Geology 5640/6640 Introduction to Seismology 20 Apr 2015 © A.R. Lowry 2015 Read for Wed 22 Apr: S&W 185-198 (§3.7) Last time: Anisotropy(Cont’d) Anisotropy.

Geology 5640/6640Introduction to Seismology

20 Apr 2015

© A.R. Lowry 2015Read for Wed 22 Apr: S&W 185-198 (§3.7)

Last time: Anisotropy(Cont’d)• Anisotropy refers to directional dependence of velocity

• The fourth-order elasticity tensor, cijkl, can be expressed more succinctly as a 6x6 “Voight matrix” Cmn.

• Transverse anisotropy describes e.g. SPO in horizontally layered media, and is characterized by one P-velocity for x1 & x2 propagation, a different P-velocity for x3 propagation, and differing SV- & SH-velocities:

• Azimuthal anisotropy describes the more complicated case of azimuthally-varying P-velocity, generalizing to:€

VP12 =A

ρ; VP 3 =

C

ρ; VSH =

N

ρ; VSV =

L

ρ

€

vP θ( ) = A1 + A2 cos2θ + A3 sin2θ + A4 cos 4θ + A5 sin 4θ

Reminder: The Final Exam is posted on the course website… Due 8:30 am Fri May 1.

6640 Semester Project due-dates: • Presentations on your research results are to be given 11:30 am to 1:20 pm on Mon Apr 27; will be max 30 minutes each• Research reports are due Fri May 1 at 5 pm. No fixed length, but these should include ‡ Intro/Context (presumably including relevance to your thesis topic) ‡ Description of Math/Physics of the problem ‡ Methods Used (if any) ‡ Details of Analysis ‡ Results, Discussion, Future Work (if any)

Fluid-filled cracks in an isotropic medium will havean elasticity tensor of the form:

In this example we’ve assumed that the normals totwo-dimensional cracks parallel the x1 axis. Note how theshear modulus is reduced by the cracks. If the fluid isincompressible, P-wave velocity is unaffected, butS-wave velocity is.

Here, ε = Na3/V, with N being the number of cracks pervolume V, and a is the half-width of the cracks.

€

Cmn =

λ +2μ λ λ 0 0 0

λ λ +2μ λ 0 0 0

λ λ λ +2μ 0 0 0

0 0 0 μ 0 0

0 0 0 0 μ 1−ε( ) 0

0 0 0 0 0 μ 1−ε( )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Lin & Schmandt, Geophys. Res. Lett. 2014

Example of uppercrustal (10-16 s)Rayleigh velocityanisotropy

… Interpreted as compressional stress direction closed fractures

Lin & Schmandt, Geophys. Res. Lett. 2014

Here, observations were made usingmarine seismic observations of refraction travel-times near Hawaii.

Azimuth is measured relative to trendof magnetic isochrons in the region,so velocity peaks at 90° and 270°indicate that the fast direction is inthe direction of spreading at the timethe lithosphere was formed.

So is that more likely to representfluid-filled fractures or flow of mantleolivine?

The depth and agedependence ofanisotropy in theoceans also lendsinsight into physicalprocesses…

Important to note, forinterpreting this signal,that the direction andmagnitude ofanisotropy reflects anintegral of the strainhistory of the rocks!

€

ξ =N

L=

VSH

VSV

⎛

⎝ ⎜

⎞

⎠ ⎟

2

Too

mey

et a

l. N

atur

e 20

07

More recent data from the EastPacific Rise tell a morecomplicated (and moreinteresting) story…

With significant implications foractive- vs passive-componentsof the flow dynamics in oceanicspreading centers (not tomention segmentation ofridges, and the bathymetryand melt chemistry variationsalong a mid-ocean ridge…)

Shear Wave Splitting is commonly-used to identifyazimuthal anisotropy. Given initial signal s(t) on the radialcomponent only of the SKS arrival & angle f between radial &fast directions,

Radial:

Transverse:€

s1 t( ) = s t( )cosφ s2 t( ) = s t −δt( )sinφ

€

R t( ) = s t( )cos2 φ + s t −δt( )sin2 φ

€

T t( ) = s t( )+s t −δt( )

2

⎡

⎣ ⎢

⎤

⎦ ⎥sin2φ

Note that normallyone wouldn’t get atransversecomponent SKSarrival; only radial!but anisotropy “splits”the arriving energy incomponents & || toanisotropy axes.

In practice, try lots of rotations and t’s to find which maximizesthe radial component of amplitude.

Note gives youthe fastdirection;t describesthicknesstimes v!

Miller & Savage, Science, 2001

Time-varyinganisotropyhas beenobservede.g. beforeand afteran eruptionof MountRuapehuin NewZealand…

1994, <30 km 1998, <30 km

1994, >50 km 1998, >50 km

Attenuation and AnelasticityWave amplitudes depend on:

• Source energy

• Transmission/Reflection at interfaces (i.e., Zoeppritz’ Equations)

• Geometric Spreading: As a wavefront propagates from a finite source and encompasses a larger volume, conservation of energy requires amplitude to diminish.

• Multipathing: Focusing and defocusing of waves (analogous to mirages in the case of light).

• Scattering: Like multipathing, but this occurs when velocity heterogeneities have wavelengths of the order of the propagating wave.

• Anelasticity: Elastic energy is converted to heat during unrecoverable deformation.

Geometrical Spreading:Recall from our derivation of the wave equation in spherical coordinates that amplitudes of a spherical (body) wavefront decay as 1/r; we also noted that a (cylindrical) head wave amplitude decays as 1/ .

A surface wave on a spherefollows a ring whose circumferenceequals asin. The energy per unitwavefront decreases as

Amplitude is proportional to thesquare-root of energy, so

which is a minimum at = 90° andmaximum at = 0° & 180°!

€

r

€

1

r=

1

a sin Δ

€

A∝1

a sin Δ

Multipathing:

Seismicwavesalso canbe focusedanddefocusedby velocityvariationsin themedium.

Reverse shot from a seismic refraction profile collected on afarm near Gosport, IN.

(Note: You’ll be looking at something similar to this for yourFinal Exam…)

4 m4 m

Gosport Best Fit RMS = 1.39 ms

Note that the amplitudes for these first arrivals do not follow asimple 1/r or 1/√r decay… !

These amplitudes decay rapidly even after correcting forgeometrical spreading (due to anelastic attenuation with lowQ: We’ll come back to that shortly). But geometric plusanelastic decay modeling poorly fits arrivals where two waves

come in at aboutthe same time..,

This is also anexample ofmultipathing!

1 2 3

V = 1250 m/sf = 250 Hz

Q = 5.1(10.2)

V = 3680 m/sf = 125 Hz

Q = 3.2(6.3)X

X

5301006.6

(13.3)