Technical Appendix

v_sound = 331.3 × sqrt(1 + T/273.15) × (1 + humidity_correction)

Where:
  T = 29.44°C (85°F)
  h = 0.37 (37% relative humidity)
  P = 855.6 hPa (at 1404m elevation)

Cramer humidity correction applied for h, T, P.

Result: v = 349.7 m/s

Sensitivity:
  ∂v/∂T ≈ 0.6 m/s per °C
  At ±2°F (±1.1°C): Δv = ±0.7 m/s
  At 100m propagation: Δt = ±0.58ms

Given N receivers at positions (x_i, y_i) and a source at (x_s, y_s):

  d_i = sqrt( (x_s - x_i)² + (y_s - y_i)² )

  TDOA_ij = (d_i - d_j) / v_sound

Each TDOA defines a hyperbola with foci at receivers i and j.
With N=4 receivers and 1 reference, we get 3 independent TDOAs.
3 hyperbolas → unique 2D intersection (overdetermined: 3 equations, 2 unknowns).

Levenberg-Marquardt minimizes:

  χ² = Σ_i [ (TDOA_measured_i - TDOA_predicted_i) / σ_i ]²

Solver: Grid search (0.1m resolution) → L-M refinement → convergence check.

Reference: R2
TDOAs = onset_i - onset_R2:

  Raw TDOA(R3-R2) = 1855.0 - 1855.0 = 0.0ms   → PA-corrected: -1.68ms
  Raw TDOA(R4-R2) = 1856.5 - 1855.0 = 1.5ms   → PA-corrected: +5.03ms
  Raw TDOA(R1-R2) = 1864.5 - 1855.0 = 9.5ms   → PA-corrected: +6.55ms

Predicted distances from solution (-0.5, -2.2):

  d(R3) = sqrt((-5.00 - (-0.5))² + (-14.21 - (-2.2))²) ≈ 12.8m
  d(R2) = sqrt((-5.38 - (-0.5))² + (-14.70 - (-2.2))²) ≈ 13.4m
  d(R4) = sqrt(( 7.15 - (-0.5))² + (-15.30 - (-2.2))²) ≈ 15.2m
  d(R1) = sqrt((-11.59 - (-0.5))² + ( 8.91 - (-2.2))²) ≈ 15.7m

Algorithm:

For each iteration k = 1..10,000:
  1. Perturb each receiver position: (x_i + N(0, 0.5), y_i + N(0, 0.5))
  2. Perturb each onset time: t_i + N(0, 1.0) ms
  3. Perturb speed of sound: v + N(0, 0.7) m/s
  4. Run Levenberg-Marquardt solver
  5. Record solution (x_k, y_k)

Compute 95% confidence ellipse from covariance matrix of all (x_k, y_k).

Results:
  Mean solution: (-0.5, -2.2)m
  Semi-major 1σ ≈ 0.49m, semi-minor 1σ ≈ 0.21m (95% / 2.448)
  95% ellipse: 2.42 × 1.01m, rotated -151° from east
  Ellipse area: 7.7m²
  Convergence: 100% (all 10,000 iterations found a solution)

GCC-PHAT formula:

  R_12(τ) = IFFT[ X_1(f) · X_2*(f) / |X_1(f) · X_2*(f)| ]

Peak at τ=0 (within ±1 sample) confirms synchronization.

Sub-sample refinement via parabolic interpolation:

  τ_refined = τ_peak + 0.5 × (R[τ-1] - R[τ+1]) / (R[τ-1] - 2R[τ_peak] + R[τ+1])

Closure test (for receivers A, B, C):
  τ_AB + τ_BC + τ_CA should equal 0
  Maximum closure error: 0.03 samples = 0.6μs

All 4 triplets tested. All pass.

The solved source position predicts specific reflection arrival times based on known building geometry. Each receiver's measured reflection pattern encodes information about nearby reflective surfaces — walls, buildings, and other large-scale structures. R2's early wall reflection (+50.1ms at −3.3dBFS) is consistent with its known location near or against a building wall, producing a short-path specular return. R4's close double reflection (+247ms and +259ms, separated by only 12ms) suggests two nearby reflective surfaces at nearly equal path lengths.

Full reflection geometry verification — comparing predicted reflection delays (computed from the solved source position and known building surfaces) against measured reflection arrival times — would serve as an independent validation of the TDOA solution. If the source is where the multilateration says it is, the reflection geometry must also be consistent.