A Bayesian Attribution Analysis of Extreme Temperature Changes at Global and Regional Scales

Min-Gyu Seong aDivision of Environmental Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea

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Seung-Ki Min aDivision of Environmental Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea

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Xuebin Zhang bClimate Research Division, Environment and Climate Change Canada, Toronto, Ontario, Canada

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Abstract

Recent studies showed that anthropogenic greenhouse gas (GHG) increase is a major driver of the observed increases in extreme temperatures at global and regional scales using an optimal fingerprint (OF) method, which is a frequentist approach based on linear regression. Here, a Bayesian decision method is employed, which finds the most probable cause of the observed changes by comparing likelihoods of different forcings in view of observations. To quantify individual forcing contributions, a new modified attribution procedure based on Bayesian decision is proposed, i.e., computing the likelihood ratio [Bayes factor (BF)] between different forcings. First, the contribution of anthropogenic forcing (ANT) is measured by BF between anthropogenic-plus-natural forcing (ALL) and natural forcing (NAT) using a threshold for “substantial” evidence (lnBF ≥ 1). Similarly, the NAT contribution is assessed by BF between ALL and ANT. Further, the GHG contribution to the detected ANT is quantified by BF between ANT and anthropogenic aerosols (AA), and the AA contribution is evaluated by BF between ANT and GHG. The devised Bayesian approach is applied to HadEX3 observations and CMIP6 multimodel simulations for extreme temperature intensities (warmest day/night and coldest day/night) for global, continental, and regional domains following previous studies. Bayesian attribution results indicate that the ANT signal is detected in many continental and subregions for all extremes indices. This is generally consistent with OF-based results but with less frequent detection, indicating that the Bayesian method is slightly stricter than the OF method. However, GHG contributions to the detected ANT are identified over more subregions in the Bayesian attribution, suggesting its potential advantage over conventional methods in case of low signal-to-noise ratio and high collinearity.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher's Note: This article was revised on 2 December 2022 to include a funding acknowledgement that was omitted when originally published.

Corresponding author: Seung-Ki Min, skmin@postech.ac.kr

Abstract

Recent studies showed that anthropogenic greenhouse gas (GHG) increase is a major driver of the observed increases in extreme temperatures at global and regional scales using an optimal fingerprint (OF) method, which is a frequentist approach based on linear regression. Here, a Bayesian decision method is employed, which finds the most probable cause of the observed changes by comparing likelihoods of different forcings in view of observations. To quantify individual forcing contributions, a new modified attribution procedure based on Bayesian decision is proposed, i.e., computing the likelihood ratio [Bayes factor (BF)] between different forcings. First, the contribution of anthropogenic forcing (ANT) is measured by BF between anthropogenic-plus-natural forcing (ALL) and natural forcing (NAT) using a threshold for “substantial” evidence (lnBF ≥ 1). Similarly, the NAT contribution is assessed by BF between ALL and ANT. Further, the GHG contribution to the detected ANT is quantified by BF between ANT and anthropogenic aerosols (AA), and the AA contribution is evaluated by BF between ANT and GHG. The devised Bayesian approach is applied to HadEX3 observations and CMIP6 multimodel simulations for extreme temperature intensities (warmest day/night and coldest day/night) for global, continental, and regional domains following previous studies. Bayesian attribution results indicate that the ANT signal is detected in many continental and subregions for all extremes indices. This is generally consistent with OF-based results but with less frequent detection, indicating that the Bayesian method is slightly stricter than the OF method. However, GHG contributions to the detected ANT are identified over more subregions in the Bayesian attribution, suggesting its potential advantage over conventional methods in case of low signal-to-noise ratio and high collinearity.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher's Note: This article was revised on 2 December 2022 to include a funding acknowledgement that was omitted when originally published.

Corresponding author: Seung-Ki Min, skmin@postech.ac.kr

1. Introduction

Many studies have detected human influences on the observed increases in frequency and intensity of extreme temperatures at global to subcontinental scales (Morak et al. 2011; Zwiers et al. 2011; Min et al. 2013; Kim et al. 2016; Wang et al. 2017). Using updated observations (HadEX3; Dunn et al. 2020) and CMIP6 multimodel simulations under individual forcings (Eyring et al. 2016; Gillett et al. 2016), recent studies have further quantified anthropogenic greenhouse gas (GHG) contributions to the observed changes in extreme temperatures in separation from other forcings including anthropogenic aerosols (AA) and natural forcing (NAT; solar and volcanic activities). Hu et al. (2020) found a dominant contribution by GHG forcing to the observed frequency changes in temperature extremes at global and most continental scales with an offsetting cooling contribution by AA forcing. Seong et al. (2021) obtained the same results for the intensity changes in extreme temperatures and additionally at subcontinental regions. They also detected AA signals in some continental domains, which have induced the recent warming in Europe and cooling in Asia, reflecting the contrasting influence of aerosol reduction and increase, respectively. These studies have contributed to the stronger conclusions about human influence on climate extremes made in the Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC; Eyring et al. 2021; Seneviratne et al. 2021).

Most previous detection and attribution (D&A) studies of extreme temperature changes used a frequentist approach based on multiple linear regression, called optimal fingerprinting (OF; Hasselmann 1997; Allen and Tett 1999; Allen and Stott 2003; Ribes et al. 2013). The OF method compares an observed change pattern (usually low-frequency time series) with those from model simulations under different forcing factors by regressing the observations onto the model simulated response patterns (fingerprints). If the 90% ranges of the resulting regression coefficients (scaling factors) lie above zero, the null hypothesis is rejected, indicating that the observed change is beyond the range of natural variability (“detection”). If the scaling factor of detected signal includes unity, the signal is further assessed to be consistent with the observations in terms of amplitude (“attribution”). Assuming linear additivity of different forcing influences (e.g., Meehl et al. 2004; Shiogama et al. 2013; Ribes et al. 2015), the OF method enables one to separate a signal from other forcings and also provides a way to quantify each forcing’s contribution based on corresponding regressed terms, so-called attributable changes. Also, by checking the scaling factors, one can evaluate whether models underestimate or overestimate the observed changes. The OF method has, however, some limitations. It is difficult to interpret negative scaling factors and resulting attributable trends, which reverse signs of fingerprints to match with the observed patterns (e.g., Christidis et al. 2005; Rupp et al. 2013). More importantly, when fingerprint patterns are similar between different external forcings, multicollinearity can make the signal detection difficult (Ribes and Terray 2013; DelSole et al. 2019). A typical example is a strong similarity in temperature fingerprints between GHG and AA (i.e., steady increases versus decreases, respectively), which can widen the uncertainty ranges of the scaling factors (Jones et al. 2016; DelSole et al. 2019).

As an alternative approach, Bayesian D&A methods provide more comprehensive interpretations of data and results based on probabilistic estimations (Hasselmann 1998; Annan 2010). Some studies applied Bayesian inference to estimate the scaling factors based on the OF framework (Berliner et al. 2000; Lee et al. 2005; Katzfuss et al. 2017), as reviewed in Eyring et al. (2021). Others consider using likelihood ratios (or Bayes factors; see below) to directly evaluate the hypotheses for explaining the observed changes (Min et al. 2004; Schnur and Hasselmann 2005; Paeth et al. 2008, 2017). Several Bayesian D&A studies detected human influences on the observed warming at global and continental scales, supporting the OF-based results (e.g., Schnur and Hasselmann 2005; Min and Hense 2006, 2007). However, there have been no Bayesian attribution analyses of extreme temperature changes. More importantly, quantifying individual forcing contributions to the detected anthropogenic influences has not been implemented in the Bayesian framework.

The aim of this study is 1) to improve the Bayesian decision method to be able to measure the additional contribution of individual forcings, particularly GHG and AA, and 2) to apply the devised method to extreme temperature indices using recent observations and climate model simulations and evaluate results in comparison with OF-based ones. In the next section, observational and model datasets are described. In section 3, new Bayesian attribution method is explained. Application results to extreme temperatures are detailed in section 4, followed by conclusions and discussion in the last section.

2. Data

a. Observations

Four extreme temperature indices—annual minima/maxima of daily minimum temperatures (TNn/TNx) and daily maximum temperatures (TXn/TXx)—are considered for the period 1951–2015. Observations are obtained from HadEX3 dataset (Dunn et al. 2020), which has a 1.875° × 1.25° horizontal resolution. The data preprocessing procedure is identical to that in Seong et al. (2021). All extreme temperature indices are first transformed into probability-based indices (PI) based on generalized extreme value (GEV) distribution, which range from 0 to 1. The maximum likelihood method (Kharin and Zwiers 2005) is used to estimate GEV parameters. The influence of estimated GEV parameters on our attribution results would be negligible. As discussed in Min et al. (2013), this PI transformation is done to give roughly equal weight to all grids considering that extreme temperatures can be highly skewed locally. Further, we use large area averages of PI (see below) for detection, which would be close to Gaussian (as a consequence of the central limit theorem). A quick check using a model simulation indicates that GEV parameters remain similar across time, exerting negligible influences on area-mean PI (not shown).

Analysis grids are selected based on the data coverage, requiring observations available for longer than 46 years (70% of total period) and for at least 3 years during the recent 5 years (2011–15). TXx, which has the least data coverage among the four extreme indices, is used as a common observational mask. Then we obtain spatial averaged time series of PI anomaly relative to the 1951–2015 mean over the global domain (GLB), five continental domains [North America (NA), South America (SA), Europe (EUR), Asia (ASI), and Oceania (OCE)], and 33 subcontinental domains defined by Iturbide et al. (2020). These domains were determined based on spatial data coverage (more than 60% of grids have sufficient long-term data). To remove interannual variability noise, 5-yr averaged PI time series are finally obtained for each domain, which gives 13-dimensional vectors. Results remain unaffected when using 10-yr averaged PI (not shown), indicating the robustness to data processing, consistent with previous studies (e.g., Min et al. 2013). Refer to Seong et al. (2021) for more details.

b. Model simulations

Model simulations from phase 6 of the Coupled Model Intercomparison Project (CMIP6; Eyring et al. 2016) are used for the period 1951–2015, which includes ALL (anthropogenic-plus-natural forcing), GHG (greenhouse gas), AA (anthropogenic aerosol-only forcing), and NAT (natural-only forcing) simulations (Table 1), identical to those used in Seong et al. (2021). There are 32 runs from 7 models for GHG, AA, and NAT and 46 runs from 7 models for ALL. For ALL, “historical” simulations (1951–2014) are extended by 2015 by combining with corresponding Shared Socioeconomic Pathway (SSP) 2–4.5 simulations. Preindustrial control simulations (CTL) from 21 models are also used to estimate internal variability ranges (see below). PIs are calculated on original grids and then are interpolated onto the HadEX3 grids. Interpolated PIs are then masked with HadEX3 data coverage of TXx following Seong et al. (2021).

Table 1

CMIP6 models used in this study. Numbers indicate the number of runs for each forced experiment and the number of 65-yr chunks for CTL.

Table 1

3. Bayesian attribution method

a. Bayesian decision method

The OF method compares observed changes (d) with model simulated patterns (fingerprints, μ) based on multiple linear regression (Fig. 1a). The total least squares (TLS) method is typically employed to estimate regression coefficients, which considers errors in fingerprints as well as internal variability (Allen and Stott 2003). The regression coefficients are referred to as scaling factors and here their 90% confidence intervals are estimated using the method of Ribes et al. (2013). Detection of a signal is declared if the lower bound of its scaling factor is greater than zero. Additionally, when the range of scaling factor includes unity, it implies that the detected signal has consistent amplitude with the observations (attribution), as illustrated in Fig. 1a.

Fig. 1.
Fig. 1.

Schematic illustration of anthropogenic (ANT) and natural (NAT) forcing signal detection using the (a) optimal fingerprint and (b) Bayesian attribution techniques. In (a), regression coefficients (scaling factors) and their 90% confidence intervals (error bars) are used for detection. In (b), the Bayes factor is used for detection, calculated as the likelihood ratio of a certain forcing (ALL) to reference forcing (NAT or ANT in this example). Darker gray shading indicates stronger detection evidence (substantial, strong, or decisive; Table 2).

