the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Spatially aggregated climate indicators over Sweden (1860–2020), part 2: Precipitation
Abstract. The Swedish Meteorology and Hydrology Institute (SMHI) provides a national aggregated climate indicator from 1860 to present. We present a new method to compute the national climate indicator based on Empirical Orthogonal Functions (EOF). EOF are computed during the1961–2018 calibration period, and later applied to the full experiment period 1860–2020. This study focuses the climate indicator for precipitation; it follows the same methodology as for the national climate indicator for temperature, described in the companion article (Sturm, 2024a).
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 precipitation indicator rose from 8.37 mm.month^{1} in 1903 to 4.08 mm.month^{1} in 2010 (with respect to the mean value of 54.18 mm.month^{1} for the 1961–2018 calibration period), i.e. a 27 % increase over a century. Winter (DJF) precipitation rose by +20 mm.month^{1 }between 1890–2010, summer precipitation by +25 mm.month^{1}.
The leading EOF patterns illustrate the spatial modes of variability for climate variability. 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 precipitation.
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 temperature is presented in a companion article (Sturm, 2024a), which includes a detailed description of the datasets and methods used in this study. 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 egusphere2024940', Anonymous Referee #1, 21 May 2024
Summary: This is the second part of two manuscripts aimed at deriving spatially averaged climate indicators (temperature, precipitation) for Sweden over the 20th and 21st centuries. The method used data from long instrumental records and short highresolution reanalysis and applied a statistical method based on Empirical Orthogonal Functions.
The second manuscript is focused on seasonal and annual precipitation. The main conclusion is that the EOFbased method is superior to the more simple arithmetic average of available stations, and this improvement is more noticeable in the case of precipitation.
Recommendation: I also reviewed the first manuscript on temperature, and many of the shortcomings that were discussed in that review also applied to this second part. I guess that both manuscripts will be revised and eventually published in parallel, so I will not repeat in detail the methodological concerns that I expressed for the first manuscript. I also think that this manuscript requires considerable revisions and that there are several questionable methodological steps that need to be addressed.
Main points
1) Again, the EOFreconstruction method is the same as in the first manuscript, and it is not totally correct. It is unclear how a correct application of the method would affect the results, but certainly, a published manuscript needs to contain statistically correct steps. Here, for instance, the third line n equation 5 is not correct, as the EOF patterns are not orthogonal on the subset of available stations. They are only orthogonal on the full grid on which they were calculated. There are alternative methods to estimate the time scores a(t) when data are missing (see Storch and Zwiers, 1999).
2) Some usage of units in the text is unclear, especially when stating the precipitation temporal trends
3) The structure of the manuscript is somewhat not optimal, but this can be a matter of writing style or personal taste. I found the discussion section disorganized, including different subsections that are not really well connected. It looks rather like afterthoughts the author felt needed to be somehow included. Perhaps the authors may want to revise the structure of this section with a fresh mind after some time has passed since submission.
Also, the description of the different data sets is a bit chaotic and difficult to follow. There are different grids and station sets, and some are combinations of others. A table may help the reader quickly reference the text.
There are, however, some substantial points that do need revisions. For instance, the manuscript mentions the time scale of the indicators only almost at the end of the introduction (seasonal means. annual means). Until then, the reader does not know whether the author is referring to daily means or centennial means. Even the title could be more specific in this regard. The Mora station set is mentioned as an acronym before it is described. The manuscript should undergo careful revisions to avoid this type of inconsistencies.
Particular points
4) Neither the title nor the abstract mentions the time scale of the climate indicators. It should be clearly stated that the indicators are annual and seasonal means of precipitation.
5) 'The present study establishes that the 10year running averaged precipitation indicator rose from −8.37 mm.month−1 in 1903 to 4.08 mm.month−1 in 2010 (with respect to the mean value of 54.18 mm.month−1 for the 1961–2018 calibration period), i.e. a 27% increase over a century. Winter (DJF) precipitation rose by +20 mm.month−1 between 1890–2010, summer precipitation by +25 mm.month−1 .'
