oscillations and warming trend in global temperature time seriesabhishek/geo-project.pdf ·...

Oscillations and Warming Trend inGlobal Temperature time series

-Abhishek Bhattacharya

1

Introduction

• The goal of this project is to distinguish

a warming trend and periodic oscillations

from natural variability in global surface air

temperatures.

• First I use a Singular Value decompo-

sition (SVD) analysis of the three dimen-

sional temperature data over different sta-

tions all over the world and over differ-

ent months. That gives the leading spa-

tial patterns explaining most of the vari-

ation. Since those patterns concentrate

mostly over the North Atlantic region, I

focus on the analysis of that region. I use

the leading patterns to reconstruct the an-

nual average temperature over the North

Atlantic.

2

• Then I use Singular Spectrum Analysis

(SSA) to analyze the time series of an-

nual average surface air temperatures over

North Atlantic for the past 136 years, al-

lowing a secular warming trend and a small

number of oscillatory modes to be sepa-

rated from the noise. There is a rising

trend since 1910. The oscillations exhibit

periods of around 5 or 6 years.

3

Data Source

• The source of the climatic data is UEA

CRU Jones HAD CRUT2v temperature anomaly

data.

• The data has the following grid structure:

• Time: In months from 1/1870 to 12/2005

by 1. 1632 grid points

• Longitude: 177.5W to 177.5E by 5. 72

grid points.

• Latitude: 87.5N to 87.5S by 5. 32 grid

points.

• Temperature: In degree Celsius. Missing

value -99.99.

4

Figure 1 shows the plot of the temperature

over certain months.

Figure 1a: Jan 1870 Global Surface Air

temperature

180oW 120oW 60oW 0o 60oE 120oE 180oW 80oS

40oS

0o

40oN

80oN

−5

−4

−3

−2

−1

0

1

2

3

4

5

5

Figure 1b: Dec 2005 Global Surface Air

temperature


40oS

0o

40oN

80oN

6

Empirical Orthogonal Function (EOF)

decomposition Analysis

• Prior to the analysis, I weight the data ap-

propriately to take into account the distor-

tion in high latitudes in a Mercator projec-

tion. I apply weights to the grid points, the

weight being proportional to the cosine of

the latitude.

• Then I compute the spatial covariance ma-

trix. To have significant number of obser-

vations, I use the last 100 years: 01/1906

to 12/2005 in my covariance calculation.

Also I use only those grid points which have

at least 50 years of data. That would en-

sure that the pairs have at least 5 years of

overlap.

7

• Figure 2 shows the grid points used in my

analysis. Most of them concentrate around

Northern Atlantic (NA).

• Then I compute the eigen vectors and eigen

values of the covariance matrix.

• Figure 3 shows the spatial pattern of the

first four EOFs. All of them concentrate

on NA and have a single pole in that re-

gion. EOFs 1 and 2 are positive correlated

with space while EOFs 3 and 4 are close

to negative. They explain 22, 13, 9 and 7

percent of the variation in the data respec-

tively. Together they explain about 51% of

the variance.

• Since the EOF plots are very localized,

they explain the spatial directions well.

8

• I perform a white noise test to select the

EOFs that explain a significant amount of

variation.

• Figure 4 shows the plot of the first 40 eigen

values of the sample covariance matrix vs

the corresponding 95th percentile from a

white noise time series with same variance

structure. The plot suggests that the first

11 eigen values and hence the correspond-

ing EOF patterns are significantly above

noise level.

• I use these EOF patterns to reconstruct

the time series from 01/1870 to 12/2005.

I look at the annual average temperature

averaged over the NA for this time period.

9

• Figure 5 shows the annualized Principal

components (PC) of the 4 leading EOFs.

These represent the time pattern of those

EOFs. As the plot shows, the first 2 PCs

are negatively correlated with time, the third

is positively correlated while the fourth is

close to being uncorrelated.

• Figure 6 shows the average annual tem-

perature over NA timeseries, the true one

vs the one reconstructed using the first 11

leading patterns. They are pretty close.

• The time series shows a rising trend to-

wards the later years and some irregular os-

cillations. The next goal would be to sep-

arate the trend and oscillations from noise.

