methods and applications of stochastic time series ...€¦ · methods and applications of...

Methods and Applications of stochastic Time Series Analysis

in Climatology

Manfred Koch

Department of Geohydraulics and Engineering Hydrology, University

of Kassel, Germany

Email: [email protected]

Keywords: Climate change, hydro-climate time-series, precipitation,

streamflow, NAO, AO, wavelet analysis, Hurst parameter

AbstractThe quantitative analysis of stochastic hydro-meteorological data series and the understanding of their natural or externally-caused fluctuations on different scales, i.e. their general variability, is of fundamental importance in climate research and currently highly topical. Recent publications indicate that climate change appears to have a “break” at present-day for several years. As there has also been quite a bit of controversy recently on the proper interpretation and decadal extrapolation of the so-called “Hockey-“global temperature curve, the understanding of the sources of the variability of climate and hydrological indices time series is of utmost importance, i.e. one needs to know as to whether the fluctuations of hydro-meteorological parameters are natural or externally triggered on the climate time scale .

Here we discuss some of the features and modern methods to analyze long-term climate time series which on first sight appear to be stochastic, i.e. random, in nature. Nevertheless, to reveal certain “structure” of the time series, a time series can usually be decomposed in (1) a trend, (2) periodicities and (3) undetermined noise. Stochastic time series analysis permits then to estimate the long-term statistical characteristics, like fluctuations, correlations, periodicities, trends and possible non-stationarities in hydro-climate time series. The centre of the presentation will then on wavelet analysis and the quantification of periodicities and scale („long-memory“) – effects, as quantified by the Hurst coefficient. For the recognition of trends and possible non-stationary characteristics of the time series wavelet multi-resolution analysis (MRA) and Singular Spectrum Analysis (SSA) are powerful techniques.

Applications of some of the methods above to observed and predicted climate and hydrological time series in various regions of the world, namely, Germany, with emphasis on the Elbe and the Fulda catchment, the Andean glacier region in Peru and various regions in Thailand are discussed. The results indicate the intricacies to properly interpret the variability of such hydro-meteorological time series and to understand its underlying sources. It is found that on the medium-term (decadal) scale, teleconnective, hemispherically-acting ocean indices, such as the North Atlantic Oscillation (NAO) for Germany, and the El Nino/Southern Oscillation (ENSO) for the other regions studied, play a non-negligent role as an external triggering climate signal on the various hydro-meteorological indices investigated.

Overview

1. INTRODUCTION1.1 Climate change: Global observations and predictions 1.2 Climate change: Recent observations and controversies 1.3 Climate change: Regional predictions /Global/Europe/Germany

2. VARIABILITY OF HYDRO-METEOROLOGICAL TIME SERIES IN GERMANY

2.1 Observational indicators of possible climate change in Germany over the last century2.2 Variability of time-series and their teleconnections with large-scale circulation pattern

3. METHODS

4. VARIABILITY ANALYSIS OF 20th CENTURY PRECIPITATION OVER GERMANY4.1 Study area and data4.2 Wavelet analysis of precipitation pattern 4.3 Estimation of Hurst parameter and analysis of long-term memory with DFA4.4 Correlation of the precipitation with NAO

5. ANALYSIS OF ELBE BASIN CLIMATE AND RIVER FLOW 5.1 Motivation and approach5.2 General trends of climate variables 5.3 Modes of NAO- / AO-Indices5.4 Correlations between climate variables and NAO- / AO-Index5.5 Wavelet analysis of Elbe river discharge 5.6 Temporal variability modes (SSA and DFA)5.7 Temporal changes of the Hurst parameter (longe-range memory)

6. CONCLUSIONS

1. INTRODUCTION

1.1 Climate change: Observations and predictions

1. INTRODUCTION

1.2 Climate change: Recent observations and controversies

First two EOF modes of SSA Analysis (Stockwell, 2008)

SSA Prediction of future trend (Stockwell, 2008)

All EOF modes of SSA Analysis (Stockwell, 2008)

1. INTRODUCTION

1.3 Climate change: Regional predictions /Global

* Most GCMs predict a strong change in rainfallover the the Northern hemisphere with wetterwinters and drier summers in Northern Europe

• GCM models predict increases in both frequencyand intensity of heavy rainfall events underenhanced greenhouse conditions

• In Northern Europe changes will cause a 10 to 30 percent increase in the magnitude of rainfall eventsup to a 50 year return period by end of the century

• One of the most significant impacts of such changes may be on hydrological processes and, particularly, river flow

• Changes in seasonality and an increase in low and high rainfall extremes, such as the droughts of the1990s and floods of 2000/01 can severely affect thewater balance of river basins.

