Proof · what any way of reading one image could learn, and what it assumes
The bound, finite and stated
Can one satellite radar image reveal what is underground (whether something is there, where it is, its shape) even with its depth uncertain? Within a stated model this can be bounded for every algorithm at once. One auditable comparison, five worlds identical but for the cavity, shows how far: under the ground’s own trembling or beside a bouncing lorry, no method reading one image comes near reliable detection; nor, computed, of the structures claimed under Khafre. The model, its inputs and what is assumed are set out below; the physical inputs are models and are marked as such.
status · proved within a stated model; physical inputs not certified; the noise floor specified, not measured · benchmark revision 3, 1 October 2026
Within the model, one radar image cannot reliably reveal a shallow room’s presence, place or shape, under the ground’s own trembling or beside a bouncing lorry; at every specified noise floor, not even a detector told every scatterer can
Reliable detection means 95% found at 5% false alarms. Every method, tomograms and trained networks included, is a function of the image and inherits what the image holds, so a bound on the image bounds them all at once. The numbers above are bounds within a stated model (one geology, fully developed speckle, the declared excitation), not what a particular reader achieves; the benchmark below says what each rests on, which parts are computed, which are measured in the real image, which are specified and which are assumed. The structures claimed under Khafre, computed (P2-38: the eight shafts and the void 1,220 m down, under the ground’s own trembling): any reader of one image at most 2.4 × 10⁻⁵ above chance, the oracle 6.3 × 10⁻⁴; the motion would have to grow 3.4 × 10⁴ times, 96 after allowing a louder ground, site amplification and a cube’s larger moment.
One auditable comparison: five worlds identical but for the cavity, the same lorry beside each, the whole image read by any method
Revision 3, after its third review (only the analysis changed; revision 2 is kept at commit afee349, its errors listed in the erratum); specification in BENCHMARK.md, run P2-36. The lorry is a truck over a bump 15 m west at the FTA’s level, held at 69 Hz for the whole 25 s pass (a frequency fixed by a rule before the bounds); the quiet case is Giza’s regional trembling.
| the question | quiet: any method | lorry: any method | lorry: computed field only (70 m) | lorry: told every reflectivity |
|---|---|---|---|---|
| Is a 6 m room there? | 5.0% | 8.4% · band 9.2% | 5.6% | 18% · 23.7 dB: 29% |
| Is an L-shaped tunnel there? | 5.0% | 11% · band 12% | 6.0% | 29% · 23.7 dB: 47% |
| Which of two places, 6 m apart? | 5.0% | 9.6% · band 9.6% | 5.8% | 22% · 23.7 dB: 36% |
| A room, or an L-shaped tunnel? | 5.0% | 11% · band 11% | 6.0% | 26% · 23.7 dB: 44% |
| Is a 10 m room there? | 5.0% | 17% · band 17% | 6.7% | 47% · 23.7 dB: 74% |
| the claim under Khafre, computed (P2-38): representative structures at the claimed depths, the 2025 pass, the pyramid omitted | ||||
| Are there eight shafts, 10 m across, to 640 m? | 5.0% | not computed (under the pyramid) | — | 5.1% |
| Is there an 80 m void 1,220 m down? | 5.0% | does not reach it | — | 5.0% |
Found at 5% false alarms, at most (reliable detection needs 95%). Green: within five points of chance. “Any method”: the certificate applied one along-track line at a time, told the excitation and both worlds’ whole motion but not the ground’s grain, each line’s reference holding the lorry’s own wave; it bounds every reader of the one image, the published one included. “Band”: the largest over the sampled band (every 2 Hz). “Told every reflectivity”: an oracle no single image gives, at 18.4 dB (the site’s median backscatter measured in the image over the best noise floor ICEYE specifies for the product’s mode, documentation 6.0.8: a scenario, not a measurement; the older 6.0.0 figure gives 12.7 dB and at most 27%) and at 23.7 dB for ground at 0 dB; where its bound stops excluding (28 dB) this argument no longer excludes, which shows no detector succeeding there. The claim’s rows: under the ground’s own trembling; the oracle weighted by the image’s own brightness measured over 2 km. The lorry’s differential motion would have to grow 8 times for the single-image bound to stop excluding.
Computed, measured, specified, assumed
Computed: each world’s motion by the lab’s elastic solver within 70 m; the per-line certificate, checked against the exact divergence on lines (0.18 of it at most); the oracle’s formula pulse by pulse on fixed scenes (0.994 ± 0.006, unchanged with the lorry’s wave ten times larger).
