A Geometric Algebra-Inspired Spatio-temporal Learning Model for Sea Level Variability Reconstruction Under Incomplete Observations

Reconstructing complete sea level variability (SLV) from sparse observations is a central problem in ocean science. Accurate reconstruction critically depends on modeling cross-variable dependencies, as SLV is governed by coupled ocean processes and the associated state variables are intrinsically interdependent. Most deep learning approaches process such variables as independent channels, which may limit their ability to capture coordinated multivariate interactions under data sparsity. To address this issue, we develop a geometric algebra (GA)-inspired spatio-temporal learning model in which temperature, salinity, and horizontal velocities are encoded as coefficients associated with the basis blades of a multivector. The core module, a GA-based convolutional gated recurrent unit (GAConvGRU), replaces conventional channel-wise convolution with algebraically structured convolution within a recurrent architecture. The framework is further complemented by temporal embedding, dual-path convolution, and linear mapping to support multi-scale spatio-temporal modeling. Experiments across three coastal regions with varying levels of missing observations validate the proposed model. Ablation results show that GAConvGRU yields increasingly pronounced performance gains as data sparsity grows. Comparative evaluations against state-of-the-art methods demonstrate consistently competitive or superior reconstruction accuracy, with advantages more evident in high-missing-rate scenarios. Overall, the results indicate that GA-based multivariate interaction modeling enhances robustness in spatio-temporal learning under incomplete observations and provides an effective approach for SLV reconstruction and other geophysical applications involving strongly interdependent variables.