Polarimetric Scattering and SAR Information Retrieval E-Book

Ya-Qiu Jin

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: IEEE Press
• Sprache: Englisch

Seitenzahl: 522

Veröffentlichungsjahr: 2013

Beschreibung

Taking an innovative look at Synthetic Aperture Radar (SAR), this practical reference fully covers new developments in SAR and its various methodologies and enables readers to interpret SAR imagery

An essential reference on polarimetric Synthetic Aperture Radar (SAR), this book uses scattering theory and radiative transfer theory as a basis for its treatment of topics. It is organized to include theoretical scattering models and SAR data analysis techniques, and presents cutting-edge research on theoretical modelling of terrain surface. The book includes quantitative approaches for remote sensing, such as the analysis of the Mueller matrix solution of random media, mono-static and bistatic SAR image simulation. It also covers new parameters for unsupervised surface classification, DEM inversion, change detection from multi-temporal SAR images, reconstruction of building objects from multi-aspect SAR images, and polarimetric pulse echoes from multi-layering scatter media.

Structured to encourage methodical learning, earlier chapters cover core material, whilst later sections involve more advanced new topics which are important for researchers. The final chapter completes the book as a reference by covering SAR interferometry, a core topic in the remote sensing community.

• Features theoretical scattering models and SAR data analysis techniques
• Explains the simulation of SAR images for mono- and bi-static radars, covering both qualitative and quantitative information retrieval
• Chapter topics include: theoretical scattering models; SAR data analysis and processing techniques; and theoretical quantitative simulation reconstruction and inversion techniques
• Structured to enable both academic learning and independent study, laying down the foundations first of all before advancing to more complex topics
• Experienced author team presents mathematical derivations and figures so that they are easy for readers to understand 
• Pitched at graduate-level students in electrical engineering, physics, earth and space sciences, as well as researchers
• MATLAB code available for readers to run their own routines

An invaluable reference for research scientists, engineers and scientists working on polarimetric SAR hardware and software, Application developers of SAR and polarimetric SAR, remote sensing specialists working with SAR data – using ESA.

Leseprobe

Contents

Cover

Title Page

Copyright

Preface

Chapter 1: Basics of Polarimetric Scattering

1.1 Polarized Electromagnetic Wave

1.2 Volumetric Scattering

1.3 Surface Scattering

References

Chapter 2: Vector Radiative Transfer

2.1 Radiative Transfer Equation

2.2 Components in Radiative Transfer Equation

2.3 Mueller Matrix Solution

2.4 Polarization Indices and Entropy

2.5 Statistics of Stokes Parameters

Appendix 2A: Phase Matrix of Non-Spherical Particles

References

Chapter 3: Imaging Simulation of Polarimetric SAR: Mapping and Projection Algorithm

3.1 Fundamentals of SAR Imaging

3.2 Mapping and Projection Algorithm

3.3 Platform for SAR Simulation

References

Chapter 4: Bistatic SAR: Simulation, Processing, and Interpretation

4.1 Bistatic Mapping and Projection Algorithm (BI-MPA)

4.2 Scattering Models and Signal Model

4.3 Simulated BISAR Images

4.4 Polarimetric Characteristics of BISAR Image

4.5 Unified Bistatic Polarization Bases

4.6 Raw Signal Processing of Stripmap BISAR

References

Chapter 5: Radar Polarimetry and Deorientation Theory

5.1 Radar Polarimetry and Target Decomposition

5.2 Deorientation Theory

5.3 Terrain Surface Classification

Appendix 5A: Matrix Transformations under Various Conventions

References

Chapter 6: Inversions from Polarimetric SAR Images

6.1 Inversion of Digital Elevation Mapping

6.2 An Example of Algorithm Implementation

6.3 Inversion of Bridge Height

References

Chapter 7: Automatic Reconstruction of Building Objects from Multi-Aspect SAR Images

7.1 Detection and Extraction of Object Image

7.2 Building Reconstruction from a Multi-Aspect Image

7.3 Automatic Multi-Aspect Reconstruction (AMAR)

7.4 Results and Discussion

7.5 Calibration and Validation of Multi-Aspect SAR Data

References

Chapter 8: Faraday Rotation on Polarimetric SAR Image at UHF/VHF Bands

8.1 Faraday Rotation Effect on Terrain Surface Classification

8.2 Recovering the Mueller Matrix with Ambiguity Error ±π/2

8.3 Method to Eliminate the ±π/2 Ambiguity Error

References

Chapter 9: Change Detection from Multi-Temporal SAR Images

9.1 The 2EM-MRF Algorithm

9.2 The 2EM-MRF for Change Detection in an Urban Area

9.3 Change Detection after the 2008 Wenchuan Earthquake

References

Chapter 10: Temporal Mueller Matrix for Polarimetric Scattering

10.1 Radiative Transfer in Inhomogeneous Random Scattering Media

10.2 Time-Dependent Mueller Matrix for Inhomogeneous Random Media

10.3 Polarimetric Bistatic and Backscattering Pulse Responses

10.4 Pulse Echoes from Lunar Regolith Layer

10.5 Monitoring Debris and Landslides

10.6 Appendix 10A: Some Mathematics Needed in Derivation of Temporal Mueller Matrix

References

Chapter 11: Fast Computation of Composite Scattering from an Electrically Large Target over a Randomly Rough Surface

11.1 Bidirectional Analytic Ray Tracing

11.2 Numerical Results

References

Chapter 12: Reconstruction of a 3D Complex Target using Downward-Looking Step-Frequency Radar

12.1 Principle of 3D Reconstruction

12.2 Scattering Simulation and 3D Reconstruction

References

Index

This edition first published 2013

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Library of Congress Cataloging-in-Publication Data

Jin, Ya-Qiu.

Polarimetric scattering and SAR information retrieval / Ya-Qiu Jin and Feng Xu.

pages cm.

Includes bibliographical references and index.

ISBN 978-1-118-18813-2 (cloth)

1. Synthetic aperture radar. 2. Electromagnetic waves–Scattering. I. Title.

TK6592.S95J56 2013

621.3848′5–dc23

2012038931

ISBN: 978-1-118-18813-2

Preface

Polarimetric imagery of synthetic aperture radar (POL-SAR) technology has been one of the most important advances in spaceborne microwave remote sensing during recent decades. Remote sensing data and images have resulted from complex interactions of electromagnetic (EM) waves with natural media, that is, terrain surface, atmosphere, ocean, and so on. The great amount of remote sensing data/images presents rich sources for quantitative description and monitoring of our Earth's environments. However, information retrieval from POL-SAR remote sensing images requires a full understanding of the polarimetric scattering mechanism and SAR imaging principles. The study of EM wave scattering for remote sensing applications, especially for POL-SAR technology, has become an active and interdisciplinary area.

This book presents an account of some of the progress made in research on both the theoretical and numerical aspects of POL-SAR information retrieval.

