**Remote Sensing and Digital Image Processing**

Irena Hajnsek Yves-Louis Desnos *Editors*

# Polarimetric Synthetic Aperture Radar

Principles and Application

# Remote Sensing and Digital Image Processing

Volume 25

#### Series Editors

Freek D. van der Meer, Faculty of Geo-Information Science and Earth Observation (ITC), Department of Earth Systems Analysis, University of Twente, Enschede, The Netherlands

Anna Jarocińska, Department of Geoinformatics, Cartography and Remote Sensing, University of Warsaw, Warsaw, Poland

Remote Sensing and Digital Image Processing book series. Remote sensing is the acquisition of Physical data of an object without touch or contact. Earth observation satellites have been used for many decades in a wide field of applications. With the advancements in sensor technology, earth imaging is now possible at an unprecedented level of detail. Imaging spectrometers and thermal multispectral systems acquire detailed spectroscopic information of physical properties of the earth's surface. Dynamic processes can now be studied with interferometric systems. SAR interferometry, laser altimetry and high-resolution imaging allow a very detailed, threedimensional reconstruction of the earth surface. With the advent of multi-sensor mission comes a new era of imaging, opening the possibility of integrating date from various sensor systems. The books published in the series explore these topics in remote sensing and provide a framework for related advanced digital image processing approaches.

Series Editorial Advisory Board:

Michael Abrams, NASA Jet Propulsion Laboratory, Pasadena, CA, USA Paul Curran, City University London, UK Arnold Dekker, CSIRO Land and Water Division, Canberra, Australia Steven M. de Jong, Utrecht University, The Netherlands Michael Schaepman, University of Zurich, Switzerland

Accepted for inclusion in Scopus.

More information about this series at http://www.springer.com/series/6477

Irena Hajnsek • Yves-Louis Desnos Editors

# Polarimetric Synthetic Aperture Radar

Principles and Application

Editors Irena Hajnsek Microwaves and Radar Institute German Aerospace Center (DLR) Oberpfaffenhofen, Germany

Institute of Environmental Engineering Swiss Federal Institute of Technology Zurich (ETH Zurich) Zürich, Switzerland

Yves-Louis Desnos Science, Application and Future Technology European Space Agency Frascati, Roma, Italy

ISSN 1567-3200 ISSN 2215-1842 (electronic) Remote Sensing and Digital Image Processing ISBN 978-3-030-56502-2 ISBN 978-3-030-56504-6 (eBook) https://doi.org/10.1007/978-3-030-56504-6

# The Editor(s) (if applicable) and The Author(s) 2021. This book is published open access.

Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

# Foreword

In the last 40 years, polarimetric Synthetic Aperture Radar (SAR) has evolved from an experimental concept to the implementation of many airborne and spaceborne SAR systems. In the meantime, many hard-working and dedicated researchers have developed theories and algorithms for efficient information extraction in relevant applications. These advancements have established SAR polarimetry as one of the principal remote sensing instruments for higher resolution imaging and more precise monitoring of Earth resources.

SAR is an active radar system, transmitting and receiving microwave pulses, and provides the desirable capability of all-weather, day and night imaging. For fully polarimetric SAR, also known as quad-polarization (HH, HV, VH, and VV) SAR, scatterers' dielectric properties and structures can be detected by their scattering mechanisms. With additional interferometric measurements, topography, building, and vegetation and forest height information can be retrieved. It is no wonder that a plethora of applications has flourished ranging from agriculture, forest, ocean, city buildings, terrain, cryosphere (glacier, ice, and snow), and environmental protection. Of course, military and other applications are too numerous to mention here. Some applications are well established and others are continuously advancing. Good survey papers on applications are scarce and in pressing demand.

This book contains five comprehensive survey papers written by well-known PolSAR researchers covering the application areas of forestry, agriculture and wetland, soil moisture retrieval, cryosphere, urban, and ocean. In addition, a chapter on the principle of polarimetry is given to provide a quick reference. The forest application chapter covers the techniques of forest classification, polarimetric SAR interferometry for forest height estimation, and polarimetric SAR tomography for 3D sensing. Due to the page limit, fine details could not be given. This chapter could be easily expanded to a book on polarimetric SAR forest sensing, and so can other chapters. The ocean application chapter details oil slick and ship detection. It could be extended to include the important and interesting topics of coastal remote sensing, and ocean current and wave measurements.

In summary, this book on polarimetric SAR applications is well written and timely. We congratulate the book's editors and authors for their dedicated effort and oversight in assembling such a long overdue book on polarimetric SAR applications.

Wolfgang-Martin Boerner Ph.D., IEEE, IEICE Life Fellow, OSA, SPIE, AAAS Fellow, Professor Emeritus & Distinguished Radar Scientist, The University of Illinois at Chicago

Jong-Sen Lee, Ph.D., IEEE Life Fellow, co-author of Polarimetric Radar Imaging (jointly with Prof. Eric Pottier), published by CRC/Taylor and Francis

# Preface

Dear readers and colleagues,

Welcome to the scientific publication Principles and Applications of Polarimetric Synthetic Aperture Radar! This publication is open-access and has been co-authored by over 50 of the worlds' leading scientists, working in the relevant thematic domains. This book is also a comprehensive resource for training and education that includes theory, examples of key applications, and do-it-yourself tutorials. Following current trends in Earth Observation (EO) open science, the key application examples provided in the various chapters may be reproduced by the reader using the Synthetic Aperture Radar (SAR) data provided and the algorithms implemented in the open-source PolSARpro toolbox.

In order for the (young) readers to fully understand the background of this collective, collaborative and innovative publication, I would like to guide you through the genesis of ideas, initiatives and events that led to its preparation, which spans over the last 35 years, and can be summarised in several key periods presented below.

# Pioneering Space-Borne SAR Interferometry

The first period of SAR interferometry started with the European Remote Sensing Satellite (ERS-1), developed by the European Space Agency (ESA), which was launched in July 1991. ERS-1 was equipped with a Synthetic Aperture Radar operating at C-band and single VV polarisation. Although not known at the time (and therefore not specified in ERS-1 radar development), scientists discovered, during preparation for the ERS-1 mission's scientific exploitation, that it was possible to exploit the phase difference from repeat pass acquisitions to measure topography and ground deformation. The technique, known as SAR interferometry, was demonstrated in orbit with ERS-1 data in August 1991 by the Politecnico di Milano team, allowing to generate interferometric fringes and to measure centimetre deformation of the terrain. ERS-1 was followed by the ERS-2 mission (launched in April 1995) after which scientists proposed the first tandem experiment to ESA. The objective of this experiment, which lasted 9 months, was to create a unique data set from highly coherent interferometric pairs in order to generate digital elevation models with high accuracy.

It was during this period that ESA prepared the Advanced Synthetic Aperture Radar (ASAR) to be put on-board ENVISAT. ASAR was an active array antenna with beam steering and multi-polarisation capability (HH/HV or VV/VH) and was launched in March 2002. As a lesson learned from the ERS-1 mission, the instrument was specified to perform repeat pass interferometry and the ground processor was developed for phase preservation of the ASAR signals. ASAR was the first active array SAR instrument that operated flawlessly for 10 consecutive years in space. The ASAR instrument, thanks to its experimental alternating polarisation mode, demonstrated the usefulness and importance of multi-polarisation radar for scientific and application purposes. In addition, the ASAR instrument, although not designed for it, demonstrated the capability of wide swath ScanSAR interferometry in orbit.

# Organising Airborne Polarimetric SAR Campaigns and Scientific Studies

During the ERS and ENVISAT missions period (1991–2012), airborne campaigns continued to play an important role in supporting the development of future SAR missions – helping the validation of new space-borne SAR products as well as providing key open data to the scientific community. ESA, NASA, ONERA and DLR organised several multifrequency SAR campaigns demonstrating the interest of full polarimetric and interferometric SAR data sets for scientific research (e.g. MAESTRO, INDREX, AGRISAR, BIOSAR, AfriSAR, etc.). In 2000, reaping the full benefits of the available airborne campaign data, and in order to assess the benefits and capabilities of the newly published "polarimetric interferometry technique" by K. Papathanassiou and S. Cloude, ESA launched two SAR polarimetry studies. The objectives of these studies were to investigate radar capability to penetrate ground cover and to "see" the underlying terrain, as well as to assess the possible use of SAR imagery to characterise forest structure and height. Since then, the collection of multi-frequency and multi-baseline polarimetric SAR data via dedicated airborne campaigns continue to play an essential role in the demonstration and development of innovative algorithms for potential bio-/ geophysical product derivation and support the preparation for new space-borne missions.

# Dialoguing with POLinSAR Scientists and Training the Next Generation

In 2003, the first POLinSAR workshop was organised to gather this new community, to present the findings of ESA-funded studies of SAR Polarimetry and Polarimetric Interferometry applications, and to prepare recommendations for future research work. ESA also implemented the first recommendations from the POLinSAR community to provide a state-of-the art, opensource educational software in the field of Polarimetric SAR data analysis. The Polarimetric SAR Data Processing and Educational Toolbox (PolSARpro) was developed under ESA contract by I.E.T.R at the University of Rennes 1, under the scientific leadership of Prof. E. Pottier. To date, PolSARpro provides the most comprehensive set of functions for the scientific exploitation of fully and partially polarimetric multi-data sets, and is used worldwide.

Additionally, as a second recommendation from the POLinSAR community, a multi-year programme of advanced thematic training courses and summer schools was organised for training the next generation of ESA Principal Investigators in SAR Polarimetry. These training events host about 60–70 scientists (i.e. Ph.D. students, young postdoctoral researchers) for a one week duration, that include formal lectures on theory to advanced concepts, as well as hands-on computing exercises, exposing students to EO data and the usage of ESA toolboxes. In 2019, ESA is organising the 5th Advanced Course on Radar Polarimetry dedicated to training the next generation of Earth Observation (EO) scientists to exploit dual and fully polarimetric data for science and applications development.

At the end of 2017, ESA launched, jointly with University of Jena and DLR, the first Massive Open Online Course (MOOC) on SAR that was followed by over 6000 participants and included modules on SAR polarimetry theory and applications using open-source toolboxes. In 2019, ESA organised the 9th International Workshop on Science and Applications of SAR Polarimetry and Polarimetric Interferometry, and the re-run of the SAR MOOC with the participation of the Canadian Space Agency.

# Pioneering Space-Borne SAR Polarimetric Interferometry

Recently, satellites with SAR polarimetric capability and different frequencies have been launched (e.g. SIR-C/X-SAR, ALOS-2, RADARSAT-2 and TerraSAR-X, TanDEM-X) and operated successfully. Following the recommendations of scientists at POLinSAR, campaigns were organised by Space Agencies to collect SAR data time series over a wide range of validation sites. In addition, dedicated and successive Announcement of Opportunities to research on spaceborne SAR polarimetry at different frequencies were opened to European PIs by ESA for ALOS with JAXA (2005), with CSA for RADARSAT 2 SOAR EU (2009 and 2013), and with DLR for TerraSAR X –TanDEM-X.

Having acquired the suitable time series of space-borne data over proper validation tests sites, and benefiting from expertise developed in Europe, we were in a position to advance from demonstrations to applications and, as a result, identify key areas for which polarimetric data provides unique benefits. For instance, the polarimetric interferometric forest height measurements were demonstrated successfully using ALOS data in 2008, despite the long repeat pass intervals, and were validated using airborne data.

# Future Missions

In 2013, the BIOMASS mission became the first ESA SAR mission with fully polarimetric capabilities selected to be the 7th Earth Explorer mission for launch in 2022. The primary scientific objectives of the mission are to determine the distribution of above-ground biomass in the world forests and to measure its annual changes, to help improve our understanding of the land carbon cycle. The BIOMASS radar sensor is a fully polarimetric space-borne SAR operating at P-band. During the first year of the mission, a tomographic phase will provide a full global coverage. The remaining 4 years will be an interferometric phase, providing full global coverage every 6 months. Though the prime objective is to collect data over forest, it is also foreseen to collect additional data over land, ice caps and the ocean, as appropriate. The data collected by this mission will allow advanced retrieval concepts based on polarimetric interferometry and polarimetric tomography, developed since 2003 via several ESA and DLR studies, to be put in practice.

Other SAR satellite missions are currently under study to provide fully polarimetric capabilities in future, such as the German Tandem-L, the European Commission Copernicus Sentinel-1 Next Generation and the Copernicus L-band candidate mission. The need for polarimetric SAR satellite missions and global coverage is now being well recognised.

# From Science to Applications

In 2011, ESA launched the PolSAR-Ap project, aimed at evaluating and demonstrating the importance and unique benefits of fully polarimetric SAR data for a wide range of remote sensing applications. The project was awarded to DLR and coordinated by Prof. Irena Hajnsek. The main study objective was the identification of applications for which the availability of fully polarimetric SAR data brings unique benefits and/or significant performance improvements when compared to single or dual polarimetric data. In addition, the demonstration of the improvements induced by polarimetry was to be performed by implementing the applications algorithms, applying them on suitable space-borne SAR data (if available) and validating their performance against reference data available for each case. In order to cover a whole range of remote sensing applications, the study was organised into thematic domains: Forest, Agriculture, Cryosphere, Urban, Ocean and Hazard applications.

This publication aims to present the outcome of the PolSAR-Ap project and to introduce key research results, authored by scientific experts participating in the study. Some of the relevant applications are also implemented in the ESA open-source software PolSARpro allowing the easy generation and replication of key examples.

# Outlook

Today, ESA is developing fully polarimetric space-borne SAR missions at P-band, whilst studying L-band and C-band mission concepts and preparing for their scientific exploitation. ESA data are freely and openly available for further exploitation by the community. ESA is supporting the emerging EO "open science" practices, taking full advantage of the digital age and addressing new ways to do research for society and in society. ESA "open science" activities will include open access publications, open-source scientific toolboxes, new ways to educate via MOOCs, enhanced research using applications platforms and virtual research environments, increased collaboration using social networks, as well as innovative citizen science projects.

We sincerely wish the reader a good journey through the Principles and Applications of Polarimetric Synthetic Aperture Radar. You are kindly invited to use the sample data and the ESA toolbox for demonstration and educational purposes. We would like to acknowledge the contribution of all co-authors and special thanks to the Chapter Leads Dr. Carlos Lopez Martinez, Prof. Eric Pottier, Dr. Shane Cloude, Dr. Kostas Papathanassiou, Dr. Juan Manuel Lopez Sanchez, Prof. Maurizio Migliaccio and Dr Elise Colin Koeniger.

Sincerely,

Yves-Louis Desnos and Irena Hajnsek

# In Memoriam

This book is dedicated to our beloved colleague and friend Prof. Wolfgang Boerner, age 80, who passed away on May 25, 2018, during the development of the book. Wolfgang was one of the strongest promoters for establishing a polarimetric SAR framework and applying it to a variety of applications. We have known Wolfgang attending countless workshops, conferences and meetings, promoting radar polarimetry and networking alongside scientists and researchers all around the world, with endless energy and passion. His presentations and talks were legendary, unpredictable, and nothing short of memorable!

