Quasi-analytical approximation

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. 

Note that the QL integral equation (14.39) can be cast in the form 

Following ideas of the extended Born approximation (Habashy et al., 1993), we recall that the Green s function G (r r o ) exhibits either singularity or a peak at the point where = r. Therefore, one can expect that the dominant contribution to the integral G [As Ap ] in equation (14.43) is from some neighborhood of the point Tj — r. In fact, we can expand A (r,cu) into a Taylor series about r = r  

Solving the last equation with respect to A, we find 

Substituting expression (14.46) back into (14.38), we obtain the quasi-analytical approximation, Pqa fo the scattered acoustic wavefield  

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Thus, we can see that the difference between the tensor quasi-analytical approximation and the localized nonlinear approximation is determined by a term ... 

Now we can start iterations by the modified Born series with a quasi-analytical approximation for the anomalous field ... 

We will call the first iteration determined by expression (9.171) a modified quasi-analytical approximation (MQA) ... 

The modified conductivity matrix and vector play an important role of quasi-linear inversion, which we will discuss in the next chapter. 9.4-8 Matrix form of quasi-analytical approximation in the method... 

Finally, the quasi-analytical approximations (9.90) and (9.91) of the anomalous electric and magnetic fields in discrete form can be written as ... 

We use the following notations in the last formulae. The vectors eg and hg represent the discrete quasi-analytical approximations of the anomalous electric and magnetic fields at the observation points. Vector I is a V x 1 column vector whose elements are all unity. The N xl column vector g ([Pg.280]

Zhdanov, M. S., Dmitriev, V. L, Fang, S., and G. Hursan, 2000b, Quasi-analytical approximations and series in electromagnetic modeling Geophysics, 65, 1746-1757. 

One practical way to overcome this difficulty is to abandon the integral equation approach for nonlinear inverse problems and to consider the finite difference or finite element methods of forward modeling. We will present this approach in Chapter 12. Another way is based on using approximate, but accurate enough, quasi-linear and quasi-analytical approximations for forward modeling, introduced in Chapter 8. We will discuss these techniques in the following sections of this chapter. 

Quasi analytical approximations (9.90) and (9.91) provide another tool for fast and accurate electromagnetic inversion. This approach leads to a construction of the quasi-analytical (QA) expressions for the Prechet derivative operator of a forward problem, which simplifies dramatically the forward EM modeling and inversion for inhomogeneous geoelectrical structures. ... 

Figures 10-5 - 10-6 present the results of inversion for an xy polarized field using, respectively, the iterative Born and the diagonalized quasi-analytical (DQA) inversion methods, with focusing. Both methods produce good results. Note that inversion of theoretical MT data takes just few minutes on a personal computer. A remarkable fact is that it is possible for this polarization to separate the upper and lower parts of the dike. The position and shape of the dike are also reconstructed quite well. However, the resistivity of the dike is not recovered correctly, which is a common problem in nonlinear inversion. The first method (iterative Born inversion) underestimates the anomalous conductivity, while the second (DQA inversion) produces a more correct result. The reason for the underestimation of the conductivity by the iterative Born inversion is very simple. The linearized response, which is used in this technique, usually overestimates the anomalous field, which results in an underestimation of the conductivity in inversion. The quasi-analytical approximation produces a more accurate electromagnetic response, so the inversion works more accurately as well.

The orders of reaction, U , ivith respect to A, B and AB are obtained from the rate expression by differentiation as in Eq. (11). In the rare case that we have a complete numerical solution of the kinetics, as explained in Section 2.10.3, we can find the reaction orders numerically. Here we assume that the quasi-equilibrium approximation is valid, ivhich enables us to derive an analytical expression for the rate as in Eq. (161) and to calculate the reaction orders as ... 

While a random distribution of atoms is assumed in the regular solution case, a random distribution of pairs of atoms is assumed in the quasi-chemical approximation. It is not possible to obtain analytical equations for the Gibbs energy from the partition function without making approximations. We will not go into detail, but only give and analyze the resulting analytical expressions. 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

A further analytical approximation to Eq. (369), proposed by Miller and coworkers [84-86], demonstrates how the above semiclassical reaction rate theory approaches a quasi-classical reaction rate theory. Specifically, consider the... 

Integral Equation Solutions. As a consequence of the quasi-steady approximation for gas-phase transport processes, a rigorous simultaneous solution of the governing differential equations is not necessary. This mathematical simplification permits independent analytical solution of each of the ordinary and partial differential equations for selected boundary conditions. Matching of the remaining boundary condition can be accomplished by an iterative numerical analysis of the solutions to the governing differential equations. 

Formulae (9.90) and (9.91) give quasi-analytical (QA) solutions for 3-D electromagnetic fields. Note that the only difference between the QA approximation and the Born approximation (9.74) is in the presence of the scalar function [1 — g (r)]. That is why the computational expense of generating the QA approximation and the Born approximation is practically the same. On the other hand, it was demonstrated by Zhdanov et al. (2000), that the accuracy of QA approximation is much higher than the accuracy of the Born approximation. 

The QA solutions developed in the previous section were based on the assumption that the electrical reflectivity tensor was a scalar. This assumption reduces the areas of practical applications of the QA approximations because in this case the anomalous (scattered) field is polarized in a direction parallel to the background field within the inhomogeneity. However, in general cases, the anomalous field may be polarized in a direction different from that of the background field, which could generate additional errors in the scalar QA approximation. To overcome this difficulty, we introduce in this section a tensor quasi-analytical (TQA) approximation. The TQA approximation uses a tensor A, which permits different polarizations for the background and anomalous (scattered) fields. 

Wc call expressions (9.97) and (9.98) tensor quasi-analytical (TQA) approximations for an electromagnetic field. These approximations provide a more accurate solution for a forward problem than a scalar QA airproximation (Zhdanov ct al.,... 

Formula (14.102) gives the quasi-analytical (QA) approximation for a 3-D vector wavefield, which can be treated as an analog of the corresponding QA approximation for the electromagnetic case (Zhdanov, et ah, 2000)... 

Assuming a uniform accessibility of the tip surface, e.g., a uniform concentration of electroactive species, an analytical approximation of the tip feedback current can be derived (see Chapter 5). For convenience, we repeat the main equations here. Such a model represents a thin layer cell (TLC) with a diffusion-limiting current expressed by Eq. (8). The approximate equation for a quasi-reversible steady-state voltammogram is as follows (11) ... 

The lattice gas can be treated both by analytical approximations such as the mean-field or the quasimethods complement each other the... 

At the first stage, the Cexs T) contribution was subtracted, if necessary, from the low-temperature heat capacity values. The set of Qat(T) data obtained in this way was approximated by the method of least squares using Eq. (6) that describes the contribution of fhe lattice heat capacity in the quasi-harmonic approximation. As a result, we calculated the values of the 0D/ Ei/ E2/ E3/ and oe variable parameter. We then determined the analytic dependence of changes in fhese paramefers on the molar volume V. This allowed us to estimate the 0d/ ei/ e2/ e3/ and oc parameters for unstudied compounds. 

As a result of the above, we have to deal with a 3-dimensional system, rather than with a 4-dimensional one. Nonetheless, this system is still too complex to be analytically studied. One way to simpUly the system is to suppose that the unbound-to-bound transitions are much faster than those between the open and closed states, and use the quasi-stationary approximation introduced in Chap. 5. Let us define the probabilities that the channel is open and closed (regardless of their being bound or unbound by the regulatory molecule) as follows ... 

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