Noise in every frame is a memory of zero, and the run cannot measure it at its own pace

A camera zoomed into the cascade needs frames faster than its window remembers asked how often a particle-image camera should take frames of a turbulent field when its window sits inside the inertial range, and found that the answer is not the window’s memory. A small window keeps nearly all the variance and almost exactly a probe’s integral time, yet its correlation falls quickly at first, as the small eddies inside it forget, and a run of frames has to sample that fall. The frame interval at which a faster camera stops buying came out a third shorter than the window’s own memory said, and thirty times shorter again for a velocity gradient.

Every number in it assumed the camera recorded the field exactly. No camera does. A particle-image measurement locates a correlation peak to a fraction of a pixel, and the fraction is random: it depends on how many particles happen to sit in the interrogation window, on their images, on noise in the sensor. Each frame’s vector field carries an error, and the error in one frame knows nothing about the error in the next. That essay ended by naming the consequence. Noise of that kind is a part of the signal with a memory of zero — the fastest possible fall at the origin of the correlation — and the question was how much of it it takes before the frame interval is set by the noise rather than by the flow.

There are two answers here, and the second matters more. Noise pulls in the frame interval that stops buying by nearly the same share for every window, wide or zoomed, velocity or gradient, so a noise level stated as a share of the quantity’s own variance is all that decides it. And the standard way of learning that share, from the run’s own correlation, fails at every frame rate the run’s information calls for, because it cannot tell noise from the cascade.

What a frame with no memory does to a run #

The field is the zoomed camera’s: a two-dimensional field whose one-dimensional spectrum falls as $k^{-5/3}$ between an integral scale $\ell$ and a dissipative scale a thousand times smaller, each wavenumber forgetting exponentially over its own time $\tau(k) \propto 1/k$ above the integral wavenumber — the sweeping of small eddies past a point by the large ones — and the whole scaled so that a single probe reports an integral time $T$. Times below are in probe integral times and lengths in integral scales. A disc window of diameter $L$ averages the field in each frame, and $\rho(t)$ is the autocorrelation of that window mean.

The noise is a stated share $\nu$ of the window mean’s own variance. Writing it that way rather than as an error per vector is deliberate. A window holding $N$ independent vectors divides their noise variance by $N$, a velocity gradient formed by differencing neighbours multiplies it by roughly twice the inverse square of their spacing, and the window mean’s signal variance depends on how much of the cascade the window keeps. All three are absorbed into one number, and the result will show that one number is what counts.

With noise added, the run’s autocorrelation is one at zero lag and $\rho(t)/(1 + \nu)$ at every other lag: the noise contributes to each frame’s variance and to no frame’s covariance with another. The variance of the mean of $N$ frames taken $\Delta t$ apart is then

$$\frac{\sigma^2}{N}\left[1 + \nu + 2\sum_{j \ge 1} \rho(j\Delta t)\right],$$

and continuous filming of the noise-free field over the same time, $N\Delta t$, gives $\sigma^2 \cdot 2\int\rho\,\mathrm{d}t / (N \Delta t)$. The share of continuous filming’s independent values the noisy run collects is the ratio,

$$E(\Delta t, \nu) = \frac{2\int_0^\infty \rho\, \mathrm{d}t}{\Delta t\left[1 + \nu + 2\sum_{j\ge1} \rho(j\Delta t)\right]}.$$

With $\nu = 0$ it is the zoomed camera’s share, to a part in a million in the check below. With $\nu > 0$ two things change. As $\Delta t \to 0$ the share still tends to one, because infinitely many frames average any finite noise away; the bracket grows as $2\int\rho / \Delta t$ and swamps the $\nu$. But at every finite $\Delta t$ the share is lower, and for small $\Delta t$ it falls as $1 - \nu\Delta t / (2\int\rho)$ rather than staying flat at one. A faster camera now buys something at every frame rate, because each extra frame averages down the noise as well as sampling the flow.

The interval that stops buying falls by a share the window does not set #

The first figure is the answer to the question the zoomed camera left. Its vertical axis is the frame interval at which the run still collects nine-tenths of continuous filming’s independent values, divided by the same interval with no noise; its horizontal axis is the noise share. The three solid curves are three windows chosen to be as different as the earlier essays could make them: a velocity window one integral scale across, whose noise-free interval is 1.67 probe times; a velocity window a twentieth of that, at 0.888; and a velocity gradient in the small window, at 0.0719 — twenty-three times shorter than the wide window’s.

