Analyses and Application of Ambient Seismic Noise in Sweden

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1511 Analyses and Application of Ambient Seismic Noise in Sweden Source, Interferometry, Tomography HAMZEH SADEGHISORKHANI ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2017 ISSN ISBN urn:nbn:se:uu:diva

2 Dissertation presented at Uppsala University to be publicly examined in Hambergsalen, Geocentrum, Villavägen 16, Uppsala, Friday, 9 June 2017 at 10:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Prof. Frederik Tilmann. Abstract Sadeghisorkhani, H Analyses and Application of Ambient Seismic Noise in Sweden. Source, Interferometry, Tomography. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala: Acta Universitatis Upsaliensis. ISBN Ambient seismic noise from generation to its application for determination of sub-surface velocity structures is analyzed using continuous data recordings from the Swedish National Seismic Network (SNSN). The fundamental aim of the thesis is to investigate the applicability of precise velocity measurements from ambient noise data. In the ambient noise method, a form of interformetry, the seismic signal is constructed from long-term cross correlation of a random noise field. Anisotropy of the source distribution causes apparent time shifts (velocity bias) in the interferometric signals. The velocity bias can be important for the study area (Sweden) which has relatively small velocity variations. This work explores the entire data path, from investigating the noise-source distribution to a tomographic study of southern Sweden. A new method to invert for the azimuthal source distribution from cross-correlation envelopes is introduced. The method provides quantitative estimates of the azimuthal source distribution which can be used for detailed studies of source generation processes. An advantage of the method is that it uses few stations to constrain azimuthal source distributions. The results show that the source distribution is inhomogeneous, with sources concentrated along the western coast of Norway. This leads to an anisotropic noise field, especially for the secondary microseisms. The primary microseismic energy comes mainly from the northeast. The deduced azimuthal source distributions are used to study the level of expected bias invelocity estimates within the SNSN. The results indicate that the phase-velocity bias is less than 1% for most station pairs but can be larger for small values of the ratio of inter-station distance over wavelength. In addition, the nature of velocity bias due to a heterogeneous source field is investigated in terms of high and finite-frequency regimes. Graphical software for phase-velocity dispersion measurements based on new algorithms is presented and validated with synthetic data and by comparisons to other methods. The software is used for phase-velocity measurements, and deduced azimuthal source distributions are used for velocity-bias correction. Derived phase-velocity dispersion curves are used to construct two-dimensional velocity maps of southern Sweden at different periods based on travel-time tomography. The effect of the bias correction is investigated, and velocity maps are interpreted in comparison to previous geological and geophysical information. Keywords: Seismic interferometry, Ambient noise, Surface wave, Wave propagation, Inverse theory, Sweden Hamzeh Sadeghisorkhani, Department of Earth Sciences, Geophysics, Villav. 16, Uppsala University, SE Uppsala, Sweden. Hamzeh Sadeghisorkhani 2017 ISSN ISBN urn:nbn:se:uu:diva (

3 Dedicated to my lovely family

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5 List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II Sadeghisorkhani, H., Gudmundsson, Ó., Roberts, R., Tryggvason, A., Mapping the source distribution of microseisms using noise covariogram envelopes, Geophysical Journal International, 205(3), Sadeghisorkhani, H., Gudmundsson, Ó., Roberts, R., Tryggvason, A., Velocity-measurement bias of the ambient noise method due to source directivity: A case study for the Swedish National Seismic Network, Geophysical Journal International, 209(3), III Sadeghisorkhani, H., Gudmundsson, Ó., Tryggvason, A., GSpecDisp: a Matlab GUI package for phase-velocity dispersion measurements from ambient-noise correlations, Submitted to Computers & Geosciences. IV Sadeghisorkhani, H., Gudmundsson, Ó., Li, K. L., Tryggvason, A., Roberts, R., Högdahl, K., Lund, B., Surface wave tomography of southern Sweden from ambient seismic noise, Manuscript. Reprints were made with permission from the publishers. In addition, below is a list of other publications during my PhD which are not included in this thesis. Li, K. L., Sgattoni, G., Sadeghisorkhani, H., Roberts, R., Gudmundsson, Ó., A double-correlation tremor-location method, Geophysical Journal International, 208(2), Li, K. L., Sadeghisorkhani, H., Sgattoni, G., Gudmundsson, Ó., Roberts, R., Locating tremor using stacked products of correlations, Geophysical Research Letters, 44, doi: /2016gl Sgattoni, G., Gudmundsson, Ó., Einarsson, P., Lucchi, F., Li, K. L., Sadeghisorkhani, H., Roberts, R., Tryggvason, A., The 2011 unrest at Katla volcano: characterization and interpretation of the tremor sources, Journal of Volcanology and Geothermal Research, In Press, doi: /j.jvolgeores

6 Sadeghisorkhani, H., Gudmundsson, Ó., Roberts, R., Tryggvason, A., Comparison of microseismic Rayleigh and Love waves sources around Scandinavia, Manuscript in preparation.

7 Contents 1 Introduction Noise Interferometry: Plane-Waves Superposition Cross correlation of plane waves Comparison to a direct plane wave Effects of an uneven source distribution Other components of correlation Methods of Phase-Velocity Measurements from Noise Correlations Introduction Single-pair phase-velocity dispersion curves Image Transformation Technique Multiple Filter Analysis Bessel function matching Average phase-velocity dispersion curve of a region Slant stack Bessel function matching as a function of distance Ambient Noise Tomography Velocity maps Depth inversion Summary of Papers Paper I: Mapping the source distribution of microseisms using noise covariogram envelopes Summary Discussions and conclusions Paper II: Velocity-measurement bias of the ambient noise method due to source directivity: A case study for the Swedish National Seismic Network Summary Discussions and conclusions Paper III: GSpecDisp: a Matlab GUI package for phase-velocity dispersion measurements from ambient-noise correlations Summary Conclusions Paper IV: Surface wave tomography of southern Sweden from ambient seismic noise Summary... 49

8 5.4.2 Discussions and conclusions Sammanfattning på Svenska (Summary in Swedish) Acknowledgments References... 58

9 Abbreviations and Symbols km s km/s 2D 3D Z R T ZZ RR TT ZR RZ EGF SNSN PREM FMST FMM Kilometer Second kilometer per second Two dimensional Three dimensional Vertical component Radial component Transverse component Vertical-Vertical cross correlation Radial-Radial cross correlation Transverse-Transverse cross correlation Vertical-Radial cross correlation Radial-Vertical cross correlation Empirical Green s function Swedish National Seismic Network Preliminary reference earth model Fast Marching Surface Tomography Fast Marching Method 9

10 ω Angular frequency f Frequency T Period t Time k Wavenumber r Inter-station distance c Phase velocity v p Phase velocity τ p Phase time (r/v p ) U Group velocity v g Group velocity τ g Group time (r/v g ) C Cross correlation Γ Coherency θ Azimuth from the inter-station axis φ Same as θ or source phase (only in Chapter 3) λ Wavelength or Lame s parameter (only in Chapter 4) γ Dimensionless frequency (r/λ) ε Angular power density of noise sources η Rayleigh wave ellipticity J 0 Zero order Bessel function of the first kind J 1 First order Bessel function of the first kind J 2 Second order Bessel function of the first kind i Imaginary number ( 1) R Real part of complex numbers I Imaginary part of complex numbers T t Travel time u Slowness (one over velocity) ρ Density μ Shear modulus r 1 Horizontal Rayleigh-wave displacement eigenfunction r 2 Vertical Rayleigh-wave displacement eigenfunction α P-wave velocity β S-wave velocity 10

11 1. Introduction For a long time ambient seismic noise has been seen as a nuisance obscuring "real" seismic information originating from earthquake signals, but in fact it can be used to derive similar information as that provided by earthquakes about the earth structure. Nowadays, using ambient noise is one of the fastest growing fields in the earth sciences. It has been extensively used to study Earth s structure in different part of the world during more than a decade since its introduction to seismology. Lobkis & Weaver (2001) showed that the Green s function of a medium between two receivers can emerge from cross correlation of a random acoustic field. The idea of working with random fields has been continued in two branches in seismology. In the first branch coda waves are used, i.e., the incoming waves of an event after the clear arrivals. The coda is interpreted as back-scattering waves from numerous heterogeneities. The second branch of random fields is that of ambient noise (background noise), i.e., seisomogram records when there is no dominant source-event. These random seismic fields have been extensively used to determine earth structure. A combination of these two methods which uses the coda of correlations of the ambient noise also exist (Stehly et al., 2008). In early ambient noise studies, it has been shown that mainly surface waves emerge from the correlation of the ambient-seismic noise traces (Shapiro & Campillo, 2004; Sabra et al., 2005). The emerged surface waves can be used for structural modeling and characterization of Earth by passive imaging at varying scales (from laboratory to continental) (Gouédard et al., 2008). All applications rely on the same basic property of the surface waves, i.e., the frequency dependent phase and group velocities in heterogeneous media. The frequency dependence of velocity is called dispersion, and measuring dispersion curves is the key issue in modeling or imaging a medium using surface waves. Since surface waves propagate along the surface of a medium, their particle motions decay with depth. Therefore, high frequency (short wavelength) surface waves are sensitive to shallow zones (near the free surface), whereas low frequency (long wavelength) surface waves are more sensitive to deeper structures. Consequently, surface-wave velocities of different frequencies can be translated (inverted) to intrinsic velocities at depth and they can be used for underground material characterization. In general, the surface-wave velocities are presented as two dimensional (2D) velocity maps at various frequencies using surface-wave tomography from which three dimensional (3D) shear-wave velocity models can be derived by depth inversions on a grid of local dispersion estimates. 11

