Calculator Inputs
Pick a calculation mode. Use consistent units for I and I0. If you enter transmittance directly, I and I0 are optional.
Formula Used
Atmospheric optical depth (also called optical thickness) measures how strongly the atmosphere attenuates a light beam. A common model for direct-beam transmission is the Beer–Lambert (Bouguer) law:
If you measure transmittance T = I/I0, you can solve for optical depth:
Note: In spectral work, τ depends on wavelength. Enter values consistent with your sensor band.
How to Use This Calculator
- Select Derived from measurements if you have I and I0, or transmittance T.
- Choose air mass handling: compute from zenith angle, or enter m directly.
- Optional: enter τmol to estimate aerosol depth as τ − τmol.
- Optional: enter an effective path length to compute β = τ / L in km−1.
- Press Calculate. The results panel appears above the form.
- Use Download CSV for spreadsheets, or Download PDF for a report.
Example Data Table
Sample values for quick testing (transmittance-based). Your numbers may differ by wavelength, humidity, and aerosols.
| Case | I0 | I | T = I/I0 | Zenith (°) | Air mass (m) | Optical depth (τ) |
|---|---|---|---|---|---|---|
| Clear midday | 1000 | 780 | 0.780 | 20 | ~1.064 | ~0.235 |
| Light haze | 1000 | 620 | 0.620 | 40 | ~1.305 | ~0.367 |
| Dusty air | 1000 | 420 | 0.420 | 55 | ~1.741 | ~0.500 |
| Low sun | 1000 | 260 | 0.260 | 75 | ~3.814 | ~0.352 |
Atmospheric Optical Depth Guide
1) Optical depth in clear terms
Optical depth (τ) is a dimensionless measure of how strongly the atmosphere reduces direct-beam sunlight along a path. A value of τ = 0.20 means the vertical transmittance is e−0.20 ≈ 0.82, so about 18% of the beam is removed by scattering and absorption in a vertical column.
2) Typical ranges you can expect
In many mid‑visible bands, pristine conditions often show τ ≈ 0.05–0.15. Clean rural air commonly falls near 0.15–0.30. Urban haze can push τ into 0.30–0.60, while dust, smoke, or pollution episodes frequently exceed 0.80. The same location can shift by 0.10–0.30 within a day as aerosols and humidity change.
3) Geometry and air mass matter
The calculator uses air mass (m) to convert a slant observation into an equivalent vertical optical depth. At zenith angle 0°, m ≈ 1.00. At 60°, m is about 2.0. At 75°, m rises to roughly 3.8, meaning a much longer optical path. Small errors in zenith angle near the horizon can noticeably alter retrieved τ.
4) Wavelength and band choice
Optical depth is spectral: shorter wavelengths usually have higher molecular scattering, while aerosols can steepen or flatten the spectrum depending on particle size. If your sensor band shifts from 500 nm to 870 nm, τ can drop significantly on a clear day. Keep comparisons within the same band or apply a spectral model.
5) Turning measurements into τ
In measurement mode, you enter transmittance T = I/I0 (or I and I0) and the calculator solves τ = −ln(T)/m. For example, T = 0.62 at m = 1.30 yields τ ≈ 0.37. When T is noisy, averaging several samples improves stability and reduces random error.
6) Understanding the component view
The component mode adds Rayleigh, aerosol, ozone, and water vapor terms: τ = τR + τa + τO3 + τw + … A clear, dry day might resemble τR=0.10, τa=0.12, τO3=0.02, τw=0.03, totaling τ≈0.27. This helps separate scattering‑dominated and absorption‑dominated regimes.
7) Quality checks and uncertainty
Watch for inconsistent inputs: if computed τ is negative, T likely exceeds 1 due to calibration error or scattered light. Thin cloud often increases apparent τ but violates the direct‑beam assumption. Near sunrise and sunset, air mass models are less reliable; interpret results cautiously for zenith angles above about 80°.
8) Practical uses of optical depth
Optical depth supports visibility assessment, solar resource studies, radiative transfer inputs, and air‑quality tracking. Reporting both τ and air mass provides context for geometry. Exporting CSV/PDF results makes it easy to archive daily conditions, compare seasons, and flag high‑aerosol events for further investigation.
FAQs
1) What is atmospheric optical depth?
It is a dimensionless measure of how much the atmosphere attenuates a direct light beam. Larger τ means stronger scattering and absorption and usually lower direct-beam transmittance.
2) Is optical depth the same as aerosol optical depth?
Total optical depth includes molecules, aerosols, and absorbers. Aerosol optical depth is only the aerosol part. If you provide τmol, the calculator estimates aerosol depth as τ − τmol.
3) Why do I need air mass?
Air mass converts a slant measurement at a given zenith angle into a vertical equivalent. When the sun is low, the path is longer, so the same τ produces a larger attenuation along the beam.
4) Can τ be negative?
Physically, no. Negative τ usually means transmittance was computed above 1 due to calibration issues, sensor saturation, or scattered light reaching the detector.
5) Does τ depend on wavelength?
Yes. Molecular scattering is stronger at shorter wavelengths, and aerosol effects vary with particle size. Compare τ values only within the same sensor band, or use a spectral model.
6) How should I choose τmol?
Use a value from a radiative transfer source, a climatology table for your band, or a clear-day baseline. It should be non‑negative and correspond to your site altitude and wavelength band.
7) Do clouds affect the result?
Yes. Thin or broken clouds can increase apparent attenuation and violate the direct-beam assumption. For cloud screening, use stable clear-sky periods or auxiliary measurements like sky imagery or irradiance variability checks.
Educational use: instruments, calibration, and sky conditions can strongly affect retrieved τ.