Action items

1. Code for generation of stochastic S and P waves.
2. Code for \vec{d}_i(t) given simulated sky map of S and P waves.
3. Calculate spherical harmonic decomposition of S and P waves with mathematica notebook.
4. Calculate S and P wave sky-map with regularized matrix inversion. Demonstrate the ability to recover accurate maps and to disentangle S and P. Investigate degeneracies and how to eliminate them with different array configurations.
5. Work out formalism for Rayleigh waves (covariant S+P waves).
6. Investigate the relevance of refraction / turning depth for radiometric measurements.
7. Investigate depth dependence of surface (Love) waves.

Seismology for astrophysicists

• Love waves are just a subset of S waves. However, they are surface waves, so their amplitude varies with depth.
• When an S or P wave interacts with a surface, boundary conditions require that it reflects as a *coherent* combination of S *and* P wave. These combined SP waves are Rayleigh waves, and the fact that S and P are coherent violates the assumption in Eq. 5 in Vuk's note, where S and P are assumed to be uncorrelated. Further, the fraction of power that gets converted from S into P and vice versa is O(1). The good news is that there's a very predictable phase relationship between S and P in Rayleigh waves.
• wave speed changes with depth for both S and P waves. This refraction causes wavefronts to change orientation as they travel through the Earth. Waves bouncing off the surface curve away from the center of the Earth and head back to the surface. Seismologists refer to the “turning depth” as the depth at which waves curve back to the surface. The value of turning depth depends on frequency and geological composition.
• Inhomogeneities. Any features characterized by a size of lambda/4 can probably be safely ignored. At 1 Hz, Victor thinks this corresponds to a length scale of ~700m, and the rock should appear very uniform at that length scale. However, at 10 Hz, Victor thinks that the length scale will be ~20m (note v depends on f in rock), and we might see significant inhomogeneity at this scale.

Previous questions/investigations

• Investigate 2D version of radiometer. If we were only interested in surface waves (intrinsically a 2D problem), then we could model the change in surface waves as a function of depth with a simple formula. The angular power spectrum for surface waves is the same at all depths up to an overall factor which decays exponentially with depth. However, we're interested in 3D.

Slides

• Unordered List Itemturning depth (April 3, 2014)
• Unordered List Itemmore on refraction (April 17, 2014)