* Will influence the rate of available water resourcesand the frequency of flooding and ecologicallydamaging low-flows.

Summer precipitation change in % for2070-2100 relative to 1961-1990

1. INTRODUCTION

1.3 Climate change: Regional predictions /Europe

Changes in annual precipitation, 1850- 2003 (Baurcurves) smoothed with a 11- year window.

Changes in annual mean temperature, 1750-2003smoothed with a 11- year window

1. INTRODUCTION

1.3 Climate change: Regional predictions /Germany

Regional Climate Model REMO (Meterological Institute, Hamburg, Germany)

Temperature changes Winter precipitation Summer precipitation

2071-2100 relative to 1961-1990

1. INTRODUCTION

1.3 Climate change: Regional predictions /Germany


2.1 Observational indicators of possible climate change in Germany over the last century

Changes in annual precipitation, 1850- 2003 (Baurcurves) smoothed with a 11- year window.

Changes in annual mean temperature, 1750-2003smoothed with a 11- year window


2.2 Variability of time series and their teleconnections with large scale circulation pattern

Changes in annual precipitation, 1850- 2003 smoothed with a 11-year window

Changes in winter month NAO-index (sea- level pressure difference between Azores-anticyclone and Island-depression), 1860-2005 (Hurrel, 1995), smoothed with a 5-year window

3. METHODS

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(Torrence C., Compo, Bull. Amer. Met. Soc., 1998)

Detrended Fluctuation Analysis, DFA (Determination of the Hurst-scaling exponent H)(Hu K., Ivanov P. Chen Z., Carpena P., Stanley H. E., Phys. Rev. E 64, 2000)

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Singular Spectrum Analysis, SSA (Identification of variability modes)

(Golyandina et al., Chapman & Hall, 2001)

Scale averaged wavelet spectrum

Global wavelet spectrum

Local wavelet coefficients

Fluctuation function

H < 0.5: antipersistence

H = 0.5: Gaussian (Brownian noise)

H > 0.5: long-memory, fBn

H > 1: fBm, nonstationary

ρ(k) ~L(k) k -α , as k→∞ , α = 2- 2HAutocorrelation function

4. VARIABILITY ANALYSIS OF 20th CENTURY PRECIPITATION OVER GERMANY

4.1 Study area and data

German precipitation stations used in the study, with station heights indicated.

Germany location map


4.2 Wavelet analysis of precipitation pattern

Precipitation series and wavelet scalograms for station Schwerin, split into two time spans 1890-1940 (left) and 1950-2000 (right). Thick black-line contours in the scalograms delineate the 95% -levels of statistical significance with respect to the red-noise hypothesis.

Schwerin Schwerin



Precipitation series and wavelet scalograms

Hamburg

HohenpeißenbergHamburg

Hohenpeißenberg



Global wavelet spectrum for Hamburg. Solid thick line is the overall global spectrum, other ones represent global wavelet spectra for the specified data time windows

In the early 20th century monthly precipitation in Hamburg is characterized by high-frequency oscillations, with periods of 1, 2 and 4 years, while from the mid 20th century on, only the 1-year oscillation remains significant, and a smaller 21-year oscillation peak is recognizable. Similar for most of the other stations

Global wavelet spectrum for Hannover and Potsdam

Hamburg

Dominant periods obtained with the wavelet analysis and the Hurst parameters computed with the DFA. Vertical bars denote the portion of the variance explained by the dominant oscillations in the time-series with periods T= 3-7 yr, T=7-8 yr and T> 10 yr. Colour of the squares denotes the range of the DFA-estimated H.


4.4 Estimation of Hurst parameter and analysis of long-term memory with DFA

Note a positive correlation of H >0.5

(long-term memory) with periods

of dominant oscillations, particularly

in northern Germany

� Effects of NAO with interdecadal

dominant periods


4.4 Correlation of the precipitation with NAO

Wavelet spectrum for the winter NAO index (1860-1999)

Cross-wavelet spectrum for monthly extremal precipitation (Hannover) and the NAO index 1960-1999.

