Seismic wavefield

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. 

The simplest model of these waves is one based on acoustic principles. Assume that the earth can be treated as an acoustic medium and the influence of variations in density can be ignored. In this case the propagation of seismic waves in the earth can be described by the acoustic wave equation  

The forward problem in this case is formulated as the solution of the differential equation (1-22) with respect to F(r,t) for the given velocity c(r)  

The inverse problem consists in reconstructing the velocity distribution from the observed pressure field  

Both operators of forward A° and inverse (v4 ) problems are nonlinear operators. 

---

The analysis of a seismic wavefield can be significantly simplified in the frequency domain ... 

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

The development of detailed models of the layered elastic and density stmcture of the Earth in the 1970s and 1980s made possible the accurate forward calculation of the seismic wavefield caused by forces acting in the Earth. In particular, in the formalism of the Earth s long-period elastic normal modes, the excitation of seismic waves by body forces and moment-tensor sources is straightforward (Gilbert 1971). Linear elastic wave theory thus provides the framework in which earthquake source characteristics can be deduced from observed seismic waves. At long periods, the seismic wavefield is simple and can be explained by a small number of fundamental earthquake parameters. 

Kohler A, Ohmberger M, Scherbaum F (2010) Unsupervised pattern recognition in continuous seismic wavefield records using self-organizing maps. Geophys J Int 182 1619-1630... 

The seismic wavefield is always influenced by lateral heterogeneities in the Earth, which can disturb the plane-wave approximation due to velocity inhomogeneities. In particular local... 

Seismic arrays are also used to investigate the nature and source regions of microseisms and to locate and track volcanic tremor for analyzing complex seismic wavefield properties in volcanic areas. 

Consider a 3-D seismic model with a background (normal) slowness distribution Sb(r) and a local inhomogeneity D with an arbitrarily varying square of slowness s (r) = Sb(r)+As (r). We will examine, in parallel, two cases the propagation of the acoustic field and of the vector wavefield in this model. 

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. 

---