# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Running Fix vs Estimated Position -- Monte Carlo assessment**

**From:**Joe Schultz

**Date:**2010 Jan 16, 11:24 -0800

waldendand, here goes.

We disagreed that boat course matters. I'll approach the issue from the point of view of uncertainty in the DR position which, using your terminology, I claim will affect the uncertainty in the R-FIX and EP.

The attached Uncert_DR.pdf is my solution for the uncertainty in the DR position, starting from a FIX (simplified to the first quadrant only). If you can do a Monte Carlo simulation then I'm sure you can do a partial derivative. I simply show the equations and the answers, using software called Mathcad. If you play with this then you quickly learn a few things:

1) Boat course matters.

2) You could clock the time interval with a sandglass if you want.

3) A good speedometer is just as important as a good compass.

There are a few rules concerning this method, which is called (in some circles) "propagation of uncertainty."

1) function must be linear.

2) variables must be independent.

3) uncertainties must be independent and have the same odds.

4) uncertainties behave like standard deviations.

References:

1) Kline, S.J. and McClintock, F.A. "Describing Uncertainties in Single-Sample Experiments". Mechanical Engineering 75: 3-8, January 1953. This is the seminal article.

2) Beckwith, T.G., et. al. "Mechanical Measurements, 5th ed." ISBN 0-201-56947-7. The final (hopefully) book I taught from, in an instrumentation course for first semester seniors in mechanical engineering.

Now to integrate this into your example. I really struggled with your terminology, so I changed the plot a bit to match what we American "Bowditchers" do. Hopefully I interpreted your terminology correctly. The plot is the the attached Uncert_R_fix.gif, and was done by hand. My references are Dutton's, Bowditch, and various US Navy study guides.

My thinking, when changing your plot to something I understand:

1) Your true starting and final positions can be thought of as the output of a perfect GPS. Accordingly, your true course & speed are Course Made Good (CMG) and Speed Made Good (SMG), as output by the GPS unit (or measured from the plot). Your uncertainties (+/- 5deg and +/- 2kts) should be applied to the boat course and speed, as read on the boat compass and speedometer.

2) Your estimated course & speed are the the course and speed that are listed on my DR track. In my world the DR track is reset to a good fix - this matches your world (your true starting position). Additionally, this course & speed is what you see on the boat compass and speedometer. They do have uncertainties, which is why I moved your uncertainties here. I also halved your uncertainties in an attempt to more closely replicate the real boating world.

3) I've taken the liberty of saying the first LOP was shot at time 0000 and the second LOP was shot at time 0100.

4) If the GPS is perfect (as I said it was) then the LOPs have to go through the GPS fixes as long as the LOPs are perfect shots. I'll assume they are perfect shots, as you did.

5) Your EP is a probable position in my world, sometimes called a most probable position. It is an arbitrary decision to say that (your) EP is the closest distance to the DR position, and the schoolbooks do a terrible job of explaining why, in my opinion. I'll go with it, understanding it's an arbitrary decision.

In my plot, the right track is my interpretation of your example and hopefully you're OK with it. The left track shows what happens if the steered course was different. The only way to get the tracks "apples to apples" is to say the set/drift is constant. I used the x and y components of distance run uncertainties as boundaries for the DR uncertainty.

At any rate, I think you'll see that boat course matters. We can't say we're at the circle or square. We say we're in the "splotch" bounded by the uncertainty. The "splotch" is along the LOP because we said we took perfect shots. Changing boat course changes the size and location of the "splotch."

Do I do this in real life navigation? Not a chance - I'm not a navigation slave. But I will pull out a pencil and an old envelope, if needed to convince someone to calibrate their speedometer.

If I was doing your exercise (and I won't - it's work) then I'd use a range of set/drift combinations (and uncertainties) with a range of steered courses, and constant uncertainties in course, speed, and time run. Call the resulting EP (my language) the final true position (your language), which will also have an uncertainty. Then, by playing with set/drift uncertainties, I'd learn how accurately I need to estimate set/drift in order to get a reasonable uncertainty in (your) final true position. Then I'd have a basis for stating (your) EP should be the closest distance to the LOP. And, of course, this assumes we shoot perfect LOPs (unless we also want to include those uncertainties).

None of this is criticism of your effort, which I applaud.

Joe

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