A record is as long as its integral scales

The moment a spectrum cannot hold ends by pointing at a question it does not answer: what a single long record can establish. Every turbulence statistic on this subject — a mean, a variance, a spectrum, a structure function, an exact law’s constant — is in the end computed from a record of finite length taken at one place, and every one of them is quoted as if it were the value for the flow. The replacement of an average over many independent flows by an average over time in one flow is called ergodicity, and whether it holds is not the interesting question. It holds for any statistically steady flow. The interesting question is how long a record it takes to hold to a given accuracy, and the answer has a unit that is not seconds.

The one number a record is worth #

A signal that fluctuates about its mean remembers itself for a while: a value now tells something about the value a moment later, and nothing about the value much later. The integral time scale $T_I$ is how long that memory lasts, defined as the integral of the autocorrelation over all lags — the same integral that sets how far a parcel gets before it forgets its velocity.

For a record of length $T$ the variance of its mean is exact:

$$\operatorname{Var}(\bar x) = \frac{2\sigma^2}{T}\int_0^T\left(1 - \frac{\tau}{T}\right)\rho(\tau)\,d\tau \;\longrightarrow\; \frac{2\sigma^2 T_I}{T}.$$

The record behaves as if it held $T/2T_I$ independent samples, whatever it actually holds. A record’s length is counted in integral scales, two of them to one independent value. To know the mean to a tenth of the fluctuation’s standard deviation takes two hundred integral scales; to a hundredth, twenty thousand.

The signal used to check that here is the simplest one with a finite integral scale: a stationary Gaussian process with an exponential autocorrelation, $\rho(\tau) = e^{-|\tau|/T_I}$, generated exactly — not by filtering, which approximates, but by the first-order recursion whose stationary statistics are exactly these. The generated signal has variance 0.996, correlation at one integral scale of 0.364 against $e^{-1} = 0.368$, and flatness 3.01. Four hundred records were drawn at each of seven lengths from three to three thousand integral scales, and the scatter of their means matches the exact formula to within two per cent at three, thirty and three hundred.

Sampling faster does not make a record longer #

The practical consequence is the one people get wrong. A hot-wire anemometer can sample at a hundred kilohertz, and a record of ten seconds then holds a million samples. If the flow’s integral time scale is a tenth of a second, the record holds a hundred integral scales and is worth about fifty independent values. The other 999,950 samples add nothing to the precision of the mean.

The figure shows the crossover directly. At one sample every twenty integral scales the samples are independent, there are five of them, and the scatter of the mean is 0.44 of the standard deviation — one over the square root of five. At one sample per integral scale it is 0.137. At five per scale, 0.142, and at twenty, 0.142: the limit $\sqrt{2T_I/T}$ = 0.141 has been reached, and a hundred times more samples have bought nothing.

What a high sampling rate does buy is the small scales, which are a different question. The dissipation rate is a gradient statistic and lives at the Kolmogorov scale, and resolving it needs samples closer than the Kolmogorov time. But its precision is still set by the record’s length in integral scales, because the dissipation rate itself fluctuates over the large scales — its correlation reaches across every scale — so a record that resolves it finely and spans few integral scales measures it precisely at every instant and imprecisely on average.

A short record says the wrong thing on average #

Scatter is one error. The other is bias: the average over many short records of a statistic is not the statistic’s true value, and the direction of the error is always the same.

The variance is the simplest case. A record’s variance is computed about the record’s own mean, not the true mean, and the record’s mean has wandered towards the record’s values — so the deviations are smaller than they should be, by exactly the variance of the mean. A record three integral scales long reports 0.55 of the true variance on average; thirty reports 0.94; three hundred, 0.995. The flatness, which is a ratio of fourth to squared second moments, is biased low for the same reason and by more: 2.38 against the Gaussian’s 3 at three integral scales.

