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 2
time 3
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time 4
Time series at a fixed grid point
la
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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time 2
time 3
time n
time 4
la
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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time 2
time 3
time n
time 4
la
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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time 2
time 3
time n
time 4
la
t
Correlation map
Basic Operation with gridded data1Point Correlation Map
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time 2
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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 2
time 3
time n
time 4
time 2
time 3
time n
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
'
1
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
time 1
time 2
time 3
time 4
time 5
time N
1 2 3 4 5 ….. N time
varia
ble
Leading EOF (dominant mode)
loa
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)
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