Polarimetric Doppler Weather Radar: Principles and Applications

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Acknowledgments......Page 16
List of symbols......Page 17
Abbreviations......Page 26
1.1 Review of Maxwell’s equations and potentials......Page 29
1.1.1 Vector Helmholtz equation......Page 31
1.1.2 Scalar, vector and electric Hertz potentials......Page 32
1.2 Integral representation for scattering by a dielectric particle......Page 34
1.3 Rayleigh scattering by a dielectric sphere......Page 38
1.3.1 Extension to Rayleigh scattering by spheroids......Page 41
1.4 Scattering, bistatic, and radar cross sections......Page 42
1.5 Absorption and extinction cross sections......Page 46
1.6 Clausius–Mosotti equation and Maxwell-Garnet mixing formula......Page 47
1.7 Faraday’s law and non-relativistic Doppler shift......Page 50
1.7.1 Electromotive force......Page 51
1.7.2 Example of a moving rectangular loop......Page 52
1.7.4 Re-radiation by the loop......Page 55
1.8 Moving dielectric spheres: coherent and incoherent summation......Page 57
1.9.1 Time-correlated bistatic cross section......Page 60
1.9.2 Block diagram of a pulsed Doppler radar......Page 63
1.10 Coherent forward scattering by a slab of dielectric spheres......Page 66
1.10.1 Concept of an effective propagation constant......Page 69
Notes......Page 71
2.1.1 FSA convention......Page 73
2.1.2 BSA convention......Page 76
2.2 Reciprocity theorem......Page 78
2.3.1 Sphere......Page 80
2.3.2 Spheroid......Page 81
2.3.3 Oriented spheroid......Page 85
2.3.4 Horizontal incidence…......Page 89
2.3.6 Probability distribution function of the symmetry axis......Page 94
2.3.7 General incidence…......Page 96
2.4 Mie solution......Page 103
2.4.1 van de Hulst convention: scattering and extinction cross sections......Page 107
2.4.2 Bistatic radar cross section......Page 109
2.4.3 Radar cross section......Page 110
2.5 Mie coefficients in powers of k0a: low frequency approximation......Page 111
2.6 Numerical scattering methods for non-spherical particles......Page 113
Notes......Page 116
3.1 Polarization state of a plane wave......Page 117
3.1.1 Polarization ellipse......Page 120
3.1.2 Stokes’ vector......Page 124
3.2 Basics of antenna radiation and reception......Page 126
3.2.1 Voltage equation......Page 128
3.2.2 Polarization efficiency in terms of Chi......Page 130
3.3 Dual-polarized antennas: linear polarization basis......Page 132
3.4 Radar range equation for a single particle: linear polarization basis......Page 135
3.5 Change of polarization basis: linear to circular basis......Page 136
3.5.2 BSA scattering matrix in circular basis......Page 137
3.5.3 Radar observables in circular basis......Page 139
3.6 Radar range equation: circular basis......Page 141
3.7 Bilinear form of the voltage equation......Page 145
3.8 Polarization synthesis and characteristic polarizations......Page 147
3.8.1 Elliptical depolarization ratio......Page 149
3.8.2 Copolar and cross-polar nulls for a single particle......Page 152
3.9 Partially polarized waves: coherency matrix and Stokes’ vector......Page 154
3.9.1 Coherency matrix......Page 155
3.9.2 Stokes’ vector of partially polarized waves......Page 156
3.10 Ensemble-averaged Mueller matrix......Page 157
3.10.1 Radar observables in linear basis......Page 159
3.11.1 Mueller matrix......Page 161
3.11.2 Polarimetric covariance matrix......Page 162
3.11.3 Radar observables in linear basis......Page 164
3.12.1 "Mirror" reflection symmetry......Page 166
3.12.2 Azimuthal symmetry/polarization plane isotropy......Page 167
3.13 Covariance matrix in circular basis......Page 170
3.13.2 "Mirror" reflection symmetry…......Page 171
3.13.3 "Mirror" reflection symmetry…......Page 173
3.13.4 Radar observables in circular basis......Page 174
3.13.5 Rotating linear basis......Page 175
