Spatially shifting temporal points: estimating pooled within-time series variograms for scarce hydrological data

Abstract. Estimation of pooled within-time series (PTS) variograms is a frequently used technique for geostatistical interpolation of continuous hydrological variables in spatial data-scarce regions conditional that time series are available. The only available method for estimating PTS variograms averages semivariances, which are computed for individual time steps, over each spatial lag within a pooled time series. However, semivariances computed by a few paired comparisons for individual time steps are erratic and hence they may hamper precision of PTS variogram estimation. Here, we outlined an alternative method for estimating PTS variograms by spatializing temporal data points and shifting them. The data were pooled by ensuring consistency of spatial structure and stationarity within a time series, while pooling sufficient number of data points for reliable variogram estimation. The pooled spatial data point sets from different time steps were assigned to different coordinate sets on the same space. Then a semivariance was computed for each spatial lag within a pooled time series by comparing all point pairs separable by that spatial lag, and a PTS variogram was estimated by controlling the lower and upper boundary of spatial lags. Our method showed higher precision than the available method for PTS variogram estimation and was developed by using the freely available R open source software environment. The method will reduce uncertainty for spatial variability modeling while preserving spatiotemporal properties of data for geostatistical interpolation of hydrological variables in spatial data-scarce developing countries.