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Response to First Referee’s Comments
General Comments
Sections 3-6 have been substantially modified to make the paper more comprehensible.
Several figures have been re-plotted and figure captions changed.
1.
(Page 5) Model description. What is the height distribution of aerosol o.d.? Does
it matter much?
A total aerosol optical depth of 0.05 is considered since we are not trying to tackle cloudy
sky radiative transfer. The optical depth distribution is as shown below:
Varying the distribution had negligible impact on both the qualitative nature of the EOFs
and the error in reproducing the A band spectrum.
2.
(Page 6.1) What does ‘with appropriate scaling’ mean?
A least squares fit is done to the DISORT and TWOSTR radiances to find the linear
regression coefficients m (slope) and c (y-intercept), i.e.
twostr_fitted = m*disort + c
The scaled twostream radiances are then obtained using
twostr_scaled = (twostr-c) / m
3.
(Page 6.3) 'The difference...DISORT'. Confusing sentence. How can DISORTTWOSTR residuals lead to a DISORT frequency difference? P and R branches
are considered separately but there is no further information on the differences in
EOFs or PCs between P and R branches. How important is this?
There are two questions raised here. For the first question, let D(i) and T(i) be the
radiances produced by DISORT and TWOSTR respectively at the i-th wavenumber pixel.
The observation we were making was that D(j)-D(k) is much larger than [D(j)-T(j)][D(k)-T(k)], where j and k are any two wavenumber pixels. In other words, the variance
is much lower for the residual than for the radiances directly obtained from DISORT. In
the expansion we use for performing EOF analysis, we implicitly assume that the
radiance has a quadratic relationship with the PCs. Clearly, the smaller the variance in the
radiances, the better the approximation would be. Then, what we reconstruct are of
course the residuals as well. To get back the actual radiance, we then need to add back
the twostream radiance at the particular wavenumber. Fortunately, the twostream
calculations themselves take negligible time. Thus, our method combines the strengths of
principal component analysis and the twostream RTM.
P and R branches are considered separately due to the difference in physics between the
two regions of the spectrum. However, once this separation is done, there is no difference
from the point of view of principal component analysis, which is simply a mathematical
tool. The EOFs and PCs are not significantly different and in any case they are only
important in that they contribute to the calculation of the reconstructed radiances.
Separation of the P and R branches helps obtain an accurate spectrum using fewer EOFs,
and ‘cases’ (which are explained in the response to comment 12).
4.
(Page 6.8) Try this: nu is an index (1 to N) denoting wavenumber, and i and j are
indices denoting atmospheric levels.
We agree that the nomenclature is confusing. The index for wavenumber is changed to l.
5.
Section 4 EOFs. I had difficulty focusing on this procedure because I am
accustomed to the idea of averaging over a statistical variable. But spectral
structure is not usually considered to be statistical variable, although aerosols
might introduce some randomness. The paper should either explain the procedure
/ab initio/ or allow for reader's prejudices.
We are not the first to compute the EOFs for a spectral variable. In our manuscript, we
reference a paper (reference 18) where this has already been done.
6.
page 7. Equation 3. Should this be a dot product?
Yes, you are right. The equation has been corrected. Thanks for pointing that out.
7.
page 7. last line. od increment, single-scattering albedo and phase function
should be given appropriate indices.
The quantities specified are vectors, as indicated by the boldface.
8.
Figure 2. Is this dtau_i or cumulative tau? Neither caption nor figure axes
indicate what is being plotted. Same problem for several other figures. pi should
be pi_i.
As specified in the figure caption, it is the optical depth profile that is being plotted. The
figure caption and axes have been modified taking into account your comment.
9.
page 8.8. Very unclear. F(nu,i) has a spectrum with all of the features shown in
fig. 1a. In equation 3 it forms a dot product with a non-spectral function. Why do
the PCs in figs 3b and 4b show no small-scale spectral structure? Why is there
only one PC when there are two EOFs?
