📄 earth science

Geo-χ-Former: A Geodynamically Informed Transformer for Short-Term Polar Motion Prediction in the Excitation Domain

This paper introduces Geo-χ-Former, a geodynamically informed Transformer framework that improves 30-day short-term polar motion prediction by forecasting in the excitation domain using effective angular momentum covariates and calendar encodings, thereby reducing root-mean-square errors by 9.21% and 11.93% compared to standard IERS Bulletin A predictions.

Original authors: Junhai Huang, Boyang Zhu, Xiaodong Liu, Zhigang Wu, Wei Wang

Published 2026-09-17

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CC BY 4.0

Original authors: Junhai Huang, Boyang Zhu, Xiaodong Liu, Zhigang Wu, Wei Wang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Earth does not spin on a perfectly fixed axis. Like a slightly unbalanced top, our planet wobbles as it rotates, causing the geographic North Pole to drift slowly across the surface of the globe. This movement, known as polar motion, is a subtle but critical phenomenon. It shifts the precise location of the poles by several meters over time, a change that matters deeply for anyone relying on exact positioning, from satellite navigation systems to deep-space tracking. To keep these systems accurate, scientists must predict where the pole will be in the coming weeks. However, this task is notoriously difficult because the Earth's wobble is driven by a complex mix of forces. The atmosphere, oceans, and even the shifting weight of water on land all push and pull on the planet's rotation, creating a chaotic signal that is hard to untangle from the steady, rhythmic background of the wobble itself.

For decades, researchers have tried to forecast this movement using mathematical models that look at the history of the pole's position. These methods work well for the long, steady rhythms of the wobble, but they often struggle to catch the shorter, sharper jolts caused by weather patterns or ocean currents. A new study by a team at Sun Yat-sen University and the China Aerospace Science and Technology Corporation proposes a different way to think about the problem. Instead of trying to predict the position of the pole directly, they decided to predict the forces that cause the pole to move. By shifting their focus to the "excitation" of the wobble—the physical push and pull from the atmosphere and oceans—they created a new forecasting tool called Geo-χ-Former. This approach allows the computer to learn the relationship between the Earth's fluid layers and the resulting wobble more effectively than previous methods.

The researchers built their system on a simple but powerful idea: the Earth's wobble is a reaction to external forces. Just as a boat rocks in response to waves, the Earth's rotation axis shifts in response to the movement of air, water, and ice. The team used a sophisticated type of artificial intelligence known as a Transformer, which is excellent at finding patterns in sequences of data. However, they did not feed the model the raw coordinates of the pole's location. Instead, they first converted the observed pole movements into a representation of the forces that created them. This process, known as deriving the geodetic angular momentum function, effectively strips away the long, persistent rhythms of the wobble to reveal the shorter-term signals driven by the atmosphere and oceans. The model then learns to predict these force signals for the next 30 days, using historical data on wind, ocean currents, and water storage as clues.

Once the model predicts how these forces will change in the future, the system works backward. It takes the predicted forces and mathematically reconstructs what the pole's position will be, effectively translating the "push" back into the "rock." This two-step process—predicting the cause and then calculating the effect—proved to be more accurate than trying to guess the effect directly. When the team tested their method against the standard operational forecasts used by the International Earth Rotation and Reference Systems Service, the results were clear. Over a 30-day period, their new model reduced the average error in predicting the pole's position by about 9.2 percent for one direction and nearly 12 percent for the other. These improvements, while seemingly small in absolute terms, represent a significant leap in precision for a system that operates on the scale of milliarcseconds, a unit so small it is barely perceptible to the human eye.

The study also explored why this method works so well by peering inside the model's decision-making process. The researchers found that the artificial intelligence naturally learned to pay the most attention to the atmosphere and the oceans, which are the primary drivers of short-term polar motion. It also recognized that the movement of water on land, such as in rivers and soil, plays a distinct and complementary role. Interestingly, the model did not treat all data points equally; it learned to weigh recent history heavily while also remembering patterns that repeat over the course of a year. This ability to balance immediate conditions with seasonal rhythms allowed the system to navigate the complex interplay of forces that govern the Earth's rotation.

The success of Geo-χ-Former suggests that the future of predicting Earth's rotation lies in understanding the physical causes rather than just tracking the symptoms. By framing the problem in terms of the forces that drive the wobble, the researchers were able to bypass the limitations of older methods that relied solely on the history of the pole's position. The model's performance indicates that it successfully captures the subtle, short-term variations that other systems miss, offering a more reliable guide for the next month. While the accuracy of these forecasts still depends on the quality of the data regarding atmospheric and oceanic conditions, the new approach provides a robust framework for integrating that information. As the Earth continues its slow, complex dance, this geodynamically informed tool offers a clearer view of where the poles are heading, ensuring that our global positioning systems remain aligned with the spinning world beneath them.

Technical Summary: Geo-χ-Former

Problem Statement
Short-term polar motion (PM) prediction remains challenging because observed PM time series are dominated by persistent oscillations, specifically the Chandler wobble (~433 days) and annual variability. These dominant signals obscure shorter-timescale excitation variability driven by geophysical processes (atmospheric, oceanic, hydrological, and sea-level changes). Directly forecasting PM coordinates in the spatial domain often leads models to rely heavily on extrapolating these persistent oscillations, making it difficult to learn the subtle, non-linear relationships between PM and Effective Angular Momentum (EAM) covariates. Furthermore, existing machine learning approaches typically treat EAM data merely as auxiliary covariates or residual terms rather than integrating them into a unified physical framework.

