Shoichiro FukaoKyosuke Hamazu

Consulted by Richard J. Doviak

Radar for Meteorological and Atmospheric Observations

Radar for Meteorological and AtmosphericObservations

Shoichiro Fukao Kyosuke Hamazu

Radar for Meteorological andAtmospheric Observations

Consulted by Richard J. Doviak

123

Shoichiro FukaoProfessor EmeritusKyoto University, Kyoto, Japan

Consulted By:Richard J. DoviakNational Severe Storms Laboratory, NOAAAffiliated ProfessorThe School of Meteorology and

the Department of Electricaland Computer Engineering

The University of Oklahoma

Kyosuke HamazuMitsubishi Electric Corporation and

Mitsubishi Electric Tokki SystemsCorporation

Iga, Japan

ISBN 978-4-431-54333-6 ISBN 978-4-431-54334-3 (eBook)DOI 10.1007/978-4-431-54334-3Springer Tokyo Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013943813

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Foreword

During the past several decades an appreciable amount of research and developmenthas been focused on the use of remote sensing techniques to better our understand-ing of weather and the atmosphere. Radar has the obvious advantage of providingobservations with temporal and/or spatial continuity which is leading to improvedforecasts of weather.

Observations and interpretation of Doppler and polarimetric weather radar data,combined with in situ observations, have led to giant leaps in our understandingof the dynamics and microphysics of weather systems. Complementary to weatherradar observations are those obtained with typically longer wavelength radars (i.e.,wavelengths from meters to centimeters versus centimeters to millimeters used toobserve precipitation and clouds), observing the precipitation-free atmosphere. Theechoing mechanism at these longer wavelengths is typically Bragg scatter fromrefractive index perturbations caused by turbulent mixing, or reflection from sharpgradients in refractive index. These long-wavelength and super-powerful radars,referred to as atmospheric radars, have mapped the vertical structure of reflectivityand radial winds in the clear atmosphere from below a kilometer to well above100 km, whereas meteorological radars map the reflectivity and radial velocitiesof precipitation and cloud particles on horizontal surfaces at various heights inthe troposphere. Weather and cloud radar research has attracted the attention ofmeteorologists whereas atmospheric radar research has primarily attracted theattention of atmospheric physicists.

The authors have done a remarkable job of combing the results of research inthese two disciplines to provide readers with a comprehensive overview of theoutstanding observations that have been made with radar used as a remote sensorof weather and atmospheric phenomena. This book has a generous amount offigures that display many of the remote sensing facilities to give the reader aquick appreciation for the variety of atmospheric and meteorological radar typesaround the world, many of which are unique and interesting. Furthermore, liberalreference to publications provides readers a vast reservoir for further pursuit oftheir preferred topics of interest. In addition this book presents the fundamentalsof remote sensing so that students and professors, with a minimal background in

v

vi Foreword

physics and electromagnetic theory, and engineers in the field can better understandthe potential and limitations of radar in observing weather and the atmosphere whilelearning about the various instruments and techniques used in remote sensing. Theauthors plan to maintain a Website where comments from readers can be addressedand where supplements to the book can be found; this will help to keep the bookcurrent and up-to-date.

Norman, OK Richard J. Doviak

Preface

With the application of radar to observations of the atmosphere, various weatherphenomena and winds in the clear atmosphere can be monitored and mappedin real time. Great progress in understanding weather and the dynamics of theatmosphere has been made using radar, which brings new observational discoveriesand promotes further understanding of our environment.

Remote sensing with radar has been developed in the interdisciplinary domainsof physical science and engineering. In the past, advances in weather and the atmo-spheric sciences have developed independently because the respective engineeringefforts and scientific studies were conducted within relatively separate communities.However, the scientific and technical bases for atmospheric observations with radarcan be treated in common. We worked in academia (Fukao) and industry (Hamazu)and have collaborated to develop various types of weather and atmospheric radars.Routine discussion with our colleagues convinced us that understanding of weatherand atmospheric radars can be deepened if they are described comprehensively andsystematically in one volume using common approaches whenever possible.

This book is written for scientists, engineers, students, and other interestedmeteorological and atmospheric personnel. In this book, we try to bridge the gapin our understanding of weather and atmospheric radar. The book consists of twoparts. The first half, Chaps. 17, mainly discusses the theoretical bases of weatherand atmospheric radar, and the last half, Chaps. 812, describes actual systems andobservations with these radars. This interdisciplinary book was first published inJapanese by the Kyoto University Press in 2005. In the English version, all chaptersincluding those dealing with recent developments contain more in-depth coveragethan does the original.

Kyoto, Japan Shoichiro FukaoIga, Japan Kyosuke Hamazu

vii

Acknowledgements

We are indebted to many people in bringing this book to publication in this form.First and foremost, we wish to express our deepest gratitude to Dr. Richard J. Doviakof the National Severe Storms Laboratory of the National Oceanic and AtmosphericAdministration (NOAA) and the University of Oklahoma who has long been a closefriend of one of the authors (Fukao) and has been looked upon as mentor for theother (Hamazu). All chapters have been reviewed and edited by him. We wereable to complete the work because of his continuous suggestions and stimulatingencouragement.

We would like to express our deep gratitude to our colleagues, Drs. ToruSato, Toshitaka Tsuda, Mamoru Yamamoto, and Hiroyuki Hashiguchi of KyotoUniversity; Dr. Takuji Nakamura of the National Institute of Polar Research, Japan;Dr. Manabu D. Yamanaka of the Japan Agency for Marine-Earth Science andTechnology (JAMSTEC) and Kobe University; and Drs. Hiroaki Miyasita, AtsushiOkamura, Toshio Wakayama, and Shoji Matsuda of Mitsubishi Electric Corporationfor various suggestions through discussions. Many of them provided us with theoriginal figures that are included in the book. It is a great pleasure for us toacknowledge the valuable advice and suggestions of Emeritus Professor HisanaoOgura of Kyoto University; Dr. Hubert Luce of the Universite de Toulon et du Var;Drs. Yasushi Fujiyoshi and Takeshi Horinouchi of Hokkaido University; Dr. ToshioIguchi of the National Institute of Information and Communication Technology(NICT), Japan; Dr. Hiroshi Uyeda of Nagoya University; Dr. Masahiro Ishiharaof Kyoto University; Dr. Masayuki Maki of Kagoshima University; Dr. KoyuruIwanami of the National Research Institute for Earth Science and Disaster Prevision(NIED), Japan; and Drs. Ahoro Adachi and Hiroshi Yamauchi of the MeteorologicalResearch Institute, Japan Meteorological Agency. We would like to express our spe-cial thanks to Dr. Yoshiaki Shibagaki of Osaka Electro-Communication University;Seiji Kawamura of the NICT; Drs. Masayuki Yamamoto, Tomohiko Mitani, JunichiFurumoto, and Tatsuhiro Yokoyama of Kyoto University; and Drs. NobukyukiKawano and Akihisa Uematsu of the Japan Aerospace Exploration Agency (JAXA).In writing this book, we received warm encouragement from emeritus professorsSusumu Kato and Isamu Hirota of Kyoto University.

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x Acknowledgements

Finally, we are especially grateful for the support of our respective spouses,Keiko Fukao and Masuni Hamazu, during the writing of this book.

Kyoto, Japan Shoichiro FukaoIga, Japan Kyosuke Hamazu

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Principle of Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History of Meteorological and Atmospheric Radars . . . . . . . . . . . . . . . 21.3 Radar Frequency Bands and Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Characteristics of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Basic Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.4 Radiation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Electromagnetic Wave Propagation in the Atmosphere . . . . . . . . . . . . 232.2.1 Physical Property of the Atmosphere . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Propagation of Electromagnetic Wave. . . . . . . . . . . . . . . . . . . . . 262.2.3 Wave Path in the Spherically Stratified Atmosphere .. . . . . 272.2.4 The Profile of the Standard Atmosphere . . . . . . . . . . . . . . . . . . 30

3 Radar Measurements and Scatterer Parameters . . . . . . . . . . . . . . . . . . . . . . . 333.1 Basics of Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 System Parameters of Pulse Radars . . . . . . . . . . . . . . . . . . . . . . . 333.1.2 Characteristics of Scatter and Scatterers. . . . . . . . . . . . . . . . . . . 36

3.2 Radar Observation of Isolated Scatterers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Radar Equation for an Isolated Scatterer . . . . . . . . . . . . . . . . . . 383.2.2 Characteristics of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Radar Theory for Hard Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 Scattering by Dielectric Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Radar Equation for Distributed Hard Scatterers . . . . . . . . . . 463.3.3 Mie Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.4 The Rayleigh Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.5 Radar Reflectivity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Radar Theory for Soft Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Backscattering Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 Bragg Scatter due to Refractive Index Perturbations . . . . . 59

