Spectral decomposition

Spectral decomposition has emerged recently as an enlightening seismic attribute, producing very informative maps of thin beds, especially in clastic successions with sharp impedance contrasts (Partyka et al., 1999 ). These maps are typically interpreted qualitatively, using geomorphologic pattern-recognition, or semi-quantitatively, to infer relative thickness variation. A number of commercial and non-commercial implementations of spectral decomposition are now in use, and the technique is de riguer in the analysis of subtle stratigraphic plays and fractured reservoirs.

There is also considerable scope for applying spectral decomposition quantitatively. As shown in Figure 1, the spectrum of a thin bed has characteristic periodic notches resulting from interference between the signals reflected from the top and base of the thin bed. The spacing of the notches is the reciprocal of the two-way time thickness of the thin bed.

Methods
Spectral decomposition requires the transformation of each individual 1D seismic trace, $$s(t)$$ into a 2D time-frequency representation, $$s(\tau,\omega)$$. Many methods exist to achieve this and each has different resolution capabilities in time and frequency.

Short-time Fourier transform
The short-time Fourier transform (STFT) was the first method used to investigate the local frequency of a seismic trace. It attempts to break the trace up into many subsections whose amplitude-spectrum can be calculated by using the Fourier transform.

It begins by multiplying the seismic trace by a window, $$w(t-\tau)$$, such that:


 * $$s(t)w(t-\tau)=\begin{cases}

s(t) & \tau \approx t \\ 0 & otherwise \\ \end{cases}$$

and then taking the Fourier transform of each windowed section, $$s(t)w(t-\tau)$$, as follows:


 * $$s(\tau, \omega) = \int_{-\infty}^{\infty} s(t) w(t-\tau) e^{-i \omega t} \, dt $$

where $$t$$ is the vertical (usually 2-way) travel time as read from the seismic trace and $$\tau$$ is the location of the center of the window in the $$t$$ axis (Cohen, 1994).

There are many window functions to choose from, each of which has its own resolution capabilities and artifacts. The most popular windows include: Boxcar, Gaussian, Hann, etc.

The STFT has a very limited simultaneous resolution in the time and frequency domains. By taking windows of a smaller size in the time domain, the temporal resolution of the resulting time-frequency representation increases but its frequency resolution decreases. Larger windows in the time domain will generate time-frequency representations with lower temporal resolution but higher frequeny resolution. This is known as the uncertainty principle.

Other methods

 * Continuous wavelet transform
 * S-transform
 * Matching pursuit
 * Constrained least squares spectral analysis

Applications
There are many applications of spectral decomposition in seismic exploration. It is currently a very popular subject of ongoing research.

Qualitative layer thickness estimation
The relative thickness of stratigraphic units can be observed using spectral decomposition. As strata becomes thicker, the peak frequency of their seismic response tends to be lower, and viceversa. By taking a horizon slice of a 3D seismic volume, one can predict thickening directions by viewing the slice at progressively lower frequencies. (Partyka, 1999)

Direct hydrocarbon indication
Low frequency shadows have been observed under known hydrocarbon reservoirs (Castagna, 2003).

Quantitative layer thickness estimation
Any pair of reflection coefficients, such as the ones from the top and bottom of a stratum, can be viewed as the sum of a pair of even (same sign) and a pair of odd (different sign) reflection coefficients (Chopra, ????). As explained by Widess (1973), below the typical $$\lambda / 4$$ limit of seismic resolution the amplitude of the seismic trace as a response to a thin bed tends to zero. However, this is only in the case of equal and opposite reflection coefficients above and below the thin layer (odd pair). In practice, these reflection coefficients tend to be of different magnitude and could be of the same sign. For this reason, the reflection coefficient pair contains an even component that does not tend to zero as the bed gets thinner. By observing the amplitude spectrum of the seismic response, namely the notches, one can predict how far apart the even component of the reflection coefficient pair should be. This provides a quantitative estimate of layer thickness (Puryear and Castagna, 2008).