Ever wonder how large should your search area be? (Full Version)

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wizard1073 -> Ever wonder how large should your search area be? (5/7/2015 11:25:53 PM)

I finally found and wanted to share from an authoritative source that tied search area, time, searcher speed, detection area, target speed, and cumulative probability of detection. That book is Naval Operational Analysis (2nd ed.) from the Naval Institute Press.

On page 136, the equation for a static search area (independent of target speed) is given as:

Pd(T) = 1 - exp(-w*v*T/A)

where

Pd = Cumulative Probability of Detection
w = effective sensor sweep width
v = search vehicle speed
T = total time spent searching
A = total area to be searched

The term w*v*T/A is called the Coverage Factor. Generally, you must cover a given area more than once to improve Pd to something useful, and this factor tells you how many times you covered the total amount of area.

So, for a search vehicle traveling 180 knots, with a effective sweep width of 3 nmi, and a 80 nmi square box (6400 nmi^2), it will take 19 hours to reach a cumulative Probability of Detection of 0.8. If you have only 6 hours and a Pd=0.9 requirement, then you will need to use 4 searchers in four smaller boxes of 40 by 40 (Pd=0.87).

A special case of this is searching for a moving target that starts with an known area of uncertainty. If the area is circular with initial radius R and increases in all directions with the target's speed u, then the Coverage Factor now looks like this:

C = w*v*T / (pi*R*(R+u*T))

and

Pd(T) = 1 - exp(-C)

So, for a search vehicle traveling 180 knots, with a effective sweep width of 3 nmi, and a target with a starting uncertainty of 5 nmi and moving at 20 knots, we reach a Probability of Detection of 0.81 after 5 hours of searching, after which we should probably give up. Why?

For the moving target, the maximum Probability of Detection, assuming you had infinite time to search, is given by

Pd(T -> infinity) = 1 - exp(-w*v/(pi*R*u))

In our example, that is 0.82. You spend 17 hours (after the initial 5) at Pd=0.81 before you reach Pd=0.82. Is it worth it?

I hope this helps you intelligently design your search areas and respond to intermittent detections better. You can use this for a variety of missions, from submarine detection to aircraft detection. If the developers add Search & Rescue, then you can use it for that as well.



SeaQueen -> RE: Ever wonder how large should your search area be? (5/8/2015 5:42:35 PM)

If you'd like to see these sorts of search theoretical calculations taken up a notch, check out Search and Screening by Koopman, or some of the papers you can find online by a mathematician named Alan Washburn.



wizard1073 -> RE: Ever wonder how large should your search area be? (5/9/2015 3:23:09 PM)

I'm happy to report that the Koopman reference is the most widely used source cited by the OA book, covering chapters 5-8 and 10. I'm still digging into those for items that are of practical use to players of CMANO (such as optimal placement of the screen). I'm looking at Washburn's page on the Naval Postgraduate School right now, and I am suspicious he may have contributed to the OA book. He has a lot of downloadable papers and tools (including excel spreadsheets for computing various combat outcomes).

Thanks for the references! I will take a closer look at all of these.



magi -> RE: Ever wonder how large should your search area be? (5/9/2015 4:22:03 PM)

I am not smart enough to use this...... However it seems to me that if the target changed depth.. direction and speed.. Depth temperature salinity etc. etc.… It would have a dynamic effect on all the variables… and this is assuming that you know the target speed… Unless that is just an outside variable… And it would be more complicated if you were trying to provide coverage for a moving group… but I am certain that tactical officers would have equations they would work with… Modern warfare really is for eggheads…
In the game… I establish the patrol area size visually from experience with the units that are available to get the coverage I want.....



magi -> RE: Ever wonder how large should your search area be? (5/9/2015 5:07:03 PM)

The more I think about this… I find the more interesting it is… If we had programmed formulas that we could enter variables… This would work for a guy like me…



wizard1073 -> RE: Ever wonder how large should your search area be? (5/10/2015 7:48:14 PM)

@magi, I think you can use the equations offline with a simple calculator. Consider the following problem:

I need to establish a carrier operating area. In order to do that, I need to clear the area of enemy submarines. The carrier will arrive in 12 hours, so I have that much time to clear it. Let's say that the admiral on the carrier requires that I detect any subs in that area to a 90% level (meaning that there is only a 10% chance that an enemy sub is there undetected). I am using P-3's to do the searching, and they will be using a combination of sonobuoys and magnetic anomaly detection. Only one aircraft is available at a time.

So now I know area (A=40*40 nmi^2), search velocity (v=180 nmi/hr), search time (T=12 hr), and effective detection range against subs (w=3 nmi--I made an assumption here). Can I make the admiral happy? Yes, and with a Pd of 0.98. He will accept the risk of moving the carrier into the OPAREA.

With a little algebra, you can re-arrange the equation to answer whatever you need. In my case, I made a spreadsheet so that I can fiddle with the numbers until I get the answer I need.

If someone actually included this kind of calculation into a scenario via LUA script, the enemy commander could more dynamically use operating areas and perhaps survive better.



SeaQueen -> RE: Ever wonder how large should your search area be? (5/11/2015 12:12:34 PM)

I just play any Naval games with a spreadsheet opened. It's just too useful given the number of calculations one ought to be making to play the game well.



SeaQueen -> RE: Ever wonder how large should your search area be? (5/11/2015 2:13:31 PM)

A probability of detecting a submarine of 0.98 is not the same thing as saying you're 90% confident that there are no submarines in the area. To get to the next step you need to apply Bayes Theorem, and calculate the probability that there are no submarines in the box given that you've searched it to a 0.98 probability of detecting a submarine in the box. You'd probably need to also make some initial assumption about the distribution of the number of submarines in the area. Therein lies the uncertainty. :-)

quote:

ORIGINAL: wizard1073

I need to establish a carrier operating area. In order to do that, I need to clear the area of enemy submarines. The carrier will arrive in 12 hours, so I have that much time to clear it. Let's say that the admiral on the carrier requires that I detect any subs in that area to a 90% level (meaning that there is only a 10% chance that an enemy sub is there undetected). I am using P-3's to do the searching, and they will be using a combination of sonobuoys and magnetic anomaly detection. Only one aircraft is available at a time.

So now I know area (A=40*40 nmi^2), search velocity (v=180 nmi/hr), search time (T=12 hr), and effective detection range against subs (w=3 nmi--I made an assumption here). Can I make the admiral happy? Yes, and with a Pd of 0.98. He will accept the risk of moving the carrier into the OPAREA.




wizard1073 -> RE: Ever wonder how large should your search area be? (5/13/2015 9:58:07 PM)

@SeaQueen, I've had a nagging suspicion about that statement of mine.

Submarines usually operate in exclusive OPAREAs for safety reasons, so it is generally safe to assume that there is only one in *their* box. If my box overlaps two or more of their boxes, then there could be problems with that assumption. (If their boxes do not have a significant buffer between them, as perhaps with a barrier search by the enemy, then this is likely to be the case. Otherwise, the assumption may reasonably hold.)

So a cumulative probability of detection of *at least one* target *if one is present* is what I think is the correct statement I should have made. That means that the cumulative probability of not detecting *at least one* target *if one is present* is one minus the cumulative Pd.

What you appear to be talking about is hypothesis testing. The null hypothesis is that there are no targets in the area. The alternative hypothesis is that there is a target and I can detect it. Then we can have type 1 errors (false positive--I say there are targets when there are not) and type 2 errors (false negative--I say there are no targets when there are).

So the Bayesian approach is to test for those errors:
1) I didn't detect anything, and I have a certain cumulative Pd. Is this a false negative?
2) I did detect something, and I have a certain cumulative Pd. Is this a false positive?

The outcome of these tests is some measure of confidence in one or the other hypotheses.

Am I on the right track? I'll also grab a statistician friend of mine and have a deeper look.

I've been thinking about putting in biologics into an ASW scenario in order to stress this concept a lot more. I have only been talking about detection, and not the next step in the kill chain: classification. Classification by ordinance (a frequently used method) is expensive and risky (from a counter-detection perspective).



SeaQueen -> RE: Ever wonder how large should your search area be? (5/14/2015 2:25:28 PM)

I can't comment on what submarines actually do, but for the most part you're thinking about the math correctly.

