This paper reports on ozone trends derived from observations (SWOOSH

daatset) and three versions of the TOMCAT chemistry-transport-model

(CTM). One of the CTMs (ML-TOMCAT) has been adjusted to satellite

observations, while the other two used meteorological data from

different reanalyses, ERA5 and its predescessor ERA Interim (up to

2019). Two types of regression models are used for ozone trend

estimates before and after the peak of stratospheric halogens

occurring in the middle 1990. The first is the ordinary-least-squares

regession (OLS), the second is the ridge regression. The main

idea behind the ridge regression is to introduce an additional

contraint in the cost function that minimises the fit coefficients.

Such a regression is generally recommended to avoid overfitting.

In general the ridge regression reduces the (absolute) trends (and all

other fit coefficients) and on the other hand reduces the variances

(and correlation) between regression model and the underlying data.

The trends after 1998 in the upper stratosphere are positive and

significant in agreement with other studies (~2%/decade, e.g.

Godin-Beekmann et al., 2022). The ridge regression roughly halves

these trends. Overall the paper is well written. Some issues still

need to be addressed before acceptance of the paper.

discussion points:

l. 30, l. 34 and other places: Differences in ozone trends at 100 hPa

from OLS and ridge are laerger than 4%/decase (7%/decade in the

tropics), but the trend uncertainties are on the order of 24%. This

means that these differences are not significantly different from

zero. More relevant is the difference in the upper stratosphere

(~1%/decade vs ~2%/decade), both significant. This should be mentioned

here.

l. 37: It is not surprising that ML-TOMVCAT agrees better with SWOOSH

than the other models. Satelite corrections are derived from the same

data that are also part of SWOOSH (e.g. MLS). This should be

mentioned here and also in the main text.

l. 102: A detailed comparison between ERAI-TOMCAT and ERA5-TOMCAT has

been reported in Li et al. 2022. In this paper the model data have

been extended to 2020, however, ERAI ends in 2019 and trends are only

reported up to 2018 for the ERAI-driven model. As the differences

between both models are not discussed in detail here but well covered

in Li et al. 2022, it could be safeley omitted from this paper.

l. 161: The MLS setup is very different from Li et al. 2022. For

instance, now twelve (monthly) trend terms are used instead of one

(annual) and more proxies are used (e.g. EP flux). Please motivate why

you added more terms into the regression.

l. 165: Here you mention the use of the EP flux proxy, but its

contribution to ozone changes is not discussed in the paper. Its

contribution needs to be added in Fig. 10.

l. 170: Only years 1991 and 1992 have been removed to avoid the use of

an aerosol proxy, but Pinatubo eruption affected more years, e.g. end

of 1990, 1993 and 1994. Please comment.

l. 175: Detrending means that the long-term trends in the proxies are

moved to the linear trend terms. In Weber et al. 2022 we argued argued

that the long-term dynamic trends are largely removed by the trends

in the proxies, so that linear trends are then approximating the

ODS related trends. In your case, the linear trends are combining

dynamic and chemical trends. That should be mentioned here.

l. 178: Collinearity means that both vectors (or timeseries) are 100%

correlated, which is not the case here. What you mean is that many

proxies are highly correlated with each other. It is suggested to

avoid the term collinear throughout the text.

l. 187: "OLS will be not robust and will result in inaccurate model."

I think this is not correct. The OLS regression model will yield the

same (overall) results after orthogonalising all proxies, so OLS

remains robust (as also your results show). The ridge regression is

another representation with different constraints, but not necessarily

better than OLS. Ridge and OLS derived trends in nearly all cases

agree to within the uncertainties of the trends (Figs 2 and 3).

Suggest to omit this sentence.

l. 202: What is the training dataset? Suggest omit "to the training data"

l. 203: Omit "when the MSE reaches the minimum"; reference to

Pedregosa et al. suffices.

l. 207: "cross-valdiated MSE" needs to be explained in the text. One

may also want to state the drawback of ridge regression: the fit

residuals (correlation between model and regression) will be larger

(smaller) than that from OLS.

l. 239: different period is used for ERAI. Does that have an effect on

the trends. Shouldn't ERAI be compared with other data using the same

period. see also comment earlier

l. 241: readability of numbers in the tables will be improved if only

one digit is only shown, e.g. -3.4(2.5) instead of -3-39(2.47).

l. 249: Within the uncertainties of both regressions the trend results

(ridge and OLS) are not different from each other! I think this should

be mentioned in the main text as well (see earlier comment). Is the

annual mean the average of the twelve monthly means?