This is an example of what I think is an unclear usage of units. In the last sentence, the author propably means that winter mean precipitation rose by 75 mm over the period 18902010. This amounts to a rise of winter total precipitation of 75mm/110years = 0.6 mm per year. The formulation in the text, however, could be misinterpreted as a trend of total winter precipitation of 20mm/month, which of course physically impossible.
6) 'The EOFbased method to compute a spatially aggregated indicator for temperature is presented in a companion article (Sturm, 2024a), which includes a detailed description of the datasets and methods used in this study'.
This a repetition in the abstract
7)' In order words, this study evaluates the representativeness of the M ORA observation network for the entire Swedish territory.'
The MORA network is described later. The reader does not know at this point what MORA is.,
8) Data and methods
Are the EOFs computed separately for different seasons? What is the time step (monthly, seasonal) ? This questions are only described later. They should be described here.
9) Table1: Third column (years) is not defined in the caption. The reader has to figure out the meaning
10) 'Assuming that the daily observation errors are normally distributed around 0 (with a –conservative– estimate of measurement uncertainty of εprec = ±10 mm.day −1 for precipitation), their corresponding error for monthly means are reduced by a factor √30 √ nmonth , i.e. the average amount of daily measurements in a monthly mean. nmonth represents the number of monthly ·records in the annuallyresolved average: for annual means, nmonth = 12, for seasonal means, nmonth = 3. Hence the random noise function ε is normally distributed, with a standard deviation of σAN precipitationN= √30·10√nmonth 0.53 mm.month−1 for annual mean precipitation and σseas precipitation = 1.05 mm.month−1 for seasonal means.'
This paragraph on page 9 has several issues that need to be corrected. One is that the distribution of daily precipitation is clearly not normal, so the first condition would not be fulfilled. Fortunately, this condition is actually not necessary. The only condition required for the standard deviation of the estimator of the average of N random variables is that these random variables are independent  the distribution is irrelevant. The distribution of the average will approach the normal distribution as the number of averaged variables increases. This is the Large Numbers Theorem. However, more important here is that daily precipitation is not independent of previous or following days, and so the large Number Theorem cannot be applied directly here: an effective number of degrees of freedom needs to be estimated. This can be done from the autocorrelation function of daily precipitation.
This problem was also present in the first manuscript on temperatures. Here it is probably less important because daily precipitation tends to be less autocorrelated in time, but again, these calculations need to be correct. I guess that for daily precipitation, an autocorrelation of 2 days would be reasonable, reducing the effective number of degrees of freedom for monthly means by a factor of 3, from 30 to 10, and for annual means from 365 to 100. For temperature, this factor is probably larger.
11) 'By construction, plain (i.e. unrotated) spatial EOF patterns (without prior detrending) are expected to display a unimodal distribution in the first mode, a bimodal distribution in the second, and a trimodal in the third. This behaviour is clearly apparent in Fig. (2).'
This is not correct. There is no reason for the leading EOF to have a monopole structure. It is usually so for climate fields, but there are many examples in which this is not the case. For instance, the leading EOF of the global nondeseasonalized monthly temperature field is a dipole, since the Northern Hemisphere and the Southern Hemisphere are strongly negatively correlated
12) Figure 2,3 Which normalization, which units ? This is indirectly explained in the caption of Figure 4, but it would be clearer if the methods section explained the chosen normalization of the time components (unity) and including a colour bar with the temperature units for the spatial patterns.
13) 'This example illustrates that relatively simple linear algebra methods, such as EOF and SVD, have the potential to deliver valuable results for the analysis of climate observations. The analysis can be further refined using more advanced method, such as Principal Oscillation Patterns (POP), as presented in Storch et al. (1995).'
Again, this seems to come as an afterthought, but I do not think this is correct in this case. The POP method is designed to identify oscillations in a climate field , and not to capture the main patterns of variability, which is the important goal for climate field reconstruction. The POP method may identify an oscillatory pattern, which nevertheless may explain very little portion of the total variance and be of little use to construct a climate indicator.
