A method for estimating the effect of climate change on monthly mean temperatures: September 2023 and other recent record‐warm months in Helsinki, Finland

We describe a method for quantifying the contribution of climate change to local monthly, seasonal, and annual mean temperatures for locations where long observational temperature records are available. The method is based on estimating the change in the monthly mean temperature distribution due to climate change using CMIP6 (Coupled Model Intercomparison Project Phase 6) model data. As a case study, we apply the method to the record‐warm September 2023 in Helsinki, and then briefly examine all record‐warm months of the 21st century. Our results suggest that climate change made the record‐warm September in Helsinki 9.4 times more likely and 1.4°C warmer. Thus, the new monthly mean record in September 2023 would probably not have been set without the observed global warming. The presented and provided tool allows operational meteorologists and climatologists to monitor and report the impact of climate change on local temperatures in near real time.


| INTRODUCTION
The effects of climate change are becoming increasingly visible in people's everyday lives, as ongoing global warming is shifting the temperature distribution towards warmer conditions.This is reflected in an increase in average temperatures, but also in an increased likelihood of extremely high temperatures (Christidis et al., 2023;Fischer & Knutti, 2015;Vautard et al., 2020).According to the latest Intergovernmental Panel on Climate Change (IPCC) report (Masson-Delmotte et al., 2021), human influence has contributed to many observed changes in weather and climate extremes around the world, particularly to extreme heat and heavy rainfall.
The public frequently views extreme weather and climate events like heavy rainfall, heatwaves, or anomalously warm seasons as manifestations of human-induced climate change.Consequently, meteorologists and climate scientists are often asked-by the public, the media, and decision-makers-whether a particular weather event is related to climate change.At present, these questions cannot be answered quantitatively in Finland, except for providing rough information on the nationwide probability of occurrence based on past climatological observations.However, accurate and timely information on the impacts of climate change from national meteorological institutes would be essential for properly targeting adaptation measures and educating the public on the impacts of climate change (Maibach et al., 2022).
Extreme weather event attribution has advanced significantly in recent years (Otto, 2017;van Oldenborgh et al., 2021), and there is rising interest in turning this science into operational (Stott & Christidis, 2023).A large number of specific extreme weather events have been attributed to climate change in the recent literature (e.g., Philip et al., 2022;Tradowsky et al., 2023;Vautard et al., 2023).In addition to individual research groups, researchers in the World Weather Attribution (WWA) organisation conduct these analyses and have developed methods for conducting the research in a consistent manner (Philip et al., 2020).However, and understandably, WWA focuses only on high-impact cases at the global level.As a result, many national lower-impact cases are not analysed, even though they may be of high societal interest.
This letter will serve as the basis for an attribution tool and research aimed at (1) increasing quantitative knowledge of the impacts of climate change on extreme temperatures in Finland and in the Nordics, and (2) making the method operational, or at least semi-operational, so that meteorologists and climatologists can communicate in near-real time to the general public about the contribution of climate change to an ongoing or just observed weather event.Here, we take a first step towards this goal by providing a tool for attributing the impact of climate change on monthly and seasonal temperatures for locations with a long observational data record.The described attribution method is based on global Coupled Model Intercomparison Project Phase 6 (CMIP6) model data and local monthly mean temperature observations.As a case study, we first apply the method to the record high September 2023 mean temperature in Helsinki, as well as for all monthly mean temperature records set in the 21st century at the same station.

| DATA AND METHODS
In Räisänen and Ruokolainen (2008a), a model-based approach to adjust observations to estimate the actual present-day temperature climate was proposed.The approach was successful in its prediction of the temperature climate in land areas of the world during the period 1991-2002.Further, the same methodology was used to assess the impact of climate change on the very mild temperatures observed in Finland in December 2006 and March 2007 (Räisänen & Ruokolainen, 2008b, hereafter RR08b).
Here, we continue to use the same method with a few modifications.While Räisänen andRuokolainen (2008a, 2008b) used CMIP3 models, the current version uses the latest generation CMIP6 models.In addition, a minor revision in the estimation of variability changes and a more robust method for estimating the continuous probability distributions has been adopted.Finally, as in many other attribution methods (e.g., Paciorek et al., 2018;Yiou et al., 2017), a bootstrap approach for evaluating the uncertainty in the results has been developed.
In the following, we first describe the data sets used and then the details of the methodology.

| Local observations
The presented attribution tool uses in situ monthly mean 2-m temperatures from the Finnish Meteorological Institute weather observation network.The current version of the tool is tuned to work with monthly mean temperatures, but the same method might be applicable to daily temperatures as well.
In the case study, monthly mean observations were used from the Kaisaniemi weather station in Helsinki, which is the longest operating weather station in Finland with observations dating back to 1844.We used homogenised and urban-corrected monthly observations since 1901.The urbanisation correction is documented in Heino (1994) and the homogenisation in Tuomenvirta (2001).

| Observed global mean temperature
The method also uses the observed global mean temperature.These data come from Met Office Hadley Centre/ Climatic Research Unit global surface temperature data set, HadCRUT5 (Morice et al., 2021).We use annually averaged global mean temperature for the years 1900-2022, augmented by multi-model-simulated global mean temperature for 2023-2100.

| CMIP6 model data
We use global climate model simulations from CMIP6 models (Eyring et al., 2016).Historical simulations from 1901 to 2014 are merged with shared socioeconomic pathway scenario simulations (SSP) for 2015-2100.In this case study, we use historical and SSP2-4.5 scenario data from 29 models, but the tool can also use data for the SSP1-1.9,SSP1-2.6,SSP3-7.0, and SSP5-8.5 scenarios.The choice of scenario becomes important when using the method to estimate the probabilities of temperature extremes in the future, but is much less important for the real-time attribution of extremes.The models are listed in Table S1.

| Deriving model-adjusted pseudoobservations
The first step of the method is a detrending procedure, which produces pseudo-observations that aim to represent stationary past, present, or future climates (Figure 1, upper red box).These pseudo-observations are derived by assuming that the mean and variability of the local weather parameter change linearly with low-pass-filtered global mean temperature.Following RR08b, 11-year running mean of the global mean temperature (hereafter G 11 ) is used as its low-pass-filtered value.
As a sensitivity analysis, we also tested the 3-month mean temperature in northern Europe in place of the global mean temperature (Text S1 and Figures S1 and S2).Although this reduced the regression bias somewhat, any regionally and seasonally varying covariate would have to be chosen separately for each point and month studied.We therefore retained G 11 and left the optimal choice of the regression covariate as a topic for future research.
We express the local monthly temperature as the sum of its time-dependent expected value x t ð Þ and x 0 t ð Þ, which is the deviation from this value due to internal climate variability: The regression coefficients for x t ð Þ and the magnitude of x 0 t ð Þ were derived from the CMIP6 historical and SSP2-4.5 simulations using the time series for years 1901-2100.In the first step (Equation 2), local monthly temperatures were regressed against G 11 to determine the expected value of temperature as a function of G 11 .Then the square of the regression residuals from this first step was regressed against G 11 , to find out how the variance of temperature changes with global warming (Equation 3).
The CMIP6 multi-model mean regression coefficients B and D show that for example, in Helsinki, the September mean temperature is projected to increase 1.2 C per each 1 C of global warming, and the monthly mean temperature variability is projected to decrease 2.9% per each 1 C of global warming.While these regression coefficients could, in principle, also be estimated from observations, the CMIP6 model simulations provide a much better signal-to-noise ratio because of their larger number and longer time series.
The pseudo-observations were derived from the regression coefficients D and B according to the method in Räisänen and Ruokolainen (2008a).Below, x obs t 0 ð Þ denotes the observed value of temperature in year t 0 , and y obs t 0 , t ð Þ is the modified value derived from this observation, representing the climate of the predicted year t.In this letter, we use past, present, and future climates, that is, t ¼ 1900, t ¼ 2023, and t ¼ 2050.Here, t ¼ 1900 approximate the climate without anthropogenic influences.
As an intermediate step, we first calculate the adjusted observations z obs t 0 , t ð Þ without taking into account the change in variability: F I G U R E 1 Flowchart illustrating the methodology.SGS, stochastically generated skewed.
where ΔG 11 t 0 , t ð Þ is the change in G 11 from t 0 to t.Here, ΔG 11 is derived by combining the observed time series of G 11 with the corresponding CMIP6-simulated time series, merging the two at the last year (currently 2017) for which the observed G 11 is available.
Once these intermediate values have been calculated for all years within the baseline period, they are replaced by the final pseudo-observations y obs t 0 , t ð Þ using the estimated relative changes in variability: where Z t ð Þ is the average of z obs (or, equivalently, that of for the whole baseline period (here 1901-2023), and R t 0 , t ð Þ denotes the ratio of the variance at time t to the variance at time t 0 : The above-described procedure converts the time series of real observations x obs t 0 ð Þ to pseudo-observations y obs t 0 , t ð Þ representing stationary climate for year t, combining the time series of global mean temperature with coefficients B and D taken from the climate models.As model simulations are only used to represent the effect of climate change (rather than the absolute local temperatures), there is no need to explicitly debias the model data.Figure 2 shows the observed time series of September mean temperature in Helsinki together with pseudo-observations representing the climate at year t ¼ 2023.September 2023 was a record warm month, but the pseudo-observations suggest that Septembers 1934Septembers , 1938Septembers , and 1949 would have been even warmer in today's climate (Figure 2).

