Localized quasi-linear approximation

In a similar way we can find a localized quasi-linear approximation for the magnetic field ... 

Zhdanov, M. S., and E. Tartaras, 2002, Inversion of multi-transmitter 3-D electromagnetic data based on the localized quasi-linear approximation Geophys. J. Int., 148, No 3. 

The quasi-linear inversion, introduced above, cannot be used for interpretation of multi-transmitter data, because both the reflectivity tensor A and the material property tensor in depend on the illuminating background electromagnetic field. However, in many geophysical applications, for example, in airborne EM and in well-logging, the data are collected with moving transmitters. In this case one can build an effective inversion scheme based on the localized quasi-linear approximation, introduced in Chapter 9, which is source independent. 

Following the same principle that was used in deriving the localized quasi-linear approximation for an electromagnetic field, we can assume that the dominant contribution to the integral G [As Ap ] in equation (14.50) is from the neighborhood of the point r = r. We can expand p (r,o ) into a Taylor series about r = Fj, similar to (14.44) ... 

The solution of equation (9.115) gives us a localized electrical reflectivity tensor Al (r), which is obviously source independent. Expression (9.111) with (r) determined according to (9.115), is called a localized quasi-linear (LQL) approximation (Zhdanov and Tartaras, 2001) ... 

By full analogy with the electromagnetic case, one can consider different ways of introducing the reflectivity coefficient A. In particular, two of these solutions play an important role in inversion theory. One is the so-called quasi-analytical (QA) solution, and the other is the localized quasi-linear (LQL) approximation. In this section I will introduce the QA approximation for the acoustic wavefield. 

Localized quasi-linear inversion increases the accuracy and efficiency of wave-field data interpretation because it is based on a much more accurate forward modeling solution than the Born approximation, used in the original Bleistein method. An example of successful application of the localized QL approximation in radar-diffraction tomography can be found in (Zhou and Liu, 2000). 

A complete study of the molecular orbitals for an octahedral complex sue as [Cr(CN)6] or [Co(NH3)6] " " would require linear combinations of all the valence atomic orbitals of the metal and of the ligands. An approximation isl to take the metal valence a.o.s (nine a.o.s for a metal of the first transition series (five 3d orbitals, one 4s and three 4p orbitals)) together with six a.o.s from the ligands, one for each atom directly bonded to the metal atom. Ini general, these six a.o.s are quasi-localized molecular orbitals (see Chapter 8), which point from the ligand to the metal and have essentially non-bonding character ... 

CPR = conjugate peak refinement GDIIS = geometry direct inversion in the iterative subspace GE = gradient extremal LST = linear synchronous transit LTP = line then plane LUP = locally updated planes NR = Newton-Raph-son P-RFO = partitioned rational function optimization QA = quadratic approximation QST = quadratic synchronous transit SPW = self-penalty walk STQN = synchronous transit-guided quasi-Newton TRIM = trust radius image minimization TS = transition structure. 

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