Signal/Noise separation

Consider the recorded data to be the simple superposition of signal and noise events: $\bf d = s + n$. The so-called Wiener estimator is a filter, which when applied to the data, produces an optimal (least-squares sense) estimate of the embedded signal (). For the special case of uncorrelated signal and noise, the frequency response of this filter is  

$$\bf H = \frac{ P_s }{ P_n + P_s },$$ (177)

where $\bf P_s$ and $\bf P_n$ are the signal and noise power spectra, respectively. Define operators $\bf N$ and ${\bf S}$, as convolution with filters which decorrelate the unknown noise $\bf n$ and signal ${\bf s}$, respectively. (), for example, show that the following least-squares optimization problem is approximately equivalent to Wiener estimation:

$$\begin{eqnarray} \bf Nn &\approx& 0 \nonumber \ \bf \epsilon Ss &\approx& 0 \ \mbox{subject to} &\leftrightarrow& \bf d = s+n \nonumber\end{eqnarray}$$ (178)

Equation () is a regularized linear least-squares problem. The scalar parameter $\epsilon$ is related to the data's signal-to-noise ratio.

The conceptual model of seismic data as n locally-crossing plane waves lends itself well to parameterization by a few parameters. The multidimensional prediction-error filter (PEF) is a particularly popular option (see, for example, ()). Estimated by autoregression against the data, the PEF encodes hidden multiplicity in the data with a few filter coefficients. It has the approximate inverse spectrum of the data from which it was estimated.

By using a model of the noise to obtain a nonstationary noise PEF and deconvolving a PEF estimated from the data by the noise PEF to obtain a signal PEF (), many authors have solved equation () to successfully separate coherent noise from signal (, , , , ).

As noted by (), however, the considerable amount of parameter tuning required to create stable nonstationary PEFs (a requirement for the deconvolution step) remains a significant obstacle to their use in industrial-scale processing environments.

If the signal and noise consist of distinct slopes everywhere, then it is in theory possible to implicitly separate signal from noise in the slope domain with a two-slope estimation algorithm. Fomel uses estimated slope to construct plane-wave destructor filters which are used directly as $\bf N$ and ${\bf S}$ in equation (), without any deconvolution. The filters are guaranteed stable and insensitive to spatially aliased data. Fomel obtains an independent estimate the noise slope from a prior noise model, and then fixes the noise slope as the signal slope is estimated.

I take a slightly different tack at the problem. Like Fomel, I use my two-slope estimation technique to directly obtain signal and noise slope estimates. I also exploit a prior noise model and also a prior signal model, in cases where the signal is simpler to model than the noise. Most importantly, I find that very simple, easily-obtainable signal or noise models suffice. To overcome aliasing, I apply normal moveout (NMO) to the data. Rather than plane-wave destructor filters, I (again) use 9-point Lagrange steering filters derived by ().