Signal to noise separation

Consider the recorded data $\bf d$ to be the simple superposition of ``signal'' $\bf s$, i.e., reflection events, and ``noise'' $\bf n$, i.e., multiples: $\bf d = s + n$. 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 (least-squares sense) estimate of the embedded signal Castleman (1996). The frequency response $\bf H$ 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. Abma (1995) and Claerbout (1999) 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 non-stationary 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:

$$\bf \hat{s} = \left( \bold N^T \bold N + \epsilon^2 \bold S^T \bold S \right)^{-1} \bold N^T \bold N \ \bold d$$ (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.

Spitz (1999) showed that for uncorrelated signal and noise, the signal can be expressed in terms of a PEF, $\bf D$, estimated from the data $\bf d$, and a PEF, $\bf N$, estimated from the noise model:  

$$\bold S = \bold D \bold N^{-1} .$$ (4)

Spitz' result applies to one-dimensional PEF's in the f-x domain, but our use of the helix transform Claerbout (1998) permits stable inverse filtering with multidimensional t-x domain filters.

Substituting $\bold S = \bold D \bold N^{-1}$ and applying the constraint $\bf d = s + n$ to equation (2) gives

$$\begin{eqnarray} \bf Ns &\approx& \bf Nd \nonumber \\ \epsilon \bold D \bold N^{-1}\bf s &\approx& \bf 0. \end{eqnarray}$$ (5)

Iterative solutions to least-squares problems converge faster if the data and the model being estimated are both uncorrelated. To precondition this problem, we again appeal to the Helix transform to make the change of variables ${\bf x = Ss =} \bold D \bold N^{-1}\bf s$ or $\bf s = \bold N \bold D^{-1}\bf x$ and apply it to equation (5):

$$\begin{eqnarray} {\bf NND}^{-1}\bf x &\approx& \bf Nd \nonumber \\ \epsilon \bf x &\approx& \bf 0 \end{eqnarray}$$ (6)

After solving equation (6) for the preconditioned solution $\bf x$, we obtain the estimated signal by reversing the change of variables: $\bf \hat{s} = \bold N \bold D^{-1}\bf x$.