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1 Geophysicl Journl Interntionl Geophys. J. Int. (2016) 207, Advnce Access publiction 2016 September 25 GJI Seismology doi: /gji/ggw361 Frequency domin nlysis of errors in cross-correltions of mbient seismic noise Xin Liu, Yehud Ben-Zion nd Dimitri Zigone Deprtment of Erth Sciences, University of Southern Cliforni, Los Angeles, CA , USA. E-mil: Accepted 2016 September 23. Received 2016 September 21; in originl form 2016 Mrch 4 1 INTRODUCTION SUMMARY We nlyse rndom errors (vrinces) in cross-correltions of mbient seismic noise in the frequency domin, which differ from previous time domin methods. Extending previous theoreticl results on ensemble verged cross-spectrum, we estimte confidence intervl of stcked cross-spectrum of finite mount of dt t ech frequency using non-overlpping windows with fixed length. The extended theory lso connects mplitude nd phse vrinces with the vrince of ech complex spectrum vlue. Anlysis of synthetic sttionry mbient noise is used to estimte the confidence intervl of stcked cross-spectrum obtined with different length of noise dt corresponding to different number of evenly spced windows of the sme durtion. This method llows estimting Signl/Noise Rtio (SNR) of noise cross-correltion in the frequency domin, without specifying filter bndwidth or signl/noise windows tht re needed for time domin SNR estimtions. Bsed on synthetic mbient noise dt, we lso compre the probbility distributions, cusl prt mplitude nd SNR of stcked cross-spectrum function using one-bit normliztion or pre-whitening with those obtined without these pre-processing steps. Nturl continuous noise records contin both mbient noise nd smll erthqukes tht re inseprble from the noise with the existing pre-processing steps. Using probbility distributions of rndom cross-spectrum vlues bsed on the theoreticl results provides n effective wy to exclude such smll erthqukes, nd dditionl dt segments (outliers) contminted by signls of different sttistics (e.g. rin, culturl noise), from continuous noise wveforms. This technique is pplied to constrin vlues nd uncertinties of mplitude nd phse velocity of stcked noise cross-spectrum t different frequencies, using dt from southern Cliforni t both regionl scle ( 35 km) nd dense liner rry ( 20 m) cross the plte-boundry fults. A block bootstrp resmpling method is used to ccount for temporl correltion of noise cross-spectrum t low frequencies ( Hz) ner the ocen microseismic peks. Key words: Time-series nlysis; Interferometry; Seismic tomogrphy; Theoreticl seismology; Wve scttering nd diffrction; Wve propgtion. Extrcting empiricl Green s function between multiple pirs of sttions from mbient seismic noise cross-correltion hs been widely pplied in vrious regions nd scles (e.g. Shpiro & Cmpillo 2004;Sbret l. 2005;Bensenet l. 2007;Linet l. 2008; Hillers et l. 2014). Aki (1957) proposed tht seismic properties of the subsurfce mteril cn be obtined from the ensemble-verged cross-spectr of dt t two different loctions smpling stochstic wve field. An empiricl Green s function cn be derived from the expected cross-correltion of fully diffused sttionry noise wvefield (e.g. Lobkis & Wever 2001; Roux et l. 2005;Sánchez-Sesm & Cmpillo 2006;Gouédrd et l. 2008). In prctice, the mbient noise field my not be fully diffuse t certin frequency rnges (e.g. Sens-Schönfelder et l. 2015; Liu & Ben-Zion 2016) due to imperfectly scttered ocen microseismic noise or other effects. A non-diffuse mbient noise field my produce bised empiricl Green s function estimted from cross-correltions (Liu & Ben-Zion 2016). Furthermore, the distribution of mbient seismic noise sources Now t: Institut de Physique du Globe de Strsbourg, Université de Strsbourg, EOST, CNRS, Strsbourg, Frnce C The Authors Published by Oxford University Press on behlf of The Royl Astronomicl Society.

