal circular processes does not depend on the cross-correlation between the signals.
1.2.3.1 Real Signals
A slightly different conclusion is found for real signals. Defining the product , the autocorrelation function is
and assuming that signal samples and their delayed versions are zero-mean jointly normal random variables, property (1.29) can be used by identifying , , and . The above expression then reduces to
where . The reader may easily check that for uncorrelated (zero mean) signals this reduces to (1.62) as expected. Particularization for , knowing that both processes have zero mean, yields
which, compared to (1.64), includes an additive term that depends on the cross-correlation of the signals. Only if and are uncorrelated,
