14-dec-1997
After inserting realistic numbers for HESSI high-energy count rates, here are some simulated HESSI images, for the 500 keV continuum and the 2.223 MeV line. All of these are generated using the Maximum Entropy Method (MEM). Note that the integration times and energy bands (for the continuum images) are large, due to the small number of counts. So at high energies, you can really expect only one image per flare. Variation of the source over the long integration times may be an issue, and is not included here.
As for the Hessi transmission calculations , Model #1 refers to the grid sizes and thicknesses given in the proposal: grids 1 and 2 have 0.055mm thickness, 3, 4, 5, 7, 8 are 3mm thick, 6 is 20 mm thick, and 9 is 30 mm thick. Model #2 has the grid thicknesses set so that grids 3, 4, 5, 6 have 1 degree fields of view. Model #3 is the same as the proposed configuration, but with the thicknesses of 4 and 5 increased, to the max for a 1 degree field of view. Model #4 has the thicknesses of 4 and 5 increased to that for a 1 degee FOV, and 7 and 8 increased to 6.2 mm. Model #5 has 4, 5, 7, and 8 all having a thickness of 6.0mm.
There is still minimal interpretation here, it should be noted that I have not spent a whole lot of time optimizing the imaging program, e.g., how often the mod. patterns are sampled per rotation, how to bin photon counts in energy and time, which grids should be left out for a given source size, etc... All of this may make a difference.
Now there are two input images, the first is a 2 footpoint source, each footpoint is a gaussian of 2 arcsec width, and the footpoints are separated by 20 arcsec. The second is a 2 footpoint source, with a separation of 150 arcsec, and a strong (1/2 the total brightness) looptop source, to test for imaging of extended sources.
Note that you'll have to click on the appropriate link to look at the images, they're large, and it takes a long time to load this page if i include them inline. Some of the results are included in tables here.
Case 0: Compact Source, Ideal conditions, E = 500 keV, 10 counts/sec-keV totaled for all detectors (this is about 6 counts/sec-keV, obtained from figure D-4 in the HESSI-SMEX proposal, divided by the 60% transmission fraction expected at 500 keV), DT = 300 sec, DE = 60 keV. Zero background, but the effect of counting statistics has been included in calculating the error image. The first image is the input image, then models #1-#5. Images have 2 arcsec pixels. The effects of the thicker grids is taken into account for each model, so the total counts per image are decreased for the models with thicker grids. For this case you can sort resolve the footpoints for Model #1, Model #2 (the max-thickness model) does best. Models #3 and #4 are almost as good as Model #2, as might be expected, since the grid thicknesses of grids 4 and 5 (10 and 20 arcsec resolution) are the same as for Model #2.
The second set of images are error images. These give the expected error in the image pixels, in the same units as the images (which are counts/detector/rotation). These are calculated using a Monte Carlo technique.
In the Table, Total counts refers to the total number of counts in the image, Frac Error is the fractional error for the brightest image pixel, and Source Fraction is the relative fraction of emission in the pixels which are brightest in the input image , so it tells us how much of the brightness goes where it's supposed to be. It's a good measure of the image resolution and contrast, and it clearly shows the effect of the filter thicknesses in each model. The 2d width is the FWHM in arcseconds of a footpoint.
| Total Counts | Frac Error at Max | Source Fraction | 2d width | |
|---|---|---|---|---|
| Model #1 | 142983. | 0.056 | 0.202 | 6.629 |
| Model #2 | 117600. | 0.044 | 0.376 | 5.331 |
| Model #3 | 132596. | 0.036 | 0.347 | 5.498 |
| Model #4 | 124138. | 0.039 | 0.346 | 5.527 |
| Model #5 | 126664. | 0.036 | 0.331 | 5.557 |
Case 1: Compact Source, E = 500 keV, 10 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 60 keV. Signal to noise ratio is 14. Images have 2 arcsec pixels. The images are fuzzier, even though the signal to noise is not that high, and the fraction of emission in the brightest pixels is reduced. Model #2 still gives the best resolution.
| Total Counts | Frac Error at Max | Source Fraction | 2d width | |
|---|---|---|---|---|
| Model #1 | 142998. | 0.272 | 0.120 | 7.573 |
| Model #2 | 118065. | 0.277 | 0.252 | 6.078 |
| Model #3 | 132188. | 0.208 | 0.243 | 6.334 |
| Model #4 | 123536. | 0.236 | 0.232 | 6.363 |
| Model #5 | 126253. | 0.181 | 0.217 | 6.304 |
Case 2: Compact Source, E = 500 keV, 4 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 80 keV. Signal to noise ratio is 5.7. Images have 2 arcsec pixels. Here the count rate has been reduced by a factor of 2.5, the bandwidth can be increased somewhat to help this. Note that in real life, i would have to subtract out the 511 line emission to do this properly. For this case, you can still sort of make out the footpoints, but the images are more spread out, and the errors are much larger.
| Total Counts | Frac Error at Max | Source Fraction | 2d width | |
|---|---|---|---|---|
| Model #1 | 76563. | 0.243 | 0.067 | 14.929 |
| Model #2 | 62117. | 0.264 | 0.165 | 6.953 |
| Model #3 | 70520. | 0.212 | 0.150 | 7.602 |
| Model #4 | 65077. | 0.241 | 0.148 | 7.396 |
| Model #5 | 67691. | 0.232 | 0.143 | 7.750 |
Case 6 (Cases 3, 4 and 5 didn't make the cut.): Extended Source, E = 500 keV, 10 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 60 keV. Signal to noise ratio is 14.3. Images have 4 arcsec pixels. Here models with thicker grids 7 and 8 work well, note the shadow of the looptop in the images for Models #2 and #4, which have 3mm thcknesses for grids 7 and 8. Note that there is no width measurement quoted here, since we don't have point sources.
| Total Counts | Frac Error at Max | Source Fraction | |
|---|---|---|---|
| Model #1 | 99551. | 0.774 | 0.516 |
| Model #2 | 82003. | 0.179 | 0.715 |
| Model #3 | 92319. | 0.692 | 0.530 |
| Model #4 | 86490. | 0.508 | 0.655 |
| Model #5 | 88245. | 0.593 | 0.639 |
Case 7: Extended Source, E = 500 keV, 4 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 80 keV. Signal to noise ratio is 5.7. Images have 4 arcsec pixels. Fewer counts, more background, worse images, not profoundly surprising.
| Total Counts | Frac Error at Max | Source Fraction | |
|---|---|---|---|
| Model #1 | 53094. | 1.712 | 0.362 |
| Model #2 | 43735. | 0.083 | 0.575 |
| Model #3 | 49237. | 1.529 | 0.367 |
| Model #4 | 46128. | 0.959 | 0.495 |
| Model #5 | 47064. | 1.131 | 0.485 |
Case 10: Try the 2.2 MeV line; Line Emission, Compact Source, E = 2223 keV, 24.0 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 2.5 keV. Signal to noise ratio is 400. Images have 2 arcsec pixels. Lots of counts, Great signal to noise ratio, but no modulation for the high resolution grids equals no chance.
| Total Counts | Frac Error at Max | Source Fraction | 2d width | |
|---|---|---|---|---|
| Model #1 | 14311. | 0.373 | 0.027 | 22.788 |
| Model #2 | 11392. | 0.372 | 0.041 | 20.309 |
| Model #3 | 13489. | 0.385 | 0.030 | 19.532 |
| Model #4 | 12888. | 0.549 | 0.028 | 19.562 |
| Model #5 | 13223. | 0.555 | 0.025 | 21.185 |
Case 12: Line Emission, Extended Source, E = 2223 keV, 24.0 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 2.5 keV. Signal to noise ratio is 400. Images have 4 arcsec pixels. Lots of counts, Great signal to noise ratio, and the thick grids in Model #2 enable us to do something with extended sources. All of the other models have a bad shadow image present. This may disappear with better imaging routines, but MEM is helpless.
| Total Counts | Frac Error at Max | Source Fraction | |
|---|---|---|---|
| Model #1 | 9951. | 1.115 | 0.225 |
| Model #2 | 7967. | 1.867 | 0.460 |
| Model #3 | 9346. | 1.198 | 0.226 |
| Model #4 | 8934. | 2.919 | 0.239 |
| Model #5 | 9168. | 2.799 | 0.236 |
Case 13: Line Emission, Extended Source, E = 2223 keV, 2.40 counts/sec-keV totaled for all detectors, DT = 300 sec, DE = 2.5 keV. Signal to noise ratio is 40. Images have 4 arcsec pixels.
| Total Counts | Frac Error at Max | Source Fraction | |
|---|---|---|---|
| Model #1 | 995. | 1.364 | 0.096 |
| Model #2 | 797. | 12.316 | 0.206 |
| Model #3 | 935. | 1.780 | 0.094 |
| Model #4 | 893. | 9.534 | 0.097 |
| Model #5 | 917. | 8.511 | 0.096 |
What do we conclude from this exercise?
(1) at 500 keV, the baseline model (Model #1) will only be useful for separating footpoints, or distinguishing looptops from footpoints, for the largest flares. For smaller flares, a model with thicker grids (especially grids #4 and #5) is better. These models can distinguish footpoints of 20 arcsec resolution for weaker sources, but at the price of increased accumulation time, and larger energy bins (note that increasing the energy binsize is not necessarily an option for line emission). Even for these models, there is a limit, a low signal-to-noise ratio and photon statistics will kill you. Here this limit seems to be about a factor of 10 in flare size, where the signal-to-noise is about 1.0. For the extended sources, the models with thicker grids 7 & 8 are best. As always, Model #2 gives the best images, but Models #4 and #5, which have 6mm grid thickness, also do a good job. Models #1 and #3, which have thin grids 7 & 8 are less useful.
(2) At 2200 keV, there is no chance for any image resolution below 30 arcsec (i.e., grid 6 resolution) For any model. Only Model #2, which has grids 7 and 8 30mm thick, gives good results for the extended looptop plus 2 footpoint source. The intermediate models with 6mm grids 7 and 8 are only a marginal improvement on the baseline model for extended sources at 2.2 MeV.
A list of all of the HESSI test images using the different grid models?
Comments to: jimm@ssl.berkeley.edu
