1. Aiming Wu, William W. Hsieh
The nonlinear patterns of North American winter temperature and precipitation associated with ENSO
Aiming Wu, William W. Hsieh
Dept. of Earth & Ocean Sciences,
University of British Columbia
and
Amir Shabbar
Meteorological Services of Canada
Downsview, Ontario

2. ENSO = El Niño + Southern Oscillation
La Niña

3. Atmos. Response to ENSO is nonlinear
Composite of Z500 and tropical precipitation during El Niño (A) and La Niña (B)
(from Hoerling et al 1997 J. of Climate) + - A + -
La Niña El Niño
Sign reversed
Shifted eastward by 30-40°(asymmetric) + + B -

4. Question
If x is the ENSO index, how to derive the atmos. response y = ƒ(x) ?
linear regression (or projection)
y = a • x –x +x - - + + - +
Linear method cannot extract asymmetric patterns between –x and +x
Need a nonlinear method

5. Nonlinear projection via Neural Networks (NN projection)
x, the ENSO index
h, hidden layer
y´, output, the atmos. response
A schematic diagram
Cost function J = || y – y´ || is minimized to get optimal Wx, bx, Wh and bh (y is the observation)

6. 1st principal component (PC) of the tropical Pacific SSTA
Data
ENSO index (x)
1st principal component (PC) of the tropical Pacific SSTA
Nov.-Mar.
,monthly
SST data from ERSST-v2 (NOAA)
Linear detrend
standardized
Atmos. Fields (y)
surface air temp. (SAT) and precip.(PRCP)
From CRU-UEA (UK)
Monthly,1950–2001, 11
Nov.-Mar.; North America
Anomalies ( Clim)
Linear detrend
PRCP standardized
Condensed by PCA
10 SAT PCs (~90%) retained
12 PRCP PCs (~60%)

7. A single NN model may not be stable (or robust)
Bootstrap
A single NN model may not be stable (or robust)
Bootstrap: randomly select one winter’s data 52 times from the 52-yr data (with replacement)  one bootstrap sample
Repeat 400 times  train 400 NN models  average
of the 400 models as the
final solution
400 NN models
Give a x  NN model  y  (combined with EOFs)  atmosphere anomaly pattern associated with x

8. NN projecton in the SAT PC1-PC2-PC3 space
Green: 3-D
Blue: projected on 2-D PC plane
“C” extreme cold state; “W” extreme warm state
Straight line: linear proj.
Dots: data points

9. SAT anomalies
as ENSO index takes on its
(a) min.
(d) max.
(b) 1/2 min.
(e) 1/2 max.
(c) a-2b
(f) d-2e
Darker color  above 5% significance

10. PCA on Lin. & Nonlin. Parts of NN projection
Linear regression
NL = NN – LR
73%
27%

11.  PC1 of Lin. part vs. ENSO index  a straight line
PC1 of Nonlin. part vs. ENSO index  a quadratic curve 
A quadratic
response

12. 1 , 2 are x, x2 normalized, x is the ENSO index
A polynomial fit
1 , 2 are x, x2 normalized, x is the ENSO index
SAT

13. PRCP anomalies
as ENSO index takes on its
(a) min.
(d) max.
(b) 1/2 min.
(e) 1/2 max.
(c) a-2b
(f) d-2e
Darker color  above 5% significance

14. Lin. & nonlin. prcp. response to ENSO
LR + NL = NN
78%
22%

15. Lin. & nonlin. prcp. PC1 vs. ENSO index

16. Summary and Conclusion
N. Amer. winter climate responds to ENSO in a nonlinear fashion (exhibited by asymmetric SAT and PRCP patterns during extreme El Niño and La Niña events).
The nonlinear response can be successfully extracted by the nonlinear projection via neural networks (NN), while linear method can not.
NN projection consists of a linear part and a nonlinear part. The nonlinear part is mainly a quadratic response to the ENSO SSTA, accounting for 1/4~1/3 as much as the variance of the linear part.

17. Thank you !