2 Error nlysis for cross-spectrum 1631 cn be non-isotropic (e.g. Cmpillo 2006;Weveret l. 2009;Tsi2011), which cn lso bis empiricl Green s function estimtes. Despite those complictions, mbient noise correltions hve been pplied in numerous studies of phse/group velocity tomogrphy (Shpiro et l. 2005; Lin et l. 2008; Zigone et l. 2015), ttenution coefficient/site fctor estimtion (Prieto et l. 2009; Lin et l. 2012; Liu et l. 2015), nd body wve phses retrievl (e.g. Poli et l. 2012; Lin & Tsi 2013). Noise correltions were lso used to nlyse temporl chnges of seismic velocities (e.g. Sens-Schönfelder & Wegler 2006; Brenguier et l. 2008; Brenguier et l. 2014; Hillers et l. 2015). The erly work on convergence rte of diffused wvefield cross-correltion primrily focused on the vrince of time domin noise cross-correltion. Snieder (2004) derived the time domin vrince expression for isotropiclly distributed sctters (sources) embedded in homogenous medi. The time domin cross-correltion vrince is proportionl to the product of two utocorreltion functions from two sttions t zero lg time divided by the time-bndwidth product. Wever & Lobkis (2005) derived vrince expression tht decys like 1/(recording time) for open systems bsed on modl expnsion of diffused wvefield. Sbr et l.(2005) generlized the vrince expression to heterogeneous medi nd coloured noise spectr ssuming isotropic noise source distribution nd sttionry process. They estimted vrince by windowing the cod of cross-correltion where the empiricl Green s function equls zero. Bsed on the definition of vrince for cross-correltion, the Signl/Noise Rtio (SNR) of nrrow-bnd filtered cross-correltion cn be conveniently defined s the rtio of pek mplitude of wve pcket envelope over the squre root of vrince (e.g. Sbr et l. 2005b; Linet l. 2008; Poliet l. 2013; Zigone et l. 2015). The squre root of vrince of cod noise is simply the Root Men Squre (RMS) of the sme noise segment. These methods use only informtion from the stcked/verged cross-correltion functions, nd they do not consider temporl correltion structure nd non-sttionrity in the noise recordings. In this study we use more informtion nd propose prcticl pproch for computing verged cross-spectr nd estimting rndom errors (vrinces) with vlidtions from block bootstrp resmpling. In Section 2, we develop probbilistic description of stcked (verged) cross-spectrum bsed on ssumptions on Gussin noise model, sttionry noise rndom process nd N independent, identiclly distributed (i.i.d) rndom cross-spectrum observtions. These cn be pproximted from N evenly spced time windows extrcted from mbient noise dt. The vrince of stcked cross-spectrum is found to be equl to the product of power spectrum vlues of two sttions t the sme frequency divided by the number of observtions N (Section 2.1), which is linked to the time domin vrince expression mentioned bove. The rndom errors in phse nd cusl (or nti-cusl) mplitude bsed on stndrd errors of complex stcked cross-spectrum re derived in Section 2.2. Equtions necessry for prcticl dt processing bsed on this method re lso given in Section 2.2. The remining theoreticl sections my be useful for future work. The SNR for nrrowbnd Gussin filtered cross-correltion is estimted in time domin nd linked to the vrince nd men of stcked cross-spectrum (Section 2.3). The men nd vrince estimtions for piecewise sttionry mbient noise records re given in Section 2.4. In Section 3, we simulte synthetic noise dt t two sttions nd study the distribution nd sttistics of stcked cross-spectrum t ech frequency computed from numerous non-overlpping windows. The modified rndom cross-spectrum distributions nd cusl mplitude spectrl decy curves bsed on two widely used pre-processing techniques, one-bit normliztion nd pre-whitening, re compred to those ssocited with unprocessed dt. Synthetic dt re lso used to study phse nd mplitude errors s well s the convergence imges of SNR. In Section 4, similr error nlysis procedures re pplied to three regionl sttions ( 35 km spcing) of the Southern Cliforni regionl network crossing the Sn Andres Fult. The histogrm of rel cross-spectrum observtions t ech frequency contins outliers tht should be removed. After removing outliers, we find SNR convergence imges contining peks nd troughs t different frequencies. Block bootstrp method is pplied to the sme dtset s n independent wy of estimting the empiricl confidence intervls of stcked cross-spectrum compred with the results bsed on stndrd error of stcked cross-spectrum. The results suggest tht temporl correltions of cross-spectrum observtions mong different windows increse the vrince of stcked cross-spectrum (between 0.05 nd 0.2 Hz). In Section 5, we nlyse the errors in high frequency noise recorded by densely spced ( 20 m) rrys tht cross the Sn Jcinto Fult Zone. Removing the outliers increses the SNR, which improves both the mplitude nd phse velocity estimtions. Implictions of the results nd prospects for future work re discussed in Section 6. 2 THEORY We consider mbient noise wve field t two sttions nd b. The rndom wve field d j (ω) t ech sttion j ( j = or b) cn be expressed s the sum of correlted noise u j (ω), which stnds out through cross-correltion, nd uncorrelted noise v j (ω), which cncels out fter cross-correltion, d j (ω) = u j (ω) + v j (ω). Assuming the correlted noise field t different sites hve common noise sources, the correlted noise spectrum of sttion j t ngulr frequency ω is written s sum of contributions from numerous noise sources s k (ω), which include sctterers (discrete version of eq. 1 in Liu & Ben-Zion 2013), u j (ω) = A j (ω) k 1 jk exp [ iω jk c (ω) ] s k (ω), where jk represents distnce from noise source k to sttion j. The inverse complex phse velocity is defined s 1/ c(ω) = (1 isgn(ω)/2q(ω))/c(ω) where Q is the sptilly vrying (heterogeneous) ttenution qulity fctor nd c(ω) is the rel phse velocity. A j (ω) represents the site mplifiction fctor for sttion j. The noise sources s k (ω) re ssumed to be circulr Gussin rndom vribles tht re mutully uncorrelted such tht E[s k (ω)s l(ω)] = B k (ω)δ kl,whereb k (ω) is the spectrl density of kth noise source nd δ kl is the Kronecker delt. Other fctors in eq. (1) depending on frequency ω cn be bsorbed in the spectrl density term B k (ω) for simplicity. The noise sources distribution cn be non-isotropic. The noise sources re (1)

3 1632 X. Liu, Y. Ben-Zion nd D. Zigone ssumed sttionry nd wve components t different frequencies re not correlted (Liu & Ben-Zion 2016). As result of the summtion in eq. (1), the correlted noise spectrum u j (ω) is Gussin rndom vrible. The uncorrelted noise spectrum is lso ssumed circulr Gussin rndom vrible nd stisfies E[u k (ω)v l(ω)] = 0ndE[v k (ω)v l(ω)] = V l (ω)δ kl,wherev l (ω) is the spectrl density of uncorrelted noise t sttion l. 2.1 Stcked (verged) cross-spectrum nd its men nd vrince In frequency domin, the cross-correltion opertion cn be expressed s multipliction of the spectrums t two sttions. In relistic cses, there re only finite lengths of noise recordings vilble. Here we ssume i.i.d spectrum observtions (rndom vribles) t the sme frequency, nd denote the nth observtion of rndom spectrum on sttion j s d n j (ω). The cross-spectrum between sttions nd b from the n.th observtion is d n (ω)dn b (ω). Then the verge of N observed i.i.d cross-spectrum vlues between sttions nd b cn be formulted s R N b (ω) = 1 N d n (ω) d n b N (ω). (2) n=1 The expecttion of the verged cross-spectrum is E [ R N b (ω)] = 1 N E [ d n (ω) d n b N (ω)] = 1 N E [ u n N (ω) un b (ω)] = C b (ω) (3) n=1 n=1 where C b (ω) is the expected cross-spectrum (ensemble verge) of correlted noise recordings from two sttions nd b (e.g. Wever et l. 2009; Liu & Ben-Zion 2013; Liuet l. 2015). The uncorrelted noise terms v (ω)ndv b (ω) cncel out. The stcked (verged) cross-spectrum Rb N (ω) is complex rndom vrible tht pproches Gussin distribution for lrge number of observtions N ccording to the Centrl Limit Theorem of probbility. The vrince of Rb N (ω) is (Appendix A), Vr [ R N b (ω)] = 1 N E [ d (ω) d (ω) ] E [ d b (ω) d b (ω) ] = 1 N A (ω) A bb (ω) (4) where A (ω) nda bb (ω) re expected (ensemble-verged) power spectrum functions for sttions nd b, respectively. This expression is relted to the time domin vrince expression (e.g. Sbr et l. 2005), which is lso proportionl to the product of two utocorreltion functions divided by length of recording T (T is relted to number of observtions N; explined in the numericl simultion Section 3 using N windows). The pseudo-vrince of the stcked cross-spectrum is (Appendix A), pvr [ R N b (ω)] = 1 N E [ d (ω) d b (ω) ] E [ d (ω) d b (ω) ] = 1 N C b (ω) C b (ω) (5) which is generlly not zero. A non-zero pseudo-vrince indictes tht the complex rndom vrible Rb N (ω) is not circulr (Ollil 2008) nd therefore its rel nd imginry prts re correlted (Appendix B). 2.2 Errors of mplitude nd phse mesurements For N i.i.d cross-spectrum observtions (from finite mount of noise dt), the stndrd errors for both rel nd imginry prts of the verged cross-spectrum cn be mesured directly from the distributions of these cross-spectrum observtions. The stndrd errors describe the uncertinties in the verged cross-spectrum. The verged cross-spectrum cn be represented by sum of its expected vlue C b nd zero-men noise term n C (ω) which is complex rndom vrible, R N b (ω) = C b (ω) + n C (ω) = C b (ω) + n R (ω) + in I (ω) (6) where n R (ω) ndn I (ω) re rel nd imginry prts of the complex residul noise n C (ω), which hs the sme vrince s the verged cross-spectrum Rb N (ω), nd is relted to both the correlted noise u j nd uncorrelted noise term v j. The noise term n C (ω) pproches zero for infinite number of i.i.d cross-spectrum reliztions, which requires unlimited mount of dt. The expected cross-spectrum cn be pproximted with two wve pckets on cusl (t > = 0) nd nti-cusl (t < 0) prts propgting in opposite directions with the sme speed (simplified from eq. 1 of Liu et l. 2015), [ ωx C b (ω) = α (ω) exp i c (ω) + i π ] [ ωx + β (ω) exp i 4 c (ω) i π ] (7) 4 where α(ω) nd β(ω) re two rel mplitude terms including ttenution, site mplifiction nd fr-field noise source intensity for cusl nd nticusl prts, respectively. A phse shift of π/4 is equivlent to multiplying with i which is due to the pproximtion of Bessel functions when the inter-sttion distnce is greter thn the wvelength (Liu et l. 2015). Phse shift due to imperfectly isotropic noise sources is not considered to simplify derivtions. In eq. (7), the wve phse speed c(ω) is ssumed finite nd greter thn zero.

