Map Making with Differential Data

WMAP observes temperature differences between points separated by ~141° on the sky. Maps of the relative sky temperature are reconstructed from the difference data using a modified form of the algorithm adopted by COBE-DMR.

The algorithm WMAP uses to reconstruct sky maps from differential data is an iterative one. In the limit that the instrument noise is white (uncorrelated from one sample to the next in time) the algorithm is mathematically equivalent to a least squares fit of a set of temperature differences to a set of map pixel temperatures. However, the implementation does not require the evaluation or inversion of large matrices and has a very intuitive interpretation as follows. The actual signal WMAP measures is the temperature difference between two points on the sky, DT = T(A)-T(B), where T(A) is the temperature seen by feed A, and likewise for B. Feed A can be thought of as viewing the sky while feed B can be thought of as viewing a comparative reference signal, or vice versa. In WMAP's case, the comparative signal is a different point in the sky. If we knew the temperature T(B) we could recover T(A) using T(A) = DT+T(B), but since we don't know T(B), we use a guess in which T(B) is estimated from a previous sky map iteration. Thus the temperature in pixel i of a map is given by the average of all observations of pixel i after correcting each observation for the estimated signal seen by the opposite feed.

For this scheme to be successful it is imperative for a given pixel i to be observed with many different pixels on its ring of neighbors. Thus the method requires a carefully designed scan strategy to go with it. The strategy designed for WMAP achieves this while simultaneously avoiding close encounters with the Sun, Earth, and Moon. The algorithm has been tested with the WMAP scan strategy using an end-to-end mission simulation that incorporates a realistic sky signal, instrument noise, and calibration methods. The results of these simulations are described in detail in an Astrophysical Journal article. The main figure from that paper is reproduced here. After 40 iterations of the algorithm, the artifacts that remain in the map due to the map-making itself have a peak-peak amplitude of less than 0.2 µK, even in the presence of Galactic features with a peak brightness in excess of 60 mK.