Improving Smartphone GNSS Positioning Accuracy Using Inequality Constraints

Round 1

Reviewer 1 Report

1. page 3 line 18: need references

2. (1.) again, formulas need  numbers, not all (1.)

3. Where the ?? is vertical velocity and the ?? is the threshold. Note that this inequation is 142: the ?? need in right form

4. DSRC short for what?

5. Expressed as Eq.20.????  mean (1.)???? same error again

6. Irregular writing occured to many times

Overall:

- the writting need improve, the formulas are not in order.

- the proposed method show good result.

- careful writting and check the formulas.

- need more appropriate references.

The author need improve the writting and check it carefully. It is a rigious work. 

Author Response

Dear Reviewer:

Thanks very much for your comments. We have studied your comments carefully and have made corrections which we hope meet with your approval.

Point 1. page 3 line 118: need references

Response: The references have been added to the corrected manuscript.

Point 2. (1.) again, formulas need numbers, not all (1.)

Response: The number of formulas has been corrected in the new manuscript.

Point 3. Where the ?? is vertical velocity and the ?? is the threshold. Note that this inequation is 142: the ?? need in right form

Response: The  and  have been corrected to  and . Additionally, all variables with subs are checked and corrected in the new manuscript.

Point 4. DSRC short for what?

Response: DSRC shorts for Dedicated Short-Range Communications. And all abbreviations are checked in the new manuscript.

Point 5. Expressed as Eq.20.????  mean (1.)???? same error again

Response: The IDs of formulas has been corrected in the new manuscript. And we are so sorry about the carelessness.

Point 6. Irregular writing occurred to many times

Response: The paper has been carefully proofread and amended by professional English editing.

Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript outlines a generally interesting research, especially considering the mass market smartphone application potential, and an attempt to propose a device-independent position estimation method. However, it suffers from a number of shortcomings, as presented below.

1. Variable measurement units are not presented. For instance, velocity v, in [m/s].

2. Variable names are inconsistent, especially for those with subs. For instance, sometimes p0, and sometimes p₀ refer to the same variable, rendering confusion and inconsistency.

3. Mathematical expressions are not identified with unique numbering, instead every expression is marked as (1.).

4. Manuscript suffers from unnecessary repetitions (for example, types of constraints are repeated at least three times before they are actually defined and described).

5. Manuscript suffers from poor style.

5.1. Broken sentences can be found occasionally, not starting with the subject. (for example, L15, L47, L69, ...)

6. Variables used in text are defined poorly, and explained insufficiently. For instance, ambiguity (L110, L111).

7. Section 2.1 does not result from the authors' research, thus it requires explanations and citations for mathematical models and methods discussed.

8. L147 - L150, including mathematical expression without ID, refer to velocity, and mathematical expression states heights. Measuring units are not presented. The method for determination of constraint value is not presented.

9. L169 - L173 A method is poorly explained, and may require illustration. Additionally, an algorithm expression of the procedure would far more useful than a narrative.

10. Methods and procedures outlined in L173 - L179 may be more understandable when supported with an illustration, and an algorithm. The same applies to 2.2.3.

11. Section 3.1 needs references and citations, since it is not a contribution of the research presented.

12. Selection of Kalman filter as the foundation of the method proposed in the manuscript should be justified.

13. L300 - L308 Satellite position and clock offset calculated by the ultra-rapid precise ephemeris should be given as a reference/resource, and described mathematically and algorithmically.

14. In discussion of obtained positioning errors, the authors described some results as 'significant', without revealing the method (test) for, and the level of (statistical) significance estimates.

15. Referring to percentage as levels of improvements, manuscript fails in the presentation of the methodology of percentages determination and interpretation.

16. Presentation of methodology and results, including the proposed method performance validation, are conducted insufficiently, without the firm mathematical/theoretical framework.

17. More authoritative publications have been published, and should be listed in references.

18. Language and style should be improved by proof-reading and necessary amendments. 

Author Response

Dear Reviewer:

Thanks very much for your comments. We have studied your comments carefully and have made corrections which we hope meet with your approval.

Point 1. Variable measurement units are not presented. For instance, velocity v, in [m/s].

