Noise, error bars and what you can conclude

The gather in the header has noise on it. Picking the amplitude of the top reflection on each trace gives the scattered points in the panel below, and the line fitted through them gives an intercept and a gradient. That is the whole measurement, and it is what every AVO product in a survey is built from.

Press record it again. The rock has not changed by so much as a percent of porosity. The survey has not changed. Only the particular noise on this particular gather has changed, and the answer moves. The gray lines are the previous attempts, kept so that they accumulate into a fan.

One end of the fan is tight and the other is loose. Near zero the lines pass through nearly the same point. Out at the far offsets they spread widely. The fit is anchored at the near end and poorly constrained at the far end, and everything else in this module follows from that one picture.

Rather than a hundred presses of the button, the page repeats it directly. The two histograms below are a thousand recordings of the same rock, showing where the recovered intercept landed and where the recovered gradient landed. Both are centered on the truth. What differs is the width.

The numbers underneath give the ratio, and it is large. At a 0–30° fit the gradient's scatter is about eight times the intercept's. A narrower angle range makes the ratio grow and a wider one makes it shrink. It does not depend on the noise level at all — doubling the noise doubles both scatters and leaves the ratio exactly where it was. The ratio is a property of the geometry of the fit, not of the data quality.

The reason is in step 1's picture. The intercept is read where all the lines converge; the gradient is read from how much they fan apart. A fit that is well anchored at one end is necessarily loose about its slope.

The gradient's uncertainty depends on how far apart the angles are spread, and on nothing else about them. That makes the mute — a processing decision, often made for reasons that have nothing to do with AVO — one of the largest controls on the quality of an AVO product.

The widest angle slider moves the curve on the left. Cutting from 40° back to 20° roughly triples the uncertainty in the gradient. Cutting to 15° multiplies it by more than six. The rock is identical throughout; so is the noise on each trace.

The panel on the right shows the other lever, which is the number of traces. It helps, but only as the square root: halving the error takes four times as many traces. Adding fold is expensive and adding angle range is often free, since it is a mute parameter, which bears on what a reprocessing is asked to deliver.

Intercept and gradient are usually treated as two independent measurements. They are not. They come from one fit to one set of points, so if the noise happens to pull the fitted line up at the near end, the line has to tilt down more steeply to still pass through the far points. The two errors compensate for each other.

The cloud on the left is a thousand recordings plotted on module 06's crossplot. It forms a tilted ellipse rather than a circle, leaning the same way as the background trend, and the correlation between the two errors is −0.742.

That correlation has a consequence. A quantity measured along the long axis of that ellipse is badly determined, and one measured across it is much better determined than either the intercept or the gradient alone. Distance from the background trend is close to the second kind, which is part of why it is a more stable discriminator than the gradient by itself — though the readouts below show the gain is modest rather than dramatic.

Module 05 ended by drawing two pairs of rocks and reporting how far apart their curves got: the oil-against-brine pair separated by 0.0256 at thirty degrees and the gas-against-fizz pair by only 0.0123. Those were exact separations between exact curves. Here they are compared against the error bars a real recording carries.

The useful measure is the separation divided by its own uncertainty — how many standard deviations apart the two answers are. Below about two, the two clouds overlap enough that a single measurement cannot choose between them.

The result qualifies a claim made earlier in the set. At moderate noise the oil and brine sands separate by 1.54 standard deviations in gradient, which is suggestive and not conclusive. At high noise it drops to about half a standard deviation, which is nothing. The fizz pair does not reach one standard deviation even at moderate noise. Module 05 stated that offset breaks the first tie and not the second; with error bars included, offset weakens the first tie and leaves the second where it was.

The practical content of this module is that an AVO gradient is an estimate with a large and computable error bar, and that almost nothing in a normal workflow displays that error bar.

• Compute the error bar; it is free. The uncertainty follows from the angle range, the number of traces and an estimate of the noise. All three are known. There is no excuse for a gradient volume without an uncertainty volume beside it.
• Argue about the mute. Step 3 shows the mute deciding the quality of the answer. If AVO matters on a project, the mute is an AVO parameter and should be chosen with the AVO in mind rather than inherited from the stacking flow.
• Do not treat intercept and gradient as independent. Their errors are correlated at −0.742. Anything that combines them — a fluid factor, a projection, a crossplot cut-off — needs the covariance, not two separate error bars.
• Weight consistency over magnitude. A moderate anomaly that holds together across a mapped area is stronger evidence than a large one at a single location, because the errors this module describes are largely independent from trace to trace and the geology is not.

The error this module is built against is the confident gradient map. It is a continuous, colorful, plausible-looking volume, and at any single location a substantial part of what it shows is noise. Interpreters know this in the abstract and then pick features off it anyway, because the display gives no clue which features are supported.

The second is quieter, and it is this site's own. Module 05 reported that adding offset separates the oil sand from the brine sand, and that was true of the exact curves. Step 5 shows that at a realistic noise level the separation is 1.54 standard deviations. Anything computed from noise-free models will overstate what a method can do, and this set of modules is not exempt from that.

The rock physics and the reflection coefficients are modules 01 to 03's, unchanged. What this module adds is noise and a fit.

The uncertainties are exact, not simulated. For a least-squares straight line through points with independent noise of standard deviation σ, the uncertainty in the slope is σ/√S_xx and in the intercept σ√(1/n + x̄²/S_xx), where S_xx is the spread of the sin²θ values about their own mean. Those formulas are what the readouts use. The scatter clouds and histograms are simulated, and the test suite checks that thirty thousand simulated fits reproduce the formulas' widths to within about half a percent, and their correlation to within 0.02.

The noise model is deliberately simple. Independent, identically distributed Gaussian noise is added to each trace's picked amplitude, at 0.015 or 0.040 in reflection-coefficient units. Real gather noise is none of those things: it is correlated between neighboring traces, usually worse at far offsets than near, and often not Gaussian at all. Correlated noise is the important omission, because noise that varies smoothly with offset looks exactly like a gradient and this model cannot produce that failure.

Where this departs from production practice.

• Amplitudes are picked perfectly. The pick is taken at the true time on every trace. Real picking has to contend with residual moveout, and a small timing error at far offset turns directly into an amplitude error that is systematic rather than random.
• No NMO stretch. Far-offset wavelets are stretched to lower frequency by the moveout correction, which changes the amplitude of a tuned event with offset for reasons that have nothing to do with the rocks. On a thin bed this can dominate everything shown here.
• Noise is the same at every angle. In practice fold falls off at far offset, so the far traces — the ones the gradient depends on most — are usually the noisiest.
• The fit is unweighted and the angles are exact. Production fits are often weighted, and are usually done against straight-ray angles, which module 04 showed to be biased.
• Nothing here is spatial. The single largest defense against everything in this module is that anomalies repeat over an area while noise does not, and a single-trace analysis cannot show that.