For the upcoming scientists, Wolfgang was the one to introduce them to the famous polarimetry experts whose names could be found on papers they have read many times. He was integrating everyone into the "polarimetric family" and he appeared as the "grandfather" of this international family. In this way, he was opening a new perspective on the scientific world but also on the real world for us. He followed the progress of young scientists from workshop to workshop and from conference to conference. He praised the progress, encouraged the overcoming of problems and urged to continue to work hard. He never allowed us to resign in difficult moments. He himself, with his energetic and tireless promotion and his confidence on the importance of polarimetry, has been the best example of resilience to students and young professionals. He was a real mentor, at a time when the importance of mentoring was not yet well established. We – all his co-strugglers – got formed and stand by his idea that polarimetric radar and remote sensing collaboration will improve the world.

Oberpfaffenhofen, Germany Irena Hajnsek Frascati, Roma, Italy Yves-Louis Desnos

# Contents

# Symbols

# Basic Principles of SAR Polarimetry 1

C. López-Martínez and E. Pottier

#### Abstract

This chapter critically summarizes the main theoretical aspects necessary for a correct processing and interpretation of the polarimetric information towards the development of applications of synthetic aperture radar (SAR) polarimetry. First of all, the basic principles of wave polarimetry (which deals with the representation and the understanding of the polarization state of an electromagnetic wave) and scattering polarimetry (which concerns inferring the properties of a target given the incident and the scattered polarized electromagnetic waves) are given. Then, concepts regarding the description of polarimetric data are reviewed, covering statistical and scattering aspects, the latter in terms of coherent and incoherent decomposition techniques. Finally, polarimetric SAR interferometry and tomography, two acquisition modes that enable the extraction of the 3-D scatterer position and separation, respectively, and their polarimetric characterization, are described.

# 1.1 Theory of Radar Polarimetry

# 1.1.1 Wave Polarimetry

Polarimetry refers specifically to the vector nature of the electromagnetic waves, whereas radar polarimetry is the science of acquiring, processing and analysing the polarization state of an electromagnetic wave in radar applications. This section summarizes the main theoretical aspects necessary for a correct processing and interpretation of the polarimetric

C. López-Martínez (\*)

E. Pottier

information. As a result, the first part presents the so-called wave polarimetry that deals with the representation and the understanding of the polarization state of an electromagnetic wave. The second part introduces the concept of scattering polarimetry. This concept collects the topic of inferring the properties of a given target, from a polarimetric point of view, given the incident and the scattered polarized electromagnetic waves.

# <IP_ADDRESS> Electromagnetic Waves and Wave Polarization Descriptors

The generation, the propagation and the interaction with matter of the electric and the magnetic waves are governed by Maxwell's equations (Balanis 1989). For an electromagnetic wave that is propagating in the <sup>b</sup><sup>z</sup> direction, the real electric wave can be decomposed into two orthogonal components <sup>b</sup><sup>x</sup> and <sup>b</sup><sup>y</sup> , admitting the following vector formulation:

$$\overrightarrow{\mathbf{E}}(z,t) = \begin{bmatrix} E\_x \\ E\_\mathbf{y} \\ E\_z \end{bmatrix} = \begin{bmatrix} E\_{0x}\cos\left(\alpha t - kz + \delta\_\mathbf{x}\right) \\ E\_{0\mathbf{y}}\cos\left(\alpha t - kz + \delta\_\mathbf{y}\right) \\ 0 \end{bmatrix} \tag{1.1}$$

which may be also considered in a complex form

$$
\overrightarrow{\mathbf{E}}(z,t) = \begin{bmatrix} \underline{E}\_{\mathbf{x}} \\ \underline{E}\_{\mathbf{y}} \\ \underline{E}\_{\mathbf{z}} \end{bmatrix} = \begin{bmatrix} E\_{0\mathbf{x}}e^{j\delta\_{\mathbf{x}}}e^{-jkz}e^{j\alpha t} \\ E\_{0\mathbf{y}}e^{j\delta\_{\mathbf{y}}}e^{-jkz}e^{j\alpha t} \\ 0 \end{bmatrix},\tag{1.2}
$$

where E0<sup>x</sup> and E0<sup>y</sup> are the amplitudes of the waves in each coordinate. The electric wave in (1.1) and (1.2) presents a harmonic time dependence of the type e <sup>j</sup>ω<sup>t</sup> , where <sup>ω</sup> ¼ <sup>2</sup>π<sup>f</sup> is the angular frequency and f is the time frequency. The propagation direction of an electromagnetic wave is determined by the propagation vector <sup>b</sup><sup>k</sup> that in case of (1.1) and (1.2) is considered parallel to <sup>b</sup>z. The amplitude of the propagation vector is represented by <sup>k</sup> ¼ <sup>2</sup>π/λ, where <sup>λ</sup> is the wavelength.

Signal Theory and Communications Department, Universitat Politècnica de Catalunya , Barcelona, Spain e-mail<EMAIL_ADDRESS>
Institut d'Électronique et de Télécommunications de Rennes, University of Rennes-1, Rennes, France

Finally, δ<sup>x</sup> and δ<sup>y</sup> represent the wave phases in each component. The magnetic wave H ! ð Þ <sup>z</sup>, <sup>t</sup> can be also represented in the same form.

According to the IEEE Standard Definitions for Antennas (IEEE standard number 145 1983), the polarization of a radiated wave is defined as that property of the radiated electromagnetic wave describing a time-varying direction and relative magnitude of the electric wave vector, specifically the figure traced as a function of time by the extremity of the vector at a fixed location in space and the sense in which it is traced as observed along the direction of propagation. Hence, polarization is the curve traced out by the end point of the arrow representing the instantaneous electric wave.

Let us consider the geometric locus described by the electric wave, as a function of time, for a particular point in space, which can be assumed <sup>z</sup> ¼ <sup>z</sup>0, without loss of generality. Under these hypotheses, the wave components Ex and Ey satisfy the following equation:

$$
\left(\frac{E\_\mathbf{x}}{E\_{0\mathbf{x}}}\right)^2 - 2\frac{E\_\mathbf{x}E\_\mathbf{y}}{E\_{0\mathbf{x}}E\_{0\mathbf{y}}}\cos\left(\delta\_\mathbf{y} - \delta\_\mathbf{x}\right) + \left(\frac{E\_\mathbf{y}}{E\_{0\mathbf{y}}}\right)^2
$$

$$
=\sin\left(\delta\_\mathbf{y} - \delta\_\mathbf{x}\right).\tag{1.3}
$$

The previous equation describes an ellipse that is called polarization ellipse. As one may deduce from the previous equation, the electric wave, as a function of time, describes in the most general case an ellipse, whose shape does depend neither on time nor on space. The polarization ellipse, for some particular configurations, may reduce to a circle or to a line.

As it may be deduced from (1.3), the polarization state is completely characterized by three independent parameters: the wave amplitudes E0<sup>x</sup> and E0<sup>y</sup> and the phase difference <sup>δ</sup> <sup>¼</sup> <sup>δ</sup><sup>y</sup> δx. Figure 1.1 presents the polarization ellipse for a general polarization state. In addition to the previous three parameters, it is also possible to describe the polarization ellipse by a different set of parameters:

• Orientation or tilt angle ϕ. This angle gives the orientation of the ellipse major axis with respect to the <sup>b</sup><sup>x</sup> axis in such a way that <sup>ϕ</sup> 2 [π/2, π/2]. This angle may be obtained as follows:

$$\tan 2\phi = 2 \frac{E\_{0\mathbf{x}} E\_{0\mathbf{y}}}{E\_{0\mathbf{x}}^2 - E\_{0\mathbf{y}}^2} \cos \delta. \tag{1.4}$$

• Ellipticity angle τ. This angle represents the ellipse aperture in such a way that <sup>τ</sup> 2 [π/4, π/4]. This angle may be obtained as follows:

Fig. 1.1 Polarization ellipse

$$|\sin 2\pi| = 2\frac{E\_{0\mathbf{x}}E\_{0\mathbf{y}}}{E\_{0\mathbf{x}}^2 + E\_{0\mathbf{y}}^2}|\sin \delta|\tag{1.5}$$

$$A = \sqrt{E\_{0\mathbf{x}}^2 + E\_{0\mathbf{y}}^2} \tag{1.6}$$

• The absolute phase ζ. This phase represents the initial phase with respect to the phase origin for <sup>t</sup> ¼ 0 in such a way that <sup>ζ</sup> 2 [π, π]. This term corresponds to the common phase in δ<sup>x</sup> and δy. This absolute phase cannot be directly measured as it corresponds to the exit phase from the radar system at <sup>t</sup> ¼ 0.

Considering the previous sets of parameters describing the polarization state of a wave, one can identify some important polarization states that can be considered as canonical polarization states:

• Linear polarization state. Considering the expression for the real electric wave in (1.1), two canonical linear polarization states can be identified. Table 1.1 details the orientation and the ellipticity angles for these polarization states. These are the linear polarization states according to the <sup>b</sup><sup>x</sup> and to the <sup>b</sup><sup>y</sup> axes, respectively. The linear polarization states are characterized by presenting a phase difference of

$$
\delta = \delta\_\circ - \delta\_x = m\pi, m = 0, \pm 1, \pm 2, \dots \tag{1.7}
$$

As it may be seen, the linear nature of the polarization state is independent of the phase ζ.

• Circular polarization state. In this particular case, also two canonical circular polarization states can be defined. Table 1.1 details the orientation and the ellipticity angles for these polarization states. When the ellipticity angle takes a value of π/4, the circular polarization state is right-handed, whereas this value is equal to π/4 when it is left-handed. The circular polarization states are characterized by presenting a phase difference of

$$
\delta = \delta\_\mathbf{y} - \delta\_\mathbf{x} = m\frac{\pi}{2} \quad m = \pm 1, \,\pm 3, \,\pm 5, \dots \qquad (1.8)
$$

and equal amplitudes for the components of the electric wave <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>E</sup>0<sup>x</sup> <sup>¼</sup> <sup>E</sup>0y. Also for circular polarization states, the polarization state is independent of the absolute phase ζ.

• Elliptical polarization state. When there are not restrictions on the orientation and ellipticity angle values, the electric wave is said to present an elliptical polarization state.

As observed, the polarization ellipse may be completely described by two equivalent sets of three independent parameters: the set of wave parameters {E0x, E0y, δ} or the set of ellipse parameters {ϕ, τ, A}. In addition to these, there exist additional equivalent descriptors that are detailed in the following.

Considering (1.1), the real electric wave vector can be directly obtained from the complex electric wave vector

$$\overrightarrow{\mathbf{E}}(z,t) = \begin{bmatrix} E\_{0x}\cos\left(\alpha t - kz + \delta\_x\right) \\ E\_{0y}\cos\left(\alpha t - kz + \delta\_y\right) \end{bmatrix} = \Re\left\{ \begin{bmatrix} E\_{0x}e^{j\delta\_x} \\ E\_{0y}e^{j\delta\_y} \end{bmatrix} e^{-jkz}e^{j\alpha t} \right\} = 0$$

$$= \Re\left\{ \overrightarrow{\mathbf{E}}\left(z\right)e^{j\alpha t} \right\} \tag{1.9}$$

where <sup>ℜ</sup>{} denotes the real part. The time dependence has been removed from the wave description. This is possible as the polarization state of the wave does not change with time. In order to derive a simple and concise description of the polarization state, it is also possible to remove the space dependence of Ε !ð Þ<sup>z</sup> by considering the polarization state in a particular point of the space. Without loss of generality, this point can be <sup>z</sup> ¼ 0. Hence, <sup>Ε</sup> !ð Þ<sup>0</sup> reduces to

$$\underline{\mathbf{E}} = \stackrel{\rightarrow}{\underline{\mathbf{E}}}(0) = \begin{bmatrix} E\_{0\mathbf{x}} e^{j\delta\_{\mathbf{x}}} \\ E\_{0\mathbf{y}} e^{j\delta\_{\mathbf{y}}} \end{bmatrix}. \tag{1.10}$$

The two-dimensional complex vector Ε is referred to as the Jones vector, and it is a concise representation of a monochromatic, uniform plane wave with a constant polarization (Jones 1941a; Jones 1941b; Jones 1941c).

In the rectangular coordinate system, the Jones vector can be written as a function of the parameters that describe the polarization ellipse (Huynen 1970):

$$\underline{\mathbf{E}} = A e^{i\zeta} \begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} \cos \pi \\ j \sin \pi \end{bmatrix}. \tag{1.11}$$

The Jones vector, considering the unitary vectors <sup>b</sup><sup>x</sup> and <sup>b</sup>y, may be also expressed as

$$\underline{\mathbf{E}}\begin{Bmatrix}\hat{\mathbf{x}},\hat{\mathbf{y}},\end{Bmatrix} = A \begin{bmatrix}\cos\phi & -\sin\phi\\\sin\phi & \cos\phi\end{bmatrix} \begin{bmatrix}\cos\tau & j\sin\tau\\j\sin\tau & \cos\tau\end{bmatrix}$$

$$\times \begin{bmatrix} e^{j\zeta} & 0\\0 & e^{-j\zeta} \end{bmatrix} \hat{\mathbf{x}} \tag{1.12}$$

where the sub-index f g <sup>b</sup>x, <sup>b</sup>y, indicates that the Jones vector is expressed in the linear basis f g <sup>b</sup>x, <sup>b</sup>y, . The Jones vector describes completely the polarization ellipse shape, as well as the rotation sense of the electric wave vector. On the contrary, handedness information cannot be included within the Jones vector as propagation information has been removed. The use of the Jones vector to describe the polarization state is of enormous importance as it allows to define a polarization algebra that makes possible to perform a mathe-

Table 1.1 Geometrical parameters of the polarization ellipse for canonical polarization states in the rectangular coordinate system

matical treatment and analysis of the wave polarization. This treatment allows, for instance, the correct definition of orthogonal polarization states. Finally, Table 1.2 details the Jones vector, in the rectangular basis, i.e. E bx, by, , for some

particular polarization states.

Another equivalent description of the wave polarization state is the so-called complex polarization ratio:

$$
\rho = \frac{E\_\text{y}}{E\_x} = \frac{E\_{0\text{y}}}{E\_{0\text{x}}} e^{j\left(\delta\_\text{y} - \delta\_\text{y}\right)}.\tag{1.13}
$$

As in the case of the Jones vector, the complex polarization ratio is not able to determine the handedness of the polarization state as propagation information is removed.