Related to this comment, I wonder why these two manuscripts want to use a more complicated method than actually needed. If the target is to estimate the spatial average of temperature or precipitation, a multilinear regression, with station data as predictors and the spatial average as precipitated, can also be used: trained in the recent period (as in the calculation of the EOFs) and then applied to the longer period. The training would use the available stations for the reconstruction period so that a collection of multilinear regressions would probably be needed to account for dropping stations back in time. The EOFbased method is, in my view, also suitable when applied correctly, but in my view, it is more complex than necessary.
Citation: https://doi.org/10.5194/egusphere2024940RC1  RC2: 'Comment on egusphere2024940', Rasmus Benestad, 24 May 2024

RC3: 'Comment on egusphere2024940', Anonymous Referee #3, 24 May 2024
This paper is a continuation (second part) of a previous paper the author submitted to EGUSPHERE and which I also reviewed. The previous paper dealt with temperature.
I think that some results of the paper might be of interest to researchers in this line, however, I think that the paper suffers from significant pitfalls in its current status, so that I ask a major revision.
First of all, I understand that the author is going to uccessfully address all the methodological issues I raised in the previous paper https://egusphere.copernicus.org/preprints/2024/egusphere2024582 because many of them dealt with problems that affect the methodology used in that paper and in this one.
Following with this paper, I will first mention the MAJOR problems that need to be addressed.
1. A review of English is needed. There are many typos which should have been corrected before submission. See, for instance, page 2, line 30. "a numerical methods that aggregate" > "a numerical method that aggregates" or same page, line 36 "between to stations" > between two stations"
2. Values and uncertainties.The author estimates the trend with three significant digits, but I have seen no estimation of the uncertainty of this value. Why three significant digits and not 4, 5 or two? Considering that the erros of the individual measurements are propagated through the EOFs, the handling of missing data and so on, this would ask for a particular estimation (Monte Carlo?) of a confidence interval surrounding this value.
Continuing with this, there is a very strange value in the abstract, with a Running average precipitation indicator" 8.37 mm/month". I guess he refers to an anomaly. but this should be stated.
3. The way missing data are handled must be clearly stated. I guess that missing values are substituted by null anomalies but, in that case, I guess the author is falling into DINEOF with a single iteration. I gave the author more information on this in my previous review.
4. Clarification of the methodology. The author mentions in many instances EOF and SVD together. SVD can be used for coupled datasets or it can also be used to calculate the EOFs (the same way as alternative methologies such as QR decomposition might be used). This must be clarified. And the way the author calculated the pseudoinverse matrix (this is not explained). Again in page 9, the author seems to be surprised that SVD and EOF patterns are virtually identical. Well, they must actually be, SVD is a way (a good one, indeed) of calculating the EOFs.
5. Page 7 around line 140. The author says the dataset is centered. Are "columns" removed from the dataset? Are these columns removed after getting seasonal averages? Is the overal mean removed? This must be clarified.
6. The equation for the covariance matrix lackas a division by the number of samples.
7. Pages 7 and 8. It is custmary that vectors (spatial patterns in this paper) are represented either by mathematical boldface or by arrows on top of the vector. In this paper, both are used. The author should stick to common practice.
8. Figures can not generally be properly read. In the previous paper, I suggested many things to do in order to improve the readability of the figures. a) Fonts must be larger. b) Please, do not stack 12 panels in the same figure (24 panels in figure 4!!).... Better join figures using different line colors if they are timeseries. Do not waste too much space in color scales if using maps, share the same colormap for all the subplots in a fgure. The scale must be informative. Which are the units of the variable? How do they calculate this scaling for PCs? Are EOFs plot as regression of the original field onto the principal components? Tell us that is the case. Showinf¡g a map of the EOF loadings regridded is nice, but if you showed the placement of stations it would be better. Labels, fonts must be legible without a magnifying glass. Figures 6. and 7., for instance. labels are again unreadable. The colors used can not be seen. Use other compinetion of colors and thinner lines if the information is important, currently, we can't know, it can't be seen.