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3.4.3 Partial Reflection from a Stratified Atmosphere . . . . . . . . . . 683.4.4 Scattering by Linear Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Principle of Doppler Velocity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Doppler Velocity Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.1 Principles of Doppler Radar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.2 Measurable Limit of Doppler Velocity . . . . . . . . . . . . . . . . . . . . 794.1.3 Expansion of Doppler Velocity Measurement Range . . . . . 81

4.2 Methods of Applying Doppler Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.1 Volume Velocity Processing (VVP) Method . . . . . . . . . . . . . . 824.2.2 Velocity Azimuth Display (VAD) Method . . . . . . . . . . . . . . . . 864.2.3 Wind Observations with Bistatic Doppler Radar. . . . . . . . . . 89

4.3 Multiple Monostatic Doppler Radars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Independent Scanning Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3.2 COPLAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.3 Distance of Two Doppler Radars . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3.4 Wind Velocity Observations with Three or

More Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Reception and Processing of Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1 Receiver Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.1 Noise Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1.2 Receiver Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2 Receiver System .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2.1 Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2.2 Frequency Conversion and Phase Measurement . . . . . . . . . . 118

5.3 Characteristics of Received Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3.1 Signals Received from Precipitation Particles

and the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3.2 Probability Density Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4 Fundamentals of Radar Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.4.1 Fourier Transform and Its Characteristics . . . . . . . . . . . . . . . . . 1255.4.2 Signals in a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4.3 Power Spectral Moments and Basic Radar Parameters . . . 130

5.5 Processing of Sampled Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.5.1 Waveform of Transmitted Pulse and Series

of Signal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.5.2 Sampling of a Received Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.5.3 Processing of Discrete Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.5.4 Estimation of Mean Doppler Frequency.. . . . . . . . . . . . . . . . . . 1435.5.5 Estimation of Spectrum Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.5.6 Estimation of Spectral Parameter by Fitting . . . . . . . . . . . . . . 1465.5.7 Estimation Based on Prediction Theory . . . . . . . . . . . . . . . . . . . 149

5.6 Correlation and Accuracy of Sampled Signal . . . . . . . . . . . . . . . . . . . . . . . 1505.6.1 Correlation Function and Correlation Time . . . . . . . . . . . . . . . 1505.6.2 Coherent Integration .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

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5.6.3 Incoherent Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.6.4 Standard Deviation of Radar Reflectivity Factor . . . . . . . . . . 1595.6.5 Standard Deviation of Mean Doppler Velocity. . . . . . . . . . . . 1625.6.6 Standard Deviation of Spectrum Width . . . . . . . . . . . . . . . . . . . 164

6 Radar Observations of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.1 Parameters of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.1.1 Parameters of Drop Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.1.2 Relations Between Basic Radar Parameters

and DSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.1.3 Physical Quantities Concerned with Precipitation . . . . . . . . 1736.1.4 Radar Reflectivity Factor and Rainfall Rate . . . . . . . . . . . . . . . 176

6.2 Estimation of DSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.3 Attenuation of Radio Waves in the Atmosphere .. . . . . . . . . . . . . . . . . . . 182

6.3.1 Attenuation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.3.2 Attenuation by the Atmosphere .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.3.3 Attenuation by Water Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.4 Polarimetric Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.4.1 Generation of Dual Polarized Wave . . . . . . . . . . . . . . . . . . . . . . . 1906.4.2 Characteristics of Polarization Parameter . . . . . . . . . . . . . . . . . 1956.4.3 Shapes of Precipitation Particles and

Polarization Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2006.4.4 Attenuation Correction Using KDP . . . . . . . . . . . . . . . . . . . . . . . . . 2046.4.5 Estimates and Variances of Polarization Parameters . . . . . . 2076.4.6 Radar Rainfall Estimation Using Polarization

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2166.4.7 Estimation of Cloud Water Content . . . . . . . . . . . . . . . . . . . . . . . 2186.4.8 Hydrometeor Classification with Polarization

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7 Radar Observations of the Clear Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.1 Detectability of Atmospheric Radar Signals . . . . . . . . . . . . . . . . . . . . . . . . 223

7.1.1 Received Power and Radar Reflectivity . . . . . . . . . . . . . . . . . . . 2237.1.2 Coherent Integration in Atmospheric Radar. . . . . . . . . . . . . . . 2247.1.3 Detection of Signal in Noise Background .. . . . . . . . . . . . . . . . 226

7.2 Wind Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.2.1 DBS/VAD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.2.2 Wind Velocity Measurements from Spaced

Antenna Methods .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2307.3 Turbulence Observations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.3.1 Measurement of Momentum Flux . . . . . . . . . . . . . . . . . . . . . . . . . 2357.3.2 Estimation of the Turbulence Contribution

to Spectrum Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.3.3 Estimation of Turbulence Parameters. . . . . . . . . . . . . . . . . . . . . . 2427.3.4 Relation Between Refractive Index and

Structure Constant for Refractivity Turbulence . . . . . . . . . . . 245

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7.4 Observations of Temperature Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.4.1 Measurement of Atmospheric Temperature

with RASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.4.2 Change of Refractive Index and Radar

Equation for RASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.4.3 Bragg Condition and Background Wind . . . . . . . . . . . . . . . . . . 251

7.5 Estimation of Water Vapor Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2557.6 Radar Interferometry Techniques .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.6.1 SDI and FDI Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.6.2 Radar Imaging Techniques .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

8 Overview of Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.1 Brief Discussion on Two Types of Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

8.1.1 FMCW Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.1.2 Pulse Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2728.1.3 Echo Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2738.1.4 Scanning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

8.2 Radar Antenna .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.2.1 Radar Antenna Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.2.2 Parabolic Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.2.3 Radome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2838.2.4 Array Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2858.2.5 Measurement of Antenna Radiation Pattern. . . . . . . . . . . . . . . 295

8.3 Transmitters and Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.3.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2988.3.2 Transmitter Used for Meteorological Doppler Radar . . . . . 2998.3.3 Transmitter of Atmospheric Radar. . . . . . . . . . . . . . . . . . . . . . . . . 3078.3.4 Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3088.3.5 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

8.4 Digital Signal Processing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3218.4.1 Signal Processing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3218.4.2 Removal of Unwanted Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3248.4.3 Analog to Digital Conversion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3298.4.4 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3328.4.5 Window Function .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3338.4.6 Parameters for the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

9 Practical Meteorological Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.1 Meteorological Radars of Various Frequency Bands . . . . . . . . . . . . . . . 3419.2 Precipitation Observation Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

9.2.1 2.8-GHz Band Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3449.2.2 NEXRAD: WSR-88D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3449.2.3 5.6-GHz Band Radar: The Terminal Doppler

Weather Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3479.2.4 5.3-GHz Band Radar: The Doppler Radar for

Airport Weather in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

Contents xv

9.2.5 5.3-GHz Band Meteorological Radars in Japan . . . . . . . . . . . 3539.2.6 Radar Raingauge.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3569.2.7 9.5-GHz Band Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

9.3 Cloud and Fog Observation Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3619.3.1 35-GHz Band Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3619.3.2 35/95-GHz Multiple Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

9.4 Satellite-Borne Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3669.4.1 Tropical Rainfall Measuring Satellite . . . . . . . . . . . . . . . . . . . . . 3669.4.2 Global Precipitation Measurement Program . . . . . . . . . . . . . . 368

10 Practical Atmospheric Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36910.1 Characteristics of Atmospheric Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36910.2 Large-Scale Atmospheric Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

10.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37010.2.2 Radars with COCO Array Antenna . . . . . . . . . . . . . . . . . . . . . . . . 37210.2.3 The MU Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37410.2.4 Equatorial Atmospheric Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38010.2.5 The Antarctic Syowa MST/IS Radar: PANSY . . . . . . . . . . . . 384

10.3 Wind Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38410.3.1 The NOAA Profiler Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38410.3.2 Wind Profiler Network in Europe .. . . . . . . . . . . . . . . . . . . . . . . . . 387

10.4 Lower Troposphere Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38710.4.1 Boundary Layer Radar (BLR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38810.4.2 Turbulent Eddy Profiler (TEP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38910.4.3 Lower Troposphere Radar (LTR) . . . . . . . . . . . . . . . . . . . . . . . . . . 39110.4.4 WINDAS of Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

11 Observations by Meteorological Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39511.1 Precipitation Observation by Meteorological Radar . . . . . . . . . . . . . . . . 39511.2 Mesoscale Rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

11.2.1 Structure of Extratropical Cyclone and Front . . . . . . . . . . . . . 39711.2.2 Horizontal Structure of Precipitation . . . . . . . . . . . . . . . . . . . . . . 40111.2.3 Vertical Structure of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . 402

11.3 Typhoon.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40511.3.1 Horizontal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40511.3.2 Spatial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

