Radiance Calculation for Lambertian Surface

This problem requires calculating the radiance reaching a sensor from a Lambertian surface, considering factors like irradiance, surface reflectance, atmospheric path radiance, and atmospheric transmissivity.

Given Information

• Total Irradiance ($E$): $540 \text{ W m}^{-2}$
• Surface Reflectance ($\rho$): $20\% = 0.20$
• Path Radiance ($L_p$): $2.5 \text{ W m}^{-2} \text{ sr}^{-1}$
• Atmospheric Transmissivity ($\tau$): $80\% = 0.80$

Radiance Formula

For a Lambertian surface, the radiance ($L_{sensor}$) reaching the sensor is calculated using the following formula, which accounts for the reflected radiance transmitted through the atmosphere plus the atmospheric path radiance:

$ L_{sensor} = \left( \frac{\rho \times E}{\pi} \right) \times \tau + L_p $

Where:

• $\frac{\rho \times E}{\pi}$ represents the radiance leaving the Lambertian surface.
• $\tau$ accounts for the reduction in radiance due to atmospheric transmission.
• $L_p$ is the radiance added by the atmosphere itself.

Calculation Steps

1. Calculate the radiance leaving the surface:

   $ L_{surface\_radiance} = \frac{\rho \times E}{\pi} = \frac{0.20 \times 540 \text{ W m}^{-2}}{\pi} \approx \frac{108}{\pi} \text{ W m}^{-2} \approx 34.377 \text{ W m}^{-2} $

2. Calculate the portion of surface radiance transmitted through the atmosphere:

   $ L_{transmitted} = L_{surface\_radiance} \times \tau \approx 34.377 \text{ W m}^{-2} \times 0.80 \approx 27.5016 \text{ W m}^{-2} $

3. Add the atmospheric path radiance to find the total radiance reaching the sensor:

   $ L_{sensor} = L_{transmitted} + L_p \approx 27.5016 \text{ W m}^{-2} + 2.5 \text{ W m}^{-2} \text{ sr}^{-1} \approx 30.0016 \text{ W m}^{-2} \text{ sr}^{-1} $

4. Round the result to the nearest integer:

   The calculated radiance is approximately $30.0016 \text{ W m}^{-2} \text{ sr}^{-1}$. Rounded to the nearest integer, the value is 30.

Final Result

The radiance reaching the sensor is calculated to be approximately 30 $\text{W m}^{-2} \text{ sr}^{-1}$.