are these methods appropriate for four dimensional modeling?

On Monday 08 February 2010, Mark Connolly wrote:
   

repeat post - added signature

I am working on a thesis in soil science with four dimensional modeling
as a major component.  I am using R for the interpolation of data and
VisIT to render the data as volumes and as volumes changing in time.  I
am using Applied Spatial Data Analysis with R (Bivand, Pebesma,
Gomez-Rubio) as a reference.  I've also been using Chatfield's Analysis
of Time Series, less for time series specifically than for helping to
understand the statistical concepts.
     

Hi Mark, nice to see a fellow soil scientist on R-sig-geo.

   

My data are soil property data. A purely spatial component comprises
nine soil physical properties taken across a 12 ha agriculture field.
There are sixty locations with five measurement depths at each location
(3D).  The spatial-temporal component is the measure of volumetric soil
moisture status taken at the same 300 positions.  These data are
discrete readings taken at unequal intervals.  The readings exist for
growing seasons over three years.
     

We have a similar dataset, and have found that RST interpolation to be a
useful starting point. GRASS has functionality for interpolating volumetric
data, and numerous export routines for visualization in Paraview. Here are
some examples:

http://www-pool.math.tu-berlin.de/~soeren/grass/modules/screenshots/

   

The observation grid is half regular and half semi-random.  Each of
thirty regularly placed locations has a satellite location set in random
proximity but not too far and not too close.  The observation depths are
at 15 cm intervals starting at -15 cm and extending down to -75 cm.

My first pass through the data was mainly concerned with the mechanics
of the process.  I used IDW for interpolating each of the soil
properties through the volume (myriad packages including sp, gstat, and
others mentioned in Bivand et al), treating each property as independent
of the others.

I then went after the temporal soil moisture data.  At each of the 300
positions, I took the set of time-sequenced measures (measurement
intervals varied from days to weeks during each growing season) and
interpolated values for days not measured.  I used the Stineman
algorithm provided by the na.stinterp function from the stinepack R
package.  Once I had all days for all positions, I used IDW again to
interpolate volumes for each day.

Each volume was exported as an unstructured point grid for rendering in
VisIT.  My data analysis was limited to adjusting the IDW weighting
power such that the density distributions of the interpolated values
were similar to the density distributions of the observed values
(eyeballing overlaid plots).

My interpolation grid cell size is defined at 10x10x1, modeling 10 m by
10 m by 1 meter.  This is a little misleading.  I treat the depth
dimension as having the same units as the areal dimension, so -15 cm
becomes equivalent to -15 m.  Two reasons (the second not necessarily
defensible): I need the depth to be scaled for reasonable visualization,
and I want to decrease the influence of the depth measures on the layers
above and below each depth.
     

Again RST interpolation to voxels may be useful in this case. There has been
some talk on the GRASS mailing list about interpolation through time, but I
haven't seen anything available yet.

Cheers,
Dylan


   

In my next iteration through the data, I'd like to use the
geostatistical techniques presented in Bivand et al.  I have (just)
started with variograms with the hope of exploring the variogram in the
context of the volume.  I have run into the problem of dimensionality.
The plot functions are 2D, so projecting the variogram onto the
observation spatial volume is not working.  That got me thinking.  If I
use kriging, will I be using the influence of depth in the interpolation
model?  Is the variogram created with the influence of depth when the
spatial structure has three dimensions?  Is a better use to treat each
depth as a separate layer to be interpolated in two dimensions,
especially given that soils are often physically layered?  If three
dimensional kriging is okay, how about four dimensional to move through
a time series? Or would I be better to stick with the Stineman algorithm
for the time-series interpolation?

Any feedback on this (including any warnings that I might be abusing
things with my time series and IDW approaches) would be very much
appreciated.  I am definitely in learning mode.

Mark