We begin in Chapter 1 with some basics of polarimetric scattering, for example, the concepts of the scattering matrix, the polarization vector, the four Stokes parameters, and the Mueller matrix solutions for volumetric and surface scattering.

In Chapter 2, the vector radiative transfer (VRT) theory of natural media is presented. The VRT takes account of multiple scattering, emission, and propagation of random scattering media. Its analytical solution provides insights into the underlying mechanism of EM wave–terrain surface interaction. The components of VRT, that is, the scattering, absorption, and extinction coefficients, and the phase matrix of non-spherical scatterers, are introduced. The first-order Mueller matrix solution of VRT for a vegetation canopy model is derived. Polarization indices, eigen-analysis, and entropy are presented. The statistics of multi-look SAR images and the covariance matrix are also investigated.

Chapter 3 first introduces some fundamentals for POL-SAR technology. Based on POL-SAR imaging technology, a new approach to SAR imaging simulation for comprehensive scenarios is developed. It is called MPA (mapping and projection algorithm). In the MPA, scattering contributions from scatterers are cumulatively summed on the slant range plane (the mapping plane), while their extinction and shadowing effects are cumulatively multiplied on the ground range plane (the projection plane). Some parameterized models of random non-spherical scattering media, for example, buildings, the vegetation canopy, rough and tilted surfaces, and so on, are incorporated into the MPA simulation scheme. It finally constructs simulated POL-SAR images of a comprehensive terrain scene. The MPA presents a complete SAR raw data simulation chain, based on scattering models of natural and artificial media and a time-domain raw-data generator.

Chapter 4 deals with the simulation of bistatic POL-SAR images as well as bistatic polarimetry and signal processing. Bistatic SAR (BISAR) with separated transmitter and receiver flying on different platforms has become of great interest. Most BISAR research is focused on hardware engineering realization and signal processing algorithms, with a little on land-based bistatic experiments or theoretical modeling of bistatic scattering. The BI-MPA is developed as a fast and efficient tool for monostatic and bistatic imaging simulations. It involves the physical scattering process of multiple terrain objects. Based on simulated BISAR images, bistatic polarimetry and the unified bistatic polarization bases are discussed. Finally, raw data processing of stripmap BISAR is explored.

Chapter 5 first presents some basic concepts of radar polarimetry and target decomposition, and then develops deorientation theory for surface classification of POL-SAR images. The novel “deorientation” concept is introduced to reduce the influence of randomly fluctuating orientation and reveal the generic characteristics of the targets for better classification. The parameterized model of a random non-spherical scattering medium and its Mueller matrix solution for deorientation validation are developed. A new set of parameters of polarimetric scattering targets is defined to describe target orientation, orderly or random, and scattering surface classification.

Chapter 6 presents some inversion applications using POL-SAR images. It can be found that, as the surface is tilted, the orientation angle at the maximum of co-pol or cross-pol signature can shift from . This shift can be applied to convert the surface slopes. Making use of the Mueller matrix solution and morphological processing, the digital elevation mapping of the Earth's surface is inverted from single-pass polarimetric SAR image data. Using the deorientation theory of Chapter 5, the imaging positions of first-, second-, and third-order scattering of a bridge body can be located. Using the thinning, clustering algorithm, and Hough transform, the lines indicating different orders of scattering are used to invert the bridge height.

Chapter 7 deals with automatic reconstruction of building objects from multi-aspect SAR images. This approach is called as AMAR (automatic multi-aspect reconstruction). Reconstruction of three-dimensional objects from SAR images has become a key issue for target information retrieval. It has been noted that scattering from man-made objects produces bright spots at submeter resolution, or presents strip/block images at meter resolution. The linear profile of a building object is regarded as the most prominent characteristic. The POL-CFAR detector, Hough transform, probabilistic description of the detected façade images, and the maximum-likelihood estimation of building objects from multi-aspect observed façade images are discussed. Eventually, in association with a hybrid priority rule of inversion reliability, an automatic algorithm is designed to match multi-aspect façade images and reconstruct the building objects.

Chapter 8 discusses the Faraday rotation (FR) on a POL-SAR image at UHF/VHF bands. As the polarized electromagnetic wave propagates through the Earth's ionosphere, the polarization vectors are rotated as a result of the anisotropic ionosphere and the action of the geomagnetic field. This rotation is called the FR effect. The FR may decrease the difference between co-polarized backscattering, enhance cross-polarized echoes, and mix different polarized terms. Since the FR is proportional to the square power of the wavelength, it has an especially serious impact on SAR observations operating at a frequency lower than L band. For example, the FR angle at P band can reach dozens of degrees. In this chapter, the Mueller matrix with FR is derived, and recovered from FR. The algorithm of phase unwrapping is also studied.

Chapter 9 presents an algorithm for change detection. Multi-temporal SAR observations provide fast and practicable technical means for surveying and assessing terrain surface changes. How to detect and automatically analyze information on change in terrain surfaces is a key issue in remote sensing. In this chapter, two-threshold expectation maximum (EM) and Markov random field (MRF) algorithms (2EM-MRF) are developed to detect the change direction of backscattering enhanced, reduced, and unchanged regimes from the SAR difference image. As examples, this approach is applied to change detection of urban areas and earthquake regions.

Chapter 10 presents the temporal Mueller matrix solution for polarimetric scattering and radiative transfer as a pulse wave incident upon an inhomogeneous random medium. The Mueller matrix solution for seven scattering mechanisms describes temporal polarimetric scattering and propagation through inhomogeneous scattering media, and, as examples, is applied to image simulation of pulse echoes from the lunar regolith layer, and for monitoring debris flows and landslides.

Chapter 11 introduces a novel numerical approach, that is, the bidirectional analytic ray tracing (BART) method. The basic idea is to launch ray tubes from both the source and observation, trace among the facets, and then record the illumination areas of rays shooting on facets and edges. For each pair of forward and backward rays illuminating the same facet/edge, a scattering path from source to observation is constructed by linking the forward and backward rays, and the corresponding scattering contributions are added up to give the total scattering. This fast computation is applied to the simulation of composite scattering from an electrically large target over a randomly rough surface.

Finally, Chapter 12 deals with a numerical train of simulated polarized scattering–imaging–reconstruction for a complex-shaped electrically large object above a dielectric rough surface. Target detection and reconstruction have been of great interest in both civilian and defense applications. The step-frequency (SF) radar technique is employed to 3D SAR imaging using the downward-looking spotlight mode to alleviate the sheltering and layover effect. The BART of Chapter 11 is employed to calculate the scattering matrix of a 3D target over a rough surface. Some simulation examples of 3D imaging and reconstruction of perfectly electrically conducting targets, that is, a square frustum, and a tank-like target over a rough surface, as well as discussions on the accuracy and computational efficiency, are presented.

This book is based on research progress in our laboratories over the past decade. It is assumed that readers are familiar with material normally covered in undergraduate courses on electromagnetics and some basis of radar remote sensing. However, in each chapter, some fundamentals of these topics are given whenever needed.

