# Reading — Threat Geometry: SAM Rings, MEZ/LEZ By the end of this lesson you should be able to: 1. Draw an **aspect-dependent detection contour** for a radar site from the R⁴ law and the B-21 signature pattern. 2. Bound that contour with the **radar horizon**, and say which bound — the horizon or the R⁴ law — sets detection at a given altitude. 3. Distinguish the **MEZ** from the **LEZ**, and say what turning away does inside each. 4. Read a **layered laydown** for overlaps, gaps, and seams. ## From the chain to the map L31 wrote survivability as a chain: $P_K = P_D \cdot P_{T|D} \cdot P_{I|T} \cdot P_{H|I} \cdot P_{K|H}$, one conditional probability per link, and multiplied independent encounters into $P_S$. Every one of those links has a place on the map where it can be nonzero and a place where it cannot. A site cannot detect you where its radar cannot see you, and it cannot hit you where its missile cannot reach you. This lesson draws those two places — where you are **seen** and where you are **shot** — for one site, and then for a layered defense. The planning-chart habit is to draw each site as a circle. The circle is not wrong so much as it is one special case: it is the range at which one radar sees one aspect of your aircraft, at an altitude where the Earth does not get in the way, against a missile shot at a target flying straight at it. Change your heading, your altitude, or what you do after launch, and the ring changes shape. That is the lesson's thesis: *a threat ring is not a circle — it depends on where you point your nose and how high you fly.* ## The detection contour Start with the radar range law you have used since Block 1, written against a reference target — the form Project 3 used: $$ R_\text{det} = R_\text{ref}\left(\frac{\sigma}{\sigma_\text{ref}}\right)^{1/4}. $$ What Block 3 added is that $\sigma$ is not one number. The B-21's pattern (`B21_SignaturePattern.csv`, Project 3) gives the RCS at every **aspect** $\theta$ — the angle from the aircraft's nose to the radar, measured clockwise, $0^\circ$ nose-on and $180^\circ$ tail-on — every $5^\circ$, separately at VHF and at X-band. Now put the aircraft on a map. Measure bearings clockwise from North, as in Project 2 and L14. Let the aircraft sit at bearing $\varphi$ **from the radar**, flying heading $\psi$. From the aircraft, the radar then lies at bearing $\varphi + 180^\circ$, and the aspect it presents is that bearing measured from the nose: $$ \theta(\varphi) = \mathrm{wrap}\left(\varphi + 180^\circ - \psi\right). $$ The detection contour is the R⁴ law evaluated, bearing by bearing, at the aspect that bearing produces: $$ R(\varphi) = R_\text{ref}\left[\frac{\sigma_b\big(\theta(\varphi)\big)}{\sigma_\text{ref}}\right]^{1/4}. $$ Read it slowly, because the counterintuitive part is what stays fixed. The aircraft's heading is the same everywhere on the contour. Only the **bearing** changes, and the bearing alone decides which face the radar sees. An aircraft upstream of the site on its own track ($\varphi = \psi + 180^\circ$) is nose-on, one abeam of the site presents its beam, and one past the site presents its tail — all flying the same heading. :::{admonition} Key Concept :class: key-concept The detection contour is the R⁴ law at the aspect each bearing produces: $R(\varphi) = R_\text{ref}\,[\sigma_b(\theta(\varphi))/\sigma_\text{ref}]^{1/4}$ with $\theta = \mathrm{wrap}(\varphi + 180^\circ - \psi)$. Heading sets the shape; the circle on the chart is only the nose-on case. ::: ## The B-21's contour against Project 3's radars Put the B-21 pattern against the two radars Project 3 issued — a VHF early-warning radar (EWR) that detects $\sigma_\text{ref} = 2$ m² at 90 km, and an X-band target-tracking radar (TTR) that detects $0.03$ m² at 45 km — and read the contour at four aspects: | Radar | Nose-on ($\theta = 0^\circ$) | Beam ($90^\circ$) | Tail ($180^\circ$) | Quietest aspect | |---|---|---|---|---| | VHF EWR | 89.9 km (+3.0 dBsm) | 100.9 km (+5.0) | 89.9 km (+3.0) | 89.9 km, nose or tail | | X TTR | 45.6 km (−15.0 dBsm) | 91.0 km (−3.0) | 60.8 km (−10.0) | 21.6 km, in the quarters (e.g. $\pm 55^\circ$–$65^\circ$) | Drawn on a map for heading 090°, the VHF contour is nearly a circle and the X-band contour is a cross: 45.6 km toward the approach, needles to 91 km at the two beams, 60.8 km behind, and pinched to 21.6 km in the four quarters between them. Two lessons from Block 3 explain the difference. At VHF the wavelength is comparable to the airframe's features — the resonance region of L25 — so shaping cannot steer the echo; the pattern moves only 2 dB around the compass, and the fourth root turns 2 dB into a 12% change in range. At X-band the