Many Instruments, One Sky

More data than unknowns, and still not enough

The companion page, Reading the Sky from the Ground, asked what one ground array sees. This page asks the next question. Several instruments look at the same ionosphere: magnetometers on the ground, a magnetometer on a satellite a few hundred kilometres above the current, a radar that measures plasma velocity along its beams. Each samples the sky differently, each has its own noise, and none of them sees everything. How are they combined into one estimate of the current, and how does one say, with numbers, what that estimate can and cannot resolve?

The answer is linear inverse theory, and its central fact is easy to state and easy to forget. A problem can be overdetermined, with many more measurements than unknowns, and still be ill-posed: some combinations of the unknowns change the data so little that the noise hides them, and some change the data not at all. Adding more of the same measurement does not fix that. Adding a measurement that looks from a different direction can.

1.1The question each instrument answers

The rest of the page builds this up. Tab 2 writes down the forward models d = Gm for each instrument. Tab 3 introduces the standard basis for ionospheric currents, spherical elementary current systems, and shows exactly what the ground cannot see. Tabs 4 and 5 are the linear algebra: least squares, priors, the singular value decomposition, and the resolution matrix, which is the honest answer to “what does this estimate mean?”. Tab 6 combines instruments. Tab 7 is about checking an inversion before believing it. Tab 8 runs the inversion live on two hours of real IMAGE data from a published event.

The forward model: d = G m

Every instrument on this page is linear in the current. Write the unknown current as a vector m of amplitudes in some basis, the measurements as a vector d, and each measurement is a fixed linear combination of the amplitudes: d = Gm + e, with e the noise. Building G is physics; everything after it is linear algebra.

2.1A meridian slab

To see the instruments side by side, take the simplest geometry: currents that do not vary in the east–west direction, a thin sheet at height h = 110 km, and a meridian along which ground stations, a satellite track and a radar beam lie. The sheet is cut into 30 ribbons, 40 km wide, between 600 km south and 600 km north of the array centre. Each ribbon carries two unknowns: an eastward surface current density K_y (this one is divergence-free in this geometry) and a northward one K_x, which can only be steady if it is fed and drained by field-aligned currents (FACs) at the ribbon edges: it is the curl-free part. Coordinates are x north, y east, z down.

Ground and satellite, from K_y. By Biot–Savart, an eastward line current I at height h produces at a point a vertical distance ζ below it and a horizontal distance u north of it

Integrating over a ribbon of width x₂ − x₁ carrying K_y gives a closed form: B_x = (μ₀K_y/2π)[arctan(u₁/ζ) − arctan(u₂/ζ)] and B_z = −(μ₀K_y/4π) ln[(u₁²+ζ²)/(u₂²+ζ²)], with u_i = x − x_i. A ground station has ζ = h. A satellite at altitude 450 km has ζ = −340 km: the same formula with the sign of ζ reversed. So the horizontal component B_x reverses across the sheet, exactly so at equal distance above and below an infinite planar sheet, while the vertical component B_z does not, and at a real orbit, three times farther from the sheet than the ground, the geometry factors differ: the satellite sees a smoother, weaker version.

Satellite, from K_x. The northward current and its FACs form, in this geometry, a current system whose field is zero below the sheet (the two-dimensional form of Fukushima’s theorem, tab 3) and, above it, purely eastward and local: Ampère’s law across the sheet gives

This is the slab form of the standard single-satellite estimate of FACs from the east–west magnetic perturbation along the track.^[17] The ground has a whole column of zeros in G for every K_x: it is blind to it.

Radar, from both. A radar measures the line-of-sight component of the plasma drift v. In the F region the plasma drifts with v = E×B/B², so E = −v×B (the ideal-MHD relation). The current is tied to E through the height-integrated conductivity tensor,

with Σ_P, Σ_H the Pedersen and Hall conductances and b̂ the unit vector along B (down, in the north).^[16] Inverting (2.3) gives E from K, hence v, hence the line-of-sight velocity: a radar row of G touches both K_x and K_y, but only through an assumed conductance. Here Σ_P = 6 S, Σ_H = 12 S, |B| = 50 000 nT, and the beam points 40° east of north. The radar is a schematic instrument: no radar data appear on this page.

Spherical elementary current systems

On a sphere, any horizontal current field can be split into a divergence-free (DF) part and a curl-free (CF) part (the Helmholtz decomposition on the sphere). Amm built a basis for each part out of one elementary system per grid point, with closed-form magnetic fields. These spherical elementary current systems, SECS, are the standard way to turn scattered ground and satellite magnetometer data into a map of ionospheric current.^[1][2][21]

3.1The fields

For a DF system of amplitude I₀ at radius R (= 6371 + 110 km), the current density is J = (I₀/4πR) cot(θ/2) ê_φ, with θ the angular distance from the pole. Its field at radius r < R, with s = r/R, is^[2][21]

and above the sheet (r > R, s = R/r) B_r = (μ₀I₀/4πr)[s/√(…) − s], B_θ = −(μ₀I₀/4πr sinθ)[(1 − s cosθ)/√(…) − 1]. The page uses exactly these expressions; it also checks them, once, against a brute-force Biot–Savart integration of the current density (the readout below the interactive says how well they agree).

For the CF system with its radial FACs, the result is simpler still and follows from Ampère’s law on a circle of constant θ. Below the ionosphere the circle bounds a cap through which no current flows, so B = 0 exactly. Above it, the cap is crossed by the incoming pole current and by a fraction (1 − cosθ)/2 of the outgoing distributed FAC, which gives a purely toroidal (horizontal, east–west around the pole) field^[1]

The invisibility belongs to the whole system: horizontal CF current and its FACs. A horizontal CF current without its FACs is not invisible from the ground. It is not a physical steady current on its own (charge would pile up at the pole), but as a mathematical source it produces a ground field of the same order as the DF system; the toggle in the interactive shows it.

3.2Why the ground sees the DF part

Equation (3.3) is Fukushima’s theorem in SECS form:^[9] with radial field lines, the ground is exactly blind to the CF part with its FACs, and every ground datum is a function of the DF amplitudes alone. With real, inclined field lines (about 12° from vertical over Fennoscandia) the cancellation is no longer exact and the ground sees mainly, not only, the DF part; see tab 2 of Reading the Sky from the Ground. Two further cautions. First, the DF part is not the same thing as the Hall current: they coincide only if the conductance is uniform. With conductance gradients, Hall currents acquire a divergence and Pedersen currents a curl, and the DF part that the ground sees is a mixture.^[20][21] Second, the “equivalent current” mapped from ground data is the DF part of the real current, and nothing more. The checkbox below removes the FACs. The horizontal CF current alone then has a field at the ground, and in closed form: the outgoing FAC spread over the sphere is spherically symmetric and has no field, so the horizontal part is (3.3) minus the field of the incoming line current along the pole axis from R to infinity, B_φ^line = −(μ₀I₀/4πρ) [1 − (R − z)/√(ρ² + (R − z)²)] with ρ = r sinθ and z = r cosθ. A horizontal CF current without its FACs is not invisible.

The numbers make the point. Set the DF amplitude to zero and the CF amplitude to 150 kA: the ground readout says the largest ground field is 0 nT, not by approximation but by construction of the formulas, while the satellite sees an east–west field of tens to hundreds of nT. Tick the “without its FACs” box and the ground lights up. A satellite pass therefore adds information the ground cannot provide: the CF part, and, because it sits on the other side of the sheet, a different weighting of the DF part. Combining ground and satellite SECS in one inversion is the subject of tab 6.^[22][21]

Least squares, the SVD and priors

Divide each row of G and d by that datum’s standard deviation (from here on G and d mean these whitened versions). Least squares minimises ‖Gm − d‖². The singular value decomposition shows why that alone fails, and what to add.

4.1Normal equations and the SVD

The right singular vectors v_i are patterns of current; s_i is how strongly each pattern shows up in the data. Patterns with s_i = 0 form the null space: they can be added to any solution without changing the data at all (the CF columns of the ground block in tab 2 are the plainest example). Patterns with small but non-zero s_i are worse in practice: their coefficient u_i^Td/s_i divides the noise, which has unit size in every direction u_i, by a small number.^[15][3] For a current sheet seen from the ground, the singular values fall roughly exponentially with the spatial frequency of the pattern, because a field continued downward through a height h loses its structure of wavenumber k as e^−kh.

4.2Damping and smoothness as priors

Tikhonov regularisation adds a penalty: minimise ‖Gm − d‖² + λ‖Lm‖², so

With L = I (ridge or damped least squares), the solution is (4.1) with each term multiplied by a filter factor f_i = s_i²/(s_i² + λ): patterns the data see well pass, poorly seen ones are suppressed. With L a difference operator, rough patterns are penalised instead of large ones. The Bayesian reading makes the choice concrete:^[19] (4.2) is the posterior mean (and maximum) for Gaussian noise of unit variance and a Gaussian prior m ~ N(0, (λL^TL)⁻¹). The damping is a statement about how large or how rough you believed the current to be before you looked; its posterior covariance is C = (G^TG + λL^TL)⁻¹.

4.3Choosing λ: the L-curve and GCV

Plot the model norm ‖Lm_λ‖ against the residual norm ‖Gm_λ − d‖ on log axes as λ varies: the curve is usually L-shaped, and its corner balances fitting the data against fitting the noise.^[11] Generalised cross-validation picks the λ that would best predict a left-out datum, averaged over all data, without actually leaving any out:^[10]

Since the noise here is known (whitened to unit variance), a third rule is available: the discrepancy principle, choose λ so that ‖Gm − d‖² ≈ n_d. None of the three is a law; they usually agree within a factor of a few, and when they do not, that disagreement is itself information.^[12]

Things to try. Slide λ down to the left end: the estimate oscillates wildly while the residual barely improves, the classic noise amplification of (4.1). Slide it right: the estimate becomes a smooth, too-small bump and the residual grows past √n_d. In the right-hand panel, the data coefficients |u_i^Td| follow the singular values down and then level off at about 1: that floor is the noise, and the components below it carry no information (the discrete Picard condition^[12]). The filter factors switch off close to where the coefficients hit that floor. Switch the prior to smoothness: the narrow westward channel is penalised much more heavily than the broad jet, because it is rough, which is exactly what that prior says you believe.

Resolution and leakage

Any linear estimate is m̂ = G^♯d for some matrix G^♯, the generalised inverse actually used. Substitute d = Gm + e: m̂ = G^♯G m + G^♯e. The matrix R = G^♯G says exactly how the estimate is built from the truth.

For the undamped pseudo-inverse G⁺ = V_pΣ_p⁻¹U_p^T (keeping the p non-zero singular values), R = V_pV_p^T: a projection, equal to the identity only when the null space is empty. For the damped estimator (4.2), G^♯ = (G^TG + λL^TL)⁻¹G^T, and R = (G^TG + λL^TL)⁻¹G^TG is never the identity: damping buys stability by blurring, and R is the receipt.^[15][3] N, the data resolution matrix, says how much each datum predicts itself; its trace is the effective number of parameters the data determine.

Two readings of R. A row k is the averaging kernel of the estimate at cell k: m̂_k = Σ_j R_kjm_j + noise, a weighted average of the true current over the map. This is Backus and Gilbert’s idea: the data never give you the value at a point, only a known average around it.^[4] A column j is the point-spread function: what a unit current at cell j alone becomes in the estimate. The part of the column outside cell j is leakage: true current in one place reported in another. For ridge damping, R is symmetric and the two readings coincide; for a smoothness prior they do not.

5.1Posterior covariance, and what it does not include

In the Bayesian reading, the posterior covariance C = (G^TG + λL^TL)⁻¹ gives the uncertainty of each cell, and it grows wherever stations are missing. A frequentist would quote instead the covariance of the estimate under repeated noise, G^♯G^♯T, which is smaller: it leaves out the bias (R − I)m. The two are related by C = G^♯G^♯T + (I − R) (λL^TL)⁻¹(I − R)^T: the second term is the prior’s estimate of what the blurring hides.^[19] Neither covariance includes errors in G itself: the thin-sheet height, radial field lines, ignored induction. Those enter as bias, and only the checks of tab 7 can bound them.

Joint inversion: stacking instruments

Joint inversion is, formally, nothing more than stacking: put every instrument’s rows into one G, every datum into one d, and weight each block by its own noise covariance. The substance is in the weights, in the parts of m each instrument reaches, and in the assumptions some instruments bring with them.

6.1What each instrument adds

The satellite adds the CF part through B_y = μ₀K_x (2.2), along its track only, and it measures the DF part again from the other side of the sheet, with a different geometric weighting: the altitude separation between ground and satellite is what lets a combined inversion separate currents in the sheet from currents elsewhere, the principle behind ground–satellite SECS and the published frameworks below.^[22] The radar adds E, and links E to the current only through (2.3). This is where the conductance assumption enters, and the only place: the magnetometer rows of G contain no conductance at all. If the assumed Σ_H is wrong, the radar rows are wrong, and the inversion will fit them by distorting the current.

