22-EFT.WP.Metrology.Instrument v1.0 | Chapter 12 — Multi-Instrument Harmonization and Cross-Comparison (RRT / Interlab)

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Chapter 12 — Multi-Instrument Harmonization and Cross-Comparison (RRT / Interlab)

One-Sentence Objective
Establish a closed loop for harmonizing measured quantities and correcting biases across multiple instruments and laboratories via RRT/Interlab, and publish traceable consensus values, bias maps, and uncertainties.

I. Scope and Objects

1. Scope
   • Consistency assessment and correction for instruments of the same measurand across brands/models/labs, covering one-off RRT and rolling Interlab programs.
   • Single-point and multi-point calibrations; static and dynamic measurements (including arrival time T_arr).
   • Outputs: consensus value, per-instrument deviations and uncertainties, En and z indices, correction mappings, and conformity results.
2. Objects
   • Instrument set inst ∈ {1..M}, laboratories lab ∈ {1..L}, and measurement channels/conditions cond.
   • Raw results y_{i,lab,k}, reference or standard value Y_ref (possibly unknown; estimated via consensus).
   • Uncertainty & environment: u_i, U_i = k * u_i, RefCond. Timebase: tau_mono, ts, offset/skew/J.
3. Outputs
   {Y_cons, U_cons}, per-instrument bias b_i, repeatability s_i, En_i, z_i, correction map map_i(x), and manifest.instrument.harmon.*.

II. Terms and Variables

• Weighting & random effects: w_i = 1 / u_i^2, tau^2 (between-instrument variance).
• Bias parameters: b_i (bias), g_i (gain), linear map map_i(x) = g_i * x + b_i.
• Harmonization metrics: En_i = ( y_i - Y_ref ) / sqrt( U_i^2 + U_ref^2 ), z_i = ( y_i - Y_cons ) / u_i*.
• Interlab statistics: Mandel h_i, k_i; Youden (two-condition) point pairs.
• Two arrival-time forms: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ), with difference delta_form.

III. Postulates P712-*

• P712-1 (Dimensional conservation): before comparison, fitting, or publication, enforce check_dim( y - Y ) and normalize units.
• P712-2 (Unified timebase): perform all cross-instrument comparisons on tau_mono, publish on ts, and carry offset/skew/J.
• P712-3 (Dual arrival-time forms): when paths/arrival time are involved, compute both forms in parallel and record delta_form.
• P712-4 (Explicit uncertainty): use U = k * u_c and declare k and nu_eff.
• P712-5 (Traceability closure): each instrument must have a valid cal_cert_id and a closed SI_link.
• P712-6 (Environmental alignment): anchor on RefCond; when misaligned, apply corr_env(x; RefCond) and persist the correction.

IV. Minimal Equations S712-*

• S712-1 (Fixed-effects consensus)
  Y_cons = ( ∑ w_i * y_i ) / ( ∑ w_i ), U_cons = sqrt( 1 / ∑ w_i ).
• S712-2 (Random-effects consensus)
  w_i* = 1 / ( u_i^2 + tau^2 ), Y_cons(RE) = ( ∑ w_i* * y_i ) / ( ∑ w_i* ), U_cons(RE) = sqrt( 1 / ∑ w_i* ).
  Estimate tau^2 via DL (DerSimonian–Laird) or REML.
• S712-3 (Linear harmonization map)
  For instrument i: y_i = g_i * Y_cons + b_i + e_i.
  Estimate hat{g_i}, hat{b_i} by least squares or ODR; publish map_i(x) = hat{g_i} * x + hat{b_i}.
• S712-4 (Consistency statistics)
  En_i = ( y_i - Y_ref ) / sqrt( U_i^2 + U_ref^2 ) with |En_i| ≤ 1 as the usual pass band.
  z_i = ( y_i - Y_cons ) / u_i*; compute h_i, k_i per Interlab definitions to detect location/variance anomalies.
• S712-5 (Two-condition Youden plot)
  With two references A, B: for instrument i, point ( y_i^A - Y_cons^A , y_i^B - Y_cons^B ); clustering along the main diagonal indicates systematic bias, orthogonal spread indicates random error.
• S712-6 (Time/path harmonization constraint)
  For T_arr comparisons: record and assert delta_form ≤ tol_Tarr.