3.13.6 Covariance matrix for a two-component mixture of hydrometeor types......Page 178
3.14.2 "Mirror" reflection symmetry…......Page 179
3.14.3 Case of…......Page 181
3.14.4 Asymmetry ratio......Page 182
3.14.5 Framework for classification of hydrometeor types......Page 184
Notes......Page 186
4 Dual-polarized wave propagation in precipitation media......Page 188
4.1 Coherent wave propagation......Page 189
4.1.1 Oriented particles (Beta = 0º case): circular polarization......Page 191
4.1.2 Uniformly canted particles: linear polarization......Page 194
4.1.3 Uniformly canted particles: circular polarization......Page 195
4.2 Oguchi’s solution......Page 199
4.3 Radar range equation with transmission matrix: linear polarization basis......Page 204
4.3.1 Uniform oblate raindrops (zero canting angle)......Page 206
4.3.2 General form of S with a diagonal transmission matrix......Page 207
4.4 Radar range equation with transmission matrix: circular polarization basis......Page 212
4.5 Transmission-modified covariance matrix......Page 220
4.5.1 Transmission-modified covariance matrix: circular basis......Page 223
4.6 Relation between linear and circular radar observables in the presence of propagation effects......Page 227
4.7 Measurements in a "hybrid" basis......Page 232
4.7.1 Errors caused by non-diagonal S......Page 234
4.7.2 Error in Z due to a non-zero mean canting angle......Page 235
Notes......Page 237
5.1 Review of signals and systems......Page 239
5.1.1 Fourier transform......Page 240
(iii) Frequency shifting......Page 241
5.1.3 Transmission of signals through linear systems......Page 243
5.1.4 Gaussian-shaped pulse and filters......Page 244
5.2 Received signal from precipitation......Page 245
5.3 Mean power of the received signal......Page 250
5.4 Coherency matrix measurements......Page 261
5.5.1 Range–time autocorrelation......Page 263
5.5.2 Sample–time autocorrelation due to transmit pulse train......Page 266
5.6 Spaced-time, spaced-frequency coherency function......Page 271
5.7 Sampling the received signal......Page 274
5.7.1 Discrete time processing......Page 277
5.7.2 Power spectral density......Page 280
5.8 Noise in radar systems......Page 285
5.9 Statistical properties of the received signal......Page 290
5.9.1 Probability distribution functions......Page 292
5.9.2 Joint distribution of dual-polarized signals......Page 295
5.10 Estimation of mean power......Page 299
5.11 Doppler spectrum (or power spectral density) and estimate of mean velocity......Page 302
5.11.1 Mean of the Doppler velocity spectrum......Page 304
5.11.2 Mean velocity estimation from sampled signals......Page 305
5.11.3 Pulse pair estimate......Page 307
5.11.4 Estimation of Sigma......Page 309
(i) Averaging......Page 311
(ii) Windowing......Page 312
(i) Maximum likelihood spectral estimation......Page 313
(ii) Maximum entropy spectral estimation......Page 314
5.12 Example of received signal statistics and spectral estimation......Page 315
Notes......Page 321
6.1 General system aspects......Page 322
6.1.1 Polarization-agile/single-receiver system......Page 326
6.1.2 Polarization diversity systems......Page 333
6.1.3 Polarization-agile/dual-receiver systems......Page 337
6.2 Antenna performance characteristics......Page 345
6.2.1 Evaluation of antenna performance assuming spherical particles fill the beam......Page 348
6.2.2 Formulation of radar observables in the presence of system polarization errors......Page 353
6.3 Radar calibration......Page 360
6.3.1 Absolute calibration using a metal sphere......Page 362
6.3.2 Polarization-agile/single-receiver radar......Page 363
6.3.3 Polarization diversity systems......Page 367
6.3.4 Polarization-agile/dual-receiver systems......Page 368
6.4.1 Covariance matrix of the received signal vector......Page 370
