1. Cross-spectral Analysis Investigates the dependence or relationship between two time series. Begin by reviewing cross-correlation function – how similar two time series x, y are L = lag N = # of elements in time series Covariance or cross-covariance Cross-correlation -1 < ψ xy < 1

2. Ensenada de la Paz

3. Year Day Current velocity in Ensenada de La Paz

4. Current velocity and water level (red) in Ensenada de La Paz Year Day

5. Current velocity and water level (red) in Ensenada de La Paz Year Day

6. Cross-correlation between current velocity and water level in Ensenada de La Paz

8. A generalization of: * is the complex conjugate Complex Cross-correlation

9. L  w 1 w 2 Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz

12. Coherence spectrum (coherency)  = 0.1, 0.05, 0.01 X is the Fourier Transform = running mean

13. Current velocity and water level (red) in Ensenada de La Paz Year Day

14. cpd  = 0.1, 0.05, 0.01 Coherence squared window size= 2

15.  = 0.1, 0.05, 0.01

16. JAMES RIVER DATA

17. Coherence 2 Phase (deg) 95% 99% Window size= 10

18. Coherence 2 Phase (deg) 95% 99% Window size= 24

20. Coherence 2 Phase (deg) 95% 99% Window size= 10

21. Coherence 2 Phase (deg) 95% 99% Window size= 10

22. Coherence 2 Phase (deg) 95% 99% Window size= 24

23. Coherence 2 Phase (deg) 95% 99% Window size= 24