Change of Support in the Atmospheric Sciences from First Principles: The Discrete Gaussian Model, Why It Outperforms Common Practice, and Worked Examples

Every comparison of a station with a grid box, of a satellite footprint with a gauge, or of an hourly value with a daily one is a change of support. The variable is the same, but its distribution is not. Block averages have smaller variance, different skewness, thinner tails and different exceedance frequencies than the point values they summarize. These differences come from geometry and spatial correlation alone, before any instrument or model error. The atmospheric sciences mostly handle this with shortcuts. Common ones are treating the station as grid-box truth, applying a single areal reduction factor, shrinking variance about the mean, adding Gaussian representativeness noise, quantile mapping, and gridding station-level indices. Each matches at most one feature of the block distribution. None gives the distribution of a point value given the block value, which is the quantity forecasters, verifiers and impact modellers need. This paper derives change of support from first principles and presents the discrete Gaussian model (DGM) of geostatistics as a practical, general solution. The method transforms the point variable to a Gaussian score and expands it in Hermite polynomials. A single support coefficient r, estimated from the station network’s own correlogram, then gives three things in closed form: the block distribution, the joint point–block law, and the conditional distribution of a point given a block. A second correlation, ρ, extends it to a forecast or model block. The method applies to any variable with a monotone transform to a Gaussian score, including temperature, wind speed, humidity, aerosol concentration, radiation and precipitation with its dry-day atom. Four worked examples follow: a Gaussian temperature case done by hand; a Monte Carlo test for wind speed and a strongly skewed variable, where the DGM is compared with the common alternatives against a known truth; a hand-worked conversion of a grid-box rain forecast into point rain probabilities; and a real application to GraphCast and ECMWF HRES day-1 to day-3 rainfall over the US Midwest in 2020. There the DGM cut the continuous ranked probability score of raw GraphCast by about 35% with one fitted parameter. A step-by-step recipe and a discussion of limitations are included.