## [ai-geostats] Re: Kriging Small Blocks

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• Jul The warning about kriging small blocks is about small relative to the sampling density. For example, less than about one-third of the grid spacing. The
Message 1 of 5 , Jul 19, 2004
Jul

"small" relative to the sampling density. For example,
less than about one-third of the grid spacing.

The warning is the same as the one about 'point'
kriging (mapping) The map is too smooth - or, at
least, a lot smoother than the real surface would be.
High value areas will be under-estimated and low value
areas will be over-estimated.

If your objective in kriging is to obtain general maps
of an area with an idea of where the highs and lows
are, then ordinary kriging is sufficient. The over-
and under- estimations cancel out on average.

In mining applications, where block kriging
originated, most applications require a 'cutoff',
where values below a certain value are not included in
the 'plan'. In this case, mapping or estimating small
blocks will result in an over-estimation of 'payable'
ground and an under-estimation in average value.

In pollution or environmental applications, the areas
at risk will be under-estimated as will the true
toxicity or risk factors.

There are two major ways round this problem:

(1) use a non-linear kriging approach such as
disjunctive kriging or the multivariate gaussian. Ed
Isaacs and Mohan Srivastava's book is th ebest
reference for the latter. Rivoirard's book for DK.

(2) simulation. There are lots of simulation methods
around, which allow you to 'put back the roughness'
and get an idea how bad the problem might be. GSLib is
pretty good on this.

Isobel
http://geoecosse.bizland.com/course_brochure.htm

If, as in mining, you wish to apply some sort

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• Nicolau I was talking about kriging before cutoff is applied. If the cutoff is applied to the block estimates my comments stand. If you aply the cutoff to your
Message 2 of 5 , Jul 19, 2004
Nicolau

I was talking about kriging before cutoff is applied.
If the cutoff is applied to the block estimates my
first and then krige, you get the opposite problem,
because you will over-estimate every value and
under-estimate the tonnage.

My point (1) is that, if you wish to avoid conditional
bias in your kriging, you could consider using a
non-linear kriging method such as those mentioned. I
have no experience with either, since I follow a
different route in the correction of conditional bias
in mineral resource estimation.

Isobel
http://geoecosse.bizland.com/whatsnew.htm

--- nicolau.barros@... wrote: > Isobel,
>
> So for mining purposes can't we just krige before
> applying the cut-off
> criteria? I mean, for most mining applications one
> will prefer to have a
> more realistic geologic block model and will always
> have the chance to
> evaluate his/her panels under the appropriate
> cut-off criteria, but applying
> that criteria after estimating small blocks, right?
>
> below? Thanks for
> indicating the literature.
>
> Thanks
>
> Nicolau Barros
> Engineer
> Mine Planning and Production Control Department
> Mineração Rio do Norte S.A.
> nicolau.barros@...
> +55 (93) 549 8215
>
> Esse e-mail e possíveis anexos podem possuir
> informações confidenciais e de
> interesse somente do destinatário. Portanto, se você
> recebeu esta mensagem
> por engano, favor comunicar imediatamente o
> remetente e deletá-la logo em
> seguida. Esteja ciente que o uso indevido do
> conteúdo das informações em
> questão é estritamente proibido.
> Confidentiality
> This message and any possible attached files may
> contain confidential
> information and only for interest of the intended
> recipient. If you have
> sender and delete the
> message immediately. Be aware that the unauthorized
> use of the
> above-mentioned information is strictly forbidden.
>
> -----Mensagem original-----
> De: Isobel Clark [mailto:drisobelclark@...]
> Enviada em: segunda-feira, 19 de julho de 2004 05:23
> Para: Julhendra_Solin@...
> Cc: ai-geostats@...
> Assunto: [ai-geostats] Re: Kriging Small Blocks
>
> Jul
>
> "small" relative to the sampling density. For
> example,
> less than about one-third of the grid spacing.
>
> The warning is the same as the one about 'point'
> kriging (mapping) The map is too smooth - or, at
> least, a lot smoother than the real surface would
> be.
> High value areas will be under-estimated and low
> value
> areas will be over-estimated.
>
> If your objective in kriging is to obtain general
> maps
> of an area with an idea of where the highs and lows
> are, then ordinary kriging is sufficient. The over-
> and under- estimations cancel out on average.
>
> In mining applications, where block kriging
> originated, most applications require a 'cutoff',
> where values below a certain value are not included
> in
> the 'plan'. In this case, mapping or estimating
> small
> blocks will result in an over-estimation of
> 'payable'
> ground and an under-estimation in average value.
>
> In pollution or environmental applications, the
> areas
> at risk will be under-estimated as will the true
> toxicity or risk factors.
>
> There are two major ways round this problem:
>
> (1) use a non-linear kriging approach such as
> disjunctive kriging or the multivariate gaussian. Ed
> Isaacs and Mohan Srivastava's book is th ebest
> reference for the latter. Rivoirard's book for DK.
>
> (2) simulation. There are lots of simulation methods
> around, which allow you to 'put back the roughness'
> and get an idea how bad the problem might be. GSLib
> is
> pretty good on this.
>
> Isobel
> http://geoecosse.bizland.com/course_brochure.htm
>
> If, as in mining, you wish to apply some sort
>
>
>
>
>
>
___________________________________________________________ALL-NEW
> Yahoo!
> Messenger - sooooo many all-new ways to express
> yourself
> http://uk.messenger.yahoo.com
>