12 In this thesis, I concentrate on the surface-wave modeling at a scale of hundreds of kilometers in the crust and uppermost mantle using the Swedish National Seismic Network (SNSN) stations. For surface-wave modeling on this scale, the period (one over frequency) should be larger than 2 s. Microseisms are the most energetic signal of ambient noise in the periods between approximately 4 and 30 s. Microseisms are generated by the interaction of oceans and the solid Earth and propagate to the continent (Hasselmann, 1963; Stehly et al., 2006). Hence, microseisms are generated near to the Earth s surface and they mainly contain fundamental mode surface waves. The advantages of using the ambient-noise method compared to classical earthquake-based surface-wave studies are: 1- the ambient-noise method is not limited to regions with high seismicity, and 2- it is applicable to short periods compared to teleseismic studies. Therefore, it can be used for high-resolution seismic tomography of the crust in any region. This makes the ambient-noise method a well suited tool for Scandinavia. The availability of continuous recordings of noise and powerful computers make the ambient-noise method a robust technique to investigate Earth s elastic structures. In this method, dispersion curves between station pairs are measured, and the velocity structure of the Earth is calculated in a two-step procedure. First, 2D velocity maps are constructed by travel-time tomography at individual frequencies. Second, local estimates of dispersion curves at a grid of geographical coordinates are inverted for the velocity structure of Earth as it varies with depth. Combining many local depth profiles yields a three-dimensional picture. The theoretical justification of the ambient-noise method is that the wave field is diffuse or the noise-source distribution is uniform. In practice these conditions are rarely satisfied. This results in velocity bias. The introduced bias may not be significant in some circumstances as reported by some studies (e.g., Yao & van der Hilst, 2009; Tsai, 2009; Froment et al., 2010). However, assessing the reliability of the velocity measurements from the ambientnoise method is important, especially in a region like Sweden where the Earth structure is fairly homogeneous and the source distribution is very anisotropic. Therefore, studying the ambient-noise source distribution and the consequent velocity bias prior to velocity measurements and tomography is important. This thesis contains a comprehensive summary of four papers that are dedicated to the ambient-noise method from the source distribution to tomography. It consists of five chapters. In the second chapter, the fundamental theory of ambient-noise interferometry is described based on superposition of plane waves. The third chapter summarizes different methods to measure phase-velocity dispersion curves from ambient-noise correlations. The fourth chapter focuses on surface-wave tomography and depth inversion. Finally, a summary of the four included papers in this thesis is presented in chapter five. 12

13 In paper I, a method to map the ambient-noise sources is introduced. It is tested synthetically and applied to the covariograms of one year of data (2012) from the SNSN stations. Also, the seasonal variations of the source distribution around Scandinavia are investigated. In this method, the envelopes of full-length covariograms in the time domain are inverted for azimuthal distribution of the noise sources in selected geographical sub-groups of the SNSN stations. Finally, another inversion is formulated to map the spatial distribution of noise sources using the previously calculated azimuthal-source distributions in all sub-groups. To calculate the azimuthal source distribution, a 2D forward calculation based on the plane-wave assumption is derived. Then, an iterative, linearized inversion is formulated. The forward calculation involves bandpass filtering of the covariograms, and therefore the inversion gives the source distribution in different period ranges. The method introduced in this paper is applied to data that retain physical units (they are not amplitude normalized in any way), and therefore, the final-inversion results have physical units that may be used to study the noise-generation excitation processes in absolute terms. Paper II focuses on the bias of velocity measurements introduced by nonuniform source distributions. In the first part of paper II, the effects of nonuniform source distributions are studied synthetically assuming 2D plane waves, and they are explained conceptually in terms of two regimes, a highfrequency and a finite-frequency regime. The velocity bias is quantified and a basic scaling relation is established for the finite-frequency regime in terms of inter-station distance over wavelength. For the high-frequency regime, the results are compared to a theoretical derivation by Weaver et al. (2009), and the high-frequency definition is established in terms of the first Fresnel angle. In the second part of paper II, the expected bias of velocity measurements in Sweden, based on the noise-source model provided in paper I, is analyzed. The very anisotropic source distribution as seen from the SNSN stations results in predictions of relatively strong bias in the area at relevant frequencies and inter-station distances. This bias may not be small compared to the level of heterogeneity in the Baltic shield when the inter-station distance over wavelength is small, the incoming energy along the inter-station axis is low and/or highly variable. In paper III, a graphical user interface (GUI) package is presented to measure phase-velocity dispersion of surface waves from the noise-correlation traces. The GSpecDisp package provides an interactive environment for the dispersion measurements and presentation of the results. Two algorithms are used for the phase-velocity dispersion measurements in the package; (1) average velocity of a region based on the approach proposed by Prieto et al. (2009), and (2) single-pair phase velocity based on the method proposed by Ekström et al. (2009). The first algorithm provides a reference, data-driven dispersion curve that can be used for automatic selection of dispersion curves calculated with the second method. Both measurements are carried out by 13

14 matching the real part of cross-correlation spectra with an appropriate Bessel function in the frequency domain, whereas the GSpecDisp inputs are timedomain cross-correlations. To the best of my knowledge, GSpecDisp is the only available program that can measure the phase velocity of Rayleigh and Love waves from all possible components of the noise correlation tensor. It can also measure the phase-velocity dispersion up to the period corresponding to an inter-station distance of one wavelength, because there is no need for a high-frequency approximation. Paper IV is an application of the ambient-seismic noise method to determine velocity structure of the crust in southern Sweden. Phase-velocity dispersion curves of 630 station pairs from the SNSN are used to construct 2D velocity maps at periods between 3 and 30 seconds based on travel-time tomography. The software package, GSpecDisp presented in paper III, is used to measure phase-velocity dispersion curves. Also, the phase-velocity bias due to the anisotropy of the ambient-noise source distribution is corrected (based on methods presented in paper I and II). The effect of the bias removal on the velocity maps at different periods is investigated by comparison to the same tomographic maps without bias removal. Finally, the results are interpreted in comparison to other geological and geophysical information. At the shortest periods, there is good agreement with features of the gravity field, seismicity distribution, and surface geology. The final tomographic models provide new insights into the structure of the entire crust and may help constrain the depth extent of common features in the gravity field and other surface observables. 14

15 2. Noise Interferometry: Plane-Waves Superposition This chapter describes the fundamental theory of the ambient-noise method based on the superposition of plane waves coming from all azimuths. The assumption is that the noise field consists of 2D uncorrelated surface waves from large distances. The theoretical explanations in this chapter are the basis of papers I to III. Other derivations of ambient-noise theory are not considered here. First, I explain the superposition of plane waves, and then its difference from the direct plane wave is described. After explaining the concepts, we can investigate the effects of non-uniform source distributions on the constructed cross correlations. Finally, the theoretical representation of the correlation tensor is demonstrated. 2.1 Cross correlation of plane waves Consider a geometry of two stations as presented in Fig. 2.1, where the energy comes from a distant source. The inter-station distance and the medium velocity (assumed constant) are denoted by r and c, respectively. Therefore, the difference in travel time to these two stations is Δt = r cosθ/c. Assuming that the sources are not correlated, the cross correlation of the signals at the two stations is simply the sum of the cross-correlation function for each relevant direction. Therefore, assuming a homogeneous source field and ignoring a scalar amplitude coefficient, the cross-correlation in the frequency domain is given by integration of all plane waves over angle 2π C(ω)= exp(i ωr cosθ) dθ, (2.1) 0 c where ω stands for angular frequency. The exact solution of this integral is simple and is equal to the zeroth-order Bessel function of the first kind 2π C(ω)= exp(i ωr 0 c cosθ) dθ = 2π J 0( ωr ). (2.2) c This is an important result because there is no further simplification or assumption needed to get eq. (2.2), and it shows that the cross correlation of uniformly distributed plane waves for a homogeneous medium is proportional to the imaginary part of the 2D Green s function (Sánchez-Sesma et al., 2006). 15

