the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Technical Note: A bestpractice approach to calculating the Southern Annular Mode index
Abstract. The Southern Annular Mode (SAM) strongly influences climate variability in the Southern Hemisphere. The SAM index describes the phase and magnitude of the SAM and can be calculated by measuring the difference in mean sea level pressure (MSLP) between mid and highlatitudes. This study investigates the effects of calculation methods and data resolution on the SAM index, and subsequent interpretations of SAM impacts and trends. We show that the normalisation step that is traditionally used in calculating a nondimensional SAM index leads to substantial differences in the magnitude of the SAM index calculated at different temporal resolutions, and that the equal weighting given to MSLP variability at the mid and high southern latitudes artificially alters temperature and precipitation correlations and the interpretation of climate change trends in the SAM. These issues can be overcome by instead using a dimensional formulation of the SAM based on MSLP anomalies, resulting in consistent scaling and variability of the SAM index calculated at daily, monthly and annual data resolutions. The 10 dimensional version of the SAM index has improved representation of SAM impacts in the high southern latitudes, including the asymmetric (zonal wave3) component of MSLP variability, whereas the increased weighting given to midlatitude MSLP variability in the nondimensional SAM incorporates a stronger component of tropical climate variability that is not directly associated with SAM variability. We conclude that a bestpractice approach of calculating the SAM index as a dimensional index derived from MSLP anomalies would aid consistency across climate studies and avoid potential ambiguity in the SAM 15 index, including SAM index reconstructions from paleoclimate data, and enable more consistent interpretations of SAM trends and impacts.
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CC1: 'Comment on cp202364', Elio Campitelli, 23 Aug 2023
This is an nice paper. I have to admit that I'd always thought that these indices were always computed using the raw anomalies and share the authors' confusion on why this method is used. (I prefer EOFbased methods). MSLP difference is, to me, the selfevidently physically relevant variable. The authors here also demonstrate empirically other benefits.
The results in Figure 7 are particularly intriguing, since I would've not expected for the signaltonoise ratio to change so much.
I do think that having a dimensionless index is useful, especially for comparing models among themselves and model with observations. So I'd use a dimensionless SAM defined as:
(P'_40  P'_65)/sd(P'_40  P'_65)
That is: the (dimensional) difference between MSLP anomalies divided by the standard deviation of such difference computed in the reference period.
This index is essentially the same as the dimensional index in this paper but scaled by a constant factor, so it should share all its good properties of plus it has the advantage of being easy to compare between models.
Some comments below:
Figure 3b shows that the correlation of the annual nondimensional SAM indices is essentially 1. A correlation of 0.9998 between indices is basically perfect and is much greater than the correlation of SAM indices computed using different datasets, different reference periods, different latitude bands, different levels, or slightly different methods. Therefore, they are basically equivalent except for a constant scaling factor (shown as the slope). Being dimensionless quantities, this scaling factor is not relevant. Particularly, it's not relevant for correlations.
This can be seen in the correlation differences shown in the bottom left panel of Figures 5 and 6, which are tiny (on the order of 0.01). These differences are probably not statistically significant and, IMHO, most definitely not physically significant. They are likely much smaller than the standard error of their estimations or the difference between various reasonable SAM indices.
Figure 3d: Why are not the three time series exactly equal? The monthly mean of the difference between daily values must be exactly the same as the difference between monthly mean values. If they are not equal, what's their correlation?
Figure 4: What's the correlation between the dimensional and nondimensional indices at the annual, monthly and daily resolution? It would be helpful to have a number attached to these differences.
Figure 5: There's a strange artefact in the topleft panel near the dateline. A vertical discontinuity in the shading. The artefact might also be present in the other panels but harder to see.
Figure 5 and 6: Would it be better to plot the difference in r2 between each panel? Right now, to interpret the figure one needs to look also at the sign of the correlation in the original panels. By plotting the difference in r2, negative values would be directly interpretable as a decrease in the strength of the (linear) relationship and vice versa.
Are the differences in correlation statistically significant?
Figure 4 and 7: How are these plots created? By definition, the units of the dimensional and nondimensional indices are not the same so their alignment should be arbitrary (in other words, the y axis has two different units). Did you standardise both indices to show them on the same scale?
Figure 7: It's surprising, at least to me, that both indices are almost exactly the same before ~1950. Why is that? Was there a change in the ratio of the standard deviation of MSLP between 40ºS and 60ºS?Citation: https://doi.org/10.5194/cp202364CC1 
AC1: 'Reply on CC1', Laura Velasquez Jimenez, 10 Oct 2023
Dear Elio, thank you for taking the time to read our preprint and provide comments. We appreciate your feedback and have responded below.
The results in Figure 7 are particularly intriguing, since I would've not expected for the signaltonoise ratio to change so much.
Response: Thank you. Yes, while the differences in SAM index methods are small over the observational period, the impact of these differences becomes more evident when looking at longerterm climate change signals.
I do think that having a dimensionless index is useful, especially for comparing models among themselves and model with observations. So I'd use a dimensionless SAM defined as:(P'_40  P'_65)/sd(P'_40  P'_65)
That is: the (dimensional) difference between MSLP anomalies divided by the standard deviation of such difference computed in the reference period. This index is essentially the same as the dimensional index in this paper but scaled by a constant factor, so it should share all its good properties of plus it has the advantage of being easy to compare between models.Response: Thank you for this suggestion. We will test this approach when we prepare a revised version of the manuscript. While we can see that this would provide a way of standardising the SAM index produced between different climate models, it is the actual pressure difference that is the physically relevant variable, and this standardisation step would obscure actual differences in the representation of the SAM between different climate models.
Some comments below:
Figure 3b shows that the correlation of the annual nondimensional SAM indices is essentially 1. A correlation of 0.9998 between indices is basically perfect and is much greater than the correlation of SAM indices computed using different datasets, different reference periods, different latitude bands, different levels, or slightly different methods. Therefore, they are basically equivalent except for a constant scaling factor (shown as the slope). Being dimensionless quantities, this scaling factor is not relevant. Particularly, it's not relevant for correlations.Response: Yes, the difference in the nondimensional SAM index calculated different resolution is most evident in the scaling factor. You are correct that this should not be particularly relevant for a dimensionless quantity, but it is an issue that has resulted in confusion across previous literature. For example, for many years it was thought that there were large differences in the magnitude of SAM changes reconstructed over the last millennium. This can be traced back to the resolution of instrumental MSLP data used to calculated the observational SAM index that is used as the target for the paleoclimate reconstruction. Because the indices produced are dimensionless it is very hard to trace that this is where the discrepancy between reconstructed values originates from. Our study (Figure 3d) shows that dimensional SAM index that retains pressure units avoids introducing any scaling factors that in the past have resulted in ambiguities between SAM studies.
We also demonstrate that while the difference between the nondimensional SAM indices is the scaling factor, there are differences that are relevant for spatial correlations that are introduced by the nondimensional SAM giving equal weighting to MSLP changes in the mid and high latitudes (Figure 5). Hence, we do not agree with the statement that differences in the SAM indices are not relevant for correlations.
This can be seen in the correlation differences shown in the bottom left panel of Figures 5 and 6, which are tiny (on the order of 0.01). These differences are probably not statistically significant and, IMHO, most definitely not physically significant. They are likely much smaller than the standard error of their estimations or the difference between various reasonable SAM indices.
Response: We agree that the differences in spatial correlations from different resolutions of the SAM index (bottom row of Figures 5 and 6) are small and over the observational period will not be physically significant. However, our point here is not to assess significance, it is merely to show that there are some differences in spatial correlations that are introduced solely by data resolution when using a nondimensional SAM index. In contrast, these differences are essentially completely avoided when using a dimensional SAM index.More importantly in Figure 5 and 6 we demonstrate that there are much larger differences in spatial correlations when using a dimensional or nondimensional SAM index (far right column of Figures 5 and 6). This is due to the nondimensional SAM index giving an equal weighting to pressure changes at the mid and high latitudes, when in reality the MSLP variability and trends at the high latitudes have a larger magnitude than those at the mid latitudes. In this case the differences in correlation values are on the order of 0.1, which will influence interpretations of statistical significance in some regions.
Figure 3d: Why are not the three time series exactly equal? The monthly mean of the difference between daily values must be exactly the same as the difference between monthly mean values. If they are not equal, what's their correlation?
Response: This is a good point, and we will check this when revising the manuscript. We will also add a supplementary table with all correlation and slope values for the different methods and resolution of the SAM indices. We expect that the very slight differences in Figure 3d are related to the averaging of MSLP at different resolutions. For the daily SAM index each of the 365/366 days in the year has equal weighting when then averaging to an annual SAM value. For the monthly SAM index the different numbers of days per year gives very slightly different weighting to how the data for each day contributes to the calculated annual SAM value.Figure 4: What's the correlation between the dimensional and nondimensional indices at the annual, monthly and daily resolution? It would be helpful to have a number attached to these differences.
Response: We will add a supplementary table with all correlation and slope values for the different methods and resolution of the SAM indices.
Figure 5: There's a strange artefact in the topleft panel near the dateline. A vertical discontinuity in the shading. The artefact might also be present in the other panels but harder to see.
Response: Thank you for spotting this. We will fix this for all panels. It looks like it may be an edge effect in the plotting where the edges of the correlation matrix meet.
Figure 5 and 6: Would it be better to plot the difference in r2 between each panel? Right now, to interpret the figure one needs to look also at the sign of the correlation in the original panels. By plotting the difference in r2, negative values would be directly interpretable as a decrease in the strength of the (linear) relationship and vice versa. Are the differences in correlation statistically significant?Response: This interpretation of what we have plotted isn’t correct. We have already made this adjustment for correlation sign in calculating the correlation differences, but we will clarify this in our figure caption text. The process involved turning the correlation values into absolute values so that the difference between the correlations could be used to determine which version of the SAM index resulted in stronger correlations (positive values are stronger in one version, negative values are stronger in the other version). This is what is shown by the shading on the difference plots. The stippling on the difference plots is used to refer back to where correlations values were negative prior to being made into absolute values. For grid cells where the correlation values were a different sign between the two versions of the SAM, these correlation values were very close to zero, and so were set to zero for plotting the correlation differences.
Since we are not testing a hypothesis in our study, we did not conduct a significance analysis. We are simply demonstrating that differences in data resolutions and calculation methods can carry over into subsequent analysis (such as spatial correlations). While these differences are small over the historical era, they become more consequential as the climate change influence on the SAM becomes stronger this century (Figure 7).
Figure 4 and 7: How are these plots created? By definition, the units of the dimensional and nondimensional indices are not the same so their alignment should be arbitrary (in other words, the y axis has two different units). Did you standardise both indices to show them on the same scale?
Response: Yes, we agree that it would be better to plot Figures 4 and 7 with two yaxes – one for the dimensional SAM and the other for the nondimensional SAM. We will also use the correlation slopes between the two versions of the SAM indices during the 19611990 reference interval to determine the best scaling to use for these two yaxes so as to objectively show the similarities between the SAM indices and when differences between them develop.
Figure 7: It's surprising, at least to me, that both indices are almost exactly the same before ~1950. Why is that? Was there a change in the ratio of the standard deviation of MSLP between 40ºS and 60ºS?Response: We weren’t sure if you were referring to the similarities between the dimensional and nondimensional SAM indices before 1950, or the identical SAM indices between panel a and b? Both panels use the same historical simulation so panels a and b are identical until 2014 when the SSP future scenarios begin. The very close similarities between the dimensional and nondimensional SAM indices before 1950 are consistent with what we show in Figure 4, where the historical differences are small (but not altogether negligible in some years). As the climate change influence on the SAM develops during 20^{th} and 21^{st} centuries the differences between the nondimensional and dimensional SAM indices increases. We will check the moving standard deviation of the MSLP at 40°S and 65°S to see how this evolves through time and if this information should be added to Figure 7. Thank you for suggesting to look at this.
Citation: https://doi.org/10.5194/cp202364AC1

AC1: 'Reply on CC1', Laura Velasquez Jimenez, 10 Oct 2023

RC1: 'Comment on cp202364', Anonymous Referee #1, 25 Sep 2023
This paper explores how the normalization component of calculating the Southern Hemisphere Annular Mode (SAM) impacts its magnitude and relationships with climate.
I thought that a lot of the manuscript was ‘stating the obvious’: e.g., the dimensional SAM produces consistent indices across different temporal resolutions. Having said that, no doubt there are many researchers who have correlated something against the SAM without fully understanding how it was produced and how that might impact their findings (e.g., the incorrect Dätwyler methodology). So, in that sense, I think it is a useful addition to the literature, perhaps particularly for the paleoclimate community.
I have a few issues with some of the language used, both the tone and some of the terms used.
One might argue that what the authors are calculating is simply the mean SLP difference between 40°S and 65°S and that this is not the SAM at all. This has been done for other climate indices, such as the NAO, when it is termed the ‘natural NAO’. So, maybe this could be referred to as the ‘natural SAM’ to distinguish it? I agree that in many cases this might be a more useful metric than the SAM (see also Elio Campitelli’s comment).
The normalization is done, as it is for many climate indices such as the NAO or SOI, to adjust for seasonal differences in both the average and in the yeartoyear range of variability at each of the latitudes, so that each latitude always contributes equally to the index. For certain studies this may be an important criterion. You could argue that the ‘natural SAM’ is basically a SLP index at the higher latitude, because of the significantly larger variability there, and maybe if you are interested in how the SAM impacts lower latitudes this is going to swamp the local signal in SLP variability.
I don’t like the use of ‘bestpractice’ and ‘biases’, which comes across as arrogant (how about ‘alternative methodology’ and ‘differences’?): present the positives of your method of producing the ‘natural SAM’ (which I agree are several) and let the reader decide if they want to use it instead of the standard method. There are also other issues to think about when calculating the SAM, such as using SLP versus say 700hPa geopotential height as used by the Climate Prediction Center to calculate their SAM/AAO:
(https://www.cpc.ncep.noaa.gov/products/precip/CWlink/daily_ao_index/history/method.shtml).
I also concur with Elio Campitelli’s comment that a dimensionless SAM is useful when comparing models, which may have differing biases in representing the SH extratropical SLP field.
Specific points
Given that many authors prefer to use an EOFbased SAM index, it would be helpful to do a comparison with this method of deriving the SAM too.
I also looked at Fig. 7 and wondered why the differences between the two SAM indices are so much less prior to the reference interval than afterwards. It would be helpful if the authors could investigate this a little more: does it say something about temporal changes in circulation or is it simply biases in the historical model fields?
Citation: https://doi.org/10.5194/cp202364RC1 
AC2: 'Reply on RC1', Laura Velasquez Jimenez, 10 Oct 2023
Response to comments by Reviewer #1:
This paper explores how the normalization component of calculating the Southern Hemisphere Annular Mode (SAM) impacts its magnitude and relationships with climate.
I thought that a lot of the manuscript was ‘stating the obvious’: e.g., the dimensional SAM produces consistent indices across different temporal resolutions. Having said that, no doubt there are many researchers who have correlated something against the SAM without fully understanding how it was produced and how that might impact their findings (e.g., the incorrect Dätwyler methodology). So, in that sense, I think it is a useful addition to the literature, perhaps particularly for the paleoclimate community.
Response: Thank you for taking the time to review our manuscript. We agree you’re your sentiments above and this is why we have submitted this as a technical note, rather than a research paper.
I have a few issues with some of the language used, both the tone and some of the terms used.
One might argue that what the authors are calculating is simply the mean SLP difference between 40°S and 65°S and that this is not the SAM at all. This has been done for other climate indices, such as the NAO, when it is termed the ‘natural NAO’. So, maybe this could be referred to as the ‘natural SAM’ to distinguish it? I agree that in many cases this might be a more useful metric than the SAM (see also Elio Campitelli’s comment).
Response: Thank you for this suggestion. We propose to update our terminology to refer to the normalised SAM index (currently referred to as the nondimensional SAM index in our manuscript) and the natural SAM index (currently referred to as the dimensional SAM index).
The normalization is done, as it is for many climate indices such as the NAO or SOI, to adjust for seasonal differences in both the average and in the yeartoyear range of variability at each of the latitudes, so that each latitude always contributes equally to the index. For certain studies this may be an important criterion. You could argue that the ‘natural SAM’ is basically a SLP index at the higher latitude, because of the significantly larger variability there, and maybe if you are interested in how the SAM impacts lower latitudes this is going to swamp the local signal in SLP variability.
Response: We will add some additional text to the discussion that covers this. Indeed, Figure 5 and 6 do show that the influence of the normalised SAM index versus that natural SAM index emphasises different climate impacts in different regions (e.g. the normalised SAM index tends to have stronger correlations in the low latitudes).
