Relation between Geoidal separation and antenna altitude in GGA sentence?

See this scholarly discussion at from ESRI Understanding Our World

Looking at NMEA Revealed,

We have for description of the field altitude:

Antenna Altitude above/below mean-sea-level (geoid) (in meters)

And for description of the field geoidal separation:

Geoidal separation, the difference between the WGS-84 earth ellipsoid and mean-sea-level (geoid), "-" means mean-sea-level below ellipsoid

Let's note (as in answer by @mkennedy):

h = height above ellipsoid
H = elevation, orthometric height (i.e. antenna altitude in GPGGA)
N = geoidal separation (some books call this the geoidal height)

And consider N to be negative. Then according to catdb.org, the mean-see-level is below ellipsoid ("-"), so H > h (H is height from mean see level, h is height from ellipsoid). So H = h + D with D positive. As N is negative we must take D = - N.

So the correct formula is:

h = N + H

-=geoid is below WGS-84 ellipsoid would suggest that the relationship is

H = h + N

See also The geoid undulation. Note that in the article the definitions of H and h are reversed.

The geoidal separation just reports the height difference between the ellipsoidal surface and the geoid model's surface. Natively, GNSS calculates ellipsoidal height (height above ellipsoid surface) but it's usually more useful to have a geoidal height, approximately a height above mean sea level.

The geoidal height is often called an altitude or elevation.

The relationship is

h = N + H

where 

h = height above ellipsoid
H = elevation, orthometric height
N = geoidal separation (some books call this the geoidal height)

The GPS/GNSS device will have some geoid model that it's using to calculate the N value which is the 2nd "z" value given in a GGA NMEA sentence. The first is the calculated orthometric height value, H. Here's the layout of the GGA sentence:

eg3. $GPGGA,hhmmss.ss,llll.ll,a,yyyyy.yy,a,x,xx,x.x,x.x,M,x.x,M,x.x,xxxx*hh
1    = UTC of Position
2    = Latitude
3    = N or S
4    = Longitude
5    = E or W
6    = GPS quality indicator (0=invalid; 1=GPS fix; 2=Diff. GPS fix)
7    = Number of satellites in use [not those in view]
8    = Horizontal dilution of position
9    = Antenna altitude above/below mean sea level (geoid)
10   = Meters  (Antenna height unit)
11   = Geoidal separation (Diff. between WGS-84 earth ellipsoid and
       mean sea level.  -=geoid is below WGS-84 ellipsoid)
12   = Meters  (Units of geoidal separation)
13   = Age in seconds since last update from diff. reference station
14   = Diff. reference station ID#
15   = Checksum

from Glenn Bladdeley GPS NMEA sentence information