The mathematics of roving searches

[TorpX]

It is commonly believed that cruising around at good speed is essential to searching on patrol. It seems to have become accepted wisdom, that this not only produces better chances for finding a contact, but that the odds are dramatically improved by doing so.

Since this topic comes up periodically, I decided to try to figure out a proper solution to the question of how much continuous cruising helps ones chances of finding a contact.

First off, we need to make some general assumptions about the conditions. I am going to assume that we know the direction of the prevailing shipping traffic, and that the odds of a target transiting through any portion of our search area is the same as for any other area of equal size. That is, that shipping is equally distributed (which it would be, at least as far as we know). I consider it fairly obvious that the best "search pattern" is one that cuts across the greatest number of potential target routes in a given length of time; that is, moving perpendicular to traffic. This means that fancy box or 'x' patterns are less effective, and moving parallel to the direction of traffic is just a waste of fuel.

In the diagram below, we have our sub moving east to west, and enemy traffic moving north to south. We could change the directions any number of ways; the important thing is that our sub is moving perpendicular to shipping traffic. The black lines at either side is the search limits we have marked off for our sub. It will go east to west until it reaches the west limit, then reverse course and go east until it reaches the east limit, and so on. The gray circle is the sub's detection bubble. Any target that comes within that space will be detected. The red arrows represent potential target ships. At first blush, we might be tempted to think that all, or most, of these targets would be detected, but this isn't the case. If we were to stay in place, we would expect to find #6 and #7. Moving to the west, we might hope to find #1 to #7, but it is possible, even likely, that #1, #2 and #3 would get by before we get far enough west, and #7 and the rest would be safe because we will be over to the west as they go by. So the question is how many targets are we likely to detect, compared to the same situation, where we are sitting still?

We might try to plot out the route and timing of every potential target, dividing the sea lane into 1 nm strips, but life is short and there must be a better way to go about this.

If we adopt a different frame of reference, the problem is soluble. Below, I have diagramed the movement of our sub, using an enemy ship as a frame of reference, instead of a fixed earth frame. The sub appears to move in a zig-zag pattern, though it is really still going east-west. The sub is starting at point A moving to B, reversing course until C, and going east once more to D. At D the sub is again in the middle of it's search zone and the cycle will repeat. I'll explain the reason this is important later. Essentially, we have used vector addition to subtract the target speed, from both sub and targets.

Note that, using this frame of reference, the target ships do not move. Any target that falls within the searched area, will be located, any that are outside will not be. I call the areas computed in this frame, pseudo-areas. I shaded the first segment, AB, manila so we can see that it is a rectangular segment, with a missing semi-circle at one end, and a extra semi-circle at the other. The remaining segments are light green. The concave area at A is there because we only want the newly searched area after the sub passes A. Since this geometrically is equivalent to the semi-circle past B, we can see that the area of this segment is equal to the length of A to B times the width of the segment (2r). Also, it is easy to conclude that the segment CD is the same length as AB and the segment BC is twice AB. So if we multiply the area of segment AB by 4 we should have our total pseudo-area searched by our moving sub.

But wait a minute; there are parts of this area that would be counted twice if we do that. See the wedge shaped area in brown (near C). There is also an equivalent area near B. For accurate results, we mush subtract the areas there, lest we overestimate our search ability. This stems from the fact that when we reverse course, we will be searching an area that we already searched minutes before, so we have to expect some loss of efficiency there. This is why we must be careful to have the sub move a complete cycle; otherwise, we could not properly account for the overlapping sections.

To obtain a ratio for the probability of finding a contact by a moving sub relative to that by a stationary sub, we need to know r, the detection radius of our sub, w, the distance from the center of the search zone to either edge, and b. b is the angle between the horizontal and the line AB. The angle ABC is 2b, as is angle BCD. It depends on the relative speeds of the target and sub.

