A Bayesian Model for Area and Point Predictions - A Case Study of Predictions of Annual Precipitation and Runoff in the Voss Area.

In this work we perform predictions of annual precipitation and runoff by spatial interpolation. For this purpose, we utilise both point observations of precipitation and/or area observations of runoff from several years. We suggest a statistical model for annual precipitation and runoff consisting of two spatial terms: One spatial term that is common for all years which models the climatology in the area of interest, and one spatial term for year-to-year variation. The model is set up as a Bayesian hierarchical model of three levels, and we use informative priors based on information from the available observations. A stochastic partial differential equation (SPDE) approach to spatial modelling is used to make inference and predictions less computationally expensive. The model is implemented by using the R-package R-INLA, and we demonstrate how R-INLA can be used for making predictions and drawing inference from a model based on both point observations (e.g of precipitation) and area observations (e.g of runoff). The statistical model is tested through a case study of catchments located around Voss in Norway and through simulation studies. The main focus is on the predictive performance. In particular we explore how the predictive performance is affected by having a spatial varying climate effect in the model. We find that the spatial predictions of runoff and precipitation often are uncalibrated if the spatial differences in the observed annual precipitation are stable from one year to another. The consequence of this model property is that an observation design that produces accurate predictions one year, also will produce accurate predictions other years. Further we compare the predictive performance for annual runoff when using observation samples consisting of (1) only observations of runoff, (2) only observations of precipitation and (3) observations of both runoff and precipitation. The results from the simulation studies did not favour one of the observation types (runoff and/or precipitation), and both observation types can produce accurate predictions of annual runoff depending on the underlying climatology. For the real dataset we saw that observation samples of only runoff produced the most accurate predictions. Observation samples of only precipitation were not suitable for runoff predictions for the real dataset and led to large biases between the true observations and the predicted values