Koopman analysis of sea surface temperature with a signature kernel

We develop a trajectory-based Koopman method for sea surface temperature (SST) that lifts annual SST segments using a signature kernel—a reproducing kernel Hilbert space kernel that compares paths via iterated-integral features—and learns the one-year shift operator. By operating on annual trajectory segments rather than instantaneous fields, the method encodes finite-time history, which helps to capture memory effects in SST-only evolution. The resulting operator improves the out-of-sample multiyear forecast skill relative to a climatology baseline and reveals coherent spectral modes. We implement the approach via kernel extended dynamic mode decomposition (kEDMD) on signature-kernel Gram matrices, yielding a single pipeline for forecasting and spectral diagnostics of high-dimensional SST dynamics.