structural imaging using scattered teleseismic body waves

STRUCTURAL IMAGING USING SCATTERED TELESEISMIC BODY WAVES

Michael BostockDepartment of Earth & Ocean

SciencesThe University of British Columbia

ESIW-UCBerkeley, June 23, 2011

LECTURE - OUTLINE

IntroductionGeometrical ConsiderationsSource-signature separation &

DeconvolutionOne-dimensional studiesMulti-dimensional studies

INTRODUCTION

high, frequency 0.1-4 Hz teleseismic body wave scattering

highest potential resolving capability of any component within the global seismic wave train

close analogy to reflection seismics

RECEIVER FNS vs SEISMIC REFLECTION - SIMILARITIES

> near-vertical wave propagation

> sub-horizontal stratification

> modest velocity contrasts

> single-scattering (Born) approximation

GLOBAL vs EXPLORATION SEISMOLOGY - GEOMETRIES

GLOBAL vs EXPLORATION SEISMOLOGY - 2

> exploration studies have adopted acoustic approximation to model pure P-P back-scattering interactions (explosive source / vertical sensors)

> computationally and practically expedient

GLOBAL vs EXPLORATION SEISMOLOGY - 3

> global studies rely principally on elastic, forward-scattering interactions

GEOMETRICAL CONSIDERATIONS

> Key definitions

> Teleseismic P

> Teleseismic S

> Other phases

KEY DEFINITIONS

> Teleseismic wave: body wave recorded at epicentral distance > 30 degrees

> Incident wave: contribution associated with primary body-wave phase reflected/ converted if at all only at Earth’s surface and/or core-mantle boundary (e.g. P, pP, PP, S, pS, PKP, SKS, ScS)

CANDIDATE INCIDENT WAVES

MORE KEY DEFINITIONS

> Scattered wave: contribution to teleseismic wavefield generated through scattering of incident wave from receiver-side structure

> Source: source signature and scattering from source-side structure

TELESEISMIC P - 1

most generally useful phase in receiver function studies

propagation in lower mantle ( ) simple vs propagation in transition zone ( ) that gives rise to triplicated interfering phases

UPPER MANTLE TRIPLICATIONS

> Erdogan & Nowack, 1993, PAGEOP, 141, 1-24

TELESEISMIC P - 2

for slowness is single valued, monotonically decreasing function of (0.08 s/km to 0.04 s/km between 30 and 100 degrees

near vertical propagation, less probability of critical reflection

wavefront curvature small; adopt plane wave approximation

DEPTH PHASE COMPLICATIONS

Depth phases dealt with in 2 ways:

a) at shallow depths, depth phases have slowness similar to incident wave; consider part of source

b) at greater depths, slowness differences increase but interference reduced through larger time separation and short source time functions

Transition occurs at depths between 100-200 km

AK135 (Kennett) - EVENT DEPTH : 100 km

TELESEISMIC STraditionally less useful owing to:

a) more limited distance (slowness) range

b) larger slownesses, closer to criticalc) interference between S, SKS, ScS

between 70-90 degreesd) variable source-side polarization

imprinte) lower frequency contentf) higher signal-generated noise levels

Farra & Vinnik, 2000, 141, 699-712

S-TELESEISMIC RECEIVER FUNCTIONS

QuickTime™ and a decompressor

are needed to see this picture.

S - Receiver Functions & the LAB

Rychert et al, 2007

renewed interest in S owing to its utility in identifying shallow mantle discontinuities unobscured by the crustal reverberations

RECEIVER FUNCTIONS FROM REGIONAL P

Park & Levin, 2001, GJI, 147, 1-11

Levin & Park, 2000, Tectonophysics, 323, 131-148

PKP - RECEIVER FUNCTIONS

SOURCE SIGNATURE SEPARATION & DECONVOLUTION

CONVOLUTIONAL MODEL - 1

> : observed displacement seismogram

> : effective source (includes source-side scattering)

> : Green’s function (receiver side response to an impulsive plane wave with horizontal slowness )

