Chapter 13 — Application Scenarios & Cases
- I. One-Sentence Objective
One-sentence goal: Along the full chain SeaProfile → n_eff → T_arr / ΔT_arr for the layered Energy Sea, provide six core, ready-to-deploy scenarios (A…F) plus one end-to-end integrated case with explicit inputs/outputs, stepwise workflows, interface mappings, and pass criteria/falsification lines—so engineering implementations are auditable, reproducible, and comparable. - II. Scope & Non-Goals
- Coverage: scenario goals; alignment between data contracts and the interface family; audit checkpoints for the two formulations/differencing/energy consistency/thin–thick consistency; logging and publication conventions; common risks and safeguards.
- Non-Goals: no repetition of derivations from Chapters 3–12 or detailed error analyses; no device-level hardware or external observing pipelines.
- III. Minimal Terms & Symbols
- Layers & profiles: SeaProfile, W_k(chi), Xi_k(chi), Sigma_sea, Delta_k, intersections { ell_i }.
- Propagation & formulations: n_eff(x,t,f) (with n_eff ≥ 1), c_ref, T_arr (constant-factored/general).
- Observables & differencing: T_arr_obs(f, gamma), Delta_T_arr_obs(f1,f2, gamma).
- Energy triplet: R_sea, T_trans, A_sigma with R_sea + T_trans + A_sigma = 1.
- Consistency: two-formulation eta_T, thin–thick tau_switch; name isolation: T_fil ≠ T_trans, n ≠ n_eff.
- IV. Scenario A | Layer-Parameter Identification (SeaProfile Inversion)
Goal: From multi-band, cross-angle arrival-time data, invert layer parameters { chi_k, Delta_k, sigma_k, a_{lm}^{(k)} } and RTParams curves.
Inputs - Observations: T_arr_obs(f_m, gamma_a) with uncertainties; f_grid; coords_spec/units_spec/metric_spec; a calibrated or to-be-calibrated c_ref.
- Priors: Phi_T, grad_Phi_T or measurable surrogates; an initial SeaProfile.
Outputs - theta_hat (layer parameters) with covariance; RTParams (R_sea(f), T_trans(f), A_sigma(f)); consistency metrics eta_T, tau_switch, and the lower-bound margin.
Workflow (template interface family → suggested I60-*) - I.Path.Capture / I.Path.Segment → capture_path, detect_sea_intersections.
- I.Build.Phi / I.Build.Neff → apply_sea_matching, estimate_neff_sea.
- I.Fit.Profile → fit_sea_profile to obtain theta_hat; I.RT.Estimate → estimate_RT_sea.
- I.Consistency.DualMode | ThinThick → check_dual_arrival_consistency, consistency_thin_vs_thick_LS.
- I.Report.Emit → archive report and logs.
Pass criteria: |Residual| ≤ GB; eta_T ≤ threshold; tau_switch ≤ limit; R_sea + T_trans + A_sigma = 1; n_eff^± ≥ 1.
Falsification lines: stable n_eff < 1 or eta_T / tau_switch over threshold with no successful backtracking. - V. Scenario B | Cross-Layer Propagation Estimation (Inner/Layer/Outer Segmentation)
Goal: Under explicit Region_in / layer / out segmentation, compute cross-layer T_arr and audit interface matching and energy consistency.
Inputs/Outputs - Inputs: SeaProfile, path set { gamma_a } with { ell_i }, Phi_T/grad_Phi_T, c_ref, mode.
- Outputs: per-segment T_arr_i and synthesized T_arr_total; residuals for R_sea/T_trans/A_sigma and one-sided reports.
Workflow - apply_sea_matching → estimate_neff_sea to obtain n_eff^±.
- segment_integrals (and interface_correction_sea if required).
- estimate_RT_sea to audit energy consistency.
Pass/Fail: energy consistency and one-sided limits pass/fail; endpoints explicitly included; no cross-interface interpolation. - VI. Scenario C | In-Band Differencing to Isolate the Path Term
Goal: On the same path, use Delta_T_arr_obs to isolate n_path and fit the in-band order.
Inputs/Outputs - Inputs: Observations, f_grid, gamma, SeaProfile or n_eff.
- Outputs: n_path parameters (c_m) and a differencing consistency report (correlation, slope, out-of-band residuals).
Workflow - Reuse the same { gamma[k], Δell[k] } and segmentation/correction configuration.
- Compute Delta_T_arr_mod (with identical configuration for the two formulations).
- Evaluate correlation/slope and leakage; update c_m or backtrack path/band layout.
Pass/Fail: in-band linear region/specified order satisfied; out-of-band residuals folded into u_sys; otherwise falsify differencing assumptions or band configuration. - VII. Scenario D | Thin/Thick Decision & Switching
Goal: Decide the layer-band formulation at run time based on Delta_k / L_char and tau_switch, maintaining robustness and traceability.
Workflow - Pre-evaluation: consistency_thin_vs_thick_LS on representative paths and step-size tiers.
- Online: record eta_w; in the threshold neighborhood, dual-run and compare tau_switch.
- Policy: if tau_switch > limit → lock thick-layer chain; backtrack endpoint tolerances and SeaProfile.
- Archiving: log_artifacts_LS with switching rationale and difference curves.
Pass/Fail: after switching, eta_T, the lower bound, and energy consistency pass; persistent failures falsify the thin-layer approximation or profile settings. - VIII. Scenario E | Long-Term Drift Monitoring & Guarding (Streaming)
Goal: Monitor c_ref(t), n_common(x,t), and layer-parameter drift; auto-alert and maintain formulation consistency.
Workflow - Periodic calibration calibrate_c_ref to update c_ref(t).
- Sliding-window fit_sea_profile and (optional) decompose_n_eff to refresh slow variables.
- Guarding: compute GB = k_guard · u_c, eta_T, tau_switch; trigger backtracking and alerts if thresholds are exceeded.
- Archiving: log_artifacts_LS with metric time series and environmental blocks.
Pass/Fail: drift within band / out-of-band but unexplained by environment falsifies the current calibration. - IX. Scenario F | Risk Assessment & Guardband Setting
Goal: Before deployment, assess tail risk and set GB and operational thresholds.
Workflow - propagate_uncertainty_MC to generate T_arr / Delta_T_arr distributions.
- Evaluate n_eff clamping trigger rate, Delta_T_sigma trigger statistics, and leakage ratio.
- Set k_guard and GB and fix them in the report.
Pass/Fail: coverage probability meets target / tail risk too high → adjust paths/bands or raise data thresholds. - X. End-to-End Case | Imaging, Inversion & Echo Verification for a Non-Spherical Layer
Goal: With chi_k(θ,φ) non-sphericity, complete layer-parameter imaging, arrival-time inversion, energy consistency, and echo-order joint verification.
Steps - Prep & metrology: declare_tw_contract (or LS Contract), calibrate_c_ref, capture_path, detect_sea_intersections.
- Assembly & segmentation: apply_sea_matching, estimate_neff_sea, segment_integrals (and interface_correction_sea if needed).
- Inversion: fit_sea_profile to obtain theta_hat; estimate_RT_sea to audit energy consistency.
- Consistency: check_dual_arrival_consistency, consistency_thin_vs_thick_LS.
- Echo verification: simulate_multipath_LS aligned to observations; extract Delta_T_echo(k).
- Reporting & archiving: emit_measurement_report; persist hashes and falsification samples.
Pass criteria: residuals within GB; two-formulation agreement; tau_switch passed; energy consistency; echo orders matched. - XI. Logging & Minimal Records (Universal Across Scenarios)
- Physics & geometry: hash(SeaProfile), hash(grad_Phi_T), hash(n_eff), hash(gamma), Sigma_sea labels and { ell_i } tolerances.
- Formulations & thresholds: mode, eps_T, eta_T, eta_c, eta_w, tau_switch, lower-bound margin T_arr − L_path/c_ref.
- Energy & differencing: residuals for R_sea/T_trans/A_sigma, Delta_T_sigma trigger counts and magnitudes, Delta_T_arr linear region and leakage ratio.
- Uncertainty & reproduction: u_stat, u_sys, u_c, k, seed, coords_spec/units_spec/metric_spec, SolverCfg and hash manifest.
- XII. Minimal Data-Contract Checklist
- Contract: spec_version, coords_spec, units_spec, metric_spec, mode, tolerances:{ eps_T, eta_T, eta_w, tau_switch }.
- SeaProfile: model, parameters, hash(SeaProfile); Sigma_sea metadata.
- Path / Observations / RTParams: required fields and units; timestamps in ISO-8601.
- Report / Log: metrics, falsification samples, replay handles, and hashes.
- XIII. Interface & Implementation Mapping (Scenario → Template Family → Suggested I60-*)
- Scenario A: I.Path.Capture | Segment → I.Build.Phi | Neff → I.Fit.Profile → I.RT.Estimate → I.Consistency.* → I.Report.Emit.
- Scenario B: I.Interface.ApplyMatching → I.Build.Neff → I.Path.Segment | InterfaceCorrection → I.RT.Estimate.
- Scenario C: I.Arrival.Delta → I.Consistency.DualMode → I.Report.Log.
- Scenario D: I.Consistency.ThinThick → I.Report.Log.
- Scenario E: I.Calibration.Cref → I.Fit.Profile (sliding window) → persist guarding metrics.
- Scenario F: I.Uncertainty.MC → I.Report.Emit.
- XIV. Pass Criteria & Falsification Lines (Summary)
- Pass: T_arr ≥ L_path / c_ref; eta_T ≤ threshold; R_sea + T_trans + A_sigma = 1; n_eff^± ≥ 1; tau_switch ≤ limit; Delta_T_arr in-band linear region satisfied.
- Falsification: any single failure after excluding implementation error; three consecutive, independent reproducible falsifications on the same dimension trigger a review of postulates and profiles.
- XV. Deliverables
- Scenario-specific workflow checklists and parameter templates (A…F).
- Formulation/consistency/energy audit templates: eta_c, eta_T, tau_switch, and residual dashboards.
- End-to-end case reproducibility package structure: data/code/parameters/SolverCfg/random seed/hash manifest and replay scripts.
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