Geovariances: Where no one has gone before

Isatis offers a large panel of estimation methods. They can be distinguished by:

• The model to which the field is associated (if any)
• The algorithm used to answer the question (generality, efficiency, speed...)

In addition to the model, all the algorithms depend on a neighborhood (recipe for selecting a part of the information available for the estimation of a target point).

Model-Free Methods

A variety of algorithms (with a limited number of parameters) are provided, such as:

• Inverse distance
• Closest neighbour
• Least square fit (order 2)
• Moving average
• Moving median
• Moving projected slope
• Discrete spline
• Kriging with a linear variogram (no fitting)
• Kriging with a spline generalized covariance (no fitting)

Kriging

• This well-known method provides the Best Linear Unbiased Estimator of the target variable. For each target point, it provides:
  
  
  • The estimated value
  • The corresponding standard deviation which measures the quality of the estimation
  • Several additional outcomes: Sum of Weights, Sum of Positive Weights, Lagrange Parameter, Variance of Z*, Correlation of Regression Z/Z*, Covariance of Regression Z/Z*, Slope of Regression Z/Z*
    
  • For each target point, this method not only gives the estimated value of the variable, but any function derived linearly from the variable, such as:
    
  • The punctual value
  • Its mean value over a cell
  • Its derivatives (gradients)
  • Its convoluted value
  • Its derivatives
    
• In Isatis, kriging algorithms cope with:
  
  
  • Multivariate cases
  • Stationary hypothesis
  • Intrinsic hypothesis
  • Non-stationary hypothesis
    
• Specific options of kriging deal with various types of data, such as:
  
  
  • Data provided with measurement errors
  • Data corresponding to the information convoluted and combined with an instrumental noise
  • Data provided through inequalities (Conditional Expectation with Inequalities)
  • Data consisted in the depth information and its gradient
  • Data consisted in a variable and its laplacian
  • Sparse accurate data and a related indirect exhaustive variable: external drift, collocated cokriging.
    
• Distinctive methodologies making use of kriging techniques have also been developed for specific issues:
  
  
  • Polygon Kriging for global estimations
  • Universal Kriging when the variable may be considered as the sum of a stationary residual and a drift
  • Disjunctive Kriging and Uniform Conditioning for non linear issues and global respurce estimation
  • Lognormal Kriging for highly skeewed distributions
  • Indicator Kriging for the estimation of discrete variables
  • Factorial Kriging Analysis to extract components of a model
  • Confidence Intervals to assess the confidence intervals within the frame of the Discrete Gaussian Model, based on simple kriging of gaussian values.
    
• Isatis allows cross-validation (blind test) :
  
  Cross-validation is used for double checking the quality of the model with regard to the data. Each sample is considered in turn as a target and estimated by kriging using the remaining information. The experimental error between the estimation and the true value is compared to the theoretical error predicted by the model through the standard deviation of the estimation.