Hard X-ray Imaging Using a Microwave MEM Program

The Ramaty High Energy Spectroscopic Imager (RHESSI) has been observing flares since February 2002 (Lin et al 2002). RHESSI is the first HXR imager to use Fourier-based methods with high spectral (~ 1 keV) energy resolution in the 3 keV - 17 MeV range, although previous lower-resolution Fourier-based imagers -- Hinotori and Yohkoh/HXT have provided abundant proof of principle of this method of imaging. The raw amplitudes and phases of RHESSI's modulation profiles are analogous to the amplitudes and phases derived from radio interferometers. After calibration, these amplitudes and phases become device-independent (Hurford, Schmahl & Schwartz, 2005), and are precisely equivalent to radio visibilities.

Mapping with visibilities has a rich history in radio astronomy. Visibilities themselves are samples of the Fourier transform of the flare source for which a given image is desired. Given dense sampling of the u,v plane (u and v are the coordinates of the Fourier plane), the inverse Fourier transform yields a map. Since the u,v sampling is not usually dense, and often quite sparse, the map so produced is called the "dirty map." Various methods have been devised to remove the sidelobes produced by sparse sampling in the dirty map and we focus on the Maximum Entropy Method (MEM).

Other methods of HXR imaging have been developed for RHESSI ( Schwartz et al, 2002 ). Among them are Back-projection, Clean, and Pixons, and these can provide validation and testing of the MEM images.

1.1 RHESSI Visibilities

Each amplitude (Aj) and phase (φj) computed from RHESSI modulation profiles are combined into visibilities (Vj) in the standard way:

Vj = Aj * exp(i φj)         (1)

Each visibility Vj is derived from a single cell (uj,vj) in the Fourier plane. These (u,v) cells are taken from roll/phase bins obtained over multiple rotations (typically, 5-20) corresponding to time averages of 20-80s. (See Hurford, Schmahl & Schwartz 2005 for details)

kj = 1/(4.5 * 3(j-1)/2) arcsec-1,   j = 1,2,...,9        (2)

The smallest kj, for the coarsest subcollimator 9, provides an amplitude closest to the total flux, and the smallest kj, for the finest collimator 1, can in principle provide information about spatial scales of ~2.3 arcsec. For the best (u,v) coverage, one must use as many of the (u,v) circles as possible, consistent with the minimization of "over-resolution" and avoidance of RHESSI's spin axis. Typically, this means using circles 3-9, although a few compact flares permit the use of circles 1-9, while some extended sources require restricting the range to 4-9. (We address this matter below.)

1.2 The Maximum Entropy Method

MEM imaging has been widely used in a variety of fields and was first developed by Jaynes (1957, 1968). The basis of the method is to maximize the information entropy H while minimizing the χ2 -- a measure of the goodness-of-fit to the observations -- and maintaining the correct value of the flux. In practice, following Cornwell & Evans 1985, one maximizes the objective function:

J = H - α * χ2 - β * F         (3)

over the parameter space of all possible images. Here α, β are Lagrange multipliers, χ2 is the statistical measure of goodness of fit to the data, and F is the flux, i.e. the sum of the pixel brightnesses. Differentiating equation (1) with respect to the pixel brightnesses and setting that gradient equal to zero gives

grad J = grad H - α * grad χ2 - β = 0        (4)

Bong et al (2004 a , b) have developed a MEM program, ( SSMEM), to map multi-frequency Owens Valley Solar Array data. In its original form, it is a 3-D MEM, using not only u and v but frequency ν.

The solar group at New Jersey Institute of Technology (NJIT) has provided a 2-D version of SSMEM to the RHESSI team and dubbed it MEM NJIT. We have computed visibilities and input them into the MEM NJIT program. We have selected 25 flares and mapped them using the MEM and Clean algorithms. The maps shown below demonstrate the many similarities and differences of the methods.

Clean and mem_njit_compared

To be continued...


Last modified: Fri Nov 18 10:39:23 EST 2005 http://web.njit.edu/~leejw/ssmem.html