Below Tuning, Thickness and Contrast Are the Same Number

Equinor says artificial intelligence gave it a tenfold increase in seismic interpretation capacity, and that data from two million square kilometres of the Norwegian continental shelf were interpreted using AI during 2025 [1]. Hege Skryseth, the company's executive vice president for technology, digital and innovation, puts the same figure as analysing seismic data ten times faster, alongside better well and field development planning and more efficient operations [1]. ADNOC and AIQ ran a 90 day proof of concept on their ENERGYai system and reported "70% improvement in accuracy in major seismic interpretation aspects" [2], with no stated baseline for that percentage. Petrobras, working with Emerson E&P Software on the Santos Basin pre-salt, went the other way: rather than claim resolution, it priced it, building an uncertainty envelope around the top reservoir surface from the resolution limits Widess set out in 1973, by combining dominant frequency and interval velocity into a single map [3].

Three companies, three postures towards the same physical quantity. Two of them are counting how much interpretation they can produce. The third is counting how much of the answer the data cannot contain.

That third quantity is the one that decides whether a net pay model is worth anything, and it has a hard edge in it.

What one trace can carry

Take the simplest case in the literature and the one every wedge model draws: a thin bed of velocity $v$ embedded in surroundings such that the top and base reflection coefficients are equal in strength and opposite in sign. Convolve a zero-phase wavelet with that two-spike series and the composite trace is a difference of two copies of the wavelet, separated by the two-way time through the bed.

$$c(t) \;=\; r\,\bigl[\,w(t)\;-\;w(t-\Delta t)\,\bigr], \qquad \Delta t \;=\; \frac{2b}{v}$$

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Widess showed in 1973 that as the bed thins towards one eighth of the predominant wavelength, this composite converges on the derivative of the input zero-phase wavelet. De Voogd and den Rooijen derived the same result quantitatively ten years later, with transmission loss and internal multiples carried rather than ignored, and found the reflected pulse takes the shape of the time derivative of the incident wavelet with an amplitude proportional to the two-way travel time in the layer. Chung and Lawton, whose 1990 CREWES report is the accessible account of both, state the shared conclusion plainly: the maximum amplitude is linearly proportional to the bed thickness and to the reflection coefficient, and inversely proportional to the predominant wavelength, for this opposite-polarity equal-strength case [4].

Read that sentence as an equation and the problem is already visible.

$$A_{\max} \;\propto\; \frac{r\,b}{\lambda}$$

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Two unknowns on the right, one measurement on the left. The trace does not carry $r$ and $b$ separately. It carries their product. Everything else in that expression, the wavelength, is something you can measure from the data itself.

Chung and Lawton go further and derive the general case, where the two reflection coefficients are not equal and opposite. Their result is a second-order polynomial rather than a straight line, and they draw the practical conclusion for the reader in the seat: calibrating amplitude to infer thickness on a linear relationship gives erroneous results unless the lower reflection coefficient is the negative of the upper one [4]. So the clean linear case this piece works through is the friendliest one available. Real beds are worse.

Above the limit, the expression changes

Now let the bed thicken past a quarter of the predominant wavelength, the limit Chung and Lawton use. They note that various authors define that limit by different criteria, and that Kallweit and Wood showed in 1982 that Ricker's 1953 zero-curvature criterion generalises to this two-term case [4]. The top and base arrivals separate in time. Thickness now comes from picking two events and differencing them.

$$b \;=\; \frac{v\,\Delta t}{2}$$

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Look at what is not in that expression. There is no $r$. A zero-phase wavelet is symmetric about its centre, so scaling it does not move the time of its peak. Get the reflection coefficient wrong by any amount you like and the arrival times are where they were. The error you make in thickness comes from how well you can pick a time, and from the velocity that converts time to metres, and from nothing else.

This is the point that gets lost when resolution is discussed as though it were a continuous quality that degrades as beds thin. It is not one quantity getting gradually worse. It is one estimator handing over to a different estimator, and the two are sensitive to different things.

