  This paper explores optimal methods for obtaining one-dimensional (1D) powder
pattern intensities from two-dimensional (2D) planar detectors with good
estimates of their standard deviations. We describe methods to estimate
uncertainties when the same image is measured in multiple frames as well as
from a single frame. We show the importance of considering the correlation of
diffraction points during the integration and the re-sampling process of data
analysis. We find that correlations between adjacent pixels in the image can
lead to seriously overestimated uncertainties if it is neglected in the
integration process. Off-diagonal entries in the variance-covariance (VC)
matrix are problematic as virtually all data processing and modeling programs
cannot handle the full VC matrix. We show the off-diagonal terms come mainly
from the pixel splitting algorithm used as the default integration algorithm in
many popular 2D integration programs, as well as from re-binning and
re-sampling steps later in the processing. When the full VC matrix can be
propagated during the data reduction, it is possible to get accurate refined
parameters and their uncertainties at the cost of increasing computational
complexity. However, as this is not normally possible, we explore the best
approximate methods for data processing in order to estimate uncertainties on
refined parameters with the greatest accuracy from just the diagonal variance
terms in the VC matrix.
