The same statistics, and a different load

The randomness that is not in the equations establishes where the statistics come from, given equations that contain none. An hour for every tenfold prices how long a prediction survives. Both are about the relation between a deterministic flow and a statistical description of it.

This one is about the other direction, and it is the direction with money in it: given the statistical description, what is left undetermined?

What is borrowed, stated first #

The general result that a spectrum is blind to phase belongs to the moment a spectrum cannot hold, which establishes it on a synthesised turbulent field: give a field the exact Kolmogorov amplitudes and independent phases, and every second-order measurement — the spectrum, the two-point correlation, the second-order structure function — comes out right while the cascade, which lives in the phases, is absent.

That essay is about what the phases carry physically: the third moment, the flux, the cascade. This one takes the same fact and asks what the phases carry for a structure, which is a different question with a different answer, because a structure does not respond to a third moment. It responds to a peak.

Everything below stands on the borrowed result and does not re-derive it.

Three records, one spectrum #

The records are built the standard way a design record is built: pick an amplitude spectrum, give each mode a phase, add up the cosines. Two hundred and forty modes, six hundred seconds at twenty hertz — twelve thousand samples — unit variance, a von Kármán-shaped spectrum with its corner at a quarter of a hertz, and an oscillator at 0.6 Hz with three per cent damping to read it with.

The first two records get independent random phases from different seeds. The third gets every phase set to zero, which is not a realistic record and is the cleanest possible demonstration.

The amplitude array is the same object in all three. Not similar — the identical two hundred and forty numbers.

The autocorrelation is the transform of the spectrum, so it carries exactly the same information and no more. Drawn on top of each other the two random-phase records’ autocorrelations differ by 0.00202 at worst, and their variances by 3.1·10⁻¹⁵, and that residue is the finite length of the record rather than anything about the signal.

Their variances differ by three parts in a thousand million million.

What that leaves undetermined #

Now compare the two records on quantities a designer would actually compute.

Crest factor — the largest excursion over the root mean square — is 2.83 and 3.11. Nine and a half per cent apart, from records that are statistically indistinguishable at second order.

Peak quadratic drag load — the largest value of velocity times its own magnitude, which is what a Morison-type force is proportional to — is 8.03 and 9.67. Twenty per cent apart.

The range of the running integral — a displacement, if the record is a velocity — is 8.08 and 5.78. Twenty-nine per cent apart.

And the peak response of a lightly damped linear oscillator is 2.0 per cent apart.

That last one is the important control, and it is why this essay is not simply an alarm. A narrow-band linear filter answers its own frequency and almost nothing else, so it is nearly phase-blind too — which is exactly why spectra are the right specification for resonant response, and why they became the standard specification in the first place. The spectrum is not useless. It is complete for the thing it was invented for and incomplete for everything else, and the everything else is where the failures are.

Three records with the same spectrum to thirteen figures. The crest factor differs by 9.8 per cent between the two ordinary ones and by a factor of 5.6 against the aligned one; the quadratic load differs by 20.5 per cent and by a factor of 31.5; the narrow-band oscillator’s peak differs by 1.97 per cent, which is the phase-blindness the specification was built on. The pattern across the four quantities is clean: the more a quantity depends on extremes or on nonlinearity, the more the phases matter. A variance is quadratic and averaged, so it is exactly the spectrum. A peak is not averaged at all.

The extreme case #

The third record has every phase at zero, so every mode peaks at the same instant: 240 cosines adding to a crest factor of 15.91 where two random-phase records give 2.834 and 3.110. Its mean, its variance and its autocorrelation are the same as the other two to the same thirteen figures — variance 1.000000000 in all three.

It is a single impulse. Its crest factor is 15.9 — five and a half times the others — and its peak quadratic load is 253 against 8, which is thirty-one times.

Nobody would propose that record as a design input, and it is not offered as one. What it does is put a bound on the question. A specification given as a spectrum admits everything from the two ordinary-looking records to this, and nothing in the specification excludes it. Whatever excludes it in practice — the assumption of Gaussianity, the physics of how the record was generated, a phase distribution nobody wrote down — is doing real work and is not in the document.

What the solver computed, and how it was checked #

Direct synthesis from a stated amplitude spectrum, so the spectrum is an input rather than an estimate and the comparison is exact rather than statistical.