airframe is many wavelengths across, shaping works, and — in the spirit of L26's planform alignment — the energy is gathered into a few known spikes: a small one on the nose, large ones on the beams, and a deep −28 dBsm floor across the quarters between them. Two consequences follow. First, the quietest way to approach this X-band radar is not exactly nose-on but some $55^\circ$ off it, in the quarter the designers emptied. Second, the dashed nose-on ring that a planning chart (or the Final Project starter) draws is honest only along one line; everywhere else the real contour is inside it (the quarters) or far outside it (the beams). Final Project task T1 asks for the contour, at least nose-on and a worst case, for exactly this reason. ## The radar horizon The R⁴ law assumes the radar has a line of sight to you. Over a curved Earth it may not. With the standard 4/3-Earth allowance for atmospheric refraction, the distance to the horizon from a height $h$ in meters is $4.12\sqrt{h}$ km. L11 used that one-ended form for the listener at altitude — a B-21 at 40,000 ft sees about 455 km. A radar-to-aircraft path has two ends, each with its own horizon, and the path clears the Earth if the two tangent distances cover it: $$ d = 4.12\left(\sqrt{h_r} + \sqrt{h_t}\right)\ \text{km}, \qquad h_r, h_t \text{ in meters}. $$ Here $h_r$ is the radar's antenna (mast) height and $h_t$ the aircraft's altitude. Detection needs **both** a strong enough echo and a line of sight, so the effective contour is the smaller of the two bounds at every bearing: $$ R_\text{eff}(\varphi) = \min\big(R(\varphi),\, d\big). $$ For the B-21 at 12,000 m against a 10 m mast, $d = 4.12(\sqrt{10} + \sqrt{12000}) = 464$ km — the horizon is irrelevant, and the signature decides everything. Come down, and $d$ collapses as the square root of altitude. The two bounds cross where $d = R$: $$ h^\ast = \left(\frac{R}{4.12} - \sqrt{h_r}\right)^2 . $$ Below $h^\ast$ the horizon binds; above it the R⁴ law does. For the X-band TTR on a 10 m mast, seeing the B-21 nose-on at 45.6 km, $h^\ast = (45.6/4.12 - \sqrt{10})^2 = 62.5$ m. At 30 m the horizon is at 35.6 km and the Earth, not the signature, decides; at 150 m the horizon is out at 63.5 km and the signature is back in charge. The same mast against the VHF EWR's 89.9 km crosses at about 350 m. That pairing is the useful insight. Low observability and low altitude are not independent tricks: the stealthier the aircraft, the shorter its R⁴ ranges, and the lower it has to fly before the horizon adds anything. Terrain-following buys the most against the long-range, low-band layers — the ones a stealthy aircraft cannot shrink — and very little against a TTR that already sees it late. Two refinements push the same way. A taller mast pushes the radar's term out (the defender's lever), and real terrain is a nearer horizon than the smooth Earth: a ridge between you and the site masks you sooner than the formula says. That is what terrain masking is. :::{admonition} Key Concept :class: key-concept Detection needs line of sight: $R_\text{eff} = \min(R, d)$ with $d = 4.12(\sqrt{h_r} + \sqrt{h_t})$ km. Which bound binds depends on altitude, crossing at $h^\ast = (R/4.12 - \sqrt{h_r})^2$ — and the quieter your signature, the lower that crossing sits. ::: ## Seen is not shot: engagement zones Detection gives the site a track. To hit you it must also reach you, and a missile's reach depends on what you do after it launches. This lesson uses the simplest model that captures that — a teaching model, not a planning tool: the missile flies at a constant **average** speed $v_m$ for at most $t_\text{max}$ seconds, the target flies at speed $v_t$, and the geometry is one-dimensional along the line between them. If you fly straight at the site, you and the missile close the gap together, and the longest launch range that still intercepts within $t_\text{max}$ is the **missile engagement zone** radius: $$ R_\text{MEZ} = (v_m + v_t)\,t_\text{max}. $$ If instead you turn directly away at launch, the missile must chase you, gaining only $v_m - v_t$ per second. The range inside which it catches even a fleeing target is the **lethal (no-escape) zone**: $$ R_\text{LEZ} = (v_m - v_t)\,t_\text{max}. $$ With $v_m = 1.0$ km/s, $v_t = 0.25$ km/s and $t_\text{max} = 40$ s, the MEZ is 50 km and the LEZ 30 km. Take two aircraft closing on the site, one at 40 km and one at 20 km; both are inside the 50 km reach. Now let both turn directly away at launch. The first needs the missile to make up 40 km at 0.75 km/s, which takes 53 s — longer than it can fly — so it escapes. The second needs 27 s, and it is caught. Between the LEZ and the MEZ, the turn is the countermeasure; inside