The Pennsylvania State University. The Graduate School. College of Engineering CRUST AND UPPER MANTLE STRUCTURE OF WEST ANTARCTICA

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering CRUST AND UPPER MANTLE STRUCTURE OF WEST ANTARCTICA FROM A SPARSE SEISMIC ARRAY A Thesis in Acoustics by Tongjun Cho 006 Tongjun Cho Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 006

2 The thesis of Tongjun Cho was reviewed and approved* by the following: Sridhar Anandakrishnan Associate Professor of Geosciences Thesis Adviser Chair of Committee Charles J. Ammon Associate Professor of Geosciences Joseph L. Rose Paul Morrow Professor of Engineering Science and Mechanics Bernhard R. Tittmann Schell Professor of Engineering Science and Mechanics Anthony A. Atchley Professor of Acoustics Head of the Graduate Program in Acoustics *Signatures are on file in the Graduate School.

3 ABSTRACT The West Antarctic Rift System is poorly understood relative to the other major rift systems of the world because of the remoteness of Antarctica and because the West Antarctic Ice Sheet covers most of the rift. In this paper, the crust and upper mantle structures of the interior of the West Antarctic Rift System were investigated using broadband seismic data from five widely spaced stations installed on the West Antarctic Ice Sheet. The crust and upper mantle seismic wave velocity structures were modeled using receiver functions and the inter-station surface wave dispersion curves. A new surface waveform extraction technique involving ambient seismic noise field correlation was applied for refining the surface wave dispersion measurements. A method for removing the ice layer reverberation effects on the receiver functions was designed and applied to retrieve the receiver function for crust and mantle-only structure. The mantle anisotropy of the rift region was measured by shear wave splitting analysis. The crust and uppermost shear wave velocity structure modeled in this study show that the crust is relatively thin, which we hypothesize is due to extension of the West Antarctic Rift System. In support of this hypothesis, I show that the measured anisotropy indicates that the fast axis direction is rift orthogonal. I model an approx. 100km thick anisotropic layer in the upper mantle, which may represent preservation of fabric from the major Late Cretaceous extension between East and West Antarctica. iii

4 TABLE OF CONTENTS List of Figures.... vi List of Tables.... xi Acknowledgements... xii Chapter 1 INTRODUCTION Overview Seismic Waves and Earth Structures.. 4 Chapter RECEIVER FUNCTION ANALYSIS Introduction to Receiver Function Analysis A Method of Removing the Ice layer Reverberation Effect on the Receiver Functions Application of the Deconvolution Method to Removing Ice Layer Effects on the Raw Field Receiver Functions Inversion of the Ice Layer-Removed Receiver Functions Chapter 3 SURFACE WAVE DISPERSION AND THE SHALLOW VELOCITY STRUCTURE OF THE WAIS REGION Dispersion of Surface Waves by Earthquakes Generation of Surface Waves Surface Wave Dispersion Relation for the Vertically Heterogeneous Medium Two-Station Group Velocity Dispersion Multiple Filter Analysis iv

5 3.1.5 Phase Match Filtering Data and Results of the Dispersion Measurements Dispersion measured in the Seismic Noise Field Ambient Seismic Noise Correlation between two Receivers Mathematical Review Correlation Function and its Time Derivative Application to the Field Data Inversion of the Dispersion Data for the Velocity Structure Linearized Inversion (surface wave and receiver function) Parameter Space Inversion Algorithm Estimation of the Velocity Models and Discussions Chapter 4 MANTLE ANISOTROPY MEASURED BY SHEAR WAVE SPLITTING Anisotropy and Shear Wave Splitting Inverse Method for Shear Wave Splitting Estimation Data and Measurements Noise Filtering and Phase Degrading Shear Wave Splitting Estimation for the WARS Region Geologic Analysis for the Anisotropy beneath the WARS Chapter 5 CONCLUSIONS.. 1 REFERENCES v

6 List of Figures Figure Study region and the locations of the 5 stations used in this study. Solid lines are the inferred boundaries of the WARS. Southern flank of the WARS is marked by the Transantarctic Mountains and Northern flank partly consists of Marie Byrd Land. The shading indicates the elevation of bedrock beneath the WAIS... 3 Figure 1..1 Propagation paths of body waves and a surface wave used in this study... 5 Figure.1.1 Receiver function and the corresponding ray path for the different phase arrivals... 8 Figure..1 Illustration of the convolution approximating the three-layer receiver function Figure.. Ice layer over crust layer half space model (model 1), crust layer over mantle layer half space model (model ), combined model with three layers (model 3). sij is the arrival time lag for which i is the model number and j = 1 for P-S arrival and j = for P-P-P-S arrival Figure..3 Synthetic receiver functions used to test the convolution effect. Model 1 (middle), model (top) and model 3 (bottom) in Figure..4 were used to compute each receiver function Figure..4 Models used to compute the synthetic receiver functions in Figure Figure..5 Comparison between the receiver function for the model with three layers and the convolution of the receiver function for the top layer - second layer half space model and the receiver function for the second layer - third layer half space model. The Gaussian filter parameter is 0.5, 1,, 5 from top to bottom Figure..6 Comparison between the receiver function for the crust-mantle half space model and the ice-removed receiver function Figure..7 Inversion results for the ice-removed receiver function and the receiver function for the original crust over mantle half space model.. 19 Figure..8 Inverted shear velocity structure models. (thick solid line - the velocity model from the receiver function for the original crust over mantle half space model; thin solid line - the velocity model from the ice-removed receiver function; dotted line - the true velocity model) Figure..9 Plots of tests for sediments layer effect on the deconvolution method. Ice and sediment-removed receiver functions (left panel) and the inversion results of the deconvolution with the ice-sediment-crust receiver function (right panel)....1 vi

7 Figure.3.1 Ice-removed receiver functions for BYRD for varying ice thickness (.45km,.km, and 1.5km). The measured thickness at BYRD is.km Figure.3. The raw receiver function for BYRD (top), the synthetic receiver function for ice over crust half space model (middle) and the ice-removed receiver function (bottom) Figure.3.3 Comparison between the synthetic receiver function for the model with three layers (.km ice layer - 6km crust mantle half space) and the raw receiver function at BYRD.. 6 Figure.3.4 Comparison between the ice-removed BYRD receiver function and the synthetic receiver function for 6km crust over mantle- half space model Figure.3.5 Receiver functions deconvolved with different ice and sediments combinations (MBL) Figure.3.6 Synthetic experiment for the deconvolution with ice-sediment-crust model receiver function Figure.3.7 Receiver functions deconvolved with different ice thickness model receiver functions (ISDE) Figure.3.8 Plots for the deconvolution test for SDM.. 30 Figure.4.1 The ice-removed receiver function inversion results for BYRD (top) and comparison between the estimated shear velocity models from the iceremoved receiver function inversion (thick line) and the surface wave dispersion inversion (thin line) (bottom) Figure.4. Inversion results for the receiver function deconvolved with the 1.85km ice and 0.km sediment model receiver function at MBL Figure Vertical and radial components of Rayleigh waveforms recorded at station MBL (event 01015) Figure 3.1. Single station group velocity estimation by multiple filter technique (event 99058) Figure Single station group velocity estimation by multiple filter technique (event 00344) Figure Single station group velocity estimation by multiple filter technique (event 00345) Figure Single station group velocity estimation by multiple filter technique (event 01015) Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering (bottom) and the great circle path (top) (event 99058) vii

8 Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering (bottom) and the great circle path (top) (event 00344) Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering (bottom) and the great circle path (top) (event 00345) Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering (bottom) and the great circle path (top) (event 01015) Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). ISDE-BYRD path (89km), event Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). BYRD-MTM path (396km), event Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). MBL-ISDE path (31km), event Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). ISDE-SDM path (91km), event Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). SDM-ISDE path (91km), event Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). ISDE-MBL path (31km), event Figure 3..1 Comparison of two methods for extracting Green s function between two receivers. In the top set of diagram, a delta function at A will be observed as a Green s function at B. A surface wave train at A will be observed at B also, but will be modified. We can calculate the Green s function from the two waveforms. In the lower panels, I show the noise-correlation method where the correlation of ambient noise at A and B results in the Green s function Figure 3.. Comparison between the cross correlation function and its time derivative for the noise field method. Bottom is the superposition of the two above viii

9 Figure 3..3 Comparison between the group velocities computed by the cross correlation function and its time derivative for the noise field method Figure 3..4 Ensemble averaged cross correlation functions computed in the random ambient noise field (period -1999, Feb) Figure 3..5 Ensemble averaged cross correlation functions computed in the random ambient noise field (period -1999, Mar) Figure 3..6 Ensemble averaged cross correlation functions computed in the random ambient noise field. (period -1998, Dec, 1999, Jan, 000, Feb) Figure 3..7 Cross correlations between MTM and the other 4 stations. Note that the distances are matched with the correlation time lags Figure 3..8 Comparisons of the band-pass filtered two noise field cross-correlation functions computed from different month period (BYRD-ISDE path (top), MTM-SDM path (bottom)). The last plot in each path is the superposition of the two above for comparison purpose Figure 3..9 Group velocity estimations by the waveforms extracted by noise field correlation, BYRD-ISDE path (1999, Feb, 1999 Mar) Figure Group velocity estimations by the waveforms extracted by noise field correlation, MTM-BYRD path (1999, Feb, 1999, Mar).. 77 Figure Group velocity estimations by the waveforms extracted by noise field correlation, MBL-ISDE (1999, Mar), ISDE-SDM (001, Feb) Figure 3..1 Group velocity estimations by the waveforms extracted by noise field correlation, MBL-BYRD (1999, Feb, 1999, Mar) Figure Group velocity estimations by the waveforms extracted by noise field correlation, MTM-ISDE (1999, Feb, 1999, Mar). 80 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (ISDE-BYRD). 88 Figure 3.3. Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (BYRD-MTM) Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (MBL-ISDE) Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (ISDE-SDM) Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (SDM-ISDE)... 9 ix

10 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (ISDE-MBL) Figure Estimated shear velocity models for the four regions in the WARS. 94 Figure Phase velocity profiles for hexagonal anisotropy (titanium) and orthorhombic anisotropy (olivine). The distances from origin are the velocities Figure 4.1. Illustration of shear wave splitting in hexagonal anisotropic medium Figure Patterns of the detectable and the non-detectable shear wave splitting measurements. Detectable case (A,B,C,D - MBL, event 99315) and nondetectable case (E,F - BYRD, event 99345). A. Superposition of the fast and slow components (before and after correction). B. Radial and transverse components (before and after correction). C. Particle motion diagrams (before and after correction). D. Contour plot of the energy on corrected transverse components and 95% confidence region (dotted line). E. Typical nondetectable observation for contour plot of the energy on corrected transverse components. F. Radial and transverse components for the non-detectable observation (before and after correction) Figure Noise filtering aspect for phase degrading. Without noise filter (left) and with noise filter (right) (MBL, event 99315) Figure 4.4. Noise filtering aspect for effective noise filtering. Without noise filter (left) and with noise filter (right) (MBL, event 99067) Figure % confidence bounds variation for the estimated fast axis direction and the time delay. Phase degrading case (top) and effective noise filtering case (bottom) Figure Averaged shear wave splitting parameters (the fast polarization direction and the time delay) at the four stations. The fast polarization directions at four stations are almost orthogonal to the inferred strike of the rift and parallel to the North- South elongated extension between East and West Antarctica. The time delay is represented by the length of the line Figure The fast polarization direction φ and the time delay δt as a function of back azimuth at MBL Figure The fast polarization direction φ and the time delay δt as a function of back azimuth at SDM Figure The fast polarization direction φ and the time delay δt as a function of back azimuth at BYRD x

11 List of Tables Table 1 List of events used in observational surface wave dispersion measurements... 4 Table Summary of the random noise cross correlation measurements Table 3 Standard errors of the estimated velocity models Table 4 List of events used in shear wave splitting measurements.. 10 Table 5 The results of all shear wave splitting measurements Table 6 The weighted averages of the splitting parameters xi

