1. Direct Migration of Passive Seismic Data Brad Artman 1, Deyan Draganov 2, Kees Wapenaar 2, Biondo Biondi 1 1 Stanford Exploration Project, Geophysics, Stanford University, 94305, USA 2 Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands General relations between the reflection and the transmission responses of a 3-D inhomogeneous medium [4] have been developed to forward model the transmission coda, suppress multiples, provide the basis of seismic interferometry, and forward model the reflection response from the transmission response. The final relation, exploited for acoustic daylight imaging, shows that by cross-correlating traces of the transmission response of a medium one can synthesize the reflection response collected in a conventional active source experiment. Having generated shot-gathers in this manner, they can be processed with conventional techniques to enhance signal, remove artifacts, or create a migrated image. Here, we present the theory of direct migration of the measured transmission response wavefields without the need to first generate the reflection response wavefields by cross-correlation. Removing this step allows for significant time savings when dealing with these inherently large data sets and produces an equivalent image. Apart from computational advantages, moving the modeling of the reflection response from the transmission response down to the image point during migration opens the possibility of more advanced imaging conditions in the future. Introduction In 2002, Wapenaar showed that for a 3-D inhomogeneous medium we can calculate the reflection response of the medium from its transmission response. Consider a 3-D, inhomogeneous, source-free domain D (see Figure 1) embedded between plan-parallel boundaries  D 0 and  D m. Just above  D 0 there is a free surface. The medium below the lower boundary  D m is assumed to be homogeneous. The medium is considered lossless so as to neglect the dynamics of attenuation. Passive seismic imaging formulae Fig. 1: Domain D, situated between plan-parallel surfaces  D 0 and  D m with its reflection (left) and transmission response (right). For this configuration we can write the following relation for the responses in D: In equation (1) R(x A,x B,t) denotes the reflection response of the domain D including all internal and free-surface multiples recorded at point x A in the presence of a source at point x B. T(x A,x,t) denotes the transmission response of the domain D including all internal and free-surface multiples recorded at point x A in the presence of a source at point x. x H,A symbolizes the horizontal coordinates of the point A. This relation shows that by cross-correlating the transmission responses measured at points x A and x B from sources in the subsurface we can simulate the reflection response (and less importantly its time-reversal) measured at x A from an impulsive source at surface location x B. Notice that there is no requirement to know when the subsurface sources activate in relation to each other or ‘time zero’ of the passive experiment. The zero lag of the cross-correlation effectively establishes ‘time zero’ of the simulated reflection experiment. If the integral over the boundary  D m, on which the subsurface sources are situated, is discretised and the subsurface sources are assumed white and completely uncorrelated in time, equation (1) can be rewritten as where T obs is the collection of observed traces that include all subsurface sources N i : In the derivation of equations (1) and (2) the evanescent fields were neglected and it was also assumed that the medium below the sources is homogeneous. While this formalism does not handle evanescent fields, Draganov (2003) shows that the homogenaity requirement is not restrictive. Neglecting the acausal term on the LHS, equation (2) shows that the reflection response of the subsurface can be directly modeled from the transmission response. The delta function is analagous to the direct arrival in the conventional experiment and can be muted from the output of the cross-correlations. Cross-correlation of each trace, acting as a source location, with every other trace results in N 2 traces of output. This volume of data is N simulated shot-gathers with N traces. Transmission wavefields, T obs, are very long records to include as many possible subsurface noise sources. The ouput of the cross-correlations however can be significantly shorter. Only as many lags (multiplied by the time discritization) to include the two-way-travel time of the deepest reflector of interest need be saved. Equations (3) and (4) are traces from the passive recording of the ambient seismic energy within the subsurface domain D observed at the surface points A and B repsectively. In equation (2) the noise sources are distributed along the surface  D m. However, the correlation process eliminates the extra travel times related to different source depths. This allows the sources to be randomly distributed at or below the surface  D m (see Figure 2). Fig. 2: Transmission response T obs recorded at points x A and x B at the free surface in the presence of white noise sources below the boundary  D m. (a) Shot-gathers by cross-correlation Fig. 4: (a) Modeled shot-gather over a mult-layered earth. (b) Cross-correlation of modeled transmission data to simulate same shot-gather using 100 minute input. (c) Simulated gather using 20 second input. Figure 3 shows how the quality of modeled shot-gathers decreases (by (N t ) 1/2 ) when shorter transmission records are processed. Eleven sub-surface impulses convolved with long random sequences were summed to generare the transmission wave-fields. The short records used in panel (c) results in almost loosing the reflection hyperbola. Even the longer record in panel (b) is poor compared to the real shot-gather in panel (a). Both modeled gathers will improve with increased coverage of the bottom of the earth model with source locations. In a real experiment increased recording time results in both capture of increasing quantities of ambient noise as well as the entire wave-train of long-coda incident energy. (a)(b)(c) Intuition via incident plane-wave (d) (b)(a) (c) Reflector Incident plane wave Downward reflection Free-surface reflection Free-surface reflection Fig. 3: Incident plane wave source passing through single reflector earth model at successive time steps (a-c). Panel d shows transmission wavefield recorded at the surface.

