Review of Ray Theory Applications in Modelling and Imaging - Norsar
Review of Ray Theory Applications in Modelling and Imaging - Norsar
Review of Ray Theory Applications in Modelling and Imaging - Norsar
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REVIEW OF RAY THEORY APPLICATIONS IN MODELLING AND<br />
IMAGING OF SEISMIC DATA<br />
HÅVAR GJØYSTDAL, EINAR IVERSEN, RENAUD LAURAIN, ISABELLE LECOMTE, VETLE VINJE, KETIL ÅSTEBØL *<br />
ABSTRACT<br />
Throughout the last twenty years, 3D seismic ray modell<strong>in</strong>g has developed from<br />
a research tool to a more operational tool that has ga<strong>in</strong>ed grow<strong>in</strong>g <strong>in</strong>terest <strong>in</strong> the<br />
petroleum <strong>in</strong>dustry. Various areas <strong>of</strong> application have been established <strong>and</strong> new ones are<br />
under development. Many <strong>of</strong> these applications require a modell<strong>in</strong>g system with flexible,<br />
robust <strong>and</strong> efficient modell<strong>in</strong>g algorithms <strong>in</strong> the core. The present paper reviews the basic<br />
elements <strong>of</strong> such a system, based on the ‘open model’ concept <strong>and</strong> the ‘wavefront<br />
construction’ technique. In the latter, Červený’s dynamic ray trac<strong>in</strong>g is an <strong>in</strong>tr<strong>in</strong>sic part.<br />
The modell<strong>in</strong>g system can be used for generat<strong>in</strong>g ray attributes <strong>and</strong> synthetic<br />
seismograms for realistic 3D surveys with tens <strong>of</strong> thous<strong>and</strong>s <strong>of</strong> shots <strong>and</strong> receivers.<br />
Moreover, some other types <strong>of</strong> application areas are illustrated: Production <strong>of</strong> Green’s<br />
functions for prestack depth migration <strong>and</strong> hybrid modell<strong>in</strong>g (comb<strong>in</strong>ed ray <strong>and</strong> f<strong>in</strong>itedifference<br />
modell<strong>in</strong>g), attribute mapp<strong>in</strong>g <strong>and</strong> illum<strong>in</strong>ation analysis, both for survey<br />
plann<strong>in</strong>g <strong>and</strong> <strong>in</strong>terpretation. F<strong>in</strong>ally, the concepts <strong>of</strong> ‘isochron rays’ <strong>and</strong> ‘velocity rays’<br />
related to seismic isochrons have been <strong>in</strong>troduced recently, with very <strong>in</strong>terest<strong>in</strong>g future<br />
applications.<br />
1. INTRODUCTION<br />
Twenty years ago a small group <strong>of</strong> geophysicists at NORSAR started to develop a new<br />
s<strong>of</strong>tware package, with the ultimate goal <strong>of</strong> simulat<strong>in</strong>g seismic waves <strong>in</strong> 3D geological<br />
models. Some activity on seismic modell<strong>in</strong>g had also been performed prior to that time −<br />
some experiments had been done with various 3D model representation techniques, <strong>and</strong><br />
the first 3D k<strong>in</strong>ematic ray paths had already been tediously calculated by the old IBM 360<br />
<strong>and</strong> drawn on the shaky drum plotter. But at that time, the amplitudes were absent.<br />
The new s<strong>of</strong>tware project was sponsored by the service company Geco (at that time<br />
‘the Geophysical Company <strong>of</strong> Norway’), hav<strong>in</strong>g ambitions <strong>of</strong> becom<strong>in</strong>g a technology<br />
leader on seismic acquisition, process<strong>in</strong>g, <strong>and</strong> <strong>in</strong>terpretation, <strong>and</strong> clearly see<strong>in</strong>g the need<br />
for an advanced tool <strong>in</strong> 3D modell<strong>in</strong>g. In particular, the grow<strong>in</strong>g development <strong>of</strong> 3D<br />
seismic <strong>in</strong>terpretation stations throughout the eighties made the way short to the<br />
construction <strong>of</strong> 3D models. This potentially short l<strong>in</strong>k between <strong>in</strong>terpretation <strong>and</strong><br />
modell<strong>in</strong>g, with the possibility <strong>of</strong> <strong>in</strong>tegrat<strong>in</strong>g velocity estimation (e.g. tomography) <strong>and</strong><br />
time-to-depth conversion (map migration) <strong>in</strong> the process, was one <strong>of</strong> the ma<strong>in</strong><br />
motivations.<br />
From the very beg<strong>in</strong>n<strong>in</strong>g it was quite clear that the wave simulation method on which<br />
the new modell<strong>in</strong>g system was to be built, would have to be ray trac<strong>in</strong>g. With the<br />
* NORSAR, P.O. Box 51, N-2027 Kjeller, Norway<br />
Stud. geophys. geod., 46 (2002), 113−164 113<br />
© 2002 StudiaGeo s.r.o., Prague
H. Gjøystdal et al.<br />
computer speed <strong>in</strong> those days, ray techniques were <strong>in</strong> fact the only practically applicable<br />
ones <strong>in</strong> general 3D structures, <strong>and</strong>, even more important, these techniques possessed<br />
a number <strong>of</strong> properties that were perfectly suited for <strong>in</strong>tegrat<strong>in</strong>g modell<strong>in</strong>g <strong>and</strong><br />
<strong>in</strong>terpretation: the ability <strong>of</strong> calculat<strong>in</strong>g the wave field as labelled ray contributions (or<br />
events), where event attributes like traveltimes, amplitudes, <strong>in</strong>cident angels, etc., could be<br />
obta<strong>in</strong>ed <strong>in</strong> an explicit manner. In addition, the attributes could serve as a basis for<br />
generat<strong>in</strong>g ray theoretical seismograms, which e.g. could be used for evaluat<strong>in</strong>g <strong>and</strong><br />
test<strong>in</strong>g <strong>of</strong> 3D process<strong>in</strong>g schemes. The advantage <strong>of</strong> be<strong>in</strong>g able to explicitly extract the<br />
various wave propagation effects, like geometrical spread<strong>in</strong>g factors,<br />
reflection/transmission effects at <strong>in</strong>terfaces, phase changes at caustics, splitt<strong>in</strong>g <strong>in</strong> various<br />
branches, P- <strong>and</strong> S-wave modes, etc., was very much appreciated. Another important<br />
characteristic <strong>of</strong> the ray method is the ability to locate the propagation <strong>of</strong> energy <strong>in</strong> the<br />
model, mak<strong>in</strong>g the method particularly suited for applications <strong>in</strong> tomographic <strong>in</strong>version.<br />
Moreover, the <strong>in</strong>tuitive nature <strong>of</strong> ray trac<strong>in</strong>g to visualize wave propagation would make<br />
such a modell<strong>in</strong>g system an ideal ‘educational tool’, <strong>in</strong> order to better underst<strong>and</strong> the<br />
complex features <strong>of</strong> seismic reflections <strong>in</strong> 3D complex media.<br />
Two basic issues were strongly addressed <strong>in</strong> the first specification <strong>and</strong> design phase <strong>of</strong><br />
the 3D seismic modell<strong>in</strong>g system:<br />
− a general <strong>and</strong> flexible model representation scheme for the 3D structural (layered)<br />
model with seismic properties, <strong>and</strong><br />
− an efficient ray trac<strong>in</strong>g algorithm that could be <strong>in</strong>tegrated with the model<br />
representation, calculat<strong>in</strong>g both the k<strong>in</strong>ematic <strong>and</strong> dynamic properties <strong>of</strong> the wave<br />
field along various ray paths between given source <strong>and</strong> receiver positions.<br />
With respect to the first issue, it very soon became clear that we had to start more or<br />
less on scratch. No published methods were found that met our needs, even if some ideas<br />
could be obta<strong>in</strong>ed from the relatively new area <strong>of</strong> 3D computer graphics.<br />
With respect to the second issue we were far luckier. In our earlier experiments we had<br />
used theory from the book <strong>of</strong> Červený et al. (1977), e.g. for cod<strong>in</strong>g the<br />
reflection/transmission coefficients at an <strong>in</strong>terface, <strong>and</strong> later Červený <strong>and</strong> Hron (1980)<br />
turned out to be a major reference. The formulation <strong>of</strong> their dynamic ray tracer fit<br />
perfectly <strong>in</strong>to our modell<strong>in</strong>g scheme, <strong>and</strong> conta<strong>in</strong>ed all the necessary <strong>in</strong>gredients we<br />
searched for: calculation <strong>of</strong> rays as well as ray attributes like wavefront curvatures <strong>and</strong><br />
amplitudes along the rays, <strong>in</strong>clud<strong>in</strong>g both the travell<strong>in</strong>g through the cont<strong>in</strong>uous properties<br />
<strong>of</strong> the layers <strong>and</strong> the conversions at the model <strong>in</strong>terfaces. The methods were clearly <strong>and</strong><br />
explicitly formulated, so the way from the mathematics to computer code was mostly<br />
straightforward. In particular the ray-centered coord<strong>in</strong>ate system was very appropriate to<br />
use <strong>and</strong> easy to implement. It is true to say that these basic works <strong>of</strong> Červený (<strong>and</strong> <strong>of</strong><br />
course also later works) formed one <strong>of</strong> the key elements for our work with modell<strong>in</strong>g<br />
s<strong>of</strong>tware throughout the follow<strong>in</strong>g twenty years. Even after convert<strong>in</strong>g from s<strong>in</strong>gle twopo<strong>in</strong>t<br />
ray trac<strong>in</strong>g <strong>in</strong> the eighties to the wavefront construction technique <strong>in</strong> the n<strong>in</strong>eties,<br />
Červený’s formulas still constitute the very ‘heart’ <strong>of</strong> the system.<br />
Until some few years ago we had to realize that the use <strong>of</strong> 3D seismic modell<strong>in</strong>g as an<br />
<strong>in</strong>dustrial tool was rather limited. Throughout the eighties most applications <strong>of</strong> our<br />
modell<strong>in</strong>g systems were devoted to R&D topics, such as produc<strong>in</strong>g synthetic 3D data for<br />
114 Stud. geophys. geod., 46 (2002)
<strong>Review</strong> <strong>of</strong> <strong>Ray</strong> <strong>Theory</strong> <strong>Applications</strong> <strong>in</strong> Modell<strong>in</strong>g <strong>and</strong> Imag<strong>in</strong>g <strong>of</strong> Seismic Data<br />
test<strong>in</strong>g new process<strong>in</strong>g methods, or check<strong>in</strong>g the validity <strong>of</strong> assumptions <strong>and</strong><br />
approximations adopted <strong>in</strong> order to use more simplified models <strong>in</strong> e.g. migration <strong>and</strong><br />
time-to-depth conversion. There seemed to be a relatively high threshold for us<strong>in</strong>g such<br />
‘advanced’ tools on a more rout<strong>in</strong>e basis <strong>in</strong> operational departments <strong>in</strong> the petroleum<br />
<strong>in</strong>dustry. One <strong>of</strong> the reasons was def<strong>in</strong>itely that the establishment <strong>of</strong> 3D ray trac<strong>in</strong>g<br />
models <strong>of</strong> real complex structures required considerable manual work, <strong>and</strong> the two-po<strong>in</strong>t<br />
ray trac<strong>in</strong>g process for real 3D surveys was very computer <strong>in</strong>tensive, both with respect to<br />
time <strong>and</strong> memory.<br />
We clearly realized two basic th<strong>in</strong>gs: firstly, that the new generation <strong>of</strong> modell<strong>in</strong>g<br />
tools had to be more adapted to the structure (topology) <strong>of</strong> the real <strong>in</strong>terpretation data, to<br />
facilitate a flexible model build<strong>in</strong>g, <strong>and</strong> secondly, that the ray calculation process had to<br />
be far more efficient for simulat<strong>in</strong>g surveys with thous<strong>and</strong>s <strong>of</strong> shots <strong>and</strong> receivers. This<br />
led to the development <strong>of</strong> a new model representation, the open model concept, <strong>and</strong> the<br />
wavefront construction method throughout the n<strong>in</strong>eties. In the last years we have<br />
experienced a cont<strong>in</strong>uous growth <strong>of</strong> advanced 3D seismic modell<strong>in</strong>g tools <strong>in</strong> the seismic<br />
<strong>in</strong>dustry.<br />
Two rather ‘classical’ applications <strong>of</strong> the NORSAR ray trac<strong>in</strong>g systems througout<br />
many years have been simple forward modell<strong>in</strong>g to compute synthetic seismograms, <strong>and</strong><br />
various approaches with<strong>in</strong> tomography <strong>and</strong> velocity <strong>in</strong>version, e.g. as described <strong>in</strong><br />
Gjøystdal <strong>and</strong> Urs<strong>in</strong> (1981), Iversen et al. (1994), <strong>and</strong> Iversen <strong>and</strong> Gjøystdal (1996).<br />
Methods for estimat<strong>in</strong>g the velocity field <strong>in</strong> macro models now plays an important role <strong>in</strong><br />
3D prestack depth migration techniques, e.g. see Faye <strong>and</strong> Jeannot (1986), Iversen et al.<br />
(2000), Br<strong>and</strong>sberg-Dahl et al. (2001).<br />
The purpose <strong>of</strong> this paper is to address some important <strong>and</strong> <strong>in</strong>terest<strong>in</strong>g new areas <strong>of</strong><br />
application <strong>of</strong> a modern 3D seismic modell<strong>in</strong>g system, as encountered dur<strong>in</strong>g our last<br />
years’ activity for the petroleum <strong>in</strong>dustry. For the sake <strong>of</strong> completeness, we will first give<br />
a short descriptive review <strong>of</strong> the basic methods <strong>of</strong> model representation <strong>and</strong> wavefront<br />
construction to produce st<strong>and</strong>ard ray (event) attributes <strong>and</strong> synthetic seismograms. We<br />
have then selected a number <strong>of</strong> examples where ray trac<strong>in</strong>g based modell<strong>in</strong>g has been<br />
applied <strong>in</strong> practical studies, <strong>in</strong>clud<strong>in</strong>g hybrid modell<strong>in</strong>g (comb<strong>in</strong><strong>in</strong>g ray trac<strong>in</strong>g <strong>and</strong> f<strong>in</strong>itedifference<br />
schemes), imag<strong>in</strong>g (prestack depth migration), <strong>and</strong> seismic attribute<br />
mapp<strong>in</strong>g/illum<strong>in</strong>ation analysis. F<strong>in</strong>ally, we describe two recently <strong>in</strong>vented ray concepts<br />
(‘isochron rays’ <strong>and</strong> ‘velocity rays’), which we believe will have very <strong>in</strong>terest<strong>in</strong>g<br />
application potentials <strong>in</strong> the future.<br />
2. MODELS FOR RAY TRACING<br />
An essential part <strong>of</strong> a ray trac<strong>in</strong>g system is the subsurface model <strong>in</strong> which to propagate<br />
the rays. We will here discuss three different aspects <strong>of</strong> the model <strong>in</strong> ray trac<strong>in</strong>g:<br />
1. What the rays need to know about the model.<br />
2. The limitations imposed on the model by the ray trac<strong>in</strong>g method.<br />
3. Model representation for seismic ray trac<strong>in</strong>g.<br />
Stud. geophys. geod., 46 (2002) 115
H. Gjøystdal et al.<br />
2.1. What the rays need to know<br />
The task <strong>of</strong> the model is to serve the rays with some basic <strong>in</strong>formation about the<br />
subsurface. What is requested is well def<strong>in</strong>ed <strong>and</strong> quite simple: At any location <strong>in</strong> the<br />
model, the model must provide the ray-tracer with a few parameters concern<strong>in</strong>g the bulk,<br />
material properties, e.g., the value <strong>and</strong> spatial derivatives <strong>of</strong> isotropic P- <strong>and</strong> S-velocity<br />
functions, anisotropy parameters, density, <strong>and</strong> attenuation factors. The tracer must also get<br />
<strong>in</strong>fo about discont<strong>in</strong>uities <strong>in</strong> the properties, that is, about the subsurface <strong>in</strong>terfaces. It must<br />
be <strong>in</strong>formed if there are any potential ray-<strong>in</strong>terface <strong>in</strong>tersections <strong>in</strong> the vic<strong>in</strong>ity, <strong>and</strong>, if<br />
any, the <strong>in</strong>terface normal <strong>and</strong> curvature must be provided.<br />
2.2. <strong>Ray</strong> trac<strong>in</strong>g model limitations.<br />
The ray trac<strong>in</strong>g theory imposes some limitations on the subsurface model (Červený,<br />
1985): To ensure valid rays, the model parameters must be smooth <strong>and</strong> vary slowly. The<br />
limitations are rather vague, as it is hard to set quantitatively what ‘vary slowly’ means.<br />
What really counts is the relative model smoothness with<strong>in</strong> the Fresnel volume around<br />
each ray (Červený <strong>and</strong> Soares, 1992). It is difficult to use this criterion to prepare a valid<br />
model, as the validity depends not only on model parameters, but also on the f<strong>in</strong>al rays<br />
between sources <strong>and</strong> receivers. In practice we stick to a simpler, ad hoc procedure <strong>and</strong><br />
consider the smoothness as a ray-<strong>in</strong>dependent model property. We l<strong>in</strong>k the smoothness <strong>in</strong><br />
a rough manner to the seismic experiment by smooth<strong>in</strong>g properties <strong>and</strong> <strong>in</strong>terfaces so they<br />
vary with longer wavelength than the dom<strong>in</strong>at<strong>in</strong>g wavelength <strong>in</strong> the recorded seismic<br />
signals (Červený, 2001, p.608).<br />
2.3. Model representation for seismic ray trac<strong>in</strong>g.<br />
Different model representations have been developed for seismic ray trac<strong>in</strong>g. The<br />
simplest one is the 1D model with a vertical sequence <strong>of</strong> constant material properties<br />
separated by plane, horizontal layers. A more flexible model is the ‘layer cake’, which is<br />
a similar sequence, but it varies laterally as the properties <strong>and</strong> <strong>in</strong>terfaces are represented as<br />
grids. More comprehensive seismic model representations for ray trac<strong>in</strong>g <strong>in</strong>clude<br />
CAD/CAM-like solid modell<strong>in</strong>g representation (Gjøystdal et al., 1985), <strong>and</strong> models based<br />
on surface patches (Pereyra, 1992). Very advanced model builders <strong>and</strong> ray tracers are<br />
found <strong>in</strong> other fields, like computer graphics, CAD/CAM, <strong>and</strong> the enterta<strong>in</strong>ment <strong>in</strong>dustry,<br />
<strong>and</strong> there are also well-established general purpose model builders <strong>in</strong> the geoscience<br />
<strong>in</strong>dustry, see for example Mallet et al. (1989). However, as discussed below, seismic ray<br />
modell<strong>in</strong>g has some quite specific characteristics <strong>and</strong> sets certa<strong>in</strong> requirements which <strong>in</strong><br />
general are not fully met by models made for other purposes.<br />
In our present 3D modell<strong>in</strong>g system (V<strong>in</strong>je et al., 1999) we have designed <strong>and</strong> tuned<br />
the subsurface model representation specifically to seismic ray modell<strong>in</strong>g. It is designed<br />
for the external model data we might expect, <strong>and</strong> it takes <strong>in</strong>to account the characteristics<br />
<strong>of</strong> ray trac<strong>in</strong>g. An example <strong>of</strong> a 3D ray trac<strong>in</strong>g model is shown <strong>in</strong> Figure 1.<br />
116 Stud. geophys. geod., 46 (2002)
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Fig. 1. Example <strong>of</strong> a 3D ray trac<strong>in</strong>g model; the SEG/EAGE Salt Model (Am<strong>in</strong>zadeh et al., 1997).<br />
A typical subsurface data set <strong>in</strong> seismic modell<strong>in</strong>g is characterized by:<br />
− A number <strong>of</strong> <strong>in</strong>terfaces, <strong>of</strong>ten <strong>of</strong> complex nature<br />
− Material properties vary <strong>in</strong> a general way <strong>in</strong>side the layers.<br />
− Often <strong>in</strong>complete data, imply<strong>in</strong>g that parts <strong>of</strong> <strong>in</strong>terfaces <strong>and</strong> properties are miss<strong>in</strong>g.<br />
<strong>Ray</strong> trac<strong>in</strong>g requires (see Subsections 2.1 <strong>and</strong> 2.2):<br />
− Frequent <strong>and</strong> fast property evaluation <strong>and</strong> ray-<strong>in</strong>terface <strong>in</strong>tersection.<br />
− Cont<strong>in</strong>uous <strong>and</strong> smooth derivatives <strong>of</strong> properties <strong>and</strong> <strong>in</strong>terfaces.<br />
To honour the above po<strong>in</strong>ts we have made the follow<strong>in</strong>g model representation:<br />
In the model there are just two types <strong>of</strong> geometrical elements: <strong>in</strong>terfaces <strong>and</strong><br />
properties. An <strong>in</strong>terface describes the location <strong>of</strong> the discont<strong>in</strong>uity between two different<br />
layers (volumes, blocks) <strong>in</strong> the subsurface. A property represents a cont<strong>in</strong>uous, bulk,<br />
material property <strong>in</strong> a layer, such as isotropic P-velocity. The model is a simple system <strong>of</strong><br />
po<strong>in</strong>ters from <strong>in</strong>terfaces to properties, see Figure 2.<br />
Each <strong>in</strong>terface is represented as a triangular network with explicit normals <strong>and</strong><br />
curvatures associated with every node. This representation comb<strong>in</strong>es the flexibility <strong>of</strong><br />
triangular networks with an approximate solution for the smoothness required <strong>in</strong> ray<br />
trac<strong>in</strong>g. Strictly speak<strong>in</strong>g, dynamic ray trac<strong>in</strong>g requires <strong>in</strong>terfaces with cont<strong>in</strong>uous <strong>and</strong><br />
smooth 1st <strong>and</strong> 2nd derivatives. However, it is very difficult to make a surface<br />
representation that comb<strong>in</strong>es these properties with the flexibility to represent the complex<br />
<strong>in</strong>terfaces prevalent <strong>in</strong> seismic models. To get around this problem, we have ‘de-coupled’<br />
the <strong>in</strong>terface slightly from its derivatives. As a part <strong>of</strong> the <strong>in</strong>terface construction, the<br />
normals <strong>and</strong> curvatures <strong>in</strong> each node are estimated from the shape <strong>of</strong> the surround<strong>in</strong>g<br />
network <strong>and</strong> stored <strong>in</strong> the node. When the normals <strong>and</strong> curvatures <strong>in</strong> an arbitrary po<strong>in</strong>t are<br />
requested, they are <strong>in</strong>terpolated from the pre-computed values associated with the<br />
surround<strong>in</strong>g nodes, by a procedure similar to the one described <strong>in</strong> Mallet et al. (1997).<br />
With this representation normals <strong>and</strong> curvatures are cont<strong>in</strong>uous <strong>and</strong> smooth, but they are<br />
not necessarily strictly the ‘true’ ones for the <strong>in</strong>terface.<br />
Stud. geophys. geod., 46 (2002) 117
H. Gjøystdal et al.<br />
Fig. 2. The model structure. From each side <strong>of</strong> each <strong>in</strong>terface there is a po<strong>in</strong>ter to the bulk<br />
properties that shall be used for rays traced out from that side <strong>of</strong> the <strong>in</strong>terface. Pi, Si, <strong>and</strong> Di are the<br />
P-wave, S-wave, <strong>and</strong> density functions for the same material ‘i’. Each <strong>in</strong>terface is labelled by an<br />
<strong>in</strong>dex ‘j’ (e.g., I1, I2,...).<br />
Fig. 3. <strong>Ray</strong> trac<strong>in</strong>g <strong>in</strong> an open model. Notice the hole <strong>in</strong> the white <strong>in</strong>terface. <strong>Ray</strong> ‘A’ through<br />
the hole is rejected. It starts out <strong>in</strong> properties #3 below the top <strong>in</strong>terface, while at the next ray<strong>in</strong>terface<br />
<strong>in</strong>tersection at the bottom <strong>in</strong>terface, it is expected to arrive through properties #2. <strong>Ray</strong> ‘B’<br />
is accepted, as the properties at the departure from an <strong>in</strong>terface matches the properties at the arrival<br />
po<strong>in</strong>t on the next one.<br />
118 Stud. geophys. geod., 46 (2002)
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The properties are much simpler to represent than the <strong>in</strong>terfaces. Each property for<br />
each layer is a separate, smooth, <strong>and</strong> slowly vary<strong>in</strong>g function, <strong>and</strong> is represented by a tricubic<br />
spl<strong>in</strong>e.<br />
In real case modell<strong>in</strong>g projects there may be small or even big gaps <strong>in</strong> the <strong>in</strong>terpreted<br />
<strong>in</strong>terfaces where data are miss<strong>in</strong>g for some reason. To allow for ray trac<strong>in</strong>g <strong>in</strong> these<br />
<strong>in</strong>complete data, we have developed a concept where the model is ‘open’, see Åstebøl<br />
(1994) <strong>and</strong> V<strong>in</strong>je et al. (1999). A consistency check sorts out <strong>and</strong> rejects rays that have<br />
passed through the <strong>in</strong>valid, <strong>in</strong>complete parts <strong>of</strong> the model, see Figure 3. The open model<br />
concept saves a lot <strong>of</strong> edit<strong>in</strong>g that would otherwise have been required to fill <strong>in</strong> the<br />
miss<strong>in</strong>g parts. The <strong>in</strong>-fill also <strong>in</strong>troduces uncerta<strong>in</strong>ties <strong>in</strong> the modell<strong>in</strong>g, as ray trac<strong>in</strong>g then<br />
is allowed where very little is known about the subsurface.<br />
3. WAVEFRONT CONSTRUCTION (WFC)<br />
3.1. Background<br />
The wavefront construction (WFC) method was <strong>in</strong>troduced for 2D models <strong>in</strong> a paper<br />
by V<strong>in</strong>je et al. (1993b). Dur<strong>in</strong>g the n<strong>in</strong>eties the method was further developed to<br />
accommodate 3D models (V<strong>in</strong>je et al., 1996a,b; V<strong>in</strong>je et al., 1999). Based on the latter<br />
papers we review here the basic pr<strong>in</strong>ciples <strong>and</strong> properties <strong>of</strong> WFC.<br />
WFC is a ray field technique based on conventional seismic ray theory, as described<br />
by Červený <strong>in</strong> a long list <strong>of</strong> papers, lecture notes, <strong>and</strong> books (e.g., Červený, 1985, 2001).<br />
Instead <strong>of</strong> construct<strong>in</strong>g the ray field ray-by-ray, as it is done <strong>in</strong> classic <strong>in</strong>itial-value ray<br />
trac<strong>in</strong>g or two-po<strong>in</strong>t ray trac<strong>in</strong>g, WFC is build<strong>in</strong>g up the ray field wavefront-by-wavefront.<br />
The propagation from one wavefront to the next is based on trac<strong>in</strong>g <strong>of</strong> a large number <strong>of</strong><br />
short ray segments. The basic idea <strong>in</strong> WFC is to ma<strong>in</strong>ta<strong>in</strong> a uniform density <strong>of</strong> ray<br />
segments on the wavefront dur<strong>in</strong>g its propagation. To control the ray density it is now<br />
common to use criteria based on i) the distance between neighbor<strong>in</strong>g rays <strong>and</strong> ii) the<br />
separation <strong>of</strong> wavefront normals related to such rays. Application <strong>of</strong> these criteria on<br />
a wavefront will lead to creation <strong>of</strong> new rays by <strong>in</strong>terpolation.<br />
The WFC method was developed as a consequence <strong>of</strong> the problems aris<strong>in</strong>g <strong>in</strong> twopo<strong>in</strong>t<br />
ray trac<strong>in</strong>g when attempt<strong>in</strong>g to f<strong>in</strong>d all rays connect<strong>in</strong>g a source po<strong>in</strong>t with<br />
a receiver po<strong>in</strong>t. Traditionally this is done us<strong>in</strong>g shoot<strong>in</strong>g or bend<strong>in</strong>g algorithms. In<br />
a variant <strong>of</strong> the shoot<strong>in</strong>g method described by Červený (1985), a fan <strong>of</strong> rays is traced from<br />
the source po<strong>in</strong>t, <strong>and</strong> paraxial extrapolation is used to estimate values at the receiver<br />
po<strong>in</strong>t(s). The ma<strong>in</strong> problem with these approaches is the lack <strong>of</strong> control <strong>of</strong> divergence<br />
between the rays <strong>in</strong> the search fan. Therefore the trade-<strong>of</strong>f between efficiency <strong>and</strong><br />
reliability/robustness may be unfavorable, especially for complicated 3D models.<br />
Dur<strong>in</strong>g the last ten years WFC has proven to be a very powerful tool <strong>in</strong> seismic<br />
modell<strong>in</strong>g <strong>and</strong> imag<strong>in</strong>g. In Kirchh<strong>of</strong>f type migration or similar (see Section 5) estimation<br />
<strong>of</strong> the Green’s functions by forward modell<strong>in</strong>g is <strong>of</strong>ten the bottleneck <strong>of</strong> the whole<br />
procedure. Also <strong>in</strong> velocity estimation <strong>and</strong> <strong>in</strong> survey plann<strong>in</strong>g forward modell<strong>in</strong>g plays an<br />
important role. The advantage <strong>of</strong> the WFC method is that it can do forward modell<strong>in</strong>g <strong>in</strong><br />
complex 3D models <strong>in</strong> reasonable time, <strong>and</strong> still be able to f<strong>in</strong>d all, or a selection <strong>of</strong>,<br />
reflected <strong>and</strong> transmitted events <strong>in</strong> the receiver po<strong>in</strong>ts.<br />
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The orig<strong>in</strong>al WFC technique was developed for isotropic models. Development to take<br />
<strong>in</strong>to account anisotropy has been reported recently (Gibson, 2000; Mispel <strong>and</strong> Williamson,<br />
2001).<br />
3.2. Representation <strong>of</strong> wavefronts<br />
In the papers by V<strong>in</strong>je et al. (1996a,b) it was essential to f<strong>in</strong>d a simple <strong>and</strong> efficient<br />
numerical representation <strong>of</strong> the 3D wavefront (WF). In general, the representation <strong>of</strong> the<br />
WF <strong>in</strong> 3D is somewhat more complicated than <strong>in</strong> 2D, where the rays are situated side by<br />
side along the 2D WF. The term ‘neighbor<strong>in</strong>g’ or ‘adjacent’ ray is more difficult to def<strong>in</strong>e<br />
on a 3D WF where the rays are distributed <strong>in</strong> two directions. Some k<strong>in</strong>d <strong>of</strong> network<br />
connect<strong>in</strong>g the rays on such a 3D WF must be def<strong>in</strong>ed. This <strong>in</strong>ternal order<strong>in</strong>g <strong>of</strong> po<strong>in</strong>ts<br />
(i.e. the <strong>in</strong>tersection po<strong>in</strong>ts between rays <strong>and</strong> WFs) <strong>and</strong> their <strong>in</strong>ternal connection l<strong>in</strong>es can<br />
be termed as the WF topology. As demonstrated by V<strong>in</strong>je et al. (1996a,b)<br />
a triangular network has both a simple topology <strong>and</strong> the ability to adjust to the stretch<strong>in</strong>g<br />
<strong>and</strong> twist<strong>in</strong>g <strong>of</strong> the WF dur<strong>in</strong>g the propagation through the medium. The processes <strong>of</strong><br />
check<strong>in</strong>g, <strong>in</strong>terpolation <strong>and</strong> estimation <strong>of</strong> receiver parameters are made fairly simple us<strong>in</strong>g<br />
this topology.<br />
The ma<strong>in</strong> steps <strong>in</strong> WFC are<br />
3.3. Propagation <strong>of</strong> wavefronts<br />
− Generation <strong>of</strong> an <strong>in</strong>itial WF,<br />
− Propagat<strong>in</strong>g the WF one time step,<br />
− Controll<strong>in</strong>g the density <strong>of</strong> ray segments on the WF,<br />
− Interpolation to f<strong>in</strong>d arrivals <strong>in</strong> the receivers.<br />
Fig. 4. The icosahedron (a) describes the basic network <strong>of</strong> the po<strong>in</strong>t source. In (b) four repeated<br />
<strong>in</strong>terpolations have taken place on the icosahedron. This leads to a polyhedron that consists <strong>of</strong> 2562<br />
nodes (rays), 7680 sides <strong>and</strong> 5120 triangles.<br />
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In order to start WF propagation, an <strong>in</strong>itial (source) WF is necessary. We need<br />
a triangular grid with sides <strong>and</strong> triangles as equal as possible, <strong>and</strong> <strong>in</strong> this respect, an<br />
icosahedron is especially well suited. The icosahedron consists <strong>of</strong> twelve vertices, thirty<br />
sides <strong>and</strong> twenty regular triangles. We let twelve rays from the source po<strong>in</strong>t pass through<br />
the vertices <strong>of</strong> the icosahedron thus creat<strong>in</strong>g the desired triangular network. The angle<br />
between neighbor<strong>in</strong>g rays <strong>in</strong> this network is then approximately 63.43°. In order to get<br />
a denser sampl<strong>in</strong>g <strong>of</strong> the take<strong>of</strong>f directions we perform <strong>in</strong>terpolation <strong>of</strong> new rays on the<br />
triangular network (the icosahedron). The <strong>in</strong>terpolation is repeated until the criteria for the<br />
density <strong>of</strong> ray segments are fulfilled. In Figure 4 the icosahedron itself <strong>and</strong> the source WF<br />
after four <strong>in</strong>terpolations are plotted.<br />
Assume that a WF represented as a triangular network is present at a time t. For<br />
propagation <strong>in</strong> an isotropic medium (V<strong>in</strong>je et al., 1996a,b) each node (i.e., the<br />
<strong>in</strong>tersections between rays <strong>and</strong> WF) is characterized by a set <strong>of</strong> parameters like:<br />
− Position<br />
− <strong>Ray</strong> tangent<br />
− Take<strong>of</strong>f direction <strong>of</strong> the ray at the source<br />
− Wave mode (e.g. P or S <strong>in</strong> an isotropic medium)<br />
− Unit ray normal <strong>in</strong> the ray-centered coord<strong>in</strong>ate system<br />
− Dynamic parameters such as the P- <strong>and</strong> Q-matrix (Červený, 1985) <strong>and</strong> zero-order ray<br />
amplitude coefficient(s).<br />
Given this set <strong>of</strong> parameters for all nodes at time t, all the rays <strong>in</strong> the WF are traced<br />
a small time step ∆t so that a new WF at time t + ∆t is created. A part <strong>of</strong> the complete<br />
wave field at three successive time steps is plotted <strong>in</strong> Figure 5. The WFs for two time<br />
steps (t <strong>and</strong> t + ∆t) are kept <strong>in</strong> memory to facilitate <strong>in</strong>terpolation <strong>of</strong> new rays <strong>and</strong><br />
estimation <strong>of</strong> event data <strong>in</strong> receivers. All the rays <strong>in</strong> the network are equipped with a userdef<strong>in</strong>ed<br />
ray code which determ<strong>in</strong>es the specific sequence <strong>of</strong> transmissions <strong>and</strong>/or<br />
reflections at the <strong>in</strong>terfaces <strong>in</strong> the model.<br />
Fig. 5. The WF is propagated through the model by trac<strong>in</strong>g all the rays from the old to the new<br />
WF. The WFs for two successive time steps (t <strong>and</strong> t + ∆t) are kept <strong>in</strong> memory.<br />
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Fig. 6. Simple ray cell with an <strong>in</strong>terior receiver. Seismic data are estimated <strong>in</strong> the receiver by<br />
<strong>in</strong>terpolation from the three rays r1 , r2 <strong>and</strong> r3 .<br />
When divergence is present <strong>in</strong> the wave field, parts <strong>of</strong> the WF will be stretched dur<strong>in</strong>g<br />
its propagation through the medium. Both the distance <strong>and</strong> the angular difference <strong>of</strong> ray<br />
tangents between neighbor<strong>in</strong>g rays <strong>in</strong>crease, <strong>and</strong> <strong>in</strong>terpolation is needed <strong>in</strong> order to<br />
ma<strong>in</strong>ta<strong>in</strong> some pre-def<strong>in</strong>ed sampl<strong>in</strong>g density <strong>of</strong> the WF. Several <strong>in</strong>terpolation criteria has<br />
been proposed s<strong>in</strong>ce the first publications on WFC (V<strong>in</strong>je et al. 1992, 1993a, 1993b). For<br />
example, Lambaré‚ et al. (1994) <strong>in</strong>troduced <strong>in</strong>terpolation criteria based on the paraxial ray<br />
approximation (Červený, 1985). Furthermore, V<strong>in</strong>je (1997) <strong>in</strong>troduced an <strong>in</strong>terpolation<br />
criterion based on so-called probe rays that gives control <strong>of</strong> the relative error <strong>of</strong> any<br />
parameter propagated along the WF, but with the cost <strong>of</strong> <strong>in</strong>creased CPU time.<br />
In order to f<strong>in</strong>d arrivals at the receivers we need a procedure able <strong>of</strong> transferr<strong>in</strong>g the<br />
seismic parameters from the mov<strong>in</strong>g WF <strong>in</strong>to each receiver. The volume between two<br />
consecutive WFs (i.e., the old <strong>and</strong> the new WF) is divided <strong>in</strong>to ray cells. These are prism<br />
shaped bodies bounded by three rays <strong>and</strong> the triangle that connects them on each <strong>of</strong> the<br />
two WFs. The six extreme boundaries <strong>of</strong> the spatial coord<strong>in</strong>ates (x, y, z) <strong>of</strong> the ray cell are<br />
used to construct a box that completely conta<strong>in</strong>s the ray cell. Then the subset <strong>of</strong> the<br />
receivers located with<strong>in</strong> the box is found. Some <strong>of</strong> these receivers may also be located<br />
with<strong>in</strong> the ray cell itself. A ray cell with an <strong>in</strong>terior receiver is shown <strong>in</strong> Figure 6. The<br />
seismic parameters for this receiver are found by <strong>in</strong>terpolation from the three rays <strong>of</strong> the<br />
ray cell us<strong>in</strong>g the ray cell coord<strong>in</strong>ates u, v, <strong>and</strong> t*, where (u, v) are the so-called<br />
barycentric coord<strong>in</strong>ates <strong>of</strong> the footpo<strong>in</strong>t <strong>of</strong> the receiver at the base <strong>of</strong> the ray cell, <strong>and</strong> t* is<br />
the traveltime from the base <strong>of</strong> the ray cell to the receiver.<br />
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Fig. 7. An example <strong>of</strong> WF construction <strong>in</strong> an open model. A WF (at 0.9 s) is reflected from the<br />
deepest <strong>in</strong>terface <strong>in</strong> the seismic model. The <strong>in</strong>terface is plotted <strong>in</strong> black while the WF is plotted <strong>in</strong><br />
white. The <strong>in</strong>terface conta<strong>in</strong>s two undef<strong>in</strong>ed areas (holes). Note the hole <strong>in</strong> the WF were it reaches<br />
one <strong>of</strong> the undef<strong>in</strong>ed areas <strong>of</strong> the <strong>in</strong>terface.<br />
3.4. Reflection <strong>and</strong> transmission at <strong>in</strong>terfaces<br />
The open model (see Section 2) is a practical concept for representation <strong>of</strong> realistic<br />
geological models. As shown <strong>in</strong> Figure 7, WF construction can be performed <strong>in</strong> an open<br />
model. The triangular network form<strong>in</strong>g the <strong>in</strong>terfaces possesses the same local<br />
characteristics as the WFs, <strong>and</strong> this can be utilized <strong>in</strong> the process <strong>of</strong> reflect<strong>in</strong>g or<br />
transmitt<strong>in</strong>g a ray at an <strong>in</strong>terface. Each time an <strong>in</strong>tersection between a ray <strong>and</strong> a triangle <strong>of</strong><br />
an <strong>in</strong>terface is detected, the follow<strong>in</strong>g parameters are obta<strong>in</strong>ed for the ray/<strong>in</strong>terface<br />
<strong>in</strong>tersection po<strong>in</strong>t:<br />
− Spatial position: Located <strong>in</strong> the plane between the three vertices <strong>of</strong> the triangle.<br />
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− Interface normal: Interpolated between the three <strong>in</strong>terface normals <strong>in</strong> the vertices <strong>of</strong><br />
the triangle.<br />
− Interface curvature matrix with correspond<strong>in</strong>g local coord<strong>in</strong>ates: Interpolated between<br />
the three curvature matrices <strong>in</strong> the vertices <strong>of</strong> the triangle.<br />
This is sufficient <strong>in</strong>formation to reflect or transmit the ray at the <strong>in</strong>terface, calculat<strong>in</strong>g<br />
both k<strong>in</strong>ematic <strong>and</strong> dynamic parameters.<br />
3.5. WFC <strong>and</strong> multi-arrivals<br />
We consider a model with a s<strong>in</strong>gle <strong>in</strong>terface separat<strong>in</strong>g two homogeneous isotropic<br />
half-spaces. In this model WFC is used for computation <strong>of</strong> reflected contributions<br />
correspond<strong>in</strong>g to a straight receiver l<strong>in</strong>e (see Figure 8a-b). The curvatures <strong>of</strong> the <strong>in</strong>terface<br />
give rise to triplication <strong>of</strong> the depart<strong>in</strong>g WFs <strong>and</strong> thus to multi-arrivals at the receivers<br />
(more than one event per receiver). Examples <strong>of</strong> multi-arrivals for the source <strong>in</strong> Figure 8b<br />
are plotted <strong>in</strong> Figure 9. Four event parameters for the receiver l<strong>in</strong>e are shown: traveltime,<br />
take<strong>of</strong>f angle from the source, amplitude coefficient <strong>and</strong> ray direction at the receivers. The<br />
bow-tie <strong>in</strong> the lower part <strong>of</strong> Figure 9a is the traveltime <strong>of</strong> flank energy which could not be<br />
present <strong>in</strong> any 2D modell<strong>in</strong>g. These arrivals are nicely unfolded <strong>in</strong>to an oval curve <strong>in</strong> the<br />
take<strong>of</strong>f direction plot <strong>in</strong> Figure 9b. In Figure 10, the error (<strong>in</strong> milliseconds) <strong>of</strong> the<br />
traveltime calculations are plotted. The figure shows the difference between the<br />
traveltimes computed by the WF construction <strong>and</strong> the traveltime <strong>of</strong> rays traced from the<br />
source po<strong>in</strong>t to each receiver position. An error <strong>of</strong> maximum 0.2 − 0.3 ms, as showed <strong>in</strong><br />
the figure, is sufficiently small for most geophysical applications. Us<strong>in</strong>g the traveltime<br />
<strong>and</strong> amplitude data from the receivers, a synthetic seismogram is computed (Figure 11).<br />
A 50 Hz Berlage pulse was used <strong>in</strong> the generation <strong>of</strong> the traces. Amplitude variations as<br />
well as phase shifts may be observed <strong>in</strong> the seismogram.<br />
3.6. Computation efficiency − parallel WFC.<br />
The CPU time <strong>of</strong> the WF construction method depends on factors such as the<br />
complexity <strong>of</strong> the model, the sampl<strong>in</strong>g <strong>of</strong> the wavefield (controlled by the user def<strong>in</strong>ed ray<br />
density <strong>and</strong> time-step length), total time propagated, <strong>and</strong> number <strong>of</strong> receivers. A general<br />
observation regard<strong>in</strong>g the CPU time is that the efficiency <strong>in</strong>creases considerably with<br />
number <strong>of</strong> receivers <strong>in</strong> the model compared to traditional two-po<strong>in</strong>t search strategies. To<br />
make simulation <strong>of</strong> large realistic 3D surveys feasible (tens <strong>of</strong> thous<strong>and</strong>s <strong>of</strong> shots <strong>and</strong><br />
receivers), the survey may be automatically divided <strong>in</strong>to a number <strong>of</strong> <strong>in</strong>dependent subtasks,<br />
with a number <strong>of</strong> shots <strong>in</strong> each, which can be distributed to a number <strong>of</strong> computers<br />
or nodes <strong>in</strong> a network. Each <strong>of</strong> these parallel WFC jobs may apply the pr<strong>in</strong>ciple <strong>of</strong> shot<br />
similarity, tak<strong>in</strong>g advantage <strong>of</strong> the fact that with<strong>in</strong> a given arrival branch <strong>of</strong> the wavefronts<br />
hitt<strong>in</strong>g the receivers, the correspond<strong>in</strong>g ray take<strong>of</strong>f directions will generally vary little<br />
from one shot to the next. This makes it possible to use a strategy <strong>of</strong> send<strong>in</strong>g out<br />
wavefronts only <strong>in</strong> preselected directions for most <strong>of</strong> the shots <strong>in</strong> the survey. Only for<br />
a smaller number <strong>of</strong> ‘master shots’, full wavefronts have to be propagated. This option<br />
may decrease the CPU time drastically.<br />
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Fig. 8. WFs <strong>and</strong> re-traced rays to a straight receiver l<strong>in</strong>e <strong>in</strong> a model with a double s<strong>in</strong>usoidal<br />
reflector. (a) The source po<strong>in</strong>t is located <strong>in</strong> the left part <strong>of</strong> the model at position (0.2, 1.25, 0.2) km.<br />
(b) The source po<strong>in</strong>t is at position (1.25, 1.25, 0.2) km.<br />
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Fig. 9. A selection <strong>of</strong> the parameters recorded <strong>in</strong> for the shot <strong>and</strong> receivers <strong>in</strong> Figure 8b. The<br />
figure shows traveltime (a) <strong>and</strong> log10 <strong>of</strong> the modulus <strong>of</strong> the amplitude coefficient (c) as a function<br />
<strong>of</strong> the distance along the receiver l<strong>in</strong>e. In (b) <strong>and</strong> (d), take<strong>of</strong>f angles from the source <strong>and</strong> ray tangent<br />
angles at the receivers are shown.<br />
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Fig. 10. Error <strong>in</strong> traveltime correspond<strong>in</strong>g to the traveltime curve <strong>in</strong> Figure 9a. The difference<br />
between the result <strong>of</strong> WF construction <strong>and</strong> the direct trac<strong>in</strong>g between the source <strong>and</strong> the receivers<br />
has been plotted.<br />
Fig. 11. Synthetic seismogram correspond<strong>in</strong>g to the event data shown <strong>in</strong> Figure 9. A 50 Hz<br />
Berlage pulse was used when the traces were computed.<br />
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4. HYBRID RAY TRACING / FINITE-DIFFERENCE MODELLING<br />
Beside ray trac<strong>in</strong>g (RT) for 2D or 3D wave propagation modell<strong>in</strong>g <strong>of</strong> the earth’s<br />
structures, f<strong>in</strong>ite-difference (FD) techniques are widely used. Both RT <strong>and</strong> FD have<br />
specific advantages <strong>and</strong> severe drawbacks. They are fundamentally different but they do<br />
<strong>in</strong>tend to solve the same problem, i.e., to simulate acoustic/elastic wave propagation <strong>in</strong> the<br />
earth. Unfortunately, the trend is <strong>of</strong>ten to use only one method <strong>and</strong> ignore the other, while<br />
<strong>in</strong> fact they do complete each other. Each method needs the model to be represented <strong>in</strong><br />
a specific way. However, there are very few seismic models which are optimally suited<br />
for RT or FD. A possible solution would therefore be to apply RT modell<strong>in</strong>g <strong>in</strong> some parts<br />
<strong>of</strong> the models where it is appropriate, <strong>and</strong> FD modell<strong>in</strong>g <strong>in</strong> the rest. We will illustrate such<br />
a hybrid comb<strong>in</strong>ation <strong>of</strong> RT <strong>and</strong> FD to model the wavefield scattered by a chosen target<br />
area <strong>of</strong> the model <strong>and</strong> orig<strong>in</strong>ally developed for oil exploration with seismic modell<strong>in</strong>g <strong>of</strong><br />
reservoirs (Lecomte, 1996; Hokstad et al., 1998; Gjøystdal et al., 1998). FD modell<strong>in</strong>g is<br />
applied at the target itself, which may be too complex to be properly described by RT<br />
models. The rest <strong>of</strong> the propagation, prior to <strong>and</strong> after the FD modell<strong>in</strong>g, is performed by<br />
RT through the calculation <strong>of</strong> Green’s functions (GFs). But before illustrat<strong>in</strong>g the results<br />
<strong>of</strong> this hybrid approach, let us first review the characteristics <strong>and</strong> limitations <strong>of</strong> both RT<br />
<strong>and</strong> FD modell<strong>in</strong>g <strong>and</strong> compare them.<br />
4.1. Why hybrid modell<strong>in</strong>g?<br />
There are many versions <strong>of</strong> both RT <strong>and</strong> FD techniques on the market so we will not<br />
focus on a specific one <strong>in</strong> each case but analyze their overall characteristics. The ma<strong>in</strong><br />
philosophy <strong>of</strong> a hybrid concept is <strong>in</strong>deed to take advantage <strong>of</strong> each class <strong>of</strong> methods,<br />
while try<strong>in</strong>g to significantly reduce their <strong>in</strong>herent drawbacks.<br />
<strong>Ray</strong> trac<strong>in</strong>g (RT) modell<strong>in</strong>g<br />
In ray theory, an asymptotic approximation <strong>of</strong> the exact solution <strong>of</strong> the wave equation<br />
is obta<strong>in</strong>ed by search<strong>in</strong>g for a displacement solution <strong>in</strong> the form <strong>of</strong> a ray series (Červený et<br />
al., 1977). The most common approach is to keep only the first-order term after <strong>in</strong>sert<strong>in</strong>g<br />
this solution <strong>in</strong> the wave equations, thus apply<strong>in</strong>g a high frequency approximation. We<br />
then end up with two separate equations: the Eikonal equation for the traveltime <strong>and</strong> the<br />
first transport equation for the amplitudes. This means <strong>in</strong> theory that the results are valid<br />
for wavelengths significantly smaller than any physical scale <strong>in</strong> the model. In practice RT<br />
is <strong>of</strong>ten more robust than predicted <strong>and</strong> may work well for structures down to one fifth <strong>of</strong><br />
the wavelength. Typical RT models are described by smooth <strong>in</strong>terfaces <strong>and</strong> smooth<br />
velocity fields (see Section 2). Furthermore, RT is event-oriented, imply<strong>in</strong>g that the user<br />
may choose the type <strong>of</strong> waves to be modelled <strong>in</strong> the various layers <strong>and</strong> the type <strong>of</strong><br />
reflection/transmission to occur at <strong>in</strong>terfaces. For this reason RT modell<strong>in</strong>g has become a<br />
useful tool for analysis <strong>of</strong> wave propagation <strong>and</strong> identification <strong>of</strong> arrivals.<br />
The major drawback <strong>of</strong> RT is the high frequency assumption, which may lead to<br />
<strong>in</strong>correct amplitudes for the detailed structures <strong>in</strong> oil exploration. Diffractions can be<br />
modelled approximately, but many other wave phenomena are miss<strong>in</strong>g <strong>in</strong> RT modell<strong>in</strong>g,<br />
such as surface waves, evanescent waves, head waves, <strong>and</strong> so forth. There are also<br />
difficulties <strong>in</strong> the vic<strong>in</strong>ity <strong>of</strong> caustic po<strong>in</strong>ts where RT usually gives artificially high<br />
128 Stud. geophys. geod., 46 (2002)
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amplitudes. The efficiency <strong>of</strong> RT modell<strong>in</strong>g is very model dependent <strong>and</strong> difficult to<br />
predict. Moreover, it may be difficult to know <strong>in</strong> advance what waves to model, except for<br />
general classes such as primary reflections.<br />
In prestack depth migration (see Section 5), where GFs are to be calculated on a dense<br />
grid, classic ray trac<strong>in</strong>g is def<strong>in</strong>itely not efficient enough. A more attractive approach is<br />
the WFC technique exposed <strong>in</strong> Section 3. Efficient GFs calculations are <strong>in</strong>deed a necessity<br />
for RT-FD hybrid modell<strong>in</strong>g, especially <strong>in</strong> 3D, even though the hybrid method requires<br />
a much smaller number <strong>of</strong> calculation po<strong>in</strong>ts than prestack depth migration.<br />
F<strong>in</strong>ite-difference (FD) modell<strong>in</strong>g<br />
While RT is utiliz<strong>in</strong>g an asymptotic solution <strong>of</strong> the wave equation, FD modell<strong>in</strong>g is<br />
based on direct discretization <strong>of</strong> the govern<strong>in</strong>g differential equations. To enhance accuracy<br />
the model parameters, the wavefields, <strong>and</strong> the stress components are sampled on staggered<br />
grids (Virieux, 1984; Virieux, 1986). The length <strong>of</strong> the spatial differential operators may<br />
vary from one po<strong>in</strong>t on each side <strong>of</strong> the po<strong>in</strong>t where the derivatives are to be estimated<br />
(2nd order scheme) to as many po<strong>in</strong>ts as one wishes (Holberg, 1987). The differential<br />
operator <strong>in</strong> time is usually <strong>of</strong> 2nd order. The wavefield obta<strong>in</strong>ed by FD modell<strong>in</strong>g can be<br />
stored either as a snapshot, i.e., at a given time iteration for the whole grid, or as<br />
seismograms at a set <strong>of</strong> receivers, imply<strong>in</strong>g that the wavefield is recorded time step after<br />
time step at specific locations.<br />
F<strong>in</strong>ite-difference calculation are constra<strong>in</strong>ed by two conditions. The stability condition<br />
controls that the time sampl<strong>in</strong>g ∆t is not too large with respect to the smallest grid<br />
sampl<strong>in</strong>g ∆h follow<strong>in</strong>g a rule <strong>of</strong> the form:<br />
Vmax∆t ≤ α<br />
∆h<br />
where Vmax is the largest velocity <strong>in</strong> the model, usually a P-wave velocity. The other<br />
constra<strong>in</strong>t is the dispersion condition which assure that the smallest wavelength is not<br />
aliased due to a too coarse grid, i.e., a m<strong>in</strong>imum number <strong>of</strong> grid po<strong>in</strong>ts per wavelength is<br />
def<strong>in</strong>ed:<br />
ν<br />
Vm<strong>in</strong><br />
≤ Nm<strong>in</strong><br />
max∆h<br />
Vm<strong>in</strong> is the smallest velocity <strong>in</strong> the model, usually a S-wave velocity if the calculation is<br />
elastic, <strong>and</strong> νmax is the highest frequency <strong>of</strong> the signal. Both α <strong>and</strong> Nm<strong>in</strong> depend on the<br />
f<strong>in</strong>ite-difference scheme <strong>and</strong> more especially on the length <strong>of</strong> the spatial differential<br />
operators.<br />
FD modell<strong>in</strong>g is easy to use <strong>and</strong> always provide a complete wavefield, <strong>in</strong>clud<strong>in</strong>g all<br />
types <strong>of</strong> waves. The major drawback is the cost <strong>of</strong> such modell<strong>in</strong>g which requires both<br />
large computer memory <strong>and</strong> high CPU time, especially <strong>in</strong> 3D. In the latter case, rout<strong>in</strong>e<br />
modell<strong>in</strong>g <strong>of</strong> complete surveys are absolutely out <strong>of</strong> question. Besides the cost <strong>of</strong> FD<br />
modell<strong>in</strong>g, one possible problem is the grid representation <strong>of</strong> the model which may not be<br />
able to represent properly a detailed geological structure. This is especially true when long<br />
spatial differential operators are used to <strong>in</strong>crease the efficiency <strong>of</strong> the calculation by<br />
Stud. geophys. geod., 46 (2002) 129<br />
(1)<br />
(2)
H. Gjøystdal et al.<br />
choos<strong>in</strong>g a coarser grid sampl<strong>in</strong>g. Actual <strong>in</strong>terfaces <strong>of</strong> the model will take the form as<br />
‘step-stairs’, <strong>and</strong> this may <strong>in</strong>duce annoy<strong>in</strong>g artificial diffractions. A classic example is<br />
mar<strong>in</strong>e seismics with a gently vary<strong>in</strong>g sea floor. The artificial diffractions generated at<br />
that level may be very strong compared to the deeper scattered wavefields. There are<br />
some f<strong>in</strong>ite-difference schemes tak<strong>in</strong>g <strong>in</strong>to account a given <strong>in</strong>terface to correct this effect<br />
but this significantly complicates the code <strong>and</strong> <strong>in</strong>creases the CPU time. The borders <strong>of</strong> the<br />
grid are also troublesome due to generation <strong>of</strong> artificial reflections. They are generally<br />
conditioned to be absorb<strong>in</strong>g by us<strong>in</strong>g an attenuation zone or apply<strong>in</strong>g one-way paraxial<br />
wave equations (Clayton <strong>and</strong> Enquist, 1977), or a comb<strong>in</strong>ation <strong>of</strong> these two techniques.<br />
But the attenuation zones significantly <strong>in</strong>crease the size <strong>of</strong> the grid, which is especially<br />
prejudicial <strong>in</strong> 3D. The one-way paraxial wave equations used to cancel reflections on the<br />
sides require less memory but the absorption depends on the <strong>in</strong>cidence angle. Even with<br />
good absorb<strong>in</strong>g boundaries, artificial reflections generated by the direct wave may be <strong>of</strong><br />
the same order <strong>of</strong> magnitude as the scattered wavefields <strong>of</strong> <strong>in</strong>terest, especially for the deep<br />
structures.<br />
RT/FD comparisons<br />
Table 1 summarizes for each modell<strong>in</strong>g approach its characteristics, advantages, <strong>and</strong><br />
drawbacks. It is easy to see that the RT <strong>and</strong> FD approach complete each other very well,<br />
i.e., a drawback <strong>of</strong> FD is <strong>of</strong>ten an advantage <strong>of</strong> RT <strong>and</strong> vice versa. For <strong>in</strong>stance, the wave<br />
selectivity <strong>of</strong> RT is not available with FD, which is act<strong>in</strong>g as a ‘black-box’. At the same<br />
time, this wave selectivity <strong>in</strong> RT obliges the user to def<strong>in</strong>e a priori all possible significant<br />
waves if wish<strong>in</strong>g a complete calculation, while FD does that automatically. Figure 12<br />
shows the result <strong>of</strong> modell<strong>in</strong>g performed with both methods for the same model, survey<br />
<strong>and</strong> source pulse. The model is not complicated <strong>in</strong> terms <strong>of</strong> the number <strong>of</strong> layers but does<br />
show strong fault<strong>in</strong>g. In the RT modell<strong>in</strong>g we requested the direct transmitted wave <strong>and</strong><br />
all primary P-wave reflections, as well as diffractions correspond<strong>in</strong>g to the direct wave.<br />
The diffractions were obta<strong>in</strong>ed us<strong>in</strong>g the theory <strong>of</strong> Klem-Musatov (1994). The FD<br />
modell<strong>in</strong>g was run with absorb<strong>in</strong>g conditions at the surface <strong>in</strong> order to avoid multiples<br />
(the only possibility to reject arrivals <strong>in</strong> FD). To illustrate the wave propagation with<strong>in</strong><br />
RT, direct transmitted wavefronts calculated by the WFC method <strong>and</strong> raypaths from twopo<strong>in</strong>t<br />
RT are superimposed on the model (Figure 12a, top). The ‘holes’ appear<strong>in</strong>g on the<br />
wavefronts when cross<strong>in</strong>g the faulted area due to diffractions (not shown here) or overcritical<br />
<strong>in</strong>cidences (head waves are miss<strong>in</strong>g). A snapshot <strong>of</strong> the FD wavefield at 1.5 s <strong>of</strong><br />
propagation clearly shows a very complicated pattern where the different arrivals are<br />
difficult to analyze (Figure 12b, top). In the FD snapshot we can identify head waves <strong>and</strong><br />
diffractions <strong>in</strong> addition to reflections <strong>and</strong> pure transmission. The RT seismograms<br />
(Figure 12a, bottom) are not complete but give rather good results for the direct<br />
transmitted waves for most <strong>of</strong> the receivers. Later arrivals (primary reflections) are more<br />
discont<strong>in</strong>uous due to the complexity <strong>of</strong> the structure <strong>and</strong> diffractions would be necessary<br />
to avoid sharp cut<strong>of</strong>fs <strong>of</strong> the reflected phases. Other phases may eventually give<br />
significant arrivals, but it is difficult to def<strong>in</strong>e them a priori. On the other h<strong>and</strong>, the FD<br />
seismograms appear to be more complete <strong>and</strong> cont<strong>in</strong>uous, but wave identification is very<br />
difficult. Note the rather strong arrivals at about 2 s for the deepest receivers (Figure 12b,<br />
bottom), or equivalently, consider the phase identified with white arrows on the snapshot<br />
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(Figure 12b, top). These arrivals correspond to an artificial box-side reflection not totally<br />
suppressed by the chosen one-way wave equation absorb<strong>in</strong>g conditions (Reynolds, 1978).<br />
Without a ray analysis, it is very difficult to tell wether or not such arrivals are real. The<br />
CPU cost <strong>of</strong> RT here was very small (below 1 m<strong>in</strong>.) compared to FD (about one hour),<br />
though add<strong>in</strong>g diffractions <strong>in</strong> RT may significantly <strong>in</strong>crease this cost. It is anyway a good<br />
habit to first try RT to better constra<strong>in</strong> wave propagation <strong>in</strong> a model, then apply FD if<br />
CPU <strong>and</strong> memory resources are available, <strong>and</strong> f<strong>in</strong>ally re-use RT to analyze the FD results.<br />
Fig. 12. RT/FD modell<strong>in</strong>g comparisons <strong>in</strong> a complex model <strong>and</strong> with VSP acquisition. (a) RT<br />
modell<strong>in</strong>g with direct transmitted wavefronts <strong>and</strong> a few selected raypaths for both direct<br />
transmission <strong>and</strong> primary reflections (top) <strong>and</strong> seismograms (bottom). (b) FD modell<strong>in</strong>g with<br />
snapshot <strong>of</strong> the wavefield at 1.5 s (top) <strong>and</strong> seismograms (bottom). One artificial left box-side<br />
reflection is identified on both the snapshot (white arrows) <strong>and</strong> the seismograms (dashed l<strong>in</strong>e).<br />
Stud. geophys. geod., 46 (2002) 131
Table 1.<br />
Type <strong>of</strong><br />
modell<strong>in</strong>g<br />
<strong>Ray</strong><br />
trac<strong>in</strong>g<br />
F<strong>in</strong>itedifference<br />
H. Gjøystdal et al.<br />
Model Arrivals Efficiency Accuracy<br />
- <strong>in</strong>terfaces<br />
separat<strong>in</strong>g layers<br />
BUT<br />
- smoothness is<br />
required<br />
- how smooth?<br />
- parameter grids<br />
BUT<br />
- ‘step-stairs’<br />
troubles<br />
- f<strong>in</strong>est detail =<br />
sampl<strong>in</strong>g<br />
- full selectivity<br />
BUT<br />
- what waves to<br />
select?<br />
- no head waves, no<br />
surface waves,<br />
approximate<br />
diffractions<br />
- complete wavefield<br />
BUT<br />
- what is what?<br />
- reflections from the<br />
borders.<br />
- fast<br />
BUT<br />
- model dependent<br />
- decreases with<br />
<strong>in</strong>creas<strong>in</strong>g<br />
complexity,<br />
no. <strong>of</strong> layers, arrivals<br />
- model <strong>in</strong>dependent<br />
BUT<br />
- slow: model size,<br />
model sampl<strong>in</strong>g <strong>and</strong><br />
frequency dependent<br />
- accurate for smooth<br />
models<br />
BUT<br />
- high-frequency<br />
approximation<br />
- caustic problems<br />
- can be controlled<br />
BUT<br />
- stability <strong>and</strong> dispersion<br />
need to be verified,<br />
- constra<strong>in</strong>ed sampl<strong>in</strong>g<br />
<strong>in</strong> time <strong>and</strong> space<br />
4.2. Where <strong>and</strong> how perform a hybrid RT-FD modell<strong>in</strong>g?<br />
In the follow<strong>in</strong>g we consider the idea <strong>of</strong> apply<strong>in</strong>g RT <strong>and</strong> FD successively <strong>in</strong> a hybrid<br />
comb<strong>in</strong>ation. Orig<strong>in</strong>ally developed for typical oil/gas reservoirs found e.g. on the<br />
Norwegian cont<strong>in</strong>ental shelf (mar<strong>in</strong>e seismics), the hybrid technique is suited for<br />
structures with a rather gentle <strong>and</strong> smooth overburden over a more complex <strong>and</strong> possibly<br />
faulted area <strong>in</strong>clud<strong>in</strong>g a reservoir. The overburden will then be the RT modell<strong>in</strong>g doma<strong>in</strong><br />
<strong>and</strong> the target (i.e., the reservoir) will be the FD modell<strong>in</strong>g doma<strong>in</strong>. There is however no<br />
theoretical limitation <strong>of</strong> the hybrid concept to that type <strong>of</strong> structures only. Alternatively,<br />
we could imag<strong>in</strong>e other type <strong>of</strong> structures where the complexity is close to the surface,<br />
while the target itself could be properly described with RT models. This may be more the<br />
case <strong>in</strong> l<strong>and</strong> seismics, for <strong>in</strong>stance <strong>in</strong> Canada.<br />
As orig<strong>in</strong>ally proposed by Lecomte (1996) <strong>in</strong> acoustics, <strong>and</strong> later further developed <strong>in</strong><br />
elastics (Hokstad et al., 1998), the hybrid modell<strong>in</strong>g scheme proceeds <strong>in</strong> three steps as<br />
illustrated <strong>in</strong> Figure 13, assum<strong>in</strong>g that a target has been selected. RT <strong>and</strong> FD is then totally<br />
de-coupled, i.e., the processes are applied successively with communication via well<br />
def<strong>in</strong>ed data sets (e.g. disk files).<br />
First step: The <strong>in</strong>cident wavefield − RT<br />
The WFC technique <strong>of</strong> RT is used to efficiently compute the GFs connect<strong>in</strong>g any shot<br />
<strong>of</strong> the survey to a coupl<strong>in</strong>g l<strong>in</strong>e (2D) or surface (3D). The overburden should fit for RT<br />
modell<strong>in</strong>g, i.e., with proper smoothness <strong>of</strong> <strong>in</strong>terfaces <strong>and</strong> property fields, <strong>and</strong> with no<br />
major fault<strong>in</strong>g which could generate holes <strong>in</strong> the wavefield. At that stage, we benefit from<br />
the wave selectivity <strong>of</strong> RT, i.e., the user chooses what arrivals to send to the target.<br />
Furthermore, we will not suffer from any artificial noise <strong>in</strong> FD such as diffractions at the<br />
sea floor or box-side reflections. Note that <strong>in</strong> most cases the user selects the direct<br />
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transmitted P-wave as the <strong>in</strong>cident wave <strong>and</strong> direct transmission is less sensitive to small<br />
scale fluctuations <strong>of</strong> the <strong>in</strong>terfaces or parameter fields than reflections. Once the GFs are<br />
calculated, any wavelet can be convolved to generate a wavefield <strong>in</strong>cident to the coupl<strong>in</strong>g<br />
l<strong>in</strong>e (Figure 13, step 1). This wavefield can be re-used for any model <strong>of</strong> the target as long<br />
as the velocity structure is not changed above <strong>and</strong> at the coupl<strong>in</strong>g l<strong>in</strong>e. This <strong>in</strong>creases<br />
efficiency for repeated modell<strong>in</strong>g (Gjøystdal et al., 1998).<br />
Second step: The scattered wavefield − FD<br />
The <strong>in</strong>cident wavefield can now propagate through the target via a local FD modell<strong>in</strong>g<br />
(Figure 13, step 2). The goal here is to record the scattered energy com<strong>in</strong>g back from the<br />
structures <strong>in</strong> the target <strong>in</strong> order to send further this energy towards the receivers <strong>in</strong> the<br />
third step. A special treatment must be applied to prevent the <strong>in</strong>cident wavefield to<br />
propagate upwards, act<strong>in</strong>g as an artificial reflection from the coupl<strong>in</strong>g l<strong>in</strong>e. Complete<br />
boundary conditions may be used to constra<strong>in</strong> the propagation downwards (Hokstad et al.,<br />
1998). Alternatively, the scattered field from a background target model, with no<br />
scatter<strong>in</strong>g structures below the coupl<strong>in</strong>g l<strong>in</strong>e, may be subtracted. The ga<strong>in</strong> <strong>in</strong> cost, with<br />
respect to a classic global FD modell<strong>in</strong>g, is directly proportional to the reduction <strong>in</strong> size <strong>of</strong><br />
the FD-grid times the reduction <strong>in</strong> number <strong>of</strong> time iterations as the <strong>in</strong>terest<strong>in</strong>g time<br />
w<strong>in</strong>dow is shorter. We will here benefit from FD modell<strong>in</strong>g <strong>and</strong> get a complete scattered<br />
wavefield, which was however generated from the <strong>in</strong>cident field explicitly chosen at the<br />
first stage. We thus ga<strong>in</strong> partly the flexibility <strong>of</strong> RT <strong>in</strong> choos<strong>in</strong>g waves: we can for<br />
<strong>in</strong>stance study the difference <strong>in</strong> illum<strong>in</strong>ation <strong>of</strong> a target by primary reflections <strong>and</strong> sea<br />
floor generated multiples. The local FD modell<strong>in</strong>g can be easily applied <strong>in</strong> parallel for<br />
different target models, <strong>and</strong> distributed to a network <strong>of</strong> computers without requir<strong>in</strong>g huge<br />
capacity <strong>in</strong> CPU <strong>and</strong> memory.<br />
Fig. 13. Scheme <strong>of</strong> the three-step process <strong>of</strong> the hybrid RT-FD modell<strong>in</strong>g. (1) Incident field<br />
calculation us<strong>in</strong>g RT calculated GFs. (2) Scattered field calculation <strong>in</strong> the target with a local FD.<br />
(3) Recorded field calculation at the receivers us<strong>in</strong>g Kirchh<strong>of</strong>f <strong>in</strong>tegral <strong>and</strong> RT calculated GFs.<br />
Stud. geophys. geod., 46 (2002) 133
H. Gjøystdal et al.<br />
Fig. 14. Example <strong>of</strong> the three-step process (acoustic). (1) Incident field to the local FD box with<br />
coupl<strong>in</strong>g at the top. (2) Scattered field com<strong>in</strong>g out from the local FD. (3) Recorded field after<br />
wavefield extrapolation <strong>of</strong> the scattered wavefield us<strong>in</strong>g Kirchh<strong>of</strong>f <strong>in</strong>tegral.<br />
Third step: The recorded wavefield − RT<br />
In the last step, the scattered wavefield emerg<strong>in</strong>g from the local FD-grid is propagated<br />
further towards the receivers by us<strong>in</strong>g the Kirchh<strong>of</strong>f <strong>in</strong>tegral <strong>and</strong> GFs connect<strong>in</strong>g the<br />
coupl<strong>in</strong>g l<strong>in</strong>e to these receivers (Figure 13, step 3). We then aga<strong>in</strong> benefit from RT<br />
selectivity by choos<strong>in</strong>g the paths to follow, i.e., direct transmitted P or direct transmitted<br />
S, <strong>and</strong> so on.<br />
To summarize the hybrid RT-FD modell<strong>in</strong>g, Figure 14 shows a model <strong>in</strong>spired from<br />
exist<strong>in</strong>g structures <strong>in</strong> the North Sea, with the wavefield attached to each step <strong>of</strong> the<br />
process, the coupl<strong>in</strong>g l<strong>in</strong>e be<strong>in</strong>g at the top <strong>of</strong> the FD-grid <strong>in</strong> that example (not required <strong>in</strong><br />
general). In order to check the validity <strong>of</strong> the results, we can compare seismograms<br />
obta<strong>in</strong>ed with RT (Figure 15a), FD (Figure 15b), <strong>and</strong> the hybrid modell<strong>in</strong>g (Figure 15c).<br />
The latter gave a more complete result than RT <strong>and</strong> a better result than the pure FD<br />
(notice the box-side reflections appear<strong>in</strong>g <strong>in</strong> Figure 15b), but at a much cheaper price. The<br />
cost <strong>of</strong> hybrid modell<strong>in</strong>g is about 20% <strong>of</strong> the global FD at the most, although the<br />
parameters were not optimally chosen because we kept the same grid sampl<strong>in</strong>g for the<br />
pure (global) FD <strong>and</strong> the local FD (<strong>in</strong>herent <strong>in</strong> the hybrid approach). In practice, the<br />
optimum grid sampl<strong>in</strong>g depend on the smallest velocity, which is <strong>of</strong>ten located close to<br />
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Fig. 15. Comparison <strong>of</strong> RT, FD <strong>and</strong> hybrid RT-FD results. (a) Classic two-po<strong>in</strong>t RT modell<strong>in</strong>g (no<br />
diffractions). (b) Classic acoustic FD modell<strong>in</strong>g <strong>of</strong> the whole model. (c) Hybrid RT-FD modell<strong>in</strong>g<br />
<strong>of</strong> the target.<br />
the surface. Thereby, the grid sampl<strong>in</strong>g can be coarser for a deep local FD modell<strong>in</strong>g than<br />
for a global FD modell<strong>in</strong>g. Alternatively, one may wish to use the hybrid RT-FD<br />
modell<strong>in</strong>g for detailed models <strong>of</strong> the target, i.e., with sampl<strong>in</strong>g down to just a few meters,<br />
especially vertically. This may be necessary for a proper representation <strong>of</strong> the model as<br />
deduced from well <strong>in</strong>formation. For global FD the cost is then very high because this<br />
sampl<strong>in</strong>g is to be used all over the grid.<br />
Studies carried out the last years show that, even <strong>in</strong> 2D, the hybrid RT-FD modell<strong>in</strong>g<br />
concept may be very powerful for detailed reservoir modell<strong>in</strong>g, especially for 4D studies<br />
with repeated modell<strong>in</strong>g where only the target is to be modified (only the local FD is<br />
affected).<br />
5. PRESTACK DEPTH MIGRATION AND RAY THEORY<br />
Integral approaches us<strong>in</strong>g pre-calculated Green’s functions (GFs) are popular<br />
techniques <strong>in</strong> seismic imag<strong>in</strong>g s<strong>in</strong>ce the eighties (Schneider, 1978; Beylk<strong>in</strong>, 1985;<br />
Bleiste<strong>in</strong>, 1987; Miller et al., 1987). Though significant differences exist <strong>in</strong> the way the<br />
problem is solved, all <strong>in</strong>tegral approaches are usually regrouped under the general term <strong>of</strong><br />
Kirchh<strong>of</strong>f migration (Claerbout, 1985). Some do rely on the Kirchh<strong>of</strong>f <strong>in</strong>tegral, which is<br />
used for backward wavefield extrapolation from the receiver l<strong>in</strong>e/surface. This<br />
Stud. geophys. geod., 46 (2002) 135
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extrapolated wavefield is then correlated to the propagated source wavefield to extract the<br />
reflectivity at a po<strong>in</strong>t (Schneider, 1978). Note that the ‘true’ Kirchh<strong>of</strong>f <strong>in</strong>tegral, strictly<br />
speak<strong>in</strong>g, applies only <strong>in</strong> the constant-velocity case (Claerbout, 1985). Moreover, this<br />
<strong>in</strong>tegral is <strong>in</strong> practice not used directly as it requires knowledge <strong>of</strong> both pressure <strong>and</strong><br />
particle velocity data on a closed surface (Berkhout, 1980). But, assum<strong>in</strong>g a horizontal <strong>and</strong><br />
flat record<strong>in</strong>g l<strong>in</strong>e/surface, <strong>Ray</strong>leigh <strong>in</strong>tegrals are derived from the Kirchh<strong>of</strong>f <strong>in</strong>tegral <strong>and</strong><br />
the <strong>in</strong>tegration is easier, with <strong>of</strong>ten further simplifications where near-field <strong>and</strong> secondorder<br />
terms are neglected. The other type <strong>of</strong> methods (Beylk<strong>in</strong>, 1985; Bleiste<strong>in</strong>, 1987;<br />
Miller et al., 1987), based on <strong>in</strong>version theory <strong>and</strong> Born approximation, takes also the<br />
form <strong>of</strong> ‘a Kirchh<strong>of</strong>f migration <strong>of</strong> frequency filtered <strong>and</strong> spatially weighted traces, with an<br />
<strong>in</strong>tegral kernel dictated by <strong>in</strong>version theory’ (Bleiste<strong>in</strong>, 1987). These techniques are<br />
sometimes called Born migration, though aga<strong>in</strong> usually considered as belong<strong>in</strong>g to the<br />
Kirchh<strong>of</strong>f migration family. In recent years, the term <strong>of</strong> diffraction-stack (DS) migration<br />
has also emerged <strong>in</strong> the Kirchh<strong>of</strong>f migration family, attention be<strong>in</strong>g focused on preserv<strong>in</strong>g<br />
the amplitudes dur<strong>in</strong>g the migration by properly remov<strong>in</strong>g geometrical spread<strong>in</strong>g,<br />
transmission loss <strong>and</strong> caustics effects (Hubral et al., 1991; Schleicher et al., 1993;<br />
Hanitzsch et al., 1994). At last, another <strong>in</strong>tegral approach, called generalized diffraction<br />
tomography (GDT) was developed under the Born or Rytov approximations (Devaney<br />
<strong>and</strong> Zhang, 1991; Gelius et al., 1991), for heterogeneous background <strong>in</strong> opposition to the<br />
classic diffraction tomography techniques (Devaney, 1982; Wu <strong>and</strong> Toksöz, 1987; Lo <strong>and</strong><br />
Toksöz, 1988), <strong>and</strong> was orig<strong>in</strong>ally used for transmission-mode geometry <strong>in</strong> cross-well<br />
tomography. GDT is a less well-known technique <strong>in</strong> seismic imag<strong>in</strong>g but Hamran <strong>and</strong><br />
Lecomte (1993) developed a local plane-wavenumber approach <strong>of</strong> GDT <strong>and</strong> showed its<br />
applications <strong>in</strong> reflection-mode geometry (Lecomte, 1999; Lecomte et al., 2001).<br />
All Kirchh<strong>of</strong>f migration techniques, either ‘true’ Kirchh<strong>of</strong>f like diffraction-stack, or<br />
Born-type, like GDT, require pre-calculated GFs as provided by ray techniques. The<br />
traveltime is the major parameter necessary for migration. For amplitude-preserv<strong>in</strong>g<br />
migration, amplitudes are also necessary though very <strong>of</strong>ten, as a first choice, the data are<br />
rout<strong>in</strong>ely corrected for a simple spherical divergence <strong>in</strong> order to get similar strength for all<br />
reflectors. The Kirchh<strong>of</strong>f techniques may <strong>of</strong>ten need also to use ray direction <strong>in</strong>formation.<br />
This is the case for the obliquity or directivity factor, i.e., the cos<strong>in</strong>e <strong>of</strong> the angle between<br />
the direction <strong>of</strong> propagation at the receiver <strong>and</strong> the vertical axis (horizontal record<strong>in</strong>g<br />
l<strong>in</strong>es). <strong>Ray</strong> directions are also necessary <strong>in</strong> the Beylk<strong>in</strong> determ<strong>in</strong>ant, i.e., the Jacobian <strong>of</strong><br />
the transformation from image to surface coord<strong>in</strong>ates, where the <strong>in</strong>tegration takes place <strong>in</strong><br />
Born-type approaches (Zhang et al., 2000). Even after tak<strong>in</strong>g <strong>in</strong>to account all the<br />
weight<strong>in</strong>g factors <strong>and</strong> ‘true-amplitude’ (TA) corrections, we may not retrieve directly<br />
from the migrated section the true reflectivity (so the term TA migration is mislead<strong>in</strong>g).<br />
Other effects rema<strong>in</strong>, like resolution (Lecomte <strong>and</strong> Gelius, 1998; Gelius <strong>and</strong> Lecomte,<br />
2000) <strong>and</strong> illum<strong>in</strong>ation (Rosl<strong>and</strong> <strong>and</strong> Drivenes, 2000), without speak<strong>in</strong>g <strong>of</strong> possible<br />
coherent noise <strong>in</strong>terfer<strong>in</strong>g with the structure <strong>and</strong> events not correspond<strong>in</strong>g to the<br />
calculated GFs. As far as illum<strong>in</strong>ation is concerned, a technique was proposed by V<strong>in</strong>je<br />
(2000), where ray trac<strong>in</strong>g is used to correct vary<strong>in</strong>g illum<strong>in</strong>ation <strong>of</strong> a target reflector after<br />
PSDM. <strong>Ray</strong> trac<strong>in</strong>g modell<strong>in</strong>g can also be used <strong>in</strong> a survey plann<strong>in</strong>g stage, <strong>in</strong> order to<br />
control illum<strong>in</strong>ation before any data acquisition <strong>and</strong> PSDM. So, prior, dur<strong>in</strong>g <strong>and</strong> after<br />
migration, ray trac<strong>in</strong>g is a powerful technique which <strong>of</strong>ten helps greatly the user to<br />
136 Stud. geophys. geod., 46 (2002)
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underst<strong>and</strong> what happens <strong>in</strong> seismic imag<strong>in</strong>g. In the follow<strong>in</strong>g we shall illustrate the role<br />
<strong>of</strong> various ray trac<strong>in</strong>g parameters (traveltime, ray direction, amplitude) <strong>in</strong> PSDM.<br />
5.1. Traveltimes <strong>and</strong> imag<strong>in</strong>g<br />
Figure 16a shows a simple model we will use to illustrate the role <strong>of</strong> traveltimes <strong>in</strong><br />
PSDM. This model <strong>in</strong>cludes a salt body <strong>in</strong>trud<strong>in</strong>g a few layers <strong>of</strong> constant velocity <strong>and</strong><br />
with a water layer at the top. The survey is a mar<strong>in</strong>e one with a boat shoot<strong>in</strong>g along<br />
a 2 km l<strong>in</strong>e, every 50 m, <strong>and</strong> with a 1 km streamer (25 m sampl<strong>in</strong>g). The target chosen for<br />
PSDM studies is superimposed. For the GF calculation, a model was created which does<br />
not conta<strong>in</strong> the salt body but is an extrapolation <strong>of</strong> the actual model from the left to the<br />
right side as we will only focus on the left side <strong>of</strong> the salt (Figure 16b). Primary P-wave<br />
reflections from all <strong>in</strong>terfaces were generated by dynamic ray trac<strong>in</strong>g, <strong>and</strong> synthetic<br />
seismograms were calculated by convolv<strong>in</strong>g RT results with a 20 Hz zero-phase Ricker<br />
wavelet (Figure 16c). To emphasize the effect <strong>of</strong> traveltime <strong>in</strong> PSDM, we did not apply<br />
neither the geometrical spread<strong>in</strong>g factor nor the reflection/transmission effects. For the<br />
largest <strong>of</strong>fset <strong>of</strong> the shot record<strong>in</strong>g <strong>of</strong> Figure 16c, 5 arrivals are identified by their raypaths<br />
as given <strong>in</strong> Figure 16d. Around ‘arrival 5’ <strong>in</strong> Figure 16c, there are <strong>in</strong> fact other arrivals<br />
correspond<strong>in</strong>g to a reflection at the left flank <strong>of</strong> the salt, illum<strong>in</strong>ated by the primary<br />
P-wave reflection at the reflector at about 2 km <strong>in</strong> depth (arrival 3 is part <strong>of</strong> this primary<br />
reflection). If we now consider only one trace, i.e., the 1 km <strong>of</strong>fset <strong>of</strong> the shot <strong>of</strong><br />
Figure 16c, traveltime <strong>in</strong>formation <strong>of</strong> GFs will be used to map back to depth the energy <strong>of</strong><br />
the seismograms along isochrons (Figure 16e). As will be discussed further <strong>in</strong> Section 7,<br />
an isochron is a surface <strong>of</strong> constant two-way traveltime correspond<strong>in</strong>g to a s<strong>in</strong>gle shot <strong>and</strong><br />
a s<strong>in</strong>gle receiver. Apply<strong>in</strong>g now DS to the chosen trace, we get the PSDM section <strong>of</strong><br />
Figure 16f where we easily identified our 5 arrivals <strong>of</strong> Figure 16c <strong>and</strong> 16d. Their<br />
respective energy has been mapped along the isochrons <strong>of</strong> Figure 16e, as expected, <strong>and</strong><br />
with the same amplitude all along each isochron <strong>in</strong> our case. Referr<strong>in</strong>g to Figure 16d, we<br />
see however that only arrival 2 corresponds to a reflection <strong>in</strong> our target. Reflections 1,3,4<br />
<strong>and</strong> 5 are all associated with part <strong>of</strong> reflectors outside our target but the correspond<strong>in</strong>g<br />
isochrons cross the latter because one trace is not a sufficient <strong>in</strong>formation to locate energy.<br />
Basically, a PSDM section is formed by superposition <strong>of</strong> such elementary ‘one-trace’<br />
images, the idea be<strong>in</strong>g that, all together, constructive <strong>in</strong>terference will occur at the actual<br />
loci <strong>of</strong> reflections, while the useless part <strong>of</strong> the isochrons will be cancelled by destructive<br />
<strong>in</strong>terference.<br />
Figure 17 shows what happens when several traces are migrated. As very <strong>of</strong>ten done,<br />
data can be migrated <strong>in</strong> constant-<strong>of</strong>fset sections, like <strong>in</strong> Figure 17a (zero-<strong>of</strong>fset) <strong>and</strong> 17b<br />
(1 km <strong>of</strong>fset). The three reflectors <strong>of</strong> the target are com<strong>in</strong>g forward, though we still see<br />
some ‘background pattern’ similar to the isochrons <strong>of</strong> Figure 16f <strong>in</strong> both cases. This is<br />
<strong>of</strong>ten called alias<strong>in</strong>g noise, which is due to a too coarse sampl<strong>in</strong>g between the shots: the<br />
<strong>in</strong>terference is not perfect between the isochrons <strong>and</strong> they do not cancel each other where<br />
they should. By migrat<strong>in</strong>g all shots <strong>and</strong> <strong>of</strong>fsets together, the PSDM section gets much<br />
clearer (Figure 17c), though we can still see what rema<strong>in</strong>s <strong>of</strong> an isochron correspond<strong>in</strong>g to<br />
the salt reflection.<br />
Stud. geophys. geod., 46 (2002) 137
H. Gjøystdal et al.<br />
Fig. 16. Salt model. (a) P-velocity from 1.5 km/s (top) to 4.0 km/s (salt) <strong>and</strong> target location.<br />
(b) Model used for Green’s function calculation. (c) Shot record<strong>in</strong>g at 0 km with <strong>of</strong>fsets from 0 to<br />
1 km every 25 m (right side). (d) Primary reflections at 1 km <strong>of</strong>fset with identification <strong>of</strong> 5 arrivals<br />
(see also c). (e) Scatter<strong>in</strong>g traveltime isochrons for 1 km <strong>of</strong>fset. (f) PSDM section for 1 km <strong>of</strong>fset<br />
with identification <strong>of</strong> the same arrivals as <strong>in</strong> (c) <strong>and</strong> (d).<br />
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Fig. 17. PSDM sections for a survey <strong>of</strong> 41 shots from 0 to 2 km every 50 m, <strong>and</strong> same <strong>of</strong>fset range<br />
as <strong>in</strong> Figure 16. (a) zero-<strong>of</strong>fset section. (b) 1 km <strong>of</strong>fset. (c) All shots <strong>and</strong> <strong>of</strong>fset.<br />
In Kirchh<strong>of</strong>f migration techniques, the choice <strong>of</strong> the traveltimes to use is <strong>of</strong>ten a critical<br />
issue, especially because one arrival per grid po<strong>in</strong>t, <strong>and</strong> only one, is the preferred solution<br />
for the sake <strong>of</strong> efficiency. So, should we use first-arrival (but may be associated with very<br />
low energy), highest-amplitude arrival, shortest-path arrival? There is no absolute answer<br />
to these questions because, ideally, we should use all arrivals to get closer to<br />
the − smooth − result <strong>of</strong> wave equation migration, but aga<strong>in</strong> at the cost <strong>of</strong> efficiency. That<br />
is why some people claim that we will soon be ‘beyond’ Kirchh<strong>of</strong>f imag<strong>in</strong>g, argu<strong>in</strong>g that<br />
the computers are develop<strong>in</strong>g so fast that wave-equation migration will take over. But<br />
Kirchh<strong>of</strong>f migration will get faster as well, so may be we can use all arrivals then?<br />
Gaussian-beam techniques <strong>in</strong> RT may also be a better solution than the ‘s<strong>in</strong>gle-raypath’<br />
solutions provided by most GFs calculators, i.e., with events tak<strong>in</strong>g <strong>in</strong>to account the<br />
surround<strong>in</strong>g structures <strong>in</strong> ‘energy tubes’ where the Fresnel zone is considered. This may<br />
<strong>in</strong>duce a smooth<strong>in</strong>g <strong>of</strong> the GFs (especially their amplitudes) which may greatly improve<br />
the quality <strong>of</strong> PSDM sections, giv<strong>in</strong>g smoother images, like <strong>in</strong> wave-equation migration?<br />
This is def<strong>in</strong>itely an <strong>in</strong>terest<strong>in</strong>g topic <strong>of</strong> research, which is <strong>in</strong>deed a fundamental question<br />
for RT techniques: how does the actual propagation <strong>of</strong> the waves compare to the ray<br />
approach? RT techniques will however always be much more <strong>in</strong>formative about what<br />
happens when perform<strong>in</strong>g migration than FD techniques used for wave-equation<br />
migration, which act as black-boxes. As <strong>in</strong> the previous section we f<strong>in</strong>d here <strong>in</strong>deed the<br />
same arguments both for <strong>and</strong> aga<strong>in</strong>st each technique. One should not systematically reject<br />
one method <strong>in</strong> favor <strong>of</strong> the other one, s<strong>in</strong>ce they may <strong>in</strong>deed complete each other.<br />
Figure 18 is an example illustrat<strong>in</strong>g the flexibility <strong>of</strong> RT techniques <strong>in</strong> case <strong>of</strong> complicated<br />
illum<strong>in</strong>ation <strong>of</strong> a structure. Let us consider the same shot gather as <strong>in</strong> Figure 16c <strong>and</strong><br />
migrate it with direct transmitted waves both from shots <strong>and</strong> towards receivers<br />
(Figure 18a). In agreement with the raypaths showed above, we retrieve only a small part<br />
<strong>of</strong> the salt <strong>in</strong> its upper concave part. Note that the small illum<strong>in</strong>ated zone predicted by rays<br />
is <strong>in</strong> fact larger due to Fresnel zone effects <strong>and</strong> poor vertical resolution. If we now<br />
consider GFs comb<strong>in</strong><strong>in</strong>g the primary reflections at the reflector at about 2 km <strong>in</strong> depth<br />
Stud. geophys. geod., 46 (2002) 139
H. Gjøystdal et al.<br />
Fig. 18. Salt flank imag<strong>in</strong>g with shot at 0 km. (a) <strong>Ray</strong>s attached to primary reflections from salt<br />
(top) <strong>and</strong> PSDM section us<strong>in</strong>g only direct transmission GFs both from shot <strong>and</strong> towards receivers<br />
(bottom). (b) Salt flank illum<strong>in</strong>ated from a primary reflection (top), which is used for GFs<br />
calculation from the shot to get the PSDM section (bottom) while direct transmission is still used<br />
towards the receivers.<br />
from the shots <strong>and</strong> direct transmission towards the receivers, the new PSDM section is<br />
totally different, show<strong>in</strong>g now the nearly vertical left flank <strong>of</strong> the salt. Is there any waveequation<br />
migration technique able to take <strong>in</strong>to account such vertical structures? This is<br />
<strong>in</strong>deed the weakness <strong>of</strong> these approaches, so far, rely<strong>in</strong>g onto paraxial approximations <strong>of</strong><br />
the wave-equation <strong>and</strong> limited dip ranges.<br />
5.2. <strong>Ray</strong> directions <strong>and</strong> imag<strong>in</strong>g<br />
While traveltimes play a very important role <strong>in</strong> migration, by locat<strong>in</strong>g properly <strong>in</strong><br />
depth the recorded seismic energy, other RT parameters are also significant <strong>and</strong> very<br />
<strong>in</strong>formative. We chose here to illustrate the concept <strong>of</strong> the scatter<strong>in</strong>g wavenumber vector<br />
K def<strong>in</strong>ed by:<br />
K = kr −ks<br />
140 Stud. geophys. geod., 46 (2002)<br />
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<strong>Review</strong> <strong>of</strong> <strong>Ray</strong> <strong>Theory</strong> <strong>Applications</strong> <strong>in</strong> Modell<strong>in</strong>g <strong>and</strong> Imag<strong>in</strong>g <strong>of</strong> Seismic Data<br />
Fig. 19. Scatter<strong>in</strong>g wavenumber K. (a) Def<strong>in</strong>ition. (b) K is perpendicular to a reflector if reflection<br />
is attached to it. These two angles are equal if P-wave to P-Wave, respectively S-wave to S-wave,<br />
scatter<strong>in</strong>g, but different if P- to S-wave <strong>and</strong> vice versa.<br />
where ks <strong>and</strong> kr are the two local wavenumbers given <strong>in</strong> Figure 19a, i.e., the wavenumber<br />
attached to the <strong>in</strong>cident field at the considered image po<strong>in</strong>t, respectively the wavenumber<br />
attached to the scattered field at the same po<strong>in</strong>t. This vector, or more precisely its<br />
opposite, was already part <strong>of</strong> the early Born approaches <strong>in</strong> Kirchh<strong>of</strong>f migration (Beylk<strong>in</strong>,<br />
1985). In GDT approaches (Hamran <strong>and</strong> Lecomte, 1993), K is a key parameter which<br />
completely describes the illum<strong>in</strong>ation <strong>and</strong> the resolution at each image po<strong>in</strong>t. Seismic<br />
imag<strong>in</strong>g can <strong>in</strong>deed be assimilated to a generalized Fourier transform (FT) <strong>and</strong> the local<br />
coverage with respect to length <strong>and</strong> orientation <strong>of</strong> K vectors <strong>in</strong>dicates how much<br />
<strong>in</strong>formation we have about the local scatter<strong>in</strong>g properties <strong>of</strong> the structure. The more K<br />
coverage we have, the better is the possibility for illum<strong>in</strong>ation <strong>and</strong> sharp resolution<br />
(Lecomte <strong>and</strong> Gelius, 1998). Furthermore, while most classic <strong>in</strong>tegral approaches apply<br />
a coord<strong>in</strong>ate transformation from the scatter<strong>in</strong>g wavenumber back to the shot <strong>and</strong> receiver<br />
l<strong>in</strong>es, a much simpler approach is to stay <strong>in</strong> a local coord<strong>in</strong>ate system (Figure 19b). This<br />
system is simply polar <strong>in</strong> 2D <strong>and</strong> spherical <strong>in</strong> 3D, <strong>and</strong> this is very similar to what is done<br />
<strong>in</strong> classic spotlight-mode synthetic aperture radar (SAR) imag<strong>in</strong>g (Lecomte et al., 2001).<br />
In a local 2D coord<strong>in</strong>ate system, K is def<strong>in</strong>ed by an orientation angle, <strong>and</strong> a length<br />
function <strong>of</strong> the frequency <strong>and</strong> the angle θ (Figure 19b). If there was a reflection attached<br />
to the wavenumber couple (ks, kr), K is perpendicular to that reflector.<br />
To illustrate the importance <strong>of</strong> the scatter<strong>in</strong>g wavenumber vector, determ<strong>in</strong>ed by RT<br />
from ray directions at the image po<strong>in</strong>t, a realistic model <strong>of</strong> the Gullfaks oil field on the<br />
Norwegian cont<strong>in</strong>ental shelf has been used (Figure 20). Synthetic data correspond<strong>in</strong>g to<br />
the reservoir target <strong>in</strong>dicated <strong>in</strong> Figure 20a, with an oil/water contact, were generated by<br />
the hybrid RT/FD modell<strong>in</strong>g method exposed earlier. Three data set were used to generate<br />
PSDM sections: zero-<strong>of</strong>fset (Figure 20b), far-<strong>of</strong>fset (Figure 20c) <strong>and</strong> all data (Figure 20d).<br />
On the right side <strong>of</strong> the constant-<strong>of</strong>fset sections, the local scatter<strong>in</strong>g wavenumber coverage<br />
at two po<strong>in</strong>ts <strong>of</strong> the target are given. In each case, the white arrow <strong>in</strong>dicates the mean<br />
scatter<strong>in</strong>g wavenumber, i.e., the reflector normal which is best illum<strong>in</strong>ated, while the grey<br />
arrow is the perpendicular to the actual local reflector. In the zero-<strong>of</strong>fset case, reflector 1<br />
is rather poorly illum<strong>in</strong>ated (grey arrow near the border <strong>of</strong> the K coverage <strong>and</strong> far from<br />
Stud. geophys. geod., 46 (2002) 141
H. Gjøystdal et al.<br />
Fig. 20. Illum<strong>in</strong>ation <strong>and</strong> resolution. (a) Model <strong>of</strong> the Gullfaks field on the Norwegian cont<strong>in</strong>ental<br />
shelf with survey (161 shots every 25 m, 100 <strong>of</strong>fsets from 0. to 2475 m) <strong>and</strong> PSDM target. (b) Zero<strong>of</strong>fset<br />
PSDM section with superimposed Po<strong>in</strong>t-Spread functions (PSFs, 100×100 m 2 ) at some<br />
locations (left). Scatter<strong>in</strong>g wavenumber coverage at two reflectors (right). (c) Far-<strong>of</strong>fset PSDM<br />
section. Same as for (b). (d) All shots <strong>and</strong> <strong>of</strong>fsets section with superimposed PSFs. Courtesy <strong>of</strong><br />
Statoil.<br />
the white arrow) while reflector 2 is clearly better illum<strong>in</strong>ated. This is confirmed by the<br />
PSDM section which shows a strong reflector 2 <strong>and</strong> weak reflector 1. This effect is not<br />
only due to the impedance contrast <strong>and</strong> does really reflect the differences <strong>in</strong> illum<strong>in</strong>ation,<br />
as confirmed by the far-<strong>of</strong>fset section which shows now the opposite, i.e., a strong<br />
reflector 1 <strong>and</strong> a non-exist<strong>in</strong>g reflector 2. The latter result is aga<strong>in</strong> <strong>in</strong> agreement with the<br />
illum<strong>in</strong>ation analysis provided by the local K coverages. Note a little ‘alias<strong>in</strong>g-noise’ on<br />
the zero-<strong>of</strong>fset section due to the same effect mentioned <strong>in</strong> Figure 17a.<br />
The local K coverage does not only give <strong>in</strong>formation about illum<strong>in</strong>ation but also about<br />
the expected resolution <strong>of</strong> the <strong>in</strong>version process which is seismic imag<strong>in</strong>g. The FT <strong>of</strong> the<br />
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K coverage is <strong>in</strong>deed giv<strong>in</strong>g us local 2D resolution functions <strong>and</strong> a few <strong>of</strong> them are<br />
superimposed on the PSDM sections <strong>of</strong> Figure 20. These resolution functions are more<br />
precisely here po<strong>in</strong>t-spread functions (PSFs), correspond<strong>in</strong>g to the FT <strong>of</strong> the whole K<br />
coverage <strong>and</strong> <strong>in</strong> agreement with a Born approach where the scatter<strong>in</strong>g structures are seen<br />
as (weak) diffractors, radiat<strong>in</strong>g energy <strong>in</strong> all directions. These PSFs give a good idea <strong>of</strong><br />
the resolution across the reflectors <strong>and</strong> the zero-<strong>of</strong>fset PSDM section is ‘sharper’ than the<br />
far-<strong>of</strong>fset one because the K vectors are longer at zero-<strong>of</strong>fset (θ is close to zero <strong>in</strong><br />
Figure 19b, see K coverage <strong>in</strong> Figure 20b <strong>and</strong> 20c). The PSFs give however a poorer<br />
representation <strong>of</strong> the resolution tangent to the reflector as they do not take <strong>in</strong>to account<br />
that the reflected energy is preferentially radiated along K perpendicular to the reflector<br />
(Snell’s law). To better illustrate this, synthetic PSDM sections generated by convolv<strong>in</strong>g<br />
the reflectivity <strong>of</strong> the target (Figure 21a) <strong>and</strong> either PSFs (Figure 21b, left) or a dipfiltered<br />
version <strong>of</strong> PSFs (Figure 21b, right), called reflector-spread functions (RSFs,<br />
Gelius <strong>and</strong> Lecomte, 2000), can be compared with the correspond<strong>in</strong>g far-<strong>of</strong>fset section<br />
(Figure 21c). The RSFs do a much better job, smooth<strong>in</strong>g the reflectivity function<br />
‘laterally’ as we expect the seismic wave to do due to the Fresnel zone. Note that we did<br />
not take <strong>in</strong>to account here the illum<strong>in</strong>ation effect, i.e., we should have weighted each<br />
image po<strong>in</strong>t by an illum<strong>in</strong>ation factor to really get a synthetic PSDM section closer to the<br />
actual one. This synthetic image does not reproduce the possible alias<strong>in</strong>g noise. However,<br />
such a simple synthetic PSDM image, us<strong>in</strong>g only ray directions from the GFs, gives<br />
a good <strong>in</strong>dication <strong>of</strong> the expected resolution <strong>of</strong> a target <strong>and</strong> could help an <strong>in</strong>terpreter to<br />
decide wether or not picked events on PSDM sections are realistic.<br />
Fig. 21. Simple synthetic imag<strong>in</strong>g. (a) Normal-<strong>in</strong>cidence reflectivity <strong>of</strong> the target. (b) Synthetic<br />
PSDM images obta<strong>in</strong>ed by convolv<strong>in</strong>g reflectivity <strong>and</strong> resolution functions with, PSFs (left) <strong>and</strong><br />
Reflector-Spread functions (RSFs, right) for the far <strong>of</strong>fset. (c) Far-<strong>of</strong>fset PSDM section for<br />
comparison (same as <strong>in</strong> Figure 20c).<br />
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5.3. Amplitudes <strong>and</strong> imag<strong>in</strong>g<br />
As already mentioned, DS techniques are called true-amplitude preserv<strong>in</strong>g migration<br />
because the geometrical spread<strong>in</strong>g, transmission losses <strong>and</strong> caustics effects between<br />
shot/receiver <strong>and</strong> the considered image po<strong>in</strong>t are taken <strong>in</strong>to account. Still, efforts should<br />
be focused onto gett<strong>in</strong>g smoother amplitudes with RT techniques, where the Fresnel zone<br />
is considered. This would certa<strong>in</strong>ly yield smoother PSDM sections, like <strong>in</strong> wave-equation<br />
migration where the smooth<strong>in</strong>g is obta<strong>in</strong>ed directly from wave propagation <strong>of</strong> b<strong>and</strong>limited<br />
signals. But even if we get better amplitudes for such corrections, we saw <strong>in</strong> the<br />
previous examples two other effects which are due to the migration technique itself, i.e.,<br />
vary<strong>in</strong>g illum<strong>in</strong>ation <strong>and</strong> limited resolution. Figure 17a-c clearly shows a stronger salt<br />
reflector because its shape, with rather strong local <strong>and</strong> concave curvature, ‘traps’<br />
reflections on a more concentrated area than for flat reflectors (see also Figure 18a, top).<br />
We rem<strong>in</strong>d that <strong>in</strong> this example, the data did not conta<strong>in</strong> any amplitude effect, <strong>in</strong>clud<strong>in</strong>g<br />
the reflection coefficient at the illum<strong>in</strong>ated reflector, so this amplitude variation is due to<br />
illum<strong>in</strong>ation effects. The ‘alias<strong>in</strong>g-noise’ <strong>of</strong> constant-<strong>of</strong>fset sections (Figure 17a-c) will<br />
also degrade the amplitudes along a reflector. Note that all these effects are also occurr<strong>in</strong>g<br />
<strong>in</strong> wave-equation migration! But, with RT approaches, there is some hope to correct for<br />
that, either follow<strong>in</strong>g the approach <strong>of</strong> V<strong>in</strong>je (2000), which simulates DS <strong>and</strong> <strong>in</strong>cludes the<br />
illum<strong>in</strong>ation corrections <strong>in</strong> the migration, or by work<strong>in</strong>g directly with the seismic<br />
record<strong>in</strong>g, i.e., by analyz<strong>in</strong>g amplitudes <strong>in</strong>dividually without suffer<strong>in</strong>g from the<br />
summation process performed by migration. PSDM is then more a technique which helps<br />
to complete the velocity model by <strong>in</strong>clud<strong>in</strong>g reflectors to the smooth velocity field used<br />
for the GFs calculation (we need to get the best structural image). Then RT can be used<br />
to model amplitudes <strong>of</strong> s<strong>in</strong>gle events <strong>and</strong> compare them to the actual measured amplitude<br />
<strong>in</strong> order to extract angle-dependent reflectivity. It is also important for such reflectivity<br />
analyses to be sure to use the proper data, i.e., data associated to the same scatter<strong>in</strong>g<br />
angle θ. There aga<strong>in</strong>, the local ray directions will help to sort the data for AVA studies. In<br />
the example <strong>of</strong> Figure 20, the zero-<strong>of</strong>fset section was really associated to values <strong>of</strong> θ very<br />
close to zero, while the far-<strong>of</strong>fset section was associated to θ values vary<strong>in</strong>g from 24° to<br />
44°! At last, we saw that a PSDM section is a filtered version <strong>of</strong> the reflectivity by<br />
resolution functions (PSFs or RSFs, Figure 21), also determ<strong>in</strong>ed only from local ray<br />
directions, which are therefore a key <strong>in</strong>formation for analyz<strong>in</strong>g migration (as it is <strong>in</strong> any<br />
<strong>in</strong>version process, <strong>in</strong>deed!). When consider<strong>in</strong>g the RSFs <strong>of</strong> Figure 21, we have nearly<br />
lateral resolution <strong>of</strong> about 100 m (the nearly vertical one is better). This means that the<br />
migrated value at one po<strong>in</strong>t results from constructive <strong>in</strong>terference <strong>of</strong> reflected energy on<br />
a 100 m wide zone along the reflector. How can we retrieve ‘true’ amplitudes from PSDM<br />
sections without tak<strong>in</strong>g all these effects (illum<strong>in</strong>ation, resolution, coherent-noise<br />
<strong>in</strong>terference) <strong>in</strong>to account? Even nice, smooth PSDM sections, provided by wave-equation<br />
migration, will not give true amplitudes because they suffer from the same effects. But RT<br />
analyses may help to isolate the different effects <strong>and</strong> get closer to the true reflectivity.<br />
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Fig. 22. (a) Illustration <strong>of</strong> a 3D survey from an attribute mapp<strong>in</strong>g case. Shot l<strong>in</strong>es <strong>and</strong> streamer<br />
configuration are <strong>in</strong>dicated. Only the target reflector is shown, although the overburden consists <strong>of</strong><br />
a number <strong>of</strong> layers. (b) Amplitude density map for the target reflector <strong>in</strong> (a). Depth contours are<br />
<strong>in</strong>dicated by dashed l<strong>in</strong>es.<br />
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6. SEISMIC ATTRIBUTE MAPPING BY COMPREHENSIVE 3D MODELLING<br />
6.1. Mapp<strong>in</strong>g <strong>of</strong> ray related seismic attributes<br />
A goal <strong>of</strong> a seismic survey is to achieve the best possible illum<strong>in</strong>ation <strong>of</strong> the<br />
subsurface geologic structures. Until the last few years, conventional seismic survey<br />
design has relied mostly on a simplified process, assum<strong>in</strong>g that uniform midpo<strong>in</strong>t<br />
coverage will lead to uniform illum<strong>in</strong>ation <strong>of</strong> the subsurface. Such simple survey designs<br />
are effective <strong>in</strong> areas with flat or gently dipp<strong>in</strong>g layer<strong>in</strong>g. But <strong>in</strong> areas <strong>of</strong> complex<br />
geometry (such as salt structures) or rough topography on the target reflector, the<br />
wavefield becomes more distorted, result<strong>in</strong>g <strong>in</strong> non-regular illum<strong>in</strong>ation patterns on the<br />
reflector, even if the midpo<strong>in</strong>t coverage is regular (Bear et al., 1999; Sassolas et al., 1999;<br />
Pereyra et al., 1999; Rosl<strong>and</strong> <strong>and</strong> Drivenes, 2000). The computation <strong>and</strong> visualization <strong>of</strong><br />
maps show<strong>in</strong>g different seismic attributes computed by ray trac<strong>in</strong>g <strong>in</strong> true depth at<br />
selected target horizons can help plann<strong>in</strong>g a survey, <strong>in</strong>terpret<strong>in</strong>g data after e.g. prestack<br />
depth migration. Typical attributes to be mapped are reflection po<strong>in</strong>t density <strong>and</strong><br />
<strong>in</strong>tegrated amplitude (also called amplitude density), but any other parameter related to<br />
the ray paths can be mapped, such as reflection angle or phase shifts <strong>in</strong> the seismic pulse.<br />
Figure 22 shows an example from the simulation <strong>of</strong> a 3D survey <strong>in</strong> an area where<br />
a model was constructed from seismic <strong>in</strong>terpretation maps (Rosl<strong>and</strong> <strong>and</strong> Drivenes, 2000).<br />
A mar<strong>in</strong>e survey <strong>of</strong> 12.000 shots was simulated, hav<strong>in</strong>g 10 streamers with 1200 receivers.<br />
The shot l<strong>in</strong>es are shown together with the target reflector. The overburden (not shown)<br />
consisted <strong>of</strong> 5-6 <strong>in</strong>terfaces with constant or lateral velocity variation <strong>in</strong> each layer. The<br />
calculated amplitude density map <strong>of</strong> the primary reflections shows a large variation <strong>in</strong><br />
amplitude density, which <strong>in</strong> this case turns out to be closely related to the topography <strong>of</strong><br />
the target.<br />
6.2. Synthetic data from 2D model<br />
In order to study the relationship between the illum<strong>in</strong>ation <strong>and</strong> amplitude maps<br />
calculated from ray trac<strong>in</strong>g <strong>and</strong> the amplitudes appear<strong>in</strong>g <strong>in</strong> prestack depth migrated<br />
(PSDM) data, we carried out a relatively simple 2D model experiment (see Laura<strong>in</strong> <strong>and</strong><br />
V<strong>in</strong>je, 2001). The model was designed to comb<strong>in</strong>e the effects <strong>of</strong> both a syncl<strong>in</strong>e with<br />
a steep flank <strong>and</strong> a lens (e.g. a salt body). A full f<strong>in</strong>ite-difference data set was generated<br />
<strong>and</strong> migrated us<strong>in</strong>g a PSDM algorithm. Then, the correspondence between PSDM root<br />
mean square amplitude (RMS amplitude) pr<strong>of</strong>iles along the target horizon <strong>and</strong> hit density<br />
<strong>and</strong> amplitude illum<strong>in</strong>ation maps were studied. Figure 23a displays the model we used for<br />
generat<strong>in</strong>g a synthetic mar<strong>in</strong>e data set with classic 2D f<strong>in</strong>ite-difference (FD) code. The<br />
upper layer represents water (velocity 1.5 km/s). The sediment layer, delimited by the sea<br />
bottom <strong>and</strong> the target horizon, is affected by a vertical gradient function (from 2.0 to<br />
3.0 km/s at the deepest po<strong>in</strong>t). In the sediment layer, a salt lens with constant velocity <strong>of</strong><br />
4.0 km/s has been <strong>in</strong>troduced to study the effects <strong>of</strong> shadow zones <strong>in</strong>duced on the target<br />
horizon. The data acquisition scheme was designed as a mar<strong>in</strong>e survey with a nom<strong>in</strong>al<br />
fold <strong>of</strong> 10 <strong>in</strong> b<strong>in</strong> cells <strong>of</strong> 5 m size. Shots were located on the surface, every 50 m between<br />
3.0 <strong>and</strong> 14.5 km. For each shot, 100 receivers with <strong>of</strong>fsets from 10 m to 1000 m <strong>and</strong><br />
a 10.0 m <strong>in</strong>terval were used (Figure 23a). An example <strong>of</strong> constant-<strong>of</strong>fset FD section is<br />
shown <strong>in</strong> Figure 24.<br />
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Fig. 23. (a) Model used for generation <strong>of</strong> synthetic data. Shot po<strong>in</strong>ts are located between 3 km <strong>and</strong><br />
14.5 km along the surface. The target horizon is <strong>in</strong>dicated. (b) Velocity model used for perform<strong>in</strong>g<br />
the depth migration. Target areas are limited by the black boxes.<br />
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Fig. 24. Constant-<strong>of</strong>fset section for <strong>of</strong>fset 10.0 m.<br />
6.3. Prestack depth migration<br />
The f<strong>in</strong>ite-difference data were migrated for two local targets us<strong>in</strong>g a un-weighted<br />
prestack depth migration algorithm. For each po<strong>in</strong>t <strong>of</strong> the def<strong>in</strong>ed target, the migration<br />
process performs a sum <strong>of</strong> amplitudes with wide aperture <strong>and</strong> unit weights along<br />
diffraction-traveltime curves. The traveltime maps required for this process were<br />
generated with a first arrival Eikonal solver <strong>in</strong> a background velocity model where the<br />
target <strong>in</strong>terface has been removed (Figure 23b). The first target area (denoted as target 1)<br />
covers the syncl<strong>in</strong>e. The second target area (denoted as target 2) has been chosen beneath<br />
the salt lens <strong>in</strong> order to study the <strong>in</strong>duced focus<strong>in</strong>g/de-focus<strong>in</strong>g effects. Migrated sections<br />
us<strong>in</strong>g the source-receiver pairs as <strong>in</strong>put (every shot, all <strong>of</strong>fsets, traveltimes up to 8.5 s) are<br />
displayed <strong>in</strong> Figures 25 <strong>and</strong> 26.<br />
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Fig. 25. Migrated section <strong>of</strong> target 1.<br />
Fig. 26. Migrated section <strong>of</strong> target 2. The thick black l<strong>in</strong>e superimposed on the horizontal axis<br />
represents the salt lens extension.<br />
6.4. Illum<strong>in</strong>ation maps<br />
In order to generate illum<strong>in</strong>ation maps <strong>of</strong> the target horizon, a full data set was<br />
modeled by wavefront construction (V<strong>in</strong>je et al., 1996a,b). Reflection po<strong>in</strong>ts on the target<br />
horizon <strong>and</strong> amplitudes <strong>of</strong> the correspond<strong>in</strong>g rays from each shot to each receiver <strong>in</strong> the<br />
survey were computed. To be comparable to the f<strong>in</strong>ite-difference data set, ray trac<strong>in</strong>g was<br />
done with cyl<strong>in</strong>drical sources. The wavefront construction was performed <strong>in</strong> a 2.5D model<br />
extrapolated from the 2D model (see Figure 27) us<strong>in</strong>g the same survey as used for the<br />
f<strong>in</strong>ite-difference data.<br />
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Fig. 27. 2.5D model used to generate illum<strong>in</strong>ation maps. Shots are along the surface (black l<strong>in</strong>e).<br />
<strong>Ray</strong>paths for one shot-receiver couple are <strong>in</strong>dicated. Reflection po<strong>in</strong>ts are superimposed as a l<strong>in</strong>e<br />
along the target horizon.<br />
To perform the calculation <strong>of</strong> the illum<strong>in</strong>ation maps, the surface <strong>of</strong> the reflector was<br />
divided <strong>in</strong> b<strong>in</strong> cells <strong>of</strong> given (constant) area. In each cell, the follow<strong>in</strong>g quantities were<br />
computed:<br />
− the hit density (i.e. number <strong>of</strong> reflection po<strong>in</strong>ts per unit area) def<strong>in</strong>ed as<br />
N<br />
D = (4)<br />
B<strong>in</strong> Cell Area<br />
where N is the number <strong>of</strong> reflection po<strong>in</strong>ts <strong>in</strong> the b<strong>in</strong> cell.<br />
− the illum<strong>in</strong>ation amplitude modulus (i.e. sum <strong>of</strong> the amplitude modulus per unit area)<br />
def<strong>in</strong>ed as<br />
∑<br />
Ai<br />
i<br />
A =<br />
B<strong>in</strong> Cell Area<br />
where Ai is the complex amplitude coefficient <strong>in</strong> the receiver for ray number i<br />
reflect<strong>in</strong>g <strong>in</strong> the b<strong>in</strong> cell.<br />
6.5. Amplitude illum<strong>in</strong>ation <strong>and</strong> migration<br />
Even us<strong>in</strong>g a fairly small aperture (1 km), the survey illum<strong>in</strong>ates the whole target<br />
horizon as may be seen by the reflection po<strong>in</strong>ts on the target horizon <strong>in</strong> Figure 27. The<br />
steepest part <strong>of</strong> the syncl<strong>in</strong>e represents a low illum<strong>in</strong>ated zone (x = 3.4 km). This is<br />
obvious on both hit density <strong>and</strong> amplitude illum<strong>in</strong>ation curves (see Figure 28). RMS<br />
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amplitude along the migrated horizon (Figure 29) is smoother than the illum<strong>in</strong>ation<br />
amplitude curve, even if the general variations show some similarities.<br />
Under the salt lens (black l<strong>in</strong>e <strong>in</strong> Figures 30 <strong>and</strong> 31), the hit density returns to normal<br />
(compared to the hit density before x = 10.0 km) whereas the amplitude illum<strong>in</strong>ation<br />
rema<strong>in</strong>s lower due to the energy losses occurr<strong>in</strong>g when the wavefield is transmitted<br />
through the salt. This effect is also noticeable on the RMS amplitude extracted from the<br />
migrated section (Figure 31). Near the left edge <strong>of</strong> the salt lens (x = 10.7 km <strong>in</strong> Figure 30),<br />
there is a low-amplitude zone where very little energy is reflected even if the illum<strong>in</strong>ation<br />
density is high. This low-amplitude zone is also visible at x = 10.7 km <strong>in</strong> the RMS section<br />
<strong>in</strong> Figure 31, but <strong>in</strong> this case the low-amplitude zone is larger. This is not an <strong>in</strong>tr<strong>in</strong>sic<br />
limitation <strong>of</strong> the PSDM process, but rather a side effect <strong>in</strong>troduced when us<strong>in</strong>g first arrival<br />
traveltime maps. The Eikonal solver computes first arrival traveltimes, which may not<br />
represent the most energetic part <strong>of</strong> the wavefield (Geoltra<strong>in</strong> <strong>and</strong> Brac, 1993). In this<br />
case, the first arrival is associated to a low energy event pass<strong>in</strong>g through the salt. The<br />
consequence is that the imag<strong>in</strong>g will take place through the salt giv<strong>in</strong>g lower amplitudes<br />
<strong>in</strong> the PSDM image from about 9.5 to 11 km. This effect could be corrected by us<strong>in</strong>g only<br />
the traveltimes for the most energetic events <strong>in</strong>stead, or all arrivals. Generally speak<strong>in</strong>g,<br />
RMS amplitude pr<strong>of</strong>iles along both migrated horizons (Figures 29 <strong>and</strong> 31) are smoother<br />
than the correspond<strong>in</strong>g illum<strong>in</strong>ation <strong>and</strong> amplitude pr<strong>of</strong>iles. The PSDM process is<br />
performed on a f<strong>in</strong>ite-difference data set on which Fresnel zone effects <strong>in</strong>troduce<br />
a smooth<strong>in</strong>g (Bear et al., 1999; Thore <strong>and</strong> Juliard, 1999). Moreover, the PSDM algorithm<br />
performs another smooth<strong>in</strong>g when stack<strong>in</strong>g amplitudes along the diffraction-traveltime<br />
curves. Therefore, RMS amplitude variations are not as drastic as they appear on the<br />
illum<strong>in</strong>ation maps.<br />
Fig. 28. Hit density <strong>and</strong> amplitude illum<strong>in</strong>ation for the syncl<strong>in</strong>e (target 1). Sampl<strong>in</strong>g <strong>in</strong>terval is<br />
25.0 m.<br />
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Fig. 29. RMS amplitude along the migrated horizon for the syncl<strong>in</strong>e (target 1).<br />
Fig. 30. Hit density <strong>and</strong> amplitude illum<strong>in</strong>ation for the area under the salt lens (target 2). Sampl<strong>in</strong>g<br />
<strong>in</strong>terval is 25.0 m. The black l<strong>in</strong>e on the horizontal axis represents the salt lens extension.<br />
Fig. 31. RMS amplitude along the migrated horizon under the salt lens (target 2). The black l<strong>in</strong>e on<br />
the horizontal axis represents the salt lens extension.<br />
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6.6. Conclusion<br />
PSDM <strong>in</strong>troduces a smear<strong>in</strong>g <strong>of</strong> the recorded traces along isochrons tangential to the<br />
reflectors. This is not taken <strong>in</strong>to account <strong>in</strong> the illum<strong>in</strong>ation maps, as they are made<br />
without any knowledge <strong>of</strong> the source pulse, us<strong>in</strong>g a simple process <strong>of</strong> add<strong>in</strong>g amplitudes<br />
<strong>in</strong> b<strong>in</strong> cells. However, the synthetic example studied here shows that the variation <strong>of</strong> the<br />
illum<strong>in</strong>ation amplitude is comparable to the PSDM amplitude. Important effects like<br />
focus<strong>in</strong>g/de-focus<strong>in</strong>g <strong>and</strong> the reduction <strong>of</strong> amplitude due to energy losses <strong>in</strong> the<br />
overburden is seen on the illum<strong>in</strong>ation amplitude pr<strong>of</strong>iles.<br />
Compar<strong>in</strong>g amplitude maps from prestack depth migrated data performed on real <strong>and</strong><br />
modelled data, Rosl<strong>and</strong> <strong>and</strong> Drivenes (2000) showed the importance <strong>of</strong> the reflector<br />
geometry <strong>in</strong> the amplitude distribution.<br />
7. ISOCHRON RAYS AND VELOCITY RAYS<br />
Conventional rays are associated to propagation <strong>of</strong> wavefronts. Familiar examples <strong>in</strong><br />
nature are propagation <strong>of</strong> water waves, electromagnetic waves, <strong>and</strong> seismic waves. The<br />
physical wavefronts are isochrons correspond<strong>in</strong>g to constant traveltime. Here, we shall<br />
refer to this traveltime as a one-way time. However, <strong>in</strong> the context <strong>of</strong> prestack depth<br />
migration <strong>of</strong> seismic data an isochron is commonly def<strong>in</strong>ed as a surface <strong>of</strong> potential<br />
scatter<strong>in</strong>g po<strong>in</strong>ts <strong>in</strong> the subsurface, all correspond<strong>in</strong>g to the same two-way time. A ray<br />
from shot to receiver can then be divided naturally <strong>in</strong> two parts; a shot ray <strong>and</strong> a receiver<br />
ray. Furthermore, isochrons may be considered as wavefronts <strong>of</strong> a propagat<strong>in</strong>g nonphysical<br />
wave, <strong>and</strong> recent work by Iversen (2001a,b,c, 2002) has shown that ray systems<br />
can be established for such propagation. Seismic ray theory (Červený, 1985, 2001) applied<br />
to the conventional rays from shot <strong>and</strong> receiver is a fundamental build<strong>in</strong>g block <strong>in</strong> the new<br />
theory, which <strong>in</strong>cludes two basic ray concepts: the isochron ray <strong>and</strong> the velocity ray.<br />
As with conventional rays, the position x <strong>and</strong> slowness vector p related to isochrons<br />
shall be expressed as functions<br />
x= x( γ ,, tm)<br />
, p = p( γ ,, tm)<br />
(6)<br />
Here, γ is a two-dimensional vector conta<strong>in</strong><strong>in</strong>g two <strong>in</strong>dependent parameters (γ1, γ2)<br />
describ<strong>in</strong>g any isochron at two-way time t = constant. The quantity m denotes one <strong>of</strong> the<br />
various parameters compris<strong>in</strong>g the velocity model. When the velocity model <strong>and</strong> the<br />
vector γ is fixed, the po<strong>in</strong>ts x are located along a trajectory, def<strong>in</strong>ed by Iversen (2001a,b)<br />
as an isochron ray. Similarly, if γ <strong>and</strong> t are fixed, <strong>and</strong> the model parameter m is allowed to<br />
vary, we get a velocity ray (Iversen, 2001a,c). Examples <strong>of</strong> isochron rays <strong>and</strong> velocity<br />
rays <strong>of</strong> type PP are shown <strong>in</strong> Figure 32, for a homogeneous velocity model. The notation<br />
‘PP’ means that each shot ray or receiver ray corresponds to a purely transmitted P-wave.<br />
The term ‘velocity ray’ was <strong>in</strong>troduced by Adler et al. (1997). From a system <strong>of</strong> velocity<br />
rays we can obta<strong>in</strong> certa<strong>in</strong> surfaces for which the model parameter m is constant. In<br />
accordance with the work on seismic image waves by Hubral et al. (1996), such surfaces<br />
can be referred to as image wavefronts.<br />
In conventional ray theory a ray is uniquely specified by the ray parameter vector γ,<br />
while a po<strong>in</strong>t x on the ray is uniquely specified by the extended ray parameter vector<br />
Stud. geophys. geod., 46 (2002) 153
γ<br />
⎡ 1 ⎤<br />
ˆ = ⎢γ⎥ ⎢ 2 ⎥<br />
⎢⎣ t ⎥⎦<br />
H. Gjøystdal et al.<br />
γ , (7)<br />
where t is the one-way time from the start po<strong>in</strong>t to x. In the new ray theory it has been<br />
essential to establish similar unique relationships. This problem has been approached by<br />
express<strong>in</strong>g the ray parameters for the isochron ray <strong>in</strong> terms <strong>of</strong> the ray parameters for the<br />
shot <strong>and</strong> receiver rays (Iversen 1996, 2001a, 2001b), i.e.,<br />
γ = γ( γ , γ ) . (8).<br />
s r<br />
A perturbation <strong>of</strong> the first order <strong>in</strong> γ is therefore<br />
where<br />
δ δ δ<br />
γ = S γs + R γ r , (9)<br />
∂ γ<br />
S ≡<br />
T<br />
∂ s<br />
γ ;<br />
∂ γ<br />
R ≡<br />
T<br />
∂ γr<br />
Any isochron ray <strong>and</strong> velocity ray will then be required to fulfil<br />
respectively.<br />
. (10)<br />
d<br />
0<br />
dt =<br />
γ<br />
d<br />
, 0<br />
dm =<br />
γ<br />
, (11)<br />
7.1. A system <strong>of</strong> ODEs for isochron rays<br />
The process for generation <strong>of</strong> an isochron ray can be organized with systems <strong>of</strong><br />
ord<strong>in</strong>ary differential equations (ODEs) on two levels. The upper level conta<strong>in</strong>s the ma<strong>in</strong><br />
system for numerical <strong>in</strong>tegration <strong>of</strong> isochron rays (Iversen, 2001b):<br />
d<br />
dt =<br />
x d<br />
V , ˆ<br />
dt =<br />
p<br />
MV , (12)<br />
dγˆ<br />
dt<br />
s<br />
= Qˆ V ,<br />
−1<br />
s<br />
dγˆ<br />
dt<br />
r<br />
= Qˆ V . (13)<br />
The system consists <strong>of</strong> twelve equations <strong>in</strong> the case <strong>of</strong> general propagation <strong>in</strong> a 3D<br />
velocity model. When ray propagation is conf<strong>in</strong>ed to a plane, the number <strong>of</strong> equations can<br />
be reduced to eight. S<strong>in</strong>ce we are not aware <strong>of</strong> an Eikonal equation for isochron rays, the<br />
derivation <strong>of</strong> the system <strong>of</strong> ODEs <strong>in</strong> (12) − (13) is directly based on properties <strong>of</strong> the shot<br />
<strong>and</strong> receiver ray, us<strong>in</strong>g paraxial ray theory.<br />
We shall expla<strong>in</strong> the various quantities on the left-h<strong>and</strong> side (LHS) <strong>and</strong> right-h<strong>and</strong> side<br />
(RHS) <strong>of</strong> (12) − (13). t denotes the two-way time, x is the position vector describ<strong>in</strong>g<br />
po<strong>in</strong>ts on the isochron ray, <strong>and</strong> p is the isochron slowness vector. This vector is normal to<br />
154 Stud. geophys. geod., 46 (2002)<br />
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r
<strong>Review</strong> <strong>of</strong> <strong>Ray</strong> <strong>Theory</strong> <strong>Applications</strong> <strong>in</strong> Modell<strong>in</strong>g <strong>and</strong> Imag<strong>in</strong>g <strong>of</strong> Seismic Data<br />
the isochron at x, <strong>and</strong> it can be obta<strong>in</strong>ed simply by summ<strong>in</strong>g the two slowness vectors <strong>of</strong><br />
the shot <strong>and</strong> receiver ray, i.e.,<br />
p = ps+ p r . (14)<br />
The two sets <strong>of</strong> equations <strong>in</strong> (13) have been <strong>in</strong>cluded to ensure cont<strong>in</strong>uous updat<strong>in</strong>g <strong>of</strong> the<br />
extended ray parameter vectors γ ˆ s <strong>and</strong> γˆ<br />
r <strong>of</strong> the shot ray <strong>and</strong> the receiver ray,<br />
respectively.<br />
On the RHS <strong>of</strong> (12) − (13), V can be <strong>in</strong>terpreted as a group velocity vector for<br />
isochron rays. Observe that V is the only quantity on the RHS depend<strong>in</strong>g on the<br />
parameterization <strong>of</strong> the isochron ray field. The vectors p <strong>and</strong> V generally po<strong>in</strong>t <strong>in</strong> different<br />
directions even if the velocity model is isotropic.<br />
Furthermore, the 3×3 matrix ˆM conta<strong>in</strong>s the second derivatives <strong>of</strong> two-way time with<br />
respect to spatial coord<strong>in</strong>ates. It can be obta<strong>in</strong>ed from the sum<br />
Mˆ = Mˆ ˆ<br />
s + M r , (15)<br />
where the matrices on the RHS conta<strong>in</strong>s second derivatives with respect to one-way time.<br />
Follow<strong>in</strong>g st<strong>and</strong>ard paraxial ray theory (Červený, 2001) we can express the one-way<br />
ˆM -matrices by the products<br />
ˆ ˆ ˆ 1<br />
s s s −<br />
M = P Q ,<br />
ˆ ˆ ˆ 1<br />
r r r −<br />
M = P Q (16)<br />
In (13) <strong>and</strong> (16) ˆ Q s <strong>and</strong> Q ˆ<br />
r are 3×3 geometrical spread<strong>in</strong>g matrices correspond<strong>in</strong>g to the<br />
shot <strong>and</strong> receiver ray. The matrices P ˆ<br />
s <strong>and</strong> Pˆ<br />
r can be referred to as ‘slowness spread<strong>in</strong>g’<br />
matrices.<br />
The upper level <strong>of</strong> the ray trac<strong>in</strong>g process is model <strong>in</strong>dependent. On the lower level the<br />
shot <strong>and</strong> receiver ray are re-traced accord<strong>in</strong>g to the <strong>in</strong>put <strong>in</strong> γ ˆ s <strong>and</strong> γˆ<br />
r , us<strong>in</strong>g<br />
conventional k<strong>in</strong>ematic <strong>and</strong> dynamic ray trac<strong>in</strong>g. Thereby we obta<strong>in</strong> all the quantities<br />
p , ˆ , ˆ<br />
s Qs P s , p ˆ ˆ<br />
r, Qr, Pr , (17)<br />
needed to evaluate the RHS <strong>of</strong> the system (12) − (13). It should be noted that (12) − (13)<br />
has redundant <strong>in</strong>formation, so by reduc<strong>in</strong>g the number <strong>of</strong> equations the <strong>in</strong>tegration <strong>of</strong><br />
ODEs can be done more efficient. On the other h<strong>and</strong>, redundancy can be valuable for<br />
error control.<br />
It is important to note that the two-way time t is not always suited as the <strong>in</strong>dependent<br />
<strong>in</strong>tegration variable. Dur<strong>in</strong>g its propagation, isochron rays can become tangential to the<br />
‘wavefront’, which means that the length <strong>of</strong> the group velocity vector blows to <strong>in</strong>f<strong>in</strong>ity.<br />
For such situations we recommend a special version <strong>of</strong> (12) − (13) where t is a dependent<br />
variable <strong>and</strong> distance is the <strong>in</strong>dependent variable along the ray. After trac<strong>in</strong>g a given<br />
distance the ray can then be divided <strong>in</strong> portions along which the two-way time varies<br />
monotonously, followed by resampl<strong>in</strong>g to the prescribed output times.<br />
Stud. geophys. geod., 46 (2002) 155
H. Gjøystdal et al.<br />
Fig. 32. (a) PP isochron rays <strong>in</strong> a homogeneous medium with νP = 2.5 km/s. The <strong>in</strong>crement<br />
between isochrons is 0.5 s. (b) PP velocity rays, with the isochron t = 2.5 s act<strong>in</strong>g as a source for the<br />
propagation. The <strong>in</strong>crement between image wavefronts is 0.1 km/s.<br />
156 Stud. geophys. geod., 46 (2002)
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Fig. 33. <strong>Ray</strong> perturbation due to two different conditions. (a) The extended ray parameter vector<br />
is fixed. (b) The ray po<strong>in</strong>t x is fixed.<br />
7.2. Perturbation <strong>of</strong> conventional rays<br />
Figure 33a shows a conventional ray with ray parameters γ, <strong>and</strong> a po<strong>in</strong>t x on the ray<br />
with parameters given by ˆγ . We have also <strong>in</strong>dicated a wavefront through x <strong>and</strong> the<br />
associated slowness vector p. If a model parameter is perturbed, say m is changed to<br />
m + δ m, the ray will be perturbed as a result <strong>of</strong> this, but the perturbed ray can take various<br />
forms, depend<strong>in</strong>g on specified conditions. Figure 33a shows a perturbed ray (dashed) for<br />
Stud. geophys. geod., 46 (2002) 157<br />
ˆγ
H. Gjøystdal et al.<br />
the condition that the ray parameters shall be kept fixed. Also <strong>in</strong>dicated is the perturbed<br />
wavefront, the perturbed ray po<strong>in</strong>t x + δ x, <strong>and</strong> the perturbed slowness vector p + δ p. The<br />
po<strong>in</strong>ts x <strong>and</strong> x + δ x then correspond to the same vector ˆγ , <strong>and</strong> the collection <strong>of</strong> all such<br />
po<strong>in</strong>ts forms a velocity ray (grey), i.e., a velocity ray for fixed one-way time. Such rays<br />
are described to the first order by derivatives d x / d m <strong>and</strong> d p / d m, which can be obta<strong>in</strong>ed<br />
by the ray perturbation method (see, e.g., Farra <strong>and</strong> Madariaga, 1987; Farra <strong>and</strong> Le<br />
Bégat, 1995).<br />
Another condition that can be enforced on ray perturbations is that the ray po<strong>in</strong>t x shall<br />
be fixed (see Figure 33b). When we perturb the model, the perturbed ray must go through<br />
x, <strong>and</strong> this will impose changes upon the ray parameters (from γ to<br />
( x<br />
γ+ δ γ<br />
)<br />
), the one-<br />
way time (from t to<br />
( x<br />
t+ δ t<br />
)<br />
), <strong>and</strong> the slowness vector (from p to p+<br />
δ<br />
( x<br />
p<br />
)<br />
). The<br />
connection between the two cases is given by<br />
( x<br />
dp )<br />
dp<br />
= − ˆ dx<br />
M ,<br />
dm dm dm<br />
d<br />
( x<br />
γˆ<br />
)<br />
ˆ −1<br />
dx<br />
=−Q<br />
. (18)<br />
dm dm<br />
On the LHS we have the derivatives for fixed x, on the RHS the derivatives for fixed ˆγ .<br />
The quantities ˆM <strong>and</strong> Q ˆ are st<strong>and</strong>ard 3×3 matrices for second derivatives <strong>of</strong> one-way<br />
time <strong>and</strong> geometrical spread<strong>in</strong>g, respectively. The fixed po<strong>in</strong>t derivatives <strong>in</strong> (18) are <strong>of</strong><br />
particular importance <strong>in</strong> the generation <strong>of</strong> velocity rays for fixed two-way time.<br />
7.3. A system <strong>of</strong> ODEs for velocity rays<br />
First-order approximations <strong>of</strong> velocity rays have been described by Iversen (1996,<br />
2001a). The first-order theory <strong>in</strong> the latter papers were extended to yield ‘exact’ velocity<br />
rays <strong>in</strong> Iversen (2001c), utiliz<strong>in</strong>g the follow<strong>in</strong>g system <strong>of</strong> ODEs:<br />
d<br />
dm =<br />
x<br />
U ;<br />
( x<br />
dγˆ ˆ )<br />
s dγs<br />
ˆ −1<br />
= + QsU ;<br />
dm dm<br />
( x<br />
dp dp<br />
)<br />
=<br />
dm dm<br />
ˆ<br />
+MU<br />
( x<br />
dγˆ ˆ )<br />
r dγr<br />
= + ˆQ<br />
dm dm<br />
−1<br />
r<br />
, (19)<br />
U . (20)<br />
The system comprises twelve equations, which can be reduced to eight <strong>in</strong> the case <strong>of</strong> 2D<br />
ray propagation. The vector U is tangent to the velocity ray, <strong>and</strong> thus we can say that this<br />
vector has a similar role with respect to velocity rays as the group velocity vector V has to<br />
isochron rays. Furthermore, U is the only quantity on the RHS that depends on the<br />
parameterization.The derivatives for a fixed po<strong>in</strong>t <strong>in</strong> (19) − (20) can be obta<strong>in</strong>ed from<br />
<strong>in</strong>dependent perturbation <strong>of</strong> the shot <strong>and</strong> receiver ray, us<strong>in</strong>g the formulas <strong>in</strong> (18).<br />
As with isochron rays the process for generation <strong>of</strong> exact velocity rays can be<br />
organized on two levels. The upper level conta<strong>in</strong>s the ma<strong>in</strong> system for velocity rays <strong>in</strong><br />
(19) − (20), while the lower level <strong>in</strong>cludes two systems <strong>of</strong> ODEs for conventional rays,<br />
one for the shot ray, another for the receiver ray. The <strong>in</strong>formation needed from the latter<br />
rays is<br />
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, ˆ , ˆ<br />
dxsdps ps Qs P s,<br />
, , ˆ ˆ dxr dpr<br />
pr, Qr, P r,<br />
, . (21)<br />
dm dm<br />
dm dm<br />
We can obta<strong>in</strong> these quantities by conventional k<strong>in</strong>ematic ray trac<strong>in</strong>g, dynamic ray<br />
trac<strong>in</strong>g, <strong>and</strong> solution <strong>of</strong> the ray perturbation equations. As <strong>in</strong> the case <strong>of</strong> isochron rays, the<br />
upper level <strong>of</strong> the process is model <strong>in</strong>dependent.<br />
7.4. <strong>Applications</strong><br />
Isochron rays can be used for visualization <strong>of</strong> geometric effects <strong>in</strong> prestack depth<br />
migration, <strong>and</strong> for map migration <strong>of</strong> events from the time doma<strong>in</strong> to the depth doma<strong>in</strong>.<br />
Isochron rays also have a potential with respect to generation <strong>and</strong> propagation <strong>of</strong> the<br />
boundaries <strong>of</strong> Fresnel volumes. An approximate method for the latter purpose, us<strong>in</strong>g<br />
paraxial ray theory, was described by Červený <strong>and</strong> Soares (1992). An <strong>in</strong>terest<strong>in</strong>g future<br />
task is to <strong>in</strong>vestigate whether isochron rays also can be beneficial <strong>in</strong> prestack depth<br />
migration.<br />
Velocity rays can be used for sensitivity studies related to the depth doma<strong>in</strong> <strong>of</strong> the<br />
seismic data. It is important to note that velocity rays can be used not only to propagate<br />
geometry, but even to propagate complete seismic images as a function <strong>of</strong> a velocity<br />
parameter (Adler, 2000). Furthermore, it has been demonstrated recently that velocity<br />
rays, whether exact or approximate, can be used with success for estimation <strong>of</strong> the<br />
parameters <strong>of</strong> an anisotropic velocity model (Iversen et al., 2000).<br />
8. CONCLUSION<br />
We have clearly observed that the <strong>in</strong>dustry’s <strong>in</strong>terest <strong>in</strong> apply<strong>in</strong>g advanced seismic<br />
modell<strong>in</strong>g tools is grow<strong>in</strong>g. A number <strong>of</strong> application areas are develop<strong>in</strong>g, <strong>and</strong> we have<br />
tried to illustrate some <strong>of</strong> them <strong>in</strong> this paper: Prestack depth migration, hybrid RT/FD<br />
modell<strong>in</strong>g, <strong>and</strong> survey plann<strong>in</strong>g/attribute mapp<strong>in</strong>g. Particularly, the production <strong>of</strong> Green's<br />
functions from source <strong>and</strong> receiver positions to a dense 3D grid <strong>of</strong> imag<strong>in</strong>g po<strong>in</strong>ts<br />
(probably millions <strong>of</strong> po<strong>in</strong>ts) <strong>in</strong> the subsurface has turned out to be an <strong>in</strong>dustrial<br />
application, <strong>and</strong> several process<strong>in</strong>g companies have already <strong>in</strong>cluded wavefront<br />
construction <strong>in</strong> their imag<strong>in</strong>g process. As opposed to the st<strong>and</strong>ard first arrival Eikonal<br />
solver, implementation <strong>of</strong> dynamic ray trac<strong>in</strong>g <strong>in</strong> wavefront construction gives<br />
possibilities <strong>of</strong> choos<strong>in</strong>g arrivals based on more flexible criteria (e.g. the most energetic<br />
arrival or even multi-arrivals) <strong>and</strong> <strong>of</strong> us<strong>in</strong>g illum<strong>in</strong>ation <strong>and</strong> amplitude <strong>in</strong>formation <strong>in</strong> the<br />
imag<strong>in</strong>g process. It has been documented that a st<strong>and</strong>ard Kirchh<strong>of</strong>f migration does not<br />
generally preserve amplitude (reflection coefficient) at the target reflector. We are<br />
currently work<strong>in</strong>g on<br />
a new concept (V<strong>in</strong>je, 2000), us<strong>in</strong>g 3D dynamic ray attributes to correct amplitudes <strong>in</strong><br />
common <strong>of</strong>fset Kirchh<strong>of</strong>f migrated data (reflector oriented amplitude recovery).<br />
Moreover, it should be added that the general wavefront construction method is currently<br />
extended to anisotropic media. This will def<strong>in</strong>itely have potentials to improve imag<strong>in</strong>g <strong>in</strong><br />
areas where such effects are present (Iversen et al., 2000).<br />
Stud. geophys. geod., 46 (2002) 159
H. Gjøystdal et al.<br />
With respect to modell<strong>in</strong>g <strong>of</strong> synthetic seismic data from large 3D multi-streamer<br />
surveys, wavefront construction has proved to be a robust <strong>and</strong> efficient technique. Us<strong>in</strong>g<br />
the option <strong>of</strong> distribut<strong>in</strong>g the job to a number <strong>of</strong> computers (st<strong>and</strong>ard workstations or<br />
nodes <strong>in</strong> a network), <strong>and</strong> tak<strong>in</strong>g advantage <strong>of</strong> the pr<strong>in</strong>ciple <strong>of</strong> ‘shot similarity’, tens <strong>of</strong><br />
thous<strong>and</strong>s <strong>of</strong> shots can be simulated with<strong>in</strong> reasonable computer times (although CPU<br />
time is drastically dependent <strong>of</strong> model complexity <strong>and</strong> smoothness. For complex detailed<br />
reservoir target zones overlaid by a simpler, more ray trac<strong>in</strong>g friendly overburden, the<br />
newly developed hybrid modell<strong>in</strong>g technique has turned out to be very feasible.<br />
Especially for repeated modell<strong>in</strong>g, e.g. when chang<strong>in</strong>g properties <strong>in</strong> the target zone while<br />
keep<strong>in</strong>g the overburden fixed (4D surveys, study<strong>in</strong>g effects <strong>of</strong> fluid changes), hybrid<br />
modell<strong>in</strong>g could be a powerful tool, also adapted for parallel process<strong>in</strong>g.<br />
F<strong>in</strong>ally, for future applications <strong>in</strong> prestack depth imag<strong>in</strong>g <strong>and</strong> time-to-depth map<br />
migration, the new concept <strong>of</strong> ‘isochron rays’ is considered as a promis<strong>in</strong>g tool. Also with<br />
respect to the calculation <strong>and</strong> propagation <strong>of</strong> the boundaries <strong>of</strong> Fresnel volumes, this new<br />
technique may have a potential, <strong>in</strong> a certa<strong>in</strong> way connect<strong>in</strong>g the ma<strong>in</strong> signal frequency to<br />
the otherwise frequency <strong>in</strong>dependent ray attributes. This may e.g. open for an efficient<br />
Fresnel zone dependent smooth<strong>in</strong>g <strong>of</strong> the ray amplitudes, reduc<strong>in</strong>g the drawbacks <strong>of</strong> the<br />
extremely local behavior <strong>of</strong> ray amplitudes. Similarly, ‘velocity rays’ can facilitate<br />
sensitivity studies related to the depth doma<strong>in</strong> <strong>of</strong> seismic data, such as propagat<strong>in</strong>g seismic<br />
images when velocity parameters change, for example <strong>in</strong> an anisotropic medium.<br />
Acknowledgement: The authors want to direct a general thank to Geco <strong>and</strong> Statoil for support<strong>in</strong>g a<br />
number <strong>of</strong> projects on 3D modell<strong>in</strong>g developments <strong>and</strong> applications at NORSAR throughout the<br />
eighties <strong>and</strong> n<strong>in</strong>eties. Thanks are also given to the sponsors <strong>of</strong> the hybrid modell<strong>in</strong>g project<br />
(HybriSeis Consortium): Eni Agip, Norsk Hydro, Statoil, S<strong>in</strong>tef Petroleum <strong>and</strong> NORSAR, <strong>and</strong> to<br />
the Research Council <strong>of</strong> Norway for support<strong>in</strong>g a number <strong>of</strong> modell<strong>in</strong>g activities. A special thank to<br />
Statoil Research for permission to use the Gullfaks modell<strong>in</strong>g example, <strong>and</strong> to Enterprise Oil for<br />
show<strong>in</strong>g some results from an attribute mapp<strong>in</strong>g project.<br />
F<strong>in</strong>ally, our deepest gratitude <strong>and</strong> respect goes to pr<strong>of</strong>essor Červený for devot<strong>in</strong>g his pr<strong>of</strong>essional<br />
life to ray theory. By his work, major theoretical advances have been made with<strong>in</strong> numerous<br />
research topics. In particular, he has been equipped with the ability <strong>of</strong> giv<strong>in</strong>g the theory a form<br />
suited for numerical implementation.<br />
Received: January 2, 2002; Accepted: March 5, 2002<br />
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