2.1 GG2020 approach

Following GG2020, Warming is defined as an increasing trend in certain characteristics of the temperature distribution. More precisely:

Definition 1. (Warming): Warming is defined as the existence of an increasing trend in some of the distributional characteristics measuring the central tendency or position (quantiles) of the temperature distribution.

An example is a deterministic trend with a polynomial function for certain values of the β parameters C_t = β₀ + β₁t + β₂t² + … + β_kt^k, as well as, an integrated process of order one (I(1)). An I(1) process is the accumulation of an I(0) process. Our definition of an I(0) process follows [18]. A stochastic process Y_t that satisfies Y_t − E(Y_t) is called I(0) if converges for |z| < 1 + δ, for some δ > 0 and , where the condition ε_t ∼ iid(0,σ²) with σ² > 0 is understood.

In GG2020 temperature is viewed as a functional stochastic process X = (X_t(ω), t ∈ T), where T is an interval in , defined in a probability space (Ω, ℑ, P). A convenient example of an infinite-dimensional discrete-time process consists of associating with a sequence of random variables whose values are in an appropriate function space (see [19]). This may be obtained by setting (1) so X = (X_t, t = 0, 1, 2, …, T). If the sample paths of ξ are continuous, then we have a sequence X₀, X₁, …. of random variables in the space C[0, N]. The choice of the period or segment t will depend on the situation in hand. In our case, t will be the period of a year, and N represents cross-sectional units or higher-frequency time series.

We may be interested in modeling the whole sequence of G functions, for instance the sequence of state densities (f₁(ω), f₂(ω), …, f_T(ω)) as in [20, 21] or only certain characteristics (C_t(w)) of these G functions, for instance, the state mean, the state variance, the state quantile, etc. These characteristics can be considered time series objects and, therefore, all the econometric tools already developed in the time series literature can be applied to C_t(w). This resembles the quantile curve estimation analyzed in [22, 23]. With this characteristic approach we go from Ω to , as in a standard stochastic process, passing through a G functional space:

Going back to the convenient example and abusing notation (X_t(n) = X_tn and n a natural number), the stochastic structure can be summarized in the following array:

(2)

Throughout this paper, similarly to the assumptions made in [20, 23, 24], we assume that in each period, t, the stochastic functional process X = (X_t(ω), t ∈ T) satisfies certain regularity conditions (local stationarity, strong mixing conditions, etc.) and there are sufficient temporal or cross-sectional observations (N → ∞) for these characteristics to be estimated consistently. The main characteristics analyzed in this paper are moments and quantiles (q_τ, 0 < τ < 1) of the temperature distribution.

After the development of the proposed theoretical core, we are in a position to design tools to approach the empirical strategy. The following subsection describes each of them.

2.2 Empirical tools: Definitions and tests

One of the objectives of this section is to provide a simple test to detect the existence of a general unknown trend component in a given characteristic C_t of the temperature process X_t. To do this, we need to convert Definition 1 into a more practical definition.

Definition 2. (Practical trend definition): A characteristic C_t of a functional stochastic process X_t contains a trend if in the LS regression, (3) β = 0 is rejected.

GG2020 shows that a simple t − test (H₀ : β = 0, H₁ : β ≠ 0) that asymptotically follows a normal distribution under H₀ is able to detect most of the existing deterministic trends (polynomial, logarithmic, exponential, etc) and also the trends generated by any of the three standard persistent processes considered in the literature (see [25]): (i) fractional or long-memory models (1/2 < d < 3/2); (ii) near-unit-root AR models; and (iii) local-level models.

Several remarks are relevant with respect to this definition: (i) regression (3) has to be understood as the linear LS approximation of an unknown trend function h(t) (see [26]); (ii) the parameter β is the plim of ; (iii) if the regression (3) is the true data-generating process, with u_t ∼ I(0), then the OLS estimator is asymptotically equivalent to the GLS estimator (see [27]) and the t−test(β = 0) is N(0,1); (iv) in practice, in order to test β = 0, it is recommended to use a robust HAC version of t_{β = 0} (see [28]); and (v) this test only detects the existence of a trend but not the type of trend. Notice also that in (3) we could be totally agnostic about u_t being I(0) or I(1). In this case following [29] we can estimate the model by Feasible Generalized Least Squares and construct a similar t-stat of β = 0 that still will follow a N(0,1). This method depends on a tuning parameter (how closes is u_t of being I(1)). To avoid that, in this paper, we follow an alternative approach. We pre-test the temperature data for unit roots, once they are rejected we proceed as if u_t in 3 is I(0).