They lie almost on top of each other. At a noise share of a tenth the interval falls to 0.78 of its noise-free value for the wide window, 0.80 for the small one and 0.82 for the gradient. At three-tenths, to 0.51, 0.56 and 0.59. At a noise share equal to the signal, to 0.20, 0.24 and 0.24. The dashed curve is a field with one exponential memory, for which the share has a closed form,

$$E = \frac{2}{\Delta t\,[\nu + \coth(\Delta t / 2)]},$$

in units of its own integral time; it gives 0.77, 0.48 and 0.18 at the same three noise shares and sits just below the rest.

So the noise sets its own scale, and the window’s memory sets only the unit it is measured in. That is not what one might have guessed from the zoomed camera’s result, where the window decided everything: a gradient’s frames had to come thirty times faster than a probe’s because its correlation fell so quickly. Here, whatever that fall is, a given share of noise costs the same fraction of it. The reason is visible in the small-interval form of the share. When the noise dominates the shortfall, $E \approx 1/(1 + \nu\Delta t / 2\int\rho)$, and the interval at which that is nine-tenths is $2\int\rho / 9\nu$. Divided by the noise-free interval, that is a ratio of two times the same window sets, and across these three windows the ratio $2\int\rho / 9\Delta t_0$ lies between 0.21 and 0.27. The window changes both numerator and denominator; it changes their ratio by little.

That ratio also answers the zoomed camera’s question in the form it was asked. The noise takes over from the flow when $\nu$ passes about $2\int\rho / 9\Delta t_0$: at a noise share near a quarter of the window mean’s variance, for every window here. Below it the frame interval is the flow’s, pulled in a little; above it the interval falls as $1/\nu$ and is the noise’s. At ten times the variance the interval is a fortieth to a fiftieth of its noise-free value, for all three.

Slower frames now lose what the noise-free run kept #

The second figure draws the share itself for the small velocity window, against the frame interval in probe times, at a noise share the dial sets. With no noise it is the zoomed camera’s curve: flat at one out to about a tenth of a probe time, past nine-tenths until 0.888, and falling after. With noise the curve loses its flat top. At a noise share of a tenth, a run with frames 0.03 probe times apart collects 0.998 of continuous filming’s information and one with frames 0.3 apart collects 0.970, where the noise-free run collected 0.983 there. At a noise share equal to the variance, the same two runs collect 0.986 and 0.863. At three times the variance, 0.959 and 0.694.

The shape answers a practical worry. On a noise-free field the flat top of the curve was the argument for a slow camera: frames closer than a tenth of a probe time gave almost nothing the slower run did not have, so a run of a given length could be stored at a tenth of the frames without loss. With noise, the flat top tilts, and storing fewer frames now costs information in proportion to the noise. The loss is still small while the noise is small — three parts in a thousand at a tenth of the variance — but it is no longer zero, and it grows linearly with the interval rather than waiting for the flow’s memory.

There is a corollary that is easy to get backwards. Faster frames do not reduce the noise in any one frame; they reduce the noise in the run’s mean, by averaging more independent errors. If the quantity wanted is a mean over the run — a mean velocity, a Reynolds stress, a spectrum — faster frames buy a better estimate. If the quantity wanted is the field in a single frame, they buy nothing at all, and the noise in that frame is whatever the interrogation left in it.

The noise is a step, and the cascade’s first fall looks like one #

All of this assumes the noise share is known. In practice it is not stated; it has to be measured, and the measurement most often made is from the run itself. The third figure shows the idea. With noise the autocorrelation is one at zero lag and drops at once to the noise-free value divided by $1 + \nu$: a step at the origin of height $\nu/(1 + \nu)$, which is 0.091 for a noise share of a tenth. Extrapolate the correlation measured at the first few non-zero lags back to zero, and the gap between that intercept and one is the noise.

