23-EFT.WP.Metrology.PathCorrection v1.0 | Chapter 4 — Overview of Medium Models and Construction of n_eff

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Chapter 4 — Overview of Medium Models and Construction of n_eff

One-Sentence Goal
Unify refractive-index and propagation-speed conventions across electromagnetic and acoustic media; construct path-wise n_eff(ell; band, pol, RefCond) along gamma(ell); compute T_arr with the two formulations and publish uncertainties.

I. Scope and Objects

1. Applicable Objects
   • Unified representation and switching of propagation-medium models for electromagnetic waves (RF/optical) and acoustics (air/water/solid).
   • Construction of path-dependent effective refractive index n_eff(ell), segmented mixing, and discrete integration.
   • Parallel computation of the two arrival-time formulations and binding/traceability to environmental reference conditions RefCond.
2. Outputs
   Field representations n_field(x) or n_field(h), sampled sequence n_eff(ell), dual-form T_arr and delta_form, uncertainty U_Tarr, and a manifest of medium and parameter versions.
3. Out of Scope
   Ray tracing and geometric configuration (see Chapter 3); time-base and sync semantics (see TimeBase and Sync).

II. Terms and Variables (Units and Dimensions)

1. Media and Bands
   band ∈ {RF, optical, acoustic}, pol ∈ {H, V, RHC, LHC, …} (electromagnetics); mode ∈ {air, water, solid} (acoustics).
2. Refractive Index and Speed
   • n(x, lambda, env) (dimensionless), n_g(lambda) = n(lambda) - lambda * ( d n / d lambda ) (group index), c(x) = c_ref / n(x).
   • Acoustic equivalent: c_w(T,S,p) with n_w = c_ref / c_w.
3. Environment and Fields
   env = {P, T, RH, N_e, S, p, ...}, TEC = ( ∫ N_e d s ).
4. Path and Integration
   • gamma(ell), ell ∈ [0, L_gamma], L_gamma = ( ∫_gamma 1 d ell ).
   • Two formulations: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ), T_arr = ( ∫ ( n_eff / c_ref ) d ell ), delta_form = | form1 - form2 |.
5. Metrology and Uncertainty
   u(x) (standard uncertainty), U = k * u_c (expanded uncertainty), check_dim(expr).

III. Axioms P804-* (Non-Negotiables)

• P804-1 (Explicit Kind): n_kind ∈ {phase, group} must match the measurement convention (phase measurements use n, pulse arrivals use n_g).
• P804-2 (Binding to RefCond): Any n(x) must declare RefCond and its sources (model/observations/reanalysis), including versioning and interpolation strategy.
• P804-3 (Dual Forms in Parallel): Compute T_arr via both formulations and persist delta_form ≤ tol_Tarr.
• P804-4 (Segmented Multi-Media): Model each medium segment m independently; integrate piecewise along gamma = ⋃_m gamma_m and stitch results.
• P804-5 (Dimensional Homogeneity): Apply unit/dimension checks to every field entering equations; implicit unit conversions are forbidden.
• P804-6 (Dispersion Consistency): When dispersion matters, provide the functional or tabulated n(lambda) and derivative conventions.
• P804-7 (Uncertainty Publication): U_Tarr must be published with explicit propagation path and coverage factor k.
• P804-8 (Path Consistency): Sample points of n_eff(ell) must align with the geometric discretization (the same ell grid); if not, provide a mapping.
• P804-9 (Time Semantics): All windows are computed on tau_mono; publish on ts and record offset/skew/J.
• P804-10 (Physical Bounds): n and env must lie within physically reasonable domains; out-of-range values trigger de-weighting or fallback strategies.

IV. Minimal Equations S804-* (Models and Construction)

1. S804-1 (Path-Effective Refractive Index)
   • n_eff(ell; band, pol, RefCond) = n( x = gamma(ell), lambda|band, env(x) ).
   • Segmented average: bar{n}_m = ( 1 / L_m ) * ( ∫_{gamma_m} n d ell ).
2. S804-2 (Group Index)
   n_g(lambda) = n(lambda) - lambda * ( d n / d lambda ).
3. S804-3 (Two Arrival-Time Formulations)
   • T_arr_form1 = ( 1 / c_ref ) * ( ∫_0^{L_gamma} n_eff(ell) d ell ).
   • T_arr_form2 = ( ∫_0^{L_gamma} ( n_eff(ell) / c_ref ) d ell ).
   • delta_form = | T_arr_form1 - T_arr_form2 |.
4. S804-4 (Tropospheric Approximation, RF/Optical)
   • n - 1 = 1e-6 * N(P, T, e, ... ), where N = alpha_1 * ( P / T ) + alpha_2 * ( e / T^2 ) + ... (coefficients follow the model version).
   • Tropospheric delay: T_tropo = ( 1 / c_ref ) * ( ∫ ( n - 1 ) d ell ).
5. S804-5 (Ionospheric Dispersion, RF)
   T_iono = ( K_iono / f^2 ) * ( ∫ N_e d s ) / c_ref = ( K_iono * TEC / f^2 ) / c_ref (K_iono is a constant by convention).
6. S804-6 (Acoustic Media)
   c_w = c_w(T, S, p), n_w = c_ref / c_w, T_acoustic = ( ∫ ( 1 / c_w ) d ell ).
7. S804-7 (Gradient Refraction and First-Order Bending)
   d/ds ( n * r_hat ) = grad n; for layered n(h): d theta / d s ≈ ( 1 / n ) * ( ∂ n / ∂ h ).
8. S804-8 (Uncertainty Propagation)
   u^2( T_arr ) ≈ ( 1 / c_ref^2 ) * ( ∫ u^2( n_eff(ell) ) d ell ) + u^2_{geom} + u^2_{num}.
9. S804-9 (Multi-Segment Stitching)
   T_arr = ∑_m ( 1 / c_ref ) * ( ∫_{gamma_m} n_eff d ell ), L_gamma = ∑_m L_m.
10. S804-10 (Dispersion Consistency Check)
    For multi-wavelength/frequency {lambda_k}, verify monotonicity non_decreasing( n_g(lambda_k) ) in the expected dispersion range.

