Bleistein inversion

Let us assume again that the wavespeed, c, the pressure field, p, and the source field, /, are functions of the vertical coordinate 2 only. We also assume that the wavespeed in a 1-D model can be represented as follows  

The 1-D field p may describe, for example, plane acoustic wave propagation in the vertical direction in the medium. This field, according to (13.57), satisfies the equation 

We assume also that p(z, z, oj) is bounded for all z and satisfies the 1-D radiation condition 

The total field in this model, as in the general 3-D case, can be represented as the sum of the incident and the scattered fields  

The incident field p (z, z, u) satisfies the 1-D Helmholtz equation with the background velocity distribution 

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In the section on Bleistein inversion and later on -we will direct the vertical axis z downward. 

Comparing the leading order term in (15.75) with the step value in (15.74), we see that the inverse formula provides the correct estimate of a step, b, with the accuracy determined by the leading term in the representation (15.75). However, Bleistein inversion is based on a Born approximation formula which is valid only to leading-order in 6cg. Thus the theoretical inversion result for this simple model fits well the basic assumption of the Bleistein method. 

Applying the Bleistein inverse formula to this data, we obtain 4... 

Thus, to transform the inverse formula for the anomalous square slowness into the inverse formula for corresponding reflectivity function distribution we have to multiply the observed scattered field by the factor —IIlj/cq) (—Cq/4) = ta co/2. We will use this simple rule in subsequent sections on Bleistein inversion as well. 

As we have discussed already in the previous section on Bleistein inversion, the goal of inversion in many applications is the reconstruction of the reflectivity function, which provides information about the distribution of the reflecting boundaries in the medium under investigation. This problem can also be addressed by using the Kirehhoff approximation (14.66), which we reproduce here for convenience ... 

Thus, localized QL inversion is reduced to Bleistein inversion with respect to the material property function mi (x, t/, z) and then to a simple correction of this inversion result by solving the minimization problem (15.161) and applying the algebraic transformation (15.162). 

Bleistein Cohen Stockwell Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion... 

The inverse source problem is more ambiguous because there usually exists a source distribution that generates a zero external field. This type of source is called a nonradiating source. The detailed analysis of the analytical properties of the nonradiating source was given in a classical paper by Bleistein and Cohen (1976). Following this paper, we can easily demonstrate this fact as applied to inverse source problems for an acoustic field. 

Bleistein, N., and J. Cohen, 1976, Non-uniqueness in the inverse source problem in acoustics and electromagnetics J. Math. Physics, 18, 194-201. 

Bleistein, N, and S. H. Gray, 1985, An extension of the Born inversion method to a depth dependent reference profile Geophys. Prosp., 33, 999-1022. 

Bleistein, N., Cohen, J. K., and J. W. Stockwell, Jr., 2001, Mathematics of multidimensional seismic imaging, migration, and inversion Springer, New York, Berlin, London, Tokyo, 510 pp. 

The observed seismic data are usually bandlimited, because the frequency range is limited by the natural time constant of the source process, by receiver and survey design, by seismic preprocessing required for noise removal, etc. (see Aki and Richards, 2002, and Bleistein et al., 2001 for further discussions). The bandlimiting is an important characteristic of observed seismic data which should be taken into account, especially if we use the high frequency asymptotics for data inversion. 

Substituting formula (15.72) into inverse formula (15.63) of the Bleistein method, we obtain... 

Comparing formula (15.74) for the predicted anomalous square slowness As (2) with the expression (15.66) for the original As (2), we find that the inverse result has the step at the right depth, but the magnitude of the step is not exact and the entire curve As (2) is shifted with respect of the true curve As (2) by a constant (2A/cq). The shift of the inversion curve actually depends on the value of the observed scattered field at zero frequency, which can never be practically determined in a real-world experiment (see Bleistein et al., 2001, for more details). The discrepancy in the step sizes can be easily resolved if we evaluate the reflection coefficient value for a small step b, taking into account formula (15.68)... 

Localized quasi-linear inversion based on the Bleistein method We have noticed already in electromagnetic sections of the book that the quasi-linear inversion, introduced above, cannot be used for interpretation of multi-source data, because both the reflectivity coefficient A and the material property parameter m depend on the illuminating incident wavefield. However, in many geophysical applications, for example in seismic exploration or in cross-well tomography, the data are collected using 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 (Zhou and Liu, 2000 Zhdanov and Tartaras, 2002). 

This inversion scheme can be used for a multi-source technique because A and mi are source independent. It reduces the original nonlinear inverse problem to the same Born-type inversion we discussed in the previous sections. That is why some methods of Born-type inversion can be applied in this situation as well. In particular, the Bleistein method can be very useful in combination with localized QL inversion. 

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). 

Thus, modern developments in theoretical geophysics have led to dissolving the difference between these two approaches to interpretation of seismic data (Bleistein et ah, 2001). In this section of the book I will discuss the basic ideas underlying the principles of wavefield migration, and will show how these principles are related to the general inversion technique developed in the previous sections. 

N. Bleistein, J.K. Cohen, and J.W. Stockwell (2001) Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. Springer, New York. 

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