For all these reasons, in the empirical applications we implement Definition 2 by estimating regression (3) using OLS and constructing a HAC version of t_β=0 [30]. In the definition of C_t we can consider any distributional characteristics as time series objects. This set includes not only the quantiles that constitute the distribution, but also other characteristics that may be of interest, enabling a much more comprehensive analysis of temperature than is achievable with the mean alone.

These linear trends can be common across characteristics indicating similar patters in the time evolution of these characteristics.

Definition 3. (Co-trending): A set of m distributional characteristics (C_1t,C_2t,…,C_mt) do linearly co-trend if in the multivariate regression (4) all the slopes are equal, β₁ = β₂ = … = β_m.

This definition is slightly different from the one in [31].

This co-trending hypothesis can be tested by a standard Wald test (HAC version) with H₀ : β₁ = β₂ = … = β_m and H1 : ∃β_i ≠ β_j, i ≠ j ∈ 1, 2, …, m. This test follow asymptotically a distribution.

When m = 2 an alternative linear co-trending test can be obtained from the regression i ≠ j i, j = 1, …, m by testing the null hypothesis of β = 0 vs β ≠ 0 using a simple t_{β = 0} test.

Notice that this test does not suffer of spurious co-trending. In a situation where C_it and C_jt were two non-cointegrated I(1) processes, the test would reject the null hypothesis of β = 0 and conclude that both characteristics do not co-trend.

Climate classification is a tool used to recognize, clarify and simplify the existent climate heterogeneity in the Globe. It also helps us to better understand the Globe’s climate and therefore to design more efficient global warming mitigation policies. The prevalent climate typology is that proposed by [32] and later on modified in [33]. It is an empirical classification that divides the climate into five major types, which are represented by the capital letters A (tropical zone), B (dry zone), C (temperate zone), D (continental zone), and E (polar zone). Each of these climate types except for B is defined by temperature criteria. More recent classifications can been found in the AR6 of the IPCC (2021, 2022) but all of them share the spirit of the original one of [32].

The climate classification we propose in this section is also based on temperature data and it has three simple distinctive characteristics:

• It considers the whole temperature distribution and not only the average
• It has a dynamic long-run nature: it is based on the evolution of the trend of the temperature quantiles (lower and upper)
• It can be easily tested.

Definition 4. (Warming Typology): We define four types of warming processes:

• W0: There is no trend in any of the quantiles (No warming).
• W1: All the location distributional characteristics have the same positive trend (dispersion does not contain a trend)
• W2: The Lower quantiles have a larger positive trend than the Upper quantiles (dispersion has a negative trend)
• W3: The Upper quantiles have a larger positive trend than the Lower quantiles (dispersion has a positive trend).

Climate is understood, unlike weather, as a medium and long-term phenomenon and, therefore, it is crucial to take trends into account. Notice that this typology can be used to describe macroclimate as well as microclimate locations.

The literature on global and local warming that solely examines the trend behavior of the central part of the temperature distribution (mean or median) overlooks valuable information that can be utilized to describe the entire warming process. It is possible that by analyzing only the mean temperature of a region, one may conclude that there is no significant warming (no trend in the mean), while clear warming (trend) exists in other parts of the temperature distribution. The proposed warming typology takes into account the trend behavior of the entire temperature distribution, not just the mean temperature of a region. This approach offers a more comprehensive understanding of the warming process and is valuable for characterizing and measuring climate change heterogeneity. However, this typology does not address the intensity of the warming process and its dynamics. Part of this intensity is captured in the following testable definitions of warming acceleration and warming amplification.