the LEZ, it is too late. Turning away has a price, of course: it stops your progress toward the target, and it shows the radar your tail, which on the B-21's X-band pattern is louder than your nose. Every heading between those two extremes gives an intermediate reach. Resolve the target's velocity along the line of sight and the one-dimensional approximation gives $$ R(\alpha) \approx \left(v_m + v_t\cos\alpha\right) t_\text{max}, $$ where $\alpha$ is the angle between your velocity and the direction to the site: $\alpha = 0$ gives the MEZ, $\alpha = 180^\circ$ the LEZ, and abeam the reach is $v_m t_\text{max}$. Notice what $\alpha$ is. It is the same angle, measured the same way, as the aspect $\theta$ that set your signature — the site's missile and its radar both care about which way your nose points relative to them. On a map at a fixed heading the envelope is an off-centre, nearly circular curve bulging toward the direction you come from. (The exact constant-speed intercept triangle gives $t_\text{max}\left[v_t\cos\alpha + \sqrt{v_m^2 - v_t^2\sin^2\alpha}\right]$: identical at $0^\circ$ and $180^\circ$, and only about 3% shorter abeam at these speeds, so the radial form is a fair teaching approximation.) Every real missile also has a **minimum range** $R_\text{min}$ inside which it cannot arm, guide or turn in time — a hole in the middle of every zone. A word on the vocabulary, because it is a small lesson in reading sources. In Joint doctrine a *missile engagement zone* is a volume of **airspace** in which engagement responsibility rests with surface-to-air missiles — an airspace-control measure (JP 3-01, *Countering Air and Missile Threats*; the *DoD Dictionary of Military and Associated Terms*). Threat analysis and mission planning borrow the same letters for the kinematic reach of a site, which is the sense used here. Air-to-air literature calls the inner, no-escape region the *no-escape zone*. And real engagement envelopes come from fly-out models that depend on altitude, launch conditions and the target's maneuver; Ball's *The Fundamentals of Aircraft Combat Survivability Analysis and Design* treats them as part of susceptibility. The constant-speed model captures the one idea this lesson needs — that the reach depends on what the target does — and nothing more. :::{admonition} Key Concept :class: key-concept $R_\text{MEZ} = (v_m + v_t)t_\text{max}$ is the reach against a target flying straight at the site; $R_\text{LEZ} = (v_m - v_t)t_\text{max}$ is where even turning directly away fails. Between them, the turn is the countermeasure; inside the LEZ there is no escape. In between, $R \approx (v_m + v_t\cos\alpha)t_\text{max}$. ::: ## The lethal zone Put the two halves together. A site can shoot you only where you are inside its tracking radar's effective contour **and** inside its missile's envelope, **and** outside its minimum range. For the B-21 at altitude against the TTR, nose-on, the radar sees it at 45.6 km while the missile could reach 50 km: the **track**, not the missile, sets the first shot. That is what low observability buys in engagement terms — it pulls the detection contour inside the kinematic envelope, so the missile's reach goes partly unused. At low level, add the horizon to the same minimum. This intersection is the region the Final Project's hazard model integrates over: the length $\ell_i$ of route inside site $i$'s lethal zone. ## Layered defenses: overlaps, gaps, and seams No site defends alone. L7 laid out the layers: early-warning radars that detect and cue at long range, acquisition radars that build the tracks, and the SAM sites whose TTRs track and whose missiles reach. On a laydown map those layers produce three features worth naming: - An **overlap** is a point inside two or more lethal zones — depth for the defender, and double exposure for you. - A **gap** is a region inside the early-warning coverage but outside every lethal zone: you are seen, cued and reported, but not shot. - A **seam** is the boundary where one site's coverage hands off to the next — where a track can drop between radars and a cue can arrive late. Seams are where gaps open when the geometry shifts, and L33 will find them again in the information geometry of the adversary's passive network. The lesson's figure and demo lay out one notional example to scale: a VHF EWR whose coverage spans the whole first line, two SAMs 100 km apart forming that line, and a third SAM behind and south of them. At heading 090°, each of the first two SAMs sees an aircraft between them at its beam. The TTR's beam spike reaches 91 km, but the missile's envelope abeam is only 40 km, so each lethal zone reaches 40 km toward the