ECE 495 — Electromagnetic Warfare and Survivability
Block 4: Aircraft Survivability and Mission Planning
Lesson 31: Survivability and the PK Chain
Lesson 31: Survivability and the PK Chain
Juan Jurado, Col, USAF, Ph.D.
Permanent Professor and Head
Electrical and Computer Engineering
Fall 2026
Permanent Professor and Head
Electrical and Computer Engineering
Fall 2026
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Block 4 · Lesson 31 · Where We Left Off
Where We Left Off
Three blocks, three toolkits: B1 the threat's physics, B2 the RF fight, B3 the signatures.
Each one bought a way to break a link of the L1 kill chain. None of them told you how much a mission's worth of links adds up to.
Today we put one number on the whole engagement, then on the whole route, so the three toolkits can finally be compared on one scale.
Block 4 · Lesson 31 · Objectives
Lesson Objectives
By the end of this lesson, you will be able to:
- Define survivability, susceptibility, and vulnerability (Ball)
- Decompose one engagement's $P_K$ along the L1 kill chain
- Combine independent encounters into a mission survival probability $P_S$
- Convert $P_S$ to additive risk, $-\ln P_S$ — the cost a route planner can sum
Survivability is a product — every block bought a lever on one link.
Block 4 · Lesson 31 · Roadmap
Block 4: Every Lesson Adds a Layer
One capstone, built a lesson at a time — the Final Project is issued today.
| Lesson | What it adds to the Final Project |
|---|---|
| L31 · PK chain | the survival metric — per-site chain to route $P_S$ (T2) · today |
| L32 · Threat geometry | the threat map — detection contours, MEZ and LEZ (T1) |
| L33 · Info geometry | their track on you — the passive network's error along your route (T4) |
| L34 · Path planning | the route — Dijkstra on a risk graph, and the fuel-limited choice (T3) |
| L35 · Fusion | the pop-up threat — localize it on the next leg (T6) · checkpoint |
| L36 · SEAD & SOJ | the support package — a standoff jammer and one SEAD nomination (T5) |
| L37 · Modern trends | limitations and future threats |
| L38–L41 | work day, peer review, revision, the final paper |
Block 4 · Lesson 31 · Survivability
Survivability: Avoid the Hit, or Withstand It
Ball: survivability is the ability to avoid or withstand a hostile environment.
Susceptibility · $P_H$
- Can they hit you?
- Everything before the hit: find, hold, name, and engage you
Vulnerability · $P_{K|H}$
- Does a hit kill you?
- What the airframe does with the damage a hit leaves
$$P_S = 1 - P_K = 1 - P_H \, P_{K|H}$$
Block 4 · Lesson 31 · The PK Chain
The L1 Kill Chain, Now in Probabilities
Detectfind it
→
Trackhold it
→
Identifyname it
→
Engagecommit it
→
Killthe outcome
$P_D$
$P_{T|D}$
$P_{I|T}$
$P_{H|I}$
$P_{K|H}$
susceptibility · $P_H$
vulnerability
$$P_K = P_D \, P_{T|D} \, P_{I|T} \, P_{H|I} \, P_{K|H} = P_H \, P_{K|H}$$
Each link is conditional: the chance it succeeds, given every link before it did.
Block 4 · Lesson 31 · The PK Chain
Break a Link, Shrink a Factor
- L1: break any link and the chain fails — one factor at zero zeroes $P_K$
- Real levers shrink a link: a signature cut lowers $P_D$, a decoy lowers $P_{H|I}$
- The product holds only if each factor is conditioned on the links before it
- In a defended zone $P_{I|T}$ is often near 1: anything unknown is hostile
So which link is worth shrinking first? Hold your answer — the type-along settles it.
Block 4 · Lesson 31 · The Levers
Which Block Bought Which Lever
| Link | Levers you already own | Block |
|---|---|---|
| Detect · $P_D$ | range law and threshold; RCS, IR signature, EMCON; noise jamming | B1 · B3 · B2 |
| Track · $P_{T|D}$ | deception jamming, chaff, the Doppler notch | B2 · B1 |
| Identify · $P_{I|T}$ | emission discipline — give them nothing to classify | B2 · B3 |
| Engage · $P_{H|I}$ | flares, DIRCM, towed decoys, seeker-aware tactics | B3 · B2 |
| Kill · $P_{K|H}$ | airframe design — Ball's vulnerability toolkit, named but not taught | — |
| Exposure · shots | route, standoff jamming, SEAD — how often the chain runs | B4 |
Block 4 adds no new link. It decides how many times the threat gets to run the chain.
Block 4 · Lesson 31 · The Levers
Ball's Two Toolkits
Susceptibility reduction · $P_H$
- Threat warning
- Noise and deception jamming
- Signature reduction
- Expendables — chaff, flares, decoys
- Threat suppression
- Tactics
Vulnerability reduction · $P_{K|H}$
- Component redundancy, with separation
- Component location
- Passive damage suppression
- Active damage suppression
- Component shielding
- Component elimination or replacement
The test: does the lever change whether they hit you, or what a hit does to you?
