Distance constraints, time-averaged

The constraint on a hydrogen-hydrogen distance derived from an NOE signal is reformulated with the requirement that a time average of the distance, d(t), should be smaller than the upper limit u. In equation (8), part of Edist of equation (5) is rewritten for a single distance constraint with n = 2 ... 

That assumption simplifies the analysis of primitive path rearrangement. Local path jumps now correspond to random flips in the Orwoll-Stockmayer chain model, and we can apply these results directly. For our case the local jump distance is the path step length a, and the average time between jumps is 2r , where r is the mean waiting time for release of a constraint which allows a length preserving jump. The average number of such suitably situated constraints per cell is z(z < zo), and we assume for simplicity that all cells have z such constraints. 

System assumptions that should be valid for such applications include fluid flow in the porous media is isotropic and adsorption is fast, reversible, and linear (cf. Freeze and Cherry 1979). Given these constraints, the comparative transport of a conserved (nonadsorbed) tracer, such as Br , and an adsorbed or retarded species, such as Am, can be described as shown in Fig. 10.29. A comparison of migration distances of the two species after time t, is made at concentrations where C(measured)/Co(initial) = 0.5 for the conserved and adsorbed species. The migration distance X of the conserved species after time r is a measure of the average groundwater velocity (U), or X = vt. Similarly, the migration distance of the adsorbed species (X,) i related to its velocity of movement (v ) by Xf = vj. The retardation factor (/tj for the adsorbed species is then given by... 

As mentioned earlier, in the event that interresidue NOEs arise from averaging of conformations, the use of ISPA distances as constraints may lead to virtual conformations. Moreover, the ISPA assumption that no spin diffusion is occurring is generally true only when the NOE mixing times are extremely short. In... 

The fluctuation forces also cause the curvilinear diffusion along the primitive path. However, because of the constraint effect of the tube, the positive and negative fluctuation forces perpendicular to the primitive path cancel each other out when averaged over a time period longer than required for a segment to travel over an entanglement distance, and thus make no net contribution to the curvihnear diffusion and the translational diffusion of the center of mass. Thus, in the study of the cmvilinear diffusion, Eq. (9.A.3 ) should be used, instead of Eq. (9.A.3). 

In other words, a velocity can be defined only with respect to a given point in space-time and relative to a particular observer. Since one cannot measure velocity at an infinitesimal point in time or space, any measurement of a rate or a velocity requires averaging the change in the quantity of interest at some point in space over a period of time as well as averaging it over a volume at some point in time. The period of time over which the quantity is averaged determines the time scale and the volume determines the space (or distance scales). We note that the measurement precision of the resulting rate or velocity is directly proportional to the time scale and inversely proportional to the distance scales. But since these time and distance scales may be arbitrarily determined (within certain limiting constraints), the observer can influence the measurement precision and thus influence the power required to measure... 

Thus, for typical values of average P velocities, 8-10 times the S - P arrival-time difference gives a reasonable estimate of the epicentral distance from the station. Figure 1 shows that if there are several stations, the intersection of the circles drawn around the stadmis with the corresponding radius of the epicentral distance will define the most likely position of the epicenter. Note that because the velocity model is never perfect and the arrival times suffer from measurement errors, the circles do not intersect in a point The area of the intersection is an indication of epicenter uncertainty. One of the major limitations of the method is that it gives httle constraint on the source depth. Furthermore, measurement error... 

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