Abstract: This talk will describe recently developed data-adaptive harmonic decomposition (DAHD) technique that estimates power and phase spectra of multivariate datasets.Central to the approach is spectral analysis of a new class of integral operators whose kernels are built from correlation functions in a fundamentally different way than in Principal Component Analysis and its time-delayed embedding generalizations. At a practical level, DAHD relies on eigendecomposition of block-Hankel matrix formed from time-lagged cross-correlations in the data. DAHD spatio-temporal eigenmodes form an orthogonal set of oscillating functions that come in pairs and in exact phase quadrature for a given temporal Fourier frequency.Time evolution of DAHD modes can be efficiently modeled and predicted by a universal family of frequency-based low-order stochastic differential equations involving a fixed set of predictor functions, namely coupled stochastic oscillators made of multilayer Stuart-Landau models (MSLMs). DAHD-MSLM results will be presented for a wide range of challenging geophysical applications, such as space physics, Arctic sea ice, and oceanic turbulence.