Geostatistics


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地球統計学


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

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

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

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
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
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