11.4 Cumulus Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40911.4.1 Multicell Thunderstorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40911.4.2 Ordinary Thunderstorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41011.4.3 Tornado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41111.4.4 Downburst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

11.5 Polarimetric Radar Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41711.5.1 Polarimetric Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41711.5.2 Attenuation Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41911.5.3 Radar Rainfall Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42211.5.4 Hydrometeor Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

xvi Contents

11.6 Clear Air Observations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42611.6.1 High Power Large Radar Observation .. . . . . . . . . . . . . . . . . . . . 42611.6.2 FMCW Radar Observation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

11.7 Cloud and Fog Observations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42811.7.1 Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42811.7.2 Fog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

11.8 Retrieval of Heating Distribution in a Cloud . . . . . . . . . . . . . . . . . . . . . . . . 430

12 Observations by Atmospheric Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43512.1 Wind Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43512.2 Mesoscale Convective System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

12.2.1 Cold Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43812.2.2 Tropical Cyclone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44112.2.3 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44412.2.4 Precipitating Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44612.2.5 Orographic Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44812.2.6 Echoes from Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

12.3 Atmospheric Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45112.3.1 Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45212.3.2 Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45312.3.3 Critical Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45512.3.4 Gravity Wave Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45612.3.5 Momentum Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45812.3.6 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46012.3.7 Wave Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

12.4 Boundary Layer and Equatorial Atmosphere . . . . . . . . . . . . . . . . . . . . . . . 46312.4.1 Boundary Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46312.4.2 Equatorial Atmosphere .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46512.4.3 Atmospheric Temperature and Water Vapor Content . . . . . 467

12.5 Beam Swinging and Radar Imaging Techniques . . . . . . . . . . . . . . . . . . . 46912.5.1 Scattering Layer Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

12.6 Wind Profiler Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48212.6.1 Quality Control and Actual Operation .. . . . . . . . . . . . . . . . . . . . 48212.6.2 Application for Short-Term Forecasting .. . . . . . . . . . . . . . . . . . 483

A Mie Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

B Autocovariance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493B.1 Mean Doppler Frequency .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493B.2 Doppler Frequency Spectrum Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Contents xvii

C The Fast Fourier Transform (FFT) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 497C.1 Decimation-in-Time (DIT) FFT Algorithm .. . . . . . . . . . . . . . . . . . . . . . . . 497C.2 Decimation-in-Frequency (DIF) FFT Algorithm . . . . . . . . . . . . . . . . . . . 499

D Radar Equation for RASS Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

List of Symbols

a Attenuation rate [m1], mean radius of the Earth (6370 km), semi-majoraxis diameter of spheroid rain drop

ae Effective Earth radiusaT Temperature lapse rateA Attenuation coefficient [dB km1], physical antenna apertureA Vector potentialAe Effective antenna apertureb Semi-minor axis diameter of spheroid rain dropB Frequency bandwidth of the receiverB Magnetic flux densityBf Filter bandwidthBn Noise bandwidthc Speed of light (in vacuum) [m s1]ca Sound velocityca Apparent sound velocitycs True sound velocityC2n Refractive index structure constant [m2/3]Cp Specific heat capacity at constant pressure ( 1004) [J K1 kg1]d Distance between successive element antennasD Detectability of radar signal, diameter of raindrop, wind direction,D Electric flux densityD0 Median volume diameterDa Antenna diameter, distance of two separated antennasDm Mass weighted mean drop diameterDr Dynamic range of A/D conversionDrmax Maximum dynamic rangee Partial pressure of water vapor [hPa]E Total energy of a receiver input signal, withstand voltage [V mm1]E Electric field strengthE0 Incident electric fieldEa Array factor

xix

xx List of Symbols

Es Scattered electric fieldf Radar frequency (transmitted frequency) [Hz]f0 Carrier frequency [Hz]fc Frequency of coherent oscillator (COHO)fd Doppler frequency (Doppler shift)fdmax Maximum measurable Doppler frequencyfi Inertial frequencyfN Nyquist frequencyfp Pulse repetition frequencyfs Frequency of stabilized local oscillator (STALO), sampling frequencyF Noise figureFr Froude Numberg Antenna gain at the direction of the maximum radiation pattern (main lobe)

in linear unit, radiation pattern of the element antenna (or element pattern),gravitational acceleration

gat Transmission gain of the RASSgD Directivity of antennaG Antenna gain in decibelh Altitude (height from sea level), beam height, mountain heightH Magnetic field strengthH1 Scale height (7.3 km)ii Unit vector along the radar beam directionI Electric current, in-phase component of the complex signalIa Acoustic intensity [W m2]j Imaginary unit ( j2 =1)J Electric current densityk Boltzmann constant (= 1.38 1023 J K1), radar wave number

(=

= 2/ )ka Imaginary part of the complex refractive index, wave number of acoustic

wave

ks Scattering vector wave numberK Thermodynamic temperature measured in kelvins, vertical eddy diffusivityKDP Specific differential phase [deg km1]l Autocorrelation time lag, length of short dipole (differential antenna), loss

value in a true numberl Separation of the scatterer from the volume centerl0 Inner scale of turbulencelK Kolmogoroff microscaleL Loss value in decibelLB Maximum scale of eddy in the inertial subrange (or buoyancy lengthscale)Ldr Linear depolarization ratio in linear unitLDR Linear depolarization ratio in decibelm Complex refractive index of drop (or particle), modified refractive index,

vertical wavenumbermn The nth moment of drop size distribution

List of Symbols xxi

M Mean molecular weight of the atmosphere, number of DFT or FFTpoints, number of signal samples along sample time axis (total numberof samples), refractive modulus,

MB Total number of points of periodogram (FFT points)Mcoh Number of coherent integrationMI Number of independent samplesMinc Number of incoherent integrationMn Refractive index gradient (=dn/dz)Ms Total number of actual signal samplesMv Total water vapor content [kg mm3]n Refractive indexnr Real part of the complex refractive indexN Bit length, Brunt Vaisala frequency, number of element antenna, number

of raindrops, number of range samples, Nyquist numberN0 Parameter of drop size distribution (intercept parameter)N(D) Drop size distribution (DSD)Ne Density of free electron [m3]NT Total number of raindropsp Atmospheric pressure [hPa]P Breakdown power, total electric powerP Dielectric polarizationPa Transmitted power from sound wave sourcePar Received power backscattered from sound wave surfacePr Received signal powerPs Scattered powerPt Transmitted power, peak transmitted powerPV Dipole momentq Humidity mixing ratio [kg kg1]qe Linear density of meteor trail [m1]Q Quadrature phase component of the complex signalr Distance between the radar and the scatterer, rangera Maximum observable rangerR Distance between bistatic scatterer and receiverrT Distance between transmitter and bistatic scattererR Gas constant, rainfall rate,Rd Transmitters duty cycleRf Flux Richardson numberRi Richardson numberRR Radiation resistance of short dipoleRsp Specific constant of drying air (= 287 J K1 kg1)s f Frequency stabilitys Backscattering matrix of the linear polarization waveS Power density, signal powerS Complex Poynting vectorSi Incident power density

xxii List of Symbols

SN Power spectral density of noiseSs Scattered power densitySS Power spectral density of signalSw Vertical shear [s1]SNR Signal-to-noise ratiot TimeT Atmospheric temperature [K], noise temperature [K], period of gravity

wave, pulse repetition time (PRT) [s], time periodT0 Room temperature (290 K)Tc Correlation timeTe Equivalent input noise temperatureTi Input noise temperature, independent sample timeTs Sample time interval (sampling interval), sky noise temperatureTsys System noise temperatureTv Temperature of moist atmosphereTW Window widthu East-west (zonal) windu Mean zonal windu Horizontal IGW perturbation from turbulenceU Horizontal wind speedv Phase velocity of electromagnetic wavev Fluctuation component of wind perpendicular to the direction of wave

travelv Wind vector (vx, vy, vz)vd Doppler velocityvd Mean Doppler velocityvh Horizontal wind velocityvN Nyquist velocity (Nyquist limit)vr Radial velocityV Radar resolution volumeV6 Resolution volume circumscribed by the 6 dB contour of radar parametersVD Volume of raindropw Vertical wind velocity (or vertical component of wind velocity; vz)w Vertical IGW perturbation from turbulencewT Terminal velocity of precipitation (fall speed)W Cloud water content (or water content in unit volume) [g m3]WB Bandwidth of the signalz Altitude, height from sea level [km]Z Radar reflectivity factorZdr Differential reflectivity in linear unitZDR Differential reflectivity in decibelZe Equivalent radar reflectivity factorZi Radar reflectivity factor for ice particles Azimuth angle of the baseline formed between two antennas in SDI Bistatic angle