CONVOLUTIONAL MODEL - 2

> observation index; source index (implicitly assumed)

> separation of is canonical problem in seismology

> first step is ``modal decomposition’’: isolation of P, Sv, Sh contributions

MODAL DECOMPOSITION> renders wavefield minimum phase

> 3 approaches:

1. Cartesian Decomposition

2. Covariance Eigenvector Decomposition

3. 1-D Slowness Decomposition

CARTESIAN DECOMPOSITION> for steeply propagating teleseismic waves, modal components are approximately separated on vertical and horizontal components

> Langston, 1979, JGR, 94, 1935-1951



EIGENVECTOR DECOMPOSITION

> rotate the particle motions to a coordinate system where maximum linear polarization is mapped to one component

> accomplished through diagonalization of the displacement covariance matrix, e.g.

> approximate (cf. plane waves in 1-D media)

> Vinnik, 1977, PEPI, 15, 39-45

P

S

PPP PPS PSS P PS

OBLIQUE RAYPATHS/POLARIZATIONS IN 1-D

1-D SLOWNESS DECOMPOSITION> assume 1-D media, then

where F is a fundamental matrix (e.g. Kennett, 1983, CUP)

> at free surface traction vanishes, so recast to recover upgoing wavefield

> leads to definition of free surface transfer matrix which for isotropic media is

> requires a priori knowledge of surface velocities and horizontal slowness; can be assessed by examination of first motions at time 0

> Kennett, 1991, GJI, 104 153-163; cf. Svenningsen and Jacobsen, GRL, 31, doi:2004GL021413

1-D SLOWNESS DECOMPOSITION

SCATTERING GEOMETRY

PS

PPP PPS PSS P PS

> consider plane P wave incident from below

> receiver side scattering includes forward and back scattering, P and S

> legs ending in P and S isolated through modal decomposition

MINIMUM PHASE 1> 1-D, 2-layer isotropic model, impulsive source

> 1-D slowness decomposition exact

> note dominance of direct wave at early time

> assert that P-component is minimum phase

RECEIVER FUNCTIONS & DECONVOLUTION

Since P-component is minimum phase and dominated by direct wave at time 0, it can be used as an estimate of source time function and deconvolved from Sv, Sh components to produce estimate of S-contributions to Earth’s Green’s function

GREEN’S FN vs RECEIVER FN

> Receiver function is a leading order approximation to Green’s function

> P-component captures direct wave

> S-component captures 1st order scattered wavefield

IMPROVED RX FNS - MOTIVATIONS-component of P receiver function comprises conversions sensitive to combinations of , e.g.

P-component of P Green’s function contains information on , e.g.

improved estimate of Green’s function would allow tighter constraints to be applied to lithological interpretation, and would narrow the gap between active and passive source studies

Improved representation of Earth’s Green’s Function involves blind deconvolution

WATER-LEVEL DECONVOLUTION > water-level deconvolution introduced by Clayton & Wiggins

> for small c approaches deconvolution, large c approaches scaled cross-correlation

> similar to damped least squares solution

SIMULTANEOUS DECONVOLUTION

> when large numbers of seismograms representing a single receiver/Green’s function are available, perform simultaneous, least-squares deconvolution

> advantageous due to fact that smaller sum of spectra in denominator reduce likelihood of spectral zeros allowing for smaller values of water level parameter to be used

> Gurrola et al, 1995, GJI, 120, 537-543

EXAMPLES - STATION HYB

Kumar & Bostock, 2006, JGR, 111, doi:10.1029/2005JB004104

1-D INVERSION

> 3 categories:

1. Least-squares inversion

2. Monte Carlo / Directed Search

3. Inverse Scattering

LEAST-SQUARES INVERSION

> receiver function inversion cast in standard inverse theory framework

> less expensive than MC/DS methods and makes less stringent demands on data than inverse scattering methods

> data insufficiency compensated for by regularization (e.g. damping)

> like MC/DS methods LS involves model matching so there is no formal requirement that data are delivered as Green’s functions (e.g. receiver function is adequate); only a forward modelling engine is strictly required

