1. SILSOE RESEARCH INSTITUTE Using the wavelet transform to elucidate complex spatial covariation of environmental variables Murray Lark

2. SILSOE RESEARCH INSTITUTE Geostatistical analysis: Our data are realizations of coregionalized random variables, Z u (x) and Z v (x) with auto– and cross– variograms:

3. SILSOE RESEARCH INSTITUTE From Atteia et al. (1984)

4. SILSOE RESEARCH INSTITUTE Lag distance /km

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6. Assumptions intrinsic stationarity, including the requirement that the variogram may be defined as a function of lag only: A motivation for considering the wavelet transform.

7. SILSOE RESEARCH INSTITUTE The wavelet transform. The basis functions (wavelets) have a narrow support and so provide a local analysis

8. SILSOE RESEARCH INSTITUTE A complete analysis is obtained by translation and dilation of a basic (mother) wavelet The wavelet transform.

9. SILSOE RESEARCH INSTITUTE The wavelet transform.

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11. Using the Adapted Maximal Overlap Discrete Wavelet Transform (Lark and Webster, 2001).

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13. AMODWT partitions variance and covariance by scale.

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15. Wavelet correlations of N 2 O emissions and soil organic carbon content

16. SILSOE RESEARCH INSTITUTE Wavelet correlations of N 2 O emissions and soil pH

17. SILSOE RESEARCH INSTITUTE N 2 O emission rate Soil OC content

18. SILSOE RESEARCH INSTITUTE N 2 O emission rate as measured N 2 O emission rate predicted by a mechanistic model

19. SILSOE RESEARCH INSTITUTE Conclusions. 1.The wavelet transform allows us to identify scale- and location-dependency in the relationships between variables. 2.No assumptions of stationarity are invoked. 3.The analysis can give insight into spatially complex relationships and into the performance of process models.