Measured in the image, over a specified floor: the ground’s backscatter is measured; the noise is not (neither product carries a floor), so the signal-to-noise ratio takes ICEYE’s best specified figure for the product’s mode, 18.4 dB, the most favourable to a detector; the image’s darkest cells bound it only from below (5.9 dB).
Assumed: the scattered wave beyond 70 m, carried undiminished to the image’s edge (98% of the lorry case’s signal; measured falling as r-0.4 out to 70 m; five times larger, still at most 31%); the lorry’s level held all pass; one geology; fully developed speckle (bright points are covered by the oracle only).
Achieved: the score test, told the pattern, on synthesised images with one noise law, its threshold from 400 calibration grounds and read on 100 + 100 others: with the difference amplified 800 times it finds 73% (95% interval 63–81%) at 15% false alarms, AUC 0.91 (predicted 0.93). The threshold targeted 5% false alarms but gave 15% (9–24%) on the held-out grounds, which remains unexplained: the discrimination is supported by the AUC, calibration at a 5% operating point is not established. At the real level 7% at 15% false alarms (p 0.98), AUC 0.47: at chance.
Read as images: a detector told the lorry finds every world once the difference is large enough; our implementation of the published method, with its stated readout, finds none at any level
| the cavity’s difference | told the lorry: names the world (chance 33%) | room or room 6 m east | room or L tunnel | our implementation of the published method, blind: names | its presence AUC |
|---|---|---|---|---|---|
| as it is | 33% (p = 1) | 8 of 16 | 8 of 16 | 33% (p = 1) | 0.52 |
| amplified 200 times (a diagnostic) | 63% (p < 0.001) | 11 of 16 | 12 of 16 | 33% (p = 1) | 0.53 |
| amplified 1000 times (a diagnostic) | 100% (p < 0.001) | 16 of 16 | 16 of 16 | 33% (p = 1) | 0.52 |
P2-37: the benchmark’s worlds synthesised as images on the real dwell, 8 independent grounds (each its own speckle, trembling and lorry timing; the grounds are the experimental units), the room, the L tunnel and the room 6 m east on each, with controls; p from the true worlds permuted within grounds. The told detector succeeding at the positive control shows the images hold recoverable information, which the tested tomogram-and-readout pipeline does not recover; p = 1 means a reader named the same world for every image of a ground, its answer unchanged by the cavity. A diagnostic of our implementation of the published method (from the paper, not the authors’ software) with a frozen plan-map readout, on a smaller scene than P2-36’s, without receiver noise and with the motion field truncated at 70 m.
Why it holds: six finite steps, none assuming a resonance peak or a particular algorithm
Processing cannot add distinguishability
For any test, detection rate − false-alarm rate ≤ TV(P₁, P₀), and for equal priors the best accuracy is ½ + TV/2. Any channel applied the same way under both hypotheses, a sub-aperture bank, a tomogram, a renderer, a trained network, contracts TV. So 95% found at 5% false alarms needs TV ≥ 0.9, and a proved TV below it excludes that target for every method at once.
Proof
Split p₁ − p₀ into positive and negative parts of equal mass TV. A test 0 ≤ t ≤ 1 integrates to at most TV against it; the indicator of the positive part attains it. A processor K pulls back to the test y ↦ E[t(K(y))], so it can do no better on K’s output.
The exact Gaussian divergence, bounded
For proper complex Gaussians, KL = tr(C₀⁻¹C₁) − d − log det(C₀⁻¹C₁) + δm*C₀⁻¹δm. With E the whitened covariance change and its eigenvalues ej > −1, KL = Σ(ej − log(1 + ej)) + δm*C₀⁻¹δm, and since e − log(1 + e) ≤ e²/(2 min(1 + e, 1)), for equal means (speckle is zero-mean; otherwise the mean term is added) and ‖E‖ < 1, KL ≤ ‖E‖F²/(2(1 − ‖E‖)). Pinsker and Bretagnolle–Huber turn KL into TV. No expansion in the displacement is made.
