Review of Ray Theory Applications in Modelling and Imaging - Norsar

REVIEW OF RAY THEORY APPLICATIONS IN MODELLING AND 

IMAGING OF SEISMIC DATA 

HÅVAR GJØYSTDAL, EINAR IVERSEN, RENAUD LAURAIN, ISABELLE LECOMTE, VETLE VINJE, KETIL ÅSTEBØL * 

ABSTRACT 

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, Červený’s 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 Green’s 

functions for prestack depth migration and hybrid modelling (combined ray and finitedifference 

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. 

1. INTRODUCTION 

Twenty years ago a small group of geophysicists at NORSAR started to develop a new 

software package, with the ultimate goal of simulating seismic waves in 3D geological 

models. Some activity on seismic modelling had also been performed prior to that time − 

some experiments had been done with various 3D model representation techniques, and 

the first 3D kinematic ray paths had already been tediously calculated by the old IBM 360 

and drawn on the shaky drum plotter. But at that time, the amplitudes were absent. 

The new software project was sponsored by the service company Geco (at that time 

‘the Geophysical Company of Norway’), having ambitions of becoming a technology 

leader on seismic acquisition, processing, and interpretation, and clearly seeing the need 

for an advanced tool in 3D modelling. In particular, the growing development of 3D 

seismic interpretation stations throughout the eighties made the way short to the 

construction of 3D models. This potentially short link between interpretation and 

modelling, with the possibility of integrating velocity estimation (e.g. tomography) and 

time-to-depth conversion (map migration) in the process, was one of the main 

motivations. 

From the very beginning it was quite clear that the wave simulation method on which 

the new modelling system was to be built, would have to be ray tracing. With the 

* NORSAR, P.O. Box 51, N-2027 Kjeller, Norway 

Stud. geophys. geod., 46 (2002), 113−164 113 

© 2002 StudiaGeo s.r.o., Prague

H. Gjøystdal et al. 

computer speed in those days, ray techniques were in fact the only practically applicable 

ones in general 3D structures, and, even more important, these techniques possessed 

a number of properties that were perfectly suited for integrating modelling and 

interpretation: the ability of calculating the wave field as labelled ray contributions (or 

events), where event attributes like traveltimes, amplitudes, incident angels, etc., could be 

obtained in an explicit manner. In addition, the attributes could serve as a basis for 

generating ray theoretical seismograms, which e.g. could be used for evaluating and 

testing of 3D processing schemes. The advantage of being able to explicitly extract the 

various wave propagation effects, like geometrical spreading factors, 

reflection/transmission effects at interfaces, phase changes at caustics, splitting in various 

branches, P- and S-wave modes, etc., was very much appreciated. Another important 

characteristic of the ray method is the ability to locate the propagation of energy in the 

model, making the method particularly suited for applications in tomographic inversion. 

Moreover, the intuitive nature of ray tracing to visualize wave propagation would make 

such a modelling system an ideal ‘educational tool’, in order to better understand the 

complex features of seismic reflections in 3D complex media. 

Two basic issues were strongly addressed in the first specification and design phase of 

the 3D seismic modelling system: 

− a general and flexible model representation scheme for the 3D structural (layered) 

model with seismic properties, and 

− an efficient ray tracing algorithm that could be integrated with the model 

representation, calculating both the kinematic and dynamic properties of the wave 

field along various ray paths between given source and receiver positions. 

With respect to the first issue, it very soon became clear that we had to start more or 

less on scratch. No published methods were found that met our needs, even if some ideas 

could be obtained from the relatively new area of 3D computer graphics. 

With respect to the second issue we were far luckier. In our earlier experiments we had 

used theory from the book of Červený et al. (1977), e.g. for coding the 

reflection/transmission coefficients at an interface, and later Červený and Hron (1980) 

turned out to be a major reference. The formulation of their dynamic ray tracer fit 

perfectly into our modelling scheme, and contained all the necessary ingredients we 

searched for: calculation of rays as well as ray attributes like wavefront curvatures and 

amplitudes along the rays, including both the travelling through the continuous properties 

of the layers and the conversions at the model interfaces. The methods were clearly and 

explicitly formulated, so the way from the mathematics to computer code was mostly 

straightforward. In particular the ray-centered coordinate system was very appropriate to 

use and easy to implement. It is true to say that these basic works of Červený (and of 

course also later works) formed one of the key elements for our work with modelling 

software throughout the following twenty years. Even after converting from single twopoint 

ray tracing in the eighties to the wavefront construction technique in the nineties, 

Červený’s formulas still constitute the very ‘heart’ of the system. 

Until some few years ago we had to realize that the use of 3D seismic modelling as an 

industrial tool was rather limited. Throughout the eighties most applications of our 

modelling systems were devoted to R&D topics, such as producing synthetic 3D data for 

114 Stud. geophys. geod., 46 (2002)

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

testing new processing methods, or checking the validity of assumptions and 

approximations adopted in order to use more simplified models in e.g. migration and 

time-to-depth conversion. There seemed to be a relatively high threshold for using such 

‘advanced’ tools on a more routine basis in operational departments in the petroleum 

industry. One of the reasons was definitely that the establishment of 3D ray tracing 

models of real complex structures required considerable manual work, and the two-point 

ray tracing process for real 3D surveys was very computer intensive, both with respect to 

time and memory. 

We clearly realized two basic things: firstly, that the new generation of modelling 

tools had to be more adapted to the structure (topology) of the real interpretation data, to 

facilitate a flexible model building, and secondly, that the ray calculation process had to 

be far more efficient for simulating surveys with thousands of shots and receivers. This 

led to the development of a new model representation, the open model concept, and the 

wavefront construction method throughout the nineties. In the last years we have 

experienced a continuous growth of advanced 3D seismic modelling tools in the seismic 

industry. 

Two rather ‘classical’ applications of the NORSAR ray tracing systems througout 

many years have been simple forward modelling to compute synthetic seismograms, and 

various approaches within tomography and velocity inversion, e.g. as described in 

Gjøystdal and Ursin (1981), Iversen et al. (1994), and Iversen and Gjøystdal (1996). 

Methods for estimating the velocity field in macro models now plays an important role in 

3D prestack depth migration techniques, e.g. see Faye and Jeannot (1986), Iversen et al. 

(2000), Brandsberg-Dahl et al. (2001). 

The purpose of this paper is to address some important and interesting new areas of 

application of a modern 3D seismic modelling system, as encountered during our last 

years’ activity for the petroleum industry. For the sake of completeness, we will first give 

a short descriptive review of the basic methods of model representation and wavefront 

construction to produce standard ray (event) attributes and synthetic seismograms. We 

have then selected a number of examples where ray tracing based modelling has been 

applied in practical studies, including hybrid modelling (combining ray tracing and finitedifference 

schemes), imaging (prestack depth migration), and seismic attribute 

mapping/illumination analysis. Finally, we describe two recently invented ray concepts 

(‘isochron rays’ and ‘velocity rays’), which we believe will have very interesting 

application potentials in the future. 

2. MODELS FOR RAY TRACING 

An essential part of a ray tracing system is the subsurface model in which to propagate 

the rays. We will here discuss three different aspects of the model in ray tracing: 

1. What the rays need to know about the model. 

2. The limitations imposed on the model by the ray tracing method. 

3. Model representation for seismic ray tracing. 

Stud. geophys. geod., 46 (2002) 115


2.1. What the rays need to know 

The task of the model is to serve the rays with some basic information about the 

subsurface. What is requested is well defined and quite simple: At any location in the 

model, the model must provide the ray-tracer with a few parameters concerning the bulk, 

material properties, e.g., the value and spatial derivatives of isotropic P- and S-velocity 

functions, anisotropy parameters, density, and attenuation factors. The tracer must also get 

info about discontinuities in the properties, that is, about the subsurface interfaces. It must 

be informed if there are any potential ray-interface intersections in the vicinity, and, if 

any, the interface normal and curvature must be provided. 

2.2. Ray tracing model limitations. 

The ray tracing theory imposes some limitations on the subsurface model (Červený, 

1985): To ensure valid rays, the model parameters must be smooth and vary slowly. The 

limitations are rather vague, as it is hard to set quantitatively what ‘vary slowly’ means. 

What really counts is the relative model smoothness within the Fresnel volume around 

each ray (Červený and Soares, 1992). It is difficult to use this criterion to prepare a valid 

model, as the validity depends not only on model parameters, but also on the final rays 

between sources and receivers. In practice we stick to a simpler, ad hoc procedure and 

consider the smoothness as a ray-independent model property. We link the smoothness in 

a rough manner to the seismic experiment by smoothing properties and interfaces so they 

vary with longer wavelength than the dominating wavelength in the recorded seismic 

signals (Červený, 2001, p.608). 

2.3. Model representation for seismic ray tracing. 

Different model representations have been developed for seismic ray tracing. The 

simplest one is the 1D model with a vertical sequence of constant material properties 

separated by plane, horizontal layers. A more flexible model is the ‘layer cake’, which is 

a similar sequence, but it varies laterally as the properties and interfaces are represented as 

grids. More comprehensive seismic model representations for ray tracing include 

CAD/CAM-like solid modelling representation (Gjøystdal et al., 1985), and models based 

on surface patches (Pereyra, 1992). Very advanced model builders and ray tracers are 

found in other fields, like computer graphics, CAD/CAM, and the entertainment industry, 

and there are also well-established general purpose model builders in the geoscience 

industry, see for example Mallet et al. (1989). However, as discussed below, seismic ray 

modelling has some quite specific characteristics and sets certain requirements which in 

general are not fully met by models made for other purposes. 

In our present 3D modelling system (Vinje et al., 1999) we have designed and tuned 

the subsurface model representation specifically to seismic ray modelling. It is designed 

for the external model data we might expect, and it takes into account the characteristics 

of ray tracing. An example of a 3D ray tracing model is shown in Figure 1. 



Fig. 1. Example of a 3D ray tracing model; the SEG/EAGE Salt Model (Aminzadeh et al., 1997). 

A typical subsurface data set in seismic modelling is characterized by: 

− A number of interfaces, often of complex nature 

− Material properties vary in a general way inside the layers. 

− Often incomplete data, implying that parts of interfaces and properties are missing. 

Ray tracing requires (see Subsections 2.1 and 2.2): 

− Frequent and fast property evaluation and ray-interface intersection. 

− Continuous and smooth derivatives of properties and interfaces. 

To honour the above points we have made the following model representation: 

In the model there are just two types of geometrical elements: interfaces and 

properties. An interface describes the location of the discontinuity between two different 

layers (volumes, blocks) in the subsurface. A property represents a continuous, bulk, 

material property in a layer, such as isotropic P-velocity. The model is a simple system of 

pointers from interfaces to properties, see Figure 2. 