This book is intended as a textbook for a graduate course on advanced topics in electromagnetic scattering and SAR remote sensing, and is also a useful reference for researchers working on electromagnetic scattering and POL-SAR microwave remote sensing with various applications. New advances on POL-SAR, such as POL-INSAR, tomography SAR, inverse SAR, counter-SAR, and so on, remain for further study.

This work was partly supported by several projects of the National Science Foundation of China.

Ya-Qiu Jin and Feng XuShanghai and RockvilleOctober 2012

1

Basics of Polarimetric Scattering

1.1 Polarized Electromagnetic Wave

1.1.1 Jones Vector and Scattering Matrix

Radar transmits an electromagnetic (EM) wave and receives the scattered wave from the target. The polarization of an EM wave is defined as the time–space variation of the electric field vector in the plane perpendicular to its propagation direction. As depicted in Figure 1.1, let us first define a coordinate system (), where denotes the propagation direction, and denote the directions of vertical and horizontal linear polarizations, respectively. (In this book, unless otherwise specified, vectors are denoted as bold letters, for example, , and matrices are bold letters with a bar, for example, . A letter with a hat (e.g., ) denotes a unit vector.) A completely polarized monochromatic EM wave can be represented by two orthogonal components along (Kong, 2005)

Figure 1.1 (a) Polarized EM wave and polarization vector. (b) Coordinate system of the polarization vector

(1.1)

where are the corresponding vertical and horizontal components of the electric field, k is the wavenumber and is the angular frequency. In addition, we have the following relationships, which can be readily derived:

(1.2)

In spherical coordinates, where the source is located at the origin, we have .

Imagine when an EM wave passes a certain position in space; the tip of the electric field vector will form a certain trajectory. The shape of such a trajectory is determined by the amplitude ratio and phase difference between the two orthogonal components, that is, the polarization ratio,

(1.3)

The polarization of an EM wave is named after the shape of such a trajectory, that is, linear polarization (when ), circular polarization (when , and ) or elliptical polarization (in general). The trajectories of different polarizations are depicted accordingly in the plane as in Figure 1.2a. In fact, any polarization trajectory can be expressed using the following ellipse equation:

(1.4)

Figure 1.2 (a) Plane of polarization ratio. (b) Parameters of elliptical polarization

There are three geometric parameters of such an elliptical trajectory: amplitude , elliptical angle , and orientation , as illustrated in Figure 1.2b, that is

(1.5)

where the sign of indicates the rotation direction, that is, counter-clockwise (+) or clockwise (−), the elliptical angle determines the shape, and indicates orientation.

We can write Equation 1.1 in the form of a complex vector,

(1.6)

which is often referred to as the Jones vector. It can be found from Equation 1.3 that the polarization ratio is in fact the ratio of these two elements

(1.7)

The Jones vector can also be expressed using the elliptical parameters,

(1.8)

where is the absolute phase. It can be seen from Equation 1.8 that transformation of the Jones vector would correspondingly modify polarization characteristics. This is important in radar polarimetry, as will be discussed later in Chapter 5.

A completely polarized wave incident upon a point target or deterministic simple target would produce a completely polarized scattered wave, which is often called coherent scattering. As a polarized wave incident on a scatterer of volume and dielectric constant , the wave equation can be written as (Jin, 1994)

(1.9)

where the time-harmonic factor is () and is the distance from the scattering point to the observation point. Solving Equation 1.9 under the far-field approximation yields the scattering wave

(1.10)

Here, the scattering field is now expressed by a 2 × 2 (dimensional) complex matrix, that is, the scattering matrix, . Each element of the scattering matrix , represents the complex scattering coefficient under q-polarization incidence and p-polarization scattering.

For a single scattering, its scattering matrix has to be obtained by first calculating the internal field in Equation 1.10 and then performing the integration. Explicit solutions exist for many canonical problems. For example, a small scatterer with its size a far smaller than the wavelength, that is, , can be solved by Rayleigh approximation. A non-spherical particle such as an ellipsoid, a needle- or a disk-like particle with its smallest dimension d far smaller than the wavelength, that is, , can be solved using the Rayleigh–Gans approximation (Ishimaru, 1978, 19911978, 1991). A scatterer of perfectly spherical geometry can be solved using Mie theory. Explicit expressions for also exist for symmetric geometries such as cylinders, spheroids, and so on. Analytic calculations of the scattering matrix for these examples under the Rayleigh–Gans approximation are given later in this chapter. For a more complete derivation, readers are referred to the literature focusing on EM theory (e.g., Jin, 1994).

For a large scattering body with complicated geometry, we have to resort to numerical methods such as the method of momentss (MoM), finite element method (FEM), and finite difference time difference (FDTD), or high-frequency methods such as geometrical optics and physical optics. Chapter 11 will introduce a high-frequency method for calculation of scattering from very complex targets in natural environments.

In practice, fully polarimetric radar measures the entire scattering matrix , while single-polarization radar (single-pol) or dual-polarization radar (dual-pol) can only measure one or two elements of the scattering matrix. Conventionally, co-pol refers to scattering with the same polarization for incidence and scattering, while cross-pol refers to different polarization for incidence and scattering.

Another important concept is the scattering coefficient or radar cross-section (RCS). It is defined as the power ratio of scattering versus incidence as if the scattered field were uniform in all directions:

(1.11)

The RCS is more useful for single-polarization radar where phase information is not critical.

1.1.2 Stokes Vector and Mueller Matrix

As opposed to coherent scattering for a single deterministic target, incoherent scattering is from random or distributed targets, for example, random discrete particles, random media, and randomly rough surfaces. Incoherent scattering is studied from a statistical perspective. That is based on the second-order moment of the random field , which is in the unit of power.

Incoherent scattering produces a partially polarized wave, which is usually described using the Stokes vector , which consists of four Stokes parameters as follows:

(1.12)

where denotes conjugate, denotes wave impedance, and denotes the ensemble average. Stokes law gives

(1.13)

where the equality only holds for a completely polarized wave. For a completely polarized wave, only three of the Stokes parameters in Equation 1.12 are independent. This corresponds to the above-mentioned three geometric parameters of elliptical polarization:

(1.14)

With given intensity , any complete polarization can be mapped into a point on a sphere in 3D space. The sphere with radius is referred to as the Poincaré sphere. Partial polarizations will be distributed inside the Poincaré sphere. Interestingly, two angles defining a 3D vector pointing to any point on the Poincaré sphere are equal to the parameters of the elliptical polarization on that point. Thus, we can draw the distribution of polarizations on the Poincaré sphere as in Figure 1.3. It can be directly mapped to the polarization ratio plane in Figure 1.2.

Figure 1.3 Poincaré sphere

Another commonly used form of Stokes parameters are defined as

(1.15)

These two definitions of Stokes parameters can be transformed into one another.

