1. Empirical Models of SEASONAL to decadal variability and predictability
Matt Newman and Mike Alexander
CIRES/University of Colorado and NOAA/ESRL/PSD

2. 2010-2060 “A1B” tropical trends, same model, different ensemble members

3. Outline of Talk
Multivariate red noise: a basic model of Pacific climate variability
Applied to:
Tropics
Pacific Basin (PDO)
Decadal forecasts of global surface temperature anomalies

4. Some requirements for empirical climate models
Capture the evolution of anomalies
Growth/decay, propagation
need anomaly tendency: dynamical model
Can relate to physics/processes and estimate predictability?
Limited data + Occam’s razor = not too complex
How many model parameters are enough?
Problem: is model fitting signal or noise?
Test on independent data (or at least cross-validate)
Testable
Is the underlying model justifiable?
Where does it fail?
Can we understand where/why it succeeds? (no black boxes)
Previous success of linear diagnosis/theory for climate suggest potential usefulness of linear empirical dynamical model

5. Two types of linear approximations
“Linearization” : amplitude of nonlinear term is small compared to amplitude of linear term
Then ignore nonlinear term
“Coarse-grained” : time scale of nonlinear term is small compared to time scale of linear term
Then parameterize nonlinear term as (second) linear term + unpredictable white noise: N(x) ~ Tx + ξ
For example, surface heat fluxes due to rapidly varying weather driving the ocean might be approximated as

6. “Multivariate Red Noise” null hypothesis
Noise/response is local (or an index)
For example, air temperature anomalies force SST
use univariate (“local”) red noise:
dx/dt = bx + fs where x(t) is a scalar time series, b<0,
and fs is white noise
Noise/response is non-local: patterns matter
For example, SST sensitive to atmospheric gradient
use multivariate (“patterns-based”) red noise:
dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,
and Fs is white noise (maps)
Note that B is a matrix and x and Fs are vectors
If B is not symmetric* (*nonnormal), transient anomaly growth is possible even though exponential growth is not
How can we determine B?

7. Linear inverse model (LIM)
“Inverse method” – derive B from observed statistics
If the climate state x evolves as
dx/dt = Bx + FS
then τ0-lag and zero-lag covariance are related as
C(τ0) = G(τ0) C(0) = exp(Bτ0) C(0) [where C(τ) = <x(t+τ)x(t)T>].
LIM procedure:
Prefilter data in EOF space (since B = logm [C(τ0)C(0)-1]/τ0 )
Determine B from one training lag τ0.
Test for linearity
For much longer lags τ, is C(τ) = exp(Bτ) C(0) ?
This “τ-test” is key to LIM.
Cross validate hindcasts (withhold 10% of data)

8. Linear inverse model (LIM), cont.
“Inverse method” – derive B from observed statistics
If the climate state x evolves as
dx/dt = Bx + FS
then ensemble mean forecast at lead τ is
x(τ) = exp(Bτ) x(0) .
Eigenmodes of B are all damped but can be either stationary or propagating* (*Bei = λiei , where λi can be complex) & not orthogonal.
When B is “nonnormal” (dynamics are not symmetric) transient “optimal” anomaly growth can occur* (*DG(τ)vi = σiui, where D is a norm), leading to greater predictability.

9. ENSO flavors
Newman, M., S.-I. Shin, and M. A. Alexander, 2011: Natural variation in ENSO flavors. Geophys. Res. Lett., L14705, doi: /2011GL

10. “Multivariate Red Noise” null hypothesis
dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,
and Fs is white noise (maps)
Determine B and Fs using “Linear Inverse Model” (LIM)
x is SST/20 C depth/surface zonal wind stress seasonal anomalies in Tropics, (Newman et al. 2011, Climate Dynamics)
prefiltered in reduced EOF space (23 dof)
LIM determined from specified lag (3 months) as in AR1 model
Extension of work by Penland and co-authors (e.g. Penland and Sardeshmukh 1995)

11. Verifying Multivariate Red Noise: compare observed and LIM-predicted lag-covariances and spectra
Note that LIM entirely determined from one-season lag statistics

12. Multivariate red noise captures “optimal” evolution of ENSO types
SST: shading
Thermocline depth: contours
Zonal wind stress: arrows

13. Optimal structures are relevant to observed EP and CP ENSO events
Composite:
Six months after
a > ± 1 sigma
projection (blue dots) on either the first or second optimal initial condition, constructed separately for warm and cold events
Green dots represent
mixed EP-CP events

14. Multidecadal variations of CP/EP ENSOs driven by noise
“Increasing CP/EP Cases” : Adjacent 60-yr segments where
CP/EP ratio increases
r(Nino3,Nino4) decreases
24000 yr LIM “model run”: dx/dt = Bx + Fs
Values determined over 30-yr intervals spaced 10 years apart

