Frequency attributes: spectral decomposition

Iso-frequency volumes: the simplest spectral attribute

Pick a frequency — say 30 Hz. For each voxel $(i, x, t)$, take a small time window centred at $t$ and compute how much of the trace’s energy in that window is concentrated at 30 Hz. The result is a new volume the same shape as the input, where every voxel says "the 30 Hz content here is X." That is the iso-frequency volume at 30 Hz.

Slice it like any other volume. A 30 Hz iso-frequency time slice is a map of "where is 30 Hz energy strongest." Repeat at 15 Hz and at 50 Hz, and you have a frequency-stratified description of the same data: the same physical rock seen through three different acoustic lenses.

The math: a windowed single-frequency probe

For each voxel, project the windowed trace onto a complex sinusoid at the chosen frequency:

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where $W$ is the window length, $f_s$ is the sample rate, and $w(k)$ is a tapering window (we use a Hann taper, which goes smoothly to zero at the window edges). The complex exponential is just a compact way of writing "multiply by $\cos(2\pi f k / f_s)$ and by $\sin(2\pi f k / f_s)$, square each sum, add them." The squared magnitude is the power at frequency $f$.

Why the Hann taper? Without tapering, you implicitly multiply the trace by a rectangular window, which has very high spectral leakage — energy from one frequency spills into neighbouring frequencies and the iso-frequency picture is blurry. Hann concentrates the response cleanly at the chosen frequency and suppresses the side lobes by orders of magnitude. A small change in the formula, a huge improvement in the picture.

The widget below uses exactly this formula on the F3 dataset. Try it.

Exercise — The frequency story of F3

• The widget starts on the spectral attribute at 30 Hz on a time slice. The bright zones are where 30 Hz energy is strongest.
• Slide the Freq slider down to 15 Hz. Notice the pattern shifts — different regions are bright now. At low frequency, the iso-frequency map emphasizes thicker, more massive units (a 15 Hz wave has a wavelength of ~200 m at 3000 m/s velocity, so it "cares about" features on the 50-metre scale).
• Slide up to 60 Hz. The pattern shifts again, often becoming finer and more textured. High frequencies illuminate thin beds whose thickness is around the tuning thickness for 60 Hz (~12 m at the same velocity).
• Now scroll the Time slider while holding the Freq fixed at 50 Hz. Watch how the high-frequency content evolves with depth. Most volumes show progressive loss of 50 Hz energy at deeper times — the wavelet "reddens" with depth as high frequencies attenuate.
• Switch the View dropdown to Inline and scroll the inline slider. Spectral attributes also work as vertical sections — an inline at 30 Hz shows where 30 Hz energy is laterally concentrated and how it varies with depth. Useful for tracking a specific stratigraphic interval.

What different frequencies see

The connection to physics is the same one as §1.7’s tuning curve. A bed of thickness $h$ is "tuned" to the frequency whose quarter-wavelength matches that thickness:

$f_{\text{tune}} \approx \dfrac{V}{4 h}$

where $V$ is the rock’s P-wave velocity. So:

• A 50 m bed in 3000 m/s rock tunes at 15 Hz — the 15 Hz iso-frequency map will brighten at this bed.
• A 25 m bed in the same rock tunes at 30 Hz.
• A 12 m bed tunes at 60 Hz.

So sliding the Freq slider is like changing the bed thickness you’re asking about. This is the essence of spectral decomposition: different frequencies highlight different bed thicknesses. A channel of one specific thickness will appear most clearly on the iso-frequency map at its tuning frequency.

RGB blending: the iconic spectral image

Once you have iso-frequency volumes at three frequencies (low, middle, high), you can render them as the red, green, and blue channels of a color image. The result, often called a "frequency RGB blend," displays the spectral signature of every location in one picture:

• Red regions are dominated by the low frequency — thick beds, attenuated zones, gas shadows.
• Green regions dominate the mid frequency — typical reservoir sands, regional reflectors.
• Blue regions dominate the high frequency — thin beds, sharp interfaces.
• White regions have balanced energy across all three (the rock is reflecting all frequencies equally).

Frequency RGB images of channel systems are some of the most striking visualizations in geophysics: meandering fluvial channels light up in colour patterns that delineate their stratigraphic architecture beautifully. Production tools (Petrel, OpendTect, GeoTeric, etc.) all support this kind of blend natively. Building it in a browser widget is straightforward but adds enough UI complexity that we leave it to a future section; the single-frequency view above already shows you the underlying mechanism.

Window length: the time-frequency trade-off

Every spectral measurement faces the uncertainty principle. You cannot simultaneously have arbitrarily good resolution in time AND in frequency. The product of the two is bounded:

$\Delta t \cdot \Delta f \gtrsim 1$

(In radians, with appropriate constants — the exact value depends on the window shape.)

Practically:

• Short window (16–24 samples). Good time localization — you can pin the spectral content to a narrow depth interval. Frequency resolution is poor: the "30 Hz" measurement actually averages across maybe 25–35 Hz. Use when you want to see a specific stratigraphic interval clearly.
• Long window (48–80 samples). Good frequency resolution — you can distinguish 30 Hz from 35 Hz. Time resolution is poor: each measurement averages across a broader depth range. Use when you want to characterize the spectrum of a thicker geological unit.
• Sweet spot. 32–40 samples (~128–160 ms) is a good default for most exploration-scale work.

Common spectral pitfalls

• The bandwidth is what it is. Probing at 5 Hz on a survey whose lowest usable frequency is 8 Hz returns mostly noise. Probing at 100 Hz on a survey that’s been low-passed at 70 Hz returns near-zero everywhere. Always check the dataset’s usable bandwidth (look at a single-trace amplitude spectrum) before interpreting iso-frequency volumes — the answer at out-of-band frequencies is "nothing meaningful."
• Tuning vs presence. A bright iso-frequency response could mean (a) there’s a real interface that resonates at this frequency, or (b) there’s thin-bed tuning that happens to enhance this frequency. The two have different geological meanings. Disambiguate by checking adjacent frequencies (true reflectors brighten at multiple frequencies; tuning artifacts brighten at one specific frequency that matches $V/4h$).
• Window-edge artifacts. Like every windowed attribute, spectral power near the top and bottom of the trace uses an incomplete window. Hann tapering minimizes this but doesn’t eliminate it. Don’t over-interpret the outermost $W/2$ samples.
• Apparent frequency shifts from offset/azimuth. Pre-stack data has azimuthal frequency variations from incidence-angle effects. Spectral attributes on stacked data smooth these out, but stacked-data spectral content itself is partly an averaging artefact, not a pure rock property.

When spectral attributes are most useful

• Channel detection. Fluvial and turbidite channels have characteristic thicknesses, so they tune at characteristic frequencies. Iso-frequency time slices at the channel’s tuning frequency reveal the channel geometry beautifully — often more clearly than amplitude maps.
• Thin-bed mapping. Below seismic resolution, individual beds can’t be picked, but their tuning effects shift the local spectrum. Iso-frequency maps detect these systematic spectral shifts even when the beds themselves are invisible.
• Direct hydrocarbon indicators. Some gas-charged sands produce a "low-frequency shadow" — a localized brightening of the low-frequency end of the spectrum directly below the reservoir. Spectral decomposition makes this visible.
• Reservoir characterization. Combining iso-frequency information with amplitude and coherence builds the full multi-attribute picture that modern reservoir interpretation depends on. §6.5 takes up combining attributes systematically.