9) Table 3. Some characters are written using yellow. I can't read that over white background
10) Page 20, line 397. The EOF method deviates from SMHIref. This probably means that the method is very sensitive to the available stations. A sensitivity analysis using subsets of the available data should be performed.
Suggestions.
Figure 1. Add a continentality index in a lower panel, as mentioned in my previous review
Citation: https://doi.org/10.5194/egusphere2024940RC3
Status: closed (peer review stopped)

RC1: 'Comment on egusphere2024940', Anonymous Referee #1, 21 May 2024
Summary: This is the second part of two manuscripts aimed at deriving spatially averaged climate indicators (temperature, precipitation) for Sweden over the 20th and 21st centuries. The method used data from long instrumental records and short highresolution reanalysis and applied a statistical method based on Empirical Orthogonal Functions.
The second manuscript is focused on seasonal and annual precipitation. The main conclusion is that the EOFbased method is superior to the more simple arithmetic average of available stations, and this improvement is more noticeable in the case of precipitation.
Recommendation: I also reviewed the first manuscript on temperature, and many of the shortcomings that were discussed in that review also applied to this second part. I guess that both manuscripts will be revised and eventually published in parallel, so I will not repeat in detail the methodological concerns that I expressed for the first manuscript. I also think that this manuscript requires considerable revisions and that there are several questionable methodological steps that need to be addressed.
Main points
1) Again, the EOFreconstruction method is the same as in the first manuscript, and it is not totally correct. It is unclear how a correct application of the method would affect the results, but certainly, a published manuscript needs to contain statistically correct steps. Here, for instance, the third line n equation 5 is not correct, as the EOF patterns are not orthogonal on the subset of available stations. They are only orthogonal on the full grid on which they were calculated. There are alternative methods to estimate the time scores a(t) when data are missing (see Storch and Zwiers, 1999).
2) Some usage of units in the text is unclear, especially when stating the precipitation temporal trends
3) The structure of the manuscript is somewhat not optimal, but this can be a matter of writing style or personal taste. I found the discussion section disorganized, including different subsections that are not really well connected. It looks rather like afterthoughts the author felt needed to be somehow included. Perhaps the authors may want to revise the structure of this section with a fresh mind after some time has passed since submission.
Also, the description of the different data sets is a bit chaotic and difficult to follow. There are different grids and station sets, and some are combinations of others. A table may help the reader quickly reference the text.
There are, however, some substantial points that do need revisions. For instance, the manuscript mentions the time scale of the indicators only almost at the end of the introduction (seasonal means. annual means). Until then, the reader does not know whether the author is referring to daily means or centennial means. Even the title could be more specific in this regard. The Mora station set is mentioned as an acronym before it is described. The manuscript should undergo careful revisions to avoid this type of inconsistencies.
Particular points
4) Neither the title nor the abstract mentions the time scale of the climate indicators. It should be clearly stated that the indicators are annual and seasonal means of precipitation.
5) 'The present study establishes that the 10year running averaged precipitation indicator rose from −8.37 mm.month−1 in 1903 to 4.08 mm.month−1 in 2010 (with respect to the mean value of 54.18 mm.month−1 for the 1961–2018 calibration period), i.e. a 27% increase over a century. Winter (DJF) precipitation rose by +20 mm.month−1 between 1890–2010, summer precipitation by +25 mm.month−1 .'
This is an example of what I think is an unclear usage of units. In the last sentence, the author propably means that winter mean precipitation rose by 75 mm over the period 18902010. This amounts to a rise of winter total precipitation of 75mm/110years = 0.6 mm per year. The formulation in the text, however, could be misinterpreted as a trend of total winter precipitation of 20mm/month, which of course physically impossible.
6) 'The EOFbased method to compute a spatially aggregated indicator for temperature is presented in a companion article (Sturm, 2024a), which includes a detailed description of the datasets and methods used in this study'.
This a repetition in the abstract
7)' In order words, this study evaluates the representativeness of the M ORA observation network for the entire Swedish territory.'