other and a 20 km gap opens between them. The third SAM overlaps the second by about 10 km. The gap is not a property of the map alone. Fly north instead and the first SAM sees your nose (lethal to 45.6 km) while the second sees your tail (lethal only to the 30 km LEZ): the gap widens to 24 km. Cross at 045° and both SAMs see a quiet quarter of the X-band pattern: each zone shrinks to 21.6 km and the gap opens to 57 km. Every feature of a laydown — every overlap, gap and seam — is heading-dependent, which is exactly why route planning in L34 has to be done on a graph whose edge costs know which way you are flying. ::::{admonition} Type-along :class: type-along One radar that sees a 1 m² target at 80 km, looking at an aircraft whose RCS at this aspect is 0.5 m², from a 12 m mast. Three altitudes. Type this and see which bound sets the detection range at each: ```matlab % L32 -- which bound binds: the horizon or the R^4 law? Rref = 80; sref = 1; % radar: 80 km on a 1 m^2 target sig = 0.5; % RCS it sees at this aspect [m^2] hr = 12; % radar mast height [m] ht = [30 150 600]; % three target altitudes [m] Rdet = Rref*(sig/sref)^(1/4); % R^4 law: one range [km] dhor = 4.12*(sqrt(hr) + sqrt(ht)); % 4/3-Earth horizon [km] for k = 1:3 if dhor(k) < Rdet, lim = 'horizon'; else, lim = 'R^4'; end fprintf('%3d m: horizon %5.1f R^4 %4.1f -> %4.1f km (%s)\n', ... ht(k), dhor(k), Rdet, min(dhor(k), Rdet), lim); end ``` Then answer these two: 1. At 150 m the horizon still wins, by 2.6 km. **What altitude flips it?** 2. **Raise the mast to 30 m. Which row changes its answer?** :::{admonition} Solution :class: dropdown ```text 30 m: horizon 36.8 R^4 67.3 -> 36.8 km (horizon) 150 m: horizon 64.7 R^4 67.3 -> 64.7 km (horizon) 600 m: horizon 115.2 R^4 67.3 -> 67.3 km (R^4) ``` 1. The bounds cross where $d = R$: $h^\ast = (R/4.12 - \sqrt{h_r})^2 = (67.27/4.12 - \sqrt{12})^2 = (16.33 - 3.46)^2 \approx 165$ m. Below about 165 m the Earth sets this radar's detection range; above it the signature does. 2. The middle row. With $\sqrt{30} = 5.48$, the 150 m horizon moves out to $4.12(5.48 + 12.25) = 73.0$ km, past the 67.3 km R⁴ range, so the R⁴ law now binds there. The 30 m row's horizon rises only to 45.1 km — still the horizon — and the 600 m row was never horizon-limited. A taller mast is the defender's answer to a low flyer, but it buys only the altitudes near the crossing. ::: :::: ::::{admonition} Quick problem — Low and Inside :class: example-problem A low-level ingress against one SAM site: its TTR, and the missile behind it. - TTR mast height: 9 m - Aircraft altitude: 64 m - TTR detection range (R⁴ law, this aspect): 52 km - Missile average speed $v_m$: 0.9 km/s - Maximum time of flight $t_\text{max}$: 35 s - Aircraft speed $v_t$: 0.3 km/s **(a)** The radar horizon to the aircraft, and which bound sets its detection range.\ **(b)** The MEZ and the LEZ.\ **(c)** Launched on at 30 km, it turns directly away. Escape? At 18 km? :::{admonition} Solution :class: dropdown **(a)** $d = 4.12(\sqrt{9} + \sqrt{64}) = 4.12 \times 11 =$ **45.3 km**. That is inside the 52 km R⁴ range, so **the horizon binds**: the site first detects the aircraft at 45.3 km. **(b)** $R_\text{MEZ} = (0.9 + 0.3) \times 35 =$ **42 km**, and $R_\text{LEZ} = (0.9 - 0.3) \times 35 =$ **21 km**. **(c)** At 30 km the aircraft is between the LEZ and the MEZ. Chasing it, the missile closes at $0.9 - 0.3 = 0.6$ km/s and needs $30/0.6 = 50$ s, more than its 35 s — **it escapes**. At 18 km it is inside the LEZ: $18/0.6 = 30$ s is within 35 s — **no escape**. So what: at low level the Earth, not the signature, sets first detection (45.3 km), and the missile's reach then sets the first shot (42 km). Inside 21 km no turn helps; between 21 and 42 km the turn is the countermeasure. ::: :::: ## Wrap-Up A threat ring is the R⁴ law evaluated at the aspect each bearing produces, so your heading, through the signature pattern, gives it its shape — for the B-21 against an X-band TTR, a cross rather than a circle. The radar horizon, $d = 4.12(\sqrt{h_r} + \sqrt{h_t})$, bounds it from outside, and altitude decides which bound binds; the stealthier the aircraft, the lower the crossing. A missile's reach depends on what you do after launch: the MEZ against a closing target, the LEZ where even turning directly away fails, and a heading envelope in between that shares its angle with the signature. Intersect the two and you have the lethal zone; lay several sites together and you have overlaps, gaps and seams, every one of them heading-dependent. That picture is Final Project task T1. Next, **L33 — Information Geometry** asks how well the adversary's network can locate you along a route, and where its seams are.