Incoherent scattering of random or distributed targets has a depolarization effect, which renders the scattered wave to be a partially polarized wave. Using Stokes vectors to represent the incident and scattered wave, respectively, the incoherent scattering is then represented using a 4 × 4 real Mueller matrix

(1.16)

The Mueller matrix is in fact the second-order moment of the scattering matrix. From Equations 1.10 and 1.15, we have

(1.17)

Some useful conclusions about the Mueller matrix are given here. For coherent scattering, only seven of the 16 elements of the Mueller matrix are independent, which corresponds to seven degrees of freedom in the scattering matrix (the ensemble average eliminates the absolute phase). For most natural targets, co-polarization and cross-polarization are not correlated, which makes the eight elements in the top-right and bottom-left quarters approach zero. For the remaining eight elements, the top-left quartet reflect the average power of the two polarizations and the bottom-right quartet reflect the correlation between different polarizations.

From the Mueller matrix, we can also derive polarization signatures of the target, which are defined as follows:

co-polarization signature

(1.18)

cross-polarization signature

(1.19)

where the superscript T denotes transpose. Note that polarization signatures , are in fact the co-polarized (co-pol) and cross-polarized (cross-pol) scattering coefficients under elliptical polarization incidence of the parameters . Substituting or into Equation 1.19, it gives the horizontally or vertically co-pol and cross-pol scattering coefficients.

In addition, we have the polarization degree defined as

(1.20)

and the co-pol/cross-pol phase difference as

(1.21)

and denotes the element of the Mueller matrix . Clearly, we have the inequality , where when the scattered Stokes intensity is completely polarized. The quantity denotes a point inside the Poincaré sphere, and the radius of the Poincaré sphere is equal to I.

Table 1.1 gives few examples of the scattering matrix of ideal scattering targets. The corresponding polarization signatures are given in Figure 1.4.

Figure 1.4 Co-pol and cross-pol signatures of ideal targets

Table 1.1 Example scattering matrices of ideal scattering targets.

Sphere, Plane, or Corner reflectorDiplaneDiplane (45° tilted)Dipole (vertically oriented)Dipole (45° tilted)Helix (right-handed)Helix (left-handed)

1.2 Volumetric Scattering

1.2.1 Small Particle under Rayleigh–Gans Approximation

Small-particle scattering is common in natural environments, and thus is very useful for remote sensing studies. For modeling scattering of the vegetation canopy, we mainly use the Rayleigh–Gans approximation of non-spherical particles including spheroids, disk-like and needle-like particles (Ishimaru, 1978; Jin, 1994).

Under the Rayleigh approximation, the term in Equation 1.10 becomes , that is, it ignores the coherent effect of different elements inside the particle. The internal field can be obtained using electrostatic principles. Hence, the scattering matrix can be calculated.

For non-spherical particles whose smallest dimension meets the condition , we can still use the Rayleigh–Gans approximation, where the internal field is approximated as . However, the term is kept in order to account for the coherent effect of different elements, that is

(1.22)

Notice that the above integral is in fact the Fourier transform of in the direction . If is limited to within a small region, the scattering is smooth along the wavenumber domain , that is, it is a smooth angular distribution, which is close to Rayleigh scattering. On the other hand, the scattering will focus on a narrow forward direction, with a complicated angular distribution. In this sense, it resembles the relationship between time-domain and frequency-domain signals.

Given a small ellipsoid of semi-axes and volume , the term in Equation 1.22 can be obtained as

(1.23)

where denotes the local coordinates of the ellipsoid, is expressed as

(1.24)

and can be expressed in the same way as with replaced by , respectively. It can be found that .

The scattering matrix is thus derived as

(1.25)

Note that the imaginary part of is included so that the scattering matrix can be accurate enough when later used to calculate the extinction coefficient under the optics theorem in Chapter 2.

Assuming that the ellipsoid is rotationally symmetric, that is, , which is the case for most particles in the natural environment, then the scattering matrix can be expanded to

(1.26)

Note that the orientation of the particle's symmetry axis is represented by .

Calculating the integral of Equation 1.24, we have

(1.27)

where the two cases correspond to prolate and oblate ellipsoids, respectively.

For a disk-like particle, with radius , height , and volume , and given , the integral of Equation 1.22 can be derived as

(1.28)

(1.29)

where denotes the first-order Bessel function of the first kind.

For a needle-like particle with radius , length , and volume , and given , the integral in Equation 1.22 can be derived as

(1.30)

(1.31)

Under extreme conditions, and , Equations 1.29 and 1.31 are consistent with Equation 1.27.

These expressions are critical in computing the scattering matrix, phase matrix, and extinction coefficient, and for modeling the scattering behavior of random particles in the vegetation canopy.

1.2.2 Slim Cylinder

A finite-length slim cylinder is often used to model the trunk, branches or twigs of trees. Its scattering can be solved as in Equation 1.10

(1.32)

where the internal field term can be approximated as if it was an infinitely long cylinder, for which the explicit solution can be found. The internal field of an infinitely long cylinder can be found, for example, in Wait (1986) and Karam and Fung (1988). Substituting into Equation 1.32 and completing the integral, it yields the explicit expression of the scattering field and hence the scattering matrix (Karam and Fung, 1988).

Given the cylinder parameters of length and radius , the dielectric constant , and the fact that its axis is aligned with , the scattering matrix for a cylinder can be derived as

(1.33)

(1.34)

(1.35)

(1.36)

where

(1.37)

(1.38)

(1.39)

(1.40)

and

(1.41)

Note that and stand for the nth-order Bessel function and its derivative, respectively; and and are the nth-order Hankel function of the second kind and its derivative, respectively.

It can be seen from Equations 1.35 and 1.36 that there is no cross-polarized backscattering, that is, under the condition .

Interestingly, it can be found that, if and , then in Equation 1.38; if , then , which means that the scattered wave propagates along the cone surface forming angle with respect to axis .

In the general case when the cylinder orientation differs from the global coordinate , then the above expressions should be defined in its local coordinates (). The scattering field in Equation 1.32 and the scattering matrix in Equations 1.33–1.36 have to be further converted to the global coordinate system via coordinate conversions, which will be introduced in subsection 1.3.7.

1.3 Surface Scattering

1.3.1 Plane Surface

Surface scattering is the most common scattering in the natural environment. The most basic model would be an infinite perfectly flat surface. Its scattering will concentrate on the specular direction, that is, reflection, which is directly solved as the Fresnel reflection coefficient.

As shown in Figure 1.5, a plane surface has a reflection matrix and a transmission matrix written as

Figure 1.5 Reflection and transmission of a plane surface

(1.42)

where denote the Fresnel coefficients of vertical and horizontal polarization

(1.43)

where is the incident angle, and is the relative dielectric constant. The reflection angle and transmission angle are related to the incident angle according to Snell's law:

(1.44)

For incoherent scattering, the Mueller matrix of plane surface reflection and transmission can be easily derived from Equation 1.17 as

(1.45)

(1.46)

Note that the transmission Mueller matrix, Equation 1.46, contains a factor of because the Stokes vector includes wave impedance and the transmitted wave and the incident wave are in different media. Since the Mueller matrix is a real matrix, only the real part of is used here. Also note that the same letters are used to denote both scattering and Mueller matrix, simply because they are in fact the same reflection or transmission but in different forms.

