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Re: The Darn Old Cocked Hat - the sequel 1
From: Hanno Ix
Date: 2013 Mar 15, 15:36 -0700
From: Hanno Ix
Date: 2013 Mar 15, 15:36 -0700
Richard:
I do not know of much of the systematic errors in CelNav.
Personally, I am trying to understand the random errors and their minimization in the DOCH.
Of course, your remarks will be of importance for those who include systematic errors.
Also, you said this:
The important
point to remember is that the relative effects
of normal random errors in any one direction are proportional to their squares.
of normal random errors in any one direction are proportional to their squares.
Am I correct when I understand you this way:
Errors of angular measurement affect azimuth errors
in the sqaure of the distance.
I believe that is right.
John:
The focus of your remark is right on.
BTW: what you are showing in your (beautiful) diagram is not the probability of
the absolute error, i.e of the distance to TL which is what I personally am trying to grasp.
The absolute error is always positive, has only one coordinate and is measured in units of distance.
You are showing the probability-density of the positions. Amongst
the differences are the
nr of coordinates, two, that positions have.
I asked y'all before:
May I please urge you to make in Excel two cols of Gaussian random numers,
mean = 0, σ = 1 and calculate in the third col the square root of the sum
of squares of the first two.
That sqrt is called RMS error. Also make histograms. You might be surprised.
To do this takes less that 10 min!
From: Richard B. Langley <lang@unb.ca>
To: hannoix@att.net
Sent: Friday, March 15, 2013 10:49 AM
Subject: [NavList] Re: The Darn Old Cocked Hat - the sequel 1
To: hannoix@att.net
Sent: Friday, March 15, 2013 10:49 AM
Subject: [NavList] Re: The Darn Old Cocked Hat - the sequel 1
The two distributions may not have the same standard deviation and are not necessarily independent; there may be some correlation between them. It might be worth having a look at Appendix Q of Bowditch (Navigational Errors). Not sure which editions Appendix Q appeared in -- it is at least in the 1977 edition. The following two paragraphs are particularly germane:
On 2013-03-15, at 10:17 AM, John Karl wrote:
"The value of the most probable position determined as suggested above depends
on the degree to which the various errors are in fact normal, and the accuracy with
the likely error of each is established. From a practical standpoint, the second
is largely a matter of judgment based upon experience. It might seem that inter-
pretation of results and establishment of most probable position is a matter of judgment
anyway, and that the procedure outlined above is not needed. If a person will follow
this procedure while gaining experience, and evaluate his results, the judgment he
develops should be more reliable than if developed without benefit of a knowledge of
the principles involved. You said this:
"Systematic errors are treated differently. Generally, an attempt is made to discover
the errors and eliminate them or compensate for them. In the case of a position deter-
mined by three or more lines of position resulting from readings with constant error,
the error might be eliminated by finding and applying that correction (including sign)
which will bring all lines through a common point."
on the degree to which the various errors are in fact normal, and the accuracy with
the likely error of each is established. From a practical standpoint, the second
is largely a matter of judgment based upon experience. It might seem that inter-
pretation of results and establishment of most probable position is a matter of judgment
anyway, and that the procedure outlined above is not needed. If a person will follow
this procedure while gaining experience, and evaluate his results, the judgment he
develops should be more reliable than if developed without benefit of a knowledge of
the principles involved. You said this:
"Systematic errors are treated differently. Generally, an attempt is made to discover
the errors and eliminate them or compensate for them. In the case of a position deter-
mined by three or more lines of position resulting from readings with constant error,
the error might be eliminated by finding and applying that correction (including sign)
which will bring all lines through a common point."
The following item, although specific to GPS, might
also be of general interest:
-- Richard Langley
On 2013-03-15, at 10:17 AM, John Karl wrote:
Hanno, et al.,
We don't need numerical simulation to see that the probability around a ship's location, which is determined by two independent normal distributions of lat(x) & lon(y), is another normal distribution in the distance from the ship, r:
Prob = exp(x/s)*2 exp(y/s)*2 = exp[(x/s)*2 + (y/s)*2] = exp(r/s)*2
which, of course, is maximum at r = 0. Plot is attached with ship at 15,15.
JK
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| Richard B. Langley E-mail: lang---ca |
| Geodetic Research Laboratory Web: http://www.unb.ca/GGE/ |
| Dept. of Geodesy and Geomatics Engineering Phone: +1 506 453-5142 |
| University of New Brunswick Fax: +1 506 453-4943 |
| Fredericton, N.B., Canada E3B 5A3 |
| Fredericton? Where's that? See: http://www.fredericton.ca/ |
-----------------------------------------------------------------------------
| Richard B. Langley E-mail: lang---ca |
| Geodetic Research Laboratory Web: http://www.unb.ca/GGE/ |
| Dept. of Geodesy and Geomatics Engineering Phone: +1 506 453-5142 |
| University of New Brunswick Fax: +1 506 453-4943 |
| Fredericton, N.B., Canada E3B 5A3 |
| Fredericton? Where's that? See: http://www.fredericton.ca/ |
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