the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Spatially aggregated climate indicators over Sweden (1860–2020), Part 1: Temperature
Abstract. Climate indicators are useful tools to synthesise climate information from multiple station timeseries into a single national indicator. The method applied should be spatially representative and robust over time. We introduce a new method, based on Empirical Orthogonal Functions (EOF) during the calibration period 1961–2018, in order to reconstruct the climate indicator for temperature in Sweden for the full 1860–2020 period of available observations.
The new method delivers results in good overall agreement with the reference method (i.e. arithmetic mean from selected stations in the reference network). Discrepancies are found prior to 1900, primarily related to the reduced number of active stations: the robustness of the indicator estimation is assessed by an ensemble computation with added random noise, which confirms that the ensemble spread increases significantly prior to 1880.
The present study establishes that the 10year running averaged temperature indicator rose from −1.03 °C in 1903 to +1.19 °C in 2010 (with respect to a mean value of 4.64 °C over the 1961–2018 calibration period), i.e. an increase by +2.22 °C in a century. The temperature difference between 1860 and 2020 was largest for winter (DJF) averages (+3 °C) and minimal for summer (+2 °C).
The leading EOF patterns illustrate the spatial modes of variability for climate variability, with a predominantly homogeneous, monomodal distribution for temperature. For precipitation, the first EOF pattern displays more pronounced regional features (maximum over the West coast), which is completed by a northsouth seesaw pattern for the second EOF. We illustrate that EOF patterns calculated from observation data reproduce the major features of EOF calculated from GridClim, a gridded dataset over Sweden, for annual and seasonal averages. The leading EOF patterns vary significantly for seasonal averages (DJF, MAM, JJA, SON) for temperature.
Finally, future developments of the EOFmethod are discussed for calculating regional aggregated climate indicators, their relationship to synoptic circulation patterns and the benefits of homogenisation of observation series.
The EOFbased method to compute a spatially aggregated indicator for precipitation is presented in a companion article (Sturm, 2024a). The code and data for this study is available on Zenodo (Sturm, 2024b).
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Status: closed (peer review stopped)

RC1: 'Comment on egusphere2024582', Anonymous Referee #1, 13 Apr 2024
Summary. Official climate indicators to characterise longterm climate change in Sweden are compiled by averaging station data belonging to a suitable network. The manuscript explores an alternative method to define longtime series climate indicators based on deriving spatial patterns of Empirical Orthogonal Functions during a recent period and reconstructing the past temperature field from the available more sparse observations of longer time series. The main conclusion of the manuscript is that the alternative method yields very similar results to the conventional method but has some additional advantages.
Recommendation: The manuscript's objective is sound, but it requires deep revisions, both in its structure and in the methods. Overall, the manuscript, especially the description of the data sets and station networks, is disorganized and confusing—it can be better presented to help the reader. Also, the last disunion section contains parts that would better fit the introduction, and the introduction fails to provide a suitable context about reconstruction methods for the reader.
Perhaps more importantly, the EOFbased method contains some technical errors that need to be addressed, although I believe these changes would not strongly impact the final results.
Main points:
1) The definition of climate indicators is actually a field reconstruction method, for which many alternatives exist in the literature. Estimating area averages and spatially resolved fields based on point observations has a long tradition in climatology and palaeoclimatology. For instance, the EOFbased method presented here has clear similarities with the hockeystick reconstruction method by Mann et al. (Nature,1998), but also with other variants, for instance, principal Components Regression by Luterbacher et al. (2002,10.1007/s0038200101966). Concerning more recent periods, the GISS, BEST or HadCRUT data sets use different methods to estimate area averages that could also be applied here. The introduction should provide more context to the reader to frame the present method in that context. Perhaps the author may want to check the paper by Smerdon ( 10.1002/wcc.149), although that paper is focused on paleoclimate reconstruction methods.
Also, reconstruction methods may target area averages (an easier task) or spatially resolved fields. Obviously, the latter are more challenging but provide more information. In the present study, the objective is clearly a spatially aggregated climate indicator, yet the author has chosen to apply a climate field reconstruction. The introduction should also explain the reasons for this choice, which, in principle, is not optimal for the target at hand.
2) Methodological aspects. As explained in a more detailed way below, the application of the EOFbased method is not totally correct. For instance, the method described here reconstructs the expansion temporal coefficients by projecting the available station observations onto the EOF patterns. However, when restricted to the location of the available observations, the EOF patterns are not orthogonal anymore they are only orthogonal over the full set of stations in which they were calculated. Thus, the way to 'reconstruct' the expansion coefficients is another one in the case of gappy data, as described in the book by von Storch and Zwiers (1999) Statistical Analysis in Climate Research. I believe that this methodological error is not determinant and that the application of the correct method will yield similar results. However, the methods need to be correctly applied in a published paper.
