Detector Response Matrix (DRM) Implementation
The DRM is the matrix that transforms a binned counts spectrum from true energies to measured energies:
\begin
n_
= \sum_k D_
n_k
\end
where
$n_
$
are the counts in measured energy bin
$k^\prime$
and
$n^k$
are the counts in true energy bin
$k$
. The DRM calcuation follows that performed in gtrspgen
:
\begin
D_
= \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)}
\end
Here
$D(E^\prime; E, \theta, \phi$)
is the energy dispersion function,
$A(E, \theta, \phi)$
is the effective area, and
$lt(\theta, \phi)$
is the integrated livetime as a function of detector coordinates associated with the specified sky position.
$E_k$
is the logarithmic center of the
$k$
th true energy bin. The integral over measured energy is taken over the width of the
$k^\prime$
th bin.
In princple, 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:
