A Statistical Approach on Estimations of Climate Change Indices by Monthly Instead of Daily Data

Abstract

Climate change indices (CCI) profoundly contribute to understanding the climate and its change. They are used to present climate change in an easy-to-understand and tangible way, thus, facilitating climate communication. Most of these indices are calculated by daily data but there are also many valuable data sets that consist solely of a monthly temporal frequency. In this paper, we present a method that enables the estimation of specific CCIs from monthly instead of daily data, allowing the expression and examination of data sets consisting solely of monthly parameters through climate change indices. Therefore, we used the ERA5 Land data supplemented by a CMIP6 ssp5-8.5 climate projection to train multiple regression models with different regression functions and selected the best fitting for every grid point. Using a climate projection as a supplement in training the regression functions accounts for climate change and empowers the method’s application in future climate periods. The method includes a simple bias adjustment (delta change). Its output is regridded to ERA5 Land’s $0.1^{\circ }$ grid, adapting it to the local environment and offering better application in areas with complex terrain using coarse data. Furthermore, the presented method and its regression coefficients can be created from any data set, allowing an even higher spatial resolution than ERA5 Land’s. While the method performs best for the temperature-related indices in warm temperate climates, it works uniformly well for the precipitation-related index maximum consecutive dry days on a global scale.

1. Introduction

Over the last few decades, extensive studies have been conducted to understand Earth’s climate, but it is still not fully understood and remains the aim of extensive research [1,2]. Climate models have been used to estimate the future impacts of climate change, including the effects on humans and ecology [3,4,5]. These impact assessments provide an overall better understanding of climate change and its effects, suggesting adaptation options and evaluating the impact of mitigation measures [6,7,8,9,10].

To investigate some of the above-mentioned aspects, means in climate are frequently used, while climate change indices (CCI) are used to understand the variability and extremes in the climate [3,11,12,13]. The joint CCl/CLIVAR/JCOMM expert team on climate change detection and indices (ETCCDI) defined 27 core indices that also cover extremes in climate and are calculated based on daily data. They were developed to describe the local climate but have proven to also describe the change in climate by comparing different periods, thus, making them a valuable tool for climate communication [14,15,16]. CCIs can be used to examine extremes from two perspectives: changes in frequency for a certain magnitude of extremes or changes in the magnitude for a particular return period. Both perspectives are related to each other but answer different questions. While the change in probability of an extreme event affects rare events stronger, it does not mean that the change in the magnitude of its return period is as sensitive as well [17]. While changes in magnitude are shown to behave linearly, changes in frequency tend to behave non-linearly for some indices [18,19]. However, using these indices on in situ measurements poses some challenges, such as the scarcity of measurement in some regions, leaving unobserved regions without representation, or the homogeneity of measurements [20,21,22].

With the advancements in computational power over the last decades, more and higher resolved data sets have emerged, both for observations, such as gridded observational and reanalysis data sets, as well as regional climate models [23,24,25,26]. These data sets include vast amounts of information and have been widely analysed using the CCI [27,28,29,30,31]. However, even in times of global high-resolution data, valuable data sets may consist solely of monthly values, such as HISTALP [32,33], making the calculation of CCIs based on the common methodology using daily data impossible. Therefore, we present a method that allows us to examine and use this additional source of information to further describe, analyse and understand the climate and its change.

The primary objective of the method presented in this paper is to accurately predict selected CCIs based on their corresponding monthly meteorological parameters. This approach enables the expression and examination of climate change through CCIs, even for data sets with monthly temporal frequency. The proposed method consists of local regression functions whose coefficients are trained by the global ERA5 Land data [34], supplemented with the climate change signal of a selected CMIP6 ssp585 far future climate projection [35], also taking climate change into account. When using the proposed method on coarser data, such as that provided by global climate models, a localisation is applied implicitly by using the data’s anomalies and imprinting them on the training data’s monthly climatic means. Subsequently, this will result in the training data’s spatial resolution as well, in this case, ERA5 Land’s $0.{01}^{\circ }$. This method not only enables us to estimate CCIs from monthly data but also to compare different data sets, as their resulting outputs end up on the same grid. However, it is also possible to adapt the regression functions’ coefficients to any available gridded training data set allowing higher resolutions than the $0.{01}^{\circ }$ from ERA5 Land’s grid, which can be of value especially in complex terrain.