Is the uncertainty of the annual trend the standard deviation from

taking the mean from the monthly values or are the uncertainties from

the individual months are error-propagated into the annual mean? Please

explain.

l. 265: mention here that the large differences in trends are within the

uncertainties of the individual trends (see above)

l. 335: In the lower stratosphere ridge and OLS are not reliable and

fail to capture the large variability. In addition, the data quality

of satelites is lower in this region. So the "linear relationship" is

not the issue here

l. 340: "These differences between OLS- and ridge- based ozone profile

trends imply that Ridge regression to some extent has improved the

reliability of the model in the presence of multi-collinearity." This

is not generally true as discussed above. Again: Differences between OLS and

ridge-based trends are within the uncertainties of the individual

trends.

l. 346: "Considering the nonlinear effect, the monthly terms of QBO

proxies are used for regression analyses" I do not understand what is

meant to be said here. Statement can be omitted.

l. 355: "corresponds to the more positive ozone trends in both

simulations". To me it is not clear how long-term ozone trends can be

associated with QBO (contains only periodic changes after detrending)

l. 358: "... may account for the more positive ozone trends", see

previous comment

l. 363; How is the anomaly defined (amplitude, i.e. max minus minimum

response relative to the long term ozone mean ozone times the sign of

the fit coefficient?). Please specifiy.

l. 388: ozone trends are only shown below 60degs, but solar response

up to 90degs. Ozone is not well sampled above 50-60 degs in the early

period by SWOOSH. Is the solar response a result from a fit solely

limited to the late period after 1998? Why are ozone trends above 60

degs not shown?

l. 395: use only single digits (see earlier comments). Is the table

needed as the numbers can be derived from Fig. 9?

l. 409: see commnts to l. 388. please add the results of the EP flux

proxy (I guess it is the vertical component of the EP flux)

l. 425: "The negative AO (AAO) indices in the extratropics ...". This

is evident in the models and ML-Tomcat above 60 degs but not in

SWOOSH. Can this be explained? Are the regressions above 60degs

problematic?

l. 444: "it is inappropriate to use the same regression model for all

locations" Not clear what is meant here, you mean you cannot use a

ridge regression with a contant tuning parameter or you mean OLS. As

discussed earlier I do not think that the use of OLS is inaproppriate.

l. 456: "The largest difference between OLS and Ridge regression

methods appears in the tropical lower stratosphere (with ~7 % per

decade difference at 100 hPa).", but do not forget the the trend

uncertainties for both reression are very high (~23%/decade)

technical (selected):

l. 37: change to "the SWOOSH dataset"

l. 58: "controlled by transport and" (omit "the")

l. 150: add Snow et al. 2014 (doi:10.1051/swsc/2014001) as reference

for the MgII index

l. 183: I am not sure if "objective function" is the right term,

suggest "cost function" instead.

l. 194: "as described in Hastie" (add "das described in")

l. 204: "the Python scikrit module" (add "the")

l. 220: better: "where MRE is minimum"

l. 226: "fit residuals", I guess you mean trends

l. 231: Reword: You probably mean less variability in the ridge model

and lower absolute fit coefficients in the ridge regression. Please reword.

l. 233: "insignificant due to large uncertainties )up to

24-24%/decade" (replace "with" with "due to" and remove "up to")

l. 233: "These large uncertainties" (remove "decreases and")

l. 239: "We note" (remove "should")

l. 249: change "compared between" to "derived from"

l. 256: "across all three" (remove "the")

l. 257: change "relatively" to "slightly"

l. 262: change "in the NH" to "at NH"

l. 280: "and ERA5 shows". remove "and" and start a new sentence here

l. 281: remove "more overestimated"

l. 289: change "monthly mean variations" to "seasonal variations"

l. 302: change  "... to some extent with smaller coefficients" to "absolute

ridge-based trends and fit coefficients are smaller" 

l. 310: "based on the ridge regression" (add "the")

l. 312: change "minimal" to "minimum"

l. 363: "QBO response on ozone" (add "response")

l. 373: change " there is a minimal solar cycle signal (negative

and statistically significant) at ~10 hPa" to "there is a negative

and statistically significant solar cycle response at ~10 hPa"

l. 403: "being about twice larger" (add "being")

l. 468: change "The negative AO/AAO coefficients" to "The negative

phase of AO/AAO"

There was one point I missed in my review. For trends from monthly mean ozone time series a correction is applied in the regression to account for autoregression (AR1). This correction does not change trends so much but increases the uncertainties due to the reduction of degree-of-freedom associated with AR. It can be applied to both OLS and ridge regression and should be done. If not, at least  a good reason should be given why it is not needed here.

Editor Review of the Manuscript "Stratospheric ozone trends and

attribution over 1984-2020 using ordinary and regularised multivariate

regression models" by Li et al.

As one of the reviewers of this manuscript did not commit a review

and the other review was quite positive, I decided to base the decision

of this manuscript on only one regular review and this editor review.

Although I am not an expert on regression methods, I find the paper

written clearly and understandable.  Especially, the uncertainties of

the derived ozone trends depending on the regression methods seem

important to me. Also the depiction of the contribution of the natural

processes to  ozone changes is described well.

I would, however, suggest some more discussion of the results: To me

it is not clear, in how far the the shown differences in regression

methods are now explaining the discrepancy in the lower stratosphere

that was first pointed out by Ball et al. (2020).  Besides the

variability induced by the regression method, is there a model

improvement with respect to the Multi-model-mean shown by Ball et

al. ?  Or is this only the difference between free running CCMs and

the CTM shown here.

What can be learned from the machine-learning results (ML-TOMCAT).

Does the similarity with SWOOSH suggest that the basic mechanisms are

well understood or would you expect this similarity as it is

constructed by machine-learning using the observations?

Why are the trends in the tropics so different between the two

re-analyses? Is this due to the vertical velocities?

Therefore I suggest minor revisions to include this discussion, that

would potentially bring the shown results better into the context of

the present literature.