PSF_3GAUSS

Multi-Gauss mixture models are a common way to model distributions (for source intensity profiles, PSFs, anything really), see e.g. 2013PASP..125..719H. For H.E.S.S., radial PSFs have been modeled as 1, 2 or 3 two-dimensional Gaussians \(dP/d\Omega\).

Note

A two-dimensional Gaussian distribution \(dP/d\Omega = dP/(dx dy)\) is equivalent to an exponential distribution in \(dP/x\), where \(x=r^2\) and a Rayleigh distribution in \(dP/dr\).

In this format, the triple-Gauss distribution is parameterised as follows:

\[dP/d\Omega(r, S, \sigma_1, A_2, \sigma_2, A_3, \sigma_3) = \frac{S}{\pi} \left[ \exp\left(-\frac{r^2}{2\sigma_1^2}\right) + A_2 \exp\left(-\frac{r^2}{2\sigma_2^2}\right) + A_3 \exp\left(-\frac{r^2}{2\sigma_3^2}\right) \right],\]

where \(S\) is SCALE, \(\sigma_i\) is SIGMA_i and \(A_i\) is AMPL_i (see columns listed below).

TODO: give analytical formula for the integral, so that it’s easy to check if the PSF is normalised for a given set of parameters.

TODO: give test case value and Python function for easy checking?

Note

By setting the amplitudes of the 3rd (and 2nd) Gaussians to 0 one can implement double (or single) Gaussian models as well.

Columns:

  • ENERG_LO, ENERG_HI – ndim: 1, unit: TeV
    • True energy axis
  • THETA_LO, THETA_HI – ndim: 1, unit: deg
  • SCALE – ndim: 2, unit: none
    • Absolute scale of the 1st Gaussian
  • SIGMA_1, SIGMA_2, SIGMA_3 – ndim: 2, unit: deg
    • Model parameter (see formula above)
  • AMPL_2, AMPL_3 – ndim: 2, unit: none
    • Model parameter (see formula above)

Recommended axis order: ENERGY, THETA

Header keywords:

As explained in HDU classes, the following header keyword should be used to declare the type of HDU:

Example data file: TODO