Inverse SORS

Crusem, R. and Caristan, Y., Moment ten.sor inversion, estimation, and seismic coupling variability at the French Centre d Experimentations du Pacifique. Bull Seism. Soc. Am., 82 (3) (1992) 1253-1274. 

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

The fixed nature of the collection fibres in a conventional SORS experiment limits the range of spatial offsets available. An alternative approach (inverse SORS) swaps the laser and collection fibre geometries Raman light is collected through a group of fibres (arranged in a disc shape) contained within the centre of a probed area defined by a ring-shaped laser beam (see Fig. 3.5). 

Fig. 3.5. Schematic diagram of conventional SORS and inverse SORS concepts showing Raman collection and laser beam delivery geometries (reprinted with permission from [34], Copyright (2006) The Society for Applied Spectroscopy)...

Using a ring-disk probe (also called inverse SORS) in backscattered geometry we have been able to reach bone about 6 mm below the skin [60]. Mineral/matrix ratios are accurately (<4% relative error) reproduced if a protocol for optimizing the ring/disk spacing is followed [62]. 

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

Horn (1974) / Blake (1985) Described in Section 7.2. Zeros were removed from the input by transforming each channel with data in the range [0, 1] according to y = (255,v + l)/256. The extensions of Blake (1985) are used, i.e. the threshold is applied after computing the first derivative. The threshold is set to 3. The inverse of the Laplacian is computed using an SOR iterative solver. (threshold=3)... 

Further research in the area of SORS led to the development of a more effective SORS modality, termed inverse SORS. This improved concept removes some of the artifactual limitahons inherent to the conventional SORS approach, and permits the use of higher laser powers through the provision of a wider illumination zone at the sample. This is beneficial in applications where maximum illumination intensity limits exist, such as when probing living tissue in vivo and in explosive powder environments. 

Scheme 2. Formation of colloidal crystals and their use as templates. A colloidal dispersion containing monodisperse particles undergoes controlled filtration, centrifugation, dip coating, sedimentation, or physical confinement, which results in ordered packing of the particles with void spaces between them. By infiltrating these spaces with precm-sor solution or preformed nemoparticles the hybrid material is formed. Removal of the polymer template (using solvent (toluene) orheatingtechniques) gives an inverse replica with air-fiUed, interconnected voids of monodisperse size, which is dependent on the initial particle size...

If /f > I. the opposite occurs. Micellar sy ems arc of the inverse type, and if there is enough surfactant to [ t)duce a microemulsion it will be the type encountered in Win,sor type II diagram abo e the bipha.sic region. 

The first use of SORS in the area of noninvasive probing of bones was reported by Schulmerich et al. [25]. The team succeeded in the recovery of the Raman spectra of bones transcutaneously from depths of several millimeters in animal and human cadavers. Subsequently, with improved instrumentation based around a ring illumination geometry (equivalent to inverse SORS), Schulmerich et al. [29] reached another key milestone demonstrating the feasibility of measuring Raman spectra noninvasively from a depth of 4 mm from chicken tibia with better than 8% accuracy in terms of the relative Raman band intensities between the... 

If X is a product species, Vx is negative corresponding to X increasing as the reaction progresses, whereas Vx is positive if X is a reactant. In general, the rate R is some function of the concentrations of the species in the system, and soR = f ([A], [B], [C],...). A rate equation is a specific equality between the time derivative in Equation 6.16 and the function of concentrations, R([A], [B], [C],...) it is a differential equation in time. The concentrations, for example, [A] and [B], are functions of time because as a reaction progresses, concentrations change. The units for a reaction rate are the units of concentration and inverse time. 

---