Horizontal LaPlace second term correction

"I know I am splitting hairs but humor me."

No! Sheesh.
Have a beer, Dave.

Don

OK I'll have another or maybe some whiskey or maybe I'll have some whiskey in my beer.

Dave-

I never knew that there was a simplified LaPlace.

But then I never mix whiskey and beer. I think you have to be from Pittsburgh or Youngstown to make that work.

Are you angling for one of those T-4's that's for sale? If you get one, let me know and I'll be happy to come over and keep notes for you. I'll bring the whiskey or beer. Choose one.

Carl-

Mixing whiskey and beer is not for the faint hearted, for sure. 8s start looking like 4s and vice versa. You never know what you might write down in the field book.

I was wrong. He needs to have a Doctor of Divinity.

The laplace correction is applied to the azimuth to the forepoint. The way I look at is that you have an astronomic azimuth to a star. You apply what is an astronomic angle (i.e. affected by the deflection of the vertical, although the difference is very small) to get an astronomic azimuth to the forepoint. You then apply the simple or full laplace equation to this azimuth to get a geodetic azimuth. As you mention, unless the forepoint is elevated or depressed quite a bit from the horizon, the second term is usually negligible.

The second term correction (T-t) would only need to be applied if you want a grid azimuth and the line over which you want to determine the azimuth is long (more than about 5-6 miles). Corrections for the steep angle either to the star and/or the terrestrial station is generally a function of the small errors in the collimation of the instrument. Before contemporary total stations with auto collimation this error was usually computed by having a very sensitive calibrated plate bubble (such as on a Wild T-3) or a striding level bubble (such as on a Wild T-2). The bubble corrections were applied to the observations as a part of the astro azimuth computation.

Dave K.,
Sun or Polaris (or any other star), the reduction procedure will
yield an astronomic azimuth. The Laplace correction will
provide the difference between astro and geodetic azimuth.
The second term (i.e. the "extended" part) is referring to the
zenith distance (vertical angle) of the ground target you are
seeking the azimuth to (not the star used as a reference).

John H., Yes, exactly right.

Dave D.,
The second term of the Laplace correction can be found on
page 186, eq. (5-13) of Heiskanen and Moritz. It is also
described in the documentation for the DEFLECxx products.
For example:
http://www.ngs.noaa.gov/GEOID/DEFLEC90/README.TXT

The Complete (Extended) Laplace Correction:
- Eta x Tangent(Geodetic Latitude)
- [Xi x Sine(Azimuth) - Eta x Cosine(Azimuth)] x Cotangent(Zenith Dist.)

This term arises because the instrument is leveled in the local
astronomic horizon system; but one wishes an azimuth in the local
geodetic horizon system.

You see, there is no (T-t) term. That term only arises for a map
projection, where one wishes to relate the tangent of a projected
geodesic (grid azimuth) to the rectilinear chord between the standpoint
and forepoint in the map projection.

Yes, it is true that there are bubble corrections in the astro azimuth
reduction procedure. These are accounting for the fact that the
instrument is not fully leveled. Hence, the instrument would not
actually be in the local astronomic frame. No surprise, these also
have a Cot(z) term. The astro reduction brings the instrument frame
azimuth into the local astronomic azimuth, and then Laplace brings
it into the local geodetic frame.

Hope this helps.

Actually, what also should be mentioned is that the "mislevelment" (i.e. inclination of the standing axis) can be the largest source of error in astronomic observations. Taking D & R shots does not eliminate or at all reduce this error. And the larger the vertical angle to the star, the more error there is. As Dave D mentioned, in the old days the bubble (plate level or for more accuracy, striding level) was read for each shot. The T-3 I used years ago had the value of each division determined, and this values was affixed to the instrument for use in computations.

I also used a newer T2 that had a compensator. We would point 90 left and 90 right of the star, read the vertical circle. The difference, divided by 2, was the mislevelment. This also helped to level up before the observations. We would perform this before and after each set.

The newer total stations that have dual axis compensators should nearly eliminate this error.