This is consistent with the way we have set up the problem. As is mentioned in the text,
Figs. 3b and 4b show the PCs for the case corresponding to c1 = 0.25, c2 = 0.5, c3 = 0.7
and c4 = 1 (in the P-branch). Thus, we are only considering the fine region comprising a
range of optical depths (cumulative for the lower half of the atmosphere) from 0.5 e0.25 to
0.5 e0.5, and a range of single scattering albedo (of the top layer) from 0.7 to 1. Within
this small range of optical properties, we do not expect to have further small-scale
spectral structure. Put another way, we have chosen the cases fine enough to reconstruct
the radiances to the accuracy desired.
F includes both the optical depth and the single scattering albedo profiles. The first 22
elements in each row constitute the optical depth profile for the particular wavenumber
and the rest represent the single scattering albedo profile for the same wavenumber. The
EOFs and PCs computed are for the matrix F. Hence, the first 22 elements of each EOF
are for the optical depth and the rest are for the single scattering albedo. For the figures,
we have just chosen to split each EOF into two, for ease of physical interpretation. There
is no contradiction.
10.
page 9. First line. A 'strong line' is singular in its center, and most line-wings
overlap. This statement needs to be amplified.
We are a little confused by the referee’s statement. There is no singularity at the center of
a strong line in terms of the total optical depth or single scattering albedo (see equation
A8). In any case, none of the lines in the O2 A band is completely saturated and so the
problem, if any, is theoretical. Any point in a molecular absorption band can be
considered to be some part of a strong line (center, near-wing or far-wing) or a
combination of the above (due to line overlap); the EOFs for a single strong line thus
have all the features expected in the entire band. Indeed, the results indicate that line
overlap can be accounted for, to a very good approximation, by a linear combination of
the EOFs corresponding to the individual lines.
11.
page 9. Mapping to Radiance. A much more careful explanation of this procedure
is required. What are the errors involved in a two-term expansion?
The error in the two-term expansion is O ( 3 I k Pkl3 ) . By choosing the cases such that the
change in the reflectance, for a perturbation in optical properties of magnitude one EOF,
is less than a percent, we can keep the error to a few tenths of a percent.
12.
page 10. Recovering the O2 A-band. What on Earth are these 105 'cases'? And
why does each require 9 calls? Both sections 5 and 6 require work to be done on
them.
Each ‘case’ refers to a particular choice of the parameters c1, c2, c3, and c4. In other
words, it is a range of optical properties corresponding to a single EOF calculation. As
explained in section 5, for each case a DISORT call is made for the mean optical
properties and for perturbations of magnitude one EOF (positive and negative) for each
EOF used to map back to radiance. Since we use four EOFs to reconstruct the radiance, a
total of 1+2*4 = 9 DISORT calls are required for each case.
13.
Appendix. At first sight one might expect EOF1 to have 100% of the variance.
Explaining why not would help the reader understand the whole paper.
The EOFs are a manifestation of the information content of the system. If EOF1 were to
capture 100% of the variance, it would imply that there is only 1 piece of information in
the system. There are at least 2 pieces of information in the A band spectrum, as reflected
by the first 2 EOFs capturing more than 98% of the total variance. We use 4 EOFs to
calculate the radiances with extremely high accuracy.
14.
16.5 'Remarkably... (Figs. 3a, 4a)'. What does this sentence mean? I can see no
resemblance at all between figures A1a and 3a or between A2a and 4a.
Changing the sign on the EOFs and the corresponding PCs does not alter the system. We
can thus see the ‘remarkable’ similarity by changing the sign of the EOFs in figures A1a
and A2a (while also changing the sign of the PCs in figures A1b and A2b). The figures
have been revised to incorporate this change.
15.
Figure 6 presents an unduly pessimistic picture. The lower panel shows
something close to 0/0. No wonder that figure 8 shows such an improvement.
Our analysis yields extremely precise radiances except for regions of very strong gas
absorption (saturated lines), where you observe the 0/0 phenomena. However, note that
the radiances in figure 7 are just obtained by convolving those in figure 5. So, a 0/0 in the
latter should normally be reflected in the former. The improvement shown in figure 8
then indicates that most of the errors introduced by principal component analysis are
random, and get canceled during convolution.
