1. The Fourier transform

2. Properties of Fourier transforms Convolution Scaling Translation

3. Parceval’s theorem Relates space integration to frequency integration. Decomposes variability.

4. Aliasing Observe field at lattice of spacing . Since the frequencies  and  ’=  +2  m/  are aliases of each other, and indistinguishable. The highest distinguishable frequency is , the Nyquist frequency.

5. Illustration of aliasing Aliasing applet

6. Spectral representation Stationary processes Spectral process Y has stationary increments If F has a density f, it is called the spectral density.

7. Estimating the spectrum For process observed on nxn grid, estimate spectrum by periodogram Equivalent to DFT of sample covariance

8. Properties of the periodogram Periodogram values at Fourier frequencies (j,k)  are uncorrelated asymptotically unbiased not consistent To get a consistent estimate of the spectrum, smooth over nearby frequencies

9. Some common isotropic spectra Squared exponential Matérn

10. A simulated process

11. Thetford canopy heights 39-year thinned commercial plantation of Scots pine in Thetford Forest, UK Density 1000 trees/ha 36m x 120m area surveyed for crown height Focus on 32 x 32 subset

12. Spectrum of canopy heights

13. Whittle likelihood Approximation to Gaussian likelihood using periodogram: where the sum is over Fourier frequencies, avoiding 0, and f is the spectral density Takes O(N logN) operations to calculate instead of O(N 3 ).

14. Using non-gridded data Consider where Then Y is stationary with spectral density Viewing Y as a lattice process, it has spectral density

15. Estimation Let where J x is the grid square with center x and n x is the number of sites in the square. Define the tapered periodogram where. The Whittle likelihood is approximately

16. A simulated example

17. Estimated variogram

18. Evidence of anisotropy 15 o red 60 o green 105 o blue 150 o brown

19. Another view of anisotropy

20. Geometric anisotropy Recall that if we have an isotropic covariance (circular isocorrelation curves). If for a linear transformation A, we have geometric anisotropy (elliptical isocorrelation curves). General nonstationary correlation structures are typically locally geometrically anisotropic.

21. Lindgren & Rychlik transformation