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

SILSOE RESEARCH INSTITUTE

Using the wavelet transform to elucidate complex spatial

covariation of environmental variables

Murray Lark


Geostatistical analysis:

Our data are realizations of coregionalized random variables, Zu(x) and Zv(x) with auto– and cross–variograms:


0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

From Atteia et al. (1984)


0

400

800

1200

0 0.5 1 1.5 2

Zn-Zn

0

40

80

120

0 0.5 1 1.5 2

Ni-Ni

0

0.4

0.8

1.2

Sem

ivari

ance

0 0.5 1 1.5 2

Cd-Cd

0

1

2

3

4

5

6

Cro

ss-s

em

ivari

ance

0 0.5 1 1.5 2

Cd-Ni

0

5

10

15

20

25

0 0.5 1 1.5 2

Cd-Zn

0

50

100

150

200

250

0 0.5 1 1.5 2

Ni-Zn

Lag distance /km



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.


The wavelet transform.

The basis functions (wavelets) have a narrow support and so provide a local analysis


A complete analysis is obtained by translation and dilation of a basic (mother) wavelet






Using the Adapted Maximal Overlap DiscreteWavelet Transform (Lark and Webster, 2001).



AMODWT partitions variance andcovariance by scale.



-1.0

-0.5

0.0

0.5

1.0

Wav

ele

t co

rrela

tio

n

0 50 100 150 200 250 Scale parameter /m

Wavelet correlations of N2O emissions andsoil organic carbon content


-1.0

-0.5

0.0

0.5

1.0

Wav

ele

t co

rrela

tio

n

0 50 100 150 200 250 Scale parameter /m

Wavelet correlations of N2O emissions andsoil pH


0 50 100 150 200 250 Position

256 m

128 m

64 m

32 m

16 m

8 m

N2O emission rate

Soil OC content


0 50 100 150 200 250 Position

256 m

128 m

64 m

32 m

16 m

8 m

N2O emission rate as measured

N2O emission rate predicted by a mechanistic model


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.

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

Documents

silsoe research institute

local analysis slide

soil ph slide

mechanistic model slide

basic mother wavelet

o emissions

measured n

complete analysis