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

A Bayesian decision method for climate change D&A compares the observed changes with model-simulated fingerprints in terms of distance (Fig. 1b; Min et al. 2004; Min and Hense 2006). A basic concept of Bayesian decision is to find a scenario (here, an external forcing factor like ALL, GHG, AA, or NAT) minimizing overall risk (i.e., expected loss) given the observations (d) using posterior probability (Duda and Hart 1973). Given several scenarios mi (i = 1, 2, …, N), the posterior probability of each scenario in view of observations can be calculated from the prior probability of each scenario and the likelihood based on Bayes’ theorem:
P(mi|d)=l(d|mi)P(mi)j=1Nl(d|mi)P(mj).
Here prior probability P(mi) indicates subjective belief in each scenario and the likelihood l(d|mi) represents observational probability given the scenario of consideration. Each scenario is defined using ensemble realizations (f) from climate models performed under the corresponding forcing. The likelihood is conditional on the scenario only and needs to be unconditional on f. The likelihood can be then formulated as an integral form of the multiplied conditional probability as follows:
l(d|mi)=p(d|f)p(f|mi)df.
Assuming that f and d follow multivariate Gaussian distributions, p(d|f) and p(f|mi) can be formulated as follows:
p(d|f)=12πqdetΣoexp[12(df)TΣo1(df)],
p(f|mi)=12πqdetΣiexp[12(fμi)TΣi1(fμi)],
where q is the dimension of observations and model simulations, Σo is the covariance matrix of the observed climatic variability, Σi is the covariance matrix of the internal variability of mi, and μi is the mean of the scenario mi (fingerprint vector). Here multimodel simulations for each scenario (mi) are used to estimate the mean μi and the covariance matrix Σi (see below). Multiplication of Eqs. (3) and (4), and after some algebra as detailed in Min et al. (2004), the likelihood is formulated as
l(d|mi)=1(2π)qdetAi1detΣidetΣoexp(12Λi),
where Ai is the combination of the inverse covariance matrix of model and observations ( Ai=Σi1+Σo1). The term Λi is defined as (dμi)T(Σi+Σo)1(dμi), which represents a generalized distance between observations and scenarios measured with their variability considered (Fig. 1b). Note that integrating Eq. (2) over all possible realizations f introduces the form detAi1/(detΣidetΣo) in the square root instead of det(ΣiΣo)−1.
According to the Bayesian decision rule, a decision is made by selecting the maximum posterior probability, which is calculated from Eq. (1) among defined scenarios [see Min et al. (2004) for details]. If prior probabilities are identical across scenarios, which is assumed here for simplicity as in previous studies (Schnur and Hasselmann 2005; Min and Hense 2006, 2007; Min et al. 2009), the Bayes factor (BFir) can be used as a decision function, which is defined as ratio of posterior odds to prior odds and becomes equivalent to the likelihood ratio due to the identical priors:
BFir=P(mi|d)/P(mr|d)P(mi)/P(mr)=l(d|mi)l(d|mr).
The Bayes factor BFir of the scenario mi (e.g., ALL, GHG, AA, or NAT) with respect to the reference scenario mr (usually CTL, representing internal variability) represents observational evidence against mr. Kass and Raftery (1995) suggested the descriptive scales of Bayes factor (Table 2) such that scenario mi has “substantial,” “strong,” or “decisive” evidence against the reference scenario when the logarithm of the Bayes factor is greater than or equal to 1, 2.5, or 5, respectively. These scales correspond to scenario mi being 3, 12, or 150 times more probable than the reference scenario mr.
Table 2

Descriptive scales of the Bayes factor suggested by Kass and Raftery (1995).

Table 2

In this study, possible scenarios mi include ALL, ANT, GHG, AA, NAT, and CTL; the ANT (anthropogenic) signal is estimated from ALL minus NAT assuming the linear additivity. The scenario mean (μi) for each scenario is calculated from ensemble mean of corresponding CMIP6 simulations (GHG, AA, and NAT: 32 runs, ALL: 46 runs; see Table 1). The covariance matrix for CTL (ΣCTL) is calculated using nonoverlapping chunks obtained from 21 models. In our case, mean μi are obtained by taking multimodel mean 5-yr block averaged times series of PI anomalies for 1951–2015 (13-dimensional vectors). Using 86 (first half) CTL chunks from 21 models (each consists of 13-dimensional PI time series), we calculate the 13 × 13 covariance matrix ΣCTL. Covariance matrices for scenarios are assumed to be identical to that of CTL (Σforced = ΣCTL), and the observed internal variability is also assumed to be identical to the modeled variability (Σo = ΣCTL) following previous studies. Note that, for consistency with the OF method, 172 CTL chunks are divided into 2 sets (86 chunks each). The first set is used to estimate covariance matrix (ΣCTL), and the second set is used to further assess modeled internal variability (see section 4a).

b. New Bayesian approach to identifying individual forcing contributions

Because the OF method is based on multiple linear regression, it allows separation of individual signal influences. For example, the ANT and NAT contributions can be estimated from each regressed terms in two-way regression as d = βANTμANT + βNATμNAT + ε (see Fig. 1a). Similarly, from a three-way analysis, the GHG signal can be separated from other anthropogenic (mainly AA) forcings and NAT. However, detecting a certain signal in separation from the others has not been carried out in the Bayesian decision framework. Here, a new procedure is proposed, which identifies the additional contribution of individual forcings to the detected signals.

Figure 1b shows a schematic illustration of the Bayesian attribution (BA) procedure for ANT and NAT signals. First, ALL forcing (ANT plus NAT) needs to be detected, explaining the observed temperature changes with at least “substantial” evidence [i.e., the logarithm of Bayes factor of ALL against CTL is above unity; Eq. (7)]. Next, in order to detect additional ANT contribution to the detected ALL signal, the logarithm of the Bayes factor between ALL and NAT (lnBFALL,NAT) is used, which represents the observational evidence of ALL against NAT [Eq. (8)]. If lnBFALL,NAT is above unity under the prerequisite of ALL signal detection, the additional ANT contribution is assessed to have at least “substantial” evidence. This means that ALL fingerprints are closer to observations than NAT (in terms of the generalized distance Λi in [Eq. (5)] due to the additional ANT contribution.
lnBFALL,CTL=lnl(d|mALL)l(d|mCTL)1
lnBFALL,NAT=lnl(d|mALL)l(d|mNAT)1s.t.ALLdetected
Additional NAT contributions can be measured in a similar way [Eq. (9)]. If lnBFALL,ANT ≥ 1, the additional NAT contribution is detected under the condition where the ALL signal is detected. This improvement of ALL signals against ANT indicates the NAT contribution.
lnBFALL,ANT=lnl(d|mALL)l(d|mANT)1s.t.ALLdetected
Approaches to finding additional GHG or AA contributions are similar; the only difference is the prerequisite of ANT detection based on Eq. (8). The logarithm of the Bayes factor of ANT against AA is calculated to identify additional GHG contributions. If lnBFANT,AA is above unity [Eq. (10)], an additional GHG contribution is detected, indicating that ANT detection is improved due to GHG signals, assuming that GHG and AA are dominant components of ANT as ANT ≈ GHG + AA.
lnBFANT,AA=lnl(d|mANT)l(d|mAA)1s.t.ANTdetected
Additional AA contributions are defined in the same way [Eq. (11)] as ln BFANT,GHG is above 1 under the requirement of ANT signal detection. This means that the ANT signal is more probable than GHG with at least “substantial” evidence due to AA signal contribution.
lnBFANT,GHG=lnl(d|mANT)l(d|mGHG)1s.t.ANTdetected

