ASTRONOMICAL SOCIETY OF THE PACIFICCONFERENCE SERIES

Volume 180

SYNTHESIS IMAGING IN RADIO ASTRONOMY II

A Collection of Lectures from theSixth NRAO/NMIMT Synthesis Imaging Summer School

held atSocorro, New Mexico, USA

17-23 June, 1998

Edited byG. B. Taylor, C. L. Carilli, and R. A. Perley

National Radio Astronomy Observatory, Socorro, New Mexico, USA

Synthesis Imaging in Radio Astronomy IIASP Conference Series, Vol. 180, 1999G. B. Taylor, C. L. Carilli, and R. A. Perley (eds.)

Contents

PREFACE xxxiDEDICATION xxxv

Lecture 1. Coherence in Radio Astronomy by B. G. Clark1. Introduction 12. Form of the Observed Electric Field 13. Spatial Coherence Function of the Field 44. The Basic Fourier Inversions of Synthesis Imaging 4

4.1. Measurements confined to a plane 54.2. All sources in a small region of sky 5.4.3. Effect of discrete sampling 64.4. Effect of the element reception pattern . . 6

5. Extensions to the Basic Theory 85.1. Spectroscopy 85.2. Polarimetry 9

Lecture 2. Fundamentals of Radio Interferometry by A. R. Thompson1. Introduction > 112. Response of an Interferometer i 123. Effect of Bandwidth in a Two-Element Interferometer 154. Delay Tracking and Frequency Conversion 175. Fringe Rotation and Complex Correlators 186. Phase Switching 207. Coordinate Systems for Imaging 218. Antenna Spacings and (u,v,w) Components 259. Astronomical Data from Interferometer Observations 2710. Design of Synthesis Arrays . . . 2711. The Effect of Bandwidth in Radio Images 2912. The Effect of Visibility Averaging 34

Lecture 3. The Primary Antenna Elements by P. J. Napier1. Introduction 372. Basic Antenna Formulas ' 373. General Antenna Types ' 41

3.1. Wavelength range 413.2. Types of reflector antennas 42

xv

xvi CONTENTS

4. Antenna Performance Parameters 444.1. Aperture efficiency 444.2. Pointing accuracy 514.3. Antenna polarization properties 53

Lecture 4. Cross Correlators by J. D. Romney1. Introduction 572. Interfaces 573. Background 58

3.1. Correlation and Coherence 583.2. Observations 60

4. The Correlator Frontend 604.1. Delay Compensation 614.2. Phase Compensation 61

5. The Lag Correlator 625.1. Lag Correlator: Fractional-Sample Delay 635.2. Lag Correlator: Phase Rotator Implementation 655.3. Lag Correlator: Spectral Response 66

6. The FX Correlator 676.1. FX Correlator: Fractional-Sample Delay 696.2. FX Correlator: Phase Rotator Implementation 696.3. FX Correlator: Spectral Response 706.4. FX Correlator: Segmentation 716.5. Miscellaneous FX Topics 74

7. FX and Lag Correlator Intercomparison 757.1. Cost 767.2. Summary of Other Advantages : . . 78

Lecture 5. Calibration and Editing by E. B. Fomalont & R. A. Perley1. Introduction. 792. Basic Considerations 80

2.1. Synthesis equation and phase delay 802.2. Calibration formalism 812.3. Editing 822.4. Calibration methods 822.5. Calibration of amplitude and phase 832.6. Multi-frequency and dual-polarization capability 83

3. Initial Calibrations 833.1. Antenna pointing and gain 843.2. Delay calibration 853.3. Time and place 863.4. Amplitude check 87

4. Routine Corrections 874.1. Special array systems 884.2. Externally determined parameters 884.3. Path length changes in the troposphere and ionosphere . 894.4. Absorption by the troposphere and ionosphere 91

