Back-azimuth

Most of these results are based on 1-5 SKS measurements at each station, the exception being the measurements made at the CNSN stations. At ULM, for example, the result is based on 45 measurements. Figure 9 shows that these measurements have a clear dependence on the back-azimuth to the source. Two anisotropic layers with different orientations in symmetry axes lead to a 90° periodicity in the delay time, 5t,... 

Fig. 9. Best fit two-layer model for ULM. Left shows the fast shear-wave polarization, < ), as a function of the back-azimuth (Baz) to individual events. Right shows the delay time, 5/, between the fast and slow shear waves as a function of back-azimuth. The continuous line shows the predicted variation for a two-layer model where ( ) = 85.0° and fit = 0.4 s in the upper layer and < ) = 50.0° and 5t= 1.3 sin the lower layer and for a frequency...

The array methods presented here assume that a plane wave arrives at the array. The wave must have traveled a certain distance depending on the wavelength for this assumption to be valid. The direction of a propagating elastic wave can be described by the vertical incidence angle i (Fig. 6.9, left) and the back azimuth 0 (Fig. 6.9, right) which is measured relative to the reference sensor (S4 in Fig. 6.9, right) of the array. 

Fig. 6.9. Left cross section of the incident wave front (dashed line) crossing an array of 7 sensors (SI to S7) at an incidence angle i. Right horizontal plane of an incident plane wave front (dashed line) arriving with a back-azimuth relative to the reference sensor S4 at an array of 7 sensors (SI to S7).

The slowness s is the inverse apparent velocity of the wavefront crossing the array and defined as Vapp = dxidt. The apparent velocity is a constant for a specific ray travelling through a material. If the slowness vector s = (s, Sy, is used rather than the absolute value of the slowness s, the components of the slowness can be expressed as functions of the back-azimuth and the incidence angle (Rost and Thomas 2002) ... 

Sensor arrays, as presented here, are used for the separation of coherent signals and noise. The basic method used to separate the coherent and incoherent parts of a signal is known as array beam forming . Array beam forming enables the determination of the back-azimuth of the incident wave. One sensor is chosen as a reference sensor and all parameters are taken relative to this sensor (Fig. 6.9). 

When acoustic emissions are located with a sensor array, the back-azimuths of the propagating elastic waves are determined. The true beam can only be calculated for the correct back-azimuth. However, the back-azimuth is the parameter we want to determine. Eq. 6.12 shows that any delay time for each sensor can be calculated by multiplying the coordinates of each sensor with a slowness vector. Since the slowness is a function of back-azimuth and incidence angle (Eq. 6.11) it is possible to calculate the true beam by beam forming calculations on a grid of different slowness values. More specifically, a grid of Sx and Sy values is defined and for every... 

Fig. 6.12. Calculation of the back-azimuth of the same incoming wave using the beampacking method (left) and f-k analysis (right). The cross indicates the calculated maximum energy at particular slowness values Sx and Sy. The back-azimuth is the angle between the centre of the coordinate plane and die point of maximum energy.

Fig. 7 NICISS at CujAuC 100)-c(2x2) contour plot of He 180° back scattered at Au. He intensity (white - high black - low) is plotted as a function of the angle of incidence v / (0° -90°) and azimuth eingle cp (0° - 90°) in a linear scale. Ej = 2 keV. The low index directions [011], [001], [101] are indicated, (from ref. [88]).

For consistency we go back to the problem of the twisted cell discussed in Section 8.3.2, however, the director angles cp at the boundaries will be not constant but can be changed due to elastic and external torques. Let a nematic layer be confined by two plane surfaces with coordinates zj = —plane through angle cp (there is no tilt, the angle 9 = ti/2 everywhere, and the azimuthal anchoring energy is finite). 

However, many interesting effects in ferroelectric cells may be described without account of the helicity, in the approximation of a uniform SmC structure (e.g., unwound by limiting surfaces or formed by mixtures with compensated helicity). So, in this paragraph, we ignore all the space dependent terms i.e. consider a SmC structure with azimuthal angle (p 0. Going back to Fig. 13.5 this approximation may correspond to a ferroelectric mixture with qo 0. Then the free energy is ... 

The parallel beam configuration is often used in sample stages with four circle capabilities (see Fig. 1.5). In this case, in addition to (o and 20 rotations, the sample can be rotated around the surface normal direction (defined by the z axis in Fig. 1.4) by an azimuthal angle ( ), in addition to a tilt angle / around the x axis parallel to the sample surface ( / tilts the sample back and forth relative to the difltacti(Mi plane defined by the axis x and z). The usefulness of this set up wDl become clear later in this chapter. 

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