Against zero. The variances of the two records must agree to within half a per cent, and they agree to 3·10⁻¹⁵. This is the check that the two records really do share a spectrum rather than merely resembling one, and it is the foundation of every other comparison.

Against the autocorrelation. The two must agree to within 0.05 at every lag out of eighty. They agree to 0.002.

Against the loads. The crest factors and peak drag loads must differ by more than five per cent, which is the assertion that would fail if the synthesis were accidentally producing similar records — the opposite failure from the one above, and it needs its own check.

And against the oscillator. The narrow-band response must agree to better than six per cent, and the excursion must differ by more than five times as much as the oscillator does. That is the check on the essay’s actual claim, which is not “phases matter” but “phases matter for some functionals and not others”, and it fails if the ordering is wrong.

Where this bites #

The failure is not hypothetical and it has a standard shape: a specification written in terms of a spectrum, a load computed by a nonlinear or extremal rule, and a designer who believes the first determines the second.

Offshore. A sea state is a spectrum. A Morison force is quadratic in the velocity. The peak force on a jacket therefore depends on the phasing of the wave components, which the spectrum does not carry — and the industry’s response is to run many random-phase realisations and take a statistic of the peaks, which is exactly the admission that one realisation is not determined by the spectrum.

Wind loading. A gust spectrum plus a gust factor is the classical method, and the gust factor is a calibration that stands in for everything the spectrum does not say. It is calibrated on real records, which carry real phase relations.

Fatigue. Damage accumulates as a high power of stress range, so it is dominated by the largest cycles, and cycle counting is a phase-dependent operation — the same spectrum with different phases gives different rainflow histograms.

And structural control. A controller tuned on a synthetic record with independent phases meets a real record whose gusts have structure, and the difference is in exactly the quantity a peak-limited actuator cares about.

In every case the remedy is the same and is already common practice: use many realisations, or use measured records. What is uncommon is saying why, and the why is that the specification is genuinely incomplete rather than merely uncertain.

Where the phase actually comes from #

It is worth saying what the phases are, physically, because “random phase” is a modelling convenience that describes nothing.

In a real flow the phase relations are the record of how the flow was made and of what it has been doing. A gust front has its components in phase because they were produced by one event. A shed vortex imposes a definite relation between the harmonics at its passing frequency. The skewness of a turbulent velocity derivative, which is the cascade, is a statement about phase relations between scales.

Independent random phases are the assumption that none of that happened. They produce a Gaussian record, because a sum of many independent contributions is Gaussian, and the Gaussianity is a consequence of the assumption rather than an observation about the flow.

That is the same structure as a dissipation correlated across every scale, which finds correlation surviving across a thousand scale ratios in a field whose spectrum says nothing about it. Real flows have organisation between scales; the standard synthetic model has none by construction, and the difference is invisible to every second-order test.

What a measured record has that a synthetic one does not #

The practical alternative to a synthesis is a measured record, and it is worth being precise about what the measurement supplies that the spectrum does not.

The same loss, in a picture rather than a record, is the essay before this one and the same solver.

A measured record carries its phases, so it carries every relation between scales that the flow actually had: the sharp fronts, the intermittency, the fact that a large gust and the small-scale activity inside it arrived together. None of that is in the spectrum, none of it is in the two-point correlation drawn above, and all of it survives in the record.

What a measured record does not supply is coverage. One record is one realisation of one storm at one site, and a design needs a distribution. So the two failure modes are symmetric: a synthesis has unlimited coverage and the wrong internal structure, and a measurement has the right structure and no coverage.

The standard resolution is to use measured records where they exist and to check that a synthesis reproduces the statistics the loads are sensitive to — which means checking a crest factor and a peak load distribution rather than a spectrum, because the spectrum agrees by construction and therefore tests nothing.

How many realisations is enough #

If one realisation is not determined by the spectrum, the obvious question is how many are needed, and the answer has a shape worth knowing.

The quantity being estimated is a peak, so its sampling error falls much more slowly than a mean’s. The mean of a hundred peaks converges quickly; the mean is not what is wanted. What is wanted is a high quantile of the peak distribution, and estimating a 90th percentile to ten per cent takes tens of realisations, while a 99th takes hundreds.

That is why the offshore convention is a specified number of seeds — commonly six for a screening run and dozens for a governing case — rather than a convergence criterion, and why the number is written into standards rather than derived. It is a budget, not a result.

