Introduction to Receiver Functions I.  Definition of a Receiver Function II.  Slant Stacking III.  Modeling of Receiver Functions IV.  S-wave Receiver Functions

LQ

LR

Body Waves P and S waves Particle Motion

Body Waves: Polarization

P, SV, and SH :

Seismic Body Waves

Receiver Functions

)()(

)()()()(

)()()()()()()()(

)()()(

ωω

ωωωω

ωωωωωωωω

ωω

ω

RV

RR

SV

RV

SR

RR

SV

RV

SR

RR

V

R

EE

EEEE

EESIEESI

UUR

≅⋅

⋅=

⋅⋅⋅

⋅⋅⋅==

Fundamental Assumptions

•  Plane Wave Approximation •  The PS is primarily recorded on the radial

(the vertical component is negligible)

•  These assumptions lead to the result that the receiver function *only* includes P-to-S converted energy

[ ][ ]

( )( )( )( )psppsppp

pspp

p

psp

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tips

tip

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tipp

tip

tips

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tip

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−−+−−−

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++−−−=

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<<++=++=

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ˆˆˆˆˆˆˆ1

ˆˆ1ˆ1)R(...1x)(1 :expansion binomial theusing

ˆ1

ˆˆ1)(

1 assumecan weˆˆ1)(

ˆˆ1)(

ˆ,ˆ

2

0

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r̂ˆ means waveplane

ˆˆˆ1sorder termhigher neglecting

P-wave PS-wave

What factors influence r0/z0?

Isolating Mode Conversions

Methods of Deconvoltuion

•  Spectral Division: water level technique (Burdick and Langston, 1978) •  Stacked Time Domain Deconvolution (Baker et al., 1996) •  Individual Iterative Time Domain

Deconvolution (Liggoria and Ammon, 1999)

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

Time Domain CONVOLUTIONAL MODEL

• observed displacement seismogram

•  effective source (includes source-side scattering)

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

PS-phase Move-out:

Δ−⎟⎠⎞⎜

⎝⎛ −−−

=

−−

−−

=

−=−=∴==

−−=−−=−=

−

−

pdVV

VpVVpVh

dTVVph

VVph

t

VpjVpipVjpVi

dTiV

hjV

hdTVy

Vxttt

ps

pssp

p

p

s

sPPS

Spsp

pspsPPSPPS

2222

2222

2222

11

11then

,1cos,1cos sin,sinusing

coscos

Crustal Multiples:

⎥⎦⎤

⎢⎣⎡ −+=

⎥⎦⎤

⎢⎣⎡ −+−=

−

−−

22

2222

2 pVhtt

pVpVht

sPPSPSS

psPPS

SCATTERING GEOMETRY

P S

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

Slant Stacking

Synthetic Receiver Functions:

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

Receiver Functions

Receiver Function Footprints 40 km Interface

15 km Interface

Migrating Receiver Functions

ASCENT P-wave Receiver Functions

S wave Receiver Functions

S Receiver Functions

From Farra and Vinnik, 2000

Comparisons with P Receiver Functions

Migrated S-wave Receiver Functions

Angus et al., 2005

Han et al., 2010: T43A-2158

For Next Time: Mike Pasyanos:Surface Wave Dispersion

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

( )⎥⎦⎤

⎢⎣⎡ −−−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−−

−−−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

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⎝⎛ −−−

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−−−+

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⎟⎠⎞⎜

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−+−−

⎟⎠⎞⎜

⎝⎛ −−−

=

Δ−−−

⎟⎠⎞⎜

⎝⎛ −−−

=

−−

−

−

2222

2222

22222222

2222

222222222222

2222

22222

2222

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2222

2222

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11

)1(111

11

1111

11

11

11

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1tan,

1tan

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11

11

11

pVpVhVpVpVV

VpVpVVpVpVh

VpVpVV

VpVVpVpVVpVpVVpVh

VpVp

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VpVpVV

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phVihy

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ps

psps

ppsssp

psps

ppsssppssp

ps

pssp

spps

pssp

PPS

p

p

s

s

spps

pssp

spps

pssp

PPS

[Willett and Beaumont, 1994]

[Tapponnier et al., 2001]

[Fu et al., 2008]

Background: Geodynamic models of Tibetan plateau

Unknown deep structure

Mainly north-south profiles

[Yin and Harrison, 2000]

[Kind et al., 2002]

[Zhao et al., 2010]

[Kumar et al., 2006]

Migrated S-wave Receiver Functions