General Total Least Squares Theory for Geodetic Coordinate Transformations
Round 1
Reviewer 1 Report
The paper applies the method of total least squares for calulating geodetic transformations. The method is interesting and the paper deserves to be pubished.
The description could be a bit clearer in some cases. For example, it might be worth mentioning explicitly that equations 5 and 12 define the A matrix and the y vector in the 2d and 3d cases, respectively.
For the examples, what was assumed for the a priori covaicance matrix Q of the target and source coordinates?
Author Response
Response to Reviewer 1 Comments
The paper applies the method of total least squares for calulating geodetic transformations. The method is interesting and the paper deserves to be pubished.
Point 1：
The description could be a bit clearer in some cases. For example, it might be worth mentioning explicitly that equations 5 and 12 define the A matrix and the y vector in the 2d and 3d cases, respectively.
Response 1：
Thank you for the constructive comments!
Right, we explicitly add the following details:
Eq explicitly defines the matrix and the vector in Eq for all other transformations in 2D.
Eq explicitly defines the matrix and the vector in Eq for all other transformations in 3D.
Point 2:
For the examples, what was assumed for the a priori covaicance matrix Q of the target and source coordinates?
Response 2:
The a priori covariance matrix or the dispersion matrix of the target and source coordinates is usually given by the instrument precision, or the precision determined by the adjustment in the previous period.
Reviewer 2 Report
in this work the authors establish a general theory of the total minimum squares for the datum transformation, in particular applied to TLS problems.
The algorithm design is by no means well described.
The numerical examples are not very descriptive of the application.
The conclusions do not contain any major considerations but merely summarise what has been written previously.
The paper is inconclusive and very generic
Author Response
Response to Reviewer 2 Comments
Point 1：
In this work the authors establish a general theory of the total minimum squares for the datum transformation, in particular applied to TLS problems.
The algorithm design is by no means well described.
The numerical examples are not very descriptive of the application.
The conclusions do not contain any major considerations but merely summarise what has been written previously.
The paper is inconclusive and very generic
Response 1:
Thank you for your comments! We rewrite the conclusions part and add some revision for the model adaptation and algorithm, see red part in the paper. We hope that you can satisfy our paper this time.
Reviewer 3 Report
- Why you did not send this paper to “Journal of Geodesy”? It would surely be better suited to this journal. Your previous research and most of references were also published in this journal so why not this one? Why is it important for “Applied Sciences” audience?
- The conclusion section should be extended. In current version it contains only 6 sentences and is definitely the shortest chapter in the paper, while in my opinion this chapter is the most important one.
- The paper would benefit from adding some propositions of directions for further development of model.
Author Response
Response to Reviewer 3 Comments
Point 1：
Why you did not send this paper to “Journal of Geodesy”? It would surely be better suited to this journal. Your previous research and most of references were also published in this journal so why not this one? Why is it important for “Applied Sciences” audience?
Response 1
Thank you for the constructive comments!
The reasons why we submitted the paper to “Applied Sciences” are:
1) Admittedly, Journal of Geodesy is a highly recommended journal in geodesy. However, the transformation problem is widely discussed in other scientific and engineering domain as mentioned in the introduction. Actually, most geodesists focus on the similarity transformation, which is only one kind of the transformations discussed in our paper. The other kinds of transformation are principally beyond geodesy, and may be promising for other research fields.
2) The Journal Applied Sciences facilitates expanding the non-geodetic audiences to pay attention to the topic. We believe that the paper will help them to find new interdisciplinary research problem.
3) From the technical perspective, Journal of Geodesy prefer the paper on the estimation theory. In the other words, the published papers are more theoretical and present long mathematical derivations. Our paper focus on the applications, i.e. the most important keyword in our paper is transformation instead of the total least squares. Furthermore, we provide an easily applicable algorithm implementation in the paper.
4) For the personal purpose, we need a quick acceptation of the paper before our project finishes.
Point 2:
The conclusion section should be extended. In current version it contains only 6 sentences and is definitely the shortest chapter in the paper, while in my opinion this chapter is the most important one.
The paper would benefit from adding some propositions of directions for further development of model.
Response 2:
We rewrite the conclusions and add the outlooks:
7 Conclusions and outlooks
In this contribution, we presented transformation models in the context of the TLS method, developed its algorithm based on the constrained nonlinear normal equations and showed the statistical assessment of the TLS adjustment results including the cofactor matrix of the parameter estimator and the a posterior variance factor.
In the adaptation of the transformation problems to the EIV model, we explicitly expressed the functional and stochastic models by the source and target coordinates, and emphasized the differences between the transformations distinguished only by the quadratic constraints. The adaptation in 2D and 3D are quite similar. In particular, the structure of the matrix A and the vector y need to be enlarged by the z coordinates in the assigned place. For the adaptation, it is important to note that the number of constraints equals the number of the transformation matrix minus the number of the independent parameter numbers. The number of the constraints serves holding degree of freedom in the model.
After formulating the transformation models by unconstrained or constrained EIV model, the Lagrange multipliers were applied to provide the first order necessary conditions of the TLS optimization. After some arrangements, the constrained nonlinear normal equations were established, based on which the Newton type iterative procedure can be implemented. The further advantage of the formulation of the constrained nonlinear normal equations is that one can explicitly compute the cofactor matrix and the variance factor in comparison with the other existing methods, e.g., the sequential quadratic program.
We applied the proposed algorithm to selected examples so as to present and explain the adjustment results of all transformations with regard to objective function value and the statistical characters. We showed the more constraints are available, the objective function values are larger and the cofactor of the parameter estimates are smaller. The numerical results correspond the theoretical inference.
This algorithm is not only valid for the case for many geodetic datum conversion problems, but also for other applications (photogrammetry, GIS, etc.), where the scale changes may be different or be fixed to one, which will justify the use of the suitable constraints within the EIV model.
Furthermore, we hope that the discussed model and the developed algorithm contribute its share in convincing many-not only geodetic-researchers that the benefits arising from the use of the orthogonal regression analysis outweigh the additional effort. From the methodological point of view, our TLS estimation can be generalized to any M or L types estimation, which will be promising in robustifying data processing for the large data set such as point clouds.
Round 2
Reviewer 2 Report
The structure of the algorithm has not yet been significantly described, except for a few lines in the text.
In addition, the numerical examples are not yet extensively described in the application.
Author Response
On behalf of my coauthors, we thank you very much for giving us an opportunity to revise our paper. All comments and suggestions on our manuscript are useful and constructive.
We have rewritten the section of algorithm design, and added a flow chart for explaining it.
Furthermore, we have modified the section of numerical examples. We have added the background of the dataset. In order to present the advantages of our proposed algorithm, the methods previously applied to the dataset, are also briefly discussed.
Round 3
Reviewer 2 Report
The required corrections have been made. Now the publication is clearer. Well explained and the algorithm design. The examples, previously described, are certainly well detailed. The conclusions and outlooks are also good. The article is publishable in this version only some linguistic syntax errors need to be corrected.