CONTENTS xvii

5. Bandpass Calibration 926. Editing 93

6.1. External monitoring 936.2. Scan consistency 946.3. Data displays 946.4. Shadowing 956.5. Long-term consistency 95

7. Final Calibration Using Radio Sources 957.1. Calibrator source properties 967.2. The offset term 967.3. Averaging of calibration observations 977.4. Baseline-based calibration 977.5. Antenna-based solutions for amplitude and phase 987.6. Imperfect calibrators 997.7. Directional calibration—astrometry 997.8. Checking the calibration: closure errors 1007.9. Checking the calibration: amplitude stability 1017.10. Checking the calibration: phase stability 1027.11. Calibrator choice 103

8. Application of the Calibration Values 1038.1. Gain tables 1038.2. Interpolation within the gain table 104 •8.3. Large phase-fluctuations 104

9. Polarization Calibration 1059.1. Stokes parameters 1059.2. Leakage terms 1059.3. Phase difference between polarization channels 106

10. Additional Topics 10710.1. The antenna primary beam ;10710.2. Bandwidth smearing 10810.3. Dual-frequency ionospheric calibration 10910.4. Relative vs. absolute calibration 109

Lecture 6. Polarization in Interferometry by W. D. Cotton1. Introduction I l l

1.1. Polarization Concepts I l l1.2. Polarization Measurements Using an Interferometer . . . 1131.3. Errors and Calibration 113

2. Instrumental Response to a Polarized Signal 1142.1. Feed Response * 1142.2. Interferometer Response 115

3. Source and Instrumental Polarization 1164. On-axis Calibration Strategies 118

4.1. Interaction between Polarization and Total Intensity . . . 1184.2. Circular Feeds 1184.3. Linear Feeds 120

5. Wide Field Polarization 121

xyiii CONTENTS

6. Ionospheric Faraday Rotation 1226.1. External Calibration 1236.2. Self Calibration 123

7. Imaging 124

Lecture 7. Imaging by D. S. Briggs, F. R. Schwab, & R. A. Sramek1. Fourier Transform Imaging 127

1.1. The 'direct Fourier transform' and the FFT 1282. The Sampling Function, and Weighting the Visibility Data . . . 129

2.1. The sampling function 1292.2. Weighting functions for control of the beam shape . . . . 130

3. Gridding the Visibility Data 1343.1. Gridding by convolution 1353.2. Aliasing 1423.3. Choice of a gridding convolution function 142

4. Additional Topics 1454.1. Translating, rotating, and stretching images 1454.2. Practical details of implementation 1474.3. Non-coplanar baselines 1474.4. The Problem with ID — Sidelobes 148

Lecture 8. Deconvolution by T. J. Cornwell, R. Braun, & D. S. Briggs1. Deconvolution 151

1.1. The "principal solution" and "invisible distributions" . . 1531.2. Problems with the principal solution 153

2. The CLEAN Algorithm 1542.1. The Hogbom algorithm 1542.2. The Clark algorithm 1552.3. The Cotton-Schwab algorithm 1552.4. Other related algorithms 1562.5. Practical Details and Problems of CLEAN Usage 1562.6. The use of boxes 1572.7. Number of iterations, loop gain and the beam patch size . 1582.8. The problem of short spacings 1592.9. The CLEAN beam 1592.10. Use of a priori models 1612.11. Non-uniqueness 1612.12. Instabilities 161

3. The Maximum Entropy Method (MEM) 1624. Practical Details of the Use of MEM 163

4.1. The default image (prior distribution) 1634.2. Total flux density 1634.3. Varying resolution 1634.4. Bias 1644.5. Point sources in extended emission 164

5. Comparison of CLEAN and MEM 1646. Example 166

CONTENTS xix

7. Other Methods, Including Hybrids 166

Lecture 9. Sensitivity by J. M. Wrobel & R. C. Walker1. What is Sensitivity? ". 1712. What Are Antenna Performance Measures? 1713. What is the Sensitivity of an Interferometer? 172

