P262 Limited-aperture Acquisition Compensation for Shot Profile Imaging

Transcription

1 P262 Limited-aperture Acquisition Compensation for Shot Profile Imaging C.E. Yarman* (WesternGeco-Schlumberger), R. Fletcher (WesternGeco- Schlumberger) & X. Xiao (WesternGeco-Schlumberger) SUMMARY We develop shot-profile migration weights that enhance a given imaging condition by compensating for illumination due to limited/finite aperture receiver acquisition geometry. We demonstrate our method on the Sigsbee model using reverse time-migration (RTM). Although our example is for seismic imaging, the proposed method for derivation of the weights can be extended to other imaging modalities, such as radar or other electromagnetic based imaging modalities, sonar, thermo-acoustic imaging, ultrasound or other medical imaging modalities, etc.

2 Introduction The high dimensionality and ill-posedness of the seismic imaging inverse problem in complex areas make its solution challenging. There are two main reasons for this: rst is the limited and irregular acquisition of the seismic experiment, second is the complexity of the overburden. Combined, these two eects will produce irregular illumination of the subsurface, possibly even shadow zones. Various approaches have been proposed to attempt to compensate for these two eects and form true amplitude images with shot-prole migration. Valenciano et al. (29) explicitly compute an approximation of the Hessian matrix for a least-squares inversion solution. This approach is computationally too demanding to apply in anything other than a targetorientated fashion. Slightly cheaper, but still computationally demanding, is to nd the leastsquares solution by means of an iterative solver. Many approaches attempt to provide an approximate analytic inverse of the Hessian to reduce computational costs. These approaches form images that compensate for the illumination factors (Stolk et al. (29); Kiyashchenko et al. (27) and references therein); however, none of these approaches tackles the limited-aperture receiver acquisition geometry. In this paper, we present shot-prole migration weights that enhance a given imaging condition by compensating for illumination due to limited-aperture receiver acquisition geometry. We demonstrate our method on the Sigsbee model using reverse-time migration (RTM). Method For the sake of simplicity, we will consider source-illumination-compensated imaging conditions. The method presented can be extended to other imaging conditions, such as a true-amplitude imaging condition for acoustic or elastic wave propagation, by minor modications or no modi- cation. In the case of full-receiver acquisition aperture, the source-illumination-compensated imaging condition is given by the zero time correlation of source and receiver waveelds divided by the zero-time autocorrelation of the source waveeld, which is also referred to as source illumination: I(x) = s [ S(s, x, ), R Γ(s, x, ) ] [ S(s, x, ), S(s, x, ) ] 1. (1) Here, denotes inner product with respect to frequency ω, S(s, x, ω) = G (s, x, ω) p(ω) is the source waveeld obtained by injecting a source wavelet with spectrum p(ω), and R Γ is the receiver eld obtained by injecting the reversed-time data collected over the full-receiver aperture Γ: R Γ (s, x, ω) = G (s, x, ω) d (s, r, ω) dr, (2) r Γ where G (x, y, ω) is the Green's function for a given background model, d(s, r, ω) is the recorded data at receiver r due to a source at s, denotes complex conjugation, and x is the image point. One may use either a two-way or one-way wave propagator to compute G (r, x, ω). Thus, the following method can be utilized for all shot-prole migration methods including reverse-time, one-way waveeld extrapolation, Gaussian beam and Kirchho. In practice, one often does not have the opportunity to acquire data from a full aperture but only a portion of it. In this regard, we denote the receiver aperture for each source by Γ(s) and the image formed by using the collected data by I P (x) = s w(s, x) [ S(s, x, ), R(s, x, ) ] [ S(s, x, ), S(s, x, ) ] 1, (3)

3 where w(s, x) are referred to as migration weights and R(s, x, ω) = G (s, x, ω) d (r, s, ω) dr. (4) r Γ(s) We design the weights such that I P (x) approximates I(x): [ ] [ w(s, x) G (r, x, ω) 2 p(ω) 2 dr dω r Γ r Γ(s) G (r, x, ω) 2 p(ω) 2 dr dω] 1. (5) Note that, when imaging condition (1) is replaced with any weighted imaging condition whose weights depend only on the source location and imaging coordinate, then the weights (5) can be used with minor modications or without modication. For example, in the case of trueamplitude imaging, the weights of the imaging condition include terms involving the cosine squared or cosine cubed incident angle at the imaging coordinate (see equation (1) Kiyashchenko et al. (27) and equations (27) and (27a) in Miller et al. (1987)). In practice for RTM, this cosine-related term is implemented by a Laplacian work ow that is based on equation (6) of Zhang and Sun (28). A full derivation of equation (5) and details of it's ecient calculation will be left to the presentation. Example We demonstrate our weights on the Sigsbee model using RTM, where we used a smooth version of the model for migration (Figure 1). We used the conventional RTM imaging condition that was implemented using the Laplacian work ow as described in the previous section. For a single shot, we present S(s, x, ), S(s, x, ) and the weights w(s, x), which are computed using (5), in Figure 2. Our implementation of the computation of the numerator and denominator of the weights increases the cost of RTM by a factor of 5%. We present the migrated images obtained using the conventional imaging condition with and without the receiver aperture compensated weights in Figure 3. Conclusions We present a way to design weights for receiver aperture acquisition compensation for shotprole imaging. We gave an analytic way to approximate these weights and demonstrate the approximate weights on the Sigsbee model using RTM. References Kiyashchenko, D., Plessix, R.E., Kashtan, B. and Troyan, V. [27] A modied imaging principle for true-amplitude wave-equation migration. Geophysical Journal International, 168, Miller, D., Oristaglio, M. and Beylkin, G. [1987] A new slant on seismic imaging: migration and integral geometry. Geophysics, 52, Stolk, C., de Hoop, M. and Root, T.O. [29] Reverse time migration-inversion from single-shot data. 79th SEG Annual International Meeting, Expanded Abstracts, 28(1), Valenciano, A.A., Biondi, B.L. and Clapp, R.G. [29] Imaging by target-oriented wave-equation inversion. Geophysics, 74, WCA19WCA12. Zhang, Y. and Sun, J. [28] Practical issues of reverse time migration - true-amplitude gathers, noise removal and harmonic-source encoding. 7th EAGE Conference & Exhibition, Extended Abstracts, F47.

4 x ft/sec ft/sec Figure 1 (Left) The Sigsbee stratigraphic model and (Right) the smooth model used for RTM. Figure 2 (Left) The autocorrelation S(s, x, ), S(s, x, ) of the source waveeld and (Right) the corresponding receiver aperture compensation weights.

5 Figure 3 RTM images obtained by the conventional imaging condition (Top) with and (Bottom) without receiver aperture compensation weights.