Surface Wave Dispersion Measurement with Polarization Analysis Using Multicomponent Seismic Noise Recorded by a 1-D Linear Array

Abstract

Linear arrays are popularly used for passive surface wave imaging due to their high efficiency and convenience, especially in urban applications. The unknown characteristics such as azimuth of noise sources, however, make it challenging to extract accurate phase-velocity dispersion information by employing a 1-D linear array. To solve this problem, we proposed an alternative passive surface wave method to capture the dominant azimuth of noise sources and retrieve the phase-velocity dispersion curve by polarization analysis with multicomponent ambient noise records. We verified the proposed method using synthetic data sets under various source distributions. According to the calculated dominant azimuth, it is deduced that noise sources are mainly classified as either inline or offline distribution. For inline noise source distribution, we are able to directly obtain the unbiased phase-velocity measurements; for offline noise source distribution, we should correct the velocity overestimation due to azimuthal effects using the proposed method. Results from two field examples show that the distributions of noise sources are predominantly offline. We eliminated the velocity bias caused by offline source distribution and picked phase velocities following higher amplitude peaks along the trend. After the azimuthal correction, the picked phase-velocity dispersion curves in dispersion images generated from passive source data match well with those from active source data, demonstrating the practicability of the proposed technique.

Appendix

Following Takagi et al. (2018), we assume that Rayleigh waves are incident as plane waves on each seismic station; the multicomponent wavefields are represented by a superposition of incident plane waves as follows:

$$\begin{gathered} u_{Z} \left( f \right) = \mathop \smallint \limits_{ - \pi }^{\pi } {\text{d}}\varphi A_{{\text{R}}} \left( {\varphi ,f} \right) \hfill \\ u_{{\text{N}}} \left( f \right) = \mathop \smallint \limits_{ - \pi }^{\pi } {\text{d}}\varphi iH\left( f \right)A_{{\text{R}}} \left( {\varphi ,f} \right){\text{sin}}\varphi \hfill \\ u_{{\text{E}}} \left( f \right) = \mathop \smallint \limits_{ - \pi }^{\pi } {\text{d}}\varphi iH\left( f \right)A_{{\text{R}}} \left( {\varphi ,f} \right){\text{cos}}\varphi \hfill \\ \end{gathered}$$

(A1)

where f is frequency; $u_{Z} \left( f \right)$, $u_{{\text{N}}} \left( f \right)$ and $u_{{\text{E}}} \left( f \right)$ are the Fourier spectra of the vertical, north and east wavefields, respectively; $A_{{\text{R}}} \left( {\varphi ,f} \right)$ is the Fourier spectrum of the vertical wavefield of Rayleigh wave propagating in azimuth $\varphi$; and $H\left( f \right)$ is the horizontal-over-vertical ratio of the Rayleigh waves.

Takagi et al. (2018) assume that the incident waves are uncorrelated, which means that they satisfy

$$\left\langle {A_{{\text{R}}}^{*} \left( \varphi \right)A_{{\text{R}}} \left( {\varphi ^{\prime } } \right)} \right\rangle = \frac{{\left\langle {\left| {A_{{\text{R}}} \left( \varphi \right)} \right|^{2} } \right\rangle }}{{2\pi }}\delta \left( {\varphi - \varphi ^{\prime } } \right)$$

(A2)

where ⟨⟩ denotes the ensemble average and $\delta$ is the Dirac delta function. $\left\langle {\left| {A_{{\text{R}}} \left( \varphi \right)} \right|^{2} } \right\rangle$ is the azimuthal power spectrum, representing the power distribution of Rayleigh waves as a function of propagation azimuth. The azimuthal power spectrum using a Fourier series is expressed by

$$\left\langle {\left| {A_{{\text{R}}} \left( \varphi \right)} \right|^{2} } \right\rangle = a_{{{\text{R}}0}} + 2\sum\limits_{{m = 1}}^{\infty } {\left( {a_{{Rm}} \cos m\varphi + b_{{Rm}} \sin m\varphi } \right)}$$

(A3)

where $a_{Rm}$ and $b_{Rm}$ are the Fourier coefficients.

Using Eqs. (A1) and (A2), we can write the ensemble average of the vertical-horizontal cross-spectra $u_{Z}^{*} u_{N}$

$$\begin{gathered} u_{Z}^{*} u_{{\text{N}}} = \left\langle {\mathop \smallint \limits_{ - \pi }^{\pi } \mathop \smallint \limits_{ - \pi }^{\pi } {\text{d}}\varphi {\text{d}}\varphi^{\prime}iHA_{{\text{R}}}^{*} \left( \varphi \right)A_{{\text{R}}} \left( {\varphi^{\prime}} \right)\sin \varphi^{\prime}} \right\rangle \hfill \\ \quad \quad \quad = \mathop \smallint \limits_{ - \pi }^{\pi } \mathop \smallint \limits_{ - \pi }^{\pi } {\text{d}}\varphi {\text{d}}\varphi^{\prime}iH\left\langle {A_{{\text{R}}}^{*} \left( \varphi \right)A_{{\text{R}}} \left( {\varphi^{\prime}} \right)} \right\rangle \sin \varphi^{\prime} \hfill \\ \quad \quad \quad = \frac{iH}{{2\pi }}\mathop \smallint \limits_{ - \pi }^{\pi } {\text{d}}\varphi \left\langle {\left| {A_{{\text{R}}} \left( \varphi \right)} \right|^{2} } \right\rangle \sin \varphi \hfill \\ \end{gathered}$$

(A4)

and $u_{Z}^{*} u_{E}$

$$\begin{gathered} \left\langle {u_{Z}^{*} u_{{\text{E}}} } \right\rangle = \left\langle {\mathop \smallint \limits_{{ - \pi }}^{\pi } \mathop \smallint \limits_{{ - \pi }}^{\pi } {\text{d}}\varphi {\text{d}}\varphi ^{\prime}iHA_{{\text{R}}}^{*} \left( \varphi \right)A_{{\text{R}}} \left( {\varphi ^{\prime}} \right)\cos \varphi ^{\prime}} \right\rangle \hfill \\ ~~~~~~~~~~~~~~~ = \mathop \smallint \limits_{{ - \pi }}^{\pi } \mathop \smallint \limits_{{ - \pi }}^{\pi } {\text{d}}\varphi {\text{d}}\varphi ^{\prime}iH\left\langle {A_{{\text{R}}}^{*} \left( \varphi \right)A_{{\text{R}}} \left( {\varphi ^{\prime}} \right)} \right\rangle \cos \varphi ^{\prime} \hfill \\ ~ = \frac{{iH}}{{2\pi }}\mathop \smallint \limits_{{ - \pi }}^{\pi } {\text{d}}\varphi \left\langle {\left| {A_{{\text{R}}} \left( \varphi \right)} \right|^{2} } \right\rangle \cos \varphi \hfill \\ \end{gathered}$$

(A5)

Substituting Eq. (A3) into Eqs. (A4) and (A5), we obtain using the orthogonality relation of trigonometric functions

$$\begin{gathered} \left\langle {u_{Z}^{*} u_{N} } \right\rangle = iHb_{R1} \hfill \\ \left\langle {u_{Z}^{*} u_{E} } \right\rangle = iHa_{R1} \hfill \\ \end{gathered}$$

(A6)

Equation (A6) indicates that the imaginary parts of the averaged cross-spectra between the vertical and horizontal components are proportional to the first-order Fourier coefficients of the azimuthal power spectrum of the incident Rayleigh waves.