3.1. Real and Complex Correlators 1723.2. Amplitudes and Phases 176

4. What is the Sensitivity of a Synthesis Image? 1774.1. Derivation 1774.2. Example: Stokes / 1804.3. Example: Stokes Q and U 1814.4. Brightness Temperature Regimes of Arrays 1824.5. Factors Degrading Image Sensitivity 183

5. Summary 185

Lecture 10. Self-Calibration by T. J. Cornwell & E. B. Fomalont1. Problems with Ordinary Calibration 1872. Redundant Calibration and Self-Calibration 188

2.1. Redundant calibration 1892.2. Self-calibration 1892.3. Redundant calibration or self-calibration? 191

3. Other Approaches to Phase Correction 1913.1. Closure quantities 1923.2. Adaptive optics 193

4. Why Does Self-Calibration Work? 1945. Practical Problems in Self-Calibration 194

5.1. Specifying the model 1955.2. Type of solution and weighting schemes 1955.3. Self-calibration averaging time 1965.4. Schwab's l\ and I2 solutions 1965.5. Spectral line self-calibration 1975.6. Spurious symmetrization 1975.7. Non-convergence and non-uniqueness 1975.8. Baseline-related effects . . . 198

6. Bibliography 199

Lecture 11. Spectral-Line Observing I: Introductionby D. J. Westpfahl

1. Introduction : 2011.1. The Goal of This Lecture 2011.2. The Assumptions of this Lecture 202

2. Spectral-Line Science 2032.1. Why Use Spectral-Line Mode? 2032.2. Important Centimeter-Wave Spectral Lines 204

3. Learn About the Object You Wish to Study 2064. Choosing an Interferometer Configuration 207

xx CONTENTS

5. Choosing an Observing Frequency 2075.1. Velocity Reference Frames 2085.2. The Relationship Between Frequency and Velocity . . . . 209

6. Choosing a Configuration of the Spectroscopic Correlator . . . . 2106.1. Using Lags to Define a Fourier Transform Relationship

Between Time and Frequency 2106.2. Sampling 2126.3. The Gibbs Phenomenon and Hanning Smoothing 2136.4. Channel Separation versus Channel Bandwidth 2136.5. An Example, the VLA Correlator 215

7. Sensitivity and Calibration 2177.1. Sensitivity 2177.2. Calibration 218

8. Data Reduction and Analysis 2198.1. Data Cubes 2198.2. Continuum Subtraction 2198.3. Data Analysis 220

9. Examples of Spectral-Line Data Sets 2219.1. HI Emission 2219.2. HI Absorption 2229.3. An OH Maser 224

10. Conclusion 225

Lecture 12. Spectral Line Observing II: Calibration and Analysisby M. P. Rupen

1. Introduction , 2292. Calibrating the Bandpass 229

2.1. Why is the Bandpass Not Flat? 2292.2. Splitting the Time and Frequency Dependence of the Gain 2312.3. Determining the Bandpass 2322.4. Checking and Using the Bandpass 2342.5. Complications and Subtleties 2352.6. The Bandpass and Continuum Observations 236

3. Gibbs ringing 2374. Special calibration topics 240

4.1. Self-calibration 2404.2. Strong and Ubiquitous Lines 2404.3. Joining Adjacent Frequency Bands/IFs 2404.4. Frequency, Velocity, and Doppler Tracking 241

5. Chromatic Aberration (Bandwidth Smearing) 2415.1. Single vs. Double Sideband 242

6. Continuum Subtraction 2426.1. IMAVG 2436.2. UVSUB 2436.3. UVLIN 2456.4. IMLIN 2496.5. Errors in Continuum Subtraction 249

CONTENTS xxi

7. Flagging 2508. Data Quality and the Spectral Dynamic Range 2519. Deconvolution 25210. Looking at the Images 25711. Common Analysis Problems 262

11.1. Distribution, Velocity Field, and Line Width 26211.2. Detection Experiments 26811.3. Beam Smearing 26811.4. Lines Which Track the Continuum 27011.5. Comparing Different Lines 271

