(35S) Most probable position of a vessel

[PedroLeiva]

(04-25-2017 04:46 AM)nsg Wrote:  After obtaining several lines of position (LOP) from celestial observation, navigator is faced with one more challenge -- combine all those lines into a single point, so called most probable position of his vessel, or MPP.

This program works with LOPs represented as azimuth and offset vectors
relative to same origin (such as assumed position, or D.R. position).
Each LOP with given AZ and D contributes the equation
dx*sin(AZ)+dy*cos(AZ)=D
to the system. If there are 3 or more LOPs the overdefined system can be solved for dx and dy using least squares method. The formulas work out to be

A=sum(sin(AZi)^2)
B=sum(sin(AZi)*cos(AZi))
C=sum(cos(AZi)^2)
D=sum(sin(AZi)*Di)
E=sum(cos(AZi)*Di)
DET=A*C-B*B
dx=(D*C-E*B)/DET
dy=(E*C-D*B)/DET

Here AZi,Di are azimuth vectors and offsets of LOPs, and
resulting dx and dy is an offset vector relative to same origin as
LOPs. In another words, dx and dy is a correction to previous assumed position obtained by taking into account the observations.

Operating instructions:

Switch to degrees mode.
XEQ L002 -- initializes routine, then enter

AZ_i <ENTER> D_i <R/S>

(Az_i is azimuth of i-th intercept in degrees, D_i is offset in nautical miles, positive for "towards", negative for "away")

After all pairs are entered, XEQ L003 calculates dx and dy in nm. This offset should be applied to previous assumed position to obtain a "fix".

Please, can you provide a numeric example to test the program with real case data. Thank you in advance, Pedro