ONLINE COURSE

Sample Professional Lab and Assignment Showcase

A Cyber Warfare Engineer in everything but name. Welcome to your Professional Level. This capstone focuses on adversarial analysis: end-to-end QKD telemetry triage, detector modeling, change-point detection, and key-rate bounding under realistic attacks. You’ll complete 15 labs that culminate in a final exam. Upon passing, you earn the SpecterAI Quantum Cyber Warfare Professional Engineer (SQCWPE) diploma. Expect larger datasets, multi-stage pipelines, and stricter acceptance tests. The sample lab below matches the exact structure used throughout. Remember, this is not a certificate mill—mastery is demonstrated through mathematically sound reasoning and working code.

Problem

P‑01 — C‑Lion1 Sentinel — Professional Solution Artifact

P‑01 — C‑Lion1 Sentinel — Professional Solution

Below is a complete, step‑by‑step design and analysis suitable for hand‑off to operations. Calculations are written so they can be reproduced in the companion notebook.

0) Overview and data model

We assume access to:

Coherent backscatter (DAS) cube $X_\phi(z,t)$: unwrapped phase differences over gauge length $L_g$, sampled on a range grid $z\in[0,L]$ with step $\Delta z$ and time grid $t$ with step $\Delta t$.
Stokes/SOP cube $S(z,t)\in\mathbb{R}^3$: $S_1,S_2,S_3$ at the same grid.
Auxiliary: bathymetry/route, known repeater ranges $z_r$, AIS tracks (optional), metocean data.

All processing below is single‑ended (shore station only). Launch power and LO power are configured to avoid nonlinearity and saturation. We operate a supervisory wavelength outside traffic bands.

1) Calibration and preprocessing

1.1 Clock, range, and alignment.

Calibrate the time base to GPS and range to fiducials $z_r$ by correlating known repeater signatures. Interpolate to a uniform $\Delta z$ grid (e.g., 2 m). Maintain a mapping $z\leftrightarrow(\text{lat},\text{lon})$ using the as‑laid route.

1.2 Phase unwrapping and gauge differencing.

From raw complex samples $E(z,t)$ infer wrapped phase $\phi_w(z,t)$. Use a quality‑guided unwrapping in time first (short gaps) and then in space; then compute $\Delta\phi(z,t)=\phi(z+L_g,t)-\phi(z,t)$ and scale to nanostrain equivalent via

\[ \epsilon(z,t)\;\approx\;\frac{\lambda}{4\pi n L_g}\,\Delta\phi(z,t), \]

where $\lambda$ is the wavelength and $n$ the effective index. Retain $\epsilon(z,t)$ as the DAS strain proxy.

1.3 Whitening and noise shaping.

Estimate the background power spectral density $P_\epsilon(f;z)$ from a quiet training interval. Apply a prewhitening filter $W(f;z)=\tfrac{1}{P_\epsilon(f;z)+\eta}$ (small $\eta$ for numerical stability). This levels noise across bands and stabilizes thresholds. For SOP, compute the rotation rate $\omega_S(z,t)=\lVert \dot{S}(z,t)\rVert$ using Savitzky–Golay derivatives and smooth with a 1–2 s median.

1.4 Quality masks.

DAS exhibits coherent fading—deep minima where backscatter combining is destructive. Build a quality mask $Q(z)$ by long‑term SNR or coherence; suppress decisions in bins with $Q(z)$ below threshold and interpolate across.

2) Feature extraction

Operate in sliding windows of length $T_w=10$ s with hop 2 s. For each window $W(t_0)$ at time center $t_0$ and range $z$, compute:

2.1 Band energies (DAS).

Apply a Hann taper and FFT to the whitened strain $\tilde\epsilon(z,t)$ inside $W$. Sum energy in three bands: LF $B_1: 0.1\!-\!2$ Hz (slow ramps/anchor); MF $B_2:2\!-\!20$ Hz (cable rub, hydrodynamic noise); HF‑ROV $B_3:100\!-\!800$ Hz (thrusters/gear tones). Statistic per band: \[ E_k(z,t_0)=\sum_{f\in B_k}\lvert \tilde\epsilon(z,f;t_0)\rvert^2. \]

2.2 Tonal peakiness and harmonicity.

Within $B_3$ find top 3 peaks $f_1,f_2,f_3$ and compute \[ H(z,t_0)=\frac{\sum_{i=1}^3 P(f_i)}{\sum_{f\in B_3} P(f)},\quad P(f)=\lvert\tilde\epsilon\rvert^2. \] Thrusters yield high $H$ with near‑harmonic spacing (e.g., $f_2\approx2f_1$, $f_3\approx3f_1$ or blade‑pass offsets). Turbulence is diffuse (low $H$).

2.3 SOP rotation and hysteresis.

Compute $\omega_S(z,t_0)=\frac{1}{T_w}\int_W \lVert \dot S(z,t)\rVert\,dt$ and a loop area in the $S_1\!-\!S_2$ plane: \[ A_S(z,t_0)\approx \left|\oint_W \big(S_1\,dS_2 - S_2\,dS_1\big)\right|. \] Clamp‑on or bending events produce large $A_S$ relative to background SOP flutter.

2.4 Persistence and motion.

Track persistence counters $p_k(z)$ (consecutive red windows). Estimate range velocity by cross‑correlation between adjacent windows: \[ \dot z(t_0)=\arg\max_{\delta z}\;\frac{\langle \tilde\epsilon(z,t_0),\tilde\epsilon(z+\delta z, t_0+T_h)\rangle}{\delta z}. \] Anchor drags often exhibit $|\dot z|\sim0.1\!-\!1$ m/s; stationary ROV loiter has $\dot z\approx0$.

3) Thresholds and detectors

3.1 Energy detectors.

With prewhitened noise, $E_k$ is approximately $\sigma^2\chi_\nu^2$ with $\nu$ twice the number of frequency bins in $B_k$ (after taper/overlap factors). Choose thresholds $T_k$ via $\Pr(E_k>T_k\mid H_0)=\text{Pfa}_k$ (e.g., $10^{-4}$ for red; $10^{-3}$ for yellow).

3.2 Tonal/harmonic detector.

Use z‑scores of peak power versus the local spectral median; require (i) $H>H_{\text{red}}$ (e.g., 0.45) and (ii) two peaks with near‑harmonic spacing. This flags an ROV‑tone red.

3.3 SOP detectors.

Set $T_\omega$ and $T_A$ using training background quantiles (e.g., 99.9th and 99.99th). SOP bursts above both are red.

3.4 Persistence rule.

A feature becomes operative red only if $p_k(z)\ge3$ (three consecutive windows). This mirrors the QKD triage discipline and reduces spurious calls.

4) Decision fusion

Define a tamper score per range and window: \[ \text{TS}(z,t_0)= w_1\,\mathbf{1}[E_3 \text{ red}] + w_2\,\mathbf{1}[H \text{ red}] + w_3\,\mathbf{1}[\omega_S \text{ red}] + w_4\,\mathbf{1}[A_S \text{ red}] + w_5\,\mathbf{1}[\text{LF ramp red}], \] with default $w_1=1,\;w_2=1.5,\;w_3=1,\;w_4=1,\;w_5=1$ (raise $w_2$ to favor tonal evidence).

Rule of engagement:
ALERT if (i) $p_{\text{tone}}\ge3$ and $p_{\text{SOP}}\ge3$ at (approximately) the same $z$, or TS$\ge4$ with at least two modalities red.
INVESTIGATE if any one modality persists $\ge3$ but TS<4, or the event is moving with $|\dot z|>0.25$ m/s without SOP hysteresis (anchor drag).
PASS otherwise.

Contextual overrides: if a red patch sits within ±50 m of a repeater vault or branch, lower TS by 0.5; if AIS shows a declared maintenance vessel on top of the patch, downgrade one level unless SOP is strongly hysteretic.

5) Geolocation and uncertainty

5.1 Range → coordinates.

Map $z$ to (lat,lon) with the as‑laid route. Report the center of mass of the red patch’s energy for stability.

5.2 Uncertainty budget.

Combine (i) instrumental $\pm \Delta z/2$, (ii) route error, and (iii) patch scatter in quadrature to obtain $\sigma_z$. Convert to a geographic ± ellipse using local track bearing.

5.3 Timing.

Report event start/stop with precision $T_h$ (hop). For MTTD, record detection time since onset in synthetic ground truth.

6) Worked detection on the exemplar event

Assume the synthetic tile emulates the 22:17Z event. Parameters: $\Delta z=2$ m, $L_g=10$ m, $\Delta t=1$ ms (1 kHz per range), $T_w=10$ s, hop $T_h=2$ s. Quiet background PSD from 15 min prior.

6.1 Bands.

Compute $E_1,E_2,E_3$. We observe at $z=163.4$ km: LF ramp‑like excess across three windows; HF‑ROV tones at 281 and 562 Hz with $H=0.58\!\gg\!H_{\text{red}}=0.45$; MF mild rise (not red).

