Inhomogeneities in temperature records deceive long-range dependence estimators Victor Venema Olivier Mestre Henning W. Rust Presentation is based on: Henning Rust, Olivier Mestre, and Victor Venema. Fewer jumps, less memory: homogenized temperature records and long memory Submitted to JGR-Atmospheres

Content  Long range dependence (LRD) –What it is? –Short range dependence –Why is it important  Estimating long range dependence –FARIMA modelling, Fourier analysis –Detrended Fluctuation Analysis (DFA)  The influence of inhomogeneities on LRD –Comparison of raw and homogenised data  Homogenisation produces no artefacts –Validation on artificial data

Autocorrelation function – SRD vs. LRD

Long range dependence (LRD)  Autocorrelation function  LRD: –   (  ) =  –  (  )   -α   (2-2H),  –0.5 < H < 1  Short range dependence (SRD) –   (  ) <  –  (  )  e - ,   Spectral density  LRD: –S(  )  |  | - , |  |  0 –  = 2H - 1 –0 <  < 1 –d = H - 0.5

Example long range dependence Demetris Koutsoyiannis, The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences, 47(4) August 2002.

Uncertainty in trend estimate

Inhomogeneous data and trends  LRD may lead to a higher false alarm rate (FAR) in homogenisation algorithms –Depends on physical cause of LRD  Inhomogeneities can be mistaken for a climate change signal  Inhomogeneities lead to overestimates of LRD –Artificially increase estimates of natural variability –Artificially increase the uncertainty of trend estimates

Inhomogeneous data and LRD  Most people working on LRD do not report whether their data was homogenised –Literature search: 24 articles –18 gave no information on quality –Two articles: high quality data or selected homogeneous stations –One article partially inhomogeneous –Two articles partially homogenised –One article homogenised

FARIMA - power spectrum

DFA algorithm  Cumulative sum or profile:  X t is divided in samples of length L  For every sample a linear trend is estimated and subtracted  F(L) is variance of the remaining anomaly

DFA example for one scale Peng C-K, Hausdorff JM, Goldberger AL. Fractal mechanisms in neural control: Human heartbeat and gait dynamics in health and disease. In: Walleczek J, ed. Nonlinear Dynamics, Self-Organization, and Biomedicine. Cambridge: Cambridge University Press, 1999.

DFA spectrum

Problems with DFA  H depends on subjective scaling range  No criterion for goodness of fit for DFA spectrum  Heuristic: no error estimate for H  Not robust against non-stationarities

H-estimates: raw vs. homogenised

Simulation experiment  LRD regional climate data  Added noise to obtain station data  Added inhomogeneities  Caussinus-Mestre to correct  Compared H before and after

FARIMA simulation experiment: original vs. perturbed

DFA simulation experiment: original vs. perturbed

DFA simulation experiment: original vs. homogenised

Conclusions  Inhomogeneities increase estimates of LRD –Studies on LRD should report on homogeneity –As well as other studies on slow cycles, low-frequency variability, etc.  LRD increases uncertainty of trend estimates –As well as other parameters related on slow cycles, low- frequency variability, etc.  DFA is not robust against inhomogeneities  Manuscript: venema/articles/