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Subject: Re: Accuracy of position (sextant error simulation)
From: George Huxtable (george@XXX.XXX)
Date: Thu Oct 28 1999 - 10:50:27 EDT
Oh dear. I do seem to have upset Jim Manzari (mailing, 25 Oct 99).
I'll try to explain further, in language that's intended to cast light
rather that to generate heat. In view of the emphasis in this mailing list
on navigational rather than statistical matters, I will keep the statistics
content as brief as I can.
Let me first deal with a personal comment. Jim says-
'You may have a personal prejudice against the use of statistics to
evaluate the scope of these errors, but your personal feelings won't change
the fact that the errors are probabilistic."
Well, after 40 years as a physicist in research labs, I am no stranger to
statistical techniques, and have no prejudice against them. Although I am
not a specialist in statistical method, it has been a vital tool in my
trade, throughout my career. And I have no objection to treating errors in
observations as if they are probabilistic, with a Gaussian distribution, as
Jim does. This may or may not exactly represent reality (one must consider
the facts of each case) but it's usually a fair working assumption.
What I object to, and strongly, is misuse of statistics. One has to be very
careful, in drawing conclusions from any analysis, to ensure that they are
justified by the information that has been fed in.
SEXTANT TILT AND STATISTICS.
In particular, I question Jim's conclusions, when he states-
"Misalignment of the sextant with the true vertical is THE major error to guard
against!"
and again-
"... error introduced by misalignment of sextant vertical axis with respect
to the true local vertical, this angle called "phi".
This is, by far, the largest single error and completely dominates all other
errors, barring a blunder by the navigator!!"
and later-
'In this simulation the maximum error was -32.7 arc-minutes. 98% of this
error was contributed by an extreme error in vertical alignment of the
sextant."
What I will try to show is that these "conclusions" can not be drawn from
Jim's analysis. Whether or not those statements happen to be true anyway is
a matter which I am happy to leave to others.
The analysis was made, not of a set of data taken from real observations,
but of 1000 invented sets of values, created according to a set of rules
which had been formulated by Jim Manzari. This can be a useful technique
for some purposes as long as one recognises what is being done, and its
limitations.
Jim has presumed that, at the vital moment of taking a sight, the
observer's sextant is not truly vertical. It can have a range of values, he
presumes, equally scattered on either side of the vertical. In his first
mailing, Jim did not tell us what the amount of this scatter was. That was
why I had to concentrate on the worst-case value, which was the only one
for which any details were given, in an attempt to work backwards from his
results to guess how much scatter had been assumed. Now we have been
informed-
"To model this error, I assumed a normal or Gaussian distribution for the
mis-alignment angle with a mean or average value of zero AND A VARIANCE OF
5-DEGREES."
I have capitalised the important bit to draw attention to it.
To be rather pedantic, the units of this variance should be
(degrees-squared) rather than degrees. No matter, it corresponds to a
standard deviation of root-5 degrees, or 2.24 degrees. To put it at its
simplest, this implies that the sextant tilt at the moment of taking the
sight lies, for half the observations, within 1.5 degrees either side of
the vertical, and obviously the other half lie outside those limits. This
is the Manzari Assumption.
For a tilt of only 1.5 degrees the resulting error in the measured altitude
would be significant, but not severe; about 1 minute if the altitude was 45
degrees. So we can say that only half the observations would be out by more
than a minute or so from this cause. However, the altitude error grows as
the square of the tilt, so it's clear that in cases when the tilt reaches 2
or even 3 standard deviations, (which will occasionally happen, though less
and less often for the larger values of tilt) the resulting error could
become devastatingly large, as Jim has found.
But what is this analysis based on? It's based entirely on Jim's "assumed
variance of 5 degrees" for sextant tilt? And what's THAT based on? Well,
Jim doesn't tell us. Where does his assumption come from? Is it based on
reality, or did it come out of Manzari's head? We have no way of knowing.