12. Outstanding Problems 271

Lecture 13. High Dynamic Range Imaging by R. A. Perky1. Introduction 2752. Image Fidelity vs. Image Dynamic Range 2753. The Effects of Visibility Errors on Image Dynamic Range . . . . 2774. Origin of Residual Errors for the VLA 281

4.1. Atmospheric phase errors 2814.2. (u, v) coverage errors 2814.3. 'Closure errors' 2824.4. Quantization correction ('Van Vleck correction') 2824.5. Polarization leakage 284 •4.6. Loss due to phase winding 2854.7. Noise introduced through self-calibration . 2864.8. Effects of digital representation of models 2864.9. Calculation errors 2884.10. Coverage errors 288

5. Techniques of Error Correction 2885.1. Initial editing and calibration 2895.2. Antenna-based error correction using self-calibration . . . 2895.3. Baseline-based error correction 2945.4. Coverage errors 298

Lecture 14. Image Analysis by E. B. Fomalont1. Image Modification .< 301

1.1. Smoothing or sharpening an image 3011.2. Interpolating an image . 3031.3. Primary beam correction 3041.4. Other image defects 305

2. Parameter Estimation of Discrete Components . 3052.1. Model-fitting 3052.2. Parameter adjustments 3062.3. Parameter errors 3072.4. Fitting models to the visibility data 308

3. Parameter Estimation for Extended Sources 3083.1. General problem 3083.2. The integrated intensity of an extended feature 309

xxii CONTENTS

3.3. Very extended features 3114. Image Combination, Analysis, and Errors 311

4.1. Image combination 3114.2. Linear combinations 3124.3. Nonlinear combinations 3124.4. Image errors 3134.5. Errors in image combination 3144.6. Image alignment 3154.7. Other image combinations 316

5. Selected Image Analysis Topics 3175.1. Absolute positions 3175.2. Relative positions and motions 3185.3. Intensity variations 318

Lecture 15. Error Recognition by R. D. Ekers1. Introduction 3212. Diagnosing Errors 321

2.1. Image plane or (u,v) plane? 3212.2. Short and long time-scale errors 3232.3. General forms of errors 3232.4. Real and imaginary parts of errors 326

3. Examples 3273.1. Additive errors 3273.2. Multiplicative errors 3293.3. Errors increasing with distance from the phase reference

center 3323.4. Computational errors 332

4. Diagnostic Tools '. . . . . 3334.1. Low-resolution images 3334.2. Polarization 3334.3. Fourier transforming the image 3334.4. Effective use of image displays 334

Lecture 16. Non-Imaging Data Analysis by T. J. Pearson1. Introduction 3352. Visibility Data 335

2.1. Sampling 3362.2. Calibration 3362.3. Non-Fourier Imaging 3362.4. Noise 336

3. Inspecting Visibility Data 3363.1. An Example 337

4. The Closure Quantities 3415. . Model Fitting 341

5.1. Imaging as an Inverse Process 3415.2. Uses of Model Fitting 3435.3. Practical Model Fitting 345

CONTENTS xxiii

5.4. Programs for Model Fitting 3455.5. Limitations of the Least-Squares Method 3465.6. Least-Squares Algorithms . . _. 3475.7. Error Estimation 349

6. Some Applications 3496.1. Superluminal Motion 3496.2. Gravitational Lenses . . . 3506.3. The Sunyaev-Zeldovich Effect . :. 3516.4. The Cosmic Microwave Background Radiation 351

7. Appendix 3527.1. Properties of the Fourier Transform 3527.2. Simple Models 353

Lecture 17. Special Problems in Imaging by W. D. Cotton1. Wide Field Problems 357

1.1. Bandwidth smearing (chromatic aberration) 3571.2. Time-average smearing 3601.3. Sparse fields and confusing sources 3621.4. Noncoplanar baseline effects (to-term) 3641.5. Nonisoplanatic and antenna polarization effects 3671.6. Regions larger than the primary beam 367