6.2 SOP.

$\omega_S$ peaks at 2.3 rad/s (above $T_\omega$); $A_S$ shows a clean loop in two consecutive windows, then a larger loop (suggests clamp‑on or bend with stick–slip). Persistence: $p_{\text{tone}}=3,\;p_{\text{SOP}}=3,\;p_{\text{LF}}=3.$

6.3 Decision.

$\text{TS}=1(E_3)+1.5(H)+1(\omega_S)+1(A_S)+1(\text{LF})=5.5$. Meets ALERT on both conjunction and TS rules. $\dot z\approx 0$ (stationary), favoring ROV loiter over anchor drag. The event sits within 40 m of a repeater frame—critical.

6.4 Rationale to the watch.

“ALERT: Stationary ROV‑like signature at 163.4 km from Oulu (±22 m), duration 34 s. Three consecutive windows show HF tonal energy with harmonic structure at 281/562 Hz; simultaneous SOP hysteresis loops suggest physical contact with the cable or repeater frame; a low‑frequency strain ramp indicates load. No declared maintenance. Recommend dispatch of nearest patrol asset and covert illumination restriction on repeaters 5–6.”

7) Performance estimation (Pd, Pfa, MTTD)

7.1 Monte Carlo on synthetic set.

Generate $N$ tiles per class: ROV loiter (tones + SOP), anchor drag (moving LF + impulses), trawler gear (moving HF wideband, lower harmonicity), microseism (LF only). Vary SNR to produce realistic strains (tens–hundreds of nanostrain in HF; microstrain ramps in LF). For each tile, run the detector and tally outcomes at Pfa design points (sum of per‑band PFAs ≈ $3\times10^{-4}$ per window).

7.2 Metrics.

Compute Pd by SNR bin; fit Pd(SNR) curves. Example (illustrative): at Pfa $=10^{-3}$, $\text{Pd}_{\text{ROV}}=0.94$, $\text{Pd}_{\text{anchor}}=0.87$, $\text{Pd}_{\text{trawler}}=0.81$, $\text{Pd}_{\text{microseism}}=0.09$ (correct—should be low). MTTD (median) is 12 s for ROV loiter (limited by 3‑window persistence) and 16–22 s for anchor drags. Tune persistence (2 vs 3 windows) to trade MTTD vs false alarms; three windows preferred in crisis ops.

8) Robustness and pitfalls

8.1 Fading and nulls.

DAS sensitivity varies with range due to coherent fading; SOP estimation can degrade at polarization nulls. Mitigations: spatial averaging, multitone probing, adaptive $Q(z)$ masks.

8.2 Weather coupling.

High sea states elevate LF background. Correlate with metocean and raise LF thresholds adaptively. Do not raise SOP thresholds; SOP hysteresis is less weather‑coupled.

8.3 Coexistence with traffic.

Verify no cross‑gain modulation or Raman crosstalk into traffic. Measure/log OSNR and BER on traffic before/after Sentinel enablement. Keep launch powers inside supervisory specs; capture in safety case.

8.4 Adversarial deception.

Actors may mimic trawler noise near a target. Fusion is key: without SOP hysteresis or LF ramp near a structure, do not upgrade to ALERT. Conversely, moving slowly to evade persistence is countered by multi‑scale persistence (e.g., 3 of 5 windows) and range‑stationarity checks.

9) Implementation notes for the team

Pipeline: calibrate → whiten → features → thresholds → fusion → decision; persist intermediate maps for replay.
Export a “red‑patch video”: time–range tiles with HF spectral overlays and SOP Lissajous; this is what convinces commanders.
Instrument one‑click rationale: bullet list of tripped thresholds with values.

10) Validation checklist (readiness review)

Receiver calibration plots (phase noise PSD before/after differencing; coherence vs $L_g$).
Quiet‑sea baselines: distributions of $E_k,H,\omega_S,A_S$ with thresholds labeled.
Confusion matrix over synthetic sets and any historical labeled events.
Pd/Pfa trade curves; MTTD histograms.
Localization error: injected point perturbations recovered within ±25 m.
Safety: measured OSNR/BER on traffic unaffected; laser power logs; interlocks.
SOP/damage correlation: controlled bend tests in lab segment showing loop area vs bend radius.

11) What operators actually see (UX)

Top pane: live time–range heatmap of HF energy (100–800 Hz) with red patches painted where persistence≥3. Right: SOP rotation & loop meters. Bottom: LF ramp trace and decision timeline. Clicking a red patch pops the rationale card (thresholds, tones, SOP loops, LF ramps, TS, location).

12) Limits, ethics, and next steps

This system listens to physics and does not inspect payload. It operates on supervisory channels/dark fibers with controlled powers. Limits: coherence, fading, sea state, adversary ingenuity. Extensions: bi‑end sensing; a small CNN for tonal recognition (with rule‑based fusion for transparency); multi‑cable fusion; an exercise library with controlled ROV runs.

Summary Decision Template (for “Deliverables” panel)

Calibrated features per window & range: $E_1,E_2,E_3,H,\omega_S,A_S$, persistence counters.
Decision rule (ALERT/INVESTIGATE/PASS) with rationale (“which thresholds tripped”).
Geolocation with uncertainty ellipse & MTTD.
Performance tables (Pd/Pfa, confusion matrix) on provided tiles.
Safety/Coexistence: power levels & traffic integrity verified.

Student Edition

C‑Lion1 Sentinel

1.0 Mission Brief (Scenario)

Year: 2030. Role: Lead systems engineer embedded with NATO’s Maritime Infrastructure Protection Program (MIPP). Objective: Stand up a physics‑aware monitoring layer on the C‑Lion1 subsea trunk and make minutes‑scale, defensible calls about hostile interference versus benign background. Your sensors are the cable itself (as an interferometric microphone and strain gauge) and a polarization tap; your mandate is auditable triage, not forensics after the fact.

War has pushed activity into the High North and the Baltic. Patrols shadow commercial convoys; drones skim low over the water; “research” hulls idle over repeaters. The C‑Lion1 trunk is part economic artery, part military umbilical. Tonight, Oulu’s shore station smells of coffee and printer toner. A watchstander flags you: an automated change‑point detector paints a 45‑second swell of low‑frequency strain near 163.4 km from the Finnish shore, then a burst of HF tonal energy at roughly 280–560 Hz—classic thruster offsets. In the same range patch, state‑of‑polarization (SOP) traces show a brisk rotation with a small hysteresis loop, the kind you see when something physically couples to the cable or a repeater frame. AIS playback hints at “fishing” traffic drifting up‑current. The question is brutally simple and operationally high‑stakes: do you call ALERT, INVESTIGATE, or PASS?

Constraints bite. You cannot touch revenue wavelengths, cannot risk laser over‑power into plant components, and cannot train the watch to ignore alarms. Your proof‑of‑effect must be quantitative and manager‑readable: fixed false‑alarm rates per window, measured probability of detection on labeled tiles, localization error in meters, and mean time‑to‑declare. Yet at 03:00 your tools must remain napkin‑simple: features that a junior can compute, thresholds sourced to a commissioning doc, persistence to suppress one‑off blips, and a fusion rule transparent enough to brief a colonel on a single slide.

The mission brief you are holding is written in that spirit. It gives you a complete operational narrative—from physics to features to decision ladder—without solving the lab for you. You will compute calibrated energies in three bands, an HF harmonicity statistic for thrusters, SOP rotation and loop area for contact, and a persistence count per range bin. You will then apply a guardrailed fusion into a tamper score and produce a one‑sentence console brief that cites denominators and thresholds. The rest of this chapter builds the conceptual floor you need to make that call with speed and confidence.

2.0 Historical & Conceptual Background

From payload encryption to physics listening. For decades, cable defenders leaned on cryptography and perimeter security. But attackers don’t need to read payload to damage a nation’s arteries; they can drag, bend, or clamp. That shift drew attention to the fiber itself as a sensor. Distributed acoustic sensing (DAS) treats the glass as a long interferometer by coherently probing Rayleigh backscatter; state‑of‑polarization provides a second view, sensitive to stress and twists in the cable armor. The Sentinel idea is not to peer into traffic but to “listen” for mechanical and acoustic fingerprints that ride on the glass. Detecting and classifying those signatures—then deciding quickly—is the heart of the playbook.

What “the fiber hears.” Consider three archetypes. Anchor drag: a slow LF ramp in strain as metal snags and pulls; intermittent impulses from stick‑slip; a moving, spatially narrow patch. ROV loiter: narrow HF tones with near‑harmonic spacing from thrusters; stationary in range; occasional impacts; SOP hysteresis if a manipulator brushes a frame. Trawler gear: quasi‑periodic thumps near door impact with broader spatial footprint and a Doppler‑shifted low‑frequency wash. Although the ocean is awash in noise—microseisms, turbulence, traffic—these event families differ in where their energy lives (bands), how it persists, and whether polarization loops.

Why two modalities are better than one. DAS is powerful but imperfect: coherent fading creates patchy sensitivity; some events hide in minima. SOP reacts to mechanical stress through the fiber’s birefringence tensor; clamp‑on bends and hard contacts produce abrupt rotations and, critically, hysteretic loops that are highly localized in range. Combining the two reduces false alarms because coincidence in independent sensors is rare for benign backgrounds. The design choice you inherit is deliberate simplicity: a handful of features whose physics you can explain, plus a persistence rule, beats a black‑box classifier when you’re briefing at 03:07.

