Shape Error

For 2001 June 8 version of south observatory PSFs

Expected systematic error in photometry computed from PSFs: We compute the source flux F for a single isolated point source as an estimate F^* of the length of a data vector with a certain shape, as given by the PSF, which is essentially an estimate of the scalar product

of the data vector D with a unit vector u in the direction expected for a point source, in the data space, for the given shape s. That is, we expect the data vector to be essentially Fu, a vector of length F in the direction given by the unit vector u.

But suppose the shape is incorrect, and the data are actually given by a different unit vector, u(s'), for some s' not equal to s. Then the estimate of the flux will be given by

This scalar product we can evaluate directly as the scalar product of the PSFs, each normalized to unit length.

The above graphs show the error factor,

expressed as a percent, as a function of shape difference. Each pair of PSFs is plotted as a "+".

Based on the above, it appears that the expected systematic error introduced by a shape error of even three 0.02 shape bins (ie, 0.06 error) should be less than or approximately one percent.

The scalar product is actually evaluated as a weighted sum, with weights determined using the variance map; such weighting should have little or no effect on the expectation of the estimate [6/24/02: This latter statement appears to be incorrect -- WAW]. Note that in the presence of confusing sources (ie, in deblending) the scalar product must be taken based on the component of u orthogonal to the vector representing the confusing source; in such situations the sensitivity to shape errors can be much larger.