The total field

The terms on the right side of Eq. (2.1.29) are negative or positive depending on whether they are, respectively, losses or gains. 

Upon multiplying both sides of Eq. (2.1.30) by dco / n, integrating over all solid angles, and remembering that /, 0 ix, cj) ) jAn = o(x) we infer 

Further simplifications of the terms in Eq. (2.1.32) are possible. We define an effective extinction cross section xe and an effective single scattering albedo 0 of dV by 

If the atmosphere is plane-parallel (infinitely extended in the x- and y-directions 

If the linear dimensions of dcr are made arbitrarily much larger than dz, but still small enough so that dcr remains an element of surface area, then the sides of the cylinder can be neglected relative to dcr in determining the effective cross section of dy as seen along a slant-path in the direction (/t, (f ). The geometric cross section dcr of dy is 

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It is expected that seismic iwill become even more important in determining field development strategies throughout the total field life. Indeed, many mature fields have several vintages of seismic, both 2D and 3D. 

Furthennore, the non-oscillating component of the integrand can best be sorted out by going to the complex representation of the total field, the polarization, and the susceptibility. The mathematically pure real quantities in equation (Bl.3.2) can be written in their complex representation as follows ... 

Consider all of the spectroscopies at third order s = 3). To be as general as possible, suppose the total incident field consists of the combination of three experimentally distinct fields (/ = 1, 2, 3). These can differ in any combination of their frequency, polarization and direction of incidence (wavevector). Thus the total field is written as... 

Residual Held in the space - 10-5% of the total field produced by I,. 

Certainly, Equations (1.127 and 1.128) can be derived directly from Equation (1.6), but it requires a rather cumbersome integration. This example also allows us to illustrate the fact that the gravitational field has a finite value inside any mass. With this purpose in mind, imagine that the observation point p is located at the center of a small and homogeneous sphere. Fig. 1.12a. Then, the total field can be represented as a sum ... 

Therefore, after summation the total field due to all masses in the layer is... 

It is interesting to notice that these formulas are used to calculate the Bouguer correction. Now we will study the attraction field inside a layer when the coordinate of the observation point z satisfies the condition, (Fig.. Ah) z0. Then, the total field can be presented as a sum of two fields one of them is caused by masses with thickness equal h/2 — z,and the second one is due to masses in the rest of the layer, having the thickness z + A/2, (Fig. 1.14b). In accordance with Equation (1.146) these fields are... 

The behavior of the field g caused by masses of the layer is shown in Fig. 1.14c. Thus, for negative values of z the field component g inside the layer is positive, since the masses in the upper part of the layer create a field along the z-axis, and this attraction prevails over the effect due to masses located below the observation point. At the middle of the layer, where z = 0, the field is equal to zero. Of course, every elementary mass of the layer generates a field at the plane z = 0, but due to symmetry the total field is equal to zero. For positive values of z the field has opposite direction, and its magnitude increases linearly with an increase of z. As follows from Equations (1.146-1.148) the field changes as a continuous function at the layer boundaries. 

This field is the dominant part of the total field, since the irregular part of the masses generates not more than several hundreds of milliGalls, which constitutes less than 0.1% of the resultant field. 

Here the normal potential and its first derivatives are taken at the point Ai beneath the earth s surface. Since within the interval H—h we assume that the potential varies linearly, these derivatives are equal to those on the earth s surface. At the same time, the potential of the total field at the point A we represent as... 

This means that we have made two approximations. One of them is replacement of the total field by a normal field, and the second is a shift of the interval of integration. Since the secondary field is very small, as are the intervals BBi and AAi, we may expect that the errors caused by these approximations are relatively small. Besides, the normal field varies relatively slowly within the interval BiA. Correspondingly, U(4>,h) and U(4>,0) are values of the normal potential at points Ai and Bi. From Equations (2.290 and 2.291) we have... 

First of all, we choose the parameters of the ellipsoid in such a way that the normal potential on its surface, Uq, is equal to the potential of the total field at points of the geoid, Wq. Then, Equation (2.294) is greatly simplified and we obtain... 

In the previous chapters our attention was paid mainly to the study of the figure of the earth and with this purpose in mind we represented the total field as a sum of the normal and secondary fields. In this chapter, we will discuss a completely different application of the gravity method, related to exploration geophysics, in which the gravitational field is measured in order to study lateral changes of the density near the earth s surface. By analogy, we also represent the gravitational field as a superposition of two fields ... 

Derivation of the formula for the attraction field caused by an infinitely thin line with the density X is very simple, and is illustrated in Fig. 4.5b. We will consider the field at the plane y = 0. Due to the symmetry of the mass distribution, we can always find a pair of elementary masses Xdy and —Xdy, which when summed do not create the field component gy directed along the y-axis, and respectively the total field generated by all elements of the line has only the component located in the plane y — 0. Here r is the coordinate of the cylindrical system with its origin at the point 0, and the line with masses is directed along its axis. As is seen from Fig. 4.5b the component dg at the point located at the distance r from the origin 0 is... 

The flat interface model employed by Marcus does not seem to be in agreement with the rough picture obtained from molecular dynamics simulations [19,21,64-66]. Benjamin examined the main assumptions of work terms [Eq. (19)] and the reorganization energy [Eq. (18)] by MD simulations of the water-DCE junction [8,19]. It was found that the electric field induced by both liquids underestimates the effect of water molecules and overestimates the effect of DCE molecules in the case of the continuum approach. However, the total field as a function of the charge of the reactants is consistent in both analyses. In conclusion, the continuum model remains as a good approximation despite the crude description of the liquid-liquid boundary. 

It is noteworthy that the induced energy is the dot product of the induced dipole and the static field and not the total field. The interpretation of Eq. (9-11) is that a static field is required to originate induced dipoles. 

In a magnetic resonance experiment, we apply not only a static field B0 in the z-direction but an oscillating radiation field Bx in the xy-plane, so that the total field is ... 

Let ijs now apply this concept of the RRF to the case where an rf field Hi is present. We choose a Cartesian coordinate system with tlje z axis along the dc field Hq and the y axis along the rf field Hi. The total field is given in the laboratory reference frame by... 

The nonlinear part of the susceptibility was introduced into the quasi-linear finite-difference scheme via iterations, so that at any longitudinal point, the magnitude of E calculated at the previous longitudinal point was used as a zero approximation. This approach is better than the split-step method since it allows one to jointly simulate both the mode field diffraction on irregular sections of the waveguide and the self-action effect by introducing the nonlinear permittivity into the implicit finite-difference scheme which describes the propagation of the total field. 

In terms of nonlinear dynamical systems, the second waveguide of the junction can be considered as a system that is initially more or less far from its stable point. The global dynamics of the system is directly related to the spatial transfomation of the total field behind the plane of junction. In structure A, the initial linear mode transforms into a nonlinear mode of the waveguide with the same width and refractive index. In the structure B, the initial filed distribution corresponds to a nonlinear mode of the first waveguide it differs from nonlinear mode of the second waveguide, however. The dynamics in both cases is complicated and involves nonlinear modes as well as radiation. Global dynamics of a non-integrable system usually requires numerical simulations. For the junctions, the Cauchy problem also cannot be solved analytically. 

Here E x,z) is slowly varying amplitude of the total electrical field E x, z) = E x, z) exp if3z — iujt), /3 is a parameter responsible for fast oseillations of the total field in longitudinal direction. 

As it is well known, stationary solutions to Eq.(3.2) occur at the extrema of the Hamiltonian for a given power. The solutions that correspond to global or local minimum of H for a family of solitons are stable. The representation of the output nonlinear waveguide as a nonlinear dynamical system by the Hamiltonian allows to predict, to some extent, the dynamics of the total field behind the waveguide junction. 

Such a behavior of the total field is observed provided that the beam power is smaller than a definite value P which depends on the waveguide width (for a = 1.8pm, P 8). The spatial dynamics of a light beam with P > Pi is more complicated because nonlinear self-effects in radiation field increase so that the formation of soliton-like light beams propagating in the waveguide cladding is observed. 

Fig. 4. Some highlights and details of the Fraser Lakes disooveries shown superimposed on the total field aeromagnetic image and a surface trace of the oomplexly folded EM conductor.

Fig. 5. Location of Zone B 2008 and 2009 drill holes (numbered dots) superimposed on the total field aeromagnetio image, airborne radiometric anomalies (oontoured), and EM conductor picks (dark dots). Note the highly disrupted nature of the EM oonduotor pioks in the vicinity of Zone B.

The behavior of tacl mice was also analyzed in several animal models of anxiety. The open-field test is a widely used tool for behavioral research, but less specific for the evaluation of the anxiety state of the animal, because it is a summation of the spontaneous motor and the exploratory activities, and only the latter is influenced by the anxiety level (Choleris et al. 2001). Under aversive environmental conditions (high level of illumination) the animals activity is strongly affected by the emotional state, while less aversive situations (familiar, dimly lit environment) are useful to assess the general motor activity of mice. Because rodents avoid open areas, the activity of mice in the central part of the open-field arena is inversely correlated to the anxiety level. Tad mice spent only 6.5% of their total activity in the central part, which represented 11% of the total field, indicating that they avoided this aversive area, hi contrast, tacl mice spent 13.6% of their activity in the central area (Bilkei-Gorzo et al. 2002). The increased central activity of the tad mice indicates that the test situation was anxiogenic for tad animals, but less so for the knockout mice. 

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