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## AI-GEOSTATS: Kriging with External Drift

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• Hello all, I m a bit of a rookie with GeoStatistics. I am interested in Kriging with External Drift but I am having a hard time finding information that tells
Message 1 of 13 , Dec 5, 2001
Hello all,

I'm a bit of a rookie with GeoStatistics. I am interested in Kriging with
External Drift but I am having a hard time finding information that tells
you how it works in laymans terms (without just firing a matrix at you and
leaving you to deduce what it means).

I don't think this is right but I'll give it a shot. Does it work as
follows :

1) Compute a trend for the drift variable
2) Remove the trend computed in 1) from the main variable
3) Grid the residuals from step 2)
4) Add back the trend from step 1)

I don't think this is right but if you can explain to me how the Drift is
actually applied in laymans terms it would be greatly appreciated. Also
when you do kriging with external drift do you have to model a variogram or
can a reasonable one be computed automatically, if so how would you compute
it?

Also if you have any good sources of information on Kriging with external
drift could you pass them on to me?

Thanks,

Mike

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• Mike, What you are dealing with is universal kriging, i.e., kriging in the presence of a drift. The data set that took away my geostatistical virginity was of
Message 2 of 13 , Dec 5, 2001
Mike,

What you are dealing with is universal kriging, i.e., kriging in the presence of a drift. The data set that took away my geostatistical virginity was of this same type - the Wolfcamp aquifer data. Since the
original 1985 geostat pub on this by myself and Jeff Furr, many folks have analyzed this data in numerous ways. The data is downloadable from http://uk.geocities.com/drisobelclark/briefcase.html#PG2000_demo.
Analyze it with any software that allows universal kriging (e.g., a teaching version of the software on this same page).

In essence, estimate the trend but keep in mind that universal kriging may not need as high of a trend when fitting local search areas as a global regression trend over your entire data set. Thus while a
quadratic regression may "best" fit your full data set, you might only need a linear trend when doing universal kriging since you are doing this in each smaller local search area.

With the appropriate software you are in essence taking out the trend, and fitting the semi-variogram to the residuals. But when you then do the universal kriging after deciding on a reasonable semi-variogram, it
will be done on the original data then using both the semi-variogram and trend you decided upon.

See Practical Geostatistics 2000 (ISBN 0-9703317-0-3 or 0-9703317-2-X) for additional details on how one might approach this.

I hope I am not violating any AI-GEOSTATS bylaws here. I am trying to give an answer that works for me without getting into any hot water.

Bill

Michael Dennis wrote:

> Hello all,
>
> I'm a bit of a rookie with GeoStatistics. I am interested in Kriging with
> External Drift but I am having a hard time finding information that tells
> you how it works in laymans terms (without just firing a matrix at you and
> leaving you to deduce what it means).
>
> I don't think this is right but I'll give it a shot. Does it work as
> follows :
>
> 1) Compute a trend for the drift variable
> 2) Remove the trend computed in 1) from the main variable
> 3) Grid the residuals from step 2)
> 4) Add back the trend from step 1)
>
> I don't think this is right but if you can explain to me how the Drift is
> actually applied in laymans terms it would be greatly appreciated. Also
> when you do kriging with external drift do you have to model a variogram or
> can a reasonable one be computed automatically, if so how would you compute
> it?
>
> Also if you have any good sources of information on Kriging with external
> drift could you pass them on to me?
>
> Thanks,
>
> Mike
>
> --
> * To post a message to the list, send it to ai-geostats@...
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--
William V Harper, Mathematical Sciences, Otterbein College
Towers Hall 136, Westerville OH 43081-2006 USA
614-823-1417 Fax 614-823-3201
Faculty page: http://go.to/billharper
For the best in geostatistics: http://go.to/geostatistics
Coming eventually:
http://www.wvharper.com (currently points to geostat site)

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• Hello Michael! What you are describing is kriging in the presence of a drift, e.g. kriging of a (single) nonstationary RV. You may proceed as described in
Message 3 of 13 , Dec 6, 2001
Hello Michael!
What you are describing is kriging in the presence of a drift,
e.g. kriging of a (single) nonstationary RV. You may proceed as
described in your email or use universal kriging (UK) for that.