10

Figure 2: Dec.2005 temperatures over se-

lected grids


40oS

0o

40oN

80oN

−1.5

−1

−0.5

0

0.5

1

1.5

2

11

Figure 3: The first 4 EOF patterns

3a. EOF−1 Pattern


40oS

0o

40oN

80oN 3b. EOF−2 Pattern


40oS

0o

40oN

80oN

3c. EOF−3 Pattern


40oS

0o

40oN

80oN 3d. EOF−4 Pattern


40oS

0o

40oN

80oN

−0.2

−0.1

0

0.1

0.2

0.3

12

Figure 4: Eigen value trace

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60plot of the 40 largest eigen values of the sample variance matrix, and the estimated upper quantiles

95th upper quantile from gaussian white noisetruevalue

13

Figure 5: The first 4 PC series

1900 1950 2000−5

0

5

time(years)

PC−1

1900 1950 2000−5

0

5

time(years)

PC−2

1900 1950 2000−5

0

5

time(years)

PC−3

1900 1950 2000−5

0

5

time(years)

PC−4

14

Figure 6: The true vs reconstructed an-

nual temperatures over NA

1880 1900 1920 1940 1960 1980 2000−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Annual Average temp. over N. Atlantic Region

time(years)

true time seriesreconstructed series

15

Singular Spectrum Analysis (SSA) of

annual temperatures over NA

• Now I develop a SSA of the reconstructed

time series of annual temperature data over

NA from 01/1870 to 12/2005.

• I choose an embedding dimension, m, com-

pute the covariance matrix of the embed-

ded data set, calculate its eigen values,

eigen vactors and reconstructed components

for the time series.

• I compare the eigen values with those from

a white noise process to check their signif-

icance at level 95%.

16

• Shown in Figure 7 are the plots from SSA

with m = 20 years.

• Figure 7a shows EOFs 1 and 2 patterns

over time. EOF 1 shows a declining trend

while EOF 2 shows a rising trend. To-

gether they explain about 63% of the vari-

ation. The plot suggests that these EOFs

correspond to the trend in the temperature

record.

• Figure 7b shows EOFs 3 and 4 patterns.

They are an oscillatory pair in quadrature

with each other and have a period of about

6 years. They explain about 10% of the

variation.

17

• Figure 7c shows the EOFs 5 and 6. They

also form oscillatory pairs in quadrature with

each other and have a period of about 10

years. They explain about 8% of the vari-

ation.

• In Figure 7d is shown the time series recon-

structed out of the first two EOFs. The

trend is flat until 1910 and then a rise after

that.

• Figure 8 is the plot of the eigen values of

covariance matrix and the corresponding

white noise estimates. The plot suggests

that only the first two are significantly dif-

ferent from noise values.

18

m = 20 Plots

Figure 7: SSA for m = 20.

7a: EOFs 1 and 2 Pattern; 7b: EOFs 3 and

4 patterns; 7c: EOFs 5 and 6 patterns; 7d:

Time Series reconstructed out of EOFs 1 and

2.

0 5 10 15 20−0.5

0

0.57a.

EOF−1EOF−2

0 10 20−0.5

0

0.57b.

EOF−3EOF−4

0 10 20−0.5

0

0.57c.

EOF−5EOF−6

1880 1900 1920 1940 1960 1980 2000−0.5

0

0.57d.


19

Figure 8: Eigen value trace for m=20

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Figure 8

95th upper quantile from white noisetrue value

20

Other embedding dimensions analysis.

• Figure 9 shows the plots for SSA Analysis

with m = 6.

• Again EOFs 1 and 2 correspond to trend

with similar pattern as in the m = 20 case

(Figure 7a). EOF pairs 3 & 4 and 5 &

6 are in quadrature and show oscillations

with period of about 5 to 6 years.

• However as Figure 10 shows, apart from

EOF 1, the other ones are closer to noise

floor.

21

• When m = 10 (Figure 11), EOFs 1 and

2 correspond to trend; EOFs 3 & 4 are

in quadrature with period around 6 years;

EOF 5 has period of 10 years while EOF

6 shows a noisy pattern.

• In Figure 12, again only EOF 1 is signifi-

cantly above noise floor.

• When m = 30 (Figure 13), EOFs 1 and

2 correspond to trend; EOFs 3 & 4 are

in quadrature with period around 6 years;

EOFs 5 & 6 are in quadrature with period

around 6 years. In Figure 14, EOFs 1 & 2

are significantly above noise level.

22

m = 6, 10, 30 Plots.

Figure 9: SSA for m = 6

1 2 3 4 5 6−0.5

0

0.5

19a.

EOF−1EOF−2

0 5 10−1

0

19b.

EOF−3EOF−4

0 5 10−1

0

19c.

EOF−5EOF−6

1880 1900 1920 1940 1960 1980 2000−0.5

0

0.59d.


23


0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Figure 10


24


1 2 3 4 5 6 7 8 9 10−0.5

0

0.511a.

EOF−1EOF−2

0 5 10−1

−0.5

0

0.511b.

EOF−3EOF−4

0 5 10−1

0

111c.

EOF−5EOF−6

1880 1900 1920 1940 1960 1980 2000−0.5

0

0.511d.


