Abstract Non-diffuse characteristics of the ambient seismic noise wavefield recorded at 154 broad-band stations in southern California are analysed by computing the correlation matrix of power spectral values at neighbouring frequencies. The similarities of the derived correlation matrices are compared by choosing a different reference station at a time. The data recorded by the different stations are classified into five groups using a hierarchical clustering algorithm based on the similarity of the correlation matrices. Stations belonging to different groups are clustered spatially in the Los Angeles basin, mountains and regions farther from the coast. The deviations from diffuse wavefield in the correlation matrices of representative stations from the five groups are inverted for dominant cross-frequency components. These are used to derive power spectral ratios of non-diffuse noise and fully diffuse noise. The similarity maps for different reference stations and the inverted cross-frequency components provide information on properties of the noise sources and propagation/scattering characteristics in the different regions. Interferometry, Time-series analysis, Seismic tomography, Site effects, Wave propagation, Wave scattering and diffraction 1 INTRODUCTION The condition of a fully diffuse wavefield is fundamental for extracting accurate empirical Green's functions from cross-correlations of the ambient seismic noise at pairs of stations (e.g. Weaver 1982; van Tiggelen 2003; Weaver & Lobkis 2004). A fully diffuse noise implies that the wavefield at any location is stationary and different wave modes corresponding to different frequencies are uncorrelated (e.g. Weaver 1982; Lobkis & Weaver 2001; Sánchez-Sesma et al. 2008). In practice, the condition of a fully diffuse noise wavefield may not always hold and it is important to quantify deviations from the ideal diffuse field at different locations. Several techniques have been used to assess the degree to which the ambient seismic noise field is diffuse. These include checking that the P/S wave energy ratio stabilizes at a certain limit value (e.g. Weaver 1982; Hennino et al. 2001; Margerin et al. 2009), testing that waves propagate with equal energy in different directions (Sens-Schonfelder et al. 2015), examining the azimuthal distribution of instantaneous particle polarity vector (Mulargia 2012) and analysis of power spectral ratio of vertical/horizontal components of the seismic noise data (Kawase et al. 2015). Liu & Ben-Zion (2016) showed that non-diffuse characteristics of the seismic noise field at a given location may also be analysed using the correlation matrix of power spectral values at different frequencies. The technique was demonstrated with data of several seismic stations in southern California. The results indicated that the ambient noise at these stations is not fully diffuse around the secondary ocean microseismic peak, and that stations within and outside the Los Angeles basin have some different features. The observed non-diffuse characteristics in the correlation matrix were simulated by adding cross-frequency components to a fully diffuse wavefield. Modal correlations are also observed in helioseismology (Foglizzo et al. 1998). In the present paper we generalize considerably the study of Liu & Ben-Zion (2016) by analysing data recorded by 154 southern California stations, comparing their correlation matrices with a similarity metric, classifying the observations into groups based on the similarity values, and inverting typical deviations from fully diffuse field in the different groups to dominant cross-frequency components. In the next section we first review the methodology of Liu & Ben-Zion (2016) for estimating correlations of power spectral values at neighbouring frequencies. We then apply the method to compute correlation matrices for vertical component seismograms recorded in 2014 by 154 broad-band stations in southern California, and compare the similarity of correlation matrices for every pair of stations by choosing a different reference station at a time. Using a hierarchical clustering algorithm based on similarity (dissimilarity), the results are used to classify the stations into five groups. Spatially close stations have similar correlation matrices (ambient seismic noise field characteristics) and generally belong to the same group. In Section 3, correlation matrices of representative stations from the five different groups are analysed in detail by inverting the cross-frequency components assuming Gaussian cross-frequency components with subspace projection or non-linear quasi-Newton methods. In Section 4, power ratios of non-diffuse wavefield over fully diffuse wavefield are estimated based on cross-frequency components. The analysis reveals several types of non-diffuse wavefield related to distance from the ocean and major structural units (Los Angeles basin, mountains, Mojave desert). 2 BASIC ANALYSIS Following Liu & Ben-Zion (2016), the correlation coefficient between random wavefield power spectral values at different frequencies can be derived from continuous waveform data using \begin{eqnarray} {{\rm Corr}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2},{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right]}\nonumber\\ { = \frac{{{\rm Cov}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2},{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right]}}{{\sqrt {{\rm Var}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2}} \right]{\rm Var}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right]} }}} \nonumber\\ { =\! \frac{{{\rm E}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2}{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right] \!