Patterns in the temporal complexity of global chlorophyll concentration

Satellite-based estimates of chlorophyll-a concentration ([chl-a]) are crucial for understanding oceanographic processes and planning in situ observations. Decades of research have utilized these data to analyze global trends in primary productivity, coastal runoff, sea ice extent, and harmful algal blooms. [Chl-a], the primary pigment used by phytoplankton for photosynthesis, is reliably estimated from ocean reflectance. These estimates are essential for various applications, including primary productivity estimation, ecological indicator development, long-term trend monitoring, and testing earth-system models. However, the role of [chl-a] time series complexity in explaining regional differences in global ocean color remains unexplored. The complexity of a time series can reflect various phenomena: stochasticity, measurement error, long tails in data, rapidly changing system states, or a combination of these factors. Phytoplankton exhibit chaotic dynamics, non-linear behavior, and intermittent instability, suggesting that analyzing the complexity of [chl-a] time series may reveal large-scale patterns structuring global phytoplankton communities. Various methods exist for estimating time series complexity, such as calculating fractal or Hausdorff dimension, permutation entropy, or Lyapunov exponents. While some of these analyses have been conducted locally, a global analysis is needed for a comprehensive understanding of spatio-temporal characteristics of phytoplankton blooms. This study aims to address this gap by performing a global analysis of [chl-a] time series complexity using approximately 25 years (1998-2022) of global [chl-a] observations from multiple satellite sensors (SeaWiFS, MODIS, MERIS, VIIRS, and OLCI). The merged data product from the Garver-Siegel-Maritorena Model was used at 25km resolution and daily temporal resolution.

Numerous studies have leveraged satellite ocean color data for various applications, including assessing global trends in primary productivity (Behrenfeld et al., 2006; McClain, 2009), investigating coastal runoff effects (Kudela & Chavez, 2004; Chérubin et al., 2008), monitoring sea ice extent (Brown & Arrigo, 2012; Arrigo et al., 2008), and detecting harmful algal blooms (Stumpf, 2001). Existing research has used [chl-a] time series to understand phytoplankton bloom phenology (Friedland et al., 2018, 2023), chlorophyll dynamics (Bashmachnikov et al., 2013), climate-driven [chl-a] variability (Lim et al., 2022), and the contributions of citizen science (Kirby et al., 2021). However, these studies largely focus on specific aspects of [chl-a] variability without explicitly addressing the inherent temporal complexity of these time series across global scales. Studies on the complexity of ecological time series have used various methods, including fractal dimension (Gneiting et al., 2012), permutation entropy (Bandt & Pompe, 2002), and Lyapunov exponents (Bryant et al., 1990; Wolf et al., 1985), but their application to global [chl-a] data is limited. Prior work has shown the potential of using these methods to understand local patterns in phytoplankton dynamics (He, 2021; Agarwal et al., 2021), but a global-scale investigation of [chl-a] time series complexity is lacking.

This research utilized approximately 25 years (1998–2022) of global [chl-a] observations at 25 km resolution and daily temporal resolution from a merged data product (Garver-Siegel-Maritorena Model) available from the Hermes GlobColour website. The merged product combined data from SeaWiFS, MODIS, MERIS, VIIRS, and OLCI sensors. To ensure data quality, a cutoff of 20% missing data was applied; time series with more than 20% missing days were excluded from the analysis. This ensured a minimum of approximately 1800 days of [chl-a] estimates for each time series. Two complexity metrics were employed: elasticity and fractal dimension. Elasticity, analogous to the economic concept of elasticity, measures the sensitivity of a [chl-a] time series to thresholds. It quantifies the responsiveness of the percentage change in daily [chl-a] above a threshold to changes in the threshold itself. Higher elasticity indicates a rougher time series and greater sensitivity to thresholds, reflecting greater variability in day-to-day [chl-a] change. The calculation involved creating a 1-day-lagged time series, computing the number of days where daily [chl-a] change exceeded various thresholds, and fitting a linear model to the logarithm of the relative differences in thresholds and daily changes. The absolute value of the slope of this linear model represents the elasticity. Fractal dimension, calculated using the R package 'fractaldim', quantifies the roughness of the time series. Before calculating fractal dimension, outliers exceeding 3 standard deviations from the mean [chl-a] were removed to minimize errors from likely erroneous measurements. Bootstrapping was used to fill gaps in the time series by randomly resampling values within the same time series. Both elasticity and fractal dimension were calculated annually to examine inter-annual trends. Sensitivity analyses were conducted to assess the influence of data source (merged vs. MODIS only), time series resolution (daily, weekly, monthly), and methodological choices on the complexity metrics. Traditional measures of variability (mean [chl-a], standard deviation, relative standard deviation, and average seasonal amplitude) were also compared with the calculated complexity metrics to explore their relationships. All analyses were performed in R using various packages.