Particular points
3) The title is not very informative. The reader cannot infer that an alternative method is being applied and compared to the official method
4) The introduction is very short and, as I wrote before, does not provide context. This is mainly a methodological study, and so the reader would expect a summary of the existing methods, their deficiencies and strengths and the reason the author has chosen the EOFbased method in the introduction.
5) The times scale of the targeted climate indicators is not stated after the middle of the manuscript. So the reader cannot know if this study is targeting, daily, monthly, annual means
6) Table 1 includes one entry, Gridclim, and year of maximum that are described much later in the manuscript and not in the caption. This is confusing
7) (e.g. if increasing nearby urbanisation leads to the station no longer)
This sentence seems incomplete.
8) The lower plot in Fig. (1) indicate
indicates
9) line 85 'The present study focuses on annuallyresolved climate indicators
This paragraph is a repetition of a previous paragraph.
10) subsequently downscaled at 2.5 km with MEPS
what is MEPS ?
11) 'The GRIDC LIM project (Andersson et al., 2021), performed by SMHI, combines the regional European reanalysis UERRA
(Schimanke and Service, 2019; Schimanke et al., 2019) with station observations'
How are they combined? It would be helpful to have a summary here.
12) GridClimsub is used for the gapfilling of M ORA calibration dataset (338
for temperature)
gapfilling (temporal) , what does 338 mean ? Stations?
13) ..'corresponding to the M ORA reference network SMHIref (39 for temperature)'
At this stage in the manuscript, the MORA reference network is only defined in Figure caption 1, and not in the main text, if I am not mistaken. Anyway, I had to unsuccessfully search the text several times for an explanation of MORA reference network, and so the reader will feel quite irritated.
Does 39 refers to the number of stations? if yes, please state clearly 39 stations. Table 1, however, indicates an initial number of 8 and a maximum of 35, so 39 does not match either. Table 1 seems to indicate that this reference network is not frozen in time, so I am really confused.
14) The explanation for the MORA reference network comes *after* the previous paragraph. This needs to be amended.
15) 'In the present study, the original method was emulated by selecting the individual station with the longest record in eachmetastation, without coupling and homogenisation. Hence, the emulated reference network in this study has sparser data coverage, thereby slightly different result, than the climate indicator provided by SMHI. In the results section, this dataset is referred to as GridClimsub.'
I am totally confused here. In my understanding so far, the GridCLIM data set is a gridded data set resulting from a combination of stations and reanalysis for the period 1961–2018. So, it is not per se a station data set. Why is now the emulated reference station network denoted GridCLIMsub? Are the gridcells selected according to the position of the reference network? GridCLIMsub would cover only 19612018, so how can it emulate the reference network since 1860?
In the previous section, GrdCLIMsub was defined as the subset of gridcells colocated with the calibration network, not the reference network.
The data description needs to be much better organised and presented in the right order in the manuscript. Presently, it is confusing.
16) 'Standard linear algebra methods require the dataset to be complete, without any missing values '
This is not correct. EOFs can be computed from data sets with missing values. The entries in the covariance matrix can be estimated from the pairwise common period of two series. In this case, some usual properties of the covariance matrix are not fulfilled anymore, e.g., all eigenvalues are positive. Still, this is usually not an issue if the number of gaps is not very large. This is explained in section 13.2.8 (Gappy data) of the book Statistical Analysis in Climate Research by Storch and Zwiers (1999)
17) For each station, a linear regression (in leastsquare sense, i.e. minimising ε)
Please refer here to equation (2)
18) 'By construction, each pattern in the EOF matrix is “orthogonal” to each other, in other words spatially uncorrelated'
This is incorrect. The terms Orthogonal and uncorrelated are only equivalent if the mean value is zero. However, the spatial mean of the EOFs patterns is not zero. Mathematically:
Orthogonality of eigenvectors i and j means SUM_r { x_i(r) x_j(r) } = 0
Noncorrelation means SUM_r { (x_i(r) ) (x_j(r)  ) = 0
The spatial means and are not necessarily zero
'Accordingly, all time expansion vectors (A) are temporally uncorrelated
This is correct, since the temporal expansions are all centred, since the EOFs are computed from anomalies.'
19) Reconstruction of the temporal expansion coefficients. This is a crucial step in the methodology, which is theoretically incorrect. As explained in my main point 2, the expansion coefficients cannot be computed by projecting the available station temperature anomalies onto the EOF patterns since these patterns are not orthogonal over the subset of available stations. Another method needs to be applied here, which is summarised by the following equation:
T (r,t) = SUM_i alpha_i(t) eof_i(r) + epsilon (r,t)
where r denotes the station, i the EOF number and t time. Epsilon is the residual that needs to be minimised
for each time step. This minimisation amounts to the orthogonal projection if the eof_i patterns were orthogonal over the subset of stations. But if the patterns are not orthogonal, as it is now the case, the minimisation involves a usual multivariate regression. One implication is that the expansion coefficients of a particular EOF, say alpha_i(t), depend on how many EOFs are retained: if five are retailed, the results would be different from when 10 are retained. This is also explained in section 13.2.8 (Gappy data) in von Storch and Zwiers.