16 Figure 2.1. Geometry of problem for an incoming plane wave at an angle of θ from the inter-station axis. The distance between the stations is denoted by r and velocity is c. The path difference between station 1 and 2 from the source is r cosθ. The zeroth-order Bessel function in the high-frequency and/or far-field limit (ωr/c 1) can be approximated by J 0 ( ωr c ) 2c ( ωr = ωπr cos c π ) (2.3) 4 (Abramowitz & Stegun, 1970; Boschi et al., 2012; Boschi & Weemstra, 2015). Therefore, this equation can be used to evaluate cross correlations from uniformly distributed plane waves. The time-domain cross-correlation can be calculated by taking the inverse Fourier transformation of eq. (2.1) as C(t)= 1 2π 2π 0 exp(i ωr c cosθ) exp( iωt) dθdω. (2.4) This equation can be used to generate synthetic cross-correlations in the presence of uniformly distributed noise sources. Fig. 2.2 illustrates an example of the cross correlation of isotropic incoming energy of ambient noise to a station pair with inter-station distance of 300 km and a velocity of 3 km/s at a period of 10 s. The upper panel shows the cross-correlation functions of each individual plane wave coming from 0 o to 360 o. The lower panel represents the sum of all individual cross-correlations as a function of time lag. 2.2 Comparison to a direct plane wave The plane wave that is perfectly aligned with the inter-station axis (θ = 0 o ) has an inter-station travel time that corresponds to the inter-station velocity. 16

17 Figure 2.2. Illustration of cross-correlation construction from summation of 360 plane waves emerging from all angles. The top panel shows each individual crosscorrelation at the given angle versus time lag. Each horizontal line in this image represents the individual cross-correlation function for a localized source at the corresponding azimuthal angle. The red color represents positive values while the blue color shows negative values. Small values are shown as white. The lower panel shows the interference (summation) of all the above cross correlations. Note that cross correlations are pass-band filtered. This direct plane wave represents a distant source aligned with the two stations. Such a wave of unit amplitude can be represented as cos(ωr/c) when monochromatic. This can be compared to eq. (2.3). The obtained crosscorrelation from summation of incoming plane waves from all azimuths has a π/4 phase shift compared to the direct plane wave that travels along the interstation axis (θ = 0 o ) in the asymptotic limit. The origin of this phase shift is related to the difference between the cosine function in the frequency domain (cos, i.e., the direct plane wave) and the zeroth-order Bessel function (J 0, i.e., summation of plane waves from all azimuths) at high-frequency. This phase difference is a phase advance and arises because the interference of waves from all azimuths. 17

18 Fig. 2.3 shows the positive lag of the constructed cross-correlation from uniformly distributed, narrow-band plane waves and the corresponding direct plane wave that exactly passes along the two-station axis. The π/4 phase shift or T /8 time shift (where T is period) toward shorter time-lag can be seen clearly in this figure. This phase shift must be accounted for in any phasevelocity measurement methods from the noise-correlation traces. Figure 2.3. Illustration of time shift of the direct plane wave traveling along the interstation axis (θ = 0) and the obtained cross-correlation from superimposed plane waves from all angles (interferometry). 2.3 Effects of an uneven source distribution In the previous sections, I have only considered uniformly distributed plane waves and ignored different contributions of individual plane waves in the cross-correlation construction. The general form of eq. (2.1) can be written as 2π C(ω)= P(θ,ω) exp(i ωr cosθ) dθ, (2.5) 0 c and its time-domain representation (eq. 2.4) as C(t)= 1 2π 2π 0 P(θ,ω) exp(i ωr c cosθ) exp( iωt) dθdω, (2.6) where P(θ, ω) is the power spectrum of plane-wave components varying with azimuth, θ. Then, we can follow the derivation in paper I (Sadeghisorkhani et al., 2016) and expand eq. (2.6) around a center frequency (ω o ) with a perturbation (δω) and using a zero-phase boxcar filter to get Ĉ(ω o,t)= 2δω 2π π p(ω o) ε(θ) cos(ω o (τ p cosθ t)) 0 sinc(δω(τ g cosθ t)) dθ, (2.7) 18

19 where τ p = r/v p, τ g = r/v g. v p and v g stand for the phase and group velocities, and ε(θ) is the angular power density of noise-sources. We assume that the power spectra of all noise sources are identical (P(θ,ω) =p(ω)ε(θ)). Eq. (2.7) can be used for generating synthetic cross-correlations by defining parameters of the medium and source in a desired frequency range, or it can be used to invert for noise-source density (ε(θ)) when observed crosscorrelations, phase and group velocities are known. It has been used to generate synthetic cross-correlations to test the inversion scheme developed in paper I, and also simulations of various noise-distribution scenarios in paper II to investigate velocity bias due to the anisotropic source distributions. Similar simulations are used to estimate and correct phase-velocity bias in paper IV. Figure 2.4. Same as Fig. 2.2, except that here we illustrate the effect of a non-uniform distribution on the constructed cross-correlation. A Gaussian energy anomaly (5 o width) with an amplitude 5 times larger than the background is presented at azimuth of 15 o. In the lower panel, each trace has been normalized by its maximum. Fig. 2.4 shows an example of a non-uniform source distribution. All parameters are the same as in Fig. 2.2, but a Gaussian anomaly of energy centered at 15 o compared to the inter-station axis is included. The amplitude of the 19

20 anomaly is 5 times larger than the background energy and its width is 5 o. The effect of the Gaussian anomaly can be seen as an apparent time shift (relative to a uniform source distribution) of the constructed cross-correlation at positive time lag. According to eq. (2.7), changes of the energy (amplitude) of each individual plane-wave can be seen as a variation of their weights in the summation (integration). Constructive interference of off-axis plane waves with higher weights moves the waveform towards higher velocity and manifests itself as a negative time shift. Therefore, velocity bias is unavoidable in presence of non-uniform source distributions, which is the topic of paper II. 2.4 Other components of correlation Following Haney et al. (2012), the constructed cross-correlations from a uniform source distribution in the frequency domain can be extended to other components of the correlation tensor. We have seen that the cross-correlation of the ZZ component becomes J 0. It can also be defined for other combinations of the vertical (Z), radial (R) and transverse (T) components. R and T are defined along the inter-station axis and in its perpendicular direction, respectively. The full correlation tensor for both Rayleigh and Love waves in two dimensions is ZZ ZR ZT C = RZ RR RT (2.8) TZ TR TT ε R η 2 J 0 (ωτ R ) ε R ηj 1 ( ω τ R ) 0 ε R [J 0 (ωτ R ) J 2 (ωτ R )] ε R ηj 1 ( ω τ R ) 0 = +ε L [J 0 (ωτ L )+J 2 (ωτ L )], ε L [J 0 (ωτ L ) J 2 (ωτ L )] 0 0 +ε R [J 0 (ωτ R )+J 2 (ωτ R )] where τ R = r/c R and τ L = r/c L, and c R and c L are phase velocities of the Rayleigh and Love waves respectively, and J 1 and J 2 are the first and the second order Bessel functions, respectively. η is Rayleigh wave ellipticity defined as the ratio of the vertical component over the radial component (Z/R). ε R and ε L are uniform source intensities of Rayleigh and Love waves, respectively. Here, we assume that the two wave types are not correlated with each other. Neither the Love wave on the RR component, nor the Rayleigh wave on the TT component are zero. However, J 2 is asymptotically equal to J 0, and therefore, the Rayleigh waves are dominant on RR and Love waves dominant on TT at high frequency. This is presented in Fig The upper panel shows the first three orders of Bessel functions of the first kind, and the lower panel shows J 0 + J 2 and J 0 J 2. As is explained in paper II, many aspects of the constructed cross-correlation are related to the dimensionless frequency, 20

21 1 J 0 Amplitude J 1 J J 0 + J 2 J 0 - J 2 Amplitude r/λ Figure 2.5. Different order of the Bessel function (upper plot) and combinations of J 0 and J 2 (lower plot). defined as γ = r/λ, where λ is wavelength. For this reason, the horizontal axes of Fig. 2.5 are presented in terms of r/λ. It can be seen that J 0 + J 2 is small for inter-station distances larger than 1.5 of a wavelength. All the above derivations and discussions are based on the assumption of a homogeneous medium. In a heterogeneous medium, other effects such as multi-pathing and refraction are present. Analyses based on this simplified description is, nevertheless, useful. In regions of strongly anisotropic source distributions and a relatively low level of heterogeneity, such as is the case in Scandinavia, this simplification is likely to describe the dominant effect. 21

