for detection can then be written as $$\mathcal{F}_{\rm det}(\theta) = S_{\rm th}\tau_{\rm FRB} \,\frac{\zeta}{r_1r_2g(\theta)}$$ where $g(\theta)$ is the power gain of the telescope aperture for an angular offset $\theta$ from boresight. The factors $r_1$, and $r_2$ approximately account for dilution of FRB fluence due to time integration, and SNR boost due to the number of independent epochs combined during a detection. They are respectively given by $$r_1 = \left[\frac{\mathcal{M}(\tau_{\rm FRB},\tau_{\rm ds})}{\tau_{\rm int}} \right]_0^1$$ $$r_2 = \left[\sqrt{\frac{\mathcal{M}(\tau_{\rm FRB},\tau_{\rm ds})}{\tau_{\rm int}}} \right]_1^\infty$$ where the subscript and superscript in $\left[. \right]_a^b$ represent the lower and upper bounds for the values within the square brackets, and the function $\mathcal{M}(.)$ yields the largest of its arguments. If a survey observes for $N_{\rm day}$ days, then the expected number of detections is given by. $$\label{eqn:ndet}
N_{\rm det} = N_{\rm day}\int {\rm d}\theta 2\pi\sin\theta\, \mathcal{N}\left( > \mathcal{F}_{\rm det}(\theta)\right)$$ where $2\pi\sin\theta{\rm d}\theta$ is the differential solid angle. Finally, while computing the multi-telescope detection statistics, we will assume that $g(\theta)$ is given by the Airy function: $$\label{eqn:airy}
g(\theta) = \left(2\,\frac{J_1(\pi d/\lambda \sin\theta)}{\pi d/\lambda\sin\theta} \right)^2.$$ As all surveys that we consider operate in approximately the same frequency bands, the effects of frequency-dependent scatter-broadening are absorbed into the assumed burst width.