Response to Second Referee’s Comments
General Comments
More description has been added to Sections 3-6 to make the paper more comprehensible.
The grammatical errors mentioned have been corrected. Figure captions have been
expanded.
1.
“The levels are spaced to keep the interpolation errors … to a minimum.” They
modified the ECMWF atmosphere, as is also obvious from Table 1; how was the
re-layering done?
The levels are spaced linearly in log(pressure) from 1 mbar to 1 bar.
2.
The sentence “The difference in radiance between any two wavenumbers is much
smaller ….” is troubling, since it is not explained how the difference between
calculations at different wavenumbers is used for calculational advantage.
Let D(i) and T(i) be the radiances produced by DISORT and TWOSTR respectively at
the i-th wavenumber pixel. The observation we were making was that D(j)-D(k) is much
larger than [D(j)-T(j)]-[D(k)-T(k)], where j and k are any two wavenumber pixels. In
other words, the variance is much lower for the residual than for the radiances directly
obtained from DISORT. In the expansion we use for performing EOF analysis, we
implicitly assume that the radiance has a quadratic relationship with the PCs. Clearly, the
smaller the variance in the radiances, the better the approximation would be. Then, what
we reconstruct are of course the residuals as well. To get back the actual radiance, we
then need to add back the twostream radiance at the particular wavenumber. Fortunately,
the twostream calculations themselves take negligible time. Thus, our method combines
the strengths of principal component analysis and the twostream RTM.
3.
It is never demonstrated later that PCA is a linear technique, and it is unclear
here in what regime the radiances are linear (as it turns out, they are linear in
DS-TS differences).
In our analysis, we actually use a three-term expansion in the principal component
analysis. This expansion assumes that the radiance has a quadratic relationship with the
PCs. Clearly, the smaller the variance in the radiances, the better the approximation
would be. As explained in the response to the previous comment, the variance is much
smaller for the residual; hence the choice.
4.
More troubling, case selection is not introduced, either here or in the Appendix.
The authors need to introduce it explicitly, explain how cases are used to derive
the PCs, and to clarify how PCs derived from one case relate to the 105 cases
needed to do the quantitative PC analysis. There is an overall missing description
of the method.
Each ‘case’ refers to a particular choice of the parameters c1, c2, c3, and c4. In other
words, it is a range of optical properties corresponding to a single EOF calculation. The
EOFs and PCs are then calculated using equations (5) and (6) in the revised manuscript.
Case selection is done so that wavenumbers at which the optical properties are similar are
treated together. For each case, once the EOFs and PCs are determined, a radiance
calculation is performed for the mean optical properties corresponding to that case and
for perturbations of magnitude one EOF (positive and negative) for each EOF used to
map back to the radiance. Equation (10) shows how these are then used to calculate the
actual radiance at TOA for each wavenumber.
5.
How are we to understand “in the P branch” here?
It has no special significance, just that the sample case considered corresponded to c1 =
0.25, c2 = 0.5, c3 = 0.7 and c4 = 1 (in the P branch).
6.
The fact that PC analysis is performed in logarithmic space should have been
introduced earlier, with a reference to the Appendix as to why it improves
computational efficiency (as was done for the following items).
It is now introduced in section 4, with a reference to the appendix as suggested.
7.
Explicitly define variance, referencing the covariance matrix, in an equation.
Equation 4 in the revised manuscript does that.
8.
Since perturbations are of magnitude one EOF, what are calculated are
differences, not derivatives.
The wording has been changed.
9.
Please explain further where the numbers come from: 9 calls to DS and TS for
each of the 105 cases, plus the TS calls for every wavenumber. I suspect this is
where the “difference in radiance between any two wavenumbers” comes in, but
the development needs to be much more explicit.
As mentioned in section 5, for each case a DISORT call is made for the mean optical
properties and for perturbations of magnitude one EOF (positive and negative) for each
EOF used to map back to radiance. Since we use four EOFs to reconstruct the radiance, a
total of 1+2*4 = 9 DISORT calls are required for each case.