Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

Building a 3D geological model from field and subsurface data is a typical task in geological studies involving natural resource evaluation and hazard assessment. However, there is quite often a gap between research papers presenting case studies or specific innovations in 3D modeling and the objectives of a typical class in 3D structural modeling, as more and more is implemented at universities. In this paper, we present general procedures and guidelines to effectively build a structural model made of faults and horizons from typical sparse data. Then we describe a typical 3D structural modeling workflow based on triangulated surfaces. Our goal is not to replace software user guides, but to provide key concepts, principles, and procedures to be applied during geomodeling tasks, with a specific focus on quality control.

/pdf/surface-based-3d-modeling-of-geological-structures-4g75h7dsby.pdf

Surface-Based 3D Modeling of Geological Structures

SUMMARY The fast marching method (FMM) is a grid based numerical scheme for tracking the evolution of monotonically advancing interfaces via finite-difference solution of the eikonal equation. Like many other grid based techniques, FMM is only capable of finding the first-arriving phase in continuous media; however, it distinguishes itself by combining both unconditional stability and rapid computation, making it a truly practical scheme for velocity fields of arbitrary complexity. The aim of this paper is to investigate the potential of FMM for finding later arriving phases in layered media. In particular, we focus on reflections from smooth subhorizontal interfaces that separate regions of continuous velocity variation. The method we adopt for calculating reflected phases involves two stages: the first stage initializes FMM at the source and tracks the incident wave front to all points on the reflector surface; the second stage tracks the reflected wave front by reinitializing FMM from the interface point with minimum traveltime. Layer velocities are described by a regular grid of velocity nodes and layer boundaries are described by a set of interface nodes that may be irregularly distributed. A triangulation routine is used to locally suture interface nodes to neighbouring velocity nodes in order to facilitate the tracking of wave fronts to and from the reflector. A number of synthetic tests are carried out to assess the accuracy, speed and robustness of the new scheme. These include comparisons with analytic solutions and with solutions obtained from a shooting method of ray tracing. The convergence of traveltimes as grid spacing is reduced is also examined. Results from these tests indicate that wave fronts can be accurately tracked with minimal computational effort, even in the presence of complex velocity fields and layer boundaries with high curvature. Incident wave fronts containing gradient discontinuities or shocks also pose no difficulty. Further development of the wave front reinitialization scheme should allow other later arrivals such as multiples to be successfully located.

/pdf/wave-front-evolution-in-strongly-heterogeneous-layered-media-2e8304e9mq.pdf

Wave front evolution in strongly heterogeneous layered media using the fast marching method

Seismic tomography: a window into deep Earth

https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=5237&context=facpub

Building solid models from boreholes and user-defined cross-sections

A new 3D wavefield modelling approach based on dynamic ray tracing is presented. This approach is called wavefront construction, and it can be used in 3D models with constant or smoothly varying material properties (S- and P-velocity and density) separated by smooth interfaces. Wavefronts consisting of rays arranged in a triangular network are propagated stepwise through the model. At each time step, the differences in a number of parameters are checked between each pair of rays on the wavefront. New rays are interpolated whenever this difference between pairs of rays exceeds some predefined maximum value. A controlled sampling of the wavefront at all time steps is thus obtained. Receivers are given multiple-event values by interpolation when the wavefronts pass them. The strength of the wavefront construction method is that it is robust and efficient.

Estimation of multivalued arrivals in 3D models using wavefront construction—Part I1

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.

3-D ray modeling by wavefront construction in open models

We have developed a ray tracing based wavefront (WF) construction method that finds arrivals in general 3D models consisting of blocks with smooth velocity and density variation separated by interfaces where the velocity and density change discontinuously. The method is based on the ideas in Vinje et. al. (1992) and it mimics the true wave propagation in the lense that entire WFs are propagated time step by time step creating a moving surface travelling through the model. The WF may stretch, twist and fold in any way as it moves through the medium and the spatial sampling rate of the WF is kept above a pre-defined limit. The receivers are given value as the WFs pass them, one arrival for each sheet of WF that passes. Multi-arrival travel times, amplitudes, geometric spreading etc. are thus found in an efficient way in all areas of the model that are reachable by conventional rays.

Estimation of multivalued arrivals in 3D models using wavefront construction

The numerical tracing of short ray segments and interpolation of new rays between these ray segments are central constituents of the wavefront construction method. In this paper the details of the ray tracing and ray-interpolation procedures are described. The ray-tracing procedure is based on classical ray theory (high-frequency approximation) and it is both accurate and efficient. It is able to compute both kinematic and dynamic parameters at the endpoint of the ray segments, given the same set of parameters at the starting point of the ray. Taylor series are used to approximate the raypath so that the kinematic parameters (new position and new ray tangent) may be found, while a staggered finite-difference approximation gives the dynamic parameters (geometrical spreading).

When divergence occurs in some parts of the wavefront, new rays are interpolated. The interpolation procedure uses the kinematic and dynamic parameters of two parent rays to estimate the initial parameters of a new ray on the wavefront between the two rays. Third-order (cubic) interpolation is used for interpolation of position, ray tangent and take-off vector from the source) while linear interpolation is used for the geometrical spreading parameters.

Part II: Tracing and interpolation1

During our recent work with 3-D dynamic ray-tracing and velocity inversion problems, a new 3-D model generation system has been developed using a so-called “solid modeling” technique. The term “solid modeling” refers to the fact that the logical system governing the internal geometrical properties of the model describes the model as a combination of “solids” or “volumes” in 3-D space. In each of these volumes the physical parameters (such as seismic velocity, density) vary continuously. Discontinuous changes occur only across the model interfaces separating the volumes.



The model is constructed by firstly forming a number of “simple volumes” from the given interfaces and then combining these simple volumes into more complex volumes which represent the physical volumes of the model. It is easy to make changes to the model, by adding volumes or subtracting volumes and perform more composite operations, all by simple use of Boolean expressions. Every time a model has been specified (or changed), the internal logic automatically carries out a check of physical consistency of the 3-D model space (no overlapping volumes, no holes).



By including various types of coordinate transformations, different kinds of complex structures can be handled, such as salt domes and vertical and near vertical faulting.

K. Astebol

Papers

Estimation of multivalued arrivals in 3D models using wavefront construction—Part I1

3-D ray modeling by wavefront construction in open models

Estimation of multivalued arrivals in 3D models using wavefront construction

Part II: Tracing and interpolation1

Computer representation of complex 3‐d geological structures using a new “solid modeling” technique*