1. Geology 6600/7600 Signal 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…

2. 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?

3. 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

4. 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)

5. Liu and Gurnis (2010) Forte et al. (2009) Moucha et al. (2008, 2009)

6. Becker et al. (2014) What is the origin of non-flexural topography (in the context of USArray)? Smoothed (  > 200 km) reference topography CP : Colorado Plateau CVA : Cascades Volcanic Arc cGB : central Great Basin GV : Great Valley OCR : Oregon Coastal Ranges SN : Sierra Nevada YS : Yellowstone

7. “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?

8. Isostatic topography crust,  c mantle lithosphere,  l asthenosphere  a L lclc l ridge level Isostatic contributions cf. Crough and Thompson (1977), Bird (1979), Lachenbruch and Morgan (1990) crustal layer mantle lithosphere (Removing the “flexural effects” allows approximation as “Airy”)

9. crust,  c mantle lithosphere,  l asthenosphere  a l + deflections due to present-day asthenospheric flow (“dynamic topography”) L lclc Isostatic contributions

10. L crust,  c mantle lithosphere,  l asthenosphere  a l + deflections due to present-day asthenospheric flow “Static” “Dynamic” lclc

11. 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 mean and standard deviation of all depicted fields see also Chen et al. (2013)

12. 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 asthenospheric density at fixed crustal and lithospheric density

13. Becker et al. (2014) Correlation for Airy isostasy (solid) and power spectrum (dashed) total r 2 (coherence)

14. The “total r 2 ” described here is the (now-familiar!) squared correlation coefficient between observed & predicted elevation fields: The spatial-wavelength dependent r 2 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: Multitaper Bandpass- filtered Obs Topo  Obs Grav Mod Grav  Obs Grav Mod Grav  Obs Topo

15. 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? Multitaper Bandpass- filtered Obs Topo  Obs Grav Mod Grav  Obs Grav Mod Grav  Obs Topo

16. Here, created a “faux” grid consisting of a kronecker delta function, applied the = 500 km Butterworth filter, and examined the output grid… To make sure the center point did not shift in space from the delta function.