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Surface waves (continuing)
For evanescent waves such as Rayleigh and Love waves we have seen that long wavelength waves penetrate deeper into the half space than short-wavelength waves. How exactly structure in a certain depth interval influences a wave of a particular frequency is described by a sensitivity kernel. They represent the maximum particle motion at a certain depth as a function of frequency, which can be computed from a reference Earth model.
Excitation of surface waves
Figure 1 can be used to understand in qualitative sense the excitation of surface waves by earthquakes. In general, the position of the earthquake (i.e. the depth in our case of depth- dependent media) determines which modes can be excited. A fundamental mode has no displacement deeper than a certain depth; by reciprocity, a source (assume a white spectrum of the source so that it can — in principle — excite all frequencies) that is located at those large depths will not cause displacement of that fundamental mode at the surface.
Figure 1:phase speed sensitivity kernels
Dispersion curves
We have seen that the radial variation of shear wave speed causes dispersion of the surface waves. This means that the observed surface wave dispersion contains structural information about the radial variation of seismic properties. A plot of the group or phase velocity as a function of frequency is called a dispersion curve. Their diagnostic value of 1D structure has been explored in great detail. Typically, the curves produced from observed records are matched with standard curves computed from an assumed reference Earth model that can have a structure that is characteristic for a certain type of upper mantle (e.g., old/young continents, old/young oceans, etc.). Such analyses have produced the first maps of the thickness of oceanic lithosphere which revealed the increase in thickness with increasing age of the lithosphere (or distance from the ridge. Figure 2 shows a variety of typical dispersion curves for different tectonic provinces.
1
( )
(^
/^
)
2 3 4 5 ( )
T Phase
Continental Rayleigh
Mantle Rayleigh
Mantle Love G Phase
Oceanic Rayleigh Sedimentary Rayleigh
Sedimentary Love
Group Velocity km sec2.
10 20 Period sec
30 40 50 100 200 500 1000
Oceanic Love
Continental Rayleigh Overtone
Continental Love Overtone
Continental Love
Figure by MIT OCW. Figure 2
In Figure 3, the dashed lines are travel time curves for those S/SS/SSS phases. But note that the frequency of those phases change with distance, so that the waveform changes. For instance, with increasing distance, the first arriving phase is composed of waves with larger frequencies (because they sample deeper).
( /s)
1/
Group velocity km
10 unit
unit amp.
T=14.5 s
2:
Transverse
Example of Love wave group velocity dispersion through bandpass filtering.
Unfiltered
2:
4.0 3.5 3.
Figure by MIT OCW. Figure 4: group velocity dispersion through bandpass filtering
Surface wave Tomography
- Recall from the travel time tomography:
projector (consists of ray segments)
d = Am
data
model
observations model perturbation (m-mo) δt = T – To
di = ∫ Gi
volume
( r ) μ( r ) dr
sensitivit y kernel/ Frechet derivatives
δ t = ∫ ∆ s. dl Integration over the ray path
Surface wave tomography:
-^ d^ '^ =^ A^ '^ m We have the same model as for body wave tomography.
Thus, we can combine the two:
⎛ d^ ⎞ ⎛ A^ ⎞ m ⎝
⎜ d '⎠
⎟
= ⎝
⎜ A '⎠
⎟
- When we do surface wave tomography, we measure the phase (or group) velocity: δ ci for an earthquake i.
- As with the body wave tomography, we will look at perturbations between the model and the data : 1 δ ci ( ω ) = ∫δ c ( ω , θ , ϕ ) dl ∆ (^) path
d ' = A ' m
Ray path
- Phase velocity perturbation model at a position δc (ω,θ,ϕ)
→ We can construct phase velocity maps
- Those phase velocity maps are first constructed for a particular frequency. But, because of the evanescence of the surface waves, we can see what frequency is sensitive to a particular depth. → convert ω to z
To do so, we look at the sensitivity kernel for different frequencies (figure 1). We ∂ c use ∂β
and we get:
1 ⎡⎛ ∂ c ⎞ ⎤ δ ci ( ω ) = ∆ (^) path ⎢⎣
⎝⎜^ ∂β (^) ω , z ∫ δβ^ (^ θ^ ,^ ϕ^ ,^ z ) dz ⎥^ dl ⎠⎟^ ⎥⎦
δ c ∂ c = ∂β δβ
An example of waveform modeling:
Image removed due to copyright considerations.
Please see:
Maggi, Alessia, and Keith Priestley. "Surface Waveform Tomography of the Turkish-Iranian Plateau." Geophysical Journal International 160, no. 3: 1068-1080. doi: 10.1111/j.1365-246X.2005.02505.x.
Figure 5: Sensitivity of 1-D waveform inversions to focal depths. The effects of varying focal depth for two events: (a) focal depth 2 km (b) focal depth 9 km. Fits of final synthetic (dashed) to observed (solid) seismograms for shifts in focal depths of 10-50 km are shown on the left and the corresponding inversion velocity models (thin lines) are shown on the right along with the velocity model for the correct depth (thick line). The misfit and maximum frequency achieved by the waveform inversion are denoted to the right of each waveform fit.