External drift refers to a second variable which may be used
in order to improve estimation e.g.
1st variable: rainfall data measured at a limited number of stations and
2nd variable: digital elevation data which must be known at each
grid point to be estimated.
See Goovaerts' paper for this example:
http://www.geovista.psu.edu/sites/geocomp99/Gc99/023/gc_023.htm

For UK see any standard textbook on geostats.

Regards,
Heinz Burger

*****************************************************
Visit IAMG2002: http://www.fu-berlin.de/iamg2002
*****************************************************
Dr. Heinz Burger
Freie Universitaet Berlin
- Geoinformatik -
Malteserstr. 74-100
12249 BERLIN, Germany
Tel. (49) 30-838-70561 Fax: (49) 30-838-70723
mailto: hburger@...-berlin.de
Web-Seite: http://userpage.fu-berlin.de/~hburger/hb
****************************************************

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• RE: AI-GEOSTATS: Kriging with External DriftThanks for clearing that up for me. I get confused with the terminology sometimes. I ll see if I can get a copy
Message 4 of 13 , Dec 6, 2001
RE: AI-GEOSTATS: Kriging with External DriftThanks for clearing that up for
me. I get confused with the terminology sometimes. I'll see if I can get a
copy of the book you mention to look at to see if it helps me out.

You say that KED uses a shape function for the "Drift" data. How is this
shape function computed? Isn't it just a polynomial trend? Or is it
something more complex than that? I know the bottom row of the matrix is
computed from the "drift" but I'm interested in how that bottom value is
derived.

On another note : I've run into a couple of references on Bayesian Kriging
which I get the impression is similar to KED. Can anyone tell me what the
differences are between Bayesian and KED and why you would choose one over
the other?

Thanks,

Mike

-----Original Message-----
From: WARR Benjamin [mailto:benjamin.warr@...]
Sent: Thursday, December 06, 2001 3:59 AM
To: 'Michael Dennis'
Subject: RE: AI-GEOSTATS: Kriging with External Drift

Michael,

I suggest a book by Hans Wackernagel, An Introduction to Multivariate
Geostatistics, Springer Verlag, 2nd Edition 1999.

He describes the technique thoroughly and clearly. Briefly, what you
describe is Universal Kriging (UK). KED uses a shape function, provided by
exhaustive secondary data to model a deterministic 'drift' or underlying
trend, whereas when doing UK you have to model the trend yourself using a
polynomial.

Benjamin Warr

Research Associate
Centre for the Management of Environmental Resource(CMER)
Boulevard de Constance,
77305 Fontainebleau Cedex,
France

> -----Original Message-----
> From: Michael Dennis [mailto:Mike.D@...]
> Sent: 06 December 2001 04:13
> To: AI-Geostats Mailing List
> Subject: AI-GEOSTATS: Kriging with External Drift
>
>
> Hello all,
>
> I'm a bit of a rookie with GeoStatistics. I am interested in
> Kriging with
> External Drift but I am having a hard time finding
> information that tells
> you how it works in laymans terms (without just firing a
> matrix at you and
> leaving you to deduce what it means).
>
> I don't think this is right but I'll give it a shot. Does it work as
> follows :
>
> 1) Compute a trend for the drift variable
> 2) Remove the trend computed in 1) from the main variable
> 3) Grid the residuals from step 2)
> 4) Add back the trend from step 1)
>
> I don't think this is right but if you can explain to me how
> the Drift is
> actually applied in laymans terms it would be greatly
> appreciated. Also
> when you do kriging with external drift do you have to model
> a variogram or
> can a reasonable one be computed automatically, if so how
> would you compute
> it?
>
> Also if you have any good sources of information on Kriging
> with external
> drift could you pass them on to me?
>
> Thanks,
>
> Mike
>
>
> --
> * To post a message to the list, send it to ai-geostats@...
> * As a general service to the users, please remember to post
> a summary of any useful responses to your questions.
> * To unsubscribe, send an email to majordomo@... with no
> subject and "unsubscribe ai-geostats" followed by "end" on
> the next line in the message body. DO NOT SEND
> Subscribe/Unsubscribe requests to the list
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>

[Non-text portions of this message have been removed]
• Thanks for the info. I don t mind you plugging your book, that was one of the questions I asked : What is a good reference on this subject. See this is where
Message 5 of 13 , Dec 6, 2001
Thanks for the info. I don't mind you plugging your book, that was one of
the questions I asked : What is a good reference on this subject.