12 Acknowledgements I thank my advisor, Dr. Sridhar Anandakrishnan, for his insight and guidance on this research. I also appreciate the review of this thesis by Carles J. Ammon, Joseph L. Rose and Bernhard R. Tittmann. I thank Yongcheol Park who helped me understand Geosciences. I thank Paul Winberry for the information on West Antarctica. This research was supported by National Science Foundation (research grants #OPP ). xii

13 Chapter 1 INTRODUCTION 1.1 Overview The West Antarctic Rift System (WARS) is one of the largest continental rifts on Earth but is still poorly understood relative to the other major rift systems of the world. The poor geophysical coverage of the WARS is mostly due to the remoteness of Antarctica and the large ice sheet covering 98% of the Antarctic continent. The WARS is similar in size to the Basin and Range of the western United States and the East African rifts and extends from Ellsworth Land to the Ross Embayment (Behrendt et al., 1991; Tessensohn and Worner, 1991) (Figure 1.1.1). The major lithospheric extension in the WARS is hypothesized to have occurred during the late Cretaceous between c. 105 and 85 Ma. However, the mechanism for this extension is not well known (Lawver and Gahagan, 1994; Fitzgerald, 003). Understanding of the lithospheric structure in the WARS is important because it forms the cradle of the West Antarctic Ice Sheet (WAIS) and has influence on its dynamics (Dalziel and Lawver, 001; Winberry and Anandakrishnan, 004). Seismic body wave and surface wave data contain information about geophysical properties of the Earth s interior. From 1998 to 00, the Antarctic Network of Unattended Broadband Seismometers (ANUBIS) Project was carried out in which a number of seismometers were installed on West Antarctica (Anandakrishnan et al., 000). The seismometers have recorded seismic wave data for years , and the data have been previously analyzed to investigate some aspects of the WARS. In this paper, the crust and upper mantle structure of the interior of the WARS were investigated using broadband seismic data from five stations sparsely installed on the WAIS. First, the receiver function inversion technique was used to investigate the crustal 1

14 structure with a new method that removes the ice sheet reverberation effect. Second, for the crust and uppermost mantle shear wave velocity estimation, inversion techniques using surface wave dispersion curves were used. In addition, to estimate local velocity structure, the surface wave dispersion curves were measured using the inter-station approach which allows us to localize the structure better. The measurements of the dispersion curves were made from surface waves generated by earthquakes as well as by using cross-correlations of the ambient seismic noise field. The earthquake data are dominantly intermediate period ( Hz), and can resolve structures at greater depth than can be done using cross-correlation methods. The cross-correlation method has a dominant period at Hz and can resolve shallow structure with better resolution. Third, the mantle anisotropy was measured by shear wave splitting analysis using SKS phases.

15 Figure Study region and the locations of the 5 stations used in this study. Solid lines are the inferred boundaries of the WARS. Southern flank of the WARS is marked by the Transantarctic Mountains and Northern flank partly consists of Marie Byrd Land. The shading indicates the elevation of bedrock beneath the WAIS. 3

16 1. Seismic Waves and Earth Structures Seismic waves are generated at a source, propagate through the medium (Earth s interior) and are recorded at a receiver (seismometer). A seismogram is the record of the motion of the ground at a seismometer, and contains information about the source and the properties of the path within the Earth s interior. The natural sources of seismic waves are earthquakes on faults at plate boundaries which radiate compressional (P) waves and shear (S) waves. The particle motion for P wave is in the propagation direction and for S wave, the particle motion is perpendicular to the propagation direction. The motion of P or S waves is described by wave equations in which scalar and vector potentials are used for P and S waves, respectively. By Snell s law and the fact that seismic wave velocities generally increase with depth, the ray paths of body waves bend away from the vertical as they go deeper into the Earth, eventually become horizontal, turn upward, and return to the surface (Stein and Wysession, 003). P and S waves, which travel through the Earth s interior, are called body waves and the interactions of the body waves at the surface generate a different type of wave called surface wave, which travels along the surface. The features of the seismic waves on seismograms, such as travel times, amplitudes, waveforms, eigenfrequencies, dispersion, and attenuation, can be modeled in forward problems that relate the sources and earth structure to the above features on seismograms (Stein and Wysession, 003). Therefore, the properties of the medium and the source, such as velocity structure and earthquake mechanisms, can be determined by inverse modeling. For the determination of velocity structure, travel times of body waves and dispersion of surface waves are generally used for the inverse problems. The seismic wave data used for this study has been recorded at a sparse seismic array of five seismic stations, in which the body wave travel time tomography is not effective. In this paper, the surface wave dispersion information 4

17 is used in the inverse problem for the determination of the crust and upper mantle velocity structures (Chapter 3), and the receiver function technique using body wave data for a single seismometer is used for the inverse modeling of the crustal structure (Chapter ). In addition, body wave seismograms of SKS phases are used for the mantle anisotropy determination (Chapter 4). In Figure 1..1, the propagation paths of the body waves and a surface wave used in this study are illustrated. Figure 1..1 Propagation paths of body waves and a surface wave used in this study. 5

18 Chapter RECEIVER FUNCTION ANALYSIS.1 Introduction to Receiver Function Analysis The displacement response of a 3-component seismometer at the surface for incoming compressional (P) waves can be represented as D( t) = I( t) S( t) E( t) (.1.1) where D (t) is the displacement for a given component, S (t) is the source time function, I (t) is instrument response and E (t) is the earth structure impulse response. A receiver function, E (t), exists for each of the orthogonal components (vertical, radial and tangential displacement component). However, the radial receiver function is commonly called the receiver function. The radial receiver function E R (t) is calculated by deconvolving the vertical displacement component from the radial displacement component by making the approximation that the vertical displacement response is primarily due to the instrument response and source time function (Langston, 1979) (.1.), (.1.3). I( t) S( t) D ( t) (.1.) V The approximation in (.1.) implies that we assume that the vertical displacement response acts like a Dirac delta function. The effects of later arrivals in the vertical component of displacement are neglected for this assumption (Burdick and Helmberger, 1974). 6

19 DR ( ω) ER ( ω) (.1.3) D ( ω) V The deconvolution process of (.1.3) for real seismograms is unstable because the signals contain noise and are band-limited (Langston, 1979). To prevent this problem, the so-called water-level deconvolution method and low-pass Gaussian filtering are adopted for the stabilization of the deconvolution (Helmberger and Wiggins, 1971; Langston, 1979). The resulting radial receiver function is a scaled version of P-wave multiples-removed radial displacement response that has units of 1/time (Figure.1.1) (Langston, 1979; Ammon, 1991). The receiver function can be inverted for shear wave velocities with depth allowing us to measure depth to major interfaces beneath the seismometer (Ammon et al., 1990). The inversion algorithm for receiver functions and for surface wave inversions are similar, involving the partial derivatives with respect to the Earth parameters. Therefore, I will defer discussion of the details of the inversion algorithm to Chapter 3. 7

20 Figure.1.1 Receiver function and the corresponding ray path for the different phase arrivals.. A Method of Removing the Ice layer Reverberation Effect on the Receiver Functions To explore West Antarctic subglacial crustal structure using receiver functions, the ice layer covering the crust has to be considered. One difficulty with the Antarctic data is that the acoustic impedance contrast (defined as the ratio ( ρ υ ρ1υ 1 ) at the base of the ice sheet is extremely large ( ρ υ ρ υ 4. 3, the subscripts 1 = ice, = basement rocks, ρ = density, 1 1 υ = seismic velocity). As a consequence, the amplitude of multiples within the ice sheet diminishes slowly. This violates our earlier assumption (.1.) that the vertical displacement response is a Dirac delta function. I have developed a modification of the standard receiver 8

21 function technique that accounts for the known reverberation within the ice sheet. If a receiver function for crust and mantle-only structure, without the ice layer, is obtained, we can estimate the crust-mantle velocity contrast by using standard receiver function technique. In this section, a method to remove the reverberation effect on the receiver function, which leads to the restoration of crust-mantle receiver function, is designed and tested with synthetic receiver functions. In the next section I apply the technique to real data. We begin with a simple three-layer earth model and synthesize a receiver function for a known top layer -to- second layer velocity structure. Next, we deconvolve the full receiver function using the top layer -to- second layer receiver function and obtain the receiver function for the nd layer - 3rd layer half-space model. In other words, the receiver function for a model with three layers is approximated with the convolution of the receiver function for the top layer - second layer half space model and the receiver function for the second layer - third layer half space model. This argument is mathematically described by multiplication of the two receiver functions in frequency domain. Considering the fastest three arrivals (direct P, P-S converted at interface and P-P-P-S multiple) for each receiver function, the receiver functions for the top layer - second layer half space model and the second layer - third layer half space model (Figure..) are represented, in the time and frequency domain in (..1) and (..), respectively. r1 rs 11 rs 1 r ( ) = ( ) + ( 11) + ( 1) 1 t δ t δ t s δ t s z1 r1 r1 r1 rs 11 jωs11 rs 1 jωs1 R = + + 1( ω ) 1 e e (..1) z1 r1 r1 9

22 = ) ( 1) ( ) ( ) ( 1 s t r r s t r r t z r t r s s δ δ δ + + = ) ( s j s s j s e r r e r r z r R ω ω ω (..) In (..1) and (..), i r and i z are the amplitudes of the direct P in the radial and vertical components (the subscript i represents model number). sij indicates the arrival time for which i is the model number and j = 1 for P-S arrival and j = for P-P-P-S arrival. sij r corresponds to the amplitude of the sij arrival = ) 1 ( 1 1 1) 1 ( ) 11 ( ) 11 ( ) ( ) ( s s j s s s s j s s s s j s s s s j s s s j s s j s s j s s j s e r r r r e r r r r e r r r r e r r r r e r r e r r e r r e r r z z r r R R ω ω ω ω ω ω ω ω ω ω L L (..3) In the resulting convolution, (..3), the direct-p arrivals for both model 1 and model overlap. In addition, the S-converted arrivals for the two models (the nd and 3 rd terms for model, the 4 th and 5 th terms for model 1 in..3) and the convolved arrivals (6 th to 9 th terms) for model 3 (ice-crust-mantle) also are present. Approximating the 3-layer model by a convolution of two -layer models is possible because of the large impedance contrast between the ice and the underlying crust as we now explain. For example, the PSPPS term and the PPPSPPS term in the 3-layer model are approximated by the 8 th term and the 9 th term in the convolution, respectively. In the convolution, the PP conversion is used at the crust-ice boundary for the 8 th and 9 th terms

23 instead of an SP conversion which should really be used for the 3-layer model, however, the large impedance contrast (PP~SP) does not introduce large error for the approximation. In this case, the amplitudes of the convolution terms are larger than the 3-layer model. The PSS term and the PPPSS term for the 3-layer model are approximated by the 6 th and the 7 th terms in the convolution, respectively in a similar way (PS~SS at the crust-ice boundary). In this case, the amplitudes of the convolution terms are smaller than the 3-layer model, however, the nd and 3 rd terms for model (crust-mantle model) with a slightly earlier arrival time lag (one-way S travel time in the ice layer) are summed to the 6 th and the 7 th terms. This addition rather compensates the smaller amplitudes of the 6 th and the 7 th terms. Thus, we can approximate the three-layer receiver function by a convolution of the two -layer receiver functions. The approximation of the PSS term and the PPPSS term for the 3-layer model by the convolution is graphically illustrated in Figure..1. This approximation may not be possible if the thickness of ice layer is large because, in that case, the S-converted arrivals for model are isolated from the convolved Moho multiples in the convolution. 11

24 Figure..1 Illustration of the convolution approximating the three-layer receiver function. 1

25 Figure.. Ice layer over crust layer half space model (model 1), crust layer over mantle layer half space model (model ), combined model with three layers (model 3). sij is the arrival time lag for which i is the model number and j = 1 for P-S arrival and j = for P- P-P-S arrival. 13