Definition 5. (Warming Acceleration): We say that there is warming acceleration in a distributional temperature characteristic C_t between the time periods t₁ = (1, …, s) and t₂ = (s + 1, …, T) if in the following two regressions: (5) (6) the second trend slope is larger than the first one: β₂ > β₁.

In practice, we implement this definition by testing in the previous system the null hypothesis β₂ = β₁ against the alternative β₂ > β₁ An alternative warming acceleration test can be formed by testing for a structural break at t = s. The preference for the approach of Definition 5 is due to its close alignment with the existing narrative on warming acceleration in the climate literature (see [34]).

The existence of acceleration can be considered an example of a non-linearity in the climate change process (see [10]). Warming acceleration is behind the recent surge in record-breaking heat. This acceleration has elevated the baseline for temperature fluctuations, leading to unprecedented spikes in warmth and contributing to extreme heat waves, wildfires, and other extreme climate events.

Definition 6. (Warming Amplification with respect to the mean): We say that there is a warming amplification in distributional characteristic C_t with respect the mean if in the following regression: (7) the mean slope is greater than one: β > 1.

When the mean, mean_t, and C_t come from the same distribution, we name this “inner” warming amplification. If the mean originates from an external environment, it is refereed to as “outer” warming amplification. The implications of warming amplification, particularly but not only in the Arctic, are far-reaching and have significant environmental consequences like decline of sea ice cover, retreat of the Greenland ice sheet, retreat of regional glaciers, permafrost degradation, disturbances in ecosystems, influences on extreme climate events in lower latitudes, etc. Understanding theses implications is crucial for informing climate policies and adaptation strategies on both regional and global scale. And this is why is important to have a definition of amplification that can be easily tested.

Both concepts, acceleration and amplification, introduce a quantitative dimension to the ordinarily defined classification. For example, the acceleration, which has a dynamic character, allows us to observe the transition from one type of climate to another. Amplification, on the other hand, makes it possible to compare the magnitude of the trends that define each type of climate. It should be noted that, although static in nature, it can be computed recursively at different points in time.

In the previous definitions, we categorize the warming process of different regions, which is essential for designing local mitigation and adaptation policies. However, it is also important to compare the different climate change processes of two regions in order to characterize climate heterogeneity independently of the type of warming they are experiencing. For this purpose, we propose the following definition, which is akin to the stochastic dominance concept used in the economic-finance literature.

Definition 7. (Warming Dominance (WD)): We say that the temperature distribution of Region A warming dominates (WD) the temperature distribution of Region B if in the following regression, with q_τ the quantile τ of temperature distributions of regions A or B: (8) β_τ ≥ 0 for all 0 < τ < 1 and there is at least one value τ* for which a strict inequality holds.

It is also possible to have only partial WD. For instance, in the lower or upper quantiles. The concept of WD is analyzed in full detail in [35].

3.1 The Globe

For the Globe, we utilize the Climate Research Unit (CRU) database, which provides monthly and yearly land and sea temperature data for both hemispheres from 1850 to the present, collected from different stations around the world. A recent revision of the methodology can be found in [36]. Each station temperature is converted to an anomaly, taking 1961–1990 as the base period, and each grid-box value, on a five-degree grid, is the mean of all the station anomalies within that grid box. This database (in particular, the annual temperature of the Northern Hemisphere) has become one of the most widely used to illustrate GW from records of thermometer readings. These records form the blade of the well-known “hockey stick” graph, frequently used by academics and other institutions, such as, the IPCC. In this paper, we prefer to base our analysis on raw station data, as in [4]. Similar results are obtained with stable grid anomaly data. The crucial issue in dealing with temperature data is the aggregation process. Failing to maintain consistent stations or grids across the entire sample increases the likelihood of encountering outliers (see [37] for a method to evaluate their impact on climate data) and potentially spurious unit roots in distributional characteristics such as average temperature (see [38]).

