Applied Harmonic Analysis

1. Applied Harmonic Analysis Do changes in orbital forcing drive the ice ages? and Do we understand how?

2. What is the question?

3. So what are main frequencies that we would expect to effect the Earth's climate? • 413kyrs (also eccentricity) • 100kyrs (really 95 and 125 kyrs) • 41kyrs • 19, 22 and 24 kyrs

4. What data do we have? • Specmap 18O data • What does 18O tell us?

5. What data do we have? • July 65N sunlight heat flux data • Why focus on high latitude summer light? • No real difference from signal elsewhere in high latitudes for purposes of this fit.

6. What are our questions? • With this, we ask: • Do the frequencies which dominate orbital cycles dominate past climate? • Are the strengths of the orbital variations in sunlight at the various frequencies equivalent to the strengths of the fluctuations of climate at those frequencies?

7. First, you should just look at the data, before doing statistics • But these are really important questions, so lets be more formal!

8. For the frequencies • 413kyrs (also eccentricity) • 125 kyrs • 95 kyrs • 41kyrs • 19, 22 and 24 kyrs • We want to fit y=C+A1cos(1x)+B1sin(1x)+ A2cos(2x)+B2sin(2x)+… • Why cos and sin? • Do matrix on board.

9. So here is my matlab; notice the automation %define frequencies to be included periods=[1/0.0025 1/0.008 1/0.0105 1/0.024 1/0.042 1/0.045 1/0.053]; angfreq=2*pi./periods; %do fitting %data is our y %date our x E=zeros([length(data),1+2*length(angfreq)]); E(:,1)=1; %why? for n=1:length(angfreq) E(:,2*n)=cos(angfreq(n)*date); E(:,2*n+1)=sin(angfreq(n)*date); end P=E\data; fit=E*P;

10. So what do we learn? • Much of the data is fit by these parameters • Correlation between fit and data 0.76 • Lots of unfit data (residual large)

11. When can we say a fit is succesful? • What is null model? • How many parameters for a perfect fit? • think about fitting two points with a line, etc… • should number of parameters be << N? • No, << than D.O.F.! • How many D.O.F.?

12. You can make fake data • How do you make fake data? • Same decorrelation time scale, based on data? • Results? r=0.63 +- 0.168, • Real correlation 0.75 • Not very robust!

13. So is there no link between orbital parameters and glacial climate? • Not entirely; more next class. • But for now, it is worth pointing out phase coherence. • But still not as strong a link as people make out. • However, lack of significance does not prove something wrong; it could just indicates a lack of data.

14. What are results for July 65N sunlight? • This fit is much better! • And there are more degrees of freedom! • Correlation =0.90 • How could we make fit better?

15. What are our questions? • With this, we ask: • Do the frequencies which dominate orbital cycles dominate past climate? • Are the strengths of the orbital variations in sunlight at the various frequencies equivalent to the strengths of the fluctuations of climate at those frequencies? • It seems that there variations at astronomical frequencies explains a significant amount of variation in past climate. • This is not as secure a link as commonly assumed. • Why I think it is reasonable. • However, lack of significance does not prove something wrong; it could just indicates a lack of data.

16. What are our questions? • With this, we ask: • Do the frequencies which dominate orbital cycles dominate past climate? • Are the strengths of the orbital variations in sunlight at the various frequencies equivalent to the strengths of the fluctuations of climate at those frequencies? • How do we summarize strength of A*cos(t)+B*sin(t)? • Talk about normalization. • why? • normalize 125kyr signal to 1

17. Results • Consistent with plots • Not consistent with naïve theory!

18. Conclusions • There is some evidence that climate responds to orbital cycles • This will become more clear next week. • However, the earth system must significantly alter the response, and must do so preferentially on a time scale of 100kyr. • How? Big mystery.