4 Error nlysis for cross-spectrum 1633 Substituting eq. (7) into eq. (6), we hve the stcked cross-spectrum [ ωx R N b (ω) = [α (ω) + β (ω)] cos c (ω) π ] [ ωx + n R (ω) + in I (ω) + i [β (ω) α (ω)] sin 4 c (ω) π ] 4 (8) where the rel nd imginry prts hve β(ω) + α(ω) nd β(ω) α(ω) s their expected envelope functions, respectively. Hilbert trnsform cn be used to seprte the cusl nd the nti-cusl prts becuse the nlytic form of the verged cross-spectr function is equivlent to isolting the cusl prt (t > = 0) of the verged cross-correltion. Applying frequency domin Hilbert trnsforms to the rel nd imginry prts of the stcked cross-spectrum respectively, we get their nlytic forms [ R N R (ω) = C b,r (ω) + ñ R (ω) = [α (ω) + β (ω)] exp i [ (ω) = C b,i (ω) + ñ I (ω) = [β (ω) α (ω)] exp i R N I ωx c (ω) i π 4 ωx c (ω) i π 4 i π 2 ] + ñ R (ω) ] + ñ I (ω) (9) where C b,r (ω) nd C b,i (ω) re nlytic forms of the rel nd imginry prts of the expected (men) cross-spectrum, respectively. They re derived bsed on Bedrosin s theorem (e.g. Boshsh 1992) ssuming the time domin representtions of β(ω) ± α(ω) re zero for t min[x/c(ω)] or the envelope of the rel or imginry prt cross-spectrum is smoother thn the cross-spectrum itself. The nlytic form of either rel or imginry prt of cross-spectrum is equivlent to multiplying its inverse Fourier trnsform by step function (in time domin) nd is used for obtining the envelope. The nlytic form of rndom noises for the rel nd imginry prts re ñ R (ω)ndñ I (ω), respectively ñ R (ω) = n R (ω) + ihn R (ω) ñ I (ω) = n I (ω) + ihn I (ω) (10) where Hn R (ω) ndhn I (ω) re, respectively, the Hilbert trnsforms of n R (ω) ndn I (ω). The Hilbert trnsforms Hn R (ω) ndhn I (ω) re uncorrelted with n R (ω)ndn I (ω), respectively (Appendix C). The wve pckets for cusl nd nticusl prts of the cross-spectrum cn be seprted bsed on eq. (9), 2ˆα (ω) exp [i ˆϕ] = R N R (ω) i R N I (ω) 2 ˆβ (ω) exp [i ˆϕ] = R N R (ω) + i R N I (ω) (11) From eq. (11) we cn derive the men vlues nd uncertinties for the mplitude estimtors ˆα(ω) nd ˆβ(ω) s well s the phse ngle estimtor ˆϕ = ωx/c(ω) π/4 bsed on the right hnd side of eq. (11) consisting of Hilbert trnsforms of the rel nd imginry prts of verged cross-spectrum. The mplitude uncertinties for cusl nd nticusl prts re, respectively (Appendix B) Vr [ ˆα (ω)] = [ σ 2 R (ω) + σ 2 I (ω)]/ 2 Vr[ ˆβ (ω)] = [ σ 2 R (ω) + σ 2 I (ω)]/ 2 (12) where σr 2(ω) = Vr[n R(ω)] nd σi 2(ω) = Vr[n I (ω)] re vrinces for the rel nd imginry prts of stcked cross-spectrum, respectively. Eq. (12) suggests tht the cusl nd nticusl prts hve the sme vrince. The frequency domin SNR on cusl or nti-cusl prt cross-spectrum cn be conveniently defined s men over squre root of vrince: ˆα(ω)/ Vr[ ˆα(ω)] or ˆβ(ω)/ Vr[ ˆβ(ω)]. The phse uncertinty cn be derived bsed on error propgtion in the nonliner cse (Appendix B), ˆσ 2 ϕ R (ω) = σr 2 [ (ω) ˆα (ω) + ˆβ (ω) ] 2 ˆσ 2 ϕ α (ω) = [ σ 2 R (ω) + σ 2 I (ω)]/ 4 [ ˆα (ω)] 2 ˆσ 2 ϕ β (ω) = [ σ 2 R (ω) + σ 2 I (ω)]/ 4 [ ˆβ (ω)] 2 (13) where ˆσ ϕ R 2 (ω), ˆσ ϕ α 2 (ω) nd ˆσ ϕ β 2 (ω) re the vrinces for rel prt, cusl prt nd nticusl prt of cross-spectrum, respectively. Ech vrince term in eq. (13) cn be understood s 1/SNR 2 (vrince/men 2 ) of wve envelope of the corresponding nlytic spectrum. Most studies derive phse nd group velocities from the symmetric component (rel prt) of the cross-spectrum (e.g. Lin et l. 2008;Weveret l. 2009), so the phse uncertinty derived for the rel prt cross-spectrum should be used in such cses.

