What is "dead reckoning"?

That's easy, it means figuring out where you are by keeping track of the direction and distance you have traveled from your starting point. Let's say you are in the middle of a featureless desert and you know your exact position by looking at your fiend's GPS that provides your coordinates within one-tenth of a mile. You mark this position on your chart. Now you leave your friend (and his GPS) behind and start walking straight east by looking at your compass and you count your steps. After you have counted the number of steps that you normally walk in one mile you stop. You should now be one mile straight east from your starting position so you draw a line on your chart starting from the previously marked starting position and extending straight east. You then measure one mile along this line and make a mark on the line, this is your current position and you can then read out its coordinates from your chart. Pretty simple.

It should be obvious that you can't use dead reckoning for determining your present position if you didn't know where you started from, you must have a fix. Any error in the starting position is carried forward to all subsequent dead reckoning positions that started from that fix. Any uncertainty in the starting position also carries forward and that uncertainty surrounds all subsequent dead reckoned positions and is not cured until a new fix is determined.You can never know you position from dead reckoning more accurately than you knew the starting position. For example, if you are  starting from Chicago and you want to fly to Mobile then you measure the course on your chart and find that you should fly straight south and that will take you to Mobile. But, if in fact, you were actually starting from Detroit, then flying south will NOT take you to Mobile but will take you to Tallahassee since the actual starting position, Detroit, is about 200 miles east of Chicago. Flying south from there, the track will maintain this same 200 mile constant offset to the east, the track will parallel the original planned course from Chicago to Mobile, and you will end up at Tallahassee which is also 200 miles east of the planned destination of Mobile. (See attached chart.) If you don't know where you are starting from then you can't plot a starting position on a chart and measure the course to follow to a destination by use of dead reckoning.

The next question that you have to think about is how accurately do you know your present position? First thing to realize is that since you only knew your starting position within 0.1 miles that you cannot possibly know you present position to an accuracy greater than the accuracy of your starting position. If, instead of being exactly at the coordinates read out from the GPS, that you plotted on your chart, you were actually 0.1 miles north of those coordinates when you started then you will be 0.1 miles north of the position that you marked on your chart. But that is only if you did a perfect job of walking straight east and only if you did a perfect job of counting your steps and there was no variation in your stride, not even the tinniest little bit. You think back and you remember that the compass needle wobbled around a bit and you had to go around some big rocks along the way so maybe you did not maintain a straight east path perfectly and maybe you did not walk exactly one mile. Because of these imperfections in maintaining a course and in calculating the distance traveled you realize that there is some uncertainty in the present position, you cannot be certain that you are at the exact coordinates where you marked the position on the chart. You may have wandered a little bit off to one side, to the north or to the south, and you may have stopped a little short of a mile or you may have walked a little bit further than one mile. You say to yourself "although I can't be certain that I did a perfect job of this dead reckoning I do feel very confident that I must have walked at least 0.9 miles and I know I couldn't have walked more than 1.1 miles. And although the compass needle moved around a bit (and I didn't watch it at all times) I don't think that I could have wandered more than 6 degrees off to one side or the other." So based on your thinking your are confident that you couldn't be more than one tenth of a mile from your plotted position, 10% of the distance you walked. This works because the tangent of 6 degrees is 0.1 so the possible error to the right or to the left is also 10% of the distance traveled. But don't forget the original uncertainty of 0.1 miles in the starting position which you must add to the uncertainty caused by walking. With this understanding you can draw a circle around the position you have marked your chart with a radius of 0.2 miles, one tenth of a mile from walking and one tenth of a mile uncertainty in the starting position, and be very confident that your real position must be somewhere within the circle that you have just drawn. See diagram.

You look on your chart and you see a water hole that you want to get to since you are thirsty. You measure the course from your current position and you see that it is still straight east and the distance is 3.5 miles. Since you got tired of counting your steps you decide to time this leg for one hour and ten minutes because you normally walk at three miles per hour. So again following your compass you walk east for one hour and ten minutes and then stop. You look around for that water hole and don't see it. Now you are worried since your canteen is empty. You think about it some more. You think that you just walked 3.5 miles so you figure that you should be within 10% of that distance or 0.35 miles from the water hole. But you realize that the error in the distance may be greater than 10% of the distance you walked because this time you didn't count your steps, you just timed it and maybe you didn't maintain a speed of exactly 3 miles per hour the whole time. But wait, that 0.35 uncertainty is only for this last leg and you must add the uncertainty of the starting position, 0.2 miles so now the radius of uncertainty has grown to 0.35 miles plus 0.2 miles, a total uncertainty 0.55 miles. That water hole could be more than a half mile away in almost any direction except behind you (or you would have found it by now.) You think, "in which direction should I search, should I go to the right or to the left, and it is now getting dark and I am mighty thirsty."

Good luck in finding that water hole.

And something else, if the GPS was not working right and gave you coordinates ten miles north of your actual position when you started then you would still be ten miles south of where you think you are and nowhere near that water hole.

This 10% of the distance traveled is a commonly accepted estimate of the uncertainty in dead reckoned in flight positions. See flight navigation texts about dead reckoning.

If Noonan was trying to decide whether to abandon efforts to find Howland and instead to proceed to Gardner he would have taken this knowledge of dead reckoning into consideration, that he could accumulate a DR error of 35 NM to the right or to the left while flying the 350 NM leg down to Gardner, errors large enough to keep them from spotting that island even if he had an accurate starting position. But, Noonan was aware that he didn't know his starting position accurately since, if he did, he would have found Howland. Without an accurate starting position it would not be possible to measure the correct course to follow to Gardner and the error in any assumed starting position, combined with the expect uncertainty caused by flying a dead reckoning leg, would make it obvious to Noonan that his best option was to continue to fly a search pattern and to keep looking for Howland.

Gary LaPook,
Jan 31, 2012, 10:55 PM