1. 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 refers to directional dependence of velocity The fourth-order elasticity tensor, c ijkl, can be expressed more succinctly as a 6x6 “Voight matrix” C mn. Transverse anisotropy describes e.g. SPO in horizontally layered media, and is characterized by one P-velocity for x 1 & x 2 propagation, a different P-velocity for x 3 propagation, and differing SV- & SH-velocities: Azimuthal anisotropy describes the more complicated case of azimuthally-varying P-velocity, generalizing to:

2. 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)

3. Fluid-filled cracks in an isotropic medium will have an elasticity tensor of the form: In this example we’ve assumed that the normals to two-dimensional cracks parallel the x 1 axis. Note how the shear modulus is reduced by the cracks. If the fluid is incompressible, P-wave velocity is unaffected, but S-wave velocity is. Here, ε = Na 3 /V, with N being the number of cracks per volume V, and a is the half-width of the cracks.

4. Lin & Schmandt, Geophys. Res. Lett. 2014 Example of upper crustal (10-16 s) Rayleigh velocity anisotropy

5. … Interpreted as compressional stress direction  closed fractures Lin & Schmandt, Geophys. Res. Lett. 2014

6. Here, observations were made using marine seismic observations of refraction travel-times near Hawaii. Azimuth is measured relative to trend of magnetic isochrons in the region, so velocity peaks at 90° and 270° indicate that the fast direction is in the direction of spreading at the time the lithosphere was formed. So is that more likely to represent fluid-filled fractures or flow of mantle olivine?

7. The depth and age dependence of anisotropy in the oceans also lends insight into physical processes… Important to note, for interpreting this signal, that the direction and magnitude of anisotropy reflects an integral of the strain history of the rocks!

8. Toomey et al. Nature 2007 More recent data from the East Pacific Rise tell a more complicated (and more interesting) story… With significant implications for active- vs passive-components of the flow dynamics in oceanic spreading centers (not to mention segmentation of ridges, and the bathymetry and melt chemistry variations along a mid-ocean ridge…)

9. Shear Wave Splitting is commonly-used to identify azimuthal anisotropy. Given initial signal s(t) on the radial component only of the SKS arrival & angle f between radial & fast directions, Radial: Transverse: Note that normally one wouldn’t get a transverse component SKS arrival; only radial! but anisotropy “splits” the arriving energy in components  & || to anisotropy axes.

10. In practice, try lots of rotations and  t ’s to find which maximizes the radial component of amplitude.

11. Note  gives you the fast direction;  t describes thickness times  v !

12. Miller & Savage, Science, 2001 Time- varying anisotropy has been observed e.g. before and after an eruption of Mount Ruapehu in New Zealand… 1994, <30 km1998, <30 km 1994, >50 km1998, >50 km

13. Attenuation and Anelasticity Wave 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.

14. 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 sphere follows a ring whose circumference equals asin . The energy per unit wavefront decreases as Amplitude is proportional to the square-root of energy, so which is a minimum at  = 90° and maximum at  = 0° & 180° !

15. Multipathing: Seismic waves also can be focused and defocused by velocity variations in the medium.

16. Reverse shot from a seismic refraction profile collected on a farm near Gosport, IN. (Note: You’ll be looking at something similar to this for your Final Exam…)

17. 4 m Gosport Best FitRMS = 1.39 ms

18. Note that the amplitudes for these first arrivals do not follow a simple 1/r or 1/√r decay… !

19. These amplitudes decay rapidly even after correcting for geometrical spreading (due to anelastic attenuation with low Q : We’ll come back to that shortly). But geometric plus anelastic decay modeling poorly fits arrivals where two waves come in at about the same time.., This is also an example of multipathing! 1 23 V = 1250 m/s f = 250 Hz Q = 5.1 (10.2) V = 3680 m/s f = 125 Hz Q = 3.2 (6.3) X X 530 100 6.6 (13.3)