The operational covenant: guardrails, not improvisation. Thresholds live in a commissioning document derived from quiet‑sea quantiles and validated tiles. You do not tune them mid‑alert. Doing so destroys your audit trail and biases decisions toward optimism under fatigue. The student artifact keeps that discipline: you will compute features, compare to fixed thresholds, and apply a written fusion rule. Disagree with a threshold? Capture the argument in your after‑action memo and propose a policy update—after the incident, not during it.

Geolocation and the promise of explainability. Range bins map to waypoints along the as‑laid route using repeater fiducials and bathymetry references. Report the center‑of‑mass of the red patch rather than the spikiest pixel to avoid jumps when two bins tie. Build an uncertainty ellipse that combines instrument resolution, route error, and patch scatter. When an admiral asks “Why that spot?” your answer will reference physics (thruster tones; SOP loops), policy (persistence and thresholds), and geometry (range‑to‑lat/lon), not priors and vibes.

3.0 Technical Foundations

3.1 The fiber as a distributed interferometer

Single‑mode telecom fiber contains a constellation of microscopic scatterers. Launch a narrow‑linewidth probe and heterodyne the backscatter with a local oscillator (LO) to measure the complex field as a function of range. Form phase differences between adjacent range bins over a gauge length L_g; those differences reject common‑mode LO phase noise and are proportional (under small perturbations) to axial strain integrated over the gauge.

ε (z, t) \approx \frac{λ}{4 π n L g} Δ φ (z, t)

Gauge‑differenced phase converts to a strain proxy using wavelength λ, effective index n, and gauge length L_g.

After unwrapping phase in time and space, you will operate on this strain proxy ε(z,t) in sliding analysis windows. Using a quiet training interval, estimate background power spectral density per range and flatten it with a stabilizing filter to make thresholds uniform across bands.

W (f; z) = \frac{1}{P_{ε} (f; z) + η}

Prewhitening divides by the local background PSD (with a small η for numerical safety) to stabilize downstream energy tests.

3.2 Band energies and thruster harmonicity

On whitened strain inside a window centered at t₀, accumulate energy in three bands: LF (0.1–2 Hz), MF (2–20 Hz), and HF (100–800 Hz). These choices reflect where anchor ramps, cable rub, and thrusters are most separable.

E_{k} (z, t_{0}) = \sum_{f \in B_{k}} | \tilde{ε} (z, f; t_{0}) |^2

Band energy in LF/MF/HF after whitening; high values indicate abnormal power in that band for range z.

To capture ROV‑like loiter, compute a simple harmonicity statistic in the HF band: take the three strongest peaks and compare their power to the band total. Require near‑harmonic spacing to avoid false positives on diffuse hiss.

H (z, t_{0}) = \frac{\sum_{i = 1} P^{(f_{i})}}{\sum_{f \in B_{3}} P (f)}

Harmonicity: fraction of HF power captured by the top three peaks; high H with near‑harmonic spacing flags thrusters.

3.3 State‑of‑polarization rotation and hysteresis

A polarization tap measures the Stokes vector S = (S₀,S₁,S₂,S₃) versus range. Mechanical stress rotates the SOP via local birefringence; clamp‑on bends and contact can trace loops in the (S₁,S₂) plane. You will compute an average rotation rate and an approximate loop area within each window.

ω_{S} (z, t_{0}) = \frac{1}{T_{w}} \int W_{t_{0}} ‖ \dot{S} (z, t) ‖ d t

SOP rotation rate: average speed of the Stokes trajectory within the window (rad/s).

A_{S} (z, t_{0}) \approx \oint W_{t_{0}} (S_{1} d S_{2} - S_{2} d S_{1})

SOP loop area: a hysteresis proxy; large area implies contact or a sustained bend near a structure.

3.4 Persistence and motion

Single spikes lie; persistence tells the truth. Count consecutive red windows per modality at each range. For motion, cross‑correlate neighboring windows to estimate range velocity and distinguish stationary loiter from moving drags.

\dot{z} (t_{0}) = arg max_δz \frac{⟨ \tilde{ε} (z, t_{0}), \tilde{ε} (z + δz, t_{0} + T_{h}) ⟩}{δz}

Range velocity from window‑to‑window cross‑correlation; loiter ≈ 0, anchor drags show positive drift.

3.5 Decision fusion (guardrails)

After thresholding each feature, fuse with a transparent tamper score TS that favors coincidences across modalities (especially harmonic tones + SOP hysteresis). This score feeds an action ladder: ALERT (coincident, persistent reds), INVESTIGATE (partial evidence or motion without SOP loops), and PASS (everything else). Your commissioning doc supplies exact thresholds; your job is to apply them consistently.

TS (z, t_{0}) = w_{1} [1 (E_{3} red)] + w_{2} [1 (H red)] + w_{3} [1 (ω_{S} red)] + w_{4} [1 (A_{S} red)] + w_{5} [1 (LF red)]

Tamper score uses red/green flags per feature; weights prioritize harmonic evidence together with SOP hysteresis.

Action ladder (policy): ALERT if tonal and SOP modalities are both persistent at roughly the same range, or if TS crosses a high threshold with ≥ two modalities red. INVESTIGATE for single‑modality persistence or moving events without SOP loops. PASS otherwise.

4.0 Problem Setup & Inputs

What you will receive. A strain proxy cube ε(z,t) and an SOP cube S(z,t) sampled on a uniform grid (e.g., Δz ≈ 2 m, Δt ≈ 1 ms). You will also have the as‑laid route polyline, repeater fiducials, and optional AIS/metocean snippets. You are not given labeled ground truth in the main packet; your job is to decide now, under policy guardrails, then provide the rationale a manager can read.

Operational expectations. Your solution must (i) avoid mid‑alert parameter tuning, (ii) justify thresholds by citing the commissioning doc, (iii) control per‑window false alarms at design points, (iv) show persistence counters, and (v) emit a one‑sentence decision brief along with a location and an uncertainty ellipse. Anything less fails readiness review.

5.0 Student Tasks

Calibrate and unwrap. Align range to repeater fiducials and time to GPS; unwrap phase in time then space; convert to the strain proxy via the gauge‑difference formula.
Whiten and mask. Estimate background PSD per range; build W(f;z) with a stabilizer η; apply quality masks where SNR collapses. Document your quiet interval and any assumptions.
Compute features. In windows of length T_w with hop T_h, compute LF/MF/HF energies, HF harmonicity H, SOP rotation ω_S, SOP loop area A_S, and persistence counters per range.
Threshold and fuse. Apply fixed thresholds to mark red windows. Compute the tamper score TS(z,t₀) and apply the action ladder; do not edit thresholds mid‑analysis.
Geolocate and time‑stamp. Map the range patch to (lat, lon) using the as‑laid route; compute an uncertainty ellipse that combines instrument, route, and patch scatter; record start/stop times at hop resolution.
Write the brief. Produce a one‑sentence console decision (ALERT/INVESTIGATE/PASS) that cites denominators and thresholds, followed by a bullet rationale that names which thresholds tripped and the persistence counts.

6.0 Step‑by‑Step Guidance (Non‑Solution Scaffolding)

6.1 Read the packet like weather

Start with an eyeball pass: are there narrow HF ridges? is LF breathing across many kilometers or localized? do SOP traces meander or loop? This is pre‑computation triage—use it to set your expectations and to choose sensible color scales. Don’t infer the answer here; you are simply orienting yourself before you compute.

6.2 Whiten for stability

Compute the quiet‑sea median PSD per range over a several‑minute training block that excludes the alert window. Your stabilizer η prevents division by tiny PSD minima. Verify that band‑energy histograms look near‑stationary across range after whitening. If a few ranges remain unstable due to deep fading, trust your quality mask and let the fusion lean more on the surviving modality.

6.3 Tune nothing, explain everything

Document where your thresholds came from (e.g., “99.9th percentile of LF energy across training tiles”). If you are tempted to lower a threshold “just this once,” stop and write an after‑action memo instead. Triage is a contract with your future self: repeatable, auditable, and boring in the best way.

6.4 Persistence before fusion

Mark red windows and build your persistence counters per modality. This one step collapses a thousand transient annoyances. Only then compute TS. A modal red that never persists is not allowed to drive the ladder; that is how you keep false alarms livable under sea‑state swings.

6.5 Action ladder with context

If tonal and SOP reds overlap in range and persist, ALERT is often automatic. If only one red persists with motion in range, favor INVESTIGATE; if everything is green or non‑persistent, PASS. Apply contextual modifiers last: AIS‑declared maintenance atop a patch should downgrade, while proximity to a repeater may upgrade urgency by shortening response time.

6.6 Geolocate with humility

For the center, use the energy centroid of the range patch. For uncertainty, combine (i) instrument resolution (±Δz/2), (ii) route mismatch, and (iii) patch scatter. Express the result as a ± ellipse aligned with local track bearing. That figure is what a patrol navigator actually needs.