15. LIM can provide realistic synthetic data Nino 3
LIM can provide realistic synthetic data Nino 3.4 times series: DJF (gray) and 25-yr running mean (black) Multi-proxy reconstruction (Emile-Geay 2012), one of 100 LIM realizations, forced CCSM4 show decadal signal, CCSM4 control does not

16. pacific sst empirical models
Newman, M., 2007: Interannual to decadal predictability of tropical and North Pacific sea surface temperatures. J. Climate, 20,
Alexander, M. A., L. Matrosova, C. Penland, J. D. Scott, and P. Chang, 2008: Forecasting Pacific SSTs: Linear Inverse Model Predictions of the PDO. J. Climate, 21,
Newman, M., D. Smirnov, and M. Alexander, 2012: Relative impacts of tropical forcing and extratropical air-sea coupling on air/sea surface temperature variability in the North Pacific. In preparation.

17. PDO depends on ENSO (Newman et al. 2003)
Forecast: PDO (this year) = .6PDO(last year) + .6ENSO(this year)
“reddened ENSO”
r=.74

18. “Multivariate Red Noise” null hypothesis
dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,
and Fs is white noise (maps)
Determine B and Fs using “Linear Inverse Model” (LIM)
x is SST seasonal anomalies in the Pacific (30°S-60°N), (Alexander et al. 2008, J. Climate)
prefiltered in reduced EOF space (13 dof)
LIM determined from specified lag (3 months) as in AR1 model
Skill in predicting Nino3.4 and PDO > 0.6 for 1 year forecasts when initialized in late winter

19. Diagnosing coupling
Use slightly different LIM by separating Tropics and North Pacific:
Define xtropics =SST/20 C depth/surface zonal wind stress
TNorthPac =SST (20ºN-60ºN)
Coupling effects are determined by zeroing out the appropriate submatrices within B.

20. Diagnosing coupling
Use slightly different LIM by splitting Tropics and North Pacific:
Define xtropics =SST/20 C depth/surface zonal wind stress
TNorthPac =SST (20ºN-60ºN)
Coupling effects are determined by zeroing out the appropriate submatrices within B.
Decouple Tropics from North Pacific, then recalculate statistics given same noise

21. Impact of tropical coupling on SST variability
Variance
6 month lag covariance
LIM
Uncoupled
East Pacific SST variability almost entirely due to tropical forcing.
In WBC, most variability is independent of the Tropics.

22. Dominant “internal” North Pacific SST mode
Compute new EOFs from covariance matrix determined from uncoupled LIM

23. “Multivariate Red Noise” null hypothesis
dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,
and Fs is white noise (maps)
Determine B and Fs using “Linear Inverse Model” (LIM)
x is SST annual mean (July-June) anomalies in Tropics and North Pacific, (Newman 2007)
prefiltered in reduced EOF space (10 dof)
LIM determined from specified lag (1 year) as in AR1 model

24. Components of the PDO
Leading eigenmodes of B, with time series ( )
Eigenmodes represent:
Trend
“Pacific Multidecadal Oscillation” (PMO)
“Decadal ENSO”
Almost all long range skill contained in first 2 eigenmodes

25. “PMO”
“Decadal ENSO”
“Interannual ENSO”
Reconstructed PDO
PDO
“Regime shifts”
Constructing the PDO from a sum of three red noise processes Time series show projection of each mode onto the PDO
PDO = PMO
+Decadal ENSO +Interannual ENSO

26. Decadal forecasts of global surface temperature
Newman, M., 2012: An empirical benchmark for decadal forecasts of global surface temperature anomalies. J. Climate, in review (minor revision).

27. Multivariate red noise surface temperatures
dx/dt = Bx + Fs
Determine B and Fs using “Linear Inverse Model” (LIM)
x is SST/Land (2m) temperature, 12-month running mean anomalies, (Newman 2012)
prefiltered in reduced EOF space (20 dof)
LIM determined from specified lag (12 months) as in AR1 model

28. Years 2-5
Years 6-9
Decadal skill for forecasts initialized LIM has clearly higher skill than damped persistence, comparable skill to CMIP5 CGCM decadal “hindcasts”
LIM

29. PDO hindcast skill – something more?

30. Leading eigenmodes of B, with time series (1900-2008)
Eigenmodes represent:
Trend
Atlantic Multidecadal Oscillation (AMO)
Pacific Multidecadal Oscillation (PMO)
Almost all skill contained in these 3 eigenmodes
Enhanced LIM PDO skill due to PMO

31. Conclusions North Pacific Climate Variability
Sum of “reddened” ENSO + northwest Pacific-based (KOE?) variability
Coupled GCMs may underpredict the second process
LIM is a good model of climate
Captures statistics of anomaly evolution and makes forecasts
Serves as a benchmark for numerical models
Can diagnose dynamical relationships between different variables/locations and how they provide/limit predictability
Can generate long runs of realistic synthetic “data”
Consistent with apparent “regime shifts” with limited predictability
Uses of climate variability
for scenario building to test sensitivity of ecosystem
to make predictions of ecosystem