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 < ψxy < 1

2. Ensenada de la Paz

3. Current velocity in Ensenada de La Paz
Year Day

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

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

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

9. Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz
 w1w2 L

10. Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz

11. Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz

12. Coherence spectrum (squared coherency or coherence-squared function)
X is the Fourier Transform
< > = running mean
 = 0.1, 0.05, 0.01

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

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

15.  = 0.1, 0.05, 0.01
window size= 25

16. JAMES RIVER DATA

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

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

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

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

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

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