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 (RKHS) 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 capture memory effects in SST-only evolution. The resulting operator improves out-of-sample multi-year forecast skill relative to a climatology baseline and reveals coherent spectral modes. We implement the approach via kernel extended dynamic mode decomposition (EDMD) on signature-kernel Gram matrices, yielding a single pipeline for forecasting and spectral diagnostics of high-dimensional SST dynamics.