17.2 Image formation in a mosaic

Some important mosaic properties can be understood by analyzing the combination of the data directly in the $uv$ plane. This analysis was first proposed by [Ekers & Rots 1979]. The reader is also referred to [Cornwell 1989]. We consider a source with a brightness distribution $I(x,y)$, where $x$ and $y$ are two angular coordinates. The ``true'' visibility, i.e. the Fourier Transform of $I$, is noted $V(u,v)$. An interferometer baseline, with two identical antennas whose primary beam is $B(x,y)$, measures a visibility at a point $(u,v)$ which may be written as:

$$V_{\rm mes}(u,v) = \ensuremath{\int\!\!\int}_{-\infty}^{+\infty} B(x,y) I(x,y) {\rm e}^{\displaystyle -2i\pi(ux+vy)} dx dy$$ (17.2)

If the observation is performed with a phase center in $(x=0,y=0)$ but with a pointing center in $(x_p,y_p)$, the measured visibility (whose dependence on $(x_p,y_p)$ is here explicitly indicated) is:

$$V_{\rm mes}(u,v,x_p,y_p) = \ensuremath{\int\!\!\int}_{-\infty}^{+\infty} B(x-x_p,y-y_p) I(x,y) {\rm e}^{\displaystyle -2i\pi(ux+vy)} dx dy$$ (17.3)

Using the symmetry properties of the primary beam $B$, this last relation can be rewritten:

$$V_{\rm mes}(u,v,x_p,y_p) = B(x_p,y_p) * {\cal F}(u,v,x_p,y_p)$$ (17.4)

where $*$ denotes a convolution product and the function $\cal F$ is defined as:

$${\cal F}(u,v,x_p,y_p) = I(x_p,y_p) {\rm e}^{\displaystyle -2i\pi(ux_p+vy_p)}$$ (17.5)

Now, let's imagine an ideal ``on-the-fly'' mosaic experiment: for a given, fixed, $(u,v)$ point, the pointing direction is continuously modified, and the variation of the visibility $V_{\rm mes}$ with ($x_p$, $y_p$) can thus be monitored. The Fourier Transform of these data with respect to $(x_p,y_p)$ would give (from Eq. 17.4):

$$\left[{\rm FT_p}(V_{\rm mes})\right](u_p,v_p) = T(u_p,v_p) V(u+u_p,v+v_p)$$ (17.6)

where:

• ${\rm FT_p}$ denotes the Fourier Transform with respect to $(x_p,y_p)$.

• $(u_p,v_p)$ are the conjugate variables to $(x_p,y_p)$.

• $\left[{\rm FT_p}(V_{\rm mes})\right](u_p,v_p)$ is the Fourier Transform of the observations.

• $T(u_p,v_p)$ is the Fourier Transform of the primary beam $B(x_p,y_p)$. $T$ is thus the transfer function of each antenna. For a dish of diameter $\cal D$, $T(u_p,v_p) = 0$ if $(u_p^2+v_p^2)^{1/2} > {\cal D}/\lambda$.

• $V(u+u_p,v+v_p)$ is the Fourier Transform of ${\cal F}(u,v,x_p,y_p)$ with respect to $(x_p,y_p)$. Indeed, ${\cal F}$ is the product of the sky brightness distribution (whose Fourier Transform is $V$) by a phase term (see Eq. 17.5). Hence, its Fourier Transform is $V$ taken at a shifted point.

For $\sqrt{u_p^2+v_p^2} < {\cal D}/\lambda$, we can thus derive:

$$V(u+u_p,v+v_p) = \frac{\left[{\rm FT_p}(V_{\rm mes})\right](u_p,v_p)}{T(u_p,v_p)}$$ (17.7)

This relation illustrates an important property of the experiment we have considered. The observations were performed at a given $(u,v)$ point but with a varying pointing direction. Eq. 17.7 shows that is possible to derive from this data set the visibility $V(u+u_p,v+v_p)$ at all $(u_p,v_p)$ which verify $(u_p^2+v_p^2)^{1/2} < {\cal D}/\lambda$. In other terms, the measurements have been done at $(u,v)$ but the redundancy of the observations allows to compute (through a Fourier Transform and a division by the antenna transfer function) the source visibility at all the points of a disk of radius ${\cal D}/\lambda$, centered in $(u,v)$.

Interpretation

In very pictorial terms, one can say that the adjacent pointings reinforce each other and thereby yield an estimate of the source visibility at unmeasured points. Note however that the resulting image quality is not going to be drastically increased: more information can be extracted from the data, but a much more extended region has now to be mapped^17.2. The redundancy of the observations has only allowed to rearrange the information in the $uv$-plane. This is nevertheless extremely important, as e.g. it allows to estimate part of the missing short-spacings (see below).

How is it possible to recover unmeasured spacings in the $uv$-plane? It is actually obvious that two antennas of diameter $\cal D$, separated by a distance $\cal B$, are sensitive to all the baselines ranging from ${\cal B}-{\cal D}$ to ${\cal B}+{\cal D}$. The measured visibility is therefore an average of all these baselines: $V_{\rm mes}$ is actually the convolution of the ``true'' visibility by the transfer function of the antennas. This is shown by the Fourier Transform of Eq. 17.2, which gives: $V_{\rm mes} = T*V$. Now, if the pointing center and the phase center differ, a phase gradient is introduced across the antenna apertures, which means that the transfer function is affected by a phase term. Indeed, the Fourier Transform of Eq. 17.3 yields:

$$V_{\rm mes}(u,v) = \left[T(u,v) {\rm e}^{\displaystyle -2i\pi(ux_p+vy_p)}\right]* V(u,v)$$ (17.8)

Hence, the measured visibilities are (still) a linear combination of the ``true'' visibilities. Measurements performed in various directions $(x_p,y_p)$ give many such linear combinations. One can thus expect to derive from this linear system the initial visibilities, in the baseline range from ${\cal B}-{\cal D}$ to ${\cal B}+{\cal D}$. Eq. 17.7 just shows that a Fourier Transform allows to do that operation.

Field spacing in a mosaic

In the above analysis, a continuous drift of the pointing direction was considered. However, the same results can be reached in the case of a limited number of pointings, provided that classical sampling theorems are fulfilled. We want to compute the visibility in a finite domain, which extends up to $\pm {\cal D}/\lambda$ around the nominal $(u,v)$ point, and therefore the pointing centers have to be separated by an angle of $\lambda/2{\cal D}$ radians (see [Cornwell 1988]). In practice, the (gaussian) transfer function of the millimeter dishes drops so fast that one can use without consequences a slightly broader, more convenient, sampling, equal to half the primary beam width (i.e. $1.2 \lambda/2{\cal D}$).

Mosaics and short-spacings

As with any other measured point in the $uv$ plane, it is possible to derive visibilities in a small region (a disk of diameter ${\cal D}/\lambda$) around the shortest measured baseline. This is the meaning of the statement that mosaics can recover part of the short-spacings information: a mosaic will include ($u,v$) points corresponding to the shortest baseline minus ${\cal D}/\lambda$.

In practice, however, things are more complex. First, we have to deal with noisy data. As a consequence, it is not possible to expect a gain of ${\cal D}/\lambda$: the transfer function $T$ which is used in Eq. 17.7 is strongly decreasing, and thus signal-to-noise ratio limits the gain in the $uv$ plane to a smaller value, typically ${\cal D}/2\lambda$ ([Cornwell 1988]). This is still a very useful gain: for the Plateau de Bure interferometer, this corresponds to a distance in the $uv$ plane of 7.5 m$/\lambda$, while the shortest (unprojected) baseline is 24 m$/\lambda$. Secondly, the analysis described above would be rather difficult to implement with real observations, which have a limited number of pointing centers and different $uv$-coverages. Instead, one prefers to combine the observed fields to directly reconstruct the sky brightness distribution. The resulting image should include the information arising from the redundancy of the adjacent fields, among them part of the short-spacings. However, the complexity of the reconstruction and deconvolution algorithms that have to be used precludes any detailed mathematical analysis of the structures in the maps. For instance, the (unavoidable) deconvolution of the image can also be interpreted as an interpolation process in the $uv$ plane (see [Schwarz 1978] for the case of the CLEAN algorithm) and its effects can thus hardly be distinguish from the intrinsic determination of unmeasured visibilities that occur when mosaicing.