The layer too thin to see

What happens where the medium changes found that every boundary between two media returns part of a wave that crosses it, in proportion to the mismatch in their impedances. The layer that makes a reflection vanish put two such boundaries a quarter of a wavelength apart and made their reflections cancel, the principle of every antireflection coating. Both treated a wave of one frequency, steady and endless. This essay sends a pulse instead, and asks a question the continuous wave cannot pose: when two boundaries are close together, can their echoes be told apart?

The question is the whole business of reflection seismology. A survey fires a pulse of sound into the ground — a vibrating truck, an air gun towed behind a ship — and records the echoes from every boundary between rocks of different impedance, for kilometres down. The record is read as a picture of the layers. The thickest layers are no trouble: their top and base return separate echoes, and the time between them, with the speed of sound in the rock, gives the thickness. The layers that hold oil and gas, or that seal it in, are often thin, and whether a layer twenty metres thick can be seen, measured and mapped from echoes of a pulse a hundred metres long is a question of resolution that the drawings answer.

Two echoes that merge #

The drawing is the classic test of a survey’s resolution: a wedge, a layer that thins steadily to nothing, sounded at intervals along its length. The pulse is a Ricker wavelet, a single central lobe with a smaller lobe of opposite sign on each side, the shape many seismic sources approximately make; at thirty hertz in rock where sound travels at three thousand metres a second its wavelength is a hundred metres. The layer has a higher impedance than the rock around it, so its top reflects the pulse with one sign and its base with the opposite sign.

Where the layer is a hundred metres thick the two echoes are separate and the thickness is plain. As it thins they approach, overlap and merge. Around twenty metres the merged echo is at its strongest, forty-five per cent stronger than either face’s alone: the base’s echo, inverted and delayed, lines up its lobes with the top’s and reinforces them. Below that thickness the combined echo stops changing shape. It keeps one fixed form and simply weakens as the layer thins, reaching nothing where the wedge ends.

The fixed form has a simple explanation. When the delay between the two echoes is short compared with the pulse, the sum of the pulse and a slightly delayed, inverted copy of it is the delay times the pulse’s rate of change: the echo of a thin layer is the derivative of the pulse, scaled by the layer’s two-way time. Norman Widess pointed this out in 1973 in a paper whose title asked how thin a bed is too thin to see. A thin layer is a differentiator. Its echo’s shape is fixed by the pulse, and only its size depends on the thickness.

That makes the strength of a thin layer’s echo a measurement of its thickness, provided the contrast is known. Interpreters exploit exactly this. Where a well has been drilled through a thin reservoir, its logs give the rocks’ impedances and the layer’s true thickness at one point; the echo’s strength there calibrates the relation, and away from the well the strength of the same echo along the section is read as a map of how the layer thickens and thins. The method is only as good as the assumption that the contrast stays the same across the map, and a change in what fills the rock’s pores — gas instead of water — changes the contrast and the echo’s strength without changing the thickness at all. A brighter echo may be a thicker layer or a different fluid, and deciding which is much of the art of reading a section.

Loudest at a fifth of a wavelength #

Plotted against thickness, the strength of the combined echo rises from one — two separate echoes of unit size — to a peak near a fifth of a wavelength and then falls towards zero in proportion to the thickness. The peak is called the tuning thickness, and it is where a thin layer is most conspicuous in a seismic section: layers near it stand out brighter than their contrast alone would justify, and interpreters learn to distrust a bright spot that might be a tuned thin layer rather than a strong one.

The tuning peak is the quarter-wave layer in disguise. In an antireflection coating a quarter-wave layer between media arranged to have reflections of the same sign makes them cancel; here the reflections have opposite signs, so the same half-wavelength round-trip difference makes them add. For a pulse rather than a single frequency the maximum comes at a slightly different thickness, about a fifth of the central wavelength, because the pulse contains a spread of frequencies whose quarter-wave thicknesses differ.

Where timing runs out #

The practical question is not whether the layer shows but whether its thickness can be read. The drawing measures what an interpreter would measure: the time between the echo’s strongest peak and strongest trough, taken as the two-way time through the layer. Above about a quarter of a wavelength it follows the truth. Below, it stops: every thin layer shows the same separation, a third of a period, which is the spacing between the positive and negative lobes of the pulse’s derivative, not anything about the layer.

