地统计内插方法_克里金插值(Kriging)

8.2 Random Field
Statistical Moments • The first moment is the mean or expected value of a random field, which describes its trend
• The second moment is the autocorrelation function
•Histogram: graphical representation of frequency table.– Common to use a constant class width –The height of each bar corresponds to the frequency •Cumulative frequency table and histogram -record the total number of values below certain cutoffs.
• The correlation of a random field can be measured with the covariance
• also called covariance function
8.2 Random Field
Variogram While the correlation (or similarity) between U(xi) and U(xj) is expressed with the covariance, their dissimilarity (or, variability) may be measured by their average squared difference:
Variogram When the random field U is second-order stationary (with constant mean and variance, and covariance dependent on separation vector),
Then,
8.2 Random Field
8.2 Random Field
Insight to Variogram
h C0 w[1 eh/ a2 ]
h
C0
w 1 exp
h12 a12
h22 a22
Isotropic Variogram
Anisotropic Variogram
8.2 Randoe γ(xi,xj) is the so called semi-variogram, or simply variogram. It can be shown that if the variance and covariance of U(x) exist, one has
8.2 Random Field
When is Geostatistics needed?
• How to construct high-resolution DEM that are consistent with data of various types and scales? • Where to locate the next observation points? How many to locate?
标题
Geographic Information System
Chap 8. Spatial Statistics in GIS
标题
Geographic Information System
8.1 Introduction 8.2 Random Field 8.3 Simple Kriging
8.1 Introduction
Variogram
8.2 Random Field
Variogram
8.2 Random Field
Real Example
8.2 Random Field
Real Example
h C0 w[1 eh/ a2 ]
式中：C0为块金值；C0+w为基台值；h为样本点的空间距离；α 为变程。
Sill
8.2 Random Field
Variogram Explanation to Nugget: ➢ Ideally the nugget should be zero, because the samples from the nearly same point should have practically the same value.
event falls within a certain range • The elevation in an area
8.1 Introduction
Some common summary statistics: –Measurements of location: mean, median –Measurements of spread: variance, standard deviation, coefficient of variation –Measurements of shape: coefficient of skewness
•One should decide on the following factors and determine the corresponding parameters: –Selected separation lags –Directions –The number of pairs of data at each lag
•What are the elevation values in an unmeasured location? •How to infer elevation values from other measurements in the same and other locations?
8.1 Introduction
–Range (变程）: the distance at which the variogram levels off. It implies that beyond the range, the samples are no longer correlated.
8.2 Random Field
Variogram Some quantities of a variogram: –Sill （基台值）: the value of the variogram where it levels off. –Nugget （块金值）: the value of the variogram at lag h=0
Spatial heterogeneity/ variability
Low variance
High variance
8.1 Introduction
When is Geostatistics needed?
To answer various estimation problems such as
8.2 Random Field
Variogram •In practice, a variogram is almost always computed with measurements from a discrete number of points such as well locations
•Example with data on grid: –25 data points on a 5x5 grid of separation distance 10m
8.2 Random Field
Variogram
8.2 Random Field
Variogram
8.2 Random Field
8.1 Introduction
8.1 Introduction
8.1 Introduction
8.1 Introduction
8.1 Introduction
8.1 Introduction
8.2 Random Field
A random variable (stochastic variable) is defined by a set of possible values (the sample space) and a probability distribution over this set. A random function, U(x), is an indexed collection of random variables. Also called random field or stochastic process.
–k(x) as permeability along a randomly packed sand column –z(x) as elevation in an area A sample function or realization of a random field: one particular trial or experiment out of many; The ensemble of the random field
x/km
Elevation/m
8.2 Random Field
N-point CDF & PDF
•The statistical properties of a random field are completely defined with all the multivariate (n-point) PDF or CDF. •In reality, it is however extremely complicated and costly to infer the multivariate PDFs.