## AI-GEOSTATS: Estimation in caves

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• All, I have an interesting problem that is a bit off the usual geostat track. The problem is the estimation of the amount of tracer deposited in a cave
Message 1 of 2 , Sep 12, 2003
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All, I have an interesting problem that is a bit off the usual geostat
track. The problem is the estimation of the amount of tracer deposited in a
cave system. The cave system is composed of multiple rooms and passages
connecting the rooms. Air flow and transport models can be used to model
the deposition of the tracer, but are difficult to set up and calibrate and
I'd like to pursue a more geostatistical approach to the problem (if
possible).

However, a straight forward application of kriging the amount of deposition
based on a number of samples will not work for at least two reasons:
1) Euclidean distance is not very meaningful as two rooms in the system may
only be separated by a 10 meter thick wall, yet the tortuous air flow path
from one room to the other may be over 500 meters. Therefore the
connections between sample points resemble something like a connected graph
ala graph theory. But perhaps it is possible to use this information to
remap the cave system into some sort of "connection space" and build
variograms and do kriging in that space before remapping to the actual
coordiante system (?)
2) If it were possible to develop a covariance matrix using the sample data
in some transformed coordinate system, it would not be symmetric. Due to
the air flow patterns in the cave system, point B may be "downwind" of point
A and there is a B->A connection, but there is no A->B connectivity. In the
parlance of graph theory, this would be a "directed graph".

I've found work across several different fields where the "best" places to
take a sample in such a system can be determined using graph theory coupled
with linear programming and/or heurisitc optimization techniques. I have
not come across any work where estimations are made in the system based on a
finite number of existing samples, except for those that resort to physics
based models (i.e., flow and transport).

If anyone has pondered this problem before and can point me towards any
publications, I would be very appreciative.

thanks

Sean

Sean A. McKenna Ph.D.
Geohydrology Department
Sandia National Laboratories
PO Box 5800 MS 0735
Albuquerque, NM 87185-0735
ph: 505 844-2450

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• Hi Sean, Your problem seems to bear some similarity with modeling of river networks where meaningful distances cannot be defined in the Euclidian space and
Message 2 of 2 , Sep 14, 2003
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Hi Sean,

Your problem seems to bear some similarity with modeling of
river networks where meaningful distances cannot be defined
in the Euclidian space and downstream/upstream relationships
need to be fulfilled.
A few days ago, Pascal Monestiez from INRA Avignon gave a
talk on Geostatistical modelling of spatial processes on trees:
applications to drainage networks, which might be of interest
to you... and there should be other related papers in the literature.

Cheers,

Pierre
<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

Dr. Pierre Goovaerts
President of PGeostat, LLC
Chief Scientist with Biomedware Inc.
710 Ridgemont Lane
Ann Arbor, Michigan, 48103-1535, U.S.A.

E-mail: goovaert@...
Phone: (734) 668-9900
Fax: (734) 668-7788
http://alumni.engin.umich.edu/~goovaert/

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

On Fri, 12 Sep 2003, McKenna, Sean A wrote:

> All, I have an interesting problem that is a bit off the usual geostat
> track. The problem is the estimation of the amount of tracer deposited in a
> cave system. The cave system is composed of multiple rooms and passages
> connecting the rooms. Air flow and transport models can be used to model
> the deposition of the tracer, but are difficult to set up and calibrate and
> I'd like to pursue a more geostatistical approach to the problem (if
> possible).
>
> However, a straight forward application of kriging the amount of deposition
> based on a number of samples will not work for at least two reasons:
> 1) Euclidean distance is not very meaningful as two rooms in the system may
> only be separated by a 10 meter thick wall, yet the tortuous air flow path
> from one room to the other may be over 500 meters. Therefore the
> connections between sample points resemble something like a connected graph
> ala graph theory. But perhaps it is possible to use this information to
> remap the cave system into some sort of "connection space" and build
> variograms and do kriging in that space before remapping to the actual
> coordiante system (?)
> 2) If it were possible to develop a covariance matrix using the sample data
> in some transformed coordinate system, it would not be symmetric. Due to
> the air flow patterns in the cave system, point B may be "downwind" of point
> A and there is a B->A connection, but there is no A->B connectivity. In the
> parlance of graph theory, this would be a "directed graph".
>
> I've found work across several different fields where the "best" places to
> take a sample in such a system can be determined using graph theory coupled
> with linear programming and/or heurisitc optimization techniques. I have
> not come across any work where estimations are made in the system based on a
> finite number of existing samples, except for those that resort to physics
> based models (i.e., flow and transport).
>
> If anyone has pondered this problem before and can point me towards any
> publications, I would be very appreciative.
>
> thanks
>
> Sean
>
>
>
> Sean A. McKenna Ph.D.
> Geohydrology Department
> Sandia National Laboratories
> PO Box 5800 MS 0735
> Albuquerque, NM 87185-0735
> ph: 505 844-2450
>
>
>
> --
> * 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
>

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