The echo hidden in random noise

The plate that carries two kinds of sound ends by asking whether a plate could be read from its own noise — whether the random vibrations it carries all the time could be made to reveal the waves it would carry if struck. The question has a definite answer, and the answer is yes. It was found in ultrasonics in 2001, applied to the Earth in 2005, and it turned seismometers into instruments that need no earthquakes.

The principle is easiest to see for surface waves in the ground, the waves that the wave a surface is enough to hold shows are trapped along the boundary of a solid, travelling at about three kilometres a second in the Earth’s crust and spreading only in two dimensions, so that they carry far. The Earth is never still at those wavelengths. Ocean waves striking coasts, and the interference of ocean waves travelling in opposite directions, press on the sea floor and launch surface waves with periods between about five and twenty seconds, the microseisms. Every seismometer on every continent records them continuously. They come from storms thousands of kilometres away, from every direction, and they look like noise.

Noise from everywhere, and the part that counts #

Consider two seismometers, A and B, sixty kilometres apart, with noise sources scattered round them in every direction. Each source sends waves to both stations, and its waves reach the two at different times, by an amount that depends on where the source is.

A source directly in line with the stations, beyond A, sends waves that pass A and then travel the full sixty kilometres to B: they reach B twenty seconds after A. A source off to the side, equidistant from both, sends waves that reach them simultaneously. Sources in between produce delays in between. The key observation is how the delay changes with the source’s position. Near the line through the stations, moving a source sideways barely changes the delay, because the delay is at its maximum there. Off to the side, a small move changes the delay a great deal.

Correlating the two records means multiplying them at each pair of times separated by a given lag and averaging. A single source produces a correlation that is the noise’s own short wavelet, with a period of ten seconds, placed at that source’s delay. Many sources produce many wavelets at many delays. Those from the sides, whose delays vary rapidly from one to the next, have wavelets at all phases, and they cancel. Those near the line all have nearly the same delay, the maximum, and their wavelets add up. This is the method of stationary phase, the same reasoning by which only the zones of a wavefront near the straight path contribute to what arrives at a point in Huygens’s construction.

A pulse that comes out of the average #

The figure shows the result for sources spread evenly round a ring.

The averaged correlation has two clear pulses, at plus and minus twenty seconds, and comparatively little elsewhere. The pulse at plus twenty seconds is contributed by sources beyond A, whose waves pass A first and B twenty seconds later; the one at minus twenty by sources beyond B. Together they are what a seismometer at B would record if a sharp source fired at A, and what one at A would record from a source at B — the wave travelling between the two stations, in both directions. In the language of wave theory it is the Green’s function between A and B, the response at one point to an impulse at the other.

That is remarkable for two reasons. No wave ever travelled from A to B in the experiment; every source was elsewhere. And the noise at each station, taken alone, contained no sign of the other station’s existence. The information about the ground between them was in the relation between the two records, in their correlation, and only in its average over many sources.

The pulse that grows out of the hours #

The averaging is not free. Real noise is random, and a single stretch of it contains chance correlations that have nothing to do with the stations.

In one window of about a quarter of an hour, with thirty sources firing random noise from random directions, the correlation is ragged and the pulses at twenty seconds are not distinguishable from the background. Averaged over ten windows they stand out; over a hundred they dominate. The background is made of chance correlations, which are random in sign and shrink as the square root of the number of windows averaged; the pulse is the same in every window and does not shrink. So the clarity of the result grows as the square root of the recording time. Seismologists typically stack a month of continuous records for each pair of stations, longer for long periods, and process the records first to suppress the large transient signals of earthquakes, which would otherwise dominate the average.

The same arithmetic governs every measurement of a weak signal in noise, and its appearance here is an example of the correlation that survives what the phase does not. Hanbury Brown and Twiss correlated the intensity fluctuations of starlight at two telescopes and recovered the size of a star from noise whose phase was entirely random; ambient-noise seismology correlates the motion itself, and recovers the medium between the stations. Both work because the correlation of two random signals carries information that neither signal carries alone.

When the noise comes from one side #

The derivation required noise from all directions, or at least from both ends of the line between the stations. Real noise is not so obliging.

The microseisms are strongest from the oceans, and their sources move with the storms. If all the noise comes from A’s side, only the pulse at plus twenty seconds survives: waves always pass A before B, and the correlation contains travel in one direction only. That is still enough to measure the travel time. If the noise comes from a single coast off the line, the coherent contribution comes from the part of that coast nearest the line, and the waves cross the stations at a slant: the delay is shorter than the true travel time, and a wave speed read from it is too fast. In the figure, an arc forty degrees off the line gives a speed four per cent too high. Correcting for the directional distribution of the noise, or averaging over seasons in which the storms move round, is part of the practical method.

A useful check is symmetry. If the two pulses, causal and acausal, have equal amplitude, the noise was balanced; if one is much stronger, the noise came preferentially from one end, and the asymmetry measures the direction of the noise sources. Seismologists have used exactly this to map where the ocean’s microseisms come from through the year.

The speed read off pairs of quiet stations #

With many stations, every pair gives a travel time, and together they give a map.

For pairs at different separations, the lag of the correlation pulse grows in proportion to the distance, and the slope of the line is the inverse of the wave speed. Surface waves are dispersive — longer periods reach deeper into the Earth and travel faster where the deeper rock is faster — so measuring the speed at several periods probes several depths. In 2005 Nikolai Shapiro, Michel Campillo and colleagues correlated a month of noise between pairs of the more than sixty broadband stations in southern California and produced maps of surface-wave speed that showed the sediment basins under Los Angeles as slow regions and the mountain ranges as fast ones, in more detail than earthquakes alone had given. Since then the method has been applied across continents, under oceans with sea-floor instruments, and at the scale of a single volcano.

It has a further, unique advantage: it can be repeated. The correlation between two stations can be computed for every day, and if the medium between them changes — if a volcano’s magma chamber swells, if the rock is cracked by an earthquake and slowly heals, if groundwater levels rise — the travel time changes by a tiny amount that the daily correlations can track. Changes of a part in ten thousand in seismic speed have been followed before eruptions and after large earthquakes, with no active source at all.

A clock read from the noise itself #

The correlation’s two pulses carry a check on the instruments as well as information about the ground. If both stations’ clocks are right, the causal and acausal pulses sit at equal and opposite lags, whatever the noise’s direction does to their amplitudes. If one station’s clock drifts, both pulses shift together by the same amount, and the midpoint between them moves away from zero. Sea-floor seismometers, which cannot receive satellite time signals under several kilometres of water and keep time with internal clocks for a year at a time, drift by seconds over a deployment; the noise correlations between them and their neighbours have been used to measure and correct those drifts after recovery, to a precision of tens of milliseconds.

How precisely a lag can be measured is set by the pulse’s own width, which is set by the band of periods in the noise. Sharpness has to be paid for gives the rule: a pulse built from frequencies spread over a band of width Δf can be no shorter than about 1/Δf, and a ten-second microseism band a few hundredths of a hertz wide gives pulses tens of seconds long. Locating their centres to a fraction of a second is possible only because the averaging has removed the background, and the precision improves with the square root of the recording time, like the pulse’s clarity. Changes of a part in ten thousand in seismic speed, over a separation of sixty kilometres, are changes of two milliseconds in a twenty-second travel time — measured routinely now, by stacking long enough.

Guided waves, wherever there is a guide #

The method needs waves that travel between the receivers, and it works best when the waves are guided, so that they carry far without spreading in three dimensions. Surface waves in the crust are one such guide. The ocean has another: the sound channel at about a kilometre’s depth, where the speed of sound is lowest and sound is trapped and carried across ocean basins, the effect the channel with no walls explains. Correlating the ambient noise of shipping and breaking waves between hydrophones in the channel recovers the acoustic travel time between them, which depends on the water’s temperature; ocean acoustic thermometry by noise has been demonstrated across tens of kilometres and proposed across basins.

