Skip to content

Modifying the thick and uniform PBFs

Jump to bottom
Bradley Meyers edited this page Dec 11, 2019 · 3 revisions

Williamson (1972) derived the thick screen and uniform media pulse broadening functions from ray optics principles. In some cases, the elongated tails of these distributions are difficult to reconcile with data. Instead, Williamson also suggested that it would be possible to have rise-times of the scattered pulse be defined by these functions, but the decay defined as the usually expected "thin screen", after an appropriate delay (p68, after Fig. 9). Specifically, the time at which the one-sided exponential decay kicks in (after the PBF peak) is:

  • thickdel for the thick screen model, and
  • uniformdel for the uniform medium model

Initially, a first approximation to implementing this is to just cut the thick/uniform PBF at the appropriate time and then append a scaled thin screen with the same scattering time scale such that at large times the PBF decays to zero. This will introduce discontinuities in the PBFs, which is not desirable. Instead, in tauclean, we make use of a "joining" function, that smoothly transitions between the two regimes.

In this case, we model the modified PBFs as:

combinedpbfs

where transtime and peaktime is the peak time of the relevant PBF (which can be derived analytically). In this case, pbf1 is either the thick or uniform screen, and pbf2 is a thin screen (one-sided exponential) with the same scattering time. The factor, equation, defines how smoothly the functions join -- smaller values make a smoother transition, larger values drive the tanh component of the function towards a step (i.e. Heaviside step function) and thus make the transition more abrupt. This particular functional form was chosen based on this StackExchange post, specifically because it is infinitely smooth.

Generally, the amount you will want to smooth depends on the value of tau, thus we scale the smoothing factor as:

smoothfact

which is somewhat arbitrary, but has empirically worked in our tests. Now, the modified PBF has the rise-time defined in the original PBF with a tail described by a thin screen exponential decay, and is normalised to have unit area so the fluence is conserved.

Clone this wiki locally