5 1634 X. Liu, Y. Ben-Zion nd D. Zigone 2.3 SNR on nrrow-bnd time domin cross-correltion Assuming phse velocity c(ω) nd mplitude terms α(ω) ndβ(ω) on cusl nd nticusl prts re constnt in nrrow-bnd, the time domin nlytic cross-correltion function of nrrow-bnd filtered eq. (8) cn be written s inverse Fourier Trnsform of the filtered spectrum, R b (t) = F { 1 Ã (ω ω 0 ) R N b (ω)} = F { 1 Ã (ω ω 0 )(C b (ω) + n C (ω)) } (14) where the Gussin filter is Ã(ω) = exp( 2 ω 2 ) with the filter width prmeter nd > 0. Applying convolution theorem nd ssuming the unfiltered stcked cross-spectrum stisfy Hermitin symmetry, the time domin nrrow-bnd nlytic cross-correltion becomes, R b (t) = F { 1 Ã (ω ω 0 ) } t F 1 {C b (ω) + n C (ω)} ω0 ( = exp (iω 0 t) A (t) t {α (ω 0 ) δ t x ) [ exp i π ] ( + β (ω 0 ) δ t + c (ω 0 ) 4 x ) [ exp i π ] } + n C (t; ω 0 ), (15) c (ω 0 ) 4 where A(t) = exp( t 2 /(4 2 ))/(2 π) is the inverse Fourier trnsform of the filter Ã(ω), nd t denotes convolution in time domin. Eq. (15) indictes tht the cusl prt pek mplitude of the nlytic nrrow-bnd filtered cross-correltion t frequency ω 0 is equl to α(ω 0 )/(2 π), nd similrly for the nti-cusl prt the pek mplitude is β(ω 0 )/(2 π). The RMS estimtion of noise in the rel time domin cross-correltion Re[ R b (t)] is (Appendix D), 1 RMS n,td (ω 0 ) = 4 2π E [ n C (ω 0 ) nc (ω 0) ] = 1 σr 2 (ω 0) + σi 2 (ω 0) 2 (16) 2π which is relted to the filter width. The nrrower the Gussin filter bndwidth is, the lrger nd consequently the smller the RMS vlue become. The noise in the cross-correltion function is ssumed sttionry. Bsed on eqs (15) nd (16), the conventionl SNR mesurement in time domin on cusl prt cn be relted to (Appendix D), SNR α (ω 0 ) = 1 2 π α (ω 0 ) σ 2 R (ω 0 ) + σ 2 I (ω 0). This suggests tht the SNR mesured from time domin nrrow-bnd filtered cross-correltion depends on the filter width prmeter, such tht if one increses the filter width (decreses ), the SNR in time domin increses. Theoreticlly, the time domin SNR is different from frequency domin SNR (defined below eq. 12) becuse it depends on the bndwidth of the nrrowbnd Gussin filter. In relistic cses, however, the noise in the cross-correltion function is not sttionry nd the RMS noise estimtion is less thn in eq. (16). The RMS noise vlue lso depends on the length/position of the noise window, the til of the Gussin filter in time domin nd the finite length of the cross-correltion. 2.4 Stcking of piecewise sttionry cross-spectrum The ctul seismic noise dt re not sttionry nd the noise records during different time periods cn hve different men nd vrince vlues. Assuming tht the noise recording within fixed length of time (dy/month/etc., for simplicity we use dy in the following) is sttionry, the men nd vrince for either rel or imginry prt cross-spectrum X j t single frequency ω of jth dy re, m j = E [ X j ] σ 2 j = Vr [ X j ]. They cn be estimted from the smple men nd vrince of observtions t the sme frequency nd dy. If ech dy cn be divided into M non-overlpping nd uncorrelted time windows, the verge cross-spectrum of M i.i.d observtions nd the stndrd error from jth dy is, respectively, X j = 1 M M X j (t k ) = m j k=1 σ 2 j,stde = 1 M Vr [ ] 1 X j = M σ 2 j where t k is the time of the kth time window in dy. Let X S be the verged stck of N dys of cross-spectrum, X S = 1 N N X j. j=1 (17) (18) (19) (20)

6 Error nlysis for cross-spectrum 1635 Figure 1. () Geometry for numericl simultion of mbient noise t two sttions. The intersttion distnce is 50 km nd the intersttion ttenution is 30. The bckground ttenution is 500. The fr field noise source distribution is uniform in ll directions nd source receiver distnce is 8. (b) Schemtic plot for evenly spced windows on noise recording. Window length is 100 s nd gp is 20 s. (c) 3600 independent cross-spectrum curves computed from 3600 windows in 5 d. (d) Stcked cross-spectr with error brs from 3600 observtions (windows). (e) Histogrms of the rel prt nd joint rel & imginry prts cross-spectrum from 3600 observtions. (f) Stcked cross-correltion of 3600 windows from 5 d of dt. Then the men nd vrince of X S re, respectively m S = E [X S ] = 1 N N j=1 m j σ 2 S = Vr [X S] = 1 N 2 N j=1 σ 2 j,stde = 1 1 NM N N j=1 σ 2 j = 1 NM σ 2. (21) where σ 2 = N σ j 2 /N is the verge vrince of N different dys nd NM is the totl number of windows in N dys. The men vlue of the j=1 stcked cross-spectrum of N dys is the verge of N dily men vlues, nd the stcked vrince is the N-dy verged dt vrince σ 2 divided by totl number of observtions NM. There re two wys to reduce the vrince of stcked cross-spectrum X S : (1) increse the totl number of observtions NM or (2) remove or rescle the dily segments of very lrge or smll vrinces in order to minimize σ 2 reltive to the bsolute vlue of the men m s. 3 NUMERICAL SIMULATIONS We perform numericl simultions of finite-length (25 d) sttionry rndom noise records t two sttions (Fig. 1) nd compute stcked (verged) cross-spectrum of the two sttions. The stndrd errors of the rel nd imginry prts of ech stcked cross-spectrum vlue re computed nd the errors on mplitude nd phse re estimted bsed on the stcked cross-spectrum vrinces.