Within the presented paper, the method is explained by first describing the used climate change indices, followed by the applied regression functions, then the methodological approach and, finally, the used data. Consequently, two indices were selected and described in detail within the Section 3. The method is applied to two use cases, presenting some application possibilities, and in the last part of the paper, the method’s strengths and weaknesses are discussed and conclusions are drawn.

2. Method

2.1. Climate Change Indices

Climate change indices are a profound way to convey climate change in an easy to understand way and to effectively quantify the related changes [20,21,22]. They are developed to give a simple representation of climate change’s influence on temperature, as reflected in indices such as summer or heat days, or on intense precipitation events, as characterized by indices such as maximum precipitation in five days. More complex indices, such as drought indices, which are based on multiple parameters including temperature and soil humidity, have also been developed. However, the calculation of all these indices invariably requires daily bias-adjusted data.

To make climate change detection objective and internationally comparable, the joint CCl/CLIVAR/JCOMM Expert Team (ET) on Climate Change Detection and Indices (ETCCDI) [14] has defined a set of 27 climate change indices. This paper and its method focus on simple peak over threshold indices, which are listed in

Table 1. Temperature and precipitation climate indices for which a mathematical expression is prepared and used to estimate the indices by monthly temperature means and precipitation sums.

Label	Index Name	Index Definition	Units
$FD$	Frost days	Let $T{N}_{ij}$ be daily minimum temperature on day i in year j. Count the number of days where $T{N}_{ij}<0{}^{\circ }$C.	days
$ID$	Ice days	Let $T{X}_{ij}$ be daily maximum temperature on day i in year j. Count the number of days where $T{X}_{ij}<0{}^{\circ }$C.	days
$SU25$	Summer days	Let $T{X}_{ij}$ be daily maximum temperature on day i in year j. Count the number of days where $T{X}_{ij}\ge 25{}^{\circ }$C.	days
$HD30$ 1	Heat days	Let $T{X}_{ij}$ be daily maximum temperature on day i in year j. Count the number of days where $T{X}_{ij}\ge 30{}^{\circ }$C.	days
$TR20$	Tropical nights	Let $T{N}_{ij}$ be daily minimum temperature on day i in year j. Count the number of days where $T{N}_{ij}\ge 20{}^{\circ }$C.	days
$RX1day$	Maximum 1-day precipitation	Let $R{R}_{ij}$ be the daily precipitation amount on day i in period j. The maximum 1-day value for period j are $RX1da{y}_j=max\left(R{R}_{ij}\right)$.	mm
$RX5day$	Maximum 5-day precipitation	Let $R{R}_{kj}$ be the precipitation amount for the 5-day interval ending k, period j. Then maximum 5-day values for period j are $RX5da{y}_j=max\left(R{R}_{kj}\right)$.	mm
$MCDD$	Maximum consecutive dry days	Let $R{R}_{ij}$ be the daily precipitation amount on day i in period j. Count the largest number of consecutive days where $R{R}_{ij}<1$ mm.	days

¹ The index $HD30$ is not part of the 27 core indices defined by ETCCDI. Still we consider it important and therefore included it in our study.

and are part of the indices defined by the ETCDDI.

2.2. Applied Indices

From the 27 climate indices defined by the ETCCDI [14], several have been tested but only the indices listed in Table 1 are indicating a reasonable connection to their monthly meteorological parameter and, hence, are suitable to deliver a robust estimate of the indicator with the help of a regression model.

Only the regression results for the indices $SU25$ and $MCDD$ are presented in detail in the results in Section 3. The results of the remaining indices, listed in Table 1, are given in Appendix A and are considered in the discussion.

2.3. Regression Functions

The primary objective of this method is to accurately predict selected CCIs by their related monthly meteorological parameters, such as temperature or precipitation, enabling us to calculate CCIs also for data sets with solely a monthly temporal frequency. Therefore, it is vital to identify a mathematical expression precisely describing the connection between the CCI and their monthly meteorological parameters as exactly as possible and, for the sake of interpretation and usability, as simply as possible.

