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1
Source
c 1 Head wave 𝒑 = (^) 𝒄𝟏𝟐
Post-critical reflection
c 2
Direct wave 𝒑 = (^) 𝒄𝟏𝟏
n=
n=… n=
n=
p=
p I II
𝟏 𝒄𝟐
𝟏 𝒄𝟐
Continents: Quick review. Ground Roll/Love waves. Group velocity and phase velocity. Dispersion curves.
Last lecture we looked at…
Figure 1: Ray paths for a layer over a half-space model. The head wave exists whenc 2 > c 1.
Ground roll/Love Waves
The dispersion relationship for the ground roll is given by
Figure 2: Ground roll dispersion relationship
tan ωH (^) c^1 1
2 − p^2 =
ρ (^2) c^1 22
2 −pn^2
ρ (^1) c^1 12
2 −pn^2
Receiver
H
2
A finite number of modes have discrete phase velocities. The fundamental mode (n=0), represents a direct wave; higher modes depend on the wave frequency. The left hand side of equation (1) shows the frequency relation. The spacing between the curves depends on frequency; the higher the frequency the closer the curves are as a result of the decreasing number of higher modes.
I: Pre-critical ( 𝐩 < (^) 𝐜𝟏𝟐 )
This is also known as the “leaky mode” because energy is lost due to wave transmission into the half space.
The reflection coefficient is
R =
z 1 cos i 1 − z 2 cos i 2 z 1 cos i 1 + z 2 cos i 2
where z is the impedance and is given by
z =
c p
and the transmission coefficient is
T =
2 z 1 cos i 1 z 1 cos i 1 + z 2 cos i 2
This mode is frequency independent.
II: 𝐢 > 𝐢𝐜 Post-critical (^) 𝐜𝟏𝟐 < 𝐩 < (^) 𝐜𝟏𝟏 (post critical reflector and “transmission”)
This is also called the “locked mode” because all of the energy stays in the upper layer. R ∈ ℂ , T ∈ ℂ. This mode is frequency dependent.
In the layer (0 < z < H), the displacement is given by
Un x, z, t = A ∗ cos ω
c 1
2 − pn^2 z exp i kn x − ωt. (5)
The term pn can be obtained from the dispersion relation from equation (1). It represents the
slowness of n th higher mode (overtone), pn = (^) c^1 n
. The exponential term in equation (5) represents
4
The higher the mode, the better the sensitivity is to the deeper structure. There are distinct modes, which are understood through interference boundary conditions, thus constraining which combinations can be used. Some of the modes will not propagate as a wave because there is no constructive interference.
Frequency-Wavenumber domain (ω-k)
ω – k domain is obtained by taking the Fourier transform from space-time (x,t) domain. In the ω– k space, we can analyze the previous study of surface wave dispersion relation in equation
(1). A constant phase velocity can be represented by a straight line c = ω k. The fundamental mode
and higher modes can also be represented.
Figure 4: Surface wave dispersion in the ω – k domain
Higher frequency makes the mode approach c 1 asymptotically (i.e. no sensitivity). Conversely, smaller frequency makes the mode approach c 2 asymptotically, which means low frequency waves are more sensitive to half space and evanescent waves at low frequency sample deep in the half space. We can consider three cases: fixed frequency, fixed wavenumber, and fixed wave speed.
(1) At fixed frequency, waves propagate along different directions consistent with each discrete
wavenumber. Discrete wavenumber represents discrete phase speed, cn = (^) kωn. The smaller
wavenumber has the higher phase velocity. The lowest velocity corresponds to fundamental mode and high velocity comes from higher modes.
k
c 2 =ω/k 1 c 1 =ω/k 2
Fundamental mode
1 st^ higher mode
2 nd^ higher mode
3 rd^ higher mode
ω fi x
k 2 k 1 k 0
kfix
ω 0
ω 2 ω 1
cfix
ω
5
(2) At fixed wavenumber, waves propagate along a fixed direction but with a discrete number of wave frequencies. The higher frequency has the higher phase velocity. The lowest velocity comes from fundamental mode and higher modes give higher velocities.
(3) At fixed wave speed, there are infinite numbers of modes. For a given distance, waves of low frequency of the fundamental mode can arrive at the same time as higher modes at a higher wave frequency. Note that the phase velocity is described by a straight line.