The integral scale is the worst, and the reason is an identity. The usual estimate integrates the record’s sample autocorrelation over lag. But a record’s deviations from its own mean sum to exactly zero, and the sample autocovariance summed over every lag is the square of that sum, divided by the record’s length. So:

The integral of a record’s autocorrelation over all lags is zero, for every record, whatever the flow. It was checked here to three parts in a hundred trillion. So the integral scale cannot be computed from a record by integrating its autocorrelation; it can only be computed by integrating it to somewhere and stopping. The usual convention stops at the first zero crossing, and that convention is what the bias figure measures: on a short record the sample autocorrelation crosses zero early, because the record’s mean has absorbed part of the correlation, and the integral to that crossing is too small. The integral scale reported from a record is partly a property of where the analysis chose to stop.

The unit is the hardest thing to measure #

The reason the integral scale is so hard is that it is an integral over the tail of the autocorrelation, and the tail is exactly where a finite record’s autocorrelation is least reliable. At a lag of a few integral scales the true correlation is a few per cent and the record’s estimate of it has a scatter of order $\sqrt{2T_I/T}$ — which for a record of a thousand integral scales is also a few per cent. The estimate is integrating noise as large as the signal over a range that contributes a sizeable fraction of the answer, and no choice of where to stop removes both the bias of stopping early and the noise of stopping late.

That leaves an awkward circle. Every error above is counted in integral scales, so knowing how long a record must be requires knowing the integral scale — and the integral scale is the statistic the record determines worst. Its relative scatter is 37 per cent after a hundred integral scales, 17 after a thousand and ten after three thousand, where the variance’s is 14, 4 and 2.5. A record long enough to pin the variance to a per cent leaves the integral scale uncertain by several.

The circle is broken in practice by not needing the integral scale precisely. An error estimate is itself an estimate, and knowing it to ten per cent is usually enough: the record is twice as long as needed or half as long, and the conclusion rarely turns on which. What the circle does forbid is quoting the integral scale of a flow from a single short record as if it were a property of the flow, and then using it to decide that the record was long enough.

How long real records are #

The formula turns familiar record lengths into numbers of independent values, and some of them are uncomfortable.

In a laboratory wind tunnel the integral time scale is of order ten milliseconds, and a minute of hot-wire record spans six thousand of them. That is three thousand independent values, a mean known to under two per cent of the fluctuation, and a variance to under three. Laboratory statistics are well converged because the flows are small and fast, not because the experimenters sampled quickly.

In the atmospheric surface layer the integral time scale is tens of seconds to a couple of minutes, depending on height and stability, and the conventional averaging period for a flux measurement is thirty minutes. That is between fifteen and ninety integral scales: a mean known only to between fifteen and thirty-five per cent of the fluctuation, and a flux — which is the mean of a product of two fluctuating quantities, and converges like a variance — to similar accuracy. The thirty-minute convention is a compromise with the other problem in the paragraph on stationarity below: a longer average would converge better and would span the change of the day’s weather.

A direct numerical simulation of turbulence, run for twenty large-eddy turnover times, spans of order twenty to forty integral scales, which is ten to twenty independent values of any large-scale statistic. Its small- scale statistics, which live at the Kolmogorov scale and are averaged over the whole of a large domain at every instant, are far better converged than its large-scale ones. A simulation’s spectrum is known well at high wavenumber and poorly at the lowest, which is the opposite of what its resolution suggests, and the range a real Reynolds number does not have has to be read with that in mind at its left-hand end.

Reading the error off the record itself #

The circle in the previous section has a practical way out that does not need the integral scale at all. Cut the record into blocks of equal length, average each block, and compute the scatter of the block averages. While the blocks are shorter than an integral scale, neighbouring block averages are correlated and the scatter understates the error of the whole record’s mean. Once the blocks are several integral scales long, the block averages are nearly independent, and the scatter of the block means, divided by the square root of the number of blocks, becomes the error of the overall mean — and stops changing as the blocks lengthen further.

The same curve also detects the case no formula here covers. If the flow drifts during the record — a slow change of mean speed as a tunnel warms, or of stability as the sun moves — the scatter of the block means keeps rising as the blocks lengthen instead of levelling off, because the longest blocks are sampling different flows. A curve with no plateau is therefore either a record too short or a flow that is not steady, and the two can be told apart by repeating the record, which is the one remedy neither formula nor analysis can replace.

