content_image/300192.jpg
average monthly precipitation in winter for location s and year t via 

<img src='content_image/300192.jpg'>

in which, for example, Y$_{Dec}$ ( s , t ) represents the total December precipitation in year t at location s . The atmosphere’s short memory suggests Y ( s , t ) is independent from Y ( s , t $^{′}$) for t ̸ = t $^{′}$, which is confirmed in an exploratory analysis of Colorado precipitation. Winter precipitation is important to estimate at long time scales because it strongly influences Colorado’s annual water supply. 

We formulate the problem of estimating precipitation as a need to estimate entire precipi- tation fields when only covariates are available. We build the RESP model (3) with historical data to estimate a statistical relationship between average monthly winter precipitation in Colorado and land and sea surface temperatures. We discuss inference for the RESP model to illustrate that it can estimate teleconnection patterns without specifying teleconnection indices a priori (Section 3.4.1). A leave-one-out cross-validation study validates the model’s effectiveness (Section 3.4.2), especially in relation to other common downscaling methods (Section 3.3). Although beyond the scope of this study, a next step for future work would be to apply the RESP model to simulated GCM output. 

## 3.1 Data 

The ERA-Interim reanalysis dataset provides reconstructions of historical sea surface tem- peratures and local covariates (Dee et al., 2011). The response, precipitation, comes from the PRISM dataset (Daly et al., 2008). We limit our study period to 1981 through 2013 because earlier records of large scale climate are less complete. Both datasets are reanalysis products, which are necessary because working directly with observations can be challenging. Raw data may be from various sources and are often spatially sparse and temporally incom- plete. Reanalysis products use statistical techniques and physical relationships to reproduce 

