# 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: sr^(-1)
• 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

• HDUDOC = ‘https://github.com/open-gamma-ray-astro/gamma-astro-data-formats
• HDUVERS = ‘0.2’
• HDUCLASS = ‘GADF’
• HDUCLAS1 = ‘RESPONSE’
• HDUCLAS2 = ‘PSF’
• HDUCLAS3 = ‘FULL-ENCLOSURE’
• HDUCLAS4 = ‘PSF_3GAUSS’