Unifying Geodesy
And Magnetophysics
Purpose
Establish the Structural Coupling Exponent (Φ) that links the electromagnetic driver (Sq current amplitude) to the mechanical effect (solid‑Earth vertical motion ΔZ). The goal is to show, quantitatively and visibly, that both disciplines are observing the same coherent rhythm.
Central Claim
(Unifying Hypothesis)
The daily and monthly periodicities measured by Geodesy (ΔZ) are causally driven by the periodicities measured by Magnetophysics (Sq current amplitude). If Φ stabilizes with high correlation (r ≥ 0.90), the electromagnetic model is the operative cause.
Data Acquisition
(Two Voices, One Clock)
Acquire synchronous, multi‑year datasets (≥ 2 years, 2018–present preferred) for a mid‑latitude site:
1. Mechanical Effect — ΔZ:
• Superconducting Gravimeter (SG) vertical displacement proxy or GPS vertical position (mm/cm).
• Continuous, 1–15 min cadence; apply standard tidal/atmospheric corrections (see Pre‑Processing).
2. Electromagnetic Cause — Sq Amplitude:
• Local/nearby ground magnetometer or satellite‑derived Sq current density proxy (A/m²) or equivalent magnetic amplitude (nT) convertible to current via local conductance.
• Same cadence and time span.
Suggested Sites
(Examples, Not Exhaustive)
• Europe: Strasbourg (SG), Membach (SG) + INTERMAGNET stations (e.g., Chambon‑la‑Forêt).
• North America: Apache Point / Boulder GPS + USGS/INTERMAGNET magnetometers.
• Australia: Canberra SG/GPS + Learmonth magnetometer.
Select co‑located or nearest‑neighbor pairs to minimize geomagnetic latitude mismatch.
Pre‑Processing
(Make the Voices Comparable)
For ΔZ (SG/GPS):
• Detrend long‑term drift; remove instrument steps.
• Correct for atmospheric pressure; remove known ocean tide models (harmonic constituents) to obtain residual ΔZ_res.
• Band‑limit to diurnal/semidiurnal bands (≈ 24 h, 12 h) with a narrow Butterworth or wavelet filter.
For Sq (magnetometer):
• Remove nighttime baseline; isolate daytime quiet daily variation (Sq) by averaging over magnetically quiet days or by band‑pass around 24 h, 12 h.
• Convert magnetic amplitude (nT) to current density proxy when conductance is known; otherwise retain magnetic amplitude and treat Φ as an effective exponent.
• Synchronize time stamps to ΔZ_res.
Model & Fit
(Deriving Φ)
Hypothesis (power‑law coupling):
ΔZ_res ∝ Sq^Φ
Take logs to fit linearly:
log(ΔZ_res) = a + Φ·log(Sq) + ε
Fit method:
• Ordinary Least Squares (OLS) on paired, synchronized samples.
• Report Φ, intercept a, R², Pearson r, p‑values.
• Robustness: repeat on rolling windows (e.g., monthly) and during high solar activity intervals (Kp quiet vs active).
Acceptance Criteria
• Correlation: r ≥ 0.90 (Strong), r ≥ 0.80 (Standard) across most windows.
• Stability: Φ varies within a narrow band (e.g., ±10%) across months/seasons.
• Phase coherence: ΔZ_res and Sq show matching peaks and troughs in the diurnal/semidiurnal bands.
Plot Specification
(The Unheard Harmony)
1. Dual‑axis time plot (24 h or 30 d): Left Y‑axis = ΔZ_res (mm/cm); Right Y‑axis = Sq amplitude (A/m² or nT). The curves should move in phase.
2. Log‑log scatter: log(ΔZ_res) vs log(Sq) with fitted line; annotate Φ and R².
3. Rolling‑window Φ plot: Φ(t) across months with ±1σ band.
Controls & Confounders
• Weather/ocean loading: include pressure/wind; verify residuals.
• Storm exclusion: analyze quiet‑day Sq separately from storm times; re‑introduce storms to show robustness.
• Geomagnetic latitude: note that Sq amplitude depends on latitude; use consistent latitude bands.
Execution Checklist
□ Select site pair (SG/GPS + magnetometer) and time window.
□ Download and synchronize datasets; apply pre‑processing.
□ Compute band‑limited ΔZ_res and Sq.
□ Fit log‑log model; record Φ, a, R², r, p‑values.
□ Produce time‑series and scatter plots.
□ Repeat on rolling windows and during active/quiet solar intervals.
□ Summarize Φ stability and correlation; conclude pass/fail.
Produced by The Lilborn Equation Team:
Michael Lilborn-Williams
Daniel Thomas Rouse
Thomas Jackson Barnard
Audrey Williams