The mathematics of roving searches

[merc4ulfate]

@ Rockin Robbins

"They're like raindrops. And we're trying to walk through a rainfall getting as wet as possible"

That is absolutely beautiful

Let us now go and seek the rain of war in hopes that we do not drown in it's sorrow.

[ColonelSandersLite]

Poetics aside, wouldn't one just stand in the rain to get as wet as possible?

[TorpX]

Ever ride a bike in the rain, you get wet fast.



I approached the problem from the standpoint of 'search efficiency' (meaning the ratio of contact of a moving sub compared to a stationary one), as this seemed easier than figuring on complete coverage in a certain zone, or specific probabilities.


I thought about putting together a simple program that would track randomly generated ships through an area, and tally the number that were 'detected', but this seems like more trouble. If one wanted to know how well particular search patterns worked, with traffic on multiple axes, it might be necessary to do that kind of thing.

[ColonelSandersLite]

Ever just stand in the rain? You get wet pretty fast. Just saying. I've never really seen a study on it, but I suspect that how wet you get is more of a function of time rather than speed.

Originally Posted by TorpX

I thought about putting together a simple program that would track randomly generated ships through an area, and tally the number that were 'detected', but this seems like more trouble. If one wanted to know how well particular search patterns worked, with traffic on multiple axes, it might be necessary to do that kind of thing.

I've been thinking about doing just that as well. I'm reasonably certain that the math is solvable but the problem is pretty complex and I'm not a mathematician. I do know enough math and have put just enough thought into it to understand just how complex calculating the probabilities is though. I strongly suspect that it would be easier and faster for me to come up with a computer simulation that would provide a reasonable approximation.

[ColonelSandersLite]

Actually, Check this out, it might actually answer the question but I suspect a big flaw


If we assume that like the rain in this math problem:

We must assume that you have no prior knowledge besides the reasoned deduction that you are patrolling a likely transit area and it's general direction (we'll use north-south for example sake).

Therefore, the only direction worth moving is perpendicular to the transit area (east-west in this example).

Without prior knowledge to the contrary, we must also assume that traffic flow is statistical uniform. I.E. The odds of a contact being 20 miles due north of you at the moment are the same as the odds the contact being there 4 hours later.

At any given time, a target could be anywhere in the patrol zone that is outside of your current sensor range.

If all of the above is ture you need to be moving e-w but loiter time needs to be maximised. I.E. Gas milage is unimportant, but rather consumption rate is.



The suspected flaw:
"Without prior knowledge to the contrary, we must also assume that traffic flow is statistical uniform."
When you move through an area, we know that no traffic moving at x speed can be in certain locations. For example, a 10 knot target could not have moved all the way through an area you searched with SJ-1 radar half an hour ago if you are cruising at 10 knots. This means that you do have some prior knowledge of where targets are not at any given time. Let's call these areas cavities.

In the question of rain on a person, rain falls at a relative velocity that the cavity is insignificant. I suspect that the relative velocities of ships means that the cavities are potentially quite significant when trying to form a statistical understanding.

[aanker]

I could have sworn it was in, 'Clear The Bridge', however I have read so many books - it could have been a different Skipper.

Whomever it was wrote that upon reaching his assigned area, he tried 'All Stop' and performed high-scope & radar searches at that location all day. The next day he would move 20 nm, and conduct the high-scope search at the new location all day, sail 20 nm the next day, and repeat the procedure until he found a contact.

Sadly, unlike SH1, SH4 doesn't model the larger horizon gained by using 'high-scope' searches, although in real life many patrolled using the high-scope farther horizon advantage.

What SH4 does correctly is model the almost daily position reports the Japanese (and Germans) were required to report. These reports would be intercepted and decoded becoming the ULTRA reports our Commanders received, and that We receive in-game as the red boxes with directional tails on our chart/map screen, along with the position report messages. This is why it doesn't bother me that those position reports are on the Nav map, although of course there are too many in Stock... (as was the case in SH1)

Unlike the Axis, US boats did as little communicating as possible - mostly none - because they were the 'silent service'. The few US wolfpack missions required some communication between the boats, and a small handful of boats reported that they were under attack, however most did not even report that. After a period of time they would be declared overdue and lost at sea.

This topic, "The mathematics of roving searches" is very interesting and is interesting to compare with the above static search method that conserves some fuel.

I've been a fan since the excellent tutorial on how to make a torpedo attack without the use of a TDC (S-boats didn't have a TDC) written by someone whose initials are, 'Frank Kulick'.

Happy Hunting!

[Rockin Robbins]

I base my theories on those of Admiral Eugene Fluckey, who with the USS Barb found targets when nobody else did. He didn't know the shipping lanes because he couldn't open his game box and dig one up or do a Google search. In Thunder Below he goes into great detail explaining exactly what I've laid out. All things being equal, your number of contacts developed is proportional to the number of square miles of ocean surface you search in a day.

Of course that has to be modified by the length of the cruise, the amount of fuel you have and having enough torpedoes to cruise the distance without running out.

Fluckey, starting his career when boats routinely returned to base without finding a single target, set the world on fire simply by staying on the surface, covering the most ground per day and using the longest range sensors he had (unlike SH4, radars broke painfully often). He pioneered using the scope on the surface, extended to its highest position to extend the horizon enough that he could double his visual search area.

So in real life they had to use raindrop theory at best to search for targets. Because of hindsight, we might come up with better methods but they would be bogus, based on assumptions real sub skippers couldn't make.

[ColonelSandersLite]

Originally Posted by Rockin Robbins

He didn't know the shipping lanes because he couldn't open his game box and dig one up or do a Google search.

No, they had compiled inteligence reports from sighting reports and radio intercepts. This aspect is mostly overlooked and missing in the game.

Originally Posted by Rockin Robbins

So in real life they had to use raindrop theory at best to search for targets. Because of hindsight, we might come up with better methods but they would be bogus, based on assumptions real sub skippers couldn't make.

If raindrop theory is to be used, then that would advocate just moving at 1 knot to maximise loiter time. See my previous post.



I've started putting together a simple computer model to help out and give us some actual hard data to think on. I should have it done in a day or two.

[Rockin Robbins]

Originally Posted by ColonelSandersLite

No, they had compiled inteligence reports from sighting reports and radio intercepts. This aspect is mostly overlooked and missing in the game.


If raindrop theory is to be used, then that would advocate just moving at 1 knot to maximise loiter time. See my previous post.



I've started putting together a simple computer model to help out and give us some actual hard data to think on. I should have it done in a day or two.

Actually at any given time the air of a certain volume is occupied by a certain number of raindrops. In a flow that dense you encounter the same number of raindrops no matter what speed you run until you start to go faster than the time it takes the drop to fall your height. Then your horizontal velocity brings you into more raindrops than you would encounter standing still or going slower than that speed.

Let's quote Admiral Eugene Fluckey, quoting himself on page 65 of Thunder Below, in a conversation with Admiral Lockwood, who Fluckey would replace later.

"Luck is where you find it--but to find it you have to look for it. During her seventh patrol Barb was submerged every day waiting for the enemy to pass her way. It's no good. The area os search is practically nil.

"There's a big ocean out there. I search it on the surface with our high periscope up and a wide, sweeping zig plan, using as high speed as our fuel supply will allow. Now I realize that we may be sighted, depth charged, and bombed more often, but we'll find a helluva lot more targets. On our last patrol we spent only one full day submerged to check their biggest harbor.

Pretty clear that ULTRA position reports were rare. Pretty clear that Lockwood didn't micromanage his skippers. Pretty clear that he didn't mention shipping lanes. Yes, he spent every moment reading war patrol reports for the purpose of figuring out where to hunt and what mistakes to avoid. Pretty clear that Fluckey's method is what I've copied and it works. Wide, sweeping zig plan, speed highest for patrol time: 11 knots for fleet boat, longest range sensors available.

This was the most successful sub captain in WWII for innovation, turning a slow part of the war into a bonanza. Heck, he sank a train.

[ColonelSandersLite]

Originally Posted by Rockin Robbins

Actually at any given time the air of a certain volume is occupied by a certain number of raindrops.

Now *this* is the critical factor that makes things pretty complicated. At what point exactly is it? I suspect that it depends very much on a number of factors, which is why I'm just writing a computer simulation. These factors would include:

Sensor range (maybe patrolling works better with SJ-1 while lookouts work better stationary?)
Target speed (maybe patrolling gives better odds of finding ships under a certain speed while having little or no effect on targets above a certain speed.
Patrol speed (maybe patrolling at 10 knots doesn't significantly change your odds from patrolling at 3 knots?)

The easiest way to get useful info that I see is just to write a computer program to try them all like 500,000,000 times.


Oh, and I wasn't talking about radio stuff specifically. More along the lines of something as simple as taking a map and putting a pin in it for every contact report. You know where all the ports are, and often logical deductions can be made just by connecting the dots. Other efforts will provide more data to work with, but the basic concept remains the same really. In other words, if you never look anything up in your sh carreer, you get the amount on intel you personally generate (and probably don't store it all that well), whereas a sub skipper in ww2 was additionally getting intel from other sources. That being said, I don't tend to look up things very often as the information revealed can be way too precise.

[TorpX]

Originally Posted by ColonelSandersLite

Actually, Check this out, it might actually answer the question but I suspect a big flaw

That's a pretty good analogy there.

The suspected flaw:
"Without prior knowledge to the contrary, we must also assume that traffic flow is statistical uniform."
When you move through an area, we know that no traffic moving at x speed can be in certain locations. For example, a 10 knot target could not have moved all the way through an area you searched with SJ-1 radar half an hour ago if you are cruising at 10 knots. This means that you do have some prior knowledge of where targets are not at any given time. Let's call these areas cavities.

I consider this the beauty of my solution. I've used vector addition to subtract the target ships' speed. (This sort of thing was done with a 'maneuvering board' for various problems.) Working the problem this way, the ships in diagram 2 (if there are any) do not move. We need not make any assumptions of how many, or where the ships are located. All that need to be done is compare the respective areas cut out, of the moving sub and the stationary one.


Here is another way to look at it. If the sub is going W and a ship is just far enough to the E to escape detection, it only needs to get through before the sub reverses course and can reach that area again. If the width of the search zone is very narrow, the sub will reach that area faster, and it will be hard or even impossible for the ship to get through here, but that also means there is more space on either side that is not being searched. Take this to it's logical conclusion and you are back to being stationary; no ship within your detection radius will get through, but every ship on either side will.

Originally Posted by aanker

I could have sworn it was in, 'Clear The Bridge', however I have read so many books - it could have been a different Skipper.

Yes, thank-you. It's on page 54. I'm glad I'm not the only one who remembers that. I think O'Kane did a good job of explaining it.

Sadly, unlike SH1, SH4 doesn't model the larger horizon gained by using 'high-scope' searches, although in real life many patrolled using the high-scope farther horizon advantage.

Yeah, Ubisoft sure could have done better.

Originally Posted by Rockin Robbins

I base my theories on those of Admiral Eugene Fluckey, who with the USS Barb found targets when nobody else did. He didn't know the shipping lanes because he couldn't open his game box and dig one up or do a Google search. In Thunder Below he goes into great detail explaining exactly what I've laid out. All things being equal, your number of contacts developed is proportional to the number of square miles of ocean surface you search in a day.

Not to criticize Fluckey, but iirc, he did run short of fuel on one patrol, and had to go home empty handed; the point being that roaming doesn't guarantee results, and may leave you low on fuel.

Because of hindsight, we might come up with better methods but they would be bogus, based on assumptions real sub skippers couldn't make.

Not sure what you mean here. None of the math I've used requires quantum mechanics, string theory, or black magic.


I wouldn't even say that I've come up with a new method. It's more along the lines of a guideline as to what one can expect from roaming, so one can decide if it is worthwhile. In any case, O'Kane did do stationary patrolling on at least one war patrol, so it is not a gamey-hindsight deal.

[Crannogman]

IIRC, there is no gain in efficiency below 10-11kts. So you won't gain any distance by going slower than that, but you will remain on station for more time. I guess the moral may be that efficiently patrolling a barrier longer than ~35nm will allow an increasingly large fraction of shipping to escape detection. However, a barrier shorter than 35nm allows you to spot anything transitting while moving slower and staying on station longer.

[ColonelSandersLite]

Originally Posted by TorpX

I consider this the beauty of my solution. I've used vector addition to subtract the target ships' speed. (This sort of thing was done with a 'maneuvering board' for various problems.) Working the problem this way, the ships in diagram 2 (if there are any) do not move. We need not make any assumptions of how many, or where the ships are located. All that need to be done is compare the respective areas cut out, of the moving sub and the stationary one.

Yes, except I see big flaw in the reasoning here. Suppose we are patrolling roughly E-W in a hypothetical N-S shipping lane. We add a northward cant to our patrol as in your examples. This absolutely does decrease the speed vector of any ship moving north, thus giving better odds of detection. Shipping lanes go both ways though and it also has an inverse effect on any ship heading south, decreasing odds of detection. Supposing that you're trying to find a target you have some prior knowledge of (a radio reported convoy for instance), reducing the speed vector would surely help to actually locate them. Otherwise, I suspect that it doesn't actually help due to the inverse nature this has on finding targets going the other way. Though it might due to reasons I can't quite fully articulate at the moment.

Oh, been meaning to tell you that your link to table.txt above is broken.

Originally Posted by TorpX

Not sure what you mean here. None of the math I've used requires quantum mechanics, string theory, or black magic.

I wouldn't even say that I've come up with a new method. It's more along the lines of a guideline as to what one can expect from roaming, so one can decide if it is worthwhile. In any case, O'Kane did do stationary patrolling on at least one war patrol, so it is not a gamey-hindsight deal.

Gotta agree here. The actual math on this sort of thing was probably worked out a *long* time ago due to how pertinent this thinking is to every navy on the planet. The thing is that I have no idea where to find the information. I can most likely find the answer for myself more easily than I can research it.

[Crannogman]

Originally Posted by ColonelSandersLite

Yes, except I see big flaw in the reasoning here. Suppose we are patrolling roughly E-W in a hypothetical N-S shipping lane. We add a northward cant to our patrol as in your examples. This absolutely does decrease the speed vector of any ship moving north, thus giving better odds of detection. Shipping lanes go both ways though and it also has an inverse effect on any ship heading south, decreasing odds of detection. Supposing that you're trying to find a target you have some prior knowledge of (a radio reported convoy for instance), reducing the speed vector would surely help to actually locate them. Otherwise, I suspect that it doesn't actually help due to the inverse nature this has on finding targets going the other way. Though it might due to reasons I can't quite fully articulate at the moment.

I think you're missing the fact that figures 1 and 2 show the same thing - the sub is going due East & West in both, and the ships are traveling North-South. The change is this: in Fig1, the "camera" is hovering above the same spot on the earth. In Fig2, the "camera" is hovering above the same ship. Everything else in unchanged. The sub only appears to have a northward cant because the frame of reference is moving south. To a ship moving south, the same sub would appear to have a southward cant.
It's an exercise in relativity. The reason to use the ship's frame of reference is to visually display the area searched by the sub and the areas to which the sub is blind. Fig3 demonstrates that, as the speed of the sub increases relative to the speed of the ship, the gaps in its search pattern shrink until there is noplace to hide

[Rockin Robbins]

Originally Posted by TorpX

Not to criticize Fluckey, but iirc, he did run short of fuel on one patrol, and had to go home empty handed; the point being that roaming doesn't guarantee results, and may leave you low on fuel.

You can do all the right things and still lose. It's fundamental game theory. Fluckey was by far the most successful captain of his era of the war. The reason was his strategy: search the maximum number of square miles per day consistent with your mission. Could he search and come up empty? Sure. Was his strategy totally valid? Look at his results. Look at the results of all other boats working in his time frame. What was the difference? Search methods.

Originally Posted by TorpX

Not sure what you mean here. None of the math I've used requires quantum mechanics, string theory, or black magic.

It requires your "beautiful" vector subtraction of an unknown target running an unknown speed at an unknown heading. That, sir, is black magic. Your theory is based on a fallacy: that you can know the course and speed of your enemy before you ever encounter him. The only valid strategies must assume that you don't know that information.

Again, searching is a numbers game. You're a card counter at a blackjack table. Are the odds in your favor? Sure. Are you going to win every time? Don't make me laugh. But does an example of failure invalidate card counting in blackjack? Not on your life. Play long enough and you win. Trotting out an example of Fluckey not finding anything is like that. He played long enough and cleaned out the house.

Also, if you're running RSRDC, it's fatally broken. The enemy shipping is coming no matter what. You can sit there sinking ships in a single choke point for the entire war and they just keep coming. In the quest for historical accuracy Lurker put the war in a stratjacket. He turned a living breathing war into a wind-up clock. In reality, when a target was sunk the Japanese rerouted shipping to avoid the submarine. This made covering ground as I've laid out an absolute necessity if you wanted to sink more than one or two targets.

As flawed as it is, the unmodified game traffic does a much better job of portraying the situation from the sub skipper's point of view. Actually some of the middle TMO versions were even better because they had more variety in their encounters.