26 To test this mathematical argument with waveforms, synthetic receiver functions (Figure..3) for the three models (Figure..4) were computed using the program hrftn96 implemented in Computer Programs in Seismology (Herrmann and Ammon, 00). Receiver function computation in the code hrftn96 is based on the Thomson-Haskell matrix method (Thomson, 1950; Haskell, 1953). Then, the receiver function for the model with three layers was compared with the convolution of the receiver function for the top layer - second layer half space model and the receiver function for the second layer - third layer half space model (Figure..5). The arrival times of the 6 P-S converted phases in the receiver function for the model with three layers can be calculated using Snell s law, and the arrival time of PPPSPPS for the synthetic model, which is the latest arrival, is 15 seconds. In Figure..5, we can observe good agreement between the receiver function computed from the convolution and the receiver function directly obtained from the three-layer model for the first 0 seconds for Gaussian filter parameters 1, and 5. In the case of too low-pass filtered data (top plot, Gaussian filter parameter 0.5), the receiver functions from the two methods are not in a good agreement. 14

27 Figure..3 Synthetic receiver functions used to test the convolution effect. Model 1 (middle), model (top) and model 3 (bottom) in Figure..4 were used to compute each receiver function. Figure..4 Models used to compute the synthetic receiver functions in Figure... 15

28 Figure..5 Comparison between the receiver function for the model with three layers and the convolution of the receiver function for the top layer - second layer half space model and the receiver function for the second layer - third layer half space model. The Gaussian filter parameter is 0.5, 1,, 5 from top to bottom. 16

29 The deconvolution method that removes the ice layer and retrieves a receiver function for the crust-mantle only structure was tested with the same synthetic receiver functions. For the deconvolution, computer code provided by Charles J. Ammon (Ligorria and Ammon, 1999) was used. The ice-removed receiver function obtained by deconvolving the receiver function for the ice-crust-mantle half space model with the receiver function for the ice over crust half space model is compared with the receiver function for the crust-mantle half space model in Figure..6. We can observe general agreement between them that includes fit of the earliest three P-S converted arrivals with the direct P arrival for the crust over mantle half space model, which means the deconvolution method retrieved the receiver function for the crust-mantle half space model. Figure..6 Comparison between the receiver function for the crust-mantle half space model and the ice-removed receiver function. 17

30 The ice-removed receiver function and the receiver function for the original crust over mantle half space model were then inverted for the 1-D shear wave velocity structure using inversion codes of Computer Programs in Seismology (Herrmann and Ammon, 00). The inversion results are compared in Figure..7 and it shows that the ice-removed receiver function works as well as the receiver function for the original crust over mantle half space model in the inversion. The inverted shear wave velocity structure models are compared with the true model in Figure..8. The estimation of the true model with the ice-removed receiver function is in good agreement with the estimation obtained with the receiver function for the original crust over mantle half space model. 18

31 Figure..7 Inversion results for the ice-removed receiver function and the receiver function for the original crust over mantle half space model. 19

32 Figure..8 Inverted shear velocity structure models. (thick solid line - the velocity model from the receiver function for the original crust over mantle half space model; thin solid line - the velocity model from the ice-removed receiver function; dotted line - the true velocity model). 0

33 If the thin ice layer is lying directly on the crustal rocks, the ice removing deconvolution is expected to retrieve the crust-mantle structures for the Antarctic data. However, it has been reported that some locations in the WAIS (MBL, MTM, SDM) have sediment layers between the ice and the crustal rocks. In this case, the deconvolution with the ice-crust receiver function may not work properly. To test the sediment effect, the synthetic receiver function of four layers (ice-sediment-crust-mantle) model was deconvolved with the ice-sediment-crust receiver function and the ice-crust receiver function for which the ice thickness is set to (ice + sediment) thickness (Figure..9). Figure..9 Plots of tests for sediments layer effect on the deconvolution method. Ice and sediment-removed receiver functions (left panel) and the inversion results of the deconvolution with the ice-sediment-crust receiver function (right panel). 1

34 Although the acoustic impedances of the ice and the sediments are similar, the deconvolution with the ice-crust (ice + sediment thickness) does not produce the second PS arrival, while the deconvolution with the ice-sediment-crust produces the PS arrival as well as the third PPPS arrival (Figure..9). The resulting receiver function from the deconvolution with the ice-sediment-crust shows the crust-mantle structure and the inversion test shows that the deconvolution method might be applied to the structure with sediments.

35 .3 Application of the Deconvolution Method to Removing Ice Layer Effects on the Raw Field Receiver Functions The raw receiver functions at the five stations (BYRD, MBL, MTM, ISDE, SDM) were computed (Anandakrishnan and Winberry, 004) by a frequency domain deconvolution of the vertical and radial components, with water level stabilization and a Gaussian low-pass filter (Clayton and Wiggins, 1976; Ammon, 1991). Those raw receiver functions are presumed to be produced based on a model with three layers (ice, crust and mantle) for BYRD and ISDE or a model with four layers (ice, sediments, crust and mantle) for MBL, MTM and SDM. In these cases, as mentioned in section., the reverberations in the ice (sediments) layer need to be removed for the inversion process to obtain crust-mantle velocity contrast. To apply the deconvolution technique in section. to the field raw receiver functions for a specific station, we need to set proper ice and sediments layer thicknesses beneath a given station. I used the known ice layer thicknesses estimated by radio echo sounding (Rose, 1979; Drewry, 1983) and the sediments thicknesses estimated by L1 norm of the error between the synthetic and the raw receiver functions (Anandakrishnan and Winberry, 004). In addition, I varied the ice layer thickness in 50 m steps to determine which thickness gives the best ice layer removing effect, in which the earlier three S-converted arrivals are most distinguishable. The synthetic receiver functions for the ice-crust model and the ice-sediments-crust model were computed using the program hrftn96 (Herrmann and Ammon, 00) with the same Gaussian filter widths and ray parameters as were used with the raw receiver functions. The raw receiver functions were then deconvolved with the synthetic receiver functions to estimate the receiver functions for the crust-mantle half space (ice-removed) model. 3

36 Figure.3.1 shows the ice-removed receiver functions for station BYRD that were computed by deconvolution with three different ice-crust model receiver functions that are set to have different ice layer thickness. For station BYRD, the ice layer thickness is.6±0.03 km (Rose, 1979; Drewry, 1983) and the sediments layer is not present (Anandakrishnan and Winberry, 004). Note that the.km ice thickness model produced relatively sharply peaked second PS arrival in comparison with the 1.95 km and.45 km ice thickness models while the third arrival, PPPS, appears almost similar in the three different ice-removed receiver functions. This implies that the ice-removing deconvolution technique is working properly. The raw receiver function for BYRD, the synthetic receiver function for the ice over crust half space model used for the ice-removing deconvolution process and the deconvolved receiver function are presented in Figure.3.. The ice-removed receiver function is assumed to represent a crust and mantle-only model beneath BYRD. The first direct P, the second PS and the third PPPS arrivals are identified in the ice-removed receiver function and although the fourth PPSS + PSPS arrival is not clearly visible, the Moho (The crust-mantle boundary) depth and the velocity contrast at the Moho are expected to be retrieved by inversion of the ice-removed receiver function. 4

37 Figure.3.1 Ice-removed receiver functions for BYRD for varying ice thickness (.45 km,. km, and 1.5 km). The measured thickness at BYRD is. km. Figure.3. The raw receiver function for BYRD (top), the synthetic receiver function for ice over crust half space model (middle) and the ice-removed receiver function (bottom). 5

38 In figure.3.3, the synthetic receiver function for the model with three layers (. km ice layer - 6 km crust - mantle half space) is compared with the raw receiver function at BYRD. The 6 km of crustal thickness is based on the result from the inversion of the iceremoved receiver function in section.4. The results show similar waveforms (but not perfect), which implies that the velocity structure beneath BYRD, including the ice layer, can be represented by this model. Figure.3.4 is comparison of the ice-removed BYRD receiver function and the synthetic receiver function for 6 km crust over mantle half space without the ice layer. The first three arrivals (direct P, PS and PPPS) are matched in this comparison, which also supports the possibility of the method to remove the ice layer reverberation effect on the receiver functions. Figure.3.3 Comparison between the synthetic receiver function for the model with three layers (. km ice layer 6 km crust mantle half space) and the raw receiver function at BYRD. Figure.3.4 Comparison between the ice-removed BYRD receiver function and the synthetic receiver function for 6 km crust over mantle- half space model. 6

39 Station MBL has been reported to have 1.85±0.05 km ice and 60±80 m sediment (Anandakrishnan and Winberry, 004). The deconvolution with the ice-sediment-crust receiver function in section. was applied to MBL. The receiver functions at MBL deconvolved with different ice and sediment thickness combinations are presented in Figure.3.5. For a same sediment thickness of 0.6 km, 1.85 km ice model seems to give trade-off between PS and PPPS arrivals in comparison with 1.6 km and.1km models. For a same ice thickness of 1.85 km, 0. km sediment model is considered to produce more distinguishable PS and PPPS arrivals than 0.6 km and 0.3 km sediment models. The 1.85 km ice and 0. km sediment model showed most reasonable inversion result of velocity structure model, which will be presented in the next section. Figure.3.5 Receiver functions deconvolved with different ice and sediments combinations (MBL). Station MTM has been reported to have 3.±0.15 km ice and 550±50 m sediment, which are relatively thick in comparison with other stations (Anandakrishnan and Winberry, 004). In this case, the synthetic experiment for the deconvolution with ice-sediment-crust implies that the PS arrival is not properly retrieved and the deconvolution method is not applicable (Figure.3.6). It is considered that approximating the four layer receiver function 7

40 with the convolution of the ice-sediment-crust receiver function and the crust-mantle receiver function may not be possible when the thickness of ice and sediment layers is large because, in that case, the S-converted arrivals for crust-mantle model are isolated from the convolved Moho multiples as mentioned in the previous section. Figure.3.6 Synthetic experiment for the deconvolution with ice-sediment-crust model receiver function. Station ISDE has 1.±0.05 km ice thickness and no sediment layer for ISDE has been reported (Anandakrishnan and Winberry, 004). Figure.3.7 shows the receiver functions at ISDE deconvolved with the ice-crust receiver functions. In Figure.3.7, it is observed that 1.km ice model produces clearer PS arrival than other ice thicknesses but the PPPS arrival is weaker than the PS arrival and is not well distinguishable from the following fluctuations. The ice-crust interface beneath ISDE is considered not to be flat in comparison with other stations, which forms complex interface structure with lateral variations. The obscure retrieval of the PPPS arrival in the deconvolution for ISDE may be due to the lateral heterogeneity of the ice-crust interface structure. 8

41 Figure.3.7 Receiver functions deconvolved with different ice thickness model receiver functions (ISDE). Station SDM has been reported that the ice thickness is 1.0±0.03 km and the sediment thickness is 300±50 m. In the ice-sediment removing deconvolution test, the deconvolved receiver function is significantly different from the estimated crust (7 km)-mantle model (Winberry and Anandakrishnan, 004). In addition, the raw receiver function is significantly different from the synthetic four-layer receiver function (Figure.3.8). These results imply that the estimation of 300 m sediment for SDM has to be reconsidered and also suggest the complicated lateral heterogeneities in the ice-sediment structure for SDM. 9

42 Figure.3.8 Plots for the deconvolution test for SDM..4 Inversion of the Ice Layer-Removed Receiver Functions The ice-removed receiver functions at BYRD in Figure.3.1 were inverted for the 1-D shallow shear wave velocity structure and it was found that. km ice model gives the most clear crust-mantle velocity contrast in the estimated velocity models. In Figure.4.1, the inverted velocity model and the receiver function waveform fit are presented. Also, the velocity model for BYRD from the ice-removed receiver function inversion and the velocity model for ISDE-BYRD path from the inversion of the inter-station surface wave dispersion curve are compared in Figure.4.1. Although ISDE-BYRD path region and BYRD region cannot be directly compared, we can observe overall similarity including Moho contrast in this comparison. The resulted Moho depth for BYRD is approximately 6 km and this is in 30