20) ' The corresponding measurement uncertainty for monthly means are reduced by a facto'
is reduced
'The corresponding measurement uncertainty for monthly means are reduced by a factor √30 '
is reduced
Both statements assume that the daily temperatures are not serially correlated, which is clearly incorrect. The same applies to the monthly means. The number of effective degrees of freedom, and thus of the error reduction, is smaller than 30 or 12, respeectively
21) Normalization of EOF patterns and expansion coefficients.
There is an ambiguity in the text about how the EOF patterns and the corresponding expansion coefficients have been normalized. In principle, this normalization is subjective: the eof patterns can be normalized to unity, and the physical units would be carried by the expansion coefficients, or vice versa. Also, any other option between them is mathematically possible. The reason is an eigenvector of a matrix multiplied by any real number is also an eigenvector. This is irrelevant when reconstructing the temperature anomalies  since only the respective normalization of the patterns and the expansion coefficients need to be consistent. It is, however, relevant when comparing the amplitude of the EOF patterns in separate calculations for GridClimall and GridClimref, for instance. In this case, the amplitude of the of eof patterns can only be meaningfully compared if expansion coefficients have been normalised to unit standard deviation and the physical units are carried by the spatial patterns. Is this the choice adopted in this study?
22) ' For each climate indicator (temperature and precipitation),*
This study should be just about temperature. This is confusing
23) 'As a reminder, Xc represents...'
This paragraph is repetitive and not necessary in a research paper.
24) 'which becomes pregnant when only few stations are active'
pregnant is not the right word here. Relevant is a better choice
25) 4.4 Comparison to studies of historic climate variability in Fennoscandia
I did not see the relevance of this section since this study does not really analyse the historical climate variability processes, amplitudes, forcings, etc.
Citation: https://doi.org/10.5194/egusphere2024582RC1 
AC1: 'Reply on RC1', Christophe Sturm, 30 Jul 2024
Dear reviewer,
I would hereby like to express my gratitude for the relevant comments you provided, in particular for the possibility to compute the EOF of gappy dataset. I was not aware of this method, and therefore developed a (somewhat cumbersome) workaround using a gapfilled observation over a calibration period.
Following your advice, I have undertaken an indepth description of the EOF methods, in order (i) to formally explicit the method with gapfilled calibration datasets used in the original manuscript, and (ii) describe the adaptation of the method for gappy datasets.
The responses are presented in the attached PDF file. Despite the appearances, it is not a revised manuscript: it is meant to address in detail the major revisions suggested by reviewers of this article (and partly comments on the companion article, applying the same method on precipitation observations, by the same reviewers plus one). Some parts (in particular the annex in the attached file) could be "recycled" in the revised version.
Best regards, Christophe

AC1: 'Reply on RC1', Christophe Sturm, 30 Jul 2024

RC2: 'Comment on egusphere2024582', Anonymous Referee #2, 30 Apr 2024
Review of Spatially aggregated climate indicators over Sweden, Part 1 by Sturm.
I have reviewed the paper by Sturm and, although I think it has some merit, I cannot accept it in its present structure. There are some major comments that the author must address, and I also enclose some suggestions that will make the paper more readable.
Major comments.
Major1. Abstract, lines 1516. The author mentions precipitation here, but the paper presented to evaluation refers to temperature.
Major 2. There is a general lack of detail in the way missing data is treated and in the way the algorithm works in general.
In particular, the way the EOFs are computed even though missing data exist, needs to be clarified. I think this technique is the same as the one used in DINEOF but avoiding the iterations which lead to convergence in DINEOF.
Beckers, J. M., and M. Rixen, 2003: EOF Calculations and Data Filling from Incomplete Oceanographic Datasets. J. Atmos. Oceanic Technol., 20, 1839–1856, https://doi.org/10.1175/15200426(2003)020<1839:ECADFF>2.0.CO;2.
Beckers, J.M., Barth, A., and AlveraAzcárate, A. (2006): DINEOF reconstruction of clouded images including error maps – application to the SeaSurface Temperature around Corsican Island, Ocean Sci., 2, 183–199, https://os.copernicus.org/articles/2/183/2006/os21832006.html
As the author does not mention the way missing data are treated, I cannot be sure, but I guess he uses zero instead of the missing data, which corresponds to the first iteration in the estimation of the leading EOF in DINEOF. The description of the algorithm must be improved. I guess the author uses X matrices which are m x N (m times and N sites), but he is not telling us. There is a missing 1/N in the expression leading to the covariance matrix, I guess. Equation (5) is wrong from my point of view. If the author uses boldface for vectors, then arrows on top of them are not needed. Page 9, lines 188189 must define how missing values are used. Are the corresponding anomalies substituted by 0? Then, I guess this is DINEOF. However, in line 201 (page 9) the author mentions that he uses the pseudoinverse. How is he computing it? Details are needed here. I guess the author is probably truncating negative or zero singular values. In this case, I would say the results are converging into DINEOF (first iteration).
