Abstract Following Fermat's principle, the time during which the light covers its trajectory from a given point A to another given point B satisfies the condition where N denotes the index of refraction. By transformation, which is unnecessary to restate here 3 , one writes the above equation in this form: from which results the well-known differential equations 4 All of the above relate to isotropic media. In anisotropic media, N no longer denotes the index of refraction. In such media, one must distinguish between the speed of the propagation of light in the direction of the ray s and the speed in the direction normal to the wave surface q 5 . The index of refraction is inversely proportional to q, whereas N is inversely proportional to s. There is more; N depends also on direction, and it is a function of not only x, y, z but also of direction cosines If, for convenience, we denote the total variation of N becomes The three direction cosines λ, μ, v are subject to the condition however, it is unnecessary to introduce this additional condition, since the above relationship results simply from the equality which we use in the transformation of the integral (I). Let us calculate the variation of this integral under the assumption that N depends not only on coordinates, but also on direction cosines. With fixed limits, we obtain We shall transform the second (sic) 6 integral in the same fashion as in the case of isotropic media, namely, by virtue of equalities The second integral (sic) becomes

Abstract Fermat’s Principle states that for two points A and B in a velocity field, the ray path will be the trajectory between A and B along which the travel time is stationary. For many cases in isotropic media Fermat’s Principle seems intuitively obvious. For example, in a homogeneous, isotropic medium the least time travel path between two points is a straight line. In anisotropic media the results of Fermat’s Principle are less obvious and it is useful to have a rigorous proof. In this “tutorial style” paper we present a proof of Fermat’s Principle for anisotropic elastic media. The proof involves relationships between the slowness and wave surfaces. The slowness surfaces are defined by the determinant ( S ), or equivalently the eigenvalues (G m ), of the Kelvin-Christoffel matrix. The proof is given for both cases.

Abstract A derivation of Fermat’s principle for general elastic anisotropic media is presented. It is shown that Fermat’s principle breaks down at the cusps of the wave surface. Applications of Fermat’s principle should therefore be restricted to rays associated with convex slowness surfaces.

Abstract An interferometric form of Fermat’s principle is derived that allows for high-resolution estimation of the velocity distribution between deep interfaces. The data consist of reflection traveltimes from two deeply buried interfaces A and B recorded by sources and caused by receivers at the surface. Fermat’s interferometric principle is then used to kinematically redatum the surface sources and receivers to interface A so that the associated reflection times correspond to localized transit times between the A and B interfaces. The velocity model above interface A does not need to be known, so the distorting effects of the overburden and statics are eliminated by this target-oriented approach. Interferometric target-oriented tomography promises to enhance the resolution of whole-earth and exploration tomograms.

Abstract We review the equations for correlation-based redatuming methods. A correlation-based redatuming method uses natural-phase information in the data to time shift the weighted traces so they appear to be generated by sources (or recorded by geophones) shifted to a new location. This compares to model-based redatuming, which effectively time shifts the traces using traveltimes computed from a prior velocity model. For wavefield redatuming, the daylight imaging, interferometric imaging, reverse-time acoustics (RTA), and virtual-source methods all require weighted correlation of the traces with one another, followed by summation over all sources (and sometimes receivers). These methods differ from one another by their choice of weights. The least-squares interferometry and virtual-source imaging methods are potentially the most powerful because they account for the limited source and receiver aperture of the recording geometry. Interferometry, on the other hand, has the flexibility to select imaging conditions that target almost any type of event. Stationary-phase principles lead to a Fermat-based redatuming method known as redatuming by a seminatural Green’s function. No crosscorrelation is needed, so it is less expensive than the other methods. Finally, Fermat’s principle can be used to redatum traveltimes.

Abstract This book is devoted to one important aspect of development of physical foundations of the seismic method — the theory of edge diffraction phenomena. Thoese phenomena occur when conditions of the regular wave reflection/transmission change sharply. Though these phenomena drew the attention of many scientist for many decades, their real influence on the resolution ability of the seismic method was truly understood rather recently due to interpretation of seismic data in block structures. Clearly, to develop seismic method for investigation of such structures without developing the theory of edge diffraction phenomena is impossible. The latter is the aim of this book. The seismic method is based on the fundamental laws of continuum mechanics. These laws describe the behavior of wavefields on the microscopic level, i.e., in the form of differential equations of motion. Integrating these equations under some initial conditions or boundary conditions, makes possible acquisition of all necessary information on the wavefield in the given situation. However, the working base of the seismic method consists of not only the differential equations of motion themselves but of some general and simple enough consequences of their solutions, which are formulated in the form of physical principles and l aws. The latter include the concepts of wave, Fermat’s principle, the law of conservation of the energy flux, and the reflection/transmission laws. Essentially these laws and principles must form a system of concepts sufficient for the solution of some class of typical interpretation problems. In fact, these principles and laws form the physical fo ndation of the seismic method.

