Seismic Attenuation---from seismic wave travel

times to amplitudes

Haijiang Zhang

Courtesy of many different sources

Haberland et al., GRL 2003

EARTHQUAKE MAGNITUDE

Earliest measure of earthquake size

Dimensionless number measured various ways, including

ML local magnitude

mb body wave magnitude

Ms surface wave magnitude

Mw moment magnitude

Easy to measure

No direct tie to physics of faulting

(4) intrinsic attenuation: due to anelasticity

the real earth materials are always “lossy”, leading to reduced wave amplitudes, or intrinsic attenuation.

Mechanisms to lose energy:(1) Movements along mineral dislocations(2) Shear heating at grain boundariesThese are called “internal friction”.……

(3) Scattering: A way to partition energy of supposedly main arrivals into boundary or corner diffracted, scattered energy.

Attnuation Mechanisms:

(1) Geometrical spreading: wavefront spreading out while energy per unit area becomes less.

(2) Multipathing: waves seek alternative paths to the receiver. Some are dispersed and some are bundled, thereby affecting amplitudes.

Key: very wavelength dependent.

4

Geometric spreading: light moves outward from lamp in expanding spherical wave fronts. By conservation of energy, the energy in a unit area of the growing wave front decreases as r-2, where r is the radius of the sphere or distance from the lamp.

Analogous to light behavoirs…

Expect minimum at =90º, and maxima at 0º and 180º

GEOMETRIC SPREADING: SURFACE WAVES

From geometric spreading alone, expect minimum at =90º, and maxima at 0º and 180º

Also have effects of anelasticity

For body waves, consider a spherical wavefront moving away from a deep earthquake. Energy is conserved on the expanding spherical wavefront

whose area is 4 π r 2, where r is the radius of the wavefront.

Thus the energy per unit wave front decays as 1 / r2, and the amplitude decreases as 1 / r

GEOMETRIC SPREADING: BODY WAVES

MULTIPATHING

Seismic waves are focused and defocused by variations in velocity.

Idea of Ray Tubes in body waves

Ray tube size affects amplitudes, smaller area means larger amplitude

Angle dependent: ray bundle expands or contracts due to velocity structure. Also: amplitude varies with takeoff angle

Effect of Multipathing

10

Scattering:

Scattering occurs when there are velocity heterogeneities in the medium with wavelengths on the order of λ of the wave.

Snieder et al., Science, 2002

A simulation of 100 randomly-perturbed scatterers…

Moving the scatterers (here, 1/40th of the distance shown,so the perturbation is visible) mostly changes the shapesand amplitudes of the waveforms…

Snieder et al., Science, 2002

Waveformsmeasured in agranite sampleat temperaturesof 45°C (blue)and 50°C (red).

Changing the matrix velocity, by contrast, introduces a shift(delay or advance) that increases with coda time.

Scattering:Interesting aside:

Coda energy in theEarth tends toattenuate much morerapidly than on theMoon!

This is partly becauselunar regolith ishighly fractured byimpacts, but mostlybecause it containsno fluids (andconsequently muchless intrinsicattenuation!)

Intrinsic Attenuation:Intrinsic attenuation, or anelasticity,describes the process by which elastic energyin the Earth is converted to heat when theseismic wave induces unrecoverabledeformation.

To examine this, let’s consider a spring:For an idealized spring,

has solution

with oscillation frequency

More realistically though, internal friction in thespring will damp the system resulting in where is a damping factor and Q 0/

is called the quality factor.

€

m∂ 2u

∂t2+ ku = 0

Mass m

u

Spr

ing

cons

tant

k

€

u t( ) = Aeiω0 t−t0( )

€

0 =k

m

€

m∂ 2u

∂t2+γm

∂u

∂t+ ku = 0

Intrinsic Attenuation:This system has a solution with real andimaginary parts; the actual displacement is thereal part and takes the form:

i.e., a harmonic oscillator with an exponentialdecay of amplitude. Here, A0 is the initialdisplacement (at time t = 0) and

Important to note:

• High frequencies attenuate more than low

• Harmonic frequency is changed by attenuation

• Higher Q results in less change to frequencyand less intrinsic attenuation for given time

Mass m

u

Spr

ing

cons

tant

k

€

Re u t( ){ } = A0e−

ω0t

2Q cosωt

€

=0 1−1

4Q2

Generally,loss of amplitude dueto intrinsicattenuation ismuch greaterthan that dueto partitioning,spreading andthe otheramplitudeeffects wehavediscussed

Definition of the quality factor

1 E

Q 2 E

Energy dissipated in a cycle

Energy stored

€

A = A0 exp −πf

Qt

⎛

⎝ ⎜

⎞

⎠ ⎟

( ) exp( )ij

fRB f

vQ

Putting everything together

€

Aij ( f ) = Si( f )I j ( f )Gij (R)Bij ( f ,R)

Intrinsic attenuation

(1) Movements along mineral dislocations

(2) Shear heating at grain boundaries

• Affected by temperature, pressure, frequency, and medium properties

• Attenuation property is complementary to velocity.

