Download pdf file - Berkeley Seismological Laboratory

Seismology Lecture 3 EPS 122 –Spring 2017 Thursday 02/23/2017 Instructor: Barbara Romanowicz Equation of motion (Isotropic, homogeneous medium, 1D)
∂2 u
∂2 u
ρ 2 = (λ + 2µ ) 2 + f
∂t
∂x
Medium described by : (ρ,λ,µ) or (ρ ,α ,β)
P waves – velocity:
λ + 2µ
VP = α =
ρ
S waves - velocity
µ
VS =
ρ
Polarisation of P and S waves
S waves are shear waves No volume change, shear and rotaIon P waves and S waves have different velociIes Vs < Vp P waves are compressional waves cause volume change – compression And rarefacIon No rotaIon Similar to sound waves in air Ref: Stein and Wysession
Rotation of coordinate system
•  Instruments are generally oriented (Z, N, E)
•  Ray path is in the vertical plane containing the
source and the receiver
•  In the horizontal plane tangent to the earth’s
surface at the location of the station, we want
to rotate the coordinate system from (N,E) to
(L,T) where:
–  L is for “longitudinal” – indicating the sourcestation great circle direction
–  T is transverse: 90 degrees from L (conventionally
counted counterclockwise)
Shallow earthquake From Stein and Wysession, 2003 ALPHABET SOUP FOR SEISMIC WAVES
core mantle Reflected waves on the CMB and the ICB
SS
P
S
Surface
waves
Shallow earthquake From Stein and Wysession, 2003 Surface waves •  Two types –  Rayleigh waves (R) –  Love waves (G) Appear on different seismometer components Surface waves
•  The amplitude of surface waves decays with
depth and depends on the ratio of the depth
to the wavelength.
–  => longer period (i.e. longer wavelength) surface waves tell us
more about deeper parts of the mantle
–  => shorter period surface waves are appropriate for studying
the shallow parts (crust, uppermost mantle)
•  Surface waves are primarily sensitive to shear wave
velocity
•  They are dispersive (velocity depends on period)
–  Generally longer period waves arrive first
Mongolia earthquake recorded in Japan Surface wave dispersion
•  Group velocity (U) – km/s
–  Velocity of travel of energy at a particular
frequency
•  Phase velocity (C) - km/s
–  Velocity of travel of a particular phase
(peak or trough)
•  Consider the sum of two harmonic waves
with slightly different angular frequencies
and wavenumbers:
u(x, t) = cos(ω1t − k1 x) + cos(ω 2 t − k2 x)
with :
ω1 = ω + δω1;ω 2 = ω − δω ; δω << ω
k1 = k + δ k;k2 = k − δ k;k << k
•  We obtain:
u(x, t) = 2 cos(ω t − kx)cos(δω ⋅ t − δ k ⋅ x)
u(x, t) = 2 cos(ω t − kx)cos(δω t − δ kx)
carrier Envelope travels at group velocity: dω
U=
dk
Carrier travels at phase velocity: C=
ω
k
Lower frequency smaller wavenumber: varies more slowly in Ime and space: envelope RelaIon between group and phase velocity: U=
dω
dk
=> ω
C=
k
U =C+k
dC
dC
= C +ω
dk
dω
(layer over half-­‐space)