# User's Guide to RHESSI Visibilities

This write-up summarizes the properties of RHESSI visibilities and is based on a document written by Gordon Hurford. There is also a RHESSI Science Nuggets on the subject entitled RHESSI Visibilities.

## Basic Properties of RHESSI Visibilities

A RHESSI visibility is a complex observable number that can be derived from RHESSI data and which represents a calibrated measurement of a single Fourier component of the source distribution measured at a specific spatial frequency and energy- and time-range.

As a complex number, it can be represented either by its amplitude and phase, or by the corresponding Cartesian representations, variously called the X and Y, cosine and sine, or real and imaginary components. In the context of visibilities, it traditionally labeled as u and v.

$z = X + iY = r(\cos \varphi + i\sin \varphi ) = re^{i\varphi}$

As a calibrated measurement, it incorporates all known spatial calibration information so that no further reference to aspect solution, grids response, etc is needed to interpret it. Note that the energy range of a freshly-measured visibility refers to the detected energy, not photon energy. As a ‘semicalibrated’ value, it therefore incorporates the diagonal elements of the detector response, but not the off-diagonal components. With that caveat, a measured visibility has units of photons cm2 second-1.

Except for contributing to statistical errors, background is inherently removed in the process of calculating visibilities. Background (in this context) refers to any photons that have not passed through both grids (i.e. non-imaging background).

The spatial frequency at which the visibility is measured is characterized either by its Cartesian coordinates (traditionally labeled u, v) or by the polar equivalent as the magnitude of the spatial frequency and its orientation. For RHESSI, the magnitude of the spatial frequency is determined by the subcollimator number (and harmonic) and is given by 1/(angular pitch) of the subcollimator. For the first harmonic, this is equivalent to 0.5/(FWHM resolution) (about 0.22 arcsec-1 for grid 1 to 0.0027 arcsec-1 for grid 9). Note that the coarser grids have the smaller spatial frequency.

 Grid 1 2 3 4 5 6 7 8 9 Resolution (FWHM) [arcsecs] 2.26 3.92 6.79 11.76 20.36 35.27 61.08 105.8 183.2 Spatial frequency (first harmonic) [arcsec-1] 0.22124 0.12755 0.07364 0.04252 0.02456 0.01418 0.00818 0.00472 0.00273

By convention, the orientation of a RHESSI visibility is defined by its position angle in the u,v plane, measured east from solar north. As RHESSI rotates, the magnitude of the spatial frequency measured by a given subcollimator remains fixed, while the position angle of the spatial frequency decreases as the measured visibility moves in a circle in the u,v plane.

Except for the units, RHESSI visibilities are equivalent to those measured by radio interferometers, a feature which opens the way to using full-featured image reconstruction software developed for radio applications.

Since the x-ray source is real (both practically and mathematically), visibilities measured at opposite points in the u,v plane are complex conjugates. This implies that the amplitudes of visibilities measured at opposite half rotations must be equal, and the phases of visibilities measured at opposite half rotations must sum to zero. To within statistical expectations, this condition should hold independent of source morphology. If this is not the case, then either the source time variability is relevant, or there is a calibration or calculation error. It also follows that such ‘conjugate visibilities’ can (and should) be combined without losing any imaging information.

As observables, visibilities are inherently linear both in terms of their dependence on the data and in terms of their dependence on the source. For example, the (complex) sum of two visibilities (at the same spatial frequency) measured over two separate time or energy intervals should give the same result as a measurement over the combined time or energy interval. Similarly, the visibilities of a multicomponent source should equal the sum of the corresponding visibilities of the individual components.

## Some possible application of RHESSI visibilities

• Fe line imaging by suppressing the continuum
• Measurement of source sizes
• Spatial variations on sub-second timescales
• Albedo isolation and characterization
• Averaging data over long time intervals
• Imaging in ‘photon energy’ rather than ‘detected energy’
• Improved pileup corrections
• Separation of nuclear and electron contributions to gamma continuum
• Subtracting ‘pre-event’ imaging contributions

## Software Overview

In terms of function, the visibility software can be divided into three areas:

• Calculation
• Editing
• Interpretation

A typical sequence would be to calculate visibilities from level-0 data using hsi_vis_usershell; save, display, edit and/or combine visibilities depending on the goals of the analysis; and then interpret the resulting visibilities in terms of the properties of the source. To evaluate spectral or temporal differences, this process is repeated (most easily using a script). For now, the intermediary among these routines is an array of ‘visibility structures’ that contain all the information necessary to display, combine and interpret the observed visibilities. The format of the visibility array is the same for the calculation output, editing input and output and interpretation input. Visibility arrays can be SAVEd and RESTOREd as compact .sav files.

## Visibility Cook book

Imaging through the use of visibilities is now fully integrated into the RHESSI imaging object and therefore the RHESSI GUI, as such, for straightforward tasks (e.g. creating an image using visibilities), it is no longer necessary to deal with visibilities independently.

### Using Visibilities From the Command Line

Visibilities are most directly accessed through the imaging object (through the imaging algorithms which make use of visibilities such as MEM_NJIT). As such, creating an imaging object which uses any one of those imaging algorithms also creates in the process of creating an image, the visibility object. The following code creates an image and then gets the visibility object.

 time_range = ['20-Feb-2002 11:06:10.000', '20-Feb-2002 11:06:40.000']
energy_band = [12,25]

obj = hsi_image()
obj-> set, im_time_interval= time_range
obj-> set, energy_band = energy_band
obj-> set, image_algorithm= 'MEM NJIT'
obj-> set, modpat_skip= 4
obj-> set, pixel_size= [1.00000, 1.00000]
obj-> set, time_bin_def= [1.00000, 2.00000, 2.00000, 4.00000, 8.00000, 16.0000, 16.0000, \$
32.0000, 64.0000]
obj-> set, time_bin_min= 512L
obj-> set, use_phz_stacker= 1L
data = obj-> getdata()

You can now plot this image with a simple obj-> plot.

Since the image has been created the visibility object is now available and can be retrieved with

 ov=obj->get(/obj,class='hsi_visibility')

This object contains the visibility bag which get be accessed through the following standard command

 edited_visbag = ov->getdata()

which returns a total of 94 visibilities. On thing to note here is an edited visibility bag where outliers and conjugate visibilities are removed. To get all of the visibility use the following

 full_visbag = ov->getdata(/dump)

which now returns all 400 visibilities. Each visibility in the visibility bag contains the following information.

 ** Structure HSI_VIS, 11 tags, length=72, data length=64:
ISC             INT              3          ;Subcollimator index (=0,,,,8)
HARM            INT              1          ;Harmonic number (=1,2,3)
ERANGE          FLOAT     Array[2]          ;Energy range (keV)
TRANGE          DOUBLE    Array[2]          ; Time range (anytim)
U               FLOAT         0.0424057          ;u=East-west spatial frequency component (arcsec^-1)
V               FLOAT        0.00205243         ;    v=North-south spatial frequency component (arcsec^-1)
OBSVIS          COMPLEX   (      161.854,      16.3286)          ;Observed (semicalibrated) visibility (ph/cm2/s)
TOTFLUX         FLOAT           318.008          ;“Total flux” or  semicalibrated ‘DC’ term (ph/cm2/s)
SIGAMP          FLOAT           27.3827          ;Average statistical error in obsvis components (ph/cm2/s)
CHI2            FLOAT          0.578050          ;Reduced CHI^2 in fitting visibility to stacked event list
XYOFFSET        FLOAT     Array[2]          ;West, north heliocentric offset of phase center (arcsec)

## Documentation

The following is a list of currently available documentation on RHESSI visibilities.