See this is where I start getting confused with terminology. I'm talking
about KED (Kriging with External Drift) and then you start talking about
Universal Kriging. If I understand correctly they are not the same thing?
With Universal Kriging you remove the trend and Krig the residuals but in
KED you use the trend(drift) in the actual krig matrix?

Mike

-----Original Message-----
From: Isobel Clark [mailto:drisobelclark@...]
Sent: Thursday, December 06, 2001 11:18 AM
To: Michael Dennis
Cc: wharper@...
Subject: RE: AI-GEOSTATS: Kriging with External Drift

> I'll see if I can get a
> copy of the book you mention to look at to see if it
> helps me out.
>
> You say that KED uses a shape function for the
> "Drift" data.
'shape functions' can be anything you like but most
people stick to simple polynomials. To use Univeral
Kriging the only constraint is that it has to be
expressed as a linear function in the coefficients.
That is: b0 + b1 * some function + b2 * some other
function and so on, where the b's are the
coefficients.

All this is explained in detail in Chapter 12 of
Practical Geostatistics 2000, but we aren't allowed to
say that on the open list ;-)

You can try it out free (completely) with the kriging
game in my briefcase. This shows you the equations and
the terms calculated. A full tutorial on Universal
Kriging is also available in the briefcase and can be
run with the free PG2000 software.

Find them all at:

http://uk.geocities.com/drisobelclark/briefcase.html

Let us know if we can be of further help.

Isobel Clark

________________________________________________________________
Nokia 5510 looks weird sounds great.
Go to http://uk.promotions.yahoo.com/nokia/ discover and win it!
The competition ends 16 th of December 2001.

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• ... Kriging with an external drift is just an extension of universal kriging. UK assumes that one knows the shape of the trend but not its magnitude (or
Message 6 of 13 , Dec 6, 2001
On Thursday, December 6, 2001, at 04:12 AM, Michael Dennis wrote:
>
> I don't think this is right but if you can explain to me how the Drift
> is
> actually applied in laymans terms it would be greatly appreciated. Also
> when you do kriging with external drift do you have to model a
> variogram or
> can a reasonable one be computed automatically, if so how would you
> compute
> it?

Kriging with an external drift is just an extension of universal kriging.
UK assumes that one knows the shape of the trend but not its
magnitude (or coefficients). For example a linear drift could be modeled
by Mean = a + bX + cY where X and Y are the coordinates of the data.
And so on and so forth for higher order polynomial trends.

In KED, the trend shape is not defined analytically; rather, it is
assumed that
it is defined explicitly at all locations based on some densely sampled
secondary variable. However, such a secondary variable must be
smoothly varying in space, and also it must be available at all locations
of the primary data and the locations being estimated.

As in UK, the magnitude of the trend is unimportant, it is the shape
that we're interested in. An external drift that varies linearly with X
and
Y would be equivalent to UK with an analytical trend of the same
order polynomial, i.e. 1.

Regards,

Syed
Maersk Copenhagen

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• Michael Maybe we have not made this clear. Universal kriging is a two stage process. (1) Fit a trend (global) or local trends and calculate the residuals. From
Message 7 of 13 , Dec 6, 2001
Michael

Maybe we have not made this clear.

Universal kriging is a two stage process.

(1) Fit a trend (global) or local trends and calculate
the residuals. From these residuals you obtain a
de-trended (drift-less) semi-variogram.

(2) Using the semi-variogram derived in (1) together
with the established form of the trend, you krige the
complete value with trend from your original sample
values -- not from the residuals.

The only time you use the residuals is to get the
semi-variogram model.

People who refer to 'kriging with external drift' seem
to mean different things and you would need to read
each case on its own merits. I am sure there are
people out there who can point you to the best
references for that terminology.

Try the kriging game together with our Wolfcamp
tutorial which is freely available and distributable
to anyone who wants it.

Are we getting closer?
Isobel Clark
http://uk.geocities.com/drisobelclark/briefcase.html

--- Michael Dennis <Mike.D@...> wrote: >
Thanks for the info. I don't mind you plugging your
> book, that was one of
> the questions I asked : What is a good reference on
> this subject.
>
> See this is where I start getting confused with
> terminology. I'm talking
> about KED (Kriging with External Drift) and then you
> start talking about
> Universal Kriging. If I understand correctly they
> are not the same thing?
> With Universal Kriging you remove the trend and Krig
> the residuals but in
> KED you use the trend(drift) in the actual krig
> matrix?
>
> Mike
>
> -----Original Message-----
> From: Isobel Clark
> [mailto:drisobelclark@...]
> Sent: Thursday, December 06, 2001 11:18 AM
> To: Michael Dennis
> Cc: wharper@...
> Subject: RE: AI-GEOSTATS: Kriging with External
> Drift
>
>
> > I'll see if I can get a
> > copy of the book you mention to look at to see if
> it
> > helps me out.
> >
> > You say that KED uses a shape function for the
> > "Drift" data.
> 'shape functions' can be anything you like but most
> people stick to simple polynomials. To use Univeral
> Kriging the only constraint is that it has to be
> expressed as a linear function in the coefficients.
> That is: b0 + b1 * some function + b2 * some other
> function and so on, where the b's are the
> coefficients.
>
> All this is explained in detail in Chapter 12 of
> Practical Geostatistics 2000, but we aren't allowed
> to
> say that on the open list ;-)
>
> You can try it out free (completely) with the
> kriging
> game in my briefcase. This shows you the equations
> and
> the terms calculated. A full tutorial on Universal
> Kriging is also available in the briefcase and can
> be
> run with the free PG2000 software.
>
> Find them all at:
>
> http://uk.geocities.com/drisobelclark/briefcase.html
>
> Let us know if we can be of further help.
>
> Isobel Clark
>
>
________________________________________________________________
> Nokia 5510 looks weird sounds great.
> Go to http://uk.promotions.yahoo.com/nokia/ discover
> and win it!
> The competition ends 16 th of December 2001.
>
>
> --
> * To post a message to the list, send it to
> ai-geostats@...
> * As a general service to the users, please remember
> to post a summary of any useful responses to your
> questions.
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http://www.ai-geostats.org

________________________________________________________________
Nokia 5510 looks weird sounds great.
Go to http://uk.promotions.yahoo.com/nokia/ discover and win it!
The competition ends 16 th of December 2001.

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• So are you saying that the external drift variable in the matrix is just the magnitude of the the drift variable at that point? ie : 3 point kriging s = drift
Message 8 of 13 , Dec 6, 2001
So are you saying that the external drift variable in the matrix is just the
magnitude of the the drift variable at that point?

ie :

3 point kriging

s = drift variable

[ k11 k12 k13 1 s1 ] [ l1 ] [ k01 ]
[ k21 k22 k23 1 s2 ] [ l2 ] [ k02 ]
[ k31 k32 k33 1 s3 ] [ l3 ] = [ k03 ]
[ 1 1 1 0 0 ] [ u0 ] [ 1 ]
[ s1 s2 s3 0 0 ] [ u1 ] [ s0 ]

So in this 3 point Kriging case I just plug in the magnitude of my drift
variable in for s0, s1, s2,and s3?

And if you substituted a drift with a magnitude which was computed based
upon 1st order polynomial you would get the same results from this matrix as
you would by removing the 1st order polynomial trend and kriging the
residuals and adding the 1st order polynomial trend back in?

