- Email: [email protected]
When Reflections are not Hyperbolas and Reflectors are not Points
Journal of Applied Geophysics 42 Ž1999. 139–141 www.elsevier.nlrlocaterjappgeo
Editorial
When Reflections are not Hyperbolas and Reflectors are not Points Oz Yilmaz Paradigm Geophysical (UK) Ltd., 2nd Floor, Regal Court, 42 r 44 High Street, Slough, Berkshire SL1 1EL, UK
You have started a new project that involves analyzing seismic data from the Alberta Plains. The subsurface is just as flat as the surface over which you have recorded the data. So flat are the reflectors in the subsurface that you can even conduct stacking velocity analysis using shot gathers at the start of your analysis sequence. Reflection traveltimes on your common-midpoint ŽCMP. gathers ŽWere they not called common-depth-point Ž CDP. gathers as recently as yesterday?. follow the near-ideal hyperbolic moveout trajectories. So, you compute your velocity spectra and pick your moveout velocity functions with this comforting notion in mind. When stacking the data, you can almost picture the summing of amplitudes in your moveout-corrected CMP gather at the same time over all offsets, and the resulting stacked amplitude being placed at a point reflector where the CMP raypaths nicely converge. When migrating the data, again, you sum the amplitudes along a hyperbolic trajectory and place the result at the apex of your hyperbola. You conveniently associate the apex of the latter hyperbola with a point diffractor situated on the reflector. Whether it is a reflection hyperbola associated with a point reflector or a diffraction hyperbola associated with a point diffractor, the process of stacking and migrating the data involves summation of amplitudes along hyperbo-
las and placing the resulting sum to a point in the subsurface. Such is your view of the world in the Plains — you think of reflections and diffractions as hyperbolas, and reflectors and diffractors as points. You move onto your next project from the Alberta Foothills where the cascaded flanks of the Rocky Mountains rise steeply. The subsurface is just as steep as the surface over which you have recorded your data. A diffractor still is a point whether it is in the Plains or in the Foothills, and you can still think of diffractions as hyperbolas so long as they are situated below a simple overburden. Fortunately, you can also continue to think of reflections as hyperbolas. No longer, however, can you associate a reflection hyperbola on your CMP gather with a single point reflector; instead, you have reflection points dispersed along the reflector. This is when you have to introduce a new step in your conventional processing sequence — dipmoveout Ž DMO. correction after normalmoveout ŽNMO. correction to account for the reflection-point dispersal. Once the reflectionpoint dispersal is removed, the resulting stack can be considered equivalent to a zero-offset section which you can migrate, again, using the hyperbolic summation rule. You have thus been able to overcome the Foothills problem of steeply dipping events.
0926-9851r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 6 - 9 8 5 1 Ž 9 9 . 0 0 0 3 4 - 8
140
Editorial
Finally, you move further west into the Canadian Thrust Belt beyond the Foothills. The subsurface is just as complex as it appears on the surface over which you have recorded your data. Your diffractor situated below the complex overburden structure caused by overthrust tectonics stays a point; no longer, however, can you associate it with a hyperbola. Instead, when migrating the data, you have to deal with a very complex, distorted traveltime trajectory with cusps. And you can no longer think of reflections as hyperbolas either. Instead, when stacking the data, you have to deal with a nonhyperbolic moveout trajectory associated with many reflection points scattered around in the subsurface. Your stacked section no longer resembles a zero-offset section. So the simple hyperbolic and point rules of the Plains or the Foothills are no longer applicable in the Thrust Belt. To overcome the first problem — migration of data in the presence of strong lateral velocity variations associated with complex overburden structures in the Thrust Belt, you decide to do the imaging in depth instead of imaging in time as you would have done in the Plains or in the Foothills. Earth imaging in depth requires earth modeling in depth — a challenge much higher than the imaging itself. To overcome the second problem — stacking of data with nonhyperbolic moveout, you may at first constrict your offset range to minimize the traveltime and amplitude distortions during stacking. Or you may combine the first and second problems and pursue a rigorous solution by way of prestack depth migration. In brief, your complex problem requires a complex solution that involves earth modeling and imaging in depth. While many of us have been pursuing earth modeling and imaging in depth for the last 10 years, some of us have been pursuing alternative ideas. Using his knowledge in optics, Eric de Bazelaire some years ago invented a way to stack CMP data using shifted hyperbolas. Suddenly, we began to see diffractions and reflec-