The initial analysis of the indices, especially regarding temperature-related indices, and their related monthly mean temperatures, displayed a close to linear increase in the amount of events. Hence, the first and most simple function we used for the regression model is displayed by the linear function stated in Equation (1):

$$f\left(x\right)=a·x+b$$

(1)

However, the linear increase in events of the mentioned heat-related indices is twofold. For instance, the index $SU25$ indicates a low rise in the occurrence of the index at first, but as the monthly mean temperature increases, the increase in the occurrence accelerates, a stronger and final gradient establishes and a steeper linear increase for the rest of the data’s range can be observed. Hence, as the next step of complexity, we used the step-wise linear Equation (2) in our regression model. The step-wise linear equation separates the data into two sections by using the median of the monthly meteorological parameter as splitting point $\zeta$, except for the index $MCDD$ where we used the 0.25 quantile based on the experience we received from the first results. It then fits a linear function, represented by the first term in Equation (2), and adds the second linear term after the splitting point $\zeta$. The plus as the index of the second term’s brackets indicates that only the positive values emanating from the bracket’s equation are considered. This allows two different gradients in the linear graph, which is especially useful for the before-mentioned temperature-related indices as the events in the beginning only occur slowly and start rising more rapidly afterwards, see Figure 1.

$$f\left(x\right)=a·x+b·{(x-\zeta )}_{+}+c$$

(2)

Because nature seldom behaves linearly, we also used an exponential function (Equation (3)) for the regressions of the climate change indices. This equation is similar to the Poisson regression model, which is broadly used for count data [36] and easily captures exponential in- or decreases in the data’s behaviour.

$$f\left(x\right)=a·exp\left(b·x\right)$$

(3)

In precipitation count data, such as the $MCDD$, in addition to the exponential in- or decrease in events, the data often display a linear start or fade out of events. To capture this specific behaviour, we combined an exponential with a linear term, represented by Equation (4).

$$f\left(x\right)=a·exp\left(b·x\right)+c·x+d$$

(4)

The first term in Equation (4) accounts for the exponential drop in the regimes with low precipitation amounts, while the second term considers the fade out of events in the regimes with higher precipitation amounts. Because we used this equation especially for the $MCDD$, the coefficients a and d are bound to sum up to one, making sure that the result for no precipitation is a full month of events. Equation (4) is used exclusively for the index $MCDD$ instead of (3).

In all four equations, the variables $a,b,c$ and d represent the coefficients of the regression, which are determined by the method of least squares. Furthermore, for all indices, only events that occurred at least 10 times are considered, and for the temperature-related indices, zero events and full month events are excluded from the regression.

2.4. Methodological Approach

From the daily ERA5 Land data covering 1981 to 2020, the indices listed in Table 1 are calculated for each grid point and every month. Additionally, to enable the regression to also consider future temperature-related climate states, the output of the CMIP6 MPI-M MPI-ESM1.2-HR model [37,38] is used to compute the monthly indices and their meteorological counterpart for the far future period (2071–2100). This is especially necessary as some regions might experience a strong increase in temperature, which is then reflected within the indices. Therefore, altered meteorological conditions due to climate change influence the resulting regression parameters strongly. We used the model’s ssp5-5.8 scenario, which is considered the worst case scenario of the defined social economic pathways and should represent the strongest change in the climate. Because the climate model has a far coarser spatial resolution (100 km) than ERA5 Land (9 km), it is re-gridded to the latter’s grid using python’s package xESMF and its monotonic bilinear regridding method [39].

To avoid the climate model’s $BIAS$, its temperature anomalies for the far future period (2071–2100) are calculated and added to ERA5 Land’s historical monthly climatology calculated for the period 1991 to 2020. This simple method of bias adjustment fits our method sufficiently, has been used for similar applications [40] and allows us to use 70 years of data covering the current state of the climate and, with the chosen ssp5-8.5, is also a potential worst-case future climate scenario.

These 70 years of monthly indices and their attributed monthly mean temperatures are then used to perform the regressions with the above-mentioned functions for every grid point in the ERA5 Land data set.

Because of the precipitation’s chaotic nature, it is not possible to calculate reasonable daily precipitation anomalies. Hence, the regressions for the precipitation indices $MCDD$, $RX1day$ and $RX5day$ are only performed on the historical ERA5 Land period of 40 years, covering the years 1981 to 2020.

