One pass reads the road's slope as well as its spring; a second speed only doubles the road

A rough road rings the suspension well enough to read, over four hundred metres put a two-slit scroll camera on a body sprung at 1.3 hertz and damped 0.3, and drove it along a level road with nothing on it but the ordinary unevenness of a few millimetres that every road surface carries. The body rang with the road, the camera pitched with the body, and the rows of a line of posts twelve metres out wandered by a sixth of a pixel. Their spectrum along the road carried two things added together: the scroll’s own answer to the road, which a rigid vehicle would leave as well, and a broad swelling where the spring rings. Fitted against both, 400 metres of posts read to a tenth of a pixel gave the frequency to two hundredths of a hertz and the damping to about a tenth of itself.

The fit took one thing from outside the pictures: the road spectrum’s slope. Road surveys find that the power in each band of a road’s unevenness falls as the square of the wavenumber, and the fit was told so. It was not told the road’s level, but a slope that is wrong tilts the predicted spectrum against the measured one, and the earlier essay left open how much a wrong slope costs.

It also proposed a way round it. A road’s unevenness is fixed in the road, and so is the scroll’s answer to it. The spring rings at a frequency in time, which at a stated speed is a wavenumber along the road — the speed over the frequency — so the swelling moves along the spectrum when the speed changes. Two passes at two speeds would see the scroll’s part in one place and the spring’s part in two. Fitted together, they might read the spring more sharply than one pass, and they might read the road from the part that does not move, so that nobody needs to be told its slope.

Both halves of that hope can be measured, and neither survives intact. The second speed narrows the reading no more than a second pass at the same speed, or one pass twice as long. And the road’s slope does not need a second speed to be read: one pass reads it.

The swelling moves and the rest stays put #

The figure is the proposal drawn. The vehicle is the earlier essay’s, wheels 2.7 metres apart, posts every quarter metre twelve metres out, and the same 400 metres of a good road are driven twice, at eight metres a second and at fifteen. The dashed curve is a rigid vehicle’s rows at either speed, and it is one curve: a rigid body’s pitch is set by the road under its wheels, and so are the rows, wherever along the road the posts stand and however fast they went by. Its dip near 0.3 cycles a metre is the scroll’s own blind spot — a length of unevenness whose chord over the wheelbase and whose chord between the slits’ two moments happen to cancel for posts at this depth.

The spring adds a swelling to that curve at the speed over its frequency. At eight metres a second a body sprung at 1.3 hertz rings once every 6.2 metres of road, 0.163 cycles a metre; at fifteen, once every 11.5 metres, 0.087. The two solid curves are the same spring at the two speeds and they are very different: at eight the swelling sits beside the scroll’s own peak and broadens it; at fifteen it stands alone at the long-wave end and towers over the rigid curve, and near 0.22 cycles a metre the faster pass’s rows fall to a twentieth of the rigid curve, where the body’s swing against a quick road cancels the scroll’s own answer, before the two curves meet again among the short ripples.

The fit is the earlier essay’s, extended to more than one pass. It knows the scroll’s gains, the wheelbase and each pass’s speed. It blurs its own prediction by the analysis window’s response exactly as the window blurs the measurement, because a fit that does not do that reads the blur as damping. And it chooses one frequency, one damping and one level of unevenness for the road, shared by both passes, that make the two measured spectra together most likely. For this road it reads 1.288 hertz and a damping of 0.280. The dots follow the curves through both swellings and through the blind spot between them.

So the passes do what the proposal said: the scroll’s part stays and the spring’s part moves. What remains is whether that is worth anything a second pass at the same speed would not also give.

A second speed buys what a second pass buys #

Twenty good roads, each a different draw of the same spectrum, are driven five ways. The baseline is one 400-metre pass at ten metres a second, which reads the frequency to ±0.027 hertz and the damping to ±0.044 — the earlier essay’s reading on a fresh set of roads. Every other way reads 800 metres of rows in all: two passes of 400 at eight and fifteen, two at six and twenty, two at ten and ten, or one pass of 800 at ten.

Every one of them narrows the reading by about the square root of two, which is what reading twice the road should do when the reading is set by how many windows of it are averaged. The two passes at the same speed read ±0.015 hertz and ±0.031 for the damping; one pass of 800 metres, ±0.018 and ±0.021. Those two are the same measurement cut differently — eight hundred metres of rows at one speed — and their difference is the size of the noise in a spread drawn from twenty roads, which is about a sixth of itself.

The speeds do something, but not what was hoped. They share the gain out between the two readings. Eight and fifteen read the frequency to ±0.025 and the damping to ±0.024; six and twenty, ±0.033 and ±0.017. A fast pass puts the swelling at the long-wave end of the spectrum, where the window resolves only a few points across it — at twenty metres a second the spring rings at 0.065 cycles a metre, two points above the band’s lower edge — and the swelling’s position is drawn there more coarsely than at ten. Its width is another matter. A slow pass draws the swelling broad and a fast one draws it narrow, and two very different widths of the same damping, both seen, fix the damping a little better than two looks at one width.