That is the vertical resolution of the survey, and it is the same limit that decides how far apart two things have to be before a telescope can separate them: two features closer than about the width of the instrument’s response blur into one, and the blurred image has the shape of the response rather than of the objects. Seismologists quote the limit as a quarter of a wavelength — Rayleigh’s criterion transposed — or an eighth, following Widess, depending on how much they trust the amplitude. It is set by the shortest wavelength the pulse contains, and making the pulse shorter means paying in bandwidth: higher frequencies, which the ground absorbs strongly, so that deep surveys are always low in frequency and coarse in resolution.

Why a survey cannot simply use a shorter pulse #

If resolution is a quarter of a wavelength, the obvious remedy is a higher frequency. A hundred-hertz pulse in the same rock has a wavelength of thirty metres and would resolve layers seven or eight metres thick. The difficulty is that rock is not perfectly elastic. Each cycle of a seismic wave loses a fixed small fraction of its energy to internal friction — rocks have a quality factor, in the same sense as a resonance whose width is its lifetime, of a few tens to a few hundred — and so a wave loses energy in proportion to the number of cycles it has travelled, which for a given distance grows with its frequency.

The arithmetic is severe. A wave that travels for two seconds, down to a boundary three kilometres deep and back, through rock with a quality factor of a hundred, keeps a fraction $e^{-\pi f t/Q}$ of its amplitude: about a sixth at thirty hertz, but a fortieth at sixty and a five-hundredth at a hundred. The high frequencies that would sharpen the pulse are gone by the time the echo returns, and the deeper the target the lower the frequencies that survive. Shallow engineering surveys, looking tens of metres down, use hundreds of hertz and resolve layers a metre thick; oil exploration at several kilometres works with pulses that rarely carry much above sixty hertz and resolves tens of metres at best. The resolution of a survey is set by the ground it looks through as much as by the equipment.

The record as a blurred list of boundaries #

A useful way to think of a seismic trace is as a list of boundaries, each with its reflection coefficient at the time its echo returns, smeared by the pulse: in mathematical terms, the sequence of reflection coefficients convolved with the wavelet. The wedge in the first drawing is a list with two entries, and the question of resolution is whether the smeared list can be unsmeared to recover them.

Unsmearing — deconvolution — divides the echo’s spectrum by the pulse’s. Within the band where the pulse has energy this works, and processing a survey always includes a step that flattens the spectrum to make the effective pulse as short as the band allows. Outside the band there is nothing to divide by: the pulse carried no energy there, the record carries only noise, and dividing noise by nearly zero amplifies it without limit. The resolution after the most careful processing is therefore still set by the band, and the band is set by the source and the ground. A thin layer’s derivative-shaped echo is exactly what deconvolution cannot separate into two spikes, because the frequencies that would separate them are the ones the ground removed.

The same limit in other echoes #

The pulse-echo problem is not special to rock. Medical ultrasound images the body with pulses a few cycles long at a few megahertz, and its resolution along the beam is about half the pulse’s length, a few tenths of a millimetre — too coarse for the layers of a retina, fine for the wall of an artery. Ground-penetrating radar, which sends radio pulses into soil and ice to find pipes, graves and the layering of glaciers, faces the same trade between frequency and depth, because wet ground absorbs high frequencies as rock does. Optical coherence tomography, which images the layers of the retina, escapes the limit only in part: it uses light with an enormous bandwidth, so that its effective pulse is a few micrometres long, and it reads the echoes’ timing by interference rather than by a stopwatch.

In every case the tuning, the flattening of apparent thickness below a quarter wavelength, and the notches in the spectrum appear in the same form, because they are consequences of adding a pulse to a delayed copy of itself and nothing else. What differs from one application to another is only which part of the spectrum the medium lets through.

