1. Stochastic Hydrology Random Field Simulation Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. OUTLINE Definition and introduction Sequential Gaussian Simulation (SGS) Gamma random field simulation Potential applications 2/19/2016 2 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

3. What is a random field? A random variable is completely characterized by its probability density function (PDF). A set of jointly distributed random variables is characterized by their joint PDF. [Multivariate probability distribution] Random process - Time series Random field 2/19/2016 3 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

4. A random field can be defined as a set of jointly distributed random variables defined in a spatial domain (2-, 3-, or higher dimension). Examples of random fields – Spatial variation of rainfall – Variation of terrain elevation – Spatial variation of heavy metal contamination – Grey level (reflectance) of multispectral images 2/19/2016 4 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

5. Characterizing a random field 2/19/2016 5 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6. In geostatistics, spatial variation of a random field is often expressed in terms of the semi-variogram defined as 2/19/2016 6 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

7. For a stationary random field, the semi- variogram is also independent of the locations of x and, and the following relationship between the covariance function and the semi-variogram exists where represents the distance between x and and C(0) is the variance of the random variable Z(x), i.e.. 2/19/2016 7 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

8. A typical semi-variogram (a) A pure-nugget semi-variogram (b). 2/19/2016 8 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

9. Conceptual description of a gamma random field simulation approach 2/19/2016 9 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

10. 2/19/2016 10 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

11. Components of sequential random field simulation 2/19/2016 11 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

12. 2/19/2016 12 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

13. OUTLINES Definition and introduction Sequential Gaussian Simulation (SGS) Gamma random field simulation Potential applications 2/19/2016 13 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

14. Sequential Gaussian simulation 2/19/2016 14 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

15. Bivariate Normal Distribution Bivariate normal density function 2/19/2016 15 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

16. Multivariate Normal Distribution 2/19/2016 16 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

17. 2/19/2016 17 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

18. Conditional normal density 2/19/2016 18 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

19. Conditional multivariate normal density [C] 2/19/2016 19 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

20. Equation [C] lays the foundation for stochastic simulation of a Gaussian random field. Random field simulation is generally carried out by sequentially generating random number at only one target location each time. 2/19/2016 20 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

21. 2/19/2016 21 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

22. Suppose the random field simulation begins with a univariate random number generation at an initial point x o with coordinate (1,1). We then sequentially generate random numbers at other locations under the condition of previously generated random numbers. 2/19/2016 22 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

23. The simulation is conducted following a column-preference style in which random numbers at all nodes of the same line are generated sequentially and then the process proceeds to the next line. 2/19/2016 23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

24. At any stage of the simulation process, the number and locations of the conditioning variates depend on the range measured in terms of the grid interval. 2/19/2016 24 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

25. 2/19/2016 25 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

26. [C] 2/19/2016 26 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

27. OUTLINES Definition and introduction Sequential Gaussian Simulation (SGS) Gamma random field simulation Potential applications 2/19/2016 27 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

28. Covariance matrices conversion 2/19/2016 28 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

29. 2/19/2016 29 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

30. 2/19/2016 30 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

31. [B] 2/19/2016 31 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

32. Transforming Gaussian realizations to gamma realizations 2/19/2016 32 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

33. 2/19/2016 33 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

34. An approximation of the chi-squared distribution by the standard normal distribution, known as the Wilson-Hilferty approximation, is given as follows (Patel and Read, 1996) where w represents the standard normal deviate and y is the corresponding chi-squared variate. 2/19/2016 34 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

35. Transformation of the standard normal deviate w to gamma variate x can thus be derived as Equation [D] is a one-to-one mapping function between x and w. [D] 2/19/2016 35 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

36. Summary of the simulation procedures 2/19/2016 36 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

37. 2/19/2016 37 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

38. [B] 2/19/2016 38 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

39. [C] [D] 2/19/2016 39 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

40. 2/19/2016 40 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

41. Illustration of nodes with common covariance matrix for conditional simulation using a column-preference generation algorithm. Nodes marked by the same symbols have a common covariance matrix. (The range is assumed to be twice of grid interval.) 2/19/2016 41 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

42. Simulation and verification 2/19/2016 42 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

43. Since our simulation is based on a discrete network of one pixel grid interval, the random field with one- pixel range is technically completely random with no spatial correlation between any neighboring pixels, resulting in a pure nugget semi-variogram. As the range increases, the degree of spatial correlation increases. One hundred simulation runs were conducted for each scenario type with respect to specific values of range and simulation size 2/19/2016 43 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

44. For a given scenario type and a specific value of range, parameter estimation was done for each of the 100 realizations. These estimates vary among different realizations. Therefore, the sample mean and standard deviation of these realization- specific parameter estimates were calculated. 2/19/2016 44 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

45. 2/19/2016 45 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

46. 2/19/2016 46 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

47. 2/19/2016 47 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

48. 2/19/2016 48 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

49. 2/19/2016 49 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

50. Examples of simulated gamma fields 2/19/2016 50 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

51. OUTLINE Definition and introduction Sequential Gaussian Simulation (SGS) Gamma random field simulation Potential applications 2/19/2016 51 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

52. Conditional Random Field Simulation – Heavy metal contamination (HMC) in soils 2/19/2016 52 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

53. Conditional simulation of HMC The heavy metal concentration (HMC) at each 1-meter cell can be considered as one realization of the random field. In order to understand the statistical distribution of HMC at the center of each 1- m cell, conditional random field simulation was implemented in this study. 2/19/2016 53 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

54. (1) Generation of 1000 realizations for different HMC by random field simulation. (2) Estimates of HMC, conditioned on observed and downscaled HMC values, are given by the following equation: where and are respectively ordinary kriging estimates using simulated and observed HMC at observation points. 2/19/2016 54 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

55. Contaminated zone delineation using conditional distribution function of HMC Conditional random field simulation using HYDRO_GEN and OK generates 1,000 realizations with 1-m grid interval. The conditional cumulative distribution function (CCDF) of HMC at location x, i.e., was estimated using generated HMC values. 2/19/2016 55 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

56. 2/19/2016 56 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

57. Criteria for regulation 2/19/2016 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 57

58. Sampling Locations 2/19/2016 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 58

59. Areas Without Point Samples 2/19/2016 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 59

60. Exceedance Probability Map (Ni) 2/19/2016 60 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

61. Exceedance Probability Map (Cd) 2/19/2016 61 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

62. Exceedance Probability Map (Cr) 2/19/2016 62 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

63. Exceedance Probability Map (Cu) 2/19/2016 63 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

64. Exceedance Probability Map (Zn) 2/19/2016 64 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.