Response to Reviewer’s Comments on " A Linearized Approximately Spherical Two Orders of Scattering Model to Account for Polarization in Vertically Inhomogeneous Scattering-Absorbing Media"
We would like to thank the reviewer for a careful review and insightful comments. They have helped us a lot in preparing the revised manuscript. The reviewer’s comments are italicized and our responses are below the comments.
The manuscript does not contain any new theoretical results. The authors use well known Successive Orders of Scattering method restricting themselves with two orders of scattering.
While the successive orders of scattering technique is well known, its use as a correction tool for retrievals is, to our knowledge, a novel concept. In addition, our model is highly optimized and has great computational efficiency. We analytically integrate over the optical depth; the only numerical integrations are over the angular variables. Our principal aim is to account for polarization in realistic inhomogeneous atmospheres. To the best of our knowledge, the current approach in operational retrievals is either through look-up tables for polarization correction or by means of a simple single scattering approximation. We have established a method to calculate the second order of scattering in an inhomogeneous atmosphere, which has three benefits: (1) the polarization is computed with high accuracy; (2) a correction to the scalar intensity is calculated; and (3) the calculations are two orders of magnitude faster than a full vector multiple scattering calculation (and, indeed, an order of magnitude faster than a scalar multiple scattering calculation). The linearization also enables weighting functions to be computed simultaneously.
The linearization approach is straightforward and results in cumbersome expressions. Moreover, the manuscript is written very careless. Multiple repeating similar equations distr act the reader from the scientifi c content of the paper. Some variables and parameters ap pear in the text without any defi nition.
We have taken the reviewer’s suggestions, removed redundant equations and made the text more coherent.
However, I think that if done carefully, the scalar-vector intensity correction obtained employing Two Orders of Scattering Model can be of interest for readers. Unfortunately, the investigation of the accuracy of the suggested correction was limited to one comparison of the Stokes parameter I calculated using the vector multiple scattering model to the corrected scalar multiple scattering intensity which is not enough to make any conclusion on the overall performance of the suggested approach. No comparison at all was done for the weighting functions.
We have done a variety of tests to investigate the accuracy of the calculated intensity correction and the weighting functions. For Lambertian surface, we have investigated both nadir and non-nadir viewing, at two different surface albedos. See Figs. 3-6 in the revised manuscript.
Overall comments:
Both “ re fl ection phase m atrix" and “ transmission phase matrix" terms used throughout the manu script are uncommon. Moreover, “ re fl ection phase matrix" can be easily mistaken for “ re fl ection matrix". It is much more preferable to use the term "phase matrix" instead.
We agree with the reviewer. We now use the term “phase matrix”, with the angle arguments making it obvious whether it relates to reflection or transmission.
The manuscript should be reconsidered to avoid repeating similar equations.
We have done an extensive revision of the manuscript to remove redundant equations and make it more readable.
The accuracy of the scalar intensity correction should be investigated for non-Lambertian surfaces as well, especially for highly non-Lambertian re fl ections as ocean sun glint.
We have investigated the accuracy of the intensity correction for ocean glint for a variety of scenarios: different solar zenith angles, optical depths and wind speeds. Figs. 9 and 10 in the revised manuscript demonstrate that the 2OS model works very well for non-Lambertian reflecting surfaces also.
To demonstrate the impact of the sphericity some comparisons for larger solar zenith angles ( > 89 ° ) have to be done.
Fig. 12 in the revised manuscript shows the effect of sphericity. The 2OS model gives near-exact results up to about 80°; above this value, the results are then increasingly less accurate but still better than using the scalar approximation.
It would be very helpful for readers if the error of the scalar approximation i.e., (I s - I v ) / I v was given in addition to the error resulting from the intensity correction, i.e., (I s + I corr - I v ) / I v . This would clearly demonstrate an improvement obtained employing the intensity correction with respect to the scalar case.
This has now been done. All figures now compare 2OS and scalar results.
The accuracy of the Stokes parameter Q calculated employing Two Orders of Scattering Model can not be considered as su ff icient in the spectral ranges with small gaseous absorption. The accuracy of the parameter Q within spectral lines will depend on the spectral resolution. The authors should discuss the applicability of the Stokes parameter Q obtained employing Two Orders of Scattering Model.
While the 2OS model does not compute all Stokes parameters with equal accuracy, our principal aim is to correct for polarization. The maximum errors in the Stokes parameter Q are in the continuum, where the polarization is much smaller than in the cores of strong absorption lines. Hence, the 2OS model computes the polarization very accurately. We expect our model to be useful to simulate measurements from instruments that either have a depolarizer or measure either the s - or p - polarized light (or both). For instance, the Orbiting Carbon Observatory will measure only the s-polarized radiation.
I do not see any reasons to publish the results presented in Tables 3 - 8.
We originally presented the tables as reference for anyone trying to reproduce our results in the future. However, we have now removed them as per the reviewer’s suggestion.
The accuracy of the weighing functions calculated employing the polarization correction has to be investigated. Otherwise, the expressions for the weighting functions presented in the manuscript are not of interest to readers.
Figs. 7, 8 and 11 in the revised manuscript show the results of our investigation for both Lambertian and ocean glint cases, and demonstrate the improvement from using the 2OS model over the scalar approximation.
Detailed comments:
page 5, Eq. (3): The intensity correction, I cor r , is not defi ned. It would be reasonable to start from the defi nition of I cor r using, for example, an equation similar to Eq. (12).
