Geology 6600/7600
Signal Analysis
30 Oct 2015
Last time: Kalman Filtering & Example
• As with all methods that rely on information about the
statistical behavior of signals/observations,
Kalman Filtering requires that the assumed pdf and its
properties must be approximately correct…
• Example of Kalman Filtering of postseismic GPS time series
assumed stationary white noise processes for both forcing
and additive noise… Approximately correct for measurement
error (but could have been improved by using true variability
of the variance!) but definitely not true for the forcing (which
should have had decreasing variance with time).
• As a consequence, parameters that did a better job of
reducing measurement scatter during low-signal periods
also interpreted some real signal as noise during rapid
transients…
© A.R. Lowry 2015
Reminder: Your charge for
Becker, T.W., et al., Static and dynamic support of western
United States topography, Earth Planet. Sci. Lett., 2014….
Two “background” items to discuss:
• An early draft examined only global cross-correlations, but
an associate editor wanted to see wavelength-dependence
of cross-correlation. How was this addressed, and how
does that approach relate to other topics in this class?
• The associate editor asked whether the revised approach
acts as a zero-phase filter in the frequency domain. How
could you test this?
Static and dynamic
support of western U.S.
topography
Thorsten W Becker
University of Southern California, Los Angeles
Claudio Faccenna (Universita di Roma TRE)
Eugene D Humphreys (U Oregon Eugene)
Anthony R Lowry (Utah State, Logan)
Meghan S Miller (USC)
Acknowledgements: NSF, EarthScope USArray; structural
seismologists sharing their models in electronic form, in particular
B. Schmandt, W. Chen. Code from CIG and B. Steinberger, GMT
GSA Pardee Symposium: Advances in understanding
Earth structure and process from EarthScope
Denver, October 30, 2013
* Brazenly
stolen from
Thorsten’s
2013 GSA
keynote
Origin of vertical tectonics?
Lowry et al. (2000)
e.g. Crough and Thompson (1977),
Lachenbruch and Morgan (1990),
Jones et al. (1992), Chase et al. (2002)
Forte et al. (2009)
Moucha et al. (2008, 2009)
Liu and Gurnis (2010)
What is the origin of non-flexural
topography (in the context of
USArray)?
Smoothed (l > 200 km)
reference topography
CP :
CVA :
cGB :
GV :
OCR :
SN :
YS :
Colorado Plateau
Cascades Volcanic Arc
central Great Basin
Great Valley
Oregon Coastal Ranges
Sierra Nevada
Yellowstone
Becker et al. (2014)
“Smoothing” here is actually convolution with a
6 = 300 km radius Gaussian function.
• Why was this done?
• How might this have been done more efficiently in an
alternative fashion?
• Given the objective of
this calculation, how
might this have been
done more robustly?
• What might remain
problematic even if it
were done more
robustly?
(Removing the “flexural effects” allows approximation as “Airy”)
Isostatic topography
crust, rc
L
mantle
lithosphere, rl
asthenosphere
lc
ll
tˆ
tˆ = f1lc + f2ll
ridge level
Isostatic
contributions
ra - r c
f1 =
ra
ra - r l
f2 =
ra
f1 » 0.12
f1 » -0.01
ra
crustal layer
mantle
lithosphere
cf. Crough and Thompson (1977), Bird (1979), Lachenbruch and Morgan (1990)
crust, rc
L
mantle
lithosphere, rl
asthenosphere
lc
ll
tˆ
tˆ = f1lc + f2ll
Isostatic
contributions
ra - r c
f1 =
ra
ra - r l
f2 =
ra
f1 » 0.12
f1 » -0.01
ra
+ deflections due to
present-day
asthenospheric flow
(“dynamic topography”)
crust, rc
L
mantle
lithosphere, rl
asthenosphere
ra
“Static”
ˆt
tˆ = f1lc + f2ll
l
c
ll
ra - r c
f1 =
ra
ra - r l
f2 =
ra
f1 » 0.12
f1 » -0.01
+ deflections due to present-day
asthenospheric flow
“Dynamic”
Crustal thickness from receiver
function Mohos, based on USArray
Levander and Miller (2012)
mean and standard
deviation of
all depicted fields
see also Chen et al. (2013)
Lowry and Perez-Gussinye (2011)
Residual topography for
variable crustal thickness
Based on Levander and Miller (2012)
Based on Lowry and Perez-Gussinye (2011)
All residual topography models are minimized by adjusting the asthenospheric
density at fixed crustal and lithospheric density
Becker et al. (2014)
Correlation
for Airy
isostasy
total r (coherence)
(solid)
2
and
power
spectrum
(dashed)
Becker et al. (2014)
• The “total r2” described here is the (now-familiar!) squared
correlation coefficient between observed & predicted
2
N
N
elevation fields: æ N
ö
å
å å
å
å
çç N x i y i - x i y i ÷÷
Cxy2
è i=1
ø
2
i=1
i=1
rxy =
=
2 üì
2ü
ì N
N
N
N
æ
ö ïï
æ
ö ï Cxx Cyy
ï
2
2
í N x i - çç x i ÷÷ ýí N y i - çç y i ÷÷ ý
ïî i=1
è i=1 ø ïþïî i=1
è i=1 ø ïþ
å
å
• The spatial-wavelength dependent r2 was calculated in the
same fashion after band-pass filtering of the fields (using
a fourth-order Butterworth filter from 0.8l to 1.2l to yield
wavelength-dependent fields for given l, using GMT).
This was compared to multitaper coherence:
Bandpassfiltered
Obs Topo  Obs Grav
Mod Grav  Obs Grav
Mod Grav  Obs Topo
Multitaper
• Note similar but smoothed, and factor-of-2 difference in
wavelength?
Bandpassfiltered
Obs Topo  Obs Grav
Mod Grav  Obs Grav
Mod Grav  Obs Topo
Multitaper
• Note also similarity of this approach to wavelet approach!
• Guest editor queried whether this
represents a “zero-phase filter”.
Does it?
• The filtering was done in GMT,
which for users can function
rather like a block box. How might you test to be sure?
Here, created a
“faux” grid consisting
of a kronecker delta
function, applied the
l = 500 km Butterworth
filter, and examined
the output grid… To
make sure the center
point did not shift in
space from the delta
function.