Full case study: /case-studies/port-metasurface

Observable

At the current stage the observables are matrix properties derived from simulation, not deployment metrics.

The quantities I can measure directly are:

  • Condition number $\kappa(A)$
  • Singular value spectrum
  • Discrimination matrix $D_{i,j}$
  • Leakage $1 - D_{i,j}$
  • Reconstruction error under controlled noise

Bench measurements are planned but not yet completed.
All results so far are forward-model-based.

Working claim (bounded)

Boundary geometry may change the identifiability of the inverse scattering problem.

Formally, if boundary diversity changes the sensing matrix

\[A \rightarrow A_{\text{combined}},\]

and

\[\kappa(A_{\text{combined}}) < \kappa(A_{\text{flat}})\]

or reduces discrimination leakage,

then boundary diversity improves identifiability within this controlled geometry.

This does not imply deployment-level improvement in container ports.

Load-bearing constraints

Physical

Passive structures obey reciprocity and energy conservation.

They can redirect energy but cannot increase total power.

The reconstruction model assumes the Born approximation

\[\mathbf{y} = A\mathbf{x}.\]

This assumption is valid only while scattering remains weak and linear.

At 26 GHz ($\lambda \approx 11.5$ mm), small geometric shifts produce significant phase changes.
Interference patterns therefore depend sensitively on boundary orientation.

Environmental

Real port environments are non-repeatable.

Container placement, crane motion, and ground reflections vary across time.
Multiple channel realizations may produce similar link measurements.

This introduces a non-identifiability constraint:

different environments can explain the same measured improvement.

That uncertainty is why the project shifted from coverage optimization to conditioning analysis.

Modeling

The simulation domain is deliberately minimal.

  • one occluding wall
  • one hidden cylinder
  • one boundary panel
  • one probe plane

The sensing matrix

\[A \in \mathbb{R}^{4681 \times 11}\]

therefore represents a low-dimensional discrete inverse problem.

Additional modeling simplifications:

  • PEC boundaries approximate ideal phase masks
  • metasurface unit-cell physics is not yet included
  • object position is discretized into 11 candidate locations

Continuous object reconstruction is outside the current scope.

Numerical

Flat boundary baseline:

κ(A_flat) = 30.02

Tilted boundary (Config 6):

κ(A_tilted) = 46.38

Additional numerical observations:

  • First singular mode dominates variance
    (~92% flat, ~94% tilted)

  • Within-Group-B column correlations reach

0.93 → 0.99
  • Minimum singular value remains well above the noise floor
σ_min ≈ 350 V/m
√λ ≈ 2

Thus the system is not numerically unstable.

Higher per-configuration $\kappa$ does not necessarily degrade reconstruction.
The relevant question is whether multiple configurations reduce structural degeneracy when combined.

Synthetic ceiling tests (solver applied to its own columns) therefore establish an upper bound, not deployment robustness.

Fabrication (pending)

Simulations assume ideal PEC boundaries.

Real metasurfaces introduce additional effects:

  • phase quantization
  • ohmic loss
  • fabrication tolerances
  • angular sensitivity

These effects are not yet included in the inverse framework.

Primary limiting factor

Structural degeneracy in the sensing matrix.

The dominant limitation is column similarity rather than noise or rank deficiency.

Within-Group-B degeneracy produces leakage:

\[\text{leakage}_{i,j} = \frac{|x_j|}{|x_i|}\]

Baseline leakage is approximately

~10⁻³

Boundary diversity must reduce worst-case leakage to claim improved discrimination.

An additional constraint emerges from the tilted configuration:

Group membership itself changes between configurations.

Different positions become degenerate under different boundaries.

Therefore per-configuration leakage comparisons are not meaningful.
The correct comparison is

A_combined leakage vs single-configuration leakage

What this ruled out for me

  • treating simulated coverage gain as deployment evidence
  • claiming improvement without measuring conditioning
  • increasing algorithmic complexity when conditioning is moderate
  • optimizing geometry without evaluating inverse performance
  • interpreting per-config $\kappa$ as a direct performance metric

What remains unresolved

  • whether the combined matrix reduces $\kappa(A)$ below the flat baseline
  • whether boundary diversity reduces worst-case leakage
  • sensitivity of conditioning to boundary angle
  • fabrication deviations from ideal PEC behaviour
  • extension to continuous object positions
  • cross-configuration reconstruction (train on one geometry, test on another)

What would reduce uncertainty

  • Complete tilted CST sweeps (Config 6 complete)
  • Complete stepped CST sweeps (Config 7 in progress)

Then construct

\[A_{\text{combined}} = \begin{bmatrix} A_{\text{flat}} \\ A_{\text{tilted}} \\ A_{\text{stepped}} \end{bmatrix}\]

Next steps:

  • compare $\kappa(A_{\text{combined}})$ to the baseline
  • evaluate discrimination matrices element-wise
  • measure worst-case leakage reduction
  • perform cross-configuration reconstruction tests
  • validate PEC approximation with a simple fabricated boundary

Only after those steps can boundary diversity be claimed to improve identifiability.

Status

Flat boundary inverse analysis complete.
Tilted boundary (Config 6) — sensing matrix, inversion, robustness complete.
Stepped boundary (Config 7) — CST sweeps in progress.
Combined matrix analysis pending completion of Config 7.

Fabrication remains optional depending on simulation outcome.

What surprised me

I initially expected fabrication accuracy and EM modeling fidelity to dominate the difficulty.

Instead the dominant issue is identifiability.

Even with perfect simulation fidelity, improvements cannot be attributed unless conditioning changes are measured directly.

That realization shifted the project away from geometry optimization and toward inverse problem structure.

A secondary observation:

Higher per-configuration condition numbers do not necessarily degrade reconstruction.

Config 6 has

κ(A) = 46.38

yet produces the same PSF and robustness metrics as the flat baseline.

The condition number reflects the geometry’s sensitivity structure, not whether the solver can handle it.