Geology 6600/7600 Signal Analysis 26 Oct 2015 © A.R. Lowry 2015 Last time: Wiener Filtering Digital Wiener Filtering seeks to design a filter h for a linear SISO system producing an optimal output y that “looks like” a desired signal d Applications include smoothing, prediction … To design this type of filter, seek to minimize the mean-square difference between the predicted and desired signals by setting that difference (the “error”, orthogonal to the input signal: That yields: For smoothing, ~ ~

Going forward: I currently plan to cover Kalman filtering (today) Variograms, optimal interpolation (“kriging”) and likelihood functions Fourier domain approximation of potential field modeling (including continuation; applications) Deconvolution applications in flexural analysis and receiver function analysis These were chosen primarily for relevance to Brent’s research topic… If there is interest or need for additional topics let me know !

Kalman Filtering: Kalman Filtering is an optimal approach to recursive estimation of a signal x[n] from noisy measurements y[n]. Assume the following model: ~~ + a D c + Autoregressive (AR) Model of a Random Signal Measurement Model

Assume that both the input signal w and the measurement noise v are zero-mean, white noise processes: Then what are the statistical properties of x[n] ? Note that: So:

For other lags of the autocorrelation function, And so on. So in general, Note that this signal can only exist for |a| < 1. (Why?)

Designing an Estimator for x[n] : Assume a recursive estimator of the form: Here, the first term in the new estimate represents a weighted previous estimate, and the second term is the weighted current measurement sample. We want to determine the two weight parameters a[n] and b[n] (note these are time-varying!) based on minimization of the mean-square error, represented by “error power” p[n] :

Similar to the approach taken for Wiener filtering, we set derivatives (WRT unknown weights) equal to zero: The first eqn can be rewritten: And substituting y[n] = cx[n] + v[n], ~~~

Thus we have: Substituting and noting that, we get: or: By evaluating  p  n  b  n , it can be shown that the filter gain (referred to as kalman gain ) b[n] and the mean-square error power p[n] are given by: ~~~

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.