Signature kernel Koopman analysis reveals sea surface temperature dynamics

Sea surface temperature is one of the most closely watched variables in climate science, a single observable that condenses the enormous complexity of ocean–atmosphere interaction into a field that can be measured, mapped, and modeled. For decades, researchers have tried to extract predictive structure from SST records, most famously through linear inverse models that treat SST anomalies as a linear Markov process. But a fundamental problem has always lurked beneath these efforts: sea surface temperature alone is not a closed description of the climate system. The atmosphere above it, the ocean interior below it, and countless subgrid-scale processes all influence how SST evolves, yet none of them are directly observed in an SST-only record. A new study published in Earth Science Informatics by Nozomi Sugiura, Satoshi Osafune, and Shinya Kouketsu addresses this problem with a mathematically elegant strategy that reimagines what the “state” of the climate system actually is—and in doing so, delivers measurable improvements in multiyear forecasting skill alongside a rich new diagnostic window into the ocean’s oscillatory behavior.

The core insight of the new work is that when only a partial view of a complex system is available, the observed variables become history-dependent. In the language of dynamical systems, the SST-only evolution is effectively non-Markovian: knowing the temperature field today is not sufficient to determine the field next month, because hidden variables—the atmospheric winds, the deeper ocean, unresolved eddies—carry memory that leaks into the surface record. Traditional approaches either ignore this memory or approximate it with time-delay coordinates, the classic Takens embedding, in which past snapshots are stacked into a vector. Sugiura and colleagues take a different and arguably more faithful route: instead of collapsing history into an unordered vector, they treat each year of SST data as an ordered path, a continuous trajectory through a high-dimensional space that preserves the temporal sequence of monthly anomalies within the year.

Representing dynamics as paths, however, creates a new mathematical challenge. Koopman analysis, the framework the authors adopt, offers a way out of nonlinearity by shifting attention from states to observables. First proposed by Bernard Koopman in 1931, the Koopman operator is a linear operator that acts on scalar-valued functions of the state, propagating them forward in time. The remarkable property is that even when the underlying dynamics are strongly nonlinear, the evolution of observables can be exactly linear—an infinite-dimensional linearity, but one that can be approximated in finite dimensions. Extended dynamic mode decomposition, or EDMD, and its kernelized variant kEDMD have become the workhorse tools for this approximation, projecting dynamics onto a chosen function space or a reproducing kernel Hilbert space. The question is which function space to use when the states themselves are entire annual trajectories.

This is where path signatures enter the picture. Rooted in the theory of rough paths developed by Terry Lyons, the signature of a path is a canonical collection of iterated integrals that captures the temporal order, area, and higher-order interactions along a trajectory. Signatures form a feature representation so expressive that sufficiently regular functionals of paths can be approximated by linear functionals of these features. Computing signatures explicitly, however, becomes prohibitive for high-dimensional data—the number of features grows combinatorially with truncation depth—and SST fields, with their vast spatial grids, are nothing if not high-dimensional. The authors sidestep this bottleneck by kernelizing the signature: following work by Kiraly and Oberhauser, they use a signature kernel that implicitly compares two paths through their iterated-integral features, computing inner products in an RKHS without ever constructing the features themselves.

The resulting pipeline is a single, coherent procedure. Monthly SST fields are first converted into anomalies using a strictly past-only rolling climatology, ensuring that no future information contaminates the preprocessing—a crucial discipline for honest out-of-sample testing. Twelve consecutive monthly anomalies are then grouped into an annual segment and embedded as a piecewise-linear path via cumulative summation, so that the path’s increments are precisely the monthly anomalies. The one-year shift operator—the map that carries one annual path to the next—is learned by applying kEDMD to Gram and cross-Gram matrices built from the truncated signature kernel, with hyperparameters tuned through a kernel-alignment objective that measures the normalized similarity between predicted and true paths. The output is a finite-dimensional Koopman matrix whose spectrum and iterates serve double duty: multiplying it forward yields multiyear forecasts, while its eigendecomposition yields oscillatory modes with well-defined periods and amplitudes.

