# How big is the error of relative elevation between two points taken from GPS relative to taken from a topographic survey?

I am sufficiently familiar with the fact that a GPS is not supposed to be used to measure elevation of a point, since it gives the elevations relative to a reference ellipsoid, not relative to MSL of a geoid for example. That’s why you may get readings tens of meters above or below sea level while at the sea.

However I still couldn’t figure out why the GPS is considered to be equally unreliable as a replacer to a topographic survey. Consider the figure below and the following discussion: Curve $\bar{AB}$ is the reference ellipsoid and line $\bar{AB’}$ is an ideal flat region of earth’s surface. A is where the reference ellipsoid intersects earth’s surface. Assume you wish to know relief between A and B’, those two being close together (e.g. $\bar{AB’}$ = 100 meters). In a conventional topographic survey we would determine the orthometric height of point A (by some method) and then use topography to determine the height of point B’ relative to A. We can disregard the difference between orthometric height and geometric height for the purpose of this discussion. In such topographic survey both points would have zero elevation and relief would be zero because the region is flat. Now if we take the GPS reading of B’ we will have $\Delta h$ as the elevation value. Is the reasoning correct so far? Back to the figure we get:

$$\cos 1 = \frac{R}{R + \Delta h}$$ $$\Delta h=R\bigg( \frac{1}{\cos \theta}-1 \bigg)$$ Applying series expansion to $\frac{1}{\cos \theta}$ we have $$\theta = \frac{S}{R}$$ $$\Delta h = \frac{R\theta ^2}{2}$$ $$\Delta h = \frac{S^2}{2R}$$ For $S=100m$ and $R=6370km$ we have: $$\Delta h=0.8mm$$

Hence if a survey was conducted by means of a GPS rather than conventional toopographic methods, the relief between A and B' would be just 0.8mm. An insignificant value, close to the zero obtained with conventional topography. The quality of the GPS would be reasonable. Of course we would then have to also account for the geoid height and make the propper correction.

Now, I know that there are serious precision limitations to map grade GPS units, but if one uses survey grade gps units (with mm precision) what else would be limiting the use of GPS for topographic survey. I'm sure I missed something probably regarding GPS-satellite-precision issues. Can anyone help please? I am sure this would be a useful answer for other users as well.

There is no reason why survey-grade GNSS equipment cannot be used for elevation. The required accuracy, however, will determine your positioning strategy and also duration of observation collection.

You are indeed correct that heights will be ellipsoid-referenced. In this case you will benefit greatly by a corrector surface such as OSGM15 which can convert the ellipsoid heights to sea level heights. In the absence of a local/national corrector surface you may use a global geoid model such as EGM08. Note that geoid models characterise an a geopotential surface and not mean sea level directly itself.

Regarding the rest of your post - you appear to be making a rather large assumption that the relationship between the geoid and ellipsoid are parallel - unless I've really missed something.

• I have edited my question which wasn't really clear before. I was actually disregarding the geoid, because my point was about how the relative height between two points taken from a GPS should be equal to that taken from conventional topography. What about map-grade GPS? Would they work? BTW do you know why my tex math didn't render? Jan 26, 2017 at 20:45
• Sorry can't help with the tex math. Regarding two GNSS-derived heights from two different locations - the relative difference in ellipsoidal heights is effectively meaningless unless you have corrected for the geoidal variation. The ellipsoid is an arbitrary surface.
– JimT
Jan 27, 2017 at 8:45
• Meaningless? One elevation would be meaningless I get it. But if you take from two different points (relative to the same ellipsoidal reference), although their elevations is not correct relative to MSL, any elevation difference will be due to the earth's surface entirely right? So it would be good topography? Jan 27, 2017 at 13:13
• No. Any elevation difference will be due to the earth's surface and the differences between the ellipsoid and geoid at the two locations. As I said the ellipsoid is to all intents arbitrary and is not the Earth's surface.
– JimT
Jan 27, 2017 at 13:40
• Ok, the ellipsoid is arbitrary. That's not relevant to the point here. The GPS will take readings relative to the ellipsoid. If you take a reading from one point, then the other, and considering both were taken relative to the ellipsoid, if you subtract these two elevations would you not get the actual value of the relief between these two points as though it was obtained by a topographic survey? Jan 27, 2017 at 17:18