-\! {\rm E}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2}} \right]{\rm E}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right]}}{{\sqrt {{\rm Var}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2}} \right]{\rm Var}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right]} }}} \nonumber\\ {= \frac{{{{| {{\rm E}\left[ {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} ){\boldsymbol\psi} _{tr}^*( {{{\bf r}},{f_q}} )} \right]} |}^2}}}{{{\rm E}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_p}} )} |}^2}} \right]{\rm E}\left[ {{{| {{{\boldsymbol\psi} _{tr}}( {{{\bf r}},{f_q}} )} |}^2}} \right]}},}\end{eqnarray} (1) where fp and fq are two different frequencies. The random variable |${{\boldsymbol\psi} _{tr}}( {{{\bf r}},f} )$| is assumed a zero-mean complex Gaussian and represents the noise wavefield at location r and frequency f. The motivation for assuming a Gaussian distribution is that the noise spectrum is a sum of numerous independent noise sources (eq. 1 in Liu et al. 2016), which should lead according to the Central Limit theorem to a Gaussian random variable. This assumption is validated below with data of two southern California stations (Fig. 1). The expectation of the product of four Gaussian random variables is equal to |${\rm E}[ {{{| {{{\boldsymbol\psi} _{tr}}( {{\bf r},{f_p}} )} |}^2}{{| {{{\boldsymbol\psi} _{tr}}( {{\bf r},{f_q}} )} |}^2}} ] \!=\! {| {{\rm E}[ {{{\boldsymbol\psi} _{tr}}( {{\bf r},{f_p}} ){\boldsymbol\psi} _{tr}^*( {{\bf r},{f_q}} )} ]} |^2} \!+\! {\rm E}[ {{{| {{{\boldsymbol \psi} _{tr}}( {{\bf r},{f_p}} )} |}^2}} ]{\rm E}[ {{{| {{{\boldsymbol\psi} _{tr}}( {{\bf r},{f_q}} )} |}^2}} ]$|. If the Gaussian assumption is not satisfied, eq. (1) is only valid for the first line. In that case, the correlation coefficient between power spectral values at neighbouring frequencies does not equal to the absolute square of the correlation between modal frequencies. However, it still reflects non-diffuse properties of the ambient seismic noise field. Figure 1. View largeDownload slide (a) A test for Gaussian distribution of spectral values of 1 d data recorded by station OLI. Adopting a significance level of 0.01, the hypothesis of normal distribution is not rejected for p values greater than 0.01. (b) Same as in (a) for station CHF. See Figs 2 and 3 for station locations. Figure 1. View largeDownload slide (a) A test for Gaussian distribution of spectral values of 1 d data recorded by station OLI. Adopting a significance level of 0.01, the hypothesis of normal distribution is not rejected for p values greater than 0.01. (b) Same as in (a) for station CHF. See Figs 2 and 3 for station locations. Based on eq. (1), the correlation coefficient of power spectral samples at neighbouring frequencies can be computed from Fourier transforms of numerous evenly spaced windows of length T applied to the ambient noise data. For rectangular windows, the frequency resolution of correlation coefficients in neighbouring frequencies is determined by the first zero-crossing (1/T) of sinc function (Liu & Ben-Zion 2016). Assuming the power spectral sample at each frequency follows a probability distribution, outliers at each frequency can be identified using a given threshold chosen here to be 3 Median Absolute Deviation. Any windows with more than 5 per cent outliers among discrete power spectral samples within the frequency band of interest are excluded from the estimation of correlation coefficients. We apply this technique to the vertical component (BHZ channel) data of 154 broad-band stations recorded between day 50 and 300 of year 2014 in southern California. The seismic data are obtained from the Southern California Earthquake Data Center (SCEDC 2013). The instrument response is removed and the observational unit is converted to velocity for all data. The analysis uses window lengths T = 100 s separated by gaps of 10 s, and the examined frequency band is 0.05–0.6 Hz. We first perform a test to verify that the spectral values at each frequency across different windows are normally distributed. The Kurtosis-Skewness normality test (D’Agostino & Pearson 1973) is applied to 1 d data (100th day in 2014) of stations OLI (Fig. 1a) and CHF (Fig. 1b), after removing outlier windows (e.g. earthquakes, instrumental glitches and other impulsive transient sources that have strong impact on the results). Assuming a significance level of 0.01, the null hypothesis that the spectrum follows a normal distribution is rejected for p values below 0.01. The p value is the probability of observing data at least as extreme as the tested sample data due to random sampling errors assuming a null hypothesis is true. The 0.01 significance level corresponds to ≥11 per cent chance of incorrectly rejecting a true null hypothesis of normally distributed data (Colquhoun 2014). For station OLI, ∼86 per cent of real/imaginary parts of spectrum values pass the normality test. For station CHF, ∼95 per cent of real/imaginary parts of spectrum values pass the normality test. Considering the 11 per cent false rejection rate, the hypothesis that the 1 d spectral data at OLI and CHF are normally distributed cannot be rejected. We next calculate the correlation coefficient matrix of power spectral values for each station based on eq. (1) by applying the windows to the continuous noise recording after outlier window exclusion. This is illustrated in Fig. 2(a) for station CHF. All pairs of calculated correlation matrices are compared using a similarity value defined as the scalar correlation coefficient between the elements of two matrices. Since we focus on non-diffuse characteristics of the data associated with correlations between different frequencies, we remove the main diagonal (all 1s) of each correlation matrix and its nearest neighbours that are affected by the main diagonal's sinc functions within the resolution limit 1/T. Figure 2. View largeDownload slide (a) Correlation matrix of vertical component ambient seismic noise power spectral values at station CHF with a frequency resolution 0.01 s. (b) Dissimilarity map of 154 broad-band stations in southern California for reference station OLI (circled triangle). Topography is shown in grey scale with darker colour corresponding to higher elevation and thin black lines mark fault traces. Colours of stations show their dissimilarity with respect to the reference station. Stations in the Los Angeles basin are very similar. (c) Dissimilarity map for reference station CHF (circled triangle). (d) Dissimilarity map for reference station IRM (circled