ASTRONOMICAL SOCIETY OF THE PACIFIC CONFERENCE SERIES Volume 82 VERY LONG BASELINE INTERFEROMETRY AND THE VLBA Proceedings of a Summer School held in Socorro, New Mexico 23-30 June 1993 NRAO Workshop No. 22 Edited by J. A. Zensus, P. J. Diamond, and P. J. Napier Vll
Very Long Baseline Interferometry and the VLBA ASP Conference Series, Vol. 82, 1995 J. A. Zensus, P. J. Diamond, and P. J. Napier (eds.) Contents FIGURES TABLES PREFACE DEDICATION CONTRIBUTORS xxi xxv xxvii xxix xxxi I Basic Theory 1 1 Theory 3 B. G. Clark 1.1 Introduction Basis and Textbooks 3 1.2 Electromagnetic Field 4 1.3 The Spatial Coherence Function of the Field 5 1.4 Synthesis Imaging 6 1.5 Spectroscopy 8 1.6 Polarization Imaging 9 1.7 Design of Radio Interferometers 10 1.8 Element Gain 12 1.9 Calibrating Element Gains 13 1.10 Living with Uncalibrated Gains 14 1.11 Conclusion 16 2 Theory of Correlation 17 J. D. Romney 2.1 Introduction 18 2.2 Interfaces 18 2.3 Background 19 2.3.1 Correlation and Coherence 19 2.3.2 VLBI Observations 21 2.4 The Correlator Frontend 22 2.4.1 Delay Compensation 23 2.4.2 Phase Compensation 23 2.5 The Lag Correlator 23 2.5.1 Fractional-Sample Delay 25 ix
CONTENTS 2.5.2 Phase Rotator Implementation 26 2.5.3 Spectral Response 27 2.6 The FX Correlator 27 2.6.1 Fractional-Sample Delay 29 2.6.2 Phase Rotator Implementation 29 2.6.3 Spectral Response 30 2.6.4 Segmentation 31 2.6.5 Miscellaneous FX Topics 33 2.7 FX and Lag Correlator Intercomparison 34 2.7.1 Cost 35 2.7.2 Summary of Other Advantages 36 Imaging Concepts 39 T. Cornwell 3.1 Introduction 39 3.2 Fourier Inversion: Theory 40 3.3 The Problem with /^-Sidelobes 41 3.4 Computation 42 3.4.1 The Convolution Equation 44 3.4.2 The CLEAN Algorithm 45 3.4.3 The Maximum Entropy Algorithm 46 3.5 Comparison of the CLEAN and MEM Algorithms 47 3.5.1 Non-Negative Least-squares 49 3.6 Origin of Phase Errors 50 3.7 Phase Closure 51 3.8 Self-calibration 53 3.9 Comparison of Phase Closure and Self-calibration 54 II The Very Long Baseline Array 57 4 VLBA Design 59 P. J. Napier 4.1 Introduction 59 4.2 VLBA Design Goals 60 4.2.1 Imaging with High Resolution and High Dynamic Range. 60 4.2.2 High Sensitivity 62 4.2.3 Spectral Coverage From 300 MHz To 45 GHz 62 4.2.4 Spectral Line Imaging Capability 63 4.2.5 Support of Research in Geodesy and Astrometry 63 4.2.6 User-Friendly and Reliable Operation 64 4.2.7 Full-Time Operation with Low Down Time 64 4.2.8 Compatibility with Other VLBI Antennas 65 4.3 Antennas, Optics, and Feeds 66
CONTENTS. xi 4.3.1 VLBA Sites 66 4.3.2 Antennas 67 4.3.3 Feed System 68 4.4 Conclusion 72 5 VLBA: Antenna to Data Formatter 73 A. R. Thompson 5.1 Introduction 73 5.2 Feeds and Polarizers 75 5.3 Simultaneous, Multi-Frequency Observations 76 5.4 Front Ends for Bands Below 1 GHz 76 5.5 Front Ends for Bands Above 1 GHz 77 5.6 IF and Baseband Signals in the Station Building 80 5.7 Digital Sampling 82 5.8 Local Oscillator and Timing System 83 5.9 Calibration System 87 5.10 Stability and Accuracy of Measurements 88 5.11 Observing Bands and Radio Interference 90 