through the model, and the up-going wave field is obtained from the de-propagation process (back propagated in time) of the recorder wave field into the model ...
Improve Quality of Depth Image Using Reverse Time Migration
Improve d Quality of Depth Images Using Reverse Time Migration André Bulcão* and Djalma Manoel Soares Filho, Petrobras Research Center, Webe João Mansur, Federal University of Rio de Janeiro Summary This paper present s the results of a new scheme that improves the quality of the depth image obtained using Reverse Time Migration with acoustic full two way wave equation. In this scheme - with each time step – in both down-going and up-going wave fields a new procedure to split the wave fields is applied. As a result, most of the energy that is traveling in the upward direction is eliminated and only the energy in the downward direction remains. This avoids the artifacts caused by the reflected waves [Fletcher et all, 2006] which are traveling in the upward direction. The extension for 3D problems is straightforward. As will be shown with synthetic examples the quality of the depth images is greatly improved, without applying any kind of filter. This fact makes the output ideal for AVO (Amplitude Versus Offset) analysis, because the amplitudes associated with reflection coefficients are preserved. Introduction In Reverse Time Migration (RTM) using the full two-way wave equation, there are artifacts in the depth image associated with the wave field extrapolation, that were reflected during the depropagation process. There are several different works, in this field, that apply distinct versions of the one way wave equation to make the wave field separation, which extrapolates it only in the downward direction. The use of this kind of equation avoids the appearance in the image the artifacts related with reflected waves. However, with this approximation there are some types of waves that can not be easily represented with these one-directional models, such as turning waves [Xu et all, 2006]. In the specific case of imaging structures near salt flanks and salt bodies, there are, ni the specialized literature, several works in which better quality images of these structures are obtained using turning waves [Zhang et all, 2006 and Xie et all, 2006]. Due to impedance variations in the seismic model, these types of waves have the propagation direction changed as they travel across the model. The simplest example of a model where this phenomenon occurs is a model where there is a linear increase in the velocity as the depth increases (situation characteristic of seismic model found in the Gulf of Mexico).
In this paper the full acoustic two-way wave equation is applied to obtain the complete wave field during the extrapolation process. A new scheme to split the wave field is applied on each time step in both downward and upward directions. However, there are the presence and the interactions of all different kinds of waves during the extrapolation process. In Reverse Time Migration the depth image is formatted using the call image condition. In this work two types of image conditions were used to generate the depth images: the excitation time image condition [Chang et all, 1986] and the correlation image condition [Claerbout, 1985]. In the case of correlation between the down-going and upgoing wave fields, in order to generate the depth image, the expression that simply computes the zero lag correlation was applied. Wave Field Separation Although the main aim of this work is to show the quality improvement of the depth image, in this section some results of the new wave separation scheme used to split the full two-way wave field are presented. The proposed methodology for scalar wave fields is similar to the use of the pointing vector to obtain the propagation direction (equation 1) [Cunha 1992; and Yoon et all 2004]. This measurement is more commonly used in electromagnetic problems, and was successfully applied by the authors with the same purpose as in the case of elastic wave fields [Bulcão 2004].
S j = −viτ ij
Eq. 1
Where: S is the pointing vector; v is the particle velocity and τ is the stress tensor. In the new proposed methodology for scalar wave fields, before equation 1 was applied, the following expression (equation 2) was used with the same intention as a means to improve the efficiency of the wave field separation in the downward direction [Robinson, 1999]:
u& + = 1 2 (u& − cu j )
Eq. 2
Where: u is the acoustic wave field, c is the propagation velocity and u+ represents the wave field in the downward direction.
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Improve Quality of Depth Image Using Reverse Time Migration
Some results of the pointing vector scheme are presented in figure 1 in comparison with the proposed scheme, considering the propagation process from the source point through the model. In these figures, the interfaces of the homogeneous layer of the velocity model are shown together with the wave field. The following comments are made according to the results shown in figure 1: • Where there is no interaction in the wave field between the waves that travel in the downward and upward directions, both methods present equivalent results; • The proposed scheme presents better results where there are interactions between waves traveling in the downward and up ward directions, and the amplitudes in these directions are better preserved because of the formulation employed. Other results showing the separation wave field scheme will be presented in the next sections. Reverse Time Migration In Reverse Time Migration the image condition is based on the fact that at the reflectors both wave fields, the downgoing and the up-going are in-phase. In the case of conventional data set, the down-going wave field is produced by the propagation process from the source point through the model, and the up-going wave field is obtained from the de-propagation process (back propagated in time) of the recorder wave field into the model (using the receiver locations to prescribe the energy registered as a source point). Basically, the Reverse Time Migration algorithm is divided in three parts. Depending on the specific scheme employed these parts can be computed independently or not: i) Propagation process, simulation of the wave field that propagates from the source point t hrough the model; ii) De-propagation process, extrapolation of the recorder wave field (seismogram), which is prescribed in the reverse order at the receiver location; iii) Application of the imaging condition. These parts are computed separately by apply ing the excitation time imaging condition, and to relate the time between the source point to the reflector for each point of the model, the travel time matrix is employed. This matrix is obtained using one criterion during the propagation process [Loewenthal et all 1991] or it can be approached using ray tracing, Eikonal equation, or other procedures. Using the excitation time imaging condition, in this work, the criterion to obtain the travel time matrix and the image
