## 1. Introduction

According to instrumental observations [

1], the global mean surface temperature (GMST) has risen at an increasing rate throughout the instrumental period, 1850–2017, the Common Era (CE). This rise, however, has been interrupted by three pauses of no, or even negative, growth. These are the periods 1870–1915, 1940–1970, and 1998–2014. The last hiatus was the shortest and least spectacular of these, and the reason for the great attention it has received is primarily that it is the first pause observed since global warming became a scientific and political issue in the 1980s. The existence of the most recent pause has been challenged in several recent papers, partly justified by new corrected estimates of the GMST which point out biases in older estimates (see, for instance, the review by Jones (2016) [

2]. The most prominent of these biases is gradual introduction of new measurement methods of sea surface temperature (SST) since the outbreak of World War III [

3,

4]. These recent corrections, however, are quite modest and do not remove the visual impression of a deviation throughout the 1998–2014 interval from the long-term temperature trend of the longer 1970–2017 interval. Whether or not this deviation deserves to be classified as a hiatus is a matter of definition and is not the subject of the present paper.

Multiple linear regression was applied to explain the GMST satellite record for the 1979–2011 period by Foster and Rahmstorf (2011) [

5]. Their emphasis was on eliminating the effects of volcanic and solar forcing, and the El Niño Southern Oscillation (ENSO), to recover the anthropogenic trend in the data. Lean and Rind (2008) [

6] applied such regression with the volcanic, solar, and anthropogenic forcing signals, and, in addition, the multivariate ENSO index, as predictors for the instrumental GMST for the 1889–2006 period. This description captured the initial part of the hiatus for the 1998–2006 period, but not the pauses during the 1870–1915 and 1940–1970 intervals.

Other authors (e.g., Zhou et al. (2012) [

7] and Canty et al. (2013) [

8]) have also used the Atlantic Multidecadal Oscillation (AMO) index as a predictor in addition to ENSO and the various forcing components. Zhou et al. (2012) replaced the anthropogenic forcing by a linear trend and used total solar irradiance (TSI), volcanic aerosol optical depth, and ENSO as predictors. The resulting explained signal contains an unexplained oscillation with period of approximately 60 years which is roughly in phase with the AMO. By introducing AMO as a predictor, most of this oscillation is explained.

Recently Folland et al. [

9] published a comprehensive study considering a substantial number of predictors and regression against several GMST datasets with monthly resolution and a number of reconstructions of forcing and indices for various modes. The predictors corresponding to components of the radiative forcing were filtered by convolving them with a response kernel composed of two exponential functions with time constants of 4–5 years and 200 years. This form of the linear GMST response function has been found from experiments with Earth system models [

10]. A special feature of that study was to analyze time intervals of pauses and strong warming in isolation, and then perform the regression analysis limited to those particular intervals with a variable number of predictors customized for each interval.

The Folland et al. paper reflects a trend in attribution research towards higher model complexity and the use of more data with higher resolution. However, increased model complexity is not without problems. One important issue is overfitting: a large number of predictors with sufficiently diverse spectral structure will typically produce high explained variance even when no causal connection between predictors and observation exists. Analysis of shorter time intervals in isolation also increases the chance of overfitting since the regression coefficients are allowed to be different in different intervals, thus increasing the effective number of predictors.

What is gained by analyzing data with monthly rather than annual resolution? As recognized by Folland et al., the GMST responds on time scales of a few years up to centuries, i.e., the climate system acts as a low-pass filter to remove forced variability on monthly time scales. This means that temporal structure on time scales shorter than this cannot be attributed to the variability of radiative forcing. Climatic modes used as predictors are related to interannual, decadal, or multidecadal dynamical mechanisms, and thus, the monthly variability of indices characterizing these modes should be perceived and treated as noise. If inclusion of this noise in the predictors and observations increases the explained variance, it is an indication that both predictors and observations are subject to the same noise. For instance, a particular climate mode index and the GMST can be influenced by the same large-scale weather events which are unrelated to any of the climate modes used as predictors. This leads to a spurious correlation between the mode predictor and the GMST which will attribute more of the variance to this mode. In fact, the mode dynamics may be completely unrelated to the GMST, but the correlated high-frequency components will still attribute part of the GMST to this mode.

