To date, however, there are few published reports on the application of 3-D synthetic seismograms to the modelling of 3-D field data. The ray method has its advantages and disadvantages. As we can sec, transport equations 2. This book will prove to be an invaluable advanced textbook and reference volume in all academic institutions in which seismology is taught or researched. Such an approach is advantageous in that the simplest form of spreading formulation is directly implemented by interpreting the formula geometrically, without explicitly solving the 19 equations simultaneously for each ray eqs B3.
The solution with the minus sign — corresponds to the spherical wave with an expanding wavefront for t increasing , propagating away from the origin of coordinates to infinity. For T' x, and T' x, different from T x, along E. They are also called physical rays, as opposed to the mathematical ray O. Velocity estimation for Rainier Mesa, Nevada. In the frequency domain, the ray series corresponds to a series in inverse powers of frequency co. The proof for S waves is analogous.
Thus, in 2-D with a line source, we need to distinguish between exact and asymptotic computations, even in homogeneous media. Synthetic Examples One of the main advantages of the present algorithm is its ability to handle arbitrary recording geometries when 3-D effects must be considered. Such models have been commonly used is seismic exploration. The Gaussian curvature Ks is invariant so that we can retain it. This approach has mostly been used in the seismological literature; see Savarenskiy and Kirnos 1955 , Bullen 1965 , Pilant 1979 , Aki and Richards 1980 , and Bullen and Bolt 1985 among others. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The diagonal elements in the table are always positive for a solid medium, but the off-diagonal elements may be arbitrary positive, zero, negative.
The second approach consists of approximate ray computations through an actual, nonaveraged structure. The book has a tutorial character: derivations start with a relatively simple problem, in which the main ideas are easier to explain, and then advance to more complex problems. For general transient signals, we speak of quasi-elliptical polarization, again in the plane of the wavefront. See also Babich 1956 , Babich and Alekseyev 1958 , Karal and Keller 1959 , Alekseyev and Gel'chinskiy 1959 , Alekseyev, Babich, and Gel'chinskiy 1961 , Cerveny and Ravindra 1971 , Cerveny, Molotkov, and Psencik 1977 , Cerveny and Hron 1980 , and Achenbach, Gautesen, and McMaken 1982 , among others. We shall now describe how to determine e'2 and e2 for an arbitrarily selected reflected wave, corresponding to the standard choice. The 2-D Green functions may find applications in 2-D computations similar to the 3-D Green functions in 3-D models.
This means that the S plane waves are not, in general, linearly polarized. We can then use 3. For the acoustic case, we shall use two approaches to determine the Green function: the exact approach and the approximate approach. Most of the derived equations in the book are expressed in algorithmic form and may be used directly for computer programming. We remind the reader that U is a constant vector and T x, is a linear function of coordinates in the case of plane waves.
Ray Tracing Systems in Inhomogeneous Isotropic Media -- 3. For an incident P wave, this possibility is always simpler. Thus, the relation is valid more generally, even for highfrequency seismic body waves propagating in anisotropic, smoothly inhomogeneous media. In this case, however, the representation theorem 2. Study of this text will help the reader to gain a much better understanding of the nature of the ray solutions and to prepare to implement the relevant equations. In the same way as in the cell approach, an alternative option is to divide the whole structure or the whole layer into formal fictitious sublayers and to specify a simple velocity-depth distribution within each sublayer. They remain valid even for inhomogeneous plane waves.
The upper sign — corresponds to the transmitted wave, the lower sign + refers to the reflected wave, and e is the orientation index; see 2. This plane is perpendicular to the remaining eigenvector. We can investigate the process of reflection and transmission in exactly the same way as in the acoustic case. We have considered only real-valued bulk modulus k. The algorithm may fail when the ray hits a caustic or a shadow zone. The approach, however, is not as strict as other approaches, as we simulate a smooth medium by a system of thin homogeneous layers and then decrease the thicknesses of the layers to zero.
See also Babich 1961a , Cerveny 1972 , Cerveny, Molotkov, and Psencik 1977 , and Gajewski and Psencik 1987a. The relation between rj and V is unique, and one quantity can be simply expressed in terms of the other. Let usfirstdiscuss the complex-valued roots. Seismic Rays and Travel Times -- 3. Assume that the signal under consideration is effectively concentrated in time interval tmm 'max, relation 2. Propagator Matrices of Dynamic Ray Tracing Systems -- 4. Then, the higher order terms of the ray series calculated in this way compensate term A in 2.
G i and G o are the corresponding in-plane and out-of-plane geometric spreadings. But what about term V 2 F in 2. It is a useful summary of an important topic, built on the insights one of the foremost experts in ray theory. This approach, based on the solution of the eigenvalue problem for matrix F J ; , will be applied in Section 2. First-Order Born Approximation Let us consider a structural model M.