The whole image given the room’s largest phase
If ‖C₀−½ΔA‖F ≤ q, ρ = 2q + q² bounds ‖E‖ exactly (E = DB* + BD* + DD*), so KL ≤ ρ²/(2(1 − ρ)) and TV ≤ min(√(KL/2), √(1 − e−KL)). A phase-only change obeys |e−iΦ − 1|² ≤ Φ², so q² ≤ Σ|T₀|²Φ² / λmin(C₀) ≤ m P²/λmin: the peak phase P at every one of the image’s 1.31 billion pixels, over its 703 million retained bins. A background motion bounded the same way proves λmin(C₀) ≥ [max(0, 1 − √Qbg)]² (plus the receiver noise’s variance, where it is counted) = 0.0559.
A valid reference under any shaking
Widen the image’s range band to every bin (the real image is a function of the widened one, so this can only add information): each along-track line is then independent, and KL adds over lines. On each line, C₀ = B₀B₀* + σ²I with B₀ = E∘e−iφ₀, E orthonormal, so λmin(C₀) ≥ σ² + [max(0, 1 − ‖B₀ − E‖F)]² (where the motion’s term fails, the receiver noise alone still floors it) with that line’s own motion in B₀: the lorry’s wave, the ground’s trembling at its worst realisation, world 0’s own cavity. The whole image’s floor failed under the lorry (its common phase energy over 1.3 billion pixels exceeds one); one line’s is at least 0.90. Theorem C on each line, summed, bounds every reader of the one image. Tested against the exact divergence on random lines and on the benchmark’s own (0.18 of the certificate).
Told everything but the answer
Given the reflectivity, the background motion, the shaking and both models, the raw echoes are Gaussian with the receiver noise’s covariance and means that differ by the echo of the room-specific motion alone: TV = erf(√d/2) exactly, d the noise-normalised energy of that difference. Averaged over what the oracle was told (only independent uniform phases needed), TV ≤ erf(√(E d)/2); the background motion cancels; 95% at 5% needs E d ≥ 5.41. Checked against pulse-by-pulse echoes on the real dwell: 0.993 ± 0.025 of the formula. The bound is on the average over scenes; for one scene, by Markov’s inequality, d ≤ t E d with probability at least 1 − 1/t (the spread measured on fixed scenes is a few per cent, P2-29, P2-36). Every realisation of a random excitation, rather than its expectation, costs at most √(2n) in amplitude for n equally loud independent directions. A mixture of alternatives (any prior over rooms and layouts) is no easier to detect than the prior’s average of its members, TV(P₀, Σ wᵢPᵢ) ≤ Σ wᵢ TV(P₀, Pᵢ): a bound that holds for every member holds for the mixture, while one hard member does not make a mixture hard (one hard pair defeats a uniform guarantee, not every prior). N images, where their independence is justified, raise the divergence N times, so the ceiling √(KL/2) grows at most √N times; the TV itself need not follow that ceiling.
Two worlds a method cannot tell apart
For two admissible worlds at depths Δ apart, any depth estimate has equal-prior mean absolute error at least (Δ/2)(1 − TV). One indistinguishable pair is enough to rule out reliable depth over any class that contains it.
What speckle adds
For speckle, motion shared by the scene leaves the image’s distribution unchanged, so only what differs from place to place counts: the Fisher information F = Σ|Φ̃(f − f′; f) − Φ̃(f − f′; f′)|², with a power-series remainder ε bounding the rest, ‖E‖ ≤ √F + ε (P2-25). This is far tighter than the certificate, and assumes the reference covariance is the identity, where the certificate proves its floor with the background in it.
Theorems A–E are from an independent derivation made for this lab and checked in its code (sim/sarsim/finite.py, with its tests); the tight layer is sim/sarsim/information.py. Background: Duchi’s lecture notes on total variation, data processing, Pinsker and two-point testing.
The question is broader than detection: presence, horizontal location, shape, relative vertical geometry and absolute depth are scored apart
Does one complex SAR acquisition hold reproducible information about what is actually underground, where it is or its shape, that survives controls and predicts withheld ground, even if its depth stays uncertain? A method that drew a branching tunnel or a room’s layout at one of two possible depths would be a major success, so an uncalibrated or aliased depth scale does not by itself dismiss a method. Each claim is judged on its own evidence.
