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

Stochastic Hydrology

Random Field Simulation

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

OUTLINE• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications

05/04/23 2Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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


• 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


Characterizing a random field


• In geostatistics, spatial variation of a random field is often expressed in terms of the semi-variogram defined as

., ,))()((21),( 2 xxxZxZExx


• 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. .

)()0()( hCCh || xxh x

2Z

x


A typical semi-variogram (a) A pure-nugget semi-variogram (b).


Conceptual description of a gamma random field simulation approach



Components of sequential random field simulation



OUTLINES• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications


Sequential Gaussian simulation


Bivariate Normal Distribution• Bivariate normal density function

1

21

21

2

1)(),(

zz

ZXY

T

ezfyxf


Multivariate Normal Distribution



Conditional normal density• Conditional normal density

2

22

|

1

)()(

21exp

)1(2

1

)|(

Y

XX

YY

Y

XY

xy

xyYf

)(| yf xXY


Conditional multivariate normal density

)()()(21

2/1*2/21|

*1

1**1

21 ||)2(1)|(

ww

pWW ewwf

)( 221

22121|*

21 wwW

121

221211|*

21

wW

[C]


• 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.



• 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.


• 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.


• 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.



)( 221

22121|*

21 wwW

121

221211|*

21

wW

22

12

12

11

1,12,11,1

1,22221

1,11211

qqqq

q

q

W

CCC

CCCCCC

)()()(21

2/1*2/

21|

*1

1**1

21

||)2(1

)|(

ww

p

WW

e

wwf

[C]


OUTLINES• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications


Covariance matrices conversion




32 62

933

UVYXUVYX

UVYXYXYXYXXY

CCBB

CCACCAAA

4

61

X

XA 3

66

XX

XB

2

631

X

XC

4

61

Y

YA 3

66

YY

YB 2

631

Y

YC

[B]


Transforming Gaussian realizations to gamma realizations


dyydgygfyf XY)())(()(

11

21

2)()( )2(

1y

Y eyyf

21

2)(1 yey


• 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.

3

91

9112

wy


• 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.

3

91

911

2

wyx [D]


Summary of the simulation procedures



32 62

933

UVYXUVYX

UVYXYXYXYXXY

CCBB

CCACCAAA

4

61

X

XA 3

66

XX

XB

2

631

X

XC

4

61

Y

YA 3

66

YY

YB 2

631

Y

YC

[B]


)()()(21

2/1*2/

21|

*1

1**1

21

||)2(1

)|(

ww

p

WW

e

wwf

[C]

3

91

911

2

wyx [D]



• 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.)

W


Simulation and verification


• 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


• 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.







Examples of simulated gamma fields


OUTLINE• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications


Conditional Random Field Simulation – Heavy metal conta

mination (HMC) in soils


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.


• (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.

)]()([)()( ** xZxZxZxZ SSc

)(* xZ S )(* xZ


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.

)(xFZ



Criteria for regulation

05/04/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

57

Sampling Locations

05/04/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 58

Areas Without Point Samples

05/04/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 59

Exceedance Probability Map (Ni)


Exceedance Probability Map (Cd)


Exceedance Probability Map (Cr)


Exceedance Probability Map (Cu)


Exceedance Probability Map (Zn)


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

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