Continuum Subtraction

Quite often spectral line observations include continuum flux density present in the band. The continuum in the band can arise due to a variety of reasons. Ionized Hydrogen regions, for e.g., give rise to the radio recombination lines of Hydrogen due to bound-bound transitions and the radio continuum due to thermal bremsstrahlung. Galaxies can have strong non-thermal radio continuum as well as 21-cm-line emission and/or absorption. In addition, any absorption spectral line experiment involves a bright continuum background source. In these and similar situations, detecting a weak spectral line in the presence of strong continuum contribution can be very difficult. Depending on the complexity of the angular distribution of the continuum flux density and that of the spectral feature this task might almost become impossible.

The basic problem here is one of spectral dynamic range (SDR). The spectral dynamic range is the ratio of the weakest spectral feature that can be detected to the continuum flux density in the band. This is limited by the residual errors which arise due to a variety of reasons like, for e.g., the instrumental variations, the atmospheric gain changes, the deconvolution errors, etc.. Of these, the multiplicative errors limit the SDR depending on the continuum flux density in the band. Thus, if the multiplicative errors are at 1% level, and , if the continuum flux density in the band is 10 Jy, no spectral line detection is possible below 100 mJy. On the other hand, a continuum subtraction (if successful) will lead to a situation where the SDR is decided by the peak spectral line flux density rather than the continuum flux density. Apart from the continuum flux density any other systematics which have a constant value or a linear variation across frequency will be subtracted out in the continuum subtraction procedure. This can lead to improvements in the SDR by several orders of magnitude.

There are several methods for subtracting the continuum flux density from a spectral line data. It is beyond the scope of this lecture to discuss all of these. A brief mention will be made of one of these simpler methods to illustrate some of the principles involved. In this method, which has been called $'$visibility-based subtraction$'$, a linear fit to the visibilities as a function of frequency is performed for every sample in time. This best-fit continuum can then be subtracted from the original visibilities. The resulting data can be Fourier transformed to produce continuum-free images. This method works quite well if the continuum emission is spread over a sufficiently small field of view. This limitation can be understood in the following way. Consider a two-element interferometer separated by d. Let each of the elements of the interferometer be pointing towards $\theta_{0}$ which is also the fringe tracking (phase tracking) center. The phase difference between $\theta_{0}$, and an angle $\theta$ close to this, is $\phi = 2 \pi \nu d (sin(\theta)-sin(\theta_{0}))/c$, where, $\nu$ is the observing frequency, and c is the velocity of light. For the present purpose of illustration, assume that $\theta$ is in the plane containing the pointing direction ($\theta_{0}$) and d. The visibilities from a source at $\theta$ will have the form A$_{\nu}$cos($\phi$) and A$_{\nu}$sin($\phi$), where, A$_{\nu}$ is the amplitude of the source at $\nu$. Writing $\nu = \nu_{0} + \delta\nu$, and $\theta = \theta_{0} + \delta\theta$, where, $\nu_{0}$ is the frequency of the center of the band, it can be shown that the frequency-dependent part of the phase is $\phi_{\nu} = 2 \pi \delta\nu d cos(\theta_{0}) \delta\theta / \nu_{0} \lambda_{0}$, where, $c = \nu_{0} \lambda_{0}$. It is easy to see that the variation of visibilities as a function of frequency is linear if $\phi \ll 2 \pi$. This implies that $\delta\nu \delta\theta / (\nu_{0} \theta_{syn}) \ll 1$, where, $\theta_{syn} = \lambda_{0} /d$. Thus, this method of continuum subtraction works if most of the continuum is within $\nu_{0}/\delta\nu$ synthesized beams from the phase tracking center.