Effect of finite resolution

Although this conclusion was arrived at intuitively, the answer proves to be approximately correct. The next few paragraphs will show a more rigorous mathematical verification of this conclusion. 

By restricting the maximum retardation of the interferogram to A centimeters, we are effectively multiplying the complete interferogram (between 8 = oo and 8 = +oo) by a truncation function, D(5), which is unity between 8 = —A and +A, and zero at aU other points, that is. 

In view of the shape of this function, D(5) is often called a boxcar truncation function. By analogy to Eq. 2.13, the spectmm in this case is given by the equation 

It can be shown that the Fourier transform (FT) of the product of two functions is the convolution of the FT of each function [4]. The effect of multiplying S(8) by the boxcar function D(8) is to yield a spectmm on Fourier transformation that is the convolution of the FT of S(8) measured with an infinitely long retardation and the FT of D(5). The FTof 5(8) is the tme spectmm, B(v), while theFTofD(5),/(v), is given by 

To compute the convolution of these two functions, Eq. 2.19 requires that/(v) be reversed left to right [which is trivial in this case, since/(v) is an even function], after which the two functions are multiplied point by point along the wavenumber axis. The resulting points are then integrated, and the process is repeated for all possible displacements, v, of/( relative to B v). One particular example of convolution may be familiar to spectroscopists who use grating instruments (see Chapter 8). When a low-resolution spectrum is measured on a monochromator, the true spectrum is convolved with the triangular slit function of the monochromator. The situation with Fourier transform spectrometry is equivalent, except that the true spectrum is convolved with the sine function/(v). Since the Fourier transform spectrometer does not have any slits,/(v) has been variously called the instrument line shape (ILS) Junction, the instrument function, or the apparatus function, of which we prefer the term ILS function. 

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When the Michelson interferometer with finite aperture is not properly adjusted nonlinear phase errors arise These phase errors are no longer linearly dependent on the wave number v, and they cause an asymmetric distortion of the interferogram (Figs, 40b and 41). It should be noted that all illustrations in connection with errors (Figs. 39, 40 and 41) have been produced by computer simulation (cf. Appendix 1). In order to make the essential features as clear as possible the effects of finite resolution etc. are left out where they have not necessarily to be included. In these cases, the resolution width /d is given in the figure (Figs. 39a—c). In Fig. 41, the error correction is demonstrated with finite... 

Sometimes, the autoionized lines for very large values of n near the ionization threshold present a regular pattern characterized by the appearance of fringes . This phenomenon has been attributed to a stroboscopic effect between the periodic orbital motion of the Rydberg electron and that of the ion rotation (in Na2, Labastie, et al, 1984, in Li2, Schwarz, et al., 1988) and an interference effect resulting from the relative phases of the 1-mixing matrix elements and transition moments (in NO, Fredin, et al., 1987), or an effect of finite resolution (in HC1, Drescher, et al., 1993). 

While in the ideal case S(g) according to Eq. (57) clearly reflects the singular behavior at T due to substrate inhomogeneity and/or limited resolution the actual behavior of scattering data is quite smooth see Fig. 13b for an example. A detailed analysis of finite resolution effects on the structure factor of two-dimensional lattice gas models has been presented by Bartelt et af. If the... 

The basic probe system has been refined to avoid the effects of finite input resistance as much as possible and to increase accuracy, sensitivity and spatial resolution in field and charge investigations. Three versions are briefly described below. 

The theory behind the selection of mask patterns for coded mask telescopes in the idealised case is relatively well understood, as are the techniques available to analyse the data. The effect of finite detector resolution is, however, only one of many complicating effects which are encountered in practice. Design choices for new generations of coded mask telescope will involve the exploration of these as well as the considerations discussed above. 

In order to achieve the optimum resolution accompanied with a reasonable analyzed depth, the type of incident ions and their energy, the geometry of the experiment, and the detection system must each be optimized. Composto et al. demonstrated the effect of finite instrumental resolution by the comparison of calculated volume fraction profiles with the ideal profiles (Figure 23.30) [116]. 

In Section 6.1 we show the effects of finite spectral resolution and other instrument characteristics on the recording of the emerging radiation field. Infrared data from the terrestrial planets, that is Venus, Earth, and Mars, are treated in a comparative way in Section 6.2. Emphasis is given to an understanding of the physical principles that cause the structure in the measured spectra. The spectra of the giant planets are discussed in Section 6.3, again in a comparative manner. Section 6.4 is devoted to Titan as a satellite with a deep atmosphere it is in a class by itself The last section in this chapter (6.5) is concerned with astronomical bodies without substantial atmospheres. Mercury, the Moon, and lo are most interesting examples of this class of objects. The numerical treatment of information retrieval is postponed until Chapter 8. 

Finite resolution and partial volume effects. Although this can occur in other areas of imaging such as MRS, it is particularly an issue for SPECT and PET because of the finite resolution of the imaging instruments. Resolution is typically imaged as the response of the detector crystal and associated electron to the point or line source. These peak in the center and fall off from a point source, for example, in shapes that simulate Gaussian curves. These are measures of the ability to resolve two points, e.g. two structures in a brain. Because brain structures, in particular, are often smaller than the FWHM for PET or SPECT, the radioactivity measured in these areas is underestimated both by its small size (known as the partial volume effect), but also spillover from adjacent radioactivity... 

Another issue for quantification is the effect of the finite resolution of PET/SPECT relative to small structures being imaged. This can result in the so called Partial Volume Effect which is characterized by an underestimation of the true radioactivity. This is an area of increasing research, but is not a major limitation when there is not a large difference between patients and controls in the volume of... 

Figure 27.2 shows measurements and predictions of CO2 and CO mole fractions. The sharp peaks in the mole fraction profiles of these species cause gradient-broadening errors near the flame sheet. The measurements also show the effects of broadening of the profiles caused by finite spatial resolution of the probe. However, overall the comparison between measurements and predictions is as good as any reported in the literature or better. 

Also adhesion between the tip and sample can cause deformation of the sample. Several theories have been developed to include the effect of adhesive forces. In the JKR theory adhesion forces outside the contact area are neglected and elastic stresses at the contact line are infinite [80]. Even under zero load, the adhesion force results in a finite contact radius a=(9jtR2y/2 E)1/3 as obtained from Eq. 7 for F=0. For example, for a tip radius R=10 nm, E=lGPa, typical surface energy for polymers y=25 mN/m, and typical SFM load F=1 nN, the contact radius will be about a=9.5 nm and 8=9 nm, while under zero load the contact radius and the deformation become a=4.5 nm and 8=2 nm, respectively. The experiment shows that under zero load the contact radius for a 10 nm tungsten tip and an organic film in air is 2.4 nm [240]. The contact radius caused only by adhesion is almost five times larger than the Hertzian diameter calculated above. It means, that even at very small forces the surface deformation as well as the lateral resolution is determined by adhesion between the tip and sample. 

Detector saturation can effect both quantitative and qualitative data analysis, and each of these effects should be appreciated. The effect on sample quanti-tation is intuitive, where for instance a twofold increase in sample concentration produces a less than twofold increase in response. This will cause a flattening of calibration curves at higher concentrations. For API techniques, source saturation (or ion suppression) is another source of response saturation independent of detector saturation. Detector saturation can also effect qualitative measurements such as mass accuracy and isotope ratio calculations. In the former, when a mass spectral peak that has some finite resolution stalls to saturate the detector the peak-top calculations that provide the m/ measurement of the peak will become ambiguous. Likewise, it is possible that as one isotope of an ion starts to saturate the detector, adjacent isotopes in the distribution will still provide a linear response. The result of this is that incorrect isotope ratios will be obtained. Changes in relative isotope ratios of individual spectra across a chromatographic peak is an indicator of possible detector saturation. 

In this paragraph the wall function concept is outlined. The wall functions are empirical parameterizations of the mean flow variable profiles within the inner part of the wall boundary layers, bridging the fully developed turbulent log-law flow quantities with the wall through the viscous and buffer sublayers where the two-equation turbulence model is strictly not valid. These empirical parameterizations thus allow the numerical flow simulation to be carried out with a finite resolution within the wall boundary layers, and one avoids accounting for viscous effects in the model equations. Therefore, in the numerical implementation of the k-e model one anticipates that the boundary layer flow is not fully resolved by the model resolution. The first grid point or node used at a wall boundary is thus placed within the fully turbulent log-law sub-layer, rather than on the wall itself [95]. In effect, the wall functions amount to a synthetic boundary condition for the k-e model. In addition, the limited boundary layer resolution required also provides savings on computer time and storage. 

For sufficiently small widths of entrance and exit shts, the instrument fine-shape function would be that of the diffraction grating (see Fig. 8). In the case of a continuous spectrum, the effect of the line-shape function and of the finite resolution is that each spectral element of infinitesimally small width produces such a line-shape function, and the recorded spectrum 7obs (v) is the superposition of all these. This means in practice that the ideal spectrum I (v) is scanned with this function or "spectral window and that 7obs(v) contains contributions from the range v Av (see Fig. 8). Therefore, it is often called scanning function or window... 

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