Two published frameworks do this at scale. AMIE (assimilative mapping of ionospheric electrodynamics) expands the electric potential in basis functions, ties every instrument to it through a conductance model, and solves a Bayesian least-squares problem with a prior covariance.^[16] Lompe (local mapping of polar ionospheric electrodynamics) does the same on a regional grid with SECS-like basis functions, combining ground and space magnetometers, radar velocities and other data, with explicit priors and resolution analysis.^[14] What follows is the same idea in the slab of tab 2.

Things to try. Ground alone: the DF half is resolved near the stations, the CF half has R = 0 and its estimate is the prior (zero) with the prior’s uncertainty: the data say nothing. Add the satellite: the CF diagonal lights up along the whole track and the DF band tightens. Then mis-weight the satellite: tell the inversion its noise is five times smaller than it is and watch its χ² per datum stay near 1 while the ground’s grows: the inversion now fits the satellite’s noise and distorts the current to do it. Finally, use ground and radar only, and move the assumed Σ_H away from 12 S: the CF estimate is biased, and nothing in the error bars says so, because the posterior covariance is conditional on G being right.

Checking an inversion

Resolution and covariance are computed from the model. They are only as good as the model. The checks below test the whole chain: they plant a current whose answer is known, invert it exactly as real data would be inverted, and compare. Nothing else says what a method can recover.

7.1Synthetic recovery: a drifting vortex

Plant a Hall-current vortex, a Gaussian blob of DF current, drifting east or west at a chosen speed across the IMAGE array. Forward-model it to the 23 stations with its own fine grid, add noise, invert each minute with the SECS grid of tab 5, and compare the recovered vortex with the planted one: its position, its peak current, and the speed one would infer from the recovered positions.

7.2Residuals, held-out stations and the prior

Three further checks, on the same planted vortex. Residual whiteness: if G and the noise model are right, the whitened residuals (d − Gm̂)/σ should look like independent unit-variance noise: no pattern across stations, χ² per datum near 1 minus the fraction of the data spent on parameters. Over-damping shows up as smooth, structured residuals; an unmodelled source as residuals concentrated at a few stations. Held-out stations: invert without one station and predict what it should have measured; a model that predicts its held-out data is worth more than one that only fits the data it was given. The leave-one-out error can be computed for every station and every λ without re-inverting, from d_k − G_km̂_(−k) = (I − H_kk)⁻¹(d_k − G_km̂), where H_kk is the station’s 3 × 3 block of N.^[10] Sensitivity to the prior: change the prior and see what changes; a feature that comes and goes with the prior is a feature of the prior.

The honest way to state what a method can recover is a table of such tests: for each planted speed, size and location, how well the method recovered it. Notice in particular what the recovered speed does as you change the damping. With too little damping the estimate fits the noise, small spurious amplitudes scattered over the grid pull the weighted centre toward the middle of the network, and the inferred drift is biased low: for the default vortex, about 0.3 instead of 0.5 km/s at log₁₀ λ = −2, against 0.49 km/s near χ² ≈ 1. Shrink the vortex to 60 km instead and χ² climbs above 2 at any damping: a current narrower than the pole spacing cannot be represented, and the misfit says so. That bias is a property of the array and the estimator, not of the sky, and it is only visible because the answer was known.

A real interval: 18 January 2008, 04–06 UT

Two hours of real 1-min IMAGE magnetometer data from the morning sector, the interval of a published study of equivalent currents associated with Pc5 pulsations during substorms.^[13] The page inverts it live, minute by minute, for the DF part with SECS on the same pole grid as tabs 5 and 7, from the horizontal components only. Nothing here goes beyond that paper or reproduces its analysis; it is the method of this page applied to public data.

References and glossary

Every DOI below was resolved through doi.org when this page was written (November 2026). Items with no DOI carry full bibliographic data and say so.

