Advances in sensor technology enable environmental monitoring programmes to record and store measurements at a high temporal resolution, enhancing the capacity to detect and understand short duration changes that would not have been apparent in the past with monthly, fortnightly or even daily sampling. However, there are various challenges in terms of the processing and analysis of these environmental high-frequency data due to their complex behavior over the different timescales and the strong correlation structure that persists over a large number of lags. Here, we explore the complexities of modeling high-frequency data which arise from environmental applications. With increasing understanding of the importance of surface waters as a source of atmospheric CO 2 we consider a high-resolution sensor-generated time series of the over-saturation of CO 2 , EpCO 2 , in a small order river system. We will present advanced statistical approaches to analyze and model the data, which include visualization tools for exploratory analysis, wavelets and additive models. These methods reveal the complex dynamics of EpCO 2 over different timescales, and the multivariate relationships of EpCO 2 with hydrology and temporal autocorrelation structures, which are time and scale dependent.Handling Editor: Bryan F. J. Manly. Electronic supplementary materialThe online version of this article

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.

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In several applications, data are collected as functions observed at a set of locations of a region, inducing spatially correlated functional data. The analysis of such data is performed by adapting and extending the spatial statistical tools for functional data. The variogram is a very popular measure for describing the spatial covariance structure of a spatial process. This article focuses on the extension of variograms for functional data and provides a review for the computation and estimation of different types of functional variograms.