V. Construction Workflow M84-1 n_eff_pipeline (Ready → Model → Integrate → Check → Persist)

• Ready
  Inputs: gamma(ell), band, pol, n_kind, RefCond; load medium models and versions, interpolation and extrapolation strategies.
• Model
  Acquire/assimilate env(x) (observations/reanalysis/instruments); build n_field(x) or n(h); compute n_g(lambda) when needed.
• Sample & Discretize
  Sample n_eff(ell) on the ell grid consistent with geometry; record interpolation error res_interp and gap flag m ∈ {0,1}.
• Integrate & Stitch
  Compute T_arr_form1 and T_arr_form2, obtain delta_form; accumulate T_arr and L_gamma by medium segments.
• Uncertainty
  From u(env) and model sensitivities derive u(n_eff); evaluate U_Tarr per S804-8; record numerical error u_num.
• Checks
  Thresholds: delta_form ≤ tol_Tarr, valid domains for n/n_g, res_interp ≤ tol_interp; dispersion and multi-frequency consistency.
• Persist
  manifest.path.medium.*: model_id/version, RefCond, band/pol/n_kind, delta_form, U_Tarr, metrics.*, signature.

VI. Contracts and Assertions C84-*

• C84-kind: n_kind matches the measurement convention (pulse/phase); otherwise flag violation and block publication.
• C84-delta: delta_form ≤ tol_Tarr.
• C84-domain: n_min ≤ n_eff(ell) ≤ n_max (per medium/band registry); TEC, P/T/RH within reasonable domains.
• C84-disp: Monotonicity/convexity of n_g(lambda) aligns with model priors.
• C84-miss: Gap ratio missing_ratio ≤ tol_missing; otherwise de-weight or fall back.
• C84-unc: U_Tarr ≤ U_budget; if exceeded, raise observation weight or shorten the publication window.
• C84-repeat: For repeated construction with identical inputs, |T_arr' - T_arr| ≤ tol_repeat (reproducibility).

VII. Implementation Bindings I84-* (Interface Prototypes and Invariants)

• select_n_model(band, pol, refcond, policy) -> model_id
  Invariant: returned model and version are registry-traceable.
• build_n_field(env_grid, model_id, band, pol) -> n_field
  Invariant: unit(n)=1; boundary and interpolation strategies are persisted.
• ng_from_dispersion(n_lambda_tab, lambda) -> n_g
  Invariant: numerical derivative is stable; check_dim passes.
• sample_along_path(n_field, gamma, grid) -> n_eff_samples
  Invariant: sample points are co-sourced with the ell grid.
• integrate_arrival_two_forms(n_eff_samples, c_ref) -> {T1, T2, delta_form}
  Invariant: delta_form is repeatable; numerical precision is reported.
• propagate_uncertainty(n_eff_samples, u_env, c_ref) -> U_Tarr
  Invariant: coverage factor k and confidence level are explicit.
• emit_medium_manifest(meta) -> manifest.path.medium
  Invariant: includes model/version/hash, RefCond, metrics, signature.

VIII. Cross-References

• Geometry and coordinates: Chapter 3.
• Rays and refracted paths: Chapter 6.
• Dual arrival-time and publication contracts: Chapters 2 and 8.
• Time base / synchronization: TimeBase Chapters 5–7; Sync Chapters 6–9.
• Cleaning and compliance modules: Methods.Cleaning Chapters 6/10.

IX. Quality Metrics and Risk Control

1. Core SLIs
   delta_form, U_Tarr, res_interp, missing_ratio, dispersion_consistency, model_switch_rate.
2. Risks and Fallbacks
   • Model out-of-domain or large gaps: fall back to lower-order layered n(h) or historical assimilation.
   • U_Tarr over budget: shorten the window or increase observation coverage.
   • Dispersion inconsistency: switch n_kind or change model version; trigger audit and phased release.

Summary
This chapter standardizes cross-medium n_eff construction, provides the S804-* equations, the M84-1 pipeline, and the C84-* contracts, forming a traceable chain from env → n_field → n_eff(ell) → T_arr. It establishes a solid baseline for path correction, arrival-time metrology, and cross-volume alignment.