Global wavelet spectra of the NAO (green line) and the AO

5. ANALYSIS OF ELBE BASIN CLIMATE AND RIVER FLOW

5.1 Motivation, data and approach

Data:

Climate time series (1951-2000)

(P, T, p, h, R, C, W)

Discharge time series

NH circulation indices (NAO, AO)

Elbe river basin with topography


5.2 General trends of climate variables

1950-2000 trends of major climate indices within the Elbe basin


5.3 Modes of NAO- / AO-Indices

Normalized global wavelet spectra of NAO-index (solidgreen line) and AO-Index (solid black line) with the corresponding 95% confidence levels assuming a red noise background (dashed lines).

SAWS of the NAO-Index (solid green line) and the AO-Index (solid black line) with the corresponding 95% confidence level assuming a red noise background: a) 2-14 yr. SAWS; b) 6-14 yr. SAWS.

2-14 yr. SAWS

6-14 yr. SAWS


5.4 Correlations between climate variables and NAO- / AO-Indices

NAO-Index/Precipitation AO-Index/Precipitation

NAO-Index/clouds AO-Index/clouds


5.5 Wavelet analysis of Elbe river discharge

Monthly extremal discharge time-series and wavelet spectra for Dresden and Neu Darchau. Black contours around the peaks denote the 95% confidence levels against red noise.


5.5 Wavelet analysis of Elbe river discharge

Normalized global wavelet spectra for 8 Elbe River gages:

green dashed line corresponds to southern gage (Dresden)and thick red line to northern gage (Neu Darchau)

Average normalized global wavelet power at scales s>2yr. for mean monthly discharge series versus gage altitude.

Scale-averaged wavelet spectra (SAWS) for Dresden for the large-scale band (sj = 1.8; sn = 15.6 yr) (solid line) and the annual-cycle band (sj = 0.97; sn = 1.01 yr.) (dashed line).


5.6 Temporal variability modes (SSA and DFA)

Major variability modes of extreme monthly basin precipitation and Elbe River discharge at Neu Darchau


5.6 Temporal variability modes (SSA and DFA)

SSA modes (Dresden 1852-2000, max. monthly flows).


5.7 Temporal changes of the Hurst parameter (long-range memory)

Note a continuous increase of the Hurst parameter H from about H~0.5 (white noise) at the

beginning of the 20th century, to H> 0.80 at its end, with values for the average H for the

total record somewhere in between. Whereas H up to the middle of the last century is

0.71<H<0.76, it jumps up to H >0.80 after that time.

�� Hypothesis of a sizeable change in the scaling- and long-range-dependence property of

the Elbe discharge somewhere in the middle of the last century should be accepted.

6. CONCLUSIONS

• Although hydro-meteorological time series are essentially stochastic in nature, their variability exhibits somewhat deterministic trends and periodicities acting on different time scales.

• The separation of these deterministic features from the random noise is of fundamental importance in climatology but subject to a huge amount of controversy (e.g. the “Hockey”-curve).

• Methods of stochastic time series analysis permit the decomposition of a series in (1) a trend, (2) periodicities and (3) noise, but different methods show different results which puts an unbiased interpretation in question

• Application of the various stochastic analysis methods to the analysis of possible climate change in Germany over the 20th century with respect to alterations in precipitation and streamflow is carried out.

• Wavelet analysis results in the frequency- time localization of dominant oscillations in the hydro-climate indices as well as in the NAO index on the decadal scale, particularly in the second half of the 20th century.

• Similarity of the NAO index variability with that of the precipitation and with that of the Elbe river discharge itself is a clear indicator that German long-term hydro-climate is affected remotely by this hemispherical index.

• Long-range memory of the hydro-climatic time-series is determined from the computation of the Hurst parameter H and reveal long- term persistence of precipitation across the country and of Elbe river discharge.

• H turns out to be time-dependent, with a continuous increase of H over the studied time intervals, from about H~0.5 (white noise) at the beginning, to H> 0.80 (long-range memory) at the end of the 20th century.

• Notwithstanding uncertainties in the exact time-localization of these nonstationarities of H, the idea of a certain change of the Elbe discharge’s scaling property, occurring in the middle of the last century is to be accepted.

• Not clear whether this is just an inter-decadal intermittency phenomenon or a hint of a long-term climatic trend.

• Presently applications of the various techniques to other regions in the world (Thailand, Peru, Ethopia, Hongkong)

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