Talbot Algorithm with a Trapezoidal Wave in the 2.5D Airborne Transient Electromagnetic Method in Marine Investigations
Round 1
Reviewer 1 Report
The paper presents an interesting approach for Marine investigations. I do not have any concerns about the presented article.
- Abstract- Please be more specific on the main results demonstrate what do you mean by good performance
- Keywords – Please put them in alphabetical order
- Introduction- could be expanded to further demonstrate the research problem from marine environment point of view. Why is it important to investigate marine environment?
- Conclusion should have a section of future work how the demonstrated simulated model will be evaluated in real case scenario
Author Response
Response to Reviewer 1 Comments
Dear Reviewer,
Thank you very much for your professional comments and respectable effort for improving the quality of our Applied Science manuscript entitled “Talbot Algorithm with Trapezoidal Wave in 2.5D Airborne Transient Electromagnetic Method in Marine Investigation” [applsci-700523]. We have considered all your comments carefully and revised our manuscript accordingly. For ease of understanding, we have organized the comments and listed them one by one. The associated responses are shown as follows.
Point 1: Abstract-Please be more specific on the main results demonstrate what do you mean by good performance. 

Response 1: Thank you for your helpful suggestions.
The G-S and Talbot algorithms were applied for a homogeneous sea half-space model and a two-layered seawater model. The relative errors were below 1% for most of the observation period when the voltage responses were compared by the analytic solution and the conventional solution respectively. And the voltage response of the Talbot algorithm had an accurate result when the number of terms N was small. Then these two algorithms were applied to a rugged seabed and a subsea tunnel. The voltage response of different values for the number of terms kept the same and with the increasing height of the loop source, the voltage response curve gradually became flat. It was validated that the Talbot algorithm was more accurate and robust.
1) Line 16 to 18, “Further, the Talbot algorithm is selected for the inverse Laplace transform as it can yield an accurate response with small summation terms than the commonly used Gaver–Stehfest (G-S) algorithm.” has been rewritten.
2) Line 19 to 20, “The voltage response is obtained when the loop source has different flight heights. The results confirm that the Talbot algorithm with a trapezoidal waveform is more reliable and robust for complex marine geological models and is expected to provide an effective approach for marine explorations.” has been rewritten.
Point 2: Keywords-Please put them in alphabetical order.
Response 2: Thank you for your careful reading of our manuscript.
According to your comment, we have corrected the keywords in alphabetical order as “airborne transient electromagnetic method (TEM); inverse Laplace transform algorithm; marine investigation; Talbot algorithm; trapezoidal waveform”.
Point 3: Introduction-could be expanded to further demonstrate the research problem from marine environment point of view. Why is it important to investigate marine environment?
Response 3: Thank you very much for your instructive suggestion.
Seawater is a low-resistivity medium, which has strong mask effect on EM waves. And the marine environment is complex. Some shelf areas, in addition to holding abundant oil and gas deposits, is an additional concern because it is so shallow. And these areas also are rich in various rocks, coral reefs and other deposits, where survey ships with the towed configuration is not able to arrive at. The airborne TEM system is effective and flexible covering large areas. The development of airborne TEM for marine investigations still need more research. We have added the relevant statement in the following:
1)Line 32 to 33, “And due to rocks or the depth of seawater, many shallow areas are inaccessible by exploration ships.”
2) Line 34 to 37, “In practice, the experiments conducted at sea need consider not only the complicate environment but also other factors such as manpower and costs. Thus, numerical design studies and accurate date interpretation are needed to reinforce before launching a sea-going measurement.”
3) Line 44 to 46, “Moreover, many marine activities, such as surveys of seafloor topography and subsea targets, can be viewed as 2.5D models.”
4) Line 61 to 63, “As seawater is a high-conductivity medium, the mask effect causes EM waves to yield a lower penetration depth and attenuate quickly in seawater. The response of the target in the sea is small and needs long time to be sampled for analysing.”
Point 4: Conclusion should have a section of future work how the demonstrated simulated model will be evaluated in real case scenario.
Response 4: Thank you for your valuable comment.
