international summer school on turbulence diffusion 2006 multifractal analysis in b&w soil...

$: International Summer School on Turbulence Diffusion 2006 Multifractal Analysis in B&W Soil Images Ana M. Tarquis anamaria.tarquis@upm.es Dpto. de Matemática$
International Summer School on Turbulence Diffusion 2006

Multifractal Analysis in B&W Soil Images

Ana M. Tarquis

[email protected]

Dpto. de Matemática Aplicada

E.T.S.I. Agrónomos

Universidad Politécnica de Madrid


INDEX

• Problem: motivation and start point.

• Fractals and multifractals concepts.

• Porosity images: resolved?

• Configuration Entropy

• Griding Methods


CONSERVATION OF NATURAL RESOURCES

• Agriculture : soil degradation and water contamination.

• Sustainable agriculture

• Quantification of soil quality index?


Soil structure • Water, solutes and gas transport• Soil resistance• Roots morphology• Microorganism populations

PORE AND SOIL MATRIX GEOMETRY


• Fractal structure: structured distribution of pore (and/or soil) in the space such that at any resolution the set is the union of similar subset to the whole.


Measure techniques

• The number-size relation is used normally to measure the fractal dimension of the defined measure (number of white or black pixels), or counting objects:

• Or covering the object with regular geometric elements of variable size:

DN )(

DkrrN )(


“ Box-Counting”

-m = fractal dimension, D

1 n

Black, white or interface


• Multifractal analysis consider the number of black pixels in each box (pore density=m).


• Multifractal: density has an structured distribution in the space such that at any resolution the set is the union of similar subsets to the whole. But the scale factor at different parts of the set is not the same.

• More than one dimension is needed => the measure consider (M) is characterized by the union of fractal sets, each one with a fractal dimension.


1 n

q

δn

1ii

iqii

m

mPδq,

δlog

δ1,logδ1,limD

δn

1iii

0δ1

Dq

q

)log()],(log[

lim)( 0 q

q

)log()],(log[

lim)1(

1)1(

)(0

qqq

qDq


Numerical Analysis of Multifractal Spectrum on 2-D Black and White Images

p1 p2

p3 p4

p1 p2

p3 p4

p1p2 p1p4

p1p1 p1p3

p2p2

p2p4

p2p1

p2p3

p3p2

p3p4

p3p1

p3p3 p4p1p4p2

p4p1p4p4

p1p2 p1p4

p1p1 p1p3

p2p2

p2p4

p2p1

p2p3

p3p2

p3p4

p3p1

p3p3 p4p1p4p2

p4p1p4p4

1 2 3 … 8

256

1

8

1

}4,3,2,1{.i

ik

kji mmjwithpmm

m (number of black pixels)m (number of black pixels)

p1 p2

p3 p4

p1 p2

p3 p4

p1p2 p1p4

p1p1 p1p3

p2p2

p2p4

p2p1

p2p3

p3p2

p3p4

p3p1

p3p3 p4p1p4p2

p4p1p4p4

p1p2 p1p4

p1p1 p1p3

p2p2

p2p4

p2p1

p2p3

p3p2

p3p4

p3p1

p3p3 p4p1p4p2

p4p1p4p4

1 2 3 … 8

256

1

8

1

}4,3,2,1{.i

ik

kji mmjwithpmm

m (number of black pixels)m (number of black pixels)


RANDOM AND MULTIFRACTAL IMAGES

• In this way a hierarchical probability tree was built generating an image of 1024x1024 pixels (ten subdivisions), as the soil images are normally analyzed.

• Probabilistic parameters are: { p1, p2, p3, p4 }• Random images : p1= p2 = p3 = p4 = 25%• Multifractal images: p1= 50%, p2= 5%, p3= 25%

and p4= 20% (by random arrangements or not).


Random

multifractal


-40.0

-32.0

-24.0

-16.0

-8.0

0.0

8.0

16.0

-12 -8 -4 0 4 8 12

q

(q)

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

-12 -8 -4 0 4 8 12

q

Dq

p1=0.40 p2=0.10

p1=0.33 p2=0.17

Generalized dimensions (Dq) obtained for two different distributions based on Stanley and Meakin (1988) formulas with

their respective -(q) curves.