LEAST-SQUARES IMPLEMENTATION

> string receiver function or series of receiver functions end-to-end in vector d in either time or frequency domains and write as:

where is (non-linear) forward modelling operator operating on elasticity c

> can be represented through layer matrix methods (Haskell, 1962, JGR, 67, 4751-4767 - exact but expensive) or ray methods (Langston, 1977, BSSA, 67, 1029-1050 - cheap but incomplete)


> address non-linearity in inverse problem by expanding receiver function as Taylor series about starting model

> rearrange, discard non-linear terms and write in matrix form as

where is data residual vector

is sensitivity matrix


> solve in standard fashion with desired regularization, e.g.

and iterate until convergence to address non-linearity

> since receiver functions are sensitive to short-wavelength structure, generally taken to represent slowly/rapidly varying component of velocity model, respectively

> receiver functions often combined with surface wave dispersion data to constrain long wavelength structure

LEAST SQUARES EXAMPLE

Julia et al., 2000, GJI, 143, 99-112

MONTE CARLO / DIRECTED SEARCH

> feasible owing to high-performance computing and relatively few model parameters in 1-D inversions

> no need for derivative ( ) calculation, meaning one can define an arbitrary measure of misfit

> global in nature and so less apt to identify local misfit minima as solutions

> pure Monte Carlo rarely used since inefficient, rather use directed search algorithms:

1. genetic algorithms2. nearest neighbour

GENETIC ALGORITHMS> begin with a population of models generated through an initial (uniform or random) sampling of model space

> employ evolutionary analogy wherein model parameters are encoded within binary strings (``chromosomes’’)

> model population allowed to evolve through iterations (``generations’’) by stochastic model selection based on goodness of fit, by recombination of models (through ``chromosomal splicing’’), and by random ``mutation’’

> Goldberg, 1989, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA.

GA EXAMPLE

Clitheroe et al., 2000, JGR, 105, 13697-13713

NEIGHBOURHOOD ALGORITHM

> begin with a population of models generated through an initial (uniform or random) sampling of model space

> employ an adaptive Voronoi cellular network to drive parameter search

> each iteration randomly samples the model space within cells occupied by the fittest models of the previous iteration

> algorithm focusses increasingly on regions of model space that come closer to satisfying data

> affords opportunity for both qualitative or quantitative appraisal of model space

> Sambridge, 1999, GJI, 138, 479-494

NA -SAMPLING

Sambridge, 1999, GJI, 138, 479-494

Initial randomsampling

True Misfit Function

Sampling after 500

points

Sampling after 100

points

NA -INVERSION

Sambridge, 1999, GJI, 138, 479-494

1-D BORN INVERSION

> inverse scattering relies fundamentally on explicit description of scattering process

> theoretical basis for understanding classic ``delay and sum’’ studies (e.g. Vinnik, 1977, PEPI, 15, 39-45)

> begin with Lippman-Schwinger equation

> Hudson & Heritage, 1981, GJRAS, 66, 221-240

1-D BORN INVERSION> equation again cast in terms of material property perturbations:

> superscript 0 denotes reference medium, denotes perturbation and receiver coordinate is

> similar decomposition for total wavefield:

where incident wavefield is solution for reference medium

> is Green’s function for reference medium

MEDIUM DECOMPOSITION

> decomposition of medium into reference and perturbations

1-D BORN INVERSION> linearize equation through Born approximation, ie, set

so that

> this step analogous to linearization of forward modelling operator in least-squares optimization

> for 1-D, consider only variations in depth and employ 1-D, high frequency asymptotic forms for fields:

1-D BORN INVERSION

> modal expansion in permits examination of different P, S scattering interactions

> delay times given by :

> amplitude of incident wavefield allows either direct upgoing wave or free-surface reflection, e.g., for r=2

1-D BORN INVERSION

> inserting asymptotic forms into Born integral leads to:

where

> note linear relation between scattered field and material property perturbations

