How a ground magnetometer array sees ionospheric currents, why so many things make it look as if something is moving, and how to test for motion honestly
A magnetometer on the ground measures the vector magnetic field, about 50 000 nT at auroral latitudes, and almost everything interesting is a disturbance of a few to a few hundred nT on top of it. Those disturbances are made by electric currents 100–120 km overhead, by currents far out in the magnetosphere, and by currents they induce in the ground and the sea. An array of such stations is a crude camera for the currents. This page is about what that camera can and cannot see.
Units and scale. 1 nT = 10−9 T. A sheet of current carrying 1 A per km of width overhead produces about 0.63 nT at the ground (tab 2), so a 100-nT disturbance means of order 150 A/km, a few hundred thousand amperes across an electrojet.
Sampling. Modern fluxgates sample at 1 s or faster. Data centres distribute 1-s, 10-s and 1-min values; the standard 1-min value is a filtered mean centred on the minute.[18] The IMAGE network states its timing accuracy as about the sampling interval, 10 s, and notes that averaging differs between stations, so apparent inter-station time differences can exceed 10 s.[10] Remember this number: it sets a floor under every time delay on this page.
Bands. Continuous pulsations are classed by period (Pc1–Pc5, Pc5 = 150–600 s) and irregular ones as Pi1, Pi2.[19] With 1-min data the Nyquist period is 2 min: Pc3 and most Pc4 and Pi2 are either removed by the averaging filter or aliased.
The IMAGE network runs from northern Germany and the Baltic to Svalbard, with a dense north–south chain near 20–27° E.[40] The map shows the stations used on this page, at their published geographic coordinates; colour is corrected geomagnetic (CGM) latitudeA latitude defined by tracing the real (not dipole) field line from the station to the magnetic equator plane and back to a dipole. It organises auroral phenomena better than geographic latitude. The values here are those published by FMI for epoch 2001. as published by FMI.[9] Hover or tab to a station.
The array — hover a station
Projection: sinusoidal about 21° E (east–west distances true at each latitude); graticule every 2° latitude and 5° longitude. No coastline is drawn: the stations, not the map, are the instrument. The four Svalbard stations sit 400–800 km north of the mainland chain across a gap with only Bear Island in it; later tabs let you include or drop them.
Here are four stations along a north–south line. The same wiggle reaches each one later than the one before. Did something move northward?
Delays across an array are cheap to measure and easy to misread. The rest of the page builds the physics that makes such patterns (tabs 2–5), then the signal processing that estimates a velocity from them (tabs 6–7), then the discipline that keeps the estimate honest (tab 8).
Before any physics, a reminder about the instrument. Drag the period of a sinusoidal pulsation and choose a sampling interval. The dots are what the archive holds; the faint curve is what happened.
Sampling and aliasing
Averaging over the sampling interval acts as a low-pass filter: it shrinks short-period signals (the gain of a T‑second boxcar at period P is |sinc(T/P)|) before they can alias, which is why 1-min means largely hide Pc3 rather than aliasing it. Point samples with no prefilter alias freely. Real 1-min products use a tapered (Gaussian-type) filter rather than a boxcar.[18]
Map of the page. Currents → why the ground sees mostly the Hall current, rotated 90°. Induction → the ground is a conductor that adds its own phase. Waves → standing waves whose phase slides with latitude. Look-alikes → ten mechanisms that create delays across an array. Beam and f-k → the estimators. Sensitivity → how to know what the estimator can and cannot tell you.
The ionosphere is a conductor in a strong magnetic field, which makes its conductivity a tensor. Between about 90 and 150 km electrons gyrate freely while ions are held by collisions with the neutral gas; that difference is what lets a horizontal electric field drive a current perpendicular to itself.
