  We consider statistics of spatial averages estimated by weighting
observations over an arbitrary spatial domain using identical and independent
measuring devices, and derive an account of bias and variance in the presence
of missing observations. We test the model relative to simulations, and the
approximations for bias and variance with missing data are shown to compare
well even when the probability of missing data is large. Previous authors have
examined optimal averaging strategies for minimizing bias, variance and mean
squared error of the spatial average, and we extend the analysis to the case of
missing observations. Minimizing variance mainly requires higher weights where
local variance and covariance is small, whereas minimizing bias requires higher
weights where the field is closer to the true spatial average. Missing data
increases variance and contributes to bias, and reducing both effects involves
emphasizing locations with mean value nearer to the spatial average. The
framework is applied to study spatially averaged rainfall over India. We use
our model to estimate standard error in all-India rainfall as the combined
effect of measurement uncertainty and bias, when weights are chosen so as to
yield minimum mean squared error.