I don’t like the use of ‘bestpractice’ and ‘biases’, which comes across as arrogant (how about ‘alternative methodology’ and ‘differences’?): present the positives of your method of producing the ‘natural SAM’ (which I agree are several) and let the reader decide if they want to use it instead of the standard method. There are also other issues to think about when calculating the SAM, such as using SLP versus say 700hPa geopotential height as used by the Climate Prediction Center to calculate their SAM/AAO:
(https://www.cpc.ncep.noaa.gov/products/precip/CWlink/daily_ao_index/history/method.shtml).
Response: We will update the title and wording in the revised manuscript to reflect this comment. Thank you.
I also concur with Elio Campitelli’s comment that a dimensionless SAM is useful when comparing models, which may have differing biases in representing the SH extratropical SLP field.
Response: We will mention this in the introduction when describing the EOF method in our revised manuscript. We will also add a sentence noting that where gridded data fields aren’t available then the Marshall SAM index presents an alternate way of quantifying SAM variability.
Specific points
Given that many authors prefer to use an EOFbased SAM index, it would be helpful to do a comparison with this method of deriving the SAM too.
Response: We will carry out this analysis when revising our paper and look to add information about how our findings do or don’t also related to EOFversions of the SAM index.
I also looked at Fig. 7 and wondered why the differences between the two SAM indices are so much less prior to the reference interval than afterwards. It would be helpful if the authors could investigate this a little more: does it say something about temporal changes in circulation or is it simply biases in the historical model fields?
Response: Yes, we will investigate this further when revising the manuscript, including by looking at the moving standard deviation of MSLP at 40°S and 65°S (see response posted to Elio’s comment on Figure 7).
Citation: https://doi.org/10.5194/cp202364AC2

AC2: 'Reply on RC1', Laura Velasquez Jimenez, 10 Oct 2023

RC2: 'Comment on cp202364', Anonymous Referee #2, 01 Jan 2024
I think this is a good study, as referee 1 already mentioned, in terms of clearing up questions about the obvious. The writing is very concise and the figure representation is sufficient.
The questions/comments by Elio Campitelli and referee 1 and the replies by the authors have cleared up most of my questions as well.
As referee1 already mentioned, the authors' method seems to be clearly better when using gridded data, but the reality is that the EOF method is preferred when gridded data is available. It would be nice to see a comparison with an EOFbased index. Also, it's been a while, so I'm not entirely sure, but in a personal conversation with Gong, I remember he said that normalization was done to account for situations where there is not a lot of data, or where there is a lot of uncertainty in the MSLP data, especially where there are only a few in situ observations or paleoclimate proxies.
Would the authors' method be more appropriate for situations where there are only a few observations in each latitude band and the number of samples in each of the two latitude bands is significantly different? Could you discuss this?
Citation: https://doi.org/10.5194/cp202364RC2 
AC3: 'Reply on RC2', Laura Velasquez Jimenez, 15 Jan 2024
I think this is a good study, as referee 1 already mentioned, in terms of clearing up questions about the obvious. The writing is very concise and the figure representation is sufficient.
Response: Thank you for reviewing our manuscript. We appreciate your feedback and have responded to your comments below.
The questions/comments by Elio Campitelli and referee 1 and the replies by the authors have cleared up most of my questions as well.
As referee1 already mentioned, the authors' method seems to be clearly better when using gridded data, but the reality is that the EOF method is preferred when gridded data is available. It would be nice to see a comparison with an EOFbased index. Also, it's been a while, so I'm not entirely sure, but in a personal conversation with Gong, I remember he said that normalization was done to account for situations where there is not a lot of data, or where there is a lot of uncertainty in the MSLP data, especially where there are only a few in situ observations or paleoclimate proxies.
Would the authors' method be more appropriate for situations where there are only a few observations in each latitude band and the number of samples in each of the two latitude bands is significantly different? Could you discuss this?
Response: As previously suggested, we will carry out additional analysis related to EOFversions of the SAM index and include the findings in the revised manuscript.
In response to the reviewer's comment on data availability we will also carry out additional tests for our revised manuscript. We propose to include a comparison of the SAM index produced by different methods when using only the limited number of station sites that are used for the Marshall observational SAM index. We will further carry out sensitivity testing using the ERA5 reanalysis data with varying numbers of input data points in each latitude band so assess whether normalisation is an appropriate method for dealing with data inhomogeneity and/or scarcity.
Citation: https://doi.org/10.5194/cp202364AC3

AC3: 'Reply on RC2', Laura Velasquez Jimenez, 15 Jan 2024
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