The step, in one line

Suppose you are below the limit and you invert amplitude for thickness using a reflection coefficient that is wrong by the fraction $e$. In the linear regime the amplitude you measured is $k\,r\,b$ and the thickness you report is that amplitude divided by $k\,r\,(1+e)$, so the answer is the true thickness divided by one plus the error.

$$\frac{\hat b - b}{b} \;=\; \frac{1}{1+e} \;-\; 1 \;=\; -\,\frac{e}{1+e}$$

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Three things about that expression are worth saying out loud. It does not contain the thickness, so a relative error of this size is paid at every thin bed in the survey. It does not contain the noise level, the trace count, the well count or any hyperparameter, so no amount of data reduces it. And it is not symmetric: a contrast overestimated by 20% buys a thickness error of 16.7%, while a contrast underestimated by 20% buys 25%, because the same amplitude has to be explained by a thicker bed.

Above the limit the corresponding expression is a pick tolerance rather than a bias. At the quarter wavelength exactly, it takes a form worth noticing.

$$\left.\frac{\delta b}{b}\right|_{\,b=\lambda/4} \;=\; \frac{v\,\delta t / 2}{v/4f} \;=\; 2\,f\,\delta t$$

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The velocity cancels. At 30 Hz with a one sample pick error on a 2 ms record that is 12%, and it falls as the bed thickens, because the numerator is a fixed number of metres. Raise the frequency and the quarter wavelength moves down in metres, which is what everybody wants, but the relative worth of a pick at that new limit goes up in proportion. At 60 hertz you buy a tuning thickness half the size and a pick tolerance twice as large at it.

The bench

The exhibit below computes the wedge rather than drawing it. 22 traces, each one a zero-phase Ricker convolved with the two-spike series above, the spikes separated by the two-way time through a bed that thickens left to right. The two vertical rules are the eighth-wavelength Widess limit and the quarter-wavelength limit. Move the frequency or the velocity and the metres the quarter-wavelength rule stands for run from 8.3 to 150, because the wavelength is the velocity over the frequency, while the thickness axis rescales to follow, so both rules stay in the left half of the thickness axis rather than running off it. They do not hold still while they do it: the axis climbs a ladder of fixed rungs, so a rule steps sideways whenever the axis changes rung. The two right-hand panels share that axis, so their curves slide and stretch with it too. What the frequency and the velocity cannot touch is what those curves are functions of, and that is the problem rather than the drawing: the composite amplitude is a function of thickness over wavelength alone, and the error curve a function of that ratio and of the contrast error, and of nothing else. Those two panels carry the composite peak amplitude those same traces produce, and the error in the thickness the bed reports when the amplitude is inverted with a reflection coefficient that is wrong by a stated fraction.

dominant frequency: 30.0 Hz (10 to 60)interval velocity: 3000 m/s (2000 to 6000)reflection-coefficient error: +20.0 % (-40 to 40)bed thickness: 8.0 m (0.5 to 60)

What the bench actually shows

Set the contrast error to plus 20%, which is not a pessimistic number for a lithology whose impedance contrast you are inferring rather than logging, and drag the bed thickness along the wedge.

At a bed of half a metre the readout is minus 16.7%, which is the algebra above to the decimal the plate prints. At the Widess limit, 12.5 metres at the opening settings, it is minus 23.5%, because the tuning curve has started to bend and the same fractional amplitude error now maps to a larger fractional thickness error. At 18 metres it is minus 34.9%. Cross the amber rule at 25 metres and the readout is zero. Not small. Zero, because the branch that computes it does not reference the contrast error at all.

The step at the quarter wavelength is the finding, and the property that matters about it is what it is made of. The whole amber curve is a function of the bed thickness divided by the wavelength and of the contrast error, and of nothing else. Move the frequency from 10 to 60 hertz and the wavelength, the Widess limit and the quarter-wavelength limit under the plate all change, while the amber curve stays the same function of thickness over wavelength, because its argument is a ratio and not a length. It still moves on the plate, which draws it against metres rather than against the ratio. The pick tolerance does not move with the frequency at all: it is the velocity times a fixed time, and no frequency enters it. That is measurable rather than rhetorical: across five very different frequency and velocity pairs evaluated at matched fractions of their own tuning thickness, the exhibit's error values agree to floating-point round-off.

So the step is not a noise floor and it does not respond to the things a machine learning team can buy. Add wells, add traces, add parameters, add augmentation. The step is where it was.

Two more things the bench will do if you push it.

Drag the contrast error negative and a region appears in which the readout says there is no solution. That is not a numerical failure. If you assume a contrast smaller than the truth, the amplitude you actually measured can exceed the largest amplitude any bed of that contrast can produce, and no thickness explains the observation. At minus 20%, at the opening frequency and velocity, that begins at 11.1 metres, which is below the Widess limit. A model that reports a thickness there is reporting a number the physics does not contain.

And the shaded strip between 19.5 metres and the rule is the honest caveat. The composite amplitude does not rise all the way to the quarter wavelength. It maximises when the two-way delay through the bed reaches $\sqrt{6}/(2\pi)$ of the dominant period, which works out to 0.78 of the quarter wavelength, and it turns over after that. In that strip an amplitude-to-thickness calibration is two-valued, so it returns the wrong root even when the contrast is exactly right. The exhibit switches estimators at the quarter wavelength because that is the limit the literature states. In a real interpretation the handover is a zone rather than an edge, and that zone is inside the shading.

Petrobras already prices this, and does not call it a model problem

The most useful public treatment of this quantity we found is not from a machine learning group. Paes and colleagues at Petrobras, with Emerson E&P Software, built two salt velocity models for a Santos Basin pre-salt field, one with a constant 4,500 metres per second through the whole salt section and one carrying the velocity variation the stratified evaporites impose, and ran 300 Monte Carlo realisations of the top reservoir surface against each [3].

The step that matters here is how they bounded the surface. The envelope around the top reservoir, the volume inside which the interpreter cannot resolve where the surface is, was built from the resolution limits Widess introduced, by combining the dominant frequency and interval velocity attributes into one map, then weighted by an RMS amplitude map so that low signal-to-noise areas widen it [3]. The resulting envelope ran from 70 metres, 35 metres either side of the pick, to 200 metres, 100 metres either side [3].

The consequences they report are modest in percentage and large in volume. The spread between pessimistic and optimistic realisations came out close to 1.6% for the constant-velocity scenario and 2% for the stratified one, and those percentages correspond to a gross rock volume swing of roughly three billion cubic metres, on a reservoir covering about 1,000 square kilometres with an oil column between 300 and 400 metres [3]. Scenario two came out about 0.5% more optimistic than scenario one [3].

Nobody in that paper is arguing about a model. They are treating the resolution limit as a parameter of the problem, assigning it a number, and propagating it. That is what a network sitting on the same data has to do too, and almost none of them do.

The example the CREWES report picks, and a small arithmetic note

Chung and Lawton motivate their whole study with a field case: the Bluesky Formation in the Waskahigan area of West Central Alberta, a unit whose thickness rarely exceeds 10 metres, with a P-wave velocity of about 4,600 metres per second and a peak frequency of 35 Hz [4]. They state its resolution limit, a quarter of the predominant wavelength, as 25 metres, and conclude that the top and bottom interfaces cannot be resolved [4].

Set the exhibit to 4,600 metres per second and 35 Hz and it prints a dominant wavelength of 131.4 metres and a quarter-wavelength limit of 32.9 metres. That is the arithmetic on the paper's own stated numbers, and the paper prints 25. We are not going to guess which of the two figures came from a different velocity, and it changes nothing about the argument, because a 10 metre formation is far below either. It is worth flagging only because a reader who checks the exhibit against the source will find the gap, and because checking a resolution limit against its own inputs takes about four seconds and should be a reflex.