Abstract In the first part of this paper geometrical optics is generalized to include diffracted rays, by means of an extension of Fermat’s principle. Various properties of these rays which follow from this principle are given. In particular, a law of diffraction at an edge is presented. These properties suffice for explicit construction of the diffracted rays. In the second part it is shown how to construct an electromagnetic field by means of these rays. This construction is based on Luneburg’s method for the usual geometrical optics rays and upon a theorem concerning the asymptotic behavior of the field at high frequencies near an edge or vertex. The field so constructed is exactly the leading part of the asymptotic expansion of the actual field for high frequencies.

The geometrical theory of diffraction is an extension of geometrical optics which accounts for diffraction. It introduces diffracted rays in addition to the usual rays of geometrical optics. These rays are produced by incident rays which hit edges, corners, or vertices of boundary surfaces, or which graze such surfaces. Various laws of diffraction, analogous to the laws of reflection and refraction, are employed to characterize the diffracted rays. A modified form of Fermat’s principle, equivalent to these laws, can also be used. Diffracted wave fronts are defined, which can be found by a Huygens wavelet construction. There is an associated phase or eikonal function which satisfies the eikonal equation. In addition complex or imaginary rays are introduced. A field is associated with each ray and the total field at a point is the sum of the fields on all rays through the point. The phase of the field on a ray is proportional to the optical length of the ray from some reference point. The amplitude varies in accordance with the principle of conservation of energy in a narrow tube of rays. The initial value of the field on a diffracted ray is determined from the incident field with the aid of an appropriate diffraction coefficient. These diffraction coefficients are determined from certain canonical problems. They all vanish as the wavelength tends to zero. The theory is applied to diffraction by an aperture in a thin screen diffraction by a disk, etc., to illustrate it. Agreement is shown between the predictions of the theory and various other theoretical analyses of some of these problems. Experimental confirmation of the theory is also presented. The mathematical justification of the theory on the basis of electromagnetic theory is described. Finally, the applicability of this theory, or a modification of it, to other branches of physics is explained.

Abstract An interferometric form of Fermat’s principle and travel- time tomography is used to invert ground-penetrating radar (GPR) data for the subsurface velocity distribution. The input data consist of GPR traveltimes of reflections from two bur’ied interfaces, A (reference) and B (target), where the data are excited and recorded by GPR antennas at the surface. Fer- mat’s interferometric principle is then used to redatum the surface transmitters and receivers to interface A so the associ’ated reflection traveltimes correspond to localized transit times between interfaces A and B . The overburden velocity model above interface A is not required. The result after to- mographic inversion is a high-resolution estimate of the ve’locity between interfaces A and B that does not depend on the velocity model above interface A . A motivation for introduc’ing interferometric traveltime tomography is that typical lay’er-stripping approaches will see the slowness error increase with depth as the layers are inverted. This suggests that near- surface statics errors are propagated and amplified with depth. In contrast, the interferometric traveltime tomography method largely eliminates statics errors by taking the differ’ence between reflection events that emanate from neighbor’ing layer interfaces. Slowness errors are not amplified with depth. However, the method is sensitive to the estimation ac’curacy for the geometry of the reference interface. Both syn’thetic and real field data are used successfully to validate the effectiveness of this interferometric technique.

ABSTRACT Seismic tomography methods that use waves originating outside the volume being studied are subject to bias caused by unknown structure outside this volume. The bias is of the same mathematical order and similar magnitude as the local-structure effects being studied; failure to account for it can significantly corrupt derived structural models. This bias can be eliminated by adding to the inverse problem three unknown parameters specifying the direction and time for each incident wave, a procedure analogous to solving for event locations in local-earthquake and whole-mantle tomography. The forward problem is particularly simple: The first-order change in the arrival time at an observation point resulting from a perturbation to the incident-wave direction and time equals the change in the time of the perturbed incident wave at the point where the unperturbed ray entered the study volume. This consequence of Fermat’s principle apparently has not previously been recognized. Published teleseismic tomography models probably contain significant artifacts and need to be recomputed using the more complete theory.