43 consistence with the 7 km estimation by minimizing L1 norm of the error between modeled and recorded receiver function (Winberry and Anandakrishnan, 004). The inversion results of the receiver function deconvolved with the 1.85 km ice and 0. km sediment model receiver function at MBL are presented in Figure.4.. In the results, the inverted velocity structure in the crustal layer shows a similar pattern observed in the synthetic experiments in Figure..9 and the velocity contrast at 5 km depth is presumed. This result is also consistent with the Moho depth estimation by minimizing L1 norm of the error between modeled and recorded receiver function (Winberry and Anandakrishnan, 004). The inversion of the ice-removed receiver functions for ISDE in Figure.3.7 did not estimate the crust-mantle velocity contrast. It is considered that the unreasonable inversion results for ISDE are due to the obscure retrieval of the PPPS arrival as mentioned in section.3. The standard errors in the estimated velocity models (BYRD and MBL) based on the residuals of the waveform fit are around 0.01 km/s for all depths. 31

44 Figure.4.1 The ice-removed receiver function inversion results for BYRD (top) and comparison between the estimated shear velocity models from the ice-removed receiver function inversion (thick line) and the surface wave dispersion inversion (thin line) (bottom). 3

45 Figure.4. Inversion results for the receiver function deconvolved with the 1.85 km ice and 0. km sediment model receiver function at MBL. 33

46 Chapter 3 SURFACE WAVE DISPERSION AND THE SHALLOW VELOCITY STRUCTURE OF THE WAIS REGION 3.1 Dispersion of Surface Waves by Earthquakes Generation of Surface Waves P or SV wave incident on a free surface at postcritical incidence angles produces evanescent SV or P wave propagating along the surface, respectively. For the evanescent waves generated from the postcritical incidence, vertical slowness becomes purely imaginary and the amplitude decays exponentially away from the interface. Stress free conditions at the surface preclude the existence of purely P or SV evanescent wave on the surface boundary. However, simultaneous, coupled evanescent P and SV waves satisfy the free-surface boundary condition, yielding a new form of wave solution. Lord Rayleigh explored this system in 1887 and found the existence of a coupled P-SV wave traveling along the surface with a velocity lower than the shear velocity and with amplitudes decaying exponentially away from the surface. Resultant elliptical particle motion of Rayleigh waves is neither purely shear nor purely compressional although shear motion has greater influence over Rayleigh wave behavior. Because of the elliptical displacement behavior, Rayleigh wave s strongest motion is detected in vertical component of the seismogram rather than radial component (Lay and Wallace, 1995). While Rayleigh waves can only be generated at the free surface, irrespective of whether the medium is vertically homogeneous or heterogeneous, Love waves are not generated for a homogeneous half space and require a vertically heterogeneous velocity structure. The simplest structure for Love wave generation is a layer above a half space where the shear 34

47 35 velocity for the layer is lower than that of the half space. Again, considering a postcritical angle of incidence for SH waves, the SH wave is trapped in the layer due to the total refection from the surface and the base of the layer, and evanescent SH wave exponentially decays away from the interface in a manner similar to Rayleigh waves Surface Wave Dispersion Relation for the Vertically Heterogeneous Medium The term dispersion is commonly used for the physical phenomenon where different frequency components propagate at different speeds. Surface wave dispersion in the seismic wave field is due to the variation in seismic velocity with depth within the Earth. The particle motion of surface waves exponentially decays with depth for all frequency components. However, lower frequency components sample the Earth to greater depth than higher frequency component, and hence sample shear wave velocity β over a greater depth range. Therefore, if β changes with depth, surface waves disperse. A surface wave system in a vertically heterogeneous medium where the elastic moduli are arbitrary functions of depth can be represented by a coupled motion-stress vector equations for surface waves (Aki and Richards, 1980). The motion-stress vector equations for a Rayleigh wave is given by [ ] [ ] [ ] = ) ( 0 ) ( ) ( ) ( 0 0 ) ( ) ( ) ( ) ( ) ( ) ( 4 ) ( ) ( 0 0 ) ( ) ( ) ( 0 ) ( 0 r r r r k z z z z k z z z z z z k z z z z z k z k r r r r dz d ρ ω µ λ λ ρ ω µ λ µ λ µ µ λ µ λ λ µ (3.1.1)

48 where, radial displacement = r k z e i( kx ωt ) 1 (,, ω) vertical displacement = r k z e i( kx ωt) (,, ω) horizontal shear stress ( σ ) = ZX vertical normal stress ( σ ) = ZZ k = wave number ω = angular frequency µ (z), λ (z) = Lamé constants r ir i( kx ωt ) 3 ( k, z, ω) e i( kx ωt) 4 ( k, z, ω) e ρ (z) = density The motion-stress vector equations can be treated as a dispersion relation or period equation for the vertically heterogeneous medium. For a given frequency ω, the non-vanishing solutions of the motion-stress vector equations exist for a set of wave numbers k (ω ) for the n. The phase velocity is then given by c ( ω) = ω / k ( ω). The subscript n represents modes of the solution. The solutions for displacement and stress corresponding to given c n (ω) represent z dependence profile of the mode at that frequency. If all modes exist at all frequencies, we have n ( number of ω) different z dependence profile for the wave motion. Generally, lowering ω for the same n extends the displacement deeper (this produces the dispersion) and for the sameω, higher mode results in deeper wave motion profile than lower mode. This means higher modes contains the deeper structure information. If the z dependence of the elastic moduli for any vertically heterogeneous structure is known, we can calculate the surface wave dispersion curves for all modes by solving the motionstress vector equation. The motion-stress vector equations can be solved in integration approach using numerical methods such as that of Runge-Kutta (Aki and Richards, 1980) or by the propagator matrix method (Thomson, 1950; Haskell, 1953; Gilbert and Backus, 1966; Aki and Richards, 1980) for which the structure is approximated by a stack of layers over a half-space. In the integration approach, the motion stress vector system is integrated upward n n n 36

49 from the bottom with the trial value of k for a givenω. The trials of k are repeated to find k for which the stress at the surface becomes zero and the resulting k is one of the eigenvalue solutions. In the propagator matrix method, the potential amplitudes vector in the lower half-space are related to the motion-stress vector at the surface through a product of the propagator matrix and inverse of the so-called solution matrix. The conditions of no upgoing waves in the half-space and the stress free surface give an equation that includes ω and k in algebraic manipulation for the system and the solutions are found by common numerical methods. Then we are to estimate the physical properties of the Earth interior by inverting the observed dispersion data Two-Station Group Velocity Dispersion The group velocity at a given frequency ω is the propagation speed of a wave packet that is 0 constructed from frequency band ω whose center frequency is ω 0. The propagation speed of the peaks and troughs in the wave packet is termed phase velocity for the frequency ω. Generally seismic sources excite a continuous frequency spectrum and group velocities 0 are observed in addition to the phase velocities for surface waves. Both group and phase velocities disperse and can be used to invert for the Earth parameters. For single-station measurements, the group velocities are easier to measure than the phase velocities because an accurate knowledge of the earthquake s focal mechanism is not necessary. However, for the phase velocity measurements, the initial phase of the wave at the earthquake is needed. The group velocities are the distance divided by the arrival time of the wave packet for a given period. The phase velocities are calculated from Fourier phase spectrum at the receiver, which leaves the only unknown, c n(ω). If two stations are located on the same great circle path, measurements using two 37

50 stations are more convenient than the single station method. The group velocities are then the distance between two stations divided by the time lag of the wave packet arrivals and the phase velocities are again calculated by taking the Fourier phase difference between the two stations. If the receivers form an array for a single earthquake event, a different method using p ω stacking can be applied (McMechan and Yedlin, 1981; Herrmann and Ammon, 00). The group and phase velocity dispersion curves contribute generally the same information about an earth model (Der et al., 1970; Wiggins, 197). In this study, to explore the subglacial velocity structure beneath the West Antarctic Ice Sheet, the two station Rayleigh wave group velocity dispersions were measured from the earthquakes data because those are to be combined with the dispersions measured from the noise correlation method, which generally are interpreted as group velocities. The group velocity estimation for both the single and two station methods was done by multiple filter analysis (Dziewonski et al., 1969). The phase match filter technique (Herrin and Goforth, 1977) was used for the mode isolation to obtain accurate time differences for the wave packet arrivals between the two stations. The two techniques are implemented in Computer Programs in Seismology (Herrmann and Ammon, 00) that is used for this study and I briefly describe the two techniques in the next two sections Multiple Filter Analysis The role of Multiple Filter Analysis (MFA) (Dziewonski et al., 1969) is to estimate the group velocity dispersion curves. A suite of narrow-band filters in the frequency domain is applied to each Rayleigh wave seismogram. A Hilbert transform is then applied to each filter output, resulting in the wave packet envelope for each frequency band. The normalized wave packet envelope in the time domain at each frequency band is plotted and contoured (Figure 3.1.), 38

51 and maxima are picked for the group velocity in the velocity-frequency coordinates. The time corresponding to the peak amplitude of the envelope for each frequency band in the frequency-time (or velocity) contour plot can now be interpreted as the surface wave group velocities as a function of frequency. A Gaussian filter G (ω ) (3.1.) is used for the narrowband filters. ( ω ω ) G( ω) = a exp ω 0 = 0 otherwise 0 at ω ω ω ω + ω 0 c 0 c (3.1.) ω = ω c 0 π a The choice of parameter a is crucial to a good estimate of group velocity and spectral amplitude. For good amplitude estimate, a increases with increasing source to receiver distance (Herrmann and Ammon, 00). For a distance range of approx. 000 km, a ~50 is used. This value defines a filter width that is half of the central frequency. In the wave packet envelope contour plot, modes are represented as different maximum contour region, and we can isolate a specific mode by applying phase matched filtering techniques Phase Match Filtering The phase match filter (Herrin and Goforth, 1977) is defined as a filter whose phase is identical to the phase of a signal. If we have signals that are contaminated with unwanted signals such as noises, multiples or higher modes in the surface waveforms, the desired signal can be extracted from the interfered signal using the phase-matched filter. 39

52 In the frequency domain, cross-correlation between a signal s(t) and a filter f (t) can be represented as s ( t) f ( t) s( ω) f ( ω) exp( i( σ ( ω) φ( ω)) (3.1.3) If σ ( ω) = φ( ω), the correlation produces an even function in the time domain like the autocorrelation function of s (t). To use this property, the signal is correlated with a trial filter whose Fourier phase spectrum is set to that of the desired signal. A white spectrum can be adopted as the amplitude spectrum of the trial filter, which is a compromise between improving S/N and improving time resolution for the output. The relation between group delay and Fourier phase of the signal is given as (3.1.4) (Papoulis, 196), dφ( ω) t g dω ω 1 0 ( ω) = ( ) 1 φ ω = t ω ω g ( )d (3.1.4) and the phase spectrum for the trial filter can be calculated from the desired group velocities in MFA. Next, in the time domain, a window is applied to the correlation function and signals outside the window are rejected because the unmatched components reside outside the window. The resulting correlation function is transformed to frequency domain and the phase of the correlation function is assessed. If there exist some non-zero phase for the correlation function, it represents the phase difference between the filter and the first extracted signal. The phase difference is then used to modify the group delay of the trial filter and consequently the phase spectrum of the filter is corrected through (3.1.4). This process may be repeated until we observe zero phase in the correlation function for the frequency range of interest. When the phases are matched between the filter and the signal, the desired signal can be extracted as we have determined its phase spectrum, and the amplitude spectrum is 40

53 directly obtained from that of the correlation function. Phase matched filter can therefore be employed to refine the group velocity curve and to extract any desired signal from an interfered wave train. In this study, the main role of PMF is the mode isolation from the surface waveforms Data and Results of the Dispersion Measurements The inter-station method requires that the two stations are located on the same great circle path as the earthquakes. Among the earthquakes that satisfy the above condition, wellisolated events with adequate path lengths (less than 10,000km), magnitude greater than M 5 and shallow origin depth were picked and the corresponding vertical Rayleigh wave signals were selected from the ANUBIS (Antarctic Network of Unattended Broadband Seismometers) data recorded at the five stations (Anandakrishnan et al., 000). Earthquakes used in this study for the observational inter-station dispersion measurements are listed in Table 1. Figure is an example of the Rayleigh wave signals generated by an earthquake s on Feb. 7, 1999 originated at Balleny Islands Region with M s 5.3, 500 km distance and 10 km origin depth. 41

