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# Characterize the magnetic signal generated in the magnetosphere from geomagnetic observations

Yalei Shi, Vincent Lesur, Erwan Thébault

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Yalei Shi, Vincent Lesur, Erwan Thébault. Characterize the magnetic signal generated in the magnetosphere from geomagnetic observations. Swarm 10 Year Anniversary & Science Conference, Apr 2024, Copenhagen, Denmark. http://hal.science/hal-04588665

HAL Id: hal-04588665

https://hal.science/hal-04588665

Submitted on 27 May 2024

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# Characterize the magnetic signal generated in the magnetosphere from geomagnetic observations

Yalei Shi, Vincent Lesur, Erwan Thébault

► To cite this version:

Yalei Shi, Vincent Lesur, Erwan Thébault. Characterize the magnetic signal generated in the magnetosphere from geomagnetic observations. Swarm 10 Year Anniversary & Science Conference, Apr 2024, Copenhagen, Denmark. [hal-04588665](https://hal.science/hal-04588665v1)

HAL Id: hal-04588665

https://hal.science/hal-04588665v1

Submitted on 27 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

Public Domain
# Characterize the magnetic signal generated in the magnetosphere from geomagnetic observations

Yalei SHI¹, Vincent LESUR¹, Erwan THEBAULT²

¹Université Paris Cité, Institut de physique du globe de Paris, CNRS F-75005 Paris, France
²Université Clermont-Auvergne, Laboratoire Magmas et Volcans, CNRS F-63170 Aubière, France

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

## Introduction

The main contribution to the geomagnetic field is from the core, but fields generated from the magnetosphere during magnetically quiet periods play an important role. They may have geomagnetic contributions up to hundreds of nanoteslas (nT) during magnetic storm. To study the geomagnetic field, the contributions from different sources need to be separated and modeled. In current models of the geomagnetic field derived from satellite data, the magnetospheric components are poorly described because of the limitation of satellite data spatiotemporal resolution. The poorly described magnetospheric component limits the resolution of other geomagnetic contributing sources, especially the contributions from the core (Thebault et al. [2011]) and the lithosphere (Lesur et al. [2013]). It is therefore important and necessary to describe and model precisely the magnetospheric components. We describe here an approach for their modeling based on Kalman filter approach and magnetic observatory data. These provide a data set with a temporal resolution particularly well suited to characterize rapidly varying magnetospheric signals.

## Objective

* Study the magnetospheric field up to spherical harmonic (SH) degree 6 (L=6) through normal distributions (N($\mu$ = m, $\sigma^2$ = $C_m$)) with a time resolution of an hour, based on a Kalman filter and correlation-based modeling.

## Test models with synthetic data

Synthetic data at each observatory include only magnetospheric field signal and its response in the mantle:

1. generation of a random hourly sequence during the whole 2021 year for each Gauss coefficient $q_l^m$ up to degree L=6

2. removal of the principal trend by PCA and normalization of the random distribution of $q_l^m$ by the variance trends from the IGRF-13 field power spectrum

3. generation of $i_l^m$ by multiplying the normalized $q_l^m$ with a 1D electrical conductivity model (Verhoeven, Thebault, Saturnino, Houliez, and Langlais [2021])

Night synthetic data
(23:00 – 05:00 LT)

Full-year synthetic data

<figure><img src="image_8.png" /><figcaption>**Figure 2:** The left: results by fitting only night synthetic data. The right: results by fitting full-year synthetic data.</figcaption></figure>

Top: The diagonal elements of the model covariance matrix

Bottom: Power spectrum of SH Gauss coefficient as a function of SH degree

<figure><img src="image_9.png" /><figcaption>**Figure 3:** Top: View of a comparison during the first 20 days between synthetic Gauss coefficient (blue) and the estimated $q_1^0$ (orange and green) by fitting only synthetic data at night (23:00 – 05:00) with error bar 3σ (lavender). Bottom: the number of used data at each hour</figcaption></figure>

## Future works

* The hourly model needs to be extended to cover 1999 to 2024

* Improvement of hourly model: observatory data including day side should be used. The ionospheric contribution distribution of each component at each observatory should be co-estimated, and a correlation of ionospheric contribution among observatories should be added as a priori information

* Statistics from theoretical or semi-empirical models to separate the different contributions in the magnetosphere could be used, in order to develop a better understanding of the evolution of magnetospheric sources during solar cycles

## References

1. Thebault E., Vervelidou F., Lesur V., & Harmoudi M. (2011). The pros and cons of along-track satellite analysis in planetary magnetisme. In 25th IUGG General Assembly.

2. Lesur V., Rother M., Vervelidou F., Harmoudi M., Thebault E. (2013). Post-processing scheme for modelling the lithosphere magnetic field. Solid Earth, 4(1):105-118.

