Geology 6600/7600Signal Analysis

30 Oct 2015

© A.R. Lowry 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…

Reminder: Your charge forBecker, T.W., et al., Static and dynamic support of westernUnited 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.

topographyThorsten W Becker

University of Southern California, Los AngelesClaudio 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)

Liu and Gurnis (2010)

Forte et al. (2009)

Moucha et al. (2008, 2009)

Becker et al. (2014)

What is the origin of non-flexural topography (in the context of

USArray)?

Smoothed ( l > 200 km)reference topography

CP : Colorado PlateauCVA : Cascades Volcanic ArccGB : central Great BasinGV : Great ValleyOCR : Oregon Coastal RangesSN : Sierra NevadaYS : Yellowstone

“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?

Isostatic topography

crust, rc

mantlelithosphere, r

l

asthenosphere

ra

L

lc

ll

ridge level

Isostaticcontributions

cf. Crough and Thompson (1977), Bird (1979), Lachenbruch and Morgan (1990)

crustal layer mantlelithosphere

(Removing the “flexural effects” allows approximation as “Airy”)

crust, rc

mantlelithosphere, r

l

asthenosphere

ra

ll

+ deflections due to present-day asthenospheric flow

(“dynamic topography”)

L

lc

Isostaticcontributions

L

crust, rc

mantlelithosphere, r

l

asthenosphere

ra

ll

+ deflections due to present-day asthenospheric flow

“Static”

“Dynamic”

lc

Crustal thickness from receiver function Mohos, based on USArray

Levander and Miller (2012) Lowry and Perez-Gussinye (2011)

mean and standard deviation of

all depicted fields

see also Chen et al. (2013)

Becker et al. (2014)

Based on Levander and Miller (2012) Based on Lowry and Perez-Gussinye (2011)

Residual topography for variable crustal thickness

All residual topography models are minimized by adjusting the asthenosphericdensity at fixed crustal and lithospheric density

Becker et al. (2014)

Correlationfor Airy isostasy (solid)

and

power spectrum(dashed)

total r2 (coherence)

• The “total r2” described here is the (now-familiar!) squared correlation coefficient between observed & predicted elevation fields:

• 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 0.8 to 1.2 to yield wavelength-dependent fields for given , using GMT). This was compared to multitaper coherence:

MultitaperBandpass-filtered

Obs Topo Obs GravMod Grav Obs GravMod Grav Obs Topo

• Note similar but smoothed, and factor-of-2 difference in wavelength?

• 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?

MultitaperBandpass-filtered

Obs Topo Obs GravMod Grav Obs GravMod Grav Obs Topo

Here, created a“faux” grid consistingof a kronecker deltafunction, applied the = 500 km Butterworthfilter, and examinedthe output grid… Tomake sure the centerpoint did not shift inspace from the deltafunction.