The decision on which of the regression model functions fits the index best is based on the monthly root mean squared error $RMS{E}_{monthly,hist}$, covering the historical period 1991 to 2020. We used the period 1991 to 2020 because it represents the latest 30 year reference period in climate defined by the World Meteorological Organization (WMO) and makes the results easier to compare. Using the $RMSE$ as a measure of performance is broadly implemented and discussed by [41]. We use the $RMSE$ because it includes the $BIAS$ and the data’s variance in its metrics. Calculated from the monthly data, as seen in this method, also reflects the seasonal variability throughout the year and how well the regression functions can capture it. An overview of the above-described method is depicted in Figure 2, additionally showing a schematic description on the steps of how to apply the method.

The indices’ climatological $BIA{S}_{Clim,hist}$, calculated for the historical period from 1991 to 2020 is additionally presented to give a better understanding of the regression’s performance. For the climatology of the index $MCDD$, consecutive events below 5 days are neglected because these are not significant but create noise in the climatology, see also Appendix A.1. However, it is important to note that they are considered in the regression.

Precipitation in global climate models only partly represents reality, which is why often relative changes are used to analyse precipitation [42]. The $MCDD$’s relative climate change signal (CCS), on the one hand, is calculated for the far future period (2071–2100) directly from the ssp5-8.5 climate model and, on the other hand, also reproduced with each of the regression functions. The comparison shows if the regression functions can reproduce the relative CCS and which of the functions performs best when carrying them out.

2.5. Used Data

The ERA5 Land data set used for the regression is a recalculated and down-scaled version of ERA5’s surface data. It uses ERA5’s meteorological fields as forcing and enhances its spatial resolution from 31 to 9 km and its temporal resolution from daily to hourly values, with additional lapse-rate correction accounting for the different altitudes in the models. Observations only indirectly influence the results by ERA5’s forcing where a 4D-Var data assimilation and a simplified extended Kalman filter is used [34], making it one of the state-of-the-art reanalysis data sets. It is constantly updated, starts in 1950 and is available up to two to three months before the present; although, for this work, the period 1981 to 2020 is used. According to Keller’s [25] comparison of four global reanalysis data sets, ERA5 correlates best with temperature-related indices in Europe and is only outperformed by MERRA-2 [43] regarding the bias. The best performance with respect to precipitation-based indices varies between the four data sets. However, for North America, ERA-Interim is best for temperature-related indices, except for $ID$, which is better represented in ERA5, but ERA5 never performs best for precipitation-related indices. Nevertheless, both ERA5 and ERA5 Land parameters show very good agreement with in situ measurements located in Italy [44].

As mentioned above, the application of the regression on a future projection is performed with a global earth system climate model from the Coupled Model Intercomparison Project (CMIP) phase 6 (from now on called CMIP6) [45]. We chose the Deutsches Klimarechenzentrum’s (DKRZ) MPI-M MPI-ESM1.2-HR model for the historical [37] and the shared socioeconomic pathway 5 8.5 (ssp5-8.5) output [38], which comes with a native nominal spatial resolution of 100 km and a daily temporal resolution. The choice of the model is due to its good alignment with ERA-Interim [46], which is superseded by ERA5, having, therefore, strong similarities [47], but also because it is part of the Climate Data Store’s (CDS) quality controlled subset.

Because the indices are strongly dependent on their climatic environment, the general estimation of the different regression functions’ performance concerning the five basic Köppen–Geiger climate zones is performed using the updated data produced by [48]. The data are available with a spatial resolution of 0.5${}^{\circ }$ and are representative of the period 1951 to 2000. The changes in the last 20 years were neglected in this paper.

Geosphere Austria’s HISTALP data set [32,33] is used to show a use case of calculating CCI from a data set with exclusively monthly parameters. HISTALP’s spatial resolution is 5 × 5’ and covers the European Alp domain starting even before the pre-industrial period in the year 1780 to 2014 and, therefore, represents an especially interesting data set. Additionally, the data set is split into six climatic zones.

3. Results

For the indices summer days $SU25$ and maximum consecutive dry days $MCDD$ the performance of the different regression functions is presented in detail in this section. The indices’ climatologies are displayed and calculated according to the selected best regression equation, as well as their $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$. For all of the indices listed in Table 1, the mean climatological $BIA{S}_{Clim,hist}$ and monthly $RMS{E}_{monthly,hist}$ with respect to the five basic Köppen–Geiger climate zones are shown, allowing a general understanding of the different regression function’s behaviour and performance in the climate zones.

