The mathematics of roving searches

[ColonelSandersLite]

So here's my take on this, and I would encourage further thought or debate on its merits.

So suppose, for example sake, that you are patrolling a suspected shipping lane that runs north-south.

Consider, that nearly every merchant target moves at a speed of 11 knots or less (usually slower).

Let's further assume that your sensors have a reliable detection range of 13 miles (this is my experience for SJ-1 radar in TMO).

Let's also assume that you want to patrol at a cruising speed of 10 knots.



So, to ensure that no target heading north or south gets past you, the best course of action is to patrol strictly perpendicular to the shipping lane.

If your sensor has a detection radius of 13 miles, it will take a 11 knot target 141 minutes to pass through it completely, meaning that you must be back at your original position within 140 minutes or so.

During your patrol, you must turn 180 degrees twice, taking a total of 7 minutes, leaving a total cruising time of 133 minutes. (may vary by boat, mods, etc. accurate for a balao doing 10 knots in tmo though)

Each leg of the cruise then becomes a 66 minute cruise at 10 knots, for about 11 nautical miles each way.


In this case, it is certain that no target moving at a speed no greater than 11 knots can pass through a 11X26 mile box (286 square nautical miles). Additionally, on the west and east ends of the box, there is a 13 mile radius semicircle of detection. A 11 knot target may not pass through most of either of these areas without detection either. The sole part of the patrol area that a target may pass through undetected would be a segment of the circle defined by a 13 mile cord running north-south at the far end of the circle, an area of 88 miles. This works out to guaranteed detection of any 11 knot targets passing through an area of 641 square nautical miles (with an additional less than 100% chance of detection over 176 square miles as below) vs 531 if you had remained stationary or a 17% increase.

Inside of this small segment, the odds of detection depend on target speed, and are directly proportional to the resulting length of chord. For example, the odds of detecting an 11 knot target passing through a north-south chord of 6.5 miles would be 50%.

I thought it might be helpful to include a diagram of what I'm talking about.





With other sensors, the amount of time you can spend moving east-west will be reduced proportionally. For example, suppose that your detection radius is half of the SJ-1 range, or 6.5 miles. This would reduce the cruising time of each patrol leg by half.

If you are willing to accept a less than 100% chance of detecting a 11 knot target, lets say guaranteeing detection of 9 knot targets instead, the amount of time spent on each leg of the patrol would then be increased proportionally. That would increase the length of each leg to 13.75 miles, with a 100% chance of detecting targets doing 9 knots.

[Fearless]

All great examples but how does it apply to a simulation that spawns ships at certain points on the map?

[Crannogman]

I was not aware that ships spawn randomly - I thought their spawning and routing was encoded in the campaign files

[ColonelSandersLite]

Crannogman has it right. The ships aren't just a random distribution. They spawn at and despawn at ports. Not actually at the docks, but fairly close.

The big exception is the set piece battles, like midway. They spawn a *long* way from the battle, but not actually in ports. The do go to ports when their part is done though. I'm just guessing here, but I suspect that getting the timing of the ship movements for the battles down perfectly was just too much work when spawning them all the way back at port.

If you want to see for yourself, open up the .mis files with the editor. They are in "\sh4\Data\Campaigns\Campaign". The yellow diamonds are spawn points.

As an aside, I have never gone out of my way to watch any of those set piece battles. I really should one of these days. The only time I've actually seen one by pure chance was in SH3+GWX. I was harbor raiding during the invasion of norway and all of a sudden, some german surface vessels showed up and starting shooting everything. At first I thought they where enemy destroyers bearing down on me and shooting at my periscope the way destroyers do in these games...

[ColonelSandersLite]

Oh, I may have a critical error in the post #4 above. I'll have to think on it some more later though. Gonna head to the range. I may get a chance to think it through tonight, but it might be tomorrow before I get to it.

Edit: I see RR posted something while I was typing that. He agrees with me, so I know it's got to be wrong .

[Rockin Robbins]

Originally Posted by ColonelSandersLite

Oh, I may have a critical error in the post #4 above. I'll have to think on it some more later though. Gonna head to the range. I may get a chance to think it through tonight, but it might be tomorrow before I get to it.

Edit: I see RR posted something while I was typing that. He agrees with me, so I know it's got to be wrong .

I can see that it's only good for a choke point 37nm wide. That's still a useful tool though at that. If a ship enters your box and you're at the other end it's an hour before you're to the near end to him. He's still in the 100% detection box so you've got him unless you see something I missed.

[TorpX]

Ok, I haven't read all the comments and criticisms yet. I want to finish the problem and add a little math first. I'll try to get back to the comments, as time permits.



I've computed some figures for typical (and maybe not so typical ship/sub speeds.

table1.txt

The 3 knot speed is typical of economical searched cruising, 6 kn. is good for a S-boat, 9 is economical for a fleetboat, and faster speeds to see how more intensive searches might fare.


You might ask, could we go faster and eliminate the gaps between segments? The answer is yes; if our sub goes fast enough, we could theoretically, detect every ship making a transit through this search zone (for our assumed conditions).

To show how fast we would have to go, it helps to reconfigure the diagram. Below, we see the segments of our search are much closer together (like a coilbound spring); so much so that there are no spaces uncovered between the segments. I didn't shade the overlaps, but you can easily see that there would be a large 'wedge area' or amount of overlap here.

