Geostatistics

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地球統計学

#contents

* 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
*** The joint probability distribution 
$$ Pr\lbrace z_{M+1} \leq Z_{M+1} \leq z_{M+1} + dz_{M+1},..., z_N \leq Z_N \leq z_N + dz_N | z_1,...,z_N \rbrace$$
$$ = Pr\lbrace z_{M+1} \leq Z_{M+1} \leq z_{M+1} + dz_{M+1} | z_1,...,z_M \rbrace $$
$$\,\,\,\times Pr\lbrace z_{M+2} \leq Z_{M+2} \leq z_{M+2} + dz_{M+2} | z_1,...,z_M,z_{M+1} \rbrace $$
$$\,\,\,\,\vdots$$
$$\,\,\,\times Pr\lbrace z_N \leq Z_N \leq z_N + dz_N | z_1,...,z_M,z_{M+1},...,z_N \rbrace $$

** 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
           $$G(u_i,z|(n+i-1)) = G(\frac{z-z^*_{SK}(u_i)}{\sigma_{SK}(u_i)})$$
      b. Draw a simulated value z(u_i) from the conditional distribution
           $$G(u_i,z|(n+i-1))$$
      c. Add simulated value to data-set (n+i-1)
 5. End simulation
 6. Transform the entire simulation back to the original data histogram

*** The mean of each conditional distribution
$$E[Z(u_i)|Z(u_{i-1})=z_{i-1},...,Z(u_1)=z_1] = \sum_{j=1}^{i-1}\lambda_j(u_i)z(u_j) = z_{SK}^*(u_i)$$

*** The variance of each conditional distribution
$$Var[Z(u_i)|Z(u_{i-1})=z_{i-1},...,Z(u_1)=z_1] = \sigma^2_{SK}(u_i) $$

* Kriging
** Main forms of linear kriging
| &bold(){Kriging Type} | &bold(){Mean} | &bold(){Minimal Prerequisite} | &bold(){Model Name} |
| Simple Kriging(SK) | Constant, known | Covariance | Stationary |
| Ordinary Kriging(OK) | Constant, unknown | Variogram | Intrinsic |
| Universal Kriging(OK) | Varying, unknown | Variogram | UK model |

** Common parts in derivation of the Kriging equations
** Estimated value
$$Z^* = \sum_{\alpha}\lambda_\alpha Z_\alpha + \lambda_0$$
- The weights $$\lambda_\alpha$$ depend on the location $$x_0$$ where the function is being estimated.
- $$\lambda_\alpha$$, $$\lambda_0$$ are selected so as to minimize the error $$Z^* - Z_0$$, characterized by its expected mean square $$E(Z^* - Z_0)^2$$
| $$Z_*$$ | The prediction at the point  $$x_0$$ |
| $$Z_\alpha$$ | the data at the point $$\alpha$$ |
| $$\lambda_\alpha$$ | weights |
| $$\lambda_0$$ | a constant that depends on $$x_0$$ |

** Take the kriging variance as the mean square error
Originally
 $$E(Z^* - Z_0)^2 = E\bigg(\sum_{\alpha}\lambda_\alpha Z_\alpha - Z_0 \bigg)^2$$
Adding the mean term $$\mu$$,
 $$E(Z^* - Z_0)^2 = E\bigg(\sum_{\alpha}\lambda_\alpha (Z_\alpha - \mu) - (Z_0 - \mu) \bigg)^2$$
Expand it and finally written as
 $$E(Z^* - Z_0)^2 = \sum_\alpha \sum_\beta \lambda_\alpha \lambda_\beta \sigma_{\alpha\beta} - 2\sum_\alpha \lambda_\alpha \sigma_{\alpha 0} + \sigma_{00}$$

b. $$E(Z^* - Z_0)^2 = Var(Z^* - Z_0) + [E(Z^* - Z_0)]^2$$
|$$\sigma_{\alpha\beta}$$ |Covariance between two sample points $$Z(x_\alpha)$$ and $$Z(x_\beta)$$ |
|$$\mu$$| the mean value|
|$$\sigma_{\alpha 0}$$| Covariance between one sample point and the estimated point $$x_0$$|
|$$\sigma_{00}$$| Variance at the estimated point $$x_0$$|

** Simple Kriging
Take the minimum of the mean square error
 $$ \frac{\partial}{\partial \lambda_\alpha} E(Z^* - Z_0)^2 = 2\sum_\beta \lambda_\beta \sigma_{\alpha \beta} - 2\sigma_{\alpha0} = 0$$
Therefore, Simple Kriging System is
 $$ \sum_\beta \lambda_\beta \sigma_{\alpha \beta} = \sigma_{\alpha 0} $$
Simple Kriging Variance
 $$ \sigma_{SK}^2 = E(Z^* - Z_0)^2 = \sigma_{00} - \sum_\alpha \lambda_\alpha \sigma_{\alpha 0} $$