| claim | what bounds or tests it | where it stands |
|---|---|---|
| Presence: a room here or not | the three layers below, which bound every processor within the model | under the ground’s own trembling, no useful detection (5.02% found at 5% false alarms, at most, for the bench room); with a lorry beside a room under a 5 m roof, one image lets any method find at most 8.4% at 5% false alarms (an L tunnel 11%, a 10 m room 17%); a detector told every reflectivity, at the best specified noise floor, 18% (the benchmark); the structures claimed under Khafre, computed, 5.0% (P2-38) |
| Horizontal location | the same bounds: telling a room here from a room there is a test between two worlds, and TV(A, B) ≤ TV(A, none) + TV(none, B) | bounded wherever presence is; in the benchmark, the room or the same room 6 m east, with a lorry beside it: no test between the two places finds the right one more than 9.6% of the time at 5% false alarms, and with the two equally likely no method names the right place more than 52.3% of the time (½ + TV/2); any estimate’s mean error over the pair is at least 2.9 m of the 3 m that guessing leaves; in the shape test (P2-31, below), where the data hold a layout (amplified to 20 rad) a reference detector told the shaking places it 0.6 m from its centre, but at 2 rad cannot find it by scanning; the published picture read blind centres 7.1 m off even at 20 rad, the controls 6.2 m |
| Shape: a room, an L-shaped or a branching tunnel | the same inequality between any two layouts; the shape test (P2-31) on physically generated scenes, scored blind | bounded under the ground’s own trembling (P2-32); in the benchmark, the room or the L tunnel with a lorry beside them, one image at most 11% at 5% false alarms; in the shape test the reference detector names 100% of layouts at 20 rad and 89% at 2 rad when told where to look; the published picture read blind 44% at 20 rad, chance at the real level |
| Relative vertical geometry | whether deeper parts are drawn deeper, whatever the scale | not shown: the published axis is the registration’s frequency (P2-06, P2-28) |
| Absolute depth | the axis’s calibration, and whether different depths leave different marks | uncalibrated and aliased in the published method; physically, in this model, the mark’s shape changes with depth (below), so the physics does not rule depth out; but detecting a room does not imply telling its depth, and each pair of depths is bounded on its own (Theorem E, P2-29, P2-30) |
A bound on detection rate − false-alarm rate is a bound at every operating point: found ≤ false alarms + TV. Under the regional microseism envelope that leaves no prospecting value within the model, even for ranking sites (TV ≤ 1.7 × 10⁻⁴); where the upper bound is large, as for strong shaking close by, the argument does not exclude prospecting value short of 95% at 5%; that is not evidence that any method has it, which only an implemented detector succeeding on independent tests would show. The defensible conclusion is that useful recovery is excluded under these explicitly bounded conditions, not that vibration-based mapping from one image can never work.
What stays open: bright persistent ground beside strong shaking, the far field, the physical envelopes, the noise floor, and measurement
| open | why | what would close or open it |
|---|---|---|
| Bright persistent ground beside strong shaking: an important unresolved case within the benchmark | at 40 dB, which stands for a building, a corner reflector or the pyramid’s edge, the oracle’s bound is 0.951 to 1.000 for every strong pair, and the speckle layer does not apply there: a shallow room beside a lorry under a bright persistent target is excluded by no layer of this benchmark | a diffuse-plus-bright-scatterer model fitted to the image itself, with Khafre’s geometry and the specified noise |
| The rest of the real acquisition | the benchmark is one geology and five shallow layouts in fully developed speckle; a shallow room is not by itself an upper bound on larger or deeper structures; the claim is computed under the ground’s own trembling with the pyramid omitted (its load, coupling and scattering; the direction of its effect not established), its sizes and layout representative, the shafts’ column model matched to the solver empirically | the pyramid in the model; ground motion measured on the plateau during a pass |
| The envelope beyond 70 m | 97–98% of the strong case’s signal is an assumed surface wave carried undiminished to the image’s edge; the measured fall to 70 m supports it, the ×5 test is robustness, not verification | the scattered field measured to 100 m or more; attenuation measured in the plateau’s rock |
| The physical envelopes | the imprint is the lab’s linear solver in uniform bare rock (P2-04, P2-26) and, for the claim, a damped half-space at 0.2 Hz; the ambient levels are regional; other excitations of the deep structure (a lorry’s body waves, local sources) and nonlinear or parametric coupling are not modelled | ground motion measured on the plateau during a pass; a proponent’s quantitative prediction of the room-specific motion |
| The noise floor | neither product carries one, and ICEYE’s specification changed between versions (Dwell Fine at best −18 dB in documentation 6.0.0, −23.7 dB in 6.0.8); every oracle row takes the most favourable, 18.4 dB, the older figure 12.7 dB; the image’s darkest cells bound the ratio only from below (5.9 dB) | the applicable noise for this acquisition from ICEYE, or a noise-only record for the product |
| Measured validation | the geophone-and-satellite comparison (P2-22, P2-23) is simulated | shaker or reflector data with instruments on them; surveyed voids and matched controls, blind |
A method that succeeds on well-designed, withheld ground would show that the measurements or these inputs deserve revision; the bounds are written so such a revision can be made, not to protect a conclusion. The test that would settle it is frozen and blind: motion recovered from a measured moving reflector first, then withheld cavities and controls predicted with settings fixed in advance (What would settle it). What would help most from the method’s proponents is a predicted room-specific surface motion, with its source, footprint, amplitude and phase, losses and duration, which can be put straight into these formulas without releasing any software.
