What would give the best accuracy on a surveyed site

Yes, and I can Localize to the provided control. My observations using RTK on their points was only one vertical was off, out of 3. The other thing that bothers me some, is when the survey company set control, they used a Trimble R10 and that’s great, but only sat (baked) on point for like 30 seconds.

All Levels elevation measurements are relative to BM. Many of you do not consider that some of these BM are a bit old and therefore, are not accurate no because of movements of plates, subsidence, or uplifting.
Your best accuracy will come from setting a base station and running it for 4 - 6 hours. Download the Rinex file and get your accurate base using OPUS. The two other points that you mentioned are accurate can be used with the base as control points to close on.

It sounds like the Surveying practices are very loose for this project so your gear should fit nicely. I would suggest bringing up that benchmark condition to higher ups to make sure they understand what everyone is working from and that you all are on the same page. I have had construction projects that were designed off of bad benchmarks and we had to hold down the entire design during construction because they weren’t going to fix it. Complications could have easily arisen with our subcontractors if we hadn’t had coordination meetings with them all.

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