List of Symbols xxiii

Specific heat ratio of ideal gas ( 1.4 for dry air) Dry adiabatic lapse rate (= g/Cp 9.80) [K km1] Differential scattering phase, direction of horizontal wind, phase difference

between successive element antennas Resolution of the A/D converter Turbulent energy dissipation rate Permittivity [F m1]0 Permittivity in vacuum [F m1] Axis ratio b/a, where a is the semi-major axis diameter and b the semi-

minor axis diameter of a flat raindrop Radar reflectivity1 Efficiency of antennaa Antenna aperture efficiencyi Intrinsic impedance (or wave impedance) (=

/)

Zenith angle of radar beam1 One-way beamwidth between half-power points (or beam width)e Elevation angle of radar beamB One way half-power beamwidth in the E-plane [rad] Potential temperature Wave number for Bragg scattering Wave number vector for Bragg scatteringa Wave number vector for acoustic waveb Wave number corresponding to the Bragg scaleB Wave number corresponding to buoyancy lengthscale (= 2/LB) Radar wavelength [m] Parameter of drop size distribution (or slope parameter)s Structure wavelength of perturbations within inertial subrange Permeability [H m1]0 Permeability in vacuum [H m1] Kinematic viscosity (dynamic viscosity divided by the fluid density) Electric charge density [C m1], radar cross section [m2]| |2 Partial reflection coefficienta Atmospheric density [kg m3]hv Correlation coefficient between horizontally and vertically polarized wavesv Water vapor density [g m3]w Density of precipitation particles [g m3](= 106 for water) Electric conductivity [S m1]a Absorption cross sectionb Backscattering cross section f Doppler frequency spectrum width [Hz]s Scattering cross sectiont Extinction (or attenuation) cross sectionv Doppler velocity spectrum width [m s1]vn Doppler velocity spectrum width normalized with the Nyquist width Transmitted pules width [s], time lag

xxiv List of Symbols

i Independent sample timec Correlation time Angular distance from the beam axis in the H-planeh Phase delay per unit distance (one way) for horizontally polarized wave

[rad]v Phase delay per unit distance (one way) for vertically polarized wave [rad]DP Differential phase in two-way (DP = hh hh) [deg]hh Phase shift in round trip between radar and scatterer for horizontally

polarized wave [deg]vv Phase shift in round trip between radar and scatterer for vertically polarized

wave [deg] Phase of received echo signal, zenith angle in the H-plane based on radar

beam axisB One way half-power beamwidth in the H-plane [rad] Angle between the direction of polarization of the incident electric field

and the direction of scattering vector (= /2 for backscattering Differential phase of measured signals between horizontally and vertically

polarized waves [deg], scalar potential Angular frequency [rad s1]d Doppler angular frequencyi Intrinsic frequency Angular velocity of the Earths rotation (= 7.292 105 s1)

List of Abbreviations

A/D Analog to digitalAFWS Air Force Weather ServiceAGC Automatic gain controlAGL Above ground levelAMeDAS Automated Meteorological Rata Acquisition SystemAMS American Meteorological SocietyARM Atmospheric Research Measurement programATC Air traffic controlATSR Alternate transmission and simultaneous receptionBL Boundary layerBLR Boundary layer radarCAP Cooperative Agency ProfilerCAT Clear air turbulenceCCIR International Radio Consultative CommitteeCDL Coherent Doppler lidarCIRA Committee on Space Research (COSPA) International Reference

AtmosphereCOCO Coaxial-collinearCOHO Coherent oscillatorCOST European Cooperation in Science and TechnologyCRI Coherent radar imagingCST Central Standard TimeCSU Colorado State UniversityDBS Doppler beam swingingDFT Discrete Fourier transformDIF Decimation-in-frequencyDIT Decimation-in-timeDOA Direction of arrivalDPR Dual-frequency Precipitation RadarDRAW Doppler Radar for Airport WeatherDSD Drop size distribution

xxv

xxvi List of Abbreviations

EAR Equatorial Atmospheric RadarECCD Electromagnetically coupled coaxial dipoleEIK Extended interaction amplifierEST Eastern Standard TimeFAA Federal Aviation AdministrationFCA Full correlation analysisFDI Frequency domain interferometryFET Field effect transistorFFT First Fourier transformFII Frequency domain interferometric imagingFIR Finite impulse responseFMCW Frequency-modulated continuous wavesFRP Fiber-reinforced plasticFSA Full spectral analysisFWHM Full width at half maximumGMAP Gaussian model adaptive processingGMS Geostationary meteorological satelliteGMT Greenwich mean timeGPM Global Precipitation MeasurementGPS Global positioning systemGTS Global Telecommunication SystemHEMT High electric mobility transistorHS Hail signalHVPS High-Volume Particle SpectrometerI In-phaseIDFT Inverse discrete Fourier transformIF Intermediate frequencyIFFT Inverse fast Fourier transformIGW Inertia-gravity waveIIR Infinite impulse responseIR Infrared radiationIS Incoherent scatterITU International Telecommunication UnionJAFNA Joint Air Force and NASAJAXA Japan Aerospace Exploration AgencyJMA Japan Meteorological AgencyJST Japan Standard TimeKH KelvinHelmholtzKIX Kansai International AirportLAN Local area networkLDR Linear depolarization ratioLEO Low Earth orbitLHC Left-hand circularLLJ Low-level jetLNA Low noise amplifier

List of Abbreviations xxvii

LO Local frequencyLT Local timeLTR Lower Troposphere RadarM-P MarshallPalmerMEM Maximum entropy methodMESFET Metal-semiconductor FETML Multi-lagMLIT Ministry of Land, Infrastructure, Transport and TourismMLM Maximum likelihood methodMMIC Monolithic microwave integrated circuitMOPA Master oscillator and power amplifierMP Multi-parameterMPIfR Max-Planck-Institut fur RadioastronomieMPM Millimeter-wavelength propagation modelMRI Meteorological Research instituteMSM Mesoscale numerical modelMST Mesospheric-stratospheric-troposphericMU Middle and Upper atmosphereMUSIC Multiple signal classificationNASA National Aeronautics and Space AdministrationNCAR National Center for Atmospheric ResearchNIED National Research Institute for Earth Science and Disaster Pre-

ventionNOAA National Oceanic and Atmospheric AdministrationNPN NOAA Profiler NetworkNSSL National Severe Storms LaboratoryNWS National Weather ServiceORDA Open radar data acquisitionOTH Over the horizonPA Power-aperturePANSY Program of the Antarctic Syowa MST/IS RadarPBL Planetary boundary layerPBS Post beam steeringPHS Personal Handy-phone SystemPOS PositioningPPI Plan position indicatorPR Precipitation radarPRF Pulse repetition frequencyPRT Pulse repetition timePSS Post static steeringPUP Principal user processorQ Quadrature-phaseRASS Radio acoustic sounding systemRCS Radar cross sectionRDA Radar data acquisition

xxviii List of Abbreviations

rf Radio frequencyRHC Right-hand circularRHI Range height indicatorRIM Range imagingROPS Radar Observation data Processing SystemRPG Radar product generatorRPM Rotation per minuteRX ReceiverSA Spaced antennaSAD Spaced antenna driftSCSI Small Computer System InterfaceSDI Spatial domain interferometrySI Le Syste`me InternationalSNR Signal-to-noise ratioSPBS Sequential post beam steeringSSPA Solid state power amplifierST Stratospheric-troposphericSTALO Stabilized local oscillatorSTC Sensitivity time controlSTSR Simultaneous transmission and simultaneous receptionSVD Singular value decompositionT TroposphericTAI Temps Atomique InternationalTC Tropical cycloneTDWR Terminal Doppler Weather RadarTEP Turbulent eddy profilerTOGA-COARE Tropical Ocean Global AtmosphereCoupled Ocean Atmosphere

Research ExperimentTPPN Trans-Pacific Profiler NetworkTR Transmitter/receiverTRMM Tropical Rainfall Measurement MissionTWT Traveling wave tubeTX TransmitterUHF Ultrahigh frequencyUTC Coordinated Universal TimeVAD Velocity azimuth displayVCP Volume coverage patternVHF Very high frequencyVIL Vertical integrated liquidVVP Volume velocity processingWCB Warm conveyor beltWCRP World Climate Research ProgramWINDAS Wind Profiler Data Acquisition SystemWRC World Telecommunication ConferenceWMO World Meteorological Organization

Chapter 1Introduction

1.1 Principle of Radar

A variety of weather and atmospheric phenomena occur and change every momentin the Earths atmosphere. This book presents the techniques and sciences ofremote sensing various phenomena with radar. Remote sensing is a technique thatindirectly measures target without touching it directly in a distant place. Radar isan abbreviation for RAdio Detection And Ranging, which is an electronic systemthat generates electromagnetic waves in the transmitter, radiates them into spacevia antenna, receives the scattered signal returning from the target, and measuresthe position, movement of the target, etc. Usually, the same antenna is used fortransmission of the electromagnetic wave and reception of the return signal. Thetarget position is obtained according to the direction where the scattered signalreturns to the antenna, and to the distance calculated by the lapse of time that theelectromagnetic waves make in the round-trip between radar and target.