3.1. Summer Days

In this section, the results for the temperature associated index summer days $SU25$ are presented. The cities Vienna, Austria and San Diego, USA are chosen to present the local regression functions for the index $SU25$ (Figure 1). In both cities, the step-linear function was chosen according to the lowest $RMS{E}_{monthly,hist}$, which is 2.42 days for Vienna and 2.48 days for San Diego. In both cases, the selection of the step-linear function is substantiated by the lowest values in $BIA{S}_{Clim,hist}$ but also by the highest values in $R_m^2$.

According to the monthly $RMS{E}_{monthly,hist}$ of the period 1991 to 2020, the step-wise linear function is the primarily chosen equation for the index summer days $SU25$ on a global scale (Figure 3). Depending on the region and climate zone (see also Figure 4 and Figure 5), the data do not only behave linearly, which is already captured by the step-linear regression function. While it performs best in the warm temperate and arid climate zones, it is superseded by the linear regression function in some parts of the tropics and the arid climate zones close to the Equator. The exponential equation displays the best fit further to the north in the polar and snow climate zones.

The climatology based on the selected regression function produces a north–south pattern in their historical and future $BIA{S}_{Clim,hist/future}$, underestimating the occurrence of summer days around the equator, resulting in a cold bias, while overestimating its occurrence further to the north, resulting in a warm bias (Figure 6 and Figure 7). The increase in global temperature also increases the amount of summer days, with the result of amplifying the before-mentioned cold bias around the equator. The north of North America, as well as Europe and Central to Northeast Asia, show a decreasing warm bias. The North of Africa and the lower latitudes of South America and Australia are turning from a cold bias in the historical period to a warm bias in the far future period (Figure 7). In both climate periods, the monthly $RMS{E}_{monthly,hist/future}$ is mainly below four days.

For the temperature-related indices $SU25$, $HD30$ and $TR20$, the highest $BIA{S}_{Clim,hist}$ appears around the equator where the indices occurrence tends towards the full month throughout the whole year (Figure 6, Figure 7, Figure A3 and Figure A4). These mostly full month events throughout the whole year cannot be easily resolved by the functions we used for the regressions and, therefore, produce a cold bias greater than 12 days in these regions. Users of the regressions should be aware of this behaviour and take, according to their use case, the limitations of the regressions into account.

For the indices $ID$ and $FD$, there are some regions with explicit full-month events such as in the polar climate zone where some regions every day of the month qualify as ice or frost days and, therefore, cause the regression to fail. Only the polar coastal regions under the influence of the ocean show months without days qualifying as $ID$ or $FD$ and allow the regression to perform.

3.2. Maximum Consecutive Dry Days

In this section, the results of the precipitation-related index maximum consecutive dry days are presented. Comparing Figure 8 with Figure 1, it is apparent that the data have a higher variance and are not linear. This is also seen in Vienna in the lower values in $R_m^2$ of 0.48 (0.85 for $SU25$) for Equation (4), 0.34 (0.89) for the step-wise linear and −1.15 (0.88) for the linear function, indicating that the linear function cannot be used to compute the index $MCDD$ in this location. Therefore, by the $R_m^2$ values already alluded to, the best fit of Equation (4) is confirmed by the lowest value in the $RMS{E}_{monthly,hist}$ of 2.90 days, even though the $BIA{S}_{Clim,hist}$ of Equation (4) is slightly higher than the step-wise function’s. The low value in $RMS{E}_{monthly,hist}$ indicates that Equation (4) can better capture the data’s variance and, subsequently, its seasonal variability than the step-wise function, which is also insinuated by its higher value in $R_m^2$. The same pattern in $R_m^2$, $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$ can be seen for San Diego (right), although with higher values in all three metrics but one. The exponential-linear function’s (Equation (4)) $BIA{S}_{Clim,hist}$ is also lowest, even lower than in Vienna, substantiating, this time, Equation (4) as the best fit.

The observed pattern for Vienna and San Diego, resulting in the selection of Equation (4) as best fit, can be observed within most regions of the world, as displayed in Figure 9 where, in general, Equation (4) is chosen according to the lowest $RMS{E}_{monthly,hist}$. Occasionally the step-wise linear function outperforms Equation (4), visible in the North of Africa, in parts of the Antarctic and the East of Asia. The linear function is only chosen in a few grid points in North Africa. In general, it can be said that the combination of an exponential and linear function in Equation (4) describes the relationship between the monthly precipitation sum and the index $MCDD$ best.

In contrast to the temperature index $SU25$, there is no north–south pattern regarding the $BIA{S}_{Clim,hist}$ or $RMS{E}_{monthly,hist}$. The $RMS{E}_{monthly,hist}$ is lowest in the northeast of Africa, as well as in some regions of Antarctica, which can be considered very dry regions. Except for these two regions with very low values, the $RMS{E}_{monthly,hist}$, in general, is below six days and shows a relatively equal spatial distribution (Figure 10, lower right). In regions with low amounts of $MCDD$, the selected functions tend to overestimate the index, causing a dry bias in these regions, while regions with high amounts in $MCDD$ tend to have a wet bias, underestimating the index (Figure 10, upper right). The $BIA{S}_{Clim,hist}$’s range is between ±10 days and equally spatial distributed. Hence, the $RMS{E}_{monthly,hist}$’s pattern can be interpreted as the local geomorphological influence.

Because of the precipitation’s complexity and the difficulty of properly representing it in climate models, mostly relative changes are analysed and regarded in terms of future scenarios. In Figure 11, the relative CCS for the period 2071 to 2100 of the original data (upper left) and the three different regression functions (Equations (1), (2) and (4)) are displayed. The linear regression over- and underestimates the climate change signal the most, resulting in a wet bias in wet regions and a dry bias in dry regions. The step-wise linear regression already performs better but still has a strong wet bias in parts of the earth such as central Africa or Greenland. The exponential-linear regression function has mostly the same spatial distribution and quantity as the original data and seems to perform the best of the three regression functions, which is also displayed in Figure 9 where mostly Equation (4) is selected.

The future outlook, presented in Figure 12, shows an increase in $MCDD$ in already dry regions such as the Mediterranean region and Central America and a decrease in $MCDD$ in wet regions such as India and follows along the presented pattern in relative CCS in Figure 11. The strong decrease in $MCDD$ in North Africa’s desert can be explained by the fact that small changes in precipitation in this dry region already have a considerable impact.

3.3. Metrics by Updated Köppen–Geiger Climate Zones for Both Temperature and Precipitation Climate Change Indices

The $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$ of the chosen indices (Table 1) are depicted concerning the five basic Köppen–Geiger climate zones (1991–2000) to obtain a clear overview of the behaviour of the presented method under different climatic influences (Figure 4 and Figure 5).

For the $MCDD$, the exponential Equation (4) achieves the lowest $RMS{E}_{monthly,hist}$ throughout all five basic climate zones. It is always below five days with the step-wise linear Equation (2) close behind with approximately a day more in $RMS{E}_{monthly,hist}$. The linear Equation (1), however, shows clearly the highest values in $RMS{E}_{monthly,hist}$ and only performs reasonably well in very dry regions (arid and polar climate zones). This behaviour is also transferred to the climatology $BIA{S}_{Clim,hist}$ where the Equations (2) and (4) are in every climate zone below 10 days, while the linear equation achieves its best result in the polar region with around 20 days and higher than 30 days in the other climate zones. Generally, we can see that the $MCDD$ is, in both error metrics, the most constant of the peak-over-threshold indices throughout all five basic Köppen–Geiger climate zones.

For the heat-related indices $SU25$, $HD30$ and $TR20$ the step-wise linear Equation (2) replicates the seasonal variability the best as it scores the lowest $RMS{E}_{monthly,hist}$ in the equatorial, arid and warm temperate climate zones but only by a small margin compared to the other two equations. In the snow and polar climate zones, all three equations show almost the same values in $RMS{E}_{monthly,hist}$. The differences are more obvious in the $BIA{S}_{Clim,hist}$ where the step-wise linear and the linear equation are still similar in their bias in the equatorial, arid and warm temperate climate zones, but the exponential equation displays a clear negative difference.

The precipitation indices $RX1day$ and $RX5day$ demonstrate an interesting pattern in their $RMS{E}_{monthly,hist}$. The influence of their higher proportion of convective precipitation can be observed in the equatorial and warm temperate climate zones with the highest $RMS{E}_{monthly,hist}$ of the five basic climate zones. The atmospheric stable snow and polar climate zones or most dry arid climate zone display lower $RMS{E}_{monthly,hist}$ with the lowest values in the polar climate zone. For all climate zones, either the step-wise linear equation presents the lowest $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$ or the linear equation. Only for the index $RX1day$ in the equatorial climate zone did Equation (3) seem to capture the index’s behaviour best.

4. Examples of Use

This section presents two potential use cases that demonstrate the practical application of the proposed method. The first use case involves the calculation of CCI for Geosphere Austria’s HISTALP data set, which comprises parameters with solely a monthly temporal frequency. The second use case involves a comparative display of a CMIP6 ssp585 ensemble for Vienna calculated by the proposed method and, therefore, including the localisation of the data.

4.1. Calculation of CCI for HISTALP

The Geosphere Austria’s Histalp data consists solely of monthly meteorological parameters, making a direct calculation of CCI’s impossible. The timeline for Histalp’s zone east for the indices $SU25$, $ID$, $MCDD$ and $RX5day$ are presented in Figure 13, calculated by the method presented in this paper. Additionally, for reasons of readability, the timeline is smoothed by a Gaussian filter with a 10 year window averaged window with a standard deviation of 1.5. The dashed lines mark the climatology for the pre-industrial period (1851–1900) and the more present period of 1981 to 2010. Clearly, starting with the eighties, a sharp rise in summer days and a drop in ice days can be seen, indicating the influence of anthropogenic climate change. Comparing the periods for the precipitation-related indices $MCDD$ and $RX5day$ show no difference for the maximum consecutive dry days and a slight increase in the amount of maximum precipitation in five days.

For ice days and maximum consecutive dry days, the before-mentioned climatologies are again presented in Figure 14. The difference in ice days between those two periods is showing a warmer climate in the period 1981 to 2010 with higher differences in the mountains than in the surrounding plains. The spatial distribution for the index $MCDD$ shows fewer consecutive dry days north west of the Alps and more consecutive dry days in the West of Italy’s Po plain, around Turin.

4.2. CMIP6 Ensemble Calculation

For this example, we chose Vienna to calculate the index $SU25$ for a CMIP6 ssp585 model ensemble from their monthly temperature anomalies and imprinted them on ERA5 Land’s monthly climatology for 1981 to 2010, introducing a localisation of data, as is described in Section 2.4. From the 42 models we used, the model UKESM1-0-LL shows the highest amount of summer days in the far future’s climatology, calculated by Equation (2) (steplin). Although, in the first 20 years, it seems to be in the middle range of the ensemble, it is rising rapidly to the top around the year 2040. The models CAMS-CSM1-0, FGOALS-g3 and IITM-ESM seem to be the coldest models with the lowest amount of summer days according to the far future’s climatology.

According to the ssp585 ensemble, displayed in Figure 15, Vienna can expect a doubling in summer days for the far future period with an ensemble spread of around 50 days, meaning, in the worst case, even a tripling in $SU25$.

5. Discussion and Conclusions

The presented method’s main objective is to predict selected climate indices from monthly data instead of daily. This benefits data sets consisting exclusively of monthly parameters, allowing the expression and examination through climate change indices (CCI), as the example in Section 4 displays. Furthermore, if coarser data are used, they will be downscaled to ERA5 Land’s grid of $0.1^{\circ }$ and a simple bias adjustment (delta change) is implicitly [40] performed to adapt the output to the local environment that allows us to use coarse data, for example provided by global climate models, in complex terrain. Moreover, it simplifies the comparison of different data sets as they end up with the same grid.

Throughout the process of developing the method to estimate climate change indices (CCIs) from monthly data, we encountered the challenge of identifying suitable CCIs allowing the regression models to deliver robust estimations. Therefore, we compiled a list of the indices suitable for regression models and settled on the ones presented in Table 1. Despite these limitations, the temperature and precipitation indices that we included in our study cover major aspects of climate change, such as heat, droughts and extreme precipitation, but surely further investigations on the realisation of more complex indices should be performed.

To capture the different influences of climate zones and their respective climatic properties on the climate change indices, we employed different regression functions reflecting the characteristics of the investigated parameter. The strengths and weaknesses of using these different regression functions is evident from the metrics $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$, represented in Figure 5 and Figure 4, respectively, concerning the five basic Köppen–Geiger climate zones. For example, the precipitation index $rx1day$ shows a clear tendency towards the linear or step-wise linear equation in all climate zones with lowest values in $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$, except in the equatorial climate zone. For the equatorial climate zone, the exponential Equation (3) appears to capture the unstable, convective properties of this climate zone, resulting in the lowest values in $RMS{E}_{monthly,hist}$ and $BIA{S}_{Clim,hist}$.