It is all starting to make sense now (if I'm correct in what I'm saying
above). Thanks so much for your help!

Mike

-----Original Message-----
From: ai-geostats-list@... [mailto:ai-geostats-list@...]On
Behalf Of sshibli@...
Sent: Thursday, December 06, 2001 11:59 AM
To: Michael Dennis
Cc: AI-Geostats Mailing List
Subject: Re: AI-GEOSTATS: Kriging with External Drift

On Thursday, December 6, 2001, at 04:12 AM, Michael Dennis wrote:
>
> I don't think this is right but if you can explain to me how the Drift
> is
> actually applied in laymans terms it would be greatly appreciated. Also
> when you do kriging with external drift do you have to model a
> variogram or
> can a reasonable one be computed automatically, if so how would you
> compute
> it?

Kriging with an external drift is just an extension of universal kriging.
UK assumes that one knows the shape of the trend but not its
magnitude (or coefficients). For example a linear drift could be modeled
by Mean = a + bX + cY where X and Y are the coordinates of the data.
And so on and so forth for higher order polynomial trends.

In KED, the trend shape is not defined analytically; rather, it is
assumed that
it is defined explicitly at all locations based on some densely sampled
secondary variable. However, such a secondary variable must be
smoothly varying in space, and also it must be available at all locations
of the primary data and the locations being estimated.

As in UK, the magnitude of the trend is unimportant, it is the shape
that we're interested in. An external drift that varies linearly with X
and
Y would be equivalent to UK with an analytical trend of the same
order polynomial, i.e. 1.

Regards,

Syed
Maersk Copenhagen

--
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• Michael, Kriging with External Drift, as implemented in the GSLIB set of programs (Deutsch & Journel) is distinct from Universal Kriging, in that the External
Message 9 of 13 , Dec 6, 2001
Michael,

Kriging with External Drift, as implemented in the GSLIB set of programs
(Deutsch & Journel) is distinct from Universal Kriging, in that the
"External Drift" is intended to be defined by a secondary variable that you,
the user, "believe" incorporates some information of relevance to the
primary variable that you are working with.

The "External Drift" could also be a kriged array of values based on kriging
some sparse secondary variable (onto a regular grid) that you also feel
contains information of relevance. You can also specify a "trend"
mathematically, but the original intent was to incorporate a secondary
variable "relevant" to estimation of the first.

An example would be: you have sparse porosity measurements in an oil field,
but you have a regular array of 3-D seismic amplitudes covering that same
area. Your external drift variable would be seismic amplitude and your
primary variable would be porosity. Obviously the two are not precisely the
same, but "hopefully" they are related.

It is up to you, the user, to specify external drift terms that "make sense"
physically. Obviously, this is a matter of interpretation, and you are the
one responsible for justifying your choices.

I suggest you read the sections on Kriging with External Drift beginning on
page 70 and 96 of the Deutsch and Journel (1998) book, "GSLIB Geostatistical
Software Library and User's Guide," by Oxford University Press. I'm sure
there are other papers out there on this methodology, but if you are talking
about the GSLIB program, then it's best to go to the documentation
associated with that program.

Best regards,

Chris

Christopher A. Rautman, Ph. D., P.G.
Underground Storage Technology Department
Sandia National Laboratories
P. O. Box 5800; MS-0706
Albuquerque, New Mexico 87185-0706
505-844-2109; fax: 505-844-4426

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• Hi, a little off the original topic, but one question that puzzles me is the computation of the semivariogramm for Kriging with external drift. If I understand
Message 10 of 13 , Dec 7, 2001
Hi,

a little off the original topic, but one question that puzzles me is the
computation of the semivariogramm for Kriging with external drift.

If I understand correctly, a relation of type
m*(x) = a1(x) + a2(x)*s(x)
is assumed between the mean of the primary variable at location x
(drift) and secondary variable (s). Now, the a's are dependent on the
location and are not computed explicitly. But in the kriging system the
residual covarince is needed. To compute the residuals, I would need to
calculate the equation explicitly, right?