| Fitting continuous probability distributions to pseudo-observations
After obtaining the frequency distributions of pseudoobservations, the final step in the procedure is to convert these discrete distributions into continuous probability distributions.The simplest alternative, a pure Gaussian distribution, is undesirable because many weather variables are distinctively skewed and longtailed, including the monthly mean temperatures in some areas and seasons (Figures S3 and S4).Here, we use stochastically generated skewed (SGS) distributions (Sardeshmukh et al., 2015), which have Gaussian distribution as a special case.The SGS distribution is fit using the method of moments, based on the mean, standard deviation, skewness, and kurtosis of the sample data set.Thus, a continuous probability distribution for pseudo-observations is obtained (the three lowermost boxes in Figure 1).
We fit the SGS distributions using pseudoobservations for time series starting in 1901 and extending to the last year with observations available (currently 2023, except 2022 for December).The choice of 1901 as the start point of the time series is somewhat arbitrary, but based on van Oldenborgh (2021), we would assume about a 100-year time series is necessary to determine the distributional characteristics (mean, standard deviation, skewness, and kurtosis) with sufficient accuracy to fit the SGS probability distributions.

| Treatment of uncertainty
To assess model uncertainty, the procedures presented in the previous sections are repeated for different CMIP6 models.For example, in the SSP2-4.5 scenario, this gives us 29 different time series of pseudo-observations for 1901-2023 from which model-related uncertainty can be derived.
The uncertainty associated with internal climate variability is estimated using bootstrap sampling.Specifically, we sample the pseudo-observation time series 1000 times with replacement, with each sample randomly selecting 123 years from 1901 to 2023.In each time, we fit the SGS distribution to the time series.This gives a total of 29 Â 1000 = 29,000 realisations from which 5th and 95th percentiles can be calculated.

| CASE STUDY: SEPTEMBER 202MONTHLY MEAN TEMPERATURE IN HELSINKI
September 2023 was record-warm in Helsinki Kaisaniemi, with a mean temperature of 15.8 C (Figure 2).The previous record was 15.3 C in September 1934, while the highest September mean temperature earlier in the 21st century was only 14.0 C in 2006.
Two indicators often used in attribution studies are the probability ratio (PR) and the change in intensity (ΔI).The first of these, the PR, describes how much more likely an event is due to climate change.It is calculated as: Where P T ≥ T obs ð Þis the probability of temperatures of at least T obs occurring in the present climate, and P cf T ≥ T obs ð Þ is the corresponding probability in a counterfactual climate with no anthropogenic climate change (here approximated by the climate in the year 1900).
The second indicator, ΔI, describes how much climate change has affected the intensity of the event in question.In our tool ΔI is calculated by (1) finding the percentage point of the observed temperature in today's climate, and (2) finding the temperature that corresponds to that percentage point in the distribution of the year 1900 climate.
According to our central estimate, the probability of September as warm as that observed in 2023 in the 1900 climate is 0.19% (5%-95% range 0.01%-0.53%,Table 1).This central estimate corresponds to a 1-to 532-year event.
In today's climate, the probability of the observed September 2023 mean temperature is much higher than in the past.In the climate of 2023, September that is at least as warm will occur on average once every 57 years, giving an annual probability of 1.77% (0.53%-3.68%, Figure 3a and Table 1).Thus, based on Equation ( 7), climate change has increased the probability of such a warm September by about a factor of 9.4 (3.2, 96.7).The PR values for individual models are shown in Figure S5.
According to CMIP6 simulations forced by the SSP2-4.5 emission scenario, September as warm as that of 2023 would be a 1 in 18-year event in 2050, with an annual likelihood of 5.58% (1.38%-15.16%,Table 1).This underlines the fact that the extremes we are experiencing now will be substantially less unusual in the mid-century (Figure 3b).
Regarding ΔI, we estimate that September 2023 was about 1.4 C warmer (ΔI = 1.4 C, 0.8-2.0C) in Helsinki in today's climate than it would have been without human-induced climate change (Table 1 and Figure S5).

| RECORD HIGH MONTHLY MEAN TEMPERATURES IN THE 21ST CENTURY
In this section, the attribution method is applied to all the record-warm months of the 21st century in Helsinki Kaisaniemi.Here, a record-warm month refers to a month with the highest mean temperature for that calendar month over the observational period since 1901.In addition to September 2023, there are 6 such months in the 21st century: December 2006, March 2007, July 2010, May 2018, January 2020, and June 2021.Table 2 presents that the annual probabilities of such record-warm months in the near-preindustrial (year 1900) climate are all below 1%.Thus, in the absence of anthropogenic influences, their return periods would all be over 100 years.The lowest probability is found for May 2018, 0.15%, and the highest probability is for January 2020, being 0.85%.In both cases, the 20th-century temperature records were exceeded by a high margin (3.0 vs. 2.0 C in January; 14.5 vs. 13.0C in May), but the inter-annual temperature variability is much larger in January than in May.Therefore, natural climate variability could have produced the record more easily in January than in May.
Climate change has multiplied the probability of these record high temperatures, with best estimate PRs varying from 3.0 (January 2020) to 9.3 (June 2021).These PRs depend on several factors, including the modelsimulated changes in mean temperature and temperature variability, the shape of the probability distribution near its upper tail, the magnitude of interannual variability, and the extremity of the event in question.However, independent evidence for a large increase in the probability of extremely high monthly mean temperatures comes from the observations themselves.As 7 of the 12 monthly temperature records in Helsinki are from the last 23 years and only 5 from the first 157 years of the time series , these record-high values have been 10 times more frequent on a per-year basis in the 21st century than before.
The calculated present-day (2023) probabilities for these past record-warm temperatures already slightly exceed their probabilities at their year of occurrence.For example, a March mean temperature of ≥3.1 C is estimated to have become a once in 23 years' event in today's climate, compared with once in 37 years' event in 2007.
While the warmth of May 2018 will still be a relatively rare event in mid-century ( p = 3.64%), months like July 2010 would be quite common in 2050s climate (p = 13.18%,Table 2).This result highlights the rarity of the weather pattern (i.e., the strong anticyclone) behind the May 2018 heatwave (Sinclair et al., 2019).Nonetheless, under this middle-of-the-line SSP2-4.5 scenario, all these recent temperature extremes are projected to become two to three times more common from 2023 to 2050.

