I missed the point that you actually did a stratified kriging -- this is
indeed including a trend, by ways of a mean value for each stratum (z
layer).
To demonstrate that correlation in the z-direction is present and does
play a role, you'd have to set up the cross validation more clever, and
e.g. remove a complete z layer and try interpolating this layer from the
layer above and below it -- with and without correlation in the z
direction. This will require either a constant trend in z, or some trend
model in z direction. If you do LOOCV or choose folds that ignore the
layering, the information from the layer itself seems to dominate.
--
Edzer
Altfelder, Sven wrote:
Hi Edzer, hi Sebas,
Thanks a lot for the comments on my problem. Cross validation was done by comparing MSEs and the visual inspection of scatter plots. I have indeed a high signal-to-noise ratio ratio but of course I'm still in search of the method that gives the best possible results. While it's true that all methods perform similar, from looking at the MSEs and scatterplots I don't think that kriging in X direction only performs best only by chance.
If it is indeed the best method (and not only by chance) I would try to argue the way Sebas did in his mail that corresponding values in x direction can be seen as layers with individual means but whith similar spatial covariance function for all layers.
With regard to the point that trends are not that much needed when the data say it all, shouldn't I expect that ordinary kriging should perform better than ignoring the spatial correlation in z-direction because I ignore some information (based on the similar (assumed isotropic) spatial covariance function)? But maybe this depends on the relationship between the residual and trend component. If residual component is small compared to trend the latter will influence the OK estimate in a unwanted manner...
Thanks,
Sven
-----Ursprüngliche Nachricht-----
Von: seba [mailto:sebastiano.trevis...@libero.it]
Gesendet: Donnerstag, 14. Januar 2010 14:45
An: Altfelder, Sven; ai-geostats@jrc.it
Betreff: Re: AI-GEOSTATS: Awkward interpolation approach?
Hi Sven
If I understood well, practically you used ordinary kriging within
strata in which a strata is a layer at a given depth.
For example I used that techniques for 3D interpolation of pollution
data (I made a paper in mathematical geology) as
well as for oceanographic data. Whenever you have layering in your phenomena
splitting a 3d problem in a series of 2d (or if you have a cross
section from 2d to 1d) problem makes things easier to handle
and you have not to take care of trend along z.
Bye
Sebas
At 10.53 14/01/2010, Altfelder, Sven wrote:
Dear list,
I have point measurements of soil water tensions on an area of
approx. 2m (width) x 0.8 m (depth) measured on
a regular grid of 10 by 10 cm. In each of the 17 rows of this grid,
20 measurements were made. Every second row
is shifted by 5 cm with regard to the previous row giving a
chequerboard type pattern.
My goal is to interpolate this data to a 5 cm grid, which pretty
much means that I try to fill in the gaps.
The data has strong trend which is limited to the depth direction
(driving forces are water movement under
gravity and plant uptake of water).
I had to take logarithms of the data because the original data
variance is strongly dependent on the local mean.
After taking logs it looks fine (to me).
After doing this I calculated various variogramms which I fitted
using gstat in the R environment. In log space
I interpolated data using Universal Kriging (using a variogramm
calculated perpendicular to the trend)
with various polynomial trend functions in depth direction, Ordinary
kriging, Ordinary kriging of residuals
after trend removal and an addition of the trend component after
kriging, Inverse distance weighting
and finally Ordinary kriging using the variogramm calculated
perpendicular to the trend and ignoring
spatial correlation in depth direction by assuming an appropriate
anisotropy.
(basically a 1D Kriging in X-direction).
I cross validated the various procedures on data sets that were
actually measured on a
5 x 5 cm grid and were simply reduced to a 10 x 10 cm grid by
leaving out every second data point.
I achieved by the best results with last method mentioned (Ordinary kriging
using the variogramm calculated perpendicular to the trend and
ignoring spatial dependence in depth direction).
From a practitioners point of view I'm satisfied with the result.
However, after scanning the literature I've not found anybody who
has done a 2-D interpolation this awkward way. This
gives me the uneasy feeling that in a subsequent publication my
approach will not stand up against the critical
view of a real geostatistician. Is my approach suitable or should I
further explore other methods?
Any advice on this issue would be appreciated.
Thanks,
Sven
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--
Edzer Pebesma
Institute for Geoinformatics (ifgi), University of Münster
Weseler Straße 253, 48151 Münster, Germany. Phone: +49 251
8333081, Fax: +49 251 8339763 http://ifgi.uni-muenster.de
http://www.52north.org/geostatistics e.pebe...@wwu.de
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