7 1636 X. Liu, Y. Ben-Zion nd D. Zigone Figure 2. 1σ phse rndom error nd errors for its derived quntities. () Phse error computed from the symmetric component (rel prt) of cross-spectr in Fig. 1(d). (b) Phse trveltime error. (c) Phse velocity dispersion curve with 1σ confidence intervl. (d) Reltive error in phse velocity. The synthetic mbient noise records re computed using eq. (1) with dded uncorrelted noise. The percentge of uncorrelted noise increses with frequency following 1-exp( 0.35ω), which is exggerted compred with the relistic cse to illustrte how the uncorrelted noise increse the vrince by incresing the power spectrum. The ssumed inter-sttion distnce is 50 km nd the ttenution Q vlue between the two sttions is 30, which is typicl for regionl fult zone environments (Liu et l. 2015). The noise spectrum is computed by summing over rndom noise sources tht re 8 wy from the centre of the two sttions nd re ssumed i.i.d sttionry Gussin rndom processes. The inverse Fourier Trnsform of the noise cross-spectr produces the corresponding mbient noise in time domin with equl number of dt points. The Nyquist frequency is 1 Hz. The cross-correltion cn be obtined by directly cross-correlting the continuous noise records of 25 d t the two sttions since the synthetic noise dt re sttionry with no other signls (e.g. erthqukes) included in relistic continuous wveform recordings. However, we dopt n lterntive method consistent with our theoreticl results on stcked cross-spectrum with vrince estimtion. The continuous noise dt on ech sttion re divided by evenly spced windows with length of 100 s nd gp of 20 s between windows (Fig. 1b). The gp of 20 s is sufficient to ensure tht the cross-spectr observtions computed from ny two windows re independent for frequencies bove 0.1 Hz. As first test of the synthetic dt nd their connections with the theoreticl vrinces, we select 5-d long noise dt with 3600 nonoverlpping windows, which produce 3600 independent observtions of rndom cross-spectr curves (Fig. 1c). The stcked cross-spectr of the 3600 independent cross-spectr curves re shown in Fig. (1d) with error brs for rel nd imginry prts. The stndrd errors for the rel nd imginry prts t the sme frequency re very similr but the mplitude of the rel prt is much higher thn tht on the imginry prt due to the isotopic noise source distribution. The rel prt nd joint rel-imginry prts histogrms of ll observtions of cross-spectrum t 0.14 Hz re shown in Fig. (1e). The rel prt cross-spectrum histogrm hs smple men of 2.26e-5, which is well constrined becuse it is 10 times more thn its stndrd error 1.92e-6. The stcked cross-correltion function of the 5-d (3600 windows) dt is shown in Fig. (1f). The phse rndom error (Fig. 2) cn be derived from eq. (13) using stndrd errors from the stcked cross-spectrum vrinces. The phse error is inversely proportionl to the squre of the SNR so it s lrger when SNR gets lower. The phse error should be smller thn π/4, otherwise it my cuse phse unwrpping mbiguities for trveltime nd phse velocity estimtes. Assuming the phse error is much smller thn π, the error in trveltime cn be conveniently derived from the liner reltion dt = ωdϕ (Fig. 2b) nd cn be used for tomogrphy bsed on trveltime mesurements on surfce wves (e.g. Lin et l. 2009). In ddition, ssuming tht the ry pth is stright, phse velocity rndom errors (Figs 2c nd d) cn be derived from dc = c 2 dϕ/ωx nd phse velocity cn be derived bsed on eq. (7) in frequency domin or s in Lin et l. (2008) in time domin. Phse velocity rndom errors cn potentilly be used for tomogrphy bsed on stright ry pth pproximtion nd monitoring temporl velocity chnges t different frequencies. The effects of severl commonly used pre-processing steps on distributions of cross-spectrum observtions nd mplitude of stcked cross-spectrum re evluted using the 25-d long noise dt with windows. Figs 3() (c) show results for no pre-processing, one-bit normliztion nd pre-whitening, respectively. The top pnels contin the joint distributions of the rel nd imginry prts of cross-spectrum

8 Error nlysis for cross-spectrum 1637 Figure 3. Comprison mong different pre-processing techniques: no pre-processing (), one-bit normliztion (b) nd pre-whitening (c) windows (25 d) re used. (), (b) nd (c) re joint distributions for rel nd imginry prts of cross-spectrum. (d), (e) nd (f) re cusl prt cross-spectrum mplitude curves with error brs. t 0.14 Hz. The bottom pnels show the cusl prt of stcked cross-spectr mplitude decy curves computed from first line of eq. (11) with the different pre-processing steps previously mentioned. These results suggest tht both one-bit nd pre-whitening modify the distribution of cross-spectrum observtions nd produce lrger rndom errors reltive to their corresponding mplitude men vlues. Moreover, the pre-whitening step lso normlizes the bsolute squre of ech observed cross-spectrum vlue to one (Fig. 3c) R N b,whiten (ω) = 1 N N d n d n n=1 (ω) db n (ω) (ω) db n (ω). Therefore, the joint distribution of rel nd imginry prts with pre-whitening is non-zero only on the unit circle in the complex plne. The pre-whitening step significntly modifies the mplitude decy curve (Fig. 3f) nd does not llow for ccurte mplitude estimtions. The SNR of time domin nrrowbnd cross-correltion (Fig. 4) is defined like in theory Section 2.3 s the rtio between the pek of wve pcket envelope over the RMS estimtion of cod noise. The nrrow-bnd Gussin filter width is = nd the noise window is selected between 50 nd 100 s. The frequency domin SNR for cusl prt mplitude cross-spectrum (Fig. 4b) is defined below eq. (12) s the rtio of cusl mplitude over squre root of vrince. Both time nd frequency domin SNR plots show tht higher SNR re obtined when more observtions (windows) re used for ech frequency. For fixed number of observtions, the SNR lso decreses s the frequency increses becuse of the incresing high-frequency uncorrelted noise, reltive to correlted noise, dded in the forwrd noise synthetics. Higher percentge of uncorrelted noise in the mbient noise recordings results in lrger power spectrum vlues reltive to correlted noise power nd consequently slower convergence rte ccording to eq. (4). The time domin SNR is unstble nd fluctutes with incresing trend s the number of observtions increses while the frequency domin SNR increses stedily with more observtions (Figs 4 nd b). Moreover, the time domin SNR depends on the width of the Gussin filter nd the definition of cod noise window, which mke it non-unique. For comprison, the frequency domin SNR is more stble nd is unique sttisticl estimtion for ech discrete frequency vlue. The time domin cross-correltions of 25 d ( windows) re shown in Figs 4(c) nd (d) for one-bit normliztion nd pre-whitening, respectively. The frequency domin SNR with one-bit normliztion (Fig. 4e) or pre-whitening (Fig. 4f) shows similr trend of incresing SNR with more observtions, however, the mximum SNR vlues (14.5 for one bit nd 15 for pre-whitening) re much lower thn in the cse without pre-processing (Fig. 4b, mximum SNR is 18) becuse of the nonliner nture of the one-bit or pre-whitening pre-processing steps. (22) 4 APPLICATION TO REGIONAL NETWORK STATIONS To illustrte the utility of the theoreticl results, we use dt recorded by three sttions CHF, SBB2 nd LMR2 selected from the regionl CI seismic network in Southern Cliforni (Fig. 5). After removing instrument response, the continuous verticl component recordings for 250 d in the yer 2014 (dys ) re divided into 4-hour long segments. Segments dominted by significnt erthqukes re removed

