Wiener-like method

A constrained least-squares problem using PEFs gives a similar expression for the noise estimation to the Wiener method. To see this, consider the recorded data to be the simple superposition of signal and noise, that is $\bf d = s + n$.For the special case of uncorrelated signal and noise, the so-called Wiener estimator, is a filter which when applied to the data, yields an optimal (in a least-squares sense) estimate of the embedded signal Castleman (1996). The frequency response of this filter is  

$$\bf H = \frac{ P_s }{ P_n + P_s },$$ (1)

where ${\bf P_s}$ and ${\bf P_n}$ are the signal and noise power spectra, respectively.

Similarly, Abma (1995) solved a constrained least squares problem to separate signal from spatially uncorrelated noise:

$$\begin{eqnarray} \bf Nn &\approx& 0 \nonumber \\ \bf \epsilon Ss &\approx& 0 \\ \mbox{subject to} &\leftrightarrow& \bf d = s+n \nonumber\end{eqnarray}$$ (2)

where the operators ${\bf N}$ and ${\bf S}$ represent t-x domain convolution with nonstationary PEF which whiten the unknown noise ${\bf n}$ and signal ${\bf s}$, respectively, and $\epsilon$ is a Lagrange multiplier. Minimizing the quadratic objective function suggested by equation (2) with respect to ${\bf s}$ leads to the following expression for the estimated signal:

(3)

By construction, the frequency response of a PEF approximates the inverse power spectrum of the data from which it was estimated. Thus we see that the approach of equation (2) is similar to the Wiener reconstruction process. We refer to this approach as a ``Wiener-like'' method. It has been successfully used by Brown and Clapp (2000) for ground-roll attenuation and by Clapp and Brown (2000) for multiple separation.