Geology 6600/7600Signal Analysis

03 Dec 2013

© A.R. Lowry 2013

Last time: Deconvolution in Flexural Isostasy• Surface loads can be solved from observed gravity and topography provided ~(z), flexural rigidity and internal load depth zl are known a priori:

• In contrast to Tharsis, western US topography appears to be supported significantly by dynamic (i.e., sublithospheric) buoyancy

• Estimation of flexural rigidity D still relies on sometimes- questionable assumption of uncorrelated loading, so future analysis should use seismic constraints

€

1 − ρ 0 +D

gk 4

ρ1 +D

gk 4

ΔρL

ρ1 +D

gk 4

−2πGρ0dρ

dzexp −kz( )dz

0

h

∫ρ1 +

D

gk 4

−2πGΔρL

dρ

dzexp −kz( )dz

0

h

∫ρ1 +

D

gk 4

−exp −kz l( )

ΔρL

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

HI

r k ( )

WI

r k ( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥=

Hr k ( )

Br k ( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Deconvolution of source & receiverterms from distant earthquakes:

Recall that a seismogram represents a convolution of the source-time function s(t) with the Earth system response h(t) and the seismometer response i(t):

For imaging applications we would like to remove the source and receiver terms and just look at the Earth response. One approach to doing this is to isolate the impulse response to phase-conversions at impedance boundaries using teleseismic receiver functions

Nov 30 Little Cottonwood Creek seismogram for M5 earthquake in S Mexico…

€

r t( ) = s t( )⊗h t( )⊗i t( ) ⇔ R ω( ) = S ω( )H ω( )I ω( )

P S

Rayleigh

In the most commonly-used approaches to seismic receiver function analysis (e.g., Ammon, BSSA 1991; Ligorria & Ammon, BSSA 1999) the horizontal (E, N) components of a three-component seismogram are rotated into radial and transverse directions based on back-azimuth to the source event:

For a teleseismic event arriving rays are near-vertical, so the vertical component contains predominantly P-wave particle motion (with a small contribution from SV) and the radial horizontal component contains predominantly SH motion (with a small contribution from P). In an idealized (1D, isotropic) Earth, the transverse contains motion neither from primary P or converted (polarized) S!

N

E

Transverse

Radial

P

S

Radial

Vertical

Both vertical & radial components are convolved with the same source-time function and instrument response for each different phase that comes in:

Here, k represents each of N phases that originated as a P wave and, after conversion, arrived as an S wave:

€

rZ t( ) = s t( )⊗hkZ t( )⊗i t( )

k= 0

N

∑ rR t( ) = s t( )⊗hkR t( )⊗i t( )

k= 0

N

∑

(Ammon, BSSA 1991)

Thus the source and instrument response are removed from the time series by (frequency domain) division of the radial by vertical components. The resulting impulse response function is called the receiver function:

€

H ω( ) =

RkR ω( )S ω( )I ω( )

k= 0

n

∑

RkZ ω( )S ω( )I ω( )

k= 0

n

∑=

RkR ω( )

k= 0

n

∑

RkZ ω( )

k= 0

n

∑

(Ammon, BSSA 1991)

Receiver Function Estimates ofCrustal Thickness:

P Ps

Delay Time

• Deconvolve source-time function to get impulse response of phases converted at impedance boundaries

• Delay time between phase arrivals depends on crustal thickness and relative velocities of P & S phases

• EARS uses iterative time-domain deconvolution [Ligorria & Ammon, BSSA, 1999]: well-suited to automation

P PsCrust

Mantle

P Ps PpPs

PpSsPsPs

Contribution of crustal thickness (H)versus vP/vS ratio (K) to delay time isambiguous…

Resolve using reverberations, which have differing sensitivity to H and K

P Ps PpPs PpSs PsPs

Ps

PsPs

PpSs&

PpPs

Crustalthickness (H) &vP/vS ratio (K)parametersthat predict the observed phase delay times intersect ata point inparameterspaceP Ps PpPs

PpSsPsPs

H–K Stacking: [Zhu & Kanamori, JGR, 2000]

Ps

PsPs

PpSs&

PpPs

Method stacksobserved amplitudes atdelay times predicted for converted Ps phase andreverberations.

Max stackamplitude ideally revealscrustal thickness &vP/vS ratio.

H–K Stacking: [Zhu & Kanamori, JGR, 2000]

P Ps PpPs

PpSsPsPs

(EARS H–K stack for station COR)

[Crotwell & Owens,2005]

The Moho is not the only lithospheric impedancecontrast… And crustalthickness is not constant

The Problem:

(EARS H–K stack for station TA.P10A)

Despite outliers and other issues, crustal thickness & vP/vS have statistical properties consistent with a fractal surface…

CrustalThickness

(H)

vP/vS

Ratio(K)

Ro

ot

Va

rio

gra

m H

(k

m)

Ro

ot

Va

rio

gra

m

K

Station TA.O09A (Central Nevada)

Variogramscan be used toestimate a“most likely” crustalthickness and vP/vS ratiovia optimalinterpolationfrom nearbysites.

Station TA.O09A (Central Nevada)

And search for a“most likely” modelwith uncertainties.

Can also model gravity predicted foreach H & Kat the site…

Station TA.O09A (Central Nevada)

Gra

vity

Mod

el

Lik

elih

ood

F

ilter

Optimal Interp.Likelihood Filter

Combined

Unlikely stack amplitude maxima aredownweighted using likelihood statistics

Station TA.O09A (Central Nevada)