Response: All variables that appeared in the paper are carefully checked and variable units are given in the new manuscript.

Point 2: Variable names are inconsistent, especially for those with subs. For instance, sometimes p0, and sometimes p0 refer to the same variable, rendering confusion and inconsistency.

Response: All variables are checked and corrected to keep the consistency in the corrected manuscript. Especially, ‘’, ‘’, and ‘’ are corrected as ‘’, ‘’, and ‘’, respectively.

Point 3: Mathematical expressions are not identified with unique numbering, instead every expression is marked as (1.).

Response: We apologize for the serious carelessness. The IDs of all formulas are added in the corrected manuscript.

Point 4. Manuscript suffers from unnecessary repetitions (for example, types of constraints are repeated at least three times before they are actually defined and described).

Response: We recognize that some of the concepts and definitions are repeated multiple times throughout the manuscript, particularly in the section on constraints. To address this issue, we have revised the manuscript to consolidate the repetition of concepts and definitions, and to present the information in a more streamlined and cohesive manner. Specifically, we will aim to define and describe the types of constraints at the earliest possible point in the paper and ensure that they are not repeated unnecessarily.

5. Manuscript suffers from poor style.

Response: Thanks for your comment about the poor style of the paper. We recognize that the writing style may not be as clear and engaging as it could be, and we apologize for any difficulties this may have caused for readers. To address this issue, we have revised the manuscript to improve the overall style and clarity of the writing. In addition, we have checked the manuscript with a native English editing service.

Point 5.1. Broken sentences can be found occasionally, not starting with the subject. (for example, L15, L47, L69, ...)

Response: Thanks for your comment. We have carefully reviewed the manuscript and made corrections to any broken sentences or instances where the subject does not start the sentence

Point 6. Variables used in text are defined poorly, and explained insufficiently. For instance, ambiguity (L110, L111).

Response: Thanks for the comment, we have carefully reviewed the manuscript and provided a more detailed explanation of the variables used. About the ambiguity, it is an element in the GNSS carrier phase observation. This variable has been introduced in many studies and we have added references about ambiguity in the new manuscript.

Point 7. Section 2.1 does not result from the authors' research, thus it requires explanations and citations for mathematical models and methods discussed.

Response: Thanks for your comment. we have carefully reviewed the section and included appropriate citations to the original sources of the mathematical models and methods discussed. We have also provided more detailed explanations to ensure that readers can fully understand the concepts presented.

Point 8. L147 - L150, including mathematical expression without ID, refer to velocity, and mathematical expression states heights. Measuring units are not presented. The method for determination of constraint value is not presented.

Response: Thanks for this comment. We have added formula IDs in the new manuscript. The relationship between vertical velocity constraint and height constraint is expressed. The unit of variables, measurements, and constraints are given. In addition, we presented an example of a vertical velocity constraint.

Point 9. L169 - L173 A method is poorly explained, and may require illustration. Additionally, an algorithm expression of the procedure would far more useful than a narrative.

Response: Thanks for your comment, we have revised the paper and provided more explanations of direction constraining. An example of direction information obtained is given in the new manuscript and the data is downloaded from Open Street Map (OSM). In addition, the diagram about the direction constraining method is provided in the corrected manuscript.

Point 10. Methods and procedures outlined in L173 - L179 may be more understandable when supported with an illustration, and an algorithm. The same applies to 2.2.3.

Response: The algorithm diagrams of the direction constraining method and inter-distance constraining are provided in the corrected manuscript.

Point 11. Section 3.1 needs references and citations, since it is not a contribution of the research presented.

Response: Thanks for your comment, the references to Kalman filter have been added in the new manuscript.

Point 12. Selection of Kalman filter as the foundation of the method proposed in the manuscript should be justified.

Response: The reason for selecting Kalman filter (KF) as the foundation of the proposed method has been added in the corrected manuscript. The reason for selection is that Kalman filter is commonly chosen in GNSS positioning taking advantage of its maturity and computational efficiency in implementations. In addition, the KF with state constraining is also convenient to achieve, and the methods are frequently studied.

