Grade Estimation with ESTIMA

This and related topics deal with the subject of grade estimation in Studio RM. This is a complex and extensive subject, and for this reason, the subject has been broken down into smaller categories.

The following topics cover grade estimation using ESTIMA and supporting processes. ESTIMA supports univariate estimation (for multivariate functionality in Studio RM, see COKRIG). Many topics apply to both univariate and multivariate estimation.

Note: Another process - GRADE - is available in many Studio products. It is a lightweight estimation process that may be suitable for some estimation scenarios.

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Features of ESTIMA

The main features of ESTIMA are:

  • A consistent set of search volume and estimation parameters for all interpolation methods

  • Optimization of sample searching to improve speed

  • Multiple grades can be estimated in a single run

  • The same grade can be estimated by different methods

  • Different search volumes and estimation parameters can be used for the different grades

  • Rectangular or ellipsoidal search volume with anisotropy

  • A dynamic search volume allowing the volume to be increased if there are insufficient

  • samples

  • Restriction of the number of samples by octant and key field

  • Estimation by zone, with separate parameters for each zone

  • Wide selection of variogram model types for both normal and lognormal kriging

  • Automatic transformation of data if the input model is a rotated model

  • Unfolding option available for all estimation types

  • Parent cell estimation

  • Selective update of partial model

ESTIMA requires an Input Prototype Model and a set of Sample Data as input. Usually the Input Prototype Model will already contain cells and sub-cells which represent, for example, a geological structure. In this case, grade values are interpolated into the existing set of cells and sub-cells. If however an empty prototype is specified (i.e. it does not contain any cells or sub-cells), ESTIMA will create cells and sub-cells in the area around the samples as defined by the search volume.

From here on any reference to a model cell will include both cells and sub-cells. A full cell is referred to as a parent cell.

The Sample Data file contains the data which is used to estimate cell grades. At a minimum, the data must include the X, Y and Z coordinates of each sample and at least one grade value. ESTIMA requires a search volume to be defined. This is the volume, centered on the cell being estimated, which contains the samples to be used for grade estimation. More than one search volume can be defined, so that different grades can have different search volumes. The parameters describing the search volume(s) are supplied to ESTIMA from the Search Volume Parameter file.

ESTIMA also requires a set of estimation parameters to be defined for each grade to be estimated. These parameters are also supplied to ESTIMA from a file called the Estimation Parameter file. It will include items such as the estimation method, the search volume reference number the power (for Inverse Power of Distance calculations). Each cell is selected in turn from the Input Prototype Model and the samples lying within the search volume are identified. Each grade specified in the Estimation Parameter file is estimated, and the results are written to the Output Model file.

A summary of the files used by ESTIMA is as follows:

Process

Description

PROTO

Input Prototype Model

IN

Sample Data

SRCPARM

Search Volume Parameters

ESTPARM

Estimation Parameters

VMODPARM

Variogram Model Parameters

STRING

Unfolding Strings

MODEL

Output Model

SAMPOUT

Sample Output

Other information is supplied to the ESTIMA process as fields and parameters.

Estimation Methods

The estimation methods provided by ESTIMA include:

  • Nearest Neighbor

  • Inverse Power of Distance

  • Ordinary Kriging

  • Lognormal Kriging

  • Simple Kriging

  • Sichel's t Estimator

  • F value

  • Lagrange Multiplier

Kriging and Cokriging

Kriging can be understood as linear prediction or a form of Bayesian inference. Kriging starts with a prior distribution over functions. This prior takes the form of a Gaussian process: N samples from a function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location of two points.

A set of values are then observed, each value associated with a spatial location. Now, a new value can be predicted at any new spatial location, by combining the Gaussian prior with a Gaussian likelihood function for each of the observed values. The resulting posterior distribution is also a Gaussian, with a mean and covariance that can be simply computed from the observed values, their variance, and the kernel matrix derived from the prior.

From the geological point of view, Kriging uses prior knowledge about the spatial distribution of a mineral: this prior knowledge encapsulates how minerals co-occur as a function of space. Then, given a series of measurements of mineral concentrations, Kriging can predict mineral concentrations at unobserved points.

Kriging is a family of linear least squares estimation algorithms. The end result of Kriging is to obtain the conditional expectation as a best estimate for all unsampled locations in a field and consequently, a minimized error variance at each location. The conditional expectation minimizes the error variance when the optimality criterion is based on least squares residuals. The Kriging estimate is a weighted linear combination of the data. The weights that are assigned to each known datum are determined by solving the Kriging system of linear equations, where the weights are the unknown regression parameters. The optimality criterion used to arrive at the Kriging system, as mentioned above, is a minimization of the error variance in the least-squares sense.

Kriging only considers a correlogram while co-kriging takes a secondary variable (which exhibits some correlation with the primary variable) into acount using the cross-correlogram. Cokriging can perform advantageously (that is, improve the interpolation) if:

  • the primary variable is considerably undersampled
  • variogram models differ in their shape

Cokriging is an interpolation technique that allows one to better estimate map values by kriging if the distribution of a secondary variate sampled more intensely than the primary variate is known. If the primary variate is difficult or expensive to measure, then cokriging can greatly improve interpolation estimates without having to more intensely sample the primary variate. One example of a method used that employs cokriging is Uniform Conditioning.

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