7.0 Worked Reading Prompts

Physics prompt: In two paragraphs, defend gauge differencing as a way to suppress common‑mode LO drift while keeping sensitivity to axial strain. What assumptions control the proportionality constant?
Signal‑processing prompt: Compare band sums on whitened versus raw strain. Why do thresholds derived on raw data “wander” across range?
Pattern prompt: Explain why harmonicity plus SOP hysteresis is much harder to fake than a single LF ramp. Give one benign counterexample and how persistence defeats it.
Policy prompt: Argue for or against including MF energy in the ladder for your theater. What false alarms did MF reduce or introduce in your experiments?

8.0 Real‑World Variations & Case Notes

8.1 Sea state and metocean

Microseisms lift LF floors across hundreds of kilometers during storms. Correlate LF thresholds with metocean measures to avoid alert floods. Do not over‑react to MF; it often reflects hydrodynamic hiss rather than mechanical contact. HF tonal evidence is surprisingly robust when ships idle near frames—even in foul weather.

8.2 Coherent fading and quality masks

Coherent backscatter lives and dies on interference. Deep minima are inevitable. Do not hide them; mask them. Plot the mask atop your energy map in your slide deck so leaders see where decisions were disallowed by physics, not by software moods.

8.3 Traffic coexistence

Sentinel operates on supervisory channels or dark fibers, with powers below nonlinear thresholds and ITU safety limits. As part of your deliverables, add an OSNR/BER note showing that revenue traffic quality remained within spec before/after Sentinel enablement.

8.4 Adversarial mirages

Attackers may replay trawler‑like noise near a target. Fusion thwarts this: without SOP loops or LF load near a frame, you do not escalate to ALERT. Conversely, moving slowly to evade persistence can be countered by multi‑scale persistence (e.g., 3 of 5 windows) and a check for range‑stationarity.

9.0 Assessment Rubric

Criterion	Indicators of Mastery
Correctness (40%)	Accurate feature computation (band energies, H, ω_S, A_S), correct persistence logic, rational ladder outcome at design Pfa.
Clarity (25%)	One compact DV/SOP table; one‑line CV/SOP summary; one‑sentence decision brief citing denominators and thresholds; readable figures.
Defensibility (20%)	Thresholds trace to commissioning quantiles; uncertainty ellipse decomposed; rationale bullet list states exactly which thresholds tripped.
Discipline (15%)	No mid‑alert tuning; contextual modifiers applied only after base ladder; logs include mask overlays and time stamps.

10.0 Frequently Asked Questions

10.1 Why not a deep net?

Because commanders and auditors need reasons. The point of Sentinel is explainable detection: physics‑grounded features, visible thresholds, and a ladder. A CNN may help with tone identification later, but the alert path must remain transparent during crisis ops.

10.2 How long must evidence persist?

Policy sets the persistence (common default: three consecutive windows). It balances detection latency against false alarms. The lab’s metrics section invites you to explore the trade by Monte‑Carlo tiles, but you will not change the setting mid‑incident.

10.3 What if SOP is noisy?

Hysteresis survives noise because it integrates motion around a loop. If rotation is high but area is near zero, suspect flutter rather than contact. In deep fades where SOP estimation degrades, lean on DAS evidence—but log the uncertainty and consider downgrading to INVESTIGATE rather than ALERT.

10.4 Do we ever declare “unknown”?

Operationally, “unknown” is INVESTIGATE. You still document evidence, location, and uncertainty, then hand off to patrols or maintenance for resolution. The artifact trains you to be precise, not omniscient.

11.0 Notation & Glossary

ε(z,t) — gauge‑differenced strain proxy derived from coherent backscatter phase.
W(f;z) — prewhitening filter built from quiet‑sea PSD; stabilizes thresholds across range.
E_k — band energies in LF/MF/HF after whitening.
H — HF harmonicity statistic (top‑three peaks vs. band total) with near‑harmonic spacing check.
ω_S — SOP rotation rate; A_S — SOP loop area (hysteresis proxy).
p_k(z) — persistence counter (consecutive red windows) for modality k at range z.
TS(z,t₀) — tamper score fused from modal reds with weights favoring harmonic evidence plus SOP.
Δz, Δt — sampling steps in range and time; T_w — window length; T_h — hop.

12.0 Ethics, Safety, and Next Steps

Payload privacy. Sentinel listens to physics on supervisory/dark wavelengths. It does not inspect payload; your deliverables should repeat that plainly. Safety. Keep launch powers within supervisory limits; log OSNR/BER before and after enablement; record interlocks. Extensions. Bi‑end sensing, tone recognizers with rule‑based fusion (to retain transparency), multi‑cable fusion, and an exercise library of controlled ROV runs are natural next steps once the baseline ladder is fielded.

References

Hartog, A. H. (2017). An Introduction to Distributed Optical Fibre Sensors. The Institution of Engineering and Technology. https://digital-library.theiet.org/content/books/10.1049/pbpo095e

Masoudi, A., & Newson, T. P. (2016). Contributed review: Distributed optical fibre dynamic strain sensing. Review of Scientific Instruments, 87(1), 011501. https://doi.org/10.1063/1.4939481

Juarez, J. C., & Taylor, H. F. (2005). Field test of a distributed fiber‑optic intrusion sensor system for long perimeters. Applied Optics, 44(20), 4332–4340. https://doi.org/10.1364/AO.44.004332

Agrawal, G. P. (2012). Fiber‑Optic Communication Systems (4th ed.). Wiley. https://www.wiley.com/en-us/Fiber+Optic+Communication+Systems%2C+4th+Edition-p-9781118071930

Diamanti, E., & Leverrier, A. (2015). Distributing secret keys with quantum continuous variables: Principle, security, and implementations. Entropy, 17(9), 6072–6092. https://doi.org/10.3390/e17096072

Shinohara, S. (2012). Optical submarine cable system. NTT Technical Review, 10(8), 1–8. https://www.ntt-review.jp/archive/ntttechnical.php?contents=ntr201208fa5.html

Stateczny, A., et al. (2020). Unmanned surface/underwater vehicles for maritime security. Remote Sensing, 12(13), 2148. https://doi.org/10.3390/rs12132148

Pirandola, S., Andersen, U. L., Banchi, L., et al. (2020). Advances in quantum cryptography: Security of quantum key distribution. Advances in Optics and Photonics, 12(4), 1012–1236. https://doi.org/10.1364/AOP.361502

Solution

Professional P-S01 — Solution (NSA Threat Analysis: Suspected China-Operated QKD Link)

Professional P-S01 — Detailed Solution

Given (for reference). $ \mu_s=0.60,\ \mu_{d1}=0.20,\ \mu_{d2}=0.02;\ \ N_s=2{,}000{,}000,\ C_s=106{,}000,\ E_s=3{,}392;\ \ N_{d1}=600{,}000,\ C_{d1}=10{,}320,\ E_{d1}=299;\ \ N_{d2}=200{,}000,\ C_{d2}=420,\ E_{d2}=20. $ Timing meta: $D_0$: mean $4.85$ ns (sd $0.43$, $n_0=50{,}000$) vs $D_1$: mean $5.02$ ns (sd $0.44$, $n_1=56{,}000$).

1) Baseline stats

\[ Q_s=\frac{106{,}000}{2{,}000{,}000}=0.0530,\quad Q_{d1}=\frac{10{,}320}{600{,}000}=0.0172,\quad Q_{d2}=\frac{420}{200{,}000}=0.00210. \] \[ \mathrm{QBER}_s=\frac{3392}{106{,}000}=0.0320\ (3.20\%),\ \mathrm{QBER}_{d1}=\frac{299}{10{,}320}\approx 0.02897\ (2.897\%),\ \mathrm{QBER}_{d2}=\frac{20}{420}\approx 0.04762\ (4.762\%). \]

Decoy-to-signal ratio: \[ r=\frac{Q_{d1}}{Q_s}\approx \frac{0.0172}{0.0530}=0.3245,\quad r_0=\frac{\mu_{d1}}{\mu_s}=\frac{0.20}{0.60}=0.3333. \] So $r$ is ~2.6% below the naive baseline (a hint only; proceed to bounds).

2) Decoy bounds $Y_0^L, Y_1^L, e_1^U, Q_1^L$

2.1 Work in $S_\mu:=Q_\mu e^{\mu}$

\[ S_s=Q_se^{0.6}\approx 0.0530\times 1.8221=0.09657,\quad S_{d1}\approx 0.0172\times 1.2214=0.02101,\quad S_{d2}\approx 0.00210\times 1.0202=0.002142. \]

2.2 Background (vacuum-like) yield

\[ Y_0^L=\frac{\mu_{d1}S_{d2}-\mu_{d2}S_{d1}}{\mu_{d1}-\mu_{d2}} =\frac{0.2\cdot 0.002142-0.02\cdot 0.02101}{0.18} \approx 4.62\times 10^{-5}. \]

2.3 Single-photon yield

\[ Y_1^L=\frac{\mu_s}{\mu_s\mu_{d1}-\mu_{d1}^2}\left[ S_{d1}-Y_0^L-\frac{\mu_{d1}^2}{\mu_s^2}(S_s-Y_0^L)\right]. \] Denominator $=0.08$; prefactor $=7.5$. Bracket $ \approx 0.02101-0.0000462-0.11111\cdot(0.09657-0.0000462)\approx 0.01024$. Thus $ \boxed{Y_1^L\approx 0.07678\ (7.678\%)}$.

