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 :



This Content Originally Published by a land surveyor to Land Surveyors United Network

Views: 395

See Your Saved Posts Timeline

Communicate ⇣

Comments Empower Your Profession!   Please Sign in or register to continue.

Follow Land Surveyors United

Reply by Carlo Alberto Birocco on February 1, 2018 at 5:27pm

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

Reply by Paul Williams on February 1, 2018 at 2:18pm
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

Where Thousands of Professional Land Surveyors, Students of Surveying and Educators are United through Collaborative Knowledge and Purpose.

Land Surveyors Forum

Survey Abstract Coordinates

Started by John J Rodriquez in Introduce Yourself to the Community. Last reply by Kevin D. Allen on Friday. 13 Replies

Control surveying problem

Started by shree ram bhandari in Land Surveying Jobs. Last reply by Billy Brooks on Saturday. 11 Replies

Magnetic North Update

Started by Billy Brooks in Surveying Best Practices. Last reply by Billy Brooks Feb 6. 2 Replies


Started by Lalith Senanayake in Surveying Related Links. Last reply by shree ram bhandari Feb 12. 4 Replies

Job Quotation

Started by Nitesh Shrestha in Land Surveying Jobs. Last reply by Nitesh Shrestha Jan 21. 3 Replies

Survey under tree canopy

Started by Massimo in Surveying Best Practices. Last reply by Maurice Heter on Saturday. 7 Replies

Mean Sea Level (MSL)

Started by Chuck Turlington in Geospatial Industry Types. Last reply by Kim Noblitt Jan 17. 6 Replies

Surface pro

Started by Malik in Surveying Field Tips and Tricks. Last reply by Kevin D. Allen Jan 7. 5 Replies


Started by Mthulisi Mloyi in Professional Advice for New Surveyors. Last reply by ⚡Survenator⌁ Jan 4. 2 Replies

Tools,Apps and Quick Guides


Help With Survey Equipment

Social Support For Equipment

Surveying Video Categories

Surveying Photo Categories

United States Hub Forums

International Surveyor Hub Forums

Resources for Company Owners

Land Surveyor Social Media

Surveyor Resources Collections

Getting Started & Newsletter

Featured Land Surveyor Hubs

Surveyor Birthdays

© 2019   © Created by Land Surveyors United   Powered by

Badges  |  Contact LSU  |  Privacy Policy  |  Terms of Service

Site Explorer