## 1. Introduction

Numerous studies suggest that radiative forcing from anthropogenic aerosols (AA) plays an important role in climate variability [see Boucher et al. (2013) for a comprehensive review]. In addition to global climate change, anthropogenic aerosols also have been implicated in multidecadal variability of the North Atlantic (Booth et al. 2012), precipitation changes over the Sahel and South and East Asia (Shindell et al. 2012), warming patterns in the Indian Ocean (Dong and Zhou 2014), southward shifts in the ITCZ (Rotstayn and Lohmann 2002), reduction of rainfall in the tropics (Ridley et al. 2015), the Pacific decadal oscillation and expansion and contraction of the tropical belt (Allen et al. 2014), and tropical Atlantic rainfall (Chang et al. 2011).

Unfortunately, the radiative forcing due to AA is highly uncertain because of uncertainties in emission levels, size distribution, optical properties, and the interaction of aerosols with clouds (Boucher et al. 2013). Current estimates of global radiative forcing suggest that the forcing due to AA is more uncertain than that due to well-mixed greenhouse gases (GHG) and hence dominates most of the uncertainty in total forcing (Myhre et al. 2013, their Fig. 8.16). Also, aerosol cooling partially cancels greenhouse warming. Thus, climate simulations could match past climate change through fortuitous cancellation of errors, allowing models with very different responses to individual forcings to be equally consistent with observations.

In principle, aerosol-forced climate changes can be estimated directly from observations using energy balance constraints, provided that such changes can be distinguished from those by other forcings. A major difference in forcing is that greenhouse gases tend to be well mixed and have relatively small spatial gradients, whereas anthropogenic aerosols are short lived and tend to be more concentrated around their (land) sources. These differences in forcing lead to differences in hemispheric gradients and land–sea gradients in the temperature and precipitation responses. Forcings also differ temporally: forcing from well-mixed greenhouse gases has increased monotonically over the past century, while forcing from aerosols has leveled in the last few decades. In addition, both forcings have distinct seasonal cycles. Despite these forcing differences, the associated responses tend to be similar to each other (Xie et al. 2013), leading to collinearity problems in separating these responses in data.

The most widely used method for estimating different forced responses from observations is optimal fingerprinting. Stott et al. (2006) used optimal fingerprinting to rescale model responses to match observations and thereby inferred changes due to aerosol and greenhouse gas forcing. Features of the response that appeared to provide the most discrimination between greenhouse warming and aerosol cooling were the differential warming rates between the hemispheres, between land and ocean, and between mid- and low latitudes. For example, Stott et al. (2006) showed that removing one or more of these features from the response vector increases uncertainty.

The purpose of this paper is to quantify the effectiveness of different variables for detecting the response to aerosol forcing and thereby identify the best surface variables for detecting aerosol cooling. A key quantity in our methodology is *potential* detectability, which measures the detectability of a forced response in a model. For a single forcing, detectability is measured by the total-to-noise ratio, defined as the ratio of the total variance (forced plus unforced) to the variance of internal variability. For two forcings, detectability is measured by the product of the total-to-noise ratio of the given forcing times a measure of multicollinearity between the two forcings (as we will show). Both quantities can be estimated from single-forcing simulations; hence, potential detectability can be estimated as soon as the single-forced simulations are available.

To ensure that the more effective variables for representing a forced response are used, we select the variables that maximize the total-to-noise ratio (or equivalently, the signal-to-noise ratio plus one). These variables maximally discriminate between a climate change signal and internal variability and can be identified using the method of Jia and DelSole (2012) (discussed shortly). However, these variables may not be the best variables to distinguish different forcings. For instance, two forcings may produce highly detectable but virtually identical responses in a given set of variables, in which case those variables could not be used to distinguish the forced responses. Our goal is to find the best variables for separating forced responses from each other and from internal variability.

