地统计内插方法_克里金插值(Kriging)

8.1 Introduction
Probability density function (PDF): p(u)– Discrete PDF: assigns a probability to each event • The outcome of flipping a coin • The number of road in an area Continuous PDF: determines the probability that an
•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
•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.
–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
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.
8.1 Introduction
Review of Probability
•Frequency table records how often (in terms of percentage) observed values fall within certain intervals or classes.
➢it can be attributed to measurement errors or the fact that the data have not been collected with a sufficiently small interval to show the underlying continuous behavior.
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
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.
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.
• 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:
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
–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
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。
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
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
•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
8.2 Random Field
Statistical Moments • The first moment is the mean or ex百度文库ected value of a random field, which describes its trend
• The second moment is the autocorrelation function
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 Random Field
Insight to Variogram
Variogram The theoretic variogram is computed as The sample variogram is computed as
where N(h) is the number of pairs with the separation lag h=xi-xj
8.2 Random Field
First view to Variogram
h C0 w[1 eh/ a2 ]
w=10
w=5
8.2 Random Field
Variogram Some quantities of a variogram:
Distance>Range (变程） 两点的高程完全独立
where 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?