- Dear Dr. Sandefur:

I thought about this problem quite a bit for work with finite populations

(lake and stream survey data that had a stratified random sampling design).

I don't think including sample weights as you have is valid (i.e, it won't

give the optimal solution). The point is, you don't know where the other

members of the population represented by the weight on a particular sample

are geographically. We found two solutions.

In one paper, we cokriged to estimate for a finite population of stream

nodes from a unequal-probability sample of nodes. Here we didn't make

any use of sample weights and compared population estimates obtained from

a Horowitz-Thompson estimator to those obtained by cokriging, which brought

in ancillary spatial information (elevation).

In another paper, we kriged by stratum (see the report on Pattern-plus

on my webpage). This means calculating separate variograms, etc.

I never finished, but I was working on creating a full variance-

covariance matrix by assuming that the strata had different sills (total

variances) but the same model form and nugget. The sample data can

then be standardized to have the same diagonal (sill) and the VC matrix can

be filled with entries from the standardized variogram (correlogram).

Then it should be possible to krig the whole system together and back

transform to get the final interpolated estimates. This should work

if you have samples that belong to sub-populations that differ in their

means and variances, but not the degree of autocorrelation. I have a FORTRAN

program written to do this that I've never gotten around to debugging, if

anyone wants to take it on :-).

Good luck, and I'd like to hear about it if you find another solution.

Yetta

At 06:27 AM 10/24/00 -0700, you wrote:>Hi-

and I want to use these weights in addition to kriging weights and also suppose (and a big suppose it is) that I have the (a) variogram(gamma). If I krige ignoring (w1 w2...) and weight the kriging weights (wk1 wk1 ...) with (w1 w2) ie

>

> Suppose I have some spatial samples with weights (w1 w2 ....)>

Sum(Value1*w1*wk1+Value2*w1*wk2+...)/Sum(w1*wk1+w1*wk2+...)

> Answer=>

unreasonable results e.g.

>I get I some cases (zero nugget variogram and some negative weights)>

and usually indicate a (local) inconsistency between the data and the variogram but I think the problem is exacerbated by not allowing for the weights w1.. in the kriging matrix

>V1 w1 wk1

>30 .1 1.2

>50 .9 -.2

> answer=90

>

>Unreasonable results are a well known result with negative weights>

WITHIN the kriging equations something like:

>My guess is that the sample weights should be coupled> w1w1Gamma11 w1w2Gamma12 ..... .....

1 wk1 w1Gamma1b> w2w1Gamma21 w2w2Gamma22 ..... ....

1 wk2 = w2Gamma2b>

...... ...... ..... ..... 1 .. .....>

1 1 ..... ......0 mu 1>

would like input on

>Before I work thru the math for my guess and code a solution I> 1) Has this problem been solved before?

solution available (3d preferred)

> 2) Is a C or Fortran or public .exe version of the>

ai-geostats@....

>thanx

>

>bob sandefur

>

>Principal Geostatistician

>Pincock Allen & Holt

>International Minerals Consultants

>274 Union Suite 200

>Lakewood CO 80228

>USA

>303 914-4467 v

>303 987-8907 f

>

>

>

>

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------------------------------------------------------

Yetta Jager

Environmental Sciences Division

Oak Ridge National Laboratory

P.O. Box 2008, MS 6036

Oak Ridge, TN 37831-6036

U.S.A.

OFFICE: 865/574-8143

FAX: 865/576-8543

Work email: jagerhi@...

Home email: hjager@...

WEBpage: http://www.esd.ornl.gov/~zij/

-----------------------------------------------------

- Dear Dr Isobel Clark,

It seems I am learning more now than what I could do sofar through reading

books/ periodicals. Thanks for your kind response.

I ahve to querries to make:

1) What is "Practical geostatsics 2000?" Is it a book published by you?

May I have the details of it please? I stay, first of all in India and

then in a remote place like Noamundi (75 year old iron ore mine producing

over 6 mtpa!) where access to internet and other modern means of

communications are yet to be made available.

2) When the directional variograms are made, one finds that sill value

do not reach the total variance of the smaples. Even then, when the

variogram parameters are used in kriging, we try to nullify the

variation in range by giving anisotropy factors in 3 dimensions. However,

the same is not accomodated for sill value in different directions as

there is provision to give only one C0 and one C1 value unless we have

nested variograms.

I hope I could explain my query.

3) Madam, will it be possible to share your worked out examples on

indicator kriging on iron ore deposits?

Thanks

P.V. Rao

---------------------------------------------------------------------On Fri, 27 Oct 2000, [iso-8859-1] Isobel Clark wrote:

> > Thanks for your advice on iron ore deposit. I have

> > a further quiery to you on the same subject.

>

> The semi-variogram should always be calculated on your

> basic core section length. Represent a block

> (discretisation) by four 'points' in the vertical

> direction when it is estimated. All software which

> does block estimation should allow you to specify the

> number of points in each direcion.

>

> If you have more than one type of ore in a particular

> block, the most reliable process is as follows:

>

> (1) krige a value for that block for each ore type

> which is present, using only the samples from that ore

> type.

>

> (2) use an indicator to krige the proportion of the

> block in each ore type. For example, if you have two

> ore types, each intersection with ore type A sould be

> given a value of 1. Intersections with B should be

> given a value of 0. Semi-variograms can then be

> constructed on these indicator values and the kriging

> result will be an estimated proportion of the block

> which is in type A.

>

> If you have more than two ore types, you can start

> with A being 1 and all others 0. Then you remove the A

> samples, call B 1 and the others 0. This is often

> called multiple nested indicators.

>

> We have used this successfully in ore deposits with

> several mineralisation types which cannot be separated

> geographically.

>

> If (as with other iron ore mines I know) your software

> is only operating in two dimensional slices, you can

> approximate the best answer by compositing only within

> the specific ore types and weighting by length or

> physical weight, in the same way that South African

> gold miners do with 'accumulation' values in reef

> deposits.

>

> I hope this helps. Indicator kriging is illustrated in

> Practical Geostatistics 2000, in Chapter 12.

>

> Isobel Clark

>

>

>

>

>

>

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