The plateau of that curve is the answer, read directly from the data. It needs no model of the correlation, it works for any stationary record, and its failure is informative: a record too short to show a plateau is a record too short for its mean to be trusted. The plateau’s height is also, turned round, an estimate of the integral scale — the spectrum at zero frequency — which is the estimator the last section of this essay proposes to compare with the others.

What the higher moments cost #

The mean and the variance converge as the square root of the number of integral scales with small coefficients. Higher moments converge at the same rate with larger ones, because they are dominated by rarer events. The flatness of a Gaussian record is known to about four per cent after a thousand integral scales; the flatness of a real turbulent velocity derivative, whose distribution has much wider tails, needs far more, because its fourth moment is carried by events that occur a few times per hundred integral scales.

That is the quantitative reason the exponents that stop being thirds are hard to measure at high order: a sixth-order structure function at a given separation is set by events that are rare even in a long record, and the error bars on anomalous exponents grow with the order for this reason as much as for any other. It is also why a spectrum and the loads it implies diverge at the extremes: a spectrum converges like a variance, and a peak does not converge like anything so polite.

What the picture cannot show #

The signal is Gaussian and its correlation is exponential, and both are choices. A turbulent velocity is close to Gaussian and its correlation is close to exponential at long lags, so the mean and variance results carry over closely; a velocity derivative is far from Gaussian, and its statistics converge more slowly than anything drawn. The exponential shape also sets the coefficient in the variance-of-variance result, which for a correlation with a different shape changes by a factor of order one.

The records are stationary. A wind-tunnel flow is, closely; the atmosphere is not, and its statistics drift with the time of day, the passage of weather and the growth of the boundary layer. A record long enough to converge in a stationary flow may be long enough to span a change in the flow itself, and then no length converges. An hour for every tenfold is the version of this that concerns prediction rather than measurement.

Where the model stops #

One point. Everything here is a time record at one location. A spatial average over a field — a particle-image velocimetry frame, a simulation snapshot — is a different sampling scheme, counted in integral length scales instead, and a field and a record of the same total size can hold very different numbers of independent values.

Linear estimators. The mean, the variance and the flatness are computed the obvious way. Estimators that use the record’s structure — fitting a model spectrum, for instance — can do better on some statistics at the price of assuming the model.

Intermittency not included. A flow that is turbulent only some of the time has a record whose statistics depend on how many turbulent episodes it contains, and that number converges more slowly than anything here.

The convention the numbers depend on #

The integral time scale is the integral of the autocorrelation from zero to infinity for the process, and from zero to the first zero crossing for a record — two definitions that agree only for long records. Variances are computed about the record’s own mean, dividing by the number of samples. Scatter is the standard deviation across records of a statistic, as a fraction of the statistic’s true value, except for the mean, whose scatter is in units of the signal’s standard deviation because its true value is zero.

Who found it, and when #

G. I. Taylor defined the integral scale and gave the variance of a time average in terms of it in the 1920s and 1930s, in the same work that produced the single-parcel dispersion law. Lumley and Panofsky’s 1964 account of atmospheric turbulence made the record-length formula a working tool, and Lenschow, Mann and Kristensen gave the systematic treatment of how long is long enough for the moments of atmospheric turbulence in 1994, including the higher moments’ larger coefficients. The zero-sum identity is older than any of them; it is what makes every stopping convention for the integral scale necessary.

Still open: a better estimate of the scale #

The first-zero convention is biased low and noisy. Two alternatives are standard: fit an exponential, or a model spectrum, to the autocorrelation near zero lag and read the integral scale off the fit; or estimate it from the spectrum’s value at zero frequency, which for a stationary process is proportional to the integral scale and which the variance of block means estimates directly. The second is the same formula as the scatter of the mean, turned round.

The calculation that would follow compares the three estimators on these records — first zero, exponential fit, block-mean variance — for bias and scatter at every length, and asks which is the best of them for a record of a given length. Beside it is the version for a field rather than a record: how many independent values a simulation snapshot of given size holds, counted in integral length scales, and whether the exact laws’ constants reported from single snapshots carry the error bars their snapshot size implies.