A B S T R A C TAs exploration targets have become deeper, cable lengths have increased accordingly, making the conventional two term hyperbolic traveltime approximation produce increasingly erroneous traveltimes. To overcome this problem, many traveltime formulas were proposed in the literature that provide approximations of different quality. In this paper, we concentrate on simple traveltime approximations that depend on a single anisotropy parameter. We give an overview of a collection of such traveltime approximations found in the literature and compare their quality. Moreover, we propose some new single-parameter traveltime approximations based on the approximations found in the literature. The main advantage of our approximations is that some of them are rather simple analytic expressions that make them easy to use, while achieving the same quality as the better of the established formulas. I N T R O D U C T I O NTraveltime approximations play a key role in the processing of reflection data. They are used in, for example, migration (Alkhalifah and Larner 1994; Vestrum, Lawton and Schmid 1999; Mukherjee, Sen and Stoffa 2001), moveout correction and velocity analysis (Tsvankin and Thomsen 1994;Alkhalifah and Tsvankin 1995;Fomel 2003) and remigration (Fomel 1994;Hubral, Tygel and Schleicher 1996;Schleicher and Aleixo 2007).The standard hyperbolic approximation (Dix 1955) of the P-wave reflection traveltime commonly used in seismic data processing is exact for a homogeneous isotropic medium and a planar reflector. It remains a good approximation for short offsets in layered media with not too strong lateral variations. However, as exploration targets have become deeper, cable lengths have increased accordingly. Increased offsets have made the conventional two term hyperbolic equation produce increasingly erroneous traveltimes.to the reflection moveout to guarantee an accurate determination of the model parameters.Many attempts have been made over the years to provide higher-order reflection moveout equations that provide good approximations for higher offsets. Working with a layered earth model, Bolshix (1956) obtained a sixth-order equation that approximates traveltime. Later, Taner and Koehler (1969) provided a high-order approximation for traveltimes based on an exact Taylor-series expansion of the traveltime. May and Straley (1979) used orthogonal polynomials to derive a high-order traveltime approximation. These approximations based on polynomials, Taylor series or orthogonal polynomials are rather inaccurate for larger offsets. Therefore, other approximations are necessary.To improve accuracy, particularly for large offsets, various authors proposed a shifted-hyperbola approximation (Malovichko 1978;Claerbout 1987;de Bazelaire 1988;Castle 1994). This equation describes a hyperbola that is symmetric about the t-axis and has asymptotes that intersect the time axis x = 0 at a time t = τ s that is different from the zero-offset traveltime τ 0 . The shifted hyperbola proposed by Claerbout (1987) contains a fre...

The image-wave equations for the problems of depth and time remigration in elliptically anisotropic media are secondorder partial differential equations similar to the acousticwave equation. The propagation variable is the vertical velocity or the medium ellipticity rather than time. These differential equations are derived from the kinematic properties of anisotropic remigration. The objective is to construct subsurface images that correspond to different vertical velocity and/ or different degrees of medium anisotropy directly from a single migrated image. In this way, anisotropy panels can be obtained in a way completely analogous to velocity panels for migration velocity analysis. A simple numerical example demonstrates the validity of the theory.

Using an integral representation for the first kind Hankel (Hankel-Bessel Integral Representation) function we obtain the so-called Basset formula, an integral representation for the second kind modified Bessel function. Using the Sonine-Bessel integral representation we obtain the Fourier cosine integral transform of the zero order Bessel function. As an application we present the calculation of the Green's function associated with a second-order partial differential equation, particularly a wave equation for a lossy two-dimensional medium. This application is associated with the transient electromagnetic field radiated by a pulsed source in the presence of dispersive media, which is of great importance in the theory of geophysical prospecting, lightning studies and development of pulsed antenna systems. Keywords: Sonine-Bessel, integral representation, dissipative wave equation.Usando uma representação integral para a função de Hankel de primeira espécie (representação integral de Hankel-Bessel) obtemos a chamada fórmula de Basset, uma representação integral para a função de Bessel modificada de segunda espécie. A partir de uma representação integral de Sonine-Bessel obtemos a transformada de Fourier em co-senos da função de Bessel de ordem zero. Como uma aplicação, apresentamos o cálculo da função de Green associada a uma equação diferencial parcial de segunda ordem, a saber, a equação da onda em um meio dissipativo de dimensão dois. Esta aplicação está associada ao campo eletromagnético transiente irradiado por uma fonte tipo pulso na presença de meios dispersivos, o qualé de grande importância na teoria de prospecção geofísica, estudos sobre luz e desenvolvimento de sistemas de antenas tipo pulso. Palavras-chave: Sonine-Bessel, representação integral, equação da onda dissipativa.

Standard real-valued finite-difference (FD) and Fourier finite-difference (FFD) migrations cannot handle evanescent waves correctly, which can lead to numerical instabilities in the presence of strong velocity variations. A possible solution to these problems is the complex Padé approximation, which avoids problems with evanescent waves by rotating the branch cut of the complex square root. We have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable FD and hybrid FD/FFD migrations for such media. Our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips. Our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60° for FD migration, as confirmed by numerical impulse responses and work with synthetic data.