Geophysical applications
of scattered data approximation (cooperation with Mitch Rauth from
Schlumberger/GeoQuest)
In geophysical applications one has large data sets, i.e.
10,000 to 1 million data points, with a very irregular sampling geometries.
For visualization or further data processing one needs data on a regular
grid. Thus the problem is to interpolate or approximate the measurements
onto the grid in an efficient way. This can be done by fast and robust
algorithms and taking additional information concerning the statistical
properties or the underlying physical process into account.
To get an overview of the concept, consider the following
examples:
You may wish to see the big pictures, showing much more
details by clicking on the thumbnails.

Gravity measurements in the Neusiedlersee area

This is a typical example of a geophysical
sampling set. The data are gravity measurements from the Neusiedlersee
region in the eastern part of Austria. The socalled Bouguer anomaly measurements
were made during a cold winter when the lake was completely frozen, allowing
the researchers to gather data on the otherwise unaccessible interior of
the lake. These samples cause the linetype features in the sampling set.
The number of samples is 989. 

Here you see a contour plot of the reconstruction
of the Bouguer anomaly with a two dimensional version of the ACTmethod,
developed at NUHAG. The original algorithm has been adapted to the geophysical
setup. It involves a weight function which resembles the shape of the gravitational
potential in the frequency domain. The incooperation of the knowledge of
the underlying physical process leads to a stable version of the algorithm
especially suited for the smooth approximation of data from potential functions. 

In order to obtain a better overview of the
approximation an overlay of the sampling set and the contour map is shown
here. The strong negative anomalies on the southern and eastern boundary
of the region are very probably spurious features arising from the periodicity
of the solution computed with ACT. This is not really a drawback as the
most interesting parts of a survey are in practice always near the center
of the region under consideration. 
Airborne magnetics in the Bad Gleichenberg area

This is a typical example of a sampling set
arising in airborne geophysics. In Austria these airborne survey are carried
out with a helicopter that is equipped with a variety of measuring devices.
This results in socalled linetype sampling geometries, which are characterized
by a very high sampling density along the profiles and no data between
the lines of flight, therefor having a strong spatial anisotropy in the
data density. 

This is not only the most difficult sampling
geometry but also a very common one, as airborne surveys allow to gather
a huge amount of data in a very short time. In this example here we have
more than 30000 sampling points.
In this image here, you see the contour map of an anomaly
of the total intensity of the earth's magnetic field. The gridding has
been done with the ACT algorithm. 

Magnetic measurements are very sensitive to
noise. Especially human artifacts, such as cars, powerlines, etc. cause
small scale anomalies with a high magnitude. A good gridding algorithm
should therefor not try to fit the data exactly but rather the error envelope
of the measurements. You can see in this figure, which is an overlay of
a 3D view of the anomaly and the sampling set, that the approximation with
ACT undershoots the small features in the eastern part of the region, which
are indeed due to technical distortions. 
