### B The PISA profiling function

The PISA profiling function is made up of three functions, a Gaussian, an exponential and a Lorentzian. Inside a radius ${R}_{c}$ fixed proportions of the Gaussian and Lorentzian functions are used, outside of ${R}_{c}$ the exponential replaces the Gaussian and is added to the continuing Lorentzian. The exponential is joined smoothly to the Gaussian. Inside of ${R}_{c}$ the function takes the form:-

$\begin{array}{c}\hfill 1\hfill \\ \hfill \overline{\pi {\sigma }^{2}\left(1+\left(\frac{\tau }{2ln\left(\frac{1}{\tau }\right)}\right)\right)}\hfill \end{array}\left(\begin{array}{c}\hfill Q\hfill \\ \hfill \overline{\left(1+\frac{{r}^{2}}{{\sigma }^{2}ln\left(2\right)}\right)}\hfill \\ \hfill \hfill \end{array}\begin{array}{c}\hfill +\left(1-Q\right)exp\left(\frac{-{r}^{2}}{{\sigma }^{2}}\right)\hfill \\ \hfill \hfill \end{array}\right)$

and outside of ${R}_{c}$ it takes the form:-

$\begin{array}{c}\hfill 1\hfill \\ \hfill \overline{\pi {\sigma }^{2}\left(1+\left(\frac{\tau }{2ln\left(\frac{1}{\tau }\right)}\right)\right)}\hfill \end{array}\left(\begin{array}{c}\hfill Q\hfill \\ \hfill \overline{\left(1+\frac{{r}^{2}}{{\sigma }^{2}ln\left(2\right)}\right)}\hfill \\ \hfill \hfill \end{array}\begin{array}{c}\hfill +\underset{\right)}{\left(1-Q\right)exp\left(\frac{-2r}{\sigma }\sqrt{ln\left(\frac{1}{\tau }}\right)}\hfill \end{array}\right)̲\hfill \tau \hfill \\ \hfill \hfill$

where:-

$\tau =$
the fraction of the peak intensity at which to change from the gaussian to an exponential function (CROSS/100),
$\sigma =$
the gaussian function sigma (GSIGM),
$Q=$
the fraction of the Lorentzian function to add to the gaussian or exponential function at each point (COMIX).

The radius at which the exponential replaces the gaussian is:-

${R}_{c}=\sigma \sqrt{ln\left(\frac{1}{\tau }\right)}$

The actual function is that which when multiplied by the integrated intensity gives the intensity at the given radius. Basically the functional forms from which the above equations are derived are:-

Gaussian — $exp\left(\frac{-{r}^{2}}{{\sigma }^{2}}\right)$
Lorentzian — $1/\left(1+\frac{{r}^{2}}{{\sigma }^{2}}\right)$
Exponential — $exp\left(-\left(\frac{r-{R}_{c}}{\sigma }\right)\right)$.