Use of Total Least Squares Adjustment in Geodetic Applications
Round 1
Reviewer 1 Report (New Reviewer)
Comments and Suggestions for AuthorsBrief summary
The article concerns the total least squares (TLS) adjustment method. The authors described the theory of the method and tested its performance on three examples of common geodetic tasks. Comparison of the results of TLS with the results of alternative methods – particularly the least squares (LS) adjustment – showed the superiority of TLS.
Broad comments:
The TLS method is relatively new and the article can be helpful for people who are starting to use it. The authors gave a detailed theoretical background but the literature study concerning the application needs to be extended. Particularly I suggest considering the following works:
Akyilmaz, O. (2007). Total least squares solution of coordinate transformation. Survey Review, 39(303), 68-80.
Fang, X. (2011). Weighted total least squares solutions for applications in geodesy.
There are also some works newer than the newest cited in the article paper of Amiri-Simkooei from 2012.
Detailed comments and minor mistakes:
1. Line 310: “…combining terrestrial tachimetric and GNSS surveying methods…” I do not know the details of the measurement procedure used for determining the geometry of the network but tacheometry is used for the land details measurement rather than for network establishing. I suppose ‘angular-linear’ will be more proper than ‘tachimetric’.
2. Line 315: Why is the text in brackets in red? Does it have any special meaning?
3. Lines 352-355: Can the conclusion be generalized or it is specific to the presented example? Did you test the method on other examples?
4. Table 10: The columns of WTLS method are corrupted (only dx is shown).
5. Line 457: “Gauss-Mark model” -> “Gauss-Markov model”
6. Lines 469-474: The fragment should be revised. In the present version, it sounds as if the TLS were the invention of the authors of the article.
Comments on the Quality of English LanguageThe word ‘determing’ appearing several times in the Discussion and Conclusion section should be corrected to ‘determining’.
Author Response
Please see the attachements.
Kind regards.
Authors
Author Response File: 
Author Response.pdf
Reviewer 2 Report (New Reviewer)
Comments and Suggestions for AuthorsThe paper discusses the use of the total least squares fitting method in geodetic applications, addressing the minimization of the residual data. The authors present a theoretical analysis of the method, its application in point regression, and the computation of transformation parameters between coordinate systems. Furthermore, the study investigates the application of the method in the S-transformation between geodetic datum-dependent solutions. Although the methodology presented has practical relevance and the structure of the article and the explanation of the methods used are clear and well-organised, some points need to be improved. Indeed, the presentation of this work needs to be improved since there are some missing or poorly motivated definitions. Please see some comments below:
1) The introductory paragraph currently covers a broad range of topics but lacks depth and technical rigour. The authors are advised to revise it for greater coherence, continuity, and technical precision. It is important to note that the issue at hand pertains to an inverse problem. Given the interdisciplinary nature of the Applied Sciences journal, the authors must delineate both the forward (direct) problem and the inverse problem. To illustrate, in the context of navigation, the forward problem involves determining one's position from a known location, given azimuth and distance, while the inverse problem involves calculating azimuth and distance between known positions. Providing a didactic explanation of the forward and inverse problems associated with the specific issue under consideration will aid readers in understanding its practical application. For instance, the authors can elucidate the mathematical definition of matrix A in the introduction, but it would be more beneficial to also discuss the effects of operator A (the forward operator) and the underlying physics behind it.
2) In the experiments outlined in Section 3.1, it is imperative to ascertain the true (reference) values of both "k" and "n." Furthermore, elucidating the estimation errors associated with each analysed method is crucial. Given the presence of noise in the "measurements" within this experiment, it is pertinent to expound on how this noise was incorporated. A more comprehensive description of the experiment is warranted, including the delineation of errors in estimates.
3) In Experiment 3.1, the similarities observed between the classical least squares approach and TLS (total least squares) raise the question: What was the underlying cause of this resemblance? Indeed, these similarities between the classical least squares approach and Total Least Squares (TLS) may be attributed to several factors. These could include the characteristics of the dataset used, the distribution of errors or noise in the measurements, the nature of the relationship being modelled, or the specific algorithmic implementation of both methods. Please, clarify these points.
4) In the equation for the straight line as indicated on line 219, it is advisable to consider representing the constant "n" with a different letter, given its prior usage elsewhere in the text.
5) The definition of equation (1) appears somewhat confusing. In actuality, "Ax" represents the modelled "measurements," while "b" signifies the observed measurements/observations. The authors have designated "b" as residuals; however, it is more accurate to assert that "Ax - b" constitutes the residuals.