triangle). Figure 2. View largeDownload slide (a) Correlation matrix of vertical component ambient seismic noise power spectral values at station CHF with a frequency resolution 0.01 s. (b) Dissimilarity map of 154 broad-band stations in southern California for reference station OLI (circled triangle). Topography is shown in grey scale with darker colour corresponding to higher elevation and thin black lines mark fault traces. Colours of stations show their dissimilarity with respect to the reference station. Stations in the Los Angeles basin are very similar. (c) Dissimilarity map for reference station CHF (circled triangle). (d) Dissimilarity map for reference station IRM (circled triangle). A dissimilarity value between any two correlation matrices is defined as one minus their similarity value. The dissimilarity metric is expected to increase with the distance between two stations. Fig. 2(b) shows dissimilarity values of all stations with respect to station OLI (circled triangle in the Los Angeles basin). Most stations in the Los Angeles basin are similar to OLI with dissimilarity less than 0.05. The dissimilarity value for stations outside the basin generally increases as the distance from the coast increases, suggesting more differences between the coastal stations and in-land stations. Fig. 2(c) displays dissimilarity values of all stations with respect to station CHF (circled triangle in the San Gabriel Mountain north of Los Angeles basin). The stations most similar (dark red) to CHF follow an elongated pattern parallel to the coast. The stations in the Los Angeles basin and those in eastern California are less similar to CHF. Fig. 2(d) provides a third example with reference station IRM (circled triangle in the Mojave desert). The pattern of dissimilarity values depending on distance and geologic structure is approximately the opposite of the case for OLI (Fig. 2b). Stations farther from IRM are less similar to IRM and stations in the Los Angeles basin are more different than those at the same distances but away from the basin. Choosing iteratively each station to be the reference, we construct 154 dissimilarity maps similar to those in Fig. 2(b). The dissimilarity maps contain information about distances from major noise sources, local scattering strength of different media and geological structures that can produce interference and amplification of noise (e.g. basin). Based on those dissimilarity maps, the 154 seismic stations are classified into different groups using the hierarchical clustering algorithm with complete linkage (e.g. Defay 1977), and choosing a cutoff dissimilarity value of 0.15 to group stations into different clusters (Fig. 3a). The resulting groups of stations are shown in Fig. 3(b) with different colours (corresponding to the colours used in Fig. 3a). The similarity value for any pair of stations within a cluster is greater than 0.85, while the dissimilarity values between different clusters are greater than 0.15. Figure 3. View largeDownload slide (a) Hierarchical clustering of 154 broad-band stations based on similarity (dissimilarity) between the correlation matrices of all pairs of stations. (b) 154 broad-band stations in southern California with colours corresponding to groups classified by the hierarchical clustering in panel (a). (c) Representative stations selected to illustrate results in subsequent plots. See the text for additional information. Figure 3. View largeDownload slide (a) Hierarchical clustering of 154 broad-band stations based on similarity (dissimilarity) between the correlation matrices of all pairs of stations. (b) 154 broad-band stations in southern California with colours corresponding to groups classified by the hierarchical clustering in panel (a). (c) Representative stations selected to illustrate results in subsequent plots. See the text for additional information. Although the stations are not grouped based on their distances, the map (Fig. 3b) shows that the variations of the non-diffuse ambient seismic noise field in southern California are spatially correlated. More specifically, the light-blue stations are mostly concentrated in the Los Angeles basin (e.g. OLI, Fig. 3c), the green stations are distributed near the coast or off-shore outside the Los Angeles basin (e.g. CHF) and the yellow (e.g. IPT), red (e.g. CLC) and purple (e.g. IRM) stations are farther from the coast. The black station VTV does not belong to any group because its dissimilarity with any other station is greater than the cutoff value 0.15. The two brown stations have dissimilarity values greater than 0.7 compared with the rest of stations, suggesting they may have some instrumental problems in the analysed frequency band. To illustrate the differences between the data recorded by the different groups we analyse the nondiffuse noise characteristics of five representative stations from the different groups plus the black station in Fig. 3(c). The correlation matrices of power spectral values at these stations are shown in Figs 4 and 5. Station OLI (Fig. 4a) in the Los Angeles basin (represents light blue stations in Fig. 3b) has higher power spectra than the other four representative stations. This is probably related to its proximity to the ocean and basin amplification of the noise energy. Station OLI also shows stronger correlated neighbouring frequencies between 0.25 and 0.6 Hz than the other four stations. Station CHF (Fig. 4b) at the San Gabriel Mountains (the green group stations in Fig. 3b are generally near the coast but outside the basin) has lower power spectra and significantly less correlated neighbouring frequencies between 0.25 and 0.6 Hz than OLI. Station IPT (Fig. 4c) is slightly farther from the coast than OLI and CHF and it has weaker but broader correlated zone between 0.25 and 0.6 Hz. Stations CLC (Fig. 5a) and IRM (Fig. 5b) are farther from the coast than IPT and they have lower power spectra curves as well as weaker and broader correlated zones between 0.2 and 0.6 Hz than the previous 3 stations. The stand along black station VTV (Fig. 5c) is located close to IPT but its averaged power spectra curve deviates from the maximum density