Block 4 · Lesson 31 · The Mission
Many Encounters, One Mission
- A route meets many shots: each site, each shot, a separate draw
- Survive all of them: the survivals multiply
- Independence is an assumption: a radar that cues every SAM couples them
Mission survival
$$P_S = \prod_i \left(1 - P_{K,i}\right)$$
Twenty shots at 4% each leave $P_S = 0.96^{20} = 0.44$. Small risks compound.
Block 4 · Lesson 31 · The Mission
Take the Log: Risk Adds
Additive risk
$$r = -\ln P_S = \sum_i -\ln\left(1 - P_{K,i}\right) \approx \sum_i P_{K,i}$$
- The log turns a product of survivals into a sum of risks, one term per encounter
- Each term $r_i$ belongs to one encounter — you can see where the risk lives
- For small $P_K$, $r_i \approx P_{K,i}$: the risk you would have guessed
L34: a shortest-path search adds edge costs — so the least total $r$ is the most $P_S$.
Block 4 · Lesson 31 · Quick Exercise
Quick Exercise / Type-along: Which Lever Pays?
Type this with me — one chain, a menu of four levers, then three encounters.
% L31 -- the chain is a product; risk is a sum
p = [0.9 0.8 0.9 0.5 0.6]; % P_D P_T|D P_I|T P_H|I P_K|H
fprintf('%-9s P_S = %.3f\n', 'baseline', 1 - prod(p));
nm = {'RAM','deception','flares','armor'};
k = [1 2 4 5]; % the link each lever cuts
v = [0.6 0.6 0.3 0.45]; % that link's new value
for i = 1:4
q = p; q(k(i)) = v(i);
fprintf('%-9s P_S = %.3f\n', nm{i}, 1 - prod(q));
end
PKi = [prod(p) 0.12 0.06]; % three independent encounters
r = -log(1 - PKi); % risk of each encounter
fprintf('P_S = %.3f -ln P_S = %.3f sum(r) = %.3f\n', ...
prod(1 - PKi), -log(prod(1 - PKi)), sum(r));Which lever wins — and what do the winners have in common?
Then: deception and armor tie, on different links. What would halving each link do?
Block 4 · Lesson 31 · Quick Problem
Quick Problem: Three Layers, One Lever
A notional penetrator crosses three layered SAMs and takes one independent shot from each.
- Links in chain order: $P_D,\ P_{T|D},\ P_{I|T},\ P_{H|I},\ P_{K|H}$
- Long-range SAM (outer): 0.8, 0.5, 1, 0.4, 0.5
- Medium-range SAM: 0.7, 0.8, 1, 0.5, 0.5
- Short-range SAM (at the target): 0.9, 0.8, 1, 0.5, 0.5
- (a) $P_K$ for each SAM; the mission $P_S$.
- (b) The additive risk $r = -\ln P_S$, and the SAM that carries the most of it.
- (c) The single link-halving that buys the most $P_S$, and the new $P_S$.
(a) 0.08, 0.14, 0.18; $P_S = 0.649$ · (b) $r = 0.433$; the short-range SAM, 0.198 · (c) any short-range link; $P_S = 0.720$
Block 4 · Lesson 31 · Exposure
Exposure: Risk by the Kilometer
- In its lethal zone — contour and MEZ — a site shoots every $\Delta s$ km of your track
- Each shot kills with $p_{\text{shot}}$: the chain after detection
- Twice the kilometers inside, twice the risk
Hazard: risk per km
$$P_{S,i} = e^{-\lambda_i \ell_i}, \quad \lambda_i = \frac{-\ln\left(1 - p_{\text{shot}}\right)}{\Delta s}$$
Final Project T2: this is your survival metric, ingress and egress. L34 turns it into an edge cost.
Block 4 · Lesson 31 · Final Project
The Final Project: Issued Today
Individual, 20%, in lieu of a final exam: route a B-21 through an integrated threat, then write it up as an IEEE journal paper.
| When | What is due |
|---|---|
| L31 · today | Issued: the handout, the scenario files, the toolkit, and the IEEE template |
| L35 | Checkpoint, required, feedback only: approach, T1 figure, lessons to cite |
| L38 | Work day |
| L39 | Complete draft at the start of class; peer review #1 in class |
| Before L40 | Peer review #2 |
| L40 | Revision work day |
| L41 | Final paper, response to reviewers, code, AI transcripts |
AI may typeset your paper and point out errors; the words and judgments are yours. Full rules in the handout.
Block 4 · Lesson 31 · Wrap-Up
Wrap-Up
Key ideas
- Susceptibility × vulnerability: $P_K = P_H\,P_{K|H}$ (Ball)
- One shot's $P_K$ is the product of the L1 chain's conditional links
- Independent encounters multiply: $P_S = \prod_i (1 - P_{K,i})$
- $-\ln P_S$ is additive risk — the cost L34 hands to a route planner
Next lesson
- L32: Threat Geometry — detection contours, MEZ and LEZ
- Work the L32 pre-flight before class
- Read the Final Project handout before L32