54 Figure Vertical and radial components of Rayleigh waveforms recorded at station MBL (event 01015) Depth Distance Event Origin Time Latitude Longitude (km) M s Path appx.(km) 1999/0/ :08: ISDE-BYRD-MTM 1999/0/ :13: ISDE-BYRD-MTM 1999/1/ :17: ISDE-BYRD 1999/1/ :7: ISDE-BYRD 1999/1/ :6: ISDE-BYRD 1999/1/ :35: ISDE-BYRD 000/01/ :00: MBL- BYRD 000/01/ :00: MBL- BYRD 000/01/ :59: MBL- BYRD 000/1/ :49: MBL-ISDE-SDM 000/1/ :17: SDM-ISDE-MBL 001/01/ :5: ISDE-MBL Table 1 List of events used in observational surface wave dispersion measurements. 4

55 The group velocities for each station were estimated using the multiple filter analysis and the phase matched filtering isolated the fundamental mode waveforms. To estimate wave packet arrival time lags between the two stations, the mode isolated waveforms for a pair of stations were cross-correlated and the multiple filter technique was re-applied to the cross correlation functions to yield the final group velocity curves between the two stations, which are expected to reflect the local seismic velocity structure. In figure , the group velocity plot for single stations from the multiple filter analysis is exemplified. The group velocities are given as a function of period (right), and the left plot is the corresponding spectral amplitude. In the velocity contour plot for ISDE-99058, the first higher mode is vaguely observed besides the fundamental mode. However, clear first higher mode signals were not detected at the other stations and thus the overtones were not used in this study. 43

56 Figure 3.1. Single station group velocity estimation by multiple filter technique (event 99058). 44

57 Figure Single station group velocity estimation by multiple filter technique (event 00344). 45

58 Figure Single station group velocity estimation by multiple filter technique (event 00345). 46

59 Figure Single station group velocity estimation by multiple filter technique (event 01015). 47

60 Figure shows the fundamental mode Rayleigh waveforms isolated by the phase matched filtering at the stations that are on the same great circle path for each event. Note that the traces are maintaining their waveforms as they propagate through the path and are slightly spread at the farther station due to dispersion. The traces are plotted in absolute time in Figure The cross correlation function between the fundamental mode waveforms for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms are shown in Figure The frequency bands of the group delay used to calculate Fourier phases for the trial phase matching filter are in the range of 10s 50s for most of the stations. Thus, the fundamental mode waveforms extracted by the phase-matched filter have the same frequency bands as the phase matched filter. Therefore, the group velocities from the correlated waveforms appear in 10s-50s frequency bands. The errors in the group velocities measurements for MFA are related to the Gaussian filter width. Since the Gaussian filter has a longer impulse response at longer periods, a reading error could be associated with a misplaced maximum - the group velocity measurements errors in MFA are computed assuming that the travel time can be mis-measured by one filter period (Herrmann and Ammon, 00). The errors in the dispersion observations will be used for computing the variances of the inverted velocity models in section

61 Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering (bottom) and the great circle path (top) (event 99058). 49

62 Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering and the great circle path (event 00344). 50

63 Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering and the great circle path (event 00345). 51

64 Figure Fundamental mode Rayleigh waveforms isolated by the phase matched filtering and the great circle path (event 01015). 5

65 Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). ISDE-BYRD path (89km), event

66 Figure The cross correlation function between the fundamental mode waveforms (up) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (below). BYRD-MTM path (396km), event

67 Figure The cross correlation function between the fundamental mode waveforms (up) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (below). MBL-ISDE path (31km), event

68 Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). ISDE-SDM path (91km), event

69 Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). SDM-ISDE path (91km), event

70 Figure The cross correlation function between the fundamental mode waveforms (top) for a pair of stations and the local group velocity dispersion estimation using the correlated waveforms (bottom). ISDE-MBL path (31km), event

71 3. Dispersion measured in the Seismic Noise Field 3..1 Ambient Seismic Noise Correlation between two Receivers Recent research in acoustics (Lobkis and Weaver, 001; Derode et al., 003; Roux and Kuperman, 004) and seismology (Campillo and Paul, 003; Shapiro and Campillo, 004; Shapiro et al., 005) suggest that a new technique using cross correlation between two receivers in diffuse wave fields can extract Green s function between the receivers. This method can be applied to ambient seismic noise field because very long seismic noise series are diffused by random distribution of ambient seismic noise sources such as oceanic microseism and atmospheric disturbances. Microseism is defined as a faint earth tremor caused by natural phenomena, such as winds and strong ocean waves. The Green s function for two seismic receivers located at the surface is controlled by surface wave propagation structure beneath the two receivers. Then surface wave dispersion information obtained from this noise correlation method can be used for seismic velocity structure estimation in the same manner as traditional observational surface wave analysis. I describe and compare the two methods as a diagram in Figure

72 Figure 3..1 Comparison of two methods for extracting Green s function between two receivers. In the top set of diagram, a delta function at A will be observed as a Green s function at B. A surface wave train at A will be observed at B also, but will be modified. We can calculate the Green s function from the two waveforms. In the lower panels, I show the noise-correlation method where the correlation of ambient noise at A and B results in the Green s function. 60

73 3.. Mathematical Review The mathematical relationship between noise correlation and Green s function was illustrated for homogeneous medium (Roux et al., 005). Here I summarize Roux s (005) long mathematical development in a more easily accessible way. In free space without attenuation, time domain single frequency Green s function between two points is given by R 1 iω ( t ) c e R (3..1) where R is the distance between two points, r1 r, and c is the sound speed and the infinite-bandwidth time domain Green s function is G 1 = R 1 R iω ( t ) c ( r, t; r,0) e dω = δ ( t ) 1 R R c (3..) The definition of cross correlation C ( ) between two points that have noise signals D ( r 1, t) and D ( r, t), respectively, is 1 t C 1 ( t) = D( r1, τ ) D( r, t + τ ) dτ (3..3) The noise field at a given station D ( r, t) (r = station location, t = time) is due to a distributed set of noise sources amplitude S ( r s, ts ) and the Green s function between r s and r, and is given by 61

74 D( r, t) = t S( rs, ts ) G( r, t; rs, t s ) drs dt s = 1 r r s S( r, t s r r c s ) dr s (3..4) If we assume the noise sources amplitude, S r s, t ) are spatially and temporally ( s uncorrelated, the ensemble average for two sources is given by (3..5). Consequently, the cross-correlation function, C ( ) is given by (3..6). In (3..5), P represents spatially and 1 t temporally constant acoustic power of the noise sources. In (3..5) and (3..6), denotes the ensemble average. S( r, t 1) S( r, t ) = Pδ ( r 1 r ) δ ( t 1 t ) (3..5) s1 s s s s s s s C 1 r r 1 s s1 1 ( t) = P δ ( rs 1 rs ) δ ( t + ) dτdrs 1drs r1 rs r rs 1 c c r r (3..6) Note that in the integral of (3..6), the variableτ for the cross correlation disappears due to the assumption of temporally uncorrelated noise sources distribution (3..5). For a signal of finite length, the integration d τ can be represented by the product of the signal duration and the noise source creation rate per unit volume per unit time. Thus, the ensemble averaged cross correlation function can be represented as an integral that involves only the noise source locations. C 1 r r r r δ (3..7) 1 s1 s1 1 ( t) = PF ( t + ) drs 1 r1 rs 1 r rs 1 c c 6

75 After integration over r s, the correlation function is reduced to one integral over noise source locations (3..7). F is the product of signal duration and noise source creation rate per unit volume per unit time in (3..7). In (3..7), the delta function contributes to the correlation function when r r 1 r1 r 1 ct (3..8) s s = Equation (3..8) gives physical insight that the cross correlation function is zero for outside of R / c t R / c, where R is the distance between two receivers. For a given time t, when R / c t R / c, the noise sources lie on a hyperboloid that groups noise sources. All of the noise sources located on the same hyperboloid contribute to the strength of the cross correlation function at a single time point. To perform the integral over dr s1 in (3..7) in association with the hyperboloid system of the noise source locations, the Cartesian coordinates is changed to the prolate spheroidal coordinates. Integration over the curvilinear coordinate ξ, which is a constant at the surface of a given ellipsoid, determines the amplitude of the cross-correlation function. Because we have assumed that the medium has no attenuation, we must use a limited ξ to prevent divergence of the cross correlation resulting in the final form of the ensemble averaged cross correlation function (3..9). { cosh( ) 1} { H ( t + R / c) H ( t R c) } C t) = π PF ξ / (3..9) 1 ( lim where H (t) is the Heaviside step function. 63

76 The forward and backward Green s functions for the two receivers are then obtained from the time derivative of the ensemble averaged cross correlation function (3..10). d C 1 dt ( t) 1 R 1 R = π PF{ cosh( ξ lim ) 1} δ ( t + ) δ ( t ) (3..10) R c R c When the medium has attenuation we can use the same approach as before, but we must change the Green s function. In this case, limiting the curvilinear coordinate ξ is not necessary. However, the time derivative of the cross correlation function does not exactly extract the Green s function when we add attenuation effect to the system. The estimated Green s function is now a low-pass filtered version of the real Green s function. The characteristics of the low-pass filter are determined by the frequency dependence of the attenuation but the physical relationship between the noise correlation function and the Green s function is still valid even if we consider attenuation for the medium. The mathematical verification of the relationship for heterogeneous medium has not been established yet. However, experimental research in ultrasonic (Weaver and Lobkis, 001), acoustics (Roux and Kuperman, 004) and geophysics (Shapiro and Campillo, 004; Shapiro et al., 005) have shown the emergence of the Green s function for heterogeneous medium from the diffuse field correlation Correlation Function and its Time Derivative Another issue of the noise correlation method for extracting Green s function is whether we can regard the correlation function itself as the estimated Green s function or we should use its time derivative. Actually, most of the experimental researches (Derode et al., 003; Roux 64

77 and Kuperman, 004; Shapiro et al., 005) adopt the noise correlation function as the estimation of Green s function rather than the time derivative because performing a time derivative causes extra undesirable noise for the data. Strictly speaking, as shown above, the Green s function is estimated by the time derivative of the correlation function. However, the fact that the overall waveform of the correlation function is maintained after time differentiating and the main difference is only π / phase shift allows us to use correlation function as the estimation of the Green s function. To test the possibility of the use of cross correlation function itself, a random seismic noise field cross correlation function computed from actual field data (the correlation function between BYRD and ISDE during 1999, Feb.) and its time derivative were compared. The two seismic receivers used for this test are located at the surface and thus the extracted Green s function is estimated as a surface waveform. Figure 3.. is the waveforms comparison between the correlation function and its time derivative normalized to the amplitude of the correlation function. We can see that the overall waveforms do not differ from each other and the main difference is the phase shift in the comparison. Additionally, the time derivative produced a little bit of extra noise to the trace. However, it does not affect the main part of the waveform. The group velocities from the cross correlation and its time derivative were then computed using multiple filter technique and compared in Figure 3..3 and it shows the cross correlation function can be replaced with its time derivative for the estimation of the Green s function dominated by surface wave propagation structure. 65

78 Figure 3.. Comparison between the cross correlation function and its time derivative for the noise field method. Bottom is the superposition of the two above. 66

79 Figure 3..3 Comparison between the group velocities computed by the cross correlation function and its time derivative for the noise field method. 67

