Background subtraction

Background subtraction estimates a source contribution by removing a sky or detector-background measurement. An independent background estimate adds photon shot noise; a mismatched background adds bias of an estimator.

If emitted power obeys $P=A{T}^4$ with known constant $A$, then small relative temperature uncertainty is one quarter of relative net-power uncertainty. For measured quantities $X,Y$ with covariance $C_{XY}$, $\mathrm{Var}(X-Y)={\sigma }_X^2+{\sigma }_Y^2-2C_{XY}$. A weak net signal obtained by subtracting a large background can therefore have poor relative precision even when both original powers are measured accurately. Correlated standard uncertainties and worst-case error bounds require different propagation rules.

Let $X\sim \mathrm{Poisson}(Q+B)$ and $Y\sim \mathrm{Poisson}(fB)$ be independent. Then $X-Y$ has source-estimation bias $(1-f)B$ and variance $Q+(1+f)B$. Its accuracy ratio $Q/\sqrt{\mathbb{E}[(X-Y-Q{)}^2]}$ tends to $R_Q/(|1-f|R_B)$ when $Q=R_{Q}T$, $B=R_{B}T$ and $T$ grows. This uses mean squared error, not variance alone, and assumes a fixed uncorrected mismatch.