Nonstationarities in hydrologic and environmental time series

Conventionally, time series have been studied either in the time domain or the frequency domain. The representation of a signal in the time domain is localized in time, i.e . the value of the signal at each instant in time is well defined . However, the time representation of a signal is poorly localized in frequency , i.e. little information about the frequency content of the signal at a certain frequency can be known by looking at the signal in the time domain . On the other hand, the representation of a signal in the frequency domain is well localized in frequency, but is poorly localized in time, and as a consequence it is impossible to tell when certain events occurred in time. In studying stationary or conditionally stationary processes with mixed spectra , the separate use of time domain and frequency domain analyses is sufficient to reveal the structure of the process . Results discussed in the previous chapters suggest that the time series analyzed in this book are conditionally stationary processes with mixed spectra. Additionally, there is some indication of nonstationarity, especially in longer time series.

Author Notes

A. Ramachandra Rao: School of Civil Engineering, Purdue University, West Lafayette, Indiana
Khaled H. Hamed: Irrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Giza, Egypt
Huey-Long Chen: Department of Environmental Engineering, Lan-Yang Institute of Technology, Touchen, Ilan, Taiwan

1. Introduction	p. 2
2. Data Used in the Book
2.1. Hydrologic and Climatic Data	p. 4
2.2. Synthetic and Observed Environmental Data	p. 5
2.2.1. Synthetic Data Sampling from Batchelor Spectrum	p. 5
2.2.2. Details of Data Generated by Sampling from the Batchelor Spectrum	p. 18
2.2.3. Synthetic Data from AR Model	p. 22
2.3. Observed Data	p. 25
2.3.1. Measured Temperature Gradient Profiles	p. 25
3. Time Domain Analysis
3.1. Introduction	p. 27
3.2. Visual Inspection of Time Series	p. 28
3.3. Statistical Tests of Significance	p. 29
3.3.1. Parametric Tests	p. 29
3.3.2. Non-parametric Tests	p. 31
3.4. Testing Autocorrelated Data	p. 32
3.5. Application of Trend Tests to Hydrologic Data	p. 35
3.5.1. Visual Inspection of Data	p. 35
3.5.2. Statistical Trend Tests	p. 36
3.5.3. Sub-period Trend Analysis	p. 46
3.6. Conclusions	p. 54
4. Frequency Domain Analysis
4.1. Introduction	p. 56
4.2. Conventional Spectral Analysis	p. 57
4.3. Multi-Taper Method (MTM) of Spectral Analysis	p. 59
4.4. Maximum Entropy Spectral Analysis	p. 63
4.5. Spectral Analysis of Hydrologic and Climatic Data	p. 64
4.5.1. Results from MEM Analysis	p. 64
4.5.2. Results from MTM Analysis	p. 76
4.6. Discussion of Results	p. 99
4.7. Conclusions	p. 113
5. Time-Frequency Analysis
5.1. Introduction	p. 115
5.2. Evolutionary Spectral Analysis	p. 116
5.3. Evolution of Line Components in Hydrologic and Climatic Data	p. 120
5.4. Evolution of Continuous Spectra in Hydrologic and Climatic Data	p. 130
5.5. Conclusions	p. 155
6. Time-Scale Analysis
6.1. Introduction	p. 160
6.2. Wavelet Analysis	p. 160
6.3. Wavelet Trend Analysis	p. 162
6.4. Identification of Dominant Scales	p. 178
6.5. Time-Scale Distribution	p. 179
6.6. Behavior of Hydrologic and Climatic Time Series at Different Scales	p. 180
6.7. Conclusions	p. 209
7. Segmentation of Non-Stationary Time Series
7.1. Introduction	p. 213
7.2. Tests based on AR Models	p. 215
7.2.1. Test 1 (de Souza and Thomson, 1982)	p. 215