Most common parameters calculated

• D0 q=0 box counting dimension

• D1 q=1 entropy dimension

• D2 q=2 correlation dimension


Singularities of the measure ()

For a given there is a fractal dimension f() of the set that support the singularity. At each area the relation number-size is applied:

dqqd )(

)()( fN

f()

)())(()( qqqfq


f()

Multifractal Spectrum

wf

w


ADS 375X250 PIXELS

VOIDSCIRCULARPOLARIZED

TRANSMITTED


INTER DENNY

ABOK MUNCHONG

1500x1000 pixels


-120

-70

-20

30

80

130

0 1 2 3 4 5 6 7

log r

log

(r

,q)

-10

-8

-6

-4

-2

0

2

4

6

8

10

¿How many points?


1,9900

1,9950

2,0000

2,0050

2,0100

2,0150

2,0200

-11 -9 -7 -5 -3 -1 1 3 5 7 9 11

0,0000

0,5000

1,0000

1,5000

2,0000

2,5000

3,0000

3,5000

-11 -9 -7 -5 -3 -1 1 3 5 7 9 11

0,0000

1,0000

2,0000

3,0000

4,0000

5,0000

-11 -9 -7 -5 -3 -1 1 3 5 7 9 11

ADS

BUSO

EHV1


We have to compare

1,5000

2,0000

2,5000

3,0000

3,5000

4,0000

4,5000

5,0000

-11 -9 -7 -5 -3 -1 1 3 5 7 9 11

ADS

BUSO

EVH1


0,0%

10,0%

20,0%

30,0%

40,0%

50,0%

60,0%

1,50 1,55 1,60 1,65 1,70 1,75 1,80 1,85 1,90 1,95 2,00

D0

Po

rosi

ty (

%)

1,0%

11,0%

21,0%

31,0%

41,0%

51,0%

61,0%

1,1000 1,3000 1,5000 1,7000 1,9000D1

Po

r. (

%)

1,0%

11,0%

21,0%

31,0%

41,0%

51,0%

61,0%

1,1000 1,3000 1,5000 1,7000 1,9000 2,1000

D2

Po

r. (

%)


Obtaining Dq por. q=-3

-20,0000

0,0000

20,0000

40,0000

60,0000

0,0000

2,0000

4,0000

6,0000

8,0000

log(r)

log(

S(r

))

lower

upper

por. q=0

-5,0000

0,0000

5,0000

10,0000

15,0000

0,0000

2,0000

4,0000

6,0000

8,0000

log(r)

log(

S(r

))

lower

upper

por. q=1

-15,0000

-10,0000

-5,0000

0,0000

5,0000

0,0000

2,0000

4,0000

6,0000

8,0000

log(r)

log(

S(r

))

lower

upper

Ehv1, porosity 46,7%


Calculating Dq

ADS, porosity 5,7%por. q=-3

-20,000000

-10,000000

0,000000

10,000000

20,000000

30,000000

40,000000

50,000000

0,000000 2,000000 4,000000 6,000000 8,000000

log(r)

log

(S(r

))

lower

upper

por. q=0

-4

-2

0

2

4

6

8

10

12

0,000000 2,000000 4,000000 6,000000 8,000000

log(r)

log

(S(r

))

lower

upper

por. q=1

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

0,000000 2,000000 4,000000 6,000000 8,000000

log(r)

log

(S(r

))

lower

upper


scale and put stde

multifractal? Yes, quite sure 10 points 10 points

difference in the difference in the number of points number of pointsfor the regression line for the regression line

multifractal? No, quite sure 8 points 8 points

scale and put stde

1,74

1,76

1,78

1,80

1,82

1,84

1,86

1,88

-5 -3 -1 1 3 5

q

Dq

1,93

1,94

1,95

1,96

1,97

1,98

1,99

2,00

-5 -3 -1 1 3 5q

Dq

1,601,651,701,751,801,851,901,952,002,05

-5 -3 -1 1 3 5q

Dq

1,601,701,801,902,002,102,202,302,40

-5 -3 -1 1 3 5q

Dq


Continuos line = random structure

Dashed line = mfract structure

Filled Square = values from image soils

1,0%

11,0%

21,0%

31,0%

41,0%

51,0%

61,0%

1,1000 1,3000 1,5000 1,7000 1,9000 2,1000

D0

Po

ros

ity

(%)

1,0%

11,0%

21,0%

31,0%

41,0%

51,0%

61,0%

1,1000 1,3000 1,5000 1,7000 1,9000 2,1000

D1

Po

ros

ity

(%)

1,0%

11,0%

21,0%

31,0%

41,0%

51,0%

61,0%

1,1000 1,3000 1,5000 1,7000 1,9000 2,1000

D2

Po

ros

ity

(%)

A

B

C


Considerations on Dq calculations

• Several authors have shown that the exact value of the generalized dimension is not an easy calculation to do . Vicsek proposed practical methods to compute the generalized dimension

• The main difficulty in using the multifractal formalism lies in the fact that the ideal limit cannot be reached in practice


RESULTS AND DISCUSSION (1)

• For all of the soil images with different porosity we obtain convincing straight-line fits to the data having all of them r2 higher than 0.98,


RESULTS AND DISCUSSION

• Finally, a comparison among the different images in each dimension is showed .

