Goldray

tl;dr

An ordinary metal detector reports one number at a time and keeps none of them. Goldray keeps all of them. A module on the shaft, the Ray Core, transmits seven tones at once through a double-D search coil and records the coil’s complex response at each tone about 94 times a second. The Goldfall app on a phone clipped to the same shaft already tracks where the coil is to about a centimetre, so every reading can be stamped with the coil’s position and orientation.

A cloud of readings taken from many positions is the raw material of synthetic-aperture imaging. Back-projection turns it into a three-dimensional map with a depth axis, and a fit at each bright spot recovers the object’s magnetic polarizability tensor across frequency, the quantity that separates a gold nugget from a nail from a magnetic stone. Towed arrays that classify unexploded ordnance already work this way. Goldray proposes doing it with one coil, a phone, a Pi Zero and a stereo audio codec.

Everything here is design and simulation. The parts have been checked against datasheets and the transmissions against the Australian class licence, the field, target, ground and tone-planning code passes tests against closed-form physics, and the full chain from raw frame to classified target runs on simulated scenes. Nothing has been soldered, wound or switched on.

One pixel, no memory

Ironstone country in Western Australia’s goldfields is hard ground for a detector. The soil carries fine magnetic iron oxides, the surface is strewn with dark lateritic pebbles that ring like metal, and the gold worth looking for is often small. A detectorist sweeping that ground hears a continuous wash of response from the soil, a rise and fall as the coil’s height changes, and every few metres a signal that might be a nugget, a boot tack or a hot rock. Deciding which comes down to listening and digging.

Inside the detector, a transmit winding drives an alternating magnetic field into the ground and a receive winding picks up whatever disturbs it. The measurement is a single complex number: the change in mutual coupling between the two windings, with a part in phase with the drive and a part a quarter-cycle behind it. Goldray’s concept document lists three lossy things a conventional detector then does with that number. It collapses it to a tone, it discards it after a few tens of milliseconds, and it never knows where the coil was when it took it. Depth, size and shape are reconstructed, if at all, in the operator’s head from how the tone changes as the coil is wiggled.

A detector is a one-pixel camera that forgets each frame as soon as it has shown it.

Goldfall, the survey app Goldray is built to serve, already removes the third limitation. It puts a phone on the shaft, uses ARCore’s visual-inertial tracking to follow the coil, and paints a hex grid of where the ground has been swept. What it plots today is crude: the app derives its signal level from the sound level at the phone’s microphone. Goldray is the hardware and signal chain meant to give that map something better to draw.

A swing as an array

Radar engineers learned long ago that a small antenna moved along a known path can stand in for a large one. Record the echoes with their positions, and the processor can ask, for every point in the scene, how well a reflector there would explain all of them at once. Goldray applies that idea to magnetic induction. Each record, which the project calls a ray, holds the complex responses at seven frequencies, their noise estimates, a timestamp from the Core’s audio clock and the coil’s six-degree-of-freedom pose. A survey is a cloud of rays, and nothing in it is thrown away.

For a compact buried object, the response at one coil position is set by three things: how strongly the transmit winding’s field reaches the object, how strongly the object’s induced field reaches the receive winding, and how the object itself responds.

Here r is the object’s position relative to the coil, h_T and h_R are the field patterns of the two windings per amp of current, known from the coil’s geometry, and M(ω) is the object’s 3 × 3 magnetic polarizability tensor. Y is measured in metres, which is the reason the project uses it: field patterns in inverse metres and a tensor in cubic metres multiply out with no stray frequency factor. M belongs to the object alone. Its eigenvalues encode size and aspect ratio, and the way they change with frequency encodes conductivity and permeability.

A single reading cannot tell a small shallow object from a large deep one, because both can produce the same number. A cloud of readings can, because the pattern a buried object traces as the coil passes over it broadens with depth in a way the geometry predicts exactly. The kernel κ = h_R · h_T, the coupling a small isotropic object would produce at each point under the coil, shows how fast that pattern changes.

Goldray’s map tier, Tier 1, is a matched filter over space. Under the swept footprint it lays tiles of voxels 5 cm across with eight depth layers spaced logarithmically from 2 to 40 cm below the ground. Each incoming ray adds to two running sums per voxel and tone: the kernel times the measured residual, and the kernel squared, each weighted by the ray’s pose confidence and inverse noise variance. Their ratio is the polarizability a target at that voxel would need; the detection statistic is the summed squared signal-to-noise ratio across tones. Because only two accumulators per voxel are kept, the image updates as the operator swings, and repeated passes add coherently, so the statistic grows as the square root of the number of passes.