7.2.2. Test 2 (Imberger and Ivey, 1991)	p. 217
7.2.3. Test 3 (Davis, Huang and Yao, 1995)	p. 217
7.2.4. Test 4 (Tsay, 1988)	p. 220
7.3. A test based on wavelet analysis	p. 222
7.4. Segmentation algorithm	p. 225
7.5. Variations of test statistics with the AR order p	p. 228
7.6. Sensitivity of test statistics for detecting change points	p. 231
7.6.1. Detection results for synthetic series from model 2.1.2	p. 232
7.6.2. Detection results for synthetic series from model 2.1.3	p. 233
7.6.3. Detection results for synthetic series from model 2.1.4	p. 238
7.6.4. Detection results for synthetic series from model 2.1.5	p. 244
7.6.5. Conclusions on performances of tests 1-5	p. 245
7.7. Performances of algorithms with and without boundary optimization	p. 246
7.7.1. Detection of non-stationary segment	p. 246
7.7.2. Detection of multi-segment series	p. 247
7.8. Conclusions about the segmentation algorithm	p. 249
8. Estimation of Turbulent Kinetic Energy Dissipation
8.1. Introduction	p. 253
8.2. Multi-taper Spectral Estimation	p. 256
8.3. Batchelor Curve Fitting	p. 257
8.4. Comparison of Spectral Estimation Methods	p. 259
8.5. Batchelor Curve Fitting to Synthetic Series	p. 262
8.5.1. Batchelor curve fitting using the first error function	p. 263
8.5.2. Batchelor curve fitting using the second error function	p. 268
8.5.3. Batchelor curve fitting using the third error function	p. 269
8.6. Conclusions on Batchelor curve fitting	p. 276
9. Segmentation of Observed Data
9.1. Introduction	p. 277
9.2. Temperature Gradient Profiles	p. 277
9.2.1. Ratios of Unresolved, Bad-Fit and Good-Fit Segments	p. 277
9.2.2. Estimated Values of [varepsilon] and x[subscript T] from Resolved Spectra	p. 281
9.2.3. Estimated Values of [varepsilon] and x[subscript T] from Profiles in the Same Lake	p. 281
9.2.4. Estimated Values of [varepsilon] and x[subscript T] from Different Lakes	p. 282
9.3. Conclusions on Segmentation of Temperature Gradient Profiles	p. 282
9.4. Hydrologic Series	p. 301
9.4.1. Stationary Segments from Hydrologic Series	p. 301
9.4.2. Change Points in Hydrologic Series	p. 308
9.5. Conclusions on Segmentation of Hydrologic Series	p. 311
10. Linearity and Gaussianity Analysis
10.1. Introduction	p. 312
10.2. Tests for Gaussianity and Linearity (Hinich, 1982)	p. 312
10.3. Testing for Stationary Segments	p. 315
10.3.1. Testing Temperature Gradient Profiles	p. 316
10.3.2. Testing Hydrologic Series	p. 316
10.4. Conclusions about Testing the Hydrologic Series	p. 316
11. Bayesian Detection of Shifts in Hydrologic Time Series
11.1. Introduction	p. 320
11.2. Data Used in this Chapter	p. 321
11.3. A Bayesian Method to Detect Shifts in Data	p. 322
11.3.1. Theory	p. 322
11.3.1.1. Parameters of the distribution and the change point n[subscript 1]	p. 328
11.3.1.2. The Unconditional Posterior Distributions of [delta], [upsilon] and [gamma]	p. 330
11.3.1.3. The Conditional Posterior Distributions of [beta subscript i], [sigma superscript 2 subscript i] and [theta subscript i]	p. 332
11.3.2. Computation Sequences	p. 333
11.4. Discussion of Results	p. 335
11.4.1. The Posterior Distribution of the Change point n[subscript 1]	p. 335
11.4.2. The Unconditional Posterior Distributions of [delta], [upsilon] and [gamma]	p. 335
11.4.3. The Conditional Posterior Distributions of [beta subscript i], [sigma superscript 2 subscript i] and [theta subscript i]	p. 336
11.5. Conclusions	p. 346
12. References	p. 347
13. Index	p. 359

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