> follow Burridge et al, 1998, GJI, 134, 757-777 to simplify representation of model parameters, define:

such that

1-D BORN INVERSION> now have matrix relation:

where

> integral can be discretized and problem solved using least-squares inversion

> alternatively assume that individual modes have been isolated, then define normalized time domain quantity

AMPLITUDE VERSUS SLOWNESS ANALYSIS

> reinsertion followed by evaluation of integral using Leibniz’ rule yields:

> note one-to-one correlation between time on normalized seismogram and depth of interest

> normalization ensures that seismogram appropriately scaled and filtered to reproduce perturbation profile

> combining many seismograms one can arrange individual in a vector to perform amplitude versus slowness inversion

AMPLITUDE VERSUS SLOWNESS ANALYSIS

> solutionis just a weighted stack of data along moveout curves

> provides justification of classic ``delay and sum’’ stack introduced by Vinnik, 1977, PEPI, 15, 39-45

> weights allow formal recovery of material property perturbations (within Born approximation and isolation of r,s)

> note that when inverting for full anisotropic tensor & density this matrix will usually be singular - apply singular value decomposition to recover resolved parameter combinations

> Bank & Bostock, 2003, JGR, 108, doi:10.1029/2002JB/001951

DELAY AND SUM - EXAMPLES

Kind & Vinnik, 1988, JG, 62, 138-147


Fee & Dueker, 2004, GRL, 31, 10.1029/2004GL20636

MULTI-DIMENSIONAL INVERSION

> several approaches to deal with lateral heterogeneity

1. 1-D Collage

2. CCP (Common-Conversion-Point) Stack

3. Formal Multi-dimensional Inversion

> Kumar et al, 2005, EPSL, 1-2, 249-257

1-D COLLAGE

S- receiver functions across North Atlantic

CCP STACKS

> project receiver function along 1-D ray path

> reasonable approximation for planar structures with small dips

> Dueker & Sheehan, 1997, JGR, 102, 8313-8327;

CCP STACKS - EXAMPLE

> Moho hole above Sierra mantle drip

> Zandt et al, 2004, Nature, 431, 41-46

MULTI-DIMENSIONAL INVERSION

> directed search methods computationally intractable at present time

> least-squares optimization at computational limits; more widespread application likely in coming years

> most efficient approach remains high-frequency asymptotic, linearized inverse scattering

LEAST-SQUARES INVERSIONS - EXAMPLES

> Frederiksen & Revenaugh, 2004, GJI, 159, 978-990

LEAST-SQUARES INVERSION - EXAMPLES

> Wilson & Aster, 2005, JGR, 110, doi:10.1029/2004JB003430

LINEARIZED INVERSE SCATTERING

> no conceptual difficulties in dealing with 2-D vs 3-D problems

> instrument availability and deployment logistics often constrain array geometries to be 2-D and densely sampled or 3-D and poorly sampled

> 2 approaches to remedy: interpolation or 2-D regularization

DATA INTERPOLATION

> Neal & Pavlis, 2001, GRL, 26, 2581-2584

2-D REGULARIZATION

> Bostock et al, 2001, JGR, 106, 30771-30782

2-D LINEARIZED INVERSE SCATTERING

> assume that structural target has single, dominant geologic strike

> as in 1-D start begin with Born (linearized wave) equation

> for simplicity assume 1-D isotropic reference medium on which 2-D perturbations are superimposed

MEDIUM DECOMPOSITION

> decomposition of medium into reference and perturbations


> Fourier transform over strike ( ) coordinate along which material properties do not vary

> since reference medium is 1-D, choose plane incident wavefield again

> choose Green’s function to correspond with line (2-D point) source

where

> insert asymptotic forms to yield

> the scattering potential is (e.g. for r=1, s=2)

> Fourier transform to yield:

> has form similar to 2-D Radon transform


GEOMETRICAL QUANTITIES

> P-to-S radiation pattern for different amplitude point scatterers, > Levander et al, 2006, Tectonophysics, 416, 167-185

2-D Radon Transform

> 2-D Radon transform pair:

> Deans, 1983, The Radon Transform and Some of its Applications, John Wiley, New York, NY