As for the targets that scatter electromagnetic waves, various types of scatterersare known, e.g., isolated objectives such as aircrafts and ships, minute distributedparticles such as precipitation and clouds, and perturbations of radio refractiveindex due to atmospheric turbulence. In this book, the properties of scattererssuch as precipitations, clouds, and fogs associated with weather, and refractiveindex perturbations caused by atmospheric turbulence are presented. The formeris mainly observed with meteorological radar (or weather radar), and the latterwith atmospheric radar. The conceptual diagrams of meteorological radar andatmospheric radar are shown in Fig. 1.1a and b, respectively. The atmosphericradars typically make observations overhead (i.e., at high elevation angles), whereasmeteorological radars typically scan the atmosphere at relatively low elevationangles. Furthermore meteorological radars typically use parabolic reflector antennaswhereas atmospheric radars use phased array antennas. Although the frequenciesadopted for meteorological and atmospheric radars are different due to the differ-ence of scattering mechanisms of the targets, many aspects of the basic configuration

S. Fukao and K. Hamazu, Radar for Meteorological and Atmospheric Observations,DOI 10.1007/978-4-431-54334-3 1, Springer Japan 2014

1

2 1 Introduction

Fig. 1.1 Conceptual diagrams of (a) meteorological radar and (b) atmospheric radar

and algorithms of signal processing of these radars are common. Therefore, thecommon components of both radars are stated as uniformly as possible, while theuncommon ones are dealt with in the individual chapters.

The upper atmosphere above the 100 km altitude is the ionosphere where theatmosphere is partially ionized. In the ionosphere, electromagnetic waves arescattered by free electrons, and so the scattering mechanism in this region is notthe same as those in other parts of the atmosphere (Gordon 1958). The scatteringis very close to incoherent scatter (IS) but extremely weakly affected via Coulombforce by ions. The radar which utilizes IS from the ionosphere is called incoherentscatter radar or IS radar (e.g., Evans 1969). The IS radar is beyond the scope of thisbook, and we briefly state in Sect. 10.2.1 that some large-scale atmospheric radarshave the capability of IS radar.

1.2 History of Meteorological and Atmospheric Radars

The atmosphere has been studied using radar since the 1920s. A basic model ofpresent pulsed meteorological radar in the microwave band was first put in practicefor precipitation observations in early 1940s.

Meanwhile, prior to the practical implementation of the meteorological radars,the scattering mechanism of electromagnetic wave had already been theoreticallyclarified by L. Rayleigh and G. Mie. Rayleigh showed that the magnitude of thebackscattering intensity due to precipitation particles in the atmosphere is propor-tional to the 6th power of the diameter of the scatterer, and inverse proportional tothe 4th power of the wavelength of the electromagnetic wave (e.g., Gunn and East1954; Battan 1973, p. 38).

Moreover, Mie showed that in the area where the diameter of the scatterer is aboutl/10 or larger the wavelength for which Rayleighs scattering theory is not applied,more rigorous scattering theory is necessary (e.g., Gunn and East 1954). Thesetheories are known as Mie scattering theorem and Rayleigh scattering theorem, andwill be discussed in Sects. 3.3.3 and 3.3.4, respectively.

1.3 Radar Frequency Bands and Usage 3

Ryde (1946) theoretically estimated the reflection intensity and the attenuationof microwave due to precipitation and cloud particles in the atmosphere basedon these theories during 1941 and 1946. His theories became the basics ofquantitative observation of weather with radar. When the 1st Conference on RadarMeteorology sponsored by the American Meteorological Society (AMS) was heldin the Massachusetts Institute of Technology (MIT) in March 1947, quantitativeobservations of precipitation had already been advanced around the United States.The relation between radar reflectivity factor and precipitation intensity and the dropsize distributions of precipitation found by Marshall and Palmer (1948) became thebeginning of the radar meteorology research.

On the other hand, the research on the communication over the horizon (OTH) us-ing the radio wave propagation through the troposphere became active in the 1950s.Understanding about clear air turbulence (CAT) and the scattering mechanism in thetroposphere have been rapidly advanced through this research, promoting researchesregarding atmospheric remote sensing (Booker and Gordon 1950).

The origin of atmospheric radar is the IS radar at Jicamarca, Peru, and manyatmospheric radars had been developed and constructed in the 1970s to 1980s.Woodman and Guillen (1974) showed, for the first time, that the wind and theturbulence in the mesosphere and the stratosphere can be observed using the ISradar at Jicamarca. Atmospheric radars are called as mesosphericstratospherictropospheric (MST) radar, stratospherictropospheric (ST) radar, tropospheric (T)radar, or boundary layer (BL) radar (or BLR), according to the observable regionfor the radars. At the early stage, MST and ST radars that use the frequency bandof 50 MHz to l GHz were primarily developed for research. Small-scale radars suchas BLR that utilize the microwave of 1.3 GHz was rapidly put to practical use soonlater.

1.3 Radar Frequency Bands and Usage

For transmission, various frequencies of the wide range from several MHz toaround 100 GHz are utilized to detect atmospheric scatterers. The principle of bothelectromagnetic wave propagation and operation of radar does not change by theoperational frequency and is common for various frequencies, although actual radarcomponents greatly vary, depending on the frequency.

In principle, longer wavelengths, i.e., lower transmitted frequencies are suitablefor detection of refractive index perturbations. Shorter wavelengths, i.e., highertransmitted frequencies are suitable for detection of minute scatterers. Figure 1.2shows the operational frequencies of actual radars and their adjoining frequencybands. The radar frequency bands with the main usages are shown in Table 1.1. Inpassage of the radar development, the individual radar frequency bands have beentraditionally named by letters as L, S, and so on as shown in Table 1.1. However,the letter expression is generally obscure, and inconvenient to show the concretefrequency. Therefore, in this book, we will express the operational frequency by1.3-GHz for instance, and not by L-band.

4 1 Introduction

Fig. 1.2 Operational frequency band of various radars and their adjoining frequency bands

The scattering characteristics of electromagnetic wave are closely dependentupon its wavelength. It means that each scattering mechanism has a best com-bination with a specific wavelength band in the remote sensing of precipitationparticles.1 The diameters of objective particles are between several m and severalmm. Thus, electromagnetic waves of wavelengths of several mm (millimeter wave)to several cm (micro wave) are adopted for these targets. As will be discussedin Sect. 6.3.3, electromagnetic waves of wavelength of less than several cm areattenuated due to precipitation along the propagation path. Thus, it is indispensableto evaluate the influence of the attenuation quantitatively.

In the remote sensing of the atmosphere, refractive index perturbations generatedfrom atmospheric turbulence and waves are the main source of scatterers. Therefractive index field has perturbations over a spectrum of spatial scales (i.e.,structure wavelengths of perturbations s), and in general large scale refractiveindex perturbations contain more intense perturbations. In the radar observationof the atmosphere, only scattering from a specific spatial scale corresponding tohalf the radar wavelength is detected (Brag scatter; Chap. 3). Therefore, using radarwavelength at twice the largest scales of perturbations, we can more easily observethe backscatter from refractive index perturbations. Practically, wavelengths up

1Here, the term precipitation particles include both precipitation particles such as raindropand hailstone, and non-precipitation particles such as cloud and fog particles, and they are notdistinguished strictly otherwise mentioned hereafter.

1.3 Radar Frequency Bands and Usage 5

Table 1.1 Radar frequency bands and applications

Letter band Frequency range ITUa regulation ApplicationsHF 330 MHz OTH, atmosphere observationVHF 30300 MHz 138144 MHz Ultra long range surveillance,

atmosphere observation216225 MHz Ditto

UHFb 3003,000 MHz 420450 MHz Ditto890942 MHz Ditto

L 12 GHz 1.2151.4 GHz Long range surveillance,air traffic control (ATC),atmosphere observation

S 24 GHz 2.32.5 GHz Middle range surveillance, ATC,2.73.7 GHz Long range weather observation

C 48 GHz 5.255.95 GHz Long range tracking,weather observation

X 812 GHz 8.510.68 GHz Short range tracking,weather observation

Ku 1218 GHz 13.414.0 GHz High resolution satellite altimeter15.717.7 GHz Ditto

K 1827 GHz 24.0524.25 GHz Airport surface surveillanceKa 2740 GHz 33.436 GHz Short range tracking,

weather observationV 4075 GHz 5964 GHz Remote sensingW 75110 GHz Remote sensingmmc 110300 GHz Remote sensinga International Telecommunication Unionb 3001,000 MHz (Skolnik 1990, p1.14). 230 MHz1 GHz band is often called P-bandc Radars of this frequency band have not been achieved yet

to several meters are used for the atmospheric observations in consideration ofthe reasonable physical size of the antenna aperture and the outer scale (i.e., thelargest scales) of turbulence. As will be discussed in Sect. 3.4.2, scattering fromperturbations of refractive index due to atmospheric turbulence is dominant at thefrequencies lower than 23 GHz, and that from precipitation particles are dominantat higher frequencies. Thus, atmospheric radars generally adopt frequencies ofseveral tens MHz to around 1 GHz, and at highest, 3 GHz. Meanwhile, meteoro-logical radars generally adopt frequencies higher than 23 GHz. At the frequencieslower than 3 GHz, the attenuation of electromagnetic waves due to precipitation andatmosphere along the propagation path is small and almost negligible.

