# Global High-Resolution Magnetic Field Inversion Using Spherical Harmonic Representation of Tesseroids as Individual Sources

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Tesseroid Forward Field

#### 2.2. Spherical Harmonic Model

#### 2.3. Projected Gradient Inversion

## 3. Data

#### 3.1. Model Geometry

#### 3.2. Magnetization Direction and Susceptibility

#### 3.3. Lithospheric Magnetic Field Models

#### 3.3.1. LCS-1

#### 3.3.2. Synthetic Observed Field

## 4. Results

#### 4.1. Synthetic Test

#### 4.1.1. A-Priori Model from a True Susceptibility Model

^{−4}) and remains the same for all further iterations.

^{−4}for the linear norm to set up a threshold for future inversions, since this value represents the error introduced by re-gridding and represents the conversion to spherical harmonics.

#### 4.1.2. Initial Guess with Averaged Susceptibility for Oceanic and Continental Crust

#### 4.2. Inversion of LCS-1

## 5. Discussion

#### 5.1. Performance of Inversion

^{−4}(iteration 9999), which is two magnitudes higher than the threshold estimated in the very first test.

#### 5.2. Use of a-Priori Model

#### 5.3. Layering of the Model

#### 5.4. Regularization

#### 5.5. Application to Airborne Data—Proof of Concept

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Susceptibility distribution in synthetic models: (

**a**) susceptibility based on a vertically integrated susceptibility model (Hemant and Maus, 2005). (

**b**) Initial guess ${\mathit{x}}_{A}^{0}$ with averaged susceptibilities from Hemant VIS-based initial guess ${\mathit{x}}_{H}^{0}$. Crustal domain boundaries are taken from the age of the oceanic crust map [2].

**Figure 3.**LCS-1 with and without remanence: (

**a**) LCS-1 with degrees from n = 16 to 89, (

**b**) LCS-1 without modelled remanent field (Figure 3b), within degrees from n = 16 to 89.

**Figure 4.**Spherical harmonic model of the remanent field derived from Reference [19]: (

**a**) all spherical harmonic degrees from n = 1 to 89, (

**b**) truncated model with degrees from n = 16 to 89.

**Figure 5.**Spherical harmonic model of the synthetic field: (

**a**) all spherical harmonic degrees from n = 1 to 89, (

**b**) truncated model with degrees from n = 16 to 89.

**Figure 6.**Synthetic dataset inversion result ${\mathit{x}}_{A}^{N}$ with the initial guess ${\mathit{x}}_{A}^{0}$: (

**a**) ${\mathit{x}}_{A}^{N}$ with N = 1000, (

**b**) ${\mathit{x}}_{A}^{N}$ with N = 10,000.

**Figure 7.**Different grids for the inversion with the initial guess ${\mathit{x}}_{A}^{0}$ with averaged values for a continental and oceanic crust (Figure 2b). (

**a**) Difference between the true model ${x}_{H}^{0}$ (model from Figure 2a, not initial guess ${x}_{A}^{0}$ from Figure 2b) and the inversion result ${x}_{A}^{N}$, where $N\text{}$ = 10,000 is the number of iterations. (

**b**) Difference between the synthetic truncated observed magnetic field component Bz (from Figure 3b) and truncated forward calculated component Bz of the inversion result ${\mathit{x}}_{A}^{N}$.

**Figure 8.**Inversion convergence (values of $\text{}Grad\left({\mathit{x}}^{k}\right)$) and spectrums of the synthetic field and the field of the inversion result. (

**a**) Values of projected gradient at each iteration (up until iteration N = 1000). (

**b**) Power spectrum of the synthetic observed field spherical harmonic model (Figure 3a) with all degrees and power spectrum of the forward calculated spherical harmonic model of the inversion result ${\mathit{x}}_{A}^{N}$ after N = 1000 (green) and N = 10,000 (red) iterations.

**Figure 9.**Inversion result and spectrums: (

**a**) Inversion result ${\mathit{x}}_{LCS}^{N}$ with the initial guess ${\mathit{x}}_{H}^{0}$ after N = 10,000 iterations. (

**b**) Power spectrum of LCS-1 and power spectrum of the forward calculated spherical harmonic model of the inversion result ${x}_{LCS}^{N}$ after N = 10,000 iterations.

**Figure 10.**Magnetic field of the tile inversion result and LCS-1 at the airborne altitude: (

**a**) forward calculated field of the inversion result at a 5-km altitude and (

**b**) LCS-1 calculated at a 5-km altitude.

**Figure 11.**Susceptibility models for Bangui area in Africa: (

**a**) Initial guess ${\mathit{x}}_{H}^{0}$. (

**b**) Inversion result ${\mathit{x}}_{LCS}^{N=10,000}$. (

**c**) Tile inversion result with resolution of 0.5°.

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**MDPI and ACS Style**

Baykiev, E.; Yixiati, D.; Ebbing, J.
Global High-Resolution Magnetic Field Inversion Using Spherical Harmonic Representation of Tesseroids as Individual Sources. *Geosciences* **2020**, *10*, 147.
https://doi.org/10.3390/geosciences10040147

**AMA Style**

Baykiev E, Yixiati D, Ebbing J.
Global High-Resolution Magnetic Field Inversion Using Spherical Harmonic Representation of Tesseroids as Individual Sources. *Geosciences*. 2020; 10(4):147.
https://doi.org/10.3390/geosciences10040147

**Chicago/Turabian Style**

Baykiev, Eldar, Dilixiati Yixiati, and Jörg Ebbing.
2020. "Global High-Resolution Magnetic Field Inversion Using Spherical Harmonic Representation of Tesseroids as Individual Sources" *Geosciences* 10, no. 4: 147.
https://doi.org/10.3390/geosciences10040147