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UNCERTAINTY ON SURFACE/VOLUME CALCULATIONS

When I started my career, I have been attracted by one (of many) question : how accurate is the surface/volume calculations knowing the uncertainties of the points that define these geometrical figures. I really wanted to know in term of standard deviation what would be the "end results".

The solution was to propagate the variances (standard deviation squared) into the functional model (the mathematical formula in short) and to retrieve the results I was looking for.

By taking the formula to determine the surface of a polygon, we have the coordinates of all points.

By differentiating that equation vs point's coordinates, we can assimilate (at the first order) the derivates dx, dy to the standard deviation of course squared. The result was a pretty elegant formulation.

I will attach the development below.

Now having the standard deviation on the surface in function of the point's standard deviations ... we can reverse the approach and start from the customer's perspective.

For a given polygon, knowing the tolerance on the surface would be given ... what would be the standard deviation of the points we have to achieve. That's optimization ! No need to survey more accurately than needed for instance.

We can use the same rigorous approach for volume calculations.

What do you think ? Is that something you are also concerned with ?

Development :

Starting with the surface formula :

Then we can arrange the terms :

And proceed with differentiation :

Which leads to the final expression in term of standard deviation on the surface.

You will recognize that is the sum of the diagonal squared which can be formalized as a "surface" criterion like the GDOP is for GNSS position ! I think it's an elegant result ...

Now the step by step calculation :

 

 

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Thoughts

  • Land Surveyor

    Hi Joel!

    I appreciate your effort to evaluate with the statistics the "quality" of the model ... but IMHO is more important "number" and "location" of the points are describing the model.

    I think that in our job (surveying) we rarely have the chance to evaluate a regular solid: more frequently we work with models (DTM)

    Few (but also many ...!) badly distributed points (although accurated ...) can not build a good model. 

    I think a "dense" survey with rtk (centimeter accuracy) data is more than adequate for most of the volume calculations that are required for excavations or quarries.

    Your method is very suitable for industrial applications: I'll keep this! Thanks

  • Land Surveyor
    This just gives me brain freeze!
    The question is why do you want to?... I mean isn't it a standard to always get more information as accurately as you can rather than what is required. I've no idea how I get a position out of the instrument... By that I mean the electronics used to measure angles and distance.. It just works and for that I'm grateful because chains and rods were a pain in the ass! Ha ha ha ha
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