\[I = I_0\,e^{-\tau m},\qquad \tau = \int\sigma\,n\,dz,\qquad m \approx \frac{1}{\cos\theta_z}\]
含义Meaning
阳光穿过大气被气体、气溶胶吸收和散射而衰减；光学厚度 τ 衡量垂直方向的消光，大气质量数 m 计入斜程。用于太阳辐射、遥感和气溶胶测量（太阳光度计）。
Sunlight is attenuated by gas and aerosol absorption and scattering; optical depth τ measures vertical extinction and air mass m the slant path. Used for solar radiation, remote sensing and sun photometry.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\tau\) | 光学厚度optical depth | 1 |
| \(\sigma\) | 消光截面extinction cross-section | cm² |
| \(n\) | 数密度number density | cm⁻³ |
| \(\theta_z\) | 太阳天顶角solar zenith angle | ° |
推导Derivation
朗伯–比尔定律的微分形式 \(dI = -\sigma nI\,ds\)，沿光路积分。
Differential Beer–Lambert along the path.
平面平行大气中斜程长度为垂直厚度的 1/cos θ。
In a plane-parallel atmosphere the slant path is 1/cos θ longer.
兰利作图：ln I 对 m 为直线，斜率 −τ，截距 ln I₀（大气外辐照度）。
Langley plot: ln I versus m gives τ and I₀.
适用条件与局限Assumptions & limitations
天顶角大于约 75° 时需考虑地球曲率与折射；多次散射使漫射光不服从简单指数律。
Curvature and refraction matter beyond ~75°; multiple scattering complicates diffuse light.