Among the atmospheric observation techniques, there is one that utilized smallfloating dust (aerosol) as small as m (109 m) as the scatterer. In this case, the laserbeam of wavelength of the same order as the size of scatterer is adopted. A laserradar is called lidar (LIght Detection And Ranging), and is used to observe the airtemperature inversion layer, the water vapor content, the atmospheric density, andso on. In recent years, a coherent Doppler lidar (CDL), which is possible to obtainDoppler velocities of aerosol, has become to practical use.

Chapter 2Electromagnetic Waves

2.1 Characteristics of Electromagnetic Waves

Electric and magnetic fields propagate through space in the form of electromagneticwaves (or radio waves). Various characteristics of the electromagnetic waves can bederived from Maxwells equations, which will be discussed for the simplest case inthe present chapter. In this chapter, we assume that the waves are propagating in asmoothly varying medium, and that there are no small scale (i.e., of the order of awavelength) perturbations of refractive index either.

2.1.1 Basic Equations

Maxwells Equations

Three basic differential equations concerned with electric and magnetic fields,which include Faradays law, Ampe`reMaxwells law, and Gausss law, are calledby a general name Maxwells equations. According to Maxwells equations, electricfield strength E and magnetic field strength H are expressed using magnetic fluxdensity B, electric current density J, and electric flux density D as

E = t B, Faradays law (2.1)

H = t D+ J, Ampe`reMaxwell law (2.2)

D = , Gausss law for electric flux density (2.3) B = 0. Gausss law for magnetic flux density (2.4)

S. Fukao and K. Hamazu, Radar for Meteorological and Atmospheric Observations,DOI 10.1007/978-4-431-54334-3 2, Springer Japan 2014

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8 2 Electromagnetic Waves

The vectors E, H, J, D, and B depend on position (x, y, z) and time (t). Inthe International System of Unit (SI unit; le Syste`me International dUnite),1 theA [ampere] is defined as the electric unit, and other quantities are derived asV [volt, m2 kg s3 A], Wb [weber, m2 kg s2 A1], and C [coulomb, s A]. Applyingthese quantities, the units of E, H, J, D are derived as [V m1], [m1 A], [m2 A],and [C m2], respectively. The symbol denotes electric charge density in the unitof [C m3], and symbols and in vector analysis indicate the differentialoperator for rotation2 and divergence,3 respectively.

In general, D and J are related by a linear function to E, and B to H.Their relations are expressed by the constitutive equations (2.5)(2.6) and Ohmslaw (2.7) as

D = E, (2.5)B = H, (2.6)J = E, (2.7)

where the permittivity [F m1], the permeability [H m1], and the electricconductivity [S m1] are medium constants. They are scalar quantities forthe isotropic4 medium, and tensor quantities for the aerotropic medium such asferromagnet and crystalline body. In the present book, they are treated as scalarquantities unless mentioned otherwise.

Wave Equation

Each of Maxwells equations (2.1) and (2.2) includes both electric and magneticfields. If they are solved as the simultaneous equations, the equation which containsonly electric or magnetic field as a variable is derivable. First, the equation whichincludes only the electric field is obtained. Differentiating (2.2) with respect to time

1The MKSA unit system whose basic units are length m [meter], mass [kg], time [s(second)], andelectric current A [ampere].2In the orthogonal coordinate system (x, y, z), the rotation of E is a vector quantity that isexpressed by

E = rotE =(Ez

y Ey z

)ix +

(Ex z

Ezx

)iy +

(Eyx

Exy

)iz,

where ix, iy, and iz are unit vectors in the x, y, and z directions, respectively.3The divergence of D is a scalar quantity that is expressed by

D = divD = Dxx +Dyy +

Dz z .

4If the permittivity and the permeability of the medium are independent of the direction of theradiation, the medium is called isotropic.

2.1 Characteristics of Electromagnetic Waves 9

t and replacing the right-hand side parameters using (2.5) and (2.7),

t H = 2 t2

D+ t J

= 2 t2

E+ t E. (2.8)

Applying the rotation operator to (2.1), and substituting (2.6) and (2.8) into it,

(E) = 2

t2E t E (2.9)

is obtained. Replacing the left-hand side of (2.9) with an identical equation of thevector operation,

(E) = ( E)2E, (2.10)

where and 2 are gradient5 and Laplacian,6 respectively, and substituting( E) = / , which is obtained from (2.3) and (2.5) under the assumption thatpermittivity is spatially uniform, into (2.10), the equation for electric field

2E 2

t2E t E =

1

(2.11)

is derived.Next, applying the similar operation to magnetic field,

2H 2

t2H t H = 0 (2.12)

5In the orthogonal coordinate system (x, y, z), the gradient of a scalar function (x,y, z) is a vectorquantity given by

= x ix +y iy +

z iz.

6For a scalar function (x,y, z), 2 is a scalar quantity given by

2 = div(grad ) = 2

x2 + 2y2 +

2 z2 .

10 2 Electromagnetic Waves

is obtained. Equations (2.11) and (2.12), which are called vector wave equations,are the general equations that describe wave propagation in a linear7 homogeneous8medium which is isotropic and nondispersive.9 Suppose a uniform and losslessdielectric medium ( = 0) which has an infinite extent. If the medium has no wavesource such as electric charge, = 0. The present book concerns the medium whichsatisfies these conditions. Thus (2.11) and (2.12) are simplified as

2E 2

t2E = 0, (2.13)

2H 2

t2H = 0. (2.14)

In a statistically homogeneous medium in which the permittivity is fluctuatingdue to, for example, turbulence, waves are scattered by refractive index perturba-tions. The wave equations for this problem are not derived in this book, and readersshould refer to Tatarskii (1971, Chap. 2) or to Doviak and Zrnic (2006, Chap. 11)for them.

Plane Wave

Electromagnetic wave is called a plane wave if it has electric and magnetic fieldswithin the plane perpendicular to the direction of wave propagation. When theelectromagnetic wave spreads radially from wave source, it is locally a plane wavewhen observed sufficiently far from the wave source. If the plane electromagneticwave shows sinusoidal oscillation with single angular frequency [rad s1](= 2 f ; f is frequency [Hz]), the time factor is expressed as e jt , where j is theimaginary unit ( j2 =1). Then, (2.13) and (2.14) lead to

2E+ k2E = 0, (2.15)2H+ k2H = 0, (2.16)

respectively, where

k =

. (2.17)

7The medium is called linear medium if proportionality holds between J and E, B and H, and Jand E, respectively, and nonlinear medium otherwise. In nonlinear medium, the permittivity, thepermeability, or the conductivity are the function of E or H.8The medium is called homogeneous medium if the permittivity and the permeability are spatiallyuniform, while inhomogeneous otherwise.9The medium with constant permittivity and permeability irrespective of frequency is callednondispersive, otherwise it is dispersive.

2.1 Characteristics of Electromagnetic Waves 11

Parameter k is called the (radar) wave number, which is related to radar wavelength [m] as k = 2/ . The equations in the form of (2.15) and (2.16) are calledHelmholtz equations. The solution of (2.15) at the position r is generally given by

E(r) = E1e jkr, (2.18)where E1 is the vector which does not depend on the position. If = 0 and = 0,substituting (2.18) into (2.1) to (2.4),

kE = H, (2.19)kH =E, (2.20)

k E = 0, (2.21)k H = 0 (2.22)

are obtained. Equations (2.21) and (2.22) mean that both E and H are orthogonalto k. Furthermore, (2.19) and (2.20) indicate that the sequence of E, H, and k arein the sense of rotation to advance a right-handed screw. The vectors E and H arein-phase, and related as

H =1i

ik E, (2.23)

where ik is the unit vector in the direction of k, and i is the intrinsic impedance (orwave impedance) given by

i =

/. (2.24)

Denoting and in a vacuum as 0 and 0, respectively, 0 8.8542 1012[F m1] and 0 = 4 107 1.2566 106 [H m1]. Therefore, the waveimpedance 0 becomes 0 =

0/0 ( 376.7 120) [].