We agree. The intensity correction is now defined in Eq. (3) of the revised manuscript.
page 5, Eq (4):
1. Matrices and are de fi ned according to Eq. (2) as the cosine and sine component of the Fourier series. What does the matrix in Eq. (4) mean? P lease give a relationship or defi ne it.
Eq. (5) in the revised manuscript gives a relationship for the reflection matrix . Relationships for and are given by Eqs. (6) in the revised manuscript. Note that the zenith angle is defined such that the downward direction is positive.
2. Eq. (4) is written without any comments on the sphericity of the medium which this equation is valid for. Therefore, the meaning of functions and is unc lear. Authors have either to defi ne these function if the equation is written for any medium geometry or use x and as defi ned by Eq. (5) and state that the equation is valid for a plane-parallel medium only.
The equation is valid for a pseudo-spherical medium. It can easily be extended for sphericity along the line of sight but the expressions become messy; so, we have decided to ignore the “outgoing” sphericity in the manuscript. The title of the paper has been changed to reflect this. Eq. (22) in the revised manuscript defines . x has been replaced by .
3. The matrix is not defined.
“Reflection” and “transmission” phase matrices are not used anymore. The angles in the argument for the phase matrix distinguish between them; all terms in Eq. (5) in the revised manuscript are defined.
4. The argument in all matrices in the right-hand side of Eq. (4) should be replaced by .
With the sign convention for the zenith angle as described above, the argument should remain . The upward-looking angles now have negative cosines.
page 6, fi rst line after Eq. (5 ): Wrong reference to Eq. (3): “For m = 0, (3) can be simplifi ed …” . It should be : “ F or m = 0, Eq. (4) can be simplifi ed ...".
Correction has been made.
page 6, Eq. (6a): This equation holds for p = 2 only. What does (p > 1) mean? I suggest to replace p with 2 here and in all further equations where p can be equal to 2 only. It will make the paper much more easy to read.
Reviewer’s suggestion has been implemented.
page 6, After Eq. (6b): The matrix is not defi ned.
See response to comment: “The matrix is not defined”.
page 6, Eqs. (6a), (6b), (7a) and (7c): I do not see the reason to repeat similar equations for m = 0 and m > 0 as well as for p = 1 and p = 2. I suggest to combine these equations into one equation as follows:
, (p = 1, 2) , (1)
introducing the source function, , for the cosine component as
, (2)
. ( 3 )
Reviewer’s suggestion has been implemented.
page 6, Eqs. (7b) and (7d) should be combined into one equation introducing the source function, , for the sine component in the same way as for Eqs. (2),(3) (local reference).
Reviewer’s suggestion has been implemented.
page 7, after Eq. (7d): The matrix is not de fi ned.
See response to comment: “The matrix is not defined”.
page 7, Eqs. (8a) - (8c): The variable is not defi ned. The single scattering albedo, , is de fi ned in Eq. (4) as a function on . Is independent on here? Please clarify.
The statement before the equation has been modified to read: “Integrating Eqs. (6) over the optical depth from to , …”. The single scattering albedo is constant within a layer. Hence, we remove the optical depth argument here.
page 7, Equations (8a), (8b), (9a), and (9b) should be combined into one equation as follows:
, (p = 1, 2) , (4)
where is given by Eqs. (2) and (3) (local reference).
Reviewer’s suggestion has been implemented.
page 8: Equations (10a) and (10b) should be combined into one equation in the same way as Eq. (4) (local reference).
Reviewer’s suggestion has been implemented.
page 8, after Eq. (10c): ” where S refers to the layer source terms." What does “ the layer" mean here? Please clarify.
The phrase has been removed from the text.
page 8, before Eq. (11a): The re fl ection function is not de fi ned.
Eq. (9) in the revised manuscript gives a relationship for the Fourier components, , of the reflection functions. We further explain that these reflection functions can be derived from the scalar equivalent of Eq. (5) in the revised manuscript.
page 10 - 11, before Sec. 3: The equations given here are only relevant if authors explain the relationship between the phase matrix given by Eq. (19) and previously used matrices. Otherwise the entire text from the fi rst line of the page 10 to the end of subsection 2.2 should be removed.
Section 2.2 has been expanded to give a derivation of the Fourier components of the phase matrix, as used throughout the text. Further, a note has been included to refer to this section the first time these matrices are encountered.
pa ge 12, line 3 before Eq. (21): “ For most cases, the average secant parameterization is su ffi cient:" This statement is misleading. First of all, the accuracy of this parameterization depends on the solar zenith angle as well as on the geometrical and optical thicknesses of a layer. Furthermore, as shown in [8] cited by authors the accuracy is not even below 5% for all relevant atmospheric conditions whereas I am not inclined to belive even 5% accuracy level to be su ffi cient. I suggest to remove this sentence or to investigate the accuracy of this parameterization more precisely.
The sentence has been removed.
page 13, first para., last line:” ... all calculations of multiple scattering source terms use this geometry." There is no defi nition for \multiple scattering source terms".
Line of sight sphericity correction has been removed. This statement does not appear in the revised manuscript.
pages 14 - 15, Eqs. (25b), (26b), (27b): Please explain why the phase matrices , and have the second argument instead of ?
For a pseudo-spherical treatment, the second argument should be. The basic idea is that all calculations, except for the direct solar beam attenuation, are done in a locally plane-parallel environment
page 24, Sec.5, line 6: “ ... with the fi rst N+1 expansion coe ffi cients, ..." N has already been used above to de fi ne a number of homogeneous layers.
L is now used to number the expansion coefficients.