** Ordinary Kriging(OK)

* Under the condition of second-order stationarity
- means: spatially constant mean and variance
- Relations of covariance, correlation and variogram
$$C(0) = Cov(Z(u), Z(u+h)) = Var(Z(u))$$
$$\rho(h) = \frac{C(h)}{C(0)}$$
$$\gamma(h) = C(0) - C(h)$$
|$$C(h)$$|Covariance|
|$$\rho(h)$$|Correlation|
|$$\gamma(h)$$|Semivariogram|

&blankimg(variogram_covariance.JPG)

** External drift kriging

* Covariance 共分散
- ２つの変数がどのくらい同じように動くか
$$Cov(x,y) = E[(x-\mu_x)(y-\mu_y)]$$
- corrleation coefficient
$$\rho_{XY} = \frac{Cov(X, Y)}{\sigma_X \cdot \sigma_Y}$$

* 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| $$g(h) = c (1.5\frac{h}{a} - 0.5(\frac{h}{a})^3)$$|
|Exponential (GSLIB)| $$g(h) = c (1 - exp(\frac{-3h}{a_p}))$$|
|Exponential (gstat)| $$g(h) = c (1 - exp(\frac{-h}{a_t}))$$|
|Gaussian| $$g(h) = c (1 - exp(\frac{-3h^2}{a^2}))$$|
where

|h|lag distance|
|a|range|
|$$a_p$$|practical range equal to the distance at which 95% of the sill has been reached|
|$$a_t$$|theoretical range|
|c|sill|
&blankimg(variogram_model.JPG)

** Covariogram $$Cov({\bf h})$$
- a function that depends only on the displacement vector h.

** Semivariogram $$\gamma({\bf h})$$
$$\gamma({\bf h}) = \frac{1}{2}Var\bigg(Z({\bf s}) - Z({\bf s}+{\bf h})\bigg)$$
|$$Z({\bf s})$$|spatial process at lcation $${\bf s}$$|
|$${\bf h}$$|the displacement vector|

** Relation between Covariogram $$C({\bf h})$$ and Semivariogram $$\gamma({\bf h})$$
$$\gamma({\bf h}) = C({\bf 0}) - C({\bf h})$$
|$$C({\bf 0})$$|the variance of spatial process|

** Empirical semivariogram
$$\hat\gamma({\bf h}) = \frac{1}{2|N(h)|}Var\sum_{N(h)}(Z(s_i) - Z(s_j))^2$$

* 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

地球統計学

#contents

* 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
*** The joint probability distribution 
$$ Pr\lbrace z_{M+1} \leq Z_{M+1} \leq z_{M+1} + dz_{M+1},..., z_N \leq Z_N \leq z_N + dz_N | z_1,...,z_N \rbrace$$
$$ = Pr\lbrace z_{M+1} \leq Z_{M+1} \leq z_{M+1} + dz_{M+1} | z_1,...,z_M \rbrace $$
$$\,\,\,\times Pr\lbrace z_{M+2} \leq Z_{M+2} \leq z_{M+2} + dz_{M+2} | z_1,...,z_M,z_{M+1} \rbrace $$
$$\,\,\,\,\vdots$$
$$\,\,\,\times Pr\lbrace z_N \leq Z_N \leq z_N + dz_N | z_1,...,z_M,z_{M+1},...,z_N \rbrace $$

** 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
           $$G(u_i,z|(n+i-1)) = G(\frac{z-z^*_{SK}(u_i)}{\sigma_{SK}(u_i)})$$
      b. Draw a simulated value z(u_i) from the conditional distribution
           $$G(u_i,z|(n+i-1))$$
      c. Add simulated value to data-set (n+i-1)
 5. End simulation
 6. Transform the entire simulation back to the original data histogram

*** The mean of each conditional distribution
$$E[Z(u_i)|Z(u_{i-1})=z_{i-1},...,Z(u_1)=z_1] = \sum_{j=1}^{i-1}\lambda_j(u_i)z(u_j) = z_{SK}^*(u_i)$$

*** The variance of each conditional distribution
$$Var[Z(u_i)|Z(u_{i-1})=z_{i-1},...,Z(u_1)=z_1] = \sigma^2_{SK}(u_i) $$

* Kriging
** Main forms of linear kriging
| &bold(){Kriging Type} | &bold(){Mean} | &bold(){Minimal Prerequisite} | &bold(){Model Name} |
| Simple Kriging(SK) | Constant, known | Covariance | Stationary |
| Ordinary Kriging(OK) | Constant, unknown | Variogram | Intrinsic |
| Universal Kriging(OK) | Varying, unknown | Variogram | UK model |

