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h2. Comparison of gtdiffrsp output with gtsrcmaps values
* The diffuse response quantities are proportional, up to an energy-dependent factor ({latex}$s(E)${latex}), to the probability densities of a given event for the corresponding source models.  If {latex}$\tilde{S}(\hat{p})${latex} is the spatial distribution of the diffuse component, then the diffuse response is\\
{latex}
\newcommand{\phat}{{\hat{p}}}
\newcommand{\phatp}{{\hat{p}^\prime}}
\newcommand{\E}{{\epsilon}}
\newcommand{\Ep}{{\E^\prime}}
\begin{eqnarray}
d_0(\Ep, \phatp) &= &\int d\phat \tilde{S}(\phat) P(\phatp; \E, \phat, t) A(E, \phat, t) D(\Ep; \E, \phat, t)\\
    &= &\int d\phat \tilde{S}(\phat) P(\phatp; \Ep, \phat, t) A(\Ep, \phat, t)
\end{eqnarray}
Here, $\phat$ and $E$ are true photon direction and energy, $t$ is the arrival time, primes indicate measured quantities, $P$ is the PSF, $A$ is the effective area, and $D$ is the energy dispersion (taken to be a delta function in energy in the second line).
{latex}
* In gtsrcmaps the spatial distribution is multiplied by the exposure, {latex}$E${latex} and convolved with the mean PSF:\\
{latex}
\newcommand{\phat}{{\hat{p}}}
\newcommand{\phatp}{{\hat{p}^\prime}}
\newcommand{\E}{{\epsilon}}
\newcommand{\Ep}{{\E^\prime}}
\begin{eqnarray}
E(\Ep, \phat) &=& \int dt A(\Ep, \phat, t)\\
P_{\rm avg}(\phatp; \Ep, \phat) &=& \frac{1}{E(\Ep, \phat)}
              \int dt A(\Ep, \phat, t) P(\phatp; \Ep, \phat, t)\\
d_1(\Ep, \phatp) &=& \int_{\Delta\phatp} \int d\phat E(\Ep, \phat) P_{\rm avg}(\phatp; \Ep, \phat) \tilde{S}(\phat)
\end{eqnarray}
$E$ is the time-integrated exposure and $P_{\rm avg}$ is the average PSF.  The integral over $\phatp$ is over the pixel size ($\Delta\phatp$).  For an inertial pointing, $E$ is just the effective area times the livetime $\Delta t$ and we should have
\begin{equation}
d_0 = \frac{d_1}{\Delta t \Delta \phatp}
\end{equation}
{latex}
The lhs of the above equation is plotted vs the rhs below for extragalactic (isotropic) and gll_iem_v02.fit diffuse models.  The red curves are best fit power-laws with the index equal to unity.  Other than a constant factor of approximately 2 in both plots (2.04, 1.95), the relation holds fairly well.  Replacing the gtdiffrsp values in the FT1 file with the gtsrcmaps values does not result in a substantial change in the fit of the two components.
!difrsp_comparison.png|thumbnail!