## Geostatistics

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## Conditional simulations

### A classification of the methods

#### Quantities

• Continuous variables
• Categorical variables
• Objects

#### Basic model type

• Diffusive model
• Jump model
• Mosaic model
• Random set model

### Sequential simulation

#### Outline of algorithms

1. Assign any hard data (n) to the grid
2. Define a random path visiting all nodes u in the grid
3. Loop over all nodes u_i
a. Construct a conditional distribution
Fz(u_i, z|(n+i-1)) = Pr(Z(u_i)<=z|(n+i-1))
b. Draw a simulated value z(u_i) from the conditional distribution
Fz(u_i, z|(n+i-1))
c. Add simulated value to data-set (n+i-1)
4. End simulation


### Sequential Gaussian simulation

#### Outline of the algorithms

1. Transform the sample data to standard normal scores
2. Assign the data (n) to the grid
3. Define a random path visiting all nodes u
4. Loop over all nodes u_i
a. Construct a conditional Gaussian distribution

b. Draw a simulated value z(u_i) from the conditional distribution

c. Add simulated value to data-set (n+i-1)
5. End simulation
6. Transform the entire simulation back to the original data histogram


## Kriging

### Main forms of linear kriging

 Kriging Type Mean Minimal Prerequisite Model Name Simple Kriging(SK) Constant, known Covariance Stationary Ordinary Kriging(OK) Constant, unknown Variogram Intrinsic Universal Kriging(OK) Varying, unknown Variogram UK model

### Estimated value

• The weights depend on the location where the function is being estimated.
• , are selected so as to minimize the error , characterized by its expected mean square
 The prediction at the point the data at the point weights a constant that depends on

### Take the kriging variance as the mean square error

Originally




Expand it and finally written as



b.
 Covariance between two sample points and the mean value Covariance between one sample point and the estimated point Variance at the estimated point

### Simple Kriging

Take the minimum of the mean square error


Therefore, Simple Kriging System is


Simple Kriging Variance



## Under the condition of second-order stationarity

• means: spatially constant mean and variance
• Relations of covariance, correlation and variogram

 Covariance Correlation Semivariogram

## Covariance 共分散

• ２つの変数がどのくらい同じように動くか
• corrleation coefficient

## Variogram バリオグラム

• 空間的相関、つまりデータが距離と方向にどのような関係を持つか

### Variogram model

 Spherical Exponential (GSLIB) Exponential (gstat) Gaussian
where

 h lag distance a range practical range equal to the distance at which 95% of the sill has been reached theoretical range c sill

### Covariogram

• a function that depends only on the displacement vector h.

### Semivariogram

 spatial process at lcation the displacement vector

### Relation between Covariogram and Semivariogram

 the variance of spatial process

## Difference between Kriging and Simulation

Kriging
• produces just one map of estimates which is best in a statistical sense
• a global estimator, in that its estimate represents all the data within a defined area
• good to show smooth variations and underlying trends

Simulation
• a local estimator
• reproduces exactly measured data
• good at showing local variability
• provides any number of statistically equivalent maps

## Glossary

• cdf: the cumulative distribution function 累積分布関数
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