The database provides data from 1850 to nowadays, although due to the high variability at the beginning of the period it is customary in the literature to begin in 1880. In this work, we have selected the stations that are permanently present in the period 1950–2019 according to the concept of the station-month unit. In this way, the results are comparable with those obtained for the benchmark country Spain. Notice that this period corresponds to the Anthropocene period of time. Although there are 10,633 stations on record, the effective number fluctuates each year and there are only 2,192 stations with data for all the years in the sample period, which yields 19,284 station-month units each year (see this geographical distribution in the map in Fig 1. In the CRU data there are 115 Spanish stations. However, after removing stations not present for the whole 1880 to 2019 period, only Madrid-Retiro, Valladolid and Soria remain. Since 1950, applying the same criteria, only 30 remain. In summary, we analyze raw land temperature observational global data (stations instead of grids) for the period 1950 to 2019, compute station-month units that remain all the time and with these build the annual distributional characteristics.

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Fig 1. Geographical distribution of stations.

A: The Globe. Selected stations, CRU data 1950–2019 B: Spain. Selected stations, AEMET data 1950–2019. Sources: A: Own elaboration based on CRU data with the toolbox Mapping under Matlab license version 2022b; B: Own elaboration with R and data from IGN and AEMET; the geographic data of the base map are obtained from the free-use tool developed by the National Geographic Institute of Spain and can be consulted at: https://comunidadsig.maps.arcgis.com/home/item.html?id=e75892d1a49646d8a29705ac6680f981.

https://doi.org/10.1371/journal.pone.0317208.g001

3.2 Spain

The measurement of meteorological information in Spain started in the eighteenth century. However, it was not until the mid-nineteenth century that reliable and regular data became available. In Spain, there are four main sources of meteorological information: the Resumen Anual, Boletín Diario, Boletín Mensual de Climatología and Calendario Meteorológico. These were first published in 1866, 1893, 1940 and 1943, respectively. A detailed explanation of the different sources can be found in [39].

Currently, AEMET (Agencia Estatal de Meterología) is the agency responsible for storing, managing and providing meteorological data to the public. Some of the historical publications, such as the Boletín Diario and Calendario Meteorológico can be found in digital format in their respective archives for whose use it is necessary to use some kind of Optical Character Recognition (OCR) software.

In 2015, AEMET developed AEMET OpenData, an Application Programming Interface (API REST) that allows the dissemination and reuse of Spanish meteorological and climatological information. To use it, the user needs to obtain an API key to allow access to the application. Then, either through the GUI or through a programming language such as Java or Python, the user can request data. More information about the use of the API and the resolution about the use of the data can be found on their webpage.

In this paper, we are concerned with Spanish daily station data, specifically temperature data. Each station records the minimum, maximum and average temperature as well as the amount of precipitation, measured as liters per square meter. The data period ranges from 1920 to 2019. However, in 1920 there were only 13 provinces (out of 52) who had stations available. It was not until 1965 that all the 52 provinces had at least one working meteorological station. Moreover, it is important to keep in mind that the number of stations has increased substantially from only 14 stations in 1920 to more than 250 in 2019. With this information in mind, we select the longest span of time that guarantees a wide sample of stations so that all the geographical areas of peninsular Spain are represented. For this reason, we decided to work with station data from 1950 to 2019. There are 30 stations whose geographical distribution is displayed in the map in Fig 1. The original daily data are converted into monthly data, so that we finally work with a total of 30x12 station-month units corresponding to peninsular Spain and, consequently, we have 360 observations each year with which to construct the annual distributional characteristics.

Table 1 contains a summary of the data detail for the Globe and Spain.

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Table 1. Data details.

https://doi.org/10.1371/journal.pone.0317208.t001

4.1 Global warming: The Globe

The cross-sectional analysis is approached under two assumptions. First, choosing a sufficiently long and representative period of the geographical diversity of the Globe and the Spanish Iberian Peninsula, 1950–2019. Second, we work with month-station units from monthly observations to construct the annual observations of the time series object from the data supplied by the stations, following a methodology similar to that carried out for the whole planet in [21]. The decision to work with monthly data instead of daily in the cross-sectional approach has been based on its compatibility with the data available for the Globe. The results with daily averages for Spain are very similar. The study comprises the steps described in the previous section. Figs 2 and 3 show the time evolution of the Global temperature densities and their different distributional characteristics from 1950 to 2019. Importance to notice that the unconditional quantiles correspond to locations-latitudes (Fig 4). This is a simple way of incorporating the spatial dimension into the analysis. In some sense, to go from analyzing the average temperature to analyze all the quantiles is like going from a zero dimension energy balance climate model to a one dimensional model. The data in the three figures are obtained from stations that report data throughout the sample period.

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Fig 2. Global annual temperature density calculated with monthly data across stations.

https://doi.org/10.1371/journal.pone.0317208.g002

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Fig 3. Characteristics of temperature data in the Globe (monthly data across stations, CRU, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.g003

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Fig 4. Geographical location by quantiles (monthly data across stations, CRU, 1950–2019).

Sources: Own elaboration based on CRU data with the toolbox Mapping under Matlab license version 2022b.

https://doi.org/10.1371/journal.pone.0317208.g004

Table 2 shows a positive trend in the mean as well as in all the quantiles. This indicates the clear existence of Global warming, more pronounced (larger trend) in the lower part of the distribution (a negative trend in the dispersion measures). There is a lack of consensus regarding the acceleration of the warming process. Some argue that warming is steadily increasing but not accelerating, while others, such as [34], assert a clear acceleration due to the effect of clean air regulations that have reduced aerosol pollution and the amount of sulfur in fuel used by ocean shipping. Our quantitative findings align with the latter perspective, indicating a clear acceleration in the warming process in the mean and in all the quantiles above q30.

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Table 2. Trend acceleration hypothesis (CRU monthly data across stations, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.t002

From the co-trending analysis (see Tables 3 and 4) we can determine the type of warming process characterizing the whole Globe. Table 3 indicates that in the period 1950–2019 the Globe experimented a W2 warming type (the lower part of the temperature distribution grows faster than the middle and upper part, implying iqr and std have a negative trend). Similar results are maintained for the period 1970–2019 (in this case only the dispersion measure std has a negative trend).

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Table 3. Co-trending analysis (CRU montly data, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.t003

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Table 4. Co-trending analysis (CRU montly data, 1970–2019).

https://doi.org/10.1371/journal.pone.0317208.t004

The asymmetric amplification results shown in Table 5 reinforce the W2 typology for the whole Globe: an increase of one degree in the global mean temperature increases the lower quantiles by more than one degree. In particular, about the Arctic Amplification (AA) (see [41] for a survey of this concept), we obtain (see Table 5) that the Arctic (q05) has warmed twice the global average. In the literature, there is little consensus on the magnitude of the recent AA. Numerous recent studies report the Arctic having warmed either almost twice [42], about twice [43], or more than twice [44, 45] as fast as the global average. Our findings also indicate that the amplification of warming extends beyond the Arctic, reaching lower latitude locations (q10, q20, and q30). They also demonstrate that Arctic warming amplification has peaked, with the amplification being larger for the period 1950–2019 than for 1970–2019. This is consistent with previous research (see [46, 47]).

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Table 5. Amplification hypotheses (CRU monthly data across stations, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.t005

In summary, the results from our various proposed tests for the evolution of the trend of the entire temperature distribution suggest that the Globe can be classified as undergoing a type W2 warming process. This type of warming may have more significant consequences for ice melting, sea level increases, permafrost, migrations, etc., compared to other warming types.

4.2 Local warming: Spain

In this section, we carry out a similar analysis to that described in the previous subsection for the Globe. The density of the data and the evolution of characteristics are displayed, respectively in Figs 5 and 6.

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Fig 5. Spain annual temperature density calculated with monthly data across stations.

https://doi.org/10.1371/journal.pone.0317208.g005

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Fig 6. Characteristics of temperature data in Spain with stations selected since 1950 (monthly data across stations, AEMET, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.g006

We find positive and significant trends in the mean, max, min and all the quantiles. Therefore from definition 1, we conclude there exists a clear local warming (see Table 6).

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Table 6. Trend acceleration hypothesis (Spain monthly data across stations, AEMET, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.t006

The recursive evolution for the periods 1950–2019 and 1970–2019 shows a clear increase in the trends of the mean, some dispersion measures and higher quantiles (see the last column of Table 6). More precisely, there is a significant trend acceleration in most of the distributional characteristics except the lower quantiles (below q20). These quantiles, q05 and q10, remain stable. The acceleration in Spain is reported to be stronger than the one experienced by the Globe. This indicates a notable change in the temperature distribution, particularly in the higher quantiles, which can have implications for local climate policies and environmental impacts.

