umasundresh

You have seen the UV index on weather apps, but where does that number actually come from? Behind it lies a beautiful piece of applied mathematics: a weighted integral over the solar spectrum.

What is ultraviolet radiation?

Sunlight is not just light. It is a spectrum, a continuous range of wavelengths. The part we can see, called visible light, spans roughly 380 to 700 nanometers (nm). Just beyond the violet edge of this range lies ultraviolet radiation, or UV, which stretches from about 100 to 400 nm. We cannot see it, but it has a strong effect on our skin and health.

UV radiation is divided into three bands:

• UV-A (315 to 400 nm): The longest wavelength in the UV range. It penetrates deeper into the skin and is linked to skin ageing and long-term damage.
• UV-B (280 to 315 nm): The most important for everyday exposure. It carries more energy and is the main cause of sunburn.
• UV-C (100 to 280 nm): The most energetic and dangerous type. Fortunately, it is almost completely absorbed by the atmosphere and does not reach the ground.

The core formula

The UV Index looks simple. A number like 5 or 10 appears on a weather app, and we treat it as just another daily indicator. But behind that number is a careful process that starts with real measurements of sunlight and ends with a compact mathematical result.

At the core is this formula:

E(λ) is the spectral irradiance at wavelength λ, in units of W·m⁻²·nm⁻¹. This is how much solar energy arrives at each narrow slice of wavelength.

S(λ) is the erythemal action spectrum, a dimensionless weight that encodes how biologically damaging each wavelength is to human skin. It was empirically measured from sunburn studies.

k_er = 40 m²·W⁻¹ is a normalising constant, chosen so that midday summer sun at mid-latitudes gives a UVI of about 10. The integral runs from about 250 nm to 400 nm; beyond that, S(λ) is essentially zero.

Where does E(λ) come from?

E(λ) is not assumed or invented. It is measured or modelled from physical reality using one of three approaches.

The most direct method is a ground-based spectroradiometer, an instrument that measures actual solar irradiance at each wavelength in real time. This gives highly accurate readings tied to the exact local conditions at the moment of measurement.

The second approach uses radiative transfer models such as SMARTS or libRadtran. These solve the radiative transfer equation using inputs like solar elevation angle, ozone column depth, aerosol loading, and altitude. They compute E(λ) at every nanometre across the UV spectrum from first principles.

The third source is satellite retrieval. Instruments such as NASA OMI and ESA TROPOMI measure backscattered UV radiation from space. Inversion algorithms then estimate the surface-level E(λ) from those observations, allowing global UV mapping even where no ground instruments exist.

The erythemal action spectrum S(λ)

The weight function S(λ) is piecewise defined according to the CIE standard:

Notice that S(λ) = 1 at the most damaging wavelengths (250 to 298 nm), then drops off exponentially as wavelength increases. By 400 nm, S(λ) is roughly 0.001, meaning UV-A contributes very little to the index compared to UV-B.

Scale and risk categories

The resulting integral is a continuous value. The WHO classifies the UVI into five risk categories:

What drives the UVI up or down?

Four physical variables have the largest effect on the UVI value:

• Solar elevation angle: The higher the sun in the sky, the shorter the atmospheric path UV radiation must travel, and the less it is scattered. A sun at 90° (directly overhead) delivers maximum UV.
• Altitude: UV intensity increases roughly 6% per kilometer of elevation, because there is less atmosphere to absorb it.
• Ozone column depth: Measured in Dobson Units (DU), stratospheric ozone absorbs UV-B strongly. A thinner ozone layer (lower DU) means a higher UVI.
• Cloud cover: Thick cloud cover can reduce UVI by up to 75%, but thin cloud or haze reduces it only marginally.

How the UV integral is computed in practice

In reality, neither satellites nor ground instruments measure radiation continuously across a smooth curve. Instead, they record energy at specific wavelength intervals, typically every 1 nanometer. The integral is therefore replaced by a Riemann sum:

Each term represents the contribution from one wavelength interval. With Δλ = 1 nm and roughly 150 steps from 250 to 400 nm, the sum converges very closely to the true integral. Satellite products from NASA OMI and ESA TROPOMI use exactly this method: spectral measurements, biological weighting, and a summation across all wavelength steps to produce the final UV Index value.

Takeaway

In theory, the UV Index comes from a smooth, elegant integral. In reality, it is more like a very patient accountant, quietly adding up about 150 tiny numbers, one wavelength at a time. And after all that careful work, it simply tells you: put on sunscreen.

Sources: WHO/UNEP/ICNIRP Global Solar UV Index (2002); CIE Standard 87-1989; NASA OMI Level 3 UVI product documentation.