Before confronting real ocean data, the team validated the approach on a controlled synthetic benchmark: the two-scale Lorenz–96 system, a standard testbed in which slow variables are coupled to fast ones, mimicking exactly the partial-observability situation of SST. Only the slow variables were used as observables and prediction targets; the fast variables acted as unresolved stochastic-like forcing, formally inducing memory and random forcing in the reduced slow-only description, in the spirit of the Mori–Zwanzig formalism. Across three coupling regimes, the signature-kernel method matched an explicit truncated-signature EDMD almost perfectly—confirming the kernel implementation—and outperformed baselines that used either block-mean states or conventional snapshot DMD, particularly at intermediate and longer lead times. The message was clear: the ordered structure within trajectory segments carries predictive information that averages and instantaneous snapshots destroy.

Applied to observed SST, the method delivered on its promise. Forecast experiments were conducted under two rigorous time-ordered protocols: leave-future-out evaluation for forecasting skill and leave-start-out splits for spectral diagnostics, both ensuring that model training never touched data from the verification period. Against a climatology baseline that simply repeats the anchor-year month-of-year climatology, the learned Koopman operator improved out-of-sample multiyear forecast skill, with performance measured both by area-weighted RMSE on anomaly fields and by a novel kernel-based path correlation, the kPC, which scores the similarity between predicted and true annual paths in the signature-kernel space. Equally important, the eigenspectrum of the learned one-year operator revealed coherent spectral modes organized in band-like structures in period–amplitude space, with representative modes at interannual and longer periods—the kind of structured oscillatory signal that climate scientists associate with phenomena such as the El Niño–Southern Oscillation and other basin-scale variability.

The contrast with the other dominant paradigm in modern geophysical forecasting—neural operators—is instructive. Architectures such as Fourier neural operators, and systems like OceanNet, which demonstrates competitive seasonal prediction for regional ocean dynamics, learn powerful nonlinear mappings between input and output fields and transfer well across grid resolutions. But their predictions emerge from compositions of linear and nonlinear layers, so no native eigenvalues, eigenfunctions, or Koopman modes fall out of the model; spectral diagnostics require additional post-processing or are simply unavailable. The signature-kernel kEDMD approach, by contrast, yields an explicit linear operator from which forecasting and spectral analysis follow from the same object. For scientists seeking not just predictions but physical interpretation—the identification of oscillatory patterns, their decay rates, their amplitudes in physical temperature units—this linearity is a decisive advantage.

The method also highlights a subtle conceptual shift in how memory is handled. Where delay embeddings represent the past as a static vector of coordinates, the path representation retains explicit order structure: the signature distinguishes a year in which temperature anomalies rose sharply in spring and plateaued in autumn from one in which they drifted gradually upward, even if the set of monthly values were identical. Higher-order signature terms are sensitive to increment magnitude as well as order, which is why the authors normalize cumulative paths by a data-dependent scale parameter before feature construction, ensuring that the kernel reflects relative temporal structure rather than absolute amplitude. In the SST experiments, the base kernel on the spatial field is a radial basis function with a bandwidth fixed by the dataset itself, tying the geometry of path space to the physical statistics of the ocean.

What emerges from this study is a template that could extend well beyond sea surface temperature. Any climate or geophysical variable observed only partially—ocean heat content, sea ice extent, atmospheric composition—suffers from the same effective non-Markovianity that has limited purely snapshot-based statistical models. By lifting the state along the time axis into path space and then lifting nonlinearity away through the Koopman operator, the signature-kernel pipeline offers a way to learn linear dynamics from data that is simultaneously memory-robust, scalable to high dimensions, and transparent to spectral interpretation. The authors’ demonstration that a single learned operator can both beat climatology at multiyear horizons and expose coherent modes of variability suggests that the trajectory-based view of climate dynamics may be more than a mathematical curiosity—it may be the form that the ocean’s memory actually takes.