readme.txt

Kriging插值matlab toolbox

The GLOBEC Kriging Software Package - EasyKrig3.0, May 1, 2004

Copyright (c) 1998, 2001, 2004. property of Dezhang Chu and Woods Hole Oceanographic Institution.  
All Rights Reserved.

1. 	INTRODUCTION
1.1 	General Information
1.1.1	About kriging

This section provides a brief theoretical background for kriging. If the user(s) is not interested in 
the theoretical background, he/she can skip this section and go to section 1.1.2 directly.

Kriging is a technique that provides the Best Linear Unbiased Estimator of the unknown 
fields (Journel and Huijbregts, 1978; Kitanidis, 1997).  It is a local estimator that can provide 
the interpolation and extrapolation of the originally sparsely sampled data that are assumed to be 
reasonably characterized by the Intrinsic Statistical Model (ISM). An ISM does not require the quantity 
of interest to be stationary, i.e. its mean and standard deviation are independent of position, but rather 
that its covariance function depends on the separation of two data points only, i.e.

        E[(z(x) - m)(z(x') - m) ] = C(h),                       (1)

where m is the mean of z(x)  and C(h) is the covariance function with lag h, with h being the distance 
between two samples x and x':

        h = || x - x' ||.                                       (2)


Another way to characterize an ISM is to use a semi-variogram,

       gamma(h) = 0.5* E[ (z(x) - z(x') )^2].                   (3)

The relation between the covariance function and the semi-variogram is

      gamma(h) =  C(0) - C(h).                                  (4)


The kriging method is to find a local estimate of the quantity at a specified location, x(L). 
This estimate is a weighted average of the N adjacent observations:

      z(x(L)) = sum( lambda(i) z(x(i)),                         (5)

where i is from 1 to N, and x(L) are the coordinates of an arbitrary point whose value is what 
we want to estimate.


The weighting coefficients lammbda(i) can be determined based on the minimum estimation variance criterion:

     See Eq.(6) in Description.doc file                         (6)

subject to the normalization condition. 						

      sum(lambda(i)) = 1,                                       (7)
      
where i is from 1 to N. Note that we don't know the exact value at  , but we are trying to find a predicted 
value that provides the minimum estimation variance. The resultant kriging equation can be expressed as  

     See Eq.(8) in Description.doc file                         (8)

where mu is the Lagrangian coefficient. In addition, we have replaced the covariance function with 
the normalized covariance function [normalized by C(0)]. Equivalently, by using  Eq. (4), the kriging 
equation can also be expressed in terms of the semi-variogram as

    See Eq.(9) in Description.doc file                          (9)

where we have used normalized semi-variogram, i.e., semi-variogram normalized by C(0) as we did in deriving Eq. (8).

Having obtained the weighting coefficients (lambda_beta) and the Lagrangian coefficient (mu) by solving either Eq. (8) or 
Eq. (9), the kriging variance, Eq. (6), can be expressed as:

    See Eq.(8) in Description.doc file                          (10)		

The above equations are the basis of the Easykrig software package.

1.1.2	Brief description of EasyKrig3.0

The EasyKrig program package uses a Graphical User Interface (GUI) to simplify the operation. It requires MATLAB 5.3 or 
higher with or without optimization toolbox (see section 2.2) and consists of five components, or processing stages: 
(1) data preparation, (2) variogram computation, (3) kriging, (4) visualization and (5) saving results. It allows the 
user to process anisotropic data, select an appropriate model from a list of  variogram models, and a choice of kriging 
methods, as well as associated kriging parameters, which are also common features of the other existing software 
packages. One of the major advantages of this program package is that the program minimizes the users' requirements to 
"guess" the initial parameters and automatically generates the required default parameters. In addition, because it 
uses a GUI, the modifications from the initial parameter settings can be easily performed. Another feature of this 
program package is that it has a built-in on-line help library that allows the user to obtain the descriptions of the
use of parameters and operation options easily.

The current EasyKrig3.0 is the upgraded version of the previous version (EasyKrig2.1). In addition to having corrected 
some programming errors in the previous version (mostly GUI related errors), there are many new features included in 
the current version: