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Outgassing of carbon dioxide (CO   $$_2$$  2 ) from river surface waters, estimated using partial pressure of dissolved CO   $$_2$$  2 , has recently been considered an important component of the global carbon budget. However, little is still known about the high-frequency dynamics of CO   $$_2$$  2 emissions in small-order rivers and streams. To analyse such highly dynamic systems, we propose a time-varying functional principal components analysis (FPCA) for non-stationary functional time series (FTS). This time-varying FPCA is performed in the frequency domain to investigate how the variability and auto-covariance structures in a FTS change over time. This methodology, and the associated proposed inference, enables investigation of the changes over time in the variability structure of the diurnal profiles of the partial pressure of CO   $$_2$$  2 and identification of the drivers of those changes. By means of a simulation study, the performance of the time-varying dynamic FPCs is investigated under different scenarios of complete and incomplete FTS. Although the time-varying dynamic FPCA has been applied here to study the daily processes of consuming and producing CO   $$_2$$  2 in a small catchment of the river Dee in Scotland, this methodology can be applied more generally to any dynamic time series.Supplementary materials accompanying this paper appear online.

Time-Varying Functional Principal Components for Non-Stationary EpCO$$_2$$ in Freshwater Systems