Filling the gap in the IERS C01 polar motion series in 1858.9-1860.9

This paper addresses the two-year gap in the $Y_p$ pole coordinate of the IERS C01 polar motion series (1858.9–1860.9) by comparing a parametric astronomical model with a data-driven Singular Spectrum Analysis (SSA) approach, ultimately recommending the SSA method as preferable due to its more complete modeling of polar motion.

The Gist

The Missing Chapter in Earth's Diary

Imagine the Earth as a giant, spinning top. Sometimes, this top wobbles a little bit as it spins. Scientists call this wobble "polar motion." To understand our planet's deep history—how its climate changes, how its core shifts, and how mass moves around the globe—scientists need a perfect, unbroken diary of these wobbles.

For over 170 years, the IERS (a global team of scientists) has been keeping this diary, known as the C01 series. It's the longest and most reliable record we have of Earth's rotation.

The Problem: A Two-Year Hole
However, there is a problem with this diary. Between the years 1858 and 1860, there is a two-year gap in one specific column of the data (called the Yp coordinate).

Why did this happen? Back then, astronomers in Washington, D.C. stopped taking measurements for a while. The other observatories (in Greenwich and Russia) were located at longitudes that made it impossible to figure out the "Y" part of the wobble without the Washington data. It's like trying to solve a puzzle with a piece missing; you can guess, but you don't have the full picture.

This gap is annoying because many mathematical tools used to study the Earth's rotation require a smooth, continuous line of data. A gap is like a pothole in a road; it forces drivers (or algorithms) to slow down, take detours, or risk crashing.

The Mission: Filling the Gap
The authors of this paper, Zinovy, Nina, and Roman, decided to act as "data detectives." Their goal was to fill in those missing two years with the most accurate guesses possible, so scientists could study the Earth's wobble without interruption.

They tried two different detective techniques:

1. The "Recipe" Approach (The Parametric Model)

Imagine you are trying to guess the flavor of a soup you can't taste, but you know the recipe. You know it has a base (bias), a spicy kick that changes slowly over time (Chandler wobble), and a seasonal herb that comes and goes every year (Annual wobble).

The scientists built a mathematical "recipe" based on what they knew about Earth's wobble. They assumed the wobble followed a predictable pattern where the "spiciness" (amplitude) grew a little bit linearly over time. They used the data from the years before and after the gap to tune this recipe, then used it to "cook up" the missing numbers for 1858–1860.

• Pros: It's based on solid physics.
• Cons: It assumes the Earth behaves exactly like the recipe says. If the Earth did something weird during those two years that the recipe didn't predict, this method might get it wrong.

2. The "Pattern Recognition" Approach (SSA)

This method is more like a jazz musician improvising. Instead of following a strict recipe, the musician listens to the whole song (the entire 1846–1900 data) and finds the repeating rhythms and melodies.

The scientists used a technique called Singular Spectrum Analysis (SSA). Think of this as a high-tech prism that breaks the Earth's wobble data into its purest colors (patterns). It looks for the main "notes" (the 1-year and 1.19-year wobbles) and the background noise.

Because this method is "data-driven," it doesn't force the Earth to follow a strict recipe. It simply asks: "Based on the rhythm of the whole song, what note should come next?" It is particularly good at handling complex, shifting patterns without needing to know the exact physics beforehand.

The Verdict: Which Detective Won?
The scientists tested both methods by creating fake gaps in the data, filling them in, and seeing how close they got to the real numbers.

• The Result: Both methods did a great job. The numbers they guessed were very close to each other, and the "error" in their guesses was about the same size as the natural uncertainty in the original 19th-century measurements.
• The Winner: They leaned slightly toward the SSA (Pattern Recognition) method. Why? Because it didn't rely on a rigid recipe. It captured the subtle, complex nuances of the Earth's movement better than the strict "recipe" model.

Why Does This Matter?
By filling in this gap, the scientists have given the world a continuous, smooth timeline of Earth's wobble from 1846 to today.

This is crucial because:

1. Long-term Trends: It helps us see slow changes in Earth's rotation that only become visible over decades or centuries.
2. Climate Science: It helps us understand how melting ice caps and shifting oceans affect the planet's spin.
3. Historical Accuracy: It allows modern computers to analyze the 19th-century data without getting confused by the missing piece.

In a Nutshell
The Earth's wobble diary had a missing chapter. The authors used two different methods—a strict physics recipe and a flexible pattern-recognition tool—to write that missing chapter back in. While both worked well, the flexible tool gave a slightly better story. Now, scientists can read the entire history of Earth's spin without skipping a beat.

Technical Summary

1. Problem Statement

The International Earth Rotation and Reference Systems Service (IERS) provides the C01 series, the longest reliable record of Earth Orientation Parameters (EOP), specifically Polar Motion (PM), dating back to 1846. This series is critical for analyzing long-term variations in Earth's rotation, such as the Chandler Wobble (CW) and Annual Wobble (AW).

However, the $Y_p$ component (one of the two pole coordinates) of the IERS C01 series contains a 2-year data gap from 1858.9 to 1860.9.

• Cause: The gap resulted from a lack of observations at the Washington Observatory during this period. The remaining observatories (Greenwich and Pulkovo) are located at longitudes ($0^\circ$ and $30^\circ$) that are poorly suited for determining the $Y_p$ coordinate (which requires longitudinal separation closer to $90^\circ$).
• Impact: The gap creates an unevenly spaced time series. This complicates standard spectral analysis and statistical modeling, which generally require continuous, regularly spaced data to avoid introducing spurious frequencies or biases.
• Goal: To reconstruct the missing $Y_p$ values using robust mathematical methods that preserve the physical structure of the polar motion signal without relying on external, unverified historical data.

2. Methodology

The authors employed two distinct approaches to fill the gap: a parametric astronomical model and a data-driven model.

A. Parametric Astronomical Model (BCA and BCA2)

This approach assumes the polar motion consists of a bias, a Chandler Wobble, and an Annual Wobble, with amplitudes changing linearly over time.

• BCA Model: A 9-parameter model including:
  • A constant bias ($a_0$).
  • Chandler Wobble (CW) with period $P_c \approx 1.19$ years and linearly changing amplitude.
  • Annual Wobble (AW) with period $P_a = 1$ year and linearly changing amplitude.
• BCA2 Model: A 13-parameter extension acknowledging a potential bifurcation in the CW spectrum (two distinct CW periods: 1.181 yr and 1.231 yr).
• Implementation: The models were fitted using Least Squares (LS) over various time intervals (12 to 24 years) centered on the gap. The missing values were computed based on these fitted parameters.

B. Data-Driven Model (Singular Spectrum Analysis - SSA)

This approach uses Singular Spectrum Analysis (SSA), a non-parametric method capable of extracting trends and oscillatory components from noisy time series without a priori assumptions about periods.

• Techniques Tested:
  1. 1D-SSA: Applied to the $Y_p$ series alone.
  2. MSSA (Multivariate SSA): Applied to the coupled $(X_p, Y_p)$ series to exploit physical correlations between coordinates.
  3. CSSA (Complex SSA): Applied to the complex series $Z = X_p + iY_p$. This is particularly effective for polar motion as it treats the coordinates as a rotating vector, naturally separating harmonics with a $\pi/2$ phase shift.
• Optimization: The authors optimized the window length ($L$) and the model order ($r$) (number of SVD components) by creating artificial gaps in the existing data and minimizing the reconstruction error (RMSE and MAE).
• Best Configuration: The optimal results were achieved using CSSA with window lengths $L=60, 119, 238$ and specific model orders ($r=4$ to $9$). The final result was an average of the seven best-performing CSSA variants.

3. Key Contributions

1. First Reconstruction: This is the first study to attempt filling the specific 2-year gap in the IERS C01 $Y_p$ series using rigorous mathematical modeling.
2. Methodological Comparison: The paper provides a comprehensive comparison between parametric (physics-based) and non-parametric (data-driven) methods for gap filling in geophysical time series.
3. Validation of SSA: The study demonstrates that Complex SSA (CSSA) is superior for this specific problem because it captures the full structure of the polar motion (including subtle frequency mixing and phase variations) without being constrained by the assumption of a single dominant CW frequency or strictly linear amplitude trends.
4. Error Analysis: The authors quantified the filling errors and demonstrated that the reconstruction accuracy is consistent with the inherent uncertainty of the 19th-century observational data.

4. Results

• Accuracy: Both methods produced results that agree within the estimated error margins of the original IERS C01 series.
  • RMSE/MAE: The filling errors for both methods were approximately 0.08"–0.09" arcseconds.
  • Consistency: This error magnitude matches the stated uncertainty of the IERS C01 $Y_p$ data for the 19th century ($0.09"$), confirming that the reconstruction does not introduce artificial precision but rather fills the gap with statistically consistent values.
• Comparison:
  • The SSA (CSSA) results were slightly more accurate (lower error) than the parametric BCA models.
  • The SSA method revealed additional minor spectral components (periods of ~0.85 yr and ~0.94 yr) that were below the noise level in standard periodograms but detectable by SSA, suggesting a more complete signal recovery.
• Final Output: The authors provide a table of the 21 reconstructed $Y_p$ values (from 1858.9 to 1860.9) derived from the average of the BCA and SSA methods, with an assigned uncertainty of $0.09"$.

5. Significance

• Enabling Continuous Analysis: By filling the gap, the IERS C01 series becomes a truly continuous, evenly spaced dataset from 1846 to the present. This allows researchers to apply standard spectral analysis techniques (like Lomb-Scargle or FFT) without the risk of spectral leakage or spurious frequencies caused by the missing data.
• Long-Term Climate and Geophysics: A continuous record is essential for studying decadal and multi-decadal variations in Earth's rotation, such as the ~66-year and ~80-year amplitude modulations of the Chandler Wobble, which are difficult to resolve accurately without the 19th-century data.
• Methodological Benchmark: The paper establishes CSSA as a robust tool for reconstructing gaps in geophysical time series, particularly when physical models are uncertain or when data is sparse.
• Future Work: The authors note that while the gap is filled, the 19th-century data remains of lower accuracy than modern data. They advocate for future efforts to mine and reprocess historical astronomical observations (declinations and latitudes) to further refine the pole coordinates for the period 1820–1860.

In conclusion, the paper successfully bridges a critical data gap in the history of Earth rotation science, providing a validated, continuous dataset that facilitates more accurate long-term studies of Earth's dynamic processes.