Globally, the elasticity of [chl-a] time series varied across ocean regions. Oligotrophic gyres exhibited the highest elasticity, while regions with high [chl-a] and strong seasonal cycles (e.g., Patagonian Shelf, Baltic Sea) had lower elasticity. This indicates that elasticity tracks the relative strength of climatic and seasonal forcing. The fractal dimension values were remarkably consistent across the global ocean (typically above 1.85), with some key regional differences. Subtropical Pacific, tropical North Atlantic, Amazon plume, and the East coast of Madagascar showed lower-than-average fractal dimensions, potentially due to physical ocean dynamics and episodic elevated [chl-a]. Sensitivity analyses revealed that the results were largely consistent across different data sources and time series resolutions. Inter-annual analyses showed significant shifts in both elasticity and fractal dimension in 2003 and 2019, while mean [chl-a] showed a general decrease until 2019 followed by a rise. The changes around 2003 and 2019 may be partly attributed to changes in satellite sensor coverage and data merging. Spatial comparisons of changes in mean [chl-a] and complexity metrics (2003–2022) revealed inconsistencies, suggesting that the complexity metrics are insensitive to changes in [chl-a] magnitude and independent of the processes controlling [chl-a] magnitude. The findings show that elasticity and fractal dimension provide independent measures of time series complexity which are insensitive to the magnitude of chlorophyll-a.

The findings demonstrate that both elasticity and fractal dimension provide valuable and independent metrics for monitoring global ocean change. Elasticity reflects the regularity of [chl-a] change and the influence of climatic and seasonal forcing, while fractal dimension captures the likelihood of anomalous events and the roughness of the time series. These metrics are insensitive to changes in [chl-a] magnitude, suggesting that they capture aspects of phytoplankton dynamics beyond simply biomass. The observed inter-annual shifts in complexity metrics highlight the importance of considering intrinsic temporal variation in [chl-a] time series when interpreting global ocean change. The consistency of complexity patterns across seemingly disparate regions suggests potential for classifying marine ecological provinces based on time series complexity, supplementing traditional methods. Future studies should investigate the ecological and environmental drivers underlying these patterns.

This study presents novel metrics (elasticity and fractal dimension) for characterizing the temporal complexity of global [chl-a] time series derived from satellite data. The results demonstrate that these metrics offer independent measures of time series complexity which are insensitive to the magnitude of chlorophyll-a. The observed patterns suggest that incorporating time series complexity into analyses of global ocean change is crucial, complementing traditional approaches. Future work should focus on identifying specific environmental and ecological drivers of these patterns and refining marine ecological province definitions using these complexity metrics.

The study's reliance on satellite-derived [chl-a] data introduces limitations related to sensor variability, algorithm uncertainties, and spatial and temporal sampling gaps. The chosen thresholds for elasticity calculations and outlier removal for fractal dimension might influence the results. Furthermore, the interpretation of complexity patterns relies on correlations and does not necessarily establish direct causal relationships with specific ecological or environmental processes. Future work could address these limitations by incorporating uncertainty estimates into the analyses and exploring alternative complexity metrics.