NavList:

The model described in this paper is strictly theoretical. The basic principle in the model is something very simple which we have covered many times in NavList discussions. As we all know, the equation for dip is
  dip = 0.97' · sqrt[ht(feet)],
where the height of eye is measured relative to the tops of the waves, and for a practical navigator, that's all you need to know. You do not need a new equation or a new table.

The model in the paper explores the issue of wave height variability given a specific probabilistic model. The model suggests that you should use the full average height H (that's trough to crest) multiplied by a somewhat larger factor than a naive expectation might imply. In practice, a navigator does not normally have access to this number H in the first place. Usually you make a rough guess by looking over the side at waves around your vessel, and you assume that waves a few miles away are similar. Then you do the calculation or enter the table with your height above that best estimate of the tops of the waves. No matter what, it's important to remember that the visible sea horizon is composed of a great many overlapping wavetops in a region about a mile deep (that is, when you're looking at the horizon, say, from a height of eye of 25 feet, you're looking through a zone that extends a mile deep from about 6 to 7 miles away from you).

When considering your height of eye, you need to make an educated guess: what do you believe might be the height of those wavetops out there a few miles away in the distance? In addition, you need to make an estimate that's appropriate for the timing of your sights. If you're in a small sailboat in big waves, you may have to wait until you ride to the crest of a wave to take a sight. And then you have to make another guess: how does your height at that time compare to the tops of the waves in the distance? Are you atop an average wave? An above average wave? And is your boat heeling? Are you on the high side or the low side?

The uncertainty of height of eye in dip calculations (or table lookups) is one of the key sources of system error in celestial navigation. There is no "silver bullet" equation that will resolve this. The other key system factor is anomalous dip caused by small variations in refraction. In both cases, the sea horizon is the primary limitation in manual celestial navigation. If we can replace it, for example using a semi-inertial system for determining the vertical, then celestial position fixes jump dramatically in accuracy. Instead of +/- a mile, it's possible to achieve +/- 10 meters or better.

Frank Reed