STRUCTURAL IMAGING USING
SCATTERED TELESEISMIC BODY WAVES
Michael Bostock
Department of Earth & Ocean Sciences
The University of British Columbia
ESIW-UCBerkeley, June 23, 2011
LECTURE - OUTLINE
Introduction
Geometrical Considerations
Source-signature separation &
Deconvolution
One-dimensional studies
Multi-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
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 S
Traditionally less useful owing to:
a) more limited distance (slowness) range
b) larger slownesses, closer to critical
c) interference between S, SKS, ScS
between 70-90 degrees
d) variable source-side polarization imprint
e) lower frequency content
f) higher signal-generated noise levels
S-TELESEISMIC RECEIVER FUNCTIONS
Farra & Vinnik, 2000, 141, 699-712
S - Receiver Functions & the LAB
 renewed interest
in S owing to its
utility in identifying
shallow mantle
discontinuities
unobscured by the
crustal
reverberations
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Rychert et al, 2007
RECEIVER FUNCTIONS FROM REGIONAL P
Park & Levin, 2001, GJI, 147, 1-11
PKP - RECEIVER FUNCTIONS
Levin & Park, 2000, Tectonophysics, 323, 131-148
SOURCE SIGNATURE SEPARATION
& DECONVOLUTION
CONVOLUTIONAL MODEL - 1
>
: observed displacement
seismogram
>
: effective source (includes sourceside scattering)
>
: Green’s function (receiver
side response to an impulsive plane wave
with horizontal slowness
)
CONVOLUTIONAL MODEL - 2
> observation index;
(implicitly assumed)
source index
> separation of
canonical problem in seismology
is
> 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
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> 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
OBLIQUE RAYPATHS/POLARIZATIONS IN 1D
P
S
PPP PPS PSS P PS
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
1-D SLOWNESS DECOMPOSITION
> 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
SCATTERING GEOMETRY
> 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
P
S
PPP
PPS PSS
P
PS
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 Scontributions 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 - MOTIVATION
S-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 crosscorrelation
> 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)
LEAST-SQUARES IMPLEMENTATION
> 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
LEAST-SQUARES
IMPLEMENTATION
> 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 algorithms
2. 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
Initial random
sampling
Sampling after
500 points
Sampling after
100 points
True Misfit
Function
Sambridge, 1999, GJI, 138, 479-494
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,
perturbation and receiver coordinate is
denotes
> similar decomposition for total wavefield:
where incident wavefield
reference medium
>
medium
is solution for
is Green’s function for reference
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
> solution
is 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
DELAY AND SUM - EXAMPLES
Fee & Dueker, 2004, GRL, 31, 10.1029/2004GL2063
DELAY AND SUM - EXAMPLES
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DELAY AND SUM - EXAMPLES
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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
1-D COLLAGE
S- receiver functions
across North Atlantic
> Kumar et al, 2005, EPSL, 1-2, 249-257
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
2-D LINEARIZED INVERSE SCATTERING
> 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
2-D LINEARIZED INVERSE SCATTERING
> 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
DIFFRACTION STACK EXAMPLES
> Kind et al, 2002, Science, 283, 1306-1309
DIFFRACTION STACK EXAMPLES
> Poppeliers & Pavlis, 2003, JGR, 108,10.1029/2001JB001583
DIFFRACTION STACK EXAMPLES
CD-ROM Cheyenne
Belt Experiment
> Levander et al, 2006, Tectonophysics, 416, 167-185
DIFFRACTION STACK EXAMPLES
CASC93 experiment
Across central
Oregon
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QuickTime™ and a
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> Rondenay et al, 2001, JGR, 106 30795-30807
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
Introduction
Geometrical considerations
Source-signature separation
Deconvolution
One-dimensional studies
Multi-dimensional studies
Beyond Born
Case study 1 - Cascadia
Case 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 freesurface 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
block diagonal in asymptotic treatments
is no longer
> 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
orders of data of form
, postulate series in
> 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 multiples
2. removal of internal multiples
3. imaging of scatterer location
4. 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, 264269 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
REFLECTION
TRANSMISSION
Kumar & Bostock, 2006, JGR, 111, doi:10.1029/2005/JB004104
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
Kumar & Bostock, 2006, JGR, 111, doi:10.1029/2005/JB004104
TRANSMISSION/REFLECTION DATA
Kumar & Bostock, 2006, JGR, 111, doi:10.1029/2005/JB004104
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 : MULTID
> 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
Weglein et al, 2003, IP, 19, R27-R83
EXAMPLE - INTERNAL
MULTIPLE ELIMATION
> real data example
Weglein et al, 2003, IP, 19, R27-R83
CONCLUSIONS
> sketched out steps toward non-linear, exact inversion of
scattered teleseismic wavefields:
1) transmission to reflection transformation
2) inverse scattering series (free-surface multiple elimination)
> multi-dimensional implementation requires:
a) complete data from 3 components qP, qS1, qS2
b) complete spatial coverage
> practical implementation will require interpolation and
regularization to deal with field data sets.
Phinney, 1964, JGR, 69, 29973017
• > 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, 10291050;
Langston, 1979, JGR, 94, 19351951;
> time-domain receiver function
> included modal decomposition
> focussed on crust and shallowmost
mantle
Owens et al, 1984, JGR, 89,
7783-7795
> onset of modern broadband seismology era
> receiver functions become staple of
lithospheric studies
Marfurt et al, 2003, The Leading
Edge, 22, 218-219
This special section on solid-earth seismology consists of
papers about studies associated with IRIS, the Incorporated
Research Institutions for Seismology. This is not an arbitrary
choice; the recent advances made by IRIS groups have begun
to have an impact on exploration seismology, particularly in
passive 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 were
aware of these advances.
FURTHER REFERENCES
Pavlis 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, Cambridge
Univ. 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
MINIMUM PHASE : 2-D
> 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 - 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 timenormalized, 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
have
and accordingly we
> 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 3component 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
TELESEISMIC S GREEN’S FUNCTIONS
> 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-PtoS)
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
>
QuickTime™ and a
decompressor
are needed to see this picture.
> Kumar et al, 2005, EPSL, 1-2, 249-257
TELESEISMIC S GREEN’S FUNCTIONS
> 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:
TELESEISMIC S GREEN’S FUNCTIONS
> 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
TELESEISMIC S GREEN’S FUNCTIONS
> 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-logspectral (quefrency) domain
> suffers from noise but may be worth reexamination in Green’s function estimation
problem
QuickTime™ and a
decompressor
are needed to see this picture.
EXAMPLES - STATION ULM
Mercier et al, 2006, Geophysics, 71(4), SI95-SI102, doi:10.1190/1.2213
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 at
base 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