Chylek et al. (2014) [

11] adopted a parsimonious approach, managing to explain 93% of the variance by employing only greenhouse gas (GHG) forcing and the AMO at annual resolution as predictors. This paper is interesting because it illustrates the potential pitfalls of regression analysis involving climate modes as predictors when the selection and form of the predictors are not sufficiently guided by physics. The AMO index with annual resolution is correlated with ENSO and the volcanic forcing signal on interannual time scales (just as the GMST is). For instance, the strong 1997 El Niño is present as a very prominent peak in the annual resolution AMO. We know that this high-frequency variability in the AMO index has no connection to the multidecadal dynamics governing the AMO, but is a reflection of the dynamics of ENSO and volcanic forcing. It is methodologically flawed to analyze AMO with annual resolution because it will incorrectly attribute the GMST variability caused by ENSO and volcanic forcing to AMO when ENSO and volcanic forcing are not used as predictors. Since the AMO also oscillates in phase with the slow 60-year oscillation observed in the GMST, it is not surprising that the explained variance can be very high with GHG forcing and annual resolution AMO as the only predictors.

The present paper also takes an approach based on the principle of parsimony, but great care is taken in the selection of predictors that are physically sound, for instance, by using low-pass filtered forcing similar to that of Folland et al. (2018), and have a form that prevents spurious attribution due to physical dependence between modes or forcings on certain time scales. The latter is prevented by using high resolution predictors for interannual modes and low-resolution (smoothed) predictors for multidecadal modes.

The objective is to explain the deviations from the expected anthropogenic warming on decadal and multidecacdal time scales. The deviations on these time scales have been used by some to cast doubt about the reality of human-induced climate change. Explaining monthly or annual variability is not the focus of interest, and hence, this kind of change is excluded from the analysis by using annual means of the data and employing a response filter that effectively eliminates these time scales from the analysis. A goal is also to find the most parsimonious description (the lowest number of predictors) that explains the most essential features of the observed GMST signal, i.e., those features necessary for concluding that the pauses under examination all can be attributed to natural variability. The main result of the paper is that we do not need a host of different data sets of high resolution and a large number of predictors to reach that conclusion, and we single out those predictors that are essential and those may be left out in the most parsimonious, yet sufficient, description.

The paper is a follow-up from Rypdal (2015) [

12] who employed a long-memory response filter to the three forcing components (solar, volcanic and anthropogenic) and applied the AMO index as a predictor in addition to the forcing components and the Niño3.4 index. This approach explains 89% of the variance in the 1880–2010 GMST and captures all three pauses. A characteristic result of all these approaches is a relatively strong AMO footprint and a weaker volcanic footprint compared to what one would expect from estimates of volcanic aerosol forcing like the ones presented in Hansen et al. (2011) [

13].

It will be demonstrated that two analysis features of many earlier papers may exaggerate the AMO footprint at the expense of the volcanic footprint. One is the the use of the forcing components themselves as predictors and not a long-memory filtered response that respects that the surface temperature responds to forcing on multiple time scales [

10,

14]. This long-range memory response to volcanic eruptions may explain part of the slow oscillation that remains unexplained when the AMO is not used as a predictor. Another feature is that the three forcing components are treated as independent predictors thus ignoring theoretical estimates of their relative magnitudes. In addition to discarding such estimates, this approach increases the number of regression coefficients from three to five, and hence, increases the risk of overfitting, i.e., high degree of explained variance from an incorrect statistical model with too many degrees of freedom.

The main purpose of the present paper is to extend the results of [

12] to the 1870–2017 period and hence, capture the “death” of the 1998–2014 hiatus, but also seeks to reduce the model complexity to avoid overfitting. The regression model is simpler than the one employed in [

12] in the sense that the long-memory temperature response to the sum of anthropogenic, volcanic, and solar forcing is treated as one predictor, rather than three independent predictors (the one-forcing model). In

Section 4.2, we also compare this to the model where the volcanic forcing is separated from the sum of solar and anthropogenic forcing as an independent predictor (the two-forcing model). The long-memory response to the ENSO and AMO indices are treated as predictors, rather than being the indices themselves. This means that, in the one-forcing model, the number of predictors are reduced from five to three, and hence, the chance of overfitting is considerably reduced. This enhances the confidence we can have in the model. In spite of the reduction of model complexity, the explained variance increases to 92%. This may be attributed to using the long-memory response to the ENSO index as predictor, rather than the index itself, but updated and longer data sets may also play roles. The effects of systematically increasing the number of predictors are studied in more detail in

Section 4.

The forcing signals used as a basis for the predictors are shown in

Figure 1.

Figure 1a displays the sum of volcanic, solar and anthropogenic forcing, and

Figure 1b displays the individual components of the forcing.

Figure 1c,d display the AMO and the ENSO indices, respectively.

Section 2.1 explains how these signals are filtered into signals that exhibit the temporal fingerprints of the GMST responses to these forcing signals. By means of a linear multiple regression, these fingerprints (or predictors) and the instrumental GMST are used to estimate regression coefficients which allow the formation of a weighted sum to represent the best fit to the instrumental data. Here, each weighted fingerprint is denoted as the footprint of the specific forcing or index, since it can be interpreted as the component of that forcing or index present in the instrumental GMST.

The main results of this analysis are presented in

Section 2 and are further discussed in

Section 3. The data sources are presented and some methodological issues further discussed in

Section 4.

## 3. Discussion

Rypdal (2015) [

12] obtained

${R}^{2}$ (89%), slightly smaller than that obtained here (92%) by using volcanic, solar and anthropogenic fingerprints as independent predictors in addition to AMO and ENSO. In other words, in that paper, five predictors gave no better results than we obtained here with three. On the other hand, the AIC found there was −244, while the present analysis found AIC = −312 due to the reduced model complexity. One of the major differences in these two analyses is that in reference [

12], the volcanic to anthropogenic footprint ratio was reduced by a factor of two compared to the fingerprint ratios, while the AMO footprint was enhanced. A plausible interpretation of this difference is that the freedom of allowing different effective sensitivities to the volcanic, solar and anthropogenic forcing components results in unphysical overfitting. The situation is complex, however, since a number of other factors were different between these analyses. For instance, the data series in the present work were longer (1870–2017 versus 1880–2010), and both the GMST time series and the forcing time series have been updated. From an information-theoretical viewpoint, the present analysis is preferable, and the data used are likely more correct. The reduced AMO footprint has the interesting implication that the slow oscillation of a period of about 60 years, giving rise to the 1870–1915 and 1940–1970 pauses, can be attributed to the clustering of volcanic eruptions. Moreover, since the AMO phase coincides with this oscillation, it opens the possibility that during the 20th century, AMO was paced by volcanic activity. The effects of long-memory filtering and an increasing number of predictors are presented systematically for the new and extended datasets in

Section 4.2.

Another conclusion that can be drawn from comparing these results with those obtained by Lean and Rind (2008) [

6] and by Rypdal (2015) [

12] using a zero-memory (instantaneous) response function is that a long-memory response is required to fully reproduce this slow oscillation as a response to the clustering of volcanic activity. Hence, it serves as a confirmation of the usefulness of the long-memory response description and emphasizes that volcanic forcing has effects not only a few years after eruptions, but also on multidecadal time scales (see also

Figure 5,

Figure 6,

Figure 7 and

Figure 8 and the discussion in

Section 4.2).

By including AMO and ENSO as a part of the deterministic forcing, and hence excluding these signals from the residual climate noise background, the present work obtains a better scaling description of this background on time scales from years to a few decades, and a memory exponent of

$\beta \approx 0.6$ (corresponding to a Hurst exponent of

$H=0.8$) has been obtained. This eliminates a problem that plagues many studies, whereby ENSO creates difficulties for computing scaling exponents (see, e.g., Fredriksen and Rypdal (2016) [

16]).

Seen under the context of the entire instrumental 1870–2017 GMST record, the most remarkable feature of the the 1998–2014 deviation from the long-term warming trend is how unremarkable it is. The duration of the period with a weaker GMST trend is considerably shorter than those of the 1870–1915 and 1940–1970 pauses, during which the trends were actually not zero, but negative. There have been no strong volcanic eruptions since the Mount Pinatubo eruption of 1993, but a particularly deep solar minimum occurred in 2009 and a weak solar cycle 24 during the following decade lead to the dip in the total forcing during the first decade of this century observed in

Figure 1a. At first sight, this might suggest that the entire hiatus could be attributed to this low solar forcing, but this is not supported by

Figure 4b, where the elimination of the ENSO footprint also essentially eliminates the reduction of the trend. On the other hand, the solar footprint may explain the slightly higher slope of the trend line in

Figure 4a (0.182 K per decade) compared to 0.166 K per decade in

Figure 4b.

Figure 3b and

Figure 7d show that AMO is responsible for a positive temperature anomaly of about 0.1 K during the 2000–2017 intreval, but this perturbation is almost constant during the period and does not significantly influence the shape of the GMST curve during this period.

It is worth mentioning, though, that other combinations of modes than AMO and ENSO have been used to explain the most recent hiatus. For instance, Steinman et al. (2015) [

18] employed a semi-empirical approach combining observations and global circulation model simulations to attribute this slowdown to a combination of the positive phase of AMO and negative trending of the Pacific multidecadal oscillation (PMO). However, the study deals with Northern hemisphere (not global) temperature, and the data series end in 2012. Hence, it does not capture the interannual variability associated with ENSO and, in particular, does not capture the “death” of the hiatus associated with the strong 2015 El Niño. The present paper shows that neither AMO, nor a combination of AMO and PMO, are necessary to fully explain the recent hiatus.

The overall conclusion is that the 1998–2014 deviation from the long-term warming trend is an unremarkable phenomenon that can be attributed to the influence of ENSO on the global mean surface temperature, with a slightly reduced warming trend due to low solar activity.