Page 10, line 219. It says “Observations” but the author means anomalies.
Page 11, lines 242243 (again, page 14, lines 290 and 299). The author mentions that SVD patterns and EOFs are virtually identical. I would say they must be, since SVD on a single dataset is equivalent to an EOF analysis. See section 4.5 in this document
https://atmos.washington.edu/~dennis/552_Notes_4.pdf
I might accept that there is a difference due to the consideration of missing data (substituted into zero??) in SVD or EOF or both, but this must be clarified. Again in page 19, line 394. The author mentions the EOFbased and SVDbased reconstructions. Which is the difference between both?
Figures 2, 3, 4 and 5 must be reworked.
First, the author must tell us how the EOFs are scaled. Are they represented in the original variable of the field? Then, the units must be stated, and this scaling must be explained.
Second, I, at least, cannot read the numbers, neither in the color scale nor in the X and Y axes of the time series.
I suggest that the figures are split, so that in one of the figures the author shows the spatial patterns and in a different one, the author shows the time series. In the time series plots, time series from EOF/GridClim and MORA can be mixed in 3 plots (instead of six subplots) by using blue and red lines for every PC. This will take less space, will allow the readers to compare the time series and will alow us to read the axes.
Quantitative results regarding the EOFs (congruence coefficient like in Cheng et al, 1995) and the PCs (R^2) are very useful in this part.
Cheng, X., G. Nitsche, and J. M. Wallace, 1995: Robustness of LowFrequency Circulation Patterns Derived from EOF and Rotated EOF Analyses. J. Climate, 8, 1709–1713, https://doi.org/10.1175/15200442(1995)008<1709:ROLFCP>2.0.CO;2.
Figures 4 and 5 are even worse. Use (if you wish) only the spatial patterns (same scaling, same scale bar) in this figure, plot the scale only one, larger, so that we can read the numbers. Convert the 12 panels of time series into a smaller number of time series, plotting them together using different colors (blue, red) in the same time series. Make these time series plots (are they necessary?) larger, with X and Y axes readable.
Figure 6. I guess the bottom panel is the same one as above but filtered. The properties of the filter are not given in detail in the text. Colours cannot be read inside the box.
Page 17, line 337 “temperature” and line 340 “precipitation”.
Page 20. Please, clarify whether the EOF basis is computed also independently for the ANN case as is done for the seasonal case.
Page 32, line 694. I don’t agree that SVD is a further development of EOFs. EOFs can be calculated by getting the eigenvalues of the (full rank) covariance matrix, by using the QR decomposition to find the eigenvectors or just by using the SVD decomposition, which is a different algorithm (that can be used for finding the EOFs and for other interesting linearalgebra operations).
Minor comments
Minor1. Abstract line #10. Is the value of the indicator significant to one hundredth of a degree? I can accept that, I am just asking the author to think twice about it.
Page 3, line 64. A verb is missing after “no longer”
Page 5. Lines 8889. What happens if one single month is missing? Is the site removed? Please, clarify.
Page 6, lines 118119. Please, state which is the method used to identify which grid cells correspond to MORA calibration. Nearest neighbour? Bilinear interpolation?
Page 7. Line 164 (+/). Please, tell us whether you are removing the whole period average from all variables (sites) or the whole period averaged from every variable. Are you removing the full average or the monthly averages for every month? This also affects page 18, line 374.
Page 11. Line 248. Represent > representS
Page 15. Line 308. “effect of timespace averaging order”. Please explain why they are different. It is not evident to me.
Suggestions for improvements, do as you wish
Figure 1. I wonder whether this figure can be improved if a second subpanel is attached to its bottom, reflecting a time series of continentality (Johansson’s or Conrad’s Continentality index, for instance) in the stations active in the dataset.
General comment. In the comparison of indices, I think that there is not any specific test against the geometry in the distribution of missing values. Does the author have any idea on whether this spatial distribution is biased in some way? Is it completely random?
Citation: https://doi.org/10.5194/egusphere2024582RC2 
AC2: 'Reply on RC2', Christophe Sturm, 30 Jul 2024
Dear reviewer,
I apologise for the delay in responding (colliding partly with summer holidays): it took me a certain amount of time to revisit the theoretical basis for EOF analysis (as presented in the annex of the attached PDF file), and to implement the new methodology.