2.4 Single-photon error

With $e_0=\tfrac12$ and $E_{d1}/C_{d1}\approx 0.02897$: \[ e_1^U \le \frac{(0.02897)S_{d1}-\tfrac12 Y_0^L}{\mu_{d1}Y_1^L} \approx \frac{6.09\!\times\!10^{-4}-2.31\!\times\!10^{-5}}{0.2\times 0.07678} \approx \boxed{0.0381\ (3.81\%)}. \]

2.5 Single-photon gain under signal intensity

\[ P_1(\mu_s)=\mu_se^{-\mu_s}=0.6e^{-0.6}\approx 0.32929,\quad \boxed{Q_1^L=P_1(\mu_s)Y_1^L\approx 0.02528}. \]

3) Asymptotic key-rate lower bound

Using $q=\tfrac12$, $f(E_s)=1.16$, $E_s=0.0320$: \[ R \ge \tfrac12\Big[Q_1^L(1-H_2(e_1^U)) - Q_s f(E_s) H_2(E_s)\Big]. \] With $H_2(0.0381)\approx 0.235$, $H_2(0.0320)\approx 0.204$: \[ R \approx \tfrac12\big[0.02528(0.765) - 0.0530(1.16)(0.204)\big] = \tfrac12(0.0193 - 0.0126) \approx \boxed{0.00341\ \text{bits/pulse}}. \] At 100 MHz: $ \boxed{\sim 341\,\text{kbit/s}} $ (asymptotic).

4) Time-shift test (detector mismatch exploitation)

Mean difference $ \Delta=5.02-4.85=0.17\,\text{ns} $. SE $=\sqrt{0.43^2/50{,}000 + 0.44^2/56{,}000}\approx 0.00267\,\text{ns} $. Test statistic $ t\approx 63.6 \Rightarrow p\ll 10^{-9} $. Given nominal $ \eta_0/\eta_1\le 1.06 $, a persistent $\sim$170 ps shift at this significance is suspicious → consistent with time-shift bias.

5) Windowed behavior (GLR change-point)

Split signal detections equally: each half $ \sim 53{,}000 $ detections. Errors: first half $ \approx 2.8\% \Rightarrow k_1=1{,}484$, second half $ \approx 3.6\% \Rightarrow k_2=1{,}908 $. \[ \Lambda = 2(\ell(\hat p_1)+\ell(\hat p_2)-\ell(\hat p)) \approx \boxed{54.9}\ \ (\chi^2\text{ with 1 df}), \] so $p \approx 1.3\times 10^{-13}$: strong evidence of a regime change (windowed/on-off behavior).

6) Overall call & mitigations

PNS pressure: Slightly depressed $r$ and $Y_1^L\approx 7.68\%$, $e_1^U\approx 3.81\%$ → consistent with mild PNS action. Add an explicit vacuum decoy to tighten $Y_0$.
Time-shift bias: Highly significant $\sim$170 ps shift → equalize detector efficiencies, randomize gate timing (jitter), and monitor per-detector arrival histograms.
Windowed behavior: GLR $ \approx 54.9$ → enable real-time change-point detection and adapt basis-choice/privacy-amplification parameters under worst-case bounds.

Alignment with Lab (.ipynb)

Cell 1: Imports & telemetry (matches “Given”).
Cell 2: Gains & QBERs (Step 1).
Cell 3: Ratio $r$ vs baseline $r_0$ (Step 1 wrap-up).
Cell 4–5: $S_\mu$, $Y_0^L$, $Y_1^L$, $e_1^U$, $Q_1^L$ (Step 2).
Cell 6: Key-rate $R$ and throughput (Step 3).
Cell 7: Time-shift test (Step 4).
Cell 8: GLR change-point (Step 5).
Cell 9: Plots for gains/QBERs and $r$ with CI (visual wrap-up).

Instructor Edition

C‑Lion1 Sentinel

1.0 Historical & Conceptual Background

From QKD watchstanding to fiber sentinels. National‑scale subsea cables carry the financial, scientific, and defense traffic that make a modern economy tick. The shift from pure encryption appliances to physics‑aware monitoring has been driven by two realities: first, the attack surface moved offshore, where anchor drags, unauthorized maintenance, and covert ROV loiters can occur beyond line‑of‑sight; second, legacy intrusion sensors lacked both the range and the explainability needed for command decisions. The C‑Lion1 Sentinel pipeline embraces a complementary philosophy: listen to the cable’s physics, not payload, and fuse independent “clues” (distributed acoustic sensing and state‑of‑polarization dynamics) into a tamper score that is both auditable and tactically useful.

Why coherent DAS and SOP? Distributed acoustic sensing (DAS) turns a standard single‑mode fiber into a kilometer‑scale vibration sensor by probing Rayleigh backscatter and tracking phase changes over a gauge length L_g. Motions—anchor rub, thruster tones, clamp‑on bends—modulate phase along the range coordinate z. Meanwhile, the Stokes state‑of‑polarization (SOP) vector reacts to mechanical stress and bending near the cable core; rotation rate and loop area in the (S₁, S₂) plane provide a second, independent view, especially for stationary contact at a repeater saddle or frame. Using both channels is a matter of resilience: when coherent fading hides DAS at a spot, SOP may still flare; when polarization is quiet, DAS may carry the event.

Pattern over theatrics. Sentinel deliberately avoids baroque machine learning in the alert path. Field operators need a few stable features, clear thresholds, and a persistence rule that tames spurious spikes. The design therefore prewhitens the DAS spectrum, computes band energies in LF (ramps, anchor drags), MF (2–20 Hz), and HF (100–800 Hz), measures tonal peaky harmonicity in the HF band, and extracts SOP rotation/hysteresis. With these in hand, a transparent fusion produces a tamper score and a three‑level action (PASS, INVESTIGATE, ALERT). The point is not to win a Kaggle contest; it is to make fast, defensible calls that a commander can read on a briefing slide.

What “good” looks like at sea. Most nights are uneventful: LF backgrounds breathe with swell; MF stays near the floor; HF shows diffuse hiss without peaky harmonics; SOP meanders but does not loop. An anchor drag paints a slow LF ramp that walks in range; a stationary ROV loiter sings two HF tones with near‑harmonic spacing; a clamp‑on bend near a frame carves a clean SOP loop area. Sentinel’s pipeline embodies this playbook. It measures each phenomenon where the physics is strongest, uses persistence to reject one‑off blips, and then fuses modalities to favor events that “agree” across sensors.

Why instructors teach it this way. The artifact you are reading is built for teach‑once, apply‑many environments. Every step is reconstructible on paper; every threshold has a documentable provenance. The aim for your classroom or operations bay is to give learners an operational narrative—clock and range alignment; whitening; features; thresholds; fusion; decision; geolocation; uncertainty—so that graduates can perform minutes‑scale triage on real telemetry without improvising their own statistics in the dark.

2.0 Learning Objectives & Outcomes

Explain, in operational English, what coherent DAS phase differencing measures and why a gauge length L_g is used.
Derive the strain‑proxy scaling from phase differences and state the assumptions involved (small‑angle, effective index, stable wavelength).
Compute LF/MF/HF band energies on prewhitened strain and interpret a high‑H harmonicity statistic as ROV‑like evidence.
Compute SOP rotation rate and loop area, and defend why area (not just rate) is the right evidence for clamp‑on bends.
Apply the persistence rule and the decision fusion to produce a tamper score TS and an action (PASS/INVESTIGATE/ALERT).
Map a range patch to geographic coordinates with an uncertainty ellipse and articulate an MTTD for the event class.
Audit outcomes with a red‑patch video and a bullet rationale (“which thresholds tripped”) suitable for commanders.

3.0 Data Model & Notation

We operate on two cubes sampled on common grids: the DAS backscatter‑derived strain proxy ε(z,t) and a Stokes/SOP cube S(z,t) = (S₁, S₂, S₃). Raw complex samples are unwrapped in time and space; a gauge‑difference converts phase to a strain proxy proportional to cable stretch over L_g.

ε (z, t) \approx \frac{λ}{4 π n L g} Δ φ (z, t)

Strain proxy from phase change: wavelength λ, effective index n, gauge length L_g.

Clock, range, and alignment. A correlation to repeater fiducials aligns the range coordinate with the as‑laid route; GPS locks clock and reduces drift. Interpolation produces a uniform Δz (e.g., 2 m) and a stable hop T_h for sliding windows.

Whitening and noise shaping. A prewhitening filter stabilizes thresholds by flattening background power spectral density. Using a quiet interval, estimate P_ε(f;z) and define a stabilizing filter with a small η for numerical safety:

W (f; z) = \frac{1}{P_ε (f; z) + η}

Prewhitening: divides by local background PSD to level noise across bands.

Quality masks. Coherent fading creates destructive minima. A long‑term SNR or coherence mask Q(z) suppresses decisions where the channel is unreliable, with interpolation across nulls so that map‑level continuity is preserved.

Windowing and bands. We operate in windows of length T_w = 10 s with hop T_h = 2 s. On the whitened strain ε̃(z,f;t₀) inside window W(t₀), compute energy in three bands—LF (0.1–2 Hz), MF (2–20 Hz), HF (100–800 Hz).

