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Data Reduction Case #1

Total Power Position Switched Single Beam Data

Data reduction can be summarized as follows:

  1. Calibration
  2. Flagging
  3. Averaging
  4. Baseline subtraction
  5. Analysis

The user should have taken spectra at two different, but nearby, positions on the sky. One of these is taken at the position of interest, and is called the "Signal" data, and the other is taken usually a few beamwidths away and is called the "Reference" data. These data may be taken with or without a calibration signal. Usually a calibration signal is switched on and off throughout the integration at some rate, say 1 Hz, and the data from each of the phases (calon and caloff) are stored in separate registers. If no calibration signal is available, absolute calibration to antenna temperature is not possible.

The user thus has data available in four registers: sig_{calon}, sig_{caloff}, ref_{calon}, ref_{caloff}. Usually, there is a set of these measurements for each of two polarizations.

Calibration

The first step in reducing the data is to calibrate it. The calibration procedure can vary significantly for different sets of observing circumstances. Calibration is the topic of a detailed document that can be found here: RonsDetailedCalibrationDocument?. Issues that may be of concern to the user include whether to smooth the Tsys spectrum and if so, to what extent to smooth it; how to apply a variable gain factor; how to handle nonlinearities in the gain; alternate strategies in using the noise tube calibration signal. In the present discussion, only a simple and common case for data calibration will be described.

First, the observer can calculate system temperature on blank sky as follows: T_{sys}(f,p) = T_{cal}(f,p) \times \frac{ref_{caloff}}{ref_{calon} - ref_{caloff}} + \frac{T_{cal}(f,p)}{2}

Note the term \frac{T_{cal}(f,p)}{2} accounts for the increase in Tsys caused by the noise tube calibration signal, which is on half of the time.

T_{cal}(f,p) is the calibration temperature of the noise tube calibration signal, usually available from engineering measurements, and is a function of frequency (f) and polarization (p). Note system temperature is also, then, a function of frequency and polarization. For many observing scenarios, a single value of system temperature can be applied across the entire bandwidth observed, and so system temperature is a single scalar value for each polarization: T_{sys}(p) = average(T_{sys}(f,p)), where the average is across frequencies (or spectral channels). If large bandwidths are used it is sometimes necessary to maintain the frequency dependence, but to avoid introducing noise it is necessary to smooth the T_{sys} spectrum before using it later in the calibration process. See the calibration documentation for details.

Next, the observer can generate antenna temperature as follows: T_{ant}(f) = T_{sys} \times \frac{sig - ref}{ref} where sig and ref are both functions of frequency, and can be defined in one simple case by sig = \frac{sig_{calon} + sig_{caloff}}{2} and ref = \frac{ref_{calon} + ref_{caloff}}{2}.

Next the observer may wish to convert the antenna temperature to flux density in Janskies. This can be accomplished either by applying telescope-specific parameters, or by observing a known astronomical flux calibrator.

Using known telescope parameters: S = T_{ant} \times K \times A_e \times S_p \times e^{\tau_0/sin(elv)} where K is a constant depending on the telescope (K = 2.8 for the GBT), A_e is the aperture efficiency at the frequency and elevation of interest, S_p is the spillover factor, typically S_p ~ 0.99, \tau_0 is the atmospheric opacity at the zenith, and elv is the elevation observed.

Using astronomical calibrators: Observe the calibrator using the identical observing configuration, and calculate the scaling factor: Factor = S_{calsource}/T_{calsource}. Then scale the data accordingly. Keep in mind that if the elevation of the calibration source differs from that of the source of interest, the gain will be different and must be corrected. Both aperture efficency and opacity factors must be compensated.

The final result of this step is a spectrum in units of K or Jy. However, it is often the case that an unknown offset and baseline shape remains in the spectrum. It will be accounted for in a later step.

Flagging

Flagging data is as much art as science. Flagging can be applied at any step in data reduction, and in some cases must necessarily be applied prior to calibration. However, immediately after calibration is also a common time to flag. Data can be flagged for RFI, spectrometer glitches, or bandpass edges, for example. The process can be automated (e.g. flag the first 10 channels in all available spectra; flag all data greater than a given intensity; flag all data which exceeds 5\sigma) or it can be done by hand (tedious, but often necessary).

Averaging

It is usually the case that there are many spectra available after calibration, and they must be averaged to proceed toward the final result. In the simplest case, averaging is simply a matter of taking the arithmetic mean of the flux in each spectral channel over all available integrations. However, often it is necessary to flag data from some spectra and not others, so flagged channels must be accounted. Also, if all integrations do not have identical integration times or system temperatures, then the average must be weighted accordingly, with a weighting factor of T_{int}/T_{sys}^2. Another option is to calculate an rms for each spectrum to be averaged, and weight according to rms.

The final spectrum can also be averaged in frequency. This is often accomplished either by applying a Hanning smoothing function, or a boxcar smoothing function.

Baseline subtraction

After calibration, it is usually necessary to subtract a baseline from the spectrum to remove any DC offset plus possible curvature in the baseline shape. Baseline subtraction can be preformed before or after averaging. The user must define a region of interest in the spectrum, to which the baseline can be fit. This region is usually as large as possible, but excludes any spectral lines in the spectrum, and often excludes the edges of the spectrum if there are edge effects. It should also exclude any flagged channels. A polynomial baseline can then be fit to the region of interest and subtracted. Other possibilities besides polynomila baselines can be made available, for example a sinusoidal function may be fit, or low frequency components of a Fourier decomposition may be subtracted.

Analysis

At this stage the user has a fully reduced representation of the spectral energy profile towards the source of interest. Analysis depends heavily on the science goals of the project, and can include:

  1. Integration of total line intensity
  2. Gaussian fitting to lines detected
  3. Calculation of statistics over regions of interest (mean, standard deviation, median, centroid, etc.)
  4. Comparison of the polarizations (e.g. subtract one from the other)
  5. Fitting a rotation curve
  6. Measurement of rotation velocities

-- JimBraatz - 15 Jul 2004

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