Statistical characteristics of surrogate data based on geophysical

measurements

Victor Venema 1, Henning W. Rust 2, Susanne Bachner 1, and

Clemens Simmer 1

1 Meteorological Institute University of Bonn2 PIK, Potsdam Institute for Climate Impact Research

2

Content Surrogate data: time series generated based on

statistical properties of measurements– Distribution and/or power spectrum

7 Geophysical time series Generated surrogates with 7 different algorithms

from their statistics Compared the measurements to their surrogates

– Increment distribution– Structure functions

3

Motivation Need time series with a known structure

– Statistical reconstruction– Bootstrap confidence intervals– Studying non-local process– …

FARIMA & Fourier methods vs. Multifractals Multifractals vs. surrogates

4Satellite pictures: Eumetsat

Motivation - generator Empirical studies

– Exact measured distribution– Measured power spectrum

Scale breaks Waves Deviations large scales …

5

7 Generators for surrogate data D: distribution

– PDF surrogates

S: spectrum– Fourier surrogates– FARIMA surrogates

– Seasonal cycle and logarithm if needed

DS: distribution + spectrum– AAFT, IAAFT, SIAAFT surrogates– FARIMA + IAAFT surrogates

– seasonal cycle and log. if needed

6

Mea

sure

men

ts400 600 800 1000 1200 1400 1600 1800

0

5

10

Time (pixel)

Val

ue (1

)

p-model

1894 1894.5 1895 1895.5 1896 1896.5 1897 1897.50

20

Time (year)R

ain

(mm

/d)

daily rain sums

1828 1828.5 1829 1829.5 1830 1830.5 1831 1831.5500

100015002000

Time (year)

Run

off (

m3 /s

)runoff Burghausen

1818 1818.5 1819 1819.5 1820 1820.5 1821 1821.52000400060008000

Time (year)

Run

off (

m3 /s

)

runoff Cologne

0 200 400 600 800 1000 1200 1400 16000

0.20.4

Time (s)

LWC

(g m

-3)

cumulus

0 100 200 300 400 500 600 700 800 900 1000

0.4

0.6

Time (s)

LWC

(g m

-3)

stratocumulus

1894 1894.5 1895 1895.5 1896 1896.5 1897 1897.5-10

01020

Time (year)

Tem

p. (°

C)

temperature

7

Surr

ogat

e ty

pes

00.20.4 Cumulus measurement

00.20.4 SIAAFT surrogate

00.20.4 IAAFT surrogate

00.20.4 AAFT surrogate

00.20.4 Fourier surrogate

LWC

(g m

-3)

00.20.4 PDF surrogate

00.20.4 FARIMA surrogate

0 200 400 600 800 1000 1200 1400 1600

00.20.4 FARIMA + IAAFT surrogate

Time (s)

DS

DS

DS

S

D

S

DS

8

Increment distribution Measurement: (t) Increment time series for lag l:

(x,l) = (t+l) - (t)

Distribution jumps sizes Next plots: l is 1 day

9

Increment distribution temperature

-15 -10 -5 0 5 10 15

10-4

10-3

10-2

10-1

100

Temperature increments (°C)

Rel

. fre

quen

cy

originalsiaaft (DS)iaaft (DS)aaft (DS)fourier (S)farima (S)farima+iaaft (DS)

-2 0 20.5

0.6

0.7

0.8

0.9

10

Increment distribution Rhine

-4000 -2000 0 2000 4000

10-4

10-2

100

Runoff increments (m3 s-1)

Rel

. fre

quen

cy

originalsiaaft (DS)iaaft (DS)aaft (DS)fourier (S)farima (S)farima+iaaft (DS)

-200 0 200

0.2

0.4

0.6

0.8

1

11

Structure functions Increment time series: (x,l)=(t+l)- (t)

SF(l,q) = (1/N) Σ ||q

SF(l,2) is equivalent to auto-correlation function Higher q focuses on larger jumps

12

Structure function Salzach

13

Structure function stratocumulus

14

RMSE 4th order structure functions Best surrogate in bold Multifractal means: power law fit

SIAAFT IAAFT AAFT Fourier PDF FARIMA FARIMA + IAAFT Multifractal Bias P-model 0.018 0.019 0.016 0.071 0.065 0.068 0.020 0.017 Rain Potsdam 0.0027 0.0021 0.0038 0.093 0.0023 0.099 0.0029 0.0020 Runoff Burghausen 0.011 0.0069 0.029 0.076 0.081 0.076 0.023 0.028 Runoff Cologne 0.016 0.016 0.025 0.064 1.6 0.043 0.034 0.19 Cumulus 0.012 0.0080 0.016 0.076 0.044 0.063 0.0070 0.029 Stratocumulus 0.018 0.017 0.018 0.026 0.49 0.042 0.038 0.028 Temperature 0.0027 0.0037 0.016 0.0062 1.7 0.0060 0.0055 0.073 Variabilty P-model 0.18 0.19 0.17 0.71 0.65 0.68 0.20 0.17 Rain Potsdam 0.032 0.028 0.048 0.92 0.036 0.99 0.037 0.020 Runoff Burghausen 0.12 0.081 0.31 0.76 0.81 0.76 0.24 0.28 Runoff Cologne 0.16 0.16 0.26 0.64 16 0.45 0.35 1.9 Cumulus 0.14 0.10 0.20 0.76 0.45 1.5 0.13 0.29 Stratocumulus 0.19 0.18 0.19 0.26 4.9 0.46 0.40 0.28 Temperature 0.034 0.040 0.17 0.066 17 0.062 0.057 0.73

15

Extension IAAFT algorithm 2D and 3D fields with PDF(z) PDF(t), i.e. distribution varies as function of

– Season, time of day– Break point

Multivariate statistics, cross correlations Increment distribution at small scales

– More accurate increment distribution– Asymmetric increment distribution (runoff)

Downscaling– Extrapolate spectrum – Iterate the original coarse mean values

16

Conclusions DS-Surrogates of geophysical reproduce

measurements accurately– spectrum– increments– structure functions

IAAFT algorithm– Flexibly– Efficiently– Many useful extensions are possible

Surrogates for empirical work Multifractals for theoretical work (use IAAFT)

17

More information Homepage

– Papers, Matlab-programs, examples http://www.meteo.uni-bonn.de/

venema/themes/surrogates/ Google

– surrogate clouds– multifractal surrogate time series

IAAFT in R: Tools homepage Henning Rust– http://www.pik-potsdam.de/~hrust/tools.html

IAAFT in Fortran (multivariate): search for TISEAN (Time SEries ANalysis)