6) The authors have not explicitly addressed the implications of outliers in the provided excerpt of the article. However, outliers can indeed introduce distortions in estimates of unknown parameters, potentially leading to inaccuracies or biases in the results. Hence, it becomes imperative to consider the presence of outliers when analysing and adjusting geodetic data, especially in applications where precision and reliability of estimates are crucial. Outliers are values that are not typical for the rest of the data in a dataset. They can have a big effect on estimates made by adjustment methods such as the Total Least Squares (TLS) method and the traditional least squares (LS) method. Robustness, often associated with a method's resistance to outliers, is a key criterion for evaluating its effectiveness. While LS is typically not robust, the question of whether TLS is robust remains open. To address this, it would be beneficial for the authors to consider the adaptation of TLS through approaches such as deformed LS methods, including generalised statistics (https://doi.org/10.1016/j.physa.2022.127554) and the least median squares (https://doi.org/10.1016/0019-0578(94)90024-8). Incorporating these perspectives into future work could enhance the robustness and reliability of geodetic data analysis, particularly in scenarios where outliers exist.
7) What does the red line in Figure 3 mean? And the red text on line 315?
8) Table 10 appears to be incomplete.
9) I envision that employing the Total Least Squares (TLS) method could yield computational advantages. Notably, it offers a robust strategy for addressing errors in both observations and unknowns concurrently, thereby enhancing the precision and dependability of parameter estimations. It would be beneficial for the authors to delve into the computational intricacies involved. A comparative analysis of various computational approaches could facilitate the identification of the most efficient one in terms of computational expenses, execution duration, and result accuracy. This discussion holds significance for data scientists engaged in big data analytics, as it aids in discerning optimal methodologies for their analytical endeavours.
10) The introduction should encompass a series of studies focusing on the conversion of geodetic data, exploring its diverse applications.
Comments on the Quality of English Language...
Author Response
Please see the attachment.
Kind regards.
Authors
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report (New Reviewer)
Comments and Suggestions for AuthorsThe authors address all the questions.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe type of this manscript is "article" but not “review”. However, the authors present a great deal of space to describe the fundemental principles on LS and TLS. In addition, the novelty of this manscript is not obvious. It is more like an excise than a research paper. The authors claim the main contribution is the testing of the TLS approach in the case of the S-transformation. Then they should focus on it, redesign and rewrite the paper.
Reviewer 2 Report
Comments and Suggestions for AuthorsIt is of great value to study total least squares adjustment and its applications in geodesy. The manuscript discusses the method of least squares with the additional condition of considering the errors in the unknowns. The results provide both theoretical and practical references. I have several comments and/or questions concerning the contents and the presentation of the manuscript which may help improve the publication.
1. Could the authors please provide more explanation on how to determine the weights of unknowns, both in the experiments in the paper and in real geodetic applications? According to my limited experience, the information of precision or weight for the unknowns usually cannot be obtained easily and exactly, especially before the adjustments. By the way, what is the relationships between sigma_e/n and p_e/n in Table 3? It does not seem to hold that (sigma_e / sigma_n) ^2 = p_n / p_e.
2. In Figure 3, the “solid line” should refer to the three short lines perpendicular to the regression line, but the regression line is also black solid line. To avoid confusion, another kind of line pattern and/or color can be used.
Comments on the Quality of English Language3. In Line 273, “listed Table 3” should be “listed in Table 3”.
Reviewer 3 Report
Comments and Suggestions for AuthorsREVISION MANUSCRIPT Applied Sciences- 2753297: “Use of total least squares adjustment in geodetic applications.”
General comments:
The authors present a study focused on the least squares method for computing the values of the unknowns under the condition of the minimum sum of squares of the residuals of the observations. In order to accomplish that, the authors also provided an overview of the theoretical foundations of the least squares method (also known as total least squares) and extensions of this method by considering the errors in the unknowns in the model matrix. The paper concludes that, it can be confirmed that the suitability of the described method for dealing with the considered computational tasks. I believe minor revisions need to be applied to the manuscript in order to consider it for publication.
Specific comments:
Comment 1: In the Introduction, it is necessary to explain in detail the differences with respect to other investigations, as well as the new advances provided by the authors' research to the state of art.
Comment 2: Data and sample size: The paper does not provide details on the size and diversity of the dataset used for analysis. A larger and more diverse dataset would enhance the robustness and generalizability of the findings.
Comment 3: What are the implications of not having the precisions for points in old coordinate system (D48/GK)? Because you claim estimating the coordinates with a centimeter accuracy or below.