At those settings the exhibit says something more useful anyway. A 10 metre Bluesky bed sits at 0.30 of the quarter wavelength. With a 20% contrast error, that is a thickness error of about 19%, paid on every Bluesky pick in the survey, invisibly, in the same direction.

What we do about this on our own work

Our own delivery has been on borehole image logs and well data rather than seismic amplitude inversion, so the honest thing to say is what transfers rather than what we have measured on a wedge. Two habits transfer.

The first is stating a claim's scope before its magnitude. We have written about retiring one of our own speed figures because a reviewer took it apart, and about the two gates every number we ship has to pass: can the scope be stated in one sentence a stranger would accept, and does it trace to a source someone outside the room can find [5]. A net pay accuracy figure that does not carry the thickness distribution it was measured on fails the first gate, and it fails it in a way that flatters the model.

The second is telling identifiability apart from difficulty. On a 14 well borehole image engagement in a Middle East carbonate field, the sharpest question in the peer review was whether depth-adjacent patches leaked across the train and test split, which is a difficulty problem and one a better split policy fixes [6]. We have also written about a shale development record in which spacing and completion intensity moved together, which is an identifiability problem, and no split policy in the world fixes it [7]. Below tuning, thickness and contrast are in the second category. The evaluation effort belongs where the failure actually is.

There is a version of this that is specific to seismic and worth naming. The current wave of deep learning resolution enhancement, including work such as the DAKD-Net domain-adaptive distillation network, trains on forward-modelled data to learn the relationship between low and high resolution volumes [8]. That is a reasonable way to build labels. It also means the contrast the network learned to associate with a given amplitude is whatever the forward model put there. Where the survey's real lithology contrast departs from the modelling assumption, the network inherits exactly the bias this piece describes, and it inherits it from its own training set rather than from its architecture.

Three questions before a net-pay model sets a development plan

What is the thickness distribution of the beds in the test set, expressed as a fraction of the quarter wavelength at the local frequency and velocity? A model reported as accurate on a thick-bed test set carries an unreported bias everywhere the reservoir is thin, and the report will not show it, because the thick-bed part of the test set is where the estimator was never in trouble.

What reflection coefficient does the workflow assume, and where did it come from? If the answer is a well tie, then the question becomes how far the wells are from the thin part of the field, because that distance is the size of the extrapolation the bias is proportional to.

And where does the interpretation cross the quarter wavelength? Not approximately: as a map, in the way Petrobras built one [3]. On one side of that surface the model's thickness error is a lithology-dependent bias that no dataset closes. On the other it is a pick tolerance that a better wavelet and a finer sample rate genuinely improve. Those are different engineering programmes, and a single accuracy number quoted over the whole survey averages them into something that describes neither.

