My question relates to the use of geostatistics in a field we call

quantitative metallography (in metallurgy).

The first paper I have seen on this subject was Sampling in Quantitative

Image Analysis. Examples of the Application of Sampling to Measurements of

the Contents of Phases in Sinter Types and of Inclusions in Steel Types',

T.Hersant and D.Jeulin, Rev. Met. Mem. Sci., 1976, Vol. 73, (7/8)pp

503-517). They use the Variogram to describe the error committed if the

phase content of X, P(x), measured at point x is attributed to point x+h.,

i.e.

2*gamma(h) =E{[P(x+h)- P(x)]^2}..................(1)

The TAS texture analyser is used to sample the phase type at a series

points, giving P(x)=1 if x pertains to phase X and P(x)=0 if it does not.

(NB from stereology a standard result is the equality of point, area and

volume fractions).

By algebraic manipulation and using this 'indicator ?' approach equation

(1) becomes in a semivariogram form :-

gamma(h)= E{P(x)^2}- E{P(x)*P(x+h)}

Do they get this because E{[P(x)]^2} = E{[P(x+h)]^2} ?)

They let C(h) = E{P(x)*P(x+h)}

This gives gamma(h)=C(0)-C(h)

where C(h) is the covariance of phase X, i.e. the probability that the two

points separated by a distance of h both belong to X.

They then state that the properties of the semivariogram are

C(0)= volume fraction of phase X, and

C(infinity)= (volume fraction of phase X)^2

Can anyone explain how these these limits arise ?

Does C(0)= Vv because E{[P(x)]^2} = (0)^2*(1- Vv) + (1)^2*(Vv) ?

However, I cannot see why C(infinity) = (Vv)^2 , the plateau of the

covariance, could anyone explain this ? This may be simple to someone who

has attended lectures in this field, is it ?

They determined the approximate formula for the extension variance (also

termed estimation variance) equations (equations 12 to 14 in the paper)

and certain ranges, but is not clear how these arise ? I believe that J.

Chone also used these ranges (for lineal, areal and volumetric phase

anlysis) in the papers which he published on this subject whilst at IRSID

around 1976-1977.

Finally, is there a less exacting introduction to the subject matter of

Chapter 9 on The Covariance in J.Serra's book on Mathematical Morphology,

that would be suitable for a non-mathematician ? Here he presents equations

which show the difference on going from a global representation of the

covariance to a local one, equations IX-1 to IX-25. Is there explanation

available of this and the derivation which is easier to follow ?

[As an aside, could I point out that I am not a mathematician, but a

metallurgist who has received mathematical training as an engineering

school undergraduate (over 20 years ago), so please excuse my ignorance.

For background reading I have looked at Chapter 9 of 'Image Analysis and

Mathematical Morphology' by J. Serra on The Covariance, but found it heavy

going (as stated above I could not follow fully equations IX-1 to IX-25,

i.e. moving from a global to a local representation of the covariance

function, which seems to be relevant), and Chapter 2 of 'Mining

Geostatistics' by Journel and Huijbregts on The Theory of Regionalised

Variables, which I was able to understand a greater proportion of (useful

for extension of variance, derivation of the general equation). For an

introduction to general geostatistics I have read 'Practical Geostatistics'

by Isobel Clark.]

Thank you for your attention.

--

*To post a message to the list, send it to ai-geostats@....

*As a general service to list users, please remember to post a summary

of any useful responses to your questions.

*To unsubscribe, send email to majordomo@... with no subject and

"unsubscribe ai-geostats" in the message body.

DO NOT SEND Subscribe/Unsubscribe requests to the list!