** Common parts in derivation of the Kriging equations
** Estimated value
$$Z^* = \sum_{\alpha}\lambda_\alpha Z_\alpha + \lambda_0$$
- The weights $$\lambda_\alpha$$ depend on the location $$x_0$$ where the function is being estimated.
- $$\lambda_\alpha$$, $$\lambda_0$$ are selected so as to minimize the error $$Z^* - Z_0$$, characterized by its expected mean square $$E(Z^* - Z_0)^2$$
| $$Z_*$$ | The prediction at the point  $$x_0$$ |
| $$Z_\alpha$$ | the data at the point $$\alpha$$ |
| $$\lambda_\alpha$$ | weights |
| $$\lambda_0$$ | a constant that depends on $$x_0$$ |

** Take the kriging variance as the mean square error
Originally
 $$E(Z^* - Z_0)^2 = E\bigg(\sum_{\alpha}\lambda_\alpha Z_\alpha - Z_0 \bigg)^2$$
Adding the mean term $$\mu$$,
 $$E(Z^* - Z_0)^2 = E\bigg(\sum_{\alpha}\lambda_\alpha (Z_\alpha - \mu) - (Z_0 - \mu) \bigg)^2$$
Expand it and finally written as
 $$E(Z^* - Z_0)^2 = \sum_\alpha \sum_\beta \lambda_\alpha \lambda_\beta \sigma_{\alpha\beta} - 2\sum_\alpha \lambda_\alpha \sigma_{\alpha 0} + \sigma_{00}$$

b. $$E(Z^* - Z_0)^2 = Var(Z^* - Z_0) + [E(Z^* - Z_0)]^2$$
|$$\sigma_{\alpha\beta}$$ |Covariance between two sample points $$Z(x_\alpha)$$ and $$Z(x_\beta)$$ |
|$$\mu$$| the mean value|
|$$\sigma_{\alpha 0}$$| Covariance between one sample point and the estimated point $$x_0$$|
|$$\sigma_{00}$$| Variance at the estimated point $$x_0$$|

** Simple Kriging
Take the minimum of the mean square error
 $$ \frac{\partial}{\partial \lambda_\alpha} E(Z^* - Z_0)^2 = 2\sum_\beta \lambda_\beta \sigma_{\alpha \beta} - 2\sigma_{\alpha0} = 0$$
Therefore, Simple Kriging System is
 $$ \sum_\beta \lambda_\beta \sigma_{\alpha \beta} = \sigma_{\alpha 0} $$
Simple Kriging Variance
 $$ \sigma_{SK}^2 = E(Z^* - Z_0)^2 = \sigma_{00} - \sum_\alpha \lambda_\alpha \sigma_{\alpha 0} $$

** Ordinary Kriging(OK)
** External Drift Kriging (KDE)


* Under the condition of second-order stationarity
- means: spatially constant mean and variance
- Relations of covariance, correlation and variogram
$$C(0) = Cov(Z(u), Z(u+h)) = Var(Z(u))$$
$$\rho(h) = \frac{C(h)}{C(0)}$$
$$\gamma(h) = C(0) - C(h)$$
|$$C(h)$$|Covariance|
|$$\rho(h)$$|Correlation|
|$$\gamma(h)$$|Semivariogram|

&blankimg(variogram_covariance.JPG)

* Covariance 共分散
- ２つの変数がどのくらい同じように動くか
$$Cov(x,y) = E[(x-\mu_x)(y-\mu_y)]$$
- corrleation coefficient
$$\rho_{XY} = \frac{Cov(X, Y)}{\sigma_X \cdot \sigma_Y}$$

* 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| $$g(h) = c (1.5\frac{h}{a} - 0.5(\frac{h}{a})^3)$$|
|Exponential (GSLIB)| $$g(h) = c (1 - exp(\frac{-3h}{a_p}))$$|
|Exponential (gstat)| $$g(h) = c (1 - exp(\frac{-h}{a_t}))$$|
|Gaussian| $$g(h) = c (1 - exp(\frac{-3h^2}{a^2}))$$|
where

|h|lag distance|
|a|range|
|$$a_p$$|practical range equal to the distance at which 95% of the sill has been reached|
|$$a_t$$|theoretical range|
|c|sill|
&blankimg(variogram_model.JPG)

** Covariogram $$Cov({\bf h})$$
- a function that depends only on the displacement vector h.

** Semivariogram $$\gamma({\bf h})$$
$$\gamma({\bf h}) = \frac{1}{2}Var\bigg(Z({\bf s}) - Z({\bf s}+{\bf h})\bigg)$$
|$$Z({\bf s})$$|spatial process at lcation $${\bf s}$$|
|$${\bf h}$$|the displacement vector|

** Relation between Covariogram $$C({\bf h})$$ and Semivariogram $$\gamma({\bf h})$$
$$\gamma({\bf h}) = C({\bf 0}) - C({\bf h})$$
|$$C({\bf 0})$$|the variance of spatial process|

** Empirical semivariogram
$$\hat\gamma({\bf h}) = \frac{1}{2|N(h)|}Var\sum_{N(h)}(Z(s_i) - Z(s_j))^2$$

* 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