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
11/30/2018
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
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

4. Examples of random fields
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
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

5. Characterizing a random field
11/30/2018
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
11/30/2018
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
11/30/2018
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).
11/30/2018
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
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

10. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

11. Components of sequential random field simulation
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

12. 11/30/2018
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
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

14. Sequential Gaussian simulation
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

15. Bivariate Normal Distribution
Bivariate normal density function
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

16. Multivariate Normal Distribution
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

17. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

18. Conditional normal density
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

19. Conditional multivariate normal density
11/30/2018
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.
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

21. 11/30/2018
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 xo with coordinate (1,1).
We then sequentially generate random numbers at other locations under the condition of previously generated random numbers.
11/30/2018
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.
11/30/2018
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.
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

25. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

26. [C]
11/30/2018
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
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

28. Covariance matrices conversion
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

29. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

30. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

31. [B]
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

32. Transforming Gaussian realizations to gamma realizations
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

33. 11/30/2018
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.
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

35. Equation [D] is a one-to-one mapping function between x and w. [D]
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]
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

36. Summary of the simulation procedures
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

37. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

38. [B]
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

39. [C]
[D]
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

40. 11/30/2018
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.)
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

42. Simulation and verification
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

43. As the range increases, the degree of spatial correlation increases.
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
11/30/2018
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.
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

45. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

46. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

47. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

48. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

49. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

50. Examples of simulated gamma fields
11/30/2018
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
11/30/2018
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
11/30/2018
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.
11/30/2018
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.
11/30/2018
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.
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

56. 11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

57. Criteria for regulation
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

58. Sampling Locations
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

59. Areas Without Point Samples
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

60. Exceedance Probability Map (Ni)
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

61. Exceedance Probability Map (Cd)
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

62. Exceedance Probability Map (Cr)
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

63. Exceedance Probability Map (Cu)
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

64. Exceedance Probability Map (Zn)
11/30/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.