The atmosphere and the Earth together are a third guide. The tone the whole Earth rings at is the Schumann resonance of the cavity between the ground and the ionosphere, excited by the world’s lightning, a diffuse source of exactly the kind the correlation method needs. And the Earth itself rings at its free oscillations, excited continuously at very low levels by the atmosphere and oceans — its hum, discovered in 1998 — whose correlations between stations on opposite sides of the planet sample the whole interior. Every guide that carries waves, and every medium in which something makes noise, contains its own Green’s function waiting in the correlations.

Why it works: a mirror in time #

There is a deep reason the correlation of noise gives the Green’s function, and it is the same reason that the mirror that sends a wave back to where it began can refocus a wave on its source. Correlating the record at B with the record at A is equivalent to time-reversing the record at A, re-emitting it from A, and letting it propagate to B. A time-reversed wave retraces its path, and the waves that arrived at A from every direction, played backwards, converge on A and then diverge from it — exactly as if A were a source. The part of the diverging wave that reaches B is the Green’s function from A to B. The noise sources all round act as a mirror enclosing the two stations, and the correlation is a computation of what that mirror would do.

The same relation, in statistical physics, is the fluctuation–dissipation theorem: the random fluctuations of a system at rest are tied exactly to how it responds when pushed. Half a kT in a piece of wire shows the electrical version, where the random voltage across a resistor carries its resistance; ambient-noise seismology is a version for waves, in which the random motion of the ground carries its response to an impulse. In both cases a system in equilibrium with its own noise already contains, in the correlations of that noise, the answer to the question of how it would respond to a push.

Other planets, other plates #

The method needs only waves, a medium and diffuse noise. In ultrasonics, where it was first demonstrated by Richard Weaver and Oleg Lobkis in 2001 on an aluminium block, the thermal vibrations of a solid at room temperature are enough to recover its elastic waves between two transducers, which suggests a way to inspect structures continuously without exciting them. In helioseismology, correlations of the Sun’s surface oscillations are used to map flows beneath its surface. On Mars, a single seismometer has used the autocorrelation of its own noise — a station correlated with itself — to find reflections from layers beneath it, the zero-separation case of the same idea. And the head waves that the wave that arrives first by going the long way uses to find the base of the crust can be recovered from noise too, at longer stations’ separations.

A ring of equal sources over a uniform crust #

The sources sit on a ring of fixed radius, all equally strong, and their waves are taken to travel at a single speed in a uniform medium with no attenuation and no geometrical spreading; real microseism sources are distributed over oceans and vary in strength, real surface waves weaken with distance and lose energy, and real crust is heterogeneous. The noise’s autocorrelation is a ten-second wavelet, and the simulated noise is white noise through a single resonant filter, standing in for the broader microseism spectrum. The stacking figure uses thirty sources per window and records of about a quarter of an hour; real processing uses continuous records, whitens their spectra and suppresses earthquakes. Pulse lags are read from the envelope of the correlation; real measurements fit dispersion curves across a range of periods. The derivation’s assumption of a diffuse field — energy flowing equally in all directions — is satisfied only approximately in the Earth, and the errors it causes are a subject of continuing study.

Still open: how diffuse is diffuse enough #

The theory guarantees the Green’s function only for a perfectly diffuse noise field, and the Earth’s noise is not one: it is dominated by sources in particular oceans at particular seasons, partly scattered by the crust’s heterogeneity into a more diffuse field. How far the travel times measured from noise are biased by the noise’s directionality, and how best to correct for it, are open questions that matter most for the smallest changes, the parts in ten thousand used to monitor volcanoes and fault zones. Whether the method’s amplitudes, and not only its travel times, can be trusted — so that noise correlations could measure how strongly the ground shakes between two points, which would be valuable for earthquake hazard — is less settled still.

The habit worth carrying away is to ask what a correlation knows that a signal does not. Two records of random noise from sources everywhere contain, in their average correlation, the wave that would travel between them — pulses at ±20 s for stations 60 km apart on rock carrying waves at 3 km/s, built from the sources in line with the stations, sharpening as the square root of the hours averaged. The Earth’s hum is a source at every seismometer, waiting to be correlated.