As mentioned in my response to the first reviewer, I am very grateful to the comments both reviewers provided. It allowed me to substantially improve the methodology by computing EOF directly on a dataset with missing values (which I previously wasn't aware of), instead of the (somewhat complicated) workaround with a gapfilled dataset over a calibration period. Since the request for major revisions were similar in both reviews (including commentaries on the companion manuscript, applying the same methodology to precipitation observations), I decided to address all comments in a PDF file, attached to my response to RC1 (cf. link below).
The full response to the requests for major revisions are presented in the PDF file attached to my response to RC1 (cf. link below). Despite its length, this response does not contain the revised manuscript, but a detailed response to issues raised by reviewers. Some parts (e.g. the annex) could however be included in the revised version, pending on the editor's recommendations.
Best regards, Christophe
Link to the PDF file in the response to RC1: https://egusphere.copernicus.org/index.php?_mdl=msover_md&_jrl=778&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=118425&c=269945&salt=1712524932683526288
Citation: https://doi.org/10.5194/egusphere2024582AC2

AC2: 'Reply on RC2', Christophe Sturm, 30 Jul 2024
Status: closed (peer review stopped)

RC1: 'Comment on egusphere2024582', Anonymous Referee #1, 13 Apr 2024
Summary. Official climate indicators to characterise longterm climate change in Sweden are compiled by averaging station data belonging to a suitable network. The manuscript explores an alternative method to define longtime series climate indicators based on deriving spatial patterns of Empirical Orthogonal Functions during a recent period and reconstructing the past temperature field from the available more sparse observations of longer time series. The main conclusion of the manuscript is that the alternative method yields very similar results to the conventional method but has some additional advantages.
Recommendation: The manuscript's objective is sound, but it requires deep revisions, both in its structure and in the methods. Overall, the manuscript, especially the description of the data sets and station networks, is disorganized and confusing—it can be better presented to help the reader. Also, the last disunion section contains parts that would better fit the introduction, and the introduction fails to provide a suitable context about reconstruction methods for the reader.
Perhaps more importantly, the EOFbased method contains some technical errors that need to be addressed, although I believe these changes would not strongly impact the final results.
Main points:
1) The definition of climate indicators is actually a field reconstruction method, for which many alternatives exist in the literature. Estimating area averages and spatially resolved fields based on point observations has a long tradition in climatology and palaeoclimatology. For instance, the EOFbased method presented here has clear similarities with the hockeystick reconstruction method by Mann et al. (Nature,1998), but also with other variants, for instance, principal Components Regression by Luterbacher et al. (2002,10.1007/s0038200101966). Concerning more recent periods, the GISS, BEST or HadCRUT data sets use different methods to estimate area averages that could also be applied here. The introduction should provide more context to the reader to frame the present method in that context. Perhaps the author may want to check the paper by Smerdon ( 10.1002/wcc.149), although that paper is focused on paleoclimate reconstruction methods.
Also, reconstruction methods may target area averages (an easier task) or spatially resolved fields. Obviously, the latter are more challenging but provide more information. In the present study, the objective is clearly a spatially aggregated climate indicator, yet the author has chosen to apply a climate field reconstruction. The introduction should also explain the reasons for this choice, which, in principle, is not optimal for the target at hand.
2) Methodological aspects. As explained in a more detailed way below, the application of the EOFbased method is not totally correct. For instance, the method described here reconstructs the expansion temporal coefficients by projecting the available station observations onto the EOF patterns. However, when restricted to the location of the available observations, the EOF patterns are not orthogonal anymore they are only orthogonal over the full set of stations in which they were calculated. Thus, the way to 'reconstruct' the expansion coefficients is another one in the case of gappy data, as described in the book by von Storch and Zwiers (1999) Statistical Analysis in Climate Research. I believe that this methodological error is not determinant and that the application of the correct method will yield similar results. However, the methods need to be correctly applied in a published paper.
Particular points
3) The title is not very informative. The reader cannot infer that an alternative method is being applied and compared to the official method
4) The introduction is very short and, as I wrote before, does not provide context. This is mainly a methodological study, and so the reader would expect a summary of the existing methods, their deficiencies and strengths and the reason the author has chosen the EOFbased method in the introduction.
5) The times scale of the targeted climate indicators is not stated after the middle of the manuscript. So the reader cannot know if this study is targeting, daily, monthly, annual means
6) Table 1 includes one entry, Gridclim, and year of maximum that are described much later in the manuscript and not in the caption. This is confusing
7) (e.g. if increasing nearby urbanisation leads to the station no longer)
This sentence seems incomplete.
8) The lower plot in Fig. (1) indicate
indicates
9) line 85 'The present study focuses on annuallyresolved climate indicators
This paragraph is a repetition of a previous paragraph.
10) subsequently downscaled at 2.5 km with MEPS
what is MEPS ?
