Statistical tools in Climatology René Garreaud

www.dgf.uchile.cl/rene

Fundamental problem in (paleo) Climatology: How to link local-scale phenomena with large

scale atmospheric patterns (if there such a link)?

Local signal

Large scale climate

Internal dynamics

Example of local scale signals: Rainfall, wind, temperature, tree-ring growth, lake level, wet deposition, charcoal accumulation, etc. Some times, we have local events: heavy rainfall, extended drought, lake desiccation, etc.

Example of large scale climate: wind patterns, SST, precipitation elsewhere, etc….

Longitude

Latit

ude

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Time series at a fixed grid point

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t

lon

Map

Gridded Analysis: set of maps with data on a regular grid (e.g., lat-lon, lon-lev)

• Spatial structure at a given time (displayed with isolines or colors)• Time series at each grid point

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t

Standard deviation

Mean field

Basic Operation with gridded dataMean and Std Deviation

Average, standard deviation and other statistical parameters (max, min, skewness, etc.) are calculated using the time series at each grid point

Example of mean fields…

Use compositing analysis to describe the spatial structure of selected climate variables during the occurrence of events at your site.

Sometime, the events are associated to extreme values of a continuous time series. In this case, you can also use regression analysis (1Point correlation map), but the later method will fail if there is no a linear relationship…

Your site specificvariable

SLP

els

ewhe

re

Your events

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t

Basic Operation with gridded dataCompositing Analysis (composite map)

All mean

Event!

Event!

Composite mean Composite anomalies

_ =

Here one can use a univariate method (e.g., t-test) at each point to test for statistical significance of the anomalies

Example of compositing analysis…time series

Example of compositing analysis…spatial fields

Cold minus Warm composites to enhance signal

Composite maps of (a,b) rainfall and (c,d) SST anomaly overlaid on vectors of wind stress anomaly during dry and wet years.

® Mathew England 2005

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Correlation map

Basic Operation with gridded data1Point Correlation Map

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Maps of V

ariable Y

Time Series of variable X

Here one can use a univariate method at each point to test for statistical significance of correlation

r : Measure the covariability of two time series (-1 to +1)r2: Fraction of variance in series Y explained by linear fit of series X

Annual mean Precip/SAT regressed uponindex of large-scale modes (50 years of data)

How is the spatial pattern of SST when there is above/below normalRainfall over the Altiplano?

Correlation map

Basic Operation with gridded dataPoint-Point Correlation Map

Maps of Variable Y

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time 3

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time 4

Maps of Varia

ble X

Here one can use a univariate method at each point to test for statistical significance of correlation

Precipitación en latitudes medias

Garreaud 2007

Large annual cycleSmall interannual variability

Small annual cycleLarge interannual variabilityIntraseasonal noise?

Nice trendModerate annual cycleSome noise…

Abrupt climate change?Strong annual cycle?

Full time seriesMix regular cycles andinterannual variability

Stgo. Rainfall

1-Point Correlation mapStgo. Rainfall – 1000 hPa air temperature

Stgo. Winter Rainfall

1-Point Correlation mapStgo. Winter Rainfall – 1000 hPa air temperature

Full time seriesAnnual cycle

Anomaly time series

mym

ym

Ny

y

ym

years

m

yeartimeyearmonthtime

xxx

xN

x

NNxx

'

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12

1

Mean annual cycle

Anomalies

Alternatively, use the anomaly filter, i.e. substract annual mean cycleCaution with seasonally dependent signal

Caution with seasonally dependent signal1 Point Correlation Map between U300’ over the Andes and Precipitation

DJF JJA

All months

Most geophysical (met) variables exhibit some spatial coherenceAdjacent grid points covary with reference point…but how far?

U300 winter

Stgo Winter Precip

EOF / PC Analysis

• Powerful method to synthesize (large) data sets• Allows to obtain the mode(s) that explain most of

the variance in the data• Each mode composed by a spatial pattern

(EOF), time pattern (PC, loading factors) and fraction of variance accounted by

• It uses linear algebra: EOF/PC are linear combination of original data

• EOF = Empirical orthogonal functions (instead of analytical orthogonal functions)

EOF / PC Analysis

• User has to decide spatial domain (large-small)• Pre-filter data (at least remove mean field)• How many modes are retained (North’s rule)• Each mode is orthogonal with the rest…not

always true in nature• Rotated EOFs / Extended EOFs

X(space)

varia

ble

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time N

1 2 3 4 5 ….. N time

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ble

Leading EOF (dominant mode)

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din

g

Principal ComponentEOF

Analysis

Original data Synthesized information

Let the data speak by itself

: fraction of variance accounted by a mode

X(space)

X(space)