WEIGHTED DIFFRACTION STACK > correspondence of scattering integral with Radon transform suggests use of inverse transform for retrieval of scattering potential, via a weighted diffraction stack

where

> Miller et al, 1987, Geophysics, 52, 943-964; Bostock et al, 2001, JGR, 106, 30771-30782

GEOMETRICAL QUANTITIES

> quantities employed in derivation of backprojection formula via generalized radon transform

MATERIAL PROPERTY RECOVERY

> solution is scattering potential

> extract by exploiting dependence on scattering angle , ie amplitude versus angle analysis

> collect all scattering potential measurements at a given imaging point in a vector and solve 3 x 3 system:

> very similar to 1-D formulation in form; main computational burden in computing ``diffraction stack’’

DIFFRACTION STACK EXAMPLES

> Revenaugh, 1995, Science, 268, 1888-1892


> Kind et al, 2002, Science, 283, 1306-1309


> Poppeliers & Pavlis, 2003, JGR, 108,10.1029/2001JB001583


> Levander et al, 2006, Tectonophysics, 416, 167-185

CD-ROM Cheyenne Belt Experiment


> Rondenay et al, 2001, JGR, 106 30795-30807

CASC93 experimentAcross central Oregon





CONCLUSIONS high-frequency, scattered teleseismic body waves

afford highest structural resolution of any component of the global seismic wavetrain

preprocessing of 3-component seismograms to effectively isolate modal contributions and remove source signature is an important pre-requisite to imaging at higher frequencies

imaging practice to date has relied heavily on asymptotic approaches; future efforts will likely focus on full wavefield inversion through non-linear optimization

LECTURES - OUTLINE

IntroductionGeometrical considerationsSource-signature separationDeconvolutionOne-dimensional studiesMulti-dimensional studies Beyond BornCase study 1 - CascadiaCase study 2 - Slave Province

BEYOND THE BORN APPROXIMATION

> tools of inverse scattering provide theoretical framework for understanding most analyses of teleseismic scattering

> most approaches either implicitly or explicitly employ Born approximation

> as instrument inventories grow, spatial sampling in field experiments becomes finer

> may be possible to exploit sampling in more ambitious and complete treatments that take us beyond linearized scattering

> investigate conceptual approach

LIPPMAN-SCHWINGER EQUATION

BORN APPROXIMATION

MOTIVATION

> two dominant, negative consequences follow from the Born approximation:

1. reference medium must be sufficiently close to real Earth to ensure that phase of wavefields is accurately represented (ie avoid cycle skipping); becomes increasingly problematic at higher frequencies

2. failure to account for higher-order scattering in the form of multiple reflection/conversion; less serious for teleseismic waves if reference wavefield includes free-surface reflections since lithospheric material property contrasts are generally small

MOTIVATION> despite feasibility of combining direct and free-surface reflected modes in linearized teleseismic scattering description; it has not been performed

> reason is partly computational, Hessian is no longer block diagonal in asymptotic treatments

> most analyses assume that only one scattering mode is present in the data (e.g. forward scattering)

such that other modes (i.e. free-surface reflections) are a source of contamination

> motivation to consider formal decomposition of into different modes comes from exploration practice

THE INVERSE SCATTERING SERIES

> follow Weglein et al, 2002, IP, 19, R27-R83 and adopt succinct operator notation that is independent of geometry and model type; Lippman-Schwinger equation is:

> first 3 quantities represent integral operators that act on a force distribution, i.e.

