Define variables and write an equation to model the relationship in each table

Show students that in order to graph those linear relationships, you had to type an algebraic rule or an equation into the graphing calculator for each. Have the students write the equations on bottom of the journal page. Do this for B, C, and D.

Define variables and write an equation to model the relationship in each table

Introduction Perhaps it would be useful here at the very beginning to clear up any possible ambiguity which arises because of the use of the title Geostatistics.

The application of this theory to problems in geology and mining has led to the more popular name Geostatistics. The contents of this book are confined to the simplest application of the Theory of Regionalised Variables, that of producing the 'best' estimation of the unknown value at some location within an ore deposit.

This technique is known as kriging. The purpose of this text is to provide a simple treatment of Geostatistics for the reader unfamiliar with the field. The subject may be discussed at a number of levels of mathematical complexity, and it is the intention here to keep the mathematics to a necessary minimum.

Some previous knowledge must be assumed on the reader's part of basic concepts of ordinary statistics such as mean, variance and standard deviation, confidence intervals and probability distributions.

Readers without this background are referred to any one of a large number of excellent basic texts.

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The application of Geostatistics to the estimation of ore reserves in mining is probably its most well known use. However, it has been emphasised time and again that the estimation techniques can be used wherever a continuous measure is made on a sample at a particular location is space or timei.

Since most applications -- and most of the author's experience -- are confined to the mining field, so will most of the examples in this book. Also, there will be a tendency to talk of 'grades' rather than 'sample values', for brevity if nothing else.

define variables and write an equation to model the relationship in each table

If the reader is interested in other fields, it suffices to replace 'grade' by porosity, permeability, thickness, elevation, population density, rainfall, temperature, fracture length, abundance or whatever. The application of statistical methods to ore reserve problems was first attempted some 30 years ago in South Africa.

The problem was that of predicting the grades within an area to be mined from a limited number of peripheral samples in development drives in the gold mines. Gold values are notoriously erratic, and when plotted in the form of a histogram show a highly skewed distribution with a very long tail into the rich grades.

Normal Gaussian statistical theory will not handle such distributions unless a transformation is applied first. Sichel applied a log-normal distribution to the gold grades and achieved encouraging results.

He then published formulae and tables to enable accurate calculation of local averages for log-normal variables, and also confidence limits on those local averages.

Three major drawbacks exist in the application of Sichel's 't' estimator. The 'background' probability distribution must be log-normal. The samples must be independent. There is no consideration taken of the position of the samples -- all are equally important. However, the technique proved very useful in the gold mines, especially since some measure of the reliability of the estimator was provided.

It also laid the base for further statistical work by providing the conceptual framework necessary, i. At this stage, it was assumed that all the samples in a given area came from the same probability distribution -- a log-normal one -- and this assumption is known in ordinary statistics as 'stationarity'.

Subsequent to this work attempts were made to incorporate position and spatial relationships into the estimation procedure. Two things seemed sensible: These were tackled in the s and early s by the introduction of Trend Surface Analysis. In South Africa, trends were picked out by forming a 'rolling mean' which produced a smoothed map so that high and low areas could be distinguished.

In the United States a 'Polynomial Trend Surface' analysis was propounded which used a statistical technique to fit a mathematical equation to describe the trend. Both methods have one thing in common -- the basic assumptions about the statistical characteristics of the deposit.

These assumptions have been extended from the 'stationarity' one, by stating that the sample value is expected to vary from area to area in the deposit.

Some areas are expected to be rich, some to be poor. This expectation can be expressed as a reasonably smooth variation, either by a smoothed map or a relatively simple equation.

Round about this trend there is expected to be random variation. That is, the value at any point in the deposit is supposed to comprise i a 'fixed' component of the trend which is probably unknownand ii a random variable following one specific distribution.

Thus the stationarity has been shifted one step; the expected grade may vary slowly, but the random component is 'stationary'.

We have also dropped the log-normality assumption. This approach is quite useful for an overview of the deposit, but, except in heavily sampled areas like the gold mines, is not really useful for local estimation.

Hypothetical sampling and estimation situation Let us consider the problem of local estimation, e. It seems reasonable to evolve an estimation procedure which gives more importance to sample 1 than to sample 5. A whole range of methods have been produced to decide on the 'weight' accorded to each sample, mostly based on the distance of the sample from the point being estimated.Practical Geostatistics () was the third (and probably shortest) book ever published on geostatistics.

Based on 7 years of teaching technical valuation to undergraduate and postgraduate mining engineers, examples were drawn from the author's consultancy experience. Free Online Scientific Notation Calculator.

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