As described above, the suggested BA analysis is hierarchically performed. That means that additional ANT and NAT contribution analysis proceeds only when ALL is detected. Only after detecting additional ANT contribution, further analysis is carried out to evaluate additional GHG and AA contributions. This hierarchical procedure is different from the OF method in which two-signal or three-signal analysis is performed independently such that GHG or AA can be detected even when ANT is not detected. We find that this different setting does not affect much our detection results (not shown). However, for fair comparisons, OF results are masked with the same precondition of ALL or ANT detection when they are compared with BA results.

4. Application results

a. Internal variability assessment

It is important to assess internal variability simulated by models because too-small modeled variability compared to observations can produce spurious detection results by increasing signal-to-noise ratios. The OF method includes the residual consistency test, which compares modeled internal variability with the observed residual variability (Allen and Tett 1999; Allen and Stott 2003). However, BA does not include such a standard procedure of model evaluation, and a separate evaluation was conducted by checking modeled variabilities in comparison with the observed (e.g., based on power spectra) (Min and Hense 2006). Here, we formulate model evaluation by comparing modeled variance with the observed variance for four extreme indices.

First, modeled variance is obtained from preindustrial runs (CTL) and the observational residual is estimated by removing ALL-explaining portions from observation based on linear regression dresid = dβμALL. It should be noted that this procedure can inflate the observed residual variability, increasing the chance of model underestimation, since the linearly scaled ALL (βμALL) will not perfectly represent the true forced changes in observations (d). Results are shown in Fig. 2 for the global and continental domains. CTL variance and its 5%–95% range are smallest at global scale as expected due to cancellations of internal variability across the large area. Cold extremes (TNn, TXn) have greater observed and CTL variances than warm extremes (TNx, TXx) in northern continents, reflecting larger influences of natural climate variabilities in cold seasons, such as El Niño–Southern Oscillation and the Arctic Oscillation (e.g., Brown et al. 2008; Lim and Schubert 2011). EUR and OCE have larger variance than other domains, indicating stronger climate variabilities. Model underestimation of observed variability is found in some cases for cold extremes, particularly TNn and TXn over ASI and TNn over OCE. In contrast, for warm extremes, models well reproduce observed variability in global and all continental regions. A couple of exceptions are seen in TNx over NA and OCE. Results from this simple analysis show that CMIP6 models can simulate internal variability with comparable amplitudes to the observations, largely consistent with those based on the residual consistency test in OF method (Seong et al. 2021). Below we apply this method and compare the cases with model underestimation with OF-based results.

Fig. 2.
Fig. 2.

(left) Time series of 5-yr averaged PI anomalies of four extreme temperature indices (TNn, TXn, TNx, and TXx) during 1951–2015 from observations (black) and ALL (green) at global (GLB) and five continental domains (NA, SA, EUR, ASI, and OCE). Residual observations (gray solid line) are obtained by removing ALL patterns from the raw observations based on linear regression. The first decade is excluded in South America due to poor spatial coverage. (right) Variances of the residual observations (triangle) and those from CTL simulations (gray circle indicates the mean; error bar shows the 5%–95% range). Asterisks and plus signs indicate that models underestimate or overestimate the observed variability, respectively.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

b. ANT and NAT signal detection

Figure 3 illustrate PI anomaly time series of observations and model simulations (ANT and NAT multimodel means) and BA results for global and continental scales. ANT shows gradual increases in all extreme temperature indices over all continental domains while NAT exhibits little trend with a local cooling response in 1990s related to the volcanic eruption, but more strongly in warm extremes. The logarithms of the Bayes factors of ANT and NAT are displayed in the right panels, where closed circles indicate detected signals with at least “substantial” evidence (i.e., ln BF ≥ 1) and open circles depict no detection. Darker gray shading represents much stronger evidence (“strong” or “decisive”) for signal detection and white areas indicate no detection. For all extreme indices, ANT signals are detected robustly with at least “strong” evidence in most regions. An additional NAT contribution is found over a limited number of domains, especially in warm extremes. Note that NAT detection is not obtained by directly comparing NAT time series with observations. Instead, comparing ALL (green dashed line) with ANT time series [Eq. (9); also as explained in Fig. 1b] corresponds to the evaluation of an “additional” NAT contribution. Some cases show negative ln BF for NAT such as TXn in ASI, which indicates that observations are closer to (with larger likelihood) ANT than ALL. This represents that additional consideration of NAT does not improve ALL signals but rather hinders ALL detection by increasing its generalized distance from observations in those cases.

Fig. 3.
Fig. 3.

(left) Time series of 5-yr averaged PI anomalies in four extreme temperature indices during 1951–2015 from observations (black) and from ANT (orange) and NAT (blue) simulations (multimodel mean) at global and continental domains. ALL results (green dashed lines) are displayed for comparison. (right) Bayes factors of ANT (orange) and NAT (blue). Light, medium, and dark gray shadings indicate detection with “substantial,” “strong,” and “decisive” evidence, respectively. Filled circles represent signal detection with at least substantial evidence while open circles indicate no detection. Asterisks indicate model underestimation of the observed residual variability (as evaluated in Fig. 2).

