Frequency attributes: spectral decomposition

Part 6, Seismic Attributes

Learning objectives

Explain what "frequency content" means for a seismic trace and why it varies with location and depth
Compute power at a chosen frequency at every voxel using a windowed Fourier-style decomposition
Read iso-frequency time slices and recognize geological features that prefer specific frequencies (thin beds, channels, fluid effects)
Choose probe frequencies and window lengths appropriately for a target geological question
Recognize the limits: low-energy zones, edge effects, frequency vs. time-resolution trade-off, and bandwidth-edge artefacts

Section 6.1 measured how strong the reflections are. This section measures what they are made of: how much of each frequency the local wavelet carries. Section 6.4 will then turn the similarity of neighbouring traces into a map of faults.

A seismic wavelet carries energy over a band of frequencies, not one. A processed marine survey typically has usable energy from about 5 Hz up to 80 or 100 Hz, and we often sum that up in a single dominant frequency, but the spectrum is the whole picture, and it is not the same everywhere in a volume:

Depth. The earth absorbs high frequencies faster than low ones, so the wavelet generally loses its high frequencies with depth. Over a short interval the layering can matter more than the depth; read a change against the layers before calling it attenuation.
Layering. A thin bed reflects from its top and its base, and the two reflections reinforce some frequencies and cancel others. Its spectrum is shaped by its thickness: that is tuning (Section 1.7) seen in frequency.
Fluids. Gas-charged rock can absorb strongly, and a low-frequency shadow beneath a gas sand, energy shifted toward the low end of the spectrum, is a known if unreliable indicator.

If every sample could be given a small spectrum instead of one number, how much 15 Hz, how much 30 Hz, how much 50 Hz is here, the volume would show structure the amplitude hides. That is spectral decomposition (Partyka, Gridley and Lopez 1999).

Iso-frequency maps

Pick a frequency, say 30 Hz. At every sample $(i, x, t)$ , take a short window of the trace centred on $t$ and measure how much 30 Hz it holds. The result is a volume the shape of the original in which every sample says how strong 30 Hz is there: the iso-frequency volume at 30 Hz.

Slice it like any other volume. A time slice of it is a map of where 30 Hz is strong at that time. Repeat at 15 and at 50 Hz and you see the same rock through three filters.

The arithmetic: a short-window Fourier transform

For each sample, project the tapered window of $W$ samples onto a complex sinusoid at the frequency $f$ :

P(t, f) = \Bigl|\sum_{k=0}^{W-1} w_k\, a(t - h + k)\, e^{-i 2\pi f k \Delta t}\Bigr|^{2}, \qquad w_k = \tfrac12 - \tfrac12\cos\frac{2\pi k}{W-1}

where $h = \lfloor W/2 \rfloor$ , $\Delta t$ is the sample interval and $w_k$ is the Hann taper, which falls smoothly to zero at both ends of the window. The complex exponential is shorthand for multiplying by $\cos(2\pi f k\Delta t)$ and by $\sin(2\pi f k\Delta t)$ , squaring each sum and adding them: $P$ is the power at $f$ . Figure 6.2 draws it as an amplitude, $A = 2\sqrt{P}/\sum_k w_k$ , the amplitude a cosine at $f$ alone would need to give the same power, in units of the cube’s RMS.

Without the taper the window would cut the trace off abruptly, and a sharp cut spreads each frequency into its neighbours: spectral leakage. The Hann taper lowers the side lobes of that spreading by a large factor, at the price of a wider main lobe.

Window length: time against frequency

A short window cannot isolate one frequency. A Hann taper of $W$ samples runs from zero to zero over $(W-1)\Delta t$ , and its main lobe falls to half its peak height at $\pm 1/((W-1)\Delta t)$ from its centre (Harris 1978), so an iso-frequency value really averages the band

f \pm \frac{1}{(W-1)\,\Delta t}

At 4 ms, 15 samples (60 ms) average about $f \pm 18$ Hz, 31 samples (124 ms) $f \pm 8$ Hz, and 63 samples (252 ms) $f \pm 4$ Hz. Narrowing the band means lengthening the window, and a long window mixes the layers over its whole length into each value. This is the uncertainty principle of signal analysis: no window is sharp in time and in frequency at once. Plate (c) of Figure 6.2 shades the band the window really averages.