tions on stacked sections that we have failed to preserve by conventional stacking. Then came on the scene Boris Gelchinsky. With a style of his own, he was trying to tell us that we do not need DMO correction nor prestack depth migration to tackle the problems we encounter in the Foothills and the Thrust Belt. Surely, by way of DMO correction we map events to common-reflection points in their unmigrated positions. Similarly, by way of prestack depth migration we map events to common-reflection points in their migrated positions. But, Boris kept on telling us that there is no such thing as a common-reflection point whether it is in the unmigrated or migrated position of the data. Instead, we should associate the recorded data with common-reflection surfaces. He insisted that when stacking the data we should not constrict the summation of amplitudes to within a single CMP gather based on a one-parameter stacking Õelocity analysis. Instead, we should focus the data to a common-reflection-surface (CRS) using multiple shots and receiÕers based on a three-parameter analysis. The three parameters are the emergence angle of a zero-offset ray at the surface, the curvature of the wavefront at the normal-incidence-point on the reflector associated with the zero-offset ray, and the curvature of the wavefront that represents the envelope of the wavefronts associated with Huygens’ secondary sources along the reflector. While you only need a one-paramater hyperbolic traveltime equation to stack a CMP gather, you need a three-parameter traveltime equation to stack a CRS gather. Such was the inception of the multifocusing method. While we understood the conceptual issue with common-reflection point stacking, most of us did not understand the alternative solutions offered by Eric and Boris to circumvent the problem. Finally, Peter Hubral, Martin Tygel, Evgeny Landa and other contributors to this special issue, with their elegant theory and keen practical insight, brought the concepts developed by Boris Gelchinsky and Eric de Bazelaire
Editorial
under the spotlight. I was very fortunate to have participated in the Karlsruhe Workshop on February 15, 1999, conducted by Peter Hubral on the broader topic of multifocusing and model-independent imaging. In this special workshop issue, you will find excellent papers that provide a sound theoretical understanding
141
of the subject accompanied with convincing case studies. The future developments must include practical refinements to various approaches to common-reflection-surface stacking and imaging, and their extensions to 3D to fully exploit their potential in areas with complex structures.
Editorial
When Reflections are not Hyperbolas and Reflectors are not Points Oz Yilmaz Paradigm Geophysical (UK) Ltd., 2nd Floor, Regal Court, 42 r 44 High Street, Slough, Berkshire SL1 1EL, UK
You have started a new project that involves analyzing seismic data from the Alberta Plains. The subsurface is just as flat as the surface over which you have recorded the data. So flat are the reflectors in the subsurface that you can even conduct stacking velocity analysis using shot gathers at the start of your analysis sequence. Reflection traveltimes on your common-midpoint ŽCMP. gathers ŽWere they not called common-depth-point Ž CDP. gathers as recently as yesterday?. follow the near-ideal hyperbolic moveout trajectories. So, you compute your velocity spectra and pick your moveout velocity functions with this comforting notion in mind. When stacking the data, you can almost picture the summing of amplitudes in your moveout-corrected CMP gather at the same time over all offsets, and the resulting stacked amplitude being placed at a point reflector where the CMP raypaths nicely converge. When migrating the data, again, you sum the amplitudes along a hyperbolic trajectory and place the result at the apex of your hyperbola. You conveniently associate the apex of the latter hyperbola with a point diffractor situated on the reflector. Whether it is a reflection hyperbola associated with a point reflector or a diffraction hyperbola associated with a point diffractor, the process of stacking and migrating the data involves summation of amplitudes along hyperbo-
las and placing the resulting sum to a point in the subsurface. Such is your view of the world in the Plains — you think of reflections and diffractions as hyperbolas, and reflectors and diffractors as points. You move onto your next project from the Alberta Foothills where the cascaded flanks of the Rocky Mountains rise steeply. The subsurface is just as steep as the surface over which you have recorded your data. A diffractor still is a point whether it is in the Plains or in the Foothills, and you can still think of diffractions as hyperbolas so long as they are situated below a simple overburden. Fortunately, you can also continue to think of reflections as hyperbolas. No longer, however, can you associate a reflection hyperbola on your CMP gather with a single point reflector; instead, you have reflection points dispersed along the reflector. This is when you have to introduce a new step in your conventional processing sequence — dipmoveout Ž DMO. correction after normalmoveout ŽNMO. correction to account for the reflection-point dispersal. Once the reflectionpoint dispersal is removed, the resulting stack can be considered equivalent to a zero-offset section which you can migrate, again, using the hyperbolic summation rule. You have thus been able to overcome the Foothills problem of steeply dipping events.