22 3. Methods of Phase-Velocity Measurements from Noise Correlations 3.1 Introduction Measurements of phase and group-velocity dispersion curves are the fundamental requirement of all surface-wave methods. Phase velocity is the speed of each harmonic, whereas group velocity is the speed of the wave packet which is created from the interference of harmonics that propagate together. Group-velocity measurement from ambient-noise correlations may be simpler than the phase velocity, but the group-velocity uncertainty is usually larger (Lin et al., 2008). Group velocity suffers a larger bias due to variations of the noise-source distribution (Tsai, 2009; Sadeghisorkhani et al., 2017). Also, the variation of the amplitude spectra of noise-correlation traces can affect the obtained group-velocity dispersion measurements. Therefore, it is generally more robust to measure the phase-velocity dispersion curve from noise correlations. Two classes of methods have been introduced to measure phase velocity. In the first class, which is usually applied to the time-domain cross-correlations, the phase velocity is measured based on the asymptotic formulation of the noise correlation (for example, eq. 2.3 for the ZZ component). In this class of methods, the two stations need to be in the high-frequency and/or the far-field limit that is usually defined as r 3λ (Yao et al., 2006). The second class of methods uses the formulation of the noise correlation explained in Section 2.4. Therefore, no high-frequency and/or far-field limitation is needed, and the measurement can theoretically be carried out above r/λ of 1 (Ekström et al., 2009). This class of methods are referred to as Spectral Dispersion. Paper III mostly deals with this class of methods. In this chapter, I describe some of the methods of phase-velocity measurements. I divide them into two categories: methods that can be used to measure phase velocity of single cross-correlations; and methods that can be applied to many cross correlations in a region to give the average phase-velocity dispersion curve. The average disperison curve can be used as a reference curve to solve the 2π ambiguity (periodicity) of the single-pair phase-velocity dispersions. The advantage of using the average dispersion curve rather than a synthetic or a reference model such as PREM (Dziewonski & Anderson, 1981) is that it is data driven. 22

23 3.2 Single-pair phase-velocity dispersion curves Image Transformation Technique Yao et al. (2006) developed a technique called the Image Transformation Technique to measure phase-velocity dispersion curves from noise-correlation traces. They use a far-field representation of the surface-wave Green s function, and compare it to the first time derivative of the noise-correlation traces (they call them empirical Green s function or EGF). The time harmonic wave of the Green s function of the surface-wave fundamental mode between station A and B at frequency ω in the far-field limit is given by R{G AB (ω) exp( iωt)} A cos ( rk AB ωt + π 4 ), (3.1) where k AB = ω/c is the average wavenumber, and A is an amplitude term. The maximum amplitude of eq. (3.1) occurs when the cosine argument is equal to zero. Therefore, the phase travel-time satisfies rω c ωt + π = 0, (3.2) 4 and the velocity can be calculated from r c(t )= t T /8, (3.3) where T = 2π/ω. To extract the phase velocity dispersion curve from the noise-correlation traces, first the EGF (the first time derivative of the noise-correlation traces) should be calculated. Second, EGFs are band-pass filtered narrowly. Then, a time-period (t - T) image is constructed. The (t - T) image is transformed to a velocity-period (c - T) image using eq. (3.3) and a spline interpolation. Then, the phase-velocity dispersion curve is picked from this image. Comparing eq. (3.1) and eq. (2.3) reveals that there is π/2 phase difference between them. The origin of this phase difference lies in the time derivative in the EGFs calculation. We then can modify the developed method by Yao et al. (2006), to not take the first derivative and instead use a modified form of eq. (3.3) as r c(t )= t + T /8. (3.4) This simplifies the process. To get this formula I only change the sign of π/4 in the argumant of the cosine function in eq. (3.1) Multiple Filter Analysis Multiple Filter Analysis is often used to measure phase-velocity dispersion curves from noise-correlation traces. It is a part of the Computer Programs 23

24 in Seismology software (Herrmann, 2013). It measures the phase-velocity dispersion by applying multiple Gaussian filters to the time-domain crosscorrelation traces, and comparing them to the Green s function between two stations at each frequency. The time domain representation of a dispersed surface wave can be written as f (t,r)= 1 A(ω, r) exp( ikr + φ) exp(iωt) dω, (3.5) 2π where r is distance, t is time, φ is source phase, and k is wavenumber. By applying a narrow-band Gaussian filter about a center frequency ω o, and after some algebra, the filtered signal is given by g(t,r)= 1 2π A(ω o) ω o π/α exp(i(ωo t k o + φ)) exp ( ) ω2 o 4α (t r/u o) 2, (3.6) where U o is the group velocity, and α is the filter width. The last term corresponds to the envelope of the signal and its maximum occurs at the group arrival time t g = r/u o. The phase at the group arrival time (t g ) will be Φ = tan 1 [I(t g,r)/r(t g,r)] = rω o /U o rω o /c + φ + N2π. (3.7) If the signal is related to the Green s function between two stations, the phase term φ must be π/4. Therefore, the phase velocity based on the multiple filter analysis can be calculated from rω o c = Φ + π/4 + rω o /U o + N2π. (3.8) Bessel function matching Following Aki (1957) s original work, Ekström et al. (2009) showed that the phase-velocity dispersion curve from noise-correlation traces can be obtained by matching the real part of the correlation spectrum as a function of frequency to the Bessel function. The matching is only applied at zero crossings, because the locations of the zero crossings are robust features and we do not need the amplitude information to obtain phase velocity. In contrast to the two previous methods that calculate phase velocity by comparing to an approximate function (cos), in this method the matching is to the exact Bessel function (J). At the zero crossings, the argument of the angular frequency of the nth observed zero crossing of the correlation spectrum (J(ω n r/c)=0) must equal the argument of the nth zero crossing of the Bessel function (J(z n )=0). Therefore, the corresponding phase velocity of ω n is obtained by c(ω n )= ω nr. (3.9) z n 24

25 All possible phase-velocity dispersion curves can be obtained by matching each ω n with the Bessel s function zero-crossings according to c m (ω n )= ω nr z n+2m, (3.10) where m takes the values 0, ±1, ±2,.... This provides all possible dispersion curves that correspond to various cycles of the surface wave. Finally, the results are limited to a range of realistic phase velocities. Implementing eq. (3.10) at all observed values of ω n gives several possible dispersion curves. The actual dispersion curve can be selected by comparison with a reference dispersion curve. As explained in Section 2.4, five elements of the correlation tensor are not zero (in the presence of a uniform sources distribution) and we can measure phase velocity based on all of them. Bessel function fitting should be done with the appropriate Bessel function in each case. J 0 is the appropriate Bessel function for the ZZ component, while J 1 is the corresponding Bessel function for the RZ and ZR components. J 0 J 2 is the appropriate Bessel function for the other diagonal components (RR and TT). Note that all of these derivations are based on assuming that the medium is homogeneous and sources are not correlated. Notice that the previous two methods presented in Sections and are only applicable to the diagonal components of the correlation tensor as they stand. To the best of my knowledge no literature considers applying these methods on the ZR and RZ components for the phase-velocity measurements. Since, J 0 and J 1 are out of phase and have a π/2 phase difference (see Fig. 2.5), the corresponding formula of the two methods must be corrected accordingly for these components (ZR and RZ). 3.3 Average phase-velocity dispersion curve of a region Slant stack The slant stack is a simple stacking method borrowed from reflection seismic processing. It can be used to estimate the average velocity of an area or a profile using an array of stations with an arbitrary geometry. An image is constructed in the period-velocity domain (T - c) based on a slant stack of filtered noise cross-correlation traces. First, the traces are narrow-band filtered. Then, the values of the traces are stacked over a range of velocities for a given frequency. The amplitude of the stack which corresponds to the correct velocity is enhanced, and the other velocities are suppressed as they do not stack coherently. Finally, phase-velocity dispersion can be picked by selecting amplitude peaks that align along the dispersion curve. The background of Fig. 3.1 is a slant-stack image of 55 station pairs in northern Sweden. 25

26 Figure 3.1. An example of estimated average phase velocity of a group of 55 station pairs in northern Sweden from slant stack (colored image) and Bessel Function Matching explained in Section (white curve) for the ZZ component. Note that to calculate travel times that correspond to different noise correlation traces, we need to account for the T /8 time shift for the diagonal components of the correlation tensor. Otherwise, the estimated average velocities are shifted according to the period. For the ZR and RZ components, an extra T /4 time shift must be included (see the last paragraph of Section 3.2.3) Bessel function matching as a function of distance Following the theory developed by Aki (1957), Prieto et al. (2009) suggested a method to invert for an average phase-velocity dispersion curve of a region using all available noise correlations. The real part of coherency (normalized cross correlation) as a function of frequency and distance is compared with the Bessel function. Here, we only discuss the ZZ correlation. For other components of the correlation tensor, the same procedure can be used by matching the suitable Bessel function. If the noise field is random and the noise sources are uniformly distributed, the real part of coherency (Γ) for the ZZ correlation is related to the zerothorder Bessel function J 0 as ( ) 2π fr R[Γ( f,r)] = J 0, (3.11) c( f ) 26

