spatial Interpolation of Penman Evapotranspiration J. B. Harcum, Jim C. Loftis ASSOC. MEMBER ASAE ABSTRACT F OR irrigation scheduling and hydrologic studies, it is often necessary to estimate reference evapotranspiration at points located some distance from a weather station. For regions which are served by weather station networks, one may interpolate, using evapotranspiration estimates from monitored locations. The approaches which are currently used for spatial interpolation include the Thiessen polygon, simple averaging, and inverse distance weighting. The present work examines the use of Kalman filtering as an alternate approach. The Kalman filter offers the advantages of considering reference evapotranspiration as a stochastic process and of accounting for measurement error and model error explicitly. The filter was found to be an acceptable algorithm for spatial interpolation of reference evapotranspiration based on diagonostic checks, lowest sum of squared error, and minimum variance estimates. INTRODUCTION A simple water balance equation is routinely used to determine changes in soil moisture storage by accounting for moisture inputs and withdrawals. For irrigation scheduling in particular, the water balance is performed on a daily basis, and crop evapotranspiration is the component which receives primary attention. In computing crop evapotranspiration, information on plant physiology, soil moisture, and weather data are incorporated into the basal crop coefficient (K^,,), the soil moisture stress coefficient (K^), and calculated reference evapotranspiration (E^^), respectively. Since plant physiology and soil moisture information are available on a field to field basis, K^^, and K^ estimates are made separately for each irrigated field in a region. However, E^^ estimates are usually made at only a few discrete locations throughout a region (weather stations). Since E^^ estimates are not made on a field to field basis, a statistical question arises. How does one estimate daily E^^ at locations where weather parameters are not measured? Three interpolation techniques have traditionally been applied to E^^. The most common technique applied to irrigation scheduling has been the Thiessen polygon method. Simple arithmetic averaging and inverse Article was submitted for publication in August, 1986; reviewed and approved for publication by the Soil and Water Div. of ASAE in January, 1987. Presented as ASAE Paper No. 86-2001. This work was funded in part by the Colorado Agricultural Experiment Station, Project #602, "Improving Irrigation System Performance." The authors are: J. B. HARCUM, Graduate Research Assistant, and JIM C. LOFTIS, Associate Professor, Agricultural and Chemical Engineering Dept., Colorado State University, Fort Collins distance interpolation have also been used to fill in missing E^j. estimates. All three techniques are reasonable for approximating E^^. at locations between weather stations. However, none of the three provide statistically optimal (minimum variance) estimates. Interpolation of other hydrologic parameters, notably precipitation (Bras and Col6n, 1978; Bras and Rodriquez-Iturbe, 1985; and Rodriquez-Iturbe and Mejia, 1974) and monthly regional E^ (Amegee, 1985), were considered in developing the approach used in this study. Results from E^^. time series analysis and estimation of soil moisture using Kalman filtering (Aboitiz et al., 1986) were also incorporated. The foregoing works provide a basis for the development of a Kalman filtering algorithm designed for spatial interpolation of E^^ within a weather station network. Kalman filtering is an estimation procedure (Jazwinski, 1970 and Gelb, 1974) which optimally combines information from a model and measurements of a stochastic process. The technique has been widely applied in the field of electrical engineering for processing of noisy measurements. Kalman filtering has, more recently, received attention in the field of hydrology as referenced above. The advantages of the technique include its ability to provide statisically optimal estimates of processes which are correlated in both time and space and to explicitly include measurement error. The kriging technique used by Amegee (1985) is closely related to Kalman filtering. However, kriging considers spatial correlation only. Correlation in time is ignored. Since monthly E^ may be considered uncorrelated in time, kriging is an appropriate technique for plotting contours of monthly values. However, daily E^^ values are serially correlated. Therefore, Kalman filtering was selected over kriging for the present purpose. The filter described herein allows for interpolation and forecasting on a real-time (daily) basis. Interpolation refers to the estimation of E^^ at a specific location where no weather station exists using E^j. observations from weather stations forming a regional network. Interpolated estimates are weighted combinations of station observations and time series model predictions (forecasts). The filter computes weights which provide statistically optimal (minimum variance) estimates. An additional advantage of the filter is that it directly provides a quantitative analysis of its estimation error in the form of a filtering variance. For a detailed description of Kalman filter operation, the reader is referred to O'Connell (1980) or Gelb (1974). The overall model of daily E^^ for the present study is composed of both a deterministic and a stochastic component. A priori spatial and seasonal variations are accounted for in the deterministic component of the model through multiple regression and Fourier series Vol. 30(l):January-February, 1987 © 1987 American Society of Agricultural Engineers 0001-2351/87/3001-0129$02.00 129