We know ozone is missing because...
Most of the UV radiation hitting the earth has wavelengths in the range of 100-300 nanometers (nm). When the wavelength is in the range of 315-400 nm, we call this "UVA." When the range is 290-315 nm, we call it "UVB." When the range is 100-280 nm, we call it "UVC." UVA is not blocked by ozone; UVB is somewhat blocked by ozone; UVC is completely blocked by ozone.
Portion of the electromagnetic spectrum showing UV, visible, and infrared light.
Using this knowledge, a scientist named G. M. B. Dobson figured out a way to measure atmospheric ozone. His "Dobson spectrometer" measures the intensity of ultraviolet radiation reaching the earth at four different wavelengths, two in the UVA range and two in the UVC range. Knowing how much UVA reaches the earth gives us a measuring stick against which we can compare the amount of UVC that reaches the earth. Ideally, no UVC should reach the earth, because it is blocked by ozone. Likewise, if all the ozone in the atmosphere were removed, then the amount of UVC reaching the earth would be the same as the amount of UVA that reaches the earth. The relative amounts of UVA and UVC reaching the earth tell us how much ozone there is in the air over our heads.
Dobson spectrometers are calibrated to convert the differences in spectrometer readings to the thickness an ozone layer would be if it were separated from everything else in the atmosphere and compressed to 1 atm pressure at 0°C. An ozone thickness of 1 mm corresponds to 100 Dobson units. The illustration, from Cambridge University in England, shows what the ozone over Labrador, Canada, would be like if the Dobson spectrometer indicated 300 DU.
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