27 where f is the frequency, r is the inter-station distance, and c( f ) is the phase velocity. A residual is defined as ( ) 2π fr ε( f )=R[Γ( f,r)] J 0, (3.12) c( f ) (Prieto et al. (2009)). Residuals are calculated for all station pairs in a region and minimized by a grid search over possible phase velocity to estimate the average phase velocities for each frequency. The white curve in Fig. 3.1 shows the average phase velocity from 55 station pairs in northern Sweden using this method. 27

28 4. Ambient Noise Tomography 4.1 Velocity maps The term "ambient-noise tomography" is used to refer to the surface-wave tomography of travel-time information from noise cross correlations between different station pairs. The only difference of this method to other surfacewave tomography methods lies in the measurement of the dispersion curves (travel-time information) which is from the noise correlations. Phase-velocity measurements from the ambient-seismic noise are described in the previous chapter. In this section, I briefly explain how various station-pair velocities in a region are combined using travel-time tomography to obtain 2D velocity maps. The average velocities between each station pair at different periods are measured by the dispersion curve measurements. Our aim is to use the average travel times along paths connecting station-pairs through the medium to recover 2D velocity structures at each period. In order to do tomography we need to solve an inverse problem where the observed travel-time data (d) are related to the velocity structure (m) through a relation with a form like d = g(m). The data misfit is defined as some norm applied to the prediction error (d obs g(m 0 )), i.e. the difference between observed data and data predicted based on a reference model, which gives an indication of how well the observed data are explained by the model. In tomography, like other inverse problems, we look for a model that minimizes the data misfit subject to a regularization that usually must be imposed. Most problems in geophysics are mixed determined, meaning that the matrix containing coefficients relating data to model parameters is ill-conditioned, and its inverse becomes numerically unstable. We can deal with such ill-conditioned systems by applying regularization. The travel time along a ray is a path integral of slowness (one over velocity) along the ray path T t = 1/v(s) ds = u(s) ds, (4.1) ray ray where s is incremental length along the ray path connecting a source to a receiver through a medium with a varying velocity of v. The ray path itself is determined by the velocity structure so that eq. (4.1) is non linear. A non-linear inverse problem is usually linearized around a reference model or a starting model by a truncated Taylor expansion as δt t = δu(s) ds, (4.2) ray 0 28

29 where ray 0 is the reference ray path. This means that the ray path between station-pairs is essentially unchanged for a small perturbation of slowness (δu(s)) during an inversion step, but the travel time is modified (δt t ). Most inverse problems require four steps to be solved, 1- model parametrization, 2- forward calculation, 3- inverse calculation, and 4- estimation of model uncertainty and resolution. In the first step, the model is divided into a set of unknown model parameters. The simplest way to parametrize the model is to divide the spatial slowness perturbation into cells of constant slowness. In the second step, a forward relation is formulated to calculate data from the model parameters. In the third step, observed data are inverted for the model parameters that minimize misfit between predicted data from the second step and observed data. In the fourth step, the robustness of the obtained model parameters is evaluated. The linearized inversion for the slowness perturbations (eq. 4.2) can be solved in two ways. If we assume that the ray paths do not change significantly by changing slowness, the problem will be linear and no iteration is needed. The paths are defined once based on the reference model and will be straight/great-circle paths when the reference model contains no lateral heterogeneity. This may be appropriate for regions with weakly heterogeneous velocity structures. If the assumption of straight/great-circle paths is not warranted, the problem can be solved iteratively, meaning that after inverting for the slowness, the ray paths are corrected accordingly. This procedure is repeated until for example there is no more improvement in the data fitting. Forward calculation is a critical part of an inverse problem. There are different ways to calculate travel times of the ray paths between station pairs. Ray tracing or wavefront tracking are usually used for this purpose. The basis of these methods is the eikonal equation which is a high frequency assumption T t = 1 v(x), (4.3) where T t is the wavefront travel-time and x is the position vector. Another form of the eikonal equation, which describes ray paths, is called the raypath equation (Lay & Wallace, 1995) d ds ( 1 v(x) dx ds ) ( ) 1 =. (4.4) v(x) For further details please refer to the review paper by Rawlinson & Sambridge (2003). For the inversion step usually gradient based methods are used. A cost function is defined and minimized through the inverse process. Our problems in geophysics and in particular travel-time tomography are mixed-determined, and therefore we need to regularize the inverse problem in order to decrease ill-conditionality of the matrix which we want to invert. Therefore, the cost 29

30 function is a combination of data misfit and different regularizations. The regularization can minimize the solution length, the solution structure or a combination of both. An additional common regularization strategy is to apply a prior model covariance to constrain the solution. The minimum length solution encourages it to be close to the reference model, while the minimum structure sloution requires it to have a minimum level of structural variations, i.e. to be smooth. The a priori model covariance prescribes soft constraints to the model in terms of its amplitude variation and scale. The weights of the different components of the cost function are specified based on a trade off between data misfit and the regularization components, i.e. damping (minimum length), smoothing (minimum structure), or model amplitude (amplitude of model covariance). Having defined these factors the inverse problem can be solved. Two approaches are often considered for the quality control of the final model which are based on the propagation of errors through the inversion process, 1- calculation of a covariance matrix and a resolution matrix based on the linearization of the final iteration of the inversion, and 2- use of synthetic tests. Indeed, the first approach is rather simplistic and when the non-linearity of a problem increases it becomes less meaningful. In the second approach different synthetic tests, for example checkerboard tests or spike tests can be used to explore resolution limitations, and model uncertainty can be studied by multiple realizations propagating random data errors through the non-linear inversion process. In paper IV, to invert for the 2D phase velocity maps for each period, we used the method of Fast Marching Surface Tomography (FMST) (Rawlinson & Sambridge, 2005). It uses the Fast Marching Method (FMM), which applies wavefront tracking for the forward calculation. The inversion process has been done iteratively, and we have found the minimum length solution for each period. The damping factors have been chosen from the knee of trade-off curves between data misfit and model roughness. Finally, we have tested the obtained 2D phase velocity maps by synthetic tests. 4.2 Depth inversion After obtaining the 2D velocity maps, a set of geographical coordinates can be selected for the depth inversion at the nodes of the velocity maps. Results of the phase-velocity maps can be gathered into local estimates of dispersion curves at each of these coordinates. Here, I briefly explain the concept of a variational principle for Rayleigh waves which is used for the depth inversion and to obtain depth sensitivity kernels. I follow Keilis-Borok et al. (1989), Herrmann (2013), Aki & Richards (2009), and Haney & Tsai (2015) in this section. 30

31 A solution to the eigenvalue problem of the Rayleigh wave should satisfy where I 0, I 1, I 2, and I 3 are I 3 = ω 2 I 0 k 2 I 1 2kI 2 I 3 = 0, (4.5) I 0 = 0 ρ(r r 2 2)dz, (4.6) I 1 = 0 [(λ + 2μ)r1 2 + μr2]dz, 2 (4.7) I 2 = 0 [ 0 (λ + 2μ) ( μr 2 dr 1 dz λr 1 ( ) 2 dr2 + μ dz ) dr 2 dz, (4.8) dz ( ) ] 2 dr1 dz, (4.9) dz in which ρ, λ, and μ are density, Lame s parameter, and shear modulus, respectively. r 1 and r 2 stand for the horizontal and vertical Rayleigh-wave displacement eigenfunctions. Eq. (4.5) leads to a general form of the eigenvalue equation for a given frequency in terms of k and eigenfunctions r 1 and r 2. To invert for depth based on eq. (4.5), the linearized relation between perturbation in phase velocity and perturbations in the material properties ρ, λ, and μ is found. The inversion is non-linear and can be solved iteratively from a starting model. The medium is discretized into a set of constant velocity and density layers with thickness of d m. The partial derivatives of the phase velocity with respect to ρ, P-wave velocity (α), and shear-wave velocity (β) in the layer m are calculated from ( ) c = α m ( ) c = β m ( ) αm ρ zm m UI 0 ( ) βm ρ zm m UI 0 z m d m z m d m [ ( r k [ r 1 1 k ] dr 2 2 dz, (4.10) dz ] ) dr dz k r 1 dr 2 dz dz, (4.11) ( ) c = 1 [ ( ) ( ) ] c c α + β c2 zm [ r 2 ρ m 2ρ α m β m 2UI 1 + r 2 ] 2 dz.(4.12) 0 z m d m The group velocity U in the above relations is determined from U = dω dk =(ki 1 + I 2 )/ωi 0. (4.13) In each step of the iterative inversion, the difference between the observed and predicted dispersion curves is calculated. The process is continued until it converges to an elastic model (shear and compressional velocities, and density) which satisfies data (the local phase-velocity dispersion curve). 31

