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Since we need to compute the total predicted counts for each component, we have the total counts spectrum in true energy space (This is easily obtained for each source by integrating each energy plane over angle in the associated source map and multiplying by the spectral function.) We can then convolve the spatially integrated true energy counts spectrum with the DRM to obtain the overall measured energy counts spectrum. We form the ratio of the convolved model counts to unconvolved model counts in each energy bin. When computing the contribution to the log-likelihood from each pixel

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, we multiply the unconvolved model counts for that pixel by the ratio of convolved to unconvolved model counts from the spatially integrated spectrum. This procedure does not really account for any differences in counts spectra that arise from spatial variation of the source spectrum (for diffuse sources), exposure variations across the map region, and the effect of the energy-dependence of the PSF on the counts spectrum (the observed counts spectrum for a point source should be softer near the source location). All of these neglected effects (they may be others I am missing) are ameliorated somewhat by the fact that this procedure has the right local effect, i.e., energy bins on the falling part of the counts spectrum tend to have their count increased while bins on the rising part have their count decreased. This procedure will probably not work very well for sharp spectral features.

Performance

{composition-setup}{}

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Composition Setup

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{deck:id=My Deck}

Deck of Cards

id	My Deck

{card:label=Power-law model counts spectra for various photon indices}

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label	Power-law model counts spectra for various photon indices

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I've simulated a single point source with indices 1.25, 1.50, 2.00, 2.50, 3.00, generating 70-100k events for each case for a week long observation with idealized +/-50 deg rocking and fit those data with and without energy dispersion handling enabled:

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MC index	edisp handling	no edisp handling
1.25	-1.253 +/- 1.86e-03	-1.260 +/- 1.86e-03
1.50	-1.505 +/- 2.19e-03	-1.516 +/- 2.19e-03
2.00	-2.007 +/- 3.20e-03	-2.031 +/- 3.20e-03
2.50	-2.525 +/- 4.17e-03	-2.557 +/- 4.03e-03
3.00	-3.036 +/- 4.97e-03	-3.063 +/- 4.74e-03

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{card:label=Exponential }

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label	Exponential

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cut-off

power-law,

various

photon

indices

For these tests, a point source with an exponentially cut-off power-law spectrum is modeled and fit for different values of the photon index. The cutoff energy is fixed at 1 GeV, and values of photon index = 1.25, 1.5, 2.0, 2.5, 3.0 are used. These tests are similar to those shown in fig. 68 of v1r0 of the LAT performance paper.

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Input XML model to gtobssim

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<source_library title="cutoff_pl_source">
  <source name="cutoff_pl_source_3">
    <spectrum escale="MeV" flux="20" particle_name="gamma">
      <SpectrumClass name="FileSpectrum"
                     params = "flux=1, specFile=Cutoff_PowerLaw_-3.00.txt"/>
      <celestial_dir ra="266.4" dec="-28.9"/>
    </spectrum>
  </source>
  <source name="cutoff_pl_source_2.5">
    <spectrum escale="MeV" flux="7.8" particle_name="gamma">
      <SpectrumClass name="FileSpectrum"
                     params = "flux=1, specFile=Cutoff_PowerLaw_-2.50.txt"/>
      <celestial_dir ra="266.4" dec="-28.9"/>
    </spectrum>
  </source>
  <source name="cutoff_pl_source_2">
    <spectrum escale="MeV" flux="3" particle_name="gamma">
      <SpectrumClass name="FileSpectrum"
                     params = "flux=1, specFile=Cutoff_PowerLaw_-2.00.txt"/>
      <celestial_dir ra="266.4" dec="-28.9"/>
    </spectrum>
  </source>
  <source name="cutoff_pl_source_1.5">
    <spectrum escale="MeV" flux="1.14" particle_name="gamma">
      <SpectrumClass name="FileSpectrum"
                     params = "flux=1, specFile=Cutoff_PowerLaw_-1.50.txt"/>
      <celestial_dir ra="266.4" dec="-28.9"/>
    </spectrum>
  </source>
  <source name="cutoff_pl_source_1.25">
    <spectrum escale="MeV" flux="0.73" particle_name="gamma">
      <SpectrumClass name="FileSpectrum"
                     params = "flux=1, specFile=Cutoff_PowerLaw_-1.25.txt"/>
      <celestial_dir ra="266.4" dec="-28.9"/>
    </spectrum>
  </source>
</source_library>

The flux for each source was adjusted so that 100000 events are generated for a week long idealized survey mode observation using P7SOURCE_V6.

The template spectra were generated with this python script, using the PLSuperExpCutoff model that is available from pyLikelihood. This same model was used in the spectral fitting to ensure consistency.
Results
P7SOURCE_V6

edisp-off
edisp handling turned off
edisp-on
edisp handling turned on
MC
edisp handling turned off, but using MCENERGY values
The red and green points substantially agree, as one would hope, and the trend is towards a harder measured spectra, as expected. However, there seems to be a residual bias in the edisp-on and MC results towards harder spectra.
DC1A (source fluxes were adjust to produce 200000 events)

Here the differences between the three cases is much smaller, but edisp-on and MC still agree better. The same residual bias persists.
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{card:label=Comparison with Instrument Performance Paper Figure}
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label Comparison with Instrument Performance Paper Figure
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Figure 68 of the instrument performance paper (v1r0) shows the "fractional" error on the photon index for a power-law and cutoff power-law model for different input indices:

Using the same data that went into those simulations, here are the comparisons using energy dispersion handling:

In each plot, the black points are obtained from refitting the data without energy dispersion handling (consistent with the paper figure), and the red points are the results obtained with energy dispersion turned on.
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{card:label=Soft }
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label Soft
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power-law
source,
isotropic
diffuse
background

The xml model that was input to gtobssim

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<source_library title="soft_source">
  <source flux="1" name="soft_source">
    <spectrum escale="MeV">
      <particle name="gamma">
        <power_law emax="1000000.0" emin="20.0" gamma="2.5"/>
      </particle>
      <celestial_dir dec="-28.9" ra="266.4"/>
    </spectrum>
  </source>
  <source name="test_isotropic">
    <spectrum escale="MeV">
      <SpectrumClass name="Isotropic" params="100, 2.1, 20., 2e5"/>
      <use_spectrum frame="galaxy"/>
    </spectrum>
  </source>
</source_library>

The model used in the binned analysis

Code Block

<?xml version="1.0" ?>
<source_library title="source library">
  <source name="test_source" type="PointSource">
<!-- point source units are cm^-2 s^-1 MeV^-1 -->
    <spectrum type="PowerLaw2">
      <parameter free="1" max="1000.0" min="1e-05" name="Integral" scale="1e-06" value="1.0"/>
      <parameter free="1" max="-1.0" min="-5.0" name="Index" scale="1.0" value="-2.0"/>
      <parameter free="0" max="200000.0" min="20.0" name="LowerLimit" scale="1.0" value="20.0"/>
      <parameter free="0" max="200000.0" min="20.0" name="UpperLimit" scale="1.0" value="2e5"/>
    </spectrum>
    <spatialModel type="SkyDirFunction">
      <parameter free="0" max="360.0" min="-360.0" name="RA" scale="1.0" value="266.4"/>
      <parameter free="0" max="90.0" min="-90.0" name="DEC" scale="1.0" value="-28.9"/>
    </spatialModel>
  </source>
  <source name="Extragalactic Diffuse" type="DiffuseSource">
    <spectrum type="PowerLaw2">
      <parameter free="1" max="1000.0" min="0.0" name="Integral" scale="1e-06" value="1.0"/>
      <parameter free="1" max="-1.0" min="-5.0" name="Index" scale="1.0" value="-2.1"/>
      <parameter free="0" max="200000.0" min="20.0" name="LowerLimit" scale="1.0" value="20.0"/>
      <parameter free="0" max="200000.0" min="20.0" name="UpperLimit" scale="1.0" value="200000.0"/>
    </spectrum>
    <spatialModel type="ConstantValue">
      <parameter free="0" max="10.0" min="0.0" name="Value" scale="1.0" value="1.0"/>
      <parameter free="0" max="10.0" min="0.0" name="Value" scale="1.0" value="1.0"/>
    </spatialModel>
  </source>
</source_library>

Distributions from simulations. The leftmost plot in each row is the parameter distribution obtained from the standard analysis with the energy dispersion handling turned off. The middle plot has energy dispersion handling turned on. The rightmost plot uses the MC energy values in the analysis and has energy dispersion handling turned off.
- Photon Index of the soft point source.
- Photon Index of the isotropic component.
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  {card:label=Exponential }
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  label Exponential
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  cut-off
  with
  isotropic
  diffuse

Input model to gtobssim:

Code Block

<source_library title="cutoff_pl_source">
  <source name="cutoff_pl_source">
    <spectrum escale="MeV" flux="1" particle_name="gamma">
      <SpectrumClass name="FileSpectrum"
                     params = "flux=1e-2, specFile=$(rootdir)/Cutoff_PowerLaw.txt"/>
      <celestial_dir ra="266.4" dec="-28.9"/>
    </spectrum>
  </source>
  <source name="test_isotropic">
    <spectrum escale="MeV">
      <SpectrumClass name="Isotropic" params="100, 2.1, 20., 2e5"/>
      <use_spectrum frame="galaxy"/>
    </spectrum>
  </source>
</source_library>

Likelihood model

Code Block

<?xml version="1.0" ?>
<source_library title="source library">
  <source name="test_source" type="PointSource">
<!-- point source units are cm^-2 s^-1 MeV^-1 -->
    <spectrum type="PLSuperExpCutoff">
      <parameter free="1" max="1e3" min="1e-5" name="Prefactor" scale="1e-07" value="1.0"/>
      <parameter free="1" max="0.0" min="-5.0" name="Index1" scale="1.0" value="-1.7"/>
      <parameter free="0" max="1000.0" min="50.0" name="Scale" scale="1.0" value="200.0"/>
      <parameter free="1" max="30000.0" min="500.0" name="Cutoff" scale="1.0" value="3000.0"/>
      <parameter free="0" max="5.0" min="0.0" name="Index2" scale="1.0" value="1.0"/>
    </spectrum>
    <spatialModel type="SkyDirFunction">
      <parameter free="0" max="360.0" min="-360.0" name="RA" scale="1.0" value="266.4"/>
      <parameter free="0" max="90.0" min="-90.0" name="DEC" scale="1.0" value="-28.9"/>
    </spatialModel>
  </source>
  <source name="Extragalactic Diffuse" type="DiffuseSource">
    <spectrum type="PowerLaw2">
      <parameter free="1" max="1e3" min="1e-5" name="Integral" scale="1e-06" value="1.0"/>
      <parameter free="1" max="-1.0" min="-5.0" name="Index" scale="1.0" value="-2.1"/>
      <parameter free="0" max="200000.0" min="20.0" name="LowerLimit" scale="1.0" value="20.0"/>
      <parameter free="0" max="200000.0" min="20.0" name="UpperLimit" scale="1.0" value="2e5"/>
    </spectrum>
    <spatialModel type="ConstantValue">
      <parameter free="0" max="10.0" min="0.0" name="Value" scale="1.0" value="1.0"/>
    </spatialModel>
  </source>
</source_library>

Distributions
- Photon index (Index1) of the exponentially cutoff power-law of the point source.
- Cutoff energy of the point source.
- Photon index of the isotropic component.
  
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