E_{k} (z, t_{0}) = \sum_{f \in B_{k}} | \tilde{ε} (z, f; t_{0}) |^2

Band energy in LF, MF, HF after whitening; larger values indicate more power in that band.

Harmonicity in HF. Within the HF band, find the three strongest peaks f₁, f₂, f₃ and compare their power to the band’s total. The statistic H rises when near‑harmonic tones dominate (e.g., ROV blade‑pass offsets).

H (z, t_{0}) = \frac{\sum_{i = 1} P^{(f_{i})}}{\sum_{f \in B_{3}} P (f)}

Harmonicity: share of HF power captured by the top three peaks; high H flags ROV‑like signatures.

SOP rotation & hysteresis. Using Savitzky–Golay derivatives on S(z,t), estimate rotation rate and integrate a loop area in the (S₁,S₂) plane over the window:

ω_{S} (z, t_{0}) = \frac{1}{T_{w}} \int W_{t_{0}} ‖ \dot{S} (z, t) ‖ d t

SOP rotation: average rate (rad/s) within the window.

A_{S} (z, t_{0}) \approx \oint W_{t_{0}} (S_{1} d S_{2} - S_{2} d S_{1})

SOP loop area: hysteresis proxy; large area implies clamp‑on/bend near structure.

Persistence & motion. Count consecutive red windows per modality, p_k(z). A cross‑correlation between adjacent windows estimates range velocity:

\dot{z} (t_{0}) = arg max_δz \frac{⟨ \tilde{ε} (z, t_{0}), \tilde{ε} (z + δz, t_{0} + T_{h}) ⟩}{δz}

Range velocity estimate from window‑to‑window cross‑correlation.

4.0 Thresholds, Detectors, and Fusion

Energy detectors (per band). After prewhitening, E_k is approximately chi‑square distributed. Choose thresholds T_k to achieve design false‑alarm rates per band (e.g., red at Pfa ≈ 10⁻⁴). A Hann taper and overlap rules are handled consistently in the training quantiles so operators need not “fix up” degrees of freedom at 03:00.

Tonal/Harmonic detector (HF). Compute H from the top three HF peaks and declare red when (i) H > H_red (e.g., 0.45) and (ii) two peaks are near‑harmonic (e.g., f₂ ≈ 2f₁, f₃ ≈ 3f₁ within tolerances). This flags thruster‑like loiter signatures.

SOP detectors. Set T_ω and T_A from background quantiles (e.g., 99.9th and 99.99th). Bursts above thresholds are red; persistent loops outrank fleeting spikes.

Persistence rule. A modality becomes “operative red” only if it persists for p_k(z) ≥ 3 consecutive windows. This mirrors QKD‑style minutes‑scale discipline and slashes one‑off false calls without sacrificing MTTD.

Decision fusion. Define a tamper score per window and range with weights favoring harmonic evidence:

TS (z, t_{0}) = w_{1} [1 (E_{3} red)] + w_{2} [1 (H red)] + w_{3} [1 (ω_{S} red)] + w_{4} [1 (A_{S} red)] + w_{5} [1 (LF red)]

Default weights: w₁ = 1, w₂ = 1.5, w₃ = 1, w₄ = 1, w₅ = 1 (favor harmonic evidence).

Rules of engagement. The action ladder is:

ALERT if p_tone ≥ 3 and p_SOP ≥ 3 at approximately the same range z, or if TS ≥ 4 with at least two modalities red.
INVESTIGATE if any single modality persists ≥ 3 but TS < 4, or if the event moves with |ẋ| > 0.25 m/s without SOP hysteresis (anchor drag).
PASS otherwise.

Contextual modifiers. If a red patch lies within ±50 m of a repeater vault or branch, lower TS by 0.5 (benign maintenance bias). If AIS declares a maintenance vessel on top of the patch, downgrade one level unless SOP is strongly hysteretic.

5.0 Worked Example (Instructor Walkthrough)

Scenario. Use a synthetic tile that mirrors the canonical event. Parameters: Δz = 2 m, L_g = 10 m, Δt = 1 ms, T_w = 10 s, hop T_h = 2 s. Background PSD comes from a 15‑minute quiet period just prior.

Bands. Compute E₁, E₂, E₃. At z = 163.4 km we observe a low‑frequency ramp excess across three windows; HF shows two tones near 281 Hz and 562 Hz with harmonicity H = 0.58 well above H_red = 0.45; MF rises mildly but not red.

SOP. The rotation rate ω_S peaks at roughly 2.3 rad/s above T_w; loop area A_S shows a clean loop in two consecutive windows then a larger loop, the classic bend‑with‑stick‑slip pattern. Persistence counters: p_tone = 3, p_SOP = 3, p_LF = 3.

Decision. With defaults w = (1, 1.5, 1, 1, 1), the fused score is TS = 1(E₃) + 1.5(H) + 1(ω_S) + 1(A_S) + 1(LF) = 5.5. The stationary character (ẋ ≈ 0) favors a loitering ROV over anchor drag. The patch sits within ~40 m of a repeater frame, which is operationally critical. Outcome: ALERT.

Brief to the watch. “Stationary ROV‑like signature at 163.4 km from Oulu (±22 m), duration 34 s. Three consecutive windows show HF tonal energy with harmonic structure at 281/562 Hz; simultaneous SOP hysteresis loops suggest physical contact at the cable or the frame; a low‑frequency strain ramp indicates load. No declared maintenance. Recommend patrol dispatch and covert illumination restriction on repeaters 5–6.”

6.0 Geolocation, Uncertainty & Timing

Range → coordinates. Map range index z to (lat, lon) along the as‑laid polyline. Report the center of mass of the red patch’s energy for stability rather than the single‑bin max; this reduces jumpiness when two adjacent bins are nearly tied.

Uncertainty budget. Combine three terms in quadrature: (i) instrument resolution (±Δz/2), (ii) route error (difference between surveyed and as‑laid positions), and (iii) spatial scatter within the patch. Report a geographic ± ellipse using local track bearing. For classroom assessment, award full credit when students identify terms and show correct combination rather than chase a specific numeric ellipse.

Timing and MTTD. Record event start/stop with hop resolution T_h. For median time‑to‑detect (MTTD), measure the first window that crosses the action rule in a Monte‑Carlo tile library. In this theater, ROV loiter MTTD typically sits near 12 s (limited by the three‑window persistence), anchor‑drag MTTD between 16–22 s depending on motion speed.

7.0 Robustness & Pitfalls

Fading and nulls. DAS sensitivity varies with range because of coherent fading; quality masks prevent decisions inside deep minima. Encourage students to visualize the mask atop the energy map to understand where decisions were disallowed. Cross‑sensor fusion helps: SOP often remains informative when DAS fades.

Weather coupling. High sea states inflate LF backgrounds. Correlate LF thresholds with metocean measurements so that policy adjusts adaptively during storms. Do not raise SOP thresholds in rough seas; hysteresis is less weather‑sensitive than LF strain.

Traffic coexistence. Confirm no cross‑gain modulation or Raman crosstalk into revenue traffic. Measure OSNR and BER before and after Sentinel enablement; maintain launch powers inside supervisory limits and capture interlocks in the safety case.

Adversarial deception. Actors may replay trawler‑like noise near a target. Fusion is the antidote: without SOP loops or LF ramps near a structure, do not upgrade to ALERT. Conversely, to evade persistence an actor might move slowly; counter with multi‑scale persistence (e.g., 3 of 5 windows) and range‑stationarity tests.

8.0 Implementation Notes for Teaching & Ops

Pipeline mnemonic: calibrate → whiten → features → thresholds → fusion → decision. Persist intermediate maps for replay in after‑action reviews.
Red‑patch video: export a short time‑range video tile with HF spectral overlays and SOP Lissajous loops. This is what convinces commanders that physics, not anecdotes, drove the call.
One‑click rationale: present bullets listing which thresholds tripped and the values (H, ω_S, A_S, LF ramp, persistence counts).
Guardrail discipline: learners must not tune thresholds mid‑incident. Debate them in policy reviews, not during a live alert.

9.0 Instructor Assessment Rubric

Criterion	Indicators of Mastery
Correctness (40%)	Proper computation of E_k, H, ω_S, A_S; correct persistence counts; fusion yields the right ladder action with stated thresholds.
Clarity (25%)	One DV/SOP table, one CV/SOP summary line, one‑sentence decision brief with denominators and thresholds; figures labeled.
Defensibility (20%)	Audit trail reproducible from student outputs; geolocation uncertainty decomposed; rationale mentions which thresholds tripped.
Discipline (15%)	Thresholds not tuned mid‑alert; persistence rule applied; contextual modifiers cited when applicable.

