The emissivity of Boltzmann's law

[leilei zhu]

Dear DART support team,
I entered a spectral curve of a surface feature into DART to simulate its broadband radiance between 8–13.5 µm, so I selected Boltzmann. However, the emissivity shown in simulation.properties.txt differs by about 0.07 from the value I obtain when integrating the curve myself. I have checked the DART manual but still do not understand how Boltzmann computes this emissivity.
Could you give me some hints?
Thank you for your time and help.
Best regards,
Leilei Zhu

[Yingjie Wang]

Dear Leilei,

The Boltzmann law (exitance = emissivity * sigma * T^4, https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#:~:text=For%20an%20ideal%20absorber%2Femitter,the%20black%20body's%20temperature%2C%20T%3A&text=%CF%83%20%3D%205.670374419...) gives exitance (W/m2) integrated for the whole spectrum and for all hemispherical directions.

The emissivity is an input for Boltzmann law, not an output. If you define a boradband in DART, the band-mean emissivity will be computed according to your input spectral emissivity. 

If you want to compute the broadband radiance for a given spectral interval and a given direction. You should use the Planck's law instead. DART provides Broadband tools for computing broadband radiance taking into account the spectral response function of sensor. 

Hope it clarifies the problem,

Best,

Yingjie

[leilei zhu]

Dear DART support team,
My ultimate goal is to calculate the broadband emissivity. I input a spectral curve for simulation because I noticed in your help documentation that the emissivity calculated using Boltzmann's law is displayed as shown in the figure below. I'm very curious to know whether you compute the broadband emissivity by dividing it into very small intervals and performing a weighted sum, or whether you use integration to obtain the emissivity for this range. Is it correct that I need to simulate the band range across many small intervals and use the Planck method for simulation? Could you provide some reliable advice?
Another issue I've noticed is that when DART identifies spectral curve ranges, it encounters errors in calculating emissivity values for bands with excessively high decimal places.
Best regards,
Leilei Zhu

[Jean-Philippe Gastellu-Etchegorry]

Dear Leilei,

Sorry to answer you so late. You use the work package WP2H in the DART User Manual. First, note that all DART simulations of DART WPs are available for scientists: https://dart.omp.eu/index.php#/doc.  However, I just note that the last update of WP2, including WP2H, is 2023! This is very old because DART changed since then. In order to correctly answer you, we can up-date this WP2, including WP2H. 

The objective of WP2H is to compare Stefan Boltzmann with the integral of Planck's law over 2 spectral bands   [1µm  99µm] and   [1µm  201µm]. The integral (actually a sum of discrete terms) is done with different numbers N of sub-intervals by a code called "BroadBand", which has also been greatly up-dated since 2023!  WP2H stresses that:

- Integrals differ depending on the spectral range considered. This logical, because Stefan Boltzmann is for an infinite spectral range. The interest of the exrcise is to quantify the error done if you consider a finite spectral range. This knowledge is useful ifto correct the measurement of a thermal infrared radiometers such as NR01 that operates over a finite spectral range with a spectral range [4µm  42µm].

- If ground emissivity is spectrally constant, then results do not depend on N (for a bare ground!). If ground emissivity is spectrally variable, then results depend on N, as expected.

Please let us know if an up-date of WP2, including WP2H, could be useful for your work. Then, we will up-date it very fast. Also, please send the folder "input" of your simulation. Also, to let us know the objective of your work would be useful to help you (sorry if you already indicated it).  

Advice: use the latest DART version DART V1446  https://sdrive.cnrs.fr/s/t9fWMnqC5QHAeW3

Best regards

DART Team