Frequency-wavenumber domain expression of the systems

Just as the inverse filtering expressions in frequency in are converted to simple scalar systems, the expressions for the signal and noise are appealing when expressed in the frequency or in the frequency-wavenumber domains. The expressions for noise can be considered scalar expressions at a constant frequency in one dimension, and at a constant frequency and a constant wavenumber in two-dimensions. For example, in one dimension, equation () becomes  

$$n(\omega) = \left( \frac{\overline{S(\omega)} S(\omega)}{\ov... ...) + \epsilon \overline{N(\omega)} N(\omega)} \right) d(\omega),$$ (55)

where $\overline{S}$ indicates the complex conjugate of S, and all values are scalars for a constant $\omega$.For values of $\omega$ where signal is not expected, the value of $\overline{S(\omega)} S(\omega)$ rises and the weighting of the data into the noise increases. For values of $\omega$ where signal is expected, $\overline{S(\omega)} S(\omega)$ falls and the weighting of the data into the noise decreases. Equation () can be recognized as an optimal or Wiener filterPress et al. (1986).

In the frequency-wavenumber, or $\omega$-k, domain, the same idea applies, except that each value of $n(\omega,k)$is evaluated for points in the $\omega$-k plane instead of at points of constant frequency. The $\omega$ in equation () becomes $(\omega,k)$, and the expressions are separated into samples of constant wavenumber (k) and constant frequency ($\omega$). This can be extended into more dimensions by specifying k₁, k₂, etc., for all the spatial directions considered. The inversion to separate signal and noise may then be thought of as decomposing the data into frequency and dip components, then distributing these components between the signal and noise as determined by the frequency and dip components of $\st S$ and $\st N$.The advantage of having more dimensions is that the noise and signal may be better distinguished as they are spread over $\omega$, k₁, k₂, and so on. Even if some overlap in the characterization of noise and signal remains, the overlap should tend to decrease as the number of dimensions increase.

Although viewing the separation of signal and noise in the frequency domains may be enlightening, for the work done in this thesis the inversions will generally be done in the time and space domain which provides the advantages of simplicity of coding, control of the filter shape, and easy windowing of data to account for non-stationarity.