Amplitude attributes: RMS, envelope, and their friends

What is a seismic attribute?

An attribute is any quantity computed from the seismic data that brings out a property of interest. The usual families:

• Amplitude attributes: magnitude, RMS, the envelope. They measure how strong the reflections are at each place (this section).
• Frequency attributes: spectral decomposition, dominant frequency, bandwidth. They measure what frequencies the local wavelet carries, which helps with thin beds and lithology (Section 6.2).
• Geometric attributes: dip, azimuth, curvature. They describe the orientation and the shape of the reflectors (Section 6.3).
• Coherence and discontinuity: measures of how alike neighbouring traces are, which bring out faults and channel edges (Section 6.4).
• Combined and classified attributes, built from the others (Section 6.5).

An attribute is derived from the seismic, so it adds no physical information the data did not already hold. What it adds is visibility: it makes one property easier for the eye to recognise, or for an algorithm to classify. A good attribute brings out something the interpreter cares about; a careless one can manufacture features that are not there.

The simplest attribute: magnitude

Take the absolute value of each sample, $|a(t)|$. Peaks and troughs both become positive, so the strength of each reflection shows without the alternation of sign that a red-blue display carries. It is a first map of where the data has energy, whatever the polarity.

Magnitude is also the bluntest amplitude attribute: it is one sample, so it falls to zero wherever the trace crosses zero, even in the middle of a strong reflection. Every other amplitude attribute deals with that in its own way.

RMS amplitude

Root-mean-square amplitude over a sliding window is the workhorse among amplitude attributes. Over the $W = 2h + 1$ samples centred on time $t$:

where $\Delta t$ is the sample interval. RMS is never negative, and it is the typical size of the samples near $t$: one reflection folded, peak and trough together, into one value. Two close relatives use the same window: the mean absolute amplitude, the mean of $|a|$ (less swayed by a single large sample), and the variance, the spread of the samples about their mean. A moving average of $|a|$ is sometimes offered as a cheap envelope; it is the mean absolute amplitude by another name, not the envelope, and Figure 6.1 lists it once, under that name.

Two choices matter:

• Window length. A window of $W$ samples spans $W\Delta t$: 13 samples at 4 ms are 52 ms. A short window, a cycle or so of the wavelet, measures single reflections; a long one averages an interval. Reservoir work usually sizes the window to the zone of interest (say 30 ms about a target horizon); regional mapping may use 60 to 100 ms to bring out broad trends.
• Window position. A centred window gives a value at every sample, looking the same distance up and down. Most reservoir workflows instead hang the window on an interpreted horizon (10 ms above to 20 ms below it, say), which is the subject of Section 6.6.

The envelope, phase and instantaneous frequency

Taner, Koehler and Sheriff (1979) treated the trace $a(t)$ as the real part of a complex analytic signal, $z(t) = a(t) + i\,\mathcal{H}a(t)$, whose imaginary part is the Hilbert transform of the trace: the same trace with every frequency shifted by a quarter cycle. Three attributes follow, at every sample:

• the envelope, or instantaneous amplitude, $e(t) = |z(t)| = \sqrt{a^2 + (\mathcal{H}a)^2}$;
• the instantaneous phase, $\varphi(t) = \arg z(t)$, so that $a(t) = e(t)\cos\varphi(t)$: $0^\circ$ at a peak, $180^\circ$ at a trough;
• the instantaneous frequency, $f_i(t) = \frac{1}{2\pi}\,\frac{d\varphi}{dt}$, how fast the phase turns.

The envelope is the smooth curve that bounds the trace on both sides. Where the trace crosses zero inside a reflection, $\mathcal{H}a$ is near its largest, so the envelope stays high: it measures the wavelet, not one point on it. Rotating the phase of the wavelet changes $a$ but not $e$, which is why the envelope is preferred for comparing amplitudes between data of different phase.

Phase carries no amplitude at all. A weak reflection runs through the same cycle of phase as a strong one, so a phase display draws the continuity of every layer with equal weight. The instantaneous frequency is easily upset: where the envelope is small the phase turns erratically, and $f_i$ can jump or even go negative, so read it where the envelope is large. Figure 6.1 therefore averages it over the window with the squared envelope as the weight, $\bar f = \sum e^2 f_i \big/ \sum e^2$, which keeps the frequency of the strong wavelets and discounts the quiet samples between them (Barnes 2007). On F3 the raw $f_i$ leaves the range 0 to 125 Hz, the Nyquist frequency, at 2% of samples; weighted over 13 samples it never does.

Figure 6.1 takes the Hilbert transform with a 63-tap filter. On F3 its envelope is within 0.5% of the cube’s RMS of an exact transform at the median sample, and within 2.3% at 95% of samples.

Common amplitude-attribute pitfalls

• Gain. RMS and the envelope are in the units of the data, so they carry every gain choice of the processing. Another processing of the same survey gives different values everywhere. Never compare raw amplitudes between datasets without normalising them.
• Tuning. Where a bed is thinner than about a quarter of a wavelength, the reflections from its top and base interfere. For a Ricker wavelet the composite amplitude peaks at about 1.45 times that of one interface, at a two-way thickness near $1/(2.6 f_p)$, and falls toward zero for thinner beds. A bright spot can be partly tuning; Section 1.7 shows the effect.
• Attenuation. The earth absorbs high frequencies along the way, and amplitudes generally weaken with depth. Compare a deep target with its own surroundings, not with shallow amplitudes.
• Edges. Near the top and the bottom of the traces the window holds fewer samples, so values there rest on less data and can be too high or too low. On inline 400 the only windows brighter than the reflection of exercise 1 lie in the last 16 ms of the traces, where the window is cut short.

When to use which amplitude attribute

• Magnitude: a quick scan for the strongest single samples; no parameters.
• RMS, short window (about one cycle, 8 to 15 samples at 4 ms): the brightness of individual reflections; the standard first amplitude map.
• RMS, long window (40 to 100 samples): interval-averaged amplitude, for regional trends and for separating bright zones from quiet ones.
• Envelope: amplitude independent of phase, for comparing data of different phase or where a phase rotation would fake a change in brightness.
• Instantaneous phase: the continuity of reflections whatever their strength, for following weak events and seeing terminations.
• Variance: how much the amplitude scatters, for chaotic zones where the spread matters more than the size.