Geodetic Math Problem

I haven't tried to work it out (maybe this evening), but I would first convert the 2 lat-lon's to SPC NAD 27, then do the BB Intersection. Convert that back to NAD 27 lat-lon and then convert to NAD 83. I don't know if this would make a difference in the final solution, but I'd stick to the original system until the last conversion.

BTW

I just looked at CorpsCon and there is no Zone 2600 for NE in NAD 27 - Zones 2601 & 2602. There is a Zone 2600 in NAD 83. Do you know off hand which is correct for NAD 27? Could this be the source of differences?

BTW

It would be just inside Zone 2602 (South Zone) for NAD27. The north and south zones were combined to be Zone 2600 for NAD83.

Jerry - I solved for the triangle and get a slightly different NAD 83(1995) position from yours:

Latitude = 41-41-00.19 N
Longitude = 102-52-53.55 W

Nebraska State Plane (meters)

N = 209,399.3
E = 260,207.5

The direction between the two chimneys at the station is 19-55-42.
The distance between the two chimneys is 237.634 m (Inverse NAD 27 positions)
The forward azimuth from the Courthouse to the High School is 74-15-18 (NAD 27)
The back azimuth from the High School to the Courthouse is 254-15-25 (NAD 27)
The forward azimuth from the station to the Courthouse is 344-48-24 (NAD 27)
The forward azimuth from the station to the High School is 4-44-06 (NAD 27)
Solving the triangle using a simple iterative process and the law of sines you get:

Forward azimuth from the Courthouse to the station is 164-48-19
Distance from the Courthouse to the station is 653.131 m
Forward azimuth from the High School to the station is 184-44-08
Distance from the High School to the station is 697.160 m

Using the Forward computation (NAD 27) I get:
From the Courthouse to the station = 41-41-00.28N, 102-52-51.49W
From the High School to the station = 41-41-00.28N, 102-52-52.15W
Average the two longitudes = 102-52-51.82

Transform the NAD 27 to NAD 83(1995)/NE HARN gives the values provided above.

For any of the handheld recreation-grade GPS receivers I've played with (up through the Garmin 60), the final answer should be in WGS84. The difference from NAD83 is on the order of a meter, which can change one count in the least significant digits.

Those receivers do not change the displayed position of a waypoint if you switch between WGS84 and NAD83, but do between those and NAD27. It comes from the old military publication that first listed dozens of local datum conversions, and at that time they did not have accurate enough measurements to see the difference between WGS84 and NAD83.

The difference between doing the intersection in NAD27 SPC and NAD83 SPC probably isn't much, but I would go with the conversion of the known positions into NAD83 first. I'd expect better accuracy between NAD83 stations than NAD27, and would get into the more accurate representation before doing the math.

If your starting with NAD 27 coordinates transforming them to NAD 83 doesn't make them more accurate, it fact probably just the opposite (although only slightly if you use NADCON). In searching the NGS database those positions don't appear anywhere so without more info it's not possible to assess their accuracy. Sometimes USC&GS/NGS would publish intersection stations to two decimal places in seconds if they were no-check, which always means use with caution. Using a typical hand-held receiver it wouldn't matter if you set it to NAD 83 or WGS 84. While Bill93 is correct that the difference between the reference systems in about 1 m, all of the cheap hand-helds use the old abridged Molodensky transformations computed by the Defense Mapping Agency (DMA) and in this case those values are all 0s. In addition, an unaugmented, point position will have a 95% accuracy of around 4-6 m, so the reference system differences are in the noise of the data.

Yes, a single position is certainly not more accurate after conversion to NAD83. But isn't NAD83 a hair more self-consistent, so that propagating positions in it would potentially have less distortion?

I agree it doesn't matter whether you set NAD83 or WGS84 on (at least older) handhelds, but since it will be operating in WGS84 either way, you should give it WGS84 numbers.

>an unaugmented, point position will have a 95% accuracy of around 4-6 m,

Use WAAS and take an average. Set a stake where it tells you at several different times/days at least a couple hours apart, and take an eyeball average of the stakes.

The positions are fictitious and were not meant to correspond to an actual position on the ground, although the basis of this problem is from a real life situation. I purposely changed the positions so no one would try to figure it out by going to points in the NGS database.

I am mostly wanting to know if this method for solving is correct and if you agree with my answer. It is a good challenge in geodesy and actually something I occasionally do with good results when searching for old monuments.

===============================================================================

Transformation #: 1 Region: Conus

Latitude Longitude

NAD 27 datum values: 41 41 1.67584 102 52 51.99441

NAD 83 datum values: 41 41 1.58331 102 52 53.73113

I converted NAD27 lat/long to grid, applied the average convergence angle in order to calculate an intersection on the grid, solved the NAD27 grid, and converted to NAD83 after. Could fine tune that by proportioning the convergence angle but those results are within ~2 seconds for forward azimuths when using the INVERSE program from http://www.ngs.noaa.gov/PC_PROD/Inv_Fwd/ and my own software to check the NAD27 solution.

Edit: Just because I had to, I proportioned to get the fine tuned results:

===============================================================================

Transformation #: 1 Region: Conus

Latitude Longitude

NAD 27 datum values: 41 41 1.67582 102 52 51.99421

NAD 83 datum values: 41 41 1.58329 102 52 53.73093