Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations.
The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of buildings or industrial installations, but shadows, together with the time, date and basic trigonometry, enable heights to be determined.
Recently, a former neighbour, Brian Ellis, presented me with a conundrum. He had a postcard showing Dún Laoghaire Harbour about 1905, with some children standing on the East Pier. For reasons of genealogy and local history, he wished to know the date and time when the photograph was taken. The sun shone brightly that morning, casting shadows across the pier. A boy in centre-image (Brian tells me his name was Reggie) was chosen, as his shadow was sharply defined.
Alignments of the spire of the Mariner’s Church with a bandstand pillar and of St Michael’s spire with a mooring bollard determined the camera location, on the upper deck of the pier. The height of the camera and its distance from Reggie could then be estimated.
A solar beam skirting Reggie’s head met the pier at the end of his shadow. This was used to compute the solar elevation. The angle between the shadow and the pier centreline determined the azimuth or compass-bearing of the sun.
The angles on the postcard are not what we require: the camera viewing angle must be used to correct the distortion of these angles by perspective; this requires some school trigonometry. Each stage of processing involves errors, which must be considered when interpreting the results.
The solar zenith angle at noon, called the declination, combined with the observed azimuth and elevation, allow us to determine the hour angle and, thence, the time. For the postcard image, the estimated time was 08:20 local solar time or 09:45 Irish Summer Time.
From the angles we can also estimate the date, but this cannot be done unambiguously. If we plot the solar angles on a polar diagram (like a map of the Arctic), there is an accessible region, determined by the latitude.
The sun passes through each point in this region twice per year, once when moving north in spring and once when moving south in autumn. Thus, for a given set of solar angles, there are two dates when these values occur. For the postcard, the calculated angles indicated dates about April 24th or August 18th, 58 days before and after the summer solstice.
The morning of April 21th was gloriously sunny. Brian and I met on the pier and, about 09:45, each took photographs while the other stood where Reggie had stood in 1905. These were cropped to correspond to the postcard image. The computed azimuth angle agreed closely for the old and new images, indicating that the time for the original photo was accurately estimated from the image: the error was probably not more than about 20 minutes.
However, the solar elevations derived from the two photos differed by about seven degrees, corresponding to an error in the date of perhaps two weeks. This is unsurprising, given the several sources of uncertainty.
We can be confident that analysts in the Pentagon have a vastly more powerful image-processing toolkit and are busy using it for the benefit of the citizens defending their homeland in Ukraine.
Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics – he blogs at thatsmaths.com