On Earth’s internal structure
• We image its internal structure from Earth’s surface.
Seismology is our best tool
(because each ray samples only a small portion of the Earth)
• Then, need to convert measured things (like seismic wave speed)
to things of interest (like density, composition and temperature).
This is not the subject today.
Depth = 100 km
Proterozoic
Depth (km)
• Two basic types of structures
are imaged with seismology:
Volumes (tomography)
Interfaces (receiver
functions)
Archean
Moho
a receiver function image
Receiver Functions
(using teleseismic earthquakes to estimate
interface depths and seismic wavespeeds)
6000
4000
2000
0
Radius (km)
Plot of teleseismic P-wave ray paths
between 30° and 90 ° that can be used
for receiver function calculation
IASP reference seismic Pand S-wave velocity model
Receiver functions make use of the direct P wave and the
trailing P-to-S conversions
The main arrivals
used in receiver functions:
P and Pd s
VP , VS
VP , VS
Because S waves travel slower
than P waves, they arrive later.
> Time delay proportional to depth
> Amplitude proportional to contrast at d
Separating the P and S waves
vertical
P
(Pd s)
vertical
radial
Pd s
radial
Note:
P wave is strongest on vertical record
Pd s is strongest on radial record
illustration of raypaths
traveled by phases recorded
in receiver functions
[only P waves are seen]
[only S waves (and direct P wave) are seen]
Making a receiver function (step 1)
start with seismic data
(recorded in V-N-E
coordinate system)
rotate the horizontal
(N and E) components to
R-T coordinate system.
This depends on station
and event location.
Note:
1. Seismogram is complicated because (a) earthquake rupture was not a simple pulse, and
(b) wave interaction with the Earth added more complexity. We want only part b.
2. The tangential energy is not zero (in a simple Earth it would be).
A problem: combined signals
=
Earthquake source
Earth structure
Seismogram
Seismogram as the convolution (“winding together”)
of the source with Earth structure
We want to deconvolve source from the
recorded seismogram so we are left with
just the signal from Earth structure
A problem: several combined signals
We want to deconvolve source and instrument response so
we are just left with the signal from Earth structure.
We need to (1) represent the x(t) * i(t) waveform, and
(2) remove (deconvolve) its effect.
A solution: use vertical seismogram
to represent source * instrument
{
deconvolve
vertical component
from
radial and tangential
components
The vertical
component IS
x(t) * i(t)
(& some noise)
to calculate radial and
tangential receiver
functions
vertical receiver
function
{
}
If decon were perfect
the vertical receiver ftn
would be
Two examples
Beck and Zandt [2002]
Transition zone structure
beneath western US
Gilbert et al. [2003]
depth (km)
Crustal structure
of the Altiplano
{
Notice
1.
RF turned sideways
2. time axis changed to depth
At what depth were arrivals
produced?
• Need to identify arrival times of converted
phases
• Migrate from time to depth by assuming a
S- and P-wave model and the ratio of VP/VS
arrival time of vertically incident
S-wave after P-wave
VP, VS
D
ΔTPd s =
D
Vs
D
Vp
1
1
ΔTPd s = ∫ D
−
dz
VS (z) VP (z)
0
Integral of S-wave
slowness minus Pwave slowness
Example
VP, VS
D
Expected arrival time for vertically traveling P and S waves in a 40
km thick layer with VS=3.5 and VP=6.4
40km
40km
ΔTPd s =
−
= 11.4s − 6.25s = 5.1s
3.5km /s 6.4km /s
Arrival time of a non-vertically incident
Pds wave
VP, VS
h
p = sin Θ / V
D
ΔTPd s =
∫
0
D
−2
2
−2
2
VS (z) − p − VP (z) − p dz
Need to know ray-parameter (p), VP, and VS
of the incident P-wave
Arrival time of a non-vertically incident
Pds wave
VP, VS
h
p = sin Θ / V
D
ΔTPd s =
∫
0
D
−2
2
−2
2
VS (z) − p − VP (z) − p dz
−2
2
−2
= 40 * 3.5 − 0.06 − 6.4 − 0.06
= 5.4 sec
2
we have done examples solving the
forward problem of calculating what
TIME an phase from a given depth
would arrive
Given depth, VP, VS, and p, we found ΔT
ΔTPd s =
∫
0
D
−2
2
−2
2
VS (z) − p − VP (z) − p dz
now we want to solve for what depth a phase
was produced depending on its arrival time
with constant VS and VP that do not depend on depth arrival time
equation becomes:
−2
−2
2
2
ΔTPd s = z( VS − p − VP − p )
solving for depth z
ΔTPd s
−2
2
−2
VS − p − VP − p
2
=z
we pick the arrival time of the Pds phase,
calculate p, and assume VP, and VS
ΔTPd s
−2
−2
2
VS − p − VP − p
2
=z
Pds arrival time = 5 s, p= 0.06 s/km and
VP=6.4 km/s and VS =3.5 km/s
€
5
−2
2
−2
3.5 − 0.06 − 6.4 − 0.06
2
=z
5
−2
2
−2
3.5 − 0.06 − 6.4 − 0.06
2
=z
37.0 km = z
how sensitive is this value of z to
our assumptions?
we keep VP constant and investigate the VP/VS
dependence on crustal thickness estimates
Vp(km/s)
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
Vp/Vs
1.70
1.72
1.74
1.76
1.78
1.80
1.82
1.84
1.86
1.88
1.90
1.92
1.94
1.96
1.98
Vs(km)
3.76
3.72
3.68
3.64
3.60
3.56
3.52
3.48
3.44
3.40
3.37
3.33
3.30
3.27
3.23
p (slowness(s/km))
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
crustal thickness (km)
43.67
42.48
41.36
40.29
39.28
38.31
37.40
36.52
35.69
34.90
34.13
33.41
32.71
32.04
31.40
arrival time of vertically incident
PpPs-wave after P-wave
Pp
Ps
PpPs
VP, VS
D
With three times we can solve for Vp, Vs and D.
From receiver functions to
Receiver-function imaging
Steps:
1. Lay out (“backproject”)
receiver functions along
their ray paths.
2. Deposit RF amplitude
at appropriate nodes.
3. Average the amplitude
values at each node.
Using these many rays (1) combats noise and (2) images
lateral variations in features
Investigation of western United States upper
mantle and transition zone structure
Gilbert et al. [2003]
Hot region?
Crustal structure across the Archean-Proterozoic
suture in the western US
49.5 km
N
S
Depth (km)
Proterozoic
Crosswhite and Humphreys (2002)
Archean