32 5. Summary of Papers Below I present a comprehensive summary of four papers about the ambientnoise method. Each paper is summarized to provide a brief explanation of the main idea and goal, the method developments and usage, and the main results and conclusions. My contributions to each paper are as follow: Paper I: My personal contributions to this paper were the implementation and coding of the forward and inverse calculation, processing and inversion of the data, estimating the average group and phase velocities among different groups of stations, preparation of the figures, and the writing the first draft of paper. Later, the text of the paper has been improved by my supervisors. The theoretical derivation has been done by my main supervisor. Also, the original idea was from him. However, the idea of the second inversion to estimate the spatial distribution of noise source was proposed and formulated by me. Paper II: I wrote the code to do simulations, and I preformed the numerical simulations. All the figures were prepared by me, and the paper was initially written by me. Later, Sections 2 and 3 were rewritten by my main supervisor. Other parts were improved with help of my supervisors. Paper III: My personal contribution to this paper was from the idea to the implementation and coding of all parts of the computer package. I also did the data processing, the figure preparation, and wrote the paper and the manual and tutorial of the computer package. The text of the paper and the two other documents were improved by my main supervisor. Paper IV: I can estimate my contribution to this paper to be around 75%. I measured more than half of the phase-velocity curves, and most parts of the other processing and tomography. I have prepared figures and written most of the text with the help of my main supervisor. 32

33 5.1 Paper I: Mapping the source distribution of microseisms using noise covariogram envelopes Previous studies of ambient noise in Scandinavia showed strong directivity of noise that points dominantly to the northeast Atlantic (Pedersen & Krüger, 2007; Köhler et al., 2011). The velocity bias due to this anisotropic source distribution may not be negligible in this relatively homogeneous area. Therefore, studying the source distribution around Scandinavia and its consequent effects on the velocity measurements is necessary. In this paper, a method to map noise sources based on an inversion of covariogram (cross-correlation) envelopes is introduced, tested by synthetic data and applied to a dataset from the SNSN to study the noise field around Scandinavia. The method is accomplished in a two-step inversion; first, an inversion for the azimuthal source distribution is applied in narrow frequency bands at a geographical group of stations, and second, combining the results from the previous step in a number of groups, another inversion is used to estimate the spatial source distribution. The first inversion is applied to time-domain covariogram envelopes, and the inversion is nonlinear. It can be solved by a linearized, damped leastsquares technique. The azimuthal inversion uses the total length of the covariograms (not only around their theoretical singularities). Therefore, we need relatively few stations to find the directions of incoming energy. Hence, we can divide the available stations in the SNSN into different groups and by combining the directions from these groups, we can constrain the source locations in different period ranges Summary Eq. (2.7) is the time-domain representation of narrow-band filtered covariograms assuming 2D plane waves. We can use this expression to invert for the azimuthal source distribution (ε(θ)). We choose to use the envelopes of covariograms since they are more stable and are related to the incoming energy to the stations. The squared envelope of the covariogram specified in eq (2.7) is e 2 (ω o,t)= δω 2 4 π 2 p2 (ω o ) 2π 2π 0 0 ε(θ) ε(φ) sinc(δω(τ g cosθ t)) sinc(δω(τ g cosφ t)) cos(ω o τ p (cosθ cosφ)) dθ dφ. (5.1) The forward problem as described in eq. (5.1) is quadratic in the density of noise-sources (ε(θ)). The inverse can be solved by an iterative, gradient-based scheme using the non-negative, damped least-squares technique. The data for the inverse problem are the squared envelopes of covariograms, and the model parameter is the source density (ε(θ)) as a function of azimuth. Since we assume incoming plane waves, there is a simple mapping between time lag 33

34 and the angle of the source from the inter-station direction (t(θ) r cos θ/c). Thus, for sources located off the inter-station axis (θ > 0) the peak of the covariogram will be closer to zero lag than when θ = 0. We use this behavior to relate the amplitudes of covariogram envelopes to the source strength at different azimuths. This is useful to constrain the source distribution when strong localized sources which produce early arrivals are present. All values of the covariograms between [ τ g,τ g ] are used. The sensitivity matrix (G) of the iterative linearized inversion at the k-th iteration can be derived from the eq. (5.1) as ( g m ) k(φ,t)= δω 2 8 2π π 2 p2 (ω o ) ε k 1 (θ) sinc(δω(τ g cosθ t)) 0 sinc(δω(τ g cosφ t)) cos(ω o τ p (cosθ cosφ)) dθ. (5.2) for each station pair and a specified φ. To implement, we first discretize the model. We choose to have the azimuthal energy density (ε(θ)) at every degree (dθ = 1 o ). Therefore, we select 181 time samples of each covariogram in the interval τ g τ τ g (according to t(θ) r cosθ/c). The total number of data is then N = 181 N pairs, where N pairs is the number of covariograms, and the number of model parameters is M = 360. Second, we solve eq. (5.2) for time lags between [ τ g,τ g ]. Then we resample at the 181 points corresponding to each degree to fill elements of the sensitivity matrix. An iterative inversion needs a starting model. The starting model is developed based on the comparison of the covariogram envelopes of all station pairs with a uniform distribution of noise sources. This data-driven approach estimates the main features of the model. To terminate iterations, differences between the root-mean-square (rms) of the model parameters of the current and previous iteration should be less than one percent. This criterion verifies the model convergence. After obtaining the azimuthal energy distribution at multiple groups of stations, we can combine the results to set up an inversion for the spatial distribution of the noise sources. By assuming 2D propagation of energy and a ray theoretical approach, we can relate the azimuthal energy density (ε(θ)) tothe spatial energy-source density (E(x,y))as E(x, y) ε(θ)= dr (5.3) path(θ) r where r is distance between the center of each spatial grid and the center of each group of stations, and path(θ) is a great circle path along any given azimuth. First, the surrounding region is discretized and then eq. (5.3) is solved with a constraint of E(x,y)=0 in the land areas and E(x,y) 0 at sea. 34

35 Figure 5.1. Envelope of filtered covariograms (ZZ component) between 2 and 25 s within all groups up to 220 km station separation. Figure 5.2. Map of the 54 SNSN stations that have enough length of data after preprocessing in They are divided into five groups, which are numbered as indicated. 35

36 70 N (5 6) sec 65 N 60 N 55 N 5 E 10 E 15 E 20 E 25 E Figure 5.3. Rose-digram representation of the azimuthal source distribution results in the period range of 5-6 s. Amplitudes have been normalized by the maximum of all groups, and therefore, comparable for all groups, which is (( nm s )2. s.day deg ). Note that in eq. (5.3), we ignore a number of wave-propagation phenomena which can affect amplitude (e.g., attenuation, focusing/defocussing due to lateral heterogeneity, scattering and modal conversion) and path geometry (e.g., lateral refraction). It only accounts for geometrical spreading of the signals in a simplistic way. For the pre-processing of the data, we use a modified form of the standard preprocessing of ambient-noise tomography suggested by Bensen et al. (2007) in order to retain physical amplitude of the covariograms. No temporal and/or spectral normalization is applied, instead the transients from earthquakes are clipped. This processing makes the absolute amplitude of covariograms comparable among all station pairs. Fig. 5.1 shows the envelope of covariograms within all groups up to 220 km distance filtered between 2-25 s. After preprocessing, 54 stations are used for the purpose of studying the sources. We divide them into five groups of between 9 and 12 stations (see Fig. 5.2). The covariograms are bandpass filtered into eight period ranges between 2 and 25 s in each group. Therefore, 40 independent inversions have to be solved. Figs 5.3 and 5.4 show the inversion results as rose diagrams at 36

37 70 N (16 20) sec 65 N 60 N 55 N 5 E 10 E 15 E 20 E 25 E Figure 5.4. Same as Fig. 5.3 but for the period range of s. The maximum amplitude for this period range is (( nm s )2. s.day deg ). the center of each of the five groups in the period ranges of 5-6 and s, respectively. All rose diagrams are normalized by the maximum value of all groups. Therefore, they are comparable in absolute terms. For the spatial source distribution, the region is discretized with a 0.5 o 1 o degree grid and we use the great circle paths to define length and direction. We find the smoothest, least-squares solution. Fig. 5.5 demonstrates the spatial source distribution in the period range of 5-6 s Discussions and conclusions A method to map the noise-source distribution is described, and its application to SNSN data is presented. It is an inversion-based method that is applied to the envelopes of filtered covariograms. Also, it is applied to the trueamplitude covariograms and yields therefore true amplitude of noise-source density. However, it can be applied to one-bit normalized covariograms (crosscorrelation). With the method that is described here and having global groups 37

