... frequency. The mathematical procedure consists in the integration of the second-minor propagator equation of Gilbert and Backus (1966). The determinant representing the secular function is computed directly rather than by summing the products of its elements. This improves both accuracy and computing time...

.... Courtillot et al (1973) solved the problem of continuation of a potential field measured on an uneven profile, using the Backus and Gilbert approach. Another reference relevant to this problem (solved by Bhattacharryya and Chan on p. 1424) is Parker and Klitgord (1972), who used the Schwartz-Christoffel...

...-dimensional layered medium. The layers overlie a homogeneous half-space, and are excited by a normally incident plane wave. Such direct solutions to the inverse problem are in contrast to the iterative inversion techniques of the Gilbert-Backus type. We have chosen to treat the inverse problem...

...~ = - R ~ j . 8 ~ j . (2.5) There exists a close analogy between the rows or columns of matrices (2.4) and the generalized Dirichlet kernels, described by Backus and Gilbert (1968); at least for one-dimensional Earth models. If the index n in the equation 2.4a is fixed (r. = r0), and m is allowed...

... Abstract In the article entitled “Propagator matrices in elastic wave and vibration problems,” by Freeman Gilbert and George E. Backus, April, 1966, p. 326–332, the second of equations (2.10) on page 327 should read differently. Society of Exploration Geophysicists 1966 ...

... and Gilbert to analyze linear problems. Backus and Gilbert (1967) show how to solve for δ m by minimizing the square of its norm using equations  12 as constraints. If the process has converged, which means that the | δ g i | have become smaller than some preset value, the last...

... pulse and 1.047 for the S pulse when G = 0. The hodographs I X3 W I 0nmomenu 2 G--4 I [ i l 2 5 K :FIG. 7. Dispersion diagram. G -- 4. Gravity excluded. 790 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA are typical for interface pulses (Gilbert, Laster, Backus and Schell, 1962). As G increases the t5...

...) = 0. Then (9) ,1 aTkU) = Jo Gkr)av'(r) dr where Gk(1)(r) is defined in terms of the given model vl(r) by equations (7) and (8). Now we can regard v'(r) rather than v(r) as the earth model. The two are equivalent since v(1) is fixed. And now the method of Backus and Gilbert (1967) is applicable...

.... Gilbert J. F. (1967) . Numerical applications of a formalism for geophysical inverse problems , Geophys. J. 13 , 247 - 276 . Backus G. E. Gilbert J. F...

... in an elastic, heterogeneous isotropic medium and deduce from it the equations for travel time and the energy per unit area of the wave front. INTRODUCTION The application of Backus and Gilbert methods for solving an inverse problem is possible only when the solution of the corresponding direct problem is known...

... of the Backus-Gilbert inversion method posed in matrix formal- isms are given by Wiggins (1972) and Jackson (1972). A brief outline of the inversion method followingJackson (1972) is given below. The fundamental equation for the general linear inverse problem is y = Ax (1) where the observations y are related...

... in a Cylindrical Low Velocity Medium, by W. C. Meecham and John DeNoyer 139 Backus, George--The Propagation of Short Elastic Surface Waves oil a Slowly Rotating Earth 823 Backus, Milo M., Freeman Gilbert, Stanley J. Luster, and Richard Schell--Observation of Pulses on an Interface 847 Barnes, D., B. Carder, and J...

... sources or large wavelengths, only the first one or two terms of the series are required. In the particular case of a simple shear dislocation source which is localized in space, only the term with n = 1 is nonzero in equation (4) and the results are equivalent to those of Gilbert (Gilbert, 1971; Gilbert...

... computed using the standard Backus-averaging technique. Wave propagation in a homogeneous VTI medium can be described by a linear ordinary differential equation, d f ( z ) d z = i ω M f ( z ) , (1) where the vector f ( z ) = ( v , τ ) T contains...

...Freeman Gilbert; George E. Backus Abstract The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order, ordinary differential equations with variable coefficients. Volterra...

.... Gilbert F. 1961 . The Rotational Splitting of the Free Oscillations of the Earth , Proc. Nat. Acad. Sci. , 47 : 362 - 371 . Backus G. 1962 . The Effect of the Earth's Rotation...

... main methods aimed at linear or linearized problems, namely the regularization methods ( Levenberg, 1944 ; Tikhonov, 1963 ; Marquardt, 1970 ), and the method of Backus and Gilbert (Backus and Gilbert, 1967, 1970 ). Each of these methods had its own way of dealing with nonuniqueness...

... References Backus G. Gilbert F. (1961) . The rotational splitting of the free oscillations of the earth . Proc. Nat. Acad. Sci. US. 47 , 362 - 371...

..., u3, "c13,z33] (12) (b) corresponding to SH waves with asFl(kl, x3, (D) col [~2, ~23] (13) and both these vectors satisfy equations of the type (8) (Gilbert and Backus, 1966). (14) We therefore consider the system of differential equations ~(zz = X(z) ~(z). The propagator matrix for equation (14...

.... Res. 81, 4381-4399. Backus, G. E. and J. F. Gilbert (1967). Numerical applicaton of a formalism for geophysical inverse problems, Geophys. J. 13, 247-276. Backus, G. E. and J. F. Gilbert (1968). The resolving power of gross earth data, Geophys. J. 16, 169-205. Backus, G. E. and J. F. Gilbert (1970...