Monday, August 31, 2009

I haven't been there - so I will just Interpolate

Though not as hazardous and short sighted as extrapolation, interpolation can be quite challenging and fraught with errors and pitfalls. Whenever data is collected on the ground it is not evenly sampled and smooth, there are system drifts as well as uneven motion of the sampling platform. In order relate this ground data to nicely gridded remote sensing data, from SAR or Optical platforms, interpolation is needed. There are lots of interpolation techniques, I will discuss some of the easier to explain ones here:
  1. TIN based - This technique relies on indentifying the 3 nearest non-planar points to the point being interpolated and fitting a planar or higher order surface to these points. The grid-point is then calculated as a point on that surface.
  2. Inverse Distance Weighted - This technique uses a weighted linear combination of all or a local subset of the known unevenly sampled points. The weights are calculated using various techniques e.g. 1/distance , 1/(sqr(distance)), exp(-sqr(distance)). If there are known characteristics in the sampling e.g. there is denser sampling in one direction, the weights can be biased to reflect this.
  3. Thin Plate Splines - Similar to the IDW techniques splines can be fitted to the full or local subset of the unevenly sampled dataset. Splines produce smooth interpolation results, but can have large spikes in sections if there are inherent discontinuities in the underlying data or large gaps.
  4. Geostatistical Techniques - Foremost of this is the Kriging technique which makes assumptions about the parametric distribution from which the sampled points are drawn and estimates the parameters of this distribution using maximum likelihood estimators. The assumption of spatial dependence may not be valid for widely spaced samples and statistical tests should be employed before the interpolation is used.
Surfer implements a gamut of interpolation techniques and provides a very useful way of testing the difference in the results obtained using different methods.
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