A physics teaching model. Everything here is textbook geophysics and electromagnetism with generic numbers. The buried items are archaeological discs, scrap, utility pipes, plates and rocks. Nothing in this page describes, models or depicts any explosive device, how one is built or triggered, or how to defeat detection. Real humanitarian demining is performed only by trained professionals following international standards (IMAS); this page is not training and no procedure here substitutes for it.
1. Induction-balance detector: the physics of the signal
A transmit coil carrying an alternating current I0 cos(ωt) creates a primary field Bp. A conducting target sees a changing flux, so by Faraday's law an EMF −M dI/dt drives eddy currents in it, and by Lenz's law those currents oppose the change. Model the target as a single loop with inductance Lt and resistance R, so τ = Lt/R:
Lt dIt/dt + R It = −M dI0/dt → It = −(M/Lt) I0 · jωτ / (1 + jωτ)
Its secondary field is received by the same coil (a reciprocal pair, so the coupling enters twice). With x = ωτ and the squared coupling coefficient k² = M²/(Lt Lc) the relative signal change is
ΔX = −k² x²/(1+x²) (screening, in phase), ΔR = +k² x/(1+x²) (loss, quadrature)
These two parts trace a semicircle in the X-R plane as x runs from 0 to infinity. The phase of the target current is θ = atan(ωτ): small or thin non-ferrous bits (small τ) sit near 90 degrees on the loss axis, big good conductors (large τ) near the screening axis. A ferrous object adds magnetisation of its iron, an in-phase term of opposite sign (ΔX > 0), so it falls on the other side of the plane; that is what a discrimination notch uses. The page integrates the L-R differential equation with RK4 and lock-in demodulation and compares it with these formulas (check 1). Thin-ring inductance L = μ0 b (ln(8b/w) − 2) and the Biot-Savart field of the coil (64 segments) give M; the coil is treated as a single-turn equivalent (the coupling coefficient does not depend on the number of turns).
2. Why depth sensitivity falls like 1/z⁶
The target is excited by the coil field B(z) and its reply is received through the same field, so the signal is proportional to B² ∝ M². On the axis of a circular coil of radius a, B(z) = μ0 I a² / (2 (a²+z²)3/2), hence
S(z) ∝ (a² + z²)−3 → d ln S / d ln z = −6 z²/(a²+z²) → −6 for z >> a
so doubling the depth costs a factor 64 in signal. A larger coil keeps more of its field at depth and so reaches deeper on large targets, while its coupling to a tiny piece (k²) is lower near the surface: small coils favour small shallow items, large coils large deep ones. The depth-law plot shows the numerically computed signal (Biot-Savart), the fitted slope between 3a and 10a, and the dipole prediction.
3. VLF and pulse induction
VLF detectors transmit a continuous sine (1-30 kHz here) and measure amplitude and phase, so they can tell targets apart by τ. Pulse-induction detectors switch the transmitter off and sample the decaying eddy current exp(−t/τ) after a delay td; a larger delay kills the response of short-τ objects but also of the soil. The page normalises the PI signal as k² (τc/τ) exp(−td/τ), τc = 100 us, and gives no phase information in PI mode.
4. Soil: magnetic viscosity and ground balance
Mineralised soil (iron oxides) is weakly magnetic with susceptibility χ. For χ << 1 the fractional inductance change of the coil is the field energy inside the ground, ΔL/L = χ ∫ground B² dV / (μ0 Lc); the page evaluates that volume integral numerically for the real coil radius and height. It falls rapidly with height, so a coil bobbing by a centimetre produces a signal that can exceed a deep target by a factor 10-100 in laterite. Ground balance subtracts the component along the soil's X-R direction (a projection), which cancels it at every height at the price of reducing the target signal by sin of the angle between the target and ground phases. Soil conductivity (wet clay) adds a resistive part, set here by a fixed R/X ratio per soil.
5. Detection statistics
With white Gaussian noise of standard deviation σ per sample and a threshold T = Q−1(Pfa) σ, the probability of detecting a steady signal S is Pd = Q((T − S)/σ). At S = T the probability is 50 %; every 6 dB above that raises it steeply. The depth and size dependence follows from section 2. The sweep map applies the same rule sample by sample, so lane spacing and speed matter: a lane spacing larger than the target's footprint leaves gaps.
6. Ground-penetrating radar
GPR sends a short electromagnetic pulse (Ricker wavelet, centre frequency f) into the ground and records the echoes. In a low-loss soil the wave travels at
v = c / √εr (dry sand εr ≈ 4: 0.150 m/ns; wet clay εr ≈ 25: 0.060 m/ns)
At an interface between media 1 and 2 the normal-incidence reflection coefficient of the electric field is Γ = (√ε1 − √ε2)/(√ε1 + √ε2); metal behaves like ε2 → infinity, Γ → −1 (inverted polarity), and a void or dry pocket gives a positive sign. The page checks this with a one-dimensional finite-difference time-domain (Yee) solver.
t(x) = (2/v) √(z² + (x − x0)²) → apex t0 = 2z/v, z = v t0 / 2
A small object scatters in all directions, so a moving antenna records it earlier and later than directly above: a hyperbola. Its apex gives depth, its curvature gives the velocity: t² = t0² + (2/v)²(x − x0)² is linear in x and x², so a least-squares fit of picked times returns v and hence εr = (c/v)² (calibration). A pipe is a line scatterer: perpendicular to the survey line it is a hyperbola, parallel to it a flat reflector, and at an angle ψ to the line it is a stretched hyperbola with apparent velocity v/sinψ. A plate is a flat reflector with edge diffractions.
Amplitudes: a point scatterer falls as r−2, a line as r−1.5, a flat reflector as 1/(2z), all times exp(−2 α r) with the exact lossy-dielectric attenuation
α = ω √(με/2) √( √(1 + (σeff/ωε)²) − 1 ), σeff = σdc + ω ε tanδ
and an antenna directivity factor cos θ. Conduction loss (wet, clayey soil) is what kills GPR; the dielectric-loss term tanδ makes attenuation grow with frequency. Depth resolution is about λ/4 = v/(4f), so a higher frequency resolves thinner layers but penetrates less: the trade-off tab plots both. The direct air/ground wave is nearly constant along the line and is removed by subtracting the mean trace.
7. Which tool for which target
Metal detectors respond to conductivity and iron but not to plastic pipes, voids or rocks; GPR responds to any dielectric contrast but is blind in conductive soil. Utility locating, archaeology and geology use both. No single method finds everything.
Simplifications
Targets are sets of independent loops (no mutual coupling or skin-effect distribution); the coil is a single-turn equivalent; ferrous magnetisation is a fixed in-phase fraction; the soil is homogeneous with a fixed X-R ratio; PI timing is idealised; GPR uses ray geometry (no full-wave scattering, no multiple reflections, no dispersion, no antenna ringing), and a flat surface; the B-scan synthesises noise and a weak direct wave. Scattering amplitudes are plausible but not calibrated to a specific antenna. Pipe, disc and scrap sizes are drawn true to scale.
References
Y. Das, J. McFee, J. Toews, G. Stuart (1990) on the electromagnetic response of a conducting sphere and metal-detector modelling; C. Bruschini (2002) on metal-detector sensor physics; D. Daniels, Ground Penetrating Radar (IET); H. Jol (ed.), Ground Penetrating Radar: Theory and Applications; A. Taflove and S. Hagness, Computational Electrodynamics (FDTD); J. D. Jackson, Classical Electrodynamics (loops, mutual inductance); R. Colani and M. Aitken (1966) on the magnetic viscosity of soils.