In detail: each layer and check, folded
Telling one layout from another is bounded the same way, and moving a reconstruction up or down cannot rescue it
the calculation, in full
If two layouts leave nearly the same image in distribution, no algorithm can tell them apart, however it reconstructs. A uniformly reliable shape mapper must separate two layouts even after the depth freedom it is allowed; the closest admissible pair of depth variants is no easier than the pair at the same depth, so the same-depth bound holds for every allowance. Each layout’s imprint is the lab’s solver (statically for the ambient levels, dynamically for a lorry close by), and the bound is on the difference between two layouts’ imprints (P2-32).
| which of the two? | tight (microseisms) | oracle (microseisms) | oracle’s bound, a lorry beside it all pass | signal over detection’s |
|---|---|---|---|---|
| A branching tunnel, or none | 1.4 × 10⁻⁹ | 5.6 × 10⁻⁷ | 0.25 at 71 Hz | 1 (detection) |
| An L-shaped tunnel, or a straight one | 2.6 × 10⁻⁹ | 3.9 × 10⁻⁷ | 0.25 at 66 Hz | 1.3× |
| A column with a spiral, or a plain column | 9.1 × 10⁻⁹ | 5.3 × 10⁻⁷ | 0.12 at 53 Hz | 4.1× |
| An L-shaped tunnel, or a branching one | 2.6 × 10⁻⁹ | 4.2 × 10⁻⁷ | 0.30 at 68 Hz | 1.2× |
| An isolated room, or none | 1.9 × 10⁻⁹ | 8.2 × 10⁻⁷ | 0.22 at 56 Hz | 1 (detection) |
Detection rate − false-alarm rate, at most. The tight bound and the certificate take each pair through empty ground, TV(A, B) ≤ TV(A, none) + TV(none, B), each layout certified against the same reference, since subtracting two layouts’ motions does not remove the reference covariance; the oracle uses the difference itself, where, told the reflectivity and the background, it is exact. The loose certificate with the background caps every pair at 5.1% found at 5% false alarms under the microseisms. “Signal over detection’s”: how many times the signal that detects the first layout is needed to tell the two apart. Shallow, bench-scale layouts in uniform rock: more favourable than anything claimed, and only these.
The shape test: where the data hold a layout, a reference detector names and places it; the published picture, read blind, was not shown to do either
Each layout’s imprint is the lab’s solver; the image is the synthesizer’s, with the unrelated ground and the shaking matched across layouts and controls (no cavity, and a motionless copy), three realisations each (P2-31). The reference detector is told the shaking: it names a layout by its statistic at the true place and places it by scanning every placement. The published method’s plan map is read blind; its change against the same ground without the cavity is a diagnostic it never has. Amplified levels are diagnostics, not predictions.
| imprint | reference: names | reference: scan peaks from the centre | our implementation of the published method, blind: names | its top places from the centre | published change (diagnostic): names | its top places |
|---|---|---|---|---|---|---|
| real level (the microseisms) | 33% | 38 m | 33% | 6.2 m | 33% | 13 m |
| 0.3 rad, amplified | 67% (p 0.06) | 41 m | 33% | 6.4 m | 67% (p 0.04) | 6.1 m |
| 2 rad, amplified | 89% (p 0.01) | 38 m | 33% | 7.5 m | 67% (p 0.04) | 6.1 m |
| 20 rad, the positive control | 100% (p 0.00) | 0.6 m | 44% | 7.1 m | 44% | 12 m |
| chance (the controls) | 33% | 38 m | 33% | 6.2 m | 33% | 13 m |
Nine images per row: three layouts on each of three independently drawn grounds; the three grounds are the independent units, and the layouts on one ground are paired comparisons. Distances are medians; p is a null significance test, the true layouts permuted within each ground over every arrangement, shown where below 0.2 and not corrected for the dozen comparisons; it is not a confidence interval for performance. The paired chance is the motionless copy’s change (the no-cavity image’s change against itself is zero). At 2 rad the reference detector’s response at the true place stands a median 1.7 spreads above the scatter over placements: enough to name a layout when told where to look, not enough to find it by scanning the image. Depth, for the published change after the predeclared family of transformations (12 candidate depths): 1.1 m from the truth at 20 rad, 1.3 m for the motionless copy; closeness alone says little.
