Improving Linear Interpolation of Missing Hydrological Data by Applying Integrated Autoregressive Models

Abstract

The application of linear interpolation for handling missing hydrological data is unequivocal. On one hand, such an approach offers good reconstruction in the vicinity of last observation before a no-data gap and first measurement after the gap. On the other hand, it omits irregular variability of hydrological data. Such an irregularity can be described by time series models, such as for instance the autoregressive integrated moving average (ARIMA) model. Herein, we propose a method which combines linear interpolation with autoregressive integrated model (ARI, i.e. ARIMA without a moving average part), named LinAR (available at GitHub), as a tool for inputing hydrological data. Linear interpolation is combined with the ARI model through linear scaling the ARI-based prediction issued for the no-data gap. Such an approach contributes to the current state of art in gap-filling methods since it removes artificial jumps between last stochastic prediction and first known observation after the gap, also introducing some irregular variability in the first part of the no-data gap. The LinAR method is applied and evaluated on hourly water level data collected between 2016 and 2021 (52,608 hourly steps) from 28 gauges strategically located within the Odra/Oder River basin in southwestern and western Poland. The data was sourced from Institute of Meteorology and Water Management (Poland). Evaluating the performance with over 100 million assessments in the validation experiment, the study demonstrates that the LinAR approach outperforms the purely linear method, especially for short no-data gaps (up to 12 hourly steps) and for rivers of considerable size. Based on rigorous statistical analysis of root mean square error (RMSE) – expressed (1) absolutely, (2) as percentages and (3) using RMSE error bars – the percentage improvement, understood as percentage difference between RMSE of linear and LinAR interpolations, was found to reach up to 10%.