# NavList:

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

**Re: Averaging**

**From:**Bill B

**Date:**2004 Oct 21, 17:22 -0500

Like a political candidate, my position keeps changing ;-) My questions were, how much error from averaging is acceptable, and is there a rule of thumb for estimating how much error there will be by looking at declination, latitude, and altitude? I think I am coming close to a crude working model (below). Living and sailing in the mid latitudes I think of LAN as the point the sun is south and at its highest, and exhibits no observable change in altitude for a brief period of time. For me the altitude seldom exceeds 74d. Note its azimuth is quickly changing however. Systematic error near LAN is small and getting smaller as the date moves away from the summer solstice and the maximum altitude falls. So this is a great time to average (for a full reduction) to determine latitude. As an example of the extreme where the declination and latitude are identical (imagine yourself on the Earth's equator at equinox and ignore change in declination over the day) the sun would rise directly in the east. Its azimuth would remain at 90d until LAN when it passed directly overhead, the become 270 at that moment. Trucking along approx. 1d in every 4 minutes all day, so a relatively large error from averaging. On the flip side of the coin, we have sunrise and sunset. Here the change in altitude can approach 1d per 4 minutes, but the change in azimuth can be very small, so a good opportunity for determining longitude. The problem being the relatively large (up to .8') shift for averaging 5 sights over 4 minutes because of its rate of change and non-linearity. Between sunrise, LAN, and sunset the "slope" or ratio between horizontal movement and vertical changes from vertical to horizontal to vertical. So error falls somewhere between. The above Herbert has addressed more mathematically than I can. To answer my own question, I came across a formula in Dutton's (article 3004) for rate of change. Delta H per minute = 15 x cosine Lat x sine Z Where Z is the azimuth angle of the body or its supplement. They also have a nomogram for graphic calculation. So it seems with a compass bearing (corrected to true) and a hand calculator or the nomogram one could determine rate of change for 4 minutes by multiplying the 1-minute figure by 4. If we assume 1d (60') change over 4 minutes produces .8' averaging error, we can mentally interpolate to estimate averaging error for the current situation. I am guessing I am guilty of again trying to treat a non-linear function as linear, and that is relationship will be explored as time permits. Then the navigator can determine whether that error is acceptable for his/her conditions. Bill