Март 2003

(письмо, присланное профессором Л.П. Литхартом с окончательным вариантом ответа на замечания рецензентов)

 

 

Reply on Comments of Reviewer 1

 

Comment 1.1

"Abstract: ... in the monostatic radar case.

I don't think the algorithm is limited to monostatic systems only. It can be used for arbitrary bistatic angle (whenever S is nonsymmetric) and so there is no need to mention monostatic alone in the abstract. It is misleading."

The reviewer is correct. According to this comment;

• we deleted in the 4^th sentence of the abstract the text "in the monostatic radar case"

• we deleted in the 5^th sentence of the abstract the word "arbitrary" and we inserted in the 5^th sentence of the abstract the text “with non-symmetric scattering matrix” after " radar objects".

As a result, the paragraph will be rewritten as:

 

Abstract - The procedure for estimating polarization invariants of the backscattering matrix in HV basis is considered for radar observation of arbitrary non-reciprocal objects. Two polarization invariants are added to the well-known six Huynen-Euler invariants. These new invariants (non-reciprocity angle and difference in absolute phases of the symmetric and anti-symmetric parts of the scattering matrix) describe the non-reciprocal properties of the object itself. With the simultaneous measurement of all 8 quadratures of the scattering matrix elements, the closed-form expressions for calculating the 8 polarization invariants are given. The derived expressions are the starting point for complete estimation of the polarization properties of radar objects with non-symmetric scattering matrix. The given approach can be used to study various polarization effects in radar remote sensing of artificial and natural objects, and also to simulate polarization measurement processes and estimation errors caused by the measurement of scattering matrix elements at different instants.

 

Comment 1.2

"Equation number on page 2. Also sign change needed in 12 or 21 element."

• The absence of equation number on page 2 and the same signs of 12 and 21 elements have been corrected. As a result all equations have been renumbered and take the form

 

 

etc.

 

Comment 1.3

"Use modulus instead of module throughout paper."

"Use moduli as the plural and not modules."

In the enclosed paper ("Algorithms_of_estimating_corrected-4.doc") we have made all corrections.

 

Comment 1.4

"Page 4. Asymmetric matrix case.

In general S will be non-symmetric for nonreciprocal problems i.e. situations where the vector reciprocity theorem is violated. This includes all the special cases mentioned by the authors. Not convinced that aside from trans-ionospheric wave propagation that such situations will be widespread in radar remote sensing problems. I agree some special cases can arise but the authors give the impression that backscatter S is nonsymmetric in many cases. This is not so and is misleading. I would prefer that they emphasise the use of the algorithm for general bistatic systems. Such systems are becoming of more interest now and so this could help raise the profile of the paper. Sticking to nonreciprocal backscatter is in my opinion too limited."

 

The authors are pleased by this comment. We emphasized the use of the algorithms for the bi-static case by changing the abstract (see answer on comment 1.1) and by inserting some texts in section 3.

 

As a result, the second paragraph of Section 3 becomes:

"The possibility of the existence of real objects, which polarization properties are described by asymmetric () scattering matrices in the monostatic radar sensing case or in bistatic configuration allow us to widen the mentioned group of Huynen-Euler parameters by additional independent polarization invariants".

 

"Slight logical inconsistency in the backscatter case. If we have a full SM radar then we are free to choose TX and RX polarisations to be different i.e. to consider mixed bases. In fact by doing this, the appropriate maximum states become singular values of the S matrix rather than eigenvalues. The authors consider the restriction of using the same basis for TX and RX. This is ok but they should close the logical loophole by mentioning that mixed bases can also be used. Their symmetric antisymmetric decomposition is useful when the same base is used but the SVD is more appropriate when separate bases are considered. Please mention this point."

 

We agree with this comment. However, we would like to make a remark. The purpose of our paper was to derive the closed-form expressions for 8 invariants for the most widespread case of “linear basis” radar measurements. Undoubtedly, the given expressions can be derived for the case of mixed bases for TX and RX. Furthermore, these expressions may also be obtained for the case of mixed polarization bases.

