Whistler-mode waves in near-equatorial THEMIS measurements: reconstruction of magnetic field spectra from electric field and plasma measurements

This paper proposes and validates a technique to reconstruct magnetic field spectral density for THEMIS spacecraft E and D using electric field and plasma measurements, thereby overcoming limitations caused by post-2017 search-coil magnetometer failures and enabling the continued use of their extensive whistler-mode wave datasets.

The Gist

The Big Picture: Fixing a Broken Radio

Imagine the Earth is surrounded by a giant, invisible ocean of magnetic energy called the magnetosphere. Inside this ocean, there are natural radio waves called whistler-mode waves. These waves are like invisible messengers that talk to high-energy electrons, sometimes speeding them up and sometimes knocking them out of the system. Understanding these waves is crucial for protecting our satellites and understanding space weather.

To study these waves, scientists use a fleet of five satellites called THEMIS. Think of THEMIS as a team of five weather reporters stationed around the Earth. Their job is to listen to these "whistler" waves using special microphones called search-coil magnetometers.

The Problem: Two Reporters Lost Their Balance

For many years, all five reporters (Satellites A, B, C, D, and E) worked perfectly. They could hear the waves coming from all directions (up, down, left, right).

However, starting around 2017, two of the reporters—Satellites D and E—broke. Their microphones stopped working correctly for the "up and down" direction. They could still hear the waves coming from the sides, but the signal from the top/bottom was weak and distorted.

This is like trying to listen to a symphony orchestra while wearing headphones that only work on the left ear. You can hear the music, but you can't tell how loud the whole band is playing, and you can't tell where the sound is coming from. Because of this, scientists couldn't use the data from Satellites D and E for the years after 2017, leaving a huge gap in their knowledge.

The Solution: A Mathematical "Patch"

The authors of this paper, Declan Frawley and his team, came up with a clever way to fix this broken data. They realized that while the microphones (magnetometers) on Satellites D and E were broken, the antennas (electric field instruments) on those same satellites were still working perfectly.

They used a three-step "recipe" to reconstruct the missing sound:

1. Find the Signal: First, they looked at the broken magnetic data just enough to identify when and where the whistler waves were happening. It's like looking at a blurry photo to see where a car is, even if you can't see the license plate clearly.
2. Switch Channels: Once they knew the waves were there, they switched to the working electric field data (the antennas) to get a clear reading of the wave's energy.
3. Do the Math: Using a known rule of physics (called the cold plasma dispersion relation), they translated the electric signal back into a magnetic signal. Think of this like using a translator app: "If the electric antenna hears this much noise, the magnetic microphone should have heard that much."

The Test: Did the Patch Work?

To see if their fix was good, they tested it on Satellite A, which never broke. They pretended Satellite A was broken, used their "patch" to guess the magnetic signal, and then compared their guess to the real, working data.

The Result: Their reconstructed data was very close to the real data. They found that their method could restore the magnetic signal to within a factor of 1.5 of the true value. In other words, if the real wave was a volume of 100, their fix guessed it was between 66 and 150. That is accurate enough to use for scientific studies.

The "Correction Factor"

Because the broken satellites (D and E) got worse over time, the scientists calculated a specific "correction number" for each year from 2015 to 2022.

• In 2016, they had to multiply the data by about 1.5 to fix it.
• By 2021, the satellites had degraded so much that they had to multiply the data by about 3.

This allows scientists to take the old, broken data from 2017–2022 and "scale it up" to get a usable picture of what was happening in space.

The Catch (Limitations)

The paper admits this method isn't perfect. It works best for waves that are traveling straight up or down (like a laser beam). If the waves are traveling at a weird angle (like a ricocheting bullet), the math gets trickier, and the estimate might be less accurate. Also, the method relies on knowing how dense the space plasma is, which is estimated from the satellite's own electrical charge—a bit like guessing the thickness of fog by looking at how much your car's headlights dim.

Summary

In short, this paper is a technical manual on how to rescue valuable space data from two broken satellites. By combining working electric sensors with broken magnetic sensors and applying some smart math, the team has allowed scientists to fill in the missing years of the THEMIS mission, ensuring we don't lose our understanding of how Earth's magnetic environment behaves.

