Contribution function

Fig. 3.12 Intensity contribution function, extracted from dynamic light scattering intensities, versus apparent hydrodynamic radius for PBO PEO PBOiz in aqueous solution at 10 °C and different concentrations (indicated) (Zhou el al. 1996c),...

Moreover, advanced evaluation methods exist to apportion PM levels measured at the receptor site to the trajectory segments (e.g. the residence time weighted concentration method [23]) or to regions (grid cells) hit by trajectory ensembles (e.g. the potential source contribution function, PSCF [24, 25]). Software tools are available which facilitate such calculations and visualisations of the results [26]. 

Fig.4 Individual Potential Source Contribution Function maps for toxaphene (A) Michigan (B) Indiana (C) Arkansas (D) Louisiana from Hoh and Hites [45]. Dark red represents PSCF values from 0.91 to 1.0, with shades of pink ranging down to 0.61 by tenths, and white represents from 0.41 to 0.60. Dark blue represents 0-0.10, with shades of blue ranging up to 0.40 by tenths. The green star in each map represents the sampling site. [Reproduced with permission from ACS]...

Fig. 18.1. Series 4 concerns the ground and metastable populations of ions in a plasma and their preparation and calculation for dynamical plasma models. It operates with GCR recombination and ionization coefficients, associated power loss coefficients and metastable fractions. The scope of series 4 is quite large extending into short wavelength filters modifying observed radiative power, astrophysical contribution function generation and parametrization of ionization and recombination...

Du, S.Y., Rodenburg, L.A. (2007) Source identification of atmospheric PCBs in Philadelphia/Camden using positive matrix factorization followed by the potential source contribution function. Atmospheric Environment, 41(38) 8596-8608. 

Energetic (enthalpic) contributions Functional form Magnitude of A7/(gas phase) Magnitude of AH(soln)... 

The reaction involves two reactants (butadiene and maleic anhydride), which both contribute functional groups to the product. The infrared spectrum of the isolated material reflects this observation. Compare the infrared spectrum of your product with that of the reference spectrum. 

The reason can be clarified wifh the concepts of weighting and contribution functions. If emission from the surface does not depend on azimuth, Eq. (2.4.1) can be written... 

This function is the gradient of the transmittance through the atmosphere in the direction /i. The weighting function is a maximvun where the transmittance is changing most rapidly. Hence, from Eq. (4.2.9), if the variation of B(z) with z is small compared with that of W z, At), the level at which Wiz-, ir) is maximum is also the level that contributes most to the outgoing intensity 7(zo> At). More often, however, the variation B(z) with z is not small, and the maximum of the contribution function,... 

The correlation between contribution functions and the observed spectrum becomes most evident when the atmosphere undergoes temperature reversals. Consider the temperature profile shown in Fig. 4.2.4. Three distinct lapse rates define the thermal structure two positive lapse rates bound a negative one in the middle. Discontinuities in temperature gradient are avoided by rounding the profile at the two extremes. 

The topmost panel in Fig. 4.2.5 shows a spectrum for which Zeff extends over the full altitude range, 0-90 km. This is demonstrated in Fig. 4.2.6, in which four contribution functions associated with four critical wavenumbers are displayed. Three of the wavenumbers (v = 367, 393, and 400 cm ) correspond to minima or maxima in the spectrum, while the fourth (v = 200 cm ) is located in the far wing. 

The atmosphere is very transparent at200 cm and almost opaque at the line center at400 cm . The corresponding contribution functions are relatively narrow, and their peaks are located at the surface and at the top of the atmosphere, respectively. As a result the calculated intensities of the spectrum at these two wavenumbers are almost equal to the Planck intensities associated with the temperature (T = 150 K) at the bottom and top of the atmosphere. 

The contribution function for v = 393 cm is associated with a peak spectral intensity at this wavenumber, and also has a maximum at z = 60 km, the altitude at which the temperature profile is maximum. However, the function itself is rather broad, and fairly large contributions to the outgoing intensity at 393 cm arise from a moderate range of altitudes centered about 60 km, over which the temperature is less than maximum. Consequently, the spectral intensity at 393 cm is only slightly greater than the Planck intensity for T = 180 K, as indicated in Fig. 4.2.5a, rather than that for T = 210 K, as implied by Fig. 4.2.4. 

A minimum in the spectrum occurs at v = 367 cm implying the associated weighting function is maximum near z = 30 km, where the temperature (and hence the Planck intensity) has a minimum. Thus the weighting function and Planck intensity tend to counteract each other, and their product results in the broad, double-peaked contribution function shown in Fig. 4.2.6. In this case the concept of an effective emission level has little meaning, since there exists a broad altitude range over which individual levels contribute about equally to the outgoing intensity. This phenomenon is characteristic of temperature minima... 

In summary, the spectral intensity at a given wavenumber can, in certain spectral regions, be closely associated with the Planck intensity of the atmosphere at a given effective emission level Zeff- In other spectral regions, especially near spectral minima, the association is not as close. To the extent that Zeff is meaningful, the emission properties of this level are governed by the optical properties and cross sections of the particles and molecules present at this level, as well as the temperature profile. The sharper the contribution function associated with Zeff, the better defined this level is. 

Fig. 4.2.6 Contribution functions at selected wavenumbers, corresponding to the atmospheric model with = 500 particles cm. The associated emission spectrum is shown in panel (a) of Fig. 4.2.5.

A set of functional derivatives (also referred to as contribution functions or kernels) for the Martian atmosphere, using selected wavenumbers in the 15 ptm CO2 absorption band, is shown in Fig. 8.2.1. 

Fig. 8.2.1 Contribution functions (functional derivatives of radiance with respect to temperature) for the 15 pirn CO2 band in the atmosphere of Mars. These functions were calculated using Eq. (8.2.11) and are labeled by wavenumber (cm ). Units for the contribution functions are W cm sr cm K (After Conrath et al, 2000.)...

An alternate approach is to apply a relaxation technique similar to the method of temperature inversion discussed in Subsection 8.2.c. In this case the number of parameters describing the gas profile is chosen equal to the number of wavenumbers for which we have measurements. The radiance at each wavenumber v, is associated with a gas mole fraction qi at the atmospheric level to which the radiance is most sensitive, i.e., near the peak of the contribution function. A first guess, q (i = 1, m), is introduced and used in the radiative transfer equation to calculate a set of radiances, /°(v,). In order to carry out the radiance calculation it is necessary to adopt some form of interpolation between the levels for which q is initially specified. An improved solution g/ is then obtained using the relaxation relation... 

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