38 Figure 5.5. Result of the spatial source distribution at the period range of 5-6 s. of stations, global maps of the frequency-dependent source distributions can be estimated. The frequency-dependence of the inversion results demonstrates that at the shorter periods the azimuthal source distribution is strongly directional, while at longer periods it is more diffuse. There is a strong dominant location of sources around 65 N and 8 E within the secondary microseismic band. In the primary microseismic band, the energy mainly comes from the NE. The spatial source distribution in the secondary microseismic band suggests that a significant part of the energy originates from shallow waters. 38

39 5.2 Paper II: Velocity-measurement bias of the ambient noise method due to source directivity: A case study for the Swedish National Seismic Network An anisotropic distribution of noise sources causes velocity bias, which can be important, especially when the level of heterogeneity of Earth is low. It has been discussed that the velocity bias has a relation with inter-station distance and frequency (e.g., Yao & van der Hilst, 2009; Froment et al., 2010; Kästle et al., 2016), but this was not fully investigated. In addition, Weaver et al. (2009) derives a formula to describe bias which is partly based on an assumption of smoothness, which is, however, not quantified. In this paper, first we study the behavior of velocity bias in the presence of a local anomaly of the source distribution using synthetic simulations. Second, we use the azimuthal distributions of noise sources presented in paper I to estimate the expected phase and group velocity bias in one of the group of the SNSN stations Summary Fig. 2.4 illustrates why a non-uniform source distribution causes a velocity bias. In this paper, we use eq. (2.7) to generate synthetic cross correlations for different scenarios of anisotropic source distributions, and compare them to the same cross correlations in presence of uniform source distributions. With this approach, we can estimate the amount of bias that is introduced by the anisotropic source distributions. Eq. (2.6) shows that the phase of a constructed cross correlation is affected by ωr/c which is proportional to γ = r/λ, where λ is wavelength. Therefore, all aspects of velocity bias depend on this parameter (γ), which we call it dimensionless frequency. In addition, if we rearrange the formula derived by Weaver et al. (2009) δv p = 1 ε (θ) v p 2τpω 2 2 ε(θ) = 1 ε (θ) 8π 2 γ 2 ε(θ), (5.4) we can see that the phase-velocity bias scales as γ 2. In this formula, τ p is the phase time, ε(θ) is the source-density function, and δv p /v p represents the relative phase-velocity bias. This formula is valid when the anisotropic source distribution is smooth (Weaver et al., 2009). In this asymptotic limit, the angular span of the phase-velocity bias is controlled by the normalized second derivative of the source distribution. Fig. 5.6 shows two examples of phase- and group-velocity bias in the presence of Gaussian anomalies with varying amplitudes as a function of azimuth and dimensionless frequency. Consider the phase-velocity bias (middle panel). We can see that the bias has a shape similar to the second derivative of the anomaly, but with opposite sign at large values of dimensionless frequency 39

40 Figure 5.6. Phase and group velocity bias in presence of Gaussian anomalies of energy as a function of azimuth and dimensionless frequency (γ). Top panel shows two 5 o wide Gaussian source anomalies with different amplitude on top of a uniform background (positive and negative anomalies), the middle and bottom panels show the corresponding phase and group velocity bias, respectively. (γ > 10), in agreement with Weaver et al. (2009) asymptotic prediction. As γ becomes smaller, more side lobes appear, become larger in amplitude and influence wider azimuths. This suggests two regimes of bias, namely a high and a finite-frequency regime. We show in this paper that the group-velocity bias can be derived from the phase-velocity bias as δv g = δv p + γ dδv p dγ. (5.5) Since, there is an extra term in the group-velocity bias, it is potentially larger (when both terms have the same sign) and has more structure. This restricts group velocity measurements to station pairs with large inter-station distances or to regions with more uniform source distributions more so than phase velocity measurements. 40

41 Figure 5.7. Phase-velocity bias in presence of Gaussian anomalies with various width. The measured bias (black line) and calculated bias using Weaver s formula (dashed white line) are shown at widths of 5, 15 and 25 degrees. The red values on the right side of each frame indicate the normalization percentiles of the curves. To evaluate the behavior of the high and finite frequency regimes, we calculate the phase-velocity bias as a function of azimuth and anomaly width for a fixed anomaly amplitude at three dimensionless frequencies (Fig. 5.7). We also compare our phase bias estimation to the prediction of Weaver et al. (2009) for three different anomaly widths in this figure. It is clear that Weaver s formula only holds in the high-frequency regime where either inter-station distance is much larger than a wavelength or the anomaly is smooth. In the finite-frequency regime, Weaver s formula predictions have larger amplitudes but smaller azimuthal span. The azimuthal pattern of the bias anomalies is controlled by the Fresnel angles in the finite-frequency regime. We can understand that when the energy-anomaly width is larger than the first Fresnel angle, we are in the high-frequency regime. 41

42 Figure 5.8. Scaling relations of the finite-frequency bias anomaly as functions of dimensionless frequency. a) absolute bias amplitude of the maximum (red dots) and minimum (blue dots); b) azimuth of the maximum (red dots) and minimum bias (blue dots); c) anomaly width where the maximum (red dots) and minimum (blue dots) bias occurs. In order to examine the scaling and behavior of the finite-frequency regime, in Fig. 5.8 we plot amplitude, azimuth and width of the bias anomalies (maximum and minimum of the phase-velocity bias) as a function of dimensionless frequency. In contrast to the high-frequency regime where the bias scales as γ 2, the amplitude of bias anomalies in the finite-frequency regime scales as γ 1. The angular range of the bias (azimuth and width of the maximum bias) in the finite-frequency regime scales as γ 1/2. We use the azimuthal source distribution results in paper I, to estimate velocity bias within the group of 11 stations in northern Sweden (group 1). We use the amplitude and azimuth uncertainty of the final model to calculate 1000 realizations of the uncertain source distribution model (shown as colored curves in the top row of Fig. 5.9). 42

43 Figure 5.9. The azimuthal source distribution (black curve) and a thousand random realizations of its amplitude and azimuth uncertainties (colored curves). Fig shows the mean values and standard deviations of the velocity bias at 11 stations (55 station pairs) where the positive and negative time lags have been averaged. Fig represents the mean values of the phase-velocity bias as a function of dimensionless frequency. If we want the phase-velocity bias to be within a specific range (e.g. ±1%), the simplest thing to do is to limit our measurements to those station-pairs which have dimensionless frequency higher than 5 (γ > 5) at 5-6 s and higher than 3 (γ > 3) at s Discussions and conclusions In this paper, the velocity bias due to a directional ambient-noise field and its distance-wavelength dependencies are studied. Two regimes of velocity bias are identified. It is shown that in the finite-frequency regime, the phasevelocity bias scales as γ 1 whereas in the high-frequency regime it scales as γ 2. The phase-velocity bias is potentially smaller than the group-velocity bias which displays more structures. The bias is measured for a group of stations to reveal the amount of expected bias within SNSN in two period ranges. The results show that the phase velocity can suffer a maximum of around 2% bias for most station pairs. For the group velocity it can be around 5%. Therefore, we decided to not measure group velocity in this fairly homogeneous crust. The velocity-bias strongly depends on γ. For instance, although the source distribution at s is smoother than at 5-6 s, the bias is larger because the wavelength is larger and, thus, dimensionless frequency is higher. One may choose not to measure station pairs with γ smaller than a certain threshold or identify station pairs that have a high level of potential bias due to their orientations and discard them. Another possibility is to correct for the source bias in the dispersion measurements. 43

44 Figure Predicted phase-velocity bias (top) and group-velocity bias (bottom) of 55 station pairs (11 stations) in northern Sweden in two period ranges based (left: 5-6 s, and right: s) on the source-distribution model in Fig The positive and negative lags of each covariogram have been averaged before bias measurements. Black dots correspond to pairs where the velocity-bias measurement can not be carried out. Figure Mean values of the phase velocity bias presented in Fig as a function of dimensionless frequency (γ = r/λ) at 5-6 s (left), and s (right). The dashed lines simply denote ±1 bias for reference. 44

45 5.3 Paper III: GSpecDisp: a Matlab GUI package for phase-velocity dispersion measurements from ambientnoise correlations Some methods of the phase-velocity dispersion measurements are described in Chapter 3. In this paper, we present a graphical program package for measurements from ambient-noise correlation traces. The package is called GSpecDisp and provides an interactive environment for the measurement based on spectral-dispersion methods (comparison to Bessel functions). Because it avoids an asymptotic approximation of the cross correlation, there is no need for a high frequency and/or a far-field condition. Therefore, dispersion can be measured up to the period which corresponds to r/λ of 1. Also, phase-velocity dispersion can be measured from all possible components of the correlation tensor. The software is designed such that its inputs are time-domain crosscorrelations. Therefore, we can apply a velocity filter before measurements. This reduces the noise in the calculated dispersions and makes a robust algorithm. The package has five independent modules for the velocity dispersion measurements and presentation of results. Since I explained the measurement methods in Chapter 3, in the following summary I only show some examples of the application of the package to real data from the SNSN. In the paper, we validate the developed algorithms by applying them to synthetic and real data, and compare with other measurement methods Summary We have seen in Chapters 2 and 3 that the formulation of the cross correlation of random fields in the frequency domain leads to Bessel functions. Therefore, phase-velocity measurements using this formulation are superior to formulations applying asymptotic expansions of the Bessel function(s). The package uses the Bessel function matchings explained in Sections and to measure the average phase velocity of a region as well as single-pair phase velocities. The average phase-velocity dispersion curve is used as a reference curve for the selection of the correct dispersion curves for single-pair crosscorrelations. Fig demonstrates an example of the method explained in Section to measure the average phase velocity in southern Sweden. The left subplot illustrates the Bessel function matching as a function of inter-station distance at a period of 33 s, and the right subplot shows a comparison of the estimated average dispersion curve to a synthetic dispersion curve for the SNSN. The average phase velocity of the region obtained from this method shows very good agreement with a synthetic dispersion curve calculated from one dimensional velocity model of the SNSN. 45

46 Figure Left: amplitude of the real part of correlation spectra (blue dots) normalized by their maximum as a function of distance at a period of 33 s in southern Sweden (630 station pairs), and fitted Bessel function (red curve). Right: estimated average velocity and a synthetic dispersion curve for Sweden. Figure Comparison of GSpecDisp (white circles) with Image Transformation Technique (background) and Multiple Filter Analysis (white crosses) at a station pair with the inter-station distance of 198 km. Fig shows an example of a single-pair phase velocity measurement with the method explained in Section and its comparison to the Image Transformation Technique (Section 3.2.1) and the Multiple Filter Analysis (Section 3.2.2). The inter-station distance of this station pair is approximately 198 km. The results match quite well, especially at the shorter periods where the two other methods meet their asymptotic requirement (distance > 3 wavelengths). Notice the variations of the dispersion curves around 10 s, where all the three methods show similar erratic behavior. 46

47 Figure Estimated average phase-velocity in northern Sweden (solid curves), and measured phase velocity dispersion curves from the TT (red dots), ZZ (magenta stars), RR (red squares), RZ (blue crosses) and ZR (black diamonds) components. Fig shows the measured phase-velocity dispersion curves from the 5 usable components of the correlation tensor for the same station pair that is presented in Fig and their comparison to the estimated average velocity dispersion curves of Rayleigh and Love waves in northern Sweden. The good agreement between the ZZ, ZR, RZ, and RR component measurements and also the agreement with the average Rayleigh and Love dispersion curves demonstrates the package s ability to measure phase-velocity dispersion curves precisely. Fig shows the measured phase velocity at a period of 8 s for the ZZ (Rayleigh) and the TT (Love) components of the correlation tensor for all station pairs in northern Sweden on a regional map. This figure is produced using the software s Map Viewer module Conclusions An interactive environment to measure, select and visualize phase-velocity dispersion curves is presented that uses non-asymptotic formulation of the correlation tensor for the measurements. It handles all processing steps from measurement to presentation of the results. The package can be used to measure phase velocity from five components of the correlation tensor. No other available package can do this to the best of my knowledge. Measurements results are consistent with the results of other methods, demonstrating the robustness and accuracy of the developed algorithms for phase-velocity dispersion measurements in this package. 47

48 Figure The measured phase velocities of ZZ and TT components at period of 8 s in northern Sweden. Notice, higher velocity for the Love waves (TT) rather than Rayleigh waves (ZZ). 48

49 5.4 Paper IV: Surface wave tomography of southern Sweden from ambient seismic noise As a continuation of my previous work, the ambient seismic noise is used to perform 2D tomography of Rayleigh waves in southern Sweden at different periods. The motivation for this work is that there is no previous study based on ambient noise in the area. We used 36 stations of the SNSN (630 station pairs) to do tomography for periods between 3 and 30 s. The reason that we only include the southern stations of the SNSN is that there is not enough path coverage in the central part of Sweden due to the narrow distribution of stations. They are mainly deployed near to eastern coast of Sweden. The second objective of this paper is to investigate the velocity bias due to inhomogeneous source distributions in tomographic inversions at different periods. In order to do this, the differences between the bias corrected and uncorrected tomographic maps are presented. As discussed in paper II, we only work on the phase velocity of Rayleigh waves, and do not use group velocity Summary Continuous recoding of vertical components from the SNSN in southern Sweden over one year (2012) are used for surface-wave tomography using ambient seismic noise. The daily traces containing earthquakes larger than magnitude 7 are rejected. First, the mean, trend, and instrument response are removed, and traces are pre-filtered between 0.01 Hz and 1 Hz. Then, they are decimated to a 2 Hz sampling frequency, and one-bit normalized. The traces are cross correlated and stacked for the entire one-year period. GSpecDisp (explained in paper III) is used to measure phase-velocity dispersion curves of the 630 station pairs. The software automatically selects the dispersion points. They are selected up to the period that corresponds to the inter-station distance being about 1.2 times the wavelength. Then, the selections are refined manually. Phase velocities at specified periods are obtained by interpolation between the selected dispersion points. Those interpolated phase velocities are used for tomography, if the two closest points to each of the specified period are selected. To estimate the phase-velocity bias and to correct, first the method (inversion) that is presented in paper I is used to estimate the azimuthal source distributions in groups 3, 4, and 5 in different period ranges (see Fig. 5.2). Source distributions are averaged to provide an estimate of the incoming energy to all of the stations. For the period range of s, the inversion is performed for all of the station pairs. Second, the approach that is explained in paper II is used to estimate phase-velocity bias. Finally, the phase-velocity measurements are corrected for the estimated bias. The bias correction is performed for all station pairs and periods. 49

50 Figure The estimated phase-velocity maps of Rayleigh waves at 3, 5, 8, and 12 s. The color bars in all panels are normalized to the same maximum velocity variation (±4.5 %). FMST is used to invert for 2D phase-velocity maps (see Section 4.1). First, phase velocities of all station pairs at each period are translated to travel times by dividing their great-circle distances to their estimated phase velocities. Second, the region is discretized with a nodal spacing of 0.25 degrees in latitude and 0.5 degrees in longitude for all of our inversions. A homogeneous velocity model, which is defined from the mean phase velocity of all station pairs, is chosen to be the starting model at each period. We regularize the inversion by damping and the damping parameter is chosen based on trade off between data misfit and model perturbation. Figs 5.16 and 5.17 show the phase-velocity maps at eight different periods. The phase-velocity variations from the mean is around 4 to 4.5 % for all periods. To investigate the effects of the bias correction on the tomographic inversion, we invert for another set of 2D phase-velocity maps with uncorrected 50

51 Figure Same as Fig but for periods of 16, 20, 24, and 30 s. The color bars in all panels are normalized to the same maximum velocity variation (±4 %). phase velocities. All other parameters are kept the same. Then, we calculate the difference between the corrected and uncorrected models. Fig shows the differential maps for the 4 longer periods ( 16 s). The overall difference is not big (usually within 1%) which suggests that the phase-velocity bias does not affect the tomographic models much. This is similar to the conclusion of Yao & van der Hilst (2009). This can be explained in terms of the averaging of the tomographic inversions. More crossing paths near a tomographic node lead to bias cancellations. To estimate the depth range corresponding to the different periods and to guide interpretation of the phase-velocity maps, we computed the sensitivity of phase velocity to shear-wave velocity based on a one-dimensional velocity model for Sweden. We use Computer Programs in Seismology software (Herrmann, 2013) for this purpose. Fig shows the depth sensitivity kernels of Rayleigh waves at different periods. 51

52 Figure Relative difference between the bias corrected and uncorrected tomographic models. Blue colors (positive values) indicate increased phase velocity after the bias correction Discussions and conclusions Ambient seismic noise is used for tomography of fundamental-mode Rayleigh waves in the south of Sweden at periods between 3 and 30 s. The depth penetration of these periods is approximately from 2 to 60 km. Longer periods see deeper. In general, the velocity variations are within ±4.5% of the average phase velocity at each period. The pattern and amplitude of the velocity variations agree well with the results of Köhler et al. (2015) where the results overlap. At the shortest period, two main high-velocity features are clear. One is approximately at (60.5 o,17 o ) in latitude and longitude, respectively, and another around (57 o,16 o ). These features are also present at 5 s. Another prominent high-velocity region is located between Gotland and the mainland, also at a period of 5 s. These high-velocity zones can be associated with Bergslagen, the southern tip of the Transscandinavian Igneous Belt (TIB), and Rapakivi 52