Geophysical and climatological observations, such as the time series of global terrestrial water storage (e.g., from GRACE and GRACE-FO or models), sea level (e.g., from altimetry mission), and sea surface temperature (e.g., from assimilative systems), contain many inherent time scales, which reflect the complex processes that cause their variations. Traditionally, parametric methods such as regression techniques have been applied to analyze these observations, for which one assumes that the observed time series consists of different parts, for example, a trend (defined as long-term evolution of the series), periodic components including seasonal cycles, and a random part, i.e., noise. Selecting appropriate base functions to meaningfully represent the behavior of observations is a difficult task in parametric techniques.
Alternatively, statistical methods can be used to extract data-adjusted spatial and temporal patterns from observations. The term “statistical decomposition” is applied for “transforming” or “separating” multivariate sampled variables (e.g., observed geophysical time series or model simulations) into “mathematical components”, which is also known as “statistical modes”.
The algorithms that are used in the statistical techniques to find such parameterisations can be categorised according to the statistical information used in their decomposition procedure, for example, (a) “second-order” and (b) “higher-order” techniques. They can also be classified, based on how the statistics are estimated, into (A) “stationary” and (B) “non-stationary” techniques.
Decomposition techniques have also been discussed under the “Blind Source Separation (BSS)” theme, which aims at recovering unobserved patterns or “sources” from observations that are a “mixture” of these sources (in the presence of noise) measured by an array of sensors. In other words, the term “data matrix (observations)” used in decomposition techniques is equivalent with the “mixture (matrix)” in BSS, and “statistical modes” are equivalent with the terms “source(s)” and “(de)mixing matrix” used in the BSS techniques. The BSS view has been applied in many disciplines, including computer science and feature recognition, biomedical sciences, brain imaging, and many other examples.
Principal Component Analysis (PCA), also called Empirical Orthogonal Function (EOF), is among the most popular second-order analysis techniques, therefore classified as (a), and often used to extract dominant orthogonal modes from datasets in various disciplines. More recently, the higher-order statistical technique of Independent Component Analysis (ICA, classified here as (b)) has been introduced decompose data sets into statistically independent components.
In the geodesy discipline, Forootan and Kusche (2012, 2013) introduced ICA for separating geophysical time series arguing that different physical processes generate statistically independent source signals that are superimposed in geophysical time series; thus, application of ICA likely helps separating, (extracting) their contribution from the total signal.
Forootan et al. (2018) introduced the Complex ICA (CICA) technique to deal with non-stationary behaviour of geophysical time series. An overview of statistical decomposition techniques can be found in Forootan (2014). For Matlab codes to implement PCA, ICA and complex ICA contact Ehsan Forootan on efo@plan.aau.dk.
An overview of statistical decomposition techniques based on Forootan (2014), page 67.
Related Publications:
Forootan, E. (2014), Statistical signal decomposition techniques for analyzing time-variable satellite gravimetry data. University of Bonn, Germany. Paper link: here. For an overview of the main results please see here.
Forootan, E., Kusche, J., Talpe, M.J., Shum, C.K., Schmidt, M. (2018), Developing a complex independent component analysis (CICA) technique to extract non-stationary patterns from geophysical time series. Surveys in Geophysics, 39 (3), doi:10.1007/s10712-017-9451-1.
Forootan, E., Kusche, J. (2013), Separation of deterministic signals, using independent component analysis (ICA). Studia Geophysica et Geodaetica, 57 (1), doi:10.1007/s11200-012-0718-1
Forootan, E., Kusche, J. (2012), Separation of global time-variable gravity signals into maximally independent components. Journal of Geodesy, 86, doi:10.1007/s00190-011-0532-5
Forootan, E., Awange, J., Kusche, J., Heck, B., Eicker, A. (2012), Independent patterns of water mass anomalies over Australia from satellite data and models. Remote Sensing of Environment, 86, doi:0.1016/j.rse.2012.05.023
Forootan, E., Rietbroek, R., Kusche, J., Sharifi, M.A., Awange, J., Schmidt, M., Omondi, P., Famiglietti, J.S. (2014), Separation of large scale water storage patterns over Iran using GRACE, altimetry and hydrological data. Remote Sensing of Environment, 140, doi:10.1016/j.rse.2013.09.025
Awange, J., Forootan, E., Kuhn, M., Kusche, J., Heck, B. (2014), Water storage changes and climate variability within the Nile Basin, between 2002 and 2011. Advances in Water Resources, 73, doi: 10.1016/j.advwatres.2014.06.010