> 4th quantity is differential operator that includes action of material property perturbations

INVERSE SCATTERING SERIES

> by successive insertion of first relation we recover forward scattering series

> note not required to solve the forward problem

> goal of inverse problem to recover , postulate series in orders of data of form

> insertion of inverse series into forward series permits term by term solution

SUBSERIES APPROACH

> Weglein et al (2003) have shown that blind application of series approach is marred by poor convergence

> opt instead for a sub series approach wherein individual terms are identified with specific tasks, and a sequential application is performed for reflection data:

1. removal of free-surface multiples2. removal of internal multiples3. imaging of scatterer location4. material property recovery

> tasks 1,2 solved; tasks 3,4 topic of current research

IMPLICATIONS FROM WEGLEIN

> two important implications for teleseismic work:

1) sequential treatment which proceeds from scattering mode decomposition (i.e. identification of forward and back scattered modes) through material property inversion is more likely to be tractable than blind application of non-linear inverse scattering series

2) it is possible to transform teleseismic transmission problem directly into reflection problem such that formulation developed for exploration purposes is then directly applicable

TRANSMISSION TO REFLECTION> most direct way of isolating different scattering modes is reformulation of transmission problem as reflection problem

> basis of concept from Claerbout, 1968, Geophysics, 33, 264-269 for 1-D problems

> for pre-critical, energy-flux normalized elastic waves in 1-D relation is:

> is a 3 x 3 matrix containing transmission response for different (qP, qS1, qS2) modes; are corresponding quantities for reflection, free-surface reflection, respectively

TRANSMISSION/REFLECTION GEOMETRIES

Kumar & Bostock, 2006, JGR, 111, doi:10.1029/2005/JB004104

REFLECTION

TRANSMISSION

TRANSMISSION TO REFLECTION> each element on LHS represents a sum of cross correlations in time domain equates to RHS as a sum of a causal function, acausal function and impulse (diagonal elements only)

> recover by applying after zeroing negative lags

> since represents 3 x 3 reflection response due to different incident wavetypes, a first order decomposition has been achieved

> see Kumar & Bostock, 2006, JGR, 111, doi:10.1029/2005/JB004104 for practical implementation

TRANSMISSION/REFLECTION SYNTHETICS


TRANSMISSION/REFLECTION DATA


INVERSE SCATTERING SERIES

> solve for in first equation, insert into second equation and solve for etc

> note that is just Born solution which may or may not be close to real Earth ( ) depending on choice of reference medium

FREE-SURFACE MULTIPLE REMOVAL

> note that effect of free surface is still included in V inasfar as and higher scattering interactions are concerned

> to remove (reflection) free-surface multiples apply inverse scattering series, in 1-D write using Kennett (1983) notation

> reorganize and solve as

> see Weglein et al, 2003, IP, 19, R27-R83 for further details on internal multiple elimination, imaging and material property inversion

EXTENSION TO MULTIPLE DIMENSIONS

> 3-D extension of transmission to reflection transform is:

> similar in form but requires spatial integration over sources at depth Z

> likewise 3-D equivalent of relation between reflection responses with and without free surface given by:

> Wapenaar et al, 2004, GJI, 156, 179-194

TRANSMISSION TO REFLECTION : MULTI-D

> Wapenaar et al, 2004, GJI, 156, 179-194

EXAMPLE - FREE SURFACE MULTIPLE

ELIMINATION

> real data example

Weglein et al, 2003, IP, 19, R27-R83

EXAMPLE - INTERNAL MULTIPLE ELIMATION

> synthetic example


EXAMPLE - INTERNAL MULTIPLE ELIMATION

> real data example


CONCLUSIONS> sketched out steps toward non-linear, exact inversion of scattered teleseismic wavefields:

1) transmission to reflection transformation2) inverse scattering series (free-surface multiple elimination)

> multi-dimensional implementation requires: a) complete data from 3 components qP, qS1, qS2b) complete spatial coverage

> practical implementation will require interpolation and regularization to deal with field data sets.

Phinney, 1964, JGR, 69, 2997-3017

•> frequency domain receiver function

•> spectral nulls relate to layer thicknesses

Baath & Stefanson, 1966, Ann. Geophys, 19, 119-130

•> noted potential importance of S-to-P conversions in determination of lithospheric structure

Vinnik, 1977, PEPI, 15, 39-45

> time-domain receiver function

> included modal decomposition

> transition zone discontinuities below NORSAR

Langston, 1977, BSSA, 67, 1029-1050;

> time-domain receiver function

> included modal decomposition

> focussed on crust and shallowmost mantle

Langston, 1979, JGR, 94, 1935-1951;

