Earth's climate system generates numerous dynamical modes of variability that manifest across different spatial and temporal scales. Classical principal component analysis (PCA) is widely used for dimensionality reduction, but it captures only synchronous linear relationships and performs poorly with propagating waves. Spectral PCA (sPCA), known since the early 1970s, allows incorporating phase information and searching for modes in specified frequency bands; however, its practical application is limited by the difficulty of robustly estimating cross-spectra and by interpretation problems when multiple modes have similar amplitudes. This work stands apart among neighboring studies on climate data analysis: nearby lie works on dimensionality reduction, medical image segmentation, and weakly supervised learning, but there are no direct analogs for rotated spectral PCA in climatology.
The authors use the continuous Morlet wavelet transform to estimate cross-spectral matrices in each frequency band. The Morlet wavelet is a complex exponential modulated by a Gaussian window, providing good frequency localization and robustness even for short time series. To improve interpretability when several eigenvectors have close eigenvalues, a rotation is proposed within the subspace spanned by those vectors. The optimality criterion is the minimization of the spatial integral of the Laplacian of the phase of the rotated vector: for a propagating wave, the phase should be a linear function of coordinates, and such a criterion selects physically meaningful modes. The method requires no parametric assumptions and is simple to implement.
On synthetic data with two waves propagating in opposite directions, classical PCA required 9 components to explain 10% of the variance, whereas sPCA with the Morlet wavelet explained 46% and 26% of the variance with the first two components and accurately recovered both waves. Estimating the propagation speed from the phase gave a value of 1.2, matching the true value. With equal wave amplitudes and 90% noise, the rotation of eigenvectors successfully separated the two modes, which were mixed before rotation. Application to Pacific sea surface temperature data from 1940 to 2018 revealed an El Niño mode at periods of 2–7 years with a correlation of 0.72 with the Niño 3.4 index. At subannual frequencies (2–4.4 cycles per year), the rotation allowed separating several extratropical modes, including a westward-propagating wave in the South Pacific at about 20 degrees of longitude per month (0.4 m/s) and a dipole in the North Pacific.
This work differs from neighboring studies in that it is the first to apply eigenvector rotation with phase regularization in the spectral domain to climate data. Unlike classical PCA, which mixes modes of different frequencies and ignores time lags, rsPCA can isolate physically interpretable dynamical modes even with equal amplitudes and high noise levels. This opens the way to more accurate diagnostics of climate models and to the identification of subseasonal teleconnections that were previously hidden.
The method can be applied to other climate variables, such as geopotential height, to reveal high-frequency modes. Other analytic wavelets may be used for non-oscillatory modes. The question of how to choose the number of principal components to rotate remains open and requires further research.
The method will affect seasonal and subseasonal forecasting, climate model diagnostics, and the analysis of extreme events.
Apply rsPCA to geopotential height and climate model ensembles to separate signal from noise.
This work is directly related to the problem of extracting dynamical modes in complex climate systems, where classical dimensionality reduction methods fail to handle asynchronous relationships and modes of similar amplitude. This is a key task for understanding internal climate variability and separating it from the anthropogenic signal.
The Morlet wavelet is named after French geophysicist Jean Morlet, who developed it in the 1980s for analyzing seismic signals, not climate data.