The boundary: the ground’s own trembling rules out useful recovery within the model, and in the benchmark so does a lorry beside a shallow room for one image; a reference image at high SNR, bright persistent points and unspecified mechanisms stay open
the calculation, in full
| conditions | presence | location and shape | absolute depth |
|---|---|---|---|
| Giza’s regional trembling, any reader, one image | excluded: ≤ 5.02% found at 5% (certificate), 4.7 × 10⁻⁷ for the oracle | excluded: each pair ≤ 5.1% at 5% | excluded with presence; aliased in the published method |
| The noisiest ground on Earth | excluded (tight 9.3 × 10⁻⁸) | excluded with presence | excluded with presence |
| A lorry beside a room all pass, known exactly, one image | excluded in the benchmark: ≤ 0.03 for a room under a 5 m roof (near chance), ≤ 0.06 for an L tunnel (95% at 5% excluded); the scattered wave carried to the image’s edge undiminished | excluded in the benchmark: ≤ 0.05 (6 m), ≤ 0.06 (shape) | about as hard as detection in this model (1.4× its signal, P2-29) |
| The same, told every scatterer’s reflectivity (the oracle) | 0.13 at the best specified 18.4 dB; its bound stops excluding only at 38 dB; P2-29’s plane wave at the worst of four sides gives 0.55 under a 5 m roof, 0.12 for the bench room. Upper bounds, not detections | 0.17 and 0.21; 0.30 for P2-32’s hardest pair | not computed |
| Any mechanism with gain, not yet specified | not covered: it must state the motion it adds over the room (source, footprint, amplitude and phase, losses, duration); the requirement curve above says how much is needed | ||
Limits within a stated measurement and physical model
The theorems and the three layers above: finite, algorithm-independent, and scoped to fully developed speckle (or any texture, for the oracle), the Doppler-to-time relation, the lab’s linear elastic imprints and the regional ambient levels.
Limits for a defined class of realistic archaeological settings
Feasible, with stronger physical validation: measured ground motion at the site during a pass, measured coupling over known voids, a qualified observation model and noise floor, and the open regime examined by experiment rather than by bound.
That every conceivable single-image application has no archaeological value
Too broad for this work: mechanisms, environments and acquisitions outside the model are not covered, and the open regime is open. The outcome sought is this boundary, not a verdict fixed in advance.
Three layers, one answer for the ground’s own trembling; under strong shaking the whole-image references fail, and the benchmark’s per-line certificate replaces them
the calculation, in full
| shaking | level | certificate (P2-30) | tight (P2-25) | oracle (P2-29) |
|---|---|---|---|---|
| Giza microseisms, regional | Kottamya, 67 km east | 1.7 × 10⁻⁴ background included | 1.3 × 10⁻⁹ static imprint, every direction | 4.7 × 10⁻⁷ |
| The world's noisiest stations | Peterson’s high-noise model | 2.9 × 10⁻³ reference covariance I | 9.3 × 10⁻⁸ static imprint, every direction | 3.4 × 10⁻⁵ |
| Giza 1-3 Hz, regional | Kottamya, 67 km east | 1.8 × 10⁻⁴ background included | 1.3 × 10⁻⁸ static imprint, every direction | 5.0 × 10⁻⁷ |
| Giza 3-8 Hz, regional | Kottamya, 67 km east | 1.6 × 10⁻⁴ background included | 2.8 × 10⁻⁸ static imprint, every direction | 4.3 × 10⁻⁷ |
| Urban background, bench room | FTA, 8–100 Hz | 2.7 × 10⁻³ surrogate: reference without the traffic’s own motion | 1.6 × 10⁻³ surrogate, likewise | 8.3 × 10⁻³ |
| A lorry over a bump 15 m away, bench room | FTA, all pass | 0.031 surrogate: reference without the lorry’s own motion | 0.023 surrogate, likewise | 0.12 |
| The same lorry, a room under a 5 m roof | FTA, all pass | 0.19 surrogate: reference without the lorry’s own motion | 0.12 surrogate, likewise | 0.55 |