Milestone M2 implemented that back-projector and checked it against a golden scene: a 2 cm sphere of gold-like conductivity 10 cm down, three raster passes at 0°, 90° and 45°, and 1 cm of correlated error in the pose the processor is given. The specification’s acceptance test asks for the peak within 2.5 cm laterally and within one depth layer. The figure below runs goldray’s own scene generator and back-projector on that scene. Along any one line of the swing the signal is a smooth bump with nothing in it to say how deep the sphere is. In the image the peak lands 1.1 cm from the sphere in plan, in the 11 cm layer.

The image is less decisive about depth than that sounds. Down the column under the peak the statistic barely falls: the 40 cm layer still scores 88 % of the peak, because a larger, deeper object explains the same bumps almost as well as the right one. Tier 1 finds the object and a best depth. Pinning the depth down, and saying what the object is, is the job of the next stage.

Tier 2, also built in M2, takes the strongest candidates and solves for them properly. For a trial position, each ray contributes one row of six real coefficients to a design matrix that is the same at every tone, so one QR factorisation gives the six unique elements of M at all seven frequencies. A Levenberg–Marquardt search moves the trial position to minimise the residual, a scheme known as variable projection. A joint diagonalisation of the real and imaginary parts at each tone yields the three principal polarizabilities as functions of frequency, the features a classifier reads.

Survey teams hunting unexploded ordnance have done this for years, with multi-coil arrays on carts and RTK satellite positioning. What Goldray changes is the scanner: one coil, a hand, and a phone for positioning, updating live.

Seven tones at once

Goldray needs all its tones present at once and phase-locked to a clock it controls, because the rays are meant to be combined coherently across seconds and passes. A 24-bit audio codec running at 192 kHz, with its digital-to-analogue and analogue-to-digital converters on one shared clock, does that by construction. One output channel drives the transmit amplifier; one input measures the transmit current across a shunt and the other the receive voltage, sampled together, so the measurement is a ratio and the codec’s own drift cancels.

The tones sit on exact bins of a 2048-sample frame, 93.75 Hz apart, so each frame of 10.67 ms holds a whole number of cycles of every tone and one FFT reads all seven with no window and no leakage between them. Choosing which bins is a small combinatorial problem with seven rules. The bins must be odd and avoid multiples of 8 and 16, which is where harmonics of 50 Hz and 60 Hz mains land on this grid. They should be spread roughly evenly on a logarithmic axis. No tone may sit on a second- or third-order intermodulation product of the others, since any nonlinearity in the amplifier would put energy exactly there. And no tone may fall where the Australian class licence, or a European cross-check of it, says not to transmit.

The specification handed the planner a test vector to match or beat: bins 17, 37, 79, 163, 331, 673 and 811, reaching 76.03 kHz. The planner in goldray.dsp.tones, a branch-and-bound search re-verified by a separate brute-force enumerator that shares no code with it, returns 17, 31, 59, 109, 211, 397 and 769. Every one is prime. The nearest of the 182 second- and third-order products falls three bins from any tone, the same clearance as the test vector, and the worst ratio between neighbouring tones is 1.94 against the vector’s 2.18. Its top tone is 72.09 kHz rather than 76.03.

The drive current per tone falls as 1/f. Into an inductive coil that gives every tone the same induced voltage, which is what plain voltage drive into an inductance would do anyway, and it puts most of the current at the bottom of the band, where large and deeply buried objects answer. The total is set by the head’s exposure cap of 0.1 A rms. With the reference coil modelled as 0.72 mH and 2.1 Ω, the composite peaks at 5.16 V, well inside the amplifier’s 9 V limit, so the cap and not the amplifier sets the drive.

Seven sine waves added together have a peak that depends on their relative phases. The amplifier has to swing the peak, while the field and the exposure limit follow the rms, so a low ratio between the two, the crest factor, buys field strength for free. Starting from Schroeder’s phase formula and polishing with seeded Nelder–Mead restarts, the planner reaches 2.059. The specification originally asked for 2 or less; the M1 work found that unreachable with seven tones on a 1/f spectrum, logged the change in the decisions file and set the bound at 2.1.

In phase and behind it

For a sphere the tensor collapses to one number, and that number has an exact closed form, which is why ball bearings are Goldray’s calibration kit. The project’s M1 code implements it in a form stable at every frequency and tests it against both limits and an independent pole series.

