It sounds to me, and I may be missing the objective, like you want to plot a concave surface. There are a number of ways using different sensors to do this ranging from somewhat expensive to I can't believe it cost that much.
Likely the least expensive sensors that will return a good accuracy and uncertainty would be Linear Potentiometers from a company like BEI. The BEI 9600 series being low end. Since you don't mention the depth this is pretty generic.
This is an example of the BEI 9600 series I am speaking of. They are inexpensive at about < $25 USD per copy.
Next there are higher end versions like the BEI 600 series and
here is an example of them. At roughly between $250 to $450 USD per copy you can see where ten of them can get a tad costly.
BEI is one example, you can also look at ETI Systems for like items as well as
Honeywell MLT Series Linear Position Transducers.
That would be an inexpensive starting point. Depending on how accurate you want to get running surface profiles you could move to LASER Interferometer sensors. I use the Micro Epsilon line
with a focus on these units but be advised these units run about $3,500 USD per copy so if you want ten it won't be cheap.
Charting the data becomes easy once you have a sensor head configured and designed. There are also likely off the shelf solutions out there unless your need is very unique. Anyway, this may give you a few ideas. Hopefully some other members have some thoughts that will meet the uncertainties you need.
Ron