1. Noise Diodes, Calibration, Baselines & Nonlinearities Ron Maddalena NRAO, Green Bank, WV Shelly Hynes Louisiana School for Math, Science and the Arts, Natchitoches, LA Charles Figura Wartburg College, Waverly, IA July 27, 2006

2. Calibration Data June 18, 2006 C-Band – Off-On Observations Multiple calibration sources, same hardware, same attenuator/filter settings Tcal calibration data – Various combinations of polarization, high/low noise diodes Data consists of: Sig on = On source, noise diode on Sig off = On source, noise diode off Ref on = Off source, noise diode on Ref off = Off source, noise diode off

3. Sig and Ref Definitions

5. Current Calibration Method So T sys loses all frequency information…

6. Astronomical T cal  = Efficiency A p =Area k =Boltzman’s Constant   = Elevation Opacity If  is unknown… Source: Johnson et.al., 2002

7. Linear Vector T cal Expressions

8. Now T sys retains the frequency structure

9. Power Characterization

10. ‘Resids’ for Spectral Processor

11. Resids vs. Power In C-Band

12. Nonlinear Theory Include a second-order correction for gain – So the temperature equations become

13. Non-Linear Solutions For Calibration

14. Nonlinear Application Evaluate C, B, T cal, using a known calibration source. Because nonlinearity is encapsulated within P’ out, we can use the previous expressions from the linear case:

15. Calibration Data June 18, 2006 C-Band – Off-On Observations Multiple calibration sources, same hardware, same attenuator/filter settings T cal calibration data – Various combinations of polarization, high/low noise diodes July 15, 2006 3C147 C-Band, low-diode, linear polarization Various attenuator settings at various places S-band S cal calibration data – Various combinations of polarization, high/low noise diodes

16. T sys Comparison Nonlin Lin

17. T cal Comparison Nonlin Lin

18. Baseline Improvement NonLin Linear Cat Traditional

19. Source of ‘Baselines’ – Traditional Assuming Linear

20. Source of ‘Baselines’ – Traditional Non-Linear

21. Time Dependence S13C11x July 15, 2006, t = 0 hours NonLin Linear Cat

22. Time Dependence S34C11x July 15, 2006, t = 2 hours NonLin Linear Cat

23. T A Comparisons t = 2 hours Red – Early Blue - Late

24. Time Dependence S69C11x July 15, 2006, t = 4 hours NonLin Linear Cat

25. T A Comparisons t = 4 hours Red – Early Black - Late

26. Time Dependence NonLin S34C11x NonLin S377C11x

27. Noise Estimates – Vector Tcal Assume Tsys = 10*Tcal, N Chan = 1, t ref = t sig Tsrc = 0 --- σ 2 =2*Tsys/(ChanWidth * t sig ) Tsrc = Tcal --- σ 2 =4.2*Tsys/(ChanWidth * t sig ) Tsrc = 2Tcal --- σ 2 =10.4*Tsys/(ChanWidth * t sig ) Tsrc = Tsys --- σ 2 =205*Tsys/(ChanWidth * t sig ) Assuming Tcal << Tsys

28. System Determination -3dB (IF) -6dB (IFR) -6dB (CR) -6dB (IF) -10dB (IF)

29. System Determination +3dB (IF) +6dB (CR) +6dB (IF) +10dB (IFR) +10dB (IF)

30. System Determination -3dB IFRack NonLin Linear Cat

31. System Determination -6dB IFRack NonLin Linear Cat

32. System Restoration NonLin Linear Cat

33. Summary Using a vector form of T cal for baselines is better than traditional, regardless of linear or non-linear assumptions. Baselines are slightly improved by the quadratic approximation, Cannot achieve good noise, good baselines, and good calibration simultaneously – Compromise!! System rebalancing restores original nonlinearity. Data taken with major ‘distortions’ to power levels can be recovered. ‘C’ remains fairly constant with time.

34. Conclusions Extended sources should not be used to determine linearities, Scals, etc. Polarized sources must be corrected for. Very bright sources cannot be handled by the 2 nd order nonlinear approximation.

35. Recommendations At a minimum, use the vector form of T cal. Use compact sources for calibrations. For many observations, the linear approximation is sufficient. Balancing often isn’t necessary and actually may be detrimental since C will change. Don’t skimp on channels – more channels, less compromises between noise and baseline

36. Catalog Calibration

37. Non-Linear Derivation P refoff = A + BP out + CP out 2 (Eq 1 ) P refon = A + B (P out +P cal ) + C (P out +P cal ) 2 (Eq 2 ) P sourceoff = A + B (P out +P source ) + C (P out +P source ) 2 (Eq 3 ) P sourceon = A + B (P out +P cal +P source ) + C (P out +P cal +P source ) 2 (Eq 4 )