Geology 5640/6640Introduction to Seismology

24 Apr 2015

© A.R. Lowry 2015

Last time: Amplitude Effects

• Multipathing describes the focusing and defocusing of waves by ray-bending in a heterogeneous medium (e.g., basin amplification).

• The Fresnel Zone corrects ray theory for multipathing (esp. arrivals within a half-period) given a finite frequency

• Scattering effects occur when velocity varies at even smaller scales where ray theory is no longer usable (a ~ )

• Scattering is responsible for the coda of earthquakes (the long jumble of disturbances that arrive long after the initial phases). Coda energy can be back-projected to the ellipse of possible scatterer locations associated to a given arrival time; interferometry can be used to image changes in velocity structure through time!

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

Intrinsic Attenuation:

The wave equation of course is different than that for a simple mass-spring system…

For a plane wave, the Amplitude A decreases as

where a is the absorption coefficient, a = / 2Qc

Q is generally called Quality factor… Low attenuation implies high Q. For anelastic attenuation, elastic parameters and hence velocity c are complex-valued, i.e. c = cR + icI, and

€

A =A0e−ax

€

⇒ A =A0e−

ω0x

2Qc

€

Q =cR

2cI

So amplitude decay is greater for greater distance x and frequency 0; less with higher intrinsic quality factor Q and velocity c.

Q differs for P-wave (Qp) and S-wave (Qs).

For sediments,5 < Qp < 300 (lower if porosity is high)5 < Qs < 100

For crystalline rocks at shallow depth/low temperature,100 < Qp < 800

€

⇒ A =A0e−

ω0x

2Qc

But Q variations can be imaged, and provide additional usefulinformation about a medium!

1 2 3

V = 1250 m/sf = 250 Hz

Q = 5.1

V = 3680 m/sf = 125 Hz

Q = 3.2

X

X

5301006.6

Seismic attenuation can be particularly useful forimaging of anomalies in melt fraction and free watercontent in the mantle…

And for resolving potentialambiguities in interpretationof velocity images. Example: Seismic parametersfrom the Rio Grande Rift region of Colorado (all at 100km depth)

Pasyanos, BSSA, 2013

Pn-QP

Pn-QP

Sn-QS

Sn-QS

MeasuredModeled