  Remote sensing observations, products and simulations are fundamental sources
of information to monitor our planet and its climate variability. Uncovering
the main modes of spatial and temporal variability in Earth data is essential
to analyze and understand the underlying physical dynamics and processes
driving the Earth System. Dimensionality reduction methods can work with
spatio-temporal datasets and decompose the information efficiently. Principal
Component Analysis (PCA), also known as Empirical Orthogonal Functions (EOF) in
geophysics, has been traditionally used to analyze climatic data. However, when
nonlinear feature relations are present, PCA/EOF fails. In this work, we
propose a nonlinear PCA method to deal with spatio-temporal Earth System data.
The proposed method, called Rotated Complex Kernel PCA (ROCK-PCA for short),
works in reproducing kernel Hilbert spaces to account for nonlinear processes,
operates in the complex kernel domain to account for both space and time
features, and adds an extra rotation for improved flexibility. The result is an
explicitly resolved spatio-temporal decomposition of the Earth data cube. The
method is unsupervised and computationally very efficient.We illustrate its
ability to uncover spatio-temporal patterns using synthetic experiments and
real data. Results of the decomposition of three essential climate variables
are shown: satellite-based global Gross Primary Productivity (GPP) and Soil
Moisture (SM), and reanalysis Sea Surface Temperature (SST) data. The ROCK-PCA
method allows identifying their annual and seasonal oscillations, as well as
their non-seasonal trends and spatial variability patterns.