None of this is a separation of the spring from the road. A second pass at another speed is a second pass, and a fleet that drives one stretch twice could choose its speeds by which of the two readings it wants sharper — a slow and a fast pass for a damper being watched, two passes at a middling speed for the frequency — but it gets roughly the same total either way. A third ray is worth what its picture is worth found a third camera on a point adding what its own picture could add and nothing for its being a third; a second speed is the same kind of witness.

A wrong slope tilts the whole reading #

That leaves the second half of the hope: that two speeds free the reading from the road’s slope. Before asking whether they do, it is worth knowing how badly the slope matters.

The roads here are exactly the standard, a slope of 2, and the fit is told slopes from 1.6 to 2.5. Told 2, it reads the damping as 0.305. Told 1.6, it reads 0.465 — half again too much. Told 2.5, it reads 0.186, nearly half too little. The frequency moves less but in step: 1.348 hertz when the slope is taken as 1.6, 1.270 when it is taken as 2.5. Every tenth of slope moves the damping by about a tenth of itself, which for this reading is far more than its own scatter of ±0.021.

The mechanism is easy to see on the spectrum. A road spectrum is a straight line on logarithmic axes, and its slope says how much more power the long swells carry than the short ripples. A fit told the slope is gentler than it is expects too little power at long wavelengths; the measured rows have more there, and the only part of the model free to supply it is the spring’s swelling, which sits at the long-wave end. So the fit widens the swelling to cover the deficit, and a wider swelling is a heavier damping. Told the slope is steeper than it is, the fit expects too much power at long wavelengths and narrows the swelling to give some back.

Roads do differ. Surveys find slopes from about one and a half to three, the long swells varying most between roads, and a road with a slope of 2.5 is an ordinary country road, not an exotic one. A reading that is right only on the standard road is a reading of the assumption as much as of the spring, which is the case the proportion is the assumption made of a rectangle read from a photograph: the answer was a function of where the picture’s centre was taken to be, and the photograph had never been asked.

One pass is asked for its own slope #

The light point in the figure above is the answer to that, and it does not involve a second speed. It is the same single 800-metre passes, read by a fit that is not told the slope at all and chooses it alongside the frequency, the damping and the level. It reads the damping as 0.290 ± 0.033, the frequency as 1.299 ± 0.019, and the slope as 2.05 ± 0.09.

On roads that are not the standard the single pass does the same. Twenty roads of slope 1.6 read 1.65 ± 0.08; twenty of slope 2, 2.05 ± 0.09; twenty of 2.5, 2.57 ± 0.10. On all three the frequency reads 1.299 or 1.300 to within ±0.02 and the damping 0.288 to 0.291 to within ±0.034. The slope is read high by about five hundredths on every road — a small, steady lean towards a steeper road, which carries the damping a hundredth low with it.

The reason one pass can do this is the reason the earlier essay could read the spring from the spectrum’s shape rather than its peak. Three different shapes are added in the rows’ spectrum, and they are not alike. The road’s unevenness is a straight line on logarithmic axes. The scroll’s own answer to the road is a hump at six-metre unevenness with a blind spot past it, fixed by the posts’ depth and the wheelbase. The spring’s swelling sits at the speed over its frequency with a width set by its damping. A straight line cannot imitate a hump with a notch, nor a swelling, and the fit is not asked to tell any of them from the others by position alone; it has their whole shapes over a decade of wavenumber. The second speed would have shown the swelling moving against a still road, but the road’s shape was never what was hidden. Only its tilt was assumed, and the tilt of a straight line is something a single spectrum of more than a decade draws plainly.

What freeing the slope costs is a little of the damping’s precision: ±0.033 against ±0.021 with the slope known and right. The slope and the damping both change how the spectrum leans across the swelling, so a fit that has to find both can trade a little of one for the other. The frequency, which is the swelling’s position rather than its lean, costs nothing measurable. A reading with ±0.033 that is right on every road is worth more than one with ±0.021 that is right only on the road it was told about.

Two speeds told a wrong slope pull the damping most of the way back #

The two speeds are not useless against a wrong slope. They are a weaker remedy for it than freeing the slope. On a road of slope 1.6 told 2, a single pass reads the damping as 0.203; two passes at eight and fifteen, told the same wrong slope, read 0.288. On a road of 2.5 told 2, one pass reads 0.508 and two speeds 0.336. The second speed has taken nearly nine tenths of the error away on the gentle road and more than four fifths on the steep one.

