Question

The total irradiance on a Lambertian surface is 540 \(W\,m^{-2}\). The reflectance of the surface is 20%. The path radiance from the atmosphere towards the sensor is 2.5 \(W\,m^{-2}\,sr^{-1}\). The transmissivity of the atmosphere is 80%. The radiance reaching the sensor is ________ \(W\,m^{-2}\,sr^{-1}\) (Rounded off to the nearest integer).

Hint:

Use the radiative transfer relation for at-sensor radiance: reflected surface radiance (reflectance times irradiance divided by pi), attenuated by atmospheric transmissivity, plus the path radiance.

Answer 1

Correct Answer: 30

Step 1: Write the radiance leaving a Lambertian surface.
For a Lambertian (perfectly diffuse) surface, the reflected radiance is uniform in all directions, and the total irradiance \(E\) falling on it is spread over a hemisphere of solid angle \(\pi\) steradians. If the surface reflectance is \(\rho\), the radiance leaving the surface is \[ L_{surface} = \frac{\rho E}{\pi} \]

Step 2: Substitute the given values.
Here \(\rho = 20\% = 0.20\) and \(E = 540\ W\,m^{-2}\), so \[ L_{surface} = \frac{0.20\times540}{\pi} = \frac{108}{\pi} = 34.377\ W\,m^{-2}\,sr^{-1} \]

Step 3: Account for atmospheric attenuation of the surface-leaving radiance.
As this radiance travels up through the atmosphere to the sensor, it is attenuated by the atmospheric transmissivity \(\tau = 80\% = 0.80\), giving a received component of \[ L_{surface}\times\tau = 34.377\times0.80 = 27.502\ W\,m^{-2}\,sr^{-1} \]

Step 4: Add the path radiance.
The atmosphere itself scatters sunlight directly into the sensor's line of sight without ever touching the ground; this extra term is the path radiance \(L_p = 2.5\ W\,m^{-2}\,sr^{-1}\), and it adds directly since it already represents radiance arriving at the sensor. The total radiance at the sensor is \[ L_{sensor} = \frac{\rho E}{\pi}\tau + L_p \]

Step 5: Compute and round off.
\[ L_{sensor} = 27.502 + 2.5 = 30.002\ W\,m^{-2}\,sr^{-1} \] Rounded to the nearest integer, \(L_{sensor} = 30\ W\,m^{-2}\,sr^{-1}\).

\[ \boxed{L_{sensor} \approx 30\ W\,m^{-2}\,sr^{-1}} \]