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The DRM is the matrix that transforms a binned counts spectrum from true energies to measured energies:

{latex}
\begin{equation}
n_{k^\prime} = \sum_k D_{kk^\prime} n_k  \nonumber
\end{equation}
{latex}

where

{latex}$n_{k^\prime}${latex}

{latex}$k^\prime${latex}

$n^k${latex}$n^k${latex}

$k${latex}$k${latex}

{latex}
\begin{equation}
D_{kk^\prime} = \frac{\int d\theta d\phi \left[\int_{\Delta E_{k^\prime}} dE^\prime D(E^\prime; E_k, \theta, \phi)\right] A(E_k, \theta, \phi) lt(\theta, \phi)}{\int d\theta d\phi A(E_k, \theta, \phi) lt(\theta, \phi)}  \nonumber
\end{equation}
{latex}

Here

{latex}$D(E^\prime; E, \theta, \phi$){latex}

{latex}$A(E, \theta, \phi)${latex}

{latex}$lt(\theta, \phi)${latex}

{latex}$E_k${latex}

$k${latex}$k${latex}

{latex}$k^\prime${latex}

In principle, the DRM should be evaluated at each sky pixel position in the binned counts map, but we make the approximation that the DRM does not change much over the counts map region and just evaluate it at the map center and assume it applies everywhere on the map. This is supported by these plots of the energy dispersion evaluated at various points on the sky for a one-day survey mode integration:

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1. To save execution time, the log-likelihood is evaluated using only the pixels that have non-zero data counts (as opposed to model counts). This can be seen in the expression for the Poisson log-likelihood,
   

   Latex
   Wiki Markup
   {latex} \begin{equation} \log{\cal L} = \sum_j \left[n_j\log\theta_j - \theta_j\right] \nonumber \end{equation} {latex}

   
   where

   Wiki Markup
   Latex
   {latex}$n_j${latex}

   is the observed count in pixel unmigrated-inline-wiki-markup{

   Latex
   }$j${latex}

   and

   Wiki Markup
   Latex
   {latex}$\theta_j${latex}

   is the model counts for that pixel. In the current implementation, this means that model counts are only computed for those non-zero pixels. For sparse datasets (e.g., short integration times or at higher energies), this speeds up the calculation substantially: at least an order of magnitude for 1 day integrations. Applying the DRM to each pixel counts spectrum would require the model calcuation to be made for every pixel in the counts cube, even if it is empty. However, assuming binned analysis is generally used for longer observations, and most pixels are occupied, this may not be a real limitation.
2. A more serious problem is applying the DRM convolution to the true counts spectrum at each pixel location. If the number of energy bins is

   Wiki Markup
   Latex
   {latex}$n_e${latex}

   , the convolution is an

   Wiki Markup
   Latex
   {latex}${\cal O}(n_e^2)${latex}

   operation. For

   Wiki Markup
   Latex
   {latex}$n_e = 30${latex}

   , this results in about a factor of 900 slow down in computing the expected pixel counts that go into the log-likelihood.

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

$j${latex}$j${latex}

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