80 3..4 Application to the Field Data To explore crust and uppermost mantle velocity structure between the stations deployed for ANUBIS network, the noise correlation method was applied to the ANUBIS field data. To construct an ensemble, the cross correlation was performed separately in a window and then the ensemble averaged cross correlation function was calculated by stacking every single correlation function. For the windowing, if the window length is longer than R, where R is c the distance between the two receivers and c is the surface wave velocity, then any window length can be used for the ensemble construction. For the seismic array used in this study, the largest distance between receivers is 904 km and the surface wave velocity is greater than km/sec from global averages, therefore the length of window should be at least 15 minutes. Considering computational efficiency, 1.8-hour and 3.6-hour windows were tested for this study and it was found that the resulted ensemble averages do not differ from each other. To prevent the effects of the magnitude of noise source power, 1-bit digitization was performed to the noise signals before the cross correlations (Shapiro and Campillo, 004). One monthlong ensemble averaged cross correlation function for every possible pair of station records were computed from 9 months of noise signals that do not contain any significant waveforms created by earthquakes (Figure ). In Figure 3..7, the cross correlations between MTM and the other 4 stations are plotted in ± 500 seconds window. We can observe the distances between the stations are matched with the correlation time lags. 68

81 Figure 3..4 Ensemble averaged cross correlation functions computed in the random ambient noise field (period -1999, Feb). 69

82 Figure 3..5 Ensemble averaged cross correlation functions computed in the random ambient noise field (period -1999, Mar). 70

83 Figure 3..6 Ensemble averaged cross correlation functions computed in the random ambient noise field (period -1998, Dec, 1999, Jan, 000, Feb). 71

84 Figure 3..7 Cross correlations between MTM and the other 4 stations. Note that the distances are matched with the correlation time lags. In figure , we can observe good and bad noise levels, and also some one-way propagations and some two-way propagations (Table ). We can hypothesize that the twoway propagation occurs when the ambient noise sources have spatially uniform distribution and the one-way propagation is due to the fact that the noise sources are one- side distributed. Figure 3..8 shows comparisons between the band-pass filtered two cross correlation functions calculated from different month period for two different paths. For each station pair, the cross correlations computed from the two different months are the same, which ensures that the extracted wave form from one month cross correlation represents Green s functions between the stations. Note that the arrival time of the envelope for MTM-SDM pair is approximately 3 times longer than BYRD-ISDE pair. The distance between MTM and SDM 7

85 is 904 km and the distance between BYRD and ISDE is 90 km. Figure 3..8 Comparisons of the band-pass filtered two noise field cross-correlation functions computed from different month period (BYRD-ISDE path (top), MTM-SDM path (bottom)). The last plot in each path is the superposition of the two above for comparison purpose. 73

86 File Path Noise Number Correlation grade of days direction Noise A Noise B 98.dec stk mtm mbl C dec stk mtm sdm B 4 + mtm sdm 98.dec stk mbl sdm D jan stk mtm mbl B 3 + mtm mbl 99.jan stk mtm sdm A 3 + mtm sdm 99.jan stk mbl sdm C feb stk mtm mbl B 1 + mtm mbl 99.feb stk mtm byrd A 1 + mtm byrd 99.feb stk mtm isde A 1 + mtm isde 99.feb stk mtm sdm A 1 + mtm sdm 99.feb stk mbl byrd B mbl byrd 99.feb stk mbl isde C feb stk mbl sdm D feb stk byrd isde A 1 + byrd isde 99.feb stk byrd sdm B 1 + byrd sdm 99.feb stk isde sdm B isde sdm 99.mar stk mtm mbl D mar stk mtm byrd B 4 + mtm byrd 99.mar stk mtm isde B mtm isde 99.mar stk mtm sdm D mar stk mbl byrd B mbl byrd 99.mar stk mbl isde B 4 + mbl isde 99.mar stk mbl sdm D mar stk byrd isde A 4 + byrd isde 99.mar stk byrd sdm D mar stk isde sdm C apr stk mbl isde A mbl isde 00.jan.feb stk byrd sdm B (- stronger) byrd sdm 00.jan.feb stk mbl byrd A (+ stronger) mbl byrd 00.jan.feb stk mbl sdm C feb stk isde sdm B isde sdm Table Summary of the random noise cross correlation measurements. The noise grade A means higher SNR and D means lower SNR. 74

87 The correlation functions were then used to estimate group velocity dispersion curves for surface wave in the same manner as inter-station analysis. Figure shows the group velocity plots estimated by multiple filter technique picked from cross-correlation functions with good SNR; showing the -D amplitudes contour as a function of velocity (right) and period and the spectral amplitude (left). The velocity curves from the different month period for the same path also show consistency and ensure the stabilization in the noise field method. As shown in the plots, the signals are composed of relatively higher frequency component than the frequency components of observational inter- station signals. This means that the ambient seismic noises are not excited with lower frequency bands that can be observed in seismic waves generated by earthquakes. The predominant frequencies of microseisms recorded at the stations located in North America have been reported as in the range of Hz (Alvizuri, 004; Higuera-Diaz, 004). For some paths, higher modes besides the fundamental mode may be observed in the group velocity separations. The fact that waveforms extracted by the noise correlation method contain high frequency information can be a compensation for the observational dispersion analysis because in case of surface waves emitted from earthquakes, velocity information for high frequency bands is easily lost during the long path because of intrinsic attenuation and scattering. It thus is considered that combining the two dispersion curves from the observational method and the noise correlation is a reasonable approach to find shallow seismic velocity structure down to about 50 km. 75

88 Figure 3..9 Group velocity estimations by the waveforms extracted by noise field correlation, BYRD-ISDE path (1999, Feb, 1999, Mar). 76

89 Figure Group velocity estimations by the waveforms extracted by noise field correlation, MTM-BYRD path (1999, Feb, 1999, Mar). 77

90 Figure Group velocity estimations by the waveforms extracted by noise field correlation, MBL-ISDE (1999, Mar), ISDE-SDM (001, Feb). 78

91 Figure 3..1 Group velocity estimations by the waveforms extracted by noise field correlation, MBL-BYRD (1999, Feb, 1999, Mar). 79

92 Figure Group velocity estimations by the waveforms extracted by noise field correlation, MTM-ISDE (1999, Feb, 1999, Mar). 80

93 3.3 Inversion of the Dispersion Data for the Velocity Structure Linearized Inversion (surface wave and receiver function) Once we have measured surface wave dispersion curves, the observed dispersion data can be inverted to the Earth model parameters using the dispersion relation. Our goal of the inversion is to find the Earth parameters that produce the same dispersion curves as the observed dispersion curves. For a vertically heterogeneous structure, the dispersion relation is described by the motion-stress vector equation, and the predicted dispersion curves that correspond to any given Earth parameters set can be calculated by solving the motion-stress vector equation. We vary the Earth parameters and repeat solving motion-stress vector equation to compare the observed and computed dispersion curves until we find the parameters for which the calculated dispersion curves fit the observation. The propagator matrix method is used in the computer codes used in this study (Herrmann and Ammon, 00). Consequently, the structure model is represented as a stack of layers above a half space. Rayleigh wave dispersion depends weakly on P-wave velocity structure, density and attenuation coefficients compared to shear-wave velocities (Bucher and Smith, 1971; Bache et al., 1978). Thus, we can linearize the problem to a reasonable approximation by considering shear wave velocities as the sole unknown parameter. For this linearization, Poisson s ratio σ is fixed in the layers for calculation of P-wave velocities (3.3.1) and Nafe-Drake relation between density and P-wave velocity calculates density. σ = λ ( λ + µ ) = 1 ( V p V s ) ( V V ) p s V p 1 Vs = V p Vs 1 (3.3.1) 81

94 Parameter Space Inversion Algorithm The dispersion relation system is given by ),,, ( 1 n m m m F U L = (3.3.) where n m is the Earth parameters (shear velocities in each layer in this study), U is predicted dispersion curves (phase or group velocities) that corresponds to a set of n m and F represents functional relating n m to U. For receiver function inversion, U will be the waveform of the receiver function. If we linearize ),,, ( 1 1 n m n m m m m m F L by expanding it in a Taylor series and discarding all second and higher order terms; n n n n n m m F m m F m m F m m m F m m m m m m F = L L L ),,, ( ),,, ( (3.3.3) Let s consider trial Earth parameters model { } m n m m,,, 1 L, and say ),,, ( 1 1 n m n m m m m m F L fits the observed dispersion curve. Then, n n n n n m m F m m F m m F m m m F m m m m m m F D = = L L L ),,, ( ),,, ( (3.3.4) n n n m m F m m F m m F m m m F D = L L ),,, ( (3.3.5)

95 83 where D is the observed dispersion data. Equation (3.3.5) can be written in a matrix form as Ax y = (3.3.6) where = = = n n k k k n n k k m m m m F m F m F m F m F m F m F m F m F F F F D D D M L M L L M M Ax y (3.3.7) k is the number of periods or frequencies for the dispersion curve and n is the number of layers. In (3.3.6), the unknown is x and if we solve it in a least square approach, the least square solution of the equation (3.3.6), x, can be determined from the singular value decomposition analysis and { } m n m n m m m m + + +,,, 1 1 L is the estimated model parameters that fits the observed dispersion data. Equation (3.3.6) is iteratively solved to attain to an accurate fit to the observed data by minimizing the norms of i m, n i,, 1 L = (Aki and Richards, 1980; Herrmann and Ammon, 00). In the singular value decomposition T V U A Λ =, the matrix V, whose columns are the eigenvectors of A A T, can be decomposed into p V, the eigenvector matrix for the non-zero eigenvalues, and 0 V, the eigenvector matrix for the zero eigenvalues. The matrix T PV P V is known as the resolution matrix and the columns of the resolution matrix are termed resolving kernels. The resolving kernel tells us that how a unit perturbation of the model parameter at a certain layer for the true model affects other layers through the inversion. The resolution matrix blurs the true model and when the resolving kernel is sharply peaked at the

96 corresponding depth, the resolution of the inversion is best. The uncertainties of the estimated model parameters can be determined from the errors in the dispersion observations and the resolution of the solution. There is a trade-off between the resolution and the variances of the solution due to the errors in data. A poor resolution reduces the variances of the model parameter estimates, and vice versa. In order to keep the variances of the solution small, some strategies sacrificing the resolution are generally adopted. In a strategy used by Wiggins, eigenvectors with small eigenvalues were eliminated to keep the variance below a certain level (Wiggins, 197). In the Levenberg-Marquadt method (Levenberg, 1944; Marquadt, 1963), the diagonal elements of the matrix Λ are modified to λ i λ i + Φ, where λ i = non-zero eigenvalue, i = 1, L, p, and the contributions of eigenvectors with eigenvalues smaller than Φ are suppressed (Aki and Richards, 1980). The second method is known as the damped least square solution and Φ is called a damping parameter. The inversion codes used in this study adopt the damped least square approach (Herrmann and Ammon, 00). The partial derivatives can be computed with a finite difference approximation and it is the most expensive part of the inversion in computational efficiency. Therefore, various algorithms to reduce the computational times are adopted for the computation of the partial derivatives. For receiver function, the partial derivatives can be efficiently computed by Randall (1989) s algorithm based on the reflection matrix technique (Kennett, 1983). For surface wave dispersion inversion, the partial derivatives of phase velocity with respect to the model parameters in A are given in terms of so-called Fréchet kernels (Dahlen and Tromp, 1998), c c = K β uω β (3.3.8) 84

97 where c is the phase velocity, u is the group velocity, and K β is the Fréchet kernel for the Earth model parameter. The Fréchet kernel is the ratio of the perturbation in the eigenfrequencies to the perturbation in the Earth parameter and it is computed from the unperturbed eigenfunctions. The equation for the Fréchet kernel is derived from applying Rayleigh s principle to the surface wave system. Also, the partial derivatives of group velocity with respect to the parameters (3.3.10) (Rodi et al., 1975) can be obtained by the relationship between u and c (3.3.9). c u = ω c 1 c ω (3.3.9) u = β u u c u c + ω (3.3.10) c c β c ω β For a given period, the partial derivatives are determined at each layer depth and represent how much the phase or group velocity at that period changes for the same shear velocity perturbation at each depth. The depth at which the partial derivative peaks tells us that the phase or group velocity at that period is most sensitively affected by the shear wave speed at that depth Estimation of the Velocity Models and Discussions The higher frequency range of the dispersion obtained from the ambient seismic noise field correlation indicates it only contains information about shallow subsurface velocity structure. However, the short period dispersion from the noise correlation method can be useful when 85