How do I compute the residual semivariogramm when I don't know the
residuals? Or have I misunderstood something in the concept of external
drift?

Best regards
Andreas

--
Andreas Hartmann
RWTH Aachen, Angewandte Geophysik
Lochnerstr. 4-20
52056 Aachen, Germany
(+49) (-0) 241 8094835
mailto:Andreas@...-aachen.de
http://www.geophysik.rwth-aachen.de

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• ... you got it. Plus the constraint that the drift components in the samples have to balance with the drift component at the point being estimated. Isobel
Message 11 of 13 , Dec 7, 2001
> So are you saying that the external drift variable
> in the matrix is just the
> magnitude of the the drift variable at that point?
you got it.

Plus the constraint that the drift components in the
samples have to balance with the drift component at
the point being estimated.

Isobel

________________________________________________________________
Nokia 5510 looks weird sounds great.
Go to http://uk.promotions.yahoo.com/nokia/ discover and win it!
The competition ends 16 th of December 2001.

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• ... There is a nice section (Chapter 5.4) in Cressie s textbook (Statistics for Spatial Data) that discusses the potential bias of estimating semivariograms
Message 12 of 13 , Dec 7, 2001
On Friday, December 7, 2001, at 10:05 AM, Andreas Hartmann wrote:
>
> How do I compute the residual semivariogramm when I don't know the
> residuals? Or have I misunderstood something in the concept of external
> drift?

There is a nice section (Chapter 5.4) in Cressie's textbook (Statistics
for
Spatial Data) that discusses the potential bias of estimating
semivariograms from residuals and Matheron's formulation of
IRF-K kriging as an alternative means to krige in the presence of
a trend. The iterative fitting of the covariance is usually
non-graphical,
which is a drawback in itself, and Fig. 5.1 in the same section shows
Cressie's bold attempt at showing the pitfalls of doing such an
"automatic" fit. I have not used IRF-K kriging very much in practice,
for the above reasons. Better to live with the devil that I know, i.e.
my -- albeit biased -- residual variogram.

Regards,

Syed

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• Hi, Just to add to that Gotway and cressie I can t remember the exact citation did a large simulation study and showed that the bias from using the residual
Message 13 of 13 , Dec 8, 2001
Hi,
Just to add to that Gotway and cressie I can't remember the exact citation
did a large simulation study and showed that the bias from using the
residual variogram is very small in most cases.

Nicholas

On Fri, 7 Dec 2001 sshibli@... wrote:

>
> On Friday, December 7, 2001, at 10:05 AM, Andreas Hartmann wrote:
> >
> > How do I compute the residual semivariogramm when I don't know the
> > residuals? Or have I misunderstood something in the concept of external
> > drift?
>
> There is a nice section (Chapter 5.4) in Cressie's textbook (Statistics
> for
> Spatial Data) that discusses the potential bias of estimating
> semivariograms from residuals and Matheron's formulation of
> IRF-K kriging as an alternative means to krige in the presence of
> a trend. The iterative fitting of the covariance is usually
> non-graphical,
> which is a drawback in itself, and Fig. 5.1 in the same section shows
> Cressie's bold attempt at showing the pitfalls of doing such an
> "automatic" fit. I have not used IRF-K kriging very much in practice,
> for the above reasons. Better to live with the devil that I know, i.e.
> my -- albeit biased -- residual variogram.
>
> Regards,
>
> Syed
>
>
>
> --
> * To post a message to the list, send it to ai-geostats@...
> * As a general service to the users, please remember to post a summary of any useful responses to your questions.
> * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
> * Support to the list is provided at http://www.ai-geostats.org
>

CH3
|
N Nicholas Lewin-Koh
/ \ Dept of Statistics
N----C C==O Program in Ecology and Evolutionary Biology
|| || | Iowa State University
|| || | Ames, IA 50011
CH C N--CH3 http://www.public.iastate.edu/~nlewin
\ / \ / nlewin@...
N C
| || Currently
CH3 O Graphics Lab
School of Computing
National University of Singapore
The Real Part of Coffee kohnicho@...

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