  Numerical weather forecasts can exhibit systematic errors due to simplifying
model assumptions and computational approximations. Statistical postprocessing
is a statistical approach to correcting such biases. A statistical
postprocessing model takes input data from a numerical forecast model, and
outputs a parametric predictive distribution of a real-world observation, with
model parameters learned from past forecast-observation pairs. In this paper we
develop and discuss methods for postprocessing of gridded data. We show that
estimates of postprocessing parameters on a spatial grid can be improved by
Bayesian hierarchical modelling with spatial priors. We use the
"Max-and-Smooth" approach [Hrafnkelsson et al., 2021] to approximate a fully
Bayesian inference in two steps. First we calculate maximum-likelihood
estimates (MLEs) of postprocessing parameters at individual grid points. Second
we smooth the MLEs using a measurement error model with a spatial prior. Our
approach provides the theoretical basis for the parameter smoothing approach by
Kharin et al. [2017], and simplifies and generalises the Bayesian hierarchical
modelling approach by Moeller et al. [2015]. A new derivation of Max-and-Smooth
is presented. The method is applicable to arbitrary postprocessing models, as
illustrated on Model Output Statistics, Logistic Regression, and Nonhomogeneous
Gaussian Regression. We report consistent improvements in forecast accuracy,
calibration, and probabilistic skill in postprocessing of temperature and
precipitation forecasts.
