23-EFT.WP.Metrology.PathCorrection v1.0 | Chapter 5 — Tropospheric Delay (Water Vapor / Pressure / Temperature)

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Chapter 5 — Tropospheric Delay (Water Vapor / Pressure / Temperature)

One-Sentence Goal
Establish a unified model for tropospheric dry and wet delays, provide computation, mapping, and compliance templates from surface meteorology to slant-path delay, and persist artifacts as manifest.path.tropo.*.

I. Scope and Objects

1. Inputs
   • met = { P, T, RH or e, phi, H, doy }, where P is pressure, T temperature, RH relative humidity or e water-vapor partial pressure, phi latitude, H altitude, doy day-of-year.
   • elev elevation angle, f carrier frequency, gamma(ell) geometric path or LOS pointing.
   • RefCond reference-condition binding and data-source annotations.
2. Outputs
   • T_tropo, T_hydro, T_wet (seconds), and intermediates ZHD, ZWD, m_h(elev), m_w(elev) (meters and dimensionless).
   • Uncertainty u(T_tropo) and a compliance report.
3. Applicability and Boundaries
   • Typical free-space microwave and GNSS: f ∈ [1 GHz, 30 GHz]. Strong absorption lines and heavy-precipitation scattering/absorption are handled via alpha(f) and Chapter 7; not in this chapter’s main equations.
   • Recommended lower bound elev ≥ 5°. Extremely low elevation, strong refraction, or ducting defer to Chapter 9 (ray tracing).

II. Terms and Variables

• N, refractivity, N = ( n_eff - 1 ) * 10^6, unit(N) = "ppm", dim(N) = "1".
• P, air pressure, unit(P) = "Pa", dim(P) = "[M][L]^-1[T]^-2".
• T, absolute temperature, unit(T) = "K", dim(T) = "[Θ]".
• RH, relative humidity, unit(RH) = "1".
• e, water-vapor partial pressure, unit(e) = "Pa".
• ZHD, ZWD, zenith hydrostatic/wet delays (length), unit = "m", dim = "[L]".
• m_h(elev), m_w(elev), hydrostatic/wet mapping functions, unit = "1".
• STD, slant tropospheric delay (length), STD = m_h*ZHD + m_w*ZWD.
• T_hydro = ZHD / c_ref, T_wet = ZWD / c_ref, T_tropo = (STD / c_ref), unit = "s", dim = "[T]".
• c_ref, reference vacuum propagation upper bound, unit = "m/s".
• phi (latitude, rad), H (altitude, m), elev (elevation, rad).

III. Axioms P805-*

• P805-1 (Decomposition) — Tropospheric delay decomposes metrically into dry and wet components: T_tropo = T_hydro + T_wet, corresponding to STD = ZHD + ZWD mapped by elevation-dependent functions.
• P805-2 (Quasi-Non-Dispersive Band) — For f ∈ [1 GHz, 30 GHz] away from strong lines, the first-order frequency dependence of n_eff on delay is negligible; compute T_tropo with a non-dispersive model. Near lines, annotate and switch to the absorption–dispersion coupled model (see Chapters 7 and 8).
• P805-3 (Mapping) — There exists a monotone, physically interpretable family m_x(elev) such that STD = m_h(elev)*ZHD + m_w(elev)*ZWD, with m_x(elev) non-increasing in elev.
• P805-4 (Referencing) — In absence of local met, reanalysis or GPT-class fields are permitted (see I80-51), but data source and spatiotemporal resolution must be persisted.
• P805-5 (Path Integral) — Define tropospheric delay by T_tropo = ( ∫_{tropo} ( (n_eff - 1) / c_ref ) d ell ), with n_eff = 1 + N * 10^-6.
• P805-6 (Dual-Form Consistency) — Compute and record both formulations and their difference:
  delta_form = | ( 1 / c_ref ) * ( ∫_{tropo} (n_eff - 1) d ell ) - ( ∫_{tropo} ( (n_eff - 1) / c_ref ) d ell ) |,
  and include it in contract evaluation.

IV. Minimal Equations S805-*

• S805-1 (Refractivity, Smith–Weintraub form)
  N = k1 * ( P_dry / T ) + k2 * ( e / T ) + k3 * ( e / T^2 ),
  with typical constants k1 ≈ 77.6 K/Pa, k2 ≈ 70.4 K/Pa, k3 ≈ 3.739e5 K^2/Pa, P_dry = P - e. check_dim(N) = "1".
• S805-2 (Zenith Hydrostatic Delay, Saastamoinen form)
  ZHD = 0.0022768 * P / ( 1 - 0.00266 * cos(2*phi) - 0.00028 * H/1000 ), unit(ZHD) = "m".
  Corresponding T_hydro = ZHD / c_ref.
• S805-3 (Zenith Wet Delay, temperature–vapor form)
  ZWD = 0.002277 * ( 1255 / T + 0.05 ) * e, unit(ZWD) = "m".
  Corresponding T_wet = ZWD / c_ref.
• S805-4 (Elevation Mapping and Slant Path)
  STD(elev) = m_h(elev) * ZHD + m_w(elev) * ZWD, T_tropo = STD / c_ref.
  Simple convention: m_x(elev) = 1 / sin(elev); engineering convention: continued-fraction family
  m_x(e) = ( 1 + a_x/(1 + b_x/(1 + c_x)) ) / ( sin(e) + a_x/( sin(e) + b_x/( sin(e) + c_x ) ) ),
  with parameters a_x,b_x,c_x from the chosen model (Niell/GMF/VMF).
• S805-5 (Vapor Pressure and Humidity)
  e = RH * e_s(T), with Magnus–Tetens:
  T_C = T - 273.15, e_s(T) = 610.94 * exp( 17.625 * T_C / ( T_C + 243.04 ) ) (Pa).
  If e or PWV is directly measured, prefer measured e or site PWV regressions.
• S805-6 (Path-Integral Consistency)
  T_tropo = ( 1 / c_ref ) * ( ∫_{gamma ∩ tropo} ( n_eff - 1 ) d ell ) and
  T_tropo = ( ∫_{gamma ∩ tropo} ( ( n_eff - 1 ) / c_ref ) d ell ) are numerically equivalent;
  use delta_form to monitor numerical consistency.

V. Metrological Workflow M80-5

• Ready — Ingest met and site metadata; verify Delta_t freshness and availability; if missing, call GPT/reanalysis and annotate RefCond.source.
• Medium Quantification — From P,T,RH derive e; compute ZHD, ZWD. check_dim( ZHD - f(P,T,phi,H) ) = "[L]".
• Mapping-Function Selection — Choose model ∈ { Simple, Niell, GMF, VMF }; instantiate m_h(elev), m_w(elev).
• Composition and Conversion — Compute STD and T_tropo = STD / c_ref; provide u(T_tropo) (sensor noise, model uncertainty, elevation sensitivity).
• Checks — Compute dual-form delta_form; validate elevation, ratios, and ranges; tag anomalies and de-weight or fall back.
• Persist — Emit manifest.path.tropo = { T_tropo, T_hydro, T_wet, ZHD, ZWD, m_h, m_w, model, RefCond, u/U, delta_form, tags }.
• Monitoring — Update long-term site baselines and drift indicators (see Chapter 13 and Appendix D).

VI. Contracts and Assertions (C80-51x)

• C80-511 Freshness — age(met) ≤ Delta_t (default Delta_t ≤ 10 min); otherwise down-weight or fall back.
• C80-512 Physical Ranges — P ∈ [50, 110] kPa, T ∈ [230, 320] K, RH ∈ [0, 1], e ≥ 0; reject and fall back if out of bounds.
• C80-513 Mapping Monotonicity — d m_x / d elev ≤ 0 and m_x(elev) ≥ 1.
• C80-514 Ratio Plausibility — 0 ≤ ZWD / ZHD ≤ 0.4; flag as humidity anomaly or model mismatch if exceeded.
• C80-515 Elevation Lower Bound — elev ≥ elev_min (default 5°); below this, use Chapter 9 ray tracing or inflate uncertainty.
• C80-516 Dual-Form Difference — delta_form ≤ tol_Tarr (suggest tol_Tarr ≈ 0.05 ns, tuned to system SLO).
• C80-517 Band Compliance — If band_ok = false (near strong absorption), switch to dispersion–absorption model and record under contracts.*.
• C80-518 Precipitation/Liquid Water Tagging — When precipitation exceeds threshold, add tags += {"rain"} and hand off to Chapter 7 scattering/absorption chain.
• C80-519 Dimensional Consistency — check_dim( T_tropo ) = "[T]", check_dim( STD ) = "[L]".

VII. Implementation Bindings I80-*

• I80-51 model_troposphere(met, elev, f, model) -> { T_tropo, parts:{T_hydro,T_wet}, Z:{ZHD,ZWD}, map:{m_h,m_w}, meta:{RefCond,model,tags}, qc:{u,delta_form} }
  Invariants: non_decreasing(elev → 90°): m_x ↓, T_tropo ≥ 0, delta_form ≤ tol_Tarr.
• I80-52 compute_ZHD_ZWD(P, T, e, phi, H) -> {ZHD, ZWD}
• I80-53 mapping_params(model, phi, H, doy) -> {a_h,b_h,c_h,a_w,b_w,c_w}
• I80-54 mapping_apply(elev, params) -> {m_h, m_w}
• I80-55 assert_tropo_contracts(payload, rules) -> report
• I80-56 emit_path_manifest_tropo(payload, policy) -> manifest.path.tropo

VIII. Cross-References

• Ray paths and strong refraction: EFT.WP.Metrology.PathCorrection v1.0 Chapter 9.
• Ionosphere and dispersion: this volume Chapter 6.
• Environmental fusion and correction synthesis: this volume Chapter 11.
• Dual formulations and numerical integration: this volume Chapter 10 and EFT.WP.Methods.Cleaning v1.0.
• Time-base and sync fields: EFT.WP.Metrology.TimeBase v1.0, EFT.WP.Metrology.Sync v1.0.
• Instrument chain and delay calibration: EFT.WP.Metrology.Instrument v1.0.

IX. Quality and Risk Control

• Target SLO — p95( |error(T_tropo)| ) ≤ 0.3 ns, p99 ≤ 0.6 ns (with qualified met).
• Drift Monitoring — Expect corr( ZHD, P ) > 0.9; deviations trigger sensor audits. Use STL to gate seasonality in ZWD residuals.
• Fallback — If met unavailable or contracts fail, fall back to {Saastamoinen + GMF/GPT} and inflate u(T_tropo); tag extreme weather and widen guardbands.
• Audit & Traceability — Persist RefCond.source, model, params.hash, contracts.*, delta_form; manage version changes per Appendix F.

Summary
This chapter presents a unified modeling, mapping, and compliance framework for tropospheric dry and wet components, producing the minimal key set for manifest.path.tropo:
manifest.path.tropo = { T_tropo, T_hydro, T_wet, ZHD, ZWD, m_h, m_w, model, RefCond:{ P,T,RH or e, phi,H,doy,source }, u, U, delta_form, contracts.*, tags }.
In conjunction with Chapters 6, 9, 10, and 11, free-space links can confine medium-delay uncertainty within system SLO targets while ensuring dual-form consistency and auditable persistence.