Hyperbolic Traveltime Approximation Research Articles

Application of very fast simulated annealing and differential evolution in the search for FO-CRS wavefield attributes

The finite-offset common-reflection-surface (FO-CRS) stack method can be used to simulate any common-offset (CO) seismic section by stacking prestack seismic data along the surfaces defined by the paraxial hyperbolic traveltime approximation. In two dimensions, the FO-CRS stacking operator depends on five kinematic wavefield attributes for every time sample of the target CO section. The main problem with this method is identifying a computationally efficient data-driven search strategy for accurately determining the best set of FO-CRS attributes that produce the optimal coherence measure of the seismic signal in the prestack data. Identifying a global optimization algorithm with the best performance is a challenge when solving this optimization problem. This is because the objective function is multimodal and involves a large volume of data, which leads to high computational costs. We introduced a comparative and competitive study through the application of two global optimization algorithms that simultaneously search the FO-CRS attributes from the prestack seismic data, very fast simulated annealing (VFSA) and the differential evolution (DE). By applying this FO-CRS stack to the Marmousi synthetic seismic data set, we have compared the performances of the two optimization algorithms with regard to their efficiency and effectiveness in estimating the five FO-CRS attributes. To analyze the robustness of the two algorithms, we apply them to real land seismic data and show their ability to find the near-optimal attributes and to improve reflection events in noisy data with a very low fold. We reveal that VFSA is efficient in reaching the optimal coherence value with the lowest computational costs, and that DE is effective and reliable in reaching the optimal coherence for determining the best five searched-for attributes. Regardless of the differences, the FO-CRS stack produces enhanced and regularized high-quality CO sections using both global optimization methods.

Seismic data enhancement and regularization using finite offset Common Diffraction Surface (CDS) stack

The Common Reflection Surface stack is a robust method for simulating zero-offset and common-offset sections with high accuracy from multi-coverage seismic data. For simulating common-offset sections, the Common-Reflection-Surface stack method uses a hyperbolic traveltime approximation that depends on five kinematic parameters for each selected sample point of the common-offset section to be simulated. The main challenge of this method is to find a computationally efficient data-driven optimization strategy for accurately determining the five kinematic stacking parameters on which each sample of the stacked common-offset section depends. Several authors have applied multi-step strategies to obtain the optimal parameters by combining different pre-stack data configurations. Recently, other authors used one-step data-driven strategies based on a global optimization for estimating simultaneously the five parameters from multi-midpoint and multi-offset gathers. In order to increase the computational efficiency of the global optimization process, we use in this paper a reduced form of the Common-Reflection-Surface traveltime approximation that depends on only four parameters, the so-called Common Diffraction Surface traveltime approximation. By analyzing the convergence of both objective functions and the data enhancement effect after applying the two traveltime approximations to the Marmousi synthetic dataset and a real land dataset, we conclude that the Common-Diffraction-Surface approximation is more efficient within certain aperture limits and preserves at the same time a high image accuracy. The preserved image quality is also observed in a direct comparison after applying both approximations for simulating common-offset sections on noisy pre-stack data.

Numerical analysis of the finite-offset common-reflection-surface traveltime approximations

Based on a hyperbolic traveltime approximation that depends on three kinematic attributes, the common-reflection-surface (CRS) stacking technique was developed to simulate zero-offset (ZO) sections. In addition, following this new concept of seismic imaging, a CRS stacking extension was introduced to simulate common-offset (CO) sections from multi-coverage data through the use of a hyperbolic paraxial traveltime approximation in the vicinity of a reflected central ray with finite-offset (FO). This last traveltime approximation depends on five kinematic attributes, the so-called five-parameters-traveltime approximation, which leads to the FO CRS traveltime approximation. In this paper, we obtain a new traveltime approximation for diffraction events by reducing the original CRS traveltime approximation to four parameters. We compare both traveltime approximations (reflection and diffraction events) with traveltimes calculated by ray tracing for a synthetic model. It was verified that the hyperbolic paraxial traveltime approximation associated to a diffracted central ray coincides well with the ray theoretical traveltime in comparison with the five-parameters-traveltime approximation. The sensitivity of the FO CRS traveltime approximation was also analyzed to define the priority for searching for each one of the five parameters of the FO CRS stack.

Traveltime approximations for q‐P waves in vertical transversely isotropy media*

ABSTRACTAs 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.

Generalized Dix equations for a layered transversely isotropic medium

In standard Dix equations, normal-incidence traveltimes and rms or NMO velocities for PP reflections are used to estimate interval velocity and thickness for homogeneous isotropic horizontal layers. For a stack of homogeneous horizontal layers that are transversely isotropic with a vertical symmetry axis (VTI), the qPqP traveltimes (and the qPqSV traveltimes) depend on four parameters and the thickness of each layer. These can be estimated from the traveltimes of the qPqP and qPqSV reflected waves. The estimated qPqP traveltime parameters are P-wave normal-incidence traveltime, P-wave NMO velocity, and a heterogeneity factor entering in the shifted hyperbolic traveltime approximation or in a continued fraction traveltime approximation. The estimated qPqSV traveltime parameters are PS normal-incidence traveltime and PS NMO velocity combined with the qPqP traveltime parameters. These are used to compute S-wave normal-incidence traveltime and SV-wave NMO velocity. The estimated traveltime parameters are entered into explicit expressions for the four layer parameters and the thickness of each layer. These layer-recursive formulas, except for slightly different initial conditions, are also valid for ocean-bottom seismic data. For reflected SH-wave traveltimes, there are two elastic parameters and thicknesses per layer, and we can estimate three traveltime parameters. However, two of these traveltime parameters are functionally related, and it is impossible to estimate the three layer parameters from SH-wave traveltimes only.

Offset‐dependent geometrical spreading in a layered medium

The geometrical spreading for a point source in a horizontally layered medium has been computed by Ursin (1978) and Hubral (1978) as a Taylor series in the offset coordinate. The coefficients in the Taylor series depend on the thicknesses and the velocities of the layers. Here, I start with the exact expression for geometrical spreading and show that it can be expressed as a function of the velocity in the first layer, the offset, and the first‐ and second‐order traveltime derivatives. A shifted hyperbolic traveltime approximation (Castle, 1988) and the usual hyperbolic traveltime approximation are used to derive approximate expressions for geometrical spreading. These expressions can also be derived from a truncated Taylor series by making additional approximations, but this procedure is not so obvious.

Inversion of seismic data in the τ‐p plane

Seismic data can be transformed to the domain of intercept time and ray parameter without the artifical suppression of high frequencies caused by finite temporal and spatial sampling. These data can be analyzed to derive velocity‐depth information using the loci of the arrivals in the τ‐p plane. For limited source‐receiver offsets, the parameters of the best single ellipse approximating the true τ‐p reflection trajectory can be used to derive an approximate velocity‐depth function. We show that this is equivalent to Dix's (1955) hyperbolic traveltime approximation. When large aspect ratios are involved, the path formed by the postcritical reflection and refraction arrivals can be used directly to derive the velocity function. We consider both of these inversion methods as the initial step required to implement an inversion procedure that utilizes both pre‐ and postcritical reflection and refraction arrivals by an exact "normal moveout" correction in τ‐p. This approach is computationally efficient and is the equivalent to a ray traced normal moveout correction in X‐T.