Point 13. L300 - L308 Satellite position and clock offset calculated by the ultra-rapid precise ephemeris should be given as a reference/resource, and described mathematically and algorithmically.

Response: Thanks for your comment. We calculate the satellite position and clock offset by interpolation method using ultra-rapid ephemeris. The references about the calculation have been provided in the corrected manuscript.

Point 14. In discussion of obtained positioning errors, the authors described some results as 'significant', without revealing the method (test) for, and the level of (statistical) significance estimates.

Response: Thanks for this comment. We use the root mean square error (RMSE) to indicate the positioning accuracy, and the formula for RMSE calculation is given.

Point 15. Referring to percentage as levels of improvements, manuscript fails in the presentation of the methodology of percentages determination and interpretation.

Response: The formula of improvement percentage calculation is given in the correction manuscript, as below:

where  and  are results RMSE with and without constraining. The percentage indicates the positioning accuracy improvement after using the method.

Point 16. Presentation of methodology and results, including the proposed method performance validation, are conducted insufficiently, without the firm mathematical/theoretical framework.

Response: More explanations are given in the new corrected manuscript. Including the results analysis, the indicators of accuracy comparison, and the mathematical/theoretical framework of the proposed method.

Point 17. More authoritative publications have been published, and should be listed in references.

Response: More authoritative publications related to the study are listed in the references. Including publications on smartphone collaborative positioning (CP), the positioning model, the Kalman filter, and so on.

Point 18. Language and style should be improved by proof-reading and necessary amendments.

Response: Thanks for this comment, we have carefully revised the manuscript to improve the overall style and clarity of the writing. In addition, we have checked the manuscript with a native English editing service to further improve the paper.

Author Response File: Author Response.pdf

Reviewer 3 Report

The content of this paper is interesting and related to improving GNSS positioning of smartphones with inequality constraints. The vertical velocity, moving direction, and device distance constraining were derived.

Test data from different smartphones under static and kinematic conditions are also used to validate the proposed method. Therefore, after applying the comments below, I would like to recommend this paper for publication in the Journal of Remote Sensing.

1. There have been some researches on collaborative positioning (CP) via smartphones, which should be mentioned in the introduction.

2. Line 104: All of the formulas number are repeated, please correct. The same as below.

3. Line 125: The state noise setting for KF in Equ.(2) should be mentioned. The main contribution of this article is the application of inequality constraining. In fact, state constraints in KF can also be realized by process noise (e.g. V(t+1)=V(t)+Q). What are the differences and advantages between the inequality constraint in this article and the state constraint in KF?

4. Line 142:  The 'vu' and 'sigma-u' should be written in the same form as in the formula.

5. Line 350: How do the authors estimate the smartphone velocity and obtain its error? The velocity error in sub-figure (a) seems to be too large (over 5 m/s), please explain the cause of the error.

6. Line 399: What criteria can be used to select direction thresholds?  The results in Fig. 8 show that the proposed method is effective during linear motion. How about the performance of the direction constraint during the vehicle turning?

7. The authors show the effects of these three constraints when applied separately. If they are applied simultaneously, how should steps 3 and 10 in the flowchart work? Just a suggestion, not mandatory though.

Author Response

Dear Reviewer:

Thanks very much for your comments. We have studied your comments carefully and have made correction which we hope meet with your approval.

Point 1. There have been some researches on collaborative positioning (CP) via smartphones, which should be mentioned in the introduction.

Response: We have added more information about smartphone collaborative positioning in the introduction part, and the references about it have been supplemented.

Point 2. Line 104: All of the formulas number are repeated, please correct. The same as below.

Response: The number of formulas has been corrected in the corrected manuscript. We are so sorry about the carelessness.

Point 3. Line 125: The state noise setting for KF in Equ.(2) should be mentioned. The main contribution of this article is the application of inequality constraining. In fact, state constraints in KF can also be realized by process noise (e.g. V(t+1)=V(t)+Q). What are the differences and advantages between the inequality constraint in this article and the state constraint in KF?