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

The BA results of ANT and NAT are compared with the corresponding OF results (two-signal analysis of ANT and NAT) in Fig. 4. Orange and blue colors indicate that ANT and NAT detection occurs in each continental domain, respectively. For cold extremes, GLB, NA, and SA show identical results of ANT detection between two methods. There are some noticeable differences in other domains. For TNn, ANT is detected over EUR in BA but not in OF. When checking scaling factors for this case, the best estimate is reasonable (∼2.4) but the 90% confidence interval includes zero, indicating larger noise level. OCE shows ANT detection in OF but not in BA, which is because the requirement for ALL detection is not satisfied although ln BFALL,NAT is above 1. For TXn, ANT is detected additionally over EUR in both methods while ASI and OCE have ANT detected in single method, indicating a low signal-to-noise ratio. NAT signals are detected rarely, with no agreement between the two methods (TNn over SA and TXn over NA and EUR). Warm extremes (Figs. 4c,d) generally exhibit more detection of ANT and NAT with better consistency between two methods. Improved NAT detection in warm extremes seem to be related to stronger cooling responses to volcanic eruptions during summer seasons (e.g., Min et al. 2013; Kim et al. 2016). Both methods have ANT detected in TNx and TXx over all domains except NA where ANT detection occurs only in OF. The regions with NAT detection are also consistent between two methods, in TNx over SA and ASI and in TXx over SA. TXx for NA shows NAT detection in OF only.

Fig. 4.
Fig. 4.

Comparison of ANT (orange) and NAT (blue) signal detection between the optimal fingerprint (OF) and Bayesian attribution (BA) methods. The “substantial” scale (Table 2) is applied for Bayesian attribution. Global results are provided in the bottom left corner of each panel. Gray shading in the background map indicates the data availability of TXx fixed mask. Asterisks indicate model underestimation of the observed residual variability based on the residual consistency test in OF and the similar variance comparison in BA (as evaluated in Fig. 2). See text for details.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

The same BA analysis of ANT and NAT signals has been conducted for subcontinental domains, and results are displayed in Fig. 5 for TNx, which has the most frequent signal detection among extreme temperature indices. ANT signals are detected over many regions (25 out of total 33), with good agreement between both methods. SSA and NAU (see Fig. 5a for the subdomain names and extents) have ANT detected in OF only. A few NA subregions (CNA, NCA, and SCA) show no ANT detection, indicating low signal-to-noise ratios. The NAT signal is detected mainly in ASI subregions (RFE, TIB, EAS, and SAS) with good agreement between two methods except RAR where NAT is detected only in OF. Less frequent detection is obtained for other extreme indices (Figs. S1 and S2 in the online supplemental material). For TNn (Fig. S1a), ANT is detected in most of the subregions in BA and it becomes less detectable in OF except for NAU. ANT detection is reduced for TXn (Fig. S1b), but BA has more regions than OF except for three subregions (WNA, EEU, and SEA). TXx results (Fig. S1d) resemble TNx results but with decreased detection over NA subregions. In general, daily minimum temperature indices (TNn, TNx) tend to have more frequent ANT detection than the corresponding daily maximum temperature indices (TXn, TXx), consistent with the observed greater increases in nighttime temperatures than daytime temperature (e.g., Donat and Alexander 2012). NAT detection is very limited in cold extremes (Figs. S2a,b) with weak agreement between two methods whereas a few ASI and SA subregions exhibit NAT detection for TXx from both methods (Fig. S2d).

Fig. 5.
Fig. 5.

(a) The 33 subcontinental regions used in this study [adopted from Seong et al. (2021)]. Continental regions are depicted as different colors (NA: blue, SA: green, EUR: yellow, AFR: orange, ASI: red, and OCE: purple). Bold acronyms indicate the subcontinental regions analyzed in this study with good data coverage (more than 60% of grids have sufficient long-term data). (b) ANT and (c) NAT detection results for TNx at subcontinental domains from the optimal fingerprint (OF) and Bayesian attribution (BA) methods. Orange and blue colors indicate signal detection. The six continents are divided by black vertical lines. Asterisks represents model underestimation of the observed residual variability (as in Fig. 4).

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

c. GHG and AA signal detection

Figure 6 shows PI anomaly time series from GHG and AA with corresponding logarithm of Bayes factors at global and continental scales. GHG is characterized by steady warming patterns across all regions and extreme temperature indices. In contrast, AA exhibits a pattern of stronger cooling in the earlier decades followed by flattening in the recent decades, reflecting changes in AA emissions. Slight warming is seen over Europe after the 1980s, particularly for warm extremes, due to the reduced AA emissions. BA results indicate that GHG is detected robustly in global and all continental domains with “strong” or “decisive” evidence in many cases whereas AA is detected over global and northern continental regions in warm extremes.

Fig. 6.
Fig. 6.

(left) Time series of 5-yr averaged PI anomalies in four extreme temperature indices during 1951–2015 from observations (black) and GHG (red) and AA (purple) simulations (multimodel mean) at global and continental domains. ANT results (blue dashed lines) are displayed for comparison. (right) Bayes factors of GHG (red) and AA (purple). All other details are the same as in Fig. 3.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

Global and continental BA results for additional GHG and AA contributions are summarized in Fig. 7 in comparison with OF-based (three-signal analysis) results. For cold extremes (Figs. 7a,b), GHG (red) is detected in GLB, NA, and SA from both methods with EUR and OCE having GHG detection from one method only. AA (purple) is detected limitedly with disagreement between two methods, over ASI (TNn in OF) and NA (TXn in BA). For warm extremes (Figs. 7c,d), two methods consistently detect GHG contribution over all domains except NA and OCE. AA is also detected over GLB, EUR, and ASI in both methods, indicating robust results. Further, GHG detection results are found to be very similar to ANT results (Fig. 4), representing the dominant contribution of GHG to ANT.

Fig. 7.
Fig. 7.

Comparison of GHG (red) and AA (purple) signal detection between the optimal fingerprint (OF) and Bayesian attribution (BA) methods. All other details are the same as in Fig. 4.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

A regional-scale attribution analysis of GHG and AA reveals overall similar results to those for continental scales. Figure 8 displays results for TNx. Most subregions have GHG detection from both methods (20 out of 33). For AA, consistent detection is found over a few subregions only (MED, TIB, EAS, and SEA), indicating larger uncertainties in AA detection than GHG. When checking other extreme indices (Figs. S3 and S4), GHG is detected in many regions with more frequent detection than OF (Fig. S3). This resembles ANT detection results (Fig. S1), supporting the dominant contribution of GHG in ANT detection at regional scales. AA is detected rarely in cold extremes while a few subregions (NES, CEU, TIB, and EAS) exhibit an AA signal detected in both methods for TXx (Fig. S4).