Each interface is represented as a triangular network with explicit normals and 

curvatures associated with every node. This representation combines the flexibility of 

triangular networks with an approximate solution for the smoothness required in ray 

tracing. Strictly speaking, dynamic ray tracing requires interfaces with continuous and 

smooth 1st and 2nd derivatives. However, it is very difficult to make a surface 

representation that combines these properties with the flexibility to represent the complex 

interfaces prevalent in seismic models. To get around this problem, we have ‘de-coupled’ 

the interface slightly from its derivatives. As a part of the interface construction, the 

normals and curvatures in each node are estimated from the shape of the surrounding 

network and stored in the node. When the normals and curvatures in an arbitrary point are 

requested, they are interpolated from the pre-computed values associated with the 

surrounding nodes, by a procedure similar to the one described in Mallet et al. (1997). 

With this representation normals and curvatures are continuous and smooth, but they are 

not necessarily strictly the ‘true’ ones for the interface. 



Fig. 2. The model structure. From each side of each interface there is a pointer to the bulk 

properties that shall be used for rays traced out from that side of the interface. Pi, Si, and Di are the 

P-wave, S-wave, and density functions for the same material ‘i’. Each interface is labelled by an 

index ‘j’ (e.g., I1, I2,...). 

Fig. 3. Ray tracing in an open model. Notice the hole in the white interface. Ray ‘A’ through 

the hole is rejected. It starts out in properties #3 below the top interface, while at the next rayinterface 

intersection at the bottom interface, it is expected to arrive through properties #2. Ray ‘B’ 

is accepted, as the properties at the departure from an interface matches the properties at the arrival 

point on the next one. 



The properties are much simpler to represent than the interfaces. Each property for 

each layer is a separate, smooth, and slowly varying function, and is represented by a tricubic 

spline. 

In real case modelling projects there may be small or even big gaps in the interpreted 

interfaces where data are missing for some reason. To allow for ray tracing in these 

incomplete data, we have developed a concept where the model is ‘open’, see Åstebøl 

(1994) and Vinje et al. (1999). A consistency check sorts out and rejects rays that have 

passed through the invalid, incomplete parts of the model, see Figure 3. The open model 

concept saves a lot of editing that would otherwise have been required to fill in the 

missing parts. The in-fill also introduces uncertainties in the modelling, as ray tracing then 

is allowed where very little is known about the subsurface. 

3. WAVEFRONT CONSTRUCTION (WFC) 

3.1. Background 

The wavefront construction (WFC) method was introduced for 2D models in a paper 

by Vinje et al. (1993b). During the nineties the method was further developed to 

accommodate 3D models (Vinje et al., 1996a,b; Vinje et al., 1999). Based on the latter 

papers we review here the basic principles and properties of WFC. 

WFC is a ray field technique based on conventional seismic ray theory, as described 

by Červený in a long list of papers, lecture notes, and books (e.g., Červený, 1985, 2001). 

Instead of constructing the ray field ray-by-ray, as it is done in classic initial-value ray 

tracing or two-point ray tracing, WFC is building up the ray field wavefront-by-wavefront. 

The propagation from one wavefront to the next is based on tracing of a large number of 

short ray segments. The basic idea in WFC is to maintain a uniform density of ray 

segments on the wavefront during its propagation. To control the ray density it is now 

common to use criteria based on i) the distance between neighboring rays and ii) the 

separation of wavefront normals related to such rays. Application of these criteria on 

a wavefront will lead to creation of new rays by interpolation. 

The WFC method was developed as a consequence of the problems arising in twopoint 

ray tracing when attempting to find all rays connecting a source point with 

a receiver point. Traditionally this is done using shooting or bending algorithms. In 

a variant of the shooting method described by Červený (1985), a fan of rays is traced from 

the source point, and paraxial extrapolation is used to estimate values at the receiver 

point(s). The main problem with these approaches is the lack of control of divergence 

between the rays in the search fan. Therefore the trade-off between efficiency and 

reliability/robustness may be unfavorable, especially for complicated 3D models. 

During the last ten years WFC has proven to be a very powerful tool in seismic 

modelling and imaging. In Kirchhoff type migration or similar (see Section 5) estimation 

of the Green’s functions by forward modelling is often the bottleneck of the whole 

procedure. Also in velocity estimation and in survey planning forward modelling plays an 

important role. The advantage of the WFC method is that it can do forward modelling in 

complex 3D models in reasonable time, and still be able to find all, or a selection of, 

reflected and transmitted events in the receiver points. 



The original WFC technique was developed for isotropic models. Development to take 

into account anisotropy has been reported recently (Gibson, 2000; Mispel and Williamson, 

2001). 

3.2. Representation of wavefronts 

In the papers by Vinje et al. (1996a,b) it was essential to find a simple and efficient 

numerical representation of the 3D wavefront (WF). In general, the representation of the 

WF in 3D is somewhat more complicated than in 2D, where the rays are situated side by 

side along the 2D WF. The term ‘neighboring’ or ‘adjacent’ ray is more difficult to define 

on a 3D WF where the rays are distributed in two directions. Some kind of network 

connecting the rays on such a 3D WF must be defined. This internal ordering of points 

(i.e. the intersection points between rays and WFs) and their internal connection lines can 

be termed as the WF topology. As demonstrated by Vinje et al. (1996a,b) 

a triangular network has both a simple topology and the ability to adjust to the stretching 

and twisting of the WF during the propagation through the medium. The processes of 

checking, interpolation and estimation of receiver parameters are made fairly simple using 

this topology. 

The main steps in WFC are 

3.3. Propagation of wavefronts 

− Generation of an initial WF, 

− Propagating the WF one time step, 

− Controlling the density of ray segments on the WF, 

− Interpolation to find arrivals in the receivers. 

Fig. 4. The icosahedron (a) describes the basic network of the point source. In (b) four repeated 

interpolations have taken place on the icosahedron. This leads to a polyhedron that consists of 2562 

nodes (rays), 7680 sides and 5120 triangles. 



In order to start WF propagation, an initial (source) WF is necessary. We need 

a triangular grid with sides and triangles as equal as possible, and in this respect, an 

icosahedron is especially well suited. The icosahedron consists of twelve vertices, thirty 

sides and twenty regular triangles. We let twelve rays from the source point pass through 

the vertices of the icosahedron thus creating the desired triangular network. The angle 

between neighboring rays in this network is then approximately 63.43°. In order to get 

a denser sampling of the takeoff directions we perform interpolation of new rays on the 

triangular network (the icosahedron). The interpolation is repeated until the criteria for the 

density of ray segments are fulfilled. In Figure 4 the icosahedron itself and the source WF 

after four interpolations are plotted. 

Assume that a WF represented as a triangular network is present at a time t. For 

propagation in an isotropic medium (Vinje et al., 1996a,b) each node (i.e., the 

intersections between rays and WF) is characterized by a set of parameters like: 

− Position 

− Ray tangent 

− Takeoff direction of the ray at the source 

− Wave mode (e.g. P or S in an isotropic medium) 

− Unit ray normal in the ray-centered coordinate system 

− Dynamic parameters such as the P- and Q-matrix (Červený, 1985) and zero-order ray 

amplitude coefficient(s). 

Given this set of parameters for all nodes at time t, all the rays in the WF are traced 

a small time step ∆t so that a new WF at time t + ∆t is created. A part of the complete 

wave field at three successive time steps is plotted in Figure 5. The WFs for two time 

steps (t and t + ∆t) are kept in memory to facilitate interpolation of new rays and 

estimation of event data in receivers. All the rays in the network are equipped with a userdefined 

ray code which determines the specific sequence of transmissions and/or 

reflections at the interfaces in the model. 

Fig. 5. The WF is propagated through the model by tracing all the rays from the old to the new 

WF. The WFs for two successive time steps (t and t + ∆t) are kept in memory. 



Fig. 6. Simple ray cell with an interior receiver. Seismic data are estimated in the receiver by 

interpolation from the three rays r1 , r2 and r3 . 

When divergence is present in the wave field, parts of the WF will be stretched during 

its propagation through the medium. Both the distance and the angular difference of ray 

tangents between neighboring rays increase, and interpolation is needed in order to 

maintain some pre-defined sampling density of the WF. Several interpolation criteria has 

been proposed since the first publications on WFC (Vinje et al. 1992, 1993a, 1993b). For 

example, Lambaré‚ et al. (1994) introduced interpolation criteria based on the paraxial ray 

approximation (Červený, 1985). Furthermore, Vinje (1997) introduced an interpolation 

criterion based on so-called probe rays that gives control of the relative error of any 

parameter propagated along the WF, but with the cost of increased CPU time. 

In order to find arrivals at the receivers we need a procedure able of transferring the 

seismic parameters from the moving WF into each receiver. The volume between two 

consecutive WFs (i.e., the old and the new WF) is divided into ray cells. These are prism 

shaped bodies bounded by three rays and the triangle that connects them on each of the 

two WFs. The six extreme boundaries of the spatial coordinates (x, y, z) of the ray cell are 

used to construct a box that completely contains the ray cell. Then the subset of the 

receivers located within the box is found. Some of these receivers may also be located 

within the ray cell itself. A ray cell with an interior receiver is shown in Figure 6. The 

seismic parameters for this receiver are found by interpolation from the three rays of the 

ray cell using the ray cell coordinates u, v, and t*, where (u, v) are the so-called 

barycentric coordinates of the footpoint of the receiver at the base of the ray cell, and t* is 

the traveltime from the base of the ray cell to the receiver. 



Fig. 7. An example of WF construction in an open model. A WF (at 0.9 s) is reflected from the 

deepest interface in the seismic model. The interface is plotted in black while the WF is plotted in 

white. The interface contains two undefined areas (holes). Note the hole in the WF were it reaches 

one of the undefined areas of the interface. 

3.4. Reflection and transmission at interfaces 

The open model (see Section 2) is a practical concept for representation of realistic 

geological models. As shown in Figure 7, WF construction can be performed in an open 

model. The triangular network forming the interfaces possesses the same local 

characteristics as the WFs, and this can be utilized in the process of reflecting or 

transmitting a ray at an interface. Each time an intersection between a ray and a triangle of 

an interface is detected, the following parameters are obtained for the ray/interface 

intersection point: 

− Spatial position: Located in the plane between the three vertices of the triangle. 



− Interface normal: Interpolated between the three interface normals in the vertices of 

the triangle. 

− Interface curvature matrix with corresponding local coordinates: Interpolated between 

the three curvature matrices in the vertices of the triangle. 

