## GEOSTATS: kriging weighted values?

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• Hi- Suppose I have some spatial samples with weights (w1 w2 ....) and I want to use these weights in addition to kriging weights and also suppose (and a big
Message 1 of 7 , Oct 24, 2000
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Hi-

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

I get I some cases (zero nugget variogram and some negative weights) unreasonable results e.g.

V1 w1 wk1
30 .1 1.2
50 .9 -.2

Unreasonable results are a well known result with 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

My guess is that the sample weights should be coupled WITHIN the kriging equations something like:
w1w1Gamma11 w1w2Gamma12 ..... ..... 1 wk1 w1Gamma1b
w2w1Gamma21 w2w2Gamma22 ..... .... 1 wk2 = w2Gamma2b
...... ...... ..... ..... 1 .. .....
1 1 ..... ......0 mu 1

Before I work thru the math for my guess and code a solution I would like input on
1) Has this problem been solved before?
2) Is a C or Fortran or public .exe version of the solution available (3d preferred)

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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• 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
Message 2 of 7 , Oct 24, 2000
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Dear Dr. Sandefur:

(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-
>
> Suppose I have some spatial samples with weights (w1 w2 ....)
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
>
Sum(Value1*w1*wk1+Value2*w1*wk2+...)/Sum(w1*wk1+w1*wk2+...)
>
>I get I some cases (zero nugget variogram and some negative weights)
unreasonable results e.g.
>
>V1  w1 wk1
>30   .1  1.2
>50   .9  -.2
>
>Unreasonable results are a well known result with 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
>
>My guess is that  the sample weights should be  coupled
WITHIN  the kriging equations something like:
>  w1w1Gamma11 w1w2Gamma12 ..... .....
1               wk1              w1Gamma1b
>  w2w1Gamma21 w2w2Gamma22 ..... ....
1               wk2       =   w2Gamma2b
......                  ......                  .....  ..... 1                 ..               .....
>
1                        1                     ..... ......0                 mu                      1
>
>Before I work thru the math for my guess and code a solution  I
would like input on
>   1) Has this problem been solved before?
>   2) Is a C or Fortran or public .exe version of  the
solution available (3d preferred)
>
>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
>
>
>
>
>--
>*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.
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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/
-----------------------------------------------------

• Hi Bob, long time no see It is traditional in thin reef deposits (such as those in South Africa) to use an accumulation value which is length times grade. I
Message 3 of 7 , Oct 25, 2000
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Hi Bob, long time no see

It is traditional in thin reef deposits (such as those
in South Africa) to use an accumulation value which is
length times grade. I have also seen similar composite
values used in iron ore where a weight times grade is
used as the principle variable.

These composite values generally give acceptable
semi-variograms and can then be kriged. The result is
(of course) an accumulation value and has to be
matched with a kriged length, weight or density
variable to achieve estimates in the original grade
units.

As a simple example, in a Wits Reef length times grade
is considered to be the 'natural' variable. This is
modelled and kriged. The width of the reef is also a
natural geological variable and this can be modelled
and kriged. Final grade is then kriged accumulation
divided by kriged width. Error on this is found using
the original combination of relative variances
suggested by Matheron.

This avoids all problems of messing around with
kriging systems and/or the kriging weights.

All of the above is predicated on your "weights" being
a physical phenomenon. If you are talking about
conceptual weights, then you probably need something
like "soft kriging" with a Bayesian input.

Isobel Clark
http://uk.geocities.com/drisobelclark

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• 26th Oct., 2000 Dear Dr Isobel clark, I was made part the list of geostats from yesterday only and it is my good fortune to see your reply in my first
Message 4 of 7 , Oct 26, 2000
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26th Oct., 2000

Dear Dr Isobel clark,

I was made part the list of geostats from yesterday only and it is my
of corresponding with you on geostatistics during 1983-86.

Currently I am working with iron ore mines of Tata Steel in India. The
ore is won by opencast mining method with 12m high benches. The iron
ore consists of several types of ores each having definable quality
in terms of Fe, Al2O3 and SiO2; and physical characterisics such as
hard ore, soft ore, flaky ore, bluedust etc. making them to be
processed in wet and dry methods. In such a case, using all the iron ore
and under-estimation of high grade ores if one sees block-wise. This
type of estiamtes are not acceptable to the production staff. Hence
I used variogram parameters of the entire iron ore samples to specific
ore type estimation for 12m benches with 100mx100m blocks using the
core smaples of that specific ore type, which satisfied the production
staff. Is the procedure correct? If not, can you please suggest a better
method?

With regards

Dr P.V. Rao

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• 2th Oct., 2000 Dear Dr Isobel Clark, Thanks for your advice on iron ore deposit. I have a further quiery to you on the same subject. You may appreciate that
Message 5 of 7 , Oct 27, 2000
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2th Oct., 2000

Dear Dr Isobel Clark,

Thanks for your advice on iron ore deposit. I have a further quiery
to you on the same subject.

You may appreciate that when we take samples (let us say 3m length)
along the vertical boreholes for each 12m bench height, different
ore types with varying lengths willb e encountered. In such a case,
when the mid-slice plans are prepared, usually the maximum length of
ore (core length) is designated in the slice plans. When the variogram
has to be calculated for 12m samples how to composite the radicals
for a given ore type? if all the samples falling within 12m bench
height are taken for compositing, then the sample assay value don't
truly represent the designated ore type in the slice.

have access to internet at my place of work, which is definitely a
constraint in reviewing the published literature.

Dr P.V. RAO
Dy. Divisional Manager (planning)
Tata Steel, India
----------------------------------------------------------------------

On Thu, 26 Oct 2000, [iso-8859-1] Isobel Clark wrote:

> Dear Dr Rao
>
> How lovely to hear from you.
>
> In dealing with an area with several sub-populations
> of different geology, it is always best to use only
> the samples of the relevant geology.
>
> It would improve the estimation further if you can:
>
> (a) produce a semi-variogram model for each ore type
>
> (b) allow for the skewed nature of the data by using
> (say) a modified lognormal transformation.
>
> We have followed this pattern successfully in iron ore
> deposits in South Africa and in the USA.
>
> Let me kno wif I can be of any more help. Perhaps you
> at http://www.stokos.demon.co.uk
>
> With best regards
> Isobel Clark
>
>
>
> ____________________________________________________________
> Do You Yahoo!?
>

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• ... The semi-variogram should always be calculated on your basic core section length. Represent a block (discretisation) by four points in the vertical
Message 6 of 7 , Oct 27, 2000
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> 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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• 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
Message 7 of 7 , Oct 31, 2000
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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
>
>
>
>
>
>
> ____________________________________________________________
> Do You Yahoo!?
> --
> *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.
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> "unsubscribe ai-geostats" in the message body.
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>

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