1. International Summer School on Turbulence Diffusion 2006 Multifractal Analysis in B&W Soil Images Ana M. Tarquis anamaria.tarquis@upm.es Dpto. de Matemática Aplicada E.T.S.I. Agrónomos Universidad Politécnica de Madrid

2. International Summer School on Turbulence Diffusion 2006 INDEX Problem: motivation and start point. Fractals and multifractals concepts. Porosity images: resolved? Configuration Entropy Griding Methods

3. International Summer School on Turbulence Diffusion 2006 CONSERVATION OF NATURAL RESOURCES Agriculture : soil degradation and water contamination. Sustainable agriculture Quantification of soil quality index?

4. International Summer School on Turbulence Diffusion 2006 Soil structure Water, solutes and gas transport Soil resistance Roots morphology Microorganism populations PORE AND SOIL MATRIX GEOMETRY

5. International Summer School on Turbulence Diffusion 2006 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.whole

6. International Summer School on Turbulence Diffusion 2006 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:

7. International Summer School on Turbulence Diffusion 2006 “ Box-Counting” -m = fractal dimension, D 11 nn Black, white or interface interface

8. International Summer School on Turbulence Diffusion 2006 Multifractal analysis consider the number of black pixels in each box (pore density=m).Multifractal

9. International Summer School on Turbulence Diffusion 2006 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.

10. International Summer School on Turbulence Diffusion 2006 11 nn DqDq q

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

12. International Summer School on Turbulence Diffusion 2006 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: { p 1, p 2, p 3, p 4 } Random images : p 1 = p 2 = p 3 = p 4 = 25% Multifractal images: p 1 = 50%, p 2 = 5%, p 3 = 25% and p 4 = 20% (by random arrangements or not).

13. International Summer School on Turbulence Diffusion 2006 Random multifractal

14. International Summer School on Turbulence Diffusion 2006 Generalized dimensions (Dq) obtained for two different distributions based on Stanley and Meakin (1988) formulas with their respective -  (q) curves.

15. International Summer School on Turbulence Diffusion 2006 Most common parameters calculated D 0 q=0  box counting dimension D 1 q=1  entropy dimension D 2 q=2  correlation dimension

16. International Summer School on Turbulence Diffusion 2006 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: f(  ) 

17. International Summer School on Turbulence Diffusion 2006 f(  )  Multifractal Spectrum wfwf ww

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19. INTERDENNY ABOKMUNCHONG 1500x1000 pixels

20. International Summer School on Turbulence Diffusion 2006 ¿How many points?

21. International Summer School on Turbulence Diffusion 2006 ADS BUSO EHV1

22. International Summer School on Turbulence Diffusion 2006 We have to compare

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24. Obtaining D q Ehv1, porosity 46,7%

25. International Summer School on Turbulence Diffusion 2006 Calculating D q ADS, porosity 5,7%

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27. Continuos line = random structure Dashed line = mfract structure Filled Square = values from image soils

28. International Summer School on Turbulence Diffusion 2006 Considerations on D q 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

29. International Summer School on Turbulence Diffusion 2006 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 r 2 higher than 0.98,

30. International Summer School on Turbulence Diffusion 2006 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.

31. International Summer School on Turbulence Diffusion 2006 Configuration Entropy H(  ) The maximum value of j is  x  and the minimum value is 0 (Andraud et al., 1989)  11 ii n(  ) = boxes of size  from  = 1 to  = w /4 w N j = number of boxes with j black pixels inside

32. International Summer School on Turbulence Diffusion 2006 Configuration Entropy H(  ) The probability associated with a case of j black pixels in a box of size  (p j (  ))

33. International Summer School on Turbulence Diffusion 2006 Configuration Entropy H(  )  (pixels) H*(  ) 0 1 1 w/4 H*(L) L

34. International Summer School on Turbulence Diffusion 2006 Methods: gliding, random walks, randomly Box size Jump step length Number of jumps

35. International Summer School on Turbulence Diffusion 2006 Thank you for your attention

36. International Summer School on Turbulence Diffusion 2006 Multifractal Analysis on a Matrix Ana M. Tarquis anamaria.tarquis@upm.es Dpto. de Matemática Aplicada E.T.S.I. Agrónomos Universidad Politécnica de Madrid

37. International Summer School on Turbulence Diffusion 2006 INDEX Field Percolation Soil Roughness Satellite images Time series

38. International Summer School on Turbulence Diffusion 2006 Z= 10 cmZ = 20 cmZ = 30 cm Z = 40 cmZ = 50 cmZ = 60 cm

39. International Summer School on Turbulence Diffusion 2006 % of blue vs. depth 50% 15 cm

40. International Summer School on Turbulence Diffusion 2006 Z = 25 cm blue staining 28,95%

41. International Summer School on Turbulence Diffusion 2006 Dye Tracer Distribution

42. International Summer School on Turbulence Diffusion 2006 Multifractal Analysis of the Dye Tracer Distribution B) Generalized dimensions A) f(  ) spectrum

43. International Summer School on Turbulence Diffusion 2006 Multispectral Satellite Images

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45. 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),

46. International Summer School on Turbulence Diffusion 2006 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.

47. International Summer School on Turbulence Diffusion 2006 Soil measurements Three different soils with different textures. Three different treatments applying tillage: chisel, moldboard, seedbeds. Height measures of 2x2 m 2 plot area. Resolution of the measure each 2 cm

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49. Soil texture

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51. moldboard seedbeds chisel

52. International Summer School on Turbulence Diffusion 2006 moldboard seedbeds chisel

53. International Summer School on Turbulence Diffusion 2006 11 nn Box counting method Number of boxes depends on  i 22 33

54. International Summer School on Turbulence Diffusion 2006 MF analysis of Height Distribution (HD) Chhabra and Jenssen method

55. International Summer School on Turbulence Diffusion 2006 f(  )  Multifractal Spectrum wfwf ww

56. International Summer School on Turbulence Diffusion 2006 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 R 2 obtained were higher than 0.97

57. International Summer School on Turbulence Diffusion 2006 HD Multifractal Spectrum

58. International Summer School on Turbulence Diffusion 2006 HD Multifractal Spectrum

59. International Summer School on Turbulence Diffusion 2006 HD Multifractal Spectrum

60. International Summer School on Turbulence Diffusion 2006 Results from the multifractal analysis

61. International Summer School on Turbulence Diffusion 2006 AV= 26.98 SD =14.90 RR

62. International Summer School on Turbulence Diffusion 2006 AV= 21.89 SD =6.62

63. International Summer School on Turbulence Diffusion 2006 AV= 25.83 SD =8.92

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65. 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.

66. International Summer School on Turbulence Diffusion 2006 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.

67. International Summer School on Turbulence Diffusion 2006 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.

68. International Summer School on Turbulence Diffusion 2006 Stochastic process: fBm The minimum and maximum lag values are normally chosen. If the series is self-similar then: Hurst exponent H = 0.5 => random structure H > 0.5 => persistant structure H anti-persistant structure

69. International Summer School on Turbulence Diffusion 2006 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 (M q )

70. International Summer School on Turbulence Diffusion 2006 Generalized Hurst exponent H(q) The minimum and maximum lag values are normally chosen. If the series is self-similar or self-affine then: monotonically non-decreasing function of q

71. International Summer School on Turbulence Diffusion 2006 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)

72. International Summer School on Turbulence Diffusion 2006 Wind velocity time series

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74. Histograms of wind fluctuations

75. International Summer School on Turbulence Diffusion 2006 Structure Functions (M) for February of 2004.

76. International Summer School on Turbulence Diffusion 2006  (q) and the corresponding H(q) function

77. International Summer School on Turbulence Diffusion 2006 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.

78. International Summer School on Turbulence Diffusion 2006 Thank you for your attention