This is sufficient information to reflect or transmit the ray at the interface, calculating 

both kinematic and dynamic parameters. 

3.5. WFC and multi-arrivals 

We consider a model with a single interface separating two homogeneous isotropic 

half-spaces. In this model WFC is used for computation of reflected contributions 

corresponding to a straight receiver line (see Figure 8a-b). The curvatures of the interface 

give rise to triplication of the departing WFs and thus to multi-arrivals at the receivers 

(more than one event per receiver). Examples of multi-arrivals for the source in Figure 8b 

are plotted in Figure 9. Four event parameters for the receiver line are shown: traveltime, 

takeoff angle from the source, amplitude coefficient and ray direction at the receivers. The 

bow-tie in the lower part of Figure 9a is the traveltime of flank energy which could not be 

present in any 2D modelling. These arrivals are nicely unfolded into an oval curve in the 

takeoff direction plot in Figure 9b. In Figure 10, the error (in milliseconds) of the 

traveltime calculations are plotted. The figure shows the difference between the 

traveltimes computed by the WF construction and the traveltime of rays traced from the 

source point to each receiver position. An error of maximum 0.2 − 0.3 ms, as showed in 

the figure, is sufficiently small for most geophysical applications. Using the traveltime 

and amplitude data from the receivers, a synthetic seismogram is computed (Figure 11). 

A 50 Hz Berlage pulse was used in the generation of the traces. Amplitude variations as 

well as phase shifts may be observed in the seismogram. 

3.6. Computation efficiency − parallel WFC. 

The CPU time of the WF construction method depends on factors such as the 

complexity of the model, the sampling of the wavefield (controlled by the user defined ray 

density and time-step length), total time propagated, and number of receivers. A general 

observation regarding the CPU time is that the efficiency increases considerably with 

number of receivers in the model compared to traditional two-point search strategies. To 

make simulation of large realistic 3D surveys feasible (tens of thousands of shots and 

receivers), the survey may be automatically divided into a number of independent subtasks, 

with a number of shots in each, which can be distributed to a number of computers 

or nodes in a network. Each of these parallel WFC jobs may apply the principle of shot 

similarity, taking advantage of the fact that within a given arrival branch of the wavefronts 

hitting the receivers, the corresponding ray takeoff directions will generally vary little 

from one shot to the next. This makes it possible to use a strategy of sending out 

wavefronts only in preselected directions for most of the shots in the survey. Only for 

a smaller number of ‘master shots’, full wavefronts have to be propagated. This option 

may decrease the CPU time drastically. 



Fig. 8. WFs and re-traced rays to a straight receiver line in a model with a double sinusoidal 

reflector. (a) The source point is located in the left part of the model at position (0.2, 1.25, 0.2) km. 

(b) The source point is at position (1.25, 1.25, 0.2) km. 



Fig. 9. A selection of the parameters recorded in for the shot and receivers in Figure 8b. The 

figure shows traveltime (a) and log10 of the modulus of the amplitude coefficient (c) as a function 

of the distance along the receiver line. In (b) and (d), takeoff angles from the source and ray tangent 

angles at the receivers are shown. 



Fig. 10. Error in traveltime corresponding to the traveltime curve in Figure 9a. The difference 

between the result of WF construction and the direct tracing between the source and the receivers 

has been plotted. 

Fig. 11. Synthetic seismogram corresponding to the event data shown in Figure 9. A 50 Hz 

Berlage pulse was used when the traces were computed. 



4. HYBRID RAY TRACING / FINITE-DIFFERENCE MODELLING 

Beside ray tracing (RT) for 2D or 3D wave propagation modelling of the earth’s 

structures, finite-difference (FD) techniques are widely used. Both RT and FD have 

specific advantages and severe drawbacks. They are fundamentally different but they do 

intend to solve the same problem, i.e., to simulate acoustic/elastic wave propagation in the 

earth. Unfortunately, the trend is often to use only one method and ignore the other, while 

in fact they do complete each other. Each method needs the model to be represented in 

a specific way. However, there are very few seismic models which are optimally suited 

for RT or FD. A possible solution would therefore be to apply RT modelling in some parts 

of the models where it is appropriate, and FD modelling in the rest. We will illustrate such 

a hybrid combination of RT and FD to model the wavefield scattered by a chosen target 

area of the model and originally developed for oil exploration with seismic modelling of 

reservoirs (Lecomte, 1996; Hokstad et al., 1998; Gjøystdal et al., 1998). FD modelling is 

applied at the target itself, which may be too complex to be properly described by RT 

models. The rest of the propagation, prior to and after the FD modelling, is performed by 

RT through the calculation of Green’s functions (GFs). But before illustrating the results 

of this hybrid approach, let us first review the characteristics and limitations of both RT 

and FD modelling and compare them. 

4.1. Why hybrid modelling? 

There are many versions of both RT and FD techniques on the market so we will not 

focus on a specific one in each case but analyze their overall characteristics. The main 

philosophy of a hybrid concept is indeed to take advantage of each class of methods, 

while trying to significantly reduce their inherent drawbacks. 

Ray tracing (RT) modelling 

In ray theory, an asymptotic approximation of the exact solution of the wave equation 

is obtained by searching for a displacement solution in the form of a ray series (Červený et 

al., 1977). The most common approach is to keep only the first-order term after inserting 

this solution in the wave equations, thus applying a high frequency approximation. We 

then end up with two separate equations: the Eikonal equation for the traveltime and the 

first transport equation for the amplitudes. This means in theory that the results are valid 

for wavelengths significantly smaller than any physical scale in the model. In practice RT 

is often more robust than predicted and may work well for structures down to one fifth of 

the wavelength. Typical RT models are described by smooth interfaces and smooth 

velocity fields (see Section 2). Furthermore, RT is event-oriented, implying that the user 

may choose the type of waves to be modelled in the various layers and the type of 

reflection/transmission to occur at interfaces. For this reason RT modelling has become a 

useful tool for analysis of wave propagation and identification of arrivals. 

The major drawback of RT is the high frequency assumption, which may lead to 

incorrect amplitudes for the detailed structures in oil exploration. Diffractions can be 

modelled approximately, but many other wave phenomena are missing in RT modelling, 

such as surface waves, evanescent waves, head waves, and so forth. There are also 

difficulties in the vicinity of caustic points where RT usually gives artificially high 



amplitudes. The efficiency of RT modelling is very model dependent and difficult to 

predict. Moreover, it may be difficult to know in advance what waves to model, except for 

general classes such as primary reflections. 

In prestack depth migration (see Section 5), where GFs are to be calculated on a dense 

grid, classic ray tracing is definitely not efficient enough. A more attractive approach is 

the WFC technique exposed in Section 3. Efficient GFs calculations are indeed a necessity 

for RT-FD hybrid modelling, especially in 3D, even though the hybrid method requires 

a much smaller number of calculation points than prestack depth migration. 

Finite-difference (FD) modelling 

While RT is utilizing an asymptotic solution of the wave equation, FD modelling is 

based on direct discretization of the governing differential equations. To enhance accuracy 

the model parameters, the wavefields, and the stress components are sampled on staggered 

grids (Virieux, 1984; Virieux, 1986). The length of the spatial differential operators may 

vary from one point on each side of the point where the derivatives are to be estimated 

(2nd order scheme) to as many points as one wishes (Holberg, 1987). The differential 

operator in time is usually of 2nd order. The wavefield obtained by FD modelling can be 

stored either as a snapshot, i.e., at a given time iteration for the whole grid, or as 

seismograms at a set of receivers, implying that the wavefield is recorded time step after 

time step at specific locations. 

Finite-difference calculation are constrained by two conditions. The stability condition 

controls that the time sampling ∆t is not too large with respect to the smallest grid 

sampling ∆h following a rule of the form: 

Vmax∆t ≤ α 

∆h 

where Vmax is the largest velocity in the model, usually a P-wave velocity. The other 

constraint is the dispersion condition which assure that the smallest wavelength is not 

aliased due to a too coarse grid, i.e., a minimum number of grid points per wavelength is 

defined: 

ν 

Vmin 

≤ Nmin 

max∆h 

Vmin is the smallest velocity in the model, usually a S-wave velocity if the calculation is 

elastic, and νmax is the highest frequency of the signal. Both α and Nmin depend on the 

finite-difference scheme and more especially on the length of the spatial differential 

operators. 

FD modelling is easy to use and always provide a complete wavefield, including all 

types of waves. The major drawback is the cost of such modelling which requires both 

large computer memory and high CPU time, especially in 3D. In the latter case, routine 

modelling of complete surveys are absolutely out of question. Besides the cost of FD 

modelling, one possible problem is the grid representation of the model which may not be 

able to represent properly a detailed geological structure. This is especially true when long 

spatial differential operators are used to increase the efficiency of the calculation by 

Stud. geophys. geod., 46 (2002) 129 

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(2)


choosing a coarser grid sampling. Actual interfaces of the model will take the form as 

‘step-stairs’, and this may induce annoying artificial diffractions. A classic example is 

marine seismics with a gently varying sea floor. The artificial diffractions generated at 

that level may be very strong compared to the deeper scattered wavefields. There are 

some finite-difference schemes taking into account a given interface to correct this effect 

but this significantly complicates the code and increases the CPU time. The borders of the 

grid are also troublesome due to generation of artificial reflections. They are generally 

conditioned to be absorbing by using an attenuation zone or applying one-way paraxial 

wave equations (Clayton and Enquist, 1977), or a combination of these two techniques. 

But the attenuation zones significantly increase the size of the grid, which is especially 

prejudicial in 3D. The one-way paraxial wave equations used to cancel reflections on the 

sides require less memory but the absorption depends on the incidence angle. Even with 

good absorbing boundaries, artificial reflections generated by the direct wave may be of 

the same order of magnitude as the scattered wavefields of interest, especially for the deep 

structures. 

RT/FD comparisons 

Table 1 summarizes for each modelling approach its characteristics, advantages, and 

drawbacks. It is easy to see that the RT and FD approach complete each other very well, 

i.e., a drawback of FD is often an advantage of RT and vice versa. For instance, the wave 

selectivity of RT is not available with FD, which is acting as a ‘black-box’. At the same 

time, this wave selectivity in RT obliges the user to define a priori all possible significant 

waves if wishing a complete calculation, while FD does that automatically. Figure 12 

shows the result of modelling performed with both methods for the same model, survey 

and source pulse. The model is not complicated in terms of the number of layers but does 

show strong faulting. In the RT modelling we requested the direct transmitted wave and 

all primary P-wave reflections, as well as diffractions corresponding to the direct wave. 

