Zigzagging

1. An Exploration of Zigzagging By Brian McCue This is not a product of the Center for Naval Analyses

2. Problem Statement • Under what circumstances is, and is not, zigzagging of benefit? This question has proven difficult to answer analytically or empirically, so let us experiment to ask: • Can we find cases in which zigzagging is, and is not, of benefit?

3. Background: Intercept Kinematics - Target’s speed is triple that of the interceptor. - The interceptor is able to detect the target a long way off. - The target has no capability to make a counter-detection. - Find the intercept solution. Vt a Wt Ut a Detection radius R Vt U = speed of interceptor; V = 3U = speed of target W = interceptor’s velocity in target’s frame; Wt = R , the detection radius t = time needed to intercept (Ut)2 = (Vt/3)2 = (Wt)2 + (Vt)2 - 2 Wt Vt cos(a) cos(a) = -Wt  Ut / || Ut ||

4. Intercept Kinematics: Alternate Solution Consequence of alternate choice of root in Law of Cosines. Wt = R (still), but with a lesser W and greater t, Ut, and Vt. Wt a Ut a Detection radius R Vt Alternate intercept solution for target whose speed is triple that of interceptor Mariner’s rule-of-thumb is: “Constant bearing, decreasing range.”

5. An Important Degenerate Case If the Law of Cosines has repeated roots, the triangle is a right triangle, leading to the largest possible angle a for which an intercept can be made; this defines the amount of frontage that the interceptor can defend. Vt a 2h = 2 RU / V is the size of the “front” along which targets of this speed V can appear and be intercepted. Wt h = RU / V Ut Vt Detection radius R You can think of 2h as the “capture cross section” of the interceptor for targets of speed V. T2 = R2/(V2-U2) at the extreme

6. Rationale for Zigzagging • The preceding kinematic analysis suggests that a blind target with a speed advantage can, by course alterations, spoil an on-going intercept, should one be occurring. • This is the basis for zigzagging. • Zigzagging strategies are therefore distinguishable by their timescales.

7. Types of Zigzagging Scale of Time Zigzag so as to avoid Torpedoes Minutes Submerged intercept Hours Surface intercept (Sometimes hard to tell from evasive routing) Days

8. Zig-Zag Diagrams For Single Ships and Convoys, Royal Admiralty, 1940 • Threat levels characterized as • "Open waters where submarines have not previously been operating, but where they may appear," • "Submarine areas," and • "Specially dangerous waters.” • Changes of course every 5-10 minutes, by amounts of 10o to 80o. • Patterns given for slow ships, fast ships, and convoys

9. 2 N 1 0 2 4 6 8 10 12 14 16 18 -1 -2 Example: Admiralty Pattern #16 - “For general use in submarine areas.” - “Suitable for convoys of all speeds.” - Chart shows application of #16 to base course 90o, speed 10 knots. - Course-made-good is 17.85 nm in the pattern’s 2-hour cycle. - Purpose is evidently the avoidance of submerged intercept. Nautical Miles

10. Zigzagging Went Un-analyzed • Zigzagging has benefits, but it also has costs. • Delayed arrival, added difficulty of navigation. • Increased path length increases potential encounters. • Zigzagging’s overall worth has gone un-analyzed: • Wartime operations researchers couldn’t get data. • It’s difficult to frame analytically. • Until lately, it was too much to do in Monte Carlo. • So we will look at a model of zigzagging, try analyzing it, and then use Monte Carlo.

11. Set-up of Zigzagging Model One transiting 30-knot “target” headed Eastwards 25 10-knot “submarines” and their 20-mile detection radii 1000 Nautical miles 750 500 250 0 0 250 500 750 1000 1250 1500 1750 2000 Nautical miles

12. Comments on Realism This is an exploration of zigzagging, not a simulation of World War II, but, ... • 10 knots is about right for U-boats. • 30 knots might be fast even for a liner-troopship. • 1000 x 2000 nm is about right for Atlantic arena • Real U-boats were coordinated, but tried to attack in packs rather than ASAP.

13. 4 2 0 2 4 6 8 10 12 14 16 18 -2 -4 Transitor “Doctrine” in the Model • Zig or zag a given number of degrees North or South of the due-East base course. • Times between zigs are exponentially distributed. (The transitor doesn’t know he’s in a discrete time-step model and that the distribution is geometric.) • Bounce off North and Sout boundaries elastically. • Figure shows example with average time of 7.5 minutes and angles of +/- 21.25o --same as #16, shown again for comparison. (Model returns to baseline only by chance.) Admiralty # 16 Model

14. Submarine “Doctrine” in the Model • Uncoordinated case: • Move to intercept the target if you detect it. • If you can’t make an intercept, move to minimize the miss distance (CPA) and hope target zigs to you. • Coordinated case: • Move to intercept the target if anybody detects it. • If you can’t make an intercept, move to minimize the miss distance (CPA) and hope target zigs to you. • If contact is lost, stay on course. (Bounce off walls.)