98 dispersion is not observable for the short periods from the teleseismic measurements. Motivated by this complementarity, the dispersion curves obtained by the teleseismic observations and the noise correlation methods were combined into one dispersion curve for each station pairs and inverted for the local shear wave velocity structures. For the teleseismic dispersion curve preparation, four paths were selected (Figure 3.3.7) based on the distribution of earthquakes. Consequently, the dispersion curves using the noise field method for the same four paths were combined to the teleseismic dispersion curve. The inversion was performed using Computer Programs in Seismology (Herrmann and Ammon, 00). The estimated shear velocity models, the data fit for 4 pairs of stations and the resolving kernels are shown in Figure The standard errors of the estimated model parameters determined from the errors in the dispersion observations and the resolution of the solution as mentioned in the previous section are listed in Table 3. Standard error (km/sec) Path Average Maximum Minimum ISDE-BYRD (99058) (depth.5km) (depth 57.5km) BYRD-MTM (99058) (depth 7.5km) (depth 57.5km) MBL-ISDE (00344) (depth 7.5km) (depth 57.5km) ISDE-SDM (00344) (depth.5km) 0.08 (depth 57.5km) SDM-ISDE (00345) (depth.5km) (depth 57.5km) ISDE-MBL (01015) (depth.5km) (depth 57.5km) Table 3 Standard errors of the estimated velocity models. The largest standard errors among the paths are observed in the path MBL-ISDE (00344) and the path ISDE-MBL (01015), and the average value of the error in each layer is about 0.1 km/sec for both paths. The inverted velocity models for the 4 regions are plotted together in Figure The results in Figure show that the crustal thickness for the 4 regions is estimated as 0-30 km by the shear wave velocity gradients in the estimated models. This is 86

99 consistent with the crustal thickness estimation by minimizing L1 norm of the error between modeled and recorded receiver functions (Winberry and Anandakrishnan, 004) and indicates that the crust in this region has been thinned during rift extension. The thin crust throughout the West Antarctic Ice Sheet region is consistent with the interpretations that the main phase of the extension in the WARS between East and West Antarctica was in the late Cretaceous (Lawver and Gahagan, 1994; Fitzgerald, 003). The region D (BYRD-MTM) shows a little bit higher S-velocity model compared to the region A, B, and C. This result implies the region D is colder than the region A, B and C and is consistent with the other lower resolution velocity models of the Antarctica. (Ritzwoller et al., 001; Sieminski et al., 003). 87

100 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (ISDE-BYRD). 88

101 Figure 3.3. Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (BYRD-MTM). 89

102 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (MBL-ISDE). 90

103 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (ISDE-SDM). 91

104 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (SDM-ISDE). 9

105 Figure Estimated shear velocity model (upper left), the dispersion curve fit (upper right), and the resolving kernels (ISDE-MBL). 93

106 Figure Estimated shear velocity models for the four regions in the WARS. 94

107 Chapter 4 MANTLE ANISOTROPY MEASURED BY SHEAR WAVE SPLITTING 4.1 Anisotropy and Shear Wave Splitting Anisotropy is a property of the medium, in which elastic properties for the medium vary with direction. For an anisotropic medium the elastic stiffness tensor has 3 to 1 constants depending on the symmetry structure of the medium while an isotropic material only has constants called the Lamé parameters. Theoretically, general velocity profiles for a certain anisotropic medium can be determined by solving Christoffel s equation (4.1.1), which represents relationship between elastic constants of the material C ijkl and velocity vector V and particle displacement polarization vector p ik (Rose, 1999). ( C V V c δ ) p = 0 ijkl j l ρ (4.1.1) ik ik Christoffel s equation is equivalent to an eigenvalue problem because it has the form ( I ) X = 0 A λ. For a fixed direction of the velocity vector, the eigenvalues of the acoustic tensor C ijkl V V correspond to phase velocities while the eigenvectors specify the polarization j l of the waves. By repeating this process for every propagation direction we can get entire map of phase velocity for a given anisotropic material. The acoustic tensor is a 3 by 3 matrix and we obtain three solutions for a given propagation direction; most generally the solutions include the single quasi-p wave and two orthogonal quasi-s waves. For example, the phase velocity profiles for titanium (hexagonal anisotropy) and olivine (orthorhombic anisotropy) were computed by solving Christoffel s equation and plotted in Figure The outer ring is the P-wave velocity and the two inner rings correspond to the two orthogonal S-wave 95

108 velocities. For hexagonal anisotropy, the fast axis is along with the Z-axis and the slow axis becomes any direction in the XY plane. Figure Phase velocity profiles for hexagonal anisotropy (titanium) and orthorhombic anisotropy (olivine). The distances from origin are the velocities. Shear wave splitting is the phenomenon in which the propagating S wave energy splits into two orthogonal components corresponding to the fast and slow axes of the medium. These phases travel at different velocities and after sufficient distance can separate enough to be individually identified. Therefore, shear wave splitting occurs only when the two quasi-s waves have different speed for the given propagation direction. If there is no difference between the two quasi-s wave speeds in the propagation direction, splitting does not occur 96

109 even if the material is anisotropic (Figure 4.1.). In the example of hexagonal anisotropy in Figure 4.1.1, when the propagation direction is in the XY plane the two quasi-s waves have different speed and the splitting occurs. However, if the propagation direction is along with the Z-axis, there is no difference between the two quasi-s wave speeds and the splitting does not occur. Figure 4.1. Illustration of shear wave splitting in hexagonal anisotropic medium. 97

110 In seismology, the most commonly considered type of anisotropy is hexagonal anisotropy. Hexagonal anisotropy is divided into two categories according to the symmetry axis orientation (Stein and Wysession, 003). The case of a vertical symmetry axis is called transverse isotropy (or radial anisotropy) and the case of a non-vertical symmetry axis is called azimuthal anisotropy. Transverse isotropy has been hypothesized in the reason for discrepancies between Love and Rayleigh wave velocities (Anderson, 1961) and is often found in reflection seismic surveys of sedimentary layering in the crust (Shearer, 1999). To measure azimuthal anisotropy using horizontally traveling phases such as Pn, seismograms at many different azimuths are required. However by using shear wave splitting of a more vertically propagating phase, azimuthal anisotropy can be detected using a single seismogram, which leads to simpler and valuable way to observe azimuthal anisotropy. If we observe both transverse isotropy and azimuthal anisotropy at the same place, they might be combined to derive orthorhombic anisotropy model. 4. Inverse Method for Shear Wave Splitting Estimation If we assume a single vertically homogeneous anisotropic layer where the axis of symmetry of the anisotropy is horizontal, we can predict magnitude of shear wave splitting for a vertically propagating shear wave. For a given initial polarization angle, the projection of that angle onto the fast and slow directions of the anisotropic layer determines the amount of splitting. If the initial polarization angle is along either the fast or slow directions, we cannot detect the splitting (defined as non-detectable). If the angle between the polarization direction and the fast direction is non-zero, then the incident shear wave will propagate in quasi-shear waves at different speeds. Depending on the thickness of the anisotropic layer (and the difference in propagating velocity of the quasi-s waves), the phases will separate by an 98

111 amount δ t. The polarization directions of the two phases (the early arriving phase and the late arriving phase) are the fast and slow directions of the anisotropic layer. Note that a fundamental ambiguity arises from the magnitude of the anisotropy and the thickness of the layer the two are indistinguishable by this modeling. Then how can we estimate the parameters of shear wave splitting (fast polarization direction and time delay between faster and slower pulse) from the seismograms? The most commonly used method on a single seismogram at a station is due to Silver and Chan (1991). The method of Silver and Chan operates upon the principle that the best splitting parameters correspond to a certain inverse splitting operator that best linearizes the shear wave particle motion when the effect of propagation through the anisotropic medium is removed. For measurement of general S waves, the linearity of particle motion can be measured by eigenvalues of the covariance matrix between any two orthogonal components. For the correcting process, radial and transverse component pairs are rotated to all candidate fast and slow polarization angles then time-sifted again by all candidate time delay between faster and slower pulses. If the polarization is completely linearized after correction, the covariance matrix has one non-zero eigenvalue which corresponds to the polarization direction and one zero eigenvalue which corresponds to the direction orthogonal to the polarization direction. The eigenvalues thus represent sum of the squares of the signal for the each component. In the presence of noise, we cannot find the singular corrected covariance matrix and we seek the covariance matrix that is most nearly singular. Searching for the most nearly singular corrected covariance matrix can be done by maximizing the larger eigenvalue or, equivalently, minimizing the smaller eigenvalue. In case of splitting measurement with SKS, P-wave in the core is converted to SV wave at core mantle boundary, which means that the upgoing S-wave is radially polarized in the mantle on the receiver side of the core. In this case, the energy (sum of the squares of the 99

112 signal) on the corrected transverse component can be minimized instead of the smaller eigenvalue because it corresponds to the smaller eigenvalue that is minimized if the input polarization direction is unknown. To find the best splitting parameters, a grid search over candidate φ (fast polarization direction) and δ t (time delay between faster and slower pulse) values is used. The error analysis is performed using an inverse F-Test on the energies of corrected transverse components to construct a confidence region for each set of shear wave splitting observations. The noise in the seismogram is considered as Gaussian and thus the minimized energy on the corrected transverse component can be assumed to be Chi-square distributed. The 95% confidence region is constructed as the region within the values of corrected transverse energy E defined by T min k 1 ET ET 1 + F (0.95; k, v k) (4..1) v k (Jenkins and Watts, 1968; Bates and Watts, 1988; Silver and Chan, 1991). In (4..1), the variable for the F probability distribution is E T E E min T min T v k k, and F 1 (0.95; k, v k) is the inverse of the F distribution, which is the value of the variable corresponding to the probability 0.95 of exceeding F 1 (0.95; k, v k). k is the number of parameters ( = ) and v is the degree of freedom. According to the properties of Chisquared distribution, the degree of freedom is estimated from a sample noise process by ( E[ Y ]) Var[ Y ] v = (4..) 100

113 where Y is the sum of squares of the noise process, which is assumed to be approximately Chi-square distributed, E [ Y ] is the mean and [ Y ] Watts, 1968; Silver and Chan, 1991). var is the variance of Y (Jenkins and 4.3 Data and Measurements Shear wave splitting measurements were conducted for the West Antarctic Ice Sheet (WAIS) region in the West Antarctic Rift System (WARS). Broadband recordings from Nov to Nov. 001 at five stations installed on West Antarctica as a part of Antarctic Network of Unattended Broadband Seismometers (ANUBIS) network were used for the measurements (Anandakrishnan et al., 000). Most station uptime ranges from October to April because the power was mainly supplied by solar-energy. All events with distance greater than 90º and m b greater than 5. were searched and the isolated, well-recorded teleseismic SKS phases were selected based on the predicted SKS travel times from the IASPEI91 earth model (Kennett, 1991). 4 events were used for this study and the distance range is 90º~138º (Table 4). 101

114 Event Time Latitude Longitude Depth b Station USED 1998/1/ :47: ~6.6 SDM 1999/01/ :19: MTM,BYRD 1999/01/ :10: MTM,BYRD,MBL,SDM 1999/03/ :5: MBL 1999/03/ :05: MBL,ISDE 1999/11/ :05: MBL 1999/1/ :03: BYRD 1999/1/ :30: BYRD,MBL 1999/1/ :48: BYRD,MBL 1999/1/ :19: BYRD,MBL 000/01/ :1: BYRD 000/03/ :00: BYRD,MBL 000/06/ :57: MBL 000/06/ :46: MBL 000/06/ :31: MBL 000/06/ :3: MBL 000/08/ :7: MBL 000/09/ :49: MBL 000/1/ :11: MBL,SDM 000/1/ :13: MBL,SDM 001/01/ :57: MBL,SDM 001/01/ :30: MBL,SDM 001/01/ :0: SDM 001/01/ :33: SDM m Table 4 List of events used in shear wave splitting measurements. 10