The diffractions were obtained using the theory of Klem-Musatov (1994). The FD 

modelling was run with absorbing conditions at the surface in order to avoid multiples 

(the only possibility to reject arrivals in FD). To illustrate the wave propagation within 

RT, direct transmitted wavefronts calculated by the WFC method and raypaths from twopoint 

RT are superimposed on the model (Figure 12a, top). The ‘holes’ appearing on the 

wavefronts when crossing the faulted area due to diffractions (not shown here) or overcritical 

incidences (head waves are missing). A snapshot of the FD wavefield at 1.5 s of 

propagation clearly shows a very complicated pattern where the different arrivals are 

difficult to analyze (Figure 12b, top). In the FD snapshot we can identify head waves and 

diffractions in addition to reflections and pure transmission. The RT seismograms 

(Figure 12a, bottom) are not complete but give rather good results for the direct 

transmitted waves for most of the receivers. Later arrivals (primary reflections) are more 

discontinuous due to the complexity of the structure and diffractions would be necessary 

to avoid sharp cutoffs of the reflected phases. Other phases may eventually give 

significant arrivals, but it is difficult to define them a priori. On the other hand, the FD 

seismograms appear to be more complete and continuous, but wave identification is very 

difficult. Note the rather strong arrivals at about 2 s for the deepest receivers (Figure 12b, 

bottom), or equivalently, consider the phase identified with white arrows on the snapshot 



(Figure 12b, top). These arrivals correspond to an artificial box-side reflection not totally 

suppressed by the chosen one-way wave equation absorbing conditions (Reynolds, 1978). 

Without a ray analysis, it is very difficult to tell wether or not such arrivals are real. The 

CPU cost of RT here was very small (below 1 min.) compared to FD (about one hour), 

though adding diffractions in RT may significantly increase this cost. It is anyway a good 

habit to first try RT to better constrain wave propagation in a model, then apply FD if 

CPU and memory resources are available, and finally re-use RT to analyze the FD results. 

Fig. 12. RT/FD modelling comparisons in a complex model and with VSP acquisition. (a) RT 

modelling with direct transmitted wavefronts and a few selected raypaths for both direct 

transmission and primary reflections (top) and seismograms (bottom). (b) FD modelling with 

snapshot of the wavefield at 1.5 s (top) and seismograms (bottom). One artificial left box-side 

reflection is identified on both the snapshot (white arrows) and the seismograms (dashed line). 


Table 1. 

Type of 

modelling 

Ray 

tracing 

Finitedifference 


Model Arrivals Efficiency Accuracy 

- interfaces 

separating layers 

BUT 

- smoothness is 

required 

- how smooth? 

- parameter grids 

BUT 

- ‘step-stairs’ 

troubles 

- finest detail = 

sampling 

- full selectivity 

BUT 

- what waves to 

select? 

- no head waves, no 

surface waves, 

approximate 

diffractions 

- complete wavefield 

BUT 

- what is what? 

- reflections from the 

borders. 

- fast 

BUT 

- model dependent 

- decreases with 

increasing 

complexity, 

no. of layers, arrivals 

- model independent 

BUT 

- slow: model size, 

model sampling and 

frequency dependent 

- accurate for smooth 

models 

BUT 

- high-frequency 

approximation 

- caustic problems 

- can be controlled 

BUT 

- stability and dispersion 

need to be verified, 

- constrained sampling 

in time and space 

4.2. Where and how perform a hybrid RT-FD modelling? 

In the following we consider the idea of applying RT and FD successively in a hybrid 

combination. Originally developed for typical oil/gas reservoirs found e.g. on the 

Norwegian continental shelf (marine seismics), the hybrid technique is suited for 

structures with a rather gentle and smooth overburden over a more complex and possibly 

faulted area including a reservoir. The overburden will then be the RT modelling domain 

and the target (i.e., the reservoir) will be the FD modelling domain. There is however no 

theoretical limitation of the hybrid concept to that type of structures only. Alternatively, 

we could imagine other type of structures where the complexity is close to the surface, 

while the target itself could be properly described with RT models. This may be more the 

case in land seismics, for instance in Canada. 

As originally proposed by Lecomte (1996) in acoustics, and later further developed in 

elastics (Hokstad et al., 1998), the hybrid modelling scheme proceeds in three steps as 

illustrated in Figure 13, assuming that a target has been selected. RT and FD is then totally 

de-coupled, i.e., the processes are applied successively with communication via well 

defined data sets (e.g. disk files). 

First step: The incident wavefield − RT 

The WFC technique of RT is used to efficiently compute the GFs connecting any shot 

of the survey to a coupling line (2D) or surface (3D). The overburden should fit for RT 

modelling, i.e., with proper smoothness of interfaces and property fields, and with no 

major faulting which could generate holes in the wavefield. At that stage, we benefit from 

the wave selectivity of RT, i.e., the user chooses what arrivals to send to the target. 

Furthermore, we will not suffer from any artificial noise in FD such as diffractions at the 

sea floor or box-side reflections. Note that in most cases the user selects the direct 



transmitted P-wave as the incident wave and direct transmission is less sensitive to small 

scale fluctuations of the interfaces or parameter fields than reflections. Once the GFs are 

calculated, any wavelet can be convolved to generate a wavefield incident to the coupling 

line (Figure 13, step 1). This wavefield can be re-used for any model of the target as long 

as the velocity structure is not changed above and at the coupling line. This increases 

efficiency for repeated modelling (Gjøystdal et al., 1998). 

Second step: The scattered wavefield − FD 

The incident wavefield can now propagate through the target via a local FD modelling 

(Figure 13, step 2). The goal here is to record the scattered energy coming back from the 

structures in the target in order to send further this energy towards the receivers in the 

third step. A special treatment must be applied to prevent the incident wavefield to 

propagate upwards, acting as an artificial reflection from the coupling line. Complete 

boundary conditions may be used to constrain the propagation downwards (Hokstad et al., 

1998). Alternatively, the scattered field from a background target model, with no 

scattering structures below the coupling line, may be subtracted. The gain in cost, with 

respect to a classic global FD modelling, is directly proportional to the reduction in size of 

the FD-grid times the reduction in number of time iterations as the interesting time 

window is shorter. We will here benefit from FD modelling and get a complete scattered 

wavefield, which was however generated from the incident field explicitly chosen at the 

first stage. We thus gain partly the flexibility of RT in choosing waves: we can for 

instance study the difference in illumination of a target by primary reflections and sea 

floor generated multiples. The local FD modelling can be easily applied in parallel for 

different target models, and distributed to a network of computers without requiring huge 

capacity in CPU and memory. 

Fig. 13. Scheme of the three-step process of the hybrid RT-FD modelling. (1) Incident field 

calculation using RT calculated GFs. (2) Scattered field calculation in the target with a local FD. 

(3) Recorded field calculation at the receivers using Kirchhoff integral and RT calculated GFs. 



Fig. 14. Example of the three-step process (acoustic). (1) Incident field to the local FD box with 

coupling at the top. (2) Scattered field coming out from the local FD. (3) Recorded field after 

wavefield extrapolation of the scattered wavefield using Kirchhoff integral. 

Third step: The recorded wavefield − RT 

In the last step, the scattered wavefield emerging from the local FD-grid is propagated 

further towards the receivers by using the Kirchhoff integral and GFs connecting the 

coupling line to these receivers (Figure 13, step 3). We then again benefit from RT 

selectivity by choosing the paths to follow, i.e., direct transmitted P or direct transmitted 

S, and so on. 

To summarize the hybrid RT-FD modelling, Figure 14 shows a model inspired from 

existing structures in the North Sea, with the wavefield attached to each step of the 

process, the coupling line being at the top of the FD-grid in that example (not required in 

general). In order to check the validity of the results, we can compare seismograms 

obtained with RT (Figure 15a), FD (Figure 15b), and the hybrid modelling (Figure 15c). 

The latter gave a more complete result than RT and a better result than the pure FD 

(notice the box-side reflections appearing in Figure 15b), but at a much cheaper price. The 

cost of hybrid modelling is about 20% of the global FD at the most, although the 

parameters were not optimally chosen because we kept the same grid sampling for the 

pure (global) FD and the local FD (inherent in the hybrid approach). In practice, the 

optimum grid sampling depend on the smallest velocity, which is often located close to 



Fig. 15. Comparison of RT, FD and hybrid RT-FD results. (a) Classic two-point RT modelling (no 

diffractions). (b) Classic acoustic FD modelling of the whole model. (c) Hybrid RT-FD modelling 

of the target. 

the surface. Thereby, the grid sampling can be coarser for a deep local FD modelling than 

for a global FD modelling. Alternatively, one may wish to use the hybrid RT-FD 

modelling for detailed models of the target, i.e., with sampling down to just a few meters, 

especially vertically. This may be necessary for a proper representation of the model as 

deduced from well information. For global FD the cost is then very high because this 

sampling is to be used all over the grid. 

Studies carried out the last years show that, even in 2D, the hybrid RT-FD modelling 

concept may be very powerful for detailed reservoir modelling, especially for 4D studies 

with repeated modelling where only the target is to be modified (only the local FD is 

affected). 

5. PRESTACK DEPTH MIGRATION AND RAY THEORY 

Integral approaches using pre-calculated Green’s functions (GFs) are popular 

techniques in seismic imaging since the eighties (Schneider, 1978; Beylkin, 1985; 

Bleistein, 1987; Miller et al., 1987). Though significant differences exist in the way the 

problem is solved, all integral approaches are usually regrouped under the general term of 

Kirchhoff migration (Claerbout, 1985). Some do rely on the Kirchhoff integral, which is 

used for backward wavefield extrapolation from the receiver line/surface. This 



extrapolated wavefield is then correlated to the propagated source wavefield to extract the 

reflectivity at a point (Schneider, 1978). Note that the ‘true’ Kirchhoff integral, strictly 

speaking, applies only in the constant-velocity case (Claerbout, 1985). Moreover, this 

integral is in practice not used directly as it requires knowledge of both pressure and 

particle velocity data on a closed surface (Berkhout, 1980). But, assuming a horizontal and 

flat recording line/surface, Rayleigh integrals are derived from the Kirchhoff integral and 

the integration is easier, with often further simplifications where near-field and secondorder 

terms are neglected. The other type of methods (Beylkin, 1985; Bleistein, 1987; 

Miller et al., 1987), based on inversion theory and Born approximation, takes also the 

form of ‘a Kirchhoff migration of frequency filtered and spatially weighted traces, with an 

integral kernel dictated by inversion theory’ (Bleistein, 1987). These techniques are 

sometimes called Born migration, though again usually considered as belonging to the 

Kirchhoff migration family. In recent years, the term of diffraction-stack (DS) migration 

has also emerged in the Kirchhoff migration family, attention being focused on preserving 

the amplitudes during the migration by properly removing geometrical spreading, 

transmission loss and caustics effects (Hubral et al., 1991; Schleicher et al., 1993; 

Hanitzsch et al., 1994). At last, another integral approach, called generalized diffraction 

tomography (GDT) was developed under the Born or Rytov approximations (Devaney 

and Zhang, 1991; Gelius et al., 1991), for heterogeneous background in opposition to the 

classic diffraction tomography techniques (Devaney, 1982; Wu and Toksöz, 1987; Lo and 

Toksöz, 1988), and was originally used for transmission-mode geometry in cross-well 

tomography. GDT is a less well-known technique in seismic imaging but Hamran and 

Lecomte (1993) developed a local plane-wavenumber approach of GDT and showed its 

applications in reflection-mode geometry (Lecomte, 1999; Lecomte et al., 2001). 

All Kirchhoff migration techniques, either ‘true’ Kirchhoff like diffraction-stack, or 

Born-type, like GDT, require pre-calculated GFs as provided by ray techniques. The 

traveltime is the major parameter necessary for migration. For amplitude-preserving 

migration, amplitudes are also necessary though very often, as a first choice, the data are 

routinely corrected for a simple spherical divergence in order to get similar strength for all 

reflectors. The Kirchhoff techniques may often need also to use ray direction information. 

This is the case for the obliquity or directivity factor, i.e., the cosine of the angle between 

the direction of propagation at the receiver and the vertical axis (horizontal recording 

lines). Ray directions are also necessary in the Beylkin determinant, i.e., the Jacobian of 

the transformation from image to surface coordinates, where the integration takes place in 

Born-type approaches (Zhang et al., 2000). Even after taking into account all the 

weighting factors and ‘true-amplitude’ (TA) corrections, we may not retrieve directly 

from the migrated section the true reflectivity (so the term TA migration is misleading). 

Other effects remain, like resolution (Lecomte and Gelius, 1998; Gelius and Lecomte, 

2000) and illumination (Rosland and Drivenes, 2000), without speaking of possible 

coherent noise interfering with the structure and events not corresponding to the 

calculated GFs. As far as illumination is concerned, a technique was proposed by Vinje 

(2000), where ray tracing is used to correct varying illumination of a target reflector after 

PSDM. Ray tracing modelling can also be used in a survey planning stage, in order to 

control illumination before any data acquisition and PSDM. So, prior, during and after 

migration, ray tracing is a powerful technique which often helps greatly the user to 



understand what happens in seismic imaging. In the following we shall illustrate the role 

of various ray tracing parameters (traveltime, ray direction, amplitude) in PSDM. 

5.1. Traveltimes and imaging 

Figure 16a shows a simple model we will use to illustrate the role of traveltimes in 

PSDM. This model includes a salt body intruding a few layers of constant velocity and 

with a water layer at the top. The survey is a marine one with a boat shooting along 

a 2 km line, every 50 m, and with a 1 km streamer (25 m sampling). The target chosen for 

PSDM studies is superimposed. For the GF calculation, a model was created which does 

not contain the salt body but is an extrapolation of the actual model from the left to the 

right side as we will only focus on the left side of the salt (Figure 16b). Primary P-wave 

reflections from all interfaces were generated by dynamic ray tracing, and synthetic 

seismograms were calculated by convolving RT results with a 20 Hz zero-phase Ricker 

wavelet (Figure 16c). To emphasize the effect of traveltime in PSDM, we did not apply 

neither the geometrical spreading factor nor the reflection/transmission effects. For the 

largest offset of the shot recording of Figure 16c, 5 arrivals are identified by their raypaths 

as given in Figure 16d. Around ‘arrival 5’ in Figure 16c, there are in fact other arrivals 

corresponding to a reflection at the left flank of the salt, illuminated by the primary 

P-wave reflection at the reflector at about 2 km in depth (arrival 3 is part of this primary 

reflection). If we now consider only one trace, i.e., the 1 km offset of the shot of 

Figure 16c, traveltime information of GFs will be used to map back to depth the energy of 

the seismograms along isochrons (Figure 16e). As will be discussed further in Section 7, 

an isochron is a surface of constant two-way traveltime corresponding to a single shot and 

a single receiver. Applying now DS to the chosen trace, we get the PSDM section of 

Figure 16f where we easily identified our 5 arrivals of Figure 16c and 16d. Their 

respective energy has been mapped along the isochrons of Figure 16e, as expected, and 

with the same amplitude all along each isochron in our case. Referring to Figure 16d, we 

see however that only arrival 2 corresponds to a reflection in our target. Reflections 1,3,4 

and 5 are all associated with part of reflectors outside our target but the corresponding 

isochrons cross the latter because one trace is not a sufficient information to locate energy. 

Basically, a PSDM section is formed by superposition of such elementary ‘one-trace’ 

images, the idea being that, all together, constructive interference will occur at the actual 

loci of reflections, while the useless part of the isochrons will be cancelled by destructive 

interference. 

Figure 17 shows what happens when several traces are migrated. As very often done, 

data can be migrated in constant-offset sections, like in Figure 17a (zero-offset) and 17b 

(1 km offset). The three reflectors of the target are coming forward, though we still see 

some ‘background pattern’ similar to the isochrons of Figure 16f in both cases. This is 

often called aliasing noise, which is due to a too coarse sampling between the shots: the 

interference is not perfect between the isochrons and they do not cancel each other where 

they should. By migrating all shots and offsets together, the PSDM section gets much 

clearer (Figure 17c), though we can still see what remains of an isochron corresponding to 

the salt reflection. 



Fig. 16. Salt model. (a) P-velocity from 1.5 km/s (top) to 4.0 km/s (salt) and target location. 

(b) Model used for Green’s function calculation. (c) Shot recording at 0 km with offsets from 0 to 

1 km every 25 m (right side). (d) Primary reflections at 1 km offset with identification of 5 arrivals 

(see also c). (e) Scattering traveltime isochrons for 1 km offset. (f) PSDM section for 1 km offset 

with identification of the same arrivals as in (c) and (d). 



Fig. 17. PSDM sections for a survey of 41 shots from 0 to 2 km every 50 m, and same offset range 

as in Figure 16. (a) zero-offset section. (b) 1 km offset. (c) All shots and offset. 

In Kirchhoff migration techniques, the choice of the traveltimes to use is often a critical 

issue, especially because one arrival per grid point, and only one, is the preferred solution 

for the sake of efficiency. So, should we use first-arrival (but may be associated with very 

low energy), highest-amplitude arrival, shortest-path arrival? There is no absolute answer 

to these questions because, ideally, we should use all arrivals to get closer to 

the − smooth − result of wave equation migration, but again at the cost of efficiency. That 

is why some people claim that we will soon be ‘beyond’ Kirchhoff imaging, arguing that 

the computers are developing so fast that wave-equation migration will take over. But 

Kirchhoff migration will get faster as well, so may be we can use all arrivals then? 

Gaussian-beam techniques in RT may also be a better solution than the ‘single-raypath’ 

solutions provided by most GFs calculators, i.e., with events taking into account the 

surrounding structures in ‘energy tubes’ where the Fresnel zone is considered. This may 

induce a smoothing of the GFs (especially their amplitudes) which may greatly improve 

the quality of PSDM sections, giving smoother images, like in wave-equation migration? 

This is definitely an interesting topic of research, which is indeed a fundamental question 

for RT techniques: how does the actual propagation of the waves compare to the ray 

approach? RT techniques will however always be much more informative about what 

happens when performing migration than FD techniques used for wave-equation 

migration, which act as black-boxes. As in the previous section we find here indeed the 

same arguments both for and against each technique. One should not systematically reject 

one method in favor of the other one, since they may indeed complete each other. 

Figure 18 is an example illustrating the flexibility of RT techniques in case of complicated 

illumination of a structure. Let us consider the same shot gather as in Figure 16c and 

migrate it with direct transmitted waves both from shots and towards receivers 

(Figure 18a). In agreement with the raypaths showed above, we retrieve only a small part 

of the salt in its upper concave part. Note that the small illuminated zone predicted by rays 

is in fact larger due to Fresnel zone effects and poor vertical resolution. If we now 

consider GFs combining the primary reflections at the reflector at about 2 km in depth 



Fig. 18. Salt flank imaging with shot at 0 km. (a) Rays attached to primary reflections from salt 

(top) and PSDM section using only direct transmission GFs both from shot and towards receivers 

(bottom). (b) Salt flank illuminated from a primary reflection (top), which is used for GFs 

calculation from the shot to get the PSDM section (bottom) while direct transmission is still used 

towards the receivers. 

from the shots and direct transmission towards the receivers, the new PSDM section is 

totally different, showing now the nearly vertical left flank of the salt. Is there any waveequation 

migration technique able to take into account such vertical structures? This is 

indeed the weakness of these approaches, so far, relying onto paraxial approximations of 

the wave-equation and limited dip ranges. 

5.2. Ray directions and imaging 

While traveltimes play a very important role in migration, by locating properly in 

depth the recorded seismic energy, other RT parameters are also significant and very 

informative. We chose here to illustrate the concept of the scattering wavenumber vector 

K defined by: 

K = kr −ks 

140 Stud. geophys. geod., 46 (2002) 

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Fig. 19. Scattering wavenumber K. (a) Definition. (b) K is perpendicular to a reflector if reflection 

is attached to it. These two angles are equal if P-wave to P-Wave, respectively S-wave to S-wave, 

scattering, but different if P- to S-wave and vice versa. 

where ks and kr are the two local wavenumbers given in Figure 19a, i.e., the wavenumber 

attached to the incident field at the considered image point, respectively the wavenumber 

attached to the scattered field at the same point. This vector, or more precisely its 

opposite, was already part of the early Born approaches in Kirchhoff migration (Beylkin, 

1985). In GDT approaches (Hamran and Lecomte, 1993), K is a key parameter which 

completely describes the illumination and the resolution at each image point. Seismic 

imaging can indeed be assimilated to a generalized Fourier transform (FT) and the local 

coverage with respect to length and orientation of K vectors indicates how much 

information we have about the local scattering properties of the structure. The more K 

coverage we have, the better is the possibility for illumination and sharp resolution 

(Lecomte and Gelius, 1998). Furthermore, while most classic integral approaches apply 

a coordinate transformation from the scattering wavenumber back to the shot and receiver 

lines, a much simpler approach is to stay in a local coordinate system (Figure 19b). This 

system is simply polar in 2D and spherical in 3D, and this is very similar to what is done 

in classic spotlight-mode synthetic aperture radar (SAR) imaging (Lecomte et al., 2001). 

In a local 2D coordinate system, K is defined by an orientation angle, and a length 

function of the frequency and the angle θ (Figure 19b). If there was a reflection attached 

to the wavenumber couple (ks, kr), K is perpendicular to that reflector. 

To illustrate the importance of the scattering wavenumber vector, determined by RT 

from ray directions at the image point, a realistic model of the Gullfaks oil field on the 

Norwegian continental shelf has been used (Figure 20). Synthetic data corresponding to 

the reservoir target indicated in Figure 20a, with an oil/water contact, were generated by 

the hybrid RT/FD modelling method exposed earlier. Three data set were used to generate 

PSDM sections: zero-offset (Figure 20b), far-offset (Figure 20c) and all data (Figure 20d). 

On the right side of the constant-offset sections, the local scattering wavenumber coverage 

at two points of the target are given. In each case, the white arrow indicates the mean 

scattering wavenumber, i.e., the reflector normal which is best illuminated, while the grey 

arrow is the perpendicular to the actual local reflector. In the zero-offset case, reflector 1 

is rather poorly illuminated (grey arrow near the border of the K coverage and far from 



Fig. 20. Illumination and resolution. (a) Model of the Gullfaks field on the Norwegian continental 

shelf with survey (161 shots every 25 m, 100 offsets from 0. to 2475 m) and PSDM target. (b) Zerooffset 

PSDM section with superimposed Point-Spread functions (PSFs, 100×100 m 2 ) at some 

locations (left). Scattering wavenumber coverage at two reflectors (right). (c) Far-offset PSDM 

section. Same as for (b). (d) All shots and offsets section with superimposed PSFs. Courtesy of 

Statoil. 

the white arrow) while reflector 2 is clearly better illuminated. This is confirmed by the 

PSDM section which shows a strong reflector 2 and weak reflector 1. This effect is not 

only due to the impedance contrast and does really reflect the differences in illumination, 

as confirmed by the far-offset section which shows now the opposite, i.e., a strong 

reflector 1 and a non-existing reflector 2. The latter result is again in agreement with the 

illumination analysis provided by the local K coverages. Note a little ‘aliasing-noise’ on 

the zero-offset section due to the same effect mentioned in Figure 17a. 

The local K coverage does not only give information about illumination but also about 

the expected resolution of the inversion process which is seismic imaging. The FT of the 



K coverage is indeed giving us local 2D resolution functions and a few of them are 

superimposed on the PSDM sections of Figure 20. These resolution functions are more 

precisely here point-spread functions (PSFs), corresponding to the FT of the whole K 

coverage and in agreement with a Born approach where the scattering structures are seen 

as (weak) diffractors, radiating energy in all directions. These PSFs give a good idea of 

the resolution across the reflectors and the zero-offset PSDM section is ‘sharper’ than the 

far-offset one because the K vectors are longer at zero-offset (θ is close to zero in 

Figure 19b, see K coverage in Figure 20b and 20c). The PSFs give however a poorer 

representation of the resolution tangent to the reflector as they do not take into account 

that the reflected energy is preferentially radiated along K perpendicular to the reflector 

(Snell’s law). To better illustrate this, synthetic PSDM sections generated by convolving 

the reflectivity of the target (Figure 21a) and either PSFs (Figure 21b, left) or a dipfiltered 

version of PSFs (Figure 21b, right), called reflector-spread functions (RSFs, 

Gelius and Lecomte, 2000), can be compared with the corresponding far-offset section 

(Figure 21c). The RSFs do a much better job, smoothing the reflectivity function 

‘laterally’ as we expect the seismic wave to do due to the Fresnel zone. Note that we did 

not take into account here the illumination effect, i.e., we should have weighted each 

image point by an illumination factor to really get a synthetic PSDM section closer to the 

actual one. This synthetic image does not reproduce the possible aliasing noise. However, 

such a simple synthetic PSDM image, using only ray directions from the GFs, gives 

a good indication of the expected resolution of a target and could help an interpreter to 

decide wether or not picked events on PSDM sections are realistic. 

Fig. 21. Simple synthetic imaging. (a) Normal-incidence reflectivity of the target. (b) Synthetic 

PSDM images obtained by convolving reflectivity and resolution functions with, PSFs (left) and 

Reflector-Spread functions (RSFs, right) for the far offset. (c) Far-offset PSDM section for 

comparison (same as in Figure 20c). 



5.3. Amplitudes and imaging 

As already mentioned, DS techniques are called true-amplitude preserving migration 

because the geometrical spreading, transmission losses and caustics effects between 

shot/receiver and the considered image point are taken into account. Still, efforts should 

be focused onto getting smoother amplitudes with RT techniques, where the Fresnel zone 

is considered. This would certainly yield smoother PSDM sections, like in wave-equation 

migration where the smoothing is obtained directly from wave propagation of bandlimited 

signals. But even if we get better amplitudes for such corrections, we saw in the 

previous examples two other effects which are due to the migration technique itself, i.e., 

varying illumination and limited resolution. Figure 17a-c clearly shows a stronger salt 

reflector because its shape, with rather strong local and concave curvature, ‘traps’ 

reflections on a more concentrated area than for flat reflectors (see also Figure 18a, top). 

We remind that in this example, the data did not contain any amplitude effect, including 

the reflection coefficient at the illuminated reflector, so this amplitude variation is due to 

illumination effects. The ‘aliasing-noise’ of constant-offset sections (Figure 17a-c) will 

also degrade the amplitudes along a reflector. Note that all these effects are also occurring 

in wave-equation migration! But, with RT approaches, there is some hope to correct for 

that, either following the approach of Vinje (2000), which simulates DS and includes the 

illumination corrections in the migration, or by working directly with the seismic 

recording, i.e., by analyzing amplitudes individually without suffering from the 

summation process performed by migration. PSDM is then more a technique which helps 

to complete the velocity model by including reflectors to the smooth velocity field used 

for the GFs calculation (we need to get the best structural image). Then RT can be used 

to model amplitudes of single events and compare them to the actual measured amplitude 

in order to extract angle-dependent reflectivity. It is also important for such reflectivity 

analyses to be sure to use the proper data, i.e., data associated to the same scattering 

angle θ. There again, the local ray directions will help to sort the data for AVA studies. In 

the example of Figure 20, the zero-offset section was really associated to values of θ very 

close to zero, while the far-offset section was associated to θ values varying from 24° to 

44°! At last, we saw that a PSDM section is a filtered version of the reflectivity by 

resolution functions (PSFs or RSFs, Figure 21), also determined only from local ray 

directions, which are therefore a key information for analyzing migration (as it is in any 

inversion process, indeed!). When considering the RSFs of Figure 21, we have nearly 

lateral resolution of about 100 m (the nearly vertical one is better). This means that the 

migrated value at one point results from constructive interference of reflected energy on 

a 100 m wide zone along the reflector. How can we retrieve ‘true’ amplitudes from PSDM 

sections without taking all these effects (illumination, resolution, coherent-noise 

interference) into account? Even nice, smooth PSDM sections, provided by wave-equation 

migration, will not give true amplitudes because they suffer from the same effects. But RT 

analyses may help to isolate the different effects and get closer to the true reflectivity. 



Fig. 22. (a) Illustration of a 3D survey from an attribute mapping case. Shot lines and streamer 

configuration are indicated. Only the target reflector is shown, although the overburden consists of 

a number of layers. (b) Amplitude density map for the target reflector in (a). Depth contours are 

indicated by dashed lines. 



6. SEISMIC ATTRIBUTE MAPPING BY COMPREHENSIVE 3D MODELLING 

6.1. Mapping of ray related seismic attributes 

A goal of a seismic survey is to achieve the best possible illumination of the 

subsurface geologic structures. Until the last few years, conventional seismic survey 

design has relied mostly on a simplified process, assuming that uniform midpoint 

coverage will lead to uniform illumination of the subsurface. Such simple survey designs 

are effective in areas with flat or gently dipping layering. But in areas of complex 

geometry (such as salt structures) or rough topography on the target reflector, the 

wavefield becomes more distorted, resulting in non-regular illumination patterns on the 

reflector, even if the midpoint coverage is regular (Bear et al., 1999; Sassolas et al., 1999; 

Pereyra et al., 1999; Rosland and Drivenes, 2000). The computation and visualization of 

maps showing different seismic attributes computed by ray tracing in true depth at 

selected target horizons can help planning a survey, interpreting data after e.g. prestack 

depth migration. Typical attributes to be mapped are reflection point density and 

integrated amplitude (also called amplitude density), but any other parameter related to 

the ray paths can be mapped, such as reflection angle or phase shifts in the seismic pulse. 

Figure 22 shows an example from the simulation of a 3D survey in an area where 

a model was constructed from seismic interpretation maps (Rosland and Drivenes, 2000). 

A marine survey of 12.000 shots was simulated, having 10 streamers with 1200 receivers. 

The shot lines are shown together with the target reflector. The overburden (not shown) 

consisted of 5-6 interfaces with constant or lateral velocity variation in each layer. The 

calculated amplitude density map of the primary reflections shows a large variation in 

amplitude density, which in this case turns out to be closely related to the topography of 

the target. 

6.2. Synthetic data from 2D model 

In order to study the relationship between the illumination and amplitude maps 

calculated from ray tracing and the amplitudes appearing in prestack depth migrated 

(PSDM) data, we carried out a relatively simple 2D model experiment (see Laurain and 

Vinje, 2001). The model was designed to combine the effects of both a syncline with 

a steep flank and a lens (e.g. a salt body). A full finite-difference data set was generated 

and migrated using a PSDM algorithm. Then, the correspondence between PSDM root 

mean square amplitude (RMS amplitude) profiles along the target horizon and hit density 

and amplitude illumination maps were studied. Figure 23a displays the model we used for 

generating a synthetic marine data set with classic 2D finite-difference (FD) code. The 

upper layer represents water (velocity 1.5 km/s). The sediment layer, delimited by the sea 

bottom and the target horizon, is affected by a vertical gradient function (from 2.0 to 

3.0 km/s at the deepest point). In the sediment layer, a salt lens with constant velocity of 

4.0 km/s has been introduced to study the effects of shadow zones induced on the target 

horizon. The data acquisition scheme was designed as a marine survey with a nominal 

fold of 10 in bin cells of 5 m size. Shots were located on the surface, every 50 m between 

3.0 and 14.5 km. For each shot, 100 receivers with offsets from 10 m to 1000 m and 

a 10.0 m interval were used (Figure 23a). An example of constant-offset FD section is 

shown in Figure 24. 



Fig. 23. (a) Model used for generation of synthetic data. Shot points are located between 3 km and 

14.5 km along the surface. The target horizon is indicated. (b) Velocity model used for performing 

the depth migration. Target areas are limited by the black boxes. 



Fig. 24. Constant-offset section for offset 10.0 m. 

6.3. Prestack depth migration 

The finite-difference data were migrated for two local targets using a un-weighted 

prestack depth migration algorithm. For each point of the defined target, the migration 

process performs a sum of amplitudes with wide aperture and unit weights along 

diffraction-traveltime curves. The traveltime maps required for this process were 

generated with a first arrival Eikonal solver in a background velocity model where the 

target interface has been removed (Figure 23b). The first target area (denoted as target 1) 

covers the syncline. The second target area (denoted as target 2) has been chosen beneath 

the salt lens in order to study the induced focusing/de-focusing effects. Migrated sections 

using the source-receiver pairs as input (every shot, all offsets, traveltimes up to 8.5 s) are 

displayed in Figures 25 and 26. 



Fig. 25. Migrated section of target 1. 

Fig. 26. Migrated section of target 2. The thick black line superimposed on the horizontal axis 

represents the salt lens extension. 

6.4. Illumination maps 

In order to generate illumination maps of the target horizon, a full data set was 

modeled by wavefront construction (Vinje et al., 1996a,b). Reflection points on the target 

horizon and amplitudes of the corresponding rays from each shot to each receiver in the 

survey were computed. To be comparable to the finite-difference data set, ray tracing was 

done with cylindrical sources. The wavefront construction was performed in a 2.5D model 

extrapolated from the 2D model (see Figure 27) using the same survey as used for the 

finite-difference data. 



Fig. 27. 2.5D model used to generate illumination maps. Shots are along the surface (black line). 

Raypaths for one shot-receiver couple are indicated. Reflection points are superimposed as a line 

along the target horizon. 

To perform the calculation of the illumination maps, the surface of the reflector was 

divided in bin cells of given (constant) area. In each cell, the following quantities were 

computed: 

− the hit density (i.e. number of reflection points per unit area) defined as 

N 

D = (4) 

Bin Cell Area 

where N is the number of reflection points in the bin cell. 

− the illumination amplitude modulus (i.e. sum of the amplitude modulus per unit area) 

defined as 

∑ 

Ai 

i 

A = 

Bin Cell Area 

where Ai is the complex amplitude coefficient in the receiver for ray number i 

reflecting in the bin cell. 

6.5. Amplitude illumination and migration 

Even using a fairly small aperture (1 km), the survey illuminates the whole target 

horizon as may be seen by the reflection points on the target horizon in Figure 27. The 

steepest part of the syncline represents a low illuminated zone (x = 3.4 km). This is 

obvious on both hit density and amplitude illumination curves (see Figure 28). RMS 


(5)


amplitude along the migrated horizon (Figure 29) is smoother than the illumination 

amplitude curve, even if the general variations show some similarities. 

Under the salt lens (black line in Figures 30 and 31), the hit density returns to normal 

(compared to the hit density before x = 10.0 km) whereas the amplitude illumination 

remains lower due to the energy losses occurring when the wavefield is transmitted 

through the salt. This effect is also noticeable on the RMS amplitude extracted from the 

migrated section (Figure 31). Near the left edge of the salt lens (x = 10.7 km in Figure 30), 

there is a low-amplitude zone where very little energy is reflected even if the illumination 

density is high. This low-amplitude zone is also visible at x = 10.7 km in the RMS section 

in Figure 31, but in this case the low-amplitude zone is larger. This is not an intrinsic 

limitation of the PSDM process, but rather a side effect introduced when using first arrival 

traveltime maps. The Eikonal solver computes first arrival traveltimes, which may not 

represent the most energetic part of the wavefield (Geoltrain and Brac, 1993). In this 

case, the first arrival is associated to a low energy event passing through the salt. The 

consequence is that the imaging will take place through the salt giving lower amplitudes 

in the PSDM image from about 9.5 to 11 km. This effect could be corrected by using only 

the traveltimes for the most energetic events instead, or all arrivals. Generally speaking, 

RMS amplitude profiles along both migrated horizons (Figures 29 and 31) are smoother 

than the corresponding illumination and amplitude profiles. The PSDM process is 

performed on a finite-difference data set on which Fresnel zone effects introduce 

a smoothing (Bear et al., 1999; Thore and Juliard, 1999). Moreover, the PSDM algorithm 

performs another smoothing when stacking amplitudes along the diffraction-traveltime 

curves. Therefore, RMS amplitude variations are not as drastic as they appear on the 

illumination maps. 

Fig. 28. Hit density and amplitude illumination for the syncline (target 1). Sampling interval is 

25.0 m. 



Fig. 29. RMS amplitude along the migrated horizon for the syncline (target 1). 

Fig. 30. Hit density and amplitude illumination for the area under the salt lens (target 2). Sampling 

interval is 25.0 m. The black line on the horizontal axis represents the salt lens extension. 

Fig. 31. RMS amplitude along the migrated horizon under the salt lens (target 2). The black line on 

the horizontal axis represents the salt lens extension. 



6.6. Conclusion 

PSDM introduces a smearing of the recorded traces along isochrons tangential to the 

reflectors. This is not taken into account in the illumination maps, as they are made 

without any knowledge of the source pulse, using a simple process of adding amplitudes 

in bin cells. However, the synthetic example studied here shows that the variation of the 

illumination amplitude is comparable to the PSDM amplitude. Important effects like 

focusing/de-focusing and the reduction of amplitude due to energy losses in the 

overburden is seen on the illumination amplitude profiles. 

Comparing amplitude maps from prestack depth migrated data performed on real and 

modelled data, Rosland and Drivenes (2000) showed the importance of the reflector 

geometry in the amplitude distribution. 

7. ISOCHRON RAYS AND VELOCITY RAYS 

Conventional rays are associated to propagation of wavefronts. Familiar examples in 

nature are propagation of water waves, electromagnetic waves, and seismic waves. The 

physical wavefronts are isochrons corresponding to constant traveltime. Here, we shall 

refer to this traveltime as a one-way time. However, in the context of prestack depth 

migration of seismic data an isochron is commonly defined as a surface of potential 

scattering points in the subsurface, all corresponding to the same two-way time. A ray 

from shot to receiver can then be divided naturally in two parts; a shot ray and a receiver 

ray. Furthermore, isochrons may be considered as wavefronts of a propagating nonphysical 

wave, and recent work by Iversen (2001a,b,c, 2002) has shown that ray systems 

can be established for such propagation. Seismic ray theory (Červený, 1985, 2001) applied 

to the conventional rays from shot and receiver is a fundamental building block in the new 

theory, which includes two basic ray concepts: the isochron ray and the velocity ray. 

As with conventional rays, the position x and slowness vector p related to isochrons 

shall be expressed as functions 

x= x( γ ,, tm) 

, p = p( γ ,, tm) 

(6) 

Here, γ is a two-dimensional vector containing two independent parameters (γ1, γ2) 

describing any isochron at two-way time t = constant. The quantity m denotes one of the 

various parameters comprising the velocity model. When the velocity model and the 

vector γ is fixed, the points x are located along a trajectory, defined by Iversen (2001a,b) 

as an isochron ray. Similarly, if γ and t are fixed, and the model parameter m is allowed to 

vary, we get a velocity ray (Iversen, 2001a,c). Examples of isochron rays and velocity 

rays of type PP are shown in Figure 32, for a homogeneous velocity model. The notation 

‘PP’ means that each shot ray or receiver ray corresponds to a purely transmitted P-wave. 

The term ‘velocity ray’ was introduced by Adler et al. (1997). From a system of velocity 

rays we can obtain certain surfaces for which the model parameter m is constant. In 

accordance with the work on seismic image waves by Hubral et al. (1996), such surfaces 

can be referred to as image wavefronts. 

In conventional ray theory a ray is uniquely specified by the ray parameter vector γ, 

while a point x on the ray is uniquely specified by the extended ray parameter vector 


γ 

⎡ 1 ⎤ 

ˆ = ⎢γ⎥ ⎢ 2 ⎥ 

⎢⎣ t ⎥⎦ 


γ , (7) 

where t is the one-way time from the start point to x. In the new ray theory it has been 

essential to establish similar unique relationships. This problem has been approached by 

expressing the ray parameters for the isochron ray in terms of the ray parameters for the 

shot and receiver rays (Iversen 1996, 2001a, 2001b), i.e., 

γ = γ( γ , γ ) . (8). 

s r 

A perturbation of the first order in γ is therefore 

where 

δ δ δ 

γ = S γs + R γ r , (9) 

∂ γ 

S ≡ 

T 

∂ s 

γ ; 

∂ γ 

R ≡ 

T 

∂ γr 

Any isochron ray and velocity ray will then be required to fulfil 

respectively. 

. (10) 

d 

0 

dt = 

γ 

d 

, 0 

dm = 

γ 

, (11) 

7.1. A system of ODEs for isochron rays 

The process for generation of an isochron ray can be organized with systems of 

ordinary differential equations (ODEs) on two levels. The upper level contains the main 

system for numerical integration of isochron rays (Iversen, 2001b): 

d 

dt = 

x d 

V , ˆ 

dt = 

p 

MV , (12) 

dγˆ 

dt 

s 

= Qˆ V , 

−1 

s 

dγˆ 

dt 

r 

= Qˆ V . (13) 

The system consists of twelve equations in the case of general propagation in a 3D 

velocity model. When ray propagation is confined to a plane, the number of equations can 

be reduced to eight. Since we are not aware of an Eikonal equation for isochron rays, the 

derivation of the system of ODEs in (12) − (13) is directly based on properties of the shot 

and receiver ray, using paraxial ray theory. 

We shall explain the various quantities on the left-hand side (LHS) and right-hand side 

(RHS) of (12) − (13). t denotes the two-way time, x is the position vector describing 

points on the isochron ray, and p is the isochron slowness vector. This vector is normal to 


−1 

r


the isochron at x, and it can be obtained simply by summing the two slowness vectors of 

the shot and receiver ray, i.e., 

p = ps+ p r . (14) 

The two sets of equations in (13) have been included to ensure continuous updating of the 

extended ray parameter vectors γ ˆ s and γˆ 

r of the shot ray and the receiver ray, 

respectively. 

On the RHS of (12) − (13), V can be interpreted as a group velocity vector for 

isochron rays. Observe that V is the only quantity on the RHS depending on the 

parameterization of the isochron ray field. The vectors p and V generally point in different 

directions even if the velocity model is isotropic. 

Furthermore, the 3×3 matrix ˆM contains the second derivatives of two-way time with 

respect to spatial coordinates. It can be obtained from the sum 

Mˆ = Mˆ ˆ 

s + M r , (15) 

where the matrices on the RHS contains second derivatives with respect to one-way time. 

Following standard paraxial ray theory (Červený, 2001) we can express the one-way 

ˆM -matrices by the products 

ˆ ˆ ˆ 1 

s s s − 

M = P Q , 

ˆ ˆ ˆ 1 

r r r − 

M = P Q (16) 

In (13) and (16) ˆ Q s and Q ˆ 

r are 3×3 geometrical spreading matrices corresponding to the 

shot and receiver ray. The matrices P ˆ 

s and Pˆ 

r can be referred to as ‘slowness spreading’ 

matrices. 

The upper level of the ray tracing process is model independent. On the lower level the 

shot and receiver ray are re-traced according to the input in γ ˆ s and γˆ 

r , using 

conventional kinematic and dynamic ray tracing. Thereby we obtain all the quantities 

p , ˆ , ˆ 

s Qs P s , p ˆ ˆ 

r, Qr, Pr , (17) 

needed to evaluate the RHS of the system (12) − (13). It should be noted that (12) − (13) 

has redundant information, so by reducing the number of equations the integration of 

ODEs can be done more efficient. On the other hand, redundancy can be valuable for 

error control. 

It is important to note that the two-way time t is not always suited as the independent 

integration variable. During its propagation, isochron rays can become tangential to the 

‘wavefront’, which means that the length of the group velocity vector blows to infinity. 

For such situations we recommend a special version of (12) − (13) where t is a dependent 

variable and distance is the independent variable along the ray. After tracing a given 

distance the ray can then be divided in portions along which the two-way time varies 

monotonously, followed by resampling to the prescribed output times. 



Fig. 32. (a) PP isochron rays in a homogeneous medium with νP = 2.5 km/s. The increment 

between isochrons is 0.5 s. (b) PP velocity rays, with the isochron t = 2.5 s acting as a source for the 

propagation. The increment between image wavefronts is 0.1 km/s. 



Fig. 33. Ray perturbation due to two different conditions. (a) The extended ray parameter vector 

is fixed. (b) The ray point x is fixed. 

7.2. Perturbation of conventional rays 

Figure 33a shows a conventional ray with ray parameters γ, and a point x on the ray 

with parameters given by ˆγ . We have also indicated a wavefront through x and the 

associated slowness vector p. If a model parameter is perturbed, say m is changed to 

m + δ m, the ray will be perturbed as a result of this, but the perturbed ray can take various 

forms, depending on specified conditions. Figure 33a shows a perturbed ray (dashed) for 

Stud. geophys. geod., 46 (2002) 157 

ˆγ


the condition that the ray parameters shall be kept fixed. Also indicated is the perturbed 

wavefront, the perturbed ray point x + δ x, and the perturbed slowness vector p + δ p. The 

points x and x + δ x then correspond to the same vector ˆγ , and the collection of all such 

points forms a velocity ray (grey), i.e., a velocity ray for fixed one-way time. Such rays 

are described to the first order by derivatives d x / d m and d p / d m, which can be obtained 

by the ray perturbation method (see, e.g., Farra and Madariaga, 1987; Farra and Le 

Bégat, 1995). 

Another condition that can be enforced on ray perturbations is that the ray point x shall 

be fixed (see Figure 33b). When we perturb the model, the perturbed ray must go through 

x, and this will impose changes upon the ray parameters (from γ to 

( x 

γ+ δ γ 

) 

), the one- 

way time (from t to 

( x 

t+ δ t 

) 

), and the slowness vector (from p to p+ 

δ 

( x 

p 

) 

). The 

connection between the two cases is given by 

( x 

dp ) 

dp 

= − ˆ dx 

M , 

dm dm dm 

d 

( x 

γˆ 

) 

ˆ −1 

dx 

=−Q 

. (18) 

dm dm 

On the LHS we have the derivatives for fixed x, on the RHS the derivatives for fixed ˆγ . 

The quantities ˆM and Q ˆ are standard 3×3 matrices for second derivatives of one-way 

time and geometrical spreading, respectively. The fixed point derivatives in (18) are of 

particular importance in the generation of velocity rays for fixed two-way time. 

7.3. A system of ODEs for velocity rays 

First-order approximations of velocity rays have been described by Iversen (1996, 

2001a). The first-order theory in the latter papers were extended to yield ‘exact’ velocity 

rays in Iversen (2001c), utilizing the following system of ODEs: 

d 

dm = 

x 

U ; 

( x 

dγˆ ˆ ) 

s dγs 

ˆ −1 

= + QsU ; 

dm dm 

( x 

dp dp 

) 

= 

dm dm 

ˆ 

+MU 

( x 

dγˆ ˆ ) 

r dγr 

= + ˆQ 

dm dm 

−1 

r 

, (19) 

U . (20) 

The system comprises twelve equations, which can be reduced to eight in the case of 2D 

ray propagation. The vector U is tangent to the velocity ray, and thus we can say that this 

vector has a similar role with respect to velocity rays as the group velocity vector V has to 

isochron rays. Furthermore, U is the only quantity on the RHS that depends on the 

parameterization.The derivatives for a fixed point in (19) − (20) can be obtained from 

independent perturbation of the shot and receiver ray, using the formulas in (18). 

As with isochron rays the process for generation of exact velocity rays can be 

organized on two levels. The upper level contains the main system for velocity rays in 

(19) − (20), while the lower level includes two systems of ODEs for conventional rays, 

one for the shot ray, another for the receiver ray. The information needed from the latter 

rays is 



, ˆ , ˆ 

dxsdps ps Qs P s, 

, , ˆ ˆ dxr dpr 

pr, Qr, P r, 

, . (21) 

dm dm 

dm dm 

We can obtain these quantities by conventional kinematic ray tracing, dynamic ray 

tracing, and solution of the ray perturbation equations. As in the case of isochron rays, the 

upper level of the process is model independent. 

7.4. Applications 

Isochron rays can be used for visualization of geometric effects in prestack depth 

migration, and for map migration of events from the time domain to the depth domain. 

Isochron rays also have a potential with respect to generation and propagation of the 

boundaries of Fresnel volumes. An approximate method for the latter purpose, using 

paraxial ray theory, was described by Červený and Soares (1992). An interesting future 

task is to investigate whether isochron rays also can be beneficial in prestack depth 

migration. 

Velocity rays can be used for sensitivity studies related to the depth domain of the 

seismic data. It is important to note that velocity rays can be used not only to propagate 

geometry, but even to propagate complete seismic images as a function of a velocity 

parameter (Adler, 2000). Furthermore, it has been demonstrated recently that velocity 

rays, whether exact or approximate, can be used with success for estimation of the 

parameters of an anisotropic velocity model (Iversen et al., 2000). 

8. CONCLUSION 

We have clearly observed that the industry’s interest in applying advanced seismic 

modelling tools is growing. A number of application areas are developing, and we have 

tried to illustrate some of them in this paper: Prestack depth migration, hybrid RT/FD 

modelling, and survey planning/attribute mapping. Particularly, the production of Green's 

functions from source and receiver positions to a dense 3D grid of imaging points 

(probably millions of points) in the subsurface has turned out to be an industrial 

application, and several processing companies have already included wavefront 

construction in their imaging process. As opposed to the standard first arrival Eikonal 

solver, implementation of dynamic ray tracing in wavefront construction gives 

possibilities of choosing arrivals based on more flexible criteria (e.g. the most energetic 

arrival or even multi-arrivals) and of using illumination and amplitude information in the 

imaging process. It has been documented that a standard Kirchhoff migration does not 

generally preserve amplitude (reflection coefficient) at the target reflector. We are 

currently working on 

a new concept (Vinje, 2000), using 3D dynamic ray attributes to correct amplitudes in 

common offset Kirchhoff migrated data (reflector oriented amplitude recovery). 

Moreover, it should be added that the general wavefront construction method is currently 

extended to anisotropic media. This will definitely have potentials to improve imaging in 

areas where such effects are present (Iversen et al., 2000). 



With respect to modelling of synthetic seismic data from large 3D multi-streamer 

surveys, wavefront construction has proved to be a robust and efficient technique. Using 

the option of distributing the job to a number of computers (standard workstations or 

nodes in a network), and taking advantage of the principle of ‘shot similarity’, tens of 

thousands of shots can be simulated within reasonable computer times (although CPU 

time is drastically dependent of model complexity and smoothness. For complex detailed 

reservoir target zones overlaid by a simpler, more ray tracing friendly overburden, the 

newly developed hybrid modelling technique has turned out to be very feasible. 

Especially for repeated modelling, e.g. when changing properties in the target zone while 

keeping the overburden fixed (4D surveys, studying effects of fluid changes), hybrid 

modelling could be a powerful tool, also adapted for parallel processing. 

Finally, for future applications in prestack depth imaging and time-to-depth map 

migration, the new concept of ‘isochron rays’ is considered as a promising tool. Also with 

respect to the calculation and propagation of the boundaries of Fresnel volumes, this new 

technique may have a potential, in a certain way connecting the main signal frequency to 

the otherwise frequency independent ray attributes. This may e.g. open for an efficient 

Fresnel zone dependent smoothing of the ray amplitudes, reducing the drawbacks of the 

extremely local behavior of ray amplitudes. Similarly, ‘velocity rays’ can facilitate 

sensitivity studies related to the depth domain of seismic data, such as propagating seismic 

images when velocity parameters change, for example in an anisotropic medium. 

Acknowledgement: The authors want to direct a general thank to Geco and Statoil for supporting a 

number of projects on 3D modelling developments and applications at NORSAR throughout the 

eighties and nineties. Thanks are also given to the sponsors of the hybrid modelling project 

(HybriSeis Consortium): Eni Agip, Norsk Hydro, Statoil, Sintef Petroleum and NORSAR, and to 

the Research Council of Norway for supporting a number of modelling activities. A special thank to 

Statoil Research for permission to use the Gullfaks modelling example, and to Enterprise Oil for 

showing some results from an attribute mapping project. 

Finally, our deepest gratitude and respect goes to professor Červený for devoting his professional 

life to ray theory. By his work, major theoretical advances have been made within numerous 

research topics. In particular, he has been equipped with the ability of giving the theory a form 

suited for numerical implementation. 

Received: January 2, 2002; Accepted: March 5, 2002 

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