Nonlinear spectral analysis extracts harmonics from land-atmosphere fluxes

TL;DR

The paper tackles the challenge of extracting seasonal and multi-scale harmonic dynamics from land–atmosphere $CO_2$ fluxes using two data-driven time-series decompositions: SSA (linear) and NLSA (nonlinear diffusion-based). It demonstrates that NLSA more effectively uncovers higher-order harmonics of the seasonal cycle, particularly when high-frequency noise is mitigated by filtering, across multiple ICOS/FLUXNET sites and variables, though irregular data can suppress harmonic detection. The work provides a framework for assessing data quality and nonstationarity via harmonic extraction and discusses implications for improving land–atmosphere interaction models and potential spatio-temporal extensions. Overall, NLSA offers a more informative decomposition for capturing ecologically meaningful seasonal patterns, while SSA remains more robust to some noise and nonstationary conditions.

Abstract

Understanding the dynamics of the land-atmosphere exchange of CO$_2$ is key to advance our predictive capacities of the coupled climate-carbon feedback system. In essence, the net vegetation flux is the difference of the uptake of CO$_2$ via photosynthesis and the release of CO$_2$ via respiration, while the system is driven by periodic processes at different time-scales. The complexity of the underlying dynamics poses challenges to classical decomposition methods focused on maximizing data variance, such as singular spectrum analysis. Here, we explore whether nonlinear data-driven methods can better separate periodic patterns and their harmonics from noise and stochastic variability. We find that Nonlinear Laplacian Spectral Analysis (NLSA) outperforms the linear method and detects multiple relevant harmonics. However, these harmonics are not detected in the presence of substantial measurement irregularities. In summary, the NLSA approach can be used to both extract the seasonal cycle more accurately than linear methods, but likewise detect irregular signals resulting from irregular land-atmosphere interactions or measurement failures. Improving the detection capabilities of time-series decomposition is essential for improving land-atmosphere interactions models that should operate accurately on any time scale.

Figures (11)

• Figure 1: Net Ecosystem Exchange (NEE) measurements (signal, orange) of a deciduous broadleaf forest in central Germany (Hainich forest, DE-Hai, DBF). The seasonal cycle is approximated well by multiple harmonics (top row): four harmonics detected by NLSA, two harmonics by SSA. In contrast, the fundamental oscillation (lower row), typically detected by linear spectral analysis fails to accurately represent the variability during the summer period.
• Figure 2: Workflow. The time series $x$ (Signal) is embedded into a higher-dimensional space using Takens' delay coordinates, which yields the data matrix $X$. The dynamics are approximated by modes computed with dimension reduction techniques SSA (linear) or NLSA (nonlinear). These modes are further analyzed using FFT to identify modes representing pure harmonic oscillations. Finally, the identified harmonic oscillations are used to construct the seasonal cycle.
• Figure 3: Analysis of a regular time series with isolated quality flags: gross primary production (GPP) fluxes of a deciduous broadleaf forest (DBF) in central Germany (Hainich forest site, DE-Hai DBF). Panels: Analysis of an unfiltered (a) and filtered (b) measurement signal. Each panel column illustrates the following analysis parts. Panel signal: signal and the corresponding constructed seasonal cycle with SSA or NLSA, if harmonics are detected. The quality flag (QF) is indicated by a gray scale, i.e. lowest signal quality in black. Panel spectra: relative FFT power spectrum (left) of the signal and the SSA and NLSA constructed seasonal cycle, and the dimension reduction spectrum (right). Panel modes: shapes of the 12 most dominant modes (left) and their corresponding spectra (right). Main result: The seasonal cycle is represented by a combination of harmonics of up to fifth order, i.e. SSA detects four harmonics and NLSA five harmonics.
• Figure 4: Analysis of a time series with high frequency variability and isolated quality flags: net ecosystem exchange (NEE) fluxes of a evergreen needleleaf forest (ENF) in Eastern Germany (Tharandt forest site, DE-Tha ENF). Panels: We analyze the unfiltered (a) and the filtered (b) measurement signal. Each panel column illustrates the following analysis parts. Panel signal: signal and the corresponding constructed seasonal cycle with SSA or NLSA, if harmonics are detected. The quality flag (QF) is indicated by a gray scale, i.e. lowest signal quality in black. Panel spectra: relative FFT power spectrum (left) of the signal and the SSA and NLSA constructed seasonal cycle, and the dimension reduction spectrum (right). Panel modes: shapes of the 12 most dominant modes (left) and their corresponding spectra (right). Main result: The harmonics of up to fifth order are only detected after filtering, i.e. SSA detects only the fundamental oscillation while NLSA detects five harmonics. The amplitudes of the seasonal cycles constructed with SSA and NLSA, respectively, differ.
• Figure 5: Analysis of a time series with broadband variability and poor quality flags over extended periods of time: net ecosystem exchange (NEE) fluxes of an evergreen needleleaf forest (ENF) in western Russia (Central Forest Biosphere Reserve at Fyodorovskoe site, RU-Fyo ENF). Panels: Analysis of an unfiltered (a) and filtered (b) measurement signal. Each panel column illustrates the following analysis parts. Panel signal: signal and the corresponding constructed seasonal cycle with SSA or NLSA, if harmonics are detected. The quality flag (QF) is indicated by a gray scale, i.e. lowest signal quality in black. Panel spectra: relative FFT power spectrum (left) of the signal and the SSA and NLSA constructed seasonal cycle, and the dimension reduction spectrum (right). Panel modes: shapes of the 12 most dominant modes (left) and their corresponding spectra (right). Main result: After filtering, the time series exhibits a level of variability, which still prohibits the detection of any harmonics with NLSA. In contrast, SSA always detects the fundamental and the second harmonic oscillation.