> onset of modern broadband seismology era

> receiver functions become staple of lithospheric studies

Owens et al, 1984, JGR, 89, 7783-7795

Marfurt et al, 2003, The Leading Edge, 22, 218-219

This special section on solid-earth seismology consists ofpapers about studies associated with IRIS, the IncorporatedResearch Institutions for Seismology. This is not an arbitrarychoice; the recent advances made by IRIS groups have begunto have an impact on exploration seismology, particularly inpassive seismic imaging, imaging of converted transmissions,and velocity analysis of long-offset diving waves. Nevertheless,we feel the impact would be larger if more explorationists wereaware of these advances.

FURTHER REFERENCESPavlis G L 2005 Direct Imaging of the Coda of Teleseismic P waves. In Levander A, Nolet G (eds.) Seismic Earth: Array analysis of broadband seismograms. American Geophysical Union, Washington Vol. 157, 171-185

Kennett B L N 2002 The Seismic Wavefield Volume II: Interpretation of Seismograms on Regional and Global Scales, CambridgeUniv. Press, New York NY

S-RECEIVER FUNCTIONS

> Yuan et al, 2006, GJI, 165, 555-564

VECTORIAL DECOMPOSITION

> assume isotropic media, then

> recover P, S modes as curl-free, divergence-free components of displacement

> practically difficult since wavefield not generally sufficiently sampled to estimate spatial derivatives

MINIMUM PHASE : 1-D > 1-D, frequency domain, plane wave transmission response can be decomposed into forward and reverberation components

> reverberation component always minimum phase

> forward component usually minimum phase for realistic velocity contrasts

> product of 2 minimum phase components also minimum phase

> in multiple dimensions we must rely on Claerbout’s principle restated as:

``if scattered wavefield contains less energy at all frequencies than direct wave on P-component, then P-component is minium phase’’

> may not be valid in extreme heterogeneity or where caustics occur

> note modal decomposition improves likelihood of minimum-phase

MINIMUM PHASE : 2-D

MINIMUM PHASE - RX FNS

> minimum phase assertion bears important consequences for source removal

> original receiver function concept implicitly relies on minimum phase assertion through approximation of source time function by P (or U_z) component

> disadvantage is that all information on scattering interactions ending in a P-leg is lost

> Bostock, 2004, JGR, B03303, doi:10.1029/2005JB002783

IMPROVED RECEIVER FNS - 1> simplest approach is to stack time-normalized, P-component seismograms from same earthquake at different stations

> assume weaker ( ) scattered phases are incoherent from station to station so that stack is a scaled estimate of source time function

> main disadvantage that effect of laterally homogeneous structure (e.g. Moho) is identified with source and so absent from Green’s function

> e.g. Langston & Hammer, 2001, BSSA, 91, 1805-1951

IMPROVED RECEIVER FNS - 2

> minimum phase property implies that knowledge of amplitude spectrum alone sufficient to define time series, i.e. phase is Hilbert transform of logarithmic amplitude:

> so problem can be reduced to estimation of amplitude spectra

IMPROVED RECEIVER FNS - 3> consider cross-correlation of two P and SV component of same 3-component recording

> note source phase not present; only phase of underling Green’s function represented

> assume no common zeros/poles in Z-transforms; reconstruct shortest signal with given phase as cross correlation of Green’s function

> Hayes et al, 1980, IEEE-ASSP, 28, 672-680

IMPROVED RECEIVER FNS - 4> by deconvolution we can determine and accordingly we have

> from minimum phase condition can recover

> problem : signal reconstruction from phase is unstable in presence of noise and requires inversion of matrices with dimension of signal length

> implement as multichannel problem

IMPROVED RECEIVER FNS - 5> consider data set comprising J stations recorded at I 3-component receivers; cast convolution relation in log-spectral domain as:

> generate large system of 3IJ equations in I+3J unknowns with rank I+3J-1

> augment system with source estimates from signal reconstruction by phase:

IMPROVED RECEIVER FNS - 5 (cont’d)