Technical Summary

Technical Summary: Reconstruction of Whistler-Mode Magnetic Field Spectra from THEMIS E and D Measurements

Problem Statement
Electromagnetic whistler-mode waves are critical for the acceleration and loss of energetic electrons in Earth's outer radiation belt and magnetotail. The THEMIS mission (2007–present) has provided an extensive dataset of these waves via its Fast Fourier Transform (fff) data product, collected by search-coil magnetometers and electric field instruments. However, a significant technical limitation arose after 2017 for two of the five spacecraft (THEMIS E and D). Their search-coil magnetometers experienced signal degradation along the spacecraft spin axis (approximately the z-GSM axis), resulting in the loss of two magnetic field components. Consequently, the fff datasets from these spacecraft only reliably capture the spin-plane components, leading to a severe underestimation of total wave magnetic field amplitudes and an inability to accurately determine wave propagation directions. This degradation compromises the utility of the THEMIS E and D datasets for statistical studies of whistler-mode waves, which require accurate magnetic field intensity measurements.

Methodology
The authors propose and validate a reconstruction technique to recover the magnetic field spectral density ($B^2$) for the THEMIS E and D fff datasets using available electric field measurements and plasma parameters. The methodology proceeds as follows:

1. Wave Identification: The technique utilizes the degraded magnetic field fff data to identify the specific frequency-time domains where whistler-mode waves are present. To isolate these waves from low-intensity noise and other electrostatic modes, the authors employ a Z-score method. This statistical filter identifies outliers in the data distribution, setting a threshold (Z > +0.8) to distinguish intense wave bursts from background noise.
2. Electric Field Cleaning: Once the whistler-mode frequency domains are identified via the magnetic data, these domains are applied as a mask to the electric field fff dataset. This "cleans" the electric field spectrum, retaining only the components associated with whistler-mode waves.
3. Dispersion-Based Recalculation: The cleaned electric field spectral density is converted into magnetic field spectral density using the cold plasma dispersion relation for whistler-mode waves (Stix, 1962). This conversion requires local measurements of the electron gyrofrequency ($f_{ce}$) from flux-gate magnetometers and the electron plasma frequency ($f_{pe}$), the latter derived from spacecraft potential measurements.
4. Statistical Validation: The accuracy of the reconstruction is assessed by comparing the reconstructed magnetic field spectra against direct measurements from THEMIS A, which retained full 3D magnetic field capability throughout the study period. A correction factor, $C$, defined as $\log(B^2_{observed}/B^2_{reconstructed})$, is calculated to quantify the systematic difference between the two.

Key Results

• Reconstruction Accuracy: Comparison with THEMIS A data demonstrates that the reconstructed magnetic field spectral density is generally within a factor of 1.5 of the actually measured magnitudes for well-functioning instruments.
• Correction Factors for THEMIS E and D: Statistical analysis of ~100 hours of whistler-mode observations per year (2015–2022) reveals that the correction factor $C$ varies significantly over time for THEMIS E and D, reflecting the progressive degradation of their magnetometers.
  • For THEMIS A (control), $C$ remains stable around 0.25, implying a consistent correction factor of approximately $\times 1.8$ ($10^{0.25}$).
  • For THEMIS E and D, the mean $C$ values drift from near zero in 2015–2016 to higher values in subsequent years. For instance, in 2021, the measured magnetic field intensity for THEMIS E requires a correction factor of approximately $\times 3$ to match the reconstructed values derived from electric field data.
• Uncertainties: The authors identify several sources of uncertainty in the reconstruction:
  • Propagation Angle: The method assumes field-aligned (quasi-parallel) wave propagation. It may be less accurate for oblique waves, particularly those near the Gendrin angle or resonant cone, where the electric-to-magnetic field ratio changes significantly.
  • Dispersion Model: The use of the cold plasma dispersion relation may not adequately describe very oblique waves or environments with rarefied hot plasma.
  • Plasma Density: Estimating $f_{pe}$ from spacecraft potential introduces uncertainties in regions with rarefied hot plasma.
  • Spectral Contamination: While the Z-score method isolates waves, the electric field fff spectra still contain contributions from electrostatic waves and noise, potentially leading to an overestimation of the electric field, though this is partially balanced by the fact that fff data only captures two of the three electric field components.

Significance and Claims
The paper presents a technical solution to a specific data gap in the THEMIS mission, enabling the continued use of the THEMIS E and D datasets for statistical investigations of whistler-mode waves after 2017. The authors claim that their method successfully restores magnetic field spectral density with an accuracy sufficient for statistical analysis, provided that appropriate, time-dependent correction factors are applied.

The study does not claim to fully restore the ability to determine 3D wave propagation vectors for these spacecraft but focuses specifically on recovering the wave intensity (spectral density), which is the primary parameter for wave models and electron scattering calculations. By providing a validated reconstruction technique and a database of correction factors ($C$) for different years, the authors enable the integration of the full THEMIS mission duration (2008–2025) into studies of electron acceleration and loss in the magnetosphere. The work is framed as a technical report verifying a specific scaling technique, offering a practical tool for researchers to mitigate the impact of instrument degradation on long-term statistical studies.