Supplementary materials for this article are available at https:// doi.org/ 10.1007/ s13253-019-00365-3 . Spatiotemporal Lagged Models for Variable Rate Irrigation in Agriculture Sierra Pugh , Matthew J. Heaton, Jeff Svedin, and Neil Hansen Irrigation is responsible for 80–90% of freshwater consumption in the USA. However, excess water demand, drought, declining groundwater levels, and water quality degrada- tion all threaten future water supplies. In an effort to better understand how to efficiently use water resources, this analysis seeks to quantify the effect of soil water at various depths on the eventual crop yield at the end of a season as a lagged effect of space and time. As a novel modeling contribution, we propose a multiple spatiotemporal lagged model for crop yield to identify critical water times and patterns that can increase the crop yield per drop of water used. Because the crop yield data consist of nearly 20,000 observations, we propose the use of a nearest neighbor Gaussian process to facilitate computation. In applying the model to soil water and yield in Grace, Idaho, for the 2016 season, results indicate that soil moisture in the 0–0.3 m depth of soil was most corre- lated with crop yield earlier in the season (primarily during May and June), while the soil moisture at the 0.3–1.2 m depth was more correlated with crop yield later in the season around mid-June to mid-July. These results are specific to a crop of winter wheat under center-pivot irrigation, but the model could be used to understand relationships between water and yield for other crops and irrigation systems. Supplementary materials accompanying this paper appear online. Key Words: Distributed lag; Natural resources; Gaussian process; Bayesian. 1. INTRODUCTION Irrigation is responsible for 80–90% of freshwater consumption in the USA (Postel 1999). However, excess water demand, drought, declining groundwater levels, and water quality degradation all threaten future water supplies (Evans et al. 2013). To sustain irrigated agriculture, there is a need to increase irrigation efficiency and food production per unit of applied water (Evans et al. 2013). Much has been done to improve efficiency of irrigation water use including improvements in irrigation delivery methods, irrigation scheduling approaches, and understanding water Sierra Pugh (B ) and Matthew J. Heaton, Department of Statistics, Brigham Young University, Provo, UT, USA (E-mail: sierra.pugh@colostate.edu). Jeff Svedin and Neil Hansen, Department of Plant and Wildlife Sciences, Brigham Young University, Provo, UT, USA. © 2019 International Biometric Society Journal of Agricultural, Biological, and Environmental Statistics https://doi.org/10.1007/s13253-019-00365-3