15. But 1st, how far can we get analytically? • “Capture cross section” is 2 RU / V = 40/3 miles • There are 25 submarines in 2 million sq. nm. • Path length is 2000 / cos(zig angle). • Probability of transiting without any encounter is e-(40/3) x 25 x ( 2,000 / cos(zig angle) ) / (2,000,000) • About 70% for 0o < zig angle < 30o, degrading to 47% for 64o.

16. How far can we get analytically? (cont) • Engagement time is something like T2 = R2/(V2-U2) = 400/(900-100) = 1/2 hour = 30 min. • Probability of zigging during an engagement is something like 1-e-30/avg zig time • Probability of transiting without fatal encounter is therefore something like e-(40/3) x 25 x ( 2,000 / cos(zig angle) ) / (2,000,000) (1-exp(30/avg zig time)) • This is a lot of “something like“s! • Probability is relative insensitive to zig time.

17. Defects in Analytic Approach • “Engagement time” was an upper bound, so comparing zig time to it favors the transitor. • Analysis credited all zigs with being evasions, which they won’t be. So small zig angles get too much credit for evasion, while big zig angles’ disadvantage of long path length is taken fully into account. • No treatment of coordination of interceptors.

18. Cases to Run in the Model • All combinations of: • 3, 9, 30, 90, 300, 1000, 3000 minute average times between zigs and • 0o, 1o, 2o, 4o, 8o, 16o, 32o, 64o zig angles. • Each with: • Uncoordinated submarines. • Coordinated submarines. • 100 times each. • The target is “intercepted” if the submarine gets within a mile--because of torpedo range and time granularity (two minutes).

19. Demonstration

20. Zigzag Fitness Landscape(v. uncoordinated submarines) 3000 55-60 1000 Average time between zigs (Minutes) 65-70 60-65 300 Number of safe passages out of 100 65-70 70-75 90 Results smoothed by averaging neighbors 30 75-80 10 3 0 1 2 4 8 16 32 64 Angle of the zig-zag (degrees)

21. Zigzag Fitness Landscape(v .coordinated submarines) 3000 30-35 40- 45 45-50 35- 40 1000 50-55 Number of safe passages out of 100 55-60 Average time between zigs (Minutes) 300 60-65 40-45 Results smoothed by averaging neighbors 90 65-70 70-75 30 Admiralty plans for individual fast ships 10 75-80 3 0 1 2 4 16 32 64 Angle of the zig-zag (degrees)

22. Observations • Zigzagging can help. • High-frequency, medium-high angle, helped against the coordinated submarines • Our scenario was WW II-like, and WW II-like zigzagging worked well in it. • Zigzagging can hurt. • Low-frequency, high angle, against the coordinated submarines • High-angle, low frequency zigzagging seems to be worse than none at all against the uncoordinated subs, and no form of zigzagging is particularly better than none at all.

23. So is Coordination Bad? NO. • Zigzagging helps more if the submarines are coordinated, but • This is not to say that the submarines ought not to coordinate: even when degraded by zigzagging, their coordinated performance is no worse than their uncoordinated performance. • High-frequency, high-angle zigzagging can reduce the effectiveness of the coordinated submarines to that of the uncoordinated submarines, but not below.

24. So Is Zigzagging Good? • IT CAN BE. • We found a regime where it helped a lot: • High frequency, medium-high angle, against coordinated submarines. • It doesn’t seem to hurt unless reduced to the absurd extreme of low frequency, high angle. • There’s a cost we haven’t considered—increased transit time—that may or may not be a big consideration.

25. Questions • How much do our findings depend upon our choices of U, V, and R ? • How real is the local maximum at 2o , 300 minutes in the uncoordinated case? • What would happen if interceptors used the long intercept solution instead of the short one? • And now for your questions ...

26. 3000 45-50 1000 300 65-70 70-75 Number of safe passages out of 100 90 30 10 3 0 1 2 4 8 16 32 64 Angle of the zig-zag (degrees) Zigzag Fitness (Analytic)(v. uncoordinated submarines) 60-65 55-60 50-55 Average time between zigs (Minutes)