The Jones vectors, as well as the complex polarization ratio, are complex quantities that describe the polarization state of a wave. Sir G. Stokes introduced a wave polarization and wave amplitude description based on four real quantities in polarization wave optics (Stokes 1852). The Stokes vector, in the rectangular coordinate system, is defined as (Stokes 1852)

$$\underline{\mathbf{g}} = \begin{bmatrix} \mathbf{g}\_0 \\ \mathbf{g}\_1 \\ \mathbf{g}\_2 \\ \mathbf{g}\_3 \end{bmatrix} = \begin{bmatrix} \left| E\_x \right|^2 + \left| E\_\mathbf{y} \right|^2 \\ \left| E\_x \right|^2 - \left| E\_\mathbf{y} \right|^2 \\ 2\mathcal{R} \left\{ E\_x E\_\mathbf{y}^\* \right\} \\ -2\mathcal{I} \left\{ E\_x E\_\mathbf{y}^\* \right\} \end{bmatrix} \tag{1.14}$$

where the elements of the vector g are simply called Stokes parameters. Consequently, the Stokes vector is a fourdimensional real vector. Since the Stokes vector describes the polarization state of an electromagnetic wave, it can be directly obtained from the geometrical parameters that describe the polarization ellipse, i.e. {ϕ, τ, A}:

$$\underline{\mathbf{g}} = \begin{bmatrix} A \\ A \cos \left( 2\phi \right) \cos \left( 2\tau \right) \\ A \sin \left( 2\phi \right) \cos \left( 2\tau \right) \\ A \sin \left( 2\tau \right) \end{bmatrix}. \tag{1.15}$$

The polarization state of an electromagnetic wave is completely characterized by means of three independent parameters. These statements also hold for the Stokes parameters, since, as it may be deduced from (1.15), the following relation applies

$$\mathbf{g}\_0^2 = \mathbf{g}\_1^2 + \mathbf{g}\_2^2 + \mathbf{g}\_3^2. \tag{1.16}$$

Table 1.3 details the Stokes vector, in the rectangular basis, i.e. g bx, by, , for some particular polarization states.

#### <IP_ADDRESS> Totally and Partially Polarized Waves

Single-frequency or monochromatic waves are completely polarized, that is, the tip of the electric wave vector describes an ellipse in the plane orthogonal to the propagation direction. The shape of this ellipse, neglecting attenuation propagation effects which affect only the overall power, does not change in time or space, and hence, the wave polarization is constant. Completely polarized waves appear when the different parameters of the wave ω, E0x, E0y, δ<sup>x</sup> and δ<sup>y</sup> are constant. Nevertheless, many waves present in the nature are characterized by the fact that the previous parameters depend on time or on space randomly. Hence, the tip of the electric wave vector no longer describes an ellipse. These waves are referred to as partially polarized waves. This loss of polarization is due to the randomness of the illuminated scene, to the presence of noise, etc.

The different parameters that characterize the electric wave, i.e. ω, E0x, E0y, δ<sup>x</sup> and δy, may vary randomly. This type of variation makes the electric wave to be modulated and therefore to present a finite bandwidth, so waves can no longer be considered as being monochromatic, but polychromatic. Under this circumstance, it would be also desirable to have a complex representation of the electromagnetic wave as shown in (1.10). Nevertheless, in most of the applications, we are interested into electromagnetic waves that will only have appreciable values in a frequency range which is small compared to the mean frequency ω. Under this situation, waves are referred to as quasi-monochromatic waves. For such signals, the phase terms Θx(z, t) and Θy(z, t) change slowly when compared to the mean frequency. Then, one may represent the Jones vector of a quasi-monochromatic wave as

$$\underline{\mathbf{E}} = \begin{bmatrix} E\_x(t)e^{j\Theta\_x\left(\overrightarrow{\mathbf{r}},t\right)} \\ E\_y(t)e^{j\Theta\_\flat\left(\overrightarrow{\mathbf{r}},t\right)} \end{bmatrix}. \tag{1.17}$$

Table 1.2 Jones vector for some polarization states in the rectangular coordinate system, for <sup>A</sup> ¼ <sup>1</sup>

As one may see, the Jones vector of a quasimonochromatic electric wave depends on time and on space; thus, this vector is no longer constant. When the time dependence of the Jones vector is deterministic, the polarimetric properties of the wave also change in a deterministic way through time. In this case, the description of the wave polarization is not problematic and may be performed considering the different descriptors detailed in Sect. <IP_ADDRESS>. Nevertheless, if the time dependence is random, the analysis of the polarization state of the electromagnetic wave must be carefully addressed, as this description must take into account the stochastic nature of the electric wave.

As previously mentioned, the variation of the parameters E0x, E0y, δ<sup>x</sup> and δ<sup>y</sup> may be random, so the Jones vector will be also random. In order to characterize the polarization of the quasi-monochromatic electromagnetic wave expressed by the variable Jones vector in (1.17), it is necessary to address this characterization from a stochastic point of view. In the frame of radar remote sensing, the wave transmitted by the radar system may be considered monochromatic and hence totally polarized. Nevertheless, the scattered wave represented by the Jones vector in (1.17) results from the combination of many different waves originated by the different elementary scatterers that form the scattering media. The complex addition of these elementary waves resulting from the scattering process for one component of the electric wave can be represented as

$$Ae^{j\theta} = \frac{1}{\sqrt{N}} \sum\_{n=1}^{N} a\_n e^{j\theta\_n} \tag{1.18}$$

where A represents the total wave and ane <sup>j</sup>θ<sup>n</sup> is originated from the scattering from every elementary scatterer. Under the assumption of N, i.e. the total number of scattered waves, to be large enough and certain relations that may be established between the amplitude and the phase of the elementary waves (Chandrasekhar 1960; Goodman 1976), it is possible to demonstrate that the mean value of the electric wave and the Jones vector are zero. Consequently, the Jones vector cannot be employed to characterize the polarization state of a quasi-monochromatic wave. This characterization shall be performed considering higher statistical moments.

The second-order moments may be arranged in a vector form, giving rise to the so-called coherency vector of a quasimonochromatic vector, which is defined in the following way:

$$\mathbf{J} = E\left\{\mathbf{E}\bigotimes \mathbf{E}^\*\right\} = \begin{bmatrix} E\left\{E\_x E\_x^\*\right\} \\ E\left\{E\_x E\_y^\*\right\} \\ E\left\{E\_y E\_x^\*\right\} \\ E\left\{E\_y E\_y^\*\right\} \\ E\left\{E\_y E\_y^\*\right\} \end{bmatrix} = \begin{bmatrix} J\_{xx} \\ J\_{xy} \\ J\_{yx} \\ J\_{yy} \end{bmatrix} \tag{1.19}$$

where J stands for the temporal averaging, assuming the wave is stationary, <sup>N</sup> is the Kronecker product, () represents complex conjugation and <sup>E</sup>{} is the ensemble average. This vector is not zero for quasi-monochromatic waves. The arrangement of the second-order moments can be also done in a matrix, giving rise to the coherency matrix of the wave:

$$\mathbf{J} = E\left\{ \underline{\mathbf{E}} \cdot \underline{\mathbf{E}}^{T\*} \right\} = \begin{bmatrix} E\left\{ E\_x E\_x^\* \right\} & E\left\{ E\_x E\_y^\* \right\} \\ E\left\{ E\_\mathcal{Y} E\_x^\* \right\} & E\left\{ E\_\mathcal{Y} E\_y^\* \right\} \end{bmatrix}$$

$$= \begin{bmatrix} J\_{\rm xx} & J\_{\rm xy} \\ J\_{\rm yx} & J\_{\rm yy} \end{bmatrix} \tag{1.20}$$

where the superscript () <sup>T</sup> denotes vector transposition.

In the previous section, it was mentioned that monochromatic waves are completely polarized. This is not the case for quasi-monochromatic waves. Indeed, completely polarized waves present a polarization state that can be considered as a limit in the sense that it is constant. The opposed extreme is a completely unpolarized wave for which the polarization state is completely random. Between both extremes, waves are said to present a partial polarization state. In order to characterize the degree of polarization, one may consider the degree of polarization defined as a function of the trace of matrix J as

$$DoP = \left(1 - 4\frac{|\mathbf{J}|}{trace(\mathbf{J})}\right)^{\frac{1}{2}}.\tag{1.21}$$

#### <IP_ADDRESS> Change of Polarization Basis

As seen in Sect. <IP_ADDRESS>, an electromagnetic wave, considering the coordinate system f g <sup>b</sup>x, <sup>b</sup>y, <sup>b</sup><sup>z</sup> , that propagates in <sup>b</sup><sup>z</sup> may be decomposed as the sum of two orthogonal components. Separately, the electromagnetic wave of each component can be considered as linearly polarized. Therefore, it is possible to consider that the total electromagnetic wave results from the sum of two orthogonal linear polarized waves. Indeed, this representation must be extended in the sense that any electromagnetic wave propagating in an infinite, lossless, isotropic media can be decomposed as the sum of two orthogonal elliptically polarized waves. The advantage of this representation is that the electric wave is decomposed in a pair of orthogonal polarization states, so it is possible, through a deterministic transformation, to obtain the electric wave for any other pair of orthogonal polarization states. This process is referred to as change of polarization basis or polarization synthesis.

Given two vectors a and b, they are considered orthogonal if they verify

$$
\langle \mathbf{a}, \mathbf{b} \rangle = \mathbf{a}^T \cdot \mathbf{b}^\* = 0 \tag{1.22}
$$

that is, the scalar (Hermitian) product of both vectors is zero. In case of two electromagnetic waves, expressed in terms of the corresponding Jones vectors, they are said to be orthogonal if the scalar product of the Jones vectors is zero, considering that both Jones vectors refer to waves propagating in the same direction and sense. The polarization ellipses corresponding to two orthogonal Jones vectors presents the same ellipticity angle, opposite polarization sense and mutually orthogonal polarization axis.  In terms of (1.12), the corresponding orthogonal vector is

$$\underline{\mathbf{E}}\_{\perp} \begin{bmatrix} \overset{\frown}{\mathbf{x}, \mathbf{y}} \end{bmatrix} = A \begin{bmatrix} -\sin\phi & -\cos\phi \\ \cos\phi & -\sin\phi \end{bmatrix} \begin{bmatrix} \cos\tau & -j\sin\tau \\ -j\sin\tau & \cos\tau \end{bmatrix} \begin{bmatrix} e^{-j\zeta} & 0 \\ 0 & e^{j\zeta} \end{bmatrix} \hat{\mathbf{x}} $$

$$= A \begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} \cos\tau \ j\sin\tau \\ j\sin\tau \ \cos\tau \end{bmatrix} $$

$$\times \begin{bmatrix} e^{j\zeta} & 0 \\ 0 & e^{-j\zeta} \end{bmatrix} \hat{\mathbf{y}} \,. \tag{1.23}$$

The symbol ⊥ denotes orthogonal Jones vector.

Considering what has been indicated, an electromagnetic wave propagating in an infinite, lossless, isotropic media may be described in the following way:

$$\underline{\mathbf{E}} = E\_x \hat{\mathbf{x}} + E\_y \hat{\mathbf{y}} = E\_x \hat{\mathbf{u}}\_x + E\_y \hat{\mathbf{u}}\_y \tag{1.24}$$

where the notation referring to the unitary vectors has been generalized. If (1.23) and (1.24) are considered, it may be seen that the unitary Jones vectors corresponding to the linear orthogonal polarization states <sup>b</sup><sup>x</sup> and <sup>b</sup><sup>y</sup> are transformed to the Jones vector of any polarization state and the corresponding orthogonal Jones vector through the transformation matrix U:

$$\begin{aligned} \left\{ \hat{\mathbf{u}}, \hat{\mathbf{u}}\_{\perp} \right\} &= \begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} \cos \tau & j \sin \tau \\ j \sin \tau & \cos \tau \end{bmatrix} \begin{bmatrix} e^{-j\zeta} & 0 \\ 0 & e^{j\zeta} \end{bmatrix} \left\{ \hat{\mathbf{x}}, \hat{\mathbf{y}}, \right\} \\ &= \mathbf{U}\_{\left\{ \widehat{\mathbf{u}}, \widehat{\mathbf{u}}\_{\perp} \right\}} \left\{ \hat{\mathbf{x}}, \hat{\mathbf{y}}, \right\} \,. \end{aligned} \tag{1.25}$$

In the previous case, the matrix U bu,bu⊥ indicates the transformation matrix from the orthogonal basis f g <sup>b</sup>x, <sup>b</sup>y, to the arbitrary basis f g <sup>b</sup>u, <sup>b</sup>u<sup>⊥</sup> . Considering (1.24), the electromagnetic wave expressed in the orthogonal basis f g <sup>b</sup>u, <sup>b</sup>u<sup>⊥</sup> takes the form

$$
\underline{\mathbf{E}} = E\_u \hat{\mathbf{u}} + E\_{u\perp} \hat{\mathbf{u}}\_{\perp}.\tag{1.26}
$$

Therefore, the Jones vector in the new basis f g <sup>b</sup>u, <sup>b</sup>u<sup>⊥</sup> , expressed in terms of the Jones vector in the basis f g <sup>b</sup>x, <sup>b</sup>y, , is

$$
\begin{bmatrix} E\_u \\ E\_{u\perp} \end{bmatrix} = \mathbf{U}\_{\left\{ \widehat{\mathbf{u}}, \widehat{\mathbf{u}}\_\perp \right\}} \begin{bmatrix} E\_x \\ E\_y \end{bmatrix}. \tag{1.27}
$$

The previous equation indicates that if an electromagnetic wave has been measured in the linear orthogonal basis, it is possible to calculate the same electromagnetic wave, but measured in a different polarization basis, just multiplying by the matrix U-1 bu,bu⊥ . That is, it is possible to synthesize the electromagnetic wave for any arbitrary polarization basis just measuring it in a particular polarization basis.

Table 1.4 and Table 1.5 detail the polarization ellipse parameters, the Jones vector and the Stokes vector for different polarization states for the rotated and the linear polarization bases, respectively.

# 1.1.2 Scattering Polarimetry

The previous section was concerned with the characterization and the representation of the polarization state of an electromagnetic wave. Although this characterization is important

Table 1.4 Polarization states expressed in the rotated linear polarization basis <sup>b</sup>u<sup>π</sup>=4, <sup>b</sup>u<sup>π</sup>=<sup>4</sup> , when <sup>A</sup> ¼ <sup>1</sup>

Table 1.5 Polarization states expressed in the circular polarization basis f g <sup>b</sup>ulc, <sup>b</sup>urc , for <sup>A</sup> <sup>¼</sup> <sup>1</sup>

when a radar system is considered, as it transmits and receives electromagnetic waves, nevertheless, the interest is on the scattering process itself. The radar system transmits an electromagnetic wave, with a given polarization state, that reaches the scatterer of interest. The energy of the incident wave interacts with the scatterer, and as a result part of this energy is reradiated to the space. The way this energy is reradiated depends on the properties of the incident wave, as well as on the scatterer itself. Consequently, it is possible to infer some information of the scatterer under consideration considering the properties of the scattered electromagnetic wave with respect to the incident wave, which is basically the transmitted wave by the radar. One possibility that can be studied to characterize distant targets is to consider the change of the polarization state that a scatterer may induce to an incident wave.

In order to analyse the scattering problem, it is worth to start describing the scattering process that occurs when an incident wave reaches a flat transition between two dielectric, infinite, lossless and homogeneous media in oblique incidence. This scattering situation is exemplified in Fig. 1.2. In this case, the incident wave that propagates in the first media reaches the transition between media where part of the incident energy is scattered in the same media and part of the energy is transmitted to the second media. In order to characterize the scattering process, it is necessary to introduce the concept of plane of scattering, which is defined as the plane generated by the propagating vectors of the incident and the scattered waves.

