List of Figures

1.1 Documentary data: floods of the river Elbe during winter over the past 1000 years 8

1.2 Marine sediment core data: 518O record from Ocean Drilling Program (ODP) site 846 (eastern equatorial Pacific) within 2-4 Ma 9

1.3 Ice core data: deuterium and CO2 records from the

Vostok station (Antarctica) over the past 420,000 years 10

1.4 Ice core data: sulfate record from the NGRIP core (Greenland) over the interval from ~ 10 to ~ 110 ka 11

1.5 Ice core data: Ca concentration, dust content, electrical conductivity and Na concentration from the NGRIP core (Greenland) during the onset of Dansgaard-Oeschger (D-O) event 5 12

1.6 Tree-ring data: record of atmospheric radiocarbon content over the past 12,410 years 13

1.7 Speleothem data: oxygen isotope record from stalagmite Q5 from southern Oman over the past 10,300

years 14

1.8 Lake sediment core data: varve thickness record from Lower Mystic Lake (Boston area) over the past 1000 years 15

1.9 Climate model data: runoff from Arctic rivers 16

1.10 Measured data: surface air temperature records from Siberia and North Atlantic 17

1.11 Statistical noise distributions of selected climate time series 18

1.12 Persistence of noise in selected climate time series 19

1.13 Sampling of time series from climate archives 21

1.14 Plain-light photomicrograph from a polished section of stalagmite S3 from southern Oman 22

1.15 Spacing of selected climate time series 23

2.1 Realization of an AR(1) process 34

2.2 Autocorrelation function of the AR(1) process 35

2.3 Monte Carlo study of the bias in the autocorrelation estimation of an AR(1) process, known mean, uneven spacing 38

2.4 Regions of asymptotic stationarity for the AR(2)

process 40

2.5 Realization of an AR(2) process 40

2.6 Realization of a SETAR(2; 1,1) process 44

2.7 Detrended Fluctuation Analysis for temperature records from Siberia and North Atlantic 50

2.8 River network 53

2.9 Long-memory parameter in dependence on basin size, river Weser 54

2.10 Effective data size, mean estimation of an AR(1)

process 56

2.11 Monte Carlo study of the bias in the autocorrelation estimation of an AR(1) process, unknown mean, uneven spacing 58

2.12 Group sunspot number, 1610-1995 61

3.1 Standard error, bias and equi-tailed confidence interval 67

3.2 Lognormal density function 73

3.3 Bootstrap principle for constructing confidence intervals 77

3.4 Determination of mean CO2 levels in the Vostok record during a glacial and an interglacial 91

3.5 Hypothesis test and confidence interval 93

4.1 Linear regression models fitted to modelled Arctic river runoff 116

4.2 GLS versus OLS standard errors of linear regression estimators 120

4.3 Linear timescale model 133

4.4 Two-phase linear timescale model 135

4.5 The ramp regression model 142

4.6 Ramp regression of the marine ¿1SO record ODP 846 145

4.7 Onset of Dansgaard-Oeschger event 5, NGRIP ice core: result 148

4.8 Onset of Dansgaard-Oeschger event 5, NGRIP ice core: estimated change-points with confidence intervals 149

4.9 Onset of Dansgaard-Oeschger event 5, NGRIP ice core: sedimentation rate and ¿18O variations 149

4.10 Onset of Dansgaard-Oeschger event 5, NGRIP ice core: estimated durations with confidence intervals 149

4.11 The break regression model 150

4.12 Break change-point regression fitted to modelled

Arctic river runoff 152

4.13 Nonparametric regression of the sedimentation rate in the Vostok record 155

4.14 Nonparametric regression of the atmospheric radiocarbon record from tree-rings 155

4.15 Outlier detection 158

4.16 Extremes detection in the NGRIP sulfate record 160

4.17 Extremes detection in the Lower Mystic Lake varve thickness record 161

4.18 Trend estimation for the ¿18O record from stalagmite Q5 165

4.19 Regression models for trend estimation 167

4.20 Climate trend function comprising many jumps 168

5.1 Spectrum of the AR(1) process 180

5.2 Spectrum of the AR(2) process 181

5.3 Spectrum types 182

5.4 Welch's overlapped segment averaging 187

5.5 Tapers for spectral estimation 189

5.6 Radiocarbon spectrum, multitaper estimation 196

5.7 Bias of the Lomb-Scargle periodogram 198

5.8 Monsoon spectrum, Lomb-Scargle estimation 206

5.9 Group sunspot number spectrum 207

5.10 Monsoon spectrum, test for aliasing 212

5.11 Monsoon spectrum, influence of timescale errors 214

5.12 Wavelet 218

6.1 Distribution of the maximum of k independent standard normal variates 233

6.2 Block maxima, POT data, GEV and GP distributions 236

6.3 Elbe summer floods 1852-1999, GEV estimation applied to block maxima 242

6.4 Elbe winter floods, pseudodata generation 252

6.5 Elbe winter floods, cross-validation function 253

6.6 Elbe winter floods, bandwidth selection 254

6.7 Elbe winter floods, occurrence rate estimation 256

6.8 NGRIP sulfate record, volcanic activity estimation 258

6.9 Lower Mystic Lake varve thickness record, hurricane activity estimation 259

6.10 Density functions used in Monte Carlo experiment 260

6.11 Estimation area for extreme value time series 265

7.1 Elbe runoff 1899-1990, time series 309

7.2 Elbe runoff 1899-1990, correlations 310

7.3 Binning for correlation estimation in the presence of unequal timescales 313

7.4 Monte Carlo study of correlation estimation, generation of unequal timescales 317

7.5 Monte Carlo study of correlation estimation in the presence of unequal timescales, dependence on sample size 318

7.6 Monte Carlo study of correlation estimation in the presence of unequal timescales, dependence on persistence times 319

7.7 Monte Carlo study of synchrony Pearson's correlation coefficient for unequal timescales, dependence on percentage 320

7.8 Vostok deuterium and CO2 over the past 420 ka, correlation 322

7.9 Binormal probability density function: contour lines and marginal distributions 324

7.10 Solar cycle length and northern hemisphere land surface-air temperature anomalies, 1866-1985 334

8.1 Linear errors-in-variables regression model, OLS estimation 342

8.2 Linear errors-in-variables regression model, WLSXY

and OLS estimations 344

8.3 Geometric interpretation of WLSXY 345

8.4 Wald-Bartlett procedure 346

8.5 Pairwise-MBBres algorithm, definition of residuals 348

8.6 Northern hemisphere temperature anomalies and climate forcing, 1850-2001: data. 360

8.7 Northern hemisphere temperature anomalies and climate forcing, 1850-2001: fit 361

8.8 Bermuda air temperature and coral 518O,

1856-1920: data 365

8.9 Bermuda air temperature and coral 518O,

1856-1920: prediction 366

8.10 Vostok deuterium and CO2, timescales for lag estimation 369

8.11 Vostok deuterium and CO2, reduced sum of squares 370

8.12 Vostok deuterium and CO2, parabolic fit 371

8.13 Vostok deuterium and CO2, sensitivity study of lag estimation error 372

9.1 Hyperspace of climate parameter estimation 393

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