25


0 1 2 3 4 5 6 7 8 9 10 110

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Figure 12


26


0 5 10 15 20 25 30−0.5

0

0.513a.

EOF−1EOF−2

0 20 40−0.5

0

0.513b.

EOF−3EOF−4

0 20 40−0.5

0

0.513c.

EOF−5EOF−6

1880 1900 1920 1940 1960 1980 2000−0.5

0

0.513d.


27


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25Figure 14


28

Sensitivity of SSA to choice of

embedding dimension, m

• Now I check how robust is the SSA analy-

sis.

• As all eigen value trace plots in Figures 10

(m = 6), 12 (m = 10), 8 (m = 20) and 14

(m = 30) show, there is a sharp fall from

eigen value 1 to 2, another less sharper fall

from eigen values 2 to 3 after which the

trace gets flat.

29

• Also we see that as m increases, the trace

gets steeper. This suggests stationarity in

the data. If the series were truly station-

ary, the covariance matrix for a smaller m

can be ”roughly” embedded into that for

a larger m. Then the latter will have its

largest eigen values bigger, while the least

eigen values smaller, leading to a steeper

eigen value trace.

• Since the first two EOFs represent trend

and the next two periodicity for all dimen-

sions, I reconstruct the time series using

these EOFs for m = 5,6,10,12,20,30,50

and 100 years. In each case I compute

the correlation of the summed components

with that for m = 20.

30

• Figure 15a shows the plot of the correla-

tions. The plot shows that the correlation

is over 0.8 for m < 80. This suggests that

the different values of m less than 80 do

not substatantially alter the SSA analysis

results.

• In Figure 15b is plotted the variance ex-

plained by the first four EOFs for the above

values of m. The plot is rising which sup-

ports my earlier claim about stationarity,

and the rise is fairly smooth till m = 60

suggesting robustness of the analysis for

m < 60.

• Figure 15c shows the plot of the fractional

variance explained by the first 4 EOFs vs

the embedding dimension, m. This plot is

declining. The values are over 50% in all

case, and over 60% for m < 60.

31

Figure 15

15a: Correlation between time series constructed

using embedding dimension, m and 20; 15b:

Variance explained by 1st 4 EOFs; 15c: Frac-

tional Variance explained by the 1st 4 EOFs.

10 20 30 40 50 60 70 80 90 100

0.6

0.8

1

15a

dimension, m

corr

elat

ion

with

m=

20

20 40 60 80 1000

0.2

0.4

0.6

0.8

115b

dimension, m

var.

exp

lain

ed b

y 1s

t 4 E

OF

s

20 40 60 80 1000.5

0.6

0.7

0.8

0.9

115c

dimension, m

frac

. var

. exp

lain

ed b

y 1s

t 4 E

OF

s

32

Conclusions

• From different dimensional SSA analysis,

we observe that the EOFs 1 and 2 repre-

sent the trend in the data. As the true

and reconstructed timeseries show (Figure

7a), there has been a rise in surface air

temperatures since 1910. The estimation

of EOFs 1 and 2 is robust, in the sense

that their trend changes little with varying

dimension, m.

• EOFs 3 and 4 explain the major periodic-

ity in the data. They have a period be-

tween 5 and 6 years. However their esti-

mation is not so robust. Their trace pat-

tern varies with m. This is justified by the

fact that the corresponding eigen values

are very near to the noise floor for all m.

33

• The analysis shows no sign of bidecadal

oscillations as reported by Ghil and Vautard

[2].

• This raises the question whether a mini-

mum sample size is required to detect them.

However a sample size of 136 years seems

fairly large.

• Another reason could be, as Elsner and

Tsonis [1] point out, the years before 1881

made the difference. I get the leading pat-

terns from the EOF analysis of the covari-

ance matrix constructed using the past 100

years: 1906 to 2005 and use those to re-

construct the annual average temperature

over NA time series for the span of past

136 years (1870-2005). Perhaps including

the earliest years in calculating the EOFs

would make a difference.

34

• However that would mean that the data is

non stationary while SSA is designed mainly

for stationary data.

• This makes one doubt the existence of bidecadal

oscillations in the temperature records, and

the tools used to detect them.

35

References

[1] ELSNER, J.B. and TSONIS, A.A.(1991)

Do bidecadal oscillations exist in the global

temperature record? Nature353 551-553

[2] VAUTARD, R. and GHIL, M.(1991) Inter-

decadal oscillations and the warming trend

in global temperature time series. Nature350

324-327

[3] JONES, P.D. and MOBERG, A.(2003) Hemi-

spheric and large-scale surface air temper-

ature variations: an extensive revision and

an update to 2001. J.Climate16 206-223

36

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