1. Geology 5640/6640 Introduction to Seismology 17 Feb 2015 © A.R. Lowry 2015 Last time: Seismology as Investigative Tool Deep-Earth investigations use earthquakes as sources, three-component broadband seismometers as receivers, and analysis tools include: Normal modes Precursors Waveforms Receiver Functions Tomography Anisotropy Ambient Noise Receiver functions use transmitted and converted phase arrival times to image impedance structure Tomography uses travel-times for large numbers of criss-crossing rays to image velocity structure Anisotropy measures direction-dependence of velocity Ambient noise uses environmental Rayleigh waves Read for Wed 17 Feb: S&W 53-85 (§2.4-2.6)

2. Assignment I is now posted on the course website… Due Monday, March 2 at the beginning of class Note: Not all of the relationships you’ll need have been covered in course notes, but if you’ve been reading your text you’ll know where to look!

3. Waves in Spherical Coordinates: Up to now we’ve used  to denote the scalar displacement potential, but  shows up as a coordinate in spherical coordinates… So for now we’ll use  for potential. Then our wave equation becomes Spherical coordinates relate to Cartesian as: x = r sin  cos  y = r sin  sin  z = r cos 

4. Using these relations we can transform the Laplacian    from Cartesian to spherical coordinates: To convert to spherical coords we’ll need to use the chain rule, e.g.: If we take derivatives of the coordinate relations (last slide) we get, e.g. for ∂  ∂ x : (1)

5. We now have three equations in the three unknown derivatives; solving, we get: Then substituting (2) into (1) we get: Clearly by the time we do all three derivatives and evaluate    this will get into some heavy trig… But suffice to say, with the help of some identities this eventually leads us to: (2)

6. For a spherically symmetric solution (i.e., a point source in a constant-velocity medium), the ∂   ∂   terms are 0. Then the Laplacian is simply and our scalar potential wave equation becomes: with solutions of the form: (Note that if there is ,  -dependence, this will be  (r, , ,t) ). Recall our earlier solutions had a f(t±r/  ) dependence, so the spherical solution has an additional 1/r dependence…

7. Ray Theory represents the full wavefield in skeletal form, in terms of travel-times along propagation paths. Pros: It’s relatively simple (analogous to optics) Can be used for many problems, including: – Determination of earth structure – Locating earthquakes – Tomographic inversion for earth structure (We saw some of these Friday) Cons: Accurate only for infinite frequency waves ( = 0 ). So, It images structures on scales of order ≤ poorly (i.e., it can’t reproduce “wavefront healing” or other diffraction-related phenomena; poorly represents steep velocity gradients).