Seismology
Doctoral training on internal Earth
Part 3: Array Seismology,
Resolution issues
Christine Thomas
Barcelonnette, October 19, 20161
Array Seismology
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Seismic Arrays
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GERESS
Rost and Thomas, 2002
Seismic Arrays
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Seismic Arrays
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Rost and Thomas, 2002
Seismic Arrays
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Backazimuth and slowness
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often: S = u
Thomas, 2008
ray parameter p and horizontal slowness u:
uRv
iR
v
irp 0
0
00 sinsin
Backazimuth and slowness
Barcelonnette, October 19, 20168
at source: azimuth measured
from north to gcp
at receiver: backazimuth
(baz) measured from N to
gcp.
slowness related to angle of
incidence at array
Rost and Thomas, 2009
Horizontal slowness
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appapp
yxvv
uuucos
,sin
),(
horizontal slowness
angle of incidence
apparent velocity
Rost and Thomas, 2002
Beamforming
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Delay and sum technique
hori
N
i
i
N
i
i urtnN
tstxN
tb
11
1)()(~1
)(
Advantages
•phase with appropriate slowness amplified
•suppression of noise
noise suppression dependent of number of stations
Disadvantages
•works only for discrete values of u and baz
•waveforms must be coherent!
Beamforming
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Vespagram
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vespa=slant stack
beam trace for fixed azimuth:
N
i
uiiu ttxN
tv1
, )(1
)(
Advantages:
•only either u or baz needs to be known
•noise reduction
Disadvantages:
•structure influences results
•coherency required
•theoretical baz must be known correctly
•resolution not great for small arrays
Vespagram
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vespagram
slowness or backazimuth versus time
slownesses
Vespagram
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Rost and Thomas 02
theoretical backazimuth (or
slowness) needs to be
accurate.
otherwise changes in
slowness (or backazimuth)
due to interdependence of
slowness and backazimuth
misleading slowness
values or disappearing of
signal
correct baz
4th root stacking (vespagram)
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Nu
NuN
NuNu
i
i
NM
i
iuiNu
v
vvv
tx
txttx
Mv
,'
,'
,'
,
/1
1
,,'
)(
)()(
1
Advantages
•incoherent noise suppressed efficiently
•coherent phases amplified
•slowness enhanced
Disadvantages
•waveform distortion (non-linear technique)
•useless for waveform studies
4th root process
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traces4th root traces
linear stack4th root stack
4th root
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linear stack:
waveforms ok, other-
non-coherent phases
are also visible,
slowness resolution
poorer
4th root stack:
coherency more
important than
amplitude.
Stacked amplitude
smaller, waveforms
unusable, polarities ok.
4th root process
Barcelonnette, October 19, 201618Rost and Thomas 02
linear stack:
waveforms ok, other-
non-coherent phases
are also visible,
slowness resolution
poorer
4th root stack:
coherency more
important than
amplitude.
Stacked amplitude
smaller, waveforms
unusable, polarities ok.Rost and Thomas 02
Source stack
Barcelonnette, October 19, 201619
)()(1
),(1
jij
K
j
ji txtaK
tuS
with time delays urrjj
)( 0
Advantages
•cheap
•known structure
•identical response
Disadvantages:
•source parameter must be known
•multiples enhanced
if only one station available (with source array)
Double beam
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Rost and Thomas 02
Combines source
and receiver
stacks (e.g.,
Krüger et al.,
1993)
Better resolution
same problems as
source-stacks and
receiver stacks
Krüger et al., 1995
Phase weighted stack
Barcelonnette, October 19, 201621
)()()( tietAtS
nN
j
tiN
i
ipwsje
Nts
Ntv
1
)(
1
1)(
1)(
Another method to improve resolution:
phase weighted stack
(Schimmel and Paulssen, 1997)
complex trace
instantaneous phase
Rost and Thomas 02
"slowaz" analysis
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Simultaneous
measuring of
slowness and
backazimuth.
stacks over all
slowness and
backazimuth values
for a (small) time
window
Rost and Thomas 02
f-k (frequency wavenumber) analysis
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Jahnke 99
Advantages: can be used to find baz and u
Disadvantages: only applicable to small time windows
assumes plane wave
simultaneous
measurement of
slowness and
backazimuth.