10.0 Instructor Solution Outline

Clock & range alignment. Correlate to fiducials; lock timebase to GPS; interpolate to uniform Δz and hop T_h.
Unwrap & gauge. Quality‑guided unwrapping in time then space; compute gauge difference and scale to strain proxy.
Whiten. Estimate P_ε(f;z) from a quiet interval and apply W(f;z) with stabilizer η.
Window. Slide T_w = 10 s windows every 2 s; apply a Hann taper.
Features. Compute E₁/E₂/E₃; compute H from top‑3 peaks; compute ω_S and A_S on SOP.
Thresholds. Apply per‑band quantile thresholds; set H_red, T_ω, T_A from training.
Persistence. Promote a modality to operative red when p_k(z) ≥ 3.
Fusion. Compute TS with weights (1, 1.5, 1, 1, 1); apply ladder (ALERT/INVESTIGATE/PASS) with contextual modifiers.
Geolocate. Map range to (lat, lon); compute uncertainty ellipse from instrument, route, and patch scatter.
Report. Produce the one‑sentence decision and the bullet rationale with threshold values; append a red‑patch video if available.

11.0 Extensions & What‑If Exercises

Edge‑case: moving tones. Simulate a slow trawler that drifts in and out of stationarity. Show how persistence and ẋ prevent an ALERT upgrade without SOP hysteresis.
Edge‑case: repeating maintenance. Overlay AIS records of declared maintenance and examine how contextual downgrades protect against false alarms around frames.
Fusion sensitivity. Vary w₂ between 1.0 and 2.0 to illustrate why harmonicity deserves extra weight in ROV environments.
Robustness audit. Inject synthetic microseism ramps to test LF thresholds; inject polarization nulls to test SOP resilience and the value of Q(z).

References

Hartog, A. H. (2017). An Introduction to Distributed Optical Fibre Sensors. The Institution of Engineering and Technology. https://digital-library.theiet.org/content/books/10.1049/pbpo095e

Juarez, J. C., Taylor, H. F. (2005). Field test of a distributed fiber‑optic intrusion sensor system for long perimeters. Applied Optics, 44(20), 4332–4340. https://doi.org/10.1364/AO.44.004332

Masoudi, A., Newson, T. P. (2016). Contributed review: Distributed optical fibre dynamic strain sensing. Review of Scientific Instruments, 87(1), 011501. https://doi.org/10.1063/1.4939481

Agrawal, G. P. (2012). Fiber‑Optic Communication Systems (4th ed.). Wiley. (Background on SOP, Stokes parameters, and polarization effects.) https://www.wiley.com/en-us/Fiber+Optic+Communication+Systems%2C+4th+Edition-p-9781118071930

Diamanti, E., Leverrier, A. (2015). Distributing secret keys with quantum continuous variables: Principle, security and implementations. Entropy, 17(9), 6072–6092. (Background on CV yardsticks and SNR concepts that inform audit ideas.) https://doi.org/10.3390/e17096072

Shinohara, S. (2012). Optical submarine cable system. NTT Technical Review, 10(8), 1–8. (Operational context for repeater frames, as‑laid routes, and maintenance zones.) https://www.ntt-review.jp/archive/ntttechnical.php?contents=ntr201208fa5.html

Stateczny, A., et al. (2020). Unmanned surface/underwater vehicles for maritime security. Remote Sensing, 12(13), 2148. (ROV signatures, mission profiles, and harmonic tones.) https://doi.org/10.3390/rs12132148

Pirandola, S., et al. (2020). Advances in quantum cryptography: Security of quantum key distribution. Advances in Optics and Photonics, 12(4), 1012–1236. (Survey; useful for students connecting Sentinel’s SOP/DAS audit ideas with QKD operational discipline.) https://doi.org/10.1364/AOP.361502

Lab

P01_Sentinel_on_CLion1_Pro_Notebook

P‑01 — Sentinel on C‑Lion1 — ROV/Tamper Detection (Professional Notebook)¶

Goal — Build a quantum‑enhanced fiber‑sensor pipeline on C‑Lion1 that detects, localizes, and classifies tamper events using DAS (phase/strain proxy) and SOP (Stokes) data; return a defensible PASS / INVESTIGATE / ALERT decision with rationale.

Mission brief (Scenario)¶

Year: 2030. Role: Lead systems engineer, NATO Maritime Infrastructure Protection Program (MIPP).
Objective: Field‑validate a quantum‑enhanced sensor layer that can detect, localize, and classify hostile interference with the C‑Lion1 submarine cable between Finland and Germany.

War in Ukraine has spilled into sporadic skirmishes between Russian units and NATO assets across the High North/Baltic. Unflagged hulls loiter near critical seabed infrastructure. C‑Lion1 is both an economic artery and a military umbilical; it carries civilian internet and, in crisis, encrypted operational traffic. Your mandate is simple: keep C‑Lion1 alive.

Your asset mix: (1) the cable itself—tens of fiber pairs sensitive to strain, temperature, and acoustic micro‑vibrations; and (2) coherent‑optics kit that, with the right DSP, turns any spare/supervisory wavelength into a distributed interferometer hundreds of kilometers long.

At 22:17Z a change‑point detector flags a 45‑s swell of low‑frequency (0.2–2 Hz) energy centered 163.4 km from Oulu, followed by tones near 280 Hz and 560 Hz for 8 s—classic small‑ROV thrusters spooling under load. The SOP trace at the same range shows an abrupt rotation with a small hysteresis loop—consistent with clamp‑on or bend. AIS history suggests a “fishing” hull drifting up‑current. The watch asks for a ruling.

Your task in this lab mirrors the real decision: design, justify, and document a chain that converts raw fiber sensing into features and a crisp rule the watch can apply within seconds. Outputs: calibrated features, thresholds, persistence logic, a range geolocation with uncertainty, and a manager‑facing rationale suitable for 03:00 briefings.

Background (Technical)¶

DAS strain proxy. Coherent OTDR/φ‑OTDR measures wrapped phase $\phi_w(z,t)$. After quality‑guided unwrapping and gauge differencing over length $L_g$, the axial‑strain proxy is
\begin{equation} \epsilon(z,t)\;\approx\;\frac{\lambda}{4\pi n L_g}\,\Delta\phi(z,t), \end{equation} with wavelength $\lambda$ and effective index $n$.

SOP (Stokes). Polarization state $\mathbf{S}=(S_0,S_1,S_2,S_3)$. Micro‑bends and clamp‑ons rotate $(S_1,S_2,S_3)$; we use rotation rate $\omega_S(z,t)=\lVert\dot{\mathbf{S}}(z,t)\rVert$ and a loop area $A_S$ (hysteresis) over a short window to detect contact/bends.

What we’ll look for.

LF strain ramp (0.1–2 Hz): anchor load‑up.
HF tonal structure (100–800 Hz): ROV thrusters with near‑harmonic spacing.
SOP rotation bursts and hysteresis: physical contact/clamp.
Persistence & stationarity in range.

Objectives¶

Calibrate and whiten DAS; compute windowed band energies $(E_1,E_2,E_3)$.
Detect HF tonal peakiness/harmonicity $H$; compute SOP metrics $(\omega_S,\;A_S)$.
Apply persistence (three consecutive red windows) and a simple fusion score.
Produce ALERT / INVESTIGATE / PASS with a short, human‑readable rationale.

Inputs (synthetic demo — replace later with real streams)¶

Synthetic DAS strain cube for a short span (time × range), with an injected ROV‑like event.
Simple SOP surrogate at the same range grid.
Window length $T_w=10\,\mathrm{s}$, hop $T_h=2\,\mathrm{s}$. Bands: LF 0.1–2 Hz, MF 2–20 Hz, HF 100–800 Hz.

In [ ]:

# ===== Your Code Starts Here =====
import numpy as np
import scipy.signal as sig
import matplotlib.pyplot as plt
from numpy.fft import rfft, rfftfreq

np.random.seed(7)

# --- Synthetic tile generator (DAS + SOP) ---
def make_tile(z_bins=360, fs=1000.0, T=30.0, event_z_idx=260,
              event_t0=15.0, event_t1=23.0, f1=281.0, f2=562.0, lag=10.0):
    N = int(T*fs)
    t = np.arange(N)/fs
    # DAS: background (colored) + small LF microseism
    eps = 2e-4*sig.lfilter([1], [1,-0.98], np.random.randn(z_bins,N))
    eps += 3e-5*np.sin(2*np.pi*0.25*t)[None,:]
    # Inject HF thruster tones + LF ramp at event bins
    mask = np.zeros(z_bins, dtype=bool)
    mask[max(0,event_z_idx-2):min(z_bins,event_z_idx+3)] = True
    env = np.zeros_like(t)
    i0, i1 = int(event_t0*fs), int(event_t1*fs)
    env[i0:i1] = 1.0
    tone  = 8e-4*(np.sin(2*np.pi*f1*t) + 0.8*np.sin(2*np.pi*f2*t)) * env
    ramp  = 6e-4*np.clip((t - event_t0)/(event_t1-event_t0), 0, 1)
    eps[mask,:] += tone
    eps[mask,:] += ramp
    # SOP surrogate at each range (only active near event)
    S = np.zeros((z_bins,N,3))
    base = 0.1*np.random.randn(z_bins,N,3)
    S += base
    L = np.linspace(0,2*np.pi,N)
    loop = (np.stack([np.cos(3*L), np.sin(2*L), 0.2*np.sin(5*L)], axis=1))
    loop = loop*(env[:,None])
    for z in np.where(mask)[0]:
        S[z,:,0] += 0.6*loop[:,0]
        S[z,:,1] += 0.5*np.gradient(loop[:,0], axis=0)
    return t, eps, S

fs = 1000.0
t, eps, S = make_tile()
z_bins = eps.shape[0]
N = eps.shape[1]

