the following summarises two mails I have received from Bill Huber
) on the combined use of GIS & Geostatistics, especially
on works involving non Euclidian spaces. I thought indeed
that some transformation of the space in which the variables are
measured could be implemented in a GIS in order to help the
geostatistician to be in a situation where the hypothesis of
stationnarity can be considered as true and improve so the estimations.
I gave the example of studies in hydrology where pollutants, particles, etc.
in suspension have their spatial distribution strongly influence
by the shape of the river or a lake.
The following summarises a one-hour talk given by Bill Huber in 1995
at the American Statistical Association meeting at the University of Delaware.
Unfortunately no publication has been done because of the
lost of information due to a disk crash.
" In 1993 I studied data obtained from approximately 90 gamma/neutron
logs of borings near the Delaware river in Southern New Jersey (USA)
and used them to krige some six surfaces defining interfaces among distinct
hydrogeological layers. The overarching interpretation provided
by local geologists was that, generally, these surfaces would demarcate
"paleochannels" carved out by the river many eons ago. The paleochannels
were likely to follow the same curves as the modern channel but be
displaced laterally from it. Therefore, to achieve better estimation,
I did two things. One was to develop a non-euclidean metric based on
distances along a grid of lines parallel to the modern river bank
(the Y coordinate) and distances orthogonal to that linear grid
(the X coordinate). I referred to this informally as "straightening out
the river." Then, in addition, I had to fit some nonlinear trends to
model the cross sections of the paleochannels. After removing these
trends the hope was that the residuals would be decently modeled by a
stationary RF, which they appeared to be. Indeed, by adjusting the
non-Euclidean metric slightly, it was possible to select isotropic variograms.
I asked if the "out straightening of the river" could be somehow implemented
and automated in a GIS ?
"Yes. Ideally, using the Riemann mapping theorem, one should be able
to extend a river boundary to a conformal map of a simply connected
part of the plane where the river lies along a portion of the real axis--in
other words, it is straightened out. The inverse image of
the x,y coordinate grid then corresponds to a local orthogonal
coordinate system. I have had the vague idea that the Schwarz-Christoffel
formulas could be used to explicitly (and automatically) compute such
Another approach is to integrate an appropriate vector field.
The river boundary (or rather, a differentiable approximation to it) locally
defines an orthogonal flow which can be integrated to give
the coordinate lines perpendicular to the boundary. Distance along
those lines would provide the second coordinate. There are problems
with this approach, though, related to focal points of the river:
these are regions of relatively high curvature which would cause orthogonal
lines to converge within a short distance.
More recently, I have been seeking solutions that couple the usual
geostatistical estimators with flow models. This has the added
attraction of introducing physical knowledge into the problem
structure. I have made no significant progress.
Because of these problems, the approach I ultimately chose was
simpler: I noted that the river directions ranged within less
than a half-circle. Thus, by translating the river boundary parallel
to itself along an appropriate direction, I could guarantee that none
of the translates "doubled back" on itself relative to the direction
of translation. The resulting coordinates were not orthogonal, but
they effectively solved the problem. This procedure is readily
automated; using capabilities in ArcView + Spatial Analyst one could
even automate the process of warping point locations and vector
features according to the resulting transformation."
Same question about the transformation of more complex structures
"� the Schwarz-Christoffel approach might work. Remember, these are explicit
(integral) formulas for a conformal mapping of a closed
region bounded by a piecewise linear polygon onto the unit disk (or,
equivalently, any region conformally equivalent to the unit disk, such
as the upper half-plane)."
To conclude, some comments on the progress made in the combined use
of GIS & Geostatistics
"�.there was considerable work required to implement the transformations
of gridded data and reference data (CAD files); all this was hand-coded
in languages like Fortran and AWK and required several months of
part-time work to complete. Within less than three years I found
myself being able to perform analyses of equal complexity, with higher
accuracy, in days or even hours. The difference?
Desktop GIS. It is difficult now to imagine doing geostatistics
"� the value of the GIS in geostatistical work for me definitely
has been the data manipulation and spatial analysis rather than
the cartographic aspect (although that's nice to have). It is worth
noting, too, that with the ability to georeference spatial data and
the proliferation of useful spatial data on the internet, the geostatistician
is now for the first time *readily* able to supplement
his or her sample data with accurate, high-resolution auxiliary data (useful
for bayesian priors, covariates, and ad-hoc procedures like
the one described above) "
"The ability to bring "� "disparate data sources together and relate
them meaningfully added immeasurably to the value of the analysis. "
I don't care whether you call the software I was using GIS or something
else, but I do care that the software gave me ready access to these
data and the ability to relate them quantitatively. That's my message
to the geostatistician, however proficient, who is still not using
these GIS capabilities: you may be missing opportunities to do much richer,
more interesting, and valuable analyses."
Hope this will be as interesting for the ai-geostats readers as it was
Section of Earth Sciences
Institute of Mineralogy and Petrography
University of Lausanne
Currently detached in Italy
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