Attenuation fit normalized to a total reference rate ($N(n)=(N_{\mathrm{ref}}-B)e^{-bn}+B$)

If a measured unattenuated reference rate includes the same additive background as subsequent measurements, then $N_{\mathrm{ref}}=S_0+B$, where $S_0$ is signal alone. Substituting $S_0=N_{\mathrm{ref}}-B$ into an attenuation count model with an additive background gives $N(n)=(N_{\mathrm{ref}}-B)e^{-bn}+B$. This automatically returns the measured reference at $n=0$ and tends to $B$ at strong attenuation. Adding $B$ directly to $N_{\mathrm{ref}}{e}^{-bn}$ would count background twice at zero attenuation. A joint count likelihood can fit the reference and attenuated data with their separate exposure times, avoiding the need to regard the reference estimate as exact.