08-EFT.WP.Core.Sea v1.0 | Chapter 8 — Arrival-Time and Path Modeling

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Chapter 8 — Arrival-Time and Path Modeling

I. Objectives and Scope

• Establish a unified formulation for T_arr, the path gamma(ell) with measure d ell, and the relationship among reference speed c_ref and medium effective refractive index n_eff(x,t). Provide dual-form consistency and drift monitor delta_form, and decompose the observed arrival time into synchronization bias, front-end group delay, and detection criteria.
• Keep alignment with I80-6 estimate_toa / path_integral / enforce_arrival_time_convention, I80-5 apply_env_correction, Chapter 3 (synchronization model), Chapter 4 (group delay), and Chapter 5 (spectral features).

II. Core Objects and Notation

• Path and measure
  gamma(ell): parameterized propagation path with ell ∈ [0, L_gamma]; L_gamma = ( ∫ 1 d ell ).
  d ell: arclength measure along the path.
• Medium and reference
  n_eff(x,t): effective refractive index (equivalently a relative slowness factor); c_ref: constant reference propagation speed.
• Arrival times
  T_arr: propagation (geometric) arrival time; T_obs: observed arrival time (includes detection and system delays).
  t_emit, t_pick: emission and pick-up timestamps measured on tau_mono; T_obs = t_pick - t_emit.

III. Postulates P88- (Path and Formulation)*

1. P88-1 (Path integrability)
   Along any physical propagation path gamma(ell), n_eff(x,t) is piecewise bounded and Riemann integrable; T_arr is well-defined by path integration.
2. P88-2 (Dual-form equivalence)
   With c_ref treated as constant, define T_arr via two equivalent forms, equal in the continuous limit:
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
3. P88-3 (Observed time decomposition)
   The observation decomposes as T_obs = T_arr + T_fe + T_sync + T_det, where T_fe ≈ tau_g(H) (Chapter 4), T_sync arises from offset/skew (Chapter 3), and T_det is the detection-criterion bias.
4. P88-4 (Earliest arrival under multipath)
   With multiple paths gamma_k, the leading edge time is T_front = min_k T_arr(gamma_k). Energy arrival or correlation peaks may lag behind T_front.
5. P88-5 (Explicit environment)
   Any environmental correction embedded into n_eff MUST be recorded explicitly as corr_env(n_eff; RefCond) in the manifest.

IV. Minimal Equations S88- (Propagation, Sensitivity, and Consistency Monitor)*

• S88-1 (Dual-form baseline)
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) = ( ∫ ( n_eff / c_ref ) d ell ).
• S88-2 (Average index and path length)
  Define n_bar = ( 1 / L_gamma ) * ( ∫ n_eff d ell ), then T_arr = ( n_bar / c_ref ) * L_gamma.
• S88-3 (Difference monitor)
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
  Discretization or unit mistakes yield delta_form > 0; qualified data should satisfy delta_form <= eps_form.
• S88-4 (Discrete approximation and truncation bound)
  Piecewise-constant approximation: T_arr ≈ ( 1 / c_ref ) * ∑_i ( n_eff[i] * Delta_ell[i] ).
  If n_eff has Lipschitz constant L_n on segments and step h = max(Delta_ell[i]), the truncation error
  E_T <= ( L_n * L_gamma * h ) / ( 2 * c_ref ).
• S88-5 (Sensitivities)
  First-order sensitivity to n_eff: dT_arr/d n_eff(ell) = ( 1 / c_ref ); to path length: ∂T_arr/∂L_gamma = n_bar / c_ref.
• S88-6 (TDOA and localization primitive)
  For receivers i,j, TDOA_ij = T_arr_i - T_arr_j; if n_eff ≈ const,
  TDOA_ij ≈ ( n_bar / c_ref ) * ( L_gamma_i - L_gamma_j ).

V. Synchronization, Detection, and the Observed Time

• Sync correction
  With ts_i(t) = alpha_i * tau_mono + beta_i, apply estimated alpha_i, beta_i to map T_obs onto a common tau_mono.
• Detection criteria
  Cross-correlation peak: t_hat = argmax_tau ( ∑ x(t) * y(t+tau) );
  threshold front: t_hat = min{ t | x(t) >= theta }.
  Sampling quantization: u_fs ≈ 1 / fs; sub-sample refinement via parabolic interpolation or phase methods can reach < 1 / ( k * BW ).
• Reconstructing propagation time
  T_arr_hat = T_obs - T_fe - T_sync - T_det, with T_fe from Chapter 4 H(f) group delay.