Each entry bounds detection rate − false-alarm rate; with two equally likely cases the best accuracy exceeds a coin toss by half of it. The certificate and the tight bound hold within fully developed speckle and the Doppler-to-time relation (checked pulse by pulse, P2-25); the oracle holds for any texture with independent phases, at 18.4 dB per resolution cell (the site’s median backscatter measured in the image over the best noise floor ICEYE specifies for the product’s mode, −23.7 dB in documentation 6.0.8: a scenario, not a measurement). The ambient levels are regional, not upper limits at the pyramids. Under traffic or a lorry the ground without a room already moves with the source, so these rows’ certificate and tight bound (no such motion in the reference) are restricted surrogates, not bounds; the whole image gives no floor with the lorry’s wave in it, but one along-track line does, and the benchmark above bounds a single image that way. The oracle, whose mean echoes differ only by the added motion, stays valid, with the scattered wave carried undiminished across the scene at the worst of four sampled sides and the grid checks’ allowances. On small test images the best test comes within 1.4 times of the tight bound: these are bounds, not what a detector achieves.
What a mechanism must supply: a large enough displacement integral over the ground, whatever its pattern; smooth patterns need far more
the calculation, in full
Corrected on 30 September after two independent reviews. The solid curves were first described as the motion any mechanism must supply; they hold only for the smooth pattern computed. The requirement for every pattern is on the displacement integral E = ∫ maxt |Δulos|² dA (m⁴: an integrated squared displacement envelope, not a mechanical energy). Under strong shaking the whole-image certificate has no valid reference (the ground without a room already moves with the source), so its curve applies to quiet ground; the oracle’s applies under any shaking, and the benchmark above bounds a single image under a lorry line by line.
Solid: exact within the model for one smooth (Gaussian) pattern (P2-27), at 0.2 and 80 Hz; 2, 5 and 20 Hz lie between them in the run (20 Hz almost on the 80 Hz line). Dotted (the oracle with receiver noise, valid under any shaking) and dashed (the certificate with Giza’s background, quiet ground only): every pattern and time history, by the displacement integral, drawn as the temporal peak displacement of a motion filling the footprint evenly (a spatial rms of temporal peaks), the same at every frequency. Below a line the bound excludes 95% at 5%; between a dotted line and a solid one only the smooth pattern is excluded. Crossing a line shows neither detectability nor usefulness.
Phase modulation, nonlinear coupling, anything
The bounds use the full complex change of the image, phase included; no spectral peak is needed. Any mechanism, linear or not, reaches one image only through the motion it adds. An oracle told the reflectivity, the excitation and both models, with receiver noise at the best specified 18.4 dB per cell, cannot reach 95% at 5% while that motion’s displacement integral stays below 1.5 × 10⁻⁸ m⁴, whatever its pattern or time history, and whatever motion both worlds share, which cancels from its mean echoes; below 2.2 × 10⁻¹¹ m⁴ it stays within five points of chance. Every method told less inherits this. Under quiet ground the speckle certificate, with Giza’s background in the reference, gives 1.4 × 10⁻⁹ m⁴ without assuming receiver noise.
The pattern matters as well as the size. A Gaussian 4 m wide at 0.2 Hz, multiplied by a cosine of 4 m period along track, holds 39 times its information though nowhere larger (489 times for a 1 m period). The Gaussian’s information is 4.7 × 10⁻⁴ of the general upper bound for its envelope, 4N⟨Φ²⟩, which is a bound, not an attainable optimum; at 80 Hz it is 49%.
A change of phase alone moves the ground by at most twice the motion it acts on, 2a. To escape this exclusion, under its stated amplitude and noise, such a mechanism must change the phase over more than Emin/(2a)²: 858,457 m² under Giza’s regional microseisms, a disk of radius 523 m. That is necessary to escape this particular bound, not sufficient for detection. Whether a physically justified cavity-induced field, tails included, reaches the threshold is the table’s question below. A mechanism with gain is not excluded: it needs its output specified (source, footprint, amplitude and phase, losses, duration) before it can be tested.