values between 0.32 and 0.6 Hz due to strong outliers within this frequency band (shaded area above the averaged cross-spectra). As a result, the correlation matrix for VTV shows anomalously strong (∼0.4) correlated neighbouring frequencies (a square) between 0.32 and 0.60 Hz. Figure 4. View largeDownload slide Power spectra over 0.05–0.6 Hz and neighbouring frequency correlation matrices. (a) Results for station OLI. Top panel: stacked power spectra (red curve) and number of power spectrum measurements with a given value in each power bin (grey scale) of each discrete frequency value. The primary microseismic peak is around 0.06 Hz and secondary microseismic peak is near 0.15 Hz. Bottom panel: correlation matrix of the power spectral values. The colour scale is saturated at a correlation value of 0.4 to show the off-diagonal variations of the matrix. (b) Corresponding results for station CHF. (c) Corresponding results for station IPT. Figure 4. View largeDownload slide Power spectra over 0.05–0.6 Hz and neighbouring frequency correlation matrices. (a) Results for station OLI. Top panel: stacked power spectra (red curve) and number of power spectrum measurements with a given value in each power bin (grey scale) of each discrete frequency value. The primary microseismic peak is around 0.06 Hz and secondary microseismic peak is near 0.15 Hz. Bottom panel: correlation matrix of the power spectral values. The colour scale is saturated at a correlation value of 0.4 to show the off-diagonal variations of the matrix. (b) Corresponding results for station CHF. (c) Corresponding results for station IPT. Figure 5. View largeDownload slide Similar to Fig. 4 for stations CLC (panel a), IRM (panel b) and VTV (panel c), with stacked power spectra of distribution of power spectrum measurements at each frequency (top) and correlation matrices of power spectral values (bottom). The power spectra at station VTV deviate significantly from the stacked values between 0.32 and 0.60 Hz due to remaining strong outliers; this produces anomalously correlated neighbouring frequencies within this frequency band. Figure 5. View largeDownload slide Similar to Fig. 4 for stations CLC (panel a), IRM (panel b) and VTV (panel c), with stacked power spectra of distribution of power spectrum measurements at each frequency (top) and correlation matrices of power spectral values (bottom). The power spectra at station VTV deviate significantly from the stacked values between 0.32 and 0.60 Hz due to remaining strong outliers; this produces anomalously correlated neighbouring frequencies within this frequency band. 3 INVERSION OF CROSS-FREQUENCY COMPONENTS To further analyse the characteristics of correlation matrices for the five representative stations from the different groups, we set up an inverse problem to estimate the cross-frequency components that can produce the main observed deviations in the correlation matrices. This is based on solving iteratively the forward problem. In the forward problem, we simulate the correlation matrix from known cross-frequency components representing power spectral ratios of non-diffuse over fully diffuse noise fields. Based on Liu & Ben-Zion (2016), the normalized random noise spectra vector dN×1 of correlated frequency samples can be simulated from fully uncorrelated random vector m(K+N)×1 of zero-mean standard complex Gaussian random variables using \begin{eqnarray} {d_{N \times 1}} &=& {C_{N \times N}}{G_{N \times (N+K)}}{m_{(N+K) \times 1}} \nonumber\\ &=& {C_{N \times N}}[{{S_{N \times K}}}\,\,{{I_{N \times N}}}]{m_{(K+N) \times 1}}, \end{eqnarray} (2) where the matrix SN×K = [s1s2 …sK] contains K cross-frequency column vectors, IN×N is an identity matrix, N is the number of discrete frequencies, and the diagonal matrix CN×N rescales the variance of each resulting random spectrum sample d(fi) to unity. The ith independent cross-frequency component vector si represents the square root of its power spectral ratios relative to the fully diffuse noise at each frequency. The fully diffuse noise part is represented by the uncorrelated complex Gaussian random vector (multiplied by the identity matrix). The matrix GN×(K+N) transforms an uncorrelated random vector to a correlated random vector (normalized real noise spectra) by adding multiple cross-frequency random components (non-diffuse noise parts) to the fully diffuse noise with predictable statistics and correlation of the resulting wavefield. The correlation matrix can be computed from ensemble average of the product of normalized random spectra vectors \begin{eqnarray} {R_{N \times N}} &=& {\rm{E}}\left[ {{d_{N \times 1}}d_{N \times 1}^{T*}} \right] = {C_{N \times N}}{G_{N \times (N+K)}}G_{N \times (N+K)}^T{C_{N \times N}}\nonumber\\ & &= {C_{N \times N}}\left[ {{S_{N \times K}}S_{N \times K}^T + {I_{N \times N}}} \right]{C_{N \times N}}, \end{eqnarray} (3) where the scaling matrix |${C_{N \times N}} = {[ {diag( {{G_{N \times ( {N + K} )}}G_{N \times ( {N + K} )}^T} )} ]^{ - 1/2}}$| normalizes the diagonals of the correlation matrix to one. Based on eq. (1), each element of the correlation matrix of power spectral values estimated from noise data is equal to the absolute square of the corresponding element in RN×N. The covariance matrix before normalization can be decomposed using Singular Value Decomposition (SVD) \begin{equation}{G_{N \times (N+K)}}G_{N \times (N+K)}^T = {S_{N \times K}}S_{N \times K}^T + {I_{N \times N}} = U\Lambda {U^T},\end{equation} (4) where the columns of U are eigenvectors and Λ is a diagonal matrix of eigenvalues sorted in descending order. Each cross-frequency component vector si in matrix SN×K is a linear combination of eigenvectors in U. This suggests that to reduce redundant information, the number of cross-frequency components should be less than (or much less in reality) the number of discrete frequencies in the correlation matrix. Each cross-frequency component vector is taken to be a Gaussian function, assuming that the wave components at two neighbouring frequencies with larger frequency interval show greater differences. The jth row of the ith cross-frequency column vector si is |${s_{ji}} = {A_i}{\rm exp}[ { - {{{{( {{f_j} - {f_{Ci}}} )}^2}}/{2{\sigma}_i^2}}}]$|, where fCi is the Gaussian function centre frequency, Ai is the maximum amplitude of the ith cross-frequency vector and σi defines the width of the Gaussian function. Here fj is the frequency corresponding to the jth row in the noise spectra vector dN×1. To reduce the number of parameters, we assume that the centres [fC1 … fCK] of the Gaussian functions in K cross-frequency vectors are evenly spaced in the frequency band of interest. The parameters to be estimated are Gaussian amplitude vector A = [A1 … AK] and width vector σ = [σ1 … σK]. A non-linear function that simulates the correlation matrix based on these parameters is defined as |$R_{N \times N}^{{\rm{sim}}} = F( {{\boldsymbol A},{\boldsymbol{\sigma }}} )$|. This is computed from eq. (3) using the Gaussian cross-frequency components. The Gaussian amplitude and width parameters A* and σ* (* denotes estimated values) can be estimated by minimizing the residuals between the simulated and observed correlations matrices plus smoothing terms \begin{eqnarray} {{\boldsymbol A}^*},{{\boldsymbol \sigma} ^*} &=& {\rm argmin}\left\lbrace {{\left\| {F\left( {{\boldsymbol A},{\boldsymbol\sigma} } \right) - {R_{N \times N}}} \right\|}_2} + {\boldsymbol\lambda} \sqrt {\sum\limits_{i = 1}^{K - 1} {{{\left( {{A_{i + 1}} - {A_i}} \right)}^2}} } \right.\nonumber\\ &&\qquad\qquad+\,\left. {\boldsymbol \nu} \sqrt {\sum\limits_{i = 1}^{K - 1} {{{\left( {{\sigma _{i + 1}} - {\sigma _i}} \right)}^2}}} \right\rbrace , \end{eqnarray} (5) where ||.||2 denotes L2-norm of the residual matrix, and λ and ν are weighting factors for smoothness of Gaussian parameters among neighbouring cross-frequency vectors. The observed matrix RN×N is estimated from data by taking the square root of each element in the correlation matrix of power spectral values and skipping every other correlation values with spacing less than the frequency resolution of corresponding window length. Eq. (5) may be solved using a quasi-Newton method that computes and decreases the gradient of the objective function (right-hand side of eq. 5) at each iteration. In the following subsections we present two methods for solving eq. (5). 3.1 Subspace projection method The Gaussian width parameters can be estimated based on the eigenvectors of U in eq. (4). In practice, only the normalized correlation matrix RN×N is known. Therefore, we approximate the eigenvectors of U with those of Q, where Q contains the eigenvectors of the correlation matrix RN×N and can be directly computed based on SVD, RN × N = QΓQT. Because each cross-frequency vector belongs to the subspace spanned by the columns of U, and Q is approximately equal to U, the Gaussian width parameter σi for the ith cross-frequency vector si can be estimated by maximizing the projection of si onto the subspace of Q, which is shown in detail below. The SVD of the correlation matrix RN×N contains a few large eigenvalues and the remaining eigenvalues are negligible. The eigenvectors corresponding to the large eigenvalues are grouped in matrix Qs that spans the signal subspace. The remaining eigenvectors are grouped in matrix Qn that spans the remainder subspace. To estimate the Gaussian width parameter σi, the corresponding cross-frequency vector si is projected onto the signal subspace |${{Q_s^T{s_i}} /{\|{{s_i}} \|}}$|, where the vector si is normalized by its length ||si || to unity. By maximizing the projected length |$\| {{{Q_s^T{s_i}} /{\|{{s_i}}\|}}}\|$| of the normalized cross-frequency vector si/||si||, the Gaussian width parameter σi can be estimated using a simple Newton's method. As a practical example, the observed correlation matrix RN×N for station CHF with frequencies between 0.1 and 0.6 Hz is derived from its correlation matrix of power spectral values following the steps below eq. (5). Its eigenvectors are then computed based on SVD of RN×N (Fig. 6a). The first 10 largest eigenvalues are selected for the signal subspace with corresponding eigenvectors, while the remaining eigenvectors belong to the noise subspace and are not used. The rank of signal subspace (10) is chosen because it minimized the overall misfit between synthetic and data correlation matrices. Figure 6. View largeDownload slide Eigenvectors of correlation matrix and inversion of cross-frequency components. (a) Ten eigenvectors of the largest eigenvalues (in descending order) for station CHF. The first eigenvector accounts for nearly constant correlations in a broad frequency range, used for the extra background cross-frequency component. Eigenvectors 2–10 span the signal subspace. The Gaussian width parameter for each cross-frequency vector is estimated by projection onto the signal subspace. (b–e) Best-fitting cross-frequency component vectors for stations CHF, IPT, CLC and IRM, respectively, based on the subspace projection method. The extra background cross-frequency component is shown as the 21st vector. (f) Cross-frequency component vectors for station OLI from simultaneous inversion of Gaussian component amplitude and width parameters. See the text for additional information. Figure 6. View largeDownload slide Eigenvectors of correlation matrix and inversion of cross-frequency components. (a) Ten eigenvectors of the largest eigenvalues (in descending order) for station CHF. The first eigenvector accounts for nearly constant correlations in a broad frequency range, used for the extra background cross-frequency component. Eigenvectors 2–10 span the signal subspace. The Gaussian width parameter for each cross-frequency vector is estimated by projection onto the signal subspace. (b–e) Best-fitting cross-frequency component vectors for stations CHF, IPT, CLC and IRM, respectively, based on the subspace projection method. The extra background cross-frequency component is shown as the 21st vector. (f) Cross-frequency component vectors for station OLI from simultaneous inversion of Gaussian component amplitude and width parameters. See the text for additional information. The first eigenvector in the signal subspace corresponding to the largest eigenvalue contains a nearly constant component (∼0.15) over 0.2–0.6 Hz and a slightly smaller component (∼0.12) between 0.1 and 0.2 Hz. This is different from our assumption of a Gaussian cross-frequency component centred at each frequency with only limited bandwidth. We therefore remove this eigenvector from the signal subspace used for subspace projection in the Gaussian width estimation and consider it to be an extra background cross-frequency vector that accounts for undetected small earthquakes (outliers) and instrument/sensor noise. Because this eigenvector only has unit length, it has to multiply by a constant to become a cross-frequency component. There are 20 Gaussian cross-frequency components with centres evenly spaced between 0.1 and 0.6 Hz. The Gaussian width parameters σi are estimated with the discussed subspace projection method and the results do not depend on the initial values (between 0.01 and 0.5 Hz). The Gaussian amplitude vector A is estimated from eq. (5) by setting the initial value (which is not a sensitive parameter) to 0.3 for all components and fixing the Gaussian width vector. The weighting factor for smoothness of the Gaussian amplitude is set to λ = 0.3. The constant multiplier of the extra background cross-frequency component is simultaneously estimated with the Gaussian amplitude vector A. The estimated cross-frequency components for CHF are shown in Fig. 6(b) with the extra background cross-frequency component multiplied by its amplitude appended on the right. The quality of the result can be assessed by calculating the L2-norm of misfit, which for the observed correlation matrices varies in the range 0–50. The L2-norm misfit between the simulated and observed correlation matrices for CHF is 0.43, indicating a very good fit. There are two groups of cross-frequency components with notable amplitudes. The first group contains vectors 1–6 between 0.1 and 0.22 Hz with peak values from 0.6 to 0.9. The second group contains vectors 11–15 between 0.35 and 0.45 Hz with peak values from 0.25 to 0.35. The cross-frequency components for IPT (Fig. 6c), CLC (Fig. 6d) and IRM (Fig. 6e) are computed using the same approach and criteria as done for CHF. The amplitude of cross-frequency components generally decreases following the order CHF, IPT, CLC and IRM, which is inversely related to the distance from the coast. The group of cross-frequency vectors 11–15 between 0.35 and 0.45 Hz shows weaker peak amplitudes (0.22) for IPT than CHF, and does not exist for CLC or IRM. The amplitude of the extra background cross-frequency component also increases with the distance from the coast. This suggests that a background noise component of broadly correlated neighbouring frequencies increases its percentage as the power of ambient seismic noise recorded on the station decreases. An important advantage of this two-steps optimization method based on subspace projection and quasi-Newton inversion is insensitivity to the initial values. This technique estimates one parameter (Gaussian width or amplitude) in each step and converges fast. One drawback is that it approximates the eigenvectors of the covariance matrix with the eigenvectors of the normalized correlation matrix; if these are very different the estimations of Gaussian width based on subspace projection will not be accurate. 3.2 Simultaneous inversion of Gaussian amplitude and width The subspace projection method could fail if the eigenvectors of correlation matrix are significantly different from the eigenvectors of the covariance matrix in eq. (4). The data at station OLI provide an example where the cross-frequency components cannot be accurately estimated with that method. In such cases we can invert for both the amplitude and width of the Gaussian cross-frequency components simultaneously by applying a quasi-Newton method to eq. (5). Because this involves as twice as many parameters in each iteration than the subspace projection method, and trade-offs between Gaussian amplitude and width parameters, the simultaneous inversion of both amplitude and width parameters is much slower than the subspace projection method and more sensitive to initial conditions. For analysing the data at OLI, the initial Gaussian amplitude values are set to 0.45 and the Gaussian width parameters are set to 0.015. The smoothing factors are 0.6 and 1.0 for λ and ν, respectively. The 20 cross-frequency components at OLI are shown in Fig. 6(f) with a group of cross-frequency vectors between 11 and 17 characterized by significant Gaussian amplitudes (peak ∼0.5) and broad width between 0.25 and 0.6 Hz. As mentioned, this is likely related to basin effects that modify the ocean-generated noise. The L2-norm misfit is 0.60. Station OLI is representative for the light blue station cluster (Fig. 3b) in the Los Angeles basin region, and the significant broad-width cross-frequency components between 0.25 and 0.6 Hz exist only at these basin stations. 4 POWER RATIOS OF NON-DIFFUSE NOISE OVER FULLY DIFFUSE NOISE Because each cross-frequency component vector si is the square root of its power ratio over fully diffuse noise at corresponding frequency, the ratio of total power of non-diffuse cross-frequency components over fully diffuse noise power at frequency fj is \begin{equation}\frac{{{P_{{\rm{nondiff}}}}\,( {{f_j}})}}{{{P_{{\rm{diffuse}}}}\,( {{f_j}})}} = \sum\nolimits_{k = 1}^K {s_{jk}^2} ,\end{equation} (6) where sjk is the jth frequency of the kth cross-frequency component vector. Based on eq. (6) and the cross-frequency components in Figs 6(b)–(f), the ratios of total non-diffuse wave components power over fully diffuse noise power are shown for the five representative stations in Fig. 7. For station CHF, the non-diffuse power is ∼1.5 times the diffuse power for frequencies around 0.14–0.18 Hz and a smaller peak of power ratio 1.1 appears at ∼0.4 Hz. The power ratio for IPT is similar to that of CHF except the overall curve is shifted downwards and the smaller peak is missing, indicating relatively less non-diffuse power. This is probably due to the larger distance from the ocean. The power ratios for stations CLC and IRM share similar peak and trough frequencies, and the peak power