11) 'The GRIDC LIM project (Andersson et al., 2021), performed by SMHI, combines the regional European reanalysis UERRA
(Schimanke and Service, 2019; Schimanke et al., 2019) with station observations'
How are they combined? It would be helpful to have a summary here.
12) GridClimsub is used for the gapfilling of M ORA calibration dataset (338
for temperature)
gapfilling (temporal) , what does 338 mean ? Stations?
13) ..'corresponding to the M ORA reference network SMHIref (39 for temperature)'
At this stage in the manuscript, the MORA reference network is only defined in Figure caption 1, and not in the main text, if I am not mistaken. Anyway, I had to unsuccessfully search the text several times for an explanation of MORA reference network, and so the reader will feel quite irritated.
Does 39 refers to the number of stations? if yes, please state clearly 39 stations. Table 1, however, indicates an initial number of 8 and a maximum of 35, so 39 does not match either. Table 1 seems to indicate that this reference network is not frozen in time, so I am really confused.
14) The explanation for the MORA reference network comes *after* the previous paragraph. This needs to be amended.
15) 'In the present study, the original method was emulated by selecting the individual station with the longest record in eachmetastation, without coupling and homogenisation. Hence, the emulated reference network in this study has sparser data coverage, thereby slightly different result, than the climate indicator provided by SMHI. In the results section, this dataset is referred to as GridClimsub.'
I am totally confused here. In my understanding so far, the GridCLIM data set is a gridded data set resulting from a combination of stations and reanalysis for the period 1961–2018. So, it is not per se a station data set. Why is now the emulated reference station network denoted GridCLIMsub? Are the gridcells selected according to the position of the reference network? GridCLIMsub would cover only 19612018, so how can it emulate the reference network since 1860?
In the previous section, GrdCLIMsub was defined as the subset of gridcells colocated with the calibration network, not the reference network.
The data description needs to be much better organised and presented in the right order in the manuscript. Presently, it is confusing.
16) 'Standard linear algebra methods require the dataset to be complete, without any missing values '
This is not correct. EOFs can be computed from data sets with missing values. The entries in the covariance matrix can be estimated from the pairwise common period of two series. In this case, some usual properties of the covariance matrix are not fulfilled anymore, e.g., all eigenvalues are positive. Still, this is usually not an issue if the number of gaps is not very large. This is explained in section 13.2.8 (Gappy data) of the book Statistical Analysis in Climate Research by Storch and Zwiers (1999)
17) For each station, a linear regression (in leastsquare sense, i.e. minimising ε)
Please refer here to equation (2)
18) 'By construction, each pattern in the EOF matrix is “orthogonal” to each other, in other words spatially uncorrelated'
This is incorrect. The terms Orthogonal and uncorrelated are only equivalent if the mean value is zero. However, the spatial mean of the EOFs patterns is not zero. Mathematically:
Orthogonality of eigenvectors i and j means SUM_r { x_i(r) x_j(r) } = 0
Noncorrelation means SUM_r { (x_i(r) ) (x_j(r)  ) = 0
The spatial means and are not necessarily zero
'Accordingly, all time expansion vectors (A) are temporally uncorrelated
This is correct, since the temporal expansions are all centred, since the EOFs are computed from anomalies.'
19) Reconstruction of the temporal expansion coefficients. This is a crucial step in the methodology, which is theoretically incorrect. As explained in my main point 2, the expansion coefficients cannot be computed by projecting the available station temperature anomalies onto the EOF patterns since these patterns are not orthogonal over the subset of available stations. Another method needs to be applied here, which is summarised by the following equation:
T (r,t) = SUM_i alpha_i(t) eof_i(r) + epsilon (r,t)
where r denotes the station, i the EOF number and t time. Epsilon is the residual that needs to be minimised
for each time step. This minimisation amounts to the orthogonal projection if the eof_i patterns were orthogonal over the subset of stations. But if the patterns are not orthogonal, as it is now the case, the minimisation involves a usual multivariate regression. One implication is that the expansion coefficients of a particular EOF, say alpha_i(t), depend on how many EOFs are retained: if five are retailed, the results would be different from when 10 are retained. This is also explained in section 13.2.8 (Gappy data) in von Storch and Zwiers.
20) ' The corresponding measurement uncertainty for monthly means are reduced by a facto'
is reduced
'The corresponding measurement uncertainty for monthly means are reduced by a factor √30 '
is reduced
Both statements assume that the daily temperatures are not serially correlated, which is clearly incorrect. The same applies to the monthly means. The number of effective degrees of freedom, and thus of the error reduction, is smaller than 30 or 12, respeectively
21) Normalization of EOF patterns and expansion coefficients.