In frequency
domain.
uses wavenumber
uk
ckkkk zyx
222||
f-k (frequency wavenumber) analysis
Barcelonnette, October 19, 201624
deN
XdttbkkEN
i
kkri
oi
2
1
)(22 01
)()()(
shifted trace )()(~0uurtxtx iii
beam:
N
i
ii uurtxN
tb1
0 ))((1
)(
using Parceval’s theorem
and shift theorem
the beam energy can be written as
dBN
dttbbeamE
2
2 )(1
)()(
with:)(
10
1
)( 0 kkAeN
N
i
kkri i
the array response
function (ARF)
ARF (f-k analysis)
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Array response function of
Yellowknife array, YKA, (left)
and Gräfengberg array GRF (bottom)
ARF also serves as measure of coherency
Rost and Thomas 02
Array methods
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Thomas 2008
Array methods
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Aligned seismogram
(distance dependent)
beam
vespa
F-k
Thomas 2008
Migration
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•calculates delay times from point at depth.
•shifts traces back with these delay times
•stacks traces
•3D grids with appropriate spacing
Migration
Ampl. at b.p.Rost and Thomas, 2009
Migration
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Advantages: no plane wave
approximation
project energy back to origin
good spatial resolution (steep rays)
good depth resolution (shallow
rays)
Disadvantages: velocities must be
known exactly
poor spatial resolution (shallow
rays, high frequencies)
poor depth resolution (steep rays)
Many other migration methods exist
(e.g. Hutko et al., 2004)
Thomas et al 1999
Resolution issues
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31
The Earth
After Strobach, 1991
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32
Resolution issues/Fresnel zone
How much can a wave "see"? How accurate is the observation?
view from top
Fresnel volume
around a ray. For
reflection, the cut
through the FV is the
Fresnel zone.
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33
Resolution issues/Fresnel zone
e.g.: P-waves reflected at D”: 2 x 4 degrees (1Hz)
S-waves reflected at D”: 3.5 x 7 degrees (6 s)
PP
What does that mean? If we deal with a strictly layered (1D) Earth
our resolution is only as good as the size of the Fresnel zone!
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34
Resolution issues/Fresnel zone
The Fresnel zone can be
reduced:
One possible way is the use of
arrays - only what is common
for all rays for the array
contributes to the Fresnel
Zone
(also commonly used in
Exploration seismics!)
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35
Resolution issues/ Map views of observations
We need suitable source receiver combinations to study different
structures. This is not always possible for all the regions of the Earth!
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36
Banana-Doughnut
If dealing with
waves rather than
rays:
Sensitivity to wave
around position of
ray, no sensitivity
along the ray.
Information from
area around ray!
Not from the path
assumed for the ray.
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37
Sensitivity kernels (surface waves)
Sensitivity kernels (surface
waves) show relationship
between dispersion velocities
and Earth Structure.
With increasing period,
surface waves become
sensitive to deeper velocity
structure.
(Similar in
magneto-tellurics)
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38
Sensitivity kernels (surface waves)
Sensitivity
kernels differ
between
Rayleigh and
Love waves.
Need to calculate
kernels for best
resolution.
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39
Places of reflection
Interpretation of
reflection point in
a 1D model.
(Mid-point
between source
and receiver)
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40
Places of reflection
But for a slanted
receiver, the mid
point is not
necessarily the
reflection point
(Snell's law
applies).
This is similar to
the migration
principle:
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41
Places of reflection
But for a slanted receiver, the mid
point is not necessarily the reflection
point (Snell's law applies).
This is similar to the migration
principle:
The reflector moves "uphill" and
becomes steeper.
A way to find true reflection is
migration of waves.
=Array seismology
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42
Resolution issues/Array response function
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43
When interpreting
stacks of seismic
data the array
response function
can produce
"spatial aliasing"
This leads to
spurious energy
that should not be
interpreted.
Resolution issues/Array response function
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Resolution issues/tomography
REGULARIZATIONDamping parameter
Dat
a M
isfi
t
DampingRawlinson et al. [2010]
Regularisation:
How much can the
data misfit be
reduced without
increasing the run
time (for small
damping only
marginal changes to
misfit).
Usually point closest
to zero (in bend of L-
shaped curve) is
taken for best fit.
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45
Resolution issues/tomographySources and receivers
Ray density
Number of paths in each cell
Sensitivity
Sensitivity of dt to the velocity in each cell
sampling of the region should be
uniform, but also sensitivity of rays to
structure should be taken into account
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46
Resolution issues/tomographySources and receivers
15x15 25x25
35x35 45x45
Many crossing paths are needed to find
the true model (not always possible).
