Weekly Report 27Aug2010
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Visibility Forward Fit - The reduced chi2 and C stastistic (again)
In the following studies an eventlist file for a simulated point source at disk center was used as input to the HESSI image object. RHESSI images were reconstructed from this file using the visibility forward fit algorithm. In all cases below the model of the reconstructed image was a circular gaussian. All parameters of the model were allowed to vary for up to 2000 iterations or until the lowest reduced chi2 was achieved except for one. In the first case the full width half max (FWHM) was held fixed at values ranging from .1 arcsecs to 3 arcsecs. In the second the flux was held fixed at values ranging from 1 photon/cm2/sec to 40 photons/cm2/sec. The reduced chi2, the total C statistic, and the C statistic for each detector were calculated at each point.
Visibility Forward Fit - the reduced chi2 and C statistic for reconstructed images with the FWHM held constant
The plot below shows the reduced chi2 as a function of the FWHM. The FWHM was held at a fixed value and all other parameters were allowed to iterate until the lowest value of the reduced chi2 for the reconstructed image was reached.
The reduced chi2 has a minimum value of .80 at a FWHM of approximately 1 arcsec. Below this value the reduced chi2 for the image remains very close to minimum, When all parameters except for the FWHM are allowed to vary the values of the FWHM between ~.1 arcsec and ~1.1 arcsec all yield reduced chi2 which only differ in the third significant digit meaning that any of these could be considered the best reconstructed image for this source. Sources smaller than 1.1 arcsec all appear the same to the detectors. In order to decide between these images another restriction such as a known value of the flux would have to be imposed. Above ~1.1 arcsec the reduced chi2 rises rapidly as expected. There is very low probability that any of these reconstructed images are accurate.
The plot below shows the total C statistic as a function of the FWHM. The value of the total C statistic is it's value when the reduced chi2 has reached it's minimum.
When the size of the FWHM is below the pixel size used to create the image the total C statistic is at a value of approximately 1.09 arcsec. The lack of variation suggests that the image looks the same to all detectors. As the FWHM passes the threshold of .5 arcsec it immediately drops to its lowest value. It stays at a value of ~1.05 until the FWHM reaches ~1 arcsec. It then begins to rise as the reduced chi2 did, however the rise is not as steep.
The plot below shows the C statistic for each detector individually.
Each individual detector shows the characteristic behavior of the total C statistic for values of the FWHM below the size of the pixels used to make the image. The value of the C statistic is constant until the the FWHM becomes larger than the pixel size at .6 arcsec, at this point there is a drop in the value of the C statistic. The detectors with coarser grids in front of them (detectors 8 and 9) start with a higher C statistic in the regime where the FWHM is smaller than the pixel size and drop the most at the point were the FWHM gets larger than the pixel size, however they then start to rise dramatically immediately after this. With good statistics for detectors 8 and 9 it is not expected that the C statistic should rise. Given that the the input is a point source we would expect the C statistic to remain fairly constant as a function of the FWHM. However for these two detectors we have extremely poor statistics for the fit to the visibilities, detector 8 has only two valid visibility measurements and detector nine one. This may explain why the C statistic rises so rapidly for these detectors. The other detectors have better statistics and while they rise from their low after the transition point they do so gradually with the increase in FWHM.
Looking at the three plots it is difficult to pinpoint which value of the FWHM minimizes the reduced chi2 or the C statistics. There is a range around a FWHM of ~.9 arcsec where the measurements of the reduced chi2 for various values of the FWHM is too close to pick one as the minimum. If all other parameters are fitted one one can only pick a range of possible values for the FWHM. If the source size is small visibility forward fit is not suitable for determining the size. We need to try this with larger sources to check this result and determine the regime where this imaging algorithm is effective for determining source size.
Visibility Forward Fit - the reduced chi2 and C statistic for reconstructed images with the flux held constant
The plots below where constructed in the same manner as those where FWHM held fixed, except only the flux was held fixed for these images.
The plot below shows the reduced chi2 as a function of the flux.
In this case there is a clear minimum in the reduced chi2 at a flux value of ~20 ph/cm2/sec. The curve follows the expected pattern of having a single unique minimum.
The plot below shows the total C statistic at the points where the reduced chi2 was at a minimum for each value of the flux.
It is immediately noticeable that for values of the flux less than the value at the minimum of the reduced chi2 the total C statistic rises rapidly, however for values higher than this the total C statistic curve does not rise as rapidly. I think the reason for this can be explained by the C statistics for each detector below.
The plot below shows the C statistic foe each detector as a function of the flux.
The single detector C statistics all show a rapid rise in the C statistic for values of the flux below the point where the reduced chi2 is at a minimum. However above this point only the detectors with coarser grids in front of them show this rapid rise. The detectors with finer grids show a smaller increase. Once again I think this is due to the number of visibilities for each detector. The detectors with finer grids have an order of magnitude (~1 compared to ~10) more measurements of the visibilities. The fit of the visibilities is weighted towards these measurements so the fit is usually closer to the measured value. At values of the flux below the value at the minimum of the reduced chi2, the fit to the visibilities of the detectors with the finer grids is forced to be lower than the measured values causing a high C statistic. Above these values the fit attempts to first match the values of the detectors with finer grids and higher statistics meaning the C statistic does not increase as rapidly for the higher fluxes. The finer detectors dominate the total C statistic so it's shape is determined strongly by them.
The visibility forward fit algorithm appears to be good for determining the flux of a source. One still has to be careful using the C statistic as measure, taking note of the number of visibilities used for each detector when creating a image. At worst this method seems like a good algorithm for determining a lower limit on the flux.
IDL procedure to analyze simulated data with different image reconstruction algorithms
I've worked on an IDL procedure that creates a reconstructed image from a simulated source eventlist with an algorithm of the users choosing. Given the original data map it can then create a flux map of the root mean squares difference of the reconstructed image and the original data map. I've only been able to do this for a point source at the center of the disk because I do not have the original maps for the Gordon Emsilie simulations. Of course, I just realized I should have made a JPEG of this map and displayed it. I'll add it to this space on Sunday.
Ordered New Drives for Wilco
Researched and ordered new hard drives to add to the server wilco which we are currently using as the prepserver for images from various solar missions.
Goals for next week
- Run the above analysis of visibility forward fit with an elliptical gaussian and compare the results to those for a circular gaussian. This can be done very easily with my current scripts with minor changes.
- Use the monte carlo methods to determine the 1 sigma levels for the reduced chi2 maps shown above. I'll talk to Kim about this.
- Make my procedure for creating the difference maps of simulated data and the reconstructed RHESSI images more general. Currently it is setup specifically for the parameters I used to create the reconstructed image.
- The new RHESSI website is back online. I plan to import as many pages from the old site as I can. I'll meet with Steven Christe and Albert Shih to discuss which pages they would like to be responsible for moving and maintaining.
