Time Series & Spectral Analysis
Extracting periodicities, trends, and frequency-domain structure from oceanographic and environmental data
Transform
Transform
Decomposition
Models
Overview
Oceanographic measurements — sea surface temperature, tide gauge records, chlorophyll-a concentrations, current velocities, and acoustic backscatter — are inherently time-dependent. Time series analysis provides the mathematical framework to decompose these signals into trend, seasonal, and residual components; spectral analysis transforms data into the frequency domain to identify dominant periodicities (tidal constituents, seasonal cycles, ENSO, PDO). Wavelet analysis extends spectral methods by resolving time-frequency localization — critical for non-stationary signals like hurricane-driven wave events.
Key Concepts
Fourier Transform
The DFT decomposes a signal into sinusoidal components. The power spectral density (PSD) reveals dominant frequencies. For tidal analysis, harmonic analysis fits known tidal constituents (M2, S2, K1, O1 etc.) to observed water level records.
PSD(f) = |X(f)|² / N
Wavelet Analysis
Continuous wavelet transform (CWT) uses scaled/translated mother wavelets (Morlet, Mexican hat) to produce a time-frequency decomposition. The wavelet power spectrum shows how dominant frequencies evolve over time — revealing ENSO modulations or regime shifts.
Trend Decomposition
STL (Seasonal-Trend decomposition using LOESS) separates a time series into seasonal, trend, and remainder components. Mann-Kendall test detects monotonic trends; Sen's slope estimates the trend magnitude — standard for SST warming rate analysis.
Autocorrelation & ARIMA
Autocorrelation function (ACF) and partial ACF (PACF) reveal temporal dependence structure. ARIMA(p,d,q) models capture autoregressive, integrated, and moving-average behavior for forecasting.
Digital Filtering
Low-pass, high-pass, and band-pass filters isolate specific frequency bands. Butterworth and Lanczos filters remove tidal signals to reveal subtidal currents; running means smooth noisy satellite SST data.
Cross-Spectral Analysis
Coherence and phase spectra assess the frequency-dependent relationship between two signals (e.g., wind stress and sea level). Granger causality tests whether one time series helps predict another.
Interactive Visualizations
Synthetic Sea Surface Temperature — Time Domain & FFT Power Spectrum
Wavelet Scalogram — Time-Frequency Decomposition
STL Decomposition of Monthly Sea Level Anomaly
Key References
- Emery, W.J. & Thomson, R.E. (2014). Data Analysis Methods in Physical Oceanography, 3rd Ed. Elsevier.
- Torrence, C. & Compo, G.P. (1998). A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1), 61–78.
- Cleveland, R.B. et al. (1990). STL: A seasonal-trend decomposition procedure based on Loess. J. of Official Statistics, 6(1), 3–73.
- Pawlowicz, R. et al. (2002). Classical tidal harmonic analysis including error estimates using T_TIDE. Computers & Geosciences, 28, 929–937.
- Box, G.E.P. et al. (2015). Time Series Analysis: Forecasting and Control, 5th Ed. Wiley.