Download Seismology Lecture Notes: Tomography and Wave Propagation and more Study notes Geology in PDF only on Docsity!
CONTINUATION OF TOMOGRAPHY
Futher Comments:
- Linearization of the travel time residual
path
δ t sdl
If we assume a simple reference model we can calculate the reference ray using Eikonal equations.
Figure 1
Structures in the subsurface will affect the propagation path of a ray. Figure 1a. shows the focusing effect of a high-velocity sphere on a bundle of rays which results in a high sampling rate for the body. An example of this effect in the Earth is the high velocity of cold, subducted slabs. Figure 1b. shows the opposite situation, a low-velocity sphere. The bundle of rays tends to avoid the structure and the sphere is less well sampled. An Earth example of this is a hot, low velocity mantle plume. The bending of the rays away from a low velocity body makes it difficult to image such features.
Follow the process: m ˆ^ =∆ s ˆ a. Slowness, s, is known b. Recompute Eikonal equations
0
5
10
Depth (km)
15
20
25 5 10 15 20
5.0 7. Velocity (km/s)
Distance (km)
High-Velocity Sphere Low-Velocity Sphere
Distance (km)
0 25 0 5 10 15 20 25
0
5
10
Depth (km)
15
20
25
3.0 5. Velocity (km/s)
Figure by MIT OCW.
st (^) April 2005
c. Update the ray geometry d. Repeat the exercise
δ t = Tobs − T 3 Dref so the travel time residual becomes δ t ≅ ∫ 3 ∆ Drefpathsdl where ∆ s = s − s 3 Dref
This is normally done once or twice for global, long wavelength tomography and multiple times for detailed tomography.
- There are different strategies for Global Tomography. Each has advantages and disadvantages a. Global (e.g. spherical harmonics) vs. Local (e.g. blocks, splines) parameterizaition. Global : (+) nice theory, good for long wavelengths, facilitates spectral analysis. (-) artifacts in poorly sampled regions, not suitable for high definition imaging of, say, slabs. Local : (+) deals better with uneven data coverage, can be regionalized to obtain high definition images in densely sampled regions. (-) poor constraints on long wavelengths, many unknowns. b. Direct vs. Interative inversion algorithms Direct : (+) elegant resolution anaylsis, (-) RAM limits solution Interative : (+) “large” models, (-) indirect resolution analysis, subjection convergence criteria and regularization. c. Data : Waveforms vs. routinely processed bulletin data.
Figure 2
Image removed due to copyright considerations.
Please see: Kárason, H., and R. D. Van der Hilst. "Constraints on Mantle Convection from Seismic Tomography." In The History and Dynamics of Global Plate Motion ( Geophysical Monograph). Vol. 121. Edited by M. R. Richards, R. Gordon, and R. D. Van der Hilst. Washington, DC: American Geophysical Union, 2000, pp. 277-288.
Figure 2 illustrates the importance of cell size when approaching the question you are trying to answer. Decreasing your cell size will result in more detailed images but there is the hazard of over-parameterization of the model if the cell size is too small. You want
st (^) April 2005
Surface Waves
Figure 4
Image removed due to copyright considerations.
Figure 4. shows the vertical component waveforms for an event recorded by an array in south-east China. The first arrival is the P waves followed by the S wave. The surface waves are clearly seen as the later, high amplitude arrivals.
Image removed due to copyright considerations.
Figure 5. 12/26/04 Sumatra earthquake location.
st (^) April 2005
(from http://www.iris.iris.edu/sumatra/)
Figure 6. Low frequency, surface wave arrivals for the Sumatra earthquake. The surface waves are known as “ground roll” in the exploration industry.
Rn labeling refers to multiple global propagation paths. See Figure 7 for an explanation..
st (^) April 2005
The three-component data for an event located in Tonga is shown in Figure 9.
Figure 9
Image removed due to copyright considerations.
The vertical component (Z) shows more clearly the arrival of the compressional waves (P, PcP, PP, PPP) and coupled P+Sv waves (ScP, Rayleigh waves(LR)). The first horizontal component (N) has not recorded many P wave arrivals because this receiver was located to the west of the event, resulting in a natural polarization. This seismogram is more rich in S wave arrivals (S, ScS, SS, SSS) and Love surface waves (LQ) from SH waves interacting with the free surface. This component is equivalent to the transverse component. The second horizontal component (E) is equivalent to the radial component Notice that the Love waves arrive early than the Rayleigh waves. More on this later in the class.
st (^) April 2005
Figure 10
Love waves have a sideways motion (similar to a snake). Evanescence occurs and the amplitude decays exponentially with depth.
Rayleigh waves are analogous to ocean waves. They have retrograde or prograde motion.
LOVE WAVES
Love waves behave in the same way as ground roll or acoustic waves.
In an elastic medium V (^) p = ρ
μ
and the bulk sound speed is V (^) BSS = V (^) p^2 − 2 3
VS
In a fluid a P-wave becomes an acoustic wave. V (^) p = which is also the bulk sound ρ
speed, VBSS. Examples of acoustic waves are core waves and T-phases.
ACOUSTIC WAVES
For an S wave V =
μ s
σ
ρ
ij =^ −^ p^ δ^ ij
where p =pressure
p = −Κ∆
where:
st (^) April 2005
For a homogeneous medium ∇⎜⎜ (^) ⎟⎟ = 0 ⎝^ p ⎠ So the wave equation becomes
P &^ & = k^ ∇ 2 p ρ
where
k = c^2 ρ
If we look at displacement instead, U ≅ exp{ i ( k. x − ω t )}
We know ρ u && = −∇ p from the Equation of Motion.
Then
ρ u && = −ρω 2 u = −∇ p
u = 2 ∇ p ρω
u = ( u , u ) =
∂ p ,
∂ p (^) ⎞ x z (^) 2 ⎟ ρω (^) ⎝ ∂ x^ ∂ z^ ⎠
