Dirty Beam and Dirty Map

An aperture synthesis array measures the visibilities at discrete points in the $uv$-domain. The visibilities are Fourier transformed to get the Dirty Map and the weighted $uv$-sampling function is Fourier transformed to get the Dirty Beam using the efficient FFT algorithm.

The visibility $V$ measured by an aperture synthesis telescope is related to the sky brightness distribution $I$ as,

$$\rm {V \rightleftharpoons I},$$ (3..1)

where $\rightleftharpoons$ denotes the Fourier Transform. Since there are finite antennas in an array, $uv$-plane is sampled at discrete $uv$ points and Eq. 3.1 has to be written as,

$$\rm {V.S \rightleftharpoons I*DB (=I^d)},$$ (3..2)

where '*' operator represents convolution operation, $I^d$ is the Dirty Map, $I$ is the true brightness distribution, $DB$ is the Dirty Beam and $S$ is the $uv$-sampling function given by,

$$\rm {S(u,v)=\sum_{k} \delta(u-u_k,v-v_k)},$$ (3..3)

where $u_k$ and $v_k$ are the actual ($u,v$) points measured by the telescope. This function essentially assigns a weight of unity to all measured points and zero everywhere else in the $uv$-plane. Fourier transform of $S$, $uv$-sampling function, is referred to as the Dirty Beam. The synthesized beam of the observation is an elliptical Gaussian fitted to the main lobe of the Dirty Beam, which determines the resolution of the telescope. For a point source, flux density per synthesized beam is independent of the size of the synthesized beam, whereas, the apparent brightness of extended emission in the dirty image depends on the source structure itself and the beam width of the synthesized beam. Dirty Beam for a fully covered uv-plane will be equal to $(sin(\pi l \lambda / u_{max}) / (\pi l \lambda/u_{max})$, where, $u_{max}$ is the largest antenna spacing for which a measurement is available.

The width of the main lobe of this function is proportional to $\lambda /u_{max}$. The resolution of such a telescope is therefore roughly $\lambda /u_{max}$. For a real $uv$-coverage however, S is not flat till $u_{max}$and has holes in between representing un-sampled $(u,v)$ points. The effect of this missing data is to increase the side-lobes and make the dirty beam have large fluctuation.

The Dirty Map is a convolution of the true brightness distribution and the Dirty Beam(DB). $I^d$ is almost never a satisfactory final product since the side-lobes of $DB$ (which are due to missing spacings in the $uv$-domain) from a strong source in the map will contaminate the entire map at levels higher than the thermal noise in the map. Without removing the effect of $DB$ from the map, the effective RMS noise in the map will be much higher than the thermal noise of the telescope and will result into obscuration of faint sources in the map. The process of De-convolving is discussed in the section 3.3.2.6, which effectively attempts to estimate $I$ from $I^d$ such that $(I*DB -I^d)$ is minimized, and is consistent with the estimated noise in the map.