2D volumes

In the above equations, the same letter symbol s is used to denote the scaling operation (in Equation (6.36)), and the numerical value of the scaling factor (in Equation (6.38)). In general, the n-dimensional volumes are scaled by the n-th power s" of the scaling factor s. In the example of Figure 6.5, the dimension is n=2, and the 2D "volumes" (areas) are scaled by the squares of the scaling factors. 

An extension of the hybrid matrix method, known as the 3D hybrid-hybrid matrix method for 3D NOESY-NOESY data analysis, was proposed to generate precise and accurate distances from the 3D data while retaining computational efficiency (11). The method is based on an expression for the 3D NOE peak where the 3D volume is proportional to the product of two 2D volumes (where the effects of spin diffusion have been taken into account). 

In simulating the divisors, there is no need to simulate the entire 3D volume matrix. Only volumes needed for the deconvolution and a set of scaling volumes are simulated. The scaling set is actually the same set as those measured experimentally, with respect to the spin indices. This set is used to scale the divisors and then discarded. Once the divisors have been scaled, they are merged with the 3D experimental data and deconvolution is performed. If there is any available 2D NOESY experimental data, it is also scaled and included with the deconvoluted 2D-like data to create a hybrid-hybrid 2D volume matrix. From this point, the method... 

Figure 2. The hybrid-hybrid relaxation matrix refinement procedure for 3D NOESY-NOESY data. A starting model is used to simulate NOE data for the first iteration. Experimental 3D data is scaled and merged with simulated 3D data to produce a linear table of 3D experimental volumes and the simulated volumes needed for deconvolution. Deconvoluted 2D data is merged with any available 2D experimental data and then with a simulated, complete 2D volumes matrix. The structures resulting from the standard 2D MORASS refinement are used in subsequent iterations until convergence is reached. (Reproduced with permission from reference 24. Copyright 1996.)...

Values in parentheses are calculated from deconvoluted 2D volumes. SOURCE Adapted from ref. 24... 

The plot of deconvoluted 2D volumes derived from experimental 3D volumes versus the experimentally determined 2D volumes, shown in Figure 4, yielded a slope of 0.82. Random dispersion of the data points indicates that there is no systematic error from the deconvolution process. Figure 5 shows the number of NOE volumes measured per residue from the 2D and 3D data sets. Except for residues in the junction region, where NOE interactions are weak, the 3D NOES Y-NOESY spectra gave higher numbers of measurable NOE peaks than the 2D NOESY. We also observed a tertiary contact between the methyl group of S3-T6 and the H4 of S3-G11. This crucial NOE peak was well resolved in both 2D and 3D spectra (as 2D iij and ijj type peak), and was very useful to determine the conformation of the S3-T6 base. 

The plot of theoretically calculated 2D volumes for the final structure versus the experimental (deconvoluted 2D plus the experimentally determined 2D) volumes, gave a slope of 0.99 (Figure 6). 

Figure 4. Plot of experimentally determined 2D volumes versus the deconvoluted 2D volumes. Linear regression yields a slope of 0.82...

Figure 6. Plot of experimental 2D volumes vs. theoretical 2D volumes calculated from the final structure. The experimental 2D volumes includes deconvoluted 2D volumes (derived from experimental 3D volumes) and experimentally determined 2D volumes. The linear regression yields a slope of 0.99...

After often a lengthy period (several months) of acquisition and processing, the data may be loaded onto a seismic workstation for interpretation. These workstations are UNIX based, dual screen systems (sections on one side, maps on the other, typically) where all the trace data is stored on fast access disk, and where the picked horizons and faults can be digitised from the screen Into a database. Of vital Importance is access to all existing well data in the area for establishing the well - seismic tie. 2D data will be interpreted line by intersecting line, and 3D as a volume. 

The data volume, which can be imported as data block or single slices, can be cutted or rebinned (e.g. if the data set is very large) and interpolated (e.g. interpolating intermediate slices between measured CT cuts in the case of 2D-CT). 

Finally, the standard draft provides a detailed model of the acquisition data, which intends to describe all the possible shapes which can be taken by NDE data OD (scalar or complex), ID (sampled - cf ultrasonics A-scans - or unsampled - ef ultrasonics time/amplitude data), 2D (images) or 3D (volumes). 

External Mass Transfer and Intraparticle Diffusion Control With a linear isotherm, the solution for combined external mass transfer and pore diffusion control with an infinite fluid volume is (Crank, Mathematics of Diffusion, 2d ed., Clarendon Press, 1975) ... 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

What is the volume occupied by 5.0 g of solid silver, given the density listed in Appendix 2D ... 

The German physicist Lothar Meyer observed a periodicity in the physical properties of the elements at about the same time as Mendeleev was working on their chemical properties. Some of Meyer s observations can be reproduced by examining the molar volume for the solid element as a function of atomic number. Calculate the molar volumes for the elements in Periods 2 and 3 from the densities of the elements found in Appendix 2D and the following solid densities (g-cuU ) nitrogen, 0.88 fluorine, 1.11 neon, 1.21. Plot your results as a function of atomic number and describe any variations that you observe. 

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