Fig. 8.
Fig. 8.

(a) GHG and (b) AA detection results for TNx at subcontinental domains from the Bayesian attribution (BA) and optimal fingerprint (OF) methods. Red and purple colors indicate signal detection. Six continents are divided by black vertical lines. Asterisks indicate model underestimation of the observed residual variability.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

d. Sensitivity to Bayes factor thresholds

To check whether the threshold of the Bayes factor for detection is comparable to the 90% confidence interval of scaling factors in OF, we have repeated BA analysis for continental and subcontinental domains using varying Bayes factor thresholds from 1 to 200 and examined changes in a number of domains with signal detection in comparison with those from OF. Figure 9a shows results for ANT and NAT at global and continental (first row) and regional domains (second row), where gray shading area indicates Bayes factor ranges corresponding to the “substantial” evidence (1 ≤ lnBF < 2.5). Except for TNn, BA (solid line) tends to have less instances of ANT detection than OF (dashed line) when using the “substantial” scale. This indicates that the BA method is slightly stricter than OF, which can also be seen in NAT results. The same analysis has been done for GHG and AA (Fig. 9b). While similar numbers of GHG detection occur between the two methods for the continental scale (first row), BA gives more frequent GHG detection than OF at regional scale, particularly for cold extremes (second row). The improved detection seems to be related to the independent treatment of GHG and AA in the BA method, unlike OF, which considers GHG and AA simultaneously, hence increasing the possibility of collinearity.

Fig. 9.
Fig. 9.

(a) The number of regions where an ANT or NAT signal is detected in the BA and OF methods at global/continental scales in the first row and regional scales in the second row according to different BF values (x axis) from 1 to 200. Solid orange and blue lines indicate ANT and NAT detection results from BA method, respectively. OF results are shown as light colored horizontal dashed lines. Gray shading represents the BF range of “substantial” evidence (1 ≤ lnBF < 2.5; Table 2). (b) As in (a), but for GHG (red) and AA (purple) results. Note that the x axis is expressed as log scale.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

Regarding the potential benefits of the BA method, we have further checked the sensitivity of GHG detection probability for TNn to the correlation between GHG and AA time series (the metric for signal collinearity). We have calculated the percentage ratio of number of subcontinental regions with GHG detection to those with ANT detection (defined as detection fraction) for the OF and BA methods. In addition to signal collinearity, signal-to-noise ratios of GHG are also considered as a factor influencing the degree of detectability. Figure 10 shows distributions of detection fraction under different categories of signal collinearity (y axis) and signal-to-noise ratio of GHG (x axis), comparing OF and BA methods. In case of high signal-to-noise ratio (>3.0), OF shows a high detection rate that is largely insensitive to the GHG–AA collinearity, suggesting that the multicollinearity effect becomes weak under the strong signal-to-noise ratio condition (Fig. 10a). When signal-to-noise ratios are relatively low (<3.0), the detection fraction tends to decline as the GHG–AA correlation increases (Fig. 10a). In contrast, BA-based results exhibit no sensitivity to GHG–AA correlation, showing robust GHG detection even under a strong multicollinearity condition (Fig. 10b).

Fig. 10.
Fig. 10.

Fraction of GHG detection (%) for TNn (defined as number of subcontinental regions with GHG detection by those with ANT detection) for each category of signal-to-noise ratio of GHG (x axis) and correlation between GHG and AA time series (y axis) from the (a) OF and (b) BA methods. Numbers in each category represent the number of regions with ANT detection.

Citation: Journal of Climate 35, 24; 10.1175/JCLI-D-22-0104.1

Unlike the GHG results, the number of regions with AA detection are less in BA than in OF for both continental and regional scales, as in the case of ANT and NAT. As a whole, the “substantial” evidence category in BA method looks more rigorous than the 90% confidence level of scaling factors in the OF method. However, in the case of high collinearity under low signal-to-ratio conditions, the BA method seems to be more powerful to detect signals. The latter warrants further investigation considering idealized data having different degrees of collinearity.

5. Summary and discussion

As a frequentist method, the optimal fingerprint (OF) has been widely used to detect external influences on observed changes, which is based on multiple linear regression between observations and model simulations under different forcing factors. Recent OF-based studies have quantified greenhouse gas (GHG) and anthropogenic aerosol (AA) contributions in extreme temperature changes, demonstrating the dominant role of GHG forcing in the observed warming at continental and regional scales. A Bayesian decision method was devised for detection and attribution of external signals in the observed changes, which provides independent evidence by evaluating probability of observations (likelihood) in view of models based on the Bayes factor (BF). However, the Bayesian method has not been applied to extreme temperature changes with no evaluation of individual forcing contributions in probabilistic framework.

The present study proposes a new Bayesian attribution (BA) method, which enables one to evaluate individual forcing contributions. Under the precondition of ALL detection (lnBFALL,CTL ≥ 1), the additional anthropogenic (ANT) contribution is evaluated by comparing the Bayes factor between ALL and NAT. When lnBFALL,NAT ≥ 1, “substantial” evidence for ANT contribution is obtained, which indicates that ALL detection is improved due to ANT contribution. Similarly, additional NAT contribution is detected when lnBFALL,ANT ≥ 1 with the prerequisite of ALL detection. In a similar way, additional GHG or AA contribution to the detected ANT is measured by comparing Bayes factor of ANT with that of AA or GHG, respectively. This BA approach is applied to extreme temperature indices from HadEX3 observations and CMIP6 individual forcing simulations and results are systematically compared with OF-based results.

Results show that our BA method produces similar detection results to those based on the OF method, reaffirming the robust human influence on the observed warming of extreme temperatures at global and regional scales. In cold extremes (TNn, TXn), the GLB, NA, and SA regions show robust ANT detections obtained in both methods although there are no consistent NAT detections between them. In warm extremes (TNx, TXx), all continental regions have robust ANT detection except NA. For regional domains, two methods have similar detection results especially in warm extremes. In the analysis of GHG and AA contribution to the detected ANT, GHG is robustly detected for all extreme indices over most of continental domains. GLB, EUR, and ASI also show AA detection in warm extremes, indicating the cooling effect over ASI due to increased aerosols and the warming effect over EUR due to decreased aerosols during recent decades. At regional scales, the GHG signal is detected in many regions for warm extremes, also supporting previous OF-based results. However, more GHG detection cases were found in BA than in OF in cold extremes, which suggests a possible advantage of BA in cases of low signal-to-noise ratio and high multicollinearity. When further checking the sensitivity of BA results to different thresholds of Bayes factor, a “substantial” threshold (lnBF ≥ 1) is found to be overall comparable to (but slightly stricter than) the 90% confidence level of the scaling factor in the OF method.