In this connection, we propose:

to add the following sentence "It should be noted that the given problem can be solved for more general case of mixed bases, when TX and RX polarizations are different" to the 2^nd paragraph of Section 2.

Thus, the 2^nd paragraph of Section 2 becomes

It should be noted that the given problem can be solved for more general case of mixed bases, when TX and RX polarisations are different. However, without loss of generality, we assume that the measuring polarization basis of the radar system is linear. Besides, the radar coordinate system coincides with Cartesian coordinates XOY. The scattering matrix of a radar object can then be written in the radar's polarization basis as:

 

"Equations 16 and 25 are the two central observations in the paper. Both in fact can be related to fundamental properties of the Pauli spin matrices.

Equation 16 shows that s₃ is invariant to congruent unitary transformations and so the HV-VH term is invariant. This also secures the fact that in backscatter if the S matrix is symmetric in one base it is in all bases (reciprocity theorem).

Equation 25 also follows from the observation made on page 2 that in decomposition theory (see reference 17 for a full treatment) the s₃ term can be modelled as a target which orthogonalises all incident polarisations. Hence by definition it will not take part in copolar RCS. Please clarify this connection between equation 25 and the discussion on page 2."

 

We fully agree with the comment of the reviewer. The analysis made in the beginning of Section 4 The Non-Reciprocity Parameters is only an extended representation of the discussion in the introduction (page 2). Equation (26) is a matrix form confirmation of the statement that s₃ term corresponds to a target which orthogonalises all incident polarizations.

 

"All the authors results follow from these two properties of this single Pauli matrix. There is some subtlety to consider with the determinant of s₃ (which is why it has a j multiplier and hence has det = -1 but this is not so important for the authors developments)."

 

We agree with this statement.

 

"Full list of equations for invariants using I and Q terms is complicated but useful reference material for those wishing to calculate the parameters directly. It is not however the only way and matrix unitary transformations can also be used efficiently to compute the same parameters (the classical Huynen parameters are the most difficult to obtain, the new asymmetric parameters are relatively easy)."

 

We agree with this statement.

 

"Reference 4 please give the full citation as

A H Serbest, S R Cloude (eds), Pitman Research Notes in Mathematics Vol. 361, Longman 1996, pp 257-275"

 

We replace

 

[4] Lueneburg E. "Radar Polarimetry: A Revision of Basic Concepts". Int. Workshop on Direct and Inverse Electromagnetic Scattering, Marmara Research Center, Gebze-Turkey, 1995

 

by

 

[4] Lueneburg E. "Radar Polarimetry: A Revision of Basic Concepts". A.H. Serbest, S.R. Cloude (eds), Pitman Research Notes in Mathematics Vol. 361, Longman 1996, pp 257-275

  

"It would have been nice to see some experimental data analysis but if it is not available then some simulations could also help. With nonsymmetric matrices there are several special cases that could be employed. For example"

 

 

Concerning this comment, we propose one example to be presented at the end of section 5 after equation (51). We added there

 

It is possible to illustrate the results above with an example. Let the scattering matrix of a radar target has the form

 

 

In this case, symmetric and a skew-symmetric components of the scattering matrix are written as

 

 

Using equations (43)-(49), we find the following values of the Huynen-Euler invariants, which characterize the SM symmetric component:

m = 0.823; q = 49.34°; e = -11.637°; n = -10.061°; g = 37.769°; f = 57.353°.

Finally, we calculate (by (50) and (51)) the non-reciprocity angle and difference of absolute phases, which describe non-reciprocal properties of the radar target

z = 15.897°; h = 60.255°.

Thus, we obtain a complete group of 8 polarization invariants describing an arbitrary radar target.

 

 

Reply on Comments of Reviewer 2

 

"This paper presents a novel contribution to the polarimetry theory, which described new invariants for non-reciprocal objects in backscattering measurements."

 

Comment 2.1

"However, there are many references in Russian and conference proceedings which are not easily available for most of readers. A suggestion is to change or add similar references."

 

We do not agree with this comment.

There are only 6 references of Russian authors out of 19 references in total. Reference [2] and [3] are extremely fundamental books in Russian polarimetry and therefore cannot be replaced.

