Einar Iversen

Traveltime and amplitude estimation using wavefront construction

We have developed and implemented a new method for estimation of first and later arrival traveltimes and amplitudes in a general 2D model. The basic idea of this wavefront (WF) construction approach is to use ray tracing to estimate a new WF Erom the old one. The calculation goes along the WFs rather than along the rays.

Image-ray tracing for joint 3D seismic velocity estimation and time-to-depth conversion

Seismic time migration is known for its ability to generate well-focused and interpretable images, based on a velocity field specified in the time domain. A fundamental requirement of this time-migration velocity field is that lateral variations are small. In the case of 3D time migration for symmetric elementary waves e.g., primary PP reflections/diffractions, for which the incident and departing elementary waves at the reflection/diffraction point are pressure P waves, the time-migration velocity is a function depending on four variables: three coordinates specifying a trace point location in the time-migration domain and one angle, the so-called migration azimuth. Based on a time-migration velocity field available for a single azimuth, we have developed a method providing an image-ray transformation between the time-migration domain and the depth domain. The transformation is obtained by a process in which image rays and isotropic depth-domain velocity parameters for their propagation are estimated simultaneously. The depth-domain velocity field and image-ray transformation generated by the process have useful applications. The estimated velocity field can be used, for example, as an initial macrovelocity model for depth migration and tomographic inversion. The image-ray transformation provides a basis for time-to-depth conversion of a complete time-migrated seismic data set or horizons interpreted in the time-migration domain. This time-to-depth conversion can be performed without the need of an a priori known velocity model in the depth domain. Our approach has similarities as well as differences compared with a recently published method based on knowledge of timemigration velocity fields for at least three migration azimuths. We show that it is sufficient, as a minimum, to give as input a time-migration velocity field for one azimuth only. A practical consequence of this simplified input is that the image-ray transformation and its corresponding depth-domain velocity field can be generated more easily.

3-D ray modeling by wavefront construction in open models

A synthesis of two newly developed concepts in 3-D modeling is developed in this paper: (1) The open, (non-complete) seismic model, and (2) the ray tracing based wavefront (WF) construction method. The open model may contain interfaces with holes and other missing parts, which simplifies model building considerably because the input horizon data from standard interpretation and processing packages are often incomplete. A set of volumes is defined in the model. A volume is a logical unit that points to a set of property functions, e.g., P-velocity, S-velocity, and density. The properties are represented either as constants or as B-spline functions of the spatial coordinates (x, y) or (x, y, z). The volumes of the model are assigned to opposing sides of each interface and not to specific spatial areas of the model, which is the case in most (blocky) model representations. The interfaces are given explicitly by triangular grids where the sizes of the triangles are determined locally by the curvature of the interface. We show how modeling by WF construction is both possible and computationally efficient in open models, but only after some modifications to deal with the ambiguity of the model representation. It is not possible to find a unique volume for the spatial positions in an open model. Instead, the volumes (with associated velocities, etc.) are determined from the last interface encountered by each ray in the WF. To find an arrival in a receiver, the volume associated to the receiver has to match the volume of the WF hitting the receiver.

Improved applicability of ray tracing in seismic acquisition, imaging, and interpretation

Ray-based seismic modeling methods can be applied at various stages of the exploration and production process. The standard ray method has several advantages, e.g., computational efficiency and the possibility of simulating propagation of elementary waves. As a high-frequency approximation, the method also has a number of limitations, particularly with respect to a lack of amplitude reliability in the presence of rapid changes of the model functions representing elastic parameters and interfaces. Given the objective of improving the applicability of the standard ray method, we present a strategy that does not require specific extension to finite frequencies. Instead, we define each ray-based process as an element of a system that, as a composite process, is able to obtain better results than the ray-based process applied alone. Other elements of the composite process can be finitedifference modeling or numerical solutions for surface and volume integrals, which are basic constituents of Kirchhoff modeling and imaging. We also include among the process elements recently developed techniques for simulating the migration amplitude on a target reflector and in a local volume, e.g., a reservoir zone. The model is decomposed according to its complexity into volume elements, surface elements, or a combination. The composite process consists of a specified interaction between process elements and model elements, which fits well with the philosophy of modern software design. Model elements that will be exposed to ray-tracing algorithms may need appropriate preparation, e.g., smoothing and resampling. We demonstrate specifically, in a tutorial example, that simulating the seismic response from a reflector by ray-based composite processes can yield better results than applying standard ray tracing alone.

The isochron ray in seismic modeling and imaging

The isochron, the name given to a surface of equal two‐way time, has a profound position in seismic imaging. In this paper, I introduce a framework for construction of isochrons for a given velocity model. The basic idea is to let trajectories called isochron rays be associated with iso chrons in an way analogous to the association of conventional rays with wavefronts. In the context of prestack depth migration, an isochron ray based on conventional ray theory represents a simultaneous downward continuation from both source and receiver. The isochron ray is a generalization of the normal ray for poststack map migration. I have organized the process with systems of ordinary differential equations appearing on two levels. The upper level is model‐independent, and the lower level consists of conventional one‐way ray tracing. An advantage of the new method is that interpolation in a ray domain using isochron rays is able to treat triplications (multiarrivals) accurately, as opposed to interpolation in the dep...

Review of Ray Theory Applications in Modelling and Imaging of Seismic Data

Throughout the last twenty years, 3D seismic ray modelling has developed from a research tool to a more operational tool that has gained growing interest in the petroleum industry. Various areas of application have been established and new ones are under development. Many of these applications require a modelling system with flexible, robust and efficient modelling algorithms in the core. The present paper reviews the basic elements of such a system, based on the ‘open model’ concept and the ‘wavefront construction’ technique. In the latter, Červenýs dynamic ray tracing is an intrinsic part. The modelling system can be used for generating ray attributes and synthetic seismograms for realistic 3D surveys with tens of thousands of shots and receivers. Moreover, some other types of application areas are illustrated: Production of Greens functions for prestack depth migration and hybrid modelling (combined ray and finite-difference modelling), attribute mapping and illumination analysis, both for survey planning and interpretation. Finally, the concepts of ‘isochron rays’ and ‘velocity rays’ related to seismic isochrons have been introduced recently, with very interesting future applications.

Velocity rays for heterogeneous anisotropic media: Theory and implementation

The surface of equal two-way time referred to as the isochron is a fundamental concept in seismic imaging. The shape of an isochron depends on the source and receiver locations, on the wave type, and on the parameters constituting the seismic velocity model. A perturbation of a parameter of the velocity model forces the isochron points to move along trajectories called velocity rays, with the selected model parameter as the variable along the rays. Based on earlier work describing first-order approximations to velocity rays, I develop a general theory for velocity rays valid for 3D heterogeneous and anisotropic velocity models. By this theory, velocity rays can be obtained in a way similar to the way conventional rays are computed by numeric integration of a system of ordinary differential equations (ODEs). The process is organized with ODE solvers on two levels, where the upper level is model independent. The lower level includes conventional one-way kinematic and dynamic tracing of source and receiver rays, as well as calculation of ray perturbation quantities. Accurate velocity rays are expected to be useful for perturbation of reflectors mapped from the time domain to the depth domain, for remigration of seismic images in the depth domain, and for velocity model updating.

Amplitude, Fresnel zone, and NMO velocity for PP and SS normal-incidence reflections

Inspired by recent ray-theoretical developments, the theory of normal-incidence rays is generalized to accommodate P- and S-waves in layered isotropic and anisotropic media. The calculation of the three main factors contributing to the two-way amplitude — i.e., geometric spreading, phase shift from caustics, and accumulated reflection/transmission coefficients — is formulated as a recursive process in the upward direction of the normal-incidence rays. This step-by-step approach makes it possible to implement zero-offset amplitude modeling as an efficient one-way wavefront construction process. For the purpose of upward dynamic ray tracing, the one-way eigensolution matrix is introduced, having as minors the paraxial ray-tracing matrices for the wavefronts of two hypothetical waves, referred to by Hubral as the normal-incidence point (NIP) wave and the normal wave. Dynamic ray tracing expressed in terms of the one-way eigensolution matrix has two advantages: The formulas for geometric spreading, phase shif...

Derivatives of Reflection Point Coordinates with Respect to Model Parameters

The motivation for this paper is to provide expressions for first-order partial derivatives of reflection point coordinates, taken with respect to model parameters. Such derivatives are expected to be useful for processes dealing with the problem of estimating velocities for depth migration of seismic data.The subject of the paper is a particular aspect of ray perturbation theory, where observed parameters—two-way reflection time and horizontal components of slowness, are constraining the ray path when parameters of the reference velocity model are perturbed. The methodology described here is applicable to general rays in a 3D isotropic, heterogeneous medium. Each ray is divided into a shot ray and a receiver ray, i.e., the ray portions between the shot/receiver and the reflection point, respectively. Furthermore, by freezing the initial horizontal slowness of these subrays as the model is perturbed,elementary perturbation quantities may be obtained, comprising derivatives of ray hit positions within theisochrone tangent plane, as well as corresponding time derivatives. The elementary quantities may be estimated numerically, by use of ray perturbation theory, or in some cases, analytically. In particular, when the layer above the reflection point is homogeneous, explicit formulas can be derived. When the elementary quantities are known,reflection point derivatives can be obtained efficiently from a set of linear expressions.The method is applicable for a common shot, receiver or offset data sorting. For these gather types, reflection point perturbationlaterally with respect to the isochrone is essentially different. However, in theperpendicular direction, a first-order perturbation is shown to beindependent of gather type.To evaluate the theory, reflection point derivatives were estimated analytically and numerically. I also compared first-order approximations to ‘true’ reflection point curves, obtained by retracing rays for a number of model perturbations. The results are promising, especially with respect to applications in sensitivity analysis for prestack depth migration and in velocity model updating.

First-Order Perturbation Theory for Seismic Isochrons

A first-order perturbation theory for seismic isochrons is presented in a model independent form. Two ray concepts are fundamental in this theory, the isochron ray and the velocity ray, for which I obtain first-order approximations to position vectors and slowness vectors. Furthermore, isochron points are connected to a shot and receiver by conventional ray fields. Based on independent perturbation of the shot and receiver ray I obtain first-order approximations to velocity rays. The theory is applicable for 3D inhomogeneous anisotropic media, given that the shot and receiver rays, as well as their perturbations, can be generated with such model generality.The theory has applications in sensitivity analysis of prestack depth migration and in velocity model updating. Numerical examples of isochron and velocity rays are shown for a 2D homogeneous VTI model. The general impression is that the first-order approximation is, with some exceptions, sufficiently accurate for practical applications using an anisotropic velocity model.