{
"id": "t03_mechanism",
"order": 3,
"tag": "mechanism",
"eyebrow": "03 \u00b7 Mechanism",
"title": "Surface spacing predicts the apparent depth",
"question": "What exactly does the steering matrix respond to, if not motion?",
"finding": "In the two-scatterer examples, peak depth tracks azimuth separation at approximately 8.86 model-metres per metre. Farther scatterers can contribute outside the patch, and the sampled depth axis folds high spatial frequencies back by aliasing.",
"limitations": "Ideal point scatterers and a prescribed spectral model. The wall's comb fit depends on reflectivity: the ideal-wall fitted spacing differs from the prediction, while the randomized variants are closer. These examples identify a mechanism in this reconstruction, not the cause of a particular published feature.",
"method": "Clutter-free two-scatterer scenes at 1, 2, 3, 4 m separation and at 3 to 40 m; a 60-line periodic wall in three degrees of realism; the realistic wall at three headings with a harmonic-comb fit of the level spacing. Same bank and patch as T1; transect spectra averaged over 160 pixels.",
"figures": [
{
"file": "results/t03_mechanism/figs/t03_two_scatterers.png",
"caption": "The mechanism in its simplest form. Two point scatterers a distance \u0394x apart in azimuth, nothing else. The reference/offset shift oscillates with the sub-aperture position at a rate proportional to \u0394x, and the steering-matrix 'depth' lands at \u0394x \u00d7 \u03bb\u209b sin\u03b8 / \u03bb = 8.9 \u0394x. A pair of rocks 3 m apart is a 'structure' 27 m underground."
},
{
"file": "results/t03_mechanism/figs/t03_wall.png",
"caption": "A periodic wall. Dotted lines mark the predicted course harmonics. The fitted fundamental is about 6.9 m for the two randomized variants versus a 7.0 m prediction; the ideal-wall fit is about 9.8 m. The prediction is not equally accurate in every scene."
},
{
"file": "results/t03_mechanism/figs/t03_heading.png",
"caption": "Rotate the same wall. Fitted spacings in these three simulations track the projected-spacing prediction. This is a controlled geometry test; a comparison of real acquisitions must also account for visibility, resolution, and registration."
},
{
"file": "results/t03_mechanism/figs/t03_nonlocal.png",
"caption": "The 'structure' under a pixel is made of things tens of metres away. One scatterer at the patch centre and a second one at 3, 6, 12, 20 or 40 m. The offset band differs from the reference band by a sliver only 88 Hz wide, whose impulse response is about 80 m long, so scatterers well outside the 32-pixel patch modulate the measured shift as strongly as neighbours do. Separations beyond the depth axis' Nyquist limit fold back: the 20 m and 40 m scatterers appear at 107 m and 71 m, exactly where aliasing predicts. This is why every scene fills the whole depth axis regardless of patch size (T4)."
}
],
"metrics": {
"depth_per_metre_separation": 8.86215107245308,
"two_scatterers": [
{
"separation_m": 1.0,
"predicted_depth_m": 8.86215107245308,
"measured_depth_m": 7.73665788625154
},
{
"separation_m": 2.0,
"predicted_depth_m": 17.72430214490616,
"measured_depth_m": 16.33294442653103
},
{
"separation_m": 3.0,
"predicted_depth_m": 26.586453217359242,
"measured_depth_m": 27.508116928894363
},
{
"separation_m": 4.0,
"predicted_depth_m": 35.44860428981232,
"measured_depth_m": 36.10440346917385
}
],
"wall": [
{
"variant": "ideal: identical courses 0.79 m apart",
"comb_spacing_m": 9.774436090225564,
"predicted_spacing_m": 7.001099347237934,
"comb_score": 1.4523606563956928
},
{
"variant": "random block amplitudes (\u03c3 = 0.5 ln)",
"comb_spacing_m": 6.887218045112782,
"predicted_spacing_m": 7.001099347237934,
"comb_score": 1.383734272567367
},
{
"variant": "random amplitudes + diffuse face scatter",
"comb_spacing_m": 6.867167919799499,
"predicted_spacing_m": 7.001099347237934,
"comb_score": 1.3223366670523309
}
],
"heading": [
{
"heading_deg": 0.0,
"predicted_spacing_m": 7.001099347237934,
"comb_spacing_m": 6.887218045112782,
"comb_score": 1.383734272567367
},
{
"heading_deg": 15.0,
"predicted_spacing_m": 7.248071390880014,
"comb_spacing_m": 7.228070175438596,
"comb_score": 1.4532032161584922
},
{
"heading_deg": 30.0,
"predicted_spacing_m": 8.08417318550227,
"comb_spacing_m": 8.030075187969924,
"comb_score": 1.7374910817293938
}
],
"nonlocal": [
{
"distance_m": 3.0,
"shift_rms_millipx": 2.97069647357642,
"unaliased_depth_m": 26.586453217359242,
"aliased_prediction_m": 26.586453217359242,
"measured_depth_m": 26.648488274866413
},
{
"distance_m": 6.0,
"shift_rms_millipx": 10.1385346198171,
"unaliased_depth_m": 53.172906434718485,
"aliased_prediction_m": 53.172906434718485,
"measured_depth_m": 53.29697654973283
},
{
"distance_m": 12.0,
"shift_rms_millipx": 9.063771874603326,
"unaliased_depth_m": 106.34581286943697,
"aliased_prediction_m": 106.34581286943697,
"measured_depth_m": 106.59395309946565
},
{
"distance_m": 20.0,
"shift_rms_millipx": 12.134144072986828,
"unaliased_depth_m": 177.24302144906162,
"aliased_prediction_m": 106.34581286943697,
"measured_depth_m": 106.59395309946565
},
{
"distance_m": 40.0,
"shift_rms_millipx": 8.897343084926979,
"unaliased_depth_m": 354.48604289812323,
"aliased_prediction_m": 70.89720857962465,
"measured_depth_m": 70.4895496302918
}
],
"runtime_s": 11.974185943603516
},
"date": "2026-09-19",
"provenance": {
"python": "3.11.3",
"numpy": "2.4.6",
"source_sha256": "6e9f11cbcd20b771a05cdc99324b15d766d33cdd2b5dee5041a7ea27e8b93104"
}
}