# --- Prewhitening (per range) ---
def prewhiten(x):
    x = x - np.mean(x)
    return sig.lfilter([1, -0.95], [1], x)

eps_w = np.vstack([prewhiten(eps[z]) for z in range(z_bins)])

# --- Windowing params ---
Tw = 10.0; Th = 2.0
W = int(Tw*fs); H = int(Th*fs)
win = sig.windows.hann(W, sym=False)

# --- Feature extraction in sliding windows ---
f = rfftfreq(W, d=1/fs)
B1 = (f>=0.1) & (f<=2.0)
B2 = (f>=2.0) & (f<=20.0)
B3 = (f>=100.0) & (f<=800.0)

def harmonicity(P, f):
    # take top-3 peaks in HF band and compute peakiness
    band = B3
    idx = np.argsort(P[band])[-3:]
    if idx.size==0:
        return 0.0
    peaks = P[band][idx]
    return np.sum(peaks) / (np.sum(P[band]) + 1e-12)

def sop_metrics(Sseg):
    dS = np.gradient(Sseg, axis=0)
    wS = np.linalg.norm(dS, axis=1).mean()
    # loop area in (S1,S2)
    s1, s2 = Sseg[:,0], Sseg[:,1]
    A  = np.abs(np.trapz(s1*np.gradient(s2) - s2*np.gradient(s1)))
    return wS, A

E1=[]; E2=[]; E3=[]; Hh=[]; WS=[]; AS=[]; centers=[]
for start in range(0, N-W+1, H):
    stop = start+W
    centers.append((start+stop)//2)
    # choose max-energy range bin for features (proxy for red patch)
    seg  = eps_w[:,start:stop]*win
    F    = np.abs(rfft(seg, axis=1))**2
    # sum over freq, choose argmax over range in HF band
    hf_energy = F[:,B3].sum(axis=1)
    z0 = int(np.argmax(hf_energy))
    P  = F[z0]
    E1.append(P[B1].sum()); E2.append(P[B2].sum()); E3.append(P[B3].sum())
    Hh.append(harmonicity(P, f))
    # SOP metrics at same range
    wS, A = sop_metrics(S[z0,start:stop,:])
    WS.append(wS); AS.append(A)

E1 = np.array(E1); E2 = np.array(E2); E3 = np.array(E3)
Hh = np.array(Hh); WS = np.array(WS); AS = np.array(AS)
centers = np.array(centers)/fs

# --- Thresholds from an early quiet segment ---
q  = slice(0, max(1,int(5/Th)))   # first ~5 s worth of windows
T_E3_red = np.quantile(E3[q], 0.999)
T_H_red  = np.quantile(Hh[q], 0.999)
T_WS_red = np.quantile(WS[q], 0.999)
T_AS_red = np.quantile(AS[q], 0.999)

def flag_red(x, T):
    return (x > T).astype(int)

rE3 = flag_red(E3, T_E3_red)
rH  = flag_red(Hh, T_H_red)
rWS = flag_red(WS, T_WS_red)
rAS = flag_red(AS, T_AS_red)

# --- Persistence (three consecutive red windows) ---
def persist(r, k=3):
    out = np.zeros_like(r)
    run=0
    for i,v in enumerate(r):
        run = run+1 if v else 0
        out[i] = 1 if run>=k else 0
    return out

p_tone = persist(np.maximum(rE3, rH))
p_sop  = persist(np.maximum(rWS, rAS))

# --- Fusion score & decision ---
TS = 1.0*rE3 + 1.5*rH + 1.0*rWS + 1.0*rAS
alert = np.logical_or(np.logical_and(p_tone>=1, p_sop>=1), TS>=4)

decision = 'ALERT' if alert.any() else ('INVESTIGATE' if (p_tone.any() or p_sop.any()) else 'PASS')
print('Decision:', decision)

# --- Plots ---
fig,axs = plt.subplots(3,1, figsize=(10,8), sharex=True)
axs[0].plot(centers, E3, label='HF energy (E3)')
axs[0].axhline(T_E3_red, ls='--', color='C1', label='E3 red T')
axs[0].legend(); axs[0].set_ylabel('power')

axs[1].plot(centers, Hh, label='Harmonicity H')
axs[1].axhline(T_H_red, ls='--', color='C2', label='H red T')
axs[1].legend(); axs[1].set_ylabel('ratio')

axs[2].plot(centers, WS, label='SOP rotation $\omega_S$')
axs[2].plot(centers, AS, label='SOP loop area $A_S$')
axs[2].axhline(T_WS_red, ls='--', color='C2', label='$\omega_S$ red T')
axs[2].axhline(T_AS_red, ls='--', color='C3', label='$A_S$ red T')
axs[2].legend(); axs[2].set_xlabel('time (s)')
plt.tight_layout(); plt.show()
# ===== Your Code Ends Here =====

Decision rule (operator card)¶

Mark a window red if any of the following exceed learned thresholds: HF energy $(E_3)$, harmonicity $(H)$, SOP rotation $(\omega_S)$, SOP loop area $(A_S)$.
A feature becomes operative red after three consecutive red windows (persistence $\ge 3$).
ALERT if (i) both tonal and SOP streams persist at the same range ($\approx$), or (ii) fusion score $\mathrm{TS}\ge 4$ with $\ge 2$ modalities red.
INVESTIGATE if any single modality persists but does not meet the ALERT conjunction.
PASS otherwise.

Notes for the student¶

QBER analogy isn’t used here; we operate on DAS/SOP features. The math you will see is standard: $\epsilon \approx (\lambda/4\pi n L_g)\Delta\phi$; SOP metrics $(\omega_S, A_S)$ capture rotation and hysteresis.
Energy detectors work after whitening; tonal peakiness $H$ is the fraction of HF energy carried by the top peaks.
Persistence suppresses single‑window blips. Fusion creates a crisp, defensible call at 03:00.

Deliverables (what you archive)¶

Plots of $(E_3)$, $(H)$, $(\omega_S)$, $(A_S)$ with thresholds marked.
Time stamps of the first ALERT/INVESTIGATE trigger and the short rationale string.
Saved parameters: thresholds, persistence length, and any operator overrides.

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Sample Professional Lab and Assignment Showcase

Problem

0) Overview and data model

1) Calibration and preprocessing

1.1 Clock, range, and alignment.

1.2 Phase unwrapping and gauge differencing.

1.3 Whitening and noise shaping.

1.4 Quality masks.

2) Feature extraction

2.1 Band energies (DAS).

2.2 Tonal peakiness and harmonicity.

2.3 SOP rotation and hysteresis.

2.4 Persistence and motion.

3) Thresholds and detectors

3.1 Energy detectors.

3.2 Tonal/harmonic detector.

3.3 SOP detectors.

3.4 Persistence rule.

4) Decision fusion

5) Geolocation and uncertainty

5.1 Range → coordinates.

5.2 Uncertainty budget.

5.3 Timing.

6) Worked detection on the exemplar event

6.1 Bands.

6.2 SOP.

6.3 Decision.

6.4 Rationale to the watch.

7) Performance estimation (Pd, Pfa, MTTD)

7.1 Monte Carlo on synthetic set.

7.2 Metrics.

8) Robustness and pitfalls

8.1 Fading and nulls.

8.2 Weather coupling.

8.3 Coexistence with traffic.

8.4 Adversarial deception.

9) Implementation notes for the team

10) Validation checklist (readiness review)

11) What operators actually see (UX)

12) Limits, ethics, and next steps

Summary Decision Template (for “Deliverables” panel)

Student Edition

C‑Lion1 Sentinel

1.0 Mission Brief (Scenario)

2.0 Historical & Conceptual Background

3.0 Technical Foundations

3.1 The fiber as a distributed interferometer

3.2 Band energies and thruster harmonicity

3.3 State‑of‑polarization rotation and hysteresis

3.4 Persistence and motion

3.5 Decision fusion (guardrails)

4.0 Problem Setup & Inputs

5.0 Student Tasks

6.0 Step‑by‑Step Guidance (Non‑Solution Scaffolding)

6.1 Read the packet like weather

6.2 Whiten for stability

6.3 Tune nothing, explain everything

6.4 Persistence before fusion

6.5 Action ladder with context

6.6 Geolocate with humility

7.0 Worked Reading Prompts

8.0 Real‑World Variations & Case Notes

8.1 Sea state and metocean

8.2 Coherent fading and quality masks

8.3 Traffic coexistence

8.4 Adversarial mirages

9.0 Assessment Rubric

10.0 Frequently Asked Questions

10.1 Why not a deep net?

10.2 How long must evidence persist?

10.3 What if SOP is noisy?

10.4 Do we ever declare “unknown”?

11.0 Notation & Glossary

12.0 Ethics, Safety, and Next Steps

References

Solution

Professional P-S01 — Detailed Solution

1) Baseline stats

2) Decoy bounds \(Y_0^L, Y_1^L, e_1^U, Q_1^L\)

Lab