There is an ambiguity in the text about how the EOF patterns and the corresponding expansion coefficients have been normalized. In principle, this normalization is subjective: the eof patterns can be normalized to unity, and the physical units would be carried by the expansion coefficients, or vice versa. Also, any other option between them is mathematically possible. The reason is an eigenvector of a matrix multiplied by any real number is also an eigenvector. This is irrelevant when reconstructing the temperature anomalies  since only the respective normalization of the patterns and the expansion coefficients need to be consistent. It is, however, relevant when comparing the amplitude of the EOF patterns in separate calculations for GridClimall and GridClimref, for instance. In this case, the amplitude of the of eof patterns can only be meaningfully compared if expansion coefficients have been normalised to unit standard deviation and the physical units are carried by the spatial patterns. Is this the choice adopted in this study?
22) ' For each climate indicator (temperature and precipitation),*
This study should be just about temperature. This is confusing
23) 'As a reminder, Xc represents...'
This paragraph is repetitive and not necessary in a research paper.
24) 'which becomes pregnant when only few stations are active'
pregnant is not the right word here. Relevant is a better choice
25) 4.4 Comparison to studies of historic climate variability in Fennoscandia
I did not see the relevance of this section since this study does not really analyse the historical climate variability processes, amplitudes, forcings, etc.
Citation: https://doi.org/10.5194/egusphere2024582RC1 
AC1: 'Reply on RC1', Christophe Sturm, 30 Jul 2024
Dear reviewer,
I would hereby like to express my gratitude for the relevant comments you provided, in particular for the possibility to compute the EOF of gappy dataset. I was not aware of this method, and therefore developed a (somewhat cumbersome) workaround using a gapfilled observation over a calibration period.
Following your advice, I have undertaken an indepth description of the EOF methods, in order (i) to formally explicit the method with gapfilled calibration datasets used in the original manuscript, and (ii) describe the adaptation of the method for gappy datasets.
The responses are presented in the attached PDF file. Despite the appearances, it is not a revised manuscript: it is meant to address in detail the major revisions suggested by reviewers of this article (and partly comments on the companion article, applying the same method on precipitation observations, by the same reviewers plus one). Some parts (in particular the annex in the attached file) could be "recycled" in the revised version.
Best regards, Christophe

AC1: 'Reply on RC1', Christophe Sturm, 30 Jul 2024

RC2: 'Comment on egusphere2024582', Anonymous Referee #2, 30 Apr 2024
Review of Spatially aggregated climate indicators over Sweden, Part 1 by Sturm.
I have reviewed the paper by Sturm and, although I think it has some merit, I cannot accept it in its present structure. There are some major comments that the author must address, and I also enclose some suggestions that will make the paper more readable.
Major comments.
Major1. Abstract, lines 1516. The author mentions precipitation here, but the paper presented to evaluation refers to temperature.
Major 2. There is a general lack of detail in the way missing data is treated and in the way the algorithm works in general.
In particular, the way the EOFs are computed even though missing data exist, needs to be clarified. I think this technique is the same as the one used in DINEOF but avoiding the iterations which lead to convergence in DINEOF.
Beckers, J. M., and M. Rixen, 2003: EOF Calculations and Data Filling from Incomplete Oceanographic Datasets. J. Atmos. Oceanic Technol., 20, 1839–1856, https://doi.org/10.1175/15200426(2003)020<1839:ECADFF>2.0.CO;2.
Beckers, J.M., Barth, A., and AlveraAzcárate, A. (2006): DINEOF reconstruction of clouded images including error maps – application to the SeaSurface Temperature around Corsican Island, Ocean Sci., 2, 183–199, https://os.copernicus.org/articles/2/183/2006/os21832006.html
As the author does not mention the way missing data are treated, I cannot be sure, but I guess he uses zero instead of the missing data, which corresponds to the first iteration in the estimation of the leading EOF in DINEOF. The description of the algorithm must be improved. I guess the author uses X matrices which are m x N (m times and N sites), but he is not telling us. There is a missing 1/N in the expression leading to the covariance matrix, I guess. Equation (5) is wrong from my point of view. If the author uses boldface for vectors, then arrows on top of them are not needed. Page 9, lines 188189 must define how missing values are used. Are the corresponding anomalies substituted by 0? Then, I guess this is DINEOF. However, in line 201 (page 9) the author mentions that he uses the pseudoinverse. How is he computing it? Details are needed here. I guess the author is probably truncating negative or zero singular values. In this case, I would say the results are converging into DINEOF (first iteration).
Page 10, line 219. It says “Observations” but the author means anomalies.
Page 11, lines 242243 (again, page 14, lines 290 and 299). The author mentions that SVD patterns and EOFs are virtually identical. I would say they must be, since SVD on a single dataset is equivalent to an EOF analysis. See section 4.5 in this document
https://atmos.washington.edu/~dennis/552_Notes_4.pdf
I might accept that there is a difference due to the consideration of missing data (substituted into zero??) in SVD or EOF or both, but this must be clarified. Again in page 19, line 394. The author mentions the EOFbased and SVDbased reconstructions. Which is the difference between both?