The thickness in the spectrum #

The thickness has not vanished from the echo. It has moved. Seen in frequency rather than time, a layer multiplies the pulse’s spectrum by a factor that oscillates, $2|\sin(\pi f\tau)|$, where τ is the two-way time through the layer. At frequencies where the round trip is a whole number of periods, the two echoes cancel exactly, and the echo’s spectrum has a notch. A layer forty metres thick puts notches at 37.5 and 75 hertz; one twenty metres thick, at 75.

Where the notches fall within the pulse’s band, they carry the thickness directly, and a thin layer’s thickness can be read from them even when the timing has run out. Mapping thin layers by decomposing seismic data into frequency bands and watching where each band dims is one of the ways thin reservoirs are mapped. It is the same trade a coating makes with colour: a soap film too thin to measure with a ruler announces its thickness by which colours it removes from white light, and an oil slick’s colours are a thickness map in spectral notches.

The notches also say where the method stops. A layer much thinner than a quarter of the shortest wavelength in the pulse puts its first notch above the pulse’s band, where there is no energy to be notched. Such a layer’s spectrum is simply the pulse’s spectrum tilted towards high frequencies — the signature of differentiation — and nothing in it gives the thickness except the overall strength.

A layer and a step #

Whether a thin feature can be seen at all depends on what it separates. A layer of one rock inside another has faces of opposite sign, and as it thins their echoes cancel: it fades. A thin transition between two different rocks — say, a gradual hardening in two small steps — has faces of the same sign, and as it thins their echoes add: it grows towards twice one face’s echo, and becomes indistinguishable from a single sharp boundary with the combined contrast. The second kind of feature never becomes invisible. It becomes undetailed: the survey sees that the rock changes and cannot tell whether the change happened in one step or several over a few metres.

This distinction reaches beyond seismology. A thin gradient between two media can hide or reveal a boundary depending on how gradually it grades, and the mismatch no network can remove set limits on what any arrangement of thin layers can do across a band. In both, what a wave reports about thin structure is a statement about the whole structure’s effect at its own scale, and a wave cannot report detail finer than a fraction of its own length.

Where the model stops #

One layer, no noise. Real sections contain dozens of thin layers within one wavelength, whose echoes all overlap; the recorded trace is the rock’s sequence of reflection coefficients convolved with the pulse, and separating them is an inversion with more unknowns than the data determine, the same kind of problem that a potential field measured from outside poses. Noise sets how thin a layer can be before its derivative-shaped echo drops below it.

A known pulse. The pulse was taken to be a known, fixed Ricker wavelet. In practice its shape must be estimated from the data, it changes with depth as the ground absorbs its high frequencies, and an error in its assumed shape becomes an error in the inferred layering.

Normal incidence, elastic rock. The drawings send the pulse straight down and treat the rock as a simple fluid. Real surveys record at many angles, and the way a boundary’s reflection changes with angle carries information about the rocks’ shear properties — which is how gas-bearing sands are told from water-bearing ones — that a single normal-incidence echo does not.

What the traces do not show #

The traces are echoes from a perfect, flat layer. Real layers have rough faces, which scatter energy out of the specular echo, and pinch out sideways, so that the wedge of the first drawing is often a real feature rather than a test. They also carry the ground’s attenuation, which takes the treble out of a pulse with every kilometre it travels, so that the resolution of a survey worsens with depth for a reason unrelated to the layers themselves.

Still open: how far below the tuning thickness thickness can be read #

Methods that exploit the spectrum, the amplitude and prior knowledge of the geology together routinely claim to resolve layers well below a quarter of a wavelength, and in favourable cases, checked against wells drilled through them, they do. How far such claims can be trusted in general — how much of the apparent detail is in the data and how much is supplied by the assumptions of the inversion — is argued over layer by layer. Machine-learning methods trained on synthetic sections have added a new form of the old question, because they produce detailed pictures whose detail may come from what they were trained on rather than from what the ground returned.

The habit worth carrying away is to ask where information goes when it stops showing up where it was being looked for. A layer thinner than a quarter of a wavelength no longer separates its echoes in time, but its thickness survives in the echo’s strength and in the frequencies it cancels — so the limit of resolution is a limit of one reading of the data, and whether the other readings can be trusted depends on how much else about the layer is already known.