Bending the wrong way and imaging the right way, Ilana Erez and Luc T. Ikelle (Texas A&M University) ABSTRACT An analysis of scattering diagrams (i.e., Feynmann-like diagrams for wave scattering) of the correlation-type representation theorem has recently revealed a new type of scattering in inhomogeneous media. Unlike common scattering events, the new events are inconsistent with the current interpretation of some of the basic physical laws, such as Snell's law, just like the so-called ``negative refraction'' in optics. Yet we find them very useful, for instance, in suppressing some undesired events from scattering data, in separating reflected and refracted waves, even in imaging seismic data. We will describe the results of these applications of this new type of scattering Genesis of virtual events Representation Theorems Sommerfield’s radiation condition Convolution Correlation Diagrammatica Seismics/ Quantum Field Theory/ Optics Seismics Optics The correlation-type representation contains some “reverted” wave-propagation paths (that is, paths of waves which propagate in the opposite direction from energy flow---very much like watching a movie of wave propagation being played backward) in the construction of normal seismic events. These reversal wave- propagation paths lead to the reversal of some physical laws, such as Snell's law. There is an analogy between the way virtual reflections appear and disappear and the way virtual particles appear and disappear in quantum field theory. Virtual particles are theoretical particles that cannot be detected directly but are nonetheless a fundamental part of quantum field theory. Virtual particles are often popularly described as coming in pairs, a particle and an antiparticle, which can be of any kind. These pairs mutually annihilate each other. The virtual wave-scattering events also come in pairs in order to generate multiples and primaries. CONVOLUTION-TYPECORRELATION-TYPE Integration over the top boundary Integration over the bottom boundary

Bending the wrong way and imaging the right way (Cont.), Internal multiples, Ilana Erez and Luc T. Ikelle (Texas A&M University) Renormalization In physics, renormalization refers to a variety of theoretical concepts and computational techniques revolving either around the idea of rescaling transformation, or around the process of removing infinities from the calculated quantities. The renormalization is used here in the context of rescaling a transformation, more precisely, rescaling the crosscorrelation operation or the convolution. Data V P Without renormalization Virtual field [ K = Cross ( V,P) ] Predicted data [ W = Conv ( K,V) ] With renormalization Normalized virtual field [ K’ = Cross ( V,P’) ] Predicted data [ W = Conv ( K’,V) ] Example 1 Example 2 Data (V) Data (P) Virtual field [K = Cross (V,P)] Predicted internal multiples [W = Conv (K,V)] Data Demultiple Basic Approach Period of Multiples We know that the smallest period of free-surface multiples is the two-way traveltime in the water column. This period is a fundamental feature which allows us to distinguish between primaries and free- surface multiples. For practical purpose, we require this period to be longer the duration of source signature, to be specific about more 100 ms. In the case of internal multiples, the period of multiples can very small (less than the typical seismic temporal sampling interval, which is 4 ms) or infinity large because we have heterogeneties at almost all scale in the subsurface as well logs and core samples have shown. Actually, primaries themselves are generally an average of small period internal multiples. In other words, the period of internal multiple is not as clearly defined as that of the free-surface multiples and it is a parameter that we need to control to avoid modifying our primary signals.

Amundsen, L., 2001, Elimination of free-surface related multiples without need of the source wavelet: Geophysics, 66, Amundsen, L., L.T. Ikelle, and L.E. Berg, 2001,Multidimensional signature deconvolution and free-surface multiple elimination of 4C data: Geophysics, 66, Berkhout, A.J., and D.J. Verschuur, 2005, Removal of internal multiples with the common-focus-point (CFP) approach: Part1---Explanation of the theory: Geophysics, 70, V45--V60. Bojarski, N., 1983, Generalized reaction principles and reciprocity theorems for the wave equation, and the relationship between the time-advanced and time-retarded fields: J. Acoust. Soc. Am., 74(1), de Hoop, A.T., 1995, Handbook of radiation and scattering of waves: Academic Press, San Diego, CA. Fokkema, J.T., and P.M. van den Berg, 1993, Seismic applications of acoustic reciprocity: Elsevier Science Publ. Gangi, A.F., 1970, A derivation of the seismic representation theorem using seismic reciprocity: J. Geophys. Res., 75, Houck, A.A., J.B. Brock, and I.L. Chuang, 2003, Experimental observations of a left-handed material that obeys Snell's law: Phys. Rev. Letters, 9013, Ikelle, L.T., G. Roberts, and A.B. Weglein, 1997, Source signature estimation based on the removal of first-order multiples: Geophysics, 62, Ikelle, L.T., L. Amundsen, A. Gangi, and S. Wyatt, 2003, Kirchhoff scattering series: Insight into the multiple attenuation method: Geophysics, 68, Ikelle, L. T., 2003, Attenuation of long-period internal multiples: Expanded abstract of the first Symposium on Interpreting Reservoir Architecture using Scale-Frequency Phenomena (June 19-20, 2003), Edited by Ray Brown and Evgeni Chesnokov: The Oklahoma Geological Survey, Oklahoma City, Oklahoma Ikelle, L.T., 2004, A construct of internal multiples from surface data only: 74th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts. Ikelle, L.T., 2005, A construct of internal multiples from surface data only: the concept of virtual seismic events: Geophysical Journal International, in press. Ikelle, L.T., and L. Amundsen, 2005, An introduction to petroleum seismology: Investigations in Geophysics, Society of Exploration Geophysics, Tulsa. Ikelle, L.T., 2004, A construct of internal multiples from surface data only: 74th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts. Ikelle, L.T., 2005, New Type of Reflections in Inhomogeneous Media Is Revealed by an Analysis of Scattering Diagrams of the Correlation-Type Representation Theorem: Journal of Seismic exploration, in press. Schuster, G.T., J. Yu, J. Sheng, and J. Rickett, 2004, Interferometric/daylight seismic imaging: Geophysical Journal International, 157, Sommerfield, A., 1954, Optics: Academic Press, New York. Verschuur, D.J., and A.J. Berkhout, 2005, Removal of internal multiples with the common-focus-point (CFP) approach: Part2---Application strategies and data examples: Geophysics, 70, V61--V72. Veselago, V.G., 1968, The electrodynamics of substances with simultaneously negative values of ξ and μ: Soviet Physics Uspekhi, 10, van Manen, D.J., J. O. A. Robertsson, and A. Curtis, 2005, Modeling of Wave Propagation in Inhomogeneous Media: Physical Review Letters, 94, Wapenaar, K., 2004, Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by crosscorrelation: Physical Review Letter, 93, CONCLUSIONS We have described a construct of internal multiples which is based on the classical computational operations encountered in the construction of free- surface multiples. Our construct uses only surface data and does not require any knowledge of the subsurface. Other developments A Construct of Internal Multiples; head waves; imaging (cont’) REFERENCES (a) 1D synthetic data consisting of four primaries. We have divided these data into two parts: d 0 (x,t) and d' 0 (x,t). (b) is d 0 (x,t) and (c) d' 0 (x,t). An illustration of the construction of virtual seismic data as a multidimensional correlation of the actual d 0 (x,t) with d 0 (x,t). (a) is the actual data, (b) is d 0 (x,t), and (c) is the field of virtual seismic events. An illustration of the construction of internal multiples as a multidimensional convolution of the field of virtual events (without the apparent direct-wave arrivals) with the data d' 0 (x,t). (a) is d' 0 (x,t), (b) is d' v (x,t), and (c) is the field of predicted internal multiples d I (x,t). Notice also that the field of predicted internal multiples is displayed for a time window between 1.0 s and 3.2 s, whereas the data d' 0 (x,t) and the field of virtual events d' v (x,t) are displayed for the time window between 0.0 s and 2.2 s. a a a b b b c c c Numerical implementation strategy See Ilana Erez’ s thesis Notice that the separation of seismic data at the BIMG location does not require any special smoothing technique, as we are going to end up convolving the truncated data with the field of virtual events. This convolution allows us to smooth any rough edges that the separation of data at the BIMG location might have created. For long-offset data, some events may have their trajectories crossing the BIMG. In other words, one portion of an event may be located above the BIMG, and the other portion of the same event may be located below the BIMG. This separation is not a problem; the portion of the event located above the BIMG will be used to predict internal multiples in one iteration, and the second portion of the event located below the BIMG will be used in the next iteration to predict the second set of internal multiples associated with the event located below the BIMG. In other words, the fact that some complex events may not fall completely above the BIMG or completely below the BIMG is another reason why the iterative process is necessary. Watts and Ikelle (2005) show examples of this point for complex models containing salt bodies. Separation of reflections and headwaves Imaging of Internal multiples (Luc T. Ikelle) +              i.    (a)(b) (a) – recorded reflection and refraction (b) – autocorrelation of (a) (field of virtual events) (b) in F-K domain – easy separation of     step i.