In this paper, we present a Maximum Entropy Method (MEM) algorithm that
acts in both the spatial and spectral domains and we show first results
of applying this algorithm to data obtained with the frequency-agile
solar interferometer at Owens Valley Radio Observatory (OVRO). The
data are taken at 45 frequencies in the range and measure the
Fourier components of the brightness distribution, where each frequency
measures a different spatial component. The measured Fourier data are
used to reconstruct an image in the spatial domain, as a function of
position and frequency, to provide brightness temperature spectra at
each point which can be analyzed and interpreted in terms of physical
parameters such as coronal magnetic field strength and electron
temperature. The number of observed Fourier components can be quite
large depending on the array and the length of the observing time, but
not all Fourier components will be measured which can lead to
ambiguities in interpreting the data. Thus, one has to address the
problem of missing information or incomplete (u v) coverage. Imaging
algorithms, such as CLEAN or the standard MEM algorithm, fill in the
unmeasured Fourier components by using a priori information about
what the radio source is expected to look like. However, the existing
algorithms do not exploit the spatial information available at adjacent
frequencies in the OVRO data. These algorithms treat each frequency
separately which leads to a reconstructed image consisting of 45
independent spatial maps. We present an algorithm that obtains a
global solution to the visibilities in both the spatial and spectral
domains.
The spatial term in the traditional least-squares MEM maximizes a
spatial entropy term , where T is the map
temperature as a function of spatial position and frequency, subject to
the data constraints. This ensures that the resulting image is
spatially smooth and that it is positive everywhere. In our modified
algorithm, we include an analogous spectral entropy term defined as
, with
where T' is the temperature interpolated from the two neighboring
frequencies at the same spatial position. This term ensures that a
spectrum at a given spatial position changes only smoothly with
frequency. In this paper, the algorithm is described in one spectral
and two spatial dimensions, but is applied only for the case of one
spatial and one spectral dimension for several reasons: (1) To simplify
the development process; (2) To speed up exploration of the
relevant parameter space; (3) To make presentation of the results
simpler (we display the one spectral and one spatial dimension as 2-d
contour plots, whereas two spatial and one spectral dimensions would
require presentation of a data-cube). A subsequent paper is planned to
describe the full algorithm as applied to one spectral and two spatial
dimensions.
In Sect. 2, we describe the concept and its current implementation. In Sect. 3, we apply our 1-d algorithm to observations taken with the OVRO frequency-agile interferometer of active region AR 5417 near the solar limb on March 20, 1989 (vernal equinox) using the two 27-m antennas and the 40-m antenna arranged in a linear east-west array. The geometry of an east-west array on this date gives strictly 1-d spatial resolution, and so gives a good match to the 1-d algorithm. We compare the result of our MEM algorithm with the source structure shown on 1-d OVRO maps obtained using the conventional CLEAN algorithm, and with a reconstruction using only the spatial MEM term of our algorithm. Then, we use the reconstructed image to calculate brightness temperature spectra and to derive physical parameters.