Figures 2, 3, 4 and 5 must be reworked.
First, the author must tell us how the EOFs are scaled. Are they represented in the original variable of the field? Then, the units must be stated, and this scaling must be explained.
Second, I, at least, cannot read the numbers, neither in the color scale nor in the X and Y axes of the time series.
I suggest that the figures are split, so that in one of the figures the author shows the spatial patterns and in a different one, the author shows the time series. In the time series plots, time series from EOF/GridClim and MORA can be mixed in 3 plots (instead of six subplots) by using blue and red lines for every PC. This will take less space, will allow the readers to compare the time series and will alow us to read the axes.
Quantitative results regarding the EOFs (congruence coefficient like in Cheng et al, 1995) and the PCs (R^2) are very useful in this part.
Cheng, X., G. Nitsche, and J. M. Wallace, 1995: Robustness of LowFrequency Circulation Patterns Derived from EOF and Rotated EOF Analyses. J. Climate, 8, 1709–1713, https://doi.org/10.1175/15200442(1995)008<1709:ROLFCP>2.0.CO;2.
Figures 4 and 5 are even worse. Use (if you wish) only the spatial patterns (same scaling, same scale bar) in this figure, plot the scale only one, larger, so that we can read the numbers. Convert the 12 panels of time series into a smaller number of time series, plotting them together using different colors (blue, red) in the same time series. Make these time series plots (are they necessary?) larger, with X and Y axes readable.
Figure 6. I guess the bottom panel is the same one as above but filtered. The properties of the filter are not given in detail in the text. Colours cannot be read inside the box.
Page 17, line 337 “temperature” and line 340 “precipitation”.
Page 20. Please, clarify whether the EOF basis is computed also independently for the ANN case as is done for the seasonal case.
Page 32, line 694. I don’t agree that SVD is a further development of EOFs. EOFs can be calculated by getting the eigenvalues of the (full rank) covariance matrix, by using the QR decomposition to find the eigenvectors or just by using the SVD decomposition, which is a different algorithm (that can be used for finding the EOFs and for other interesting linearalgebra operations).
Minor comments
Minor1. Abstract line #10. Is the value of the indicator significant to one hundredth of a degree? I can accept that, I am just asking the author to think twice about it.
Page 3, line 64. A verb is missing after “no longer”
Page 5. Lines 8889. What happens if one single month is missing? Is the site removed? Please, clarify.
Page 6, lines 118119. Please, state which is the method used to identify which grid cells correspond to MORA calibration. Nearest neighbour? Bilinear interpolation?
Page 7. Line 164 (+/). Please, tell us whether you are removing the whole period average from all variables (sites) or the whole period averaged from every variable. Are you removing the full average or the monthly averages for every month? This also affects page 18, line 374.
Page 11. Line 248. Represent > representS
Page 15. Line 308. “effect of timespace averaging order”. Please explain why they are different. It is not evident to me.
Suggestions for improvements, do as you wish
Figure 1. I wonder whether this figure can be improved if a second subpanel is attached to its bottom, reflecting a time series of continentality (Johansson’s or Conrad’s Continentality index, for instance) in the stations active in the dataset.
General comment. In the comparison of indices, I think that there is not any specific test against the geometry in the distribution of missing values. Does the author have any idea on whether this spatial distribution is biased in some way? Is it completely random?
Citation: https://doi.org/10.5194/egusphere2024582RC2 
AC2: 'Reply on RC2', Christophe Sturm, 30 Jul 2024
Dear reviewer,
I apologise for the delay in responding (colliding partly with summer holidays): it took me a certain amount of time to revisit the theoretical basis for EOF analysis (as presented in the annex of the attached PDF file), and to implement the new methodology.
As mentioned in my response to the first reviewer, I am very grateful to the comments both reviewers provided. It allowed me to substantially improve the methodology by computing EOF directly on a dataset with missing values (which I previously wasn't aware of), instead of the (somewhat complicated) workaround with a gapfilled dataset over a calibration period. Since the request for major revisions were similar in both reviews (including commentaries on the companion manuscript, applying the same methodology to precipitation observations), I decided to address all comments in a PDF file, attached to my response to RC1 (cf. link below).
The full response to the requests for major revisions are presented in the PDF file attached to my response to RC1 (cf. link below). Despite its length, this response does not contain the revised manuscript, but a detailed response to issues raised by reviewers. Some parts (e.g. the annex) could however be included in the revised version, pending on the editor's recommendations.
Best regards, Christophe
Link to the PDF file in the response to RC1: https://egusphere.copernicus.org/index.php?_mdl=msover_md&_jrl=778&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=118425&c=269945&salt=1712524932683526288
Citation: https://doi.org/10.5194/egusphere2024582AC2

AC2: 'Reply on RC2', Christophe Sturm, 30 Jul 2024
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