E2e+ jkr also satisfies (2.15), and accordingly the solution of (2.15) is, in general,given by the sum of both values as

E(r) = E1e jkr +E2e+ jkr. (2.25)

Multiplying (2.25) by time factor e jt ,

E(r, t) = E1e j(tkr) +E2e j(t+kr) (2.26)

is obtained. The above equation means that waves E1 and E2 propagate to thedirections +k and k, respectively, with the phase velocity of

v =

k = 1/

, (2.27)

12 2 Electromagnetic Waves

Fig. 2.1 Horizontally (top) and vertically (bottom) polarized waves. Solid lines show electric fieldand dotted lines show magnetic field

where v is the speed of light and equal to c in a vacuum. As a result, it is derivedthat

c = 1/

00 2.9979 108 [m s1]. (2.28)

2.1.2 Polarization

When a plane electromagnetic wave propagates in the positive z direction, theelectric field vector lies in the xy plane perpendicular to the z-axis. The tip ofthe electric field traces a curve in the xy plane, corresponding to the change ofamplitude. If the curve shows a straight line, the plane wave is called linearlypolarized, while circularly polarized when the curve shows a circle. In general, thewave is called elliptically polarized when it shows an ellipse.

If the electric field of a linearly polarized plane wave oscillates in the horizontal(y) direction as shown in the upper panel of Fig. 2.1, the wave is called horizontallypolarized. On the other hand, if the wave oscillates is in the vertical (x) direction asshown in the bottom panel of the same figure, it is called vertically polarized. Thehorizontal or vertical polarization is defined with the ground as the reference. Theplanes where the electric and magnetic field vectors exist are called the E-plane andthe H-plane, respectively.

2.1 Characteristics of Electromagnetic Waves 13

Fig. 2.2 Rotation direction of circularly polarized waves viewed in the direction of propagation.(a) Right-hand circular (RHC) and (b) Left-hand circular (LHC) polarization

Here, a plane electromagnetic wave propagates in the z direction, and the unitvectors in the x and y axes are denoted ix and iy, respectively. When there are twolinearly polarized waves in the x direction and in the y direction whose phases arex and y, respectively, the sum of the two waves becomes

Ec = ixExe[ j(tkz)+ jx] + iyEye[ j(tkz)+ jy]. (2.29)

The tip of the above wave, the sum of two orthogonally polarized waves, draws anellipse and is called elliptically polarized wave. If x y = n (n = 0,1,2, ),the wave is linearly polarized. On the other hand, if Ex = Ey and x y = /2,the wave is circularly polarized. In the case of circularly and elliptically polarizedwaves the tip moves either clockwise or counterclockwise. When x y = /2,the electric field which is viewed from the tail to the z direction rotates clockwiseas time changes. As shown in Fig. 2.2a, it is called right-hand circular (RHC)polarization. On the other hand, if x y =/2, as shown in Panel (b), it rotatescounterclockwise, and is called left-hand circular (LHC) polarization. Arbitraryelliptically polarized wave can be considered as the composition of RHC and LHCwaves of different magnitude, or that of the horizontally and vertically polarizedwaves.

The circular polarized wave which is radiated to the scatterers such as rain-drops, whose reflectivity is approximately identical for horizontal and verticalpolarizations, is backscattered as circular polarized wave but in reversed rotation.On the other hand, the backscattered signal from aircrafts or vessels becomes anelliptically polarized wave. In the present book, only one polarization, horizontalor vertical, is treated for simplicity if not otherwise mentioned. Furthermore weassume that the transmitted and received signals are of the same polarization. Ifmultiple polarizations are treated as in the case of polarimetric radar, it needs topay attention to these points. The polarimetric radar will be discussed in detail inSect. 6.4.

14 2 Electromagnetic Waves

Fig. 2.3 Boundary surface between two mediums. (a) Tangential and (b) normal components

2.1.3 Reflection and Refraction

Boundary Condition

The boundary condition for propagation of electromagnetic waves at the surfacewhere the medium 1 and the medium 2 contact as shown in Fig. 2.3 is derivedbelow (e.g., Stratton 2007, Chap. 1). First, Faradays law10 is applied, as shownin Fig. 2.3a, to the closed curve C = C1 +C2 +C3 +C4 which encircles the minutearea of breadthwise a and lengthwise b in the static magnetic field. When bbecomes infinitesimally small, contribution from C3 and C4 to the line integral canbe neglected, and thus

limb0

CE dl =

C1E1 itdl+

C2E2 (itdl)

=

C1E1tdl

C2E2tdl = (E1t E2t)a = 0, (2.30)

where E1t = E1 it and E2t = E2 it are it (tangential) components of E1 and E2,respectively. From (2.30),

E1t = E2t. (2.31)That is, the tangential components of electric field Et of both sides becomes equalat the interface of two mediums. Similarly, if surface current does not exist at theinterface of the two mediums, the tangential components of magnetic field satisfy

H1t = H2t, (2.32)where H1t = H1 it and H2t = H2 it are similarly the it components of H1 and H2,respectively.

10The integral form of (2.1) is expressed as

E dl = t

SB dS, where S is encircled by an

arbitrary closed curve C. Regarding the static magnetic field in a dielectric,

E dl = 0.

2.1 Characteristics of Electromagnetic Waves 15

Secondly, Gausss law11 is applied to the minute volume with cross sectionS and height h which pass through the interface of two mediums as shown inFig. 2.3b. The surface area of the volume is S = S1 + S2 + S3, where S1, S2, and S3are areas of the bottom, the upper, and the side panels, respectively. In the limit thath becomes close to 0, contribution from S3 to the surface integral can be neglected,and consequently if there is no electric charge at the interface,

limh0

SD dS =

S1D1 (in)dS+

S2D2indS

=

S1D1ndS+

S2D2ndS = (D2n D1n)S = 0 (2.33)

is obtained, where D1n = D1 in and D2n = D2 in are the in (normal) components ofD1 and D2, respectively. From (2.33),

D1n = D2n. (2.34)The above equation means that the normal components of electric flux density inthe two mediums become equal at the interface. Similarly, the normal componentsof magnetic flux density become identical at the surface, thus

B1n = B2n, (2.35)where B1n = B1 in and B2n = B2 in are the in components of B1 and B2,respectively.

Reflection and Refraction of Plane Wave

The reflection and refraction of a plane wave at z = 0 of the interface xy between twomediums 1 and 2 shown in Fig. 2.4 is discussed. The incident wave Ei in xz-planepropagates from the medium 1 to 2. A part of the incident wave is reflected as Erback in the medium 1 and the other is refracted as Et into the medium 2. If theangles between z-axis and the traveling directions are i, r, and t, respectively,the incident wave, the reflected wave, and the refracted wave are expressed asEie jk1(xsini+zcosi), Ere jk1(xsinrzcosr), and Ete jk2(xsint+zcost), respectively,where k1 and k2 are the propagation constants of mediums 1 and 2, respectively.From the boundary condition (2.31), the tangential components of the electric fieldat the interface between mediums 1 and 2 should be continuous, that is

Eie jk1xsini +Ere jk1xsinr = Ete jk2xsint . (2.36)

11 The integral form of (2.3) is expressed as

SD dS =

VdV = Q, where space V is encircled

by an arbitrary closed surface, and Q is an electric charge. In the free space with no electric chargeinside,

SD dS = 0.

16 2 Electromagnetic Waves

Fig. 2.4 Reflection andrefraction of a plane wave atthe boundary surface betweentwo mediums

For (2.36) to be applied at any place of the interface, both phase and amplitude ofeach wave must be identical. As for the phase,

k1 sini = k1 sinr = k2 sint (2.37)is satisfied. From (2.37),

i = r, (2.38)sinisint

=k2k1

(2.39)

is derived. The above relation means that the angles of incidence and reflection areidentical, and the ratio between their sines is constant regardless of the incidentangle. This is one form of Snells law in the optics.

The refractive index is defined as the ratio of the speed of light in a vacuum tothat in a non-vacuum medium by

n cv, (2.40)

which is called as the absolute refractive index. Thus, if the (absolute) refractiveindexes of mediums 1 and 2 are n1 and n2, respectively, and the phase velocitiesv1 and v2, respectively, (2.39) is modified using (2.27) as follows. Since angularfrequency does not change as the wave passes across the interface,

sinisint

=k2k1

=v1v2

=n2n1

(2.41)

is the relative refractive index of medium 2 to medium 1. In particular, if the medium1 is vacuum and the medium 2 is the atmosphere ( = 0), the absolute refractiveindex of the atmosphere becomes

n =

2200

2/0, (2.42)

2.1 Characteristics of Electromagnetic Waves 17

where 2 and 2 are the permittivity and the permeability of the medium 2,respectively, and 2 0 for the atmosphere. If the continuity equation of the waveamplitude is applied to (2.36), the reflection coefficient Er/Ei and the transmissioncoefficient Et/Ei are derivable.