 

References to papers of Dr. Khlusov and his colleague Dr. Vorobjev (dead last year) are also obligatory, since they have led the basis for this paper. Dr. V.A. Khlusov is guest scientist at IRCTR, University Delft and co-authored this paper.

 

Comment 2.2

"Another suggestion is that it would be better to give a real example to show what they are in real polarimetric measurements or SAR imagery."

 

We added in the conclusions

 

Unfortunately, the authors could not present an example of using their approach to real polarimetric measurements or SAR imagery. The main reason is the absence of representative data of simultaneous measurement of all elements of the scattering matrix.

 

We changed in the conclusions the next sentence "The authors intend to use this approach for investigating the polarization properties of radar objects by using experimental results" into: “The authors intend to use this approach for investigating the polarization properties of radar objects in the framework of future joint research between IRCTR TUDelft (The Netherlands) and TUCSR (Russia)”

 

Reply on Comments of Reviewer 3

 

"Review of paper: TGRS-00242-2002 "Algorithms for estimating the complete group of polarization invariants of the scattering matrix (SM) based on measuring all SM elements". V. Karnichev, L. Lightart, G. Sharygin.

This paper concerns the characterization of general asymmetric scattering matrices acquired in a monostatic configuration. The authors decompose an asymmetric scattering matrix into a symmetric and a skew-symmetric components. The symmetric component is described by its 6 Huynen-Euler coefficients. The asymmetric scattering matrix is found to be invariant under a polarimetric change of basis and is fully characterized by two parameters."

 

Comment 3.1

"The Huynen-Euler parameters are generally derived from the diagonalization of the Graves matrix defined as [G]= conj([S]) . [S]. The eigenvectors of [G], which correspond to the XPOL Nulls polarization states, are then used to form a special unitary matrix that diagonalizes [S]. The Huynen-Euler parameters are then identified using the general formulation given in eq. (4). In this paper, the set of parameters is derived from a direct determination of the eigenvalues of [S]. Equations related to intermediate derivation steps should be moved to the appendix and repetitions should be avoided. The in-phase and quadrature components notations make equations significantly more complex. The use of scattering matrix elements would clarify the demonstration."

 

In our opinion, Sections 2, 3 and 4 have no equations, which are repeated.

Concerning the use of in-phase and quadrature components notation, we stated in the second paragraph of the Section 5 "We assume that all 8 quadratures of the SM elements were simultaneously measured by the radar, ...". This means that we start from 8 directly measured quantities. An important novelty in the paper is the derivation of closed-form equations for the complete invariant group as functions of the quadratures of the SM elements. The given paper is the result of a new approach being developed at IRCTR Delft University (The Netherlands) and at RIRS Tomsk State University of Control Systems and Radioelectronics (Russia).

 

Comment 3.2

"The authors may comment the eventual applications of the theoretical developments presented in the paper. In a monostatic case, the skew-symmetric scattering matrix elements were used to estimate and remove additive noise in SAR data, see Hajnsek et al., IEEE IGARSS 2001, Sidney Aus. In a bistatic case, the decomposition of an asymmetric matrix led to 9 target equations and to the definition of a specific target structure, see Germond et al. PIERS 98."

 

In our opinion, it is not correct to relate the problem of additive noise removal in SAR data with the estimation of additional parameters of non-reciprocity. The problem of SAR data relevance is a separate problem and complex also.

 

Concerning the paper of Germond et al., we agree with the reviewer. We added to the 1^st paragraph of the Introduction:

The analysis of bistatic radar polarimetry case was made and two new Euler angles, which determine the bistatic scattering matrix, were introduced in [19]

 

We added in the references

[19] Germond, A.-L., E. Pottier, J. Sailard, " Theoretical results of the Bistatic Radar Polarimetry on Canonical Targets ", Proc. of 4th International Workshop on Radar Polarimetry, IRESTE, pp.25-33, France, 1998) to the References.

 

"An application of the proposed parameterization to real or simulated data would permit to relate the different coefficients to a target geophysical properties and to evaluate the information contained in the skew-symmetric component."

 

We agree. Future experimental may prove the applicability to relate more accurately the different coefficients to the target geophysical properties.