It's a nice round number, yes. Perhaps it just seemed about right. If I had
made the same analysis, but selected a different value for variance of
tilt, I would have reached different results, and no doubt drawn different
conclusions.
So, Jim's conclusions have as much validity as his assumptions, no more, no
less. They have been dignified by putting them through a process of
statistical analysis, but that has not added credibility. I don't even
claim that the Manzari Assumption is wrong; perhaps he has happened to hit
on the right value, but who can tell? Can he convince us?
IS SEXTANT TILT AN IMPORTANT ERROR, NEEDING SEPARATE CONSIDERATION?
This is a new topic that hasn't been touched on before in this discussion.
Now, we can put statistics to one side for a while, and think hard about
the physical reality of just what a navigator does when taking a sight - a
star sight, for example.
What he DOESN'T do is to set his sextant up so its plane is as nearly
vertical as he can get it, then says to himself "OK, that'll do", and then
measures the altitude of the star above the horizon.
We all know what he does; he swings the sextant about, from one side of the
vertical to the other, so that the star image makes an arc in his view, and
waits to record the sextant reading when this arc just kisses the horizon,
as precisely as he is able to observe it. Let's call this process a "skim".
If an observer was unable to complete such a skim then he would not regard
it as a valid observation.
Ideally, in such a skim the arc of the star and the line of the horizon
should always touch as a tangent, but even in calm conditions there will
always be some error, even if it's a small one. The error could be positive
(the sextant reading set too great) in which case the star image at its
lowest point will be just below the line of the horizon. Or, equally, it
could be negative, in which case the star will never quite reach the
horizon.
In the first case (star below horizon), the star's image must actually
cross the line of the horizon at two points, corresponding to equal tilt
left and right, Perhaps one might consider then that there was some sort of
"tilt error" in making the skim; that at that tilt angle the star was
viewed as just on the horizon, but the reading would have to be somehow
corrected to get to the "true" value. I think this attitude would be wrong.
How would one deal with the opposite error, which is just as likely, when
the star image failed to reach as low as the horizon? In that case, there's
no angle at which the star is on the horizon, so what would be the sextant
tilt error in that case?
No, what the observer does, when making a skim, is that he scans over a
whole range of values of tilt to find, not a value of tilt at a particular
moment, nor a deliberate attempt to discover when the tilt is zero, but to
observe the smallest apparent-height of the image of the body above the
horizon. Automatically, that will be when the tilt is zero.
To point out the diffence, imagine you were using a pillar-sextant. This
was an instrument developed in the 18th century to make accurate altitude
measurments above the horizon from onshore (among other uses). A sextant is
mounted to rotate about a pillar which keeps it vertical, and it can't be
tilted about this vertical. A navigator would find its use frustrating,
because he couldn't tilt it about, as all his experience would train him to
do. Instead, he just has to accept that it has been preset correctly into
the vertical plane, and measure whatever altitudes the sextant reads at
that setting. This would be a case where the Manzari tilt analysis would
apply. Any error in the presetting of the vertical axis would give rise to
corresponding error of (1-cos(tilt))*altitude in the result.
However, when the observer is free to tilt to find the best altitude, it's
counter-productive to try to allow separately for tilt error. It's better
to simply consider an observer's horizon error, equally positive and
negative, in trying to kiss a star to the horizon, under calm conditions.
To this can then be added other factors such as waves on horizon,
indistinct horizon, refraction to horizon, etc., as appropriate.
There's a useful experiment could be made, in which Jim, and also perhaps
some friends, arm themselves with a sextant or sextants, perhaps on shore,
and on a day with clear and smooth horizon make many measurements in quick
succession of the altitude of the Sun or other body, preferably at its
maximum altitude so that its real altitude isn't changing. Then, a
subsequent plot of the scatter in sextant readings would be informative. If
it were highly skewed, with a long tail towards the higher side of altitude
from the mean, that could show that tilt errors were indeed important. If
it were symmetrical, it would indicate that tilt errors were unimportant. I
might even be persuaded to try it for myself.
INDEX ERROR.
