{
  "id": "r16_resolution_bound",
  "order": 26,
  "tag": "mechanism",
  "eyebrow": "16 \u00b7 The resolution bound",
  "title": "Phase precision is not the limit, and improving it changes nothing",
  "question": "SAR measures micro-motion to hundreds of micrometres from a single pass. Why is that not enough?",
  "finding": "Because precision is not the binding constraint. This dwell's own velocity floor is 77 micrometres per second, about 13 times better than the roughly 1000 micrometres per second reported for single-pass micro-motion on bridges, so this work agrees with that literature rather than disputing it. The limit is a trade between two uses of one knob. Reaching a feature of size s through the ground needs an elastic wave no longer than about twice its size, so at least Vs over 2s in frequency, and recording that frequency needs sub-apertures short enough to sample it; but azimuth resolution is inversely proportional to integration time, so every factor gained in sampling rate is paid for in lateral resolution. Requiring both leaves s squared at least equal to azimuth resolution times aperture length times shear speed, putting the smallest resolvable feature here at 52 m, and never below 27 m across every assumption worth making. That rules out every chamber actually surveyed at Giza. The Subterranean Chamber is 8 m across, needs 138 Hz to reach, and at that sampling rate each pixel spans 340 m, about 43 times wider than the chamber. The claimed 10 m shafts miss by 27 times. The one target that clears this particular bound is the claimed 79 m chamber, which is larger than 52 m, so this test does not exclude it and is not offered as if it did; that claim fails on the depth axis instead, which repeats every 27.4 m and cannot place anything at 648 m. Nothing about phase precision, coherence or signal-to-noise appears in the inequality, so improving any of them does not move it.",
  "limitations": "This is an analytic bound computed from the acquisition's own geometry, not a measurement, and it is a necessary condition rather than a sufficient one: clearing it would not by itself make the method work, since the depth axis and the coherence limit are separate failures. It assumes a half-wavelength resolution criterion, which is the standard rule but can be beaten somewhat by a well-posed inversion with strong prior information. It assumes resolution scales inversely with integration time and that non-overlapping sub-apertures give one independent sample each, both of which are standard but idealised. It says nothing about methods that do not work this way, such as multi-pass interferometry over years, ground-based geophysics, or muon tomography.",
  "method": "Aperture length, platform speed, slant range and wavelength were read from the ICEYE Giza product. Azimuth resolution was taken as the more favourable of the geometric value 0.050 m and the delivered pixel spacing 0.044 m. The bound is the square root of that resolution times the 24.8 s aperture times shear speed, evaluated at 2200 m/s and swept across 1500 to 2500 m/s and across azimuth resolutions from 0.02 to 0.5 m. The velocity floor is the measured value from R4; the comparison figure is the published single-pass bridge result.",
  "figures": [
    {
      "file": "/Users/eugene/prj/sar/results/r16_resolution_bound/r16_trade.png",
      "caption": "<b>Sampling fast enough to hear a chamber makes the pixel too wide to find it.</b> Left: the blue line is the lateral resolution left after sampling at a given rate, and the orange line is the smallest feature an elastic wave at that rate can interact with. They cross at 52 m, which is the smallest thing this acquisition could resolve by this route even with a perfect instrument. Every surveyed chamber at Giza sits far below that line. Right: the same targets, by the factor their pixel exceeds their own size once the sampling requirement is met."
    },
    {
      "file": "/Users/eugene/prj/sar/results/r16_resolution_bound/r16_assumptions.png",
      "caption": "<b>No choice of assumptions brings it down to chamber size.</b> The bound is the square root of azimuth resolution times aperture length times shear speed. Sweeping the azimuth resolution from 0.02 to 0.5 m and the shear speed across the range limestone can plausibly take, it never falls below 27 m. The shaded band is the size of the chambers actually under Giza."
