Geophones: what instruments on the ground see

Where each method stands on the one-chamber bench

Chamber recovered: mean relative P-wave speed inside the true chamber (air would read −90%); for echo imaging, where the image is brightest; for full-waveform inversion, mean relative S-wave speed. vs scatter: that anomaly over the image's own level around it. One chamber, 6 m, top 12 m down, bare limestone, noise 30 dB below each record's peak; ambient noise with an unlimited record (record length is the question, see below).

Every tomogram here is scored against a truth the method never saw

The ground is composed first, from sourced materials. The waves are computed through it by a solver that knows nothing about any inversion. The inversions see only what the sensors recorded, with noise, on a different grid. Whatever they draw is then compared with the chambers that are really there.

The solver matches exact solutions to within a fraction of a percent

Before it touches a chamber, the forward model is checked where the answer is known exactly, in Mokattam limestone (P 3,300 m/s, S 1,830 m/s).

Travel times catch the chamber's shadow from boreholes, and nothing from the surface

The one-chamber bench: a 6 m air-filled cube, its top 12 m down, in bare limestone. Every record comes from the solver above, with noise 30 dB below each record's peak; first arrivalsfirst break seismicThe first arrival of energy on a seismic record. Its timing, picked on every record, is the data of travel-time tomography. are picked by waveform correlation; the picks are inverted on a 1 m grid by a code that never sees the model. Air is 90% slower than the rock; a first arrival goes around the void instead of through it, so what travel times can see is the delay of the detour.

Crosshole rays through the chamber recover it as a slow patch of -11% (air would be -90%): first arrivals bend around a void, so travel-time tomography sees its shadow, not its emptiness. From the surface, with the direct P picked on 16 stacked blows, the same method recovers +0.0%: in uniform rock no first-arrival ray reaches 12 m.

The chamber's echo, too faint to see on any record, images it from the surface

The same hammer and geophone grids that saw nothing by travel times. Records are made twice, with and without the chamber; the difference is its echo. On a typical record that echo is 21 dB below the waves and 6 dB below the noise. Sent back through the rock by reverse-time migrationreverse-time migration seismicImaging by sending recorded waves backwards through a model of the ground and marking where they meet the waves sent forward from the source. It is the first step of full-waveform inversion., 25 shots and 81 geophones add up where the echo came from. The rock without the chamber is known exactly here, so this is the most favourable case for any waveform method: the first step of full-waveform inversionfull-waveform inversion seismicFitting whole seismic records, not just arrival times, by adjusting a model of the ground; each step is a gradient computed by propagating the misfit backwards in time., taken with a perfect starting model.

The chamber's echo is 21 dB below the records and -6 dB against their noise; migrated back through the known rock it lights up the chamber at 6.0× the image elsewhere, brightest 2.8 m from its centre.

Noise stands in for the hammer, but its image of the chamber tops out at 2.2×, however long it listens

No hammer: 48 noise sources scattered over the ground around the same geophone grid, each a stream of random vertical pushes, like footsteps and traffic. Cross-correlating the noise at two geophones gives the wave one would send the other (interferometryambient-noise interferometry seismicCross-correlating the background hum recorded at two sensors turns it into the wave that would travel from one to the other, so noise can stand in for a source.), so every geophone becomes a virtual sourcevirtual source seismicA sensor made to act as a source: the correlation of its noise record with another sensor’s is the wave a real source there would have sent. Noise sources spread over the ground give it as minus the lag derivative of the correlation.. 16 of them are used as in echo imaging: the gathers predicted without the chamber are subtracted and what remains is migrated. The rock is known exactly and the noise field is averaged over an unlimited record, the most favourable passive case; a finite record adds fluctuations whose image falls as one over the square root of its length.

Noise correlations reproduce the wave between two geophones (correlation 0.90 with a real source). With the rock known exactly and an unlimited record, the passive echo image is 2.2x brighter around the chamber than elsewhere, and its brightest point lies 7 m from the chamber's centre. No record length reaches 3x: even averaged forever, the chamber's part of the correlations images away from it at 46% of its brightness around it.

Fitting whole records from a practitioner’s start finds the chamber as a −15.8% drop in S speed, 5.7× the scatter

The same noisy surface records echo imaging migrated, now fitted whole by full-waveform inversionfull-waveform inversion seismicFitting whole seismic records, not just arrival times, by adjusting a model of the ground; each step is a gradient computed by propagating the misfit backwards in time.. Unlike echo imaging, the rock is not given: the start is what a practitioner would have, uniform rock at the P speed the surface travel times gave (3,259 m/s) and an S speed from the textbook ratio √3 (1,881 m/s, 3% fast). Gradients of P and S speed come from correlating forward and adjoint stresses, checked against finite differences; traces within 4 m of their hammer are left out, and the iterations stop at the noise level or after ten; here after ten, just above it. The records were made with the same solver and grid the inversion uses (the “inverse crime”): the most favourable case for FWI, softened only by the noise and the wrong start.