The Koopman random search equation generates the cumulative probability of detecting a single target at least once as a function of time. It's really just the cumulative exponential distribution. Essentially what it says is that you're no more likely to detect a target any place any more than any other place in the area. You could think of submarine detections the same way a physicist thinks of sub-atomic particles impinging on a detector (Poisson process).

You could think of it as the most conservative possible assumption, as opposed to the other more optimistic alternative for area search, which is "mowing the lawn" in which case your probability of detecting a submarine eventually hits one. With random search, you never actually reach 1, only approach it asymptotically.


quote:

ORIGINAL: wizard1073
...

Am I on the right track? I'll also grab a statistician friend of mine and have a deeper look.

...




wizard1073 -> RE: Ever wonder how large should your search area be? (5/20/2015 11:40:05 PM)

Got it. Thanks SeaQueen!

BTW, where did you pick this stuff up? And what else do you use spreadsheets for while playing?



SeaQueen -> RE: Ever wonder how large should your search area be? (5/21/2015 3:32:33 PM)

I learned about it from my job, although in fairness I'd already started doing some of these sorts of calculations in war games I'd always played as a hobby.

I use spreadsheets in the game for all kinds of things, from generating random numbers to calculating weapons parameters given different assumptions. It depends on the scenario, and what I'm trying to do. My ASW spreadsheet has all kinds of useful planning formulas, like barrier search equations and area search equations. I have something that calculates the "no escape" range of a weapon given various kinematic parameters. It's nothing grand, but it helps me make better decisions quickly to have a bunch of things "canned" where I can just plug in the numbers, particularly early in the game when I'm trying to formulate a plan before I hit the "go button."

quote:

BTW, where did you pick this stuff up? And what else do you use spreadsheets for while playing?



Rudd -> RE: Ever wonder how large should your search area be? (5/26/2015 12:58:48 AM)


quote:

ORIGINAL: SeaQueen

I learned about it from my job, although in fairness I'd already started doing some of these sorts of calculations in war games I'd always played as a hobby.

I use spreadsheets in the game for all kinds of things, from generating random numbers to calculating weapons parameters given different assumptions. It depends on the scenario, and what I'm trying to do. My ASW spreadsheet has all kinds of useful planning formulas, like barrier search equations and area search equations. I have something that calculates the "no escape" range of a weapon given various kinematic parameters. It's nothing grand, but it helps me make better decisions quickly to have a bunch of things "canned" where I can just plug in the numbers, particularly early in the game when I'm trying to formulate a plan before I hit the "go button."

quote:

BTW, where did you pick this stuff up? And what else do you use spreadsheets for while playing?


I'd be interested in your ASW and any other CMANO related spreadsheets, if you'd like to share/post.



SeaQueen -> RE: Ever wonder how large should your search area be? (5/27/2015 4:52:25 PM)

In general, I typically avoid sharing war gaming related materials on account of my work. I doubt I'd get in trouble for any of it, but one can never be too careful.

quote:

ORIGINAL: Rudd
I'd be interested in your ASW and any other CMANO related spreadsheets, if you'd like to share/post.




wizard1073 -> RE: Ever wonder how large should your search area be? (5/27/2015 6:28:47 PM)

The more it can be sourced to openly available works, the easier it is. @Rudd, that is part of what I am doing by going through the book and interpreting it for those who don't have it or don't have the background to understand it directly. Once the equations and algorithms are described on the forum, it shouldn't take long for an enterprising spreadsheet expert to put together some general tools. So I guess this is an open invitation.... [:D]



Zaslon -> RE: Ever wonder how large should your search area be? (6/15/2015 2:09:52 PM)

Using mathematical formulas in order to establish patterns of search and other stuff? Amazing guys [:D]



charlee22009 -> RE: Ever wonder how large should your search area be? (6/30/2015 2:04:00 AM)

This is a fantastic discussion... although the math makes my head spin.

-----Have you all seen "coordinated asw part 1" on youtube? ( I can't post links :( )

Search for it on youtube. These are old but fantastic videos and the basic principles, asaik, haven't changed much.

That whole youtube channel has good stuff...

Also watch

-----"ASW golbin on the doorstep part 1"

These should be official links in the cmano faq imho. (Giving a visceral feeling to a 2d diplay and the math involved...)



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