To achieve this level of coverage, our sub must go across the zone to the west and back again, so the detection circles at A and C are tangent. Mathematically, this means the sub must move the length of 4w in 't' hours, while the target ship moves 2r in 't' hours:

Vu = 4w / t

Vt = 2r / t

From here, all we need to do is solve the second equation for t and substitute the result in the first equation. Doing this, we obtain:

Vu = Vt * (4w / 2r)

Putting in the numbers we used for our earlier example, we find that we would need to have a speed of 4 times the target's speed, or 32 knots!

These results helps explain why sub operations in WWII didn't sink enemy ships at a blinding pace, and individual subs might spend days, or even weeks, between ship sightings.

It also shows why aircraft were such useful sub hunters. They had an enormous speed advantage over any submarine.

[Rockin Robbins]

Originally Posted by Crannogman

I was not aware that ships spawn randomly - I thought their spawning and routing was encoded in the campaign files

I didn't say they spawn randomly. I said that they were distributed more or less randomly along their routes. That they didn't proceed in line abreast as your example showed. They're like raindrops. And we're trying to walk through a rainfall getting as wet as possible. Those raindrops didn't spawn at random locations--well they did within constraints. But by the time they get to us their distribution looks random. Shipping is like that.

Now if you're coming up with a specific way to game RSRDC then, since I haven't looked at the campaign files, you might have a way. But I choose not to game the system and not to analyze the campaign files to get dates of departure, exact route and timing. The real sub skippers didn't have that information and I consider that if we use historical information to game the game we've broken the simulation irretrevably ourselves.

[Crannogman]

Originally Posted by Rockin Robbins

I didn't say they spawn randomly. I said that they were distributed more or less randomly along their routes. That they didn't proceed in line abreast as your example showed. They're like raindrops. And we're trying to walk through a rainfall getting as wet as possible. Those raindrops didn't spawn at random locations--well they did within constraints. But by the time they get to us their distribution looks random. Shipping is like that.

Now if you're coming up with a specific way to game RSRDC then, since I haven't looked at the campaign files, you might have a way. But I choose not to game the system and not to analyze the campaign files to get dates of departure, exact route and timing. The real sub skippers didn't have that information and I consider that if we use historical information to game the game we've broken the simulation irretrevably ourselves.

I was responding to a different comment about random spawning. RSRDC cannot be gamed because the shipping routes have variable waypoints (some as much as 20nm radius), and thus their heading is fairly unpredictable.

It seems that a formula will be forthcoming to calculate a minimum average speed "x" to patrol a chokepoint of width "y," variable of course on the range of your sensors. Another formula could tell you your miss percentage based upon how far below "x" you go (if, as TorpX posited, "x" is impossible or overly inefficient).

Since RSRDC includes the variable waypoints, your detection chance is probably improved since ships will generally not be sailing in a direct line from origin to destination - thus it will take them longer to transit your patrolled box

[Rockin Robbins]

Originally Posted by ColonelSandersLite

So here's my take on this, and I would encourage further thought or debate on its merits.

So suppose, for example sake, that you are patrolling a suspected shipping lane that runs north-south.

Consider, that nearly every merchant target moves at a speed of 11 knots or less (usually slower).

Let's further assume that your sensors have a reliable detection range of 13 miles (this is my experience for SJ-1 radar in TMO).

Let's also assume that you want to patrol at a cruising speed of 10 knots.



So, to ensure that no target heading north or south gets past you, the best course of action is to patrol strictly perpendicular to the shipping lane.

If your sensor has a detection radius of 13 miles, it will take a 11 knot target 141 minutes to pass through it completely, meaning that you must be back at your original position within 140 minutes or so.

During your patrol, you must turn 180 degrees twice, taking a total of 7 minutes, leaving a total cruising time of 133 minutes. (may vary by boat, mods, etc. accurate for a balao doing 10 knots in tmo though)

Each leg of the cruise then becomes a 66 minute cruise at 10 knots, for about 11 nautical miles each way.


In this case, it is certain that no target moving at a speed no greater than 11 knots can pass through a 11X26 mile box (286 square nautical miles). Additionally, on the west and east ends of the box, there is a 13 mile radius semicircle of detection. A 11 knot target may not pass through most of either of these areas without detection either. The sole part of the patrol area that a target may pass through undetected would be a segment of the circle defined by a 13 mile cord running north-south at the far end of the circle, an area of 88 miles. This works out to guaranteed detection of any 11 knot targets passing through an area of 641 square nautical miles (with an additional less than 100% chance of detection over 176 square miles as below) vs 531 if you had remained stationary or a 17% increase.

Inside of this small segment, the odds of detection depend on target speed, and are directly proportional to the resulting length of chord. For example, the odds of detecting an 11 knot target passing through a north-south chord of 6.5 miles would be 50%.

I thought it might be helpful to include a diagram of what I'm talking about.





With other sensors, the amount of time you can spend moving east-west will be reduced proportionally. For example, suppose that your detection radius is half of the SJ-1 range, or 6.5 miles. This would reduce the cruising time of each patrol leg by half.

If you are willing to accept a less than 100% chance of detecting a 11 knot target, lets say guaranteeing detection of 9 knot targets instead, the amount of time spent on each leg of the patrol would then be increased proportionally. That would increase the length of each leg to 13.75 miles, with a 100% chance of detecting targets doing 9 knots.

Excellent! That's a great analysis for detection of shipping through a choke point. And a choke point is the goal of our searching if one is attainable. Looks like a closed door to me too. Good job.