In order to examine specifically reflections at oblique angles of incidence for a general wave polarization, it is convenient to decompose the electric wave into its perpendicular and parallel components, relative to the plane of scattering. The total scattered and transmitted waves will be the vector sum from each of these two polarizations. When the wave is perpendicular to the plane of scattering, the polarization of the wave is referred to as perpendicular polarization or horizontal polarization as the electric wave is parallel to the interface. When the electromagnetic wave is parallel to the plane of scattering, the polarization is referred to as parallel polarization or vertical polarization as the electromagnetic wave is also perpendicular to the interface. As indicated in Fig. 1.2, the total incident wave E !i can be decomposed into two orthogonal components in the plane orthogonal to the incident propagation vector <sup>b</sup><sup>k</sup> i . These are the parallel E !i k and the perpendicular <sup>E</sup> !i <sup>⊥</sup> components, which can be written as

$$\mathbf{E}\_{\parallel}^{i} = E\_{\parallel}^{i} e^{-j \left< \widetilde{\mathbf{k}}^{i}, \mathbf{r} \right>} \widetilde{\mathbf{x}}^{\prime}, \tag{1.28}$$

$$\underline{\qquad}$$

$$\underline{\qquad}, \qquad \ldots, \qquad $$

$$
\stackrel{\rightarrow}{\mathbf{E}}\_{\perp}^{i} = E\_{\perp}^{i} e^{-j\left<\mathbf{k}^{\prime}, \mathbf{r}\right>} \hat{\mathbf{y}}^{\prime}. \tag{1.29}
$$

As observed, the incident wave has been defined with respect to the coordinate system <sup>b</sup><sup>x</sup> 0 , by 0 ,bz 0 in such a way

Fig. 1.2 Oblique incidence

that <sup>b</sup><sup>k</sup> i ¼ bz <sup>0</sup> . It may be shown that the scattered wave components can be written similarly

$$\left.\stackrel{\rightarrow}{\mathbf{E}}\right\|\;=E\left\|\;e^{-j\langle\mathbf{k}^\epsilon,\mathbf{r}\rangle}\hat{\mathbf{x}}^{\prime\prime},\right.\tag{1.30}$$

$$
\overrightarrow{\mathbf{E}}\_{\perp}^{s} = E\_{\perp}^{s} e^{-j\langle \mathbf{k}^{\circ}, \mathbf{r} \rangle} \hat{\mathbf{y}}^{\prime\prime}, \tag{1.31}
$$

but in this case according to <sup>b</sup>x<sup>00</sup> , by00 ,bz <sup>00</sup> n o.

Considering the equations of the incident and the scattered wave, the question rising at this point is to determine whether it is possible or not to express mathematically the scattering process that occurs at the interface between both media. First of all, it is of crucial importance to take into consideration where, in the space, the expressions of the incident and scattered waves are valid. The expressions in (1.28), (1.29), (1.30) and (1.31) make reference to uniform plane waves. In the case of the incident wave on the scatterer, such a description for the wave, i.e. the wave originated at the transmitting antenna, is only valid if the scatter is in the far-field zone of the transmitting antenna. In the case of the scattered wave, this wave admits a uniform plane wave formulation if the point where the wave is considered is in the far field of the scatterer. In both cases, the waves in the far-field zone may be considered spherical waves, which locally may be considered as uniform plane waves. Considering a spherical coordinate system centred in the scatterer and under the previous assumptions, the incident wave on the scatter can be expressed vectorially, in the far-field zone, as

$$\mathbf{E}^i = \begin{bmatrix} E^i \\ \vdots \\ E^i\_\perp \end{bmatrix}, \ \mathbf{E}^s = \begin{bmatrix} E^s \\ \vdots \\ E^s\_\perp \end{bmatrix} \tag{1.32}$$

As observed, there are different points that need to be considered in the analysis of this problem. The first one is the use of different coordinate systems to characterize, in an unambiguous way, the polarization state of the different waves involved in the scattering process. The second aspect, coupled to the previous one, is to determine the way the scatterer under study changes the different components of the wave. This section has studied this entire problem considering the analytical expressions of the waves.

#### <IP_ADDRESS> The Scattering Matrix

This section will address the generalization of the previous scattering problem, and it will introduce those concepts necessary to address it in a vector form. The first aspect that needs to be fixed is to determine the different coordinate systems necessary to characterize the scattering problem and the description of the incident and the reflected waves. In the scattering problem, three coordinate systems must be chosen. The first one is the coordinate system located at the centre of the scatterer under consideration and referred to as f g <sup>b</sup>x, <sup>b</sup>y, <sup>b</sup><sup>z</sup> . This coordinate system may be considered as a kind of absolute or global coordinate system. In addition to it, it is necessary to define two additional local coordinate systems in order to determine, in an unambiguous way, the polarization states of the incident and the scattered or reflected waves, respectively. These two coordinate systems, associated with the waves, are defined in terms of the global coordinate system.

Let us consider an object illuminated by an electromagnetic plane wave which may be described as

$$\overrightarrow{\mathbf{E}}^{i} = E\_x \hat{\mathbf{x}}^{\prime} + E\_y \hat{\mathbf{y}}^{\prime} = E\_x \hat{\mathbf{h}}\_i + E\_y \hat{\mathbf{v}}\_i \tag{1.33}$$

where the unitary vectors <sup>b</sup>x<sup>0</sup> and <sup>b</sup>y<sup>0</sup> are arbitrarily defined. Hence, the propagation direction of the incident wave is conveniently selected to be <sup>b</sup><sup>k</sup> i ¼ bz <sup>0</sup> . The incident wave reaches the object of interest and induces currents on it, which in turn reradiates a wave. This reradiated wave, as shown, is referred to as the scattered wave. In the far-field zone, the scattered wave is an outgoing spherical wave that in the area occupied by the receiving antenna can be

$$
\overrightarrow{\mathbf{E}}^{s} = E\_x \hat{\mathbf{x}}^{\prime} + E\_y \hat{\mathbf{y}}^{\prime\prime} = E\_x \hat{\mathbf{h}}\_s + E\_y \hat{\mathbf{v}}\_s. \tag{1.34}
$$

The propagation direction of the scattered wave is therefore <sup>b</sup><sup>k</sup> s ¼ bz 00 . The scattering process is finally analysed in terms of the plane of scattering, which is the plane that contains both the incident and the scattering propagating vectors. The concepts of perpendicular and parallel wave components, or horizontal and vertical wave components, are defined with respect to the plane of scattering. Consequently, and as indicated in (1.33), the perpendicular component of the wave admits to be considered as a horizontal component, i.e. <sup>b</sup>x<sup>0</sup> ¼ <sup>b</sup>hi, whereas the parallel one admits to be considered as a vertical one, i.e. <sup>b</sup><sup>y</sup> <sup>0</sup> ¼ <sup>b</sup>vi. In the case of the scattered wave, the perpendicular component of the wave admits to be considered as a horizontal component, i.e. <sup>b</sup>x<sup>00</sup> ¼ <sup>b</sup>hs, whereas the parallel one admits to be considered as a vertical one, i.e. <sup>b</sup>y<sup>00</sup> ¼ <sup>b</sup>vs.

The incident and scattered waves in (1.33) and (1.34), respectively, may be also vectorially expressed by means of the Jones vectors:

$$\underline{\mathbf{E}}^i = \begin{bmatrix} E\_h^i \\ E\_\upsilon^i \end{bmatrix}, \underline{\mathbf{E}}^s = \begin{bmatrix} E\_h^s \\ E\_\upsilon^s \end{bmatrix} \tag{1.35}$$

In the definition of the previous two Jones vectors, the coordinate systems defined previously are assumed. By using this vector notation for the electromagnetic waves, it is possible to relate the scattered wave with the one of the incident wave by means of a 2 2 complex matrix:

$$
\underline{\mathbf{E}}^s = \frac{e^{-jkr}}{r} \mathbf{S} \underline{\mathbf{E}}^i. \tag{1.36}
$$

Here, r is the distance between the scatterer and the receiving antenna, and k is the wavenumber of the illuminating wave. The coefficient 1/r represents the attenuation between the scatterer and the receiving antenna, which is produced by the spherical nature of the scattered wave. On the other hand, the phase factor represents the delay of the travel of the wave from the scatterer to the antenna. Equation (1.36) may be written as

$$
\begin{bmatrix} E\_h^\mathbf{v} \\ E\_\mathbf{v}^\mathbf{v} \end{bmatrix} = \frac{e^{-jkr}}{r} \begin{bmatrix} S\_{hh} & S\_{h\mathbf{v}} \\ S\_{vh} & S\_{v\mathbf{v}} \end{bmatrix} \begin{bmatrix} E\_h^i \\ E\_\mathbf{v}^i \end{bmatrix}. \tag{1.37}
$$

The matrix S is referred to as scattering matrix, whereas its components are known as complex scattering amplitudes. The arrangement of the scattering matrix indicates how these complex scattering amplitudes are measured. The first column of S is measured by transmitting a horizontally polarized wave and employing two antennas horizontally and vertically polarized to record the scattered waves. The second column is measured in the same form, but transmitting a vertically polarized wave.

It is worth mentioning that the scattering matrix characterizes the target under observation for a fixed imaging geometry and frequency. In addition, the four elements must be measured at the same time, especially in those situations where the scatterer is not static or fixed. If they are not measured at the same time, the coherency between the elements may be lost as the different elements may refer to a different scatterer.

As indicated, the scattering matrix represents the scattering process for particular incident and scattering directions, i.e. <sup>b</sup><sup>k</sup> i and <sup>b</sup><sup>k</sup> s , respectively. In addition to that, it is also necessary to provide the horizontal and vertical unitary vectors, for the incident and the scattered waves, as they are necessary to define the polarization states of the waves.

In the most general case, which occurs in bistatic configurations where the transmitter and receiver antennas are located in different positions, the scattering matrix contains up to seven independent parameters to characterize the scatterer under observation. These parameters are the four amplitudes and three relative phases; see (1.38). Indeed, any absolute phase in the scattering matrix can be neglected as it does not affect the received power:

$$
\begin{bmatrix} E\_h^s \\ E\_\nu^s \end{bmatrix} = \underbrace{\underbrace{e^{-jkr}e^{j\phi\_{hh}}}\_{\text{Absolute phase}}}\_{\text{Absolute phase}} \underbrace{\begin{bmatrix} |S\_{hh}| & |S\_{hr}|e^{j(\phi\_{hr}-\phi\_{hh})} \\ |S\_{rh}|e^{j(\phi\_{rh}-\phi\_{hh})} & |S\_{rv}|e^{j(\phi\_{rv}-\phi\_{hh})} \end{bmatrix}}\_{\text{Relative scattering matrix}} \begin{bmatrix} E\_h^i \\ E\_v^i \end{bmatrix}. \tag{1.38}
$$

As it was already highlighted previously, the scattering coefficients depend on the direction of the incident and the scattered waves. When considering the matrix S, the analysis of this dependence is of extreme importance since it also involves the definition of the polarization of the incident and the scattered waves. Since (1.37) considers the polarized electromagnetic waves themselves, it is mandatory to assume a frame in which the polarization is defined. There exist two principal conventions concerning the framework where the polarimetric scattering process is considered: Forward Scatter Alignment (FSA) and Backscatter Alignment (BSA); see Fig. 1.3. In both cases, the electric waves of the incident and the scattered waves are expressed in local coordinate systems centred on the transmitting and receiving antennas, respectively. All coordinate systems are defined in terms of a global coordinate system centred inside the target of interest.

The FSA convention (see Fig. 1.3), also called waveoriented since it is defined relative to the propagating wave, is normally considered in bistatic problems, that is, in those configurations in which the transmitter and the receiver are not located at the same spatial position.

The bistatic BSA convention framework (see Fig. 1.3) is defined, on the contrary, with respect to the radar antennas in accordance with the IEEE standard. The advantage of the BSA convention is that for a monostatic configuration, also called backscattering configuration, that is, when the transmitting and receiving antennas are collocated, the coordinate systems of the two antennas coincide; see Fig. 1.4. This configuration is preferred in the radar polarimetry community. In the monostatic case, the scattering matrix in the FSA convention, SFSA, can be related to the same matrix referenced to the monostatic BSA convention SBSA as follows:

Fig. 1.3 (a) FSA and (b) BSA conventions

$$\mathbf{S}\_{\rm BSA} = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \mathbf{S}\_{\rm FSA} \,. \tag{1.39}$$

As it has been mentioned previously, in the radar polarimetry community, the monostatic BSA convention (backscattering) is considered as the framework to characterize the scattering process. The reason to select this configuration is due to the fact that the majority of the existing polarimetric radar systems operate with the same antenna for transmission and reception. One important property of this configuration, for reciprocal targets, is reciprocity, which states that

$$S\_{\mathsf{h}\_{\mathsf{RSA}}} = S\_{\mathsf{v}\mathsf{h}\_{\mathsf{RSA}}},\tag{1.40}$$

$$S\_{\text{hv}\_{F\text{SA}}} = -S\_{\text{vh}\_{F\text{SA}}}.\tag{1.41}$$

Then, the formalization of the scattering process given by (1.37), in the monostatic case under the BSA convention, reduces to

$$
\begin{bmatrix} E\_h^s \\ E\_v^s \end{bmatrix} = \frac{e^{-jkr}}{r} \begin{bmatrix} S\_{hh} & S\_{hv} \\ S\_{hv} & S\_{vv} \end{bmatrix} \begin{bmatrix} E\_h^i \\ E\_v^i \end{bmatrix}. \tag{1.42}
$$

In the same sense, Eq. (1.38) takes the form

$$
\begin{bmatrix} E\_h^s \\ E\_v^s \end{bmatrix} = \underbrace{\underbrace{e^{-jkr}e^{j\phi\_{hh}}}\_{\text{Absolute phase}}}\_{\text{Absolute phase}} \underbrace{\begin{bmatrix} |S\_{hh}| & |S\_{lr}|e^{j(\phi\_{hr}-\phi\_{hh})} \\ |S\_{lr}|e^{j(\phi\_{hr}-\phi\_{hh})} & |S\_{vr}|e^{j(\phi\_{hr}-\phi\_{hh})} \end{bmatrix}}\_{\text{Relative scattering matrix}} \begin{bmatrix} E\_h^i \\ E\_v^i \end{bmatrix}. \tag{1.43}
$$

The main consequence of the previous equation is that in the backscattering direction, a given scatterer is no longer characterized by seven independent parameters, but by five. These are three amplitudes, two relative phases, and one additional absolute phase.

A central parameter when considering the scattering process occurring at a given scatterer consists of the scattered power. For single-polarization systems, the scattered power is determined by means of the radar cross section or the scattering coefficient. Nevertheless, a polarimetric radar has to be considered as a multichannel system. Consequently, in order to determine the scattered power, it is necessary to consider all the data channels, that is, all the elements of the scattering matrix. The total scattered power, in the case of a polarimetric radar system, is known as Span, being defined in the most general case as

$$\begin{split} SPAN(\mathbf{S}) &= \text{trace}\left(\mathbf{S}\mathbf{S}^{T\*}\right) \\ &= \left|\mathbf{S}\_{hh}\right|^2 + \left|\mathbf{S}\_{hv}\right|^2 + \left|\mathbf{S}\_{hv}\right|^2 + \left|\mathbf{S}\_{vv}\right|^2. \end{split} \tag{1.44}$$

In the backscattering case, due to the reciprocity theorem, the Span reduces to

$$SPAN(\mathbf{S}) = |S\_{hh}|^2 + 2|S\_{hv}|^2 + |S\_{vv}|^2. \tag{1.45}$$

The main property of the Span is that it is polarimetrically invariable, that is, it does not depend on the polarization basis employed to describe the polarization of the electromagnetic waves.

When the radar wave reaches a scatterer, part of the incident energy is reflected back to the system. If the incident wave is monochromatic, the target is unchanging and the radar-target aspect angle is constant, the scattered wave will be also monochromatic and completely polarized. Therefore, both the incident and the scattered waves can be characterized by their corresponding Jones vectors, and the scattering process can be characterized by the scattering matrix. These targets are referred to as point targets, single targets or deterministic targets, as when a radar images this type of scatterers, the scattered wave in the far-field zone appears to be originated by a single point. In other words, the target response is not contaminated by additional spurious, so it is possible to infer some information about the target from the single values of the scattering matrix. Table 1.6 shows the scattering matrix, expressed in the linear polarization basis, for some canonical bodies. These are referred to as canonical due to the simplicity of their scattering matrix.

#### <IP_ADDRESS> Scattering Polarimetry Descriptors

The scattering matrix introduced in the previous section is indeed a scattering polarimetry descriptor that could be also included in this section. Nevertheless, it merits a separate section as this matrix represents the best vehicle to introduce the description of the scattering process when polarimetry is concerned, as the scattering matrix relates the Jones vectors of the involved electromagnetic waves. Section <IP_ADDRESS> introduced additional descriptors for the polarization state of an electromagnetic wave. As a consequence, some additional descriptors for the scattering process shall be introduced in the following.

The 2 2 complex scattering matrix, as indicated, describes the scattering matrix of a given target. Table 1.6 presented several examples for some simple or canonical scatterers. Nevertheless, a real target presents always a complex scattering response as a consequence of its complex geometrical structure and its reflectivity properties. Consequently, the interpretation of this response is obscure. As it shall be presented later on, a possible solution to interpret this response is to decompose the original scattering matrix into the response of canonical mechanisms. With this idea in mind, but also with the objective to introduce a new formulism to extract physical information, it is possible to transform the scattering matrix into a scattering vector that presents a clearer physical interpretation.

The construction of a target vector k is performed through the vectorization of the scattering matrix:

$$\mathbf{k} = V(\mathbf{S}) = \frac{1}{2}\operatorname{trace}(\mathbf{S}\Psi). \tag{1.46}$$

<sup>Ψ</sup> is a set of 2 2 complex basis matrices which are constructed as an orthonormal set under a Hermitian inner product. The interpretation of the target vector k depends on the selected basis Ψ. The most common matrix bases employed in the context of the radar polarimetry are the so-called lexicographic ordering basis and the Pauli basis. The lexicographic ordering basis consists of the straightforward lexicographic ordering of the elements of the scattering matrix:

$$\Psi\_l = \left\{ 2\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, 2\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, 2\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, 2\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}. \tag{1.47}$$

The Pauli basis consists of the set of Pauli spin matrices usually employed in quantum mechanics:

$$\Psi\_p = \left\{ \sqrt{2} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \sqrt{2} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \sqrt{2} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \sqrt{2} \begin{bmatrix} 0 & -j \\ j & 0 \end{bmatrix} \right\}. \tag{1.48}$$

Note that the multiplying factor in both bases is necessary in order to keep the total scattered power constant, i.e. trace (SS<sup>T</sup> ).

The selection of the basis to vectorize the scattering matrix depends on the final purpose of the vectorization itself. When

Table 1.6 Scattering matrix for canonical bodies in the linear polarization basis <sup>b</sup>h, <sup>b</sup><sup>v</sup> n o

the objective is to study the statistical behaviour of the SAR data or the radar measurement, it is more convenient to consider the lexicographic basis due to its simplicity, as it shall be extended in the next sections. Nevertheless, when the objective is the physical interpretation of the scattering matrix, it is more convenient to consider the Pauli basis. Assuming the Pauli decomposition basis, an arbitrary 2 <sup>2</sup> scattering matrix may be written in the following terms:

$$\begin{aligned} \mathbf{S} &= \begin{bmatrix} a+b & c-jd \\ c+jd & a-b \end{bmatrix} \\ &= a \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} + c \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \\ &+ d \begin{bmatrix} 0 & -j \\ j & 0 \end{bmatrix}. \end{aligned} \tag{1.49}$$

It is worth noting that the elements a, b, c and d are complex. If one considers the decomposition of the scattering matrix as performed in (1.49), it is possible to identify the four elements of the Pauli basis with some of the scattering matrices of canonical bodies presented in Table 1.6. Therefore, the elements a, b, c and d, i.e. the elements of the target vector k, represent the contribution of every canonical mechanism to the final scattering mechanism. Therefore, the following interpretation is possible:

All in all, what has been performed in (1.49) is a target decomposition. This concept shall be analysed in depth in the next. It is also worth to notice that the different components of the Pauli basis, or scattering components, are orthogonal. This means that from a practical point of view, the separation indicated in (1.49) is possible without ambiguities.

Finally, the explicit expressions of the target vector in the lexicographic and Pauli decomposition bases, considering the expression of the scattering matrix, in the most general case are:

$$\mathbf{k}\_{l} = \begin{bmatrix} S\_{hh} \\ S\_{h\mathbf{v}} \\ S\_{vh} \\ S\_{v\mathbf{v}} \end{bmatrix}, \ \mathbf{k}\_{p} = \frac{1}{\sqrt{2}} \begin{bmatrix} S\_{hh} + S\_{\mathbf{v}\mathbf{v}} \\ S\_{hh} - S\_{\mathbf{v}\mathbf{v}} \\ S\_{h\mathbf{v}} + S\_{vh} \\ j(S\_{h\mathbf{v}} - S\_{vh}) \end{bmatrix} \tag{1.50}$$

In the backscattering case, under the BSA convention, the reciprocity property applies. Hence, the previous target vectors admit the following simplification:

$$\mathbf{k}\_{l} = \begin{bmatrix} S\_{hh} \\ \sqrt{2}S\_{hv} \\ S\_{vv} \end{bmatrix}, \mathbf{k}\_{p} = \frac{1}{\sqrt{2}} \begin{bmatrix} S\_{hh} + S\_{vv} \\ S\_{hh} - S\_{vv} \\ 2S\_{hv} \end{bmatrix} \tag{1.51}$$

The different 2 and ffiffiffi 2 p factors that appear in the definition of the target vectors are necessary in order to maintain the total scattered power or Span. As it is evident, the Span must be constant and independent from the choice of the basis in which the scattering matrix is decomposed. This is known as total power invariance.

The concept of target vector, obtained as a vectorization of the scattering matrix, makes it possible to obtain a new formulation to describe the information contained in the scattering matrix by means of the outer product of the target vector with its conjugate transpose, or adjoint vector.

For a vectorization of the scattering matrix through the lexicographic basis, in the most general case, the outer product of the target vector with its transpose conjugate klk<sup>T</sup> l leads to the matrix:

$$\mathbf{k}\_{l}\mathbf{k}\_{l}^{T\*} = \begin{bmatrix} \left| S\_{hh} \right|^2 & S\_{hh} S\_{h\nu}^\* & S\_{hh} S\_{\nu h}^\* & S\_{hh} S\_{\nu \nu}^\* \\ S\_{h\nu} S\_{h\nu}^\* & \left| S\_{h\nu} \right|^2 & S\_{h\nu} S\_{\nu h}^\* & S\_{h\nu} S\_{\nu \nu}^\* \\ S\_{\nu h} S\_{h\nu}^\* & S\_{\nu h} S\_{h\nu}^\* & \left| S\_{\nu h} \right|^2 & S\_{\nu h} S\_{\nu \nu}^\* \\ S\_{\nu \nu} S\_{h\nu}^\* & S\_{\nu \nu} S\_{h\nu}^\* & S\_{\nu \nu} S\_{\nu h}^\* & \left| S\_{\nu \nu} \right|^2 \end{bmatrix}. \tag{1.52}$$

Due to a language abuse, the matrix klk<sup>T</sup> <sup>l</sup> is sometimes referred to as covariance matrix and represented by C, but as it will be shown in Sect. <IP_ADDRESS>, the covariance matrix presents a different definition. It is worth to observe that (1.52) is a 4 4, complex, Hermitian matrix. The construction of this matrix, through the outer product of the vector k<sup>l</sup> and its transpose conjugate, makes the matrix klk<sup>T</sup> <sup>l</sup> have a rank equal to 1. Consequently, klk<sup>T</sup> <sup>l</sup> presents exactly the same information as the scattering matrix, and hence it may have up to seven independent parameters. In the case of the backscattering direction under the BSA convention, and due to the fact that the reciprocity relation applies, klk<sup>T</sup> <sup>l</sup> can be written, considering (1.51), as

$$\mathbf{k}\_{l}\mathbf{k}\_{l}^{T\*} = \begin{bmatrix} \left| S\_{hh} \right|^{2} & \sqrt{2}S\_{hh}S\_{h\nu}^{\*} & S\_{hh}S\_{\nu\nu}^{\*}\\ \sqrt{2}S\_{h\nu}S\_{hh}^{\*} & \left| S\_{h\nu} \right|^{2} & \sqrt{2}S\_{h\nu}S\_{\nu\nu}^{\*}\\ S\_{\nu\nu}S\_{hh}^{\*} & \sqrt{2}S\_{\nu\nu}S\_{h\nu}^{\*} & \left| S\_{\nu\nu} \right|^{2} \end{bmatrix}. \qquad (1.53)$$

As in the previous case, the klk<sup>T</sup> <sup>l</sup> matrix presents a rank equal to 1 as it is obtained as the outer product of a vector and its transpose conjugate. Nevertheless, in this case, the covariance matrix may present up to five independent parameters, that is, the same number of independent parameters as the scattering matrix from which it derives.

A similar procedure can be applied when the scattering matrix is obtained considering the Pauli basis. In this case, the matrix is obtained from the outer product kpkT <sup>p</sup> . Due to a language abuse, this matrix is sometimes referred to as coherency matrix and represented by T, but as it will be shown in Sect. <IP_ADDRESS>, the coherency matrix presents a different definition. Under the most general imaging configuration, considering (1.65), the coherency matrix can be written as

$$\mathbf{k}\_{p}\mathbf{k}\_{p}^{T\*} = \begin{bmatrix} \left| S\_{hh} + S\_{vv} \right|^2 & \left( S\_{hh} + S\_{vv} \right) \left( S\_{hh} - S\_{vv} \right)^\* & \left( S\_{hh} + S\_{vv} \right) \left( S\_{hr} + S\_{rh} \right)^\* & \left( S\_{hh} + S\_{vv} \right) \left( j(S\_{hr} - S\_{rh}) \right)^\* \\\\ \left( S\_{hh} - S\_{vv} \right) \left( S\_{hh} + S\_{vv} \right)^\* & \left| S\_{hh} - S\_{vv} \right|^2 & \left( S\_{hh} - S\_{vv} \right) \left( S\_{hr} + S\_{rh} \right)^\* & \left( S\_{hh} - S\_{vv} \right) \left( j(S\_{hr} - S\_{rh}) \right)^\* \\\\ \left( S\_{hr} + S\_{rh} \right) \left( S\_{hh} + S\_{vv} \right)^\* & \left( S\_{hr} + S\_{rh} \right) \left( S\_{hh} - S\_{vv} \right)^\* & \left| S\_{hr} + S\_{rh} \right|^2 & \left( S\_{hr} + S\_{rh} \right) \left( j(S\_{hr} - S\_{rh}) \right)^\* \\\\ \left( S\_{hr} - S\_{rh} \right) \left( S\_{hh} + S\_{vv} \right)^\* & j(S\_{hr} - S\_{rh}) \left( S\_{hh} - S\_{vv} \right)^\* & j(S\_{hr} - S\_{rh}) \left( S\_{hr} + S\_{rh} \right)^\* & \left| S\_{hr} - S\_{rh} \right|^2 \end{bmatrix} \, \tag{1.54}$$

As in the case of klk<sup>T</sup> <sup>l</sup> , kpk<sup>T</sup> <sup>p</sup> presents a rank equal to 1, and therefore, it may present up to seven independent parameters. Finally, if the backscattering direction is

considered under the BSA convention, the coherency matrix reduces to

$$\mathbf{k}\_{p}\mathbf{k}\_{p}^{T\*} = \begin{bmatrix} \left| S\_{hh} + S\_{vv} \right|^2 & (S\_{hh} + S\_{vv})(S\_{hh} - S\_{vv})^\* & 2(S\_{hh} + S\_{vv})S\_{hv}^\* \\ (S\_{hh} - S\_{vv})(S\_{hh} + S\_{vv})^\* & \left| S\_{hh} - S\_{vv} \right|^2 & 2(S\_{hh} - S\_{vv})S\_{hv}^\* \\ 2S\_{hv}(S\_{hh} + S\_{vv})^\* & 2S\_{hv}(S\_{hh} - S\_{vv})^\* & 4\left| S\_{hh} \right|^2 \end{bmatrix}. \tag{1.55}$$

Again, the previous matrix presents a rank equal to 1 and may have up to five independent parameters.

The lexicographic and the Pauli target vector are just a different transformation of the scattering matrix into a vector. Hence, the covariance and coherency matrices are related by the following unitary transformation in the most general configuration:

$$\begin{aligned} \mathbf{k}\_p \mathbf{k}\_p^{T\*} &= \frac{1}{2} \\ & \times \begin{bmatrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & j & -j & 0 \end{bmatrix} \mathbf{k}\_l \mathbf{k}\_l^{T\*} \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -j \\ 0 & 0 & 1 & j \\ 1 & -1 & 0 & 0 \end{bmatrix} . \end{aligned} \tag{1.56}$$

In the case of the backscattering direction under the BSA convention, the previous transformation reduces to

$$\mathbf{k}\_p \mathbf{k}\_p^{T\*} = \frac{1}{2} \begin{bmatrix} 1 & 0 & 1 \\ 1 & 0 & -1 \\ 0 & \sqrt{2} & 0 \end{bmatrix} \mathbf{k}\_l \mathbf{k}\_l^{T\*} \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & \sqrt{2} \\ 1 & -1 & 0 \end{bmatrix} . \tag{1.57}$$

As it may be seen from all this section, the matrices klk<sup>T</sup> l and kpkT <sup>p</sup> contain the same information as the scattering matrix, that is, they are rank 1 matrices. The necessity to introduce these matrices is that they will allow to define the covariance and coherency matrices.

