Seismic inversion

Returning to the seismic wavefield inverse problem, we can assume that, based on general mathematical uniqueness theorems, the seismic inverse problem. 

Overturned Surfaces in the Presence of Faulting The situation shown in Figure 6 indicates a geometry in which oil may become trapped in layers that have been faulted by a salt dome intrusion. Although the structural geology of the reservoir itself may not involve overturned surfaces, the seismic inversion requirement to describe the sound speed model in a process of optimised migration means it is necessary to model such complicated geometric features. 

The different inverse problems in the geosciences, particularly seismic inversion, resistivity inversion, and the history matching of production data are usually treated in quite different ways is this an accident of history, or should there be changes in all areas, so that a unified approach is used ... 

Of course the typical seismic trace has many hundreds of reflections in it, all the way down from the surface to the deepest times measured. These days, engineers and geologists prefer to see the seismic in terms of the acoustic impedance rather than reflection data and this can be obtained by inversion from the seismic volume. Aseismic volume is made up of hundreds of thousands of traces. 

Data reduction is done by a process called inversion. It is not possible to uniquely derive the structure of a body from first principles based on seismic data. Instead, a model of the structure must be assumed and then the predictions of the model are compared to the observations. The model is then adjusted until the predictions match the observations. The more precise the predictions, the better the model can be tested by the observations. Properties that can be investigated by helioseismic inversion include the density, pressure, sound speed, angular velocity, temperature, and composition. 

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

Khan A. and Mosegaard K. (2002) An inquiry into the lunar interior a nonlinear inversion of the Apollo lunar seismic data. J. Geophys. Res. 107, E63.1-E63.23. 

White R. S., McKenzie D., and O Nions R. K. (1992) Ocean crustal thickness from seismic measurements and rare earth element inversions. J. Geophys. Res. 97, 19683-19715. 

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. 

The problem (1.4) is called an inverse source problem. In this case an assumption is made that the model parameters (the physical properties of the medium) are known. Typical examples of this problem are the inverse gravity problem and the inverse seismological problem. In the first case, the density distribution of the rock formation is the source of the gravity field. In the second case, the goal is to find the location and type of the earthquake sources from the observed seismic field. 

Now, let us turn to the analysis of typical formulations of the forward and inverse problems for major geophysical fields gravity, magnetic, electromagnetic and seismic wave fields. 

In the case of a layered model of the earth, one can use a simple technique of geometrical seismics, which is based on studying the geometry of rays of seismic wave propagation. In more complicated geological structures, comprehensive imaging and inversion methods must be used to analyze seismic data. In order to develop these methods, one should study carefully the physics of seismic waves. 

Numerical solution of the Helmholtz equation for a given velocity distribution describes the forward ])roblem, while the inverse problem is actually aimed at determining the coefficients (velocity c(r)) for the given pres.surc field p r,uj). Both of these problems arc nonlinear. Note that often inverse seismic problems are formulated not for the velocity but for the slowness, which is the inverse velocity ... 

The mathematical formulation of seismic forward and inverse problems in the simplest case of an acoustic model in the frequency domain is given by equation (1.25), which we will repeat here for convenience ... 

In the later sections of this book we will demonstrate that the same technique can be applied to electromagnetic and seismic wave field Inversion. In these areas of geophysics, migration serves as a useful practical tool for imaging geophysical data, because of the relative numerical simplicity and transparent physical interpretation of the results of electromagnetic and seismic migration. 

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. 

Thus while the inverse operator (15.98) determines the distribution of the anomalous square slowness within the medium, formula (15.99) solves for the reflectors and the corresponding reflection coefficients. In geophysical applications, for example in seismic methods, the reflecting boundaries are the main target of exploration. That is why inversion formula (15.99) plays an important role in the interpretation of seismic data. In the next section we will show that this method can be extended to a 3-D case. This technique provides the basis for modern methods of seismic data interpretation. 

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). 

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. 

The problem of elastic field inversion is much more complicated than acoustic or vector wavefield inversion, considered in the previous sections of the book. However, the fundamental principles of elastic inversion resemble those discussed above for more simple models of seismic waves. I will present in this section a brief overview of the basic ideas underlining the elastic field inversion. 

Mora, P. R., 1987, Nonlinear two-dimensional elastic inversion of multi-offset seismic data Geophysics, 52, No. 9, 1211-1228. 

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