9 1638 X. Liu, Y. Ben-Zion nd D. Zigone Figure 4. Cusl prt SNR convergence imges for 25 d/ windows noise dt. () nd (b) re SNR convergence imges of cusl prt cross-correltion mplitude estimted in time nd frequency domins, respectively. SNR is defined s mplitude divided by corresponding squre-root of vrince. (c) nd (d) re stcked cross-correltions from windows (observtions) for one-bit normliztion nd pre-whitening, respectively. (e) Frequency domin SNR convergence imge of cusl prt cross-correltion mplitude with one-bit normliztion. (f) Frequency domin SNR convergence imge of cusl prt cross-correltion mplitude with pre-whitening. nd ny spikes more thn 4 times the stndrd devition of the noise segment re clipped to minimize the effects of smller erthqukes (Poli et l. 2013; Zigone et l. 2015). Neither one-bit normliztion nor pre-whitening is pplied in the pre-processing step to preserve the originl mplitude informtion. A bnd pss filter is pplied between 0.05 nd 0.6 Hz. The cross-correltion functions of rw 250-d stck for the three sttion pirs re shown in Fig. 5(b). The nlysis of rndom errors primrily focuses on the sttion pir CHF-SBB2. As in the numericl simultions, the non-overlpping window length is 100 s with gp of 20 s between windows (see sketch in Fig. 1b). The 250-d dt re equivlent to windows (observtions) fter erthquke segment removl. The histogrm of the rel prt cross-spectrum t 0.24 Hz is shown in Fig. 6() nd suggests the existence of mny outlier signls with different sttistics creting the long tils of the histogrm. The observtions outside of three Medin Absolute Devition (MAD) re ssumed to be outliers nd removed. After outlier removl, the new distribution of remining observtions of rel prt cross-spectrum t the sme frequency hs lrger rtio of men over stndrd error (Fig. 6b). The outliers my involve cross-spectrum of erthqukes cod or strong ocenic storms, which hve different spectrl density compred with the regulr noise energy. Therefore, it is better to exclude outliers before estimting stndrd errors from stcked cross-spectrum. The joint distribution of the rel nd imginry prts of the cross-spectrum t 0.24 Hz fter outlier removl (Fig. 6c) indictes tht the rel nd the imginry prts do not show visible correltion nd hve similr vrince. These results re consistent with the pseudo-vrince results when compring eq. (5) with eq. (B1).

10 Error nlysis for cross-spectrum 1639 Figure 5. () Mp of three sttions CHF, SBB2 nd LMR2 in CI network in Southern Cliforni (blck tringles). The blck lines indicte the fult trces. The green nd brown colours show low nd high elevtion respectively. (b) Cross-correltions of three sttion pirs computed from 250 d in Outliers for ech frequency cn be identified bsed on MAD in similr wy s in Figs 6() nd (b). The identified outliers for ech frequency cn then be ssocited with their respective noise window. For ech window, the percentge of outliers over ll frequencies in the frequency pssbnd of interest is n indictor of the mount of bnorml signl within the window. We select windows with mximum 5 per cent outliers over ll frequencies between 0.05 nd 0.6 Hz for stcking. The selected windows produce the sme number of observtions of rndom cross-spectr curves (Fig. 6d) while the unselected (outlier) windows give rndom cross-spectr curves of different sttistics (Fig. 6e). A simple stck of the selected windows produces cross-spectr with ssocited stndrd errors computed from the distribution of cross-spectrum t ech frequency (Fig. 6f). Similrly, Fig. 6(g) shows the stcked cross-spectr for outlier windows, which presents very different men vlues nd stndrd errors t ll frequencies within the pssbnd since the outlier windows follow different sttistics. The stcked cross-spectr curves derived from both selected windows nd outlier windows hve chrcteristic pek t 0.15 Hz due to the secondry ocen microseismic pek, nd the mplitude of the spectrum decys quickly bove tht pek frequency. The cross-spectr curve derived from outlier windows hs smller pek ner 0.07 Hz, probbly due to the primry microseismic pek. Dividing the stcked cross-spectrum by its stndrd error, the normlized cross-spectrum hs unit vrince. This procedure is connected to the pproch of Aki (1957) becuse the stndrd error is theoreticlly proportionl to the squre root of the product of power spectrums on two sttions (eq. 4), which is the denomintor in the normliztion formul of Aki (1957). This normliztion produces cross-spectr curve with white dditive noise nd more evenly distributed energy within the frequency bnd, which is helpful for frequency domin Hilbert trnsform (better stisfy the condition for Bedrosin s theorem) or isolting/windowing fundmentl mode surfce wve pcket in time domin. Becuse of the strong vrition of the mbient noise power spectr due to ocen microseismic peks, we normlize the stcked cross-spectrum by its stndrd error before Hilbert trnsform nd multiply bck the stndrd error fter Hilbert trnsform. The normlized cross-spectr curve for selected windows stck (Fig. 6h) shows higher SNRs thn tht for outlier windows (Fig. 6i). The time nd frequency domin SNR imges of the cusl prt cross-correltion mplitude (Figs 7 nd b, respectively) re computed in the sme wy s in the synthetic test. The colour scle represents the SNR vlues. Only the selected windows within the number of dys re used for stcking. The time domin SNR vlues re mesured with the sme Gussin filter nd noise window prmeters s in the synthetic test. The time domin SNR nlysis shows incresing SNR vlues t multiple frequencies with incresing number of dys, but the vlues re unstble nd get sturted (stop incresing fter certin number of dys) t some frequencies. In contrst, the frequency domin SNR nlysis shows consistently incresing SNR with number of dys t multiple frequencies. Both imges show generlly similr pek frequencies of SNR vlues, suggesting multiple noise sources t different frequencies other thn the min ocen microseism pek. However, the reltive mplitudes of the peks re different for time nd frequency domin SNR figures. By slicing both SNR imges t three different frequencies (0.15, 0.20 nd 0.32 Hz), the corresponding SNR vlues re plotted s functions of number of dys in Figs 7(c) nd (d) for time nd frequency domin SNR, respectively. At 0.15 nd 0.32 Hz, the SNR vlues increse with number of dys becuse they correspond to the noise peks in Figs 7() nd (b). At 0.20 Hz, however, the SNR stops incresing beyond 35 d probbly becuse of gp between two noise sources in which the noise rndom process is highly non-sttionry. By slicing the SNR imges in Figs 7() nd (b) t three different dy counts, we plot in Figs 7(e) nd (f) SNR functions of centre frequency for time nd frequency domin SNR mesurements, respectively. For 5 d stcking, the time domin SNR show nomlous pek ner 0.14 Hz (lso in Fig. 7), probbly becuse the wve pcket pek is not well defined due to the ctul low SNR vlue (the mximum fter 5 d is only 40). For 125 nd 250 d of stcking, the time domin SNR curves show similr ptterns nd SNR vlues but t some frequencies (e.g Hz, 0.24 Hz) the SNR vlues get sturted while t some other frequencies (e.g. 0.3, 0.52 Hz) the SNR vlues keep incresing up to 250 d. For frequency domin SNR (Fig. 7f), ll three SNR curves for 5, 125 nd 250 d show similr ptterns of peks nd troughs t different frequencies nd progressive increse in SNR vlues s the number of stcked dys increses. These results show more stble SNR estimtes

11 1640 X. Liu, Y. Ben-Zion nd D. Zigone Figure 6. () Rw distribution of rel prt cross-spectrum observtions from sme number of windows t 0.24 Hz for sttion pir CHF-SBB2. (b) Distribution of rel prt cross-spectrum ( observtions) for CHF-SBB2 fter outlier removl. (c) Joint distribution of rel nd imginry prts of cross-spectrum for CHF-SBB2 fter outlier removl. (d) First 2000 cross-spectr curves from selected windows. (e) First 2000 cross-spectr curves from outlier (unselected) windows. (f) Stcked cross-spectr of CHF-SBB2 for selected windows ( windows). Stndrd devition curve for rel prt cross-spectr is plotted in blck dshed line. (g) Stcked cross-spectr of CHF-SBB2 for outlier windows ( windows). Note the difference below 0.1 Hz. (h) Normlized cross-spectr of CHF-SBB2 for selected windows. The vrince of cross-spectrum t ech frequency is normlized to one. (i) Normlized cross-spectr of CHF-SBB2 for outlier windows with unit vrince.

12 Error nlysis for cross-spectrum 1641 Figure 7. () nd (b) re SNR convergence imges of cusl prt cross-correltion mplitude for 250 d estimted in time nd frequency domins, respectively. Only selected windows (defined in Fig. 6f) re used. SNR is defined s mplitude divided by corresponding squre-root of vrince. (c) nd (d) re SNR convergence curves sliced t different frequencies (0.15, 0.20, 0.32 Hz) for time nd frequency domin mesurements, respectively. (e) nd (f) re SNR versus frequency curves sliced t different dys (5, 125, 250 d) for time nd frequency domin mesurements, respectively.

13 1642 X. Liu, Y. Ben-Zion nd D. Zigone Figure 8. Block bootstrp (hourly block) estimtion of empiricl confidence intervls of 250-d stcked cross-spectrum for CHF-SBB2. () Temporl correltion of noise reveled by utocorreltion of rel prt cross-spectrum from different windows. At 0.16 Hz, there is wek correltion 0.07; t 0.20 Hz, there is nerly zero correltion (<0.01). (b) Histogrms of bootstrp verged rel prt cross-spectrum t 0.16 Hz nd 0.24 Hz, respectively. The bootstrp stndrd devition provides estimtion for the uncertinty of stcked cross-spectr. (c) Rtio of rel/imginry prt block bootstrp stndrd devition (more relistic) over stndrd error (ssuming i.i.d observtions from different windows). Bootstrp method suggests lrger vrince between 0.05 nd 0.2 Hz due to temporl correltion. The rtio pproches to one for frequency between 0.2 nd 0.6 Hz. (d) SNR convergence imge of cusl prt cross-correltion computed using hourly block bootstrp resmpling. Becuse of temporl correltion, the bootstrp SNR between 0.05 nd 0.2 Hz re lower thn the frequency domin SNR bsed on stndrd error (Fig. 7b). from frequency domin method probbly becuse the frequency domin method uses vrince of cross-spectrum in ddition to men (the stcked) cross-spectrum. We estimte the correltion between different noise windows by computing the utocorreltions of the rel prt cross-spectrum vlues for two different frequencies (Fig. 8). The temporl correltion mong windows cn dd extr cross-window (non-zero covrince) terms in clculting stndrd error for rel nd imginry prts of stcked cross-spectrum. As result, temporl correltion of cross-spectrum vlues from different time windows will increse the vrince of stcked cross-spectrum. As shown on Fig. 8(), the correltion coefficient is round 0.07 t 0.16 Hz from offset of 1 window to 49 windows, which suggest wek temporl correltion longer thn one nd hlf hours (49 windows offset with 100 s window length nd 20 s gp). At 0.20 Hz, however, the rel prt of the cross-spectrum is not correlted (correltion coefficient ±0.01 or less). For higher frequencies (between 0.2 nd 0.6 Hz), the cross-spectrum vlue does not show temporl correltions mong different windows. An hourly block bootstrp method is pplied s n independent wy to estimte the empiricl confidence intervl for the cross-spectrum derived from the sme selected windows ( ) for pir CHF-SBB2 s in the previous SNR exmple. The purpose of hourly block bootstrp is to cpture some temporl correltion mong the neighbouring windows if they exist. For ech one-hour block, cross-spectr curves from ll selected windows within the block re verged. Bootstrp resmpling is pplied by rndomly pick the sme number of blocks from vilble 1 hr blocks with replcement. At ech frequency, bootstrp men cross-spectrum observtion is computed by verging rndomly drwn block cross-spectrum vlues. There re 4000 bootstrp verged cross-spectrum observtions (Fig. 8b) nd the stndrd devition of these observtions pproximtes the uncertinty of the stcked cross-spectrum. The bootstrp histogrms (Fig. 8b) pproch to Gussin distribution tht confirm with the sttisticl property of stcked cross-spectrum in theory. The rtio of 250-d block bootstrp stndrd devition over stndrd error (which implicitly ssumes i.i.d windows) for the rel nd imginry prts of cross-spectr re plotted in Fig. 8(d). The rtio for both rel nd imginry prts re pproching to one bove 0.2 Hz, suggesting tht the stndrd error bsed on the ssumption of N i.i.d cross-spectrum observtions t the sme frequency from different windows offers relible estimtion of the dt vrince between 0.2 nd 0.6 Hz. The rtio for the rel nd imginry prts between 0.05 nd 0.2 Hz show significnt oscilltion between 2.1 nd 1 with downwrd trending, consistent with the fct tht the cross-spectrum within this frequency rnge corresponds to wek temporl correltions mong windows (Fig. 8).

14 Error nlysis for cross-spectrum 1643 Figure 9. Phse nd mplitude rndom errors (1σ ) of cross-spectr bsed on block bootstrp resmpling of selected windows. The error estimtions below 0.2 Hz re less ccurte due to temporl correltion. () Phse error curve of rel prt cross-spectr. (b) Phse trveltime error. (c) Phse velocity dispersion curve with 1σ confidence intervls. (d) Reltive error in phse velocity. (e) Cusl prt mplitude cross-spectr with error brs. (f) Zoom-in version of cusl prt mplitude cross-spectr in pnel (e). Applying block bootstrp to different number of dys (1 250), the men nd stndrd devition of bootstrp distribution for stcked cross-spectrum vlue cn be esily computed with incresing number of dys nd the SNR for the cusl prt cross-spectrum cn be derived (Fig. 8d) in similr wy s frequency domin SNR bsed on eq. (12) by replcing stndrd error with bootstrp error. For frequency bove 0.2 Hz, the bootstrp SNRs (Fig. 8d) re similr (very close) to the frequency domin SNR vlues (Fig. 7b) bsed on stndrd errors ssuming i.i.d cross-spectrum observtions t ech frequency from different windows. For frequency between 0.05 nd 0.2 Hz, however, the SNR vlues derived from bootstrp re evidently lower ( 20 per cent less) thn the vlues in Fig. 7b bsed on stndrd error, indicting temporl correltion mong the windows tht increse the vrince within this frequency rnge. Becuse the temporl correltion t some frequency (e.g Hz) is greter thn one hour, pplying block bootstrp with longer block length cn better cpture the temporl correltion t this frequency nd therefore further increse the vrince nd reduce the corresponding SNR vlue. The phse error curve (Fig. 9) is derived directly from the bootstrp vrince of the rel prt of stcked cross-spectrum ccording to eq. (13). The mximum phse error (1σ ) is 0.11 rdins, which is much less thn phse ngle period 2π. This indictes tht the phse ngle cn be relibly unwrpped without 2π mbiguities. Following similr procedure s in numericl simultion (Section 3), the phse trveltime uncertinty cn be estimted directly from the phse error. Assuming stright ry pth, the phse ngle errors cn be mpped into phse velocity uncertinties (Figs 9c nd d) nd the lrgest uncertinty is 0.4 per cent velocity vrition within 1σ intervl. The cusl prt mplitude cross-spectrum (Figs 9e nd f) is estimted bsed on eq. (11) nd the corresponding confidence intervl is computed bsed on the vrince vlues of the rel nd imginry prts of cross-spectrum in eq. (12). The mplitude errors below 0.2 Hz re underestimted becuse of temporl correltion of noise. By comprison with the bootstrp SNR figure (Fig. 8d), high SNR indicte relible estimtion of cusl prt cross-spectrum mplitude.

15 1644 X. Liu, Y. Ben-Zion nd D. Zigone 5 A P P L I C AT I O N T O D E N S E LY S PA C E D S TAT I O N S W I T H H I G H F R E Q U E N C Y D ATA As n dditionl illustrtion, we compute noise cross-correltions using dt recorded by three sttions JF00-JFS1-JFS2 of dense liner rry (JF) cross the Sn Jcinto fult seprted by bout 20 m (Fig. 10). The cross-correltion functions of first 10 d in the yer 2012 re plotted in Fig. 10(b). Evenly spced windows with length of 10 s nd gp of 1 s re used to compute cross-spectr curves within the frequency rnge of Hz. Becuse of the high vribility of the noise intensity t high frequency (Ben-Zion et l. 2015), we select only 1 d (the second dy) of 2012 nd compute cross-spectr for sttion pir JF00-JFS1. The histogrm of the rel prt cross-spectrum t 21 Hz with 7332 observtions from sme number of windows (Fig. 10c) hs rtio of men over stndrd error of After removing the outliers ccording to 3 MAD vlue nd zooming in round the selected observtions (windows), the new distribution (Fig. 10d) of rel prt cross-spectrum with 7185 observtions hs higher rtio of men over stndrd error of A selected window should contin no more thn 5 per cent of outliers from ll frequencies within Hz. There re 6220 selected windows tht produce different cross-spectr curves (Fig. 10e). Therefore, the rw stcked cross-spectr before outlier removl (Fig. 10g) hve more fluctutions nd lrger error brs reltive to spectrl Figure 10. Rndom error nlysis using 1-d high-frequency noise (10 50 Hz) on JF rry with smll sttion spcing ( 20 m). () Loction mp of dense liner JF rry cross the Sn Jcinto fult zone. (b) Cross-correltions of three sttions JF00-JFS1-JFS2 computed from the second dys in (c) Schemtic plot of evenly spced windows with 10 s window length nd 1 s gp. (d) Rw distribution of 7332 rel prt cross-spectrum observtions from sme number of windows in 1 d for sttion pir JF00-JFS1. (e) Distribution of rel prt cross-spectrum (7185 observtions) for JF00-JFS1 fter outlier removl. (f) First 2000 of 6220 selected cross-spectrum curves from 6220 windows. (g) Rw stck of cross-spectrum curves computed from ll 7332 windows in the second dy of (h) Stcked cross-spectr of 6220 selected cross-spectr curves (pnel e) from sme number of windows fter removing outliers.

16 Error nlysis for cross-spectrum 1645 Figure 11. Amplitude SNR nd phse rndom error bsed on 1-d stck of cross-spectr of JF00-JFS1. For ech pnel, the informtion derived from rw stck of cross-spectr with outliers re in blue nd those from stcked cross-spectr without outliers re in red. () Cusl prt mplitude SNR. (b) Anticusl prt mplitude SNR. (c) Phse error. (d) Phse trveltime error derived bsed phse error. (e). Phse velocity curves. (f) Phse velocity reltive error. mplitude thn the stcked cross-spectr fter outlier removl (Fig. 10h). There is n nomlous pek t 20 Hz in Fig. (10h), which might be relted to some monochromtic locl noise source t the sme frequency. Figs 11() nd (b) compre the mplitude SNR vlues bsed on rw stcked (with outliers) cross-spectr nd outlier-removed crossspectr obtined from cusl prt (nd nticusl prt) of 1-d stcked cross-spectr for pir JF00-JFS1. After outlier removl, the SNR of cusl prt mplitude is 4 10 times greter thn tht before outlier removl between 17 nd 45 Hz (Fig. 11). For nticusl prt mplitude, the outlier-removed SNR is 4 10 times better thn outlier-included SNR between 10 nd 23 Hz (Fig. 11b). In ddition, fter outlier removl the nticusl mplitude SNR decys fster thn the cusl mplitude SNR, suggesting tht different source/scttering/propgtion mechnisms for noise coming from two opposite directions my ffect the mplitude SNR differently. The outliers lso ffect phse velocity retrieved from 1-d stck of cross-spectr for the pir JF00-JFS1. The phse rndom errors (Fig. 11c) estimted from rel prt of stcked cross-spectr (Fig. 10g) with outliers re 4 times greter thn those without outliers nd show spikes greter thn π/4 t 27, nd 38 Hz, which my cuse phse unwrpping mbiguities. Bsed on phse error informtion, the phse trveltime error (Fig. 11d) nd phse velocity error (Fig. 11f) cn be derived following the sme wy s in numericl simultion (Section 3). The phse velocity dispersion curves bsed on stcked cross-spectr with nd without outliers show 1 4 per cent discrepncy between 11 nd 20 Hz (Fig. 11e), which is probbly due to systemtic errors brought by outliers cross different frequencies. Phse velocity curves with confidence intervls (Fig. 12) re estimted from the stcked cross-spectr of first month in yer The one-month stck of the pir JF00-JFS1 yields much smller phse velocity errors (Figs 12 nd b) compred to the phse velocity errors from the second dy of The spikes in the phse velocity error (Fig. 12b) with outliers re lso reduced on the one-month stcked dt. After outlier removl, however, the phse velocity errors ssocited with different frequencies re reduced by fctor of 2 6. The phse velocity error curves for two dditionl pirs, JF00-JFS2 (Figs 12c nd d) nd JFS1-JFS2 (Figs 12e nd f) show improved phse error estimtions nd less spikes fter outlier removl.