In the frame of the neutral wind, the current perpendicular to the unit field vector b̂ is
j⊥ = σP E⊥ + σH b̂ × E⊥, j∥ = σ∥ E∥(2.1) σP = (n e / B) [ νiΩi/(νi2+Ωi2) + νeΩe/(νe2+Ωe2) ], σH = (n e / B) [ Ωe2/(νe2+Ωe2) − Ωi2/(νi2+Ωi2) ](2.2)with Ω the gyrofrequencies and ν the collision frequencies with neutrals.[33] The Hall term is easy to read physically. Where electrons are magnetised (νe ≪ Ωe) they E×B-drift; where ions are collisional (νi ≫ Ωi) they do not. The electrons alone then carry a current −n e (E×B)/B2: the Hall current flows opposite to the plasma convection. The Pedersen current, along E, peaks higher, where νi ≈ Ωi. The ionosphere is thin compared with the scales of interest, so one integrates over height to get conductances ΣP, ΣH (siemens) and a sheet current J = ΣPE + ΣH b̂×E (A/m).
Conductivity profiles — a schematic model
Model, not data. An α-Chapman E layer (peak 1.5×1011 m−3 at 110 km for overhead Sun, scale height 10 km) plus a small night-time floor; O2+ ions; collision frequencies falling with height on a scale that grows from 6 km at 100 km to about 26 km at 200 km, normalised so that νi = Ωi at 125 km and νe = Ωe at 75 km; B = 50 000 nT. Real profiles need a neutral-atmosphere model and measured densities. The point is the shape: two layers, Hall below Pedersen, and conductances of a few siemens in sunlight, falling sharply at night. Measured sunlit conductances follow ΣH ≈ 1.5 (Sa cos χ)1/2 ≈ 1.7 ΣP, with Sa the 10.7-cm solar flux (Chatanika radar, χ = 45–85°).[35] The readout shows that empirical value beside the model’s.
Field-aligned currents (FACs) bring current down from the magnetosphere and it closes through the ionosphere. Fukushima showed that if the field lines are vertical and the ionospheric conductance is uniform, the magnetic field of the FACs and of the Pedersen currents that close them cancels exactly everywhere below the ionosphere.[12] The reason is a symmetry: the Pedersen current of a vertical line current spreads radially, a radially symmetric current plus its vertical feeder has a magnetic field that is purely azimuthal above the sheet and zero below it (Ampère’s law on a circle below the sheet encloses no current). With uniform conductance and an electrostatic field the Hall current is divergence-free, needs no field-aligned feeders, and is therefore the part of the system that produces a field at the ground. The ground therefore sees, to leading order, the Hall current only, and the “equivalent current” one infers from ground data is the Hall current, not the full current system.[43][20][1]
Where it fails. Field lines at 60–70° are inclined by about 10–20° from vertical, and conductances have strong gradients (the terminator, auroral arcs). Both break the cancellation, and part of the FAC/Pedersen system then leaks to the ground. The theorem is the right first approximation, not an identity.
Just below an infinite uniform sheet current J (A/m) the field is ΔB = (μ0/2) J × ẑ, with ẑ down. That has two consequences. First, the ground perturbation is the current turned 90°: an eastward current gives a northward (positive X) disturbance. Second, with J = ΣH ẑ × E (northern hemisphere, b̂ = ẑ) the vector identity (ẑ×E)×ẑ = E gives
ΔBh = (μ0 ΣH / 2) Eh (uniform ΣH, northern hemisphere)(2.3)so the horizontal ground disturbance points along the ionospheric electric field and at right angles to the plasma flow. For ΣH = 10 S and E = 20 mV/m that is about 126 nT. For waves, the ionosphere also screens small scales: a structure of horizontal wavenumber k at height h reaches the ground attenuated by about e−kh, so anything much smaller than ~100 km is invisible from the ground, and the wave field is rotated by 90° on the way down.[16]
Model the electrojet as a bundle of parallel line currents at height h. For an infinite line carrying current I along l̂, Biot–Savart gives B = (μ0I / 2π) (l̂ × ρ) / ρ2, with ρ the perpendicular vector from the line to the station. Directly under an eastward jet X is positive and largest; Z changes sign beneath it (negative, upward, on the poleward side). The ground is a conductor: a perfectly conducting layer at depth d adds a reversed image current at depth h + 2d, which boosts X and suppresses Z. The physically better “complex image” puts the image at a complex, frequency-dependent depth set by the skin depth (tab 3).[3]
Electrojet over the array — drag the jet
Drag the band (or focus it and use arrow keys). Positive current flows toward the band’s eastern end (eastward electrojet, evening sector); negative is the westward (morning, substorm) electrojet. The right panel plots the disturbance against perpendicular distance from the jet axis; dots are stations. Note how a 500-km-wide sheet looks almost the same on the ground as a line: the ground can resolve current structure only down to roughly the height of the current.