The MORA network is described later. The reader does not know at this point what MORA is.,
8) Data and methods
Are the EOFs computed separately for different seasons? What is the time step (monthly, seasonal) ? This questions are only described later. They should be described here.
9) Table1: Third column (years) is not defined in the caption. The reader has to figure out the meaning
10) 'Assuming that the daily observation errors are normally distributed around 0 (with a –conservative– estimate of measurement uncertainty of εprec = ±10 mm.day −1 for precipitation), their corresponding error for monthly means are reduced by a factor √30 √ nmonth , i.e. the average amount of daily measurements in a monthly mean. nmonth represents the number of monthly ·records in the annuallyresolved average: for annual means, nmonth = 12, for seasonal means, nmonth = 3. Hence the random noise function ε is normally distributed, with a standard deviation of σAN precipitationN= √30·10√nmonth 0.53 mm.month−1 for annual mean precipitation and σseas precipitation = 1.05 mm.month−1 for seasonal means.'
This paragraph on page 9 has several issues that need to be corrected. One is that the distribution of daily precipitation is clearly not normal, so the first condition would not be fulfilled. Fortunately, this condition is actually not necessary. The only condition required for the standard deviation of the estimator of the average of N random variables is that these random variables are independent  the distribution is irrelevant. The distribution of the average will approach the normal distribution as the number of averaged variables increases. This is the Large Numbers Theorem. However, more important here is that daily precipitation is not independent of previous or following days, and so the large Number Theorem cannot be applied directly here: an effective number of degrees of freedom needs to be estimated. This can be done from the autocorrelation function of daily precipitation.
This problem was also present in the first manuscript on temperatures. Here it is probably less important because daily precipitation tends to be less autocorrelated in time, but again, these calculations need to be correct. I guess that for daily precipitation, an autocorrelation of 2 days would be reasonable, reducing the effective number of degrees of freedom for monthly means by a factor of 3, from 30 to 10, and for annual means from 365 to 100. For temperature, this factor is probably larger.
11) 'By construction, plain (i.e. unrotated) spatial EOF patterns (without prior detrending) are expected to display a unimodal distribution in the first mode, a bimodal distribution in the second, and a trimodal in the third. This behaviour is clearly apparent in Fig. (2).'
This is not correct. There is no reason for the leading EOF to have a monopole structure. It is usually so for climate fields, but there are many examples in which this is not the case. For instance, the leading EOF of the global nondeseasonalized monthly temperature field is a dipole, since the Northern Hemisphere and the Southern Hemisphere are strongly negatively correlated
12) Figure 2,3 Which normalization, which units ? This is indirectly explained in the caption of Figure 4, but it would be clearer if the methods section explained the chosen normalization of the time components (unity) and including a colour bar with the temperature units for the spatial patterns.
13) 'This example illustrates that relatively simple linear algebra methods, such as EOF and SVD, have the potential to deliver valuable results for the analysis of climate observations. The analysis can be further refined using more advanced method, such as Principal Oscillation Patterns (POP), as presented in Storch et al. (1995).'
Again, this seems to come as an afterthought, but I do not think this is correct in this case. The POP method is designed to identify oscillations in a climate field , and not to capture the main patterns of variability, which is the important goal for climate field reconstruction. The POP method may identify an oscillatory pattern, which nevertheless may explain very little portion of the total variance and be of little use to construct a climate indicator.
Related to this comment, I wonder why these two manuscripts want to use a more complicated method than actually needed. If the target is to estimate the spatial average of temperature or precipitation, a multilinear regression, with station data as predictors and the spatial average as precipitated, can also be used: trained in the recent period (as in the calculation of the EOFs) and then applied to the longer period. The training would use the available stations for the reconstruction period so that a collection of multilinear regressions would probably be needed to account for dropping stations back in time. The EOFbased method is, in my view, also suitable when applied correctly, but in my view, it is more complex than necessary.
Citation: https://doi.org/10.5194/egusphere2024940RC1  RC2: 'Comment on egusphere2024940', Rasmus Benestad, 24 May 2024

RC3: 'Comment on egusphere2024940', Anonymous Referee #3, 24 May 2024
This paper is a continuation (second part) of a previous paper the author submitted to EGUSPHERE and which I also reviewed. The previous paper dealt with temperature.