The relative importance of spatial versus temporal structure for separating different forced responses is unclear. Part of the reason for this is that most optimal fingerprinting studies use predictors that contain both spatial and temporal structure, leaving the individual contributions unclear. The extent to which different forced responses can be separated based on time series of global mean quantities has been investigated extensively (Stern and Kaufmann 2000; Lean and Rind 2009; Zhou and Tung 2013). In this paper, we investigate the extent to which forced responses can be separated based on spatial structure alone. Thus, our approach to optimal fingerprinting differs from previous approaches in that we do not use the full space–time evolution of the climate change signal as a predictor. Instead, we use only the spatial structure of a variable, as well as its covariability with other variables, as input to fingerprinting. The output is an estimate of the time histories of the forced responses. This approach avoids making the (strong) assumption that the climate model correctly simulates the spatial–temporal evolution of the forced response.

This paper will show that the more accurate estimates of the response to aerosol forcing in model simulations are obtained using two variables rather than a single variable such as surface temperature alone. This conclusion is consistent with the fact that the joint behavior of two variables is more informative than their separate behavior. Moreover, previous studies also have found that combining different variables can improve inferences, including using joint changes of temperature at the surface and aloft (Jones et al. 2003), joint changes of temperature, diurnal temperature range and precipitation (Schnur and Hasselmann 2005), joint changes of temperature and salinity (Stott and Jones 2009; Pierce et al. 2012), joint changes of maximum amount and location of zonal mean precipitation (Marvel and Bonfils 2013), and joint changes on river flows, winter air temperature, and snowpack over the western United States (Barnett et al. 2008). While these studies illustrate the success that can be achieved using multiple variables, it is not always clear the extent to which multiple variables were required.

To avoid complications due to model error and observation error, we perform our analysis on model simulations instead of real observations. Our methodology for separating the response time series is described in section 2 and then applied to model simulations, as described in sections 3 and 4. We conclude with a summary and discussion of our results.

## 2. Methodology

As in previous studies (Bindoff et al. 2013 and references therein), we assume that a climate variable can be modeled as a linear combination of responses to external forcing, plus noise. Previous studies have shown that the linearity assumption is a good assumption for both temperature and precipitation for annual mean indices over the global domain (Marvel et al. 2015) and for 5-yr-mean indices over continental scale (Shiogama et al. 2013). We also performed more comprehensive linearity tests based on discriminant analysis methods, but these calculations are not discussed here because they merely confirm previous studies.

*n*th time step and

*s*th spatial location be denoted

*N*time steps and

*S*spatial locations. Observed changes

*S*-dimensional covariance matrix

*m*th forcing. We have obtained more accurate results by estimating the forced response by maximizing the signal-to-noise (S/N) ratio (i.e., maximizing the ratio of the variance of the forced response to the variance of internal variability). These vectors can be obtained from the method of Jia and DelSole (2012), which decomposes the forced response into a sum of components ordered by S/N ratio, where each component is a fixed pattern multiplied by an associated time series (similar to the way principal component analysis decomposes data by variance). In all cases, we find that only one component has a statistically significant S/N ratio, implying that other forced patterns, if they exist, are not detectable after the most detectable pattern has been removed. Consequently, the forced response pattern is represented as follows:

*N*-dimensional vector specifying the response time series,

*S*-dimensional vector specifying the spatial pattern, and superscript T denotes the matrix transpose. Jia and DelSole (2012) show that the pattern

*m*th forcing and anomalies are measured relative to the control mean.

*J*th-order polynomial in time:

*j*th Legendre polynomial in time

*t*. This formulation requires estimating

*J*coefficients per forcing. Legendre polynomials 1–5 are shown in Fig. 1. The polynomial representation is most questionable during volcanic eruptions, but since the dominant response to volcanic eruptions spans only a few years (Iles et al. 2013), smoothing over periods of major eruptions leads to only minor errors on multidecadal time scales.

The above regression model assumes that the spatial structure of the forced response

*m*th forced response is said to be detected if one can reject the null hypothesis

*m*th forced response vanish:

*J*degrees of freedom. This fact provides the basis for testing the null hypothesis at a prescribed significance level. The statistic

*ρ*is defined as follows:

*ρ*measures the cosine of the angle between the response vectors in “whitened space” (DelSole and Tippett 2007) and can be interpreted as a generalized spatial correlation. The squared pattern correlation

The total-to-noise ratio *ρ* can be computed immediately after the response vectors have been derived from the single-forcing runs. Accordingly, we define potential detectability to be (7) after substituting

The noise covariance matrix

In this paper, we consider detectability of not only a single variable like temperature but also combinations of variables like temperature and precipitation. Combinations of variables present no special difficulty in the above methodology. In particular, the above methodology does not involve computing EOFs, and the signal-to-noise maximization procedure is invariant to linear transformation, so the fact that the state vector contains different variables with different units is not problematic. As an example, suppose we want to detect AA-forced responses based on the first five Laplacian eigenvectors of temperature and the first five Laplacian eigenvectors of precipitation. To do this, we construct a state vector in which the first five elements give the instantaneous amplitude of the first five Laplacian eigenvectors for temperature, and the second five elements give the instantaneous amplitude of the first five Laplacian eigenvectors for precipitation. The resulting vector is 10-dimensional and evolves in time for *N* time steps, giving a *ρ* still measures the degree of multicollinearity but represents a generalized measure of pattern correlation involving temperature and precipitation.

## 3. Data

We apply the above methodology to simulations from phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012). Estimates of internal variability are derived from preindustrial control runs, which have no year-to-year variation in external forcing. The response to particular forcings are estimated from simulations that contain only anthropogenic aerosol forcing, called AA simulations, and simulations that contain all forcings except anthropogenic aerosols, called noAA simulations. In place of observations, we use simulations of the recent past (1900–2004) driven by anthropogenic and natural forcing, called historical simulations. Only two models, CSIRO Mk3.6.0 (CSIRO) and IPSL-CM5A-LR (IPSL), have both noAA and AA simulations. Thus, only these models are considered in this study. Both the CSIRO and IPSL models use the same aerosol emission inventory (Lamarque et al. 2010) over the historical period from 1850 to 2000 in decadal increments at a horizontal resolution of

## 4. Results

### a. Detection and attribution based on temperature and precipitation

The detection procedure described in section 2 differs from that of previous studies in several ways (e.g., the data are represented in terms of Laplacian eigenvectors instead of EOFs, the response patterns are obtained by maximizing detectability, and temporal smoothing is based on Legendre polynomials instead of decadal averages). Therefore, it may be instructive to illustrate the procedure by analyzing a single case in detail. Accordingly, we perform a detection analysis of AA and noAA forced responses using joint temperature and precipitation data. The detection analysis is performed on a single ensemble from a historical simulation, which serves as a surrogate for observations, thereby avoiding complications due to model error and observational error. There are 10 such ensemble members for CSIRO and 6 such ensemble members for IPSL. We first consider a state vector based on the first Laplacian eigenvector (i.e., the spatially uniform pattern) for temperature and precipitation. At one time step, this vector contains only two elements: global average temperature and precipitation. The response vectors *J. Climate*) examine this case in detail and show that the response vector effectively measures hydrological sensitivity. These two-dimensional response vectors are then used in optimal fingerprinting to estimate the respective time series from a single historical simulation. For comparison, we also consider response vectors using 20 spherical harmonics for temperature and 20 spherical harmonics for precipitation.

The estimated time series of global mean temperature change using five Legendre polynomials are shown in Fig. 3. Different curves of the same color show results for different ensemble members of the historical run (10 for CSIRO and 6 for IPSL). Figure 3 also shows global mean temperature change estimated from the AA and noAA simulations separately without temporal smoothing (A34). The top and bottom rows of Fig. 3 show results derived based on 1 and 20 spherical harmonics, respectively, while the left and right columns show results derived from CSIRO and IPSL, respectively. As is evident from the figure, the global mean temperature changes for AA and noAA are estimated very well in this “perfect model” case. In fact, the time series of global mean temperature change estimated from historical runs look essentially like smoothed versions of those obtained from the AA and noAA runs, even though the two time series were estimated independently. Moreover, global mean temperature change time series for different ensemble members are relatively close to each other, indicating that the estimates are not sensitive to sampling errors.

The impact of adding spatial gradient information in the forced response patterns is shown in the bottom row of Fig. 3. Based on 20 harmonics for temperature and 20 harmonics for precipitation, the time series estimated from the CSIRO historical run (Fig. 3, bottom left) are generally closer together (hence less sensitive to internal variability) and closer to the single-forcing runs (hence more accurate). In contrast, time series for IPSL have changed relatively little, suggesting that detectability is not strongly improved by adding spatial gradient information. Thus, the importance of spatial gradient information for separating forced responses is model dependent.

So far in this paper, optimal fingerprinting has been applied in the context of a perfect model world, in the sense that the same model generates the data and generates the forced response used in fingerprinting. In practice, the forced response vectors used in fingerprinting differ from the “true” responses that influence observations. To assess model error, one often examines the sensitivity of estimated responses to response vectors derived from different models. Accordingly, we show in Fig. 4 the change in global mean temperature in single-forcing simulations attributable to AA and noAA forced response vectors derived from different models. The top and bottom rows in Fig. 4 show results based on 1 and 20 spherical harmonics, respectively, while the left and right columns show results derived from CSIRO and IPSL single-forcing runs, respectively. Figure 4 shows that the change in global mean temperature is relatively insensitive to the model response vector. Thus, while response vectors differ between models, these differences do not lead to significantly different conclusions about climate changes attributable to AA and noAA.

The response patterns

Confidence intervals for the polynomial coefficients (in the perfect model case) are shown in Fig. 6. A forced response is said to be detected when the interval derived from the historical run does not include zero and is said to be attributable to AA or noAA forcing when the interval from the historical run overlaps with the appropriate interval from the single-forcing run. Almost all forced responses are attributable to the appropriate forcing as expected because the data come from a perfect model experiment. For both CSIRO and IPSL, the linear trend and the quadratic growth are detected (i.e., coefficients associated with the first two Legendre polynomials differ significantly from zero). Based on all variable combinations that we have analyzed, adding more Laplacian eigenvectors tends to improve the ability to detect and attribute changes in higher-order polynomials in time. However, as the order of the polynomial increases, so too does the number of fitted parameters and the likelihood of overfitting. We have found that 20 Laplacian eigenvectors produces stable results in a perfect model scenario. Since multidecadal variability is associated with polynomials of degree 3–6, these results suggest that detecting and attributing multidecadal variability to forcings in a perfect model requires spatial gradient information.

### b. Detectability under different combinations of variables, spatial structures, and seasons

*υ*th physical variable—TAS, PR, or PSL—for the

*l*th Laplacian eigenvector for time averaging period

*υ*on a global domain averaged over a year (ann), the state vector is of the following form:

Potential detectability slightly above the significance threshold implies that the signal is marginally above the noise level for detection. While such values are desirable because they meet the criterion for detection, in practice, we are interested in response vectors whose potential detectability is several times higher than the threshold because these vectors can detect a signal that is several times larger than the noise level. To be clear, we would not select the vector that maximizes detectability in the training sample because of overfitting.

Figure 7 shows that, in general, TAS and PR lead to stronger detection than PSL. PR gives stronger detection of AA than any other single variable. PSL cannot detect AA forced responses if too few spherical harmonics (

Recall that the potential detectability

*i*and

*j*on a global domain averaged over a year, the state vector is of the following form:

*x*axis while the number of spherical harmonics is shown on the upper

*x*axis of Fig. 7. Figure 7 reveals that state vectors based on TAS–PR variable pairs give the strongest potential detectability relative to other vectors of the same dimension. Interestingly, potential detectability based on TAS–PR does not increase strongly beyond the second spherical harmonic. The strong potential detectability using TAS–PR is because the associated response vectors are less collinear than other vectors (see Fig. 8). For example, other variables in CSIRO can have larger total-to-noise ratios but lower potential detectability due to high collinearity.

The above conclusions pertain to a perfect model analysis. The question arises as to whether our conclusions hold in an imperfect model setting. Figure 9 shows results similar to those in Fig. 7, except for the imperfect model case in which response vectors and noise covariances are estimated from a model that differs from the model that generated the historical single-forcing (AA and noAA) runs. As in the perfect model case, precipitation still gives the strongest potential detectability of AA, and combining precipitation with either temperature or sea level pressure gives the strongest potential detectability of all. In contrast, jumps in potential detectability occur at the second and fifth Laplacian in IPSL rather than CSIRO, implying that the jumps are due to resolving gradient information in the response vector and the noise covariance matrix rather than the single-forcing runs used. The use of seasonal variations or land–sea contrast information is more mixed in the imperfect model case; such information can slightly enhance potential detectability relative to vectors of the same dimension using annual mean information and spherical harmonics, but these increases are generally small, whereas the losses of potential detectability as a result of seasonal variations or land–sea contrast information can be substantial (especially for precipitation).

Note that TAS–PSL produces weaker potential detectability than other variable pairs. PSL is atmospheric pressure extrapolated to sea level and is computed partly from temperature data. The question arises as to whether PSL enhances potential detectability beyond TAS. In some cases, TAS and PSL are highly correlated, as shown in Fig. 10 for the global mean, which explains why the AA response is not detectable using only one Laplacian eigenvector of TAS and PSL. In all cases shown in Fig. 7, TAS always has stronger potential detectability than PSL when based on the same number of Laplacian eigenvectors. In rare cases in Fig. 9, PSL has only marginally larger potential detectability than TAS. In general, then, PSL does not enhance potential detectability significantly beyond that already available from the temperature data used to compute PSL.

The potential detectabilities shown above are computed from single-forcing runs without actually analyzing data in which both forcings are present. Realized detectability also can be computed from (7) using data from historical simulations, in which both forcings are present. A comparison between the potential detectability and realized detectability for the different vectors is shown in Fig. 11. In general, potential detectability tends to overestimate the realized detectability. This bias is presumably related to the neglect of sampling variability in the forced response patterns, an artifact of least squares that can be avoided using the error-in-variables methodology (Allen and Stott 2003). Nevertheless, the two measures tend to agree more strongly as the realized detectability increases beyond the significance threshold.

## 5. Summary and discussion

This paper showed that joint temperature–precipitation information over a global domain provides the more accurate estimate of aerosol-forced responses in climate models compared to using temperature, precipitation, or sea level pressure individually or in combination. This fact is demonstrated using a new quantity called potential detectability, which measures the extent to which a forced response can be detected in a model. Significant potential detectability is a necessary condition for detection in observations—if a forced response cannot be detected in a perfect model scenario, then it cannot provide a legitimate basis for detection and attribution in observations. Importantly, potential detectability can be evaluated from single-forcing runs. Consequently, potential detectability allows the detectability of a forced response in a model to be compared across a variety of variables so that the best choice of variables for detecting a given forced response can be identified prior to analyzing observations. When the forcing is partitioned into two response vectors, potential detectability is proportional to the product of the total-to-noise ratio of the forced response multiplied by a measure of multicollinearity between the two response vectors. This result substantially clarifies the relation between detectability, signal-to-noise ratio, and multicollinearity.

Detectability is generally enhanced by selecting response vectors that maximize signal-to-noise ratio. Such vectors can be identified using the method of Jia and DelSole (2012). Typically, only one vector is detectable for any given forcing and model. To clarify the importance of spatial versus temporal information, we use only spatial structure as input to fingerprinting and infer the associated time series from fingerprinting (which subsumes the traditional scaling factor). To do this, the data were represented in terms of a few Laplacian eigenvectors instead of EOFs. Laplacian eigenvectors enhance physical interpretation (since they often are of the form of monopoles, dipoles, tripoles, and so forth) and are data independent and hence can be compared across models. Also, the time evolution is assumed to be low frequency, in the sense that it is representable by a low-order polynomial in time. Since no temporal information is included in the response vector, the results clarify how detectability depends on spatial structure and covariability with other variables.

To illustrate the above methodology, we used it to estimate the change in global average temperature in historical simulations due to different forcings using joint temperature–precipitation response vectors. The inferred time series closely matched the time series computed from single-forcing runs, even if only the first spherical harmonic was used and even if the response vector of one model is used to infer the change in a different model. In some models, the accuracy of the estimates improves as more spherical harmonics are included in the response vector. A third-order polynomial in time was found to be sufficient for capturing most of the response over a century.

The above methodology was applied to identify the more effective variables in a model for estimating the cooling attributable to anthropogenic aerosols. We explored three different variables—surface temperature, precipitation, and sea level pressure—and various combinations of spatiotemporal information. We found that joint temperature–precipitation response vectors yielded the strongest detection of aerosol cooling, even in the absence of spatial and temporal gradient information. The best single variable for detecting anthropogenic aerosols is annual mean precipitation. We call attention to the fact that sea level pressure is computed partly from temperature data and hence is not independent of temperature. We find no evidence that sea level pressure provides any significant gain in detectability beyond that already available in the surface temperature data from which sea level pressure is derived. We also used state vectors that included seasonal variations and that separated land and ocean domains, but these vectors generally produced less detectability than vectors of the same dimension based on annual mean variables of Laplacian eigenvectors. These conclusions hold even in the imperfect model setting, in which response vectors and noise covariances are estimated from a model that differs from the model that generated the historical single-forcing runs. In one model, jumps in detectability occurred after the second and fifth spherical harmonics were included, which measure, respectively, the north–south hemispheric gradient and equator-to-pole gradient. Further comparison between perfect and imperfect model analyses indicate that these jumps are due to resolving gradient information in the response vectors and the noise covariance matrix.

As emphasized throughout this paper, our results are derived from model experiments in which errors in the forced response and observations can be neglected. There is no guarantee that response vectors that work well in models also will work well in observations. In fact, DelSole et al. (2015, manuscript submitted to *J. Climate*) applied optimal fingerprinting to observations using global mean joint temperature and precipitation information, but observed, global, long-term precipitation was inconsistent with models and even between datasets. Thus, detection analysis based on observed precipitation turns out to be difficult, despite the fact that model precipitation is the best single detection variable in a model setting. Nevertheless, the methodology illustrated here is expected to provide a useful starting point for deciding which variables to use for detection and attribution analysis of observations.

We investigated AA and noAA simulations in this paper because we were interested primarily in the response to anthropogenic aerosols. The methodology presented in this paper is not limited to the combination of single forcings of AA and noAA. In general, the potential detectability measure can be computed for any problem that can be formulated as optimal fingerprinting.

## Acknowledgments

This research was supported primarily by the National Science Foundation (CCF-1451945). Additional support was provided by the National Science Foundation (AGS-1338427), the National Aeronautics and Space Administration (NNX14AM19G), the National Oceanic and Atmospheric Administration (NA14OAR4310160), and an Office of Naval Research grant (N00014-16-1-2073). The views expressed herein are those of the authors and do not necessarily reflect the views of these agencies.

## APPENDIX

### Generalization of Optimal Fingerprinting and Derivation of Potential Detectability

This appendix shows how optimal fingerprinting can be generalized to estimate time series for forced responses and formally derives the potential detectability measure. We begin by first summarizing generalized least squares (GLS), which is a standard framework in linear regression and then discuss how optimal fingerprinting relates to GLS. This material is standard (Allen and Tett 1999; Hegerl et al. 2007; Hegerl and Zwiers 2011). Then, we show how our modified regression problem can be framed as a GLS problem and then derive potential detectability.

*K*-dimensional vector

*m*th regression coefficient is

*s*th spatial location and

*n*th time step is

*M*-dimensional vector of coefficients for the

*j*th Legendre polynomial, so that

*j*th Legendre polynomial are independent of

*j*and given by

*j*th Legendre polynomial for the

*m*th forced response is

*J*degrees of freedom. The quantity

*m*th forced response and must exceed a critical value for a forced response to be detected.

*ρ*measures the cosine of the angle between the response vectors, which in turn can be interpreted as a generalized spatial correlation. Consolidating the above results into the detection statistic in (A25) gives

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