The complex scattering matrix S is able to describe a single physical scattering process, as well as klk<sup>T</sup> <sup>l</sup> and kpk<sup>T</sup> <sup>p</sup> . All these descriptors are based on a wave representation of the data, which depend on the absolute phase from the scatterer. On the contrary, a power representation of the scattering process eliminates this dependence, as power parameters become incoherently additive parameters. In the most general case, assuming the BSA convention, one may define the 4 4 Kennaugh matrix as follows:

$$\mathbf{K} = \mathbf{A}^\* \left( \mathbf{S} \bigotimes \mathbf{S} \right) \mathbf{A}^{-1} \tag{1.58}$$

where

$$\mathbf{A} = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & j & -j & 0 \end{bmatrix}. \tag{1.59}$$

The Kennaugh matrix can be written in the following form:

$$\mathbf{K} = \begin{bmatrix} A\_0 + B\_0 & C\_\Psi & H\_\Psi & F\_\Psi \\ C\_\Psi & A\_0 + B\_\Psi & E\_\Psi & G\_\Psi \\ H\_\Psi & E\_\Psi & A\_0 - B\_\Psi & D\_\Psi \\ F\_\Psi & G\_\Psi & D\_\Psi & -A\_0 + B\_0 \end{bmatrix} \tag{1.60}$$

where

$$\begin{aligned} A\_0 &= \frac{1}{4} \left| S\_{hh} + S\_{vv} \right|^2 \\ B\_0 &= \frac{1}{4} \left| S\_{hh} - S\_{vv} \right|^2 + \left| S\_{hv} \right|^2 \\ B\_\psi &= \frac{1}{4} \left| S\_{hh} - S\_{vv} \right|^2 - \left| S\_{hv} \right|^2 \\ C\_\psi &= \frac{1}{2} \left| S\_{hh} - S\_{vv} \right|^2 \\ D\_\psi &= \mathcal{I} \left\{ S\_{hh} S\_{vv}^\* \right\} \\ E\_\psi &= \mathcal{R} \left\{ S\_{hv}^\* (S\_{hh} - S\_{vv}) \right\} \\ F\_\psi &= \mathcal{I} \left\{ S\_{hv}^\* (S\_{hh} - S\_{vv}) \right\} \\ G\_\psi &= \mathcal{R} \left\{ S\_{hv}^\* (S\_{hh} + S\_{vv}) \right\} \\ H\_\psi &= \mathcal{I} \left\{ S\_{hv}^\* (S\_{hh} + S\_{vv}) \right\} \end{aligned} (1.61)$$

In the previous definition, the sub-index ψ indicates that the different parameters are roll angle dependent, corresponding to the target rotation along the line of sight.

As detailed in Sect. <IP_ADDRESS>, the scattering matrix relates the scattered wave to the incident Jones vector. The Kennaugh matrix is related to the associated Stokes vectors defined in Sect. <IP_ADDRESS>. In the forward scattering case, where S is represented in the FSA coordinate formulation, this matrix is named the 4 4 Mueller matrix and is calculated by

$$\mathbf{M} = \mathbf{A} \left( \mathbf{S} \bigotimes \mathbf{S} \right) \mathbf{A}^{-1}. \tag{1.62}$$

The main difference of K and M, with respect to klk<sup>T</sup> <sup>l</sup> and kpk<sup>T</sup> <sup>p</sup> , is that the Kennaugh and the Mueller matrices are real matrices, whereas the covariance and coherency matrices are complex.

#### <IP_ADDRESS> Partial Scattering Polarimetry

As indicated in Sect. <IP_ADDRESS>, radar polarimetry is concerned with two types of waves. The first type is monochromatic, totally polarized electromagnetic waves where the polarization state is perfectly represented by the Jones vectors. Consequently, the scattering process can be completely represented by any of the scattering polarimetry descriptors detailed in the previous section, and especially the scattering matrix. This situation appears when the radar transmits a perfectly monochromatic wave and this wave reaches an unchanging scatterer, resulting in a perfectly polarized scattered wave. As mentioned, these targets are referred to as point targets or coherent targets. The most important point to be considered when coherent scattering is addressed is to determine the number of independent parameters necessary to represent the scattering process. That is, to determine the number of independent parameters necessary to represent the operator able to characterize the change of the polarization state of the scattered wave with respect to the incident wave that occurs in the scattering process. In a monostatic configuration, the scattering operator describing the scattering, i.e. any of the matrix operators indicated in Sections <IP_ADDRESS> and <IP_ADDRESS>, may present up to five independent parameters. In the bistatic case, these descriptors may present up to seven independent parameters.

The situation changes when the scattering properties of the target being imaged by the radar system change in time, as it would be the case for a forest being affected by the wind conditions or, for instance, when the target presents more than one scattering centre (a point at which the incident wave can be considered to be reflected). Under this situation, although the radar system transmits a perfectly polarized wave, the wave scattered by the scatterer is partially polarized. A scatterer of this category is normally referred to as distributed scatterer, depolarizing scatterer or an incoherent scattering target. The change of the polarization state of the scattered wave makes not possible to use the scattering descriptors presented in Sects. <IP_ADDRESS> and <IP_ADDRESS> to describe the scattering process, as these descriptors are not able to describe the variation of the polarization state of the scattered wave.

In the case of partially polarized waves, the description of the polarization state must be addressed through polarization descriptors relying on the second-order moments of the electromagnetic wave. If a wave is decomposed into two orthogonal components in the plane perpendicular to the propagation direction, these second-order moments refer to the power of each orthogonal component and to the correlation between them. This information is perfectly represented by the vector and the wave coherency matrix or the Stokes vector. In the case of the description of the scattering process, this information can be perfectly represented by the covariance and coherency matrices as the mean values of these matrices are not zero.

#### <IP_ADDRESS> Change of Polarization Basis

The scattering properties of a given scatterer, as demonstrated, are contained within the scattering matrix S, which, as shown previously, is measured in a particular polarization basis. Since there exist an infinite number of orthonormal polarization bases, the question rising at this point is whether it is possible or not to infer the polarimetric properties of the given target in any polarization basis from the response measured at a particular basis. This question presents an affirmative answer. The possibility to synthesize any polarimetric response of a given target from its measurement in a particular orthonormal basis represents the most important property of polarimetric systems in comparison with single-polarization systems. The most important consequence of this process is that the amount of information about a given scatterer can be increased, allowing a better characterization and study. This polarization synthesis process is based on the concept of change of polarization basis presented in Sect. <IP_ADDRESS>.

Before describing the polarization synthesis process in the backscattering direction, it is necessary to analyse the scattering process given by (1.37) with respect to the direction of propagation of the incident and the scattered waves. It must be noticed that the incident wave propagates in the direction given by the unitary vector <sup>b</sup><sup>k</sup> i , whereas the scattered one propagates in the opposite direction, given by bk i . Consequently, this difference in the propagation direction must be taken into account when defining the polarization state of the wave. Given a Jones vector propagating in the direction <sup>b</sup>k, the Jones vector of a wave presenting the same polarization state but which propagates in the direction <sup>b</sup><sup>k</sup> is obtained as

$$
\widehat{\mathbf{k}} \rightarrow -\widehat{\mathbf{k}}\ ,\ \mathbf{E}\left(\widehat{\mathbf{k}}\right) = \mathbf{E}^\*\left(-\widehat{\mathbf{k}}\right) \tag{1.63}
$$

where, as mentioned previously, the BSA convention is considered. Under this assumption, the scattering matrix is referred to as the coordinate system centred in the transmitting/receiving system. Consider a polarimetric radar system which transmits the electromagnetic waves in the following orthonormal basis f g <sup>b</sup>u, <sup>b</sup>u<sup>⊥</sup> . In this particular basis, the incident and scattered waves are related by the scattering matrix as follows:

$$\underline{\mathbf{E}}^{\rm s}\_{\left\{\widehat{\mathbf{u},\widehat{\mathbf{u}}\_{\perp}}\right\}} = \mathbf{S}\_{\left\{\widehat{\mathbf{u},\widehat{\mathbf{u}}\_{\perp}}\right\}} \underline{\mathbf{E}}^{\rm i}\_{\left\{\widehat{\mathbf{u},\widehat{\mathbf{u}}\_{\perp}}\right\}}.\tag{1.64}$$

As shown in Sect. <IP_ADDRESS>, given the Jones vector measured in a particular basis, for instance, f g <sup>b</sup>u, <sup>b</sup>u<sup>⊥</sup> , it is possible to derive it in any other polarization basis <sup>b</sup>u<sup>0</sup> , bu0 ⊥ , which may be rewritten as follows:

$$\underline{\mathbf{E}}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}\right\}} = \mathbf{U}\_{\left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\} \rightarrow \left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}^{\prime}\right\}} \underline{\mathbf{E}}\_{\left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\}}.\tag{1.65}$$

Then, the incident and the scattered waves transformed in the new basis may be considered:

$$\underline{\mathbf{E}}\_{\left\{\widehat{\tilde{\mathbf{u}}},\widehat{\tilde{\mathbf{u}}}\_{\perp}\right\}}^{i} = \mathbf{U}\_{\left\{\widehat{\tilde{\mathbf{u}}},\widehat{\tilde{\mathbf{u}}}\_{\perp}\right\} \to \left\{\widehat{\tilde{\mathbf{u}}}',\widehat{\tilde{\mathbf{u}}}\_{\perp}'\right\}} \underline{\mathbf{E}}\_{\left\{\widehat{\tilde{\mathbf{u}}},\widehat{\tilde{\mathbf{u}}}\_{\perp}\right\}}^{i},\tag{1.66}$$

$$\underline{\mathbf{E}}^{s}\_{\left\{\widehat{\tilde{\mathbf{u}}}^{\prime},\widehat{\tilde{\mathbf{u}}}^{\prime}\right\}} = \mathbf{U}\_{\left\{\widehat{\tilde{\mathbf{u}}},\widehat{\tilde{\mathbf{u}}}\_{\perp}\right\} \rightarrow \left\{\widehat{\tilde{\mathbf{u}}}^{\prime},\widehat{\tilde{\mathbf{u}}}\_{\perp}\right\}} \underline{\mathbf{E}}^{s}\_{\left\{\widehat{\tilde{\mathbf{u}}},\widehat{\tilde{\mathbf{u}}}\_{\perp}\right\}}.\tag{1.67}$$

In order to apply the transformation basis procedure to the scattered waves E<sup>s</sup> bu,bu⊥ , we need to consider that it propagates in the opposite direction as the incident wave Ei bu,bu⊥ . The transformation indicated by (1.64) assumes that the incident and the scattered waves propagate in opposite directions, but (1.66) and (1.67) assume that both waves propagate in the same direction. Consequently it is necessary to consider the transformation indicated by (1.63) in (1.67). As a result, the transformation basis procedure applies to the scattered wave as follows:

Es bu 0 ,bu 0 ⊥ ¼ <sup>U</sup> bu,bu⊥ ! <sup>b</sup><sup>u</sup> 0 ,bu 0 ⊥ Es bu,bu⊥ ð1:68Þ

where now the wave in (1.68) is assumed to propagate in opposite direction with respect to the incident wave in (1.66). Now, it is possible to introduce (1.66) and (1.68) in (1.64):

$$\mathbf{U}^\* \begin{Bmatrix} \hat{\mathbf{u}}^{\prime}, \hat{\mathbf{u}}^{\prime} \end{Bmatrix} \rightarrow \begin{Bmatrix} \hat{\mathbf{u}}, \hat{\mathbf{u}}^{\prime} \end{Bmatrix} \stackrel{\mathbf{E}^s}{=} \begin{Bmatrix} \hat{\mathbf{u}}^{\prime}, \hat{\mathbf{u}}^{\prime} \end{Bmatrix} \stackrel{\mathbf{U}}{=} \mathbf{S} \begin{Bmatrix} \hat{\mathbf{u}}, \hat{\mathbf{u}}^{\prime} \end{Bmatrix} \stackrel{\mathbf{U}}{\left\{\hat{\mathbf{u}}^{\prime}, \hat{\mathbf{u}}^{\prime} \right\}} \rightarrow \begin{Bmatrix} \hat{\mathbf{E}}^{\hat{i}} \hat{\mathbf{u}}^{\prime}, \hat{\mathbf{u}}^{\prime} \end{Bmatrix} \stackrel{\mathbf{E}^i}{=} \mathbf{S}^{\prime} \begin{Bmatrix} \hat{\mathbf{u}}^{\prime}, \hat{\mathbf{u}}^{\prime} \end{Bmatrix} \stackrel{\mathbf{J}}{\tag{1.69}}$$

As the transformation matrix U is unitary, i.e. U-1 ¼ <sup>U</sup><sup>T</sup> ,

$$\underline{\mathbf{E}}^{s}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}^{\prime}\_{\perp}\right\}} = \mathbf{U}^{T}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}\right\} \rightarrow \left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\}} \mathbf{S} \left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\} \mathbf{U} \left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}\right\} \rightarrow \left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\} \mathbf{E}^{\bar{i}}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}\right\}} \tag{1.70}$$

from where it can be clearly identified the following identity

$$\mathbf{S}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}^{\prime}\right\}} = \left(\mathbf{U}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}^{\prime}\right\} \rightarrow \left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\}}\right)^{-1} \mathbf{S}\_{\left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\}} \mathbf{U}\_{\left\{\widehat{\mathbf{u}}^{\prime},\widehat{\mathbf{u}}\_{\perp}^{\prime}\right\} \rightarrow \left\{\widehat{\mathbf{u}},\widehat{\mathbf{u}}\_{\perp}\right\}}. \tag{1.71}$$

The transformation expressed in (1.71) receives the name of con-similarity transformation. This transformation allows to synthesize the scattering matrix in an arbitrary basis bu0 , bu0 ⊥ from its measure in the basis f g <sup>b</sup>u, <sup>b</sup>u<sup>⊥</sup> .

# <IP_ADDRESS> Scatterers Characterization by Single, Dual, Compact and Full Polarimetry

The main objective behind the use of polarimetric diversity, also known as full polarimetry, when observing a particular scatterer is that this type of diversity allows a far more complete characterization of the scatterer than the characterization that could be obtained without polarimetric sensitivity, or simply single-polarization measurements. Although this improved characterization, if compared with singlepolarization data, the use of polarimetric diversity comes at a price, as the average transmitted power must be doubled and the swath width halved. In addition, a fully polarimetric SAR is technologically more complex than a singlepolarization SAR system. In order to understand the difference between these two philosophies and the improvement in the characterization of a scatterer provided by polarimetry, it is necessary to introduce two important concepts, since they will determine the way in which a target shall be characterized. It may happen the scatterer of interest to be smaller than the coverage of the radar system. In this situation, we consider the scatterer as an isolated scatterer, and from a point of view of power exchange, this target is characterized by the so-called radar cross section. Nevertheless, we can find situations in which the scatterer of interest is significantly larger than the coverage provided by the radar system. In these occasions, it is more convenient to characterize the target independently of its extent. Hence, in these situations, the target is described by the so-called scattering coefficient.

