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my question is that is there a way if you survey a land site with incorrect (ex. x100,y200,z300) coordinates and then observe only BMs with high accuracy coordinate and change all surveyed data with new observed BMs coordinates?
of course, you can do it, but for example if you use a Helmert transformation the residuals will be out away of scale
Or you can observe your points in angle and distance and when you have the accurate coordinates just calculate all your data from angle and distances. Here in Nepal we always do the survey in angle and distance because why wait for finalizing the traverse or GPS calculation.
If your question, on the other hand, is about exploring possible correlation of some sort between the two sets of coordinates, you might be able to do some stochastic modeling which is far beyond this discussion forum! This is similar to transforming between NAD27 and NAD83, where a simple shift and rotation is not possible because of the lack of accuracy with the NAD27 datum definition itself.
Hope this helps.
you will have to redo the survey since if you use the ones you took with the garmins the coordinates will still be inaccurate as per the required
Like most replied here I wouldn't even attempt to collect any info with the low grade GPS. The issue being is your low accuracy device will not have relative info within itself. So the issue being is when you localize into your BM's with the survey grade GPS you will find that none of your BM's will check.
Almost 5 meters in position and 10 meters in height. Not good for changes that you are asking. Rent a profesional equipment from the very beginning