# 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.