Relaxing the isoplanatism assumption in self-calibration; applications to low-frequency radio interferometry

1, Introduction. An assumption implicit in the usual scheme for self-calibration of radio interferometer data is one of i»oplanaii»m: that oyer each element of the array, all wavefronts arriving from different parts of the sky to which the interferometer pairs are sensitive are subject to identical tropospheric/ionospheric path delays. Approximate validity of the isoplanatism assumption is a necessary condition for the success of selfcalibration. This memorandum is an outline of a means by which the self-calibration algorithm might be modified in order to deal with the anisoplanatic case. Anisoplanatism is a severe problem with a low-frequency array, such as the one which has been proposed by R. A. Perley and W. C. Erickson [8] for construction at the VLA site. This is because, of the extreme magnitude of ionospheric effects at long wavelengths, and the large field of view of such an instrument. An initial attempt at a scheme for self-calibration of low-frequency array data is outlined in Perley and Erickson's proposal; and the need for a generalization of the self-calibration algorithm is reiterated in [2] and [4]. In § 2 is described a method of incorporating an interpolation formula in the selfcalibration solution for antenna phases. The idea is to express the phase corruption seen by a given array element, in an arbitrary direction, as a linear combination (i.e., as an interpolation) of the phase corruptions {/r}£Li toward the centers of some small number m of "isoplanatic patches". Setting m — 5 to 20, or so—with the patches judiciously centered—might suffice in a typical instance. When the source model used for self-calibration is given by a set of CLEAN point source components, it is easy to modify the solution scheme so as to yield the /{. Choice of an appropriate interpolation formula is discussed in § 3. Having obtained from the ^self-calibration solution algorithm a set of n spacevariant antenna phases, one for each antenna, the next problem is finding a way to make use of this information in mapping. The usual mapping/deconvolution schemes, such as Fourier inversion combined with CLEAN or with the maximum entropy deconvolution algorithm, are not designed to cope with space-variant effects. A means of utilizing the space-variant antenna phases in a modified, mosaicing version of the usual map/CLEAN combination is outlined in § 4. A drawback of the method described in § 2 is the increase (by a factor « m) over the usual number of solution parameters, or degrees of freedom, in the self-calibration solution algorithm. Because of this increase, a better source model, higher signal-tonoise ratio (S/N), or a larger number of antenna elements, (or a combination of all three) becomes desirable. By incorporating assumptions of spatial and temporal correlation of the antenna phases^ one may try to hold this larger number of degrees of freedom in check; this idea is pursued in §5 5-6. Perley and Erickson argue that for the proposed low-frequency array, which is designed to operate at 75 and 150 MHz, simple and accurate source models often will be available (perhaps from 327 MHz observations). And their data suggest that the spatial extent and the velocities of the ionospheric irregularities responsible for the severest phase fluctuations at 75 MHz are such that the techniques of §§ 5-6 would be useful. They report that during this summer they have found the typical case in 327 MHz VLA observations to be one of