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Smoothness and fidelity on glaciers

Whether this blog is the dullest or the second-dullest blog you have ever read will depend on whether you think shortage of wiggle room on glaciers is duller than spatial interpolation. But the two are connected, and there is a bright side to lack of wiggle room. It is what makes mapping possible and useful.

Interpolation is what we have to do when we have a geographically broad problem and measurements that don’t cover the ground densely enough. If the measurements were not correlated, reducing the wiggle room for estimating uncertainty, we couldn’t interpolate between them. We are about equally uncertain in all of the measurements, but at least the correlation lets us look at the broad picture.

We tend to trust contoured topographic maps because the height of the terrain is highly correlated over distances that are smaller than the typical distance between adjacent contours, and because they are based on dense samples of heights from air photos or satellite images. However, interpolating variables that are sparsely sampled presents problems.

Glacier mass balances tend to be well correlated over distances up to about 600 kilometres, making spatial interpolation possible. In turn that makes it possible to judge whether our sample of measurements is spatially representative, but it isn’t much help for estimating mass balance in all the regions where are no measured glaciers within 600 km.

There are several methods of interpolating to a point where there is no measurement. I like to fit polynomials — equations in the two spatial coordinates, easting and northing — to obtain smooth surfaces centred on each interpolation point.

All of the methods have one thing in common: adjustable parameters. Broadly speaking, you get a tradeoff. You can choose smoothness of the resulting map or fidelity to the raw measurements, or any combination. Some methods guarantee, mathematically, that your interpolated numbers will agree exactly with the measured numbers at the points where there are measurements. Most methods, however, acknowledge that the measurements themselves are uncertain, and allow a bit of wiggle room. To tell the truth, by twiddling the knobs on your interpolation algorithm you can seem to create as much wiggle room as you like — in other words, any shape of surface you like (almost).

There seems to be only one objective way of judging the merit of your interpolated surface: cross-validation. If you have n measurement points, redo the interpolation n times, leaving out one point each time, and calculate the typical (technically, the root-mean-square) disagreement between the omitted measurement and the corresponding interpolated estimate. The trouble with this is that it tells you nothing about how you are doing at places where there are no measurements, which is the aim of the exercise.

One thing that no interpolation algorithm can do is manufacture facts. This is an insidious problem, because it often looks as though that is what they are doing. But what are the alternatives?

The most obvious is more measurements. But we scientists always say that, don’t we, and although technology keeps advancing we still have lots of gaps. Lately I have been wondering whether we need to be more brazen about this. Yes, more measurements mean more money for scientists to spend. But suppose it were more money on an economy-altering scale. Jobs in glacier monitoring, and environmental monitoring generally, would multiply. The diversion of financial resources would mean that jobs such as making, fuelling, driving and repairing motor vehicles would dwindle. That would be a good thing, wouldn’t it?

Before it happens, though, the economists will have to work out how to fool the economy into thinking that accurate maps of glacier mass balance are worth more than motor cars.

A more practical alternative is to take the average of what you know as the best guide to what you don’t know. In fact the average is just a polynomial of order zero, that is, a limiting case of the idea of spatial interpolation. It works beyond 600 km, and indeed it works anywhere. But best does not necessarily mean good. New measurements might not change the picture much. On the other hand, they might.

Data voids mean irreducible uncertainty. It may be uncertainty we can live with, but it is also uncertainty in the face of impending trillion-dollar decisions. From that angle, a billion-dollar decision to make better maps and fewer motor cars begins to look good. In the meantime, beware of smooth maps and of maps that are slavishly faithful to the measurements. There is more going on beneath the contours than meets the eye.

Posted by Graham Cogley on April 19, 2010 3:00 PM | Permalink