Short windows (about 10 to 20 samples) locate the spectral content in time, but blur frequencies 15 to 30 Hz apart into one value.
Long windows (about 50 to 80 samples) separate frequencies a few hertz apart, but each value then describes 200 to 300 ms of layers.
Around 30 samples (about 120 ms) is a common starting point at 4 ms.

Figure 6.2 computes the spectral amplitude with the formula above, on a slice of the F3 volume at a time, and plate (c) shows the whole local spectrum at the probe, from the same window. The exercises under the figure work through it:

Exercise 1 reads the 30 Hz map on the time slice at 1052 ms against the spectrum at the probe, which peaks at 60 Hz: the map is one point of that spectrum, taken at every trace.
Exercises 2 and 3 move to 15 and then to 50 Hz. At 15 Hz the map turns into broad patches; at 50 Hz it is finer, and a fault crossing the lower right of the slice appears as a sharp line of low amplitude that 15 Hz does not show.
Exercise 4 doubles the window to 63 samples: the band narrows from $30 \pm 8$ to $30 \pm 4$ Hz and the spectrum sharpens, while each value now mixes 252 ms of layers.
Exercise 5 shows 50 Hz on inline 400, where it follows the layering rather than fading steadily with depth over these 1.2 s.

What different frequencies see

For a thin bed with opposite reflections at its top and base, separated by the two-way time $\Delta t_b = 2h/V$ , the composite spectrum is the wavelet’s spectrum times $2|\sin(\pi f \Delta t_b)|$ . It is strongest near

f_{\mathrm{tune}} = \frac{1}{2\,\Delta t_b} = \frac{V}{4h}

and has notches at multiples of $1/\Delta t_b$ , the pattern Partyka and his colleagues used to map bed thickness. With $V$ = 3000 m/s:

a 50 m bed is strongest near 15 Hz;
a 25 m bed near 30 Hz;
a 12.5 m bed near 60 Hz.

Sweeping the frequency therefore sweeps the thickness you are asking about. A feature that answers one frequency more strongly than its neighbours hints at its thickness, provided that frequency lies inside the band the data carries.

Frequency blends

Three iso-frequency volumes, low, middle and high, can be shown as the red, green and blue channels of one colour image. Where the low frequency dominates the picture is red, where the high one dominates it is blue, and where all three are balanced it is near white. Channel systems in such blends are among the most striking images in seismic interpretation. Section 6.5 builds blends of this kind.

Common spectral pitfalls

The band is what it is. At 5 Hz on a survey whose lowest usable frequency is 8 Hz, or at 100 Hz on one filtered at 70 Hz, the map is noise. Look at the spectrum first; plate (c) shows it at the probe.
Tuning or contrast. A bright iso-frequency response can be a strong reflection or a thin bed tuned to that frequency. A strong interface is bright at many frequencies; a tuned bed peaks near $V/4h$ and is weaker either side.
Edges. Within $W/2$ samples of the top or the bottom of the traces the window runs off the data. The taper reduces the error but does not remove it.
Stacked data. The spectrum of a stacked trace is partly a product of stacking (offset-dependent stretch and residual moveout lower the high frequencies), not a pure property of the rock.

When spectral attributes help most

Channels. A channel of one thickness tunes at one frequency, and its iso-frequency map can outline it more clearly than amplitude.
Thin beds. Below the resolution of the data single beds cannot be picked, but their tuning shifts the local spectrum, and iso-frequency maps follow that shift.
Low-frequency shadows. The low end of the spectrum sometimes brightens beneath a gas sand. It is a hint, never proof: tuning, processing and near-surface effects make the same pattern.
Reservoir characterisation. With amplitude and coherence, spectral content completes the multi-attribute picture of Sections 6.5 and 6.6.

Spectral decomposition is a windowed Fourier transform at every sample, and its physics is the tuning of Section 1.7. Section 6.3 turns to the geometric attributes, dip, azimuth and curvature, which describe the orientation and the shape of the reflectors rather than what they are made of.

References

Castagna, J. P., Sun, S., & Siegfried, R. W. (2003). Instantaneous spectral analysis: Detection of low-frequency shadows associated with hydrocarbons. The Leading Edge, 22(2), 120-127.
Chopra, S., & Marfurt, K. J. (2007). Seismic Attributes for Prospect Identification and Reservoir Characterization. Society of Exploration Geophysicists.
Harris, F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66(1), 51-83.
Partyka, G., Gridley, J., & Lopez, J. (1999). Interpretational applications of spectral decomposition in reservoir characterization. The Leading Edge, 18(3), 353-360.