The co-trending tests for the full sample 1950–2019 show a similar evolution of the trend for all the quantiles with a constant iqr (see Table 7). This indicates that in this period the warming process of Spain can be considered a W1 type. More recently, 1970–2019, the co-trending tests (see Table 8) indicate the upper quantiles grow faster than the lower ones. This, together with a positive trend in the dispersion measured by the iqr shows that Spain has evolved from a W1 to a W3 warming type process.

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Table 7. Co-trending analysis (Spain monthly data across stations, AEMET, 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.t007

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Table 8. Co-trending analysis (Spain monthly data across stations, AEMET, 1970–2019).

https://doi.org/10.1371/journal.pone.0317208.t008

Finally, no evidence of “inner” amplification during the period 1950–2019 is found in the lower quantiles. Regarding the upper quantiles, we found both “inner” and “outer” amplification in the second period, which supports the previous finding of a transition from type W1 to type W3 (see Table 9). With respect the “outer” amplification it is important to note that while there is no amplification with respect to the average global temperature in the median, there is amplification in the upper quantiles (approximately 1.4^oC) This is evidenced by the heatwaves experienced in the country (see [48]).

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Table 9. Amplification hypothesis (Spain monthly data, AEMET 1950–2019).

https://doi.org/10.1371/journal.pone.0317208.t009

Summing up, with our proposed tests for the evolution of the trend of the whole temperature distribution, we conclude that Spain has evolved from a W1 type to a much more dangerous W3 type. The results of acceleration and dynamic amplification reinforce the finding of this transition to type W3.

4.3 Micro-local warming: Madrid and Barcelona

The existence of warming heterogeneity implies that in order to design more efficient mitigation policies, they have to be developed at different levels: global, country, region etc. How local we need to go will depend on the existing degree of micro-warming heterogeneity. In this subsection, we go to the smallest level, climate station level. We analyze, within Spain, the warming process in two meteorological stations corresponding to two cities: Madrid (Retiro park station) and Barcelona (Fabra station). From Madrid and Barcelona there is data since 1920’s, nevertheless we began the study in 1950 for consistency with the previous analysis of Spain and the Globe. Obviously, the data provided by these stations is not cross-sectional data but directly pure time series data. Our methodology can be easily applied to higher frequency time series, in this case daily data, to compute the distributional characteristics (see Fig A1 and Fig A2 in S1 Appendix). See the application to Central England in GG2020 and in [49] to Madrid, Zaragoza and Oxford.

The results are shown in the S1 Appendix. These two stations, Madrid-Retiro and Barcelona-Fabra clearly experience two different types of warming. First, there is evidence of micro-local warming, understood as the presence of significant and positive trends, in all the important temperature distributional characteristics of both stations. The acceleration phenomenon is also clearly detected, in other words, the warming increases as time passes (see Tables A1 and A5 in S1 Appendix). Secondly, from the co-trending tests (Tables A2, A3, A6 and A7 in S1 Appendix), it can be concluded that the warming process of Madrid-Retiro is type W3 while for Barcelona-Fabra it is type W1. In both cases the warming typology is stable through both sample periods (1950–2019 and 1970–2019). Thirdly, as expected, Madrid-Retiro presents “inner” and “outer” amplification for the upper quantiles, while Barcelona-Fabra does so only for the center part of its temperature distribution (see Tables A4 and A8 in S1 Appendix).

In summary, there is clear evidence of heterogeneous warming within Spain. While Madrid, with a Cold Semi-Arid climate according to the Köppenn-Geiger classification [32, 33], exhibits a similar warming pattern to that of peninsular Spain (1970–2019) W3, Barcelona, characterized by a Mediterranean coastline climate, maintains a W1 typology. These distinct warming processes may necessitate different mitigation and adaptation policies at both the national and local levels. The differences in climate classifications based on associated warming trends highlight the need for tailored approaches to address the specific challenges posed by climate change in different regions of Spain.