It should be noted that our proposed BA approach works in a hierarchical way, starting from ALL detection and then subsequent evaluation of ANT and NAT contribution to the detected ALL. Similarly, GHG and AA contributions are evaluated with the prerequisite of ANT detection. This corresponds to a stepwise linear regression, different from the standard OF method in which multiple signals (e.g., ALL, GHG, and NAT signals) are considered simultaneously. This OF–BA discrepancy in multiple signal assessments needs to be further understood using idealized datasets. Nevertheless, it is argued that the Bayesian approach proposed here employs a useful and more general framework to ask how close the observed changes are to expected climate responses to various factors and thereby provides a different way of conducting the attribution assessment. Indeed, IPCC assessment about causes of observed changes is of Bayesian nature, although it is typically based on expert judgement rather than multiple line evidence including OF results (e.g., Seneviratne et al. 2021). In this respect, integrated approaches reconciling conventional and Bayesian approaches would be warranted in the future work (Hasselmann 1998).

Acknowledgments.

This study was supported by a National Research Foundation of Korea (NRF) grant funded by the South Korean government (MSIT) (NRF-2021R1A2C3007366) and the Human Resource Program for Sustainable Environment in the 4th Industrial Revolution Society.

Data availability statement.

Data analyzed in this study are openly available at https://www.metoffice.gov.uk/hadobs/hadex3/ (HadEX3) and https://esgf-node.llnl.gov/projects/cmip6/ (CMIP6).

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Supplementary Materials

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  • Allen, M. R., and S. F. B. Tett, 1999: Checking for model consistency in optimal fingerprinting. Climate Dyn., 15, 419434, https://doi.org/10.1007/s003820050291.

    • Search Google Scholar
    • Export Citation
  • Allen, M. R., and P. A. Stott, 2003: Estimating signal amplitudes in optimal fingerprinting, Part I: Theory. Climate Dyn., 21, 477491, https://doi.org/10.1007/s00382-003-0313-9.

    • Search Google Scholar
    • Export Citation
  • Annan, J. D., 2010: Bayesian approach to detection and attribution. Wiley Interdiscip. Rev.: Climate Change, 1, 486489, https://doi.org/10.1002/wcc.47.

    • Search Google Scholar
    • Export Citation
  • Berliner, L. M., R. A. Levine, and D. J. Shea, 2000: Bayesian climate change assessment. J. Climate, 13, 38053820, https://doi.org/10.1175/1520-0442(2000)013<3805:BCCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Brown, S. J., J. Caesar, and C. A. T. Ferro, 2008: Global changes in extreme daily temperature since 1950. J. Geophys. Res., 113, D05115, https://doi.org/10.1029/2006JD008091.

    • Search Google Scholar
    • Export Citation
  • Christidis, N., P. A. Stott, S. Brown, G. C. Hegerl, and J. Caesar, 2005: Detection of changes in temperature extremes during the second half of the 20th century. Geophys. Res. Lett., 32, L20716, https://doi.org/10.1029/2005GL023885.

    • Search Google Scholar
    • Export Citation
  • DelSole, T., L. Trenary, X. Yan, and M. K. Tippett, 2019: Confidence intervals in optimal fingerprinting. Climate Dyn., 52, 41114126, https://doi.org/10.1007/s00382-018-4356-3.

    • Search Google Scholar
    • Export Citation
  • Donat, M. G., and L. V. Alexander, 2012: The shifting probability distribution of global daytime and night-time temperatures. Geophys. Res. Lett., 39, L14707, https://doi.org/10.1029/2012GL052459.

    • Search Google Scholar
    • Export Citation
  • Duda, R. O., and P. E. Hart, 1973: Pattern Classification and Scene Analysis. John Wiley, 482 pp.

  • Dunn, R. J. H., and Coauthors, 2020: Development of an updated global land in-situ-based data set of temperature and precipitation extremes: HadEX3. J. Geophys. Res. Atmos., 125, e2019JD032263, https://doi.org/10.1029/2019JD032263.

    • Search Google Scholar
    • Export Citation
  • Eyring, V., S. Bony, G. A. Meehl, C. A. Senior, B. Stevens, R. J. Stouffer, and K. E. Taylor, 2016: Overview of the Coupled Model Intercomparison Project Phase 6 (CMIP6) experimental design and organization. Geosci. Model Dev., 9, 19371958, https://doi.org/10.5194/gmd-9-1937-2016.

    • Search Google Scholar
    • Export Citation
  • Eyring, V., and Coauthors, 2021: Human influence on the climate system. Climate Change 2021: The Physical Science Basis, V. Masson-Delmotte et al., Eds., Cambridge University Press, 423–552.

  • Gillett, N. P., and Coauthors, 2016: The Detection and Attribution Model Intercomparison Project (DAMIP v1.0) contribution to CMIP6. Geosci. Model Dev., 9, 36853697, https://doi.org/10.5194/gmd-9-3685-2016.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., 1997: Multi-pattern fingerprint method for detection and attribution of climate changes. Climate Dyn., 13, 601611, https://doi.org/10.1007/s003820050185.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., 1998: Conventional and Bayesian approach to climate-change detection and attribution. Quart. J. Roy. Meteor. Soc., 124, 25412565, https://doi.org/10.1002/qj.49712455202.

    • Search Google Scholar
    • Export Citation
  • Hu, T., Y. Sun, X. Zhang, S.-K. Min, and Y.-H. Kim, 2020: Human influence on frequency of temperature extremes. Environ. Res. Lett., 15, 064014, https://doi.org/10.1088/1748-9326/ab8497.

    • Search Google Scholar
    • Export Citation
  • Iturbide, M., and Coauthors, 2020: An update of IPCC climate reference regions for subcontinental analysis of climate model data: Definition and aggregated datasets. Earth Syst. Sci. Data, 12, 29592970, https://doi.org/10.5194/essd-12-2959-2020.