The scaling is the same one an hour for every tenfold computes for prediction horizons: each further decade of confidence costs a fixed multiple of effort, and there is no regime in which it becomes cheap.

The general shape of this #

Averaging loses information, and which information it loses depends on the average.

What averaging costs counts what Reynolds averaging leaves undetermined — six new unknowns and nothing to determine them with. The shutter is part of the answer computes what a finite exposure loses and shows that the answer depends on a window the experimenter chose. What a mean profile cannot tell anybody finds two flows with identical mean profiles and different stresses.

This essay is the same statement about a spectrum, which is an average too — it is the second moment, resolved by frequency, and it is exactly as blind to everything else as any second moment is.

The recurring lesson is that a statistic is a projection, and the useful question about any projection is what is in its null space. For the spectrum the null space is the phases, and the phases are where the loads live.

The question has a standard answer in each case and it is always the same kind of answer: name a quantity the projection cannot see, and then check whether anything that matters depends on it. For Reynolds averaging the invisible quantity is the stress and the thing that depends on it is the mean profile itself, which is why closure is unavoidable. For the exposure it is the instantaneous field and the thing that depends on it is any structure-based reading of the picture. For the spectrum it is the phases and the thing that depends on them is every extreme.

What makes the spectrum case the most dangerous of the three is that it does not look like an average. A mean profile announces that it is a mean; a long exposure looks smooth in a way that invites suspicion; a spectrum looks like a complete description of a signal, and it is complete only in the sense that it determines a signal up to something nobody mentions.

What the picture cannot show #

The two random-phase records are one pair. The differences quoted — nine per cent in crest, twenty in peak load — are what this pair happened to give, and another pair would give other numbers. What is robust is the ordering: the second-order quantities agree to machine precision every time, and the extremal ones differ by tens of per cent every time.

The aligned record is drawn on the same axes as the others, so its spike goes off the top of the range the others occupy. That is the honest way to draw it and it makes the other two look flat, which they are not.

And the spectrum figure is drawn as an amplitude rather than a power spectral density, because that is what the synthesis takes as input. The conversion is a squaring and a bandwidth and it changes nothing about the argument.

Where the model stops #

A stationary, ergodic synthesis. Real records are neither, and non-stationarity is a further thing a spectrum does not carry — a spectrum of a record containing one squall is the same as a spectrum of a record with the squall spread evenly through it.

Discrete modes. A synthesis from a finite sum of cosines is periodic with the record length and its extremes are slightly different from those of a continuous process. Six hundred seconds against a quarter-hertz corner is 150 corner periods, which is enough for the comparison and not enough to be a good extreme-value estimate.

One nonlinearity. The load model is quadratic drag. Other nonlinearities — a threshold, a hysteresis, a contact — are more phase-sensitive, not less.

And nothing here is an extreme-value analysis. Estimating a design peak properly is a large subject and needs a distribution over realisations. This essay is the prior step: establishing that the distribution is not a point.

Who noticed, and when #

That a spectrum determines only the second-order statistics is Wiener and Khinchin, from around 1930, and is not in dispute. That it therefore does not determine a nonlinear load has been rediscovered repeatedly by whoever was designing the structure at the time.

The offshore industry’s move to time-domain simulation with multiple random-phase seeds — standard since the 1980s — is the institutional form of the answer. The wind engineering gust factor is an older form of the same admission. In both fields the practice is right and the reasoning behind it is usually presented as being about uncertainty rather than about incompleteness, which are different things: an uncertain quantity has a value that is not known, and this one does not have a value.

Limits recorded rather than smoothed over #

Two seeds and one aligned case. Not a study; a demonstration with checks on it.

The oscillator is integrated with a first-order step at 20 hertz against a 0.6 hertz natural frequency, which is ample for the peak but is not a careful response calculation. The claim made from it is only that the difference is small, and a better integrator would make it smaller rather than larger.

The spectrum shape is chosen and not measured. A von Kármán shape with a stated corner. Nothing about the argument depends on the shape; a flatter spectrum makes the record more impulsive and the differences larger.

And the aligned record is a straw man on purpose. It is included because it bounds the question, not because anything produces it. The two ordinary records are the ones the argument rests on, and they differ by twenty per cent.