V. Metrology Procedure M70-12 (Plan → Acquire → Estimate → Correct → Publish)

1. Planning & readiness
   • Design RRT/Interlab: levels (points/conditions), sample sizes, rounds, RefCond, and time window Delta_t.
   • Register participating instruments, calibration certificates, and unit conventions; define pass thresholds {En ≤ 1, |z| ≤ z_thr, h/k limits}.
2. Acquisition & preprocessing
   • Collect y_{i,lab,k}, u_i; run standardize_names, repair_units, align_timebase.
   • Where environmental deltas exist, apply corr_env(x; RefCond).
3. Consensus estimation
   Compute fixed-effects Y_cons first; if heterogeneity is significant (Q > df, high I2), switch to random effects for Y_cons(RE) and estimate tau^2.
4. Bias and mapping
   Fit map_i(x) and residual structure; when necessary, use piecewise or polynomial maps while preserving monotonicity and conservation constraints.
5. Quality & outliers
   Compute En / z / h / k and multi-point consistency; trigger re-tests or isolate outlying channels when indicated.
6. Validation & replay
   Cross-validate with blind samples or external standards; replay prior rounds to assess stability and drift.
7. Publication & freeze
   Output Y_cons / U_cons / tau^2, and per instrument {map_i, En_i, z_i} with TraceID and signature; write manifest.instrument.harmon.* and freeze.

VI. Contracts & Assertions C70-12*

• dim.ok: check_dim( y - Y ) = true.
• time.ok: offset/skew/J within thresholds; when T_arr is involved, delta_form ≤ tol_Tarr.
• consensus.ok: if I2 > I2_max, switch to random effects and record tau^2.
• en.pass: |En_i| ≤ En_thr; z.pass: |z_i| ≤ z_thr.
• map.monotone: map_i'(x) ≥ 0 over the calibrated domain.
• traceability.ok: valid cal_cert_id and SI_link; RefCond persisted.

VII. Implementation Bindings I70-12* (Interface Prototypes)

• design_rrt_plan(levels, Delta_t, RefCond, thresholds) -> plan
• ingest_rrt_results(ds, plan) -> ds'
• compute_consensus(ds', model={"FE"|"RE"}) -> {Y_cons, U_cons, tau2, hetero}
• estimate_maps(ds', Y_cons, method={"OLS"|"ODR"|"piecewise"}) -> {map_i}
• interlab_stats(ds', Y_cons, U_cons) -> {En, z, h, k, youden}
• validate_harmonization(ds', maps, policy) -> report
• emit_harmon_manifest(results, policy) -> manifest.instrument.harmon
  Invariants: unit/dimension checks pass; maps are re-computable; manifests are traceable; when T_arr is present, both forms are computed and delta_form complies.

VIII. Cross-References

• Timebase & arrival time: EFT.WP.Metrology.TimeBase v1.0, Chapter 9; synchronization & link delay: EFT.WP.Metrology.Sync v1.0, Chapters 7–9.
• Cleaning & publication: EFT.WP.Methods.Cleaning v1.0, Chapter 10 and Appendix C.
• Statistical harmonization & drift: EFT.WP.Methods.CrossStats v1.0, Chapters 7 and 14.
• Related chapters in this volume: Chapter 10 (traceability & uncertainty), Chapter 11 (guardband & conformity).

IX. Quality Metrics & Risk Control

• Indicators
  En_pass_rate, z_within_rate, I2, tau2, map_stability (round-to-round change), decision_latency_p99.
• Risk control
  I2 beyond threshold triggers stratified or grouped harmonization; map drift triggers re-calibration; isolate and re-test outlier labs.
• Audit & rollback
  Retain raw data and parameters for each round; support replay by date/version; rollback to prior maps when necessary.

Summary
Through P712-* / S712-* / M70-12 / I70-12* / C70-12*, this chapter establishes a multi-instrument harmonization loop from planning and consensus estimation to bias correction and release freeze. For measurements involving time/path, it preserves dual arrival-time forms and unified timebase, and publishes results as traceable manifests with signatures.