3. Ropp G., Lesur V., Baerenzing J., & Holschneider M. (2020), Sequential modelling of the earth's core magnetic field. Earth, Planets and Space, 72(1):1-15.

4. Verhoeven, O., Thebault, E., Saturnino, D., Houliez, A., & Langlais, B. (2021). Electrical conductivity and temperature of the earth's mantle inferred from bayesian inversion of swarm vector magnetic data. Physics of the Earth and Planetary Interiors, 314, 106702.

## INTERMAGNET observatory data

<figure><img src="image_10.png" /></figure>

1. The main magnetic field contribution is removed from the hourly-mean INTERMAGNET observatory data for full-year 2021 by subtracting the MCM model (Ropp et Lesur [2020])

2. Observatory data are selected between 23:00 - 05:00 LT and during geomagnetically quiet time (Dst between -30 nT and 30 nT) to minimize the contributions of ionospheric perturbations

<caption>**Figure 1:** Distribution of the hundred or so used geomagnetic observatory positions</caption>

## Parameters

The geomagnetic signal after the selection is considered as the sum of the magnetospheric field, induced field, and crustal offsets at each observatory location:

$$B(r, \theta, \phi) = \sum_{\ell,m} q_{\ell}^{m} \left(\frac{r}{a}\right)^{l-1} Y_{\ell}^{m}(\theta, \phi) + \sum_{\ell,m} i_{\ell}^{m} \left(\frac{a}{r}\right)^{l+2} Y_{\ell}^{m}(\theta, \phi) + \sum_{i \in N_{\text{obs}}} \delta(r - r_i, \theta - \theta_i, \phi - \phi_i) O_i$$

<table><thead><tr><th></th><th>magnetosphere</th><th>induced field</th><th>crustal offset</th></tr></thead><tbody><tr><td>Coef.</td><td>q<sub>l</sub><sup>m</sup></td><td>i<sub>l</sub><sup>m</sup></td><td>O<sub>i</sub>=(O<sub>x</sub><sup>i</sup>, O<sub>y</sub><sup>i</sup>, O<sub>z</sub><sup>i</sup>)</td></tr><tr><td>m<sub>0</sub></td><td>0</td><td>0</td><td>O<sub>i</sub></td></tr><tr><td>C<sub>0</sub></td><td>V<sub>l</sub><sup>ext</sup>=R(l)=S(R/a)<sup>2l</sup></td><td>0.3<sup>2</sup>V<sub>l</sub><sup>ext</sup></td><td>~ 0</td></tr><tr><td>S</td><td>0.7.10<sup>4</sup>(nT<sup>2</sup>)</td><td>-</td><td>-</td></tr><tr><td>R</td><td>3.5.10<sup>4</sup>km</td><td>-</td><td>-</td></tr><tr><td>τ, (P = e<sup>-Δt/τ</sup>)</td><td>10 hour</td><td>10 hour</td><td>10<sup>6</sup> year</td></tr><tr><td>A</td><td>Σ<sub>l,m</sub>(r/a)<sup>l-1</sup>Y<sub>l</sub><sup>m</sup></td><td>Σ<sub>l,m</sub>(a/r)<sup>l+2</sup>Y<sub>l</sub><sup>m</sup></td><td>1</td></tr><tr><td>Nb. of par.</td><td>48</td><td>48</td><td>3×10<sup>2</sup></td></tr></tbody></table>

## Results applied to real data

<figure><img src="image_11.png" /><figcaption>**Figure 5:** Example of the comparison between CLF (Chambon-la-foret) observatory data (in blue) and smoothed modeling signal (in orange) for x component over 20 days, the dotted grey line represent the annual average residue between observatory data and modeling signal</figcaption></figure>

The distribution of residuals over year 2021 for each component at each observatory is calculated. Most of observatories have an annual average residual nearby 0 nT and an annual residual standard deviation smaller than 10 nT, except some observatories located at high latitudes shown in figure 6 where the residual standard deviation can reach 90 nT

<caption>**Figure 6:** Distribution of the 23 geomagnetic observatory positions with annual residual standard deviation over 10nT between observatory data and modeling signal</caption>

## Approach -- Kalman Filter

### Prior information

mean prior model $m_0$, prior covariance matrix $C_0$

### Analysis step

adjustment of models at the kth hour by fitting INTERMAGNET observatory data, based on the Least Square method

$$d = Am + e \\ m_k^* = m_k + (A^t C_e^{-1} A + C_k^{-1})^{-1} A^t C_e^{-1} (d - Am_k) \\ C_k^* = (A^t C_e^{-1} A + C_k^{-1})^{-1}$$