The most fundamental form to describe the interaction of an electromagnetic wave with a given scatterer is the so-called radar equation. This equation establishes the relation between the power the scatterer intercepts from the incident electromagnetic wave and the power reradiated by the same scatterer in the form of the scattered wave. The radar equation presents the following form:

$$P\_r = \frac{P\_t G\_t}{4\pi r\_t^2} \sigma \frac{A\_r}{4\pi r\_r^2} \tag{1.72}$$

where Pr represents the power detected at the receiving system. The term

$$\frac{P\_t G\_t}{4\pi r\_t^2} \tag{1.73}$$

is determined by the incident wave, and it consists of its power density expressed in terms of the properties of the transmitting system. The different terms in (1.73) are the transmitted power Pt, the antenna gain Gt and the distance between the system and the target rt. On the contrary, the term

$$\frac{A\_r}{4\pi r\_r^2} \tag{1.74}$$

contains the parameters concerning the receiving system: the effective aperture of the receiving antenna Ar and the distance between the target and the receiving system rr. The last term in (1.72), i.e. σ, determines the effects of the scatterer of interest on the balance of powers established by the radar equation. Since (1.73) is a power density, i.e. power per unit area, and (1.74) is dimensionless, the parameter σ has units of area. Consequently, σ consists of an effective area which characterizes the scatterer. This parameter determines which amount of power is intercepted from (1.73) by the scatterer and reradiated. This reradiated power is finally intercepted by the receiving system according to the distance rt. An important fact which arises at this point is the way the scatterer reradiates the intercepted power in a given direction of the space. In order to be independent of this property, the radar cross section shall be referenced to an idealized isotropic scatterer. Thus, the radar cross section of an object is the cross section of an equivalent isotropic scatterer that generates the same scattered power density as the object in the observed direction:

$$\sigma = 4\pi r^2 \frac{\left|\overrightarrow{\mathbf{E}}^s\right|^2}{\left|\overrightarrow{\mathbf{E}}^i\right|^2} = 4\pi |\mathbf{S}|^2\tag{1.75}$$

where E ! 2 represents the intensity of the electromagnetic wave and S is the complex scattering amplitude of the object. The final value of σ is a function of a large number of parameters which are difficult to consider individually: the wave frequency, the wave polarization, the imaging geometry or the geometrical structure and the dielectric properties of the scatterer. Then, the radar cross section σ is able to characterize the target being imaged for a particular frequency and imaging system configuration.

The radar equation, as given by (1.72), is valid for those cases in which the scatterer of interest is smaller than the radar coverage, that is, a point target or point scatterer. For those targets presenting an extent larger than the radar coverage, we need a different model to represent the scatterer. In these situations, a scatterer is represented as an infinite collection of statistically identical point scatterers. The resulting scattered wave E !s results from the coherent addition of the scattered waves from every one of the independent scatterers which model the extended scatterer. In order to express the scattering properties of the extended target independently of its area extent, we consider every elementary target as being described by a differential radar cross section dσ. In order to separate the effects of the target extent, we consider dσ as the product of the averaged radar cross section per unit area σ<sup>0</sup> and the differential area occupied by the target ds. Then, the differential power received by the systems due to an elementary scatterer can be written as

$$dP\_r = \frac{P\_t G\_t}{4\pi r\_t^2} \sigma^0 ds \frac{A\_r}{4\pi r\_r^2}.\tag{1.76}$$

Hence, to find the total power received from the extended target, we need to integrate over the illuminated area A0:

$$P\_r = \iint\limits\_{A\_0} \frac{P\_t G\_t}{4\pi r\_t^2} \sigma^0 \frac{A\_r}{4\pi r\_r^2} ds. \tag{1.77}$$

It must be noted that the radar equation at (1.72) represents a deterministic problem, whereas (1.77) considers a statistical problem. Eq. (1.77) represents the average power returned from the extended target. Hence, the radar cross section per unit area σ<sup>0</sup> , or simply scattering coefficient, is the ratio of the statistically averaged scattered power density to the average incident power density over the surface of the sphere of radius rr:

$$
\sigma^0 = \frac{E\{\sigma\}}{A\_0} = \frac{4\pi r\_r^2}{A\_0} \frac{E\left\{ \left| \overrightarrow{\overline{E}}^s \right|^2 \right\}}{\left| \overrightarrow{E}^i \right|^2}. \tag{1.78}
$$

The scattering coefficient σ<sup>0</sup> is a dimensionless parameter. As in the case of the radar cross section, the scattering coefficient is employed to characterize the scattered being imaged by the radar. This characterization is for a particular frequency f, polarization of the incident and scattered waves and incident and scattering directions.

As it has been shown, the characterization of a given scatterer by means of the radar cross section σ or the scattering coefficient σ<sup>0</sup> depends also on the polarization of the incident wave E !i . As one can observe in (1.75) and (1.78), these two coefficients are expressed as a function of the intensity of the incident and scattered waves. Consequently, σ and σ<sup>0</sup> shall be only sensitive to the polarization of the incident waves through the effects the polarization has over the power of the related electromagnetic waves. Hence, if we denote by p the polarization of the incident wave and by q the polarization of the scattered wave, we can define the following polarization-dependent radar cross section and scattering coefficient, respectively:

$$\sigma\_{qp} = 4\pi r^2 \frac{\left|\overrightarrow{\mathbf{E}}\_{qp}^{\circ}\right|^2}{\left|\overrightarrow{\mathbf{E}}\_{qp}^{\circ}\right|^2} = 4\pi \left|S\_{qp}\right|^2,\tag{1.79}$$

$$
\sigma\_{qp}^0 = \frac{E\{\sigma\_{qp}\}}{A\_0} = \frac{4\pi r\_r^2}{A\_0} \frac{E\left\{ \left| \overrightarrow{\mathbf{E}}\_{qp}^{\circ} \right|^2 \right\}}{\left| \overrightarrow{\mathbf{E}}\_{qp}^{\circ} \right|^2}. \tag{1.80}
$$

As it has been shown, a given target of interest can be characterized by means of the radar cross section or the scattering coefficient depending on the nature of the scatterer itself; see (1.75) and (1.78). Additionally, in (1.79) and (1.80), it has been shown that these two coefficients depend also on the polarization of the incident and the scattered electromagnetic waves. A closer look to these expressions reveals that these two real coefficients depend on the polarization of the electromagnetic waves only through the power associated with them. Thus, they do not exploit, explicitly, the vectorial nature of polarized electromagnetic waves. A SAR system that measures σ or σ<sup>0</sup> is usually referred to as single-polarization SAR systems as, normally, the same polarization is employed for transmission and for reception. In this case, the products delivered by the SAR system are real SAR images containing the information of σ or σ<sup>0</sup> .

In order to take advantage of the polarization of the electromagnetic waves, that is, their vectorial nature, the scattering process at the scatterer of interest must be considered as a function of the electromagnetic waves themselves. In Sect. <IP_ADDRESS>, it was shown that the polarization of a plane, monochromatic, electric wave could be represented by the so-called Jones vector. Additionally, a set of two orthogonal Jones vectors form a polarization basis, in which any polarization state of a given electromagnetic wave can be expressed. Therefore, given the Jones vectors of the incident and the scattered waves, E<sup>i</sup> and E<sup>s</sup> , respectively, the scattering process occurring at the target of interest is represented by the scattering matrix S. In contraposition to a single-polarization SAR system, a fully polarimetric SAR system measures the complete scattering matrix S. Therefore, the product delivered by this type of SAR systems corresponds to the 2 2 complex scattering matrix and not individual real SAR images.

As it can be observed, the polarimetric sensitivity of a measurement ranges from a complete absence of polarimetric sensitivity in the case of single-polarization SAR systems to a complete sensitivity in the case of fully polarimetric SAR systems. Polarimetric sensitivity comes to a price of a more complex system that implies, on the one hand, a heavier system and, on the other hand, the need to transmit a larger power. In addition, and due to the need to double the pulse repetition frequency to accommodate two polarizations in transmission, the radar swath is also reduced. Nevertheless, between both architectures, there exist other polarimetric radar configurations with may soften the previous limitations but at the cost of reducing the amount of acquired information.

A single-polarization or mono-polarization SAR system is composed of one transmission and one reception chain that operate at a fixed polarization. In most of the cases, both chains operate at the same polarization providing a co-pol or co-polarized channel. In the particular case of the linear polarization basis, these channels would correspond to σhh or σ<sup>0</sup> hh and σvv or σ<sup>0</sup> vv for the horizontal and the vertical polarization states, respectively. As indicated, these simple imaging radars deliver real SAR images, proportional to σ or σ0 , as products. One possibility to increase the amount of information is to consider a dual-polarized radar by including a second reception chain in the system, in such a way that it transmits in one polarization, for instance, h, and it receives simultaneously on the same polarization h and also on the orthogonal one v, leading to one co-pol and the so-called co-polarized and the cross-polarized channels, respectively. A different alternative for a dual-polarized system is to consider a transmission chain that alternates between polarizations and a single reception chain. In all these cases, the polarimetric system provides images proportional to the radar brightness.

All the previous SAR systems present the limitation that the information that may be retrieved is restricted to the information that can be extracted from the real SAR images, proportional to σ or σ<sup>0</sup> , or their different combinations. Nevertheless, this limitation is overcome by allowing two simultaneous and coherent reception channels operating at orthogonal polarizations, making it possible to measure the relative phase between them. The coherent nature of the receiving channels allows measuring the different elements of the covariance or coherency matrix. The first option that may be considered is to assume a fixed polarization in transmission and orthogonal polarizations in reception. In the case of the transmission channel, the circular polarization and the 45 linear polarizations have been proposed, whereas for reception the horizontal and vertical linear polarizations are assumed. This type of systems are collectively known as compact polarized systems as, although they allow to measure some of the elements of the covariance and coherency matrix, they do not allow to measure the complete matrices. Finally, by allowing the system to transmit alternatively between orthogonal polarizations and to receive coherently at the same two orthogonal polarizations, a system like this is able to measure coherently the scattering matrix and to produce the covariance and coherency matrices. In the case of a bistatic configuration, without any type of assumption, these

Fig. 1.5 The family of polarization diversity and polarimetric imaging radars. (Courtesy of Dr. R. K. Raney)

will be 4 4 complex matrices, whereas in the case of a monostatic configuration, these will be 3 3 complex matrices. Figure 1.5 details the complete hierarchy of polarimetric SAR systems.

# 1.2 SAR Data Statistical Description and Speckle Noise Filtering

Most of geophysical media, for instance, rough surfaces, vegetation, ice, snow, etc., have a very complex structure and composition. Consequently, the knowledge of the exact scattered electromagnetic wave, when illuminated by an incident wave, is only possible if a complete description of the scene was available. This type of description of the scatterers is unattainable for practical applications. The alternative, hence, is to describe them in a statistical form. Such scatters are named, consequently, as distributed or partial scatterers (Ulaby et al. 1986a, b).

SAR systems are mainly employed for natural scenes observation. Owing to the complexity of these scatterers, the scattered wave has also a complex behaviour. Hence, the scattering process itself needs to be analysed stochastically. Most of the techniques focused on solving the scattered wave problem trying to find, hence, the statistical moments of the scattered wave as a function of the incident wave properties, as well as the scatterer features.

In order to derive a stochastic model for the observed SAR images in the case of distributed scatterers, it is necessary to consider a model for the SAR imaging process, a model for the scattering process and a model for the distributed scatter being imaged.

The SAR imaging process is divided into two main processes. The former consists of the acquisition of the scattered data, as a result of the illuminating wave, whereas the latter comprises the focusing process. The second, which is in charge of collecting all the contributions of a particular scatterer focusing it as good as possible, tries to remove the effects of the acquisition process. The SAR impulse response, or SAR system model, embracing the acquisition, as well as the focusing processes, can be assumed to be a rectangular low-pass filter (Curlander and McDonough 1991):

$$h(\mathbf{x}, r) \propto \text{sinc}\left(\frac{\pi \mathbf{x}}{\delta\_x}\right) \text{sinc}\left(\frac{\pi r}{\delta\_r}\right). \tag{1.81}$$

In the previous equation, x and r indicate the azimuth and slant-range (simply called range in the following) dimensions of the SAR image, respectively, whereas δ<sup>x</sup> and δ<sup>r</sup> indicate the spatial resolutions in the same spatial dimensions. Finally, a SAR image, i.e. S(x, r), may be modelled as a two-dimensional low-pass filter, given by (1.81), applied to the scene's complex reflectivity σs(x, r) (Curlander and McDonough 1991):

$$S(\mathbf{x}, r) = \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \sigma\_s(\mathbf{x}', r') h(\mathbf{x} - \mathbf{x}', r - r') d\mathbf{x}' d\mathbf{r}'. \tag{1.82}$$

Since the spatial resolutions of the SAR impulse response, δ<sup>x</sup> and δr, are not zero, it is possible to introduce the concept of resolution cell as the area given by <sup>δ</sup><sup>x</sup> <sup>δ</sup><sup>r</sup> . Qualitatively, in the absence of signal re-sampling, the information contained by an image pixel is basically determined by the average complex reflectivity σs(x, r) within this resolution cell.

The resolution cell dimensions, δ<sup>x</sup> and δr, are larger than the wavelength of the illuminating electromagnetic wave λ. As a consequence, the resulting scattered wave is due to an elaborated scattering process. In order to arrive to a tractable mathematical model of the SAR image S(x, r), it is convenient to find an approximation for the scattering process within the resolution cell. The most common simplification is the Born approximation or simple scattering approximation (Ulaby et al. 1986a). Under it, first, the distributed scatterer is considered to be composed of a set of discrete scatterers characterized by a deterministic response. This scatterer model might be reasonable for those cases in which the discrete scatterer description is valid, for instance, scattering from raindrops or vegetation-covered surfaces having leaves small compared with the wavelength. On the contrary, this assumption is not valid for continuous scatterers. In these cases, it is helpful to apply the concept of effective scattering centre (Ulaby et al. 1986a), in which the continuous scatterer is analysed in a discrete way, e.g. the facet model for surface scattering (Ulaby et al. 1986a; Beckmann and Spizzichino 1987). In a second step, the scattered wave from the resolution cell is supposed to be the linear coherent combination of the individual scattered waves of each one of the discrete scatters within the cell. The main limitation of the Born approximation is that it excludes attenuation or multiple scattering in the process.

Assuming the scattered wave from any distributed scatterer to be originated by a set of discrete sources, (1.82) can be considered in its discrete form:

$$S(\mathbf{x}, r) = \sum\_{k=1}^{N} \sigma\_s(\mathbf{x}\_k, r\_k) h(\mathbf{x} - \mathbf{x}\_k, r - r\_k) \tag{1.83}$$

where the sub-index k refers to each particular discrete scatterer in the resolution cell and N is the total number of these scatterers embraced by the response of the SAR system h(x, r). Equation (1.83) can be rewritten by using

$$
\sigma\_s(x\_k, r\_k) = \sqrt{\sigma\_k} \exp\left( \begin{array}{c} j\theta\_{s\_k}' \end{array} \right), \tag{1.84}
$$

$$h(\mathbf{x} - \mathbf{x}\_k, r - r\_k) = h\_k \exp\left(\begin{array}{c} j\rho\_k \end{array}\right),\tag{1.85}$$

$$
\theta\_{s\_k} = \theta\_{s\_k}' + \varphi\_k,\tag{1.86}
$$

as follows:

$$S(\mathbf{x}, r) = \sum\_{k=1}^{N} A\_k \exp\left( \begin{array}{c} j\theta\_{s\_k} \end{array} \right) \tag{1.87}$$

where Ak <sup>¼</sup> hk ffiffiffiffiffi σk p . As observed in (1.87), the process to form a SAR image pixel consists of the complex coherent addition of the responses of each one of the discrete scatterers, which are not accessible individually. The sole available measure is the complex coherent addition itself. This coherent addition process receives the name of bi-dimensional random walk (McCrea and Whipple 1940; Doob et al. 1954).

At this point, it is necessary to consider certain assumptions about the elementary scattered waves Ak exp <sup>j</sup>θsk ð Þ in order to derive a stochastic model for the observed SAR image (Beckmann and Spizzichino 1987; Goodman 1985):

Under these conditions, (1.87) may be seen as an interference process, in which the interference itself is determined by the phases θsk . This interference can be constructive, as well as destructive. This effect can be clearly noticed in SAR images, as the amplitude or the intensity of (1.87) presents a salt and pepper or grainy aspect, as it may be observed in Fig. 1.6, which corresponds to |Shh| acquired with the RADARSAT-2 system over the city of San Francisco. Such a phenomenon is known as speckle (Goodman 1985; Lee 1981; Lopes et al. 1990; Raney 1983).

Speckle is a true electromagnetic measurement and has a complete deterministic nature, as shown in (1.87). Nevertheless, the information contained within speckle needs from two different analyses. In those cases in which there is a reduced number of discrete scatterers within the resolution cell, or its response is basically originated by a reduced set of

Fig. 1.6 RADARSAT-2 amplitude image of the scattering matrix element Shh over San Francisco (USA)

dominant ones, speckle is said to be partially developed. Hence, the interference itself, i.e. the speckle, contains information about the scattering process. On the contrary, when there is a large number of discrete scatterers in the cell, without a dominant one, the interference process becomes so complex that it does contain almost no information about the scattering process itself (Oliver and Quegan 1998). This case is called fully developed speckle (Ulaby et al. 1986a), and the complexity of the interference process is overcome by analysing it by statistical means. Hence, speckle turns to be considered as a noise-like signal (Ulaby et al. 1986b; Lopes et al. 1990).

Summarizing, due to the lack of knowledge about the detailed structure of the distributed scatterer being imaged by the SAR system, it is necessary to discuss the properties of the scattered wave statistically. The statistics of concern are defined over an ensemble of objects, all with the same macroscopic properties, but differing in the internal structure. For a given SAR system imaging a particular scatterer, e.g. a rough surface, the exact value of each pixel cannot be predicted, but only the parameters of the distribution describing the pixel values. Therefore, for a SAR image, the actual information per pixel is very low as individual pixels are random samples from distributions characterized by a reduced set of parameters.

# 1.2.1 One-Dimensional Gaussian Distribution

Considering a SAR system to be described by a rectangular low-pass filter (see (1.81)) and the distributed scatterer to be modelled by a set of discrete deterministic scatterers, by means of the single or Born scattering approximation, a SAR image, S(x, r), can be described by the model presented in (1.87).

The main parameter in the SAR image model is the number of discrete scatterers within the resolution cell, i.e. N. Depending on the nature of this parameter, different SAR image statistical models can be derived. On the one hand, if N is considered as a constant value, provided that it is large enough, (1.87) leads to the complex, zero-mean, complex Gaussian pdf model, valid for homogeneous, non-textured SAR images (Beckmann and Spizzichino 1987; Goodman 1985; Papoulis 1984). On the other hand, to consider N as a random variable makes (1.87) to lead to pdf models valid for textured areas description. In the following, the zero-mean, complex Gaussian distribution model shall be considered, although possible extensions to textured image models shall be indicated.

When the number of discrete scatterers inside the resolution cell N is large, provided that Ak cos ( jθk) and Ak sin ( jθk) satisfy the central limit theorem (Oliver and Quegan 1998), the quantities ℜ{S} and ℑ{S} are Gaussian distributed, that is, they follow a zero-mean, Gaussian probability density function (pdf). The parameters of this pdf can be obtained on the basis of the discrete scatterers model. The mean values of ℜ{S} and ℑ{S} are equal to zero, and the variances are E ℜ<sup>2</sup> f g<sup>S</sup> ¼ <sup>E</sup> <sup>ℑ</sup><sup>2</sup> f g<sup>S</sup> ¼ N <sup>2</sup> E Ak <sup>2</sup> . Besides, the symmetry of the phase distribution of the discrete scatterers produces (Beckmann and Spizzichino 1987):

$$\begin{aligned} E\{\mathcal{R}\{S\} \mathcal{I}\{S\}\} &= \sum\_{k=1}^{N} \\ &\times \sum\_{l=1}^{N} E\{A\_k A\_l\} E\{\cos\left(\theta\_k\right) \sin\left(\theta\_l\right)\} \\ &= 0. \end{aligned} \tag{1.88}$$

Under these conditions, ℜ{S} and ℑ{S}, denoted in the following by x and y, respectively, are described by means of zero-mean Gaussian pdfs:

$$p\_x(\mathbf{x}) = \frac{1}{\sqrt{\pi \sigma^2}} \exp\left(-\left(\frac{\mathbf{x}}{\sigma}\right)^2\right), \quad \mathbf{x} \in (-\infty, \infty) \quad (1.89)$$

$$p\_\mathbf{y}(\mathbf{y}) = \frac{1}{\sqrt{\pi \sigma^2}} \exp\left(-\left(\frac{\mathbf{y}}{\sigma}\right)^2\right), \quad \mathbf{y} \in (-\infty, \infty) \quad (1.90)$$

where the variance is <sup>σ</sup><sup>2</sup> <sup>¼</sup> (N/2)E{Ak 2 }. The pdfs px(x) and py( <sup>y</sup>) are denoted in the following as <sup>N</sup> 0, <sup>σ</sup><sup>2</sup> ð Þ. Consequently, a SAR image, <sup>S</sup> ¼ <sup>x</sup> <sup>+</sup> jy ¼ <sup>A</sup> exp ( <sup>j</sup>θ), is described by a zeromean, complex, Gaussian distribution, with uncorrelated real and imaginary parts, denoted next as <sup>N</sup> 0, <sup>σ</sup><sup>2</sup> ð Þ. From (1.89) and (1.90), it is straightforward to derive pA(A) and pθ(θ), where <sup>A</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>x</sup><sup>2</sup> þ <sup>y</sup><sup>2</sup> <sup>p</sup> and <sup>θ</sup> <sup>¼</sup> arctan (y/x), as:

$$p\_{A, \theta}(A, \theta) = \frac{A}{\pi \sigma^2} \exp\left(-\left(\frac{A}{\sigma}\right)^2\right) \tag{1.91}$$

$$p\_A(A) = \frac{2A}{\sigma^2} \exp\left(-\left(\frac{A}{\sigma}\right)^2\right), \quad A \in [0, \infty) \tag{1.92}$$

$$p\_{\theta}(\theta) = \frac{1}{2\pi}, \quad \theta \in [-\pi, \pi). \tag{1.93}$$

The amplitude pdf, i.e. pA(A), is known as Rayleigh distribution. In addition, if intensity, i.e. <sup>I</sup> ¼ <sup>A</sup><sup>2</sup> , is considered, the Rayleigh pdf is transformed to an exponential pdf:

$$p\_I(I) = \frac{1}{\sigma^2} \exp\left(-\frac{I}{\sigma^2}\right), \quad I \in [0, \infty). \tag{1.94}$$

On the other hand, (1.93) shows that the SAR image phase has a uniform pdf. Consequently, this phase bears no information concerning the natural scene being imaged.

Given the SAR image amplitude pdf (1.92), the amplitude mean value is equal to σ ffiffiffi π p =2, whereas the variance equals (1 - (π/4))σ<sup>2</sup> . If the coefficient of variation, defined as the standard deviation divided by the mean, is calculated, it equals ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ- 4=π 1 p . For the intensity I, it has a value equal to 1 as the mean and the variance are equal to σ<sup>2</sup> . As a consequence, the intensity of a SAR image can be modelled as the product of two uncorrelated terms (Goodman 1985; Lee 1981; Lopes et al. 1990; Raney 1983), i.e.

$$I = \sigma^0 n.\tag{1.95}$$

The deterministic-like value is given by its mean, i.e. σ<sup>0</sup> , which corresponds to the mean incoherent power of the area under study (1.78). The random process n, with mean and variance equal to 1, is characterized by an exponential pdf and is defined as the speckle noise component. As it may be observed from the model (1.94) and (1.95), if the mean value of the intensity increases, the variance increases as well. Therefore, this model is known as the multiplicative speckle noise model. In other words, the signal to noise ratio of the image cannot be improved by increasing the transmitted power, as the variance of the data will increase proportionally.

The Gaussian model for SAR data, leading to (1.91) and (1.95), is able to characterize SAR data for homogeneous areas. In this case, useful information is described by a single degree of freedom, corresponding to the mean intensity. Nevertheless, for certain types of distributed scatterers, such a simple model cannot describe all the data variability. The reason behind this limitation is that this type of scatterers need from a more sophisticated model, with more than one degree of freedom, in order to be completely described. Collectively, these models are able to describe average intensity variations, which correspond to data texture (Oliver and Quegan 1998). A variety of two-degree of freedom pdf models have been proposed in the literature, for instance: K-distribution (Kong 1990), Weibull distribution or log-normal distribution (Oliver and Quegan 1998; Papoulis 1984; Kong 1990). All these models consist of assuming the number of scatterers N, within a resolution cell, no longer as a constant, but being described also by a certain distribution. Even so, there are situations in which these two-parameter models are not able to describe the scene. Hence, the solution goes into the direction of introducing more degrees of freedom, thus resulting in more elaborate image models (Oliver and Quegan 1998; Trunk and George 1970; Trunk 1972; Schleher 1975).

# 1.2.2 Multidimensional Gaussian Distribution

The previous section was concerned with the statistical description of one-dimensional complex data acquired by a complex SAR system, i.e. a single SAR image. As shown, despite the data's complex nature, only the amplitude, or the intensity, contains useful information concerning the distributed target under analysis. The amount of information can be increased by acquiring more than one SAR image, if one or more imaging parameters, e.g. system position, acquisition time, frequency or polarization, are varied. What it is pursued, hence, is the study of the variation of the scatterer's response to changes of the SAR system parameters. The volume of information is increased as more data channels are available but also because, if available, the multidimensional data correlation structure can be also exploited to extract information about the observed scatterer (Oliver and Quegan 1998; Cloude and Pottier 1996). The following list shows the most common multidimensional SAR configurations, as well as their main applications:

to different polarizations of the incident wave. For backscattering in which waves are transmitted and collected in the same position and by considering the reciprocity theorem, PolSAR leads to three-dimensional data. On the contrary, when scattered waves are collected in a different position with respect to the transmitted one, i.e. forward scattering, PolSAR data are four-dimensional data.

From a general point of view, a multidimensional SAR system acquires a set of SAR images, represented by the complex vector

$$\mathbf{k} = \begin{bmatrix} \mathbf{S}\_1, \mathbf{S}\_2, \dots, \mathbf{S}\_m \end{bmatrix}^T \tag{1.96}$$

where m represents the number of SAR images, i.e. the data dimensionality, according to the previous description. Each element of the vector <sup>k</sup>, i.e. Si for <sup>i</sup> <sup>¼</sup> 1, 2, ..., <sup>m</sup>, represents a single complex SAR image. In PolSAR, the k vector receives the name of scattering or target vector (in the straightforward lexicographic basis), and it represents a vectorization of the scattering matrix S as detailed in Sect. <IP_ADDRESS>. The correlation structure of the vector k, provided that its m components Si <sup>N</sup> 0, <sup>σ</sup><sup>2</sup> ð Þ, is completely characterized by the Hermitian covariance matrix C, defined as follows:

$$\begin{aligned} \mathbf{C} &= E\left\{ \mathbf{k} \mathbf{k}^{H} \right\} \\ &= \begin{bmatrix} E\left\{ S\_{1} S\_{1}^{H} \right\} & E\left\{ S\_{1} S\_{2}^{H} \right\} & \cdots & E\left\{ S\_{1} S\_{m}^{H} \right\} \\ E\left\{ S\_{2} S\_{1}^{H} \right\} & E\left\{ S\_{2} S\_{2}^{H} \right\} & \cdots & E\left\{ S\_{2} S\_{m}^{H} \right\} \\ \vdots & \vdots & \ddots & \vdots \\ E\left\{ S\_{m} S\_{1}^{H} \right\} & E\left\{ S\_{m} S\_{2}^{H} \right\} & \cdots & E\left\{ S\_{m} S\_{m}^{H} \right\} \end{bmatrix} . \end{aligned} \tag{1.97}$$

In the particular case of PolSAR data, the data correlation structure can be also expressed by the Hermitian coherency matrix Τ (Cloude and Pottier 1996). Considering (1.97), one can see that the complex vector k is characterized by the following pdf (Lee et al. 1994; Tough et al. 1995):

$$p\_{\mathbf{k}}(\mathbf{k}) = \frac{1}{\pi^{m}|\mathbf{C}|} \exp\left(-\mathbf{k}^{H}\mathbf{C}^{-1}\mathbf{k}\right). \tag{1.98}$$

Hence, (1.98) represents the data pdf model for a set of m correlated SAR images, which is denoted in the following as <sup>N</sup> ð Þ <sup>0</sup>, <sup>C</sup> . Since <sup>k</sup>N~ð Þ <sup>0</sup>, <sup>C</sup> , it is completely characterized by the first and the second central moments, i.e. the mean target vector and the covariance matrix, respectively.

At this point, it is important to consider, as presented before, the issue that the mean value of the real and imaginary parts of k equals zero. The main consequence is that it prevents the possibility to extract useful information via an estimation of this mean value. For instance, this circumstance determines the way PolSAR data has to be considered when distributed scatterers are of concern. A PolSAR system measures the 2 2 complex scattering matrix <sup>S</sup>, which can be vectorized into the form presented by (1.96) (Cloude and Pottier 1996; Bebbington 1992); see also Sect. <IP_ADDRESS>. On the one hand, when this scattering vector refers to a distributed scatterer, a given sample of it has almost no information concerning the scatterer itself, as k consists of a sample of the pdf given by (1.98) (Oliver and Quegan 1998). On the other hand, if the mean value of k is estimated, it turns out to be zero. Thus, as reported in the literature, when distributed scatterers are studied, the vector k, or the scattering matrix S in the particular case of PolSAR, cannot completely describe the properties of the distributed scatterer. Therefore, it is necessary to characterize these properties by means of higher-order moments, i.e. through an estimation of the covariance matrix C, or, additionally, the coherency matrix Τ. These two matrices are derived through the outer product of the target vectors k<sup>l</sup> and kp, respectively, as indicated in Sect. <IP_ADDRESS>, so they are independent of the absolute phase of the scattering matrix S or the target vectors k<sup>l</sup> and kp. Hence, the expected value in (1.97) needs to be estimated. The process to estimate the covariance matrix C is also referred to as the polarimetric speckle noise removal process, as the objective is to remove the variability of the data making it possible to retrieve the C matrix.