C. Correction for Photometric Bias

Although the ae_better_backgrounds algorithm (§E.6) seeks to define a background region that is unbiased, i.e. one where the flux from each neighbor that we expect in the background exactly matches the contamination we expect from that neighbor, the algorithm will sometimes be forced to construct a background that has significant photometric bias over the nominal energy band. This is very undesirable because inaccuracy in the broad band background level translates to inaccuracy in the source significance statistics (§5.10) which are commonly used for pruning the source catalog, as well as inaccuracy in photometry. Thus, the algorithm includes a final step which seeks to adjust the scaling of the background spectrum so as to correct for the photometric bias. That corrected scaling is derived below.

First, note that the algorithm described above produces several important quantities for each source:

• $N$: the integer number of in-band counts observed in the background region.

• $BACKSCAL_{src}$, $BACKSCAL_{bkg}$: the measures (integrals of the exposure map) of the source and background regions. For convenience we will denote the nominal background scaling as $S = \frac{BACKSCAL_{src}}{BACKSCAL_{bkg}}$.

• $B_p$: the total number (real-valued) of counts from neighboring point sources expected to contaminate the source region, obtained by integrating models of the neighboring sources over the source region.

• $N_p$: the total number (real-valued) of counts from neighboring point sources expected to be observed in the background region, obtained by integrating models of the neighboring sources over the background region..

• $\Delta_p = B_p - N_p * S$: the estimated ``photometric bias'' of the background region (with regard to contamination from neighboring point sources), when the nominal scaling is applied. As said earlier, the algorithm seeks to drive $\Delta_p$ to zero (among other goals).

Second, we infer the number (real-valued) of background counts observed in the source region but not attributed to detected point sources as

$$N_f = \max(N - N_p, 0)$$

Since we have no model for the spatial distribution of this background component, we assume it is flat. Thus the number of counts from this background component expected in the source region is

$$B_f = N_f * S$$

Third, we wish to choose a correction to the background scaling, $c$, such that the final scaled background is unbiased with respect to our models, i.e.

$$\begin{eqnarray*} (N_p + N_f) * c * S & = & B_p + B_f \\ & = & (N_p * S + \Delta_p) + N_f * S \\ & = & (N_p + N_f)*S + \Delta_p \end{eqnarray*}$$

Solving for $c$ we have

$$c = 1 + \frac{\Delta_p}{(N_p + N_f) * S}$$

Note that when there is no photometric bias, $\Delta_p = 0$, $c=1$, and the nominal background scaling (ratio of the measures of the source and background regions) is used. When $\Delta_p$ is positive (we estimate that our background region does not contain enough power from neighboring sources), then $c>1$ and we choose to subtract more background than the nominal background scaling ($S$) would call for. When $\Delta_p$ is negative then $c<1$ and we choose to subtract less background than the nominal background scaling would call for. In the case where the estimated flat background component is zero, i.e. the model predicts more counts from neighboring point sources in the background region than were actually observed ($N_p>N$), then we must assume that $N_f$ and $B_f$ are zero, and we have

$$c = \frac{B_p}{N_p * S}$$