ratios and corresponding frequencies are lower than CHF and IPT. These results suggest that more scattering and attenuation may reduce the percentage of non-diffuse noise and shift the peak frequency to lower value. The basin station OLI shows higher overall power ratio than the other four stations. The peak power ratio is ∼1.7 between 0.14 and 0.18 Hz, and the average power ratio between 0.32 and 0.55 Hz is ∼1.1. This value is similar as the power ratio of the smaller peak at 0.4 Hz for the mountain station CHF. The comparison of results at stations OLI and CHF suggests that the basin produces reverberations of oceanic microseismic noise that is less diffuse than the noise field outside the basin. Figure 7. View largeDownload slide Power ratios of non-diffuse over fully diffuse noise components between 0.1 and 0.6 Hz derived from the cross-frequency components in Figs 6(b)–(f). Results are shown for stations CHF, IPT, CLC, IRM and OLI. Figure 7. View largeDownload slide Power ratios of non-diffuse over fully diffuse noise components between 0.1 and 0.6 Hz derived from the cross-frequency components in Figs 6(b)–(f). Results are shown for stations CHF, IPT, CLC, IRM and OLI. 5 DISCUSSION We analyse non-diffuse characteristics of the ambient seismic noise in southern California by extending the methodology of Liu & Ben-Zion (2016). The extended technique involves calculating correlation matrix of power spectral values for each station, comparing the correlation matrices by computing similarity values, classifying the results into different groups and inverting cross-frequency wave components. Dissimilarity maps made by choosing any reference station and comparing dissimilarity values to all other stations contain important information on how the seismic noise field varies geographically. The similarity clustering analysis identifies five main groups of stations located in regions with some distinct properties such as the Los Angeles basin, mountains and distance from the coast. The non-diffuse characteristics in the correlation matrices of representative stations from the five groups are used to estimate dominant cross-frequency components and the power ratios of non-diffuse over fully diffuse noise components. These features quantify properties of the noise sources at different frequencies, scattering, and subsurface structure in different regions. Following Liu & Ben-Zion (2016), the correlation matrices are computed with evenly spaced rectangular windows of the same length 100 s, and same outlier exclusion criteria are applied for all stations. The outliers have strong effects on properties of the recorded noise, including the assumed Gaussian distribution of modal coefficients. They may result from small earthquakes, recording problems and cultural noise, and can produce anomalies of power spectral values that can dominate and deteriorate the correlation matrix. One example of this is illustrated by the results for station VTV (black triangle in Fig. 3c) which are affected by outliers (Fig. 5c) that survive the exclusion criteria applied to all stations. Using more sophisticated exclusion criteria (e.g. Liu et al. 2016) can improve such results. The employed statistical outlier exclusion technique replaces the common pre-processing methods in seismic noise interferometry involving one bit normalization and spectral whitening (e.g. Bensen et al. 2007). Those two pre-processing methods only normalize the amplitudes of earthquakes and other transient signals to the same level as background noise without removing them, while transforming the waveform nonlinearly and losing important statistical and amplitude information (Liu et al. 2016). The one bit normalization, which only keeps the signs of time series signals, does not remove non-diffuse noise. If the non-diffuse noise component at time T has minus sign and the diffuse noise has positive sign and weaker amplitude, the one bit method only keeps the sign of the non-diffuse noise. On the other hand, the spectral whitening normalizes the spectral amplitudes and only keeps the phase information of spectra that are the sum of both non-diffuse and diffuse wave fields. The non-diffuse noise component can bias the resulting phase information. The 2D correlation coefficient between the correlation matrices (excluding the diagonal elements) of any two stations provides a similarity measure of the non-diffuse noise field for all the stations. The dissimilarity map is analogous to a distance/travel time map, and it shows correlations with distance and geological setting, although the dissimilarity is computed in high dimensional space (considering all elements of a correlation matrix) rather than geographical distance. A hierarchical clustering algorithm with complete linkage is used to group stations having similar correlation matrices chosen based on the dendrogram for clustering. In the application here we set a dissimilarity threshold of 0.15, equivalent to minimum similarity of 0.85 within a group. The grouping of different stations follows generally the distance from the coast, and also corresponds to locations within the Los Angeles basin, mountains and the Mojave desert. A longer distance from the ocean, which is the main source of the noise field at the analysed frequencies, can damp and scatter the seismic noise and change the non-diffuse characteristics of the recorded wavefield. In addition, interference phenomena in some geological units such as basins, mountain ranges and fault damage zones can modify locally the non-diffuse part of the wavefield by enhancing correlations in some frequency ranges. The dependencies on distance from the coast and local site effects can be separated by examining groups of stations with similar distance from the coast and different site characteristics. For example, basin stations (light blue triangles in Fig. 3b) have significant correlated frequencies between 0.25 and 0.6 Hz that may reflect amplification of noise components at these frequencies. This is in contrast with the weak off-diagonal correlations in the same frequency band for other stations with similar distance from the coast (green