Measuring the spatial lag effect of Tobler's First Law of Geography in Earth system predictive learning.

Prediction of complex geographical phenomena is central to geoscience, underpinning climate adaptation, disaster risk reduction, and resource management. A foundational principle guiding predictive learning is Tobler's First Law (TFL) of Geography. Although widely embedded in models ranging from spatial autoregression to deep learning, TFL is rarely examined in terms of its lag effect, how spatial and temporal lags jointly shape predictability. This gap constrains efforts to identify informative neighborhood sizes and memory lengths for predictive modeling. Here we introduce a predictability-based causal emergence framework that quantifies the information gain from incorporating different spatial and temporal lags. Applying this framework to global 2 m temperature reanalysis data, we find that the most informative spatiotemporal context is regionally heterogeneous under the current experimental setting. Our results provide a principled approach to refining TFL in predictive modeling, offering new insights into how spatiotemporal lag effects contribute to the emergence of predictability in complex geographical phenomena.