23-EFT.WP.Metrology.PathCorrection v1.0 | Chapter 6 — Ionospheric Delay (TEC / Dual-Frequency)

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Chapter 6 — Ionospheric Delay (TEC / Dual-Frequency)

One-Sentence Goal
Center the model on TEC, establish the first-order 1/f^2 dispersion and the dual-frequency ionosphere-free combination, provide the mapping from VTEC to STEC, the sign conventions for delays, and a compliance template, and persist artifacts as manifest.path.iono.*.

I. Scope and Objects

1. Inputs
   • Frequency & observables: f or {f1,f2}, code pseudorange P(f) (m), carrier-phase equivalent range L(f) (m).
   • Fields & mapping: TEC sources (VTEC/STEC/GIM grids), mapping ∈ {thin_shell, GIM, NeQuick, 3D}.
   • Geometry & time: gamma(ell) or LOS, elevation elev, timestamp ts, site info phi,H.
   • Reference: RefCond (TEC data source, update cadence, spatial resolution, h_iono).
2. Outputs
   • First-order ionospheric delay: T_iono_group(f), T_iono_phase(f), and slant-path length term SLD(f) (m).
   • Dual-frequency products: ionosphere-free observable obs_if, STEC_hat estimate and uncertainty.
3. Applicability and Boundaries
   Common microwave ranging bands f ∈ [1, 30] GHz, away from strong absorption lines; during strong disturbances/storms, tag and optionally include second-order terms.

II. Terms and Variables

• TEC: total electron content along the path, unit(TEC) = "el/m^2"; 1 TECU = 1e16 el/m^2.
• VTEC: zenith TEC, unit = "el/m^2"; STEC: slant TEC.
• M_iono(elev): mapping from VTEC to STEC, unit = "1".
• K_iono: first-order ionospheric constant, chosen so that SLD = ( K_iono * STEC / f^2 ) is in meters; unit(K_iono) is consistent with TEC,f.
• SLD(f): ionosphere-induced equivalent path-length change (m).
• T_iono_group = + SLD / c_ref, T_iono_phase = - SLD / c_ref, unit = "s", dim = "[T]".
• DCB_rx, DCB_tx: receiver/transmitter differential code biases, unit = "m" or "s" (see Instrument volume).
• h_iono: thin-shell height (m); Re: mean Earth radius (m).

III. Axioms P806-*

• P806-1 (Cold, Weak-Collision Plasma) — In first order, the ionosphere advances phase and retards group, with dispersion ∝ 1/f^2.
• P806-2 (Mapping) — A monotone, geometrically interpretable M_iono(elev) exists such that STEC = M_iono(elev) * VTEC, with d M_iono / d elev ≤ 0.
• P806-3 (Dual-Frequency) — If {f1,f2} are available, a linear combination cancels the first-order ionospheric term; residuals include higher-order terms, DCB, and noise.
• P806-4 (Path Integral) — TEC = ( ∫_{gamma ∩ iono} N_e d ell ), where N_e is electron density; compute T_iono with both formulations and persist delta_form.
• P806-5 (Compliance Fields) — TEC source, spatial/temporal resolution, and update cadence are mandatory records; missing fields require downgrade/fallback.
• P806-6 (Second-Order Tagging) — Under geomagnetic storms or high-latitude disturbances, enable O(1/f^3, 1/f^4) corrections and tag; never apply silently.

IV. Minimal Equations S806-*

• S806-1 (First-Order Refractive Index and Path Terms)
  n_phi(f) ≈ 1 - ( K_iono' * N_e / f^2 ), n_g(f) ≈ 1 + ( K_iono' * N_e / f^2 ).
  After path integration, obtain the length term
  SLD(f) = ( K_iono * STEC / f^2 ),
  hence
  T_iono_group(f) = + ( SLD / c_ref ), T_iono_phase(f) = - ( SLD / c_ref ).
  check_dim( SLD ) = "[L]", check_dim( T_iono_* ) = "[T]".
• S806-2 (Thin-Shell Mapping, Geometric Form)
  Let z = pi/2 - elev, psi = arcsin( ( Re / ( Re + h_iono ) ) * cos(elev) ),
  M_iono(elev) = 1 / cos( psi ) = 1 / sqrt( 1 - ( ( Re * cos(elev) / ( Re + h_iono ) )^2 ) ).
  STEC = M_iono * VTEC.
• S806-3 (Dual-Frequency Ionosphere-Free Combination)
  For any co-direction, co-epoch observable R(f) ∈ { P(f), L(f) } (meters), define
  obs_if = ( f1^2 * R(f1) - f2^2 * R(f2) ) / ( f1^2 - f2^2 ).
  The first-order ionospheric term cancels in obs_if, leaving geometry, clocks, and noise.
  Sign convention: R = P uses pseudorange; R = L uses phase equivalent range (phase must be multiplied by lambda).
• S806-4 (Estimating STEC from Dual Frequency)
  With ΔR = R(f2) - R(f1),
  STEC_hat = ( f1^2 * f2^2 / ( K_iono * ( f1^2 - f2^2 ) ) ) * sgn(R) * ( ΔR - DCB_rx - DCB_tx ),
  where sgn(P) = +1, sgn(L) = -1.
  If ΔR mixes P and L, apply the appropriate sgn and bias models to each.
• S806-5 (Two-Form Consistency Term)
  T_iono(f) = ( 1 / c_ref ) * ( ∫_{iono} ( n_eff - 1 ) d ell ) and
  T_iono(f) = ( ∫_{iono} ( ( n_eff - 1 ) / c_ref ) d ell )
  are numerically equivalent; use delta_form to monitor consistency and quadrature error.
• S806-6 (Second-Order Corrections, Optional)
  When policy triggers, include T_iono^(2) terms ∝ 1/f^3 and ∝ 1/f^4 (involving B_parallel and density gradients); implementation is bound via I80-62 and must be tagged on output.

V. Metrological Workflow M80-6

• Ready — Acquire {f or f1,f2} and P,L; align to tau_mono. Fetch TEC source or GIM grid and interpolate to the IPP.
• Map — Instantiate M_iono(elev) (given Re,h_iono or model parameters) and map VTEC → STEC.
• First-Order Solve — Compute SLD(f) = K_iono * STEC / f^2, then T_iono_group, T_iono_phase.
• Dual-Frequency & Biases — If {f1,f2} exist, estimate DCB_rx, DCB_tx (see Instrument); apply S806-4 to get STEC_hat and u(STEC_hat).
• Checks — Compute delta_form and physical-bound assertions; verify sgn(P/L) consistency and tag anomalies.
• Persist — Emit
  manifest.path.iono = { T_iono_group(f), T_iono_phase(f), SLD(f), STEC, VTEC, M_iono, model, RefCond, DCB, u/U, delta_form, tags }.
• Monitor — Maintain TEC_bias, map_residual, and storm_flag to drive second-order switches and guardbands.

VI. Contracts and Assertions (C80-61x)

• C80-611 Freshness — age(TEC) ≤ Delta_t (default Delta_t ≤ 15 min); otherwise downgrade or fall back.
• C80-612 Geometric Bound — elev ≥ elev_min (default 5°); below this, ray trace or inflate uncertainty.
• C80-613 Physical Range — VTEC ≥ 0, STEC ≥ VTEC, M_iono ≥ 1 and monotone.
• C80-614 Sign Consistency — T_iono_group ≥ 0, T_iono_phase ≤ 0; otherwise tag sign_mismatch.
• C80-615 Dual-Frequency Validity — |f1 - f2| / min(f1,f2) ≥ r_min (suggest r_min = 0.1), else the combination is ill-conditioned.
• C80-616 Bias Disclosure — Persist DCB_rx, DCB_tx; if missing, tag dcb_unmodeled on STEC_hat and inflate u.
• C80-617 Two-Form Difference — delta_form ≤ tol_Tarr (suggest ≤ 0.05 ns, tuned to system SLO).
• C80-618 Storms / High Latitudes — When storm_flag = true, enable T_iono^(2) and record strategy/parameters under contracts.*.
• C80-619 Dimensional Consistency — check_dim( SLD ) = "[L]", check_dim( T_iono_* ) = "[T]".

VII. Implementation Bindings I80-*

• I80-61 model_ionosphere(TEC, f, mapping) -> { T_iono_group, T_iono_phase, SLD, meta:{RefCond, mapping, h_iono}, qc:{u, delta_form}, tags }
  Invariants: T_iono_group ≥ 0, T_iono_phase ≤ 0, delta_form ≤ tol_Tarr.
• I80-62 estimate_TEC_dualfreq(P1,P2,L1,L2,f1,f2,DCB,policy) -> { STEC_hat, u, tags }
• I80-63 map_VTEC_to_STEC(VTEC, elev, Re, h_iono, model) -> { STEC, M_iono }
• I80-64 apply_ionofree(obs1,obs2,f1,f2) -> obs_if
• I80-65 assert_iono_contracts(payload, rules) -> report
• I80-66 emit_path_manifest_iono(payload, policy) -> manifest.path.iono

VIII. Cross-References

• Troposphere and elevation mapping: Chapter 5.
• Ray tracing and strong refraction: Chapter 9.
• Two-form numerics and delta_form: Chapter 10 and EFT.WP.Methods.Cleaning v1.0.
• Time base, sync, and DCB calibration: EFT.WP.Metrology.TimeBase v1.0, EFT.WP.Metrology.Sync v1.0, EFT.WP.Metrology.Instrument v1.0.
• Environmental fusion and policy switches: Chapter 11.

IX. Quality and Risk Control

• Target SLO — Under quiet ionosphere and qualified TEC sources: p95( |error(T_iono_group)| ) ≤ 0.3 ns, p99 ≤ 0.6 ns; raise guardbands during storm conditions.
• Drift Monitoring — Long-term bias( STEC_hat - STEC_model ) ≈ 0; keep map_residual within thresholds; on deviation, first check DCB and mapping parameters.
• Fallback — Without dual frequency or fresh TEC, fall back to broadcast or historical fields; explicitly inflate u(T_iono) and record the policy card.
• Audit & Traceability — Persist RefCond.source, mapping, h_iono, DCB, contracts.*, and delta_form; manage version changes per Appendix F.

Summary
This chapter standardizes the first-order 1/f^2 ionospheric model, VTEC → STEC mapping, and the dual-frequency ionosphere-free combination, yielding the minimal key set for manifest.path.iono:
manifest.path.iono = { T_iono_group(f), T_iono_phase(f), SLD(f), STEC, VTEC, M_iono, K_iono, model, mapping, RefCond:{source,res,Delta_t,h_iono}, DCB:{rx,tx}, u, U, delta_form, contracts.*, tags }.
Together with Chapters 5, 9, 10, and 11, the principal ionospheric term can be stably isolated on free-space links while preserving two-form consistency and meeting system SLOs.