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.
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