___________________________________________________________ALL-NEW Yahoo! Messenger - sooooo many all-new ways to express yourself http://uk.messenger.yahoo.com
• Hi I think the discussion below is missing some key points: 1. If the individual block estimates are to be used for actual selection at the time of mining,
Message 3 of 5 , Jul 19, 2004
Hi

I think the discussion below is missing some key points:
1. If the individual block estimates are to be used for actual selection at
the time of mining, then conditional bias will impact the predicted
recoveries and should be minimized.
2. However, if the block model is to be used for long term mine planning,
the preparation of production schedules etc. etc., then it is unlikely that
these same block estimates will be used for selection at the time of mining.
In this scenario, it is sufficient to know the distribution of block grades
within a mining period such as annual, semi-annual, or quarterly, etc. and
conditional bias is irrelevant.
3. Now here is the rub. One cannot accurately estimate the distribution of
block grades within a mining period without invoking conditional bias unless
each block estimate is perfect, e.g., no error!

If you read Michel Davids, "Geostatistical Ore Reserve Estimation" you will
find that he also points out this apparent contradiction (page 313 section
1. If the block grades are conditionally unbiased, then the distribution
(histogram) of block estimates is necessarily smoothed. Thus, the prediction
of in situ tones and grade above cutoff is inaccurate (biased)!
2. If the histogram of estimated block grades yields the correct in situ
proportions and grades above cutoff (for all cutoff grades), then the block
estimates are necessarily conditionally biased.

I often refer to this as the "kriging Oxymoron", and it appears to be very
poorly understood with in the geostat community. Even Dr. Krige wrongly
claims that conditional bias should be removed or minimized in a long term
mine planning model, when in fact it is irrelevant.

-----Original Message-----
From: Isobel Clark [mailto:drisobelclark@...]
Sent: Monday, July 19, 2004 8:50 AM
To: nicolau.barros@...
Cc: ai-geostats@...
Subject: [ai-geostats] Re: Kriging Small Blocks

Nicolau

I was talking about kriging before cutoff is applied.
If the cutoff is applied to the block estimates my
first and then krige, you get the opposite problem,
because you will over-estimate every value and
under-estimate the tonnage.

My point (1) is that, if you wish to avoid conditional
bias in your kriging, you could consider using a
non-linear kriging method such as those mentioned. I
have no experience with either, since I follow a
different route in the correction of conditional bias
in mineral resource estimation.

Isobel
http://geoecosse.bizland.com/whatsnew.htm

--- nicolau.barros@... wrote: > Isobel,
>
> So for mining purposes can't we just krige before
> applying the cut-off
> criteria? I mean, for most mining applications one
> will prefer to have a
> more realistic geologic block model and will always
> have the chance to
> evaluate his/her panels under the appropriate
> cut-off criteria, but applying
> that criteria after estimating small blocks, right?
>
> below? Thanks for
> indicating the literature.
>
> Thanks
>
> Nicolau Barros
> Engineer
> Mine Planning and Production Control Department
> Mineração Rio do Norte S.A.
> nicolau.barros@...
> +55 (93) 549 8215
>
> Esse e-mail e possíveis anexos podem possuir
> informações confidenciais e de
> interesse somente do destinatário. Portanto, se você
> recebeu esta mensagem
> por engano, favor comunicar imediatamente o
> remetente e deletá-la logo em
> seguida. Esteja ciente que o uso indevido do
> conteúdo das informações em
> questão é estritamente proibido.
> Confidentiality
> This message and any possible attached files may
> contain confidential
> information and only for interest of the intended
> recipient. If you have
> sender and delete the
> message immediately. Be aware that the unauthorized
> use of the
> above-mentioned information is strictly forbidden.
>
> -----Mensagem original-----
> De: Isobel Clark [mailto:drisobelclark@...]
> Enviada em: segunda-feira, 19 de julho de 2004 05:23
> Para: Julhendra_Solin@...
> Cc: ai-geostats@...
> Assunto: [ai-geostats] Re: Kriging Small Blocks
>
> Jul
>
> "small" relative to the sampling density. For
> example,
> less than about one-third of the grid spacing.
>
> The warning is the same as the one about 'point'
> kriging (mapping) The map is too smooth - or, at
> least, a lot smoother than the real surface would
> be.
> High value areas will be under-estimated and low
> value
> areas will be over-estimated.
>
> If your objective in kriging is to obtain general
> maps
> of an area with an idea of where the highs and lows
> are, then ordinary kriging is sufficient. The over-
> and under- estimations cancel out on average.
>
> In mining applications, where block kriging
> originated, most applications require a 'cutoff',
> where values below a certain value are not included
> in
> the 'plan'. In this case, mapping or estimating
> small
> blocks will result in an over-estimation of
> 'payable'
> ground and an under-estimation in average value.
>
> In pollution or environmental applications, the
> areas
> at risk will be under-estimated as will the true
> toxicity or risk factors.
>
> There are two major ways round this problem:
>
> (1) use a non-linear kriging approach such as
> disjunctive kriging or the multivariate gaussian. Ed
> Isaacs and Mohan Srivastava's book is th ebest
> reference for the latter. Rivoirard's book for DK.
>
> (2) simulation. There are lots of simulation methods
> around, which allow you to 'put back the roughness'
> and get an idea how bad the problem might be. GSLib
> is
> pretty good on this.
>
> Isobel
> http://geoecosse.bizland.com/course_brochure.htm
>
> If, as in mining, you wish to apply some sort
>
>
>
>
>
>
___________________________________________________________ALL-NEW
> Yahoo!
> Messenger - sooooo many all-new ways to express
> yourself
> http://uk.messenger.yahoo.com
>