OK, I overstated things when claiming that index error played no part. It
does play a small part, to the extent that there's a second reading of the
scale to be made, and subtracted, with the same small random errors in
reading the scale as applied to the altitude reading.
Scale calibration errors play no part in index error; those are a
separate, systematic, non-Gaussian contribution to error, which can't be
considered statistically. And, by the way, I would be amazed if a true
navigator would even consider approaching an altitude measurement and an
index check by winding the drum in different directions.
VARIABLE REFRACTION IN LIGHT PATH FROM BODY.
Jim has invented a disagreement between us here, where we are actually in
complete agreement. What I said was-
"Jim has considered refraction of the light from the body, and correctly
found that at angles above 30 degrees, variations are negligible."
The variations that I was referring to were the variations with atmospheric
conditions, just as he was. Perhaps I failed to make this clear. It was
instead interpreted as variations with altitude, which was not my intention
at all. We are in full agreement, in that variations in refraction of the
light path from the body observed, due to varying atmospheric conditions,
are quite negligible. However, this is emphatically not the case for
refraction in the light path from the horizon (see below).
REFRACTION OF THE LIGHT FROM THE HORIZON.
This refraction has a standard value, which depends on the dip (it's about
one-fifth of the dip) but may vary, and vary greatly, with local
atmospherics.
Jim has not taken this variation into account at all, assuming that its
effect is zero, for the following reasons-
"(1) Bubbles (mirages) in the air mass close to the horizon are relatively rare,
except when the sea and air temperature are markedly different, or when sights
are taken over land, or in a few locations in the world such as the Persian
Gulf; (2)I don't know how to model this kind of rare event."
Well, maybe mirages caused by bubbles of warm air are indeed rare, that's
probably true. But mirages are only the extreme cases. There must be a
whole spectrum of disturbance of this type, with the smaller disturbances
being more common, and more difficult to detect, than the larger ones. You
don't need a mirage to occur, to suffer from loss-of-precision resulting
from horizon refraction.
For the Manzari approach to be workable, horizon refraction effects have to
be of two distinct kinds; those that are major, rare, and obvious, and
those that are totally negligible: nothing in between. Does he, or anyone
else have evidence to support this unlikely contention?
He makes the patronising remark-
"George, if you had any real experience working as a navigator you would
probably know that conditions giving rise to air mass non-uniformity near the
horizon are relatively easy to detect. Experienced navigators are sensitive
to this problem whenever the air/sea temperature differential is large,
whenever the sun or the moon appear distorted when rising or setting, or
whenever a sight is taken across a land mass (island or peninsula)."
Well, firstly Jim makes much of my admitted complete lack of ocean-going
experience, but that doesn't imply that I am unaccustomed to making sextant
observations at sea. The matter of observing across a land-mass need not
concern us too much. Observations of a distorted sunrise or sunset will
have little relevance at a noonday sight, which will be taken in quite
different meteorological conditions. What exactly does "sensitive to this
problem whenever the air/sea temperature differential is large" mean? Does
Jim have a rule, I wonder, to discard sextant observations when the air/sea
differential reaches a certain value? If so what value? Extreme cases of
horizon refraction may or may not be easy to detect, but what navigator
could ever put his hand on his heart and say "today, horizon refraction is
quite negligible"?
No, the important factor is that Jim admits " I just don't know how to
model this type of rare event". Leaving aside that question of whether it's
a rare event or not, I don't think any of us know how to model it, either.
But it's there, it's a significant contribution to error, and it may be
more significant than the other contributions Jim has analysed. Really,
none of us knows.
If a reader of this list happens to live in sight of the sea, and can lay
his hands on a theodolite, perhaps he could do us all a good turn by
measuring, from day to day and in different weathers, the apparent altitude
of the horizon, whenever it's really clear. If anyone is aware of such
measurements having been made, please tell.
George Huxtable.
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george@XXX.XXX
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel, or fax, to 01865 820222 or (int.) +44 1865 820222.
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