So, we seek to compute 3 quantities:

1. the roving 'pseudo-area' (i.e. the area of the 3 segments we diagramed).
2. the 'wedge-shaped pseudo-area' (to account for the overlapping).
3. the comparable 'pseudo-area' for a stationary sub.

Note that, for a stationary sub, the graphical representation would be one large segment of the same width, 2r, going from A to D.

For number 1. we have:

Roving 'pseudo-area' = 8*r*w / cos b

where

r is the detection radius of the sub
w is the width of the search zone from the center to one edge (i.e. half of the full width)
b is the angle Arctan (Vt / Vu)
Vt is the target speed
Vu is the sub speed


For 2, the 'wedge area', we have:

[I won't go into the details of finding this. Extra credit for those who can figure this out. Note that the faster the sub goes back and forth, the greater the overlap there will be. For typical values, it shouldn't be too large.]

'wedge pseudo-area' = 2r^2 * [tan(90-b) - pi*(90-b)/180]

For the static sub computation we can calculate the time interval for the moving sub to complete one cycle, and then substitute that value in the equation for the area, to get our final result.

The time t = 4w / Vu, and the pseudo area = 2r * t * Vt.

This gives us:

Stationary 'pseudo-area' = 8*r*w * (Vt/Vu)

For a concrete example, I will use a value of 20 nm for w, 10 nm for r (this is the max. rendering distance for SH4), a Vt of 8 kn. and a Vu of 9 kn.

These values give us 41.63 degrees for b.

The Base 'pseudo-area' is 2140.7.

The 'wedge psuedo area' is 56.2.

The Stationary 'pseudo-area' is 1422.2.

Efficiency Ratio = (Roving PA - Wedge PA) / Stationary PA

So, for these numbers we have a ratio of 1.466.


In other words we can expect to obtain 46.7% more contacts under these circumstances.


How valuable is a 46% 'search bonus' for a moving sub? Well, a lot depends on how much fuel we have to play with. If we have plenty to spare, there would be little reason not to use some for roving. If, on the other hand, our supply is tight, and roving would mean cutting our patrol time in half, it isn't very good at all.



Tomorrow, I will post results for various sub and ship speeds, and add a few thoughts.

-TorpX

[Ludwig van Hursh]

*brain melts*

[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. 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. It is not only your movement that is important, but the movement of traffic into your search area. It is not coherent as you have assumed and they all don't line up side to side in a single line across your search area as you have assumed. They are distributed randomly throughout the area and are moving in randomized directions, yes, constrained by limits but random within those limits.

So while your math is admirable, your premises are questionable. The area searched each day is actually a rectangle based on the width of area searched and the end points of travel PLUS a circle of search radius around the sub. 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 say nearly because of what you're going to say next, which is that if your were entirely stationary, shipping would still enter your area because of their velocity and it would be as if you were moving against the velocity of their travel at their speed. That makes contacts found not exactly proportional to distance traveled in a day. It is conceivable if you park at a choke point you could have enough business that it wouldn't matter if you could develop more business because it takes time to deal with each contact. But ignoring random effects and choke points, your number of contacts is nearly proportional to the length of your track during each day.

Now that is contacts per day. You want best results for a cruise. Best fuel economy is 11 knots. WFO is 21 knots, which would double your results. But at WFO your fuel consumption is many times the double search coverage, therefore your patrol contacts actually drop off the proverbial cliff. Therefore, searching on the surface so you can use radar at 11 knots is by far the best way to search.

Yes, we could quantify that more precisely. We could toss the actually variable density of traffic in there to modify our path to more productive areas, and I hope we're already doing that. But the numerical results won't change our behavior at all. Our results and the calculated results will always be different from reality because our assumptions are wrong. We know that but not knowing the actual distribution of shipping, we have to generalize. We are not in total control of our destiny. We are card counters at a blackjack table, numbers on our side but knowing that we can still lose. You can do everything right and still lose. But you can only win if you do the right things......unless......

[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...

[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.

[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 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.

[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.

[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

[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.