> here

IMPROVED RECEIVER FNS - 6

> note phases of are not minimum phase but can be retrieved through application of allpass filters derived from

> details may be found in Mercier et al, 2006, Geophysics, 71(4), SI95-SI102, doi:10.1190/1.2213951

EXAMPLES - GEOMETRY

EXAMPLES - TRAVELTIMES

> two complications in extension of foregoing methodology to teleseismic S:

1. S-component of teleseismic-S-Green’s function cannot be minimum phase due to 2nd (and higher) order multiple, acausal forward scattering (e.g. S-to-P-toS)

Can be dealt with by ignoring (ie to first order minimum phase)

2. Incoming S polarization depends on source and in general is not known - to which component of S should minimum phase assumption apply?

For isotropic, 1-D media, minimum phase assumption can be made independently for both SV and SH components, e.g. S-receiver functions

TELESEISMIC S GREEN’S FUNCTIONS


>

> Kumar et al, 2005, EPSL, 1-2, 249-257




> in presence of strongly heterogeneous and/or anisotropic media, situation is more difficult - Green’s function becomes a complex function of incident wave polarization and azimuth

> at least one component of Green’s function may be decidedly non-minimum phase

> extend approach of Farra & Vinnik, 2000, GJI, 141,699-712, by writing wavefield at surface as:


> relation cast in the frequency slowness domain where half space transmission response is (notation after Kennett 1983)

> further assume that incident wavefield is linearly polarized

> SKS splitting observations indicate that off diagonal elements of U can be comparable in magnitude to diagonal elements


> if anisotropy easily modelled by receiver-side single layer, then obvious way to proceed is to find best (approximation of ) that most nearly removes elliptical particle motion

> corrected seismogram will include one minimum phase component and previous source-removal algorithm will apply

> after source removal, reapply U to recover Green’s function

> for more complicated (especially source-side) anisotropy analysis still more difficult

CEPSTRAL DECONVOLUTION

> originally devised for speech applications at MIT Lincoln Lab (Oppenheim & Schafer, 1975, Digital Signal Processing)

> introduced to seismology by Ulrych, 1971, Geophysics, 36, 650-660

> solves blind deconvolution problem by filtering (liftering) in inverse-Fourier-log-spectral (quefrency) domain

> suffers from noise but may be worth re-examination in Green’s function estimation problem



EXAMPLES - STATION ULM

Mercier et al, 2006, Geophysics, 71(4), SI95-SI102, doi:10.1190/1.2213951

OTHER FORMS OF DECONVOLUTION

> multi-taper spectral deconvolution : Park & Levin, 2000, BSSA, 90 1507-1520

> time domain deconvolution is more expensive but admits alternative regularizations e.g., Gurrola et al, 1995, GJI, 120, 537-543; Liggoria & Ammon, 1999, BSSA, 89, 1395-1400

> non-linear stacking may also be useful, e.g. Nth root (Muirhead and Datt, 1976, GJRAS, 47, 197-210), phase-weighting (Schimmel & Paulssen, 1997, GJI, 130, 497-505; (Kennett, 2000, GJI, 141, 263-269)

> for more on seismic signal processing see Rost & Thomas, 2002, Rev. Geophys., 40, 10.1029/2000RG000100

1-D INVERSION

> early studies and many studies today focus on recovery of structural information from single stations

> 1-D model interpretation especially of major discontinuities atbase of crust (Owens et al., 1984, JGR, 89, 7783-7795) and transition zone (Kind & Vinnik, 1988, ZfurG, 62, 138-147)

> more recently, studies of anisotropic structure dominantly 1-D(Bostock, 1997, Nature, 390, 392-395; Levin & Park, 1997, GJI, 131, 253-266)

MINIMUM PHASE - CAUTION

> minimum phase assertion may not always be correct

> Park and Levin, 2001, GJI, 147, 1-11

structural imaging using scattered teleseismic body waves

Documents

scattering of incident

degrees incident wave

primary bodywave phase

teleseismic wavefield

pure pp

source signature

vertical propagation

exploration studies