I think that some results of the paper might be of interest to researchers in this line, however, I think that the paper suffers from significant pitfalls in its current status, so that I ask a major revision.
First of all, I understand that the author is going to uccessfully address all the methodological issues I raised in the previous paper https://egusphere.copernicus.org/preprints/2024/egusphere2024582 because many of them dealt with problems that affect the methodology used in that paper and in this one.
Following with this paper, I will first mention the MAJOR problems that need to be addressed.
1. A review of English is needed. There are many typos which should have been corrected before submission. See, for instance, page 2, line 30. "a numerical methods that aggregate" > "a numerical method that aggregates" or same page, line 36 "between to stations" > between two stations"
2. Values and uncertainties.The author estimates the trend with three significant digits, but I have seen no estimation of the uncertainty of this value. Why three significant digits and not 4, 5 or two? Considering that the erros of the individual measurements are propagated through the EOFs, the handling of missing data and so on, this would ask for a particular estimation (Monte Carlo?) of a confidence interval surrounding this value.
Continuing with this, there is a very strange value in the abstract, with a Running average precipitation indicator" 8.37 mm/month". I guess he refers to an anomaly. but this should be stated.
3. The way missing data are handled must be clearly stated. I guess that missing values are substituted by null anomalies but, in that case, I guess the author is falling into DINEOF with a single iteration. I gave the author more information on this in my previous review.
4. Clarification of the methodology. The author mentions in many instances EOF and SVD together. SVD can be used for coupled datasets or it can also be used to calculate the EOFs (the same way as alternative methologies such as QR decomposition might be used). This must be clarified. And the way the author calculated the pseudoinverse matrix (this is not explained). Again in page 9, the author seems to be surprised that SVD and EOF patterns are virtually identical. Well, they must actually be, SVD is a way (a good one, indeed) of calculating the EOFs.
5. Page 7 around line 140. The author says the dataset is centered. Are "columns" removed from the dataset? Are these columns removed after getting seasonal averages? Is the overal mean removed? This must be clarified.
6. The equation for the covariance matrix lackas a division by the number of samples.
7. Pages 7 and 8. It is custmary that vectors (spatial patterns in this paper) are represented either by mathematical boldface or by arrows on top of the vector. In this paper, both are used. The author should stick to common practice.
8. Figures can not generally be properly read. In the previous paper, I suggested many things to do in order to improve the readability of the figures. a) Fonts must be larger. b) Please, do not stack 12 panels in the same figure (24 panels in figure 4!!).... Better join figures using different line colors if they are timeseries. Do not waste too much space in color scales if using maps, share the same colormap for all the subplots in a fgure. The scale must be informative. Which are the units of the variable? How do they calculate this scaling for PCs? Are EOFs plot as regression of the original field onto the principal components? Tell us that is the case. Showinf¡g a map of the EOF loadings regridded is nice, but if you showed the placement of stations it would be better. Labels, fonts must be legible without a magnifying glass. Figures 6. and 7., for instance. labels are again unreadable. The colors used can not be seen. Use other compinetion of colors and thinner lines if the information is important, currently, we can't know, it can't be seen.
9) Table 3. Some characters are written using yellow. I can't read that over white background
10) Page 20, line 397. The EOF method deviates from SMHIref. This probably means that the method is very sensitive to the available stations. A sensitivity analysis using subsets of the available data should be performed.
Suggestions.
Figure 1. Add a continentality index in a lower panel, as mentioned in my previous review
Citation: https://doi.org/10.5194/egusphere2024940RC3
Data sets
Spatially aggregated climate indicators over Sweden (1860–2020): scripts and data Christophe Sturm https://doi.org/10.5281/zenodo.10888129
Model code and software
Spatially aggregated climate indicators over Sweden (1860–2020): scripts and data Christophe Sturm https://doi.org/10.5281/zenodo.10888129
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