### C Explaining optimal photometry

Optimal extraction offers serveral advantages over the normal aperture method of photometry. Formally optimal extraction is equivalent to profile fitting, however it offers more robust error estimation and a freedom of the bias introduced by mis-estimating the point spread function (PSF). It has been found to offer a gain of around 10 per cent in signal-to-noise over normal aperture photometry.

A general formula for summing flux ($F$) within an aperture is

$F=\sum _{i,j}{W}_{i,j}\left({D}_{i,j}-{S}_{i,j}\right),$

where sum is over all the pixels $i$,$j$ within the aperture, where the total count in a pixel is ${D}_{i,j}$, the estimated sky level is ${S}_{i,j}$ and ${W}_{I,j}$ is the weight given to each pixel. For normal aperture photometry this is one within the aperture, and zero outside it.

Finding the optimal value for ${W}_{I,j}$ for each $i$,$j$ within the aperture is a two step process. Firstly a model profile is fitted to a nearby star (a 2-D Gaussian has proved adequate for this purpose), the resulting estimated stellar proifle ${P}_{i,j}^{E}$ is normalised to one.

As shown by Horne (PASP, 1986, 98, 609) once the estimated profile is known the best signal-to-noise is obtained for

${W}_{i,j}=\frac{{P}_{i,j}^{E}/{V}_{i,j}}{\sum _{i,j}{\left({P}_{i,j}^{E}\right)}^{2}/{V}_{i,j}},$

where ${V}_{i,j}$ is the variance for each pixel. Substituting this into our first equation we obtain the basic formula governing optimal extraction. However at this stage we make a further assumption, that the variance for each pixel is the same. For very faint stars this is clearly the case (since the counts in each pixel is dominated by the sky count), for brighter stars though this means that the extraction will be non-optimal. Hence we have that

$F=\frac{\sum _{i,j}{P}_{i,j}^{E}\left({D}_{i,j}-{S}_{i,j}\right)}{\sum _{i,j}{\left({P}_{i,j}^{E}\right)}^{2}}.$

From this we note that if the PSF is wrong there will be no systematic bias in the results, provided one is interested in the relative brightness of one star with respect to another in the same frame.

A full treatment of optimal extarction can be found in Tim Naylor’s MNRAS paper “An optimal extraction algorithm for imaging photometry” (MNRAS, 1998, 296, 339) to which the reader is directed for further information.