Nonstationarities in Hydrologic and Environmental Time Series

1. Introduction.- 2. Data Used in the Book.- 2.1. Hydrologic and Climatic Data.- 2.2. Synthetic and Observed Environmental Data.- 2.2.1. Synthetic Data Sampling from Batchelor Spectrum.- 2.2.2. Details of Data Generated by Sampling from the Batchelor Spectrum.- 2.2.3. Synthetic Data from AR Model.- 2.3. Observed Data.- 2.3.1. Measured Temperature Gradient Profiles.- 3. Time Domain Analysis.- 3.1. Introduction.- 3.2. Visual Inspection of Time Series.- 3.3. Statistical Tests of Significance.- 3.3.1. Parametric Tests.- 3.3.2. Non-parametric Tests.- 3.4. Testing Autocorrelated Data.- 3.5. Application of Trend Tests to Hydrologic Data.- 3.5.1. Visual Inspection of Data.- 3.5.2. Statistical Trend Tests.- 3.5.3. Sub-period Trend Analysis.- 3.6. Conclusions.- 4. Frequency Domain Analysis.- 4.1. Introduction.- 4.2. Conventional Spectral Analysis.- 4.3. Multi-Taper Method (MTM) of Spectral Analysis.- 4.4. Maximum Entropy Spectral Analysis.- 4.5. Spectral Analysis of Hydrologic and Climatic Data.- 4.5.1. Results from MEM Analysis.- 4.5.2. Results from MTM Analysis.- 4.6. Discussion of Results.- 4.7. Conclusions.- 5. Time-Frequency Analysis.- 5.1. Introduction.- 5.2. Evolutionary Spectral Analysis.- 5.3. Evolution of Line Components in Hydrologic and Climatic Data.- 5.4. Evolution of Continuous Spectra in Hydrologic and Climatic Data.- 5.5. Conclusions.- 6. Time-Scale Analysis.- 6.1. Introduction.- 6.2. Wavelet Analysis.- 6.3. Wavelet Trend Analysis.- 6.4. Identification of Dominant Scales.- 6.5. Time-Scale Distribution.- 6.6. Behavior of Hydrologic and Climatic Time Series at Different Scales.- 6.7. Conclusions.- 7. Segmentation of Non-Stationary Time Series.- 7.1. Introduction.- 7.2. Tests based on AR Models.- 7.2.1. Test 1 (de Souza and Thomson, 1982).- 7.2.2. Test 2 (Imberger and Ivey, 1991).- 7.2.3. Test 3 (Davis, Huang and Yao, 1995).- 7.2.4. Test 4 (Tsay, 1988).- 7.3. A test based on wavelet analysis.- 7.4. Segmentation algorithm.- 7.5. Variations of test statistics with the AR order p.- 7.6. Sensitivity of test statistics for detecting change points.- 7.6.1. Detection results for synthetic series from model 2.1.2.- 7.6.2. Detection results for synthetic series from model 2.1.3.- 7.6.3. Detection results for synthetic series from model 2.1.4.- 7.6.4. Detection results for synthetic series from model 2.1.5.- 7.6.5. Conclusions on performances of tests 1–5.- 7.7. Performances of algorithms with and without boundary optimization.- 7.7.1. Detection of non-stationary segment.- 7.7.2. Detection of multi-segment series.- 7.8. Conclusions about the segmentation algorithm.- 8. Estimation of Turbulent Kinetic Energy Dissipation.- 8.1. Introduction.- 8.2. Multi-taper Spectral Estimation.- 8.3. Batchelor Curve Fitting.- 8.4. Comparison of Spectral Estimation Methods.- 8.5. Batchelor Curve Fitting to Synthetic Series.- 8.5.1. Batchelor curve fitting using the first error function.- 8.5.2. Batchelor curve fitting using the second error function.- 8.5.3. Batchelor curve fitting using the third error function.- 8.6. Conclusions on Batchelor curve fitting.- 9. Segmentation of Observed Data.- 9.1. Introduction.- 9.2. Temperature Gradient Profiles.- 9.2.1. Ratios of Unresolved, Bad-Fit and Good-Fit Segments.- 9.2.2. Estimated Values of ? and XT from Resolved Spectra.- 9.2.3. Estimated Values of ? and XT from Profiles in the Same Lake.- 9.2.4. Estimated Values of ? and XT from Different Lakes.- 9.3. Conclusions on Segmentation of Temperature Gradient Profiles.- 9.4. Hydrologic Series.- 9.4.1. Stationary Segments from Hydrologic Series.- 9.4.2. Change Points in Hydrologic Series.- 9.5. Conclusions on Segmentation of Hydrologic Series.- 10. Linearity and Gaussianity Analysis.- 10.1. Introduction.- 10.2. Tests for Gaussianity and Linearity (Hinich, 1982).- 10.3. Testing for Stationary Segments.- 10.3.1. Testing Temperature Gradient Profiles.- 10.3.2. Testing Hydrologic Series.- 10.4. Conclusions about Testing the Hydrologic Series.- 11. Bayesian Detection of Shifts in Hydrologic Time Series.- 11.1. Introduction.- 11.2. Data Used in this Chapter.- 11.3. A Bayesian Method to Detect Shifts in Data.- 11.3.1. Theory.- 11.3.1.1. Parameters of the distribution and the change point n1.- 11.3.1.2. The Unconditional Posterior Distributions of ?, ? and ?.- 11.3.1.3. The Conditional Posterior Distributions of ?i, ?21 and ?i.- 11.3.2. Computation Sequences.- 11.4. Discussion of Results.- 11.4.1. The Posterior Distribution of the Change point n1.- 11.4.2. The Unconditional Posterior Distributions of ?, ? and ?.- 11.4.3. The Conditional Posterior Distributions of ?i,?2i and ?i.- 11.5. Conclusions.- 12. References.- 13. Index.