To ensure that the climatic influences and seasonal variability are indeed reflected in the regression, the selection of the best fitting function was carried out by their monthly $RMS{E}_{monthly,hist}$. This ensures that the predictions of monthly index values, as well as the calculated annual climatologies, are correct. When we selected the best fitting regression function according to their yearly $RMSE$, it still resulted in a very good estimate of its climatology but with the drawback of unreasonable seasonal estimations. For example, the index $ID$ was underestimated in winter but overestimated in summer, resulting in considerable monthly errors that were not perceived in the climatology. Therefore, using the monthly $RMS{E}_{monthly,hist}$ ensures the capture of seasonal variability and exact estimations of monthly values.

Regarding temperature-related indices, attention must be paid to areas where an index usually occurs throughout the whole month and year. This behaviour cannot be resolved well by the regression, as seen in the polar climate zone for indices $ID$ or $FD$ in Figure 4. Although one would think that the index is well represented in this climate zone, it has the highest values in $RMS{E}_{monthly,hist}$ across all three regression functions. The reason is the mentioned occurrence of the event throughout the whole month, making it difficult to be resolved by the regression.

Despite the regression’s mentioned flaws, comparing the results presented in Figure 15 with equilibrium climate sensitivity (ECS) values presented in [49]’s work confirms the reasonable behaviour of the regressions as the warmer models from Figure 15 such as the UKESM1-0-LL, the HadGEM3 models and the CanESM5 models have the highest ECS values. The same can be observed for the coldest models, CAMS-CSM1-0 or GFDL-ESM4, which are also concluded to have low values in ECS.

With the precipitation-related indices, the situation is somewhat different. For example, all three regression functions can reproduce the $MCDD$’s pattern of predicted climate change signals close to [50], where dry regions become drier and wet regions wetter. For most of the earth’s regions, the exponential-linear Equation (4) is performing best in terms of $BIA{S}_{CLim,hist}$ as well as $RMS{E}_{monthly,hist}$ in the function’s climatology. This is also confirmed by Figure 4 and Figure 5 where Equation (4) shows the lowest values for $MCDD$ in all five basic climate zones. Moreover, full month events are covered very well by the Equations, which is caused by the non-linear behaviour between the index and monthly precipitation sum.

Because the indices’ $RX1day$ and $RX5day$ climatology are calculated by the maximum of each year, both indices have, in general, a dry bias as the regression takes all monthly maximums into account. Especially in regions with a high fraction of convective precipitation or in regions influenced by monsoon or tropical storms, causing a high variance in the data, the resulting dry bias is pronounced. Hence, as the errors are biggest in these regions, the user should consider the regressions limitations according to her or his application. Nevertheless, for the rest of the five basic climate zones, the regression functions can reproduce the climatology well and depict a useful tool.

In general, one has to emphasise that the regression model’s output contains less information than directly calculated indices, as it is not able to reproduce the data’s full variance. Therefore, unusual processes are not represented in a realistic way. This has to be considered when using a single monthly estimation, while it has less impact when summarising the regression output to annual data and even less impact when calculating 30 year climatologies. Another aspect to be aware of, which mostly regards precipitation-related indices, is that the dynamic change in climate, such as the possible shift of stratospheric precipitation towards convective precipitation, and vice versa, or changes in the synoptic dynamics of cyclones [51], cannot be resolved with this statistical approach. Only changes that can already be observed in the monthly input data are displayed in the output of the regression. Again, the user has to be aware of the regression’s limitations and use it accordingly.

In conclusion, the proposed method enables the user to estimate CCIs from monthly data and presents a valuable asset that allows the investigation of climate change through CCIs of, formerly not usable, solely monthly data sets. As an example, one can use Histalp’s data to investigate a large ensemble of global climate models and their behaviour in the pre-industrial period with relatively low computational effort. Using three different functions per climate change index allows the regression to capture the influence of different climate zones and their properties. Furthermore, selecting the best fitting function by their monthly $RMS{E}_{monthly,hist}$ ensures the representation of the seasonal variability in the regression, resulting in the exact estimations of monthly values and, subsequently, accurate estimations of their annual climatology.