16-EFT.WP.Methods.Cleaning v1.0 | Chapter 12 Environmental Correction and Arrival-Time Harmonization

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Chapter 12 Environmental Correction and Arrival-Time Harmonization

One-Sentence Goal
Adopt RefCond as the sole publication convention, apply corr_env(x; RefCond) to environment-affected fields, and—at the path level—harmonize n_eff and T_arr with a two-form recheck.

I. Scope & Objects

1. Objects & I/O
   • Input: standardized dataset ds (see Chapter 3), containing environmental series env = {T, p, RH, ...}, path parametrization gamma(ell), time fields ts and tau_mono, and raw or inferred n_eff.
   • Output: environment-harmonized dataset ds_env (fields suffixed _ref), path-level n_eff_ref(ell), two-form arrival time T_arr_ref with difference delta_form_ref, and manifest.env.
2. Constraints & boundaries
   • Corrections change numerical values only; field semantics, units, and dimension definitions remain unchanged. Original values are preserved and traceable.
   • All windowing and interpolation run on tau_mono, while publication uses ts; synchronization parameters offset/skew/J are recorded in the manifest (see Chapter 5).

II. Terms & Variables

1. Reference condition & environmental vector
   • RefCond = { e_i_ref } = { T_ref, p_ref, RH_ref, ... }.
   • e = { e_i } is the environmental observation vector per event or along ell; each e_i must declare dim(e_i) and unit(e_i).
2. Environmental correction & harmonized fields
   • x_ref = corr_env( x_obs ; RefCond, e ).
   • phi_env(e; RefCond): dimensionless correction factor; b_env(e; RefCond): same-dimension bias.
   • n_eff_ref(ell): effective refractive-index field under RefCond.
   • T_arr_ref: arrival time under RefCond, computed in two forms.
   • delta_form_ref = | ( 1 / c_ref ) * ( ∫ n_eff_ref d ell ) - ( ∫ ( n_eff_ref / c_ref ) d ell ) |.
3. Environmental distance & uncertainty
   • D_M^2 = ( e - e_ref )^T Sigma^{-1} ( e - e_ref ) (Mahalanobis distance, preferred); or
     d_env = sqrt( ∑ w_i * ( ( e_i - e_i_ref ) / s_i )^2 ).
   • u(x) is the standard uncertainty; U = k * u_c (see Appendix E).

III. Axioms (P112-*)

• P112-01 Reference-convention axiom
  Fields exposed to consumers must be provided under RefCond; i.e., x_ref exists and is written into manifest.env.
• P112-02 Dimension-conservation axiom
  corr_env satisfies dim(x_ref) = dim(x_obs) and check_dim( x_ref - x_obs ) = 0.
• P112-03 Pathwise point correction axiom
  For path-integral quantities, corrections act pointwise inside the integral: n_eff_ref(ell) = f( n_eff_obs(ell), e(ell), RefCond ).
• P112-04 Two-form consistency under reference conditions
  Compute both forms of T_arr_ref and record delta_form_ref; assert delta_form_ref ≤ tol_Tarr by contract.
• P112-05 Traceability & versioning axiom
  Record and sign the corr_env model model_id, parameters, Sigma, and RefCond: hash_sha256(model_blob), signature.
• P112-06 Original-value retention axiom
  Original fields must not be overwritten; x_obs and x_ref coexist. Use m ∈ {0,1} to mark correction feasibility (see Chapter 7).

IV. Minimal Equations (S112-*)

• S112-01 General correction form
  x_ref = corr_env( x_obs ; RefCond, e ) = x_obs * phi_env(e; RefCond) + b_env(e; RefCond)
• S112-02 Linearized families
  Additive: x_ref = x_obs + ∑ beta_i * ( e_i - e_i_ref )
  Multiplicative: x_ref = x_obs * ∏ ( 1 + alpha_i * ( e_i - e_i_ref ) )
• S112-03 Refractive-index harmonization
  n_eff_ref(ell) = g( e(ell) ; RefCond ) or
  n_eff_ref(ell) = n_eff_obs(ell) + ∑ beta_i(ell) * ( e_i(ell) - e_i_ref )
• S112-04 Arrival time (two forms, reference conditions)
  T_arr_ref = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff_ref d ell ),
  T_arr_ref = ( ∫_{gamma(ell)} ( n_eff_ref / c_ref ) d ell ),
  delta_form_ref = | 前式 - 后式 |
• S112-05 Environmental-distance gate
  D_M^2 ≤ tol_D2 or d_env ≤ tol_env is a prerequisite for applying corrections.
• S112-06 Uncertainty propagation (linearization)
  For y = corr_env(x, e),
  u^2(y) = ( ∂y/∂x )^2 u^2(x) + ∑ ( ∂y/∂e_i )^2 u^2(e_i ) + 2 ∑_{i<j} ( ∂y/∂e_i )( ∂y/∂e_j ) cov(e_i, e_j )
  For arrival time, if n_eff_ref samples along the path are uncorrelated,
  u^2( T_arr_ref ) = ( 1 / c_ref )^2 * ( ∫ u^2( n_eff_ref(ell) ) d ell ) ; add covariance-kernel terms when correlated.
• S112-07 Environmental-field registration
  e(ell) = ( K_env ∘ align_timebase )( env_series , gamma(ell) ), where K_env denotes along-path interpolation/condensation.

V. Cleaning Process (M10-12 Environmental Harmonization)

1. Select and record reference conditions
   Define RefCond, Sigma, and thresholds { tol_env | tol_D2 }; write policy and manifest.env.ref.
2. Unit & time-base checks
   Run repair_units and check_dim on env and target fields (Chapter 4); align on tau_mono (Chapter 5).
3. Environmental-distance evaluation & applicability
   Compute D_M^2 or d_env; if over threshold, degrade: set m=0, down-weight q_score, or route to quarantine.
4. Register environment to the path
   Use K_env to obtain e(ell) on gamma(ell); for event tasks, align env(ts) to each record.
5. Apply the correction model
   For each target field, compute x_ref = corr_env( x_obs ; RefCond, e ); for path quantities, compute n_eff_ref(ell).
6. Recompute arrival time & two-form recheck
   Compute both forms of T_arr_ref and produce delta_form_ref; assert delta_form_ref ≤ tol_Tarr.
7. Uncertainty propagation
   Propagate u(x_ref) and u(T_arr_ref) via S112-06; if covariances are unavailable, apply policy defaults and record them.
8. Persistence & signing
   • Generate manifest.env: RefCond, model_id, params, Sigma, tol_*, D_M^2 statistics, delta_form_ref distribution, signature.
   • Persist original and corrected values x_obs, x_ref; ensure TraceID and version closure (see Chapter 10).

VI. Contracts & Assertions

1. Applicability & stability
   • D_M^2 ≤ tol_D2 or d_env ≤ tol_env.
   • Reuse Chapter 6 checks for non_decreasing(ell) and L_gamma.
2. Dimensions & units
   check_dim( x_ref - x_obs ) = 0, unit(x_ref) = unit(x_obs).
3. Two forms & arrival time
   • arrival_forms( delta_form_ref , tol_Tarr ).
   • | T_arr_ref - T_arr_obs | ≤ tol_Tarr_shift (if monitoring correction-induced drift).
4. Uncertainty & quality
   • u(x_ref) ≤ tol_u_x, u(T_arr_ref) ≤ tol_u_T, q_score ≥ q_min.
   • Correction failure ratio ratio(m=0) ≤ tol_fail_env.
5. Traceability & versioning
   manifest.env.model_id exists and hash_sha256(model_blob) matches; signature verification passes.

VII. Implementation Binding (I10-12)

• apply_env_correction(ds, env, RefCond, model_id) -> ds', report
  Inputs: ds, env, RefCond, model_id; outputs: ds' with _ref fields and a process report.
  Invariant: do not change unit(*) or dim(*); retain original fields.
• register_env_model(model_blob) -> model_id, params, Sigma
  Stores the model and echoes parameters; returns Sigma for D_M^2 and propagation.
• align_env_to_path(env_series, gamma, policy) -> e(ell)
  Implements K_env ∘ align_timebase with along-path interpolation and window condensation.
• recompute_arrival_ref(ds') -> T_arr_ref, delta_form_ref
  Recomputes two forms under RefCond and produces the difference metric.
• propagate_uncertainty_env(ds', fields, Sigma) -> u_report
  Performs linearized propagation and emits an uncertainty panel for key fields.
• assert_env_contract(ds', tests) -> report
  Produces pass/fail and threshold deviations per this chapter’s contracts.

VIII. Cross-References

• Schema & field units: EFT.WP.Core.DataSpec v1.0.
• Acquisition & arrival-time definitions: EFT.WP.Core.Sea v1.0.
• Execution graph & streaming: EFT.WP.Core.Threads v1.0 and this volume’s Chapter 11.
• Density & normalization conventions: EFT.WP.Core.Density v1.0.
• This volume’s Chapters 4 (units & dimensions), 5 (time base), 6 (path & arrival time), 7 (masking & imputation), and 10 (release freeze).

IX. Quality Metrics & Risk Control

1. Indicators
   • D_M^2_p95, d_env_p95, share_corrected, delta_form_ref_p95, |T_arr_ref - T_arr_obs|_p95, u(T_arr_ref)_p95.
   • Drift: drift_env = d/dt ( d_env_p50 ); alert threshold tol_drift_env.
2. Risk actions
   • If D_M^2 exceeds threshold → degrade to original-convention bypass or enlarge lateness_max to refetch environment.
   • If delta_form_ref exceeds threshold → roll back model_id to last stable version, or reduce path resolution and recompute.
   • If u(T_arr_ref) is high → enhance sensor fusion or widen the condensation window until tol_u_T is met.

Summary
This chapter integrates RefCond and corr_env(x; RefCond) into the cleansing loop, harmonizing n_eff and T_arr under reference conditions at the path level while preserving dimensions and units. Using delta_form_ref and D_M^2 as core gates, it outputs ds_env, two-form verification results, and manifest.env. Key identifiers include RefCond, model_id, D_M^2, delta_form_ref, u(T_arr_ref), signature, TraceID.