Weighting, Tapering and Beam Shaping

The shape of the Dirty Beam can be controlled by multiplying $S$ with other weighting functions. Note that the measured visibilities already carry a weight which is a measure of the signal-to-noise ratio of each measurement. Since there is no control on this weight while mapping, it is not explicitly written in any of equations here but is implicitly used.

Full weighting function $W$ as used in practice is given by

$$W(u,v)=\sum_k T_k D_k \delta(u-u_k,v-v_k).$$ (11.2.4)

The function $T_k$ is the `$uv$-tapering' function to control the shape of $DB$ and $D_k$ is the `density-weighting' function used in all imaging programs. If $S$ was a smooth function, going smoothly to zero beyond the maximum sampled $uv$-point, $DB$ would also be smooth with no side lobes (e.g. if $S$ was a gaussian). However, $S$ is collection of delta functions with gaps in between (for the missing $uv$-points not measured by the telescope) and has a sharp cut-off at the limit of $uv$-coverage. This results into $DB$ being a highly non-smooth function with potentially large side-lobes.

As is evident from the plots of $uv$-coverage, the density of $uv$-tracks decreases away from the origin. If one were to use the local average of the $uv$-points in the $uv$-plane for mapping as is done in the gridding operation described below, the signal-to-noise ratio of the points would be proportional to the number of $uv$-points averaged. Since the density of measured $uv$-points is higher for smaller values of $u$ and $v$, visibilities for shorter spacings get higher weightage in the visibility data effectively making the array relatively more sensitive to the broader features in the sky. The function $D_k$ controls the weights resulting from non-uniform density of the points in the $uv$-plane.

Both $T_k$ and $D_k$ provide some control over the shape of the Dirty Beam. $T_k$ is used to weight down the outer edge of the $uv$-coverage to decrease the side-lobes of $DB$ at the expense decreasing the spatial resolution. $D_k$ is used to counter the preferential weight that the $uv$-points get closer to the origin at the expense of degrading the signal-to-noise ratio.

$T_k$ is a smoothly varying function of ($u,v$) and is often used as $T(u_k,v_k)=T(u_k)T(v_k)$. For most imaging applications, $T(u_k,v_k)$ is a circularly symmetric gaussian. However other forms are also occasionally used.

Two forms of $D_k$ are commonly used. When $D_k=1$ for all values of ($u,v$), it is referred to as `natural weighting' were the natural weighting of the $uv$-coverage is used as it is. This gives best signal-to-nose ratio and is good when imaging weak compact sources but is undesirable for extended sources where both large scale and small scale features are present.

When $D_k=1/N_k$ where $N_k$ is a measure of the local density of $uv$-points around ($u_k,v_k$), it is referred to as `uniform weighting' where an attempt is made to assign uniform weights to the entire covered $uv$-plane. In standard data reduction packages available for use currently (AIPS, SDE, Miriad), while re-gridding the visibilities (discussed below), $N_k$ is equal the number of $uv$-points within a given cell in the $uv$-plane. However it can be shown that this can result into serious errors, referred to as catastrophic gridding error in some pathological cases. This problem can be handled to some extend by using better ways of estimating the local density of $uv$-points (Briggs, 1995).

Eq. 11.1.2, using the weighted sampling function $W$ is written as

$$(V.S.W) \rightleftharpoons (I*DB).$$ (11.2.5)

Note that $DB \rightleftharpoons S.W$, i.e. the Dirty Beam is the Fourier transform of the weighted sampling function.