It does so for the reason the proposal gave, even though the proposal’s conclusion did not follow. A wrong slope tilts the model against the measured rows, and the fit answers with a wrong swelling. With two speeds the swelling sits in two places, and a swelling widened to cover a long-wave deficit at fifteen metres a second, where it sits at the long-wave end, is a swelling at eight metres a second that sits in the middle of the spectrum, where no deficit needs covering. The two passes disagree about what the wrong damping should be, and their compromise lies nearer the truth. The price is the frequency, which the compromise moves: 1.340 hertz on the gentle road and 1.262 on the steep one, three per cent out either way, where the single pass with the wrong slope had read 1.274 and 1.355.

Left free, the slope removes the error entirely at either. A single pass reads 0.291 on the gentle road and 0.288 on the steep one, with the slope read as 1.65 and 2.57; two speeds read 0.297 and 0.296, with the slope 1.61 and 2.51. The second speed, given a free slope, reads the slope a little closer — its small lean towards steeper roads halves — and adds nothing else to a reading that was already right.

So the order of remedies is plain. Telling the fit the survey’s standard slope and trusting it is right on the standard road and wrong by up to two thirds of the damping on an ordinary one. A second speed under the same wrong assumption pulls the damping most of the way back and leaves the frequency out by a few per cent. Asking the single pass for the slope removes both errors at the cost of a third more spread in the damping. A second pass with the slope free adds what a second pass always adds.

What the second pass was worth #

The proposal rested on a true picture: the scroll’s part of the spectrum is fixed in the road and the spring’s part moves with the speed. What it took from that picture was that the road’s shape could be read only from the part that does not move, and that was the step that does not hold. The road’s shape was never as hidden as the proposal assumed. It is a straight line on logarithmic axes, two numbers, and the spring and the scroll are curves that no straight line resembles. One spectrum, drawn over a decade of wavenumber, already sets all three against one another.

The same kind of thing was found by the earlier essay about the spectrum’s peak, which was not the spring’s frequency because the scroll’s own hump was added to it: the reading had to come from the whole shape. Here the whole shape turns out to carry one more number than it was asked for. A scroll camera rings with its vehicle’s suspension read the spring from one step and needed nothing about the road but the step’s height, which it read too. A vehicle’s pitch lags the road by its wheelbase found the rows reading the chord between the wheels rather than the road; the road’s spectrum enters the rows only through that chord, and the scroll’s gain at each wavenumber is the chord’s answer to it. All of it is the point a scroll is a camera that moves started from, carried a step further: the rows record the motion of the camera, and the motion here records the road and the springs at once, and each leaves its own shape.

For a fleet the practical reading is short. Drive the stretch once at a steady middling speed; read the spring and the road’s slope together; drive it again whenever there is time, at whatever speed the traffic allows, because any second pass halves the variance. The road’s slope comes as a by-product, and a by-product of some use: a survey of road roughness is ordinarily made by a dedicated vehicle with a laser profiler, and every scroll camera on an ordinary vehicle reads the long-wave half of it from its own posts. A scroll can be asked its own radius found a scroll answering a question about its track that nobody had thought to ask it; this is another.

What the fit still takes from outside the pictures #

The road’s spectrum is one straight line. The fit now chooses the line’s slope, but a line is still what it chooses. Real roads bend: their long swells and their short ripples follow slopes that differ, and surveys sometimes describe them with two. A road bent near the spring’s swelling would give the fit a third shape it does not have a model for, and the slope it reads would be the slope of the band the swelling overlaps.

Both passes are over the same road. The fit shares the road’s level between passes. Two passes in different lanes, a metre and a half apart, see unevenness that is correlated over long wavelengths and independent over short ones; sharing the level would then be wrong at the ripple end, which is where the scroll’s own hump is.

The speeds are steady and known. A speed that wanders along a pass smears the swelling along the spectrum, as a map along, and a picture across said of a scroll’s paper; the fit assumes each pass’s speed is one number.

The posts are at one depth. The scroll’s gain, and its blind spot near 0.3 cycles a metre, belong to posts twelve metres out. A roadside at mixed depths has a blind spot at a different place for each, and the rows of a fence and a hedge do not share a spectrum.

Still open: whether the rows see a body that bounces as well as pitches #

Every reading so far has treated the body as one oscillator in pitch. A vehicle on four springs has at least two: it pitches about its middle and bounces up and down as a whole, at a somewhat different frequency, and a road’s unevenness excites both. The scroll’s rows read the camera’s pitch against the chord between the slits’ two moments; a bounce changes the eye’s height rather than its pitch, and an eye whose height wanders draws a post’s top and foot at rows that both shift, which the earlier readings of a scroll of a climbing road found the rows sensitive to only through the height climbed between the two moments.

The measurement that settles it gives the body two modes of stated frequency and damping, bounce and pitch, with a camera mounted at some distance ahead of the body’s centre so that both move its eye, drives it over a rough road, and asks whether the rows’ spectrum carries two swellings or one — whether a fit of one oscillator lands between them, as the earlier essay guessed, or on the pitch alone — and whether a fit of two reads both frequencies from a single pass, the way one pass turned out to read the road’s slope.