VI. Multipath and Scattering (Engineering Approximations)

• Energy decomposition
  h(t) = ∑_k a_k * delta( t - T_arr(gamma_k) ) (impulse-response approximation); observed y(t) = (h * s)(t) + n(t).
• Early/late windows
  Use [T_front, T_front + Delta_t_win] for leading-edge pick; choose Delta_t_win from bandwidth and expected scattering.
• Suppression strategies
  Spectral weighting or shortest-path regularization: add penalty R = lambda_path * L_gamma in estimating t_hat to bias toward shorter paths.

VII. Numerical Paths and Gridded gamma(ell)

• Grid construction
  Discretize n_eff(x,t) on voxels or meshes; approximate gamma(ell) via graph shortest path with edge weight w_e = ( n_eff_e / c_ref ) * |e|.
• Fast approximations
  Collimated approximation: straight-line gamma, T_arr ≈ ( 1 / c_ref ) * ∑ ( n_eff(x_k) * Delta_ell ).
  Curvature correction (engineering):
  Delta T ≈ ( 1 / ( 2 * c_ref ) ) * ∑ ( (∇ n_eff ⋅ n_hat) * Delta_ell^2 ).

VIII. Uncertainty and Budgeting

• Component combination (approx. independent)
  u^2(T_arr_hat) ≈ u^2(n_eff) * ( L_gamma / c_ref )^2 + u^2(L_gamma) * ( n_bar / c_ref )^2 + u^2(T_fe) + u^2(T_sync) + u^2(T_det).
• Expanded uncertainty
  U(T_arr_hat) = k * sqrt( u^2(T_arr_hat) ) (see Chapter 6 for U = k * u_c).
• QC thresholds
  Set eps_form, U_max, and a q_score floor; accept only if delta_form <= eps_form and U(T_arr_hat) <= U_max.

IX. Manifest and Observability Extensions (to Chapter 7)

• Add to object manifest
  {gamma_id, path_model, n_bar, L_gamma, T_arr_hat, U_T_arr, T_fe, T_sync, T_det, delta_form}.
• SLIs to track
  T_arr_ValidRate, P99_U_T_arr, delta_form_violations; aggregate with metric_emit over the SLA_window.

X. Workflow Mx-8 (Arrival-Time Computation and Alignment)

• Medium modeling: acquire environmental RefCond and n_eff(x,t) grid/parametric field; run corr_env(n_eff; RefCond).
• Path generation: given source/receiver coordinates, generate candidate gamma_k (line, polyline, or shortest path).
• Path integral: call I80-6 path_integral(n_eff, gamma_k, c_ref) to obtain T_arr(gamma_k); compute delta_form.
• Observation pick: call I80-6 estimate_toa(sig, method="xcorr") to get t_hat; estimate T_fe/T_det.
• Sync correction: apply Chapter 3 alpha/beta for T_sync; reconstruct T_arr_hat = T_obs - T_fe - T_sync - T_det.
• Multipath test: let T_arr* = min_k T_arr(gamma_k); if |T_arr_hat - T_arr*| > thr_path, trigger model recomputation or tag scattering.
• Uncertainty: compute U(T_arr_hat) per Section VIII; write to manifest; update SLIs and alerts.

XI. Interface Bindings (I80-6 and Constraints)

• estimate_toa(sig:any, method:str="xcorr") -> float
  Return t_pick on tau_mono; include method/meta and optional u_est.
• path_integral(n_eff:any, gamma:any, c_ref:float) -> float
  MUST compute both forms and return delta_form. If delta_form > eps_form, log raise_alert("delta_form_violation", ...).
• enforce_arrival_time_convention(trace:any) -> None
  Validate that any T_arr write carries references to {gamma, d ell, c_ref, n_eff}; reject submissions that lack these.

XII. Engineering Guidance and Defaults

• Constants and thresholds
  Choose c_ref consistent with the scenario; eps_form = 1e-12 s (suggested numeric tolerance for double-precision path sums).
  thr_path = 5 * U(T_arr_hat); Delta_t_win = 1 / BW; fs >= 4 * BW for robust xcorr interpolation.
• Sampling & filtering alignment
  With known Chapter 4 tau_g(H), include it as a fixed term in T_fe; publish H(f) version and convention.

XIII. Interlocks and Cross-Volume References

• With Chapter 3: offset/skew/J give T_sync directly; all arrival estimates use tau_mono, audit uses ts.
• With Chapter 4: anti-aliasing and group delay determine T_fe and the usable detection bandwidth; include H(f) version in manifests.
• With Chapter 5: peak width, bandwidth, and SNR drive u(T_det).
• With Chapter 7: persist gamma_id / delta_form / U(T_arr_hat) and include in SLIs.
• With Core.Threads I70-9: enforce_arrival_time_convention guarantees cross-volume unity of T_arr conventions and path bookkeeping.