115 Measurement was made in minimizing energy E T on corrected transverse component and 95% confidence region was calculated. The confidence region in plot of the energy on corrected transverse components is considered as approximately Gaussian and one half of the 95% confidence bounds was taken as 1σ uncertainties for fast direction and time delay. For grid search, o 5 for φ step size and 0.3 sec for δt step size were used. To get averaged value of the estimated splitting parameters we take weighted average (4.3.1) of detected splitting parameters that have reasonable confidence region (Bevington and Robinson, 003). φ = ( φi / σ i ) = 1, σ (1/ σ ) & φ (4.3.1) (1/ σ ) i i The average ofδt was estimated in the same manner as (4.3.1). We define a category of measurements as non-detectable when there is good SNR, but the corrected transverse energy contours are broad and ambiguous. The reasons for this are that either of the fast or slow polarization direction lies along the back azimuth or the anisotropy is zero. However, if we observe many other splitting measurements at a station, we reject the possibility of an isotropic mantle. Therefore, we assign non-detectable measurements where the back azimuth is equal to an independently measured fast slow axis as either fast or slow, but with no way to measure δ t. To get the error bound for the fast polarization angle assumption obtained from non-detectable measurements, synthetic experiments were performed. For the synthetic experiment, actual field noise samples are added to one cycle of Hz sine wave signal, which simulates the field data in regard of SNR and signal frequency. Then the fast polarization angle from a fixed back azimuth was slightly varied and the angle at which splitting was detectable is determined. It was found that the splitting is generally observable when the back azimuth is ±10 from the fast or slow axis. 103

116 Therefore, the standard deviation for the non-detectable based fast polarization direction was set to 10. The non-detectable measurements then were included in the averaging. The detectable and non-detectable measurements are exemplified in Figure In Figure 4.3.1, panel A shows superposition of fast and slow components uncorrected and corrected. After correction, anisotropy has been removed and there is no difference in arrival time between fast and slow polarization. Panel B is radial and transverse components uncorrected and corrected. It shows that the energy on transverse component has been minimized after correction. Panel C is corresponding particle motion diagrams. The elliptical particle motion due to splitting becomes linear when corrected. Panel D shows contour plot of the energy on corrected transverse components and 95% confidence region (dotted line). Dot marks measured parameters (fast polarization direction and delay time) that minimize the energy. Panel E is a typical non-detectable observation for contour plot of the energy on corrected transverse components. Panel F is radial and transverse components for the nondetectable observation. Note that no splitting effect appears in original transverse component. For this non-detectable example, back azimuth is 4 and possibly lies on slow polarization direction. 104

117 105

118 Figure Patterns of the detectable and the non-detectable shear wave splitting measurements. Detectable case (A,B,C,D - MBL, event 99315) and non-detectable case (E,F - BYRD, event 99345). A. Superposition of the fast and slow components (before and after correction). B. Radial and transverse components (before and after correction). C. Particle motion diagrams (before and after correction). D. Contour plot of the energy on corrected transverse components and 95% confidence region (dotted line). E. Typical non-detectable observation for contour plot of the energy on corrected transverse components. F. Radial and transverse components for the non-detectable observation (before and after correction). 106

119 4.4 Noise Filtering and Phase Degrading When the output of a seismometer is subject to noise, band-pass filtering is needed to reduce the noise. However, often the dominant frequencies of seismic signal coincide with those of noise. Then filtering may not improve the SNR (Signal to Noise Ratio) and rather degrade seismic signal itself so that it affects measurement results. Especially, for the shear wave splitting measurements, if the back azimuth is close to the fast or slow axis one of the polarizations is weak and we expect significant phase amplitude attenuation of the weakly polarized component. In the portion of the data signals before seismic wave arrivals, it was observed that the noise frequency bands are in the range of 0.1~0.3 Hz which coincide with phase frequencies, which are close to 0.1 Hz. To examine the aspects of phase degrading due to the noise filtering, 10 different 4-pole Butterworth filters with varying upper cutofffrequency from 0.1 to 1.0 Hz and fixed lower cutoff-frequency of 0.01 Hz were applied to the signals and the compromise between noise removal and phase degrading was carefully observed and compared. For cases where the back azimuth is well-separated from the fast or slow axis so-called detectable cases, an upper cutoff of f = Hz resulted in good measurements. However, if the back azimuth and either of the fast or slow axis are close, we observe significant phase degrading especially for the weakly polarized component with good SNR. This aspect is exemplified in Figure and In Figure 4.4.1, the measurement with Hz filtering gives clearer diagnostics than the measurement with Hz filtering. The Hz filtering produced non-detectable pattern. The back azimuth for this case is 31 and close to the computed slow axis angle of 55. However, for signals with relatively low SNR signal, it was observed that the noise filtering with upper cutoff frequency up to 0.15 Hz is effective in removing the noise to produce better diagnostics than no noise filtering. The latter case is exemplified in Figure Figure u 107

120 is the comparison of the 95% confidence bounds for the estimated fast axis direction and the time delay between the two cases. When the phase degrading effect occurs the confidence bounds converge as the upper cutoff frequency approaches 1.0 Hz. In this case noise filtering is not necessary and disguises the detectable pattern with non-detectable one. For the case that noise filtering is effective, the confidence bounds converge as the upper cutoff frequency approaches 0.1 Hz. 108

121 Figure Noise filtering aspect for phase degrading. Without noise filter (left) and with noise filter (right) (MBL, event 99315). 109

122 Figure 4.4. Noise filtering aspect for effective noise filtering. Without noise filter (left) and with noise filter (right) (MBL, event 99067). 110

123 Figure % confidence bounds variation for the estimated fast axis direction and the time delay. Phase degrading case (top) and effective noise filtering case (bottom). 111

124 4.5 Shear Wave Splitting Estimation for WARS Region MBL For station MBL, which has most number of records, four detectable and twelve nondetectable measurements were observed. From the detectable measurement we find φ =184.8±1.59 and δ t =1.±0.08 s. More than half of the back azimuths for this station are concentrated in the vicinity of 70. Based on the weighted average of the detected fast polarization direction, we assume that the non-detectable measurements with back azimuths of approx. 70 are traveling in slow polarization direction. Back azimuths of the remaining non-detectable measurements are close to the fast polarization direction of the detectable measurement. Thus, we conclude that the fast polarization direction from the detectable measurements is consistent with the non-detectable measurements. With regard to filtering effect, event is observed that as lowering upper-cutoff frequency confidence bound converges and the diagnostics becomes clear. For the other three detectable measurements (99315, 00159, 00357), best confidence bound convergence and diagnostics are observed with higher upper-cutoff frequency filtering and it produces non-detectable measurement results in case of lower cutoff frequency filtering. Back azimuths for the 4 detected measurements are close to the estimated slow axis. It means that the projection to the fast axis is weaker than to the slow axis and more easily attenuated with lower upper cutoff frequency so that it could be seen as not split. However, the SNR for is relatively low compared to the others and the noise- removing filter is effective to get reliable splitting detection. For the non-detectable cases, the multiple minima or unreasonable confidence bounds of the estimated parameters with higher upper cutoff filtering change to the clear non-detectable patterns with lower upper-cutoff frequency filtering. 11

125 BYRD For station BYRD, we obtained total eight records and observed three consistent detectable splitting measurements and three non-detectable measurements. In the detectable measurements, φ =149.63±.16 and δ t =0.75±0.06 s. All of the detectable measurements showed consistency in all the different higher cutoff filtering. The two back azimuths for nondetectable measurements (4 and 47 ) are making near 90 with the averaged detectable fast polarization direction and one (318 ) is making near 180 with the detected fast polarization direction, which supports the estimated fast polarization direction. SDM For station SDM, five measurements showed typical non-detectable splitting patterns and only two measurements showed detectable splitting pattern with reasonable confidence region. In the detectable measurements, φ =154.5±5.47 and δ t =1.84±0.35 s. All of the measurements showed similar pattern in all of the different higher cutoff filtering. The back azimuths for the two non-detectable measurements are in the vicinity of the estimated fast polarization direction and the back azimuths of the three non-detectable measurements are close to the estimated slow polarization direction. MTM and ISDE Station uptime for MTM and ISDE was relatively short and we found only two SKS phases for MTM and one for ISDE. For MTM, we made one detectable measurement with reasonable confidence region and the measured φ = 149±10 and δ t =0.9±0.4 s. SNR for this detectable measurement is relatively high and the results are consistent in all the different filtering. One more measurement for MTM, which is in low SNR, showed two minima of estimated parameters with whole energy filter and with the noise-reducing filter it turned out 113

126 to be non-detectable measurement pattern. The back azimuth of this non-detectable measurement (311 ) is 18 different from the detected fast polarization direction. The only one measurement for ISDE showed typical non-detectable splitting pattern and the back azimuth is 18. But one non-detectable measurement for ISDE is not enough to estimate splitting pattern. Because of lack of the data for these two stations, it is hard to estimate the splitting results for these two stations. However, we include the results from one detectable measurement for MTM in the map of the averaged splitting results. The results of all measurements are listed Table 5. The weighted averages of splitting parameters for the 4 stations are listed in Table 6. The weighted averages (including the nondetectable measurements) of φ and the weighted averages of δ t are plotted in Figure The estimated fast polarization directions and the delay times for MBL, SDM and BYRD are plotted as a function of back azimuths in Figure Solid circles represent detectable measurements and triangles represent non-detectable measurements. For the triangles, the fast polarization direction is plotted as back azimuth or back azimuth + 90 depending on whether the back azimuth is close to fast or slow axis estimated from the detectable measurements. Solid lines represent averaged fast and slow polarization direction from detectable and non-detectable measurements. Note that the back azimuths of the nondetectable measurements are near the estimated fast or slow polarization direction. Error bars indicate 1σ uncertainties. 114

127 station event baz Fast polarization direction dt(sec) quality (degree from N) MBL ± ±0.77 GOOD(FILTER ) MBL ± ±0. GOOD(FILTER 11) MBL ± ±0.6 GOOD(FILTER 11) MBL ± ±0.38 MEDIUM(FILTER 11) MBL Non-detectable MEDIUM(< FILTER 8) MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable GOOD MBL Non-detectable MEDIUM( up to 0.15 UCF) MBL Non-detectable MEDIUM(unreasonable c.b) MBL Non-detectable at 0.11 UCF MEDIUM BYRD ± ±0.47 GOOD(FILTER 11) BYRD ± ±0.15 GOOD(FILTER 11) BYRD ± ±0.53 MEDIUM(FILTER 11) BYRD Non-detectable at all UCF GOOD BYRD Non-detectable at all UCF GOOD BYRD Non-detectable MEDIUM (< 0.5 UCF ) SDM ±16 1.9±1.0 GOOD(FILTER 6) SDM ± ±1.0 GOOD(FILTER 6) SDM Non-detectable at all UCF GOOD SDM Non-detectable at all UCF GOOD SDM Non-detectable at all UCF GOOD SDM Non-detectable at all UCF GOOD SDM Non-detectable at all UCF GOOD MTM ±10 0.9±0.4 FILTER 11 MTM Non-detectable at all UCF GOOD Table 5 The results of all shear wave splitting measurements. Station φ (incl. non-detectable) φ (not incl. non-detectable) δ t SDM ± ± ±0.35 s MBL ± ± ±0.08 s BYRD ± ± ±0.06 s MTM 149±10 0.9±0.4 s Table 6 The weighted averages of the splitting parameters. 115

128 Figure Averaged shear wave splitting parameters (the fast polarization direction and the time delay) at the four stations. The fast polarization directions at four stations are almost orthogonal to the inferred strike of the rift and parallel to the North-South elongated extension between East and West Antarctica. The time delay is represented by the length of the line. 116