Try an envelope in the certificate
The loose certificate (Theorem C): zero-mean Gaussian texture, phase-only change, continuum area. Inputs are assumptions, not measured motion.
| the bench room’s modelled imprint | short of a smooth pattern of its peak and footprint | short of the oracle’s requirement, any pattern | short of the certificate’s (quiet ground only) | oracle’s bound on found − false alarms | outcome |
|---|---|---|---|---|---|
| static, Giza microseisms | 8.3 × 10⁸ | 4.6 × 10⁵ | 1.4 × 10⁵ | 2.8 × 10⁻⁶ | near chance |
| static, the world's noisiest stations | 1.1 × 10⁷ | 6.4 × 10³ | 2.0 × 10³ | 2.0 × 10⁻⁴ | near chance |
| static, Giza 1-3 Hz | 8.1 × 10⁷ | 4.4 × 10⁵ | 1.4 × 10⁵ | 3.0 × 10⁻⁶ | near chance |
| dynamic, truck over a bump 15 m away, 20 Hz | 1.1 × 10³ | 15 within 39 m only: 130 | — (no valid reference under the lorry) | 0.09 | 95% at 5% excluded |
| dynamic, truck over a bump 15 m away, 80 Hz | 1.0 × 10³ | 9.97 within 39 m only: 110 | — (no valid reference under the lorry) | 0.13 | 95% at 5% excluded |
| room under a 5 m roof, dynamic, truck over a bump 15 m away, 20 Hz | 210 | 3.87 within 39 m only: 28 | — (no valid reference under the lorry) | 0.33 | 95% at 5% excluded |
| room under a 5 m roof, dynamic, truck over a bump 15 m away, 80 Hz | 140 | 1.73 within 39 m only: 17 | — (no valid reference under the lorry) | 0.66 | 95% at 5% excluded |
“Short” is how many times the imprint’s amplitude would have to grow before the bound stops excluding 95% at 5%. At every operating point found ≤ false alarms + the bound: below 0.05 the reader is within five points of chance; below 0.9, 95% at 5% is excluded but weaker, possibly useful performance is not; at 0.9 or more the bound does not decide (unresolved), which is not a demonstrated detection. The static imprint is bounded over every strain direction and beyond the solver’s grid by Eshelby’s void as Okada point sources, scaled to the solver (1.62 times at most inside the grid, times 1.25), with the grid check’s 10% and a random field’s peak (3/√2 times its rms amplitude). The dynamic imprint counts the whole 39 m disk, untapered, at the worst of four sampled incidence sides (others are not computed), and beyond it the scattered wave by energy-flux conservation for a surface wave without attenuation, to the scene’s farthest corner, with the grid check’s allowance. Under the lorry the margin rests on how far the scattered wave carries: counted within 39 m it is several times larger.
Depth is carried by the mark’s shape in this model: rooms spread 15 m down leave 71% of a deeper room’s mark unmatched; the published method’s axis does not use it
the calculation, in full
An obstruction sought, and not found
Could shallower voids mimic a deeper one? Small rooms 15 m down, spread over the ground in any non-negative mix, were fitted to the surface mark of the bench room 30 m down on the lab’s solver (two solved together agree with their sum to 3.7%): 71% of the mark is left unmatched, and the bench room 15 m down alone leaves 78%. In uniform rock the mark changes shape with depth, not only size, so telling these depths apart takes only 1.4 times the signal detection takes. Theorem E's floor on any method's mean depth error, where detection just reaches 95% at 5%, is then only 1.8 m of their 15 m separation. Richer classes (layered rock, fractures, voids of other shapes) could still mimic one another; that is not tested.
Its depth follows the shaking, not the room
Run whole over the bench room 15 m down, the same room 30 m down and a granite block, each imprint boosted to 2 rad, the published focusing spreads each change over its whole axis: 16% of the 15 m room’s change lies between 12 and 18 m and 16% of the 30 m room’s between 27 and 33 m, where 15% and 15% of the axis lie. Horizontally its change does gather over the room (2.6 times elsewhere), but a random perturbation of the same pixels gathers almost as much (2.4 times): in this paired comparison the picture changes where the image changes. Its depth coordinate is an uncalibrated focus coordinate, the frequency of the registration series (P2-06): a calibration and aliasing problem, which a method that recovered the right layout would survive. It does not recover the layout: in the shape test (above) its blind map names a layout no better than guessing even where a reference detector names every one.