Reflectance Retrieval Example #1

The mls_rural_23km.adb file contains key illumination variables and this section will attempt to explain what there are, where they are located in the ADB file and how they can be used.

The output radiance for the pixel can be extracted to text in a variety of ways. One of the easiest is to use the DIRSIG image_analyze tool:

The DIRSIG radiometry engine employs a suite of radiometry solver objects to compute the radiance for a given line-of-sight (pixel). The radiance reaching a given pixel is the result of multiple solvers computing individual components including (a) the incident solar irradiance onto a surface, (b) the reflected radiance from a surface, (c) the path radiance from the atmosphere, etc. The complexity of each calculation is hard to capture in a single governing equation, but the following approximation is appropriate for the simple conditions created for this example:

\begin{equation*} L_\mathrm{pixel} = \left [ \tau_\mathrm{up}~\frac{\rho}{\pi} E_\mathrm{sun}~\tau_\mathrm{sun} \cos( \theta_\mathrm{sun} ) \right ] + \left [ \tau_\mathrm{up} \int\limits_{\theta=0}^{\pi/2} \int\limits_{\phi=0}^{2\pi} \frac{\rho}{\pi} L_\mathrm{sky}(\theta,\phi) \sin( \theta ) \cos( \theta ) d\theta d\phi \right ] + \left [ L_\mathrm{up} \right ] \end{equation*}

where L_pixel is the pixel radiance at the sensor, E_sun is the exoatmospheric solar irradiance, t_sun is the solar transmission to the ground, theta_sun is the solar declination angle, rho is the hemispherical (Lambertian) reflectance, t_up is the upward path transmission (from the ground to the sensor), L_sky(theta,phi) is the sky radiance form a specific solid angle in the hemisphere and L_up is the upward path radiance (from the ground to the sensor).

This equation has 3 major terms: (a) the directly reflected solar radiance term, (b) the diffusely reflected sky radiance term and (c) the path (upwelled) radiance term. The reflectance rho is divided by pi in all instances because the BRDF of a Lambertian reflector is the total reflectance spread across the solid angle of the hemisphere (and the solid angle of the hemisphere is pi). Because of the Lambertian reflector being used in this scenario, the reflectance does not vary with theta and phi so the reflectance and sky illumination terms can be decoupled, which will allow this term to be simplified. We still need to include the angular variability in the sky, but the total (hemispherical) sky irradiance can be computed as:

\begin{equation*} E_\mathrm{sky} = \int\limits_{\theta=0}^{\pi/2} \int\limits_{\phi=0}^{2\pi} L_\mathrm{sky}(\theta,\phi) \sin( \theta ) \cos( \theta ) d\theta~d\phi \end{equation*}

which allows us to rewrite the reflected sky term in our approximate radiative transfer equation using the total (hemispherical) reflectance and total (hemispherical) sky irradiance:

\begin{equation*} L_\mathrm{pixel} = \left [ \tau_\mathrm{up} \frac{\rho}{\pi} E_\mathrm{sun} \tau_\mathrm{sun} \cos( \theta_\mathrm{sun} ) \right ] + \left [ \tau_\mathrm{up} \frac{\rho}{\pi} E_\mathrm{sky} \right ] + \left [ L_\mathrm{up} \right ] \end{equation*}

By solving our approximate radiative transfer equation for rho, and using atmospheric terms from the ADB file with the pixel radiance from output image file, we can retrieve the reflectance of the surface:

\begin{equation*} \rho = \frac{L_\mathrm{pixel} - L_\mathrm{up}} {\tau_\mathrm{up} \frac{1}{\pi} E_\mathrm{sun} \tau_\mathrm{sun} \cos( \theta_\mathrm{sun} ) + \tau_\mathrm{up} \frac{1}{\pi} E_\mathrm{sky}} \end{equation*}

The Excel spreadsheet ReflectanceInversion.xlsx performs this retrieval on a wavelength by wavelength basis. The Excel workbook contains two sheets: one for the 2% gray scenario and one for the dry grass scenario. A screenshot of the sheet for retrieving the 2% gray reflector is shown below:

The columns in the Excel spreadsheet can be mapped to the following data sources:

In the case of the 2% Gray scenario, the retrieved reflectance matches the input value (rho = 0.02) at all wavelengths except those where the atmospheric transmission goes to zero and the calculation is undefined.

The retrieval for the dry grass scenario is also a close match. In this case the input reflectance is plotted with the retrieved reflectance and the two curves are indistinguishable except where the atmospheric transmission goes to zero and the calculation is undefined.