Response: The formula of process noise and observation noise calculation is introduced in the corrected manuscript. For the process noise, the formula is as below:

where  is an empirical variance, and we set it as  in this paper.  is process noise,  is interval sampling. For the satellite observation noise, we choose the C/N0-based stochastic model which have been introduced in Ref [1].

For the differences and advantages of using state constraining, there are three points: 1. The state constraints need an extra observation and its variance, but the inequality constraint just needs a range. For example, if we put two smartphones in a car. If we want to use the inter-smartphone distance by state constraining, we have to measure the distance. And If we use the inequality constraining, we can easily define the range as the car’s width. 2. The inequality will not perform the task when the results do not fit the given inequality information, but the state constraining will affect the estimation over the calculation. When the extra information is not accurate enough, the inequality constraining could be more stable. In contrast, if the extra information is accurate, the state constraining is better. 3. The probability distribution is different. It is equal in the range of inequality constraining. While the probability distribution of state constraining fits the normal distribution.

[1]. Bahadur B. A study on the real-time code-based GNSS positioning with Android smartphones. Measurement. 2022;194:111078. doi:10.1016/j.measurement.2022.111078

4. Line 142: The 'vu' and 'sigma-u' should be written in the same form as in the formula.

Response: The  and  have been corrected to  and . Additionally, all variables with subs are checked and corrected in the new manuscript.

5. Line 350: How do the authors estimate the smartphone velocity and obtain its error? The velocity error in sub-figure (a) seems to be too large (over 5 m/s), please explain the cause of the error.

Response: For the smartphone velocity estimation, we apply the constant acceleration (CA) motional model in the Kalman filter. In this model, the estimated state includes position, velocity and acceleration, and we can update this state in the filter with measurements.

For obtaining the velocity error, the main problem is to get the reference velocity of the smartphones. In this paper, we installed a geodetic receiver called CHCNAV I90 in the vehicle during the data collection. The velocity of this receiver and the smartphones is consistent. And then we use the measurements from CHCNAV I90 to calculate the velocity by a commercial software called CHC Geomatics Office (CGO). The result is used as the reference in smartphone velocity error calculation.

For the obvious large velocity error, the reason is that the vehicle is moved across a building-surrounded area, in where most of satellite signals are obstructed by the buildings. We explain it using the position dilution of precision (PDOP) value, and the explanation have been added in the new manuscript.

6. Line 399: What criteria can be used to select direction thresholds?  The results in Fig. 8 show that the proposed method is effective during linear motion. How about the performance of the direction constraint during the vehicle turning?

Response: In the case that the vehicle moves in a straight road, it is convenient to obtain the direction threshold by the road’s nodes that is gotten from the digital map. For instance, four nodes of Liangjiang North Road are shown in Figure 1. And the nodes’ information is given in Table 1. And the direction threshold can be calculated as .

Figure 1 The nodes of the road.

Table 1 Latitude and longitude of the nodes

In the case of vehicle turning, it is difficult to select the direction threshold. In this paper, we combine the predict yaw and empirical threshold to define the direction range, as shown in Figure 2. In this figure, the yellow points are the estimated position of the continuous epochs.  is the estimated state on last epoch,  is the estimated state on the current epoch,  is the predicted differential position,  is the predicted yaw based on , and  is the empirical direction threshold.

Figure 2 Direction threshold determination in turning

In this paper, we set the direction threshold as  to avoid the over-constraining. In most of time, the constraining will not perform the task, and the main function of the constraining is to avoid the filter divergence.

7. The authors show the effects of these three constraints when applied separately. If they are applied simultaneously, how should steps 3 and 10 in the flowchart work? Just a suggestion, not mandatory though.

Response: Thanks for your suggestion. We have provided the procedures of processing multiple constraints simultaneously. Generally, it is a iteration to achieve all inequality constraint conditions.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The author made revisions according to the review comments and can be accepted.

Reviewer 2 Report

Authors have considered comments and suggestions related to the previous version, and responded politely and efficiently to every of them with suitable action (amendments, enhancement, clarification, mathematical expression, diagram). Additionally, authors provided a particular effort in improvement of the language and style used in their manuscript.

All evidence provided lead to recommendation to accept the manuscript in question for publication in its latest revision.