Monday, September 6, 2010

Flying Bombs on London, Summer of 1944

Beginning in June, 1944, London was the target of attacks by V-1 "flying bombs" launched primarily from France. The attacks continued through the summer, ending as the launch sites were overrun by advancing allied forces.

In a short, one page, article published in the Journal of the Institute of Actuaries in 1946, R. D. Clarke presented an analysis of the distribution of V-1 impacts on London. (A pdf of the article is available here.) He showed the pattern fit a Poisson distribution extraordinarily well, refuting claims that the bombs tended to cluster geographically. Subsequently, this Poisson pattern was mentioned in Thomas Pynchon's novel Gravity's Rainbow (see the reference to page 54 here) and in Feller's classic text, An Introduction to Probability Theory and Its Applications, in 1957.

While the data table from Clarke's article has often been reprinted and used in examples for statistics classes or lectures, the raw data of individual impacts seems not to have been available. (The paper is often misrepresented on web sites, by the way, with fanciful "stories" behind the analysis and frequent mis-identification of the V-1 as the later V-2 rocket. I was surprised by how much "legend" has grown up unchecked about this paper and the data behind it. Who knew the internet contained incorrect information.)

I have reconstructed the data from the original maps in the British Archives in Kew and present above the density distribution of impacts for June through August of 1944. In the plot above, the view is from the North-East. The higher the density the greater the number of impacts. The density falls off very sharply north of the Thames, while the greatest density is slightly south and east of central London.

Below is the same density but viewed from the south-west.

It is also good to consider a contour map of the density as well. The numbers on the contours are the estimated number of impacts per square kilometer. (Clarke's analysis was based on quarter-kilometer impact data.)

Clarke's analysis was focused on the central area of higher density here and with finer geographic coordinates. Within that area his analysis found no evidence of clustering that cannot be accounted for by a Poisson process.  From the wider geographic perspective presented here, the greatest density of the attack is well defined, with impacts rising and then falling sharply along a south-east to north-west axis, roughly the flight-path of the V-1s coming from France, and falling off a bit more slowly along the south-west to north-east axis. This suggests that the range of the V-1 was somewhat more accurately controlled than was the "windage" from left to right of the flight path.

The statistician F. N. David also analyzed these impact data during the war, applying a bivariate normal distribution to the data. In Clarke's unpublished war-time analysis of 1944 he critiques her modeling, and suggests that a mixture of normals would be a better representation due to frequently changing launch points and aiming points in London.

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