Dispersion: Phase Velocity and Group velocity
Consider two harmonic waves with particular wavenumbers k 1 and k 2 ,and with particular wave frequencies ω 1 and ω2. The two waves can be represented by the following relations:
u x, t = cos k 1 x − ω 1 t + cos (k 2 x − ω 2 t) δω = ω − ω 1 = ω 2 − ω where (ω 1 < ω < ω 2 ) δk = k − k 1 = k 2 − k where k 1 < k < k 2 , (7)
using the above relations we can add the cosines and simplify to
u x, t = cos k 1 x − ω 1 t + cos (k 2 x − ω 2 t)
ei(kx^ −ωt)^ = cos kx − ωt − isin(kx − ωt) (^) (8)
u x, t = 2 ∗ cos kx − ωt cos (δkx − δωt).
The first cosine term (carrier) travels with phase velocity c = ω k , and the second (envelope or beat
pattern) travels with group velocity, u = δωδc.
Figure 5 shows the „beating‟ effect.
envelope Carrier
δω = ω − ω 1 , δk = k − k 1 δω = ω 2 − ω, δk = k 2 − k
cos k 1 x − ω 1 t
cos (k 2 x − ω 2 t)
7
Figure 7: Group and phase velocity in the frequency-wavenumber domain ω, k.
As frequency goes to infinity, group velocity is equal to phase velocity, this relation can be written as
lim n→∞ u = lim n→∞
δω δk
ω k (11)
In this case, group velocity will be equal to c 1 for high frequency waves.
Arrival time
From previous relations we know: T = (^) cx 1 ; u = x T; u = δω δk ; k = ω c ;ω = ck u = c + k (^) δδkc
k =
2 π λ u = c − λ δ δλc = group velocity
lim ω→∞ u = lim ω→∞
δω δk
ω k = c (^1) (12)
ω
k
C 2 C 1
u =
δω δk
k 0
ω
k
c 2 cn c 1
u
8
Figure 8: Evanescence of the wave
e−ηω^ z^ , e−kz^ z^ = e−
2 π λ t^. Low frequency wave is more sensitive to deep structure. Therefore, low frequency wave should arrive earlier than high frequency wave.
Looking at the fundamental mode will give us some information about shallow depths. Combing with higher modes will give even more information about what is happening at depth (see Figure 3).
Principle of the stationary phase
Only certain frequencies and directions will interfere constructively to create arrivals. If at a particular time you have wave propagation they add up to give the seismogram amplitude.
Seismogram:
Equation 14 shows plane wave superposition.
Building interference in means many combinations of ω & k will not result in displacement.
u(x, t) = A(ω, k)ei(kx^ −ωt)dωdk ωk (14)
u(x, t) = A(ω, k)ei(kx^ −ωt)dωdk ωk
d dk kx − ωt = 0
d dω kx^ −^ ωt^ =^0 ^ expressions of stationary phase
dω dk =^
x T =^ u^ group velocity
increasing λ
10
Rayleigh waves are more complicated than Love waves.
Dispersion
Phase velocity 𝑐(𝝎)
phase dispersion curves
group dispersion cruves
Oceanic crust Continental
Figure 10: Group velocity vs period
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 (Figure 11). 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
Generally, 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.
Period (T)
Contntinantal
Oceanic obs.
u
5km 35 - 40 km
11
Figure 11: Phase speed sensitivity kernels. (From "An Introdution to Seismology, Earthquakes, And Earth Structure" by Stein &Wysession (2007). )
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 12 shows a variety of typical dispersion curves for different tectonic provinces.
Figure removed due to copyright restrictions.
13
propagates mainly in the constructively interfering wave packets, which move with the group velocity. Narrow-band filtering can isolate the wave packets with specific central frequencies (see Figure 13), and the group velocity for that frequency can then be determined by simply dividing the path length along the surface by the observed travel time. This technique can be used for the construction of dispersion curves (Figure 12).
Figure 14: Group velocity dispersion through bandpass filtering (Figure 13 and 14 are adapted from Stein and Wysession, p97) Updated by: Sami Alsaadan Sources: April 11,2005 by Kang Hyeun Ji. Aaril 13, 2005 by Patricia Gregg. April 20, 2005 by Sophie Michelet. April,04,2008 lecture. “An Introdution to Seismology, Earthquakes, And Earth Structure” by Stein &Wysession (2007).