2.1.4 Radiation

In the following, the radiation field far from a wave source is derived. It is sometimesmore convenient to introduce scalar and vector potentials than to express directlythe electromagnetic field. This case is true for the radiation, and the aforementionedMaxwells equation is rewritten by using potentials below.

Vector and Scalar Potentials

Magnetic flux density B is a rotational field and satisfies the continuity givenby (2.4). The vector A defined as

B = H A (2.43)is called the vector potential. Substituting (2.43) into (2.1),

E = t (A) = (A t

)(2.44)

is obtained. It leads to

(

E+A t

)= 0. (2.45)

From the vector formula, any function with zero rotation is given by gradient ofscalar function. Applying this relation with scalar potential to (2.45),

E+A t + = 0 (2.46)

is derived. In the stationary state, E = , and hence is called a generalizedelectric potential. In the same way as the electrostatic field is only known within aconstant , either or A cannot be uniquely determined. For one solution of 0and A0, a new solution which includes arbitrary functions and ,

A = A0 , (2.47)

= 0 + t + (2.48)

18 2 Electromagnetic Waves

is derivable for the same E and B. Such a conversion between potentials is calledgauge transformation.

The radiation fields E and H of a current source J are related as

H = t E+ J, (2.49)

which is derived from (2.2) and (2.5). Substituting (2.43) and (2.46) into (2.49),

(

1 A

) t

( A t

)= J (2.50)

is obtained. The above equation can be rewritten as

A+ 2A

t2 = J t . (2.51)

Applying the vector formula (2.10) to (2.51),

2A 2A

t2 + J = (

A+ t)

(2.52)

is obtained. Here, applying Lorentz gauge,

A+ t = 0, (2.53)

the following relation

2A 2A

t2 + J = 0 (2.54)

is derived. Similarly, also satisfies the differential equation in the form

2 2

t2 += 0. (2.55)

Equations (2.54) and (2.55) are Maxwells equations using vector and scalarpotentials, which are accordingly called potential equations. Both equations aredifferential equations with respect to position r and time t, and are known to havethe following solutions,

A(r, t) = 4

V

J(r, t r/v)r

dV, (2.56)

(r, t) = 4

V

(r, t r/v)r

dV, (2.57)

2.1 Characteristics of Electromagnetic Waves 19

Fig. 2.5 The coordinates anda short dipole

where r = |r r| and v is the phase velocity of the electromagnetic wave. Hence,the above two equations mean that the potentials observed on the point r at time tare contributed by the electric current and the electric charge on the point r at timet r/v. Such potentials which are radially radiated from a wave source with phasevelocity v are called retarded potentials (e.g., Stratton 2007, p. 428).

Radiation Field

A short dipole (differential antenna) of length l (l ) is considered at the originin the rectangular coordinates (x, y, z) and polar coordinates (r, , ) as shown inFig. 2.5. The current on the dipole is assumed to be directed along z-axis and to varysinusoidally as

I = I0e jt , (2.58)

where I0 is the peak value. As the current vector is directed along z-axis, the vectorpotential A has only z-axis component and is given from (2.56) as (Ramo et al. 1965)

Az =Il4r

e jkr. (2.59)

As the polar coordinates are more convenient than the rectangular coordinates toexpress actual electromagnetic field radiated from antenna, the polar coordinatesare used to express three components of the vector potential A as follows;

Ar = Az cos =Il4r

e jkr cos , (2.60)

A = Az sin = Il4r e jkr sin , (2.61)

A = 0. (2.62)

20 2 Electromagnetic Waves

Three components of the magnetic field H are derived from the vector potential,using (2.43) and (2.60) to (2.62), as

Hr =1

r sin

[ (A sin )

A

]= 0, (2.63)

H =1r

[1

sinAr

r (rA )

]= 0, (2.64)

H =1r

[ r (rA )

Ar

]=

Il4

sin( jk

r+

1r2

)e jkr. (2.65)

Similarly, the electric field E is expressed with vector potential using (2.46)and (2.53) as

E = jk2 ( A) jA. (2.66)

From the above equation, three components of E are given by (e.g., Stratton 2007,p. 436; Ulaby et al. 1981, p. 109)

Er =Ili2

cos(

1r2

+1

jkr3)

e jkr, (2.67)

E =Ili4

sin( jk

r+

1r2

+1

jkr3)

e jkr, (2.68)

E = 0, (2.69)where i is the intrinsic impedance. The magnitude of the component of theelectromagnetic field radiated by a short dipole is given by the sum of threecomponents which are inversely proportional to distance, its square, and its thirdpower. The components which strongly depend on the distance are predominantonly in the vicinity of the current source (short dipole). The component that ispredominant in the far distance r which satisfies the following relation

kr = 2r 1 (2.70)

is called the radiation field, where is the wavelength. The radiation electric fieldis expressed from (2.58) and (2.68) as

E = j kI0li4r sin ej(tkr). (2.71)

Similarly, the radiation magnetic field is expressed from (2.58) and (2.65) as

H = j kI0l4r sin ej(tkr) =

E . (2.72)

2.1 Characteristics of Electromagnetic Waves 21

Poynting Vector

The energy which passes through a unit area in a unit time, S [kg s3, i.e.,J m2 s1 = Wm2], where J and W denote joule and watt, respectively, iscalled complex Poynting vector, and expressed as

S = 12

EH, (2.73)

where the superscript denotes complex conjugate. The coefficient 1/2 is multipliedto the product because E and H are typically expressed in terms of their amplitude(not their rms value), whereas S is the average power.

In the case of the electromagnetic field radiated by a short dipole, the Poyntingvector has only the r-component, which is radiated to the +r direction as shown by

S = 12

E H ir =k2I20 l2i sin 2

322r2 ir, (2.74)

where ir is the unit vector to r direction. The total radiation power Ps is derivedfrom (2.74) as

Ps = 2

0

(

0Re

[12

E H]

r2 sind)

d

=k2I20 l2i

322 2

0

(

0sin3 d

)d = k

2I20 l2i322

83

=i3

(I0l

)2. (2.75)

The radiation resistance RR is given by

RR =2PsI20

=2i

3

(l

)2. (2.76)

Equation (2.76) means the easiness of radiation. In other word, it is shown that thelonger the length of the antenna is, and/or the shorter the wavelength is, the easierto radiate.

Antenna Parameter

Degree of concentration of radiation power density into a specific direction ofantenna is called directivity,12 and is defined by the ratio of the radiation power

12 There are two different definitions of antenna gain, directive gain and power gain. The formeris generally called directive gain or directivity (the IEEE accepted notation), gD, while the latter iscalled antenna power gain or simply gain, g. The directivity is defined as the maximum radiationintensity relative to the average intensity, whereas the antenna power gain involves antenna losses.

22 2 Electromagnetic Waves

density in a specific direction to that power density of a non-directional antennawhich isotropically radiates power in all directions. Since the radiation powerdensity per unit area varies by distance from the source, it is common to take thepower per unit solid angle instead of that per unit area as the radiation power density.When the radiation power density is normalized with the mean radiation power perunit solid angle, the directivity gD( , ) is given by

gD( , ) = |E( , )|2

14

|E( , )|2d

=|E( , )|2

|E0|2 , (2.77)

where is the solid angle and d = sindd . If a short dipole is assumed as theantenna, E( ) sin from (2.71), and therefore, (2.77) becomes

gD( ) =sin2

14

sin2 d

=32

sin2 . (2.78)

Next, effective antenna aperture is derived from the view point of a matchingcircuit. If electric current I is supplied to the short dipole of the radiation resistanceof RR, the maximum output power is obtained under the condition that the internalresistance of the circuit is equivalent to RR. Hence, if there is no ohmic power loss,the maximum power radiated from antenna PR becomes

PR =12|I|2RR = (El)

2

8RR=

il24RR

Sr AeSr, (2.79)

where E is the electric field strength (rms value), l the length of short dipole, Sr theelectric power density E2/2i, and Ae the effective antenna aperture. From (2.76)and (2.79), Ae of the short dipole becomes

Ae =il24RR

=3 28 . (2.80)

In general, the following relation between antenna 1 of effective aperture Ae1 anddirectivity gD1 and antenna 2 of Ae2 and gD2 holds from the reciprocity theorem;

Ae1gD1

=Ae2gD2

. (2.81)

Hence, if the efficiency o