The figure carries the central constraint of hunting small gold. A non-magnetic sphere’s quadrature response peaks near the frequency set by its first eddy-current time constant, σμ₀a²/π², which grows with the square of the radius. A 1 g sphere of gold-like conductivity, 2.3 mm in radius, peaks near 6.8 kHz. A 0.1 g sphere, 1.1 mm in radius, peaks near 32 kHz, and at 72 kHz its in-phase response has reached only about 60 % of its high-frequency limit. The top of the band is where the smallest pieces of gold say anything at all, which is why the plan pushes towards 76 kHz and why losing that tone, discussed below, costs more than one seventh of the data.

Discrimination then works on shape as well as spectrum. A compact nugget has three similar eigenvalues; a nail has one dominant, permeable axis, positive in-phase at low frequency; a ring-pull or a piece of foil behaves like a shorted loop, one dominant eigenvalue along its normal with a single-pole spectrum; a magnetic stone behaves like ferrite, real, positive and nearly flat. Real nuggets are irregular alloys and will not trace a sphere’s curve, which is why the specification plans a library of computed tensors for nugget-like blobs, nails, foil, coins and bottle caps, using the open-source MPT-Calculator finite-element code, and why the classifier is to be trained on dig outcomes logged in Goldfall rather than on the closed forms alone.

A coil that cannot hear itself

The transmit field is many orders of magnitude stronger than anything a small object sends back, and the receive winding sits right next to it. A double-D coil handles that geometrically. Two D-shaped windings lie side by side with their flat edges overlapping, and at one particular overlap the transmit flux passing through the receive D sums to zero. For the idealised reference DD, two semicircles of 140 mm radius, Biot–Savart integration puts that null at 19.17 mm of overlap.

The null is sharp. Near it the coupling coefficient changes by 6.0 × 10⁻⁴ for every 0.1 mm of overlap, −64 dB of full coupling, so the printed former leaves ±3 mm of adjustment and the coil is trimmed by hand before potting, with a target residual of −40 dB. The Core then takes the leftover leakage out digitally. Its second output channel emits the same seven tones with adjustable complex weights, summed into the receive preamplifier, and during a short air null at the start of a session, with the coil held at least a metre from ground and metal, the weights are iterated until the residual is 40 dB below its un-nulled level.

Then the weights are frozen. The specification is blunt on this point: an adaptive canceller left running during a survey removes targets as readily as leakage. Slow drift after freezing is absorbed by the ground tracker’s offset term and reported, and the app prompts a re-null if it grows too large. The same machinery is meant to make a badly balanced or home-wound coil usable, which is as close as the project comes to digital tuning.

Ground as data

Mineralised soil is the main reason small gold is hard to find. Detectors deal with it through a ground balance that tracks one or two numbers and subtracts them. Goldray models the soil instead. With seven frequencies and the coil’s height known from the pose, the ground response splits into two physical terms. Viscous magnetic susceptibility, from the fine magnetic grains, gives an in-phase part that falls slowly and logarithmically with frequency and a constant quadrature part. Conductivity gives a quadrature part that rises in proportion to frequency. Each is a coil-geometry factor times a soil property, and the soil properties change slowly across the hex grid.

The specification refused to assume the sign and form of the conductive term and required it to be derived from a full layered-earth solver. M1 did that with empymod, the open-source electromagnetic modelling code, cross-checked against a magnetostatic image method and two closed-form integrals to a few parts in 10⁵. The result: at low induction number the conductive term is purely imaginary, with a positive coupling factor, the same sign as a conductive target. For coincident small loops the magnetic term falls as the inverse cube of height and the conductive term only as the inverse of height.

The DD produced one surprise. A single loop’s magnetic ground coupling keeps rising as the coil approaches the soil; the DD’s peaks near 5 cm and falls again below that. Coil bob near the surface therefore moves the DD’s ground signal less than a loop’s.

The ground state, two susceptibility parameters, a conductivity and a slow per-tone offset, is to be tracked by a Kalman filter whose process noise grows with distance moved over the ground rather than with time, since soil changes with place. Updates down-weight large innovations so a target cannot drag the ground estimate with it, and are suspended while the instant detection statistic is high. The posterior goes into the hex cell under the coil and seeds the next visit. As a by-product the app gets a mineralisation map, which a prospector may find as useful as the targets.

The linear model has a limit. In very conductive ground the induced currents interact with each other and the response stops scaling simply with conductivity. M1 measured where that happens, as the conductivity at which empymod’s exact half-space response departs from the linear term by 5 %.