Kenneth ChengMathematical modeling

Topology · percolation · observation error

Can a camera create a connectivity threshold that was never there?

Begin with a known synthetic pond field. Blur it, reduce its resolution, and displace its segmented boundary. The latent geometry stays fixed, but its measured components, holes, spanning state, and narrowest resolvable bridge may not.

Run the modelRead the research articleAll interactive models

The mathematical object

Water level builds a nested family of shapes.

A scalar topography becomes a binary pond mask by thresholding. Raising the level can add pond pixels, merge components, close ice islands, and eventually produce a path across the image.

01

Sublevel filtration

Ph={(x,y):z(x,y)≤h}P_h=\{(x,y):z(x,y)\le h\}

The control uses a rank-normalised level, so h is close to the latent pond-area fraction.

02

Connected pieces and holes

χ(Ph)=β0(Ph)−β1(Ph)\chi(P_h)=\beta_0(P_h)-\beta_1(P_h)

β0\beta_0 counts pond components; β1\beta_1 counts enclosed ice holes under the selected dual-connectivity convention.

03

Observation operator

P^=Bd ⁣(Dr(Kσ∗1Ph))⊕ε∂\widehat P=\mathcal B_d\!\left(\mathcal D_r(K_\sigma * \mathbf 1_{P_h})\right)\oplus\varepsilon_\partial

Blur and block averaging are followed by a signed boundary displacement and then boundary-label errors before topology is measured.

The lab pairs pond 4-connectivity with ice 8-connectivity by default. This complementary convention avoids counting a diagonal contact as connected in both phases at once. The alternative 8/4 convention is included as an explicit sensitivity check.

Interactive topology model

Melt-pond topology observation lab

Change water level, resolution, and segmentation error on one 96 × 96 synthetic pond field, then watch connectivity, holes, and throat width change.

This is a teaching-scale synthetic morphology experiment, not a satellite classifier, an in-situ percolation measurement, or an Arctic-climate forecast.

Smooth, rounded basins with a long correlation scale.
Observation and topology settings
Latent reference fieldThe 96 × 96 binary field before observation error
The 96 × 96 binary field before observation error
What the camera seesBlurred, downsampled, segmented, then enlarged for display
Blurred, downsampled, segmented, then enlarged for display
Observed pond fraction
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Components β₀
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Enclosed holes β₁
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Euler characteristic χ
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Directional span
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Boundary dimension
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Both-direction throat radius
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Observed grid
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Latent versus observed topology

QuantityLatentObservedChange

Fixed observation order

The same fixture moves through these stages; a control change never silently swaps the underlying field.

  1. 01Decode fixture
  2. 02Shift / mirror
  3. 03Correlate
  4. 04Rank
  5. 05Threshold
  6. 06PSF blur
  7. 07Block average
  8. 08Displace boundary
  9. 09Boundary error
  10. 10Display expand

Four investigations

Change one part of the camera at a time.

  1. Find a latent transition. Set resolution to 1, blur and errors to zero, then raise water level until the field first spans. Record whether the first span is horizontal or vertical.
  2. Coarsen without changing the pond. Hold the level and seed fixed. Move from a 96 × 96 observation to 48 × 48, 24 × 24, and 12 × 12. Watch for a span that appears, disappears, or changes direction.
  3. Separate blur from displacement. Blur may close a narrow gap; positive displacement expands the segmented pond boundary, while negative displacement erodes it. Test whether the two errors cancel or reinforce one another.
  4. Stress the convention. Switch from pond 4 / ice 8 to pond 8 / ice 4. If the verdict changes, the claimed threshold depends on digital topology rather than only physical geometry.

Model boundary

A controlled distortion experiment, not an Arctic reconstruction.

The twelve fixtures contain only deterministic synthetic 96 × 96 scalar fields: four Matérn-style topographies, four germ–grain coalescence fields, and four Ising-style morphology shifts. No satellite, drone, or field-survey pixels are shipped to the browser.

The boundary-dimension value is a finite-grid box-counting estimate, not proof of a scale-free fractal law. The throat measure is a grid-based maximum-bottleneck clearance, not a hydraulic discharge or drainage prediction. A spanning path is a geometric event inside one image window, not evidence of a system-wide physical percolation transition.

Reproducibility contract

What is fixed

  • All fields are 96 × 96, uint8, row-major, and carry SHA-256 hashes.
  • The operator order is fixed from decode through display expansion.
  • Boundary errors use an indexed integer hash, not an unrecorded browser random stream.
  • Twelve expected summaries lock Python/JavaScript-compatible topology fixtures.

Research basis

Selected starting points.

The experiment extends an established melt-pond geometry and topology literature by focusing specifically on observation-induced threshold error.

  1. Hohenegger et al. (2012), “Transition in the fractal geometry of Arctic melt ponds,” The Cryosphere.
  2. Popović et al. (2018), “Simple Rules Govern the Patterns of Arctic Sea Ice Melt Ponds,” Physical Review Letters.
  3. Offord et al. (2022), “Topological Data Analysis Detects Percolation Thresholds in Arctic Melt-Pond Evolution,” arXiv.
  4. Fuchs and Birnbaum (2023), “Orthomosaics and surface type classifications of MOSAiC Leg 4 floe (2020-06-30, 2020-07-22),” PANGAEA.
  5. Niehaus and Spreen (2022), “Melt pond fraction on Arctic sea-ice from Sentinel-2 satellite optical imagery (2017–2021),” PANGAEA.