Helmert Orthometric Corrections (LONG)

Okay...Helmert [orthometric] Corrections (NAVD88) have come up again, so for those who might be interested...here goes:

I will stipulate from the get-go that the VAST MAJORITY of leveling done by Land Surveyors (and Construction Surveyors), is of a nature that Helmert Corrections are probably NOT necessary.

That said, I still believe that Surveyors should at least be aware of the procedure for making these corrections. The following is a slightly edited version of a post I made on POB in November of 2004.

Okay, lets use a semi-theoretical example out of the NGS database. This example should clearly show the difference between "what you level" and the corrected NAVD88 Helmert Orthometric Height difference between two bench marks (an extreme example of course), and give you a feel for the computations (a Kmart Calculator is all you need).

Lets say we start at a bench mark described as follows:

BTW, all values are in Meters, Geopotential Units or Gal(gals).

S-160
NAVD88 = 2857.191m
Dyn. H = 2854.150m
Gravity = 979.4551 gals (NAVD88)
(NGS PID LO0185)

Now we grab our digital level and INVAR rod, CAREFULLY level (balanced turns etc.) 2.8 miles up the road to the next extant NGS Bench Mark (N-160).

N-160
NAVD88 = 3102.132m
Dyn. H = 3098.735m
Gravity = 979.4147 gals (NAVD88)
(NGS PID LO0181)

And lets say that we get an observed leveled difference of +244.881 meters (generated by the NGS program LVL_DH.EXE for use in our example).

Now obviously 2857.191 + 244.881 does NOT = 3102.132!

3102.132 - 2857.191 = 244.941 (difference in NAVD88 Helmert Heights)

244.941 - 244.881 = 0.060 meters (about 2 TENTHS of a foot in 2.8 miles).

Hmmmmm. How do we get from apples (difference in Helmert Heights) and oranges (leveled difference), to apples and apples, so that we can evaluate our "closure?"

Well, FIRST we need the geopotential number of our Bench Mark.

The equation to get FROM geopotential number TO the Helmert Height is (my version);

Hh = gN / (oG + 0.0000424 x Hh)

So:

gN = Hh x (oG + 0.0000424 x Hh)

Where:

Hh = the Helmert Height (in meters)
oG = the observed/modeled Gravity (in gals)
gN = the geopotential number (in Geopotential Units)

So;

gN = 2857.191 x (979.4551 + .0000424 x 2857.191)

So;

gN = 2,798,836.431 (in Geopotential Units)

Inasmuch as the Dynamic Height is equal to the geopotential number divided by the normal gravity at the 45th parallel (980.6199 gals), we can “check” this calculation by looking at the Dynamic Height on the NGS Datasheet.

Then;

dH = 2,798,836.431 / 980.6199 = 2854.150 meters
(which agrees with the NGS Datasheet for S-160, so far so good)

Okay, next we need to compute the difference in geopotential number(s) between our initial Bench Mark, and our “closing” Bench Mark.

DgN = lD x mG

Dgn = Difference in geopotential Number (in geopotential units)
lD = leveled Difference (in meters)
mG = mean Gravity [(gravity @ S-160 + gravity @ N-160)/2]

mG = (979.4551 + 979.4147)/2 = 979.4349 gals
lD = 244.881 meters (from LVL_DH.EXE, which we will pretend is OUR leveled difference)

See: http://www.ngs.noaa.gov/TOOLS/LVLDH/lvldh.shtml

So;

244.881 x 979.4349 = 239,844.998 (geopotential units)

And;

2,798,836.431 + 239,844.998 = 3,038,681.429 (in geopotential units)

Which "should" be the geopotential Number at N-160.

And;

3,038,681.429 / 980.6199 = 3098.735 meters (dynamic Height)
(which agrees with the NGS Data Sheet for N-160, OH BOY)

And;

Hh = gN/(g + 0.0000424 x aH)

where aH = the approximate Height @ N-160 (2857.191 + 244.881)
and g = the [modeled] gravity at N-160 (or the BM/TBM).

3,038,681.429 / (979.4147 + 0.0000424 x 3102.072) = 3102.132

And the Data Sheet says;

N-160
NAVD88 = 3102.132m
Gravity = 979.4147 Gal (NAVD88)
(NGS PID LO0181)

So, I guess that's how you do it! Now this last equation could (I suppose) be iterated a time or two, but I don't really think that it's necessary in most cases.

In practice, one would only have the [modeled NAVD88] gravity values at NGS Bench Marks, but “Garmin” Lat/Lons on your TBMs and BMs are good enough to generate the required gravity values using the NGS Geodetic Toolkit Program NAVDGRAV.

See: http://www.ngs.noaa.gov/TOOLS/Navdgrav/navdgrav.shtml

I would also suspect that in the above scenario, we would be setting TBMs (or Bench Marks) every so often between these two NGS Bench Marks, so you WOULD NEED NAVD88 gravity estimates (values) on EACH one.

Another BTW...NGS Datasheets return the NAVD88 Modeled Gravity in “mgal” (milligals) so you need to divide these values by 1000 in order to use the above equations. The NGS Program NAVDGRAV returns the modeled gravity values in “gals” so you won't have any problem there.

One more thing, do NOT use the NGS “Surface Gravity Prediction” software for NAVD88 Leveling computations. It is NOT the same thing. NAVD88 orthometric heights are intrinsically based on the NAVD88 Gravity Model.

Loyal