### Prediction step

prediction of the next hour's model (at the (k+1)th hour) based on the previous adjusted hourly model (at the kth hour)

$$m_{k+1} = Pm_k^* \\ C_{k+1} = PC_k P^t + C_w$$

### Smoothing step

a posteriori smoothing of the calculated series, based on conditioning rules of Gaussian distribution

$$m_k^s = m_k^* + G_k(m_{k+1}^s - m_{k+1}) \\ C_k^s = C_k^* - G_k(C_{k+1}^s - C_{k+1})G_k^t \\ G_k = C_k^* P^t C_{k+1}^{-1}$$
# Characterize the magnetic signal generated in the magnetosphere from geomagnetic observations

Yalei SHI¹, Vincent LESUR¹, Erwan THEBAULT²

¹Université Paris Cité, Institut de physique du globe de Paris, CNRS F-75005 Paris, France
²Université Clermont-Auvergne, Laboratoire Magmas et Volcans, CNRS F-63170 Aubière, France

<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

## Introduction

The main contribution to the geomagnetic field is from the core, but fields generated from the magnetosphere during magnetically quiet periods play an important role. They may have geomagnetic contributions up to hundreds of nanoteslas (nT) during magnetic storm. To study the geomagnetic field, the contributions from different sources need to be separated and modeled. In current models of the geomagnetic field derived from satellite data, the magnetospheric components are poorly described because of the limitation of satellite data spatiotemporal resolution. The poorly described magnetospheric component limits the resolution of other geomagnetic contributing sources, especially the contributions from the core (Thebault et al. [2011]) and the lithosphere (Lesur et al. [2013]). It is therefore important and necessary to describe and model precisely the magnetospheric components. We describe here an approach for their modeling based on Kalman filter approach and magnetic observatory data. These provide a data set with a temporal resolution particularly well suited to characterize rapidly varying magnetospheric signals.

## Objective

* Study the magnetospheric field up to spherical harmonic (SH) degree 6 (L=6) through normal distributions (N(μ = m, σ² = C_m)) with a time resolution of an hour, based on a Kalman filter and correlation-based modeling.

## Test models with synthetic data

Synthetic data at each observatory include only magnetospheric field signal and its response in the mantle:

1. generation of a random hourly sequence during the whole 2021 year for each Gauss coefficient $q_l^m$ up to degree L=6

2. removal of the principal trend by PCA and normalization of the random distribution of $q_l^m$ by the variance trends from the IGRF-13 field power spectrum

3. generation of $i_l^m$ by multiplying the normalized $q_l^m$ with a 1D electrical conductivity model (Verhoeven, Thebault, Saturnino, Houliez, and Langlais [2021])

Night synthetic data
(23:00 – 05:00 LT)

Full-year synthetic data

<figure><img src="image_7.png" /><figcaption>**Figure 2:** The left: results by fitting only night synthetic data. The right: results by fitting full-year synthetic data.</figcaption></figure>

Top: The diagonal elements of the model covariance matrix

Bottom: Power spectrum of SH Gauss coefficient as a function of SH degree

<figure><img src="image_8.png" /><figcaption>**Figure 3:** Top: View of a comparison during the first 20 days between synthetic Gauss coefficient (blue) and the estimated $q_1^0$ (orange and green) by fitting only synthetic data at night (23:00 – 05:00) with error bar 3σ (lavender). Bottom: the number of used data at each hour</figcaption></figure>

## Future works

* The hourly model needs to be extended to cover 1999 to 2024

* Improvement of hourly model: observatory data including day side should be used. The ionospheric contribution distribution of each component at each observatory should be co-estimated, and a correlation of ionospheric contribution among observatories should be added as a priori information

* Statistics from theoretical or semi-empirical models to separate the different contributions in the magnetosphere could be used, in order to develop a better understanding of the evolution of magnetospheric sources during solar cycles

## References

1. Thebault E., Vervelidou F., Lesur V., & Harmoudi M. (2011). The pros and cons of along-track satellite analysis in planetary magnetisme. In 25th IUGG General Assembly.

2. Lesur V., Rother M., Vervelidou F., Harmoudi M., Thebault E. (2013). Post-processing scheme for modelling the lithosphere magnetic field. Solid Earth, 4(1):105-118.

3. Ropp G., Lesur V., Baerenzing J., & Holschneider M. (2020), Sequential modelling of the earth's core magnetic field. Earth, Planets and Space, 72(1):1-15.

4. Verhoeven, O., Thebault, E., Saturnino, D., Houliez, A., & Langlais, B. (2021). Electrical conductivity and temperature of the earth's mantle inferred from bayesian inversion of swarm vector magnetic data. Physics of the Earth and Planetary Interiors, 314, 106702.

