How to coadd images? I. Optimal source detection and photometry using ensembles of images

Stacks of digital astronomical images are combined in order to increase image depth. The variable seeing conditions, sky background and transparency of ground-based observations make the coaddition process non-trivial. We present image coaddition methods optimized for source detection and flux measurement, that maximize the signal-to-noise ratio (S/N). We show that for these purposes the best way to combine images is to apply a matched filter to each image using its own point spread function (PSF) and only then to sum the images with the appropriate weights. Methods that either match filter after coaddition, or perform PSF homogenization prior to coaddition will result in loss of sensitivity. We argue that our method provides an increase of between a few and 25 percent in the survey speed of deep ground-based imaging surveys compared with weighted coaddition techniques. We demonstrate this claim using simulated data as well as data from the Palomar Transient Factory data release 2. We present a variant of this coaddition method which is optimal for PSF or aperture photometry. We also provide an analytic formula for calculating the S/N for PSF photometry on single or multiple observations. In the next paper in this series we present a method for image coaddition in the limit of background-dominated noise which is optimal for any statistical test or measurement on the constant-in-time image (e.g., source detection, shape or flux measurement or star-galaxy separation), making the original data redundant. We provide an implementation of this algorithm in MATLAB.