  The Global Historical Climatology Network-Daily database contains, among
other variables, daily maximum and minimum temperatures from weather stations
around the globe. It is long known that climatological summary statistics based
on daily temperature minima and maxima will not be accurate, if the bias due to
the time at which the observations were collected is not accounted for. Despite
some previous work, to our knowledge, there does not exist a satisfactory
solution to this important problem. In this paper, we carefully detail the
problem and develop a novel approach to address it. Our idea is to impute the
hourly temperatures at the location of the measurements by borrowing
information from the nearby stations that record hourly temperatures, which
then can be used to create accurate summaries of temperature extremes. The key
difficulty is that these imputations of the temperature curves must satisfy the
constraint of falling between the observed daily minima and maxima, and
attaining those values at least once in a twenty-four hour period. We develop a
spatiotemporal Gaussian process model for imputing the hourly measurements from
the nearby stations, and then develop a novel and easy to implement Markov
Chain Monte Carlo technique to sample from the posterior distribution
satisfying the above constraints. We validate our imputation model using hourly
temperature data from four meteorological stations in Iowa, of which one is
hidden and the data replaced with daily minima and maxima, and show that the
imputed temperatures recover the hidden temperatures well. We also demonstrate
that our model can exploit information contained in the data to infer the time
of daily measurements.