| DISCUSSION AND CONCLUSIONS
In this letter, we have documented an attribution tool, which helps meteorologists and researchers estimate the impact on climate change on monthly, seasonal, and annual mean temperatures in Finland.The tool uses observed monthly mean temperatures and CMIP6 model simulations.
An earlier version of the method has already been successfully demonstrated using CMIP3 model results and Gaussian kernel smoothing (Räisänen & Ruokolainen, 2008a, 2008b).However, the current tool has been updated and revised to use state-of-the-art CMIP6 data and more robust SGS distributions (Sardeshmukh et al., 2015), and bootstrap sampling has been added to account for the uncertainty due to internal variability.
RR08b obtained probabilities of 1.75% and 1.19% for mean temperatures in December 2006 and March 2007, respectively.These are somewhat lower values than those calculated in this study (2.78% and 2.70%, Table 1).The probability for March 2007 in RR08b (1.19%) is even below the 5%-95% confidence interval of the present method.One reason for the discrepancy may be that RR08b used raw observations, while we used homogenised and urban-corrected observations.The raw observations are slightly colder in the 20th century, reducing thus the likelihood of high temperatures in today's climate.
The implicit assumption of the method is that climate change will not change the shape of the probability distributions, except for the mean and variance.This may or may not be true and would deserve further research.Nevertheless, our supplementary analysis shows that intermodel agreement in changes in skewness and kurtosis from 1901-1950 to 2051-2100 in the SSP2-4.5 simulations is limited to specific regions such as the Arctic Ocean (Text S2 and Figure S3 and S4).
One challenge of this method, and of attribution methods in general, is obtaining a sufficient observational sample to reliably estimate the probabilities near the tails of the distribution.In order to capture long-term (multi-decadal) temperature variability, the weather station whose observations are used must have a sufficiently long observation history.If the observational record is too short and does not include enough extreme months, the shape of the tail of the SGS distribution may be incorrectly calculated, which can then affect the derived PR.According to van Oldenborgh et al. (2021), observational time series should cover at least 50 years and preferably more than 100 years.
In addition to the dependence on the length of the observational record and the potential sensitivity of the probability estimates on the curve fitting method near the tails of the distribution, two additional uncertainties in the method are worth noting.First, the estimation of the patterns of climate change (coefficients B and D in Equations 2 and 3) from model simulations alone makes the results potentially sensitive to model errors.Second, the assumption that these patterns are constant with time may also not hold exactly for large changes in climate, or when the balance of different forcing factors (e.g., global greenhouse gas forcing vs. regional aerosol forcing) changes (e.g., Wells et al., 2023).This highlights the need for a systematic comparison between different methods of extreme event attribution, not only for Finland but also in a more general context (Philip et al., 2020;van Oldenborgh et al., 2021).
We demonstrated the tool for the record high average temperature of September 2023 in Helsinki, as well as for all the other record-breaking warm months of the 21st century at the same station.In all cases, these recordhigh temperatures were found to be several times more probable in the early 21st century climate than in the near-preindustrial climate in year 1900.Regarding the most recent case, our results suggest that anthropogenic climate change made the warmth of September 2023 about nine times more likely and about 1.4 C warmer than it would have been without climate change.Thus, a new record would probably not have been set without the observed global warming.

F
I G U R E 2 Time series of observed September mean temperature in Helsinki Kaisaniemi in 1901-2023.Black line shows the actual observations, and blue dots show the pseudo-observations representing today's (2023) climate.Red error bars in pseudo-observations indicate 5th and 95th percentiles of the model ensemble.Blue-dashed line marks the 2023 monthly mean temperature, 15.8 C.

F
I G U R E 3 (a) The frequency distribution of pseudo-observations representing September monthly mean temperatures in today's (2023) climate in Helsinki (blue bars), and stochastically generated skewed (SGS) probability distribution of September pseudo-observations (blue line).In the upper left corner of the figure, the values of the four moments are annotated: mean (μ), variance (σ 2 ), skewness (γ), and kurtosis (κ).(b) SGS distributions of pseudo-observations for climates in 1900 (green line), 2023 (blue line), and 2050 (red line).Black vertical line marks the observed mean temperature in 2023.
Values in parentheses represent the 5th and 95th percentiles. Note: