Journal of Climate

Supporting Information for “Pattern Recognition
Methods to Separate Forced Responses from Internal
Variability in Climate Model Ensembles andObservations”
Robert C. J. Wills1 , David S. Battisti1 , Kyle C. Armour1 , Tapio Schneider2 ,Clara Deser3
1 University of Washington, Seattle, WA
2 California Institute of Technology, Pasadena, California
3 National Center for Atmospheric Research, Boulder, Colorado

Linear Inverse Model Optimal Perturbation Filter
Here we describe the linear inverse model (LIM) optimal perturbation filter (LIMopt),
which was identified by F17 as one of the best available method for estimating the forced
climate response from a single ensemble member. We then compare it to the LFP filtering method presented in the main text.
LIMs assume that the evolution of anomalies x (e.g., SST anomalies) are describedby a multivariate linear Markov process (i.e., a first-order Ornstein-Uhlenbeck process)dx= Lx + ξ,dt

(1)

where L is a linear operator and ξ is white noise (e.g., Penland and Sardeshmukh 1995).
The best estimate of the state vector x(t), is given by forward integration according tox(t+τ ) = G(τ )x(t), where G(τ ) = exp(Lτ ). The evolution of the state vector can be
decomposed into a sum of nonnormal eigenmodes
x(t) =

Σiuiαi(t),(2)where ui are the eigenvectors of L, estimated from the ensemble-mean covariance matrix hCi and the ensemble-mean lag-1 covariance matrix hC1 i byL=

1 ln hC1 ihCi−1 .
τ(3)Here, we have allowed for the use of multiple ensemble members, but this holds also forthe case of a single ensemble member. The time evolution αi (t) is determined from theprojection of x onto the corresponding adjoint eigenvector. The eigenmodes are either
stationary damped modes with a single spatial pattern and decay time or a damped oscillatory mode with two patterns. The first eigenmode is the one with the least damping, and will generally capture any long-term forced response in the dataset (Compo andSardeshmukh 2010; Newman 2013; F17). This framework is generally applied to seasonal
SST anomalies, with τ = 3 months; we will apply it here to seasonal surface temperature anomalies (including land and sea ice). The dimensionality is reduced by working in a truncated EOF space, in our case with 50 EOFs to account for around 77% ofthe total variance.

Corresponding author: Robert C. Jnglin Wills, rcwills@uw.edu–1–Journal of Climate

Following F17, we employ a LIM-based optimal perturbation filter (LIMopt) to identify a spatiotemporally varying estimate of the forced response in CESM-LE surface temperatures. This method was used by Solomon and Newman (2012) to isolate (and remove) ENSO variability within Pacific SSTs and by F17 to isolate the forced response
within global SSTs. The optimal initial structure Φ1 (τe ) is the pattern that evolves into
the maximum possible anomaly after time τe , taken here to be 2.5 yr (as in F17, but muchlonger than the 6-9 months used by Solomon and Newman (2012) to isolate the ENSOsignal). It is the normalized right singular eigenvector of G(τe ) = exp(Lτe ). The LIMopt
filter considers the evolution of anomalies from this optimal initial structure over a longertime period τ1 . The forced response is given byF=Σττ =0α(t − τ )G(τ )Φ1(τe),1(4)

where α(t) is determined by projection of the left singular vector Ψ1 (τe ) onto the anomaly
at that point in the iteration, i.e.,hi
τ1α(t) = x(t) − τ =0 α(t − τ )G(τ )Φ1 (τe ) Ψ1 (τe ),(5)Σwith the initial condition α(0) = x(0)Ψ1 (τe ). This method considers all anomalies thatevolve from the optimal initial structure, which allows the spatial pattern to evolve overtime.
In order to apply LIMopt to multiple ensemble members, in addition to averaging
over multiple ensemble members in the computation of hCi and hC1 i, one must replacex(t) with hx(t)i in Eq. 5. Alternatively, one could use the opposite order of operations
where the iterative procedure (Eqs. 4 and 5) is applied to each ensemble member separately and then the resulting forced response estimate F is averaged over the ensemble. However, this arrives at the same answer after a longer computation time.
We apply LIMopt to the CESM-LE seasonal surface temperature anomalies. Weuse τe = 2.5 yr (as in F17), but try two different choices of τ1 : τ1 = 20 yr (as in F17)and τ1 = 0 (which uses the optimal perturbation pattern, but skips the iterative filtering). The results are compared to LFP filtering in Fig. S1. While LIMopt provides a muchbetter estimate of the forced response than the unfiltered data (i.e., a simple ensemblemean), it generally does not perform as well as LFP filtering. In particular, it does not
improve the forced response estimate as much with the addition of ensemble members.
Focusing on the case of a single ensemble member, LFP filtering achieves a better correlation with the forced response of grid-point temperatures (Fig. S1a), global-mean surface temperature (Fig. S1c), North Atlantic SST (Fig. S1d), and US surface temperature (Fig. S1f), but LIMopt achieves a marginally better correlation with the forced response of the Pacific east-west SST difference (Fig. S1e). In terms of global-mean RMSE,
they perform about the same (Fig. S1b).
Also of note is that LIMopt with τ1 = 0 generally performs similar to or slightly
better than LIMopt with τ1 = 20 yr, indicating that the iterative filtering proceduredoes not substantially improve the forced response estimate. Simply identifying the optimal perturbation pattern is the main benefit derived from the LIMopt analysis. A potential explanation for this is that the LIMopt filter (with τ1 > 0) uses the LIM to smoothover unforced variations, but in doing so can introduce anomalies that are not actuallypresent in the data set. So while this filtering improves the estimate of the forced response
in a single ensemble member (Fig. S1, cf. F17), it can introduce small errors at the local scale, which prevent it from generalizing well to larger ensemble sizes (Fig. S1).
Here we analyzed which method works best for isolating the forced response froma single ensemble member. While LIMopt may perform better for the pattern of tropical SST change (Fig. S1e), LFP filtering performs better on average (Fig. S1a), including for global-mean surface temperature (Fig. S1c). Comparing with the work of F17,–2–Journal of Climateab0e

S/NP Filtering
Simple Ensemble MeanLFP Filtering
LIMopt ( 1 = 20 yr)

10N

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North Atlantic SST (40-60°N)

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Correlation 2

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5Correlation 2c

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Global-Mean Correlation 2

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00.2
0Figure S1.

510nE
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As in Fig. 1, except with the addition of LIMopt forced response estimates. For

legibility, S/NP filtering is omitted from panels (c)-(f).

this also means that LFP filtering performs better than a linear trend, quadratic trend,
regression against global-mean SST, or multi-variate ensemble empirical mode decomposition. Additionally, its similarity to S/NP filtering makes it easily generalizable todifferent ensemble sizes. Moreover, LFP filtering is a purely statistical method and makes
no assumptions about the nature of the underlying dynamics, as LIM-based methods do.
Overall this make LFP filtering the best available method for isolating forced responses
in small ensembles or single realizations.

References
Compo, G. P., & Sardeshmukh, P. D. (2010). Removing ENSO-related variationsfrom the climate record. J. Climate, 23 (8), 1957–1978.
Newman, M. (2013). An empirical benchmark for decadal forecasts of global surface
temperature anomalies. J. Climate, 26 (14), 5260–5269.
Penland, C., & Sardeshmukh, P. D. (1995). The optimal growth of tropical sea surface temperature anomalies. J. Climate, 8 (8), 1999–2024.
Solomon, A., & Newman, M. (2012). Reconciling disparate twentieth-century Indo-–3–

Journal of Climate

Pacific ocean temperature trends in the instrumental record.
Change, 2 (9), 691.

–4–Nature Climate