2. Time-Variable Effects 3672.1. Variable sources 3682.2. Variable sidelobes 368

Lecture 18. Bandwidth and Time-Average Smearingby A. H. Bridle & F. R. Schwab

1. Bandwidth Smearing (Chromatic Aberration) 3711.1. General description of the effect 3711.2. Square bandpass, no tapering, square (u, v) coverage . . . 3741.3. Square bandpass, circular Gaussian tapering 3751.4. Gaussian bandpass, circular Gaussian tapering 3761.5. Graphs of the main bandwidth smearing effects 376

2. Time-Average Smearing 3762.1. General description of the effect . 3762.2. Average effect on an image 379

Lecture 19. Imaging with Non-Coplanar Arrays by R. A. Perley1. Imaging in Three Dimensions - . - . . . 383

1.1. The visibility and image volumes 3831.2. Interpretation of these formulae 3851.3. Direction cosines and image coordinates 3871.4. Some fine points 388

2. Techniques for Recovering the Source Brightness 3902.1. Three-dimensional Fourier transform 3912.2. Polyhedron imaging 3932.3. Joint deconvolution 396

xxiv CONTENTS

3. Examples of Three-Dimensional Imaging 398

Lecture 20. Mosaicing with Interferometric Arrays by M. A. Holdaway1. Introduction 401

1.1. Large Sources Cause Problems 4011.2. Single Pointing Imaging of Large Sources 4021.3. Mosaicing Large Sources 403

2. Ekers and Rots and Effective {u, v) Coverage for Mosaicing . . . 4073. Mosaicing Algorithms and Total Power 409

3.1. Mosaicing Previously Deconvolved Images 4103.2. Non-linear Joint Deconvolution 4113.3. Linear Mosaic of Dirty Images with Subsequent Joint De-

convolution 4123.4. The Sault et al. Algorithm 414

4. Mosaicing as a Design Tool for Interferometric Arrays 4155. Good Mosaicing Practice 417

Lecture 2 1 . M u l t i - F r e q u e n c y S y n t h e s i s by R. J. Sault & J. E. Conway1. Introduction 4192. Spectral Variation 4203. The Linear Spectral Variation Approximation 4204. Higher Order Decompositions 4235. Calibration and Self-Calibration Issues 4256. MFS Deconvolution 426

6.1. Map and Stack 4266.2. Direct Assault 4276.3. Modified Direct Assault 4276.4. Da ta Weighting Methods 4276.5. Double Deconvolution 4286.6. Modified Double Deconvolution 429

7. Conclusion 430>,7.1. MFS and Mosaicing 4317.2. MFS and Faraday Rotat ion 431

Lecture 22. Very Long Baseline Interferometry by R. C. Walker1. Introduction 4332. VLBI Science 4343. VLBI Systems 4344. VLBI Phases 435

4.1. The Model 4364.2. Fringe Fitting 4374.3. Self-calibration 4414.4. Phase Referencing 442

5. The Life History of a VLBI Observation 4465.1. Scheduling 4475.2. Observing 4485.3. Correlation 449

CONTENTS xxv

5.4. A Priori Calibration and Flagging 4505.5. Data Based Calibration and Editing 4535.6. VLBI Imaging 456

6. Conclusions . . v~ 461

Lecture 23. Astrometry and Geodesy by E. B. Fomalont1. Introduction 4632. Astrometric Fundamentals 464

2.1. Total Phase Delay 4642.2. Fundamental Astrometric Formula 4652.3. Idealized Astrometric/Geodetic Solutions 4662.4. Astrometric Experiments 468

3. VLBI and the Group Delay 4693.1. Group Delay and Lobe Ambiguities 4703.2. Measuring the Group Delay and Bandwidth Synthesis . . 4713.3. Complications in Measuring the Group Delay 472

4. VLBI Astrometric/Geodetic Results 4744.1. Observing and Reducing a 24-hour Experiment 4744.2. Astrometry and Geodesy at the 1 mas level 4764.3. Future Research 478

Lecture 24. Spectral Line VLBI by M. J. Reid1. Spectral Line VLBI Sources 4812. Basic Concepts 4823. Spectral-line Calibration 486

3.1. Instrumental Parameters 4863.2. Doppler Tracking 4873.3. Transforming from Delays to Frequencies 4883.4. Spectral Line Amplitude Calibration 4893.5. Clock and Coordinate Corrections 4913.6. Electronic Phase Shifts 4933.7. Phase Referencing . 493

4. Fringe Rate Mapping 495

Lecture 25. VLBI Polarimetry by A. J. Kemball1. Introduction 4992. Polarization VLBI calibration . 500

2.1. Instrumental and propagation effects 5012.2. Calibration 5022.3. Feed calibration 5052.4. Absolute EVPA determination 507

3. Imaging 5074. Other polarization VLBI observing modes 508

4.1. Spectral line polarization VLBI 5084.2. Orbiting polarization VLBI 508

5. Observing and scheduling 509

xxvi CONTENTS

Lecture 26. Space Very Long Baseline Interferometry by J. S. Ulvestad1. Introduction 5132. Orbit Influence on (u,v) Coverage 514

2.1. Orbit Parameterization 5142.2. Orbit Selection 5142.3. Variations in u and v During Observation 5162.4. Orbit Effects on (u,v) Coverage in Different Directions . . 5162.5. Effects of Orbit Precession on (u,v) Coverage at Different

Times 5163. The Space Radio Telescope 518

3.1. What is the "Space Radio Telescope"? 5183.2. The Spacecraft 5203.3. Tracking Stations 5203.4. Timing and Data Link 5203.5. Orbit Determination 5233.6. Observing Constraints 5233.7. Data Calibration 524

4. Correlation and Data Processing 5264.1. Delay Model 5264.2. Correlator Inputs 5274.3. Fringe-fitting 5284.4. Coherence 529

5. Imaging and Modeling 5305.1. Data Weighting 5315.2. Self-Calibration 5315.3. Dynamic-Range Limitations 5315.4. Sample Image 532

6. Future Missions 5336.1. RadioAstron 5336.2. VSOP-2 5336.3. ARISE 534

Lecture 27. Interferometric Array Designby M. A. Holdaway & T. T. Heifer

1. Introduction 5372. Telescope Design 5373. Multi-configuration View 540

3.1. Why multiple Configurations? 5403.2. But How Many Configurations? 5413.3. Big Arrays Don't Need Big Central Holes 5423.4. Strategy to Combine Data From Multiple Configurations 542

4. Optimizing a Single Configuration: Pure Geometric Simplicity . 5434.1. Abstract Fourier Plane Distributions 5434.2. Linear Arrays 5444.3. VLA-Y and GMRT-Y 5474.4. Circular "Crystalline" Arrays 5484.5. Uniform Fourier Plane Coverage: Elastic Net 549

CONTENTS xxvii

4.6. Maximizing Brightness Sensitivity: Filled Array 5504.7. Minimizing PSF Sidelobes 5514.8. What Observations Do We Optimize Over? 5534.9. Simulations to Verify Image Quality 555

5. Complicating Factors: Dealing with Physical Reality 5555.1. Antenna Transportation 5555.2. Topography 5555.3. Overlapping Pads 5565.4. Shadowing 5565.5. Sky Coverage and Stretched Configurations for Low Ele-

vation 5565.6. Atmospheric Influence on Sensitivity 5575.7. Paired Antenna Calibration 5575.8. Redundant Arrays 557

6. Avoid Hardware Solutions to Software Problems 5587. Cases Studies 559

7.1. VLBA 5597.2. MMA 560

8. Tools for Fixing Poorly Designed Arrays 5618.1. Deconvolution 5618.2. Weighting Schemes 5618.3. Changing the Way Existing Stations are Used 561 ,8.4. Building New Stations 5628.5. Multifrequency Synthesis 562

9. General Advice : 562

Lecture 28. Millimeter Interferometryby C. L. Carilli, J. E. Carlstrom & M. A. Holdaway

1. Science with mm Interferometers 5652. Problems Unique to mm Interferometry . 5683. The Troposphere '•; 569

3.1. A General Description 5693.2. The Tropospheric Effect on Tsys 5723.3. Tsya and TO 5733.4. Absolute Gain Calibration 5743.5. The Mean Tropospheric Effect on Interferometric Phase . 5753.6. Phase Variations due to the Troposphere 5763.7. Root Phase Structure Function 5793.8. Effects of Tropospheric Phase Noise 5823.9. 'Stopping the Troposphere': Techniques to Reduce the Ef-

fects of Tropospheric Phase Noise 5844. Antennas 5955. Electronics in Millimeter Interferometry 596

5.1. The Quantum Limit 597

Lecture 29. Long Wavelength Interferometry by W. C. Erickson1. Introduction 601

xxviii __ CONTENTS

2. Features of the Radio Sky at Long Wavelengths 6022.1. The Galactic Background 6022.2. Radio Sources 6042.3. Propagation Effects 605

3. Ionospheric Effects 6063.1. Ionospheric Absorption, Scintillation, and Faraday Rotation6063.2. Ionospheric Phase Fluctuations 6063.3. Isoplanatic Patch Size 6073.4. Diurnal and Solar Cycle Variations 608

4. Long Wavelength Observing 6084.1. Angular Resolution 6084.2. Sensitivity 6094.3. Interference 6094.4. Wide Field Mapping 610

5. GPS Data 611

Lecture 30. Pulsar Observing at the VLA by T. H. Hankins1. Introduction 6132. Scientific Objectives of Pulsar Observing at the VLA 613

2.1. Time Series 6132.2. Imaging 6152.3. Spectral Analysis 616

3. Observing Issues 6164. NRAO Supported Equipment 6185. Non-supported Equipment 6216. General Procedures 6227. Documentation 623

Lecture 31. Solar System Objects by B. J. Butler & T. S. Bastian1. Introduction 6252. General considerations 626

2.1. Source motion 6262.2. Scheduling and planning 6262.3. Observing 6282.4. Calibration issues 6292.5. Imaging, deconvolution, and self-calibration 6312.6. 3-D image reconstructions 6352.7. Conversion of units and coordinates 6362.8. Data rectification to a common distance 638

3. Interferometric radar observations of solar system objects . . . . 6383.1. Telescopes, frequencies, and polarization 6403.2. Planning and scheduling 6413.3. Data reduction (calibration, imaging, self-cal) 6423.4. Unit conversion 6433.5. An example 643

4. Solar observing 6444.1. Hardware modifications 645

CONTENTS xxix

4.2. Solar data calibration 6454.3. Limitations .' 6474.4. An example 649

Lecture 32. The Hamaker-Bregman-Sault Measurement Equationby R. J. Sault & T. J. Cornwell

1. Introduction 6572. Jones Matrices 6573. Direction-Dependent Effects 6614. Mueller Matrices 6615. Limits to Polarimetric Calibration 662

5.1. Calibration with a snapshot observation of a point source 6625.2. Calibration with a long synthesis of a point source . . . . 6635.3. Requirements for complete calibration 6645.4. A practical case 664

6. Implementation in AIPS++ 665

Lecture 33. Noise and Interferometry by V. Radhakrishnan1. Preamble 6712. Information and Bandwidth 6713. The Nature of Noise 6724. Interferometers & Coherence 6735. The Similarity of Signals 6746. Wave Noise 6757. Shot Noise 6768. Intensity Interferometry 6779. Photons and Interference 67910. Uncertainty and Probability 68011. Density in Phase Space 68212. The Strength of Astronomical Signals 68313. Coherent Amplifiers 68514. Epilogue 686

INDEX 689

COLOPHON 705