## INTERMAGNET observatory data

<figure><img src="image_9.png" /></figure>

1. The main magnetic field contribution is removed from the hourly-mean INTERMAGNET observatory data for full-year 2021 by subtracting the MCM model (Ropp et Lesur [2020])

2. Observatory data are selected between 23:00 - 05:00 LT and during geomagnetically quiet time (Dst between -30 nT and 30 nT) to minimize the contributions of ionospheric perturbations

<caption>**Figure 1:** Distribution of the hundred or so used geomagnetic observatory positions</caption>

## Parameters

The geomagnetic signal after the selection is considered as the sum of the magnetospheric field, induced field, and crustal offsets at each observatory location:

$$B(r, \theta, \phi) = \sum_{\ell,m} q_{\ell}^{m} \left(\frac{r}{a}\right)^{l-1} Y_{\ell}^{m}(\theta, \phi) + \sum_{\ell,m} i_{\ell}^{m} \left(\frac{a}{r}\right)^{l+2} Y_{\ell}^{m}(\theta, \phi) + \sum_{i \in N_{\text{obs}}} \delta(r - r_i, \theta - \theta_i, \phi - \phi_i) O_i$$

<table><thead><tr><th></th><th>magnetosphere</th><th>induced field</th><th>crustal offset</th></tr></thead><tbody><tr><td>Coef.</td><td>q<sub>l</sub><sup>m</sup></td><td>i<sub>l</sub><sup>m</sup></td><td>O<sub>i</sub>=(O<sub>x</sub><sup>i</sup>, O<sub>y</sub><sup>i</sup>, O<sub>z</sub><sup>i</sup>)</td></tr><tr><td>m<sub>0</sub></td><td>0</td><td>0</td><td>O<sub>i</sub></td></tr><tr><td>C<sub>0</sub></td><td>V<sub>l</sub><sup>ext</sup>=R(l)=S(R/a)<sup>2l</sup></td><td>0.3<sup>2</sup>V<sub>l</sub><sup>ext</sup></td><td>~ 0</td></tr><tr><td>S</td><td>0.7.10<sup>4</sup>(nT<sup>2</sup>)</td><td>-</td><td>-</td></tr><tr><td>R</td><td>3.5.10<sup>4</sup>km</td><td>-</td><td>-</td></tr><tr><td>τ, (P = e<sup>-Δt/τ</sup>)</td><td>10 hour</td><td>10 hour</td><td>10<sup>6</sup> year</td></tr><tr><td>A</td><td>Σ<sub>l,m</sub>(r/a)<sup>l-1</sup>Y<sub>l</sub><sup>m</sup></td><td>Σ<sub>l,m</sub>(a/r)<sup>l+2</sup>Y<sub>l</sub><sup>m</sup></td><td>1</td></tr><tr><td>Nb. of par.</td><td>48</td><td>48</td><td>3×10<sup>2</sup></td></tr></tbody></table>

## Results applied to real data

<figure><img src="image_10.png" /><figcaption>**Figure 5:** Example of the comparison between CLF (Chambon-la-foret) observatory data (in blue) and smoothed modeling signal (in orange) for x component over 20 days, the dotted grey line represent the annual average residue between observatory data and modeling signal</figcaption></figure>

The distribution of residuals over year 2021 for each component at each observatory is calculated. Most of observatories have an annual average residual nearby 0 nT and an annual residual standard deviation smaller than 10 nT, except some observatories located at high latitudes shown in figure 6 where the residual standard deviation can reach 90 nT

<caption>**Figure 6:** Distribution of the 23 geomagnetic observatory positions with annual residual standard deviation over 10nT between observatory data and modeling signal</caption>

## Approach -- Kalman Filter

### Prior information

mean prior model $m_0$, prior covariance matrix $C_0$

### Analysis step

adjustment of models at the kth hour by fitting INTERMAGNET observatory data, based on the Least Square method

$$d = Am + e \\ m_k^* = m_k + (A^t C_e^{-1} A + C_k^{-1})^{-1} A^t C_e^{-1} (d - Am_k) \\ C_k^* = (A^t C_e^{-1} A + C_k^{-1})^{-1}$$

### Prediction step

prediction of the next hour's model (at the (k+1)th hour) based on the previous adjusted hourly model (at the kth hour)

$$m_{k+1} = Pm_k^* \\ C_{k+1} = PC_k P^t + C_w$$

### Smoothing step

a posteriori smoothing of the calculated series, based on conditioning rules of Gaussian distribution

$$m_k^s = m_k^* + G_k(m_{k+1}^s - m_{k+1}) \\ C_k^s = C_k^* - G_k(C_{k+1}^s - C_{k+1})G_k^t \\ G_k = C_k^* P^t C_{k+1}^{-1}$$