Resolution also depends on the
gridding (cells)
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Resolution issues/tomography
Rawlinson et al, 2010
Rays are not straight lines
any more but may be
deviated in strikingly
heterogeneous media. This
leads to travel time
variations for the
theoretical travel times
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48
Resolution issues/receiver function analysisDeconvolution
issues
Effects of
frequency.
Some
reflectors
may be
missed
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49
Resolution issues/receiver function analysis
How do we find reflectors in this? How believable are reflectors?
Especially in poorly sampled regions?
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What can we interpret?
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Measuring amplitudes and polarities
51Barcelonnette, October 19, 2016
What can we interpret?
52
travel times
amplitudes
polarities
waveforms
frequency content
t1 t2 t3
amplitude
frequencycontent
polarityandwaveform
traveltime
Barcelonnette, October 19, 2016
What can we interpret?
53
Amplitude
one of the most difficult measurements because
amplitude is affected by many things:
source effects
receiver effects
attenuation - intrinsic and scattering
energy partitioning
structure/topography
discontinuity versus gradient
.....
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Amplitude - Topography
54
Snell's law
Ray parameter p = sin (i)/v
Reflected waves: angle is the
same as for incident wave.
Measured w.r.t vertical.
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Amplitude - Topography
55
Snell's law
Ray parameter p = sin (i)/v
Reflected waves: angle is the
same as for incident wave.
Measured w.r.t vertical.
Focussing and defocussing
effects!
= amplitude variations!
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Amplitude - Topography
Focussing and
defocussing effects
modelled with a
simple model:
strong effects in
amplitude and more
than one apparent
reflector visible in
places.
Misinterpretations
possible
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Focussing and
defocussing effects
modelled with a
simple model:
strong effects in
amplitude and more
than one apparent
reflector visible in
places.
Misinterpretations
possible
Barcelonnette, October 19, 2016
Amplitude - Topography
58
Amplitude - Source effects
when interpreting
amplitudes and polarities:
Knowledge of the source
is important!
N
E
largest amplitude (downswing)
largest amplitude (upswing)
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Amplitude - Stacking=averaging
sum sum
stacking a number of travel
will give an average
waveform.
This does not necessarily
describe the structure
correctly as reflections in
different places are added
(without correcting for the
pace of reflection).
changes in apparent
frequency possible
v v
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60
Amplitude - Unknown mineralogy
Seismologists often work with standard Earth models (PREM, ak135...)
When interpreting depths, these are based on ak135/PREM etc values
of discontinuities. Often this means interpretations in the ol-wd-rg-
pv+mw system only
Cobden et al (2008)
showed that depending
on mineralogy, the
discontinuities change
(depth and velocity/
density increase).
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61
Amplitude - discontinuity versus gradient
Sharp discontinuities reflect (or transmit) energy different from
gradient zones. Often calculations are done with sharp discontinuities
and then amplitudes are interpreted in terms of gradients.
Measuring gradients is difficult - frequency might help (but difficult)
vel
de
pth
vel
de
pth
vel
de
pth
vel
de
pth
these gradients will produce different seismic waveforms
but the interpretation will be non-unique!
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What can we interpret?
62
travel time
travel times of seismic waves can be affected by:
structures (fast and slow velocities)
deviations from the assumed path
topography (for reflections off discontinuities)
attenuation
.....
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Travel times - velocity variations
63
+-
t+
t-
t-t
t
travel time deviations due to
velocity variations.
and due to deviations from the
great circle path.
Barcelonnette, October 19, 2016
also: need to consider out-of-plane propagation!
(Fermat's principle)
What can we interpret?
64
polarities
polarities of seismic waves can be affected by:
source effects (different parts of radiation pattern)
energy partitioning (Zoeppritz equations) due to velocity and
density changes
.....
Barcelonnette, October 19, 2016
Polarities - energy partitioning
65
Energy partitioning
We have to consider
SV
P
SV
P
SH reflected
SH transmitted
and they depend on
velocities (P and S) and
density
reflected
transmitted
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Polarities - Zoeppritz equations
66
Zoeppritz equations
wave from below
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Polarities and Amplitudes
67
Zoeppritz equations - polarity and amplitude variations due to
changes in velocity and/or density are possible
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What can we interpret?
68
travel times
amplitudes
polarities
waveforms
frequency content
t1 t2 t3
amplitude
frequency
content
polarity
and
waveformtravel
time
Barcelonnette, October 19, 2016