Geology 6600/7600
Signal Analysis
03 Dec 2013
Last time: Deconvolution in Flexural Isostasy
• Surface loads can be solved from observed gravity and
topography provided ~(z), flexural rigidity and internal
load depth zl are known a priori:

D
1  0  k 4

g

D

1  k 4

g

h d
2 G0
expkz dz
0 dz

D

1  k 4

g





2 GL 






L

D 4

1  k
 

H I k  H k 
g

W k  B k 
h d
expkz dz exp kz  I  

0 dz

l 


D 4


L

1  k

g






• In contrast to Tharsis, western US topography appears to be

supported
significantly by dynamic (i.e., sublithospheric)
buoyancy
• Estimation of flexural rigidity D still relies on sometimesquestionable assumption of uncorrelated loading, so future
© A.R. Lowry 2013
analysis should use seismic constraints
Deconvolution of source & receiver
terms from distant earthquakes:
P
S
Rayleigh
Nov 30 Little Cottonwood Creek seismogram for M5 earthquake in S Mexico…
Recall that a seismogram represents a convolution of
the source-time function s(t) with the Earth system
response h(t) and the seismometer response i(t):
rt st hti t  R SH I 

For imaging applications we would like to remove the
source and receiver terms and just look at the Earth
response. One approach to doing this is to isolate the
impulse response to phase-conversions at impedance
boundaries using teleseismic receiver functions
In the most commonly-used approaches to seismic receiver
function analysis (e.g., Ammon, BSSA 1991; Ligorria &
Ammon, BSSA 1999) the horizontal (E, N) components
of a three-component seismogram are rotated into
radial and transverse directions based on back-azimuth to
Vertical
the source event:
Radial
N
Transverse
P
E
S
Radial
For a teleseismic event arriving rays are near-vertical, so the
vertical component contains predominantly P-wave particle
motion (with a small contribution from SV) and the radial
horizontal component contains predominantly SH motion
(with a small contribution from P). In an idealized (1D,
isotropic) Earth, the transverse contains motion neither
from primary P or converted (polarized) S!
Both vertical & radial components are convolved with the
same source-time function and instrument response
for each different phase that comes in:
rZ t
N

k 0
st hkZ t i t
rR t
N

st hkR t i t
k 0
Here, k represents each of N phases that originated as a P
wave and, after conversion, arrived as an S wave:
(Ammon, BSSA 1991)
Thus the source and instrument response are removed from
the time series by (frequency domain) division of the
radial by vertical components. The resulting impulse
response function is called the receiver function:
n
H  

k 0
n

k 0
n
RkR SI 

RkZ SI 

k 0
n

k 0

(Ammon, BSSA 1991)
RkR 
RkZ 
Receiver Function Estimates of
Crustal Thickness:
Delay Time
P Ps
Crust
Mantle
P
Ps
• Deconvolve source-time function to get impulse response of
phases converted at impedance boundaries
• Delay time between phase arrivals depends on crustal
thickness and relative velocities of P & S phases
• EARS uses iterative time-domain deconvolution [Ligorria &
Ammon, BSSA, 1999]: well-suited to automation
Contribution of crustal thickness (H)
versus vP/vS ratio (K) to delay time is
ambiguous…
P Ps
PpPs
PpSs
PsPs
PpSs
PsPs
P
Ps
PpPs
Resolve using reverberations, which
have differing sensitivity to H and K
H–K Stacking:
[Zhu & Kanamori, JGR, 2000]
Ps
PpSs
&
PsPs
PpPs
PpSs
PsPs
P
Ps
PpPs
Crustal
thickness (H) &
vP/vS ratio (K)
parameters
that predict the
observed phase
delay times
intersect at
a point in
parameter
space
H–K Stacking:
[Zhu & Kanamori, JGR, 2000]
(EARS H–K stack for station COR) Method stacks
[Crotwell & Owens,
2005]
Ps
PpSs
&
PsPs
PpPs
PpSs
PsPs
P
Ps
PpPs
observed
amplitudes at
delay times
predicted for
converted
Ps phase and
reverberations.
Max stack
amplitude
ideally reveals
crustal
thickness &
vP/vS ratio.
The Problem:
(EARS H–K stack for station TA.P10A)
The Moho is
not the only
lithospheric
impedance
contrast…
And crustal
thickness is
not constant
Root Variogram H
(km)
Root Variogram
K
Crustal
Thickness
(H)
vP/vS
Ratio
(K)
Despite outliers and other issues, crustal thickness &
vP/vS have statistical properties consistent with a fractal
surface…
Station TA.O09A (Central Nevada)
Variograms
can be used to
estimate a
“most likely”
crustal
thickness
and vP/vS ratio
via optimal
interpolation
from nearby
sites.
Station TA.O09A (Central Nevada)
Can also model gravity predicted for
each H & K
at the site…
And search for a
“most likely” model
with uncertainties.
Station TA.O09A (Central Nevada)
Combined
Optimal Interp.
Likelihood Filter
Station TA.O09A (Central Nevada)
 Unlikely stack amplitude maxima are
downweighted using likelihood statistics