stochastic hydrology random field simulation professor ke-sheng cheng department of bioenvironmental...
Post on 18-Jan-2018
220 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
05/04/23 3Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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
05/04/23 4Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Characterizing a random field
05/04/23 5Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• In geostatistics, spatial variation of a random field is often expressed in terms of the semi-variogram defined as
., ,))()((21),( 2 xxxZxZExx
05/04/23 6Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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
05/04/23 7Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
A typical semi-variogram (a) A pure-nugget semi-variogram (b).
05/04/23 8Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Conceptual description of a gamma random field simulation approach
05/04/23 9Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 10Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Components of sequential random field simulation
05/04/23 11Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 12Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
OUTLINES• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications
05/04/23 13Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Sequential Gaussian simulation
05/04/23 14Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Bivariate Normal Distribution• Bivariate normal density function
1
21
21
2
1)(),(
zz
ZXY
T
ezfyxf
05/04/23 15Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Multivariate Normal Distribution
05/04/23 16Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 17Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Conditional normal density• Conditional normal density
2
22
|
1
)()(
21exp
)1(2
1
)|(
Y
XX
YY
Y
XY
xy
xyYf
)(| yf xXY
05/04/23 18Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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]
05/04/23 19Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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.
05/04/23 20Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 21Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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.
05/04/23 22Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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.
05/04/23 23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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.
05/04/23 24Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 25Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
)( 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]
05/04/23 26Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
OUTLINES• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications
05/04/23 27Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Covariance matrices conversion
05/04/23 28Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 29Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 30Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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]
05/04/23 31Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Transforming Gaussian realizations to gamma realizations
05/04/23 32Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
dyydgygfyf XY)())(()(
11
21
2)()( )2(
1y
Y eyyf
21
2)(1 yey
05/04/23 33Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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
05/04/23 34Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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]
05/04/23 35Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Summary of the simulation procedures
05/04/23 36Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 37Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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]
05/04/23 38Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
)()()(21
2/1*2/
21|
*1
1**1
21
||)2(1
)|(
ww
p
WW
e
wwf
[C]
3
91
911
2
wyx [D]
05/04/23 39Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 40Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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
05/04/23 41Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Simulation and verification
05/04/23 42Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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
05/04/23 43Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• 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.
05/04/23 44Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 45Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 46Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 47Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 48Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 49Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Examples of simulated gamma fields
05/04/23 50Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
OUTLINE• Definition and introduction• Sequential Gaussian Simulation (SGS)• Gamma random field simulation• Potential applications
05/04/23 51Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Conditional Random Field Simulation – Heavy metal conta
mination (HMC) in soils
05/04/23 52Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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.
05/04/23 53Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• (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
05/04/23 54Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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
05/04/23 55Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
05/04/23 56Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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)
05/04/23 60Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Exceedance Probability Map (Cd)
05/04/23 61Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Exceedance Probability Map (Cr)
05/04/23 62Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Exceedance Probability Map (Cu)
05/04/23 63Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Exceedance Probability Map (Zn)
05/04/23 64Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
top related