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Appendix A: Algorithms for Computing In-band Radiance

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Appendix A:  Algorithms for Computing In-band Radiance

Below are C++ functions for computing the integrated spectral radiance (W m-2 sr-1) and integrated spectral photon radiance  (photon s-1m-2 sr-1).  The functions compute the integral from the specified wavenumber to infinity for a blackbody at the input temperature.  Finite spectral regions can be computed by using this function twice-once with each end point of the spectral region.  The difference of the two gives the radiance for the spectral region.

 #include  // for “exp” function   double planck_integral (double sigma, double temperature) {   //  integral of spectral radiance from sigma (cm-1) to infinity. //  result is W/m2/sr. //  follows Widger and Woodall, Bulletin of the American Meteorological //  Society, Vol. 57, No. 10, pp. 1217   //  constants       double Planck =  6.6260693e-34 ;           double  Boltzmann = 1.380658e-23 ;       double  Speed_of_light = 299792458.0 ;       double  Speed_of_light_sq = Speed_of_light * Speed_of_light ;   //  compute powers of x, the dimensionless spectral coordinate       double c1 =  (Planck*Speed_of_light/Boltzmann) ;       double x =  c1 * 100 * sigma / temperature ;       double x2 = x *  x  ;       double x3 = x *  x2 ;   //  decide how many terms of sum are needed       double iterations = 2.0 + 20.0/x ;       iterations = (iterations<512) ? iterations : 512 ;       int iter = int(iterations) ;   //  add up terms of sum       double sum = 0  ;       for (int n=1;  n

 #include    // for “exp”  function   double planck_photon_integral (double sigma, double temperature) {   //  integral of spectral photon radiance from sigma (cm-1) to infinity. //  result is photons/s/m2/sr. //  follows Widger and Woodall, Bulletin of the American Meteorological //  Society, Vol. 57, No. 10, pp. 1217   //  constants       double Planck =  6.6260693e-34 ;           double Boltzmann = 1.380658e-23 ;       double Speed_of_light = 299792458.0 ;   //  compute powers of x, the dimensionless spectral coordinate       double c1 =  Planck*Speed_of_light/Boltzmann ;       double x =  c1*100*sigma/temperature ;       double x2 = x *  x  ;    //  decide how many terms of sum are needed       double iterations = 2.0 + 20.0/x ;       iterations = (iterations<512) ? iterations : 512 ;       int iter = int(iterations) ;   //  add up terms of sum       double sum = 0  ;       for (int n=1;  n

 Calculation of a Blackbody Radiance Units of Frequency Units of Wavelength Units of Wavenumbers Radiance: Integrating the Planck Equation In-band Radiance: Integrating the Planck Equation over a Finite Range Appendix A: Algorithms for Computing In-band Radiance Appendix B: The Doppler Effect Appendix C: Summary of Formulas References Blackbody Calculator Print Version

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Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

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light with wavenumber between σ and σ + dσ

light with wavenumber between σ and σ + dσ
Isotopes are forms of an element whose nuclei have the same atomic number, the number of protons in the nucleus,but different atomic masses because they contain different numbers of neutrons.
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Cell: model the transmission/radiance of a gas cell. Specify it's length, temperature and pressure, and the vmrs of the absorbing gases.
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Wavenumber cm-1: the number of
wavelengths of light per centimeter

LINEPAK: The GATS spectral radiance and transmission software library. Performs detailed and accurate line-by-line modeling of molecular absorption. Efficient and flexible, LINEPAK is at the heart of analysis systems for many major atmospheric remote sensing missions, including HALOE, SABER, LIMS, SOFIE, CRISTA, and CLAES.
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Tangent Path: Model the transmission or radiance of a ray that passes completely through the Earth's atmosphere but does not intersect the Earth. The path is specified by the tangent height, the height at the point of closest approach to the surface. The pressure, temperature and vmrs of absorbing gases at each altitude are chosen from a database of atmospheric states.
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Slant path: Model the transmission or radiance of a ray between two arbitrary points in the Earth's atmosphere. The points are specified by their heights and the zenith angle from one to the other.
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VMR: volume mixing ratio. The fractional number of molecules of a species in a volume.

Individual vmrs and their sum must be between 0 and 1.

If the vmrs sum to less than 1, the rest of the gas in the cell is assumed transparent.(Lineshapes for molecules with vmr less than 1 are air-broadened.)
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Temperature Offset: The model atmosphere (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust the temperature from the model value, enter a temperature offset (from -50 to 50 K). The Atmosphere Browser tool displays the temperature profiles for the model atmospheres.
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Atmosphere: An atmosphere contains profiles of temperature and gas concentrations at all altitudes. There are six system-supplied atmospheres for Earth and one for Mars. Custom atmospheres can be uploaded from the Atmosphere Browser.
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Scale Factor for Gas Concentrations: The model atmosphere (US_Standard, Tropical, etc.) determines the gas concentrations at each altitude. To adjust a gas concentration, choose a scale factor, from 0 to 1000. For example, to simulate an atmosphere with 20% more water vapor than the model, enter a scale factor of 1.2 for H2O. Note: while the model atmospheres are physically realistic, using large scale factors can produce unphysical situations where the gas abundance exceeds 100%. If this occurs, an error message will be displayed.
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The atmosphere model (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust a gas concentration, choose a scale factor other than 1 (from 0 to 1000). For example, to simulate a path with 20% more water vapor, use a scale factor of 1.2 for H2O. The Atmosphere Browser tool displays the temperature, pressure and gas mixing ratios for the model atmospheres..
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