least square adjustment of resection

Question

By setting up at an unknown point, and measuring the horizontal angles between three points with known coordinates, it is possible to calculate the coordinates of the unknown point. This process is known as resection.

If 3 known points are used then there is a unique solution. If more than 3 points are observed, then and there will be redundancy and an adjustment is required as part of the calculation. The most rigorous solution requires a least squares adjustment (LSA).

Here I am interested in latter. Can someone explain how to adjust coordinates of the unknown point using LSA? Especially a moment how the solution leads to system of linear equations? Thanks.

Answer 1

Say that the "exact" (or most probable) coordinates of the unknown point are (X,Y,Z) and that you have N points. Three points give (x,y,z) as coordinates of the unknow point. Then, what I would do in to compute (x,y,z) for each triplet of points and define the distance to (X,Y,Z) and sum these distances. Then, minimize this sum of distances with respect to (X,Y,Z). I hope you see that this is very similar to linear square adjustment; your objective function contains squares and the derivatives with respect to X , Y and Z (you want them to be zero) will be linear function. You will need to solve three linear equations for three unknowns. For sure, you must consider all the possible triplets. I solved this kind of problems many years ago and it worked very well. If this is not clear, just post and we shall continue the discussion.

Answer 2

@NurlanKenzhebekov. It depends how you define your objective function (what has to be minimized by the least square procedure). What you mentioned, which is right, is for the case when you minimize the sum of squared distances. But, you can define other objective functions, in particular to make the measurement significant (contributing in a non negligible manner to the objective function). But, whatever objective function you could select, it is obvious that the iterations will start using the average values.