Methodology
The paper proposes Geo-χ-Former, a geodynamically informed Transformer framework that shifts the forecasting task from the coordinate domain to the excitation domain. The methodology consists of four primary stages:

Excitation Derivation (Wilson Inversion): Observed PM coordinates ( $x_p, y_p$ ) are converted into the Geodetic Angular Momentum (GAM) function ( $\chi$ ) using Wilson's discrete polar motion equation. This transforms the prediction target from pole coordinates to the dynamically equivalent excitation that drives them.
Target Decomposition and Preprocessing: The derived GAM is decomposed into a slowly varying background (capturing long-term trends and offsets between GAM and summed EAM) and a short/seasonal component ( $\chi_S$ ). The model predicts only $\chi_S$ , while the background is estimated via robust local regression and extrapolated deterministically. This step removes low-frequency offsets that differ between geophysical models and observations.
Geo-χ-Former Architecture: The model employs a Transformer-based structure designed for the excitation domain:
- Variable-Wise Patch Embedding: Input sequences (historical $\chi_S$ , historical EAM, and calendar encodings) are segmented into non-overlapping patches to preserve variable identity.
- Temporal Attention with RoPE: Rotary Position Embeddings (RoPE) are applied to model temporal dependencies within each variable, capturing persistence and seasonal phase shifts.
- Structure-Aware Attention Across Variables: A cross-variable attention mechanism allows the target excitation tokens to attend to EAM covariates (Atmospheric, Oceanic, Hydrological, and Sea-level Angular Momentum) and calendar features. This layer includes an axis-aware bias to encourage alignment between target axes and corresponding EAM components.
- Future Covariate Handling: The model utilizes a "KV shift" mechanism for deterministic future covariates (like calendar encodings) and treats future EAM forecasts as auxiliary inputs available during the forecast horizon.
Reconstruction and Loss Function: The predicted short/seasonal excitation is combined with the extrapolated background to reconstruct the full GAM. This is then integrated back into PM coordinates using the Wilson forward recurrence. The training objective is a composite loss function combining a Huber loss on the physical excitation scale and a Huber loss on the reconstructed PM trajectory, ensuring the model learns both excitation dynamics and their physical manifestation in PM.

Key Contributions

Excitation-Domain Formulation: The paper introduces a framework that formulates PM prediction as a geodetic excitation problem, allowing the model to learn the physical coupling between EAM sources and Earth rotation more directly than coordinate-domain approaches.
Hybrid Physical-Data-Driven Architecture: Geo-χ-Former integrates physical constraints (Wilson equations, Liouville relationships) with deep learning, using EAM not just as features but as the primary representation of the target variable's driving forces.
Decomposition Strategy: The specific removal of a slowly varying background prior to training addresses the systematic biases often found between modeled EAM sums and geodetically inferred GAM, stabilizing the learning of short-term variability.
Comprehensive Ablation and Diagnostics: The study provides detailed analyses of model components (e.g., attention mechanisms, KV shifts) and input sensitivities (EAM source contributions), offering interpretability into how the model utilizes geophysical data.

Results
Evaluated on IERS EOP 20u24 C04 data and GFZ EAM products from 1993 to 2025, Geo-χ-Former was compared against the operational IERS Bulletin A and deep learning baselines (DLinear, LSTM, PatchTST) in both PM and excitation domains.

Performance Gains: Over a cumulative 1–30-day forecast window, Geo-χ-Former reduced the Root-Mean-Square Error (RMSE) relative to Bulletin A by 9.21% for $x_p$ and 11.93% for $y_p$ .
Domain Superiority: For all tested architectures (LSTM, DLinear, PatchTST), forecasting in the excitation domain followed by PM reconstruction yielded lower errors than direct PM coordinate prediction. The excitation-domain approach reduced cumulative 30-day RMSE by approximately 0.9 to 4.0 mas compared to the PM-domain counterparts.
Ablation Findings: Removing EAM inputs caused the largest performance degradation (+5.03% RMSE increase), confirming the value of physical covariates. Among EAM sources, Hydrological Angular Momentum (HAM) provided a distinct incremental contribution, while Atmospheric (AAM) and Oceanic (OAM) angular momentum showed the highest attention weights and forecast sensitivity.
Component Impact: Removing "attention across variables" resulted in the largest structural degradation, highlighting the importance of fusing EAM information with excitation history.

Significance
The paper claims that Geo-χ-Former addresses a central difficulty in data-driven PM forecasting: the inability of standard models to disentangle persistent oscillations from short-term excitation signals. By reformulating the problem in the excitation domain, the model leverages the physical relationship between angular momentum exchange and Earth rotation. The results demonstrate that incorporating geophysical angular momentum information within a Transformer architecture, guided by physical reconstruction constraints, yields consistent performance improvements over both operational baselines and standard deep learning models. The study concludes that this excitation-domain approach provides a structured space for fusing multi-source geophysical data, offering a pathway to more accurate short-term Earth orientation predictions.

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