6 VLBA: Formatter to Tape 93 A. E. E. R. Rogers 6.1 Introduction 94 6.1.1 Why Magnetic Tape? 94 6.1.2 VLBI Recording Systems 94 6.2 The VLBA Data Format 97 6.2.1 Longitudinal Format 97 6.2.2 Transverse Track Format 98 6.2.3 Data and Non-Data Replacement 98 6.2.4 Track Width and Track Structure 98 6.3 The VLBA Formatter 99 6.3.1 Digital Switch Board 100 6.3.2 Header Control Board 101 6.3.3 Transport Driver Board 101 6.3.4 Timing and Control Board 102 6.3.5 Data Buffer Board 102 6.3.6 Quality Analysis Board 102 6.4 Magnetic Recording Fundamentals 103 6.4.1 Recording Losses 103 6.4.2 Wavelength Response 104 6.4.3 Head-to-Tape Contact and Tribology 104 6.5 The VLBA Recorder 105 6.5.1 The Tape Transport 105 6.5.2 The Headstack 105 6.5.3 The Headstack Positioner 106
xii CONTENTS 6.5.4 Control Electronics 106 6.5.5 Signal Electronics and Eye Pattern 107 6.6 Recorder Characteristics and Limitations 108 6.6.1 Error Rate Versus Bit Density 108 6.6.2 Tape Thickness-Interchange 108 6.6.3 Head Flying 109 6.6.4 Overwrite 109 6.6.5 Tracking Errors 109 6.6.6 Headlife and Humidity 110 6.6.7 Thin Tape and Edge Damage by Frictional Heating... Ill 6.6.8 Tape Reel and Shipping Canister Ill 6.7 Future VLBI Recording Systems 112 6.7.1 Expansion Paths for the VLBA Formatter 112 6.7.2 Recorder Improvements 112 6.7.2.1 Thinner Tape 112 6.7.2.2 Higher Data Rates 113 6.7.2.3 Next Generation of Heads 113 6.7.3 Mark IV 113 6.8 Summary and Conclusfons 113 7 The VLBA Correlator 117 J. M. Benson 7.1 Introduction 117 7.2 Design Specification and History 118 7.3 Technical Description 119 7.3.1 The Playback Interface 121 7.3.1.1 The Track Recovery Card 121 7.3.1.2 The Deformatter 121 7.3.2 The Cross-Bar Switch 122 7.3.3 The FFT Engines 123 7.3.4 The Cross-Multipliers 126 7.3.5 The Correlator Backend 126 7.4 The Correlator Computers and Software 127 7.4.1 Software Architecture 127 7.4.1.1 Database Management System 127 7.4.1.2 Correlator Job Scheduling 128 7.4.1.3 Data Archiving and Distribution 128 7.5 The Correlator Model 128 7.5.1 Model Calculation 129 7.5.2 The Model Distribution 129 7.5.3 Model Accountability 130 7.5.4 The Pulsar Gate 130
CONTENTS xiii What the VLBA Can Do For You 133 R. C. Walker 8.1 Basic Capabilities of the VLBA 133 8.1.1 Special Capabilities for Imaging 135 8.1.2 Special Capabilities for Geodesy and Astrometry 137 8.1.3 Operational Advantages of the VLBA 138 8.2 Frequency Coverage 138 8.2.1 Frequency Resolution 140 8.3 (u, v) Coverage and Resolution 140 8.4 Sensitivity 145 8.4.1 System Temperature and Antenna Temperature 145 8.4.2 Sensitivity of a Single-Baseline Interferometer 147 8.4.3 Instrumental Losses 149 8.4.4 Image Sensitivity 152 8.5 Image Quality and Dynamic Range 154 8.5.1 Sample VLBA Images 156 III VLBI Data Analysis 159 9 Calibration techniques for VLBI 161 J. M. Moran and V. Dhawan 9.1 Introduction 161 9.2 Delay Calibration 165 9.3 Fringe-Rate Calibration 165 9.4 Amplitude Calibration 166 9.4.1 Antenna Calibration 166 9.4.2 Atmospheric Opacity 168 9.5 Phase Calibration 173 9.6 Coherent and Incoherent Averaging 173 9.6.1 Atmospheric Effects on Phase and Delay 178 9.7 Phased Arrays as VLBI Elements 181 9.8 The Impact of Calibration Errors in Images 183 9.9 Summary 186 10 Fringe Fitting 189 W. D. Cotton 10.1 Fringe Fitting Theory 190 10.1.1 Correlator Model Delay Errors 190 10.1.2 Fringe Fitting Techniques 191 10.1.2.1 Baseline 191 10.1.2.2 Baseline with Closure Constraints 191 10.1.2.3 Global 192 10.1.3 Solution Interval 192
xiv CONTENTS 10.1.4 Calibration Sources 193 10.1.5 Source Structure 193 10.1.6 Phase Referencing 193 10.1.7 Multi-band Data 194 10.1.7.1 Phase-Cals 194 10.1.7.2 Multi- vs. Single-band Delay 194 10.1.7.3 Sidebands 195 10.1.8 Filtering 195 10.1.8.1 Establishing a Common Reference Antenna... 195 10.1.8.2 Smoothing and Interpolating Solutions 195 10.1.9 Bandwidth Synthesis 196 lo.l.loweights 196 10.1.11 Polarization 197 10.2 Fringe Fitting Practice 197 10.2.1 Phase Slopes in Time and Frequency 197 10.2.2 Phase-Cals 198 10.2.3 Sidebands 199 10.2.4 Delay and Rate Fits 199 10.2.4.1 Signal-tg-Noise Ratios 199 10.2.4.2 Delay and Rate Windows 200 10.2.4.3 Details of Global Fringe Fitting 200 10.2.4.4 Multi- and Single-band Delays 202 10.2.4.5 Phase-Cal Errors 203 10.2.5 Calibrator Sources 203 10.2.6 Solution Interval 204 10.2.7 Weights 204 10.2.8 Source Model 205 10.2.9 Suggested Procedure 205 10.3 Bandwidth Synthesis 207 11 Spectral-Line VLBI 209 Mark J. Reid 11.1 Spectral Line VLBI Sources 209 11.2 Basic Concepts 210 11.3 Spectral-Line Calibration 214 11.3.1 Instrumental Parameters 214 11.3.2 Doppler Tracking 215 11.3.3 Transforming from Delays to Frequencies 216 11.3.4 Spectral Line Amplitude Calibration 217 11.3.5 Clock and Coordinate Corrections 219 11.3.6 Electronic Phase Shifts 220 11.3.7 Phase Referencing 221 11.4 Fringe-Rate Mapping 223
CONTENTS XV 12 VLBI Data Reduction in Practice 227 P. J. Diamond 12.1 Introduction.... ' 227 12.2 Calibration Philosophy 228 12.3 AIPS Calibration Tables and Data Structure 228 12.4 The Data Processing Path 230 12.4.1 Reading the Data 230 12.4.2 Data Examination 231 12.4.3 Lining up the Phases 234 12.4.4 Bandpass Calibration 235 12.4.5 Amplitude Calibration 236 12.4.6 Fringe Fitting 239 12.4.7 The Final Product 245 12.5 Conclusion 245 13 Practical VLBI Imaging 247 R. C. Walker 13.1 Introduction 247 13.2 Editing and Calibration 249 13.3 Controlling the Hybrid Mapping 251 13.3.1 Self-Calibration Solution Mode 253 13.3.2 Windowing 253 13.3.3 Weights 254 13.3.4 (u,v) Ranges and Tapers 255 13.3.5 Natural vs. Uniform Weighting 256 13.3.6 Cell Size and Image Size 256 13.3.7 CLEAN Components 257 13.3.8 Self-Calibration Parameters 258 13.3.9 Imaging Parameters 259 13.3.10Antenna Subsets 260 13.4 An Example 260 13.5 Difference Mapping 264 13.6 Advanced Techniques 265 14 Non-Imaging Data Analysis 267 T. J. Pearson 14.1 Introduction 268 14.2 Visibility Data 268 14.2.1 Sampling 268 14.2.2 Closure Quantities 269 14.2.3 Non-Fourier Imaging 270 14.2.4 Noise 270 14.3 Model Fitting 270 14.3.1 Imaging as an Inverse Process 270
xvi CONTENTS 14.3.2 Uses of Model Fitting 271 14.4 Practical Model Fitting 272 14.4.1 Simple Models 272 14.4.2 Inspecting Visibility Data 275 14.4.3 Programs for Model Fitting 275 14.5 Least-squares Fitting Algorithms 277 14.5.1 Non-Linear Least Squares Algorithms 280 14.5.2 Error Estimation 281 14.6 Comparison of Observations 282 A Properties of the Fourier Transform 284 A.I Addition Theorem 285 A.2 Convolution 285 A.3 Shift Theorem 285 A.4 Similarity Theorem 285 B Simple Models 285 B.I Delta Function (Point Source) 285 B.2 Gaussian 286 B.3 Uniformly Bright Disk 286 B.4 Optically Thin Sphere 286 B.5 Ring '. 286 IV Advanced VLBI Topics 287 15 Polarimetry 289 W. D. Cotton 15.1 Introduction 290 15.2 Instrumental Response 290 15.2.1 Interaction with Total Intensity Calibration 291 15.2.1.1 Phase 292 15.2.1.2 Delays and Rates 292 15.2.2 Ellipticity-Orientation Model 293 15.2.3 Leakage-Term Model 294 15.2.4 Polarization Angle 295 15.3 Phase Calibration 295 15.3.1 Linearly Polarized Feeds 295 15.3.2 Circularly Polarized Feeds 295 15.3.3 Phase Rereferencing 296 15.3.4 Phase-Cals 296 15.3.5 Fringe Fitting 297 15.3.5.1 Parallel-Hand Fringe Fits 297 15.3.5.2 Right-left Delay Calibration 297 15.3.6 Right-Left Coherence 298 15.3.7 Ionospheric Faraday Rotation 299
CONTENTS xvii 15.3.7.1 External Faraday Rotation Calibration 299 15.3.7.2 Faraday Self-Calibration 299 15.4 Instrumental Polarization Calibration 300 15.4.1 Calibrator Polarization Model 301 15.4.2 Fitting A Feed Model 302 15.4.3 Iterative Calibration 302 15.5 Polarization Angle Calibration 302 15.5.1 Unresolved Calibrator 303 15.5.2 Resolved Calibrator 303 15.6 Imaging 303 15.6.1 Asymmetric (u, v) Coverage 304 15.6.2 Complex Deconvolution 304 15.7 Spectro-Polarimetry 304 15.8 Suggested Procedure 306 16 Multi-Frequency Synthesis 309 J. E. Conway and R. J. Sault 16.1 Introduction 310 16.2 Image Fidelity 310 16.3 Multi-Frequency Synthesis 310 16.4 Spectral Effects 312 16.4.1 The Spectral Expansion 312 16.4.2 Spectral Dirty Beams 313 16.4.3 First Order Spectral Errors 313 16.4.4 Second Order Spectral Errors 315 16.5 The MFS Deconvolution Problem 317 16.5.1 Nature of The Problem 317 16.5.2 Map and Stack 318 16.5.3 Direct Assault 318 16.5.4 Data Weighting Methods 318 16.5.5 Double Deconvolution 319 16.5.6 The Sault Algorithm 321 16.6 Multi-Frequency Self-Calibration 322 16.7 Practical MFS 323 16.8 Conclusions 324 17 VLBI Phase-Referencing 327 A. J. Beasley and J. E. Conway 17.1 VLBI Phase Errors 328 17.1.1 Geometric Errors 328 17.1.2 Instrumental Errors 330 17.1.3 Neutral Atmosphere Effects 330 17.1.4 Ionospheric Effects 331 17.2 Phase-Referencing 333
