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Hi Frank,

The "practical circumstance", is: one is sailing with known course and
speed and to obtain a fix uses two altitudes with a sextant and a
lunar distance at different times ant the chronometer or on-board
watch has an error. Please Paul, correct me if I have not
misunderstood your example.

If the unknowns are: time, latitude and longitude,
the general differential equation is: dF = ∂F/∂t dt + ∂F/∂B dB + ∂F/∂L dL
and one can combine navigational data from different sources, this is,
F could be:
- Altitude
- Lunar distance
- Star-star distance
- Azimuth (measured on land with an astronomical instrument)
- Bearing
- Distance
- Horizontal angle
- Radio-bearing
...
dF = Fobserved - Fcalculated and the partial derivatives are obtained
with finite differences.

The easier way to obtain a confidence ellipse is using the covariance
matrix of the system of equations.
Metcalf (see IoN journal) use celestial and coastal lines of position
to obtain a fix, and a program is available for free at my web site.

regards
-- 
Andrés Ruiz
Navigational Algorithms
http://sites.google.com/site/navigationalalgorithms/

   

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