1.3.2 Rough Surface

In natural environments, most surfaces are randomly rough surfaces, for example, the ground surface, sea surface, and building facets. A randomly rough surface is often described by a random height profile, that is, for a 2D rough surface, and for a 3D rough surface. When calculating rough-surface scattering, the most important entity is the correlation function of the height profile and its Fourier transform, that is, the spectral density function (or spectral function).

A Gaussian rough surface is the most basic type of rough surface. It follows the Gaussian distribution function:

(1.47)

Let , the correlation function between and , be

(1.48)

where the correlation length in both x and y directions is simply assumed to be the same (it can be different as ). Then, the joint probability density function can be derived as

(1.49)

(1.50)

where is the mean-square surface slope

(1.51)

In general, a surface is considered of higher roughness if is larger, is smaller or is larger.

The spectral function of a Gaussian surface can be obtained as

(1.52)

For different types of rough surfaces, the same scattering model can be applied but with different spectral functions. The most commonly used spectral functions include the Gaussian function, exponential function, and the Pierson–Moskowitz empirical spectrum, which is often used to model the wind-driven sea surface (Fung and Lee, 1982; Huang and Jin, 1995).

The scattering model of a rough surface is illustrated in Figure 1.6. We define the random height profile function as . If a polarized wave is incident in the direction , the scattered wave in the direction can be written as

(1.53)

Figure 1.6 Scattering model of a rough surface

Following Huygens' principle (Kong, 2005), Equation 1.53 can be derived as integration of the induced electric and magnetic fields on the rough surface,

(1.54)

where denotes the dyadic Green function, are the induced electric and magnetic fields at position , and denotes the local normal vector at position , that is

(1.55)

Under the far-field approximation, this yields

(1.56)

Clearly, and are random. However, the incoherent scattering, that is, the ensemble average of the scattering wave, , is of interest to us, which is further related to the correlation function and spectral function of the rough surface.

Approximate analytical solutions to the problem of rough-surface scattering can be mainly categorized into the Kirchhoff approximation (KA) approach, which is valid for large-scale roughness, the small-perturbation approximation (SPA), suitable for small-scale roughness, the two-scale approximation (TSA) approach, which combines the former two, and the integral equation method (IEM), which are introduced in detail in the following subsections.

1.3.3 Kirchhoff Approximation

If the rough surface fluctuates on a large scale, meaning that its average radius of curvature is far larger than its wavelength, then any portion of the surface can be locally seen as a flat facet and the induced current can be approximated as the equivalent tangential field. The validity condition of KA can be summarized as

(1.57)

where

denotes the mean radius of curvature, the wavenumber , and is the dielectric constant.

For a Gaussian rough surface, this translates to

(1.58)

Using KA, Equation 1.53 can be derived as

(1.59)

where

(1.60)

(1.61)

The local Fresnel coefficients are

(1.62)

where denote the wavenumber in the upper and lower half-spaces, respectively. The local incident angle is determined by .

The integration of can be calculated using the stationary phase method (Tsang et al., 1985; Jin, 1994), that is

(1.63)

(1.64)

where , denote the stationary phase points, and in Equation 1.64 is a random term related to the rough surface.

The second-order moment of scattering fields consists of two components: coherent part (mean-square) and incoherent part (variance):

(1.65)

(1.66)

(1.67)

For a Gaussian rough surface, the random term is calculated as

(1.68)

where , and are the Dirac delta functions.

The coherent part , however, only exists in the specular direction

(1.69)

For a Gaussian rough surface, it can be explicitly derived as

(1.70)

Finally, the bistatic scattering coefficient is written as

(1.71)

where superscripts c, ic denote the coherent and incoherent parts, respectively, that is

(1.72)

(1.73)

where is the Fresnel coefficient of flat surface at incident angle , and , . Further,

(1.74)

(1.75a)

(1.75b)

(1.75c)

(1.75d)

where

(1.76)

Given , the bistatic scattering coefficient can be simplified to

(1.77)

Using Equation 1.50, the bistatic scattering coefficient of Equation 1.77 is written as

(1.78)

and the backscattering coefficient is calculated as

(1.79)

(1.80)

where is the Fresnel coefficient of the flat surface at zero degree incidence. Note that the term is in fact the square of the average slope. In the case of a Gaussian correlation function, we have . Some of the terms related to wavenumber can be expanded as

(1.81)

(1.82)

(1.83)

Likewise, the transmission coefficients under KA can be derived, for which readers are referred to Tsang, Kong, and Shin (1985, p. 89).

1.3.4 Small-Perturbation Approximation

SPA, also referred to as the small-perturbation method (SPM), is valid when the roughness is very small compared to the wavelength (Rice, 1963), that is, the root-mean-square height and mean slope is comparable to , which can be represented by the following inequalities (Chen and Fung, 1988):

(1.84)

From Huygens' principle, Equation 1.53, let us expand the incident field , the scattered field , and the surface field on a rough surface into a perturbation series of . The zero-order term of the scattered field is the specular reflection of the plane surface. The first-order term is the incoherent part of the rough-surface scattering. Higher orders corresponds to multiple scatterings occurring on the rough surface. For detailed derivation of the SPA solution, readers are referred to the literature, such as Bass and Fuks (1979), Ulaby, Moore, and Fung (1982), and Tsang, Kong, and Shin (1985).

The first-order SPA bistatic scattering coefficient can be written as

(1.85)

where is the spectral function and

(1.86)

(1.87)

(1.88)

(1.89)

For a Gaussian rough surface, it can be further expanded as

(1.90)

(1.91)

While in the backscattering direction, it simplifies to

(1.92)

(1.93)

(1.94)

where the superscript B denotes backscattering. Hence, the backscattering coefficient can be written as

(1.95)

(1.96)

(1.97)

For a Gaussian rough surface, this yields

(1.98)

(1.99)

Note that the term is the horizontal-polarization Fresnel reflection coefficient.

Interestingly, the factor in can also be written as

(1.100)

where and denote the vertically polarized Fresnel reflection coefficient and Fresnel transmission coefficient, respectively.

Clearly, the first-order SPA solution has no cross-polarization contribution: , . However, , for the second- or higher-order solutions (Tsang, Kong, and Shin, 1985).

1.3.5 Two-Scale Approximation

In reality, natural surface roughness exists at different scales and therefore cannot be dealt with by either the KA or SPA approach only. The two-scale approximation (TSA) approach is a simple combination of KA and SPA (Wu and Fung, 1972; Jin, 1994). As illustrated in Figure 1.7, the basic assumption for the TSA model is that there is small roughness riding on top of a large fluctuation.

Figure 1.7 Two-scale model: small-scale roughness riding on large-scale fluctuation

Based on the independent KA and SPA solutions, the TSA combines the two after corresponding modifications. The KA scattering coefficient has to be modified to take into account the dimming effect by small roughness. On the other hand, the SPA solution has to be modified to take into account the local slope.