6.4.2 Estimation of the covariance matrix from signal samples......Page 371
6.4.3 Estimation of covariance matrix elements under alternate polarization mode......Page 373
6.4.4 Estimation of covariance matrix elements under periodic block pulsing mode......Page 377
6.4.5 Estimators of covariance matrix elements in hybrid mode......Page 380
(i) Variance of mean power estimates......Page 381
(ii) Variance of Z......Page 382
(iii) Variance of Psi......Page 384
(iv) Variance of Upsilon......Page 385
(v) Variance of |Rhoco|......Page 386
(vi) Variance of LDR......Page 387
(vii) Variance of…......Page 388
(i) Variance of mean ower and Z estimates......Page 389
(ii) Variance in the estimates of Psi and |Rhoco|......Page 390
6.6 Estimation of specific differential phase (K)......Page 396
6.6.1 Smoothing the range profile of Psi......Page 397
6.6.2 Regression-based estimates of K......Page 402
Notes......Page 404
7 The polarimetric basis for characterizing precipitation......Page 406
7.1.1 The equilibrium shape and orientation of raindrops: implications for Z and K......Page 407
7.1.2 Raindrop oscillations: implications for Z and K......Page 418
7.1.3 Other polarimetric observables......Page 428
7.1.4 Raindrop size distributions and simulated radar parameters......Page 434
7.1.5 Vertical structure of Z and K......Page 451
7.2 Convective precipitation......Page 454
7.2.1 Ordinary convective storms......Page 457
7.2.2 Hailstorms......Page 472
7.3 Stratiform precipitation......Page 501
7.4 The estimation of attenuation and differential attenuation in rain using Phi......Page 518
7.4.1 Simple attenuation correction of reflectivity and Z......Page 520
(i) Z constraint......Page 525
(ii) Phi constraint......Page 527
(iii) Combined Phi–Z constraint......Page 534
7.5 Hydrometeor classification......Page 541
7.5.1 Empirical algorithms for classifying hydrometeor types......Page 543
7.5.2 Fuzzy logic classification......Page 544
(i) Configuration of a general fuzzy logic system......Page 545
(ii) Architecture of a fuzzy hydrometeor classifier......Page 547
(iii) Classification procedure......Page 548
(iv) Training the fuzzy hydrometeor type classifier......Page 554
(v) Examples o FHC use......Page 555
Notes......Page 560
8.1 Physically based parametric rain rate estimation algorithms......Page 562
8.1.1 R(Z, Z) algorithm......Page 565
8.1.3 R(K, Z) algorithm......Page 567
8.1.4 Example of polarimetric radar–raingage comparison......Page 568
8.2.2 W(K) algorithm......Page 572
8.3 Error structure and practical issues related to rain rate algorithms using Z, Z, and K......Page 573
8.3.1 Errors in Z......Page 574
8.3.2 Error in estimate of K......Page 576
8.3.3 Errors in Z......Page 578
8.4.1 Area–time integral method......Page 582
8.4.2 Probability matching method......Page 585
8.5 Neural-network-based radar estimation of rainfall......Page 587
8.5.1 Input data to neural network......Page 591
8.5.2 Training the network......Page 592
8.6 Some general comments on radar rainfall estimation......Page 595
Notes......Page 597
Appendix 1 Review of electrostatics......Page 598
A1.1 Permanently polarized sphere......Page 606
A1.2 Spherical cavity......Page 608
A1.3 Dielectric sphere in a uniform incident field......Page 609
Appendix 2 Review of vector spherical harmonics and multipole expansion of the electromagnetic field......Page 613
Appendix 3 T-matrix method......Page 619
Appendix 4 Solution for the transmission matrix......Page 623
A5.1 Variance of R[l]......Page 627
A5.2 Variance of R[l]......Page 628
A5.3 Variance of arg R[l]......Page 629
A5.4 Variance of magnitude of Rhoxy[l]......Page 630
A5.5 Variance of estimates under periodic block pulsing scheme......Page 632
(i) Variance of…......Page 633
(iii) Variance of Psi and |Rhoco|......Page 634
References......Page 635
Index......Page 657