condition are computed using the maximum amplitude criterion. Using the correlation imaging condition all three parts are computed together [Cunha 1992 and Bulcão 2004] and the depth image is obtained by the zero lag correlation between the down- and the up -going wave fields [Clearbout 1985]. Numerical Example – Pluto Velocity Model Figure 2 presents the PLUTO velocity model that was applied to test the principal objective of this work. In any case this velocity model was smoothed to generate the results obtained in this paper. At first this model was used to generate a synthetic seismogram, which was obtained using an acoustic modeling program, also solving the acoustic two-way wave equation. Non-reflection boundary conditions on all sides of the velocity model were applied during the modeling. As a result no multiples related with the surfaces appear in the seismic data, although, due to the strong impedance contrast at the salt boundaries, peg-legs appear in the seismic data. These signals should be removed during the processing sequence, and their presence will produce out of place reflectors in the depth image. The acquisition geometry applied was chosen in order to have receivers at all points on surface of the model, and the modeling was made with the intention to simulate a data set that would be formatted by applying an Areal Shot Profile [Berkhout 1992] or a Time Delay with a horizontal plane wave [Zhang et all, 2003]. The figure 3 presents the synthetic seismogram used as input data to the Reverse Time Migration algorithms. The figure 4 shows the snapshots of the down-going wave field with and without the proposed scheme of wave field separation. These figures clearly show the effectiveness of the proposed scheme. When this wave field is used as an input for Reverse Time Migration algorithms, applying the two types of imaging condition used in this work, there are almost no artifacts in depth images related with reflected waves, as presented in figures 5 and 6. In general the results are greatly enhanced. Conclusions The efficiency of the scheme to split the complete wave field obtained using the two-way wave equation through several synthetic examples was proved. Although only the vertical directions (upward and downward) were presented, this methodology could be easily applied to any direction. (Continued)
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Improve Quality of Depth Image Using Reverse Time Migration
Complete Wave Field (by Full Two -Way Wave Equation)
Wave Field in Downward Direction Pointing Vector Scheme
Wave Field in Downward Direction New Proposed Scheme
Figure 1 – Numerical example: the complete wave field and others schemes for wave field separations are compared.
Figure 2 – Pluto Velocity Model (VP)
Travel Time Matrix
with wave field separation
without wave field separation
Depth Image
Figure 3 – Synthetic Seismogram employed by RTM
Figure 5 – Depth images and travel time matrixes obtained using the excitation time imaging condition, with and without the proposed scheme of wave field separation
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Improve Quality of Depth Image Using Reverse Time Migration
Complete Wave Field (by Full Two -Way Wave Equation)
Wave Field in Downward Direction New Proposed Scheme
Figure 4 – Snapshots of the down-going wave field with and without the proposed scheme of wave field separation.
Image without wave field separation Image with the proposed wave field separation Figure 6 – Depth images obtained using the correlation imaging condition, with and without the proposed scheme of wave field separation
With the proposed scheme it is possible to work with the proportion of the wave field energy (on the downward or upward directions) that really matters for the depth image generation. In this work only the energy that travels in the downward direction was used to obtain the depth images, but it is also possible to employ both directions separately,
and combine them to produces image structures such as vertical salt flanks [Zhang et all, 2006]. The results presented demonstrate unquestionably the improvement in the depth image by applying the scheme to split the wave field during the Reverse Time Migration algorithm.
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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2007 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Berkhout, A. J., 1992, A real shot record technology: Journal of Seismic Exploration, 1, 251–264. Bulcão, A., 2004, Modelagem e Migração Reversa no Tempo Empregando Operadores Elásticos e Acústicos, Tese de Doutorado – Universidade Federal do Rio de Janeiro: PEC/COPPE. Chang, W. F, and G. A. McMechan, 1986, Reverse-time migration of offset vertical seismic profiling data using the excitationtime imaging condition: Geophysics, 51, 67–84. Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications, Inc. Cunha, C. A. F, 1992, Elastic modeling and migration in earth models: Ph.D. dissertation, Stanford University. Fletcher, R. P., P. L. Fowler, P. Kitchenside, and U. Albertin, 2006, Suppressing unwanted internal reflections in prestack reverse-time migration: Geophysics, 71, no. 6, E79–E82. Loewenthal D., and L. Hu, 1991, Two methods for computing the imaging condition for common-shot prestack migration: Geophysics, 56, 378–381. Robinson, E. A., 1999, Seismic inversion and deconvolution: Handbook of Geophysical Exploration, 4B, Amsterdam: Pergamon. Xie, X., and R. Wu, 2006, A depth migration method based on the full-wave reverse-time calculation and local one-way propagation: 76th Annual International Meeting, SEG, Expanded Abstracts, 2333–2337. Xu, S., and S. Jin, 2006, Wave equation migration of turning waves: 76th Annual International Meeting, SEG, Expanded Abstracts, 2328–2332. Zhang, Y., J. Sun, C. Notfors, S. Gray, L. Chernis, and J. Young, 2003, Delayed-shot 3D prestack depth migration: 73rd Annual International Meeting, SEG, Expanded Abstracts, 1027–1030. Zhang, Y., S. Xu, and G. Zhang, 2006, Imaging complex salt bodies with turning-wave one-way wave equation: 76th Annual International Meeting, SEG, Expanded Abstracts, 2323–2327.
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