0926-9851r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 6 - 9 8 5 1 Ž 9 9 . 0 0 0 3 4 - 8
140
Editorial
Finally, you move further west into the Canadian Thrust Belt beyond the Foothills. The subsurface is just as complex as it appears on the surface over which you have recorded your data. Your diffractor situated below the complex overburden structure caused by overthrust tectonics stays a point; no longer, however, can you associate it with a hyperbola. Instead, when migrating the data, you have to deal with a very complex, distorted traveltime trajectory with cusps. And you can no longer think of reflections as hyperbolas either. Instead, when stacking the data, you have to deal with a nonhyperbolic moveout trajectory associated with many reflection points scattered around in the subsurface. Your stacked section no longer resembles a zero-offset section. So the simple hyperbolic and point rules of the Plains or the Foothills are no longer applicable in the Thrust Belt. To overcome the first problem — migration of data in the presence of strong lateral velocity variations associated with complex overburden structures in the Thrust Belt, you decide to do the imaging in depth instead of imaging in time as you would have done in the Plains or in the Foothills. Earth imaging in depth requires earth modeling in depth — a challenge much higher than the imaging itself. To overcome the second problem — stacking of data with nonhyperbolic moveout, you may at first constrict your offset range to minimize the traveltime and amplitude distortions during stacking. Or you may combine the first and second problems and pursue a rigorous solution by way of prestack depth migration. In brief, your complex problem requires a complex solution that involves earth modeling and imaging in depth. While many of us have been pursuing earth modeling and imaging in depth for the last 10 years, some of us have been pursuing alternative ideas. Using his knowledge in optics, Eric de Bazelaire some years ago invented a way to stack CMP data using shifted hyperbolas. Suddenly, we began to see diffractions and reflec-
tions on stacked sections that we have failed to preserve by conventional stacking. Then came on the scene Boris Gelchinsky. With a style of his own, he was trying to tell us that we do not need DMO correction nor prestack depth migration to tackle the problems we encounter in the Foothills and the Thrust Belt. Surely, by way of DMO correction we map events to common-reflection points in their unmigrated positions. Similarly, by way of prestack depth migration we map events to common-reflection points in their migrated positions. But, Boris kept on telling us that there is no such thing as a common-reflection point whether it is in the unmigrated or migrated position of the data. Instead, we should associate the recorded data with common-reflection surfaces. He insisted that when stacking the data we should not constrict the summation of amplitudes to within a single CMP gather based on a one-parameter stacking Õelocity analysis. Instead, we should focus the data to a common-reflection-surface (CRS) using multiple shots and receiÕers based on a three-parameter analysis. The three parameters are the emergence angle of a zero-offset ray at the surface, the curvature of the wavefront at the normal-incidence-point on the reflector associated with the zero-offset ray, and the curvature of the wavefront that represents the envelope of the wavefronts associated with Huygens’ secondary sources along the reflector. While you only need a one-paramater hyperbolic traveltime equation to stack a CMP gather, you need a three-parameter traveltime equation to stack a CRS gather. Such was the inception of the multifocusing method. While we understood the conceptual issue with common-reflection point stacking, most of us did not understand the alternative solutions offered by Eric and Boris to circumvent the problem. Finally, Peter Hubral, Martin Tygel, Evgeny Landa and other contributors to this special issue, with their elegant theory and keen practical insight, brought the concepts developed by Boris Gelchinsky and Eric de Bazelaire
Editorial
under the spotlight. I was very fortunate to have participated in the Karlsruhe Workshop on February 15, 1999, conducted by Peter Hubral on the broader topic of multifocusing and model-independent imaging. In this special workshop issue, you will find excellent papers that provide a sound theoretical understanding
141
of the subject accompanied with convincing case studies. The future developments must include practical refinements to various approaches to common-reflection-surface stacking and imaging, and their extensions to 3D to fully exploit their potential in areas with complex structures.