PDF of ${\tau }_{\mathrm{obs}}-\widehat{\tau }$. Top panel: Case for FRB 20220509G as an example (see Sect. 4 for details), assuming log $(A_{\tau }\stackrel{\sim }{F}G)=-1$. The peak of the PDF is at ${\tau }_{\mathrm{obs}}-\widehat{\tau }=0$, and the standard deviation is σ_τ = 0.02 ms (see also Table 1). Bottom panel: FRBs with an upper limit of the scattering were given a flat PDF. We take FRB 20121102A as an example with the observed scattering of τ_obs < 9.6 ms (Sect. 4 and Table 1), assuming log $(A_{\tau }\stackrel{\sim }{F}G)=-1$. When $\widehat{\tau }$ is lower than 0 or greater than τ_obs, the probability is 0. When $\widehat{\tau }$ falls between 0 and τ_obs, the probability is nonzero and flat. We note that the vertical axes in both panels indicate relative quantities of the probability density.