# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: On LOPs**

**From:**Bill Murdoch

**Date:**2002 Apr 16, 20:26 EDT

In a message dated 4/15/02 10:28:59 PM Eastern Daylight Time, hprinz{at}ATTGLOBAL.NET writes:

These are the formulas. (Any mistakes in copying and changing this to survive being typed and sent over the internet are mine. I typed it in Courier.)

From B.D. Yallop and C.Y. Hohenkerk, Compact Data for Navigation and Astronomy for the Years 1991-1995.

If p1 and Z1 are the intercept and azimuth of the first observation, if p2 and Z2 are the intercept and azimuth of the second observation, and so on; form the summations

A = cos^2 Z1 +cos^2 Z2 + ...

B = cos Z1 sin Z1 + cos Z2 sin Z2 + ...

C = sin^2 Z1 + sin^2 Z2 + ...

D = p1 cos Z1 + p2 cos Z2 + ...

E = p1 sin Z1 + p2 sin Z2 + ...

F = p1^2 + p2^2 + ...

As a check A + C should equal n, the number of sights.

G = A C - B^2

The improved estimated position is (Lat, Long), (LI, BI) made by correcting the first estimated position (LF, BF)

LI = LF + dL

BI = BF + dB

where

dL = (A E - B D) / (G cos BF)

and

dB = (C D - B E) / G

The distance between the first estimated position and the improved estimated position is

d = 60 sqrt(dL^2 cos^2 BF + dB^2)

If d exceeds about 20 nautical miles, set LF = LI and BF = BI then repeat the calculation until d, the distance between the previous estimate and the improved estimate, is less than about 20 nautical miles.

If three or more position lines are obtained, an estimate of the error may be calculated. The standard deviation of the estimated position sigma in nautical miles is given by

sigma = 60 sqrt(S / (n-2))

where

S = F - D dB - E dL cos BF

sigma sub L = sigma sqrt(A / G)

sigma sub B = sigma sqrt(C / G)

In general as the number of observations increases the error in the estimated position decreases. Statistical theory shows that the estimated position has a probability P of lying within a confidence ellipse which is specified by the lengths of its axes a and b and the azimuth theta of the a-axis, where

tan 2 theta = 2 B / (A - C)

a = sigma k / sqrt(n / 2 + B / sin 2 theta)

b = sigma k / sqrt(n / 2 - B / sin 2 theta)

where

k = sqrt(-2 ln(1-P)) is a scale factor

Values of the scale factor for selected values of P are given in the table below.

Probability, P 0.39 0.50 0.75 0.90 0.95

Scale factor, k 1.0 1.2 1.7 2.1 2.4

The usual confidence limit is 95%, that is P = 0.95.

The shape of the confidence ellipse depends only upon n and the distribution of the observations in azimuth; whilst the size of the ellipse, apart from the scale factor, depends upon the errors of observation. The method assumes that the observations have equal weight. The best results will be obtained when the observations are equally spaced in azimuth. In such cases the effect of systematic errors on the final calculated position will be minimized.

Bill Murdoch

Could you post the formula you are using so we can study this dependency?

These are the formulas. (Any mistakes in copying and changing this to survive being typed and sent over the internet are mine. I typed it in Courier.)

From B.D. Yallop and C.Y. Hohenkerk, Compact Data for Navigation and Astronomy for the Years 1991-1995.

If p1 and Z1 are the intercept and azimuth of the first observation, if p2 and Z2 are the intercept and azimuth of the second observation, and so on; form the summations

A = cos^2 Z1 +cos^2 Z2 + ...

B = cos Z1 sin Z1 + cos Z2 sin Z2 + ...

C = sin^2 Z1 + sin^2 Z2 + ...

D = p1 cos Z1 + p2 cos Z2 + ...

E = p1 sin Z1 + p2 sin Z2 + ...

F = p1^2 + p2^2 + ...

As a check A + C should equal n, the number of sights.

G = A C - B^2

The improved estimated position is (Lat, Long), (LI, BI) made by correcting the first estimated position (LF, BF)

LI = LF + dL

BI = BF + dB

where

dL = (A E - B D) / (G cos BF)

and

dB = (C D - B E) / G

The distance between the first estimated position and the improved estimated position is

d = 60 sqrt(dL^2 cos^2 BF + dB^2)

If d exceeds about 20 nautical miles, set LF = LI and BF = BI then repeat the calculation until d, the distance between the previous estimate and the improved estimate, is less than about 20 nautical miles.

If three or more position lines are obtained, an estimate of the error may be calculated. The standard deviation of the estimated position sigma in nautical miles is given by

sigma = 60 sqrt(S / (n-2))

where

S = F - D dB - E dL cos BF

sigma sub L = sigma sqrt(A / G)

sigma sub B = sigma sqrt(C / G)

In general as the number of observations increases the error in the estimated position decreases. Statistical theory shows that the estimated position has a probability P of lying within a confidence ellipse which is specified by the lengths of its axes a and b and the azimuth theta of the a-axis, where

tan 2 theta = 2 B / (A - C)

a = sigma k / sqrt(n / 2 + B / sin 2 theta)

b = sigma k / sqrt(n / 2 - B / sin 2 theta)

where

k = sqrt(-2 ln(1-P)) is a scale factor

Values of the scale factor for selected values of P are given in the table below.

Probability, P 0.39 0.50 0.75 0.90 0.95

Scale factor, k 1.0 1.2 1.7 2.1 2.4

The usual confidence limit is 95%, that is P = 0.95.

The shape of the confidence ellipse depends only upon n and the distribution of the observations in azimuth; whilst the size of the ellipse, apart from the scale factor, depends upon the errors of observation. The method assumes that the observations have equal weight. The best results will be obtained when the observations are equally spaced in azimuth. In such cases the effect of systematic errors on the final calculated position will be minimized.

Bill Murdoch