What to take from this. A single station cannot tell an overhead current of one shape from another; a line of stations across the jet sees the X peak and the Z zero-crossing and locates the jet. A moving jet makes every station’s X rise and fall in sequence, which is the cleanest kind of true apparent motion an array records. Now hold on to that picture and ask how many other things produce the same sequence.
Every time-varying field from above induces currents in the Earth and the sea, and their field adds to what a station records. The induced part has its own amplitude and its own phase, which depend on period and on what lies under each station. Two stations under the same, perfectly simultaneous source can therefore record signals that are shifted in time.
In a conductor of resistivity ρ a field varying with angular frequency ω = 2π/T diffuses in and decays over the skin depth
δ = √(2ρ / ωμ0) ≈ 0.503 √(ρ T) km (ρ in Ωm, T in s)(3.1)Sea water (ρ ≈ 0.3 Ωm) has a skin depth of about 5 km at a 5-min period, deeper than most of the sea but only just. The Precambrian shield under Fennoscandia is highly resistive near the surface, and the conducting mantle lies hundreds of km down; the contrast between ocean and continent is what produces the coast effect.[37][4][31]
For a horizontally layered (1-D) Earth and a source field of horizontal wavenumber κ, the whole response is carried by one complex length, Schmucker’s transfer function C(ω), computed by the standard layer recursion from the bottom up:
kn = √(iωμ0/ρn), CN = 1/kN, Cn = (1/kn) · (knCn+1 + tanh kndn) / (1 + knCn+1 tanh kndn)(3.2)For a uniform half-space C = (1−i)δ/2: a length of order the skin depth, with a 45° phase. The ratio of internal to external field at the surface is (1−κC)/(1+κC), so the horizontal field a station records is Bx = 2Bxext/(1+κC) (twice the external field over a perfect conductor, once over an insulator), and the vertical field is |Bz/Bx| = |κC|.[37] Put a different Earth under each of two stations and the same source gives them different amplitudes and different phases.
Coastal vs inland — a two-layer model, labelled as such
The model. Inland: a resistive shield (ρ1 as set, 150 km thick) over a conducting mantle (20 Ωm). Coastal: the same with a sea-water layer (0.25 Ωm, depth as set) on top. Each station is treated as if its own layered Earth extended to infinity, which is the 1-D approximation. A real coastline is a lateral contrast; it channels induced current along the coast and produces the large vertical fields of the coast effect, which a 1-D model cannot produce at all. The numbers here are illustrative; they are not fitted to any station.
The lesson is in the time shift. A few degrees of phase at a 30-min period is a delay of tens of seconds, which is comparable to the timing accuracy of the archive and to the delays a slow-moving structure would produce across a 100-km station spacing. It is fixed by geology, so it is the same every day and in every direction: a static station-pair delay. Any velocity estimate from an array that ignores it inherits it.
The vertical field is the clearest fingerprint. Writing Bz = A Bx + B By at each period defines the tipper (A, B); its real part drawn as an arrow is the induction arrow. In the Parkinson convention the arrow points toward good conductors: at a coast, toward the sea. Analogue-model and field studies in northern Fennoscandia find the ocean effect dominating the induction vectors near the Arctic coast, with the vertical field rising sharply as the coast is approached, and local anomalies at some inland sites.[45][44] Under the electrojet itself the source is far from uniform and arrows computed from active-time data mix source and induction effects.
Classification. Induction is neither magnetospheric physics nor ionospheric electrodynamics. It is the solid Earth and the sea responding to them. It belongs on any list of things that make a ground array show station-to-station delays, and it never moves.
