Solving the Multilateration Problem without Iteration

Round 1

Reviewer 1 Report

The paper is well written and clearly shows the results, with theoretical and practical results to backup their

The comments to the paper are the following

1. In the abstract, please add a few words just to indicate that this paper is for terrestrial surveying, to make clear to who is targeted the paper
2. Please enumerate all equations in the paper, even though you don't reference them in the paper. In this way it is straightforward to point to any equation in particular
3. Center all equations in the paper, half of them are left justified and others are centered, when all of them should be centered.
4. The phrase in line 196 "Therefore, horizontal positioning can be required" does not make sense or it is uncomplete. Please rewrite it.
5. In line 263, it is written "These smaller values will generally lead to better numerical precision, which should have a good influence on improving the estimates, but that was not the case here." Could this be due to the stations being coplanar?
6. In Figure 1, why stations numbers are not numerated from 1 to 4 instead of 1,2,4,5? It looks like one station is missing in the plot.
7. I missed a small comparison against other surveying methods, in order to know if this method improves accuracy (or any other advantages). Only the example of the "interior" stations were shown as a possible disadvantage of this method.

Author Response

Please see the attached word file

Author Response File: Author Response.docx

Reviewer 2 Report

This paper deals with a new trilateration algorithm that uses an additional beacon to linearize the trilateration equation system.

The style of writing, structure, technical soundness,  fieldwork example and conclusions are fine.

Maybe, the authors should revise the use of the terms trilateration (spherical positioning, no matters the number of beacons used) and multilateration (hyperbolic positioning):

https://www.sciencedirect.com/topics/engineering/multilateration

https://www.pathpartnertech.com/triangulation-vs-trilateration-vs-multilateration-for-indoor-positioning-systems/

My major concern with this paper is the claim of using a linearization technique at the cost of having an additional beacon. I think this has to be contextualized referencing and comparing the results with pioneering studies. See for instance the following and the references included:

William Navidi, William S. Murphy, Willy Hereman Statistical Methods in Surveying by Trilateration, April 1998., Computational Statistics & Data Analysis 27:209-227 DOI: 10.1016/S0167-9473(97)00053-4  

https://www.researchgate.net/publication/224817331_Statistical_Methods_in_Surveying_by_Trilateration

 

Author Response

We addressed the reviewer concern in the following paragraph:

Our method is similar to that developed by Navidi et al. [4]. Their approach depends on a special control station that they call the reference point that is required to be at the centroid of the other control stations. They then add and subtract the coordinates of the reference point from those of the other control stations to form a third type of distance that, when added to the distances from the other control stations to the unknown point, causes most of the nonlinear terms to cancel. Placing the reference point at the centroid is highly auspicious because it causes the observation matrix to become orthogonal to the parameter vector, which eliminates the remaining nonlinear term. Ignoring numerical stability issues, our method and theirs must be equivalent: they are both least-squares estimators, and, since least-squares is a best linear unbiased estimator, any two different approaches to the same problem using least squares must be equivalent. However, the problem formulations can differ and that can lead to implementation issues. As just stated above, we recommend that the common differencing station be near the centroid, but it is not a requirement. The method of Navidi et al. requires that the reference point be at the centroid, which could be onerous depending on how difficult it is to stake out that position, assuming that it is possible at all.

 

Round 2

Reviewer 2 Report

My previous remarks have been addressed.