Key takeaways

1. 1Equinor reports a tenfold increase in seismic interpretation capacity and two million square kilometres of the Norwegian continental shelf interpreted with AI in 2025, and ADNOC and AIQ reported a 70% accuracy improvement on major seismic interpretation aspects from a 90 day ENERGYai trial with no stated baseline. Both are counts of interpretation produced, not of what the data can support.
2. 2For a thin bed with equal and opposite reflection coefficients, Chung and Lawton record the shared conclusion of Widess (1973) and de Voogd and den Rooijen (1983): the maximum amplitude is linearly proportional to both bed thickness and reflection coefficient and inversely proportional to the predominant wavelength. One measurement, two unknowns, and only their product is observable.
3. 3Above the quarter wavelength thickness is velocity times a time difference over two, an expression the reflection coefficient does not enter. The identifiability does not degrade across the limit, it switches, which is why the error curve has a step in it rather than a slope.
4. 4Below the limit a fractional contrast error e costs a thickness error of e over one plus e. That expression contains no noise term, no trace count and no well count, so the bias is not reducible by more data. It is also asymmetric: plus 20% buys 16.7%, minus 20% buys 25%.
5. 5At the quarter wavelength a time pick is worth two times frequency times the pick error, with velocity cancelling. At 30 Hz on a 2 ms record that is 12%, and raising the frequency shrinks the tuning thickness in metres while raising the relative worth of a pick at it in exact proportion.
6. 6Petrobras and Emerson built their Santos Basin pre-salt uncertainty envelope from Widess resolution limits combined with dominant frequency and interval velocity, obtaining an envelope of 70 to 200 metres, and propagated it through 600 realisations to a gross rock volume spread of about 1.6 and 2%, roughly three billion cubic metres. Resolution priced as a parameter, not claimed as a capability.
7. 7Ask for the test set's thickness distribution as a fraction of the local quarter wavelength before accepting a net pay accuracy figure. A model that scores well on thick beds carries a bias everywhere the reservoir is thin, and the bias is proportional to the lithology variation the model was bought to map.

References

[1] Journal of Petroleum Technology. Equinor Says AI Saved It $130M in 2025. 7 January 2026. https://jpt.spe.org/equinor-says-ai-saved-it-130m-in-2025

[2] ADNOC. ADNOC and AIQ Successfully Complete Trial Phase of Agentic AI Solution. 16 January 2025. https://www.adnoc.ae/en/news-and-media/press-releases/2024/adnoc-and-aiq-successfully-complete-trial-phase-of-agentic-ai-solution

[3] Paes, M., Pereira, C., Maul, A., Meneguim, T. (Petrobras), Pinto, V., Gonzalez, M., Gonzalez, G., Meyer, R. S. and Furland, S. L. (Emerson E&P Software). Gross-Rock Volume Uncertainties Based on the Integration of Velocity Model and Seismic Resolution. 16th International Congress of the Brazilian Geophysical Society, Rio de Janeiro, 19 to 22 August 2019. https://sbgf.org.br/mysbgf/eventos/expanded_abstracts/16th_CISBGf/Gross-Rock%20Volume%20Uncertainties%20Based%20on%20the%20Integration%20of%20Velocity%20Model%20and%20Seismic%20Resolution.pdf

[4] Chung, H. and Lawton, D. Some Properties of Thin Beds. CREWES Research Report 1990, paper 24. https://www.crewes.org/ForOurSponsors/ResearchReports/1990/1990-24.pdf This report is the source for every statement attributed here to Widess (1973), to de Voogd and den Rooijen (1983) and to Kallweit and Wood (1982). Widess, M. B., How thin is a thin bed?, Geophysics 38(6), 1176 to 1180, doi 10.1190/1.1440403, could not be retrieved directly: the SEG library, the SEG wiki and GeoScienceWorld all returned 403 to an unauthenticated request. Its author, title, journal, volume and pages as given here are corroborated by the reference list of [3].

[5] EarthScan. The Self-Scrub: Two Gates Every Number Should Pass Before a Reviewer Ever Sees It. https://earthscan.io/insights/withdrawn-speed-claim-killed-before-publishing

[6] EarthScan. Patch Splits or Well Splits? Evaluating Subsurface AI Honestly When You Only Have 14 Wells. https://earthscan.io/insights/patch-splits-or-well-splits-evaluating-subsurface-ai-honestly-with-14-wells

[7] EarthScan. Two Knobs, One Record: What Shale Design Models Cannot Separate. https://earthscan.io/insights/two-knobs-one-record-what-shale-design-models-cannot-separate

[8] Cai, H., Zhang, H., Zhang, L. and Cheng, S. Seismic resolution enhancement via deep Learning with Knowledge Distillation and Domain Adaptation. arXiv:2506.22018. https://arxiv.org/abs/2506.22018