triangles in Fig. 3b). For two groups of stations that are close to each other and have about the same distance from the primary noise source, the source and propagation effects are similar. Analysing data from sets of such groups of stations may allow focusing on different local scattering effects between the neighbouring groups. Separating further the effects of propagation distance and local site effects may be facilitated by detailed frequency-dependent analysis, since high frequency waves are more sensitive to the local structure. Such analyses are left for a future study. The correlation matrices of representative stations from the five groups are analysed by inverting their dominant cross-frequency components assuming evenly spaced Gaussian function centres in the relevant frequency band with different width and amplitude parameters. The subspace projection method is applied to estimate the width parameters of the Gaussian functions by maximizing the projected length of each Gaussian cross-frequency component vector. The Gaussian amplitude parameters are then estimated by fixing the width parameters. This two-steps inversion is not sensitive to the initial values of parameters. The subspace projection method assumes that the eigenvectors of the correlation matrix span the subspace for the cross-frequency vectors. This is a valid approximation for most stations analysed except the basin stations. The Gaussian width and amplitude parameters of the cross-frequency components observed at the basin station OLI are estimated jointly using a non-linear quasi-Newton method. This method is more sensitive to the initial values and trade-offs between parameters than the two-steps procedure. Based on the inverted cross-frequency components results, the power ratios of non-diffuse noise over fully diffuse noise between 0.1 and 0.6 Hz are computed for five representative stations. The higher power ratio for station OLI suggests more non-diffuse noise energy than the other four representative stations, likely due to basin amplification of ocean-based noise components of certain frequencies. The power ratio also decreases as the distance from the ocean increases, which can be explained by more scattering and attenuation for noise of ocean origin. Surface wave components at different frequencies sample different depths, so they propagate and scatter differently due to different structures at those depths. As a result, surface wave components with greater frequency separation are less similar due to their sensitivity to different depth/scale structure. For this reason we adopt the Gaussian-shaped cross-frequency component model. However, the results show a broadly-correlated (0.1–0.6 Hz) background cross-frequency component at most stations outside of the Los Angeles basin. This background cross-frequency component may result from weak transient signals produced by small earthquakes, sinc function side lobes due to the rectangle window function, atmospheric based sources (e.g. rain, temperature, barometric pressure) and instrument glitches not identified as outliers. Developing better methods for estimating correlation matrices, separating non-diffuse and fully diffuse noise components from correlation matrices, and applying the technique to additional data sets will be subjects of future work. 6 DATA AND SOFTWARE RESOURCES The continuous waveform data are obtained from the Southern California Earthquake Data Center (SCEDC, http://scedc.caltech.edu). The correlation matrices of neighbouring frequencies for 154 broad-band stations analysed in this work are computed on a cluster at USC in under 3 d. The hierarchical clustering of correlation matrices is done on a laptop computer using Matlab. The inversion of cross-frequency components is based on the Matlab function fminunc and requires 2 min for each correlation matrix. Some well documented codes on outlier exclusion and correlation matrix estimation can be obtained from the Github site (www.github.com/liuxinSCer). Other less well documented codes may be obtained from the first author upon request. Acknowledgements The study was supported by the Department of Energy (award DE-SC0016520) and the National Science Foundation (grant EAR-1551411). The paper benefitted from useful comments by N. Nakata, K. Nishida and an anonymous reviewer. REFERENCES Bensen G.D. , Ritzwoller M.H. , Barmin M.P. , Levshin A.L. , Lin F. , Moschetti M.P. , Shapiro N.M. , Yang Y. , 2007 . Processing seismic ambient noise data to obtain reliable broad-band surface wave dispersion measurements , Geophys. J. Int. , 169 ( 3 ), 1239 – 1260 . Google Scholar Crossref Search ADS Colquhoun D ., 2014 . An investigation of the false discovery rate and the misinterpretation of p-values , R. Soc. Open Sci. , 1 ( 3 ), 140 216–140 216 . Google Scholar Crossref Search ADS D’Agostino R. , Pearson E.S. , 1973 . Tests for departure from normality. Empirical results for the distributions of b2 and √b1 , Biometrika , 60 ( 3 ), 613 – 622 . 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Diffuse fields in dynamic elasticity , Wave Motion , 45 ( 5 ), 641 – 654 . Google Scholar Crossref Search ADS SCEDC , 2013 . Southern California Earthquake Data Center, Caltech Dataset, doi:10.7909/C3WD3xH. Sens-Schönfelder C. , Snieder R. , Stähler S.C. , 2015 . The lack of equipartitioning in global body wave coda , Geophys. Res. Lett. , 42 ( 18 ), 7483 – 7489 . Google Scholar Crossref Search ADS Van Tiggelen B.A. , 2003 . Green function retrieval and time reversal in a disordered world , Phys. Rev. Lett. , 91 ( 24 ), doi:10.1103/PhysRevLett.91.243904 . Weaver R.L. , 1982 . On diffuse waves in solid media , J. acoust. Soc. Am. , 71 ( 6 ), 1608 , doi:10.1121/1.387816 . Google Scholar Crossref Search ADS Weaver R.L. , Lobkis O.I. , 2004 . Diffuse fields in open systems and the emergence of the Green's function (L) , J. acoust. Soc. Am. , 116 ( 5 ), 2731 , doi:10.1121/1.1810232 . Google Scholar Crossref Search ADS © The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.
Geophysical Journal International – Oxford University Press
Published: Feb 1, 2018
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