I wondered above what the scientific basis could possibly be for infill of gaps in the sparse radar data used to determine the bedrock DEM. It seems that 'ordinary kriging' was used as the geostatistical interpolation method (even though it is inappropriate and sub-optimal here, see below). That generates an error map which I located both on M Morlighem excellent site and also at NSIDC in all its detail as a 1.4 GB netCDF download (which can be opened with Panoply freeware).
Notice the huge formal errors in interpolated bedrock elevations -- up to 600 m, which could easily amount to 25%. This is so large relative to measured bedrock elevation changes over characteristic distance scales that structural features inside the empty polygons hardly improve on guesswork. That certainly leaves paleo stream flow up in the air for gappy regions.
http://forum.arctic-sea-ice.net/index.php/topic,867.msg41632.html#msg41632http://sites.uci.edu/morlighem/dataproducts/mass-conservation-dataset/http://nsidc.org/forms/IDBMG4_or.html?major_version=1http://www.giss.nasa.gov/tools/panoply/download_mac.htmlI looked around for decent explanations of 'ordinary kriging' since the main wikipedia article falls into the statistical weeds already in its first sentence, never really spelling out the assumptions that need be met for kriging to be applicable, namely that the data points and their auto-covariance represent a random sample from some well-behaved distribution.
The article goes on to put down bézier splines but the fact is these will provide a superior fit to radar isochrons because of a better fit to the physics governing ice deformation, to which kriging is totally oblivious as it is to ore body shapes. (I've always wondered if Krige's mining company went broke following his advice.)
Bézier splines were originally introduced to describe compressible hydrodynamic flow over auto bodies (at Renault and Citroën, 1960's) so they will work better for ice. Bedrock deformation under isostatic gravitational loading and glacial erosional processes also leads to systemic effects but with different governing physics.
Kriging is sometimes described as the '
best linear
unbiased prediction'. While that sounds hard to improve on, better than reality even, it's only as good its unrealistic assumptions and only as bad as the physical insights it neglects.
Kriging is another Daisy World of the modelling community, an over-simplification that can produce highly erroneous output yet somehow get billed as 'optimal', as if statistics somehow can be a workaround to (unnecessary) laws of physics. The real appeal of kriging is its quantification of error -- wrapping output in a soothing pseudo-scientific veneer.
Thus the error map below does NOT show experimental error in our knowledge of bedrock elevation beneath the Greenland ice. Instead it shows the statistical error in an abstract inapplicable statistical model world that has REPLACED the real world and taken on a 'reality' of its own. It doesn't really matter how the bedrock map is filled in -- at the end of the day,reported error should be with respect to actual physical measurement (held-back oblique radar tracks).
As a practical matter, kriging calculates the value of an unknown interior point simply as the average of nearby points, de-weighted by their distance away. Although that is traditionally taken as 1/r, it could just as well be 1/r
2 etc depending on the context. See links below.
The coefficients of that power series expansion could be empirically determined for the case at hand (Greenland) by 'holding back' radar tracks that cut diagonally across grid squares, then varying the de-weighting until the parameters of best fit are determined. However I haven't noticed anyone 'holding back' tracks in the Greenland scientific literature to get a real grip on error.
The radar track portfolio does not remotely meet the criterion of random sample (stochastic process). First, track density is very strongly correlated with proximity to a few Greenland airports. For example, nearly all flights in northern Greenland originate and return to Thule, almost all passing over Camp Century.
Many flights follow the summit ridge, fly parallel to the coastline, or execute a tight grid just up from select marine terminating glaciers. Very few followed principle curvatures (elevation contours and flow lines). However each flight had distinct pre-specified scientific objectives. While all this is sensible, statistically it amounts to strongly biased sampling.
Geostatistics is a highly developed field because data is always too sparse yet interpolative infill is always wanted. However one size doesn't fit all, the optimal method must be adapted to actual sampling and expectations from the operative controlling physics. Above all, a feedback cycle from prediction validation (held-back data) must inform the interpolative method.
In Greenland radar data provices continuous vertical slices. That is already very different from point data (eg ice drill cores). The result is a partition of Greenland into vertically extruded polygons whose values are known on the walls.
Additionally the ice surface elevation itself is known everywhere to great accuracy so needs no interpolation, as with ice surface velocity. With some mild physics, these two caps to the polygons can significantly inform infill of interiors.
Slight ridging of the ice surface may have some information about what's underneath -- however it is largely unable to predict bottom upheavals or conformal displacement of isochrons. Conversely, roughness sometimes observed in the surface DEM may have no evident englacial counterpart.
Isochrons are exceedingly well-behaved on many radar tracks, easily fit to simple continuous equations, readily modified to reflect dimpling waves, thus making them accessible to
reliable analytic continuation to interior surfaces of the extruded polygons. Isochrons are a more favorable situation than than bedrock topography itself which is finer-grained and so harder to interpolate.
Indeed it is the mid-depth isochrons, in conjunction with ice thickness overhead, that are best suited to refining bedrock topography
by providing intermediate caps. In this view, the unusual data structure of Greenland calls for something different than 'ordinary kriging', more along the lines of incorporating polynomial trends as in 'universal kriging'.
One problem -- or is it an opportunity -- with Greenland's extruded polygons is their surfaces are generally oblique with respect to ice flow (~ surface elevation gradient) and so physically unnatural. These polygons often have 4 sides but are not generally rectilinear. (In hindsight, it might have been better to have flown more gridded flowlines and elevation contours.)
The Petermann glacier is a special case of a regular 10 km grid in which some 300 extruded polygons are both four-sided and orthogonal, with very low angles to ice flow. This area is also sliced by dozens of oblique flight lines that can be held back for validation.
Let's consider the middle isochron of the 'Three Sisters' on an opposing pair of east-west polygonal walls (flow being northward). Suppose both are fit to simple splines and the interior predicted as a parametrized deformation of one into the other that incorporates deflectional effects of the direction and magnitude of ice flow.
Repeating this process with the north-south opposing walls gives an independent prediction (anchored to the same corner boundary conditions). These two predictions then generate a difference DEM, the best least squares splinal surface fit to which is the final interior infill prediction. The hundred or so other isochronal surfaces will be a closely related one-parameter modulation of this fit.
Since dimpling of isochrons in general reflects the ice's response to passing over bedrock topography in a low pass fourier filtration sense, the interior isochron provides an inversional control on the edge interpolation of bedrock.
http://www.bisolutions.us/A-Brief-Introduction-to-Spatial-Interpolation.phphttp://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//009z00000076000000.htm