This research is focused on the Talbot algorithm with a trapezoidal wave can be suitable for the inverse Laplace transform in 2.5D airborne TEM for marine investigations. The actual geophysical model is more complex and various, it is essential to apply parameters of the actual geophysical model to numerical simulation to furtherly improve the forward calculation. The result provides an effective approach to design an airborne TEM system for marine investigations.
A paragraph has been added at the end of Conclusion: “In future work, the actual geologic model, such as seafloor topography or subsea tunnels, will be investigated by the existing airborne TEM system. Comparing the measured date with the simulation result, parameters of the simulation model will be optimized in future research. Furthermore, the forward algorithm will also be optimized. The result and data can provide a basis to develop a compact airborne TEM system for marine investigations.”
We have tried our best to improve the quality of the manuscript and made relevant modifications according to your wonderful comments and valuable suggestions. In addition, we have also provided a red-line manuscript version, in which any change has been marked in red-line accordingly.
Thank you once again and best regards.
Yours Sincerely,
Boya Li, Huotao Gao, Huaqiao Zhao, Lijuan Yang, Haitao Chen
Author Response File: 
Author Response.docx
Reviewer 2 Report
In general, the paper is well written. However, there is an issue that should be clarified:
The authors have written in Abstract: Talbot algorithm is selected for the inverse Laplace transform because it is more accurate and robust than the commonly used Gaver–Stehfest (G-S) algorithm.
In addition, they have written a similar statement in the Conclusions section.
Talbot method is high accuracy. Anyway, to support the aforementioned statements, they should add appropriate references in the paper comparing both methods.
Author Response
Response to Reviewer 2 Comments
Dear Reviewer,
Thank you very much for your professional comments and respectable effort for improving the quality of our Applied Science manuscript entitled “Talbot Algorithm with Trapezoidal Wave in 2.5D Airborne Transient Electromagnetic Method in Marine Investigation” [applsci-700523]. We have considered all your comments carefully and revised our manuscript accordingly. For ease of understanding, we have organized the comments and listed them one by one. The associated responses are shown as follows.
Point 1: In general, the paper is well written. However, there is an issue that should be clarified:
The authors have written in Abstract: Talbot algorithm is selected for the inverse Laplace transform because it is more accurate and robust than the commonly used Gaver–Stehfest (G-S) algorithm. In addition, they have written a similar statement in the Conclusions section. Talbot method is high accuracy. Anyway, to support the aforementioned statements, they should add appropriate references in the paper comparing both methods. 

Response 1: Thank you very much for your suggestions.
As frequency-time conversion in 2.5D forward calculation needs large amounts of computation, the computing accuracy has a great effect on the calculation result. The Talbot algorithm with a step wave has been applied to a homogeneous earth model in 1D forward calculation. It is more suitable than the Gaver-Stehfest (G-S) algorithm for land applications. In addition, the Talbot algorithm has also been applied to the inverse Laplace transform in other problems, such as the noise handling properties, it handles the noisy data extremely well. According to these references, we have rewritten Line 121 to 123, “The Talbot algorithm is a complex operation. From the result of numerical examples, it has the advantage of superior accuracy and weak dependence to problems under the same calculation condition [29, 40-42].” Due to your comment, two relevant reference also have added to support the statement:
1) Defreitas, C. L.; Kane, S. J. The noise handling properties of the Talbot algorithm for numerically inverting the Laplace transform. J. Algorithms Comput. Technol. 2018, 13, 1-14.
2) Wang, M.; Luo, W. New algorithm for inverse Laplace transform and its application in calculation of time domain electromagnetic response. Progress in Geophysics 2018, 33, 740–747.
According to your wonderful comments and valuable suggestions, we have made some relevant modifications to improve the quality of the manuscript. Meanwhile, we have also provided a red-line manuscript version, in which any change has been marked in red-line accordingly.
Thank you once again and best regards.
Yours Sincerely,
Boya Li, Huotao Gao, Huaqiao Zhao, Lijuan Yang, Haitao Chen
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
The authors improved the quality of the paper adding appropriate references. I recommend that the revised version of the paper can be published in Applied Science.
