# A Linear Inversion Approach to Measuring the Composition and Directionality of the Seismic Noise Field

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## Abstract

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## 1. Introduction

## 2. Formalism

#### 2.1. Body-Wave Formalism

#### 2.2. Surface-Wave Formalism

#### 2.3. Inversion Approach and Map Making

`HEALPix`[48] formalism for pixelating the two-sphere—in particular, using the

`healpy`(https://github.com/healpy/healpy accessed on 30 July 2021) python package. As such, the number of pixels is constrained to ${N}_{\mathrm{b}.\mathrm{e}.}=12{n}^{2}$, where $n={2}^{p}$ is an integer power of two.

## 3. Performance Assessment

#### 3.1. Performance Assessment Using Simulations

#### 3.2. Performance Assessment Using Homestake Data

## 4. Microseism Noise Composition at Homestake

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plane waves from a possible source direction, $\widehat{\mathrm{\Omega}}$, arrive at two sensors shown in 2D. The plane wave is described for a given wave type, m, and polarization, A, with polarization vector ${\overrightarrow{e}}_{mA}\left(\widehat{\mathrm{\Omega}}\right)$. This is a simplified 2D example meant to illustrate the notation that we use in Section 2.1; in 3D, two angles are needed to describe the propagation direction, $\widehat{\mathrm{\Omega}}$.

**Figure 2.**Left column shows the maximum likelihood recovery for a simulated P-wave injection, ${\widehat{\mathbf{P}}}_{\mathrm{ML}}$. Right column shows the model resolution matrix of the P-wave overlap reduction function, plotted as a map, $\mathcal{M}{\mathbf{P}}_{\mathrm{true}}$. Each row represents a different choice of ${s}_{\mathrm{min}}$. We see that as the spot size in the model resolution plots increases, the covariance in each pixel decreases. In these maps, 0${}^{\circ}$ in the azimuthal coordinate corresponds to eastward propagation, while 90${}^{\circ}$ corresponds to northward propagation.

**Figure 3.**An estimate of the angular resolution for the different components of the seismic field based on $\mathcal{M}{\mathbf{M}}_{\mathrm{true}}$ (see text) for P, R ${\mathrm{S}}_{\mathrm{h}}$, and ${\mathrm{S}}_{\mathrm{v}}$ waves (

**a**–

**d**). The resolution is estimated by drawing a contour around the region containing 95% of the total map power and finding the extent in the azimuth ($\varphi $) and polar ($\theta $) angle that this contour encompasses. We also show the diffraction limit estimate on each plot.

**Figure 4.**Recovery of two pressure waves is shown, with their true propagation directions indicated by stars. The contours, defined to enclose 95% of the total power, amount to the total recovered power of $2.6\times {10}^{-8}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$, as compared to the simulated $2\times {10}^{-8}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$. This discrepancy is caused by the coherence between the two signals, breaking one of the underlying assumptions of our inversion technique.

**Figure 5.**(

**a**–

**d**) R-, P-, ${\mathrm{S}}_{\mathrm{h}}$-, ${\mathrm{S}}_{\mathrm{v}}$-wave recoveries. R, P, and S${}_{h}$ waves are simultaneously simulated with different propagation directions and are successfully recovered (no ${\mathrm{S}}_{\mathrm{v}}$ wave was simulated). The total recovered power is biased: the simulated power was uniform for all three signals, while the total power in the R-wave map was $0.85\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$; for the P-wave map, it is $1.22\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$, and for the S${}_{h}$-wave map, it is $2.48\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$. Adding the power across all four maps yields a total amplitude of $4.01\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$, which is larger than what one would expect if all injections were uncorrelated. We assumed ${\lambda}_{R}=2500\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${\lambda}_{P}=5700\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, and ${\lambda}_{S}=4000\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 6.**(

**a**) Vertical-to-horizontal phase between the east and the vertical channels of the

`YATES`station is shown for 36 h starting from 2 October 2015. (

**b**) Cross-power between the vertical and east channels of

`YATES`station. The signal at 1.4–1.6 Hz appears at the 1 h mark, and its phase hovers near $\pi /2$ radians, consistent with the retrograde motion in Rayleigh waves. This signal is clearly a peak in the spectrum, as seen on the right.

**Figure 7.**Recovery of 1.5 Hz source from 2 October 03:00–04:00 UTC for Rayleigh waves (

**a**), and P waves (

**b**). We see that there is a strong peak between 0 and 30${}^{\circ}$ for R waves, while for P waves, there is no obviously preferred direction. Note that we plot the R-wave recovery in 2D, even though only one (azimuthal) angle defines the wave propagation direction. This is done as a cross-check—if incorrect R-wave phase velocity is used in the recovery, the recovered spot could appear off the x-axis in this plot.

**Figure 8.**(

**a**) Ratio of body-wave to R-wave power (blue solid) and total map power (orange, dashed). There is an obvious seasonal shift in total map power that corresponds to a decrease in the ratio of body waves to surface waves. In (

**b**), we zoom in on fifty days. There is evident variability on a ∼7-day time-scale, and anti-correlation between total map power and the ratio of body waves to R waves. The variability is likely due to storms, which cause an increase in surface wave amplitude and hence an increase in total power and corresponding decrease in the ratio of body waves to R waves.

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## Share and Cite

**MDPI and ACS Style**

Meyers, P.M.; Prestegard, T.; Mandic, V.; Tsai, V.C.; Bowden, D.C.; Matas, A.; Pavlis, G.; Caton, R.
A Linear Inversion Approach to Measuring the Composition and Directionality of the Seismic Noise Field. *Remote Sens.* **2021**, *13*, 3097.
https://doi.org/10.3390/rs13163097

**AMA Style**

Meyers PM, Prestegard T, Mandic V, Tsai VC, Bowden DC, Matas A, Pavlis G, Caton R.
A Linear Inversion Approach to Measuring the Composition and Directionality of the Seismic Noise Field. *Remote Sensing*. 2021; 13(16):3097.
https://doi.org/10.3390/rs13163097

**Chicago/Turabian Style**

Meyers, Patrick M., Tanner Prestegard, Vuk Mandic, Victor C. Tsai, Daniel C. Bowden, Andrew Matas, Gary Pavlis, and Ross Caton.
2021. "A Linear Inversion Approach to Measuring the Composition and Directionality of the Seismic Noise Field" *Remote Sensing* 13, no. 16: 3097.
https://doi.org/10.3390/rs13163097