    • Search Google Scholar
    • Export Citation
  • Jones, G. S., P. A. Stott, and J. F. B. Mitchell, 2016: Uncertainties in the attribution of greenhouse gas warming and implications for climate prediction. J. Geophys. Res. Atmos., 121, 69696992, https://doi.org/10.1002/2015JD024337.

    • Search Google Scholar
    • Export Citation
  • Kass, R. E., and A. E. Raftery, 1995: Bayes factors. J. Amer. Stat. Assoc., 90, 773795, https://doi.org/10.1080/01621459.1995.10476572.

    • Search Google Scholar
    • Export Citation
  • Katzfuss, M., D. Hammerling., and R. L. Smith, 2017: A Bayesian hierarchical model for climate change detection and attribution. Geophys. Res. Lett., 44, 57205728, https://doi.org/10.1002/2017GL073688.

    • Search Google Scholar
    • Export Citation
  • Kharin, V. V., and F. W. Zwiers, 2005: Estimating extremes in transient climate change simulations. J. Climate, 18, 11561173, https://doi.org/10.1175/JCLI3320.1.

    • Search Google Scholar
    • Export Citation
  • Kim, Y.-H., S.-K. Min, X. Zhang, F. Zwiers, L. V. Alexander, M. G. Donat, and Y.-S. Tung, 2016: Attribution of extreme temperature changes during 1951–2010. Climate Dyn., 46, 17691782, https://doi.org/10.1007/s00382-015-2674-2.

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  • Fig. 1.

    Schematic illustration of anthropogenic (ANT) and natural (NAT) forcing signal detection using the (a) optimal fingerprint and (b) Bayesian attribution techniques. In (a), regression coefficients (scaling factors) and their 90% confidence intervals (error bars) are used for detection. In (b), the Bayes factor is used for detection, calculated as the likelihood ratio of a certain forcing (ALL) to reference forcing (NAT or ANT in this example). Darker gray shading indicates stronger detection evidence (substantial, strong, or decisive; Table 2).

  • Fig. 2.

    (left) Time series of 5-yr averaged PI anomalies of four extreme temperature indices (TNn, TXn, TNx, and TXx) during 1951–2015 from observations (black) and ALL (green) at global (GLB) and five continental domains (NA, SA, EUR, ASI, and OCE). Residual observations (gray solid line) are obtained by removing ALL patterns from the raw observations based on linear regression. The first decade is excluded in South America due to poor spatial coverage. (right) Variances of the residual observations (triangle) and those from CTL simulations (gray circle indicates the mean; error bar shows the 5%–95% range). Asterisks and plus signs indicate that models underestimate or overestimate the observed variability, respectively.

  • Fig. 3.

    (left) Time series of 5-yr averaged PI anomalies in four extreme temperature indices during 1951–2015 from observations (black) and from ANT (orange) and NAT (blue) simulations (multimodel mean) at global and continental domains. ALL results (green dashed lines) are displayed for comparison. (right) Bayes factors of ANT (orange) and NAT (blue). Light, medium, and dark gray shadings indicate detection with “substantial,” “strong,” and “decisive” evidence, respectively. Filled circles represent signal detection with at least substantial evidence while open circles indicate no detection. Asterisks indicate model underestimation of the observed residual variability (as evaluated in Fig. 2).

  • Fig. 4.

    Comparison of ANT (orange) and NAT (blue) signal detection between the optimal fingerprint (OF) and Bayesian attribution (BA) methods. The “substantial” scale (Table 2) is applied for Bayesian attribution. Global results are provided in the bottom left corner of each panel. Gray shading in the background map indicates the data availability of TXx fixed mask. Asterisks indicate model underestimation of the observed residual variability based on the residual consistency test in OF and the similar variance comparison in BA (as evaluated in Fig. 2). See text for details.

  • Fig. 5.

    (a) The 33 subcontinental regions used in this study [adopted from Seong et al. (2021)]. Continental regions are depicted as different colors (NA: blue, SA: green, EUR: yellow, AFR: orange, ASI: red, and OCE: purple). Bold acronyms indicate the subcontinental regions analyzed in this study with good data coverage (more than 60% of grids have sufficient long-term data). (b) ANT and (c) NAT detection results for TNx at subcontinental domains from the optimal fingerprint (OF) and Bayesian attribution (BA) methods. Orange and blue colors indicate signal detection. The six continents are divided by black vertical lines. Asterisks represents model underestimation of the observed residual variability (as in Fig. 4).

  • Fig. 6.

    (left) Time series of 5-yr averaged PI anomalies in four extreme temperature indices during 1951–2015 from observations (black) and GHG (red) and AA (purple) simulations (multimodel mean) at global and continental domains. ANT results (blue dashed lines) are displayed for comparison. (right) Bayes factors of GHG (red) and AA (purple). All other details are the same as in Fig. 3.

  • Fig. 7.

    Comparison of GHG (red) and AA (purple) signal detection between the optimal fingerprint (OF) and Bayesian attribution (BA) methods. All other details are the same as in Fig. 4.

  • Fig. 8.

    (a) GHG and (b) AA detection results for TNx at subcontinental domains from the Bayesian attribution (BA) and optimal fingerprint (OF) methods. Red and purple colors indicate signal detection. Six continents are divided by black vertical lines. Asterisks indicate model underestimation of the observed residual variability.

  • Fig. 9.

    (a) The number of regions where an ANT or NAT signal is detected in the BA and OF methods at global/continental scales in the first row and regional scales in the second row according to different BF values (x axis) from 1 to 200. Solid orange and blue lines indicate ANT and NAT detection results from BA method, respectively. OF results are shown as light colored horizontal dashed lines. Gray shading represents the BF range of “substantial” evidence (1 ≤ lnBF < 2.5; Table 2). (b) As in (a), but for GHG (red) and AA (purple) results. Note that the x axis is expressed as log scale.

  • Fig. 10.

    Fraction of GHG detection (%) for TNn (defined as number of subcontinental regions with GHG detection by those with ANT detection) for each category of signal-to-noise ratio of GHG (x axis) and correlation between GHG and AA time series (y axis) from the (a) OF and (b) BA methods. Numbers in each category represent the number of regions with ANT detection.

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