Geology 6600/7600 Signal Analysis 28 Oct 2015 © A.R. Lowry 2015 Last time: Kalman Filtering Kalman Filtering uses recursion ; designs a filter h to extract a signal x[n] from noisy measurements y[n]. Model the signal to be estimated as autoregressive and the measurement as having additive white noise: Want to design a recursive estimator of the form: by choosing a[n], b[n] that minimize mean square error Four equations for the scalar Kalman filter are: so must know or estimate a, c,  v 2,  w 2 ~~

Reminder: Read for Friday (30 Oct): Becker, T.W., et al., Static and dynamic support of western United States topography, Earth Planet. Sci. Lett., (I will prep discussion materials… You’ll need to be prepared to discuss specifically the SIGNAL ANALYSIS methods used in the paper!) 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?

We can substitute these into our original estimation equation: to get: where the first term is the estimate without new data, and the second is the correction incorporating new data. The filter gain can be rewritten: where Then the mean-squared error can be rewritten as: These equations constitute the scalar kalman filter.

Note : As previously (e.g., with the Wiener filter and Bayesian approaches) Kalman filtering requires that you have independent information about the system (i.e., you must know a, c,  v 2,  w 2 ). Can we get these from the measurements of y ? Thus one can estimate a from the decay of R yy with l = 1…∞, &  v 2 from difference at R yy [0]. However c versus  w 2 are completely ambiguous UNLESS we can measure input w or have some other independent information! ~ ~

Examples of Kalman filtering of GPS time series (site CARI in Andaman) using a = 1, c = 1,  w = 0.3 mm…  v = 0.1 mm  v = 2 mm

Close-up of site DGLP nearby a = 1, c = 1,  w = 0.3 mm,  v = 2 mm… Includes a transient following the 2012 Indian Ocean earthquakes ~1200 km away! EastNorth