___________________________________________________________ALL-NEW Yahoo!
Messenger - sooooo many all-new ways to express yourself
http://uk.messenger.yahoo.com
• Ed I would differ from your explanation on one point. If you are merely declaring a mineral resource, i.e. mineral in the ground, then the conditional bias may
Message 4 of 5 , Jul 19, 2004
Ed

I would differ from your explanation on one point.

If you are merely declaring a mineral resource, i.e.
mineral in the ground, then the conditional bias may
not be relevant at the "pre feasibility" stage.

However, as soon as you introduce any economic or
technical parameters which entail selection, the
conditional bias makes its appearance.

In every project I have worked on, from
pre-feasibility onwards, I have been asked for a
grade/tonnage calculation - no matter how hand-waving
it may be. The grade/tonnage curve will be affected by
the conditional bias. By how much has to be assessed
at the time. Most of Chapter 3 in Practical
Geostatistics 1979 is devoted to working out what the
(theoretical) global grade tonnage curve looks like
when you adjust for the variance reduction. Even this
will differ from the curve derived from the kriged
estimates, no matter what size the block.

The problem is even greater for environmental
applications, especially toxic level risks. A 'global
view' - i.e. a map - will not identify the true peaks
because of the conditional bias. The fact that the
overall average is unbiassed is irrelevant when trying
to identify an area which is likely to be lethal.

So, there is no contradiction. Conditional bias is
unimportant (or irrelevant) until you apply some
selection criterion. Yes, we agree. However, selection
criteria can be relevant at very early stages of a
project. It depends on your objective.

Isobel
http://uk.geocities.com/drisobelclark/practica.htm for

PS: sorry I mis-spelled your name, I know it drives me
nuts when people call me 'Clarke' ;-)

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• Hi Isabel If you go to www.isaaks.com and click on Otherstuff , you will find an example where the block estimates are conditionally biased (rather severely),
Message 5 of 5 , Jul 19, 2004
Hi Isabel

If you go to www.isaaks.com and click on "Otherstuff", you will find an
example where the block estimates are conditionally biased (rather
severely), but the grade tonnage curves are right on the money. Perhaps this
will help clear the confusion. Ed

-----Original Message-----
From: Isobel Clark [mailto:drisobelclark@...]
Sent: Monday, July 19, 2004 10:47 AM
To: Edward Isaaks
Cc: ai-geostats@...
Subject: [ai-geostats] Re: Kriging Small Blocks

Ed

I would differ from your explanation on one point.

If you are merely declaring a mineral resource, i.e.
mineral in the ground, then the conditional bias may
not be relevant at the "pre feasibility" stage.

However, as soon as you introduce any economic or
technical parameters which entail selection, the
conditional bias makes its appearance.

In every project I have worked on, from
pre-feasibility onwards, I have been asked for a
grade/tonnage calculation - no matter how hand-waving
it may be. The grade/tonnage curve will be affected by
the conditional bias. By how much has to be assessed
at the time. Most of Chapter 3 in Practical
Geostatistics 1979 is devoted to working out what the
(theoretical) global grade tonnage curve looks like
when you adjust for the variance reduction. Even this
will differ from the curve derived from the kriged
estimates, no matter what size the block.

The problem is even greater for environmental
applications, especially toxic level risks. A 'global
view' - i.e. a map - will not identify the true peaks
because of the conditional bias. The fact that the
overall average is unbiassed is irrelevant when trying
to identify an area which is likely to be lethal.

So, there is no contradiction. Conditional bias is
unimportant (or irrelevant) until you apply some
selection criterion. Yes, we agree. However, selection
criteria can be relevant at very early stages of a
project. It depends on your objective.

Isobel
http://uk.geocities.com/drisobelclark/practica.htm for