|
|
Subject: Re: It works - within limits.
From: Arthur Pearson (arthurpearson@XXX.XXX)
Date: Thu Apr 11 2002 - 21:41:08 EDT
Gentlemen,
This has been a great thread but I must make a request for some remedial
education. I am generally familiar with St. Hilaire technique as the
method by which I calculate an intercept and azimuth from by assumed
position and use them to plot an LOP from a sight. I get the feeling
from this thread that the formulas for a calculated fix solution given
in the back of the Nautical Almanac are an extension of this
methodology, but I am unsure where St. Hilaire begins and ends, and
whether least squares was part of his contribution. Could someone make a
short statement of the essence of his technique for the uninitiated?
Also, Herbert states below that "When starting from a wildly wrong DR
position, St. Hilaire will get you the right fix, albeit only after 2 or
3 iterations. That's not surprising, because direct methods will not
even need a DR." What are the "direct methods".
Apologies to those of you for whom this is old hat, my introduction to
celestial was less theoretical than the discussion here so I am playing
catch baseball (or cricket?).
Thanks,
Arthur
-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@XXX.XXX] On Behalf Of Herbert Prinz
Sent: Thursday, April 11, 2002 9:18 PM
To: NAVIGATION-L@XXX.XXX
Subject: It works - within limits.
To find GMT and our position simultaneously we need the observation of
the
altitude of any two celestial bodies, the distance of the Moon from any
suitable
body, and the time intervals between these three observations.
The simplest case from a mathematical point of view is to measure the
altitudes
of the Moon and second body themselves (because they are needed anyway
for
clearing the distance), and to measure all three quantities at exactly
the same
moment. One can cheat a little on the latter by bracketing the distance
observation with the altitude observations and subsequent averaging.
While it's true that the required altitudes can be computed, we also
know that
there is no free lunch. To use computed altitudes merely means that we
have
already observed some (other) altitudes at an earlier stage (for the
purpose of
finding time and latitude). The solution by this method is, therefore, a
running
fix.
Consider the case where double altitudes of the Sun, or some such method
is used
during the day to establish local time and latitude. If a vessel sailing
from New
York towards the Azores in and out of the meanders of the Gulf Stream
observes
Sun altitudes around 10:00 and 14:00 to establish local time and then
takes a
lunar around 20:00, the dead reckoning of longitude made good between
the former
and the later observations can easily be off by 10 nm, and hence its
local time
be off by 40s. So the error in computed Moon altitude could be up to
10' of arc,
hence the error of computed parallax up to 10" of arc, which in turn
could
translate to an error of as much as 20s in GMT or 5nm in longitude for
the final
fix. This is not much in the scheme of things. Most navigators were and
will be
happy to get GMT within a minute. I am only emphasizing that this is an
additional error that does not appear if altitudes for clearing the
distance are
measured directly and that cannot possibly be detected or eliminated by
any
mathematical tricks.
Bruce Stark soft-pedals the impact of DR error on the accuracy of the
final fix
in his message "It works", of April, 2. The reason why it works for
Bruce even
"when both GMT and longitude are wildly uncertain" is that in his
example, he
does not depend on measuring local time at all; he computes it from
accurate data
and only THEN shifts assumed GMT and assumed longitude in sync with each
other,
so as to not upset their relation (defining local time). Naturally,
after 2
iterations, one gets the correct GMT and longitude from the lunar
distance. But
this is tautological. In the real world, however, local time is only as
good as
your dead reckoning since the time you established it. The "wildly
uncertain" DR
does, indeed, not matter up to the moment where we start with the first
observation for time. But any subsequent error in dead reckoning will
have its
inevitable effect on the resulting fix for GMT from the final lunar
observation.
Of course, there is nothing special about lunars here. This is a general
problem
with the running fix. Take the standard timed altitude observation of
two stars
as an example. When starting from a wildly wrong DR position, St.
Hilaire will
get you the right fix, albeit only after 2 or 3 iterations. That's not
surprising, because direct methods will not even need a DR. The same is
true for
a running fix, if and only if you are absolutely sure about your dead
reckoning
between observations. But if you screw up on the DR (e.g. by getting
into a
current) no method will tell you. All of them will result in the same
wrong fix.
There is another, minor problem with computed altitudes. I am not sure
whether
this has been mentioned already. They rely on an unverified assumption
about
there being ideal atmospheric conditions in the direction of the Moon
and second
object. But if there are unusual circumstances, the effect can result in
up to
12s error in GMT for every 0.1' deviation from standard refraction. If
one
measures the altitudes, this error is automatically eliminated, at the
cost of
only the corresponding negligible positional error. Using computed
altitudes is
thus inherently less safe than measuring them.
All numbers I gave are worst case scenarios that I could THINK of. I
never lost
time at sea and never had to depend on a lunar distance. So, I really
don't know
what I am talking about. All I have ever done were isolated experiments
of
various kinds.
Herbert Prinz
|