The difficulty is the noise-free curve beside it. For the small window it does not leave zero lag gently. It falls to 0.886 by a lag of 0.05 probe times and to 0.693 by 0.2, because the window holds small eddies whose memory under the sweeping law is short — this is the fast fall the zoomed camera found, and the reason its frames had to come faster than its memory suggested. A run whose first lag is 0.1 sees a correlation that has already dropped by nearly a fifth before its first point. Nothing in the run distinguishes the part of that drop that happened at zero lag, which is noise, from the part that happened in the first tenth of a probe time, which is flow.

Read off the run, the noise is right only at frames the run does not need #

The fourth figure does the extrapolation. For each window and frame interval it takes the noisy correlation at the first two non-zero lags, draws a straight line through them back to zero lag, and reads off the noise share the intercept implies; dashed, a parabola through the first three. The true share is a tenth, so a perfect reading is one on the vertical axis.

For the wide velocity window the method works as long as the frames come at a few hundredths of a probe time: the reading is right to a per cent at 0.03, eight per cent high at 0.1, and sixty per cent high at 0.3. The parabola holds it to one per cent out to 0.1. For the small velocity window the reading is three per cent high at 0.01, nineteen per cent at 0.03 and twice the truth at 0.1; the parabola only moves it to 1.6 times. For the gradient the reading is twelve per cent high even at frames 0.003 probe times apart, 1.8 times the truth at 0.01, 5.3 times at 0.03 and 26 times at 0.1.

Set those beside the intervals that stop buying. The small velocity window needs frames 0.71 probe times apart at a noise share of a tenth to keep nine-tenths of its information; at that interval the read-back is nine times the truth. The gradient needs frames 0.059 apart; at that interval it is twelve times the truth, and the wide window, at its own 1.31, nearly eight. A run paced for its own information cannot measure its own noise, and the error is always an overestimate, because the cascade’s fall can only ever be added to the noise’s step, never subtracted from it.

That changes what the first figure is good for. Its horizontal axis is a share the experimenter has to supply, and a share read off a run paced at the rate the figure recommends would place the run several times too far to the right — calling for frames far faster than needed, or, read the other way, discarding as noise a part of the variance that is the cascade’s smallest eddies. The second error is the worse one, because the same extrapolation is used to correct spectra and dissipation rates, and what it subtracts there is real.

Where the noise share comes from #

The figures take the share as given; a camera hands over an error in pixels. The translation goes through three steps, and each depends on the quantity measured.

A velocity vector’s error is the displacement error divided by the displacement. Sub-pixel peak fitting commonly leaves a random error of a tenth of a pixel or less on a displacement of several pixels, a few per cent of the vector. Its variance relative to the point variance of the velocity then depends on the turbulence intensity: a two per cent error on a flow with ten per cent turbulence intensity is a fifth of the fluctuation’s amplitude and so a twenty-fifth of its variance. Averaging over a window holding $N$ independent vectors divides that by $N$, and the small velocity window of this essay, a twentieth of an integral scale across, might hold only a few. Its noise share sits near the bottom of the first figure, and its interval is the flow’s.

A gradient is different in kind. It is formed by differencing neighbouring vectors a spacing $h$ apart, so its noise variance is about twice the vector’s divided by $h^2$, while its signal variance is set by the cascade’s small eddies and is finite only because of the dissipative cut-off — the point the moment a spectrum cannot hold makes about every moment weighted towards high wavenumbers. When the vector spacing approaches the dissipative scale the signal stops growing as $h$ shrinks and the noise keeps growing as $1/h^2$. A gradient measured at the resolution that makes it worth measuring can carry a noise share of order one, and the first figure says what that costs: the interval falls to a quarter of a gradient’s already short noise-free one, about two-hundredths of a probe time.

The convention: shares, probe times and a stated law #

The noise share is the noise variance over the window mean’s own noise-free variance, not over the point variance, because the window’s variance is what the window mean’s correlation is normalised by. Frame intervals are in probe integral times $T$, the integral time a single probe in the same flow reports, so the three windows can be compared on one axis. The memory law is the sweeping one, $\tau(k) \propto 1/k$ above the integral wavenumber; the zoomed camera showed the turnover law gives the same shapes less steeply. Nine-tenths of continuous filming is the threshold, as in every earlier essay on the run’s information.

The correlation the read-back extrapolates is summed over the spectrum at each lag rather than interpolated from a table, because the table’s step is coarser than the finest frame intervals here and a straight line drawn between two tabulated values would extrapolate exactly, reporting the noise perfectly at any frame rate finer than the table.

