Quasi-linear approximation

Using Cs + CP = constant, this equation relates the reaction velocity to affinity only. A plot of JrsCrs,max versus affinity shows an inflection point, representing a maximum slope. Therefore, the quasi-linear approximation between the reaction velocity and the affinity will be valid with a deviation of less than 15%. 

Quasi-linear approximation and tensor quasi-linear equation The quasi-linear (QL) approximation is based on the assumption that the anomalous field E inside the inhomogeneous domain is linearly proportional to the background field E through some tensor A (Zhdanov and Fang, 1996a) ... 

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

We call this approximation a quasi linear approximation of the first order Eq or a modified quasi-linear approximation (MQL) E)(fQ for an anomalous field ... 

Expression (9.79) for the quasi-linear approximation of the anomalous field can be written in discrete form as... 

Zhdanov, M. S. and S. Fang, 1996a, Quasi-linear approximation in 3D EM modeling Geophysics, 61, 646-665. 

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. 

Zhdanov, M, S., Fang, S., and G. Hursan, 2000, Electromagnetic inversion using quasi-linear approximation Geophysics, 65, No. 5, 1501-1513. 

One can see that the Born approximation reduces the forward modeling solution to simple quadrature calculation. However, this approximation holds only for a weaJi scatterer (when As (r) is relatively small). We can improve the accuracy of integral approximation significantly if, following the ideas described in Chapter 9 for an electromagnetic field, we introduce a quasi-linear approximation. 

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) ... 

Note that expression (14.64) resembles formula (14.53) for quasi-linear approximation. However, the reflection coefficient A now has a much clearer physical meaning. It has been demonstrated by Bleistein et al. (2001), that to leading order (high frequency asymptotics), this reflection coefficient is the same as that derived for plane waves reflected by planar interfaces in the media with a piecewise-constant distribution of the parameters. Note also that to leading order in the frequency w, the... 

Zhou, C., and L. Liu, 2000, Radar-diffraction tomography using modified quasi-linear approximation IEEE Transactions on Geoscience and Remote Sensing, 38, No. 1, 404-415. 

The process of the magnetic field influence on a developed turbulence was examined by [8],and demonstrated the possibility of using the quasi-stationary approximation for the solution of the second type problem and suggested to use quasi-linear approximations to solve the problem at Rcm = 20. One of the second type problem results were reported in [9], the modeling of a diminishing MHD turbulence by LES and DNS methods and demonstrated that the magnetic field at the initial time started to decay under the influence of the total kinetic energy. This effect is consistent with Joule dissipation. A similar picture of the decay was not reported by the authors because their main objective was the evaluation... 

We now focus on the development of a suite of quasi-linearized approximations of the SS-MRCC theory, viz. SS-MRPT and SS-MRCEPA. We would deal with the pertmrbative formulations first, and development of the SS-MRCEPA methods next. 

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