The TSA backscattering coefficient consists of two parts:

(1.101)

The TSA bistatic coefficient is written as

(1.102)

The second term represents the ensemble average of the SPA solution over the distribution of local slope caused by large-scale fluctuation, that is

(1.103)

where is included to account for the self- shadowing effect. Let , then the integration ranges of become , respectively. is the probability distribution function of local slope , which is defined as

(1.104)

Then, the local coordinates relate to global coordinates as

(1.105)

where

Furthermore, the local angles and local slopes are written as

Scattering from a randomly rough sea surface using an application of the TSA model can be found in Huang and Jin (1995).

1.3.6 Integral Equation Method

In practice, the KA is often used in circumstances of high frequency and small incident angle (<20°), while the SPA is suitable for cases where the frequency is low and the incident angle is large (20°–84°). The TSA combines the two types of scattering independently and thus is still limited by their respective constraints. Fung (1994) developed the integral equation method (IEM), which introduces a complementary field into the KA framework and takes into account the scattering caused by rapid fluctuations. Most of this book will employ the IEM solution to rough-surface scattering. Its validity condition is written as

(1.106)

The Mueller matrix form of the IEM rough-surface scattering is written as (Fung, 1994)

(1.107)

where

(1.108)

(1.109)

The KA coefficients are

(1.110)

and the complementary field coefficients are

(1.111)

where the numerous symbols employed are defined as

(1.112)

For rough-surface transmission, the Mueller matrix has exactly the same form as for scattering but with its elements replaced by (Fung, 1994)

(1.113)

(1.114)

(1.115)

(1.116)

and

(1.117)

1.3.7 Tilted Surface or Oriented Object

A rough surface may be tilted as a result of topographic fluctuation. Terrain objects or building surfaces may have different orientations in common global coordinates. Thus, a general method of coordinate conversion would be useful in adapting canonical scattering models for a tilted surface or oriented object.

As shown in Figure 1.8, the orientation of local coordinates is defined by three Euler angles (Tsang, Kong, and Shin, 1985; Jin, 1994). Through three steps of rotation, can be aligned with the global coordinates : that is, first rotate along by angle so that is located in the plane ( becomes ); then rotate along by angle so that is overlapped with ; finally, rotate along by angle so that is overlapped with , so do the other two axes.

Figure 1.8 Global coordinates and local coordinates

The coordinate conversion can be expressed in matrix form as

(1.118)

If the scatterer has rotational symmetry along ′, without loss of generality let and it is simplified to

(1.119)

In addition, the wave propagation vector defined in and the vector defined in has the following conversion:

(1.120)

Note that the polarization basis defined in two coordinate systems also differs by a certain angle . Here, it is defined as the angle required to rotate clockwise (as seen from the direction) to , that is

(1.121)

Now we can conclude the coordinate conversions of the scattering matrix , and the Mueller matrix , :

(1.122)

(1.123)

where the polarization basis rotation matrix is given as (Xu and Jin, 2006)

(1.124)

(1.125)

With the above equations, we can always calculate the new scattering matrix and Mueller matrix of a scatterer after being rotated to a new orientation.

References

Bass, F.G. and Fuks, I.M. (1979) Wave Scattering from Statistically Rough Surface, Pergamon, New York.

Chen, M.F. and Fung, A.K. (1988) A numerical study of the regions of validity of Kirchhoff and small perturbed rough surface scattering models. Radio Science, 24 (2), 163–170.

Fung, A.K. (1994) Microwave Scattering and Emission Models and Their Applications, Artech House, Norwood, MA.

Fung, A.K. and Lee, K.K. (1982) A semi-empirical sea-spectrum model for scattering coefficient estimation. IEEE Journal of Oceanic Engineering, 7 (4), 166–176.

Huang, X. and Jin, Y.Q. (1995) Scattering and emission from two-scale randomly rough sea surface with foam scatterers. IEE Proceedings H – Microwaves, Antennas, and Propagation, 142 (2), 109–114.

Ishimaru, A. (1978) Wave Propagation and Scattering in Random Media, Academic Press, New York.

Ishimaru, A. (1991) Electromagnetic Wave Propagation, Radiation, and Scattering, Prentice-Hall, Englewood Cliffs, NJ.

Jin, Y.Q. (1994) Electromagnetic Scattering Modelling for Quantitative Remote Sensing, World Scientific, Singapore.

Karam, M.A. and Fung, A.K. (1988) Electromagnetic scattering from a layer of finite-length randomly oriented dielectric circular cylinders. International Journal of Remote Sensing, 9 (6), 1109–1134.

Kong, J.A. (2005) Electromagnetic Wave Theory, EMW Publishing, Cambridge, MA.

Rice, S.O. (1963) Reflection of Electromagnetic Waves by Slightly Rough Surfaces, Interscience, New York.

Tsang, L., Kong, J.A., and Shin, B. (1985) Theory of Microwave Remote Sensing, John Wiley & Sons, Inc., New York.

Ulaby, F.T., Moore, R.K., and Fung, A.K. (1982) Microwave Remote Sensing: Active and Passive, Vol. II: Radar Remote Sensing and Surface Scattering and Emission Theory, Addison-Wesley, Reading, MA.

Wait, J.R. (1986) Introduction to Antennas and Propagation, Peter Peregrinus, London.

Wu, S.T. and Fung, A.K. (1972) A non-coherent model for microwave emission and backscattering from the sea surface. Journal of Geophysical Research, 77 (30), 5917–5929.

Xu, F. and Jin, Y.Q. (2006) Imaging simulation of polarimetric synthetic aperture radar remote sensing for comprehensive terrain scene using the mapping and projection algorithm. IEEE Transactions on Geoscience and Remote Sensing, 44 (11), 3219–3234.

2

Vector Radiative Transfer

Modeling of multiple scattering from layered random media is often studied in remote sensing. Radiative transfer theory is often used to describe absorption and single and multiple scattering of random media, such as rainfall, the vegetation canopy, snowpack, and so on. Vector radiative transfer theory is used to solve polarimetric scattering of complex or random media. In this chapter, we first introduce the radiative transfer equation and its components, as well as the boundary conditions, the Mueller matrix solutions, and so on, which lay the basis for the next few chapters. Some results of VRT for modeling the vegetation canopy and snowpack, as well as some numerical examples, are presented.

2.1 Radiative Transfer Equation

The VRT theory of random media is a physical–mathematical method for studying the scattering, multiple scattering, absorption, and transmission of polarized electromagnetic intensity by or through random scatterers or continuous random media. The radiative transfer (RT) theory was first studied to explain the appearance of the absorption and emission lines of stellar spectra (Chandrasekhar, 1960). Since then, RT theory has been extensively studied and applied in many areas: astrophysics, transport in neutron diffusion, heat transfer in engineering physics, atmospheric radiation, clouds and precipitation, light propagation through the atmosphere, and so on. RT theory has a simple physical meaning based on the principle of energy conservation, and takes account of absorption, multiple scattering, and emission. One of the advantages of RT theory is the capability to calculate multiple scattering numerically. Over recent decades, the RT theory of polarized electromagnetic intensity, that is, vector radiative transfer (VRT) theory, has been studied and applied to both active and passive remote sensing.