9.1References

1. Amm, O. (1997). Ionospheric elementary current systems in spherical coordinates and their application. J. Geomag. Geoelectr. 49, 947–955. doi:10.5636/jgg.49.947 — DF and CF SECS, CF with radial FACs (tabs 3, 5). Record checked.
2. Amm, O. & Viljanen, A. (1999). Ionospheric disturbance magnetic field continuation from the ground to the ionosphere using spherical elementary current systems. Earth Planets Space 51, 431–440. doi:10.1186/BF03352247 — DF SECS field formulas (3.1)–(3.2) (tab 3). Record checked.
3. Aster, R. C., Borchers, B. & Thurber, C. H. (2019). Parameter Estimation and Inverse Problems, 3rd edn. Elsevier. doi:10.1016/C2015-0-02458-3 — SVD, Tikhonov, resolution (tabs 4, 5). Edition DOI checked.
4. Backus, G. & Gilbert, F. (1968). The resolving power of gross Earth data. Geophys. J. R. Astron. Soc. 16, 169–205. doi:10.1111/j.1365-246X.1968.tb00216.x — averaging kernels (tab 5). Record checked.
5. Demmel, J. & Veselić, K. (1992). Jacobi’s method is more accurate than QR. SIAM J. Matrix Anal. Appl. 13, 1204–1245. doi:10.1137/0613074 — the one-sided Jacobi SVD this page computes with. Record checked.
6. Finnish Meteorological Institute. IMAGE magnetometer network: conditions of use (rules of the road). space.fmi.fi/image/www/index.php?page=rules_of_road (read 7 Nov 2026). No DOI. — acknowledgement wording and station licences (tab 8). Page read.
7. Finnish Meteorological Institute. IMAGE stations: geographical and CGM coordinates. space.fmi.fi/image/www/index.php?page=stations. No DOI. — station coordinates (all tabs), as on the companion page.
8. Friis-Christensen, E., Lühr, H. & Hulot, G. (2006). Swarm: A constellation to study the Earth’s magnetic field. Earth Planets Space 58, 351–358. doi:10.1186/BF03351933 — a low-orbit magnetometer mission of the kind the synthetic satellite imitates (tabs 2, 3, 6). Record checked.
9. Fukushima, N. (1976). Generalized theorem for no ground magnetic effect of vertical currents connected with Pedersen currents in the uniform-conductivity ionosphere. Report of Ionosphere and Space Research in Japan 30(1–2), 35–40. No DOI (pre-DOI report series). — Fukushima’s theorem (tabs 2, 3).
10. Golub, G. H., Heath, M. & Wahba, G. (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215–223. doi:10.1080/00401706.1979.10489751 — GCV (4.3), the leave-one-out identity (tabs 4, 7). Record checked.
11. Hansen, P. C. (1992). Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review 34, 561–580. doi:10.1137/1034115 — the L-curve (tab 4). Record checked.
12. Hansen, P. C. (1998). Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM. doi:10.1137/1.9780898719697 — filter factors, discrete Picard condition, parameter-choice rules (tab 4). Record checked.
13. Kauristie, K., Uspensky, M. V., Kleimenova, N. G., Kozyreva, O. V., van de Kamp, M. M. J. L., Dubyagin, S. V. & Massetti, S. (2016). Equivalent currents associated with morning-sector geomagnetic Pc5 pulsations during auroral substorms. Ann. Geophys. 34, 379–392. doi:10.5194/angeo-34-379-2016 — the published interval of tab 8. Record checked.
14. Laundal, K. M., Reistad, J. P., Hatch, S. M., Madelaire, M., Walker, S., Hovland, A. Ø., Ohma, A., Merkin, V. G. & Sorathia, K. A. (2022). Local mapping of polar ionospheric electrodynamics. J. Geophys. Res. Space Physics 127, e2022JA030356. doi:10.1029/2022JA030356 — Lompe (tab 6). Record checked.
15. Menke, W. (2018). Geophysical Data Analysis: Discrete Inverse Theory, 4th edn. Academic Press (Elsevier). doi:10.1016/C2016-0-05203-8 — generalised inverses, model and data resolution (tabs 4, 5). Edition DOI checked (the DOI record carries the short title “Geophysical Data Analysis”).
16. Richmond, A. D. & Kamide, Y. (1988). Mapping electrodynamic features of the high-latitude ionosphere from localized observations: Technique. J. Geophys. Res. 93(A6), 5741–5759. doi:10.1029/JA093iA06p05741 — AMIE; the conductance tensor linking E and K (tabs 2, 6). Record checked.
17. Ritter, P., Lühr, H. & Rauberg, J. (2013). Determining field-aligned currents with the Swarm constellation mission. Earth Planets Space 65, 1285–1294. doi:10.5047/eps.2013.09.006 — FACs from the east–west perturbation along a track (tab 2). Record checked.
18. Tanskanen, E. I. (2009). A comprehensive high-throughput analysis of substorms observed by IMAGE magnetometer network: Years 1993–2003 examined. J. Geophys. Res. 114, A05204. doi:10.1029/2008JA013682 — IMAGE network reference (tab 8). Record checked.
19. Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM. doi:10.1137/1.9780898717921 — the Bayesian reading, posterior covariance (tabs 4, 5). Record checked.
20. Untiedt, J. & Baumjohann, W. (1993). Studies of polar current systems using the IMS Scandinavian magnetometer array. Space Sci. Rev. 63, 245–390. doi:10.1007/BF00750770 — equivalent currents and their relation to the Hall current (tab 3). Record checked.
21. Vanhamäki, H. & Juusola, L. (2020). Introduction to spherical elementary current systems. In M. W. Dunlop & H. Lühr (eds.), Ionospheric Multi-Spacecraft Analysis Tools, ISSI Scientific Report Series 17, Springer, 5–33. doi:10.1007/978-3-030-26732-2_2 — SECS review, formulas, DF/Hall caveat (tab 3). Record checked (online 2019).
22. Vanhamäki, H., Amm, O. & Viljanen, A. (2003). One-dimensional upward continuation of the ground magnetic field disturbance using spherical elementary current systems. Earth Planets Space 55, 613–625. doi:10.1186/BF03352468 — ground-to-space continuation with SECS (tabs 3, 6). Record checked.
23. Weygand, J. M., Amm, O., Viljanen, A., Angelopoulos, V., Murr, D., Engebretson, M. J., Gleisner, H. & Mann, I. (2011). Application and validation of the spherical elementary currents systems technique for deriving ionospheric equivalent currents with the North American and Greenland ground magnetometer arrays. J. Geophys. Res. 116, A03305. doi:10.1029/2010JA016177 — SECS validated with synthetic recovery tests (tab 7). Record checked.

9.2Glossary

Averaging kernel

A row of the model resolution matrix: the weights with which the true model is averaged into one estimated value.

CF / DF

Curl-free and divergence-free parts of a horizontal current on a sphere (Helmholtz decomposition). The CF part needs field-aligned currents to be steady.

Conductance

Height-integrated conductivity of the ionosphere, in siemens. Pedersen (Σ_P, along E) and Hall (Σ_H, perpendicular to E and B).

Data resolution matrix

N = GG^♯: how each predicted datum is built from the observed data. Its trace is the effective number of parameters.

Discrepancy principle

Choose the damping so that the residual is as large as the known noise, no smaller.

Equivalent current

The DF current in a thin sheet that reproduces the ground magnetic field. Equals the Hall current only for uniform conductance.

FAC

Field-aligned current: current along the magnetic field, connecting the ionosphere to the magnetosphere.

Filter factor

s_i²/(s_i² + λ): how much of each singular component ridge damping lets through.

Fukushima’s theorem

Radial FACs closed by a CF horizontal current produce no magnetic field below the ionosphere (for radial field lines).

GCV

Generalised cross-validation: a rule for choosing the damping that approximates leave-one-out prediction error.

Generalised inverse

Any matrix G^♯ with m̂ = G^♯d; the pseudo-inverse G⁺ is one, the damped inverse another.

Ill-posed

A problem whose solution does not depend continuously on the data, or is not unique; discretely, a problem with a null space or very small singular values.

Inverse crime

Testing an inversion with synthetic data made by the same discretised model it inverts with, which hides discretisation error.

L-curve

Log model norm against log residual norm as the damping varies; its corner is a common choice of damping.

Leakage

True structure at one place appearing in the estimate at another: the off-diagonal part of a column of R.

Null space

Model patterns m with Gm = 0: invisible to the data, unrecoverable without a prior.

Picard condition

Data coefficients |u_i^Td| must fall faster than the singular values for the naive solution to make sense; where they flatten at the noise level, information ends.

Point-spread function

A column of R: the estimate produced by a unit spike of true model.

Posterior covariance

(G^TC_e⁻¹G + C_m⁻¹)⁻¹: the uncertainty of a Gaussian-prior estimate, conditional on G being right.

Resolution matrix

R = G^♯G: the linear map from the true model to the expected estimate.

SECS

Spherical elementary current systems: DF and CF basis currents on a sphere, one per grid pole, with closed-form magnetic fields.

SVD

Singular value decomposition G = UΣV^T; columns of V are model patterns, singular values say how strongly the data see them.

Tikhonov regularisation

Adding λ‖Lm‖² to the misfit; equivalent to a Gaussian prior.

Whitening

Dividing data and rows of G by the noise standard deviation (or by C_e^−½) so that all noise has unit variance.