Field lines are strings. Disturb one and a shear Alfvén wave runs along it, reflects off the conducting ionosphere at each end, and sets up a standing wave whose period is the field line’s own. Drive the magnetosphere at one frequency and the field line whose eigenfrequency matches responds resonantly. The ground sees an amplitude peak at one latitude and a phase that changes by about 180° across it. Nothing moves, yet a latitude chain sees a perfectly regular delay pattern.
The shear Alfvén wave travels along B at vA = B/√(μ0ρ), with ρ the plasma mass density. On a dipole field line of equatorial distance L RE,
r = L RE cos2λ, B = B0(RE/r)3√(1+3 sin2λ), ds = L RE cosλ √(1+3 sin2λ) dλ, cos2Λ = 1/L(4.1)where Λ is the latitude of the field line’s foot and B0 ≈ 31 000 nT. With the ionosphere as a node at both ends, the fundamental toroidal (azimuthally polarised) mode fits half a wavelength on the line, so in the WKB approximation its period is twice the Alfvén travel time:
T1 ≈ 2 ∫ ds / vA(4.2)The exact eigenperiods solve an ordinary differential equation along the field line;[38][8] the WKB estimate is good to tens of percent for the fundamental and is used on this page only to set the shape of the eigenfrequency–latitude curve. Longer field lines with weaker field mean the eigenfrequency usually falls with latitude, from tens of mHz near L = 3 to 1–3 mHz (Pc5) on auroral field lines.
A compressional (fast-mode) disturbance in the magnetosphere, driven for instance by Kelvin–Helmholtz waves on the magnetopause or by cavity and waveguide modes,[24] couples to the shear Alfvén wave where the driving frequency equals the local Alfvén eigenfrequency. Near that latitude the response behaves like a driven, damped oscillator, with the field line’s eigenfrequency playing the role of the spring:[39][7]
ξ(Λ) ∝ 1 / [ ωA2(Λ) − ω2 − iγω ](4.3)The amplitude peaks at the resonant latitude Λr with a width set by the damping γ (mostly Joule dissipation in the ionosphere, so larger for larger Pedersen conductance). The phase follows the oscillator: in phase with the driver on the side where the line is stiffer (ωA > ω), lagging by 90° at resonance, by 180° on the other side. The sign of the latitude phase gradient is the sign of dωA/dΛ. With the usual eigenfrequency falling poleward, the phase lag grows poleward and the oscillation appears to travel poleward; where a density drop such as the plasmapause makes the eigenfrequency rise with latitude, it appears to travel equatorward.
A field-line resonance under the chain
Model: eq. (4.3) with ωA(Λ) from the WKB integral (4.2) on a dipole, plasma mass density ρ ∝ r−1 along the line, scaled so that the eigenfrequency equals the driving frequency at Λr. That scaling stands in for the plasma mass density, which in the real magnetosphere includes heavy ions; the WKB estimate of the fundamental is too crude to turn it into a density, so none is quoted. The “rises poleward” option mirrors the eigenfrequency profile about Λr. Station latitudes are CGM latitudes used as dipole foot latitudes, and the 90° ionospheric rotation and e−kh screening, which smooth the peak seen on the ground, are left out. The lower image, latitude up and time across, is what a stack plot of a latitude chain looks like: tilted stripes, the classic look of motion.
Reading a chain. An apparent latitudinal “velocity” from an FLR is v = ω / (dφ/dx), which can be anything from a few to hundreds of km/s and depends on the period. Real motion gives a delay that does not depend on period; a resonance gives a phase that is fixed by latitude, so the delay (phase/ω) scales with the period. That, together with an amplitude peak at the place where the phase changes fastest, is the discriminator.
Pc5 waves (2–7 mHz) also have an azimuthal phase pattern eimφ. In 61 events on an east–west chain the phase velocity was about 14 km/s and independent of frequency, the azimuthal wavenumber growing with frequency as m = (1.4 ± 0.4)f + 0.26 (f in mHz), and the sign reversed abruptly 1–2 h before local noon: the phase runs westward in the morning and eastward in the afternoon, away from the pre-noon sector, consistent with a Kelvin–Helmholtz source on the magnetopause flanks.[30] It is not universal: three quiet-time events seen by radars over more than 120° of longitude propagated westward on both sides of noon.[28]
Pc5 phase around the clock
The dial is drawn in magnetic local time, so a pattern fixed relative to the Sun stands still in it while the Earth, and the array, rotate underneath. m from the empirical fit above; ground phase speed at 65° = 2πf RE cos 65°/m. The animation is slowed by a large factor.
Two things follow for an array. At 10–15 km/s a Pc5 crest crosses a 500-km east–west baseline in about 40 s, below one 1-min sample: the delay exists only as a cross-spectral phase. And the direction flips with local time, so any average over a day mixes opposite signs.
An array estimates a velocity from delays between stations. Below are eleven well-documented ways to get delays. Some involve something that really moves; many do not; several move but are not the thing one wanted to find. Each is tagged by the physics it belongs to.
Tidal winds drive a dynamo in the sunlit E region; its current system is two large vortices fixed relative to the Sun, so each station records a daily variation that changes noticeably from day to day.[46] In the ground frame a Sun-fixed pattern sweeps westward at 464 cos φ m/s: 246 m/s at 58°, 196 at 65°, 143 at 72°. Any slow structure tied to local time does the same. At auroral latitudes Sq is weak and entangled with other current systems.[46]
Solar EUV conductance switches on along the terminator, measured as ΣH ≈ 1.5 (Sa cos χ)1/2.[35] Under a steady electric field the Hall current, and so the ground field, changes as the conductance does, station by station in crossing order. The terminator line is tilted by the season, so the crossing order is oblique: a season-dependent apparent heading at Earth-rotation speed. Try it in the interactive below.
Quasi-periodic fluctuations of magnetospheric convection, coherent with the interplanetary magnetic field, drive a two-cell Hall current pattern that varies almost simultaneously from the auroral zone to the dip equator.[29] The large-scale convection itself is well catalogued.[14] The array sees a common, near-simultaneous signal whose sign can reverse across a cell boundary: after removing the array mean, the reversal is all that is left, and it can look like a pattern with a direction.
The auroral oval and its electrojets move in latitude with activity: auroral boundaries expand equatorward as the Kp index rises.[6] A jet edge sliding across a latitude chain makes X rise or fall in latitude order (compare the electrojet interactive in tab 2). Something genuinely moves, but it is a boundary of a large system, not a localised object, and it moves only in latitude.
Atmospheric gravity waves[15] launched by auroral heating or from below move the ionospheric plasma and modulate conductance.[17] Medium-scale TIDs have periods of 15–60 min and speeds of 100–250 m/s and are seen almost all the time; large-scale TIDs exceed 400 m/s with wavelengths over 1000 km and are mostly, not only, a disturbed-time phenomenon.[22] These are real moving waves, but in the neutral gas; their magnetic signature is indirect and small.
A jump in solar-wind dynamic pressure compresses the magnetosphere; the ground records a sudden impulse, explained as a preliminary and a main impulse with their own current systems.[2] A superposed-epoch study of 2058 events at high latitude finds the preliminary impulse starting 1–2 min before the low-latitude response with a 4–6 min rise, and the main impulse about 2 min after it, rising in 6–11 min, with transient current vortices.[27] The onset is simultaneous across an array; the vortices that follow can drift.
Twin vortices of Hall current, fed by a pair of field-aligned current filaments, travel tailward through the dayside high-latitude ionosphere. The first detailed case, near 08 local time, had filaments about 600 km apart east–west moving westward at about 4 km/s and was linked to a sudden solar-wind change.[11] Ground and satellite cases followed.[13] This is real motion of a localised structure — of a current system, not of plasma.
Quiet, discrete arcs are long in the east–west direction, narrow in latitude, and can persist and drift slowly for tens of minutes; they carry field-aligned currents and raise the local conductance.[21] Under a steady convection field the conductance enhancement adds a Hall current that moves with the arc. The ground signal is weak and smeared by the ionospheric screening of small scales (tab 2), so an array sees a slow latitudinal drift of a broad, low-amplitude feature.
A standing Alfvén wave with a resonant latitude: amplitude peak, 180° of phase across it, and an apparent latitudinal phase motion whose delay scales with period.[39][7] In longitude, Pc5 phase runs antisunward at ~14 km/s, flipping direction before noon.[30] Explore both in tab 4.
The same external field reaches every station at once, but each station adds the field of currents induced in the ground and sea beneath it, with its own amplitude and phase.[31][45] The result is a fixed delay between station pairs that depends on period but not on time of day or direction of the source’s motion. See the model in tab 3.
Timing accuracy of about the sampling interval, with averaging that differs between stations;[10] 1-min values that are filtered means centred on the minute;[18] baseline jumps; temperature drifts; and short-period signals such as Pi2 (40–150 s)[23] aliased by sampling (tab 1). One station out of step is enough to tilt a fitted plane. A result that does not survive dropping any single station is not a result about the sky.
The E-region terminator is where the solar zenith angle at 110 km altitude reaches 90° + arccos(RE/(RE+110 km)) ≈ 100.6°: the ionosphere is still sunlit when the ground is dark. Choose a day of the year and a time. The map shades the dark ionosphere; the readout lists the dawn crossing times at each station and fits a plane wave to them, as an array analysis would. The fitted “velocity” is perfectly coherent, slow, and has a heading that depends on the season. Nothing travels except the shadow.
Terminator crossing — season and time
Solar position from the standard low-precision formulas (declination δ = −23.44° cos[360°(d+10)/365], hour angle from UT and longitude); the equation of time (up to ±16 min) is ignored, which shifts all crossings equally and does not change the fitted velocity. Stations whose E region never crosses (polar night or midnight sun at 110 km) are greyed and left out of the fit. Heading is the direction of travel, clockwise from north.
The common thread. Every entry above produces a coherent delay pattern across an array. Several produce one that a velocity estimator will happily turn into a heading and a speed with a high coherence. An estimate is evidence of motion only after the look-alikes that fit the same pattern have been considered and excluded by something other than the estimate itself.
If a pattern translates rigidly across the array with velocity v, every station records the same waveform, shifted by a delay that is linear in its position. Undo the delays for a trial velocity and add the traces: at the right velocity they add in phase. Scanning trial velocities and measuring how well the traces add is beamforming, also called slant-stacking.
Write the velocity as a slowness vector s = v̂ / |v| (s/km), pointing along the direction of travel with magnitude 1/speed. A rigidly translating signal xj(t) = f(t − rj·s) reaches station j at position rj with delay
τj = rj · s = (rj · v̂) / |v|(6.1)The delay-and-sum beam for a trial slowness and its normalised power (semblance) are
b(t; s) = (1/N) Σj xj(t + rj·s), S(s) = Σt [Σj xj(t + rj·s)]2 / ( N ΣtΣj xj(t + rj·s)2 )(6.2)S lies between 0 and 1, reaching 1 when all aligned traces are identical. It is the time-domain form of the classical beam power used in array seismology.[36] The array is a spatial filter: its ability to separate slownesses is set by its aperture (largest baseline), its spacing (aliasing), and the sampling interval (the shortest resolvable delay).
The interactive builds synthetic 1-min (or 10-s) data on the real station geometry: a Gaussian disturbance of chosen width moving at a chosen heading and speed, passing the array centre at mid-window, with independent noise at each station. Optionally an array-wide common signal is added: the same slow variation at every station, as from a distant large-scale current. It then computes the semblance over headings and speeds and reports the peak.
Beam on a planted disturbance
How to read the map. Heading across (direction of travel, clockwise from north), speed upward on a log scale from 0.05 to 30 km/s; colour is semblance S from 0 (dark) to 1. Ring: the planted velocity. Cross: the peak. The top edge of the map is the infinite-speed limit, where all delays are zero: a signal common to the whole array scores highest there, in every heading.
Simplifications. Station positions are on a flat local projection (adequate over 2000 km for this purpose but not for precise work). The planted signal translates rigidly; real structures evolve while they move, which lowers S and is one reason a moderate semblance is not, by itself, evidence of anything. Delays use linear interpolation between samples.
The beam works in the time domain on one trial slowness at a time. Fourier-transforming each station first turns every delay into a phase, the whole array into one complex vector per frequency, and beamforming into linear algebra on the cross-spectral matrix. That opens the door to adaptive estimators such as Capon’s minimum-variance beam.
At angular frequency ω a delay τ is a phase factor e−iωτ, so a rigid translation with slowness s0 gives station spectra Xj(ω) = F(ω) e−iω rj·s0. Collect them into a vector X, estimate the cross-spectral matrix R = ⟨X XH⟩ by averaging over tapers or segments, and define the steering vector aj(s) = e−iω rj·s (equivalently wavenumber k = ωs). The conventional (Bartlett) and the Capon estimators are
PB(s) = aH R a / N2, PC(s) = 1 / ( aH (R + εI)−1 a )(7.1)Bartlett is the frequency-domain beam of tab 6; its resolution is fixed by the array response. Capon designs, for each trial slowness, the weight vector with unit gain in that direction and minimum total output power; the result is the minimum-variance estimate above, which suppresses power leaking in from other slownesses and resolves far better when the signal-to-noise ratio is good.[5] Both were developed and compared on large seismic arrays.[26][25] Capon needs R to be well conditioned; with few averages or after projecting out a subspace it is not, and one adds diagonal loading ε, which trades resolution for robustness and tends to Bartlett as ε grows.
Multitaper spectra in one paragraph. The cross-spectral matrix must be estimated from one finite record. Multiplying the record by K orthogonal tapers (Thomson used the discrete prolate spheroidal sequences, which concentrate as much energy as possible into a chosen bandwidth ±W) gives K nearly independent spectral estimates with controlled leakage; averaging them reduces the variance by about K while the bandwidth 2W sets the resolution, a choice made explicitly rather than by accident of windowing.[42][32] This page uses the closed-form sine tapers, a close relative with minimum local bias.[34]
The same kind of planted disturbance as in tab 6, now as a wave train of chosen period moving at a chosen velocity, plus two things that are not motion at all: an array-wide common signal, and a static gradient — a fixed north–south pattern of amplitude that rises and falls in time everywhere at once. Toggle common-mode removal (projection of the data onto the subspace orthogonal to the array mean, x → (I − 11T/N) x) and watch where the static pattern goes.
Polar slowness maps
Axes. The direction from the centre is the direction of travel (north up, east right); the distance from the centre is slowness, so the centre is infinite speed and the rim the slowest speed shown; rings are labelled in km/s. Colour: power in dB relative to each map’s own peak, 0 to −20 dB. Analysis: 256 one-minute samples, four sine tapers, the cross-spectral matrix averaged over the three Fourier bins nearest the chosen period, maps summed over those bins. Mainland stations.
A static pattern g(r) q(t) has no delays: every station oscillates in phase or in antiphase. Its wavenumber spectrum is the spatial spectrum of g, centred on k = 0 but broad. While the common mode is present it dominates and the maps peak at the centre (infinite speed). Project the array mean out and the zero-wavenumber part of g is gone; what remains is a north–south antisymmetric pattern, half the array against the other half. The steering vector closest to “north half +, south half −” is a wave with a wavelength of about twice the north–south extent, travelling north or south. The estimator reports it, with a speed set by that wavelength and the analysis period (v ≈ λ/T), not by anything moving. Antiphase between halves is symmetric under reversing the direction, so the map shows two opposite lobes or a ridge through the centre along the gradient; with noise, one wins.
Try: wave amplitude 0, common signal 20, gradient 15, remove common mode on. Then change the period and watch the “speed” change with it, the signature of a pattern rather than a mover.
The response of the array to a single plane wave at s0 is the beam pattern |(1/N) Σj eiω rj·(s − s0)|2, the point-spread function of every map above. Its main lobe width is about 1/(f D) in slowness for aperture D, so long periods give fat lobes. A regular grid of spacing d has exact replicas (grating lobes) every 1/(f d) in slowness: a wave of wavelength shorter than 2d is aliased. Irregular arrays smear the grating lobes into sidelobes.[36]
Array response pattern
Left: station layout (same scale for all three arrays). Right: the beam pattern in dB on the slowness plane out to 0.2 km/s. At 1–3 min periods the regular grid shows grating lobes inside the map; the IMAGE layout shows a forest of sidelobes instead.
An estimator always returns an answer. Whether the answer means anything depends on two questions that have to be asked before the data are looked at: could this array, with this sampling and noise, have recovered the thing if it were there? And how often does it report something just as convincing when nothing is there?
Four conditions bound what an array can resolve as motion. A structure of size w moving at speed v across an array of aperture D, station spacing d, sampling Δt, analysed in windows of length T:
The visible region
The bounds are order-of-magnitude, not sharp: near each line the estimate degrades gradually. The boxes show where some of the mechanisms of tab 5 sit, using published scales (TCV filament separation ~600 km at ~4 km/s; TIDs; Pc5 with m ≈ 4 at 65°, a wavelength of about 4000 km at ~14 km/s; the Sun-fixed sweep at continental scale).
The conditions above are rules of thumb. The real sensitivity of a specific pipeline, with its real geometry, filters and noise, is found by injection and recovery: plant synthetic signals with known heading and speed into data like the real data, run the identical pipeline, and record how often and how accurately it recovers them. The result is a map of where the method works. A null result is meaningful only inside the region where the sweep shows recovery; a positive result is plausible only if similar planted signals come back looking like it.
Heading × speed recovery sweep
Each cell: one planted disturbance (amplitude/noise as set, mainland array, 6-h window), the beam of tab 6 on a coarse heading × speed grid, peak taken as the estimate. Recovered = heading within ±30° and speed within a factor 1.5. Arrows show the estimated heading in each cell. One noise realisation per cell; a real sweep uses many and reports rates.
The other half is the false-alarm rate. Semblance from pure noise is not zero: with N stations and a finite record the maximum over a grid of trial velocities is biased upward by the search itself. A null distribution answers “how high does S get when there is no coherent motion?” One generic recipe keeps every trace intact and shuffles which station position each trace is assigned to. That preserves each station’s spectrum, amplitude and any common-mode content, and destroys only the geometric consistency of the delays. The observed peak is then ranked against the shuffled peaks. It is a cousin of the surrogate-data tests used for nonlinearity.[41]
Shuffled-array null
p = (1 + number of shuffles with peak ≥ observed)/(1 + shuffles). Try “common signal only”: the observed semblance is high, but so is every shuffle, because a common signal is coherent whatever the geometry, and the test correctly says “nothing geometric here”. With noise only, p is uniform between 0 and 1: about one run in twenty gives p ≤ 0.05 by chance, which is what a 5 % false-alarm rate means. A null tests one specific alternative; choose it to match the claim.
Check the test on data where the answer is known. Before trusting a pipeline on the case one cares about, run it where the answer is known: planted signals (does it find them?), shuffled or time-reversed data (does it stay quiet?), and known natural events with independent confirmation (does it agree?). Each look-alike of tab 5 is also a test case: if the pipeline reports Sq or a terminator crossing as a moving object, it will do the same in the data that matter. Decide the thresholds, the windows and the success criteria first; tune nothing on the case of interest.
Every DOI below was resolved on 5 November 2026 (doi.org redirect and Crossref record, checking author, year, title, journal and pages). Each entry says what it is used for on this page and how far it was checked: abstract checked means the statement attributed to it was compared with the abstract or an available text; record checked means the bibliographic record was verified and the attributed content is the paper’s standard, widely quoted result, not re-read for this page. Three sources have no DOI and are marked so.
Related on this annex: The Magnetosphere (animated introduction: solar wind, Birkeland currents, substorms) · Four-Way Satellite Convergence (what a constellation sees that a ground array cannot) · Next: Many Instruments, One Sky (joint inversion of ground, satellite and radar data).
All numbers on this page are either taken from the cited literature or generated live in your browser by the stated synthetic models; no measured data are shown except published station coordinates. Synthetic noise uses a seeded generator, so “New noise” gives a fresh but reproducible realisation.