xviii CONTENTS 17.3 Imaging 336 17.3.1 Residual Phase Errors 336 17.3.2 A Priori Phase Calibration 336 17.3.3 Temporal Phase Variations and Switching Time 336 17.3.4 Angular Phase Variations and Switching Angle 337 17.3.5 Calibrator Position Errors 338 17.3.6 Hybrid Self-Calibration Imaging 339 17.4 Relative Astrometry 340 17.5 Prospects For Phase-Referencing 340 17.6 Summary and Conclusions 341 18 Geodesy 345 D. B. Shaffer 18.1 A Bit of History 345 18.2 Size of Geodetic and Astrometric Effects 346 18.3 Phase and Group Delays 347 18.4 Observing Techniques and Calibrations 351 18.5 Analysis 355 18.6 Results 360 18.7 Literature 360 19 Astrometry 363 E. Fomalont 19.1 Types of Astrometry 364 19.1.1 Absolute Astrometry 364 19.1.2 Relative Astrometry 365 19.2 The Basic Array Measurement 365 19.2.1 The Total Phase Delay 365 19.2.2 The Model Phase Delay 366 19.2.3 Phase (Lobe) Ambiguities 367 19.3 Absolute Astrometry 368 19.3.1 The Fundamental Astrometric Equation 369 19.3.2 Multi-element Arrays 371 19.3.3 Common Non-dispersive Phase-Delay Terms 371 19.3.3.1 Tropospheric Refraction 372 19.3.3.2 Clock Errors 373 19.3.3.3 Instrumental Path Lengths 373 19.3.4 Common Dispersive Phase Delay Terms 373 19.3.4.1 Ionospheric Phase Delay 373 19.3.4.2 Visibility Phase 374 19.3.4.3 Phase Changes 374 19.3.5 Multi-Frequency Observations 374 19.3.6 Reference Frames and the Motion of the Earth 375 19.3.6.1 Celestial Coordinate Frame 375
CONTENTS xix 19.3.6.2 Terrestrial Coordinate Frame 376 19.3.6.3 Earth Orientation 376 19.3.6.4 Polar Motion 376 19.3.6.5 Time and Earth Rotation 377 19.3.6.6 Plate Tectonics 377 19.3.6.7 Earth Orbital Parameters 377 19.4 VLBI Observations 378 19.4.1 Phase Derivatives with Time and Frequency 378 19.4.2 Solution Using the Phase Rate 379 19.4.3 Solution Using the Group Delay 380 19.4.4 Measuring the Group Delay 380 19.4.5 Measuring the Ionosphere Delay 382 19.4.6 Relative Accuracies of Phases and Delays 383 19.4.7 Analyzing VLBI Data: SOLVE 384 19.5 Relative Astrometry 388 19.5.1 Nodding Observations 388 19.5.2 Groups of Calibrators 390 19.6 Nodding Errors and Supportive Astrometric Observations... 391 19.7 Simultaneous Observations of Source and Calibrators 391 V Other Networks and Global VLBI 395 20 EVN and Global VLBI 397 R. T. Schilizzi 20.1 Introduction 397 20.2 The World of VLBI 398 20.3 The European VLBI Network 399 20.4 (u, v) Coverage for the EVN Alone 400 20.5 (u, v) Coverage for the EVN+VLBA 401 20.6 EVN Sensitivity 401 20.6.1 Current Performance. 401 20.6.2 Expected Performance by 1997 401 20.7 EVN Upgrade 404 20.7.1 Data Processor 406 20.7.2 Correlator 406 20.7.3 User and Technical Support 407 20.8 The Asia-Pacific Telescope 407 20.9 Final Remarks 408
xx CONTENTS VI Planning a VLBI Observation 409 21 VLBI Observing Strategies 411 J. M. Wrobel 21.1 Select a Target 411 21.2 Desired Angular Resolution 412 21.3 Desired Largest Angular Scale 413 21.4 Desired Observing Frequency 413 21.5 Desired Stokes Parameters 416 21.6 Desired Recorded Bandwidth 416 21.7 Select and Evaluate a Trial VLBI Array 417 21.7.1 Special Needs Evaluation 418 21.7.2 Geometric Evaluation 418 21.7.3 Baseline Sensitivity Evaluation 419 21.7.4 Image Sensitivity Evaluation 421 21.7.5 Field-of-View Evaluation 422 21.8 Optimal Times, Seasons, or Years to Observe? 424 21.9 Submit an Observing Proposal 424 21.10Make Antenna and Tape Control Schedule Files 424 21.11Correlator Tolerance on» Target Positional Errors 425 ABBREVIATIONS 429 BIBLIOGRAPHY 431 INDEX 442 COLOPHON 453