2. Using equation (2), we can calculate the reflection response by cross-correlating one of the transmission traces (in this case the trace at horizontal position 0 m. – see right picture on Figure 4) with all the transmission traces from the transmission panel. After muting the non-causal part from the correlation result, we receive the simulated reflection response as shown on Figure 5 (left). The right picture on the same figure shows for comparison the directly modelled reflection response. Fig. 5: (left) Simulated reflection response from the left model on Figure 3; (right) directly modelled reflection response for the same subsurface model. Acknowledgments This project is supported by the Netherlands Research Centre for Integrated Solid Earth Science (ISES) and the Dutch Science Foundation (STW, grant DTN4915) References Claerbout, J.F., 1968, Synthesis of a layered medium from its acoustic transmission response: Geophysics 33, 264-269. Wapenaar, C.P.A., Thorbecke, J.W., Draganov, D., and Fokkema, J.T., 2002, Theory of acoustic daylight imaging revisited: 72-nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded abstracts ST 1.5. Comparing the two pictures we see that on the simulated reflection response appear several ghost events. The one with apex at 0.20 s. and its free-surface multiples are result from the presence of the reflector below the sources. The ghost with apex at 0.50 al;kdjfa;lksdjfl;aksdjf from the left and the right clusters only (right), i.e. there is a relatively big gap between the sources, we can hardly recognize the reflection. This is not the case when the structure is illuminated by the left and the middle white noise source clusters (left). The numerical simulations in this poster confirm relation (2) between the reflection and transmission response of a 3-D inhomogeneous losseless medium in the presence of white noise sources in the subsurface. When reflectors are present below the sources, ghost events appear in the simulated reflection response. These ghost events are strongly weakened, however, when the noise sources have random distributed depths. Big gaps in the horizontal distribution of the noise sources results in poor simulated reflection response. It is important to have the sources “exposing” the structure of interest from all the angles. The effect of internal multiples and refraction in the transmission data before the first free-surface reflection needs to be further investigated. Conclusions Fig. 6: (left) Simulated reflection response from the right model on Figure 3; (right) directly modelled reflection response for the same subsurface model. Fig. 7: Anticline subsurface model with extra reflector below the sources. (left) The sources are evenly distributed in the horizontal direction from –2800 m. till 2800 m. each 25 m. and with random depths between depth levels 700 and 800 m.; (right) simulated reflection response for the model to the left Fig. 8: (left) Simulated reflection response from the right model on Figure 3 when sources only from the left and the middle clusters are present; (right) simulated reflection response from the right model on Figure 3 when sources only from the left and the right clusters are present. Direct Migration of Passive Seismic Data Brad Artman 1, Deyan Draganov 2, Kees Wapenaar 2, Biondo Biondi 1 1 Stanford Exploration Project, Geophysics, Stanford University, 94305, USA 2 Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands