The mathematics of roving searches

[TorpX]

Originally Posted by Rockin Robbins

It is also arbitrary. Saying that an area which is searched will not be counted because we don't want it is just arbitrary. It is searched, therefore it contributes the the result.

You have to make some assumptions to solve the problem. Otherwise all you can do is speculate and theorize. I freely admit that traffic will not always be traveling alone a single axis, but frequently much of it will be. If you want to calculate a comprehensive figure for X% going N-S, and Y% going E-W, you can break the problem down in cases and do that. I am assuming a best case situation (for the sub), where they do know the axis of traffic. It is certainly possible to have the traffic on a different axis, not perpendicular, but I wanted to show what the best possible results would be.

Every area was counted, I just didn't count any area twice.

Also, the wedge areas are not researched areas, they are as if they had not been searched at all the second time through because there is a time element here as well. Since you are spending double the amount of time in the wedge and enemy traffic is moving randomly within constraints of traffic patterns there is actually a higher percentage chance of obtaining a target in the wedge than outside the wedge.

The wedge areas overlap and you must account for this or the results are not correct. You are ignoring the fact that in diagram 2, I have subtracted the vector velocity of the target from both target and sub (I.e. different frame of reference). In diagram 2 the target does not move. To say that simply moving fast without regard to this fact, would be like expecting if you went very fast in a small circle, you would still get lots of contacts. By the same token, moving fast parallel to shipping will not avail you anything.

It is not only your movement that is important, but the movement of traffic into your search area.

The expressions I have posted use the target's speed and it is expressly mentioned in the text. In fact, that is the whole basis of the computation. If you look over the results in the table, you will see, that increased sub speed does help with the chances, just not as much as some might expect. Also, a 3 kn. sub vs. a 6 kn. ship will give the same results as a 6 kn. sub and a 12 kn. ship. Iow, it is the ratio of sub speed to target speed that is important.

Assuming homogeneous distribution of shipping, your number of targets developed is directly proportional to the length of travel per day times the diameter of your search distance circle, plus the area of that search circle, half of which is added to your search area at the beginning of the day and half to the end of the day. That makes your search area a long hot dog. The longer your travel during the day the more contacts you will develop, nearly proportional to your speed.

I don't know where you get that idea. It just isn't true. I've never seen any such thing in a sub ops document, and furthermore, O'Kane downplayed the idea of searching this way. [For those who still doubt, read CLEAR THE BRIDGE.] Of course, he didn't lay out the geometry, but from what he said, it is plain he understood the concept very well.

The reason the search bands are not "hot dogs" is that if a target was in the first circle, it would have already been detected before we started. We want those targets that would have been detected from time '0' to time 'x', x being the time for the sub to make one complete cycle. The same applies to our stationary 'control' sub.


***
If the game was "Stealth Blockage Runner", instead of Silent Hunter, and we were playing a merchant skipper, diagram 2 is what it would look like using the 'God's eye view' option over a period of time, with our ship locked in the center of our map. If there were spaces between the sub's search bands (as there are in the diagram 2), we might be able to get through. It would depend on exactly where we attempted our transit. If there were no spaces, our cause would be hopeless. If we could increase speed, the diagram would change, with the space between the segments opening up and our chances improving.




ColonelSandersLite,

It looks like you anticipated my question in part 2.

It is certainly worthwhile to consider larger search zones and different parameters. I thought 40 nm was a good 'typical' idea of a sea-lane. If there were a narrow choke point that would make things easier, or one might try a larger zone. At some size, I think it gets ridiculous, though. Saying we want to search a 120 nm zone is like saying we haven't the slightest idea where the enemy is.

[ColonelSandersLite]

Yeah, I did make a major mistake above. The boundary that I had previously called the guaranteed detection line is actually the 50% line. The line I have previously called the 50% line is actually the 25% line. Notice that this would be defined by a logarithmic function. The 100% line would be where the north/south @11knts contact moves just under 26nm except for the fact that radar coverage at both ends of the patrol zone extend past this. Apparently, I need to put more thought into this problem.

[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.