What the model leaves out #

The noise is white: independent from frame to frame and with no structure in time. Real particle-image noise has a part that is not. A peak-locking bias pulls displacements towards whole pixels and persists while the displacement does, so it shares the flow’s memory and is not averaged away by more frames at all; it is a bias, not a noise, and nothing here measures it. Laser pulse jitter and sensor drift add slow components. All of these make the error look more like signal, which makes the read-back even less able to find them.

The noise is also independent of the flow. In a real run the error grows with the local gradient, since a window holding a shear deforms its particle pattern and degrades its correlation peak, and with out-of-plane motion. That couples the noise to the cascade’s small eddies — exactly the part of the signal the read-back confuses it with.

And the window mean is still a single scalar. A real run computes many quantities from one vector field, each with its own window, its own noise share and its own interval; the figures say how to choose a frame rate for one of them at a time. A run that also counts structures in each frame, rather than averaging over a window, meets the different arithmetic of a snapshot counts areas, and a flow counts its edge.

Who asked it before #

The decomposition of a measured correlation into a delta-function at the origin and a smooth remainder is old in hot-wire anemometry, where electronic noise was separated from turbulence by exactly this extrapolation, and in particle-image velocimetry it is the standard way of estimating random error from time-resolved data. The difficulty that the extrapolation absorbs the fast early part of the turbulence correlation is known; it is why dissipation estimates from noisy data are usually corrected with a model of the spectrum rather than by subtracting the read-back alone. What this calculation adds is a number for when that happens: at frames slower than a few hundredths of a probe time for a zoomed velocity window and slower than a few thousandths for a gradient, with the sweeping law setting the small eddies’ memory, which is the regime nearly every time-resolved camera actually runs in.

Its other result, that a noise share pulls in the interval by nearly the same fraction for every window, follows the earlier essays’ habit of reducing a run’s design to one number. Frames that share their eddies count as one reduced it to the integral time, the camera needs the window’s memory, not the probe’s to the window’s integral time, and the zoomed camera showed that even that number fails inside the inertial range. Here a second number, the noise share, enters, and the window’s own complication largely cancels out of its effect.

How it was checked #

The ledger gives the checks. With the noise share set to zero, the share is the zoomed camera’s to a part in a million. With one exponential memory at every scale the share matches the closed form $2/(\Delta t[\nu + \coth(\Delta t/2)])$ to a part in a hundred thousand at three pairs of interval and noise. The noise-free thresholds are the earlier essay’s quantities for these windows. The read-back’s error is zero to a tenth of a per cent as the frame interval goes to zero for the two velocity windows, which is the limit in which the extrapolation must be exact.

The calculation also rejects what it should: a frame interval of zero, which would hold infinitely many frames, and a negative noise share.

One limit was found in passing and not repaired. The zoomed camera’s tabulated correlation steps about two-hundredths of a probe time, and for the gradient, whose correlation falls a fifth in the first hundredth, linear interpolation between steps overstates the share by about a per cent near the threshold. The ratios drawn here are insensitive to it, since numerator and denominator share the table; the gradient’s absolute interval of 0.0719 probe times may be a few per cent long.

Related reading on why a record’s length is counted in these memories, not in frames, is in a record is as long as its integral scales; on what a scale estimate assumes about the spectrum’s shape, in the best estimate of a scale assumes its shape; and on the inertial range itself and how much of it a real flow has, in the range a real Reynolds number does not have. What any camera can and cannot see of a three-dimensional field is in four cameras and a field they cannot see.

Still open: a bias that remembers #

Every error here forgets at once, and that is why more frames could average it. The error that does not is peak-locking: a displacement measured from a correlation peak is pulled towards a whole number of pixels, so the error depends on the displacement itself and lives as long as the displacement does. It has the flow’s memory, which means neither more frames nor the read-back can find it, and it is distributed over the scales in proportion to how the displacement is — concentrated on the large, slow eddies whose displacements change least. The next calculation would add an error that is a stated periodic function of the window mean’s own value, and ask what share of the spectrum it corrupts and at which scales: whether a pixel-locked run overstates the inertial range’s slope or its integral scale, and how many pixels of displacement a run needs before the locking is smaller than the noise this essay has treated.