    }
  ],
  "metrics": {
    "aperture_s": 24.762808,
    "platform_speed_m_s": 7661.5329854710735,
    "slant_range_m": 606873.6299081678,
    "wavelength_m": 0.031228381041666666,
    "azimuth_resolution_geometry_m": 0.04994616815216092,
    "azimuth_pixel_m": 0.04397846814199768,
    "azimuth_resolution_used_m": 0.04994616815216092,
    "shear_speed_m_s": 2200.0,
    "shear_speed_range_m_s": [
      1500.0,
      2500.0
    ],
    "smallest_resolvable_m": 52.16297747476544,
    "smallest_resolvable_low_m": 43.07216106061448,
    "smallest_resolvable_high_m": 55.605920824307816,
    "grid_minimum_m": 27.25590284690639,
    "targets": [
      {
        "name": "Typical mastaba shaft",
        "size_m": 1.5,
        "depth_m": 12.0,
        "frequency_needed_hz": 733.3333333333334,
        "sampling_needed_hz": 1466.6666666666667,
        "sub_aperture_s": 0.0006818181818181818,
        "lateral_resolution_m": 1813.9841460219247,
        "resolution_shortfall": 1209.3227640146165,
        "resolvable": false
      },
      {
        "name": "Khufu's Subterranean Chamber",
        "size_m": 8.0,
        "depth_m": 30.0,
        "frequency_needed_hz": 137.5,
        "sampling_needed_hz": 275.0,
        "sub_aperture_s": 0.0036363636363636364,
        "lateral_resolution_m": 340.1220273791108,
        "resolution_shortfall": 42.51525342238885,
        "resolvable": false
      },
      {
        "name": "Khufu's King's Chamber",
        "size_m": 5.2,
        "depth_m": 43.0,
        "frequency_needed_hz": 211.53846153846152,
        "sampling_needed_hz": 423.07692307692304,
        "sub_aperture_s": 0.0023636363636363638,
        "lateral_resolution_m": 523.2646575063244,
        "resolution_shortfall": 100.62781875121622,
        "resolvable": false
      },
      {
        "name": "Osiris Shaft, lowest level",
        "size_m": 9.0,
        "depth_m": 30.0,
        "frequency_needed_hz": 122.22222222222223,
        "sampling_needed_hz": 244.44444444444446,
        "sub_aperture_s": 0.00409090909090909,
        "lateral_resolution_m": 302.3306910036541,
        "resolution_shortfall": 33.592299000406015,
        "resolvable": false
      },
      {
        "name": "ScanPyramids Big Void",
        "size_m": 20.0,
        "depth_m": 70.0,
        "frequency_needed_hz": 55.0,
        "sampling_needed_hz": 110.0,
        "sub_aperture_s": 0.00909090909090909,
        "lateral_resolution_m": 136.04881095164436,
        "resolution_shortfall": 6.802440547582218,
        "resolvable": false
      },
      {
        "name": "Claimed Khafre shaft",
        "size_m": 10.0,
        "depth_m": 648.0,
        "frequency_needed_hz": 110.0,
        "sampling_needed_hz": 220.0,
        "sub_aperture_s": 0.004545454545454545,
        "lateral_resolution_m": 272.0976219032887,
        "resolution_shortfall": 27.20976219032887,
        "resolvable": false
      },
      {
        "name": "Claimed Khafre chamber",
        "size_m": 79.0,
        "depth_m": 648.0,
        "frequency_needed_hz": 13.924050632911392,
        "sampling_needed_hz": 27.848101265822784,
        "sub_aperture_s": 0.03590909090909091,
        "lateral_resolution_m": 34.442736949783374,
        "resolution_shortfall": 0.43598401202257436,
        "resolvable": true
      }
    ],
    "measured_floor_um_s": 77.39,
    "literature_floor_um_s": 1000.0,
    "precision_margin_over_literature": 12.92156609381057,
    "runtime_s": 0.5734879970550537
  },
  "date": "2026-09-23"
}