Generalized internal multiple imaging

Internal multiples deteriorate the image when the imaging procedure assumes only single scattering, especially if the velocity model does not have sharp contrasts to reproduce such scattering in the Green’s function through forward modeling. If properly imaged, internal multiples (internally scattered energy) can enhance the seismic image. Conventionally, to image internal multiples, accurate, sharp contrasts in the velocity model are required to construct a Green’s function with all the scattered energy. As an alternative, we have developed a generalized internal multiple imaging procedure that images any order internal scattering using the background Green’s function (from the surface to each image point), constructed from a smooth velocity model, usually used for conventional imaging. For the first-order internal multiples, the approach consisted of three steps, in which we first back propagated the recorded surface seismic data using the background Green’s function, then crosscorrelated the back-propagated data with the recorded data, and finally crosscorrelated the result with the original background Green’s function. This procedure images the contribution of the recorded first-order internal multiples, and it is almost free of the single-scattering recorded energy. The cost includes one additional crosscorrelation over the conventional single-scattering imaging application. We generalized this method to image internal multiples of any order separately. The resulting images can be added to the conventional single-scattering image, obtained, e.g., from Kirchhoff or reversetime migration, to enhance the image. Application to synthetic data with reflectors illuminated by multiple scattering (double scattering) demonstrated the effectiveness of the approach. INTRODUCTION Internal multiples recorded in the seismic data can be an invaluable source of information in cases in which the single-scattering approximation is limited to image certain reflectors. Moreover, the multiscattered data not only improve the illumination of the target reflectors but also enhance the amplitudes in the presence of strong scattering layers (like in Saudi Arabia, where strong scattering above target reflectors weakens their image). Conventionally, to image internal multiples, one would need sharp contrasts in the velocity model to build Green’s functions that include the multiple scattering. However, often such sharp contrast information for the velocity model is not readily available. Recently, an iterative imaging procedure has been suggested by Behura et al. (2012) (see also Behura et al., 2014; Slob et al., 2014) to image all the