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Instrument Functions

Instrument Functions and Spectral Sampling

The "Monochromatic" Spectrum


At www.spectralcalc.com, every spectrum is first computed at "full" resolution.  This is essential for accurate results.  Sometimes referred to as the "monochromatic" spectrum, this has a resolution adaptively set to 20% of the narrowest line's Lorentz halfwidth.  This adequately captures all information in the spectrum.  The resolution is adaptive because the Lorentz halfwidth of the narrowest line will change with waveband, species, pressure and temperature.  Note that even when simulating spectra from a low-resolution instrument, it is necessary to first accurately compute the monochromatic spectrum, and then convolve this with the appropriate instrument response function. 

Smoothed (or "apodized") Spectra

Monochromatic spectra can have extremely fine resolution, especially for low-pressure situations where the lines are very narrow.  This can result in huge data files with upwards of hundreds of millions of points.  Any real instrument, however, will have a spectral response function that defines its resolution.  The measurements that it produces are the result of the convolution of the monochromatic spectra with the instrument spectral response function.  This convolution or smoothing process is sometimes referred to as "apodizing" the spectrum.  This process can be modeled on SpectralCalc using Instrument Functions.

One must remember that smoothed spectra should generally not be used in post-processing applications.  Smoothing with an instrument function is correctly modeled only as the last step in the chain.   For example, to model the combined spectrum of sequential absorbing paths, we should first compute and save the monochromatic transmittance spectra of each. Then (on the My Spectra Tab) we can compute their product, and finally apply the appropriate instrument function to the resulting monochromatic product spectrum.  Applying the instrument function to the individual spectra and then combining them will produce erroneous results.

Applying Instrument Functions

To apply an Instrument Function on SpectralCalc, select the type of function and specify its width W.  We provide a variety of Instrument Function types to choose from:



Here x = σ/W.  (σ  denotes wavenumber).  W is therefore the full width (as opposed to half width).  We constrain W to be no more than 250 cm‑1, and no more than half of the waveband:

where σ1 and σ2 are the lower and upper bounds of the waveband. 

Note that Instrument Functions are always computed and applied in wavenumbers.  A width specified in microns is converted to the equivalent width in wavenumbers at the center of the bandpass.  As a result, an instrument function with its width specified in microns will actually have different effective widths in different spectral bandpasses.

The relative responses of each these functions are shown in Fig. 1. The sinc function is truncated at ±2W, and the Gaussian at ±3W/2.  The triangle's extent is ±W, and the square's is ±W/2.  These functions are sampled at same resolution as the monochromatic spectrum, and normalized to give unit integrated response (i.e. scaled so that the sampled points sum to 1).  The normalization is performed numerically by summing the actual samples of the truncated functions, not by analytically computing the integrals of the underlying continuous functions.  Then the final, smoothed spectrum is computed by a straightforward convolution of the instrument function over the monochromatic spectrum.


Final Resampling

Having selected waveband limits σ1 and σ2, and an acceptable instrument width W, the smoothed spectrum is resampled with a resolution commensurate with the inputs:


This gives a resolution of 5% of the instrument halfwidth or finer.  It further ensures that the samples evenly divide the user-specified waveband, so that samples occur exactly at σ1 and σ2, and at evenly spaced intervals therein.


Fig. 1—Instrument Functions


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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

The information provided will not be shared, sold or used in any way other than to contact users to announce new features.
Radiance: radiant flux radiated per unit area, per unit solid angle, per wavenumber

light with wavenumber between σ and σ + dσ
Radiance: radiant flux radiated per unit area, per unit solid angle, per wavenumber

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.
Cell: model the transmission/radiance of a gas cell. Specify it's length, temperature and pressure, and the vmrs of the absorbing gases.

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.
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.
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.
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.)
Clicking this will display the data as text in a new browser window. Right-clicking will download the data file to your computer (recommended). These files can be extremely large depending on the spectrum simulated.
Clicking this will open a new browser window suitable for printing.
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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.
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.
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.
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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