Vector Fitting–Cauchy Method for the Extraction of Complex Natural Resonances in Ground Penetrating Radar Operations

Round 1

Reviewer 1 Report

This is an interesting approach and shows promise. However, the authors have not really addressed the key issue thats limit the general approach of target identification and recognition using complex natural resonances. The main problem is signal to noise ratio and its effect on recovery. Dudley pointed out  in Antenna and Propagation newsletter august 1988 that "in lossy mediums with frequency dependent attenuation only approximations to resonances are ever available even in the limit of vanishing noise". The authors need to address this issue in the paper and benchmark the limits of the proposed approach.  There are also some spelling errors that need correction.

Author Response

Thank you very much for the important revision of this paper. The important insights from Dudley we were missing, and have provided the paper with an enhaced explanation and focus. It has been a really good feedback.

• Corrections to the spelling errors were done.
• Dudley Antennas and Propagation article was referenced, as the key issue of signal to noise ratio was missing. This was explained in the second paragraph of the introduction and throughout the text.
• A benchmark for the increasing noise scenarios compared to the FSV reconstruction validation was included in figure 14, and in the last paragraph of section 4
• The introduction, the conclusions and in general the manuscript has mentioned addressed the SNR importance. 

Thank you very much again for this great feedback. 

Reviewer 2 Report

The paper deals with the Singularity Expansion Method (SEM) in electromagnetics and proposes a new rational function approximation scheme to extract the natural resonances of objects for Ground Penetrating Radar (GPR) applications. Numerical and experimental results are presented.

My opinion is that the paper cannot be accepted, as it needs a substantial review.

The main issue of the paper is that many elements are missing for a full understanding of the described techniques. In the following, some examples:

1) When describing the Cauchy method in Subsection 1.1, the f_i's are not defined, although I suppose they are a set of matching frequencies. Nothing is sait about how the numerator and denominator orders P and Q are chosen as well as how the overall number of matching frequencies N are selected.

2) When discussing about the Vector Fitting method in Subsection 1.2, the function f(s) is not defined. Only from Fig. 1 one understands that it is related to the input signal. Is it the transform of y(t) in eq. (1)? In Subsection 1.2, N becomes the number of complex natural frequencies, differently from Subsection 1.1. Although the underlying idea behind the Cauchy method can be understood, the rationale behind the Vector Fitting method is not clear and the Author should spend some words about it.

3) In Section 2, the role of Y(s) is not clear. Why P=(N-3)/2 and Q=P+1? Also, I believe that eq. (11) and the first equality of eq. (14) are not compatible. From eq. (11), I would say that Y(s) should be a polynomial of degree P+Q, while this does not appear from eq. (14). The acronym TLS (Total Least Square?) after eq. (11) does not appear to be defined througout the paper.

4) If I analyze Figs. 6/7 and 11/12, the two compared methods differ much for the reconstructed resonances, but there is not that much difference between the time-domain signals, especially for Fig. 6. I would deduce that the problem is ill-conditioned. I believe that the Authors should use some for of Singular Value Decomposition (SVD) regularization, as in

A. Liseno, R. Pierri, "Imaging perfectly conducting objects as support of induced currents: Kirchhoff approximation and frequency diversity," J. Opt. Soc. Am. A, vol. 19, n. 7, pp. 1308-1318, Jul. 2002.

Author Response

Thank you very much for this revision. This feedback has been very important for us.

Here, the corrections you suggested are listed:

• An explanation of the transfer function in equation (2) and the initial guess for the orders P and Q has been included in section 1.1.
• In section 1.2, f(s) ahs been defined, and a clarification has been written of N being here the number of CNRs instead of the number of samples as in Cauchy method. Some extended clarification of vector fitting method can be found in this subsection as well.
• In section 2, the role of Y(s) is explained alongside the selection of the initial system order. A description of Eq. (14), formed by Eqs. (2) and (7) with the initial poles and system order calculated in Eqs.(11), (12) and (13) was expanded. Also, Eqs. (11) and (14) were rectified as there was a LATEX error and it was supposed to be a fractions. The acronym TLS was defined in the text.
• Some comments have been added about the figures (6), (7), (11) and (12). the difference was explained and a citation of A. Liseno et. al. was pertinent, as the SVD approach to solve linearized problems.
• Also, the evidence of decreasing accuracy in the reconstruction as increasing noise is part of the received signal has been added as a figure before the conclusions, according to your recommendations.

Again, thanks for this important and necessary corrections.

Round 2

Reviewer 1 Report

The revised version is acceptable subject to a careful check of minor grammar and text errors

Author Response

Dear professor,

Thank you very much again. 

Grammar, spelling and minor errors in the text have been checked and corrected.

Regards, 

Reviewer 2 Report

The Authors have improved their manuscript, also with the discussion of a new result. The paper can be now considered for publication.

However, before final publication, a more clear introduction of problem and a general, conceptual plot (without formulas) of the solution would be necessary. The Authors may wish to better introduce the Reader with how the problem is solved in the literature, what are the numerical difficulties, how their approach helps with those and which results the Reader should expect.

Author Response

Dear professor, 

Thank you again for this valuable input. 

We have check the docuent for grammar and spelling. A diagram has been included alongside more detailed explanations of the problem solved, why it was necessary, and the numerical difficulties and approach thorughout the text.

Regards,