• In all of them, the points corresponding to porosities higher than 30% lie on the line representing the Dq calculated for the random generated images.

• Observing the difference between the fractal dimensions coming from multifractal and random images (discontinue line and continue line respectively) it is obvious that decreases when porosity increases in the images.


Configuration Entropy H()

The maximum value of j is x and the minimum value is 0 (Andraud et al., 1989)

1 i

n() = boxes of size from = 1 to = w /4

w

Nj = number of boxes with j black pixels inside



The probability associated with a case of j black pixels in a box of size (pj())

)(

)()(

n

Np jj

))(plog()(p ) ( j0j

j

x

H

) (

) ( ) (*

max

H

HH )1log( ) ( 2

max H



(pixels)

H*()

0

1

1 w/4

H*(L)

L


Methods: gliding, random walks, randomly

•Box size

•Jump step length

•Number of jumps


Thank you for your attention


Multifractal Analysis on a Matrix

Ana M. Tarquis

[email protected]

Dpto. de Matemática Aplicada

E.T.S.I. Agrónomos

Universidad Politécnica de Madrid


INDEX

• Field Percolation

• Soil Roughness

• Satellite images

• Time series


Z= 10 cm Z = 20 cm Z = 30 cm

Z = 40 cm Z = 50 cm Z = 60 cm


0

10

20

30

40

50

60

70

80

0 20 40 60 80 100Dye Tracer (%)

Dep

th (

cm

)

% of blue vs. depth

50%

15 cm


-120

-70

-20

30

80

130

0 1 2 3 4 5 6 7

log r

log

(r,q

)

-10

-8

-6

-4

-2

0

2

4

6

8

10

Z = 25 cm blue staining 28,95%


Dye Tracer Distribution

1

C1

0

50

100

150

200

250

Dye Tracer

X

Y


Multifractal Analysis of the Dye Tracer Distribution

1,830000

1,930000

2,030000

2,130000

2,230000

2,330000

-12 -9 -6 -3 0 3 6 9 12

qD

q

0,00

0,50

1,00

1,50

2,00

1,60 1,80 2,00 2,20 2,40 2,60

f(

)

A B

B) Generalized dimensions

A) f() spectrum


•Multispectral Satellite Images


0.45 m - 0.52 m

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

f(

)

Landsat-7 Ikonos (32 m)

0.52 m - 0.60 m

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

f(

)

Ikonos (32 m) Landsat-7

0.63 m - 0.69 m

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

f(

)


0.76 m - 0.90 m

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

f(

)



Soil Rougness• Roughness indices normally are based on

transects data. One of the most used is the Random Roughness (RR).

• RR is the standard deviation of the soil heights readings from the transect. This implies that there is not an spatial component.

• Several authors have applied fractal dimensions to this type of data. Burrough (1989), Bertuzzi et al. (1990), Huang and Bradford (1992),


INTRODUCTION

• The aim of this work is to study soil height readings with multifractal analysis in the context of soil roughness.

• Several soils, with different textures, with different tillage methods have been analysed to compare their multifractal spectrum.


Soil measurements

• Three different soils with different textures.

• Three different treatments applying tillage: chisel, moldboard, seedbeds.

• Height measures of 2x2 m2 plot area.

• Resolution of the measure each 2 cm


%Sand %Silt %ClayM.O. (%)

2.59

1.49 P_3

TextureCode

50.49 21.2 28.3

P_1

20.7 40.8 38.5 P_2

1.76 30.6 52.3 17.1

La Higueruela C.S.I.C.

La Higueruela C.S.I.C.

Place

E.T.S.I.A. Experimental fields

Soil texture


moldboard

seedbedschisel


moldboard

seedbeds

chisel


1

n

Box counting method

Number of boxes depends on i

20.4 20.2 13.0 58.7 20.5 21.9 22.7 22.4 22.1 21.1 22.0 21.0 22.4 21.6 37.5 21.2 20.520.8 20.0 20.6 20.6 20.0 14.8 22.3 23.4 22.7 21.5 21.9 41.3 22.2 22.0 22.2 21.3 37.127.5 19.3 20.3 22.7 20.9 21.4 21.4 23.3 22.6 21.8 21.5 21.1 21.5 21.9 22.2 22.4 20.822.1 19.3 20.1 21.6 22.0 21.5 22.0 21.9 22.4 21.5 21.6 22.2 21.6 22.1 22.6 22.3 22.123.5 19.0 19.4 20.0 21.5 20.3 23.2 20.5 22.4 22.1 21.4 21.6 21.2 21.1 21.4 22.0 22.123.0 19.1 18.3 19.5 19.8 19.8 19.9 25.0 22.0 21.8 22.1 21.1 21.0 21.9 21.6 22.0 22.620.3 18.9 20.3 19.2 19.4 19.5 19.4 24.6 23.1 21.6 22.8 21.6 21.0 22.7 21.5 22.4 22.719.7 18.8 19.4 18.8 18.3 18.2 19.9 20.1 25.5 22.9 22.4 72.9 20.4 21.3 21.8 22.9 22.519.1 18.4 19.5 19.3 18.8 18.1 18.5 18.9 18.5 24.3 23.1 23.9 19.9 20.2 22.1 22.2 22.718.2 18.4 17.9 19.8 18.9 18.4 18.7 18.6 18.7 18.6 24.5 21.7 20.0 20.5 21.9 21.8 22.317.2 18.5 17.8 18.4 19.3 18.9 18.1 18.8 19.0 18.4 18.6 18.7 20.4 21.4 21.6 21.4 22.217.3 17.1 17.7 17.8 19.4 19.0 18.4 19.6 19.3 19.5 18.3 18.2 18.3 20.2 23.2 21.0 21.917.8 17.2 17.2 17.9 18.9 19.1 19.2 19.3 19.0 18.2 19.5 18.3 18.9 13.0 22.3 20.8 21.0

20.4 20.2 13.0 58.7 20.5 21.9 22.7 22.4 22.1 21.1 22.0 21.0 22.4 21.6 37.5 21.2 20.520.8 20.0 20.6 20.6 20.0 14.8 22.3 23.4 22.7 21.5 21.9 41.3 22.2 22.0 22.2 21.3 37.127.5 19.3 20.3 22.7 20.9 21.4 21.4 23.3 22.6 21.8 21.5 21.1 21.5 21.9 22.2 22.4 20.822.1 19.3 20.1 21.6 22.0 21.5 22.0 21.9 22.4 21.5 21.6 22.2 21.6 22.1 22.6 22.3 22.123.5 19.0 19.4 20.0 21.5 20.3 23.2 20.5 22.4 22.1 21.4 21.6 21.2 21.1 21.4 22.0 22.123.0 19.1 18.3 19.5 19.8 19.8 19.9 25.0 22.0 21.8 22.1 21.1 21.0 21.9 21.6 22.0 22.620.3 18.9 20.3 19.2 19.4 19.5 19.4 24.6 23.1 21.6 22.8 21.6 21.0 22.7 21.5 22.4 22.719.7 18.8 19.4 18.8 18.3 18.2 19.9 20.1 25.5 22.9 22.4 72.9 20.4 21.3 21.8 22.9 22.519.1 18.4 19.5 19.3 18.8 18.1 18.5 18.9 18.5 24.3 23.1 23.9 19.9 20.2 22.1 22.2 22.718.2 18.4 17.9 19.8 18.9 18.4 18.7 18.6 18.7 18.6 24.5 21.7 20.0 20.5 21.9 21.8 22.317.2 18.5 17.8 18.4 19.3 18.9 18.1 18.8 19.0 18.4 18.6 18.7 20.4 21.4 21.6 21.4 22.217.3 17.1 17.7 17.8 19.4 19.0 18.4 19.6 19.3 19.5 18.3 18.2 18.3 20.2 23.2 21.0 21.917.8 17.2 17.2 17.9 18.9 19.1 19.2 19.3 19.0 18.2 19.5 18.3 18.9 13.0 22.3 20.8 21.0

2

20.4 20.2 13.0 58.7 20.5 21.9 22.7 22.4 22.1 21.1 22.0 21.0 22.4 21.6 37.5 21.2 20.520.8 20.0 20.6 20.6 20.0 14.8 22.3 23.4 22.7 21.5 21.9 41.3 22.2 22.0 22.2 21.3 37.127.5 19.3 20.3 22.7 20.9 21.4 21.4 23.3 22.6 21.8 21.5 21.1 21.5 21.9 22.2 22.4 20.822.1 19.3 20.1 21.6 22.0 21.5 22.0 21.9 22.4 21.5 21.6 22.2 21.6 22.1 22.6 22.3 22.123.5 19.0 19.4 20.0 21.5 20.3 23.2 20.5 22.4 22.1 21.4 21.6 21.2 21.1 21.4 22.0 22.123.0 19.1 18.3 19.5 19.8 19.8 19.9 25.0 22.0 21.8 22.1 21.1 21.0 21.9 21.6 22.0 22.620.3 18.9 20.3 19.2 19.4 19.5 19.4 24.6 23.1 21.6 22.8 21.6 21.0 22.7 21.5 22.4 22.719.7 18.8 19.4 18.8 18.3 18.2 19.9 20.1 25.5 22.9 22.4 72.9 20.4 21.3 21.8 22.9 22.519.1 18.4 19.5 19.3 18.8 18.1 18.5 18.9 18.5 24.3 23.1 23.9 19.9 20.2 22.1 22.2 22.718.2 18.4 17.9 19.8 18.9 18.4 18.7 18.6 18.7 18.6 24.5 21.7 20.0 20.5 21.9 21.8 22.317.2 18.5 17.8 18.4 19.3 18.9 18.1 18.8 19.0 18.4 18.6 18.7 20.4 21.4 21.6 21.4 22.217.3 17.1 17.7 17.8 19.4 19.0 18.4 19.6 19.3 19.5 18.3 18.2 18.3 20.2 23.2 21.0 21.917.8 17.2 17.2 17.9 18.9 19.1 19.2 19.3 19.0 18.2 19.5 18.3 18.9 13.0 22.3 20.8 21.0

3


MF analysis of Height Distribution (HD)

),(

),( )(q, i

i

q

q

log

,1log,lim 1

0

n

iii q

q

δlog

δq,logδq,limqf

δn

1iii

0δ

Chhabra and Jenssen method

q

δn

1ii

iqii

m

mPδq,


f()

Multifractal Spectrum

wf

w


Considerations on MF calculations

• Height readings have been corrected for slope and tillage tool marks.

• The linearity in the function were found in all cases from =1 to =64 cm.

• The range of q values used were from –5 to +5 with increments of 0.5.

• All the R2 obtained were higher than 0.97


HD Multifractal Spectrum

soil P-1

1,000

1,200

1,400

1,600

1,800

2,000

1,500 1,750 2,000 2,250 2,500

f(

)

chisel

seedbeds

moldboard



soil P-2

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 1,95 2,00 2,05 2,10

f(

)

chisel

seedbeds

moldboard



soil P-3

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 1,95 2,00 2,05 2,10

f(

)

chisel

seedbeds

moldboard


Results from the multifractal analysis

Code Treatment max f(max)

min f(min) w wf

Chisel 2.020 1.970 1.980 1.690 0.040 0.280Seedbeds 2.020 1.120 1.730 1.160 0.290 -0.040Moldboard 2.090 1.900 1.690 1.220 0.400 0.680Chisel 2.010 1.990 2.000 1.990 0.010 0.000Seedbeds 2.010 1.990 2.000 1.990 0.010 0.000Moldboard 2.010 1.970 1.990 1.970 0.020 0.000Chisel 2.003 1.990 1.998 1.990 0.006 0.000Seedbeds 2.004 1.990 1.997 1.990 0.007 0.000Moldboard 2.005 1.990 1.996 1.990 0.009 0.000

P_1

P_2

P_3


AV= 26.98 SD =14.90

1.051.151.251.351.451.551.651.751.851.952.05

1.5 1.7 1.9 2.1

f(

)random

Chisel

RR


AV= 21.89 SD =6.62

1.05

1.15

1.25

1.35

1.45

1.55

1.65

1.75

1.85

1.95

2.05

1.5 1.7 1.9 2.1

f(

)

random

Seedbeds


AV= 25.83 SD =8.92

1.05

1.15

1.25

1.35

1.45

1.55

1.65

1.75

1.85

1.95

2.05

1.5 1.7 1.9 2.1

f(

)

random

Moldboard


1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 2,00 2,10

f(

)

random S.D. 2,84

seedbeds

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 2,00 2,10

f(

)

random S.D. 5,70

moldboard

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 2,00 2,10

f(

)

random S.D. 3,14

chiselsoil P-2

soil P-2

soil P-2

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 2,00 2,10

f(

)

random S.D. 3,34

chisel

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 2,00 2,10

f(

)

random S.D. 2,15

seedbeds

1,95

1,96

1,97

1,98

1,99

2,00

2,01

1,90 2,00 2,10

f(

)

random S.D. 2,47

moldboard

soil P-3

soil P-3

soil P-3


CONCLUSIONS

• Fractal dimensions estimated from MF analyses of HD are useful descriptors.

• Multifractal parameters seem to be correlated depending on soil texture properties.

• Comparison between data structure and a random structure can be used to get a complementary index to RR.


Further research

• More work on correlating parameters from multifractal analysis to soil properties: we need to understand what represent each parameter.

• More work on application of multifractal parameters to the prediction of processes related to soil erosion.


WIND FLUCTUATIONS• The study of wind-speed (w) is aimed at greenhouse

control (heating and ventilation), since wind velocity influences both types of control. Wind increases heat losses in winter nights, so it is of interest to regulate the heating as a function of wind-speed and its realistic simulation is an important task in modeling and system design.

• To study the multifractal nature of this series and to fully characterize the dynamical system that supports it is the first step before any simulation could be successfully achieved.

• Time series data from 2004 were used in this study. Every ten minutes, the station recorded mean values of the wind velocity in m/s. Thus we handle in each yearly analysis a series of 105.408 data points, and in the monthly analysis a minimum of 4.176 values (February) and a maximum of 4.464.


Stochastic process: fBm2

)()()( ii twtwCov

•The minimum and maximum lag values are normally chosen.

•If the series is self-similar then:

22)( HCCov Hurst exponent

•H = 0.5 => random structure

•H > 0.5 => persistant structure

•H < 0.5 => anti-persistant structure


Multifractal Analysis (MF)• Multiscaling analysis determines the

dependence of the statistical moments (and not only the covariance) of the time series on the resolution with which the data are examined.

• Different moments different exponent in the increments (q).

• Structure Function (Mq)


Generalized Hurst exponent H(q)

•The minimum and maximum lag values are normally chosen.

•If the series is self-similar or self-affine then:

q

iiq twtwM )()()( 21

)()( qqq CM

monotonically non-decreasing function of q

q

qqH

)()(


CASES

• stationary processes have scale-independent increments and show invariance under translation => H(q)=0

• non-stationary and monofractal processes => constant H(q)

• non-stationary and multifractal => non constant H(q)


Wind velocity time series

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

time

win

d (

m/s

)FEBRUARY 2004

0

0.5

1

1.5

2

2.5

3

time

win

d (

m/s

)JULY 2004

0

0.5

1

1.5

2

2.5

3

3.5

4

time

win

d (

m/s

)DECEMBER 2004


-1

-0.5

0

0.5

1

1.5

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

0:00

time

win

d f

luct

uat

ion

(m

/s)

FEBRUARY 2004

-1.5

-1

-0.5

0

0.5

1

0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00

timew

ind

flu

ctu

atio

n (

m/s

)

JULY 2004

-1.5

-1

-0.5

0

0.5

1

1.5

2

0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00

time

win

d f

luct

uat

ion

(m

/s)

DECEMBER 2004


Histograms of wind fluctuations


Structure Functions (M) for February of 2004.

-6

-4

-2

0

2

4

6

8

10

12

-8 -6 -4 -2 0

ln( /max)

ln(M

q)

2

4

6

8

10

12


(q) and the corresponding H(q) functionMonth (q) H(q) july y = -0.0156x2 + 0.4004x

R2 = 0.9881

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14q

(q)


COMMENTS AND CONLUSION

• There are several steps as number of data and lag values range chosen that influence the numerical results.

• February shows a different behavior from the other months, however the q values used are much higher that it is normally found in the literature.

• July shows a clear multiscaling pattern with a non constant H(q). December shows an almost constant H(q)

• All of them, as the annual time series analysis, show an anti-persistent character.

• Structure Functions is a way to usefully characterizing this multiscale heterogeneity. Based on this modeling simulation of wind fluctuations can be done in easy way and being realistic.


Thank you for your attention

international summer school on turbulence diffusion 2006 multifractal analysis in b&w soil...

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