Haberland et al., GRL 2003

Myers et al., 1995

The key here is to correlate a decrease in Q with fluids in the crust and mantle. The fluid layer again represents melting due to subduction.

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(Romanowicz & Mitchell, 2007)

Nakajima et al., 2003, GRL

How to estimate Q

•Code wave

•Surface wave

•Body wave

•Amplitude data

•…….

Coda wave attenuation

Kumar et al., 2005

Estimating Q from body wave ----- using t*

€

Aij ( f ) = Si( f )I j ( f )Gij (R)Bij ( f ,R)

Site Response

Attenuation Model

ABCE

• Amplitude of horizontal component

• Period

2005 1 1 O=03 15 40.2 +/- 0.04s LAT=26.92 N +/- 0.30km LONG=100.31 E +/- 0.33km DEPTH= 10 km +/- 0.14km STATIONS USED = 6, STAND DEV= 2.63s ML=2.9/ 6, EYA 0.9 202 Pg 03 15 56.4 0.7 Sg 03 16 07.6 -0.1 SMN ML=3.0 0.6 0.20 SME 0.6 0.61 PZH 1.4 108 Pn 03 16 03.1 -2.3 Pg 03 16 05.0 0.9 Sg 03 16 19.4 -3.2 SMN ML=2.9 0.6 0.17 SME 0.7 0.18 XAC 2.1 347 Pn 03 16 15.7 0.5 Pg 03 16 19.6 3.0 Sg 03 16 44.5 -0.4 SMN ML=3.2 1.0 0.11 SME 1.0 0.23 TCG1 2.5 221 Pg 03 16 24.0 0.0 Sg 03 16 55.7 -2.2 SMN ML=3.7 0.6 0.52 SME 0.5 0.42 KMI 2.8 128 Pg 03 16 28.9 -1.1 Sg 03 17 07.4 -1.1 SMN ML=2.8 0.8 0.045 SME 0.7 0.047 ZOT 3.1 82 ePg 03 16 36.8 2.5 Sg 03 17 14.6 -1.5 SMN ML=2.8 1.0 0.031 SME 1.0 0.038

Amplitude Q tomography(Shunping Pei)

Theory and Method

Source

Station

Crust

Upper mantle

( ) ( ) ( ) ( ) ( ) ( , )ij j i i ij ijA f O f I f S f G R B f R

10( ) exp( ) exp( )ij

fRB f cQ R

vQ

1 /c f v 0Q Q f 1

0ln ( ) ln ( )ij ij ij i jY A f G R a b Q cR

ln ( )i ia S f 0ln ( )jb O f

( ) ( ) ( ) ( ) ( ) ( , )ij j i i ij ijA f O f I f S f G R B f R

10ln ( ) ln ( ) *ij ij ij L i jY A f G R p M a b Q cR

0ln ( ) *j Lb O f p M

10ij i j k ijk

k

Y a b Q cR

ML Tomography in North China

ABCE of 1985 to 1995(Annual Bulletin of Chinese Earthquakes)

1. Each event was recorded by at least 2 stations,2. Residuals between -2.0 ~ 2.0 ,3. Period T of the Sg wave maximum amplitude (A) is between 0.4~2.0s,4. Epicentral distance is between 100km-800km.5. Event depth is less than 20km.

10899 amplitude data1732 events91 stations

10899 Amplitude data1732 Events (+)91 Stations (▲)

Histogram of ML distribution

Result

Earthquakes with M>7

Phillips et al., 2005

Q from the tomographic inversion of 1 Hz Lg amplitude ratios

Liu et al., 2004

10ij i j k ijk

k

Y a b Q cR

10ij i j k ijk

k

Y a b Q cR

Checkerboard Test

2°× 2°

Checkerboard Test

1.5°× 1.5°

Standard deviation 0.67

Standard deviation 0.41

Summary• Crustal attenuation can be reconstructed by tomographic

imaging method using amplitude data.• Attenuation levels are correlated with regional tectonic structure. High attenuation often occurs in active tectonic areas with

significant faulting, while attenuation is low in the stable Ordos Craton.

• The estimate of attenuation shows a close correlation with topography.

Q0 is generally low in basins, whereas high Q0 mostly occurs in mountains and uplift regions where crystalline basement appears in the surface. It is possible that low Q0 in basins is caused by fluid in the upper crust, and deep sediment in basins, while high Q0 in the mountains and uplift regions results from the presence of old, dense rocks there.

(Published in BSSA(2006))