2.1.1 Specific Intensity and Stokes Vector

We define the specific intensity to express the amount of power flowing within a solid angle through an elementary area in the frequency range (see Figure 2.1):

Figure 2.1 Specific intensity

(2.1)

where θ is the angle between the propagation direction of the specific intensity and the outward normal of the area . Generally, is dependent on and . If the specific intensities for all directions at a point are the same, this point is known as “isotropic”; if the specific intensities for all directions and at all points in the medium are the same, then the medium is uniform and isotropic.

Integrating over a small frequency interval , the specific intensity at the frequency ν is

(2.2)

which is the definition of specific intensity.

The unit of the radial Poynting vector is watts per square meter per steradian (W m−2 sr−1) and has a factor of 1/r2, which is different from of Equation 2.1. The power of the spherical wave on the area is

(2.3)

From Equations 2.1 and 2.3, we have

(2.4)

The power of a plane wave on the plane area is

(2.5)

Together with Equation 2.1, it yields

(2.6)

As discussed in Chapter 1, any time-harmonic elliptically polarized electromagnetic wave with exp(−iωt) can be expressed as the summation of vertically and horizontally linear polarized waves, . These two linearly polarized waves have their own amplitudes and their phases are independent. So, we have three independent variables to describe a polarized wave.

If the electromagnetic wave is perfectly monochromatic, and are independent of time. We can define the four Stokes parameters for a completely polarized wave as

(2.7)

where is the wave impedance of the medium. All of the four Stokes parameters have the same unit as the intensity, which should be more convenient to use than the amplitude and phase with different units. We may write

(2.8)

Substituting Equation 2.8 into Equation 2.7 yields

(2.9)

Thus, only three of the four Stokes parameters are independent.

Actually, electromagnetic waves are often quasi-monochromatic with and randomly fluctuating with time, that is, partially polarized waves. Thus, , , and in the definitions of Equation 2.7 should be the averages of the stochastic process. The Stokes vector is then defined as

(2.10a)

(2.10b)

As shown in Equations 1.12 and 1.15 of Chapter 1, the first two Stokes parameters, and , sometimes are replaced by

(2.11)

When independent waves, for example, scattered waves from different scatterers, arrive at the observer, the total Stokes parameters are simply the sum of the Stokes parameters of each wave. For example,

(2.12)

This gives

(2.13)

The four Stokes parameters compose the Stokes vector – specific intensity in the VRT equation. In the infrared frequencies, it is often termed the radiance.

2.1.2 Thermal Emission and Brightness Temperature

All objects can absorb, reflect or scatter incident electromagnetic radiation, and can also emit radiation. This radiation is thermal electromagnetic emission. The emissivity e of an object is equal to its absorptivity (Peake, 1959). The absorptivity of a body is defined as the ratio of the total thermal energy absorbed by the surface to the total thermal energy incident upon it. A black body has maximum absorptivity and is a perfect emitter. The absorptivity is usually dependent upon radiation direction, frequency, and polarization.

From Planck's radiation law of quantum physics, the emission intensity of an object with emissivity is

(2.14)

where the subscript in denotes thermal emission, is the real part of the dielectric constant of the object, is Planck's constant (), B is Boltzmann's constant (), and T is the absolute temperature (K). In the Rayleigh–Jeans low-frequency limit, for example, microwave region, we can apply the approximation to yield

(2.15)

where λ is the wavelength in free space, and if we let μ = μ0. In microwave remote sensing, the source of thermal emission can be written as in Equation 2.15. If we consider the thermal emission of an elementary volume with unit length, the emissivity in Equation 2.15 is then replaced by the absorption coefficient . Thus, the Stokes vector of thermal emission of an elementary volume with unit length and physical temperature T is

(2.16)

where = diagonal matrix [].

In passive microwave remote sensing, the thermal emissions from the target and background are the sources for radiative transfer. The brightness temperature observed by a radiometer is defined as

(2.17)

where the subscript p denotes the polarization, v or h, and is the observation direction. The brightness temperature of the object is equal to the physical temperature of the black body that can emit the same amount of radiation power as the object.

2.1.3 Vector Radiative Transfer Equation

Consider the intensity change caused by scattering and absorption through a cylindrical elementary volume with a unit cross-section and length embedded with random spherical particles, as shown in Figure 2.2. From energy conservation, the intensity change can be written as a conservative form

Figure 2.2 Derivation of the RT equation

(2.18)

Equation 2.18 means that the change is due to absorption and scattering of n particles in the volume, absorption of the background medium in the volume, the thermal emission from the volume, and the contribution of multiple scattering from all other directions . The function coupling the incident with is termed the phase function. Equation 2.18 is the scalar radiative transfer (RT) equation. It can be understood that the derivation of Equation 2.18 is heuristic, and independent scattering of particles is always assumed.

It is known that if the fractional volume of scatterers is higher than 0.1, the scattering medium is seen as a dense medium, and the independent scatterings assumption for dense scatterers becomes invalid. The dense-medium RT can be found in Tsang, Kong, and Shin (1984) and Jin (1994), for example.

Define the albedo as and the extinction coefficient as for single scattering. Then, is the absorbed energy during single scattering. The phase function should be

(2.19a)

So, the term for multiple scattering in Equation 2.18 is usually written as a normalized form

(2.19b)

The simplest case is isotropic scattering, , where . For small spherical particles, it can be shown (Chandrasekhar, 1960) that

(2.20)

For a simple anisotropic scattering, it is usually assumed that

(2.21)

Generally, can be expanded in terms of Legendre polynomials

(2.22)

where are constants.

We now extend Equation 2.18 to the case of polarized electromagnetic intensity, that is, four Stokes parameters. The vector form of the RT equation is written as

(2.23)

where the extinction matrix defines the attenuation of due to scattering and absorption. The phase matrix (, ′) defines angular scattering from incident direction ′ to scattered direction . The physical meaning of the VRT equation is apparent: as wave transfers over distance in direction , the differential change is the sum of three parts, that is, the attenuation defined by , the thermal emission , and the scattering contribution from all directions toward , which is defined by (, ′).

It should be noted that all the components of the VRT equation are now in 4 × 4 matrix form. In the microwave region, we have microwave radiance , where T is absolute temperature.

2.2 Components in Radiative Transfer Equation

2.2.1 Scattering, Absorption, and Extinction Coefficients

Following Chapter 1, when a polarized EM wave is incident upon a single scatterer, Equation 1.10 gives the scattering matrix to describe the coupling relation between the incident and scattered fields as

(2.24)

where the elements of the scattering matrix, that is, , , are also referred to the scattering amplitude functions of this single scatterer.

If there are scatterers in a unit volume, the polarized electromagnetic intensity passing through the volume will lose part of this energy due to scattering. The scattering coefficients of vertical and horizontal polarizations are defined as

(2.25a)

(2.25b)

For spherical particles, clearly we have .

The absorption coefficient includes the absorption of the scatterers and the absorption of the background medium. If the fractional volume of the scatterers is and the medium occupies , then we have

(2.26)

where is the imaginary part of wavenumber of the background. The absorption coefficient of the scatterer can be obtained following the definition

(2.27)

where the integration is over the scatterer volume; is the internal field of the scatterer.

If it is difficult to solve the internal field, the optical theorem can be used to calculate the extinction coefficient as (Tsang, Kong, and Shin, 1984)

(2.28)

where the forward scattering matrix is used. Note that forward scattering means that the scattering direction is the same as the incident direction. Thus, instead of solving Equation 2.27, the absorption coefficient can now be calculated as

(2.29)

2.2.2 Extinction Matrix

However, the extinction coefficient of non-spherical scatterers is in fact a 4 × 4 matrix, generally non-diagonal. It cannot be calculated simply by the optical theorem. The electric field in propagation consists of the coherent (average) field and the incoherent (diffuse) field. The extinction can be essentially identified with the attenuation of coherent waves, whose phases preserve coherency in propagation. From the optical theorem, Equation 2.28, the intensity extinction is . Thus, as a coherent wave with vertically and horizontally polarized components , propagates along , its propagation equations of the coherent fields are written as (Oguchi, 1983; Ishimaru and Yeh, 1984; Tsang, Kong, and Shin, 1984)

(2.30a)

(2.30b)

where is obtained by the optical theorem, Equation 2.28, and denotes attenuations of the co-polarized (vv or hh) and depolarized (vh or hv) waves; and denotes the elements of the forward scattering matrix . The angular brackets, , denote the ensemble average. Using the matrix eigenvalue technique to solve Equations 2.30a and 2.30b, we obtain two characteristic propagation wavenumbers (Ishimaru and Yeh, 1984):

(2.31a)

(2.31b)

where , the sign of R thus being determined by the sign of the real part of .

The eigenvectors of Equations 2.30a and 2.30b are and with

(2.32a)

(2.32b)

If for uniformly random non-spherical particles, this leads to . Then, and of Equations 2.31a and 2.31b are simply reduced to vertically and horizontally polarized waves, respectively,

(2.33)

Making

and using Equations 2.30a and 2.30b, this yields

(2.34a)

Similarly, we have

(2.34b)

(2.34c)

(2.34d)

Equations 2.34a–2.34d yield the propagation equation of the coherent Stokes vector

(2.35)

where the subscript denotes the coherent wave. The extinction matrix in Equation 2.35 can be readily obtained from Equations 2.34a–2.34d,

(2.36)

It can be found that would reduce to a scalar if the scatterer is a perfect sphere and to a diagonal matrix if the scatterers have a uniform distribution in a medium. Note that the scattering matrix must be accurate enough in Equation 2.28, otherwise it might cause . This is the reason why an additional imaginary term is included in of Equation 1.25.

We seek the solution of Equation 2.35 with dependence and obtain four eigenvalues

(2.37)

It may be seen that and are the extinction of two characteristic waves; and and are the difference of the phase and attenuation between two waves.

The eigenvectors of Equation 2.35 can be solved by its eigenmatrix equation

(2.38)

where is a diagonal matrix with the iith element , i = 1,2,3,4, that is, . The solution of Equation 2.38 is expressed as (Tsang, Kong, and Shin, 1984)

(2.39)

where the first column is the first eigenvector corresponding to the first eigenvalue, and so on.

Only under the condition does , and then becomes a constant matrix with no dependence on :

(2.40)

and

Letting an electromagnetic wave propagate in direction through distance z in a random medium with the condition , the scattered Stokes vector can be written as

(2.41)

It can be seen that the four Stokes parameters propagating in a non-spherical particle medium have different wavenumbers. Also if , Equation 2.39 can cause coupling among the four Stokes parameters.

Note that the eigen-expansion of the extinction matrix in the form of Equations 2.39–2.41 is useful later in solving VRT equations.

2.2.3 Phase Matrix

The phase matrix in the VRT equation is a real matrix describing the coupling between and . The term phase matrix is inherited from the field of astronomy; however, it is not related to the phase in the signal at all. The phase matrix is derived as

(2.42)

where is the number of particles per unit volume.

For random scatterers, the same type of scatterers may have random distributions of orientation, size, dielectric properties, and so on. Using the n parameters to represent the stochastic properties of the scatterers, the first- and second-order moments of the scattering matrix can be written as

(2.43)

where denotes the probability distribution function (PDF). It can be very intensive to calculate the integration numerically. However, for Rayleigh–Gans non-spherical particles with random orientation, the above integration can be explicitly derived. Some expressions of the phase matrix for non-spherical particles can be found in Appendix 2A (Jin, 1994).

As a simple case, when the particle size a is much less than the wavelength, , the Rayleigh approximation can be used. The Rayleigh–Gans approximation (Jin, 1994) has been presented in Chapter 1.

If the particle is a Mie spherical large particle, disk, needle, finite cylinder, or other shape, calculations of the scattering matrix of Equation 2.24, the scattering, absorption, and extinction coefficients of Equations 2.25a, 2.25b and 2.37 and the phase matrix of Equation 2.42 can be found in the literature (e.g., Ishimaru, 1978; Waterman, 1965, 1971, 19731965, 1971, 1973; Bohren and Huffman, 1983; Tsang, Kong, and Shin, 1985; Mishchenko, Travis, and Lacis, 2002).

2.3 Mueller Matrix Solution

2.3.1 First-Order Mueller Matrix Solution

Consider an elliptically polarized wave incident on a layer of random non-spherical particles. For example, a vegetation canopy is modeled as a layer of hybrid particles consisting of leaves modeled as non-spherical particles, and trunks and branches modeled as cylinders, as shown in Figure 2.3. Non-spherical particles have a non-uniformly random orientation described by the probability density function of the Euler angle (). As a first case, we consider a flat underlying surface at z = −d; the case of a rough-surface model will be discussed in Section 2.3.2.

Figure 2.3 Scattering model of a layer of non-spherical particles

The VRT equation is

(2.44a)

(2.44b)

where

(2.45)

The function is the same as with all replaced by .

The boundary conditions for transmission and reflection are written as

(2.46a)

(2.46b)

where denotes the incident wave Stokes vector with incident angles , denotes the Dirac delta function, , and is the reflectivity of the flat surface at .

Following the iterative method of the VRT equation of spherical particles, the integral VRT equations can be obtained as

(2.47a)

(2.47b)

where the notation is a diagonal matrix with the iith element . The eigenvector matrix and eigenvalue , are given in Equations 2.39 and 2.37, respectively. The first term on the right-hand side (RHS) of Equations 2.47a and 2.47b is a coherent Stokes vector with as the zeroth-order solution. Substituting these zeroth-order solutions , into the functions and of Equations 2.44a, 2.44b and 2.45