15.4 Error Analysis

We thus succeeded to preserve a convolution equation, with the slight restrictions due to the aliasing and gridding correction. Let us explore now what errors or systematic effects may appear in the image plane.

First, consider the noise. Aliasing increases the noise level at the map edges (by noise aliasing and then by the gridding correction since this amounts to divide by the Fourier Transform of the gridding function, which is unity at the map center, but smaller at the map edges). For example, the noise increases by a factor $(\pi/2)^2$ at map corners for the Gaussian-Sinc function. Near the map center, the effect is negligible. Note that for a given field of view, the noise increase can be arbitrarily limited by making a sufficiently large image, but this has a high computational price.

Concerning errors, it is important to separate two main classes of errors.

Additive errors
The Fourier transform being linear, additive errors result in artificial structure added to the true map, e.g.
- A single spurious visibility will produce fringes in the map
- An additive real term (correlator offset), will produce a point source at the phase tracking center.

Multiplicative errors
A multiplicative term on the visibility distorts the image, since

$$V(u,v) \times \epsilon (u,v) \longleftrightarrow \hat{V}(x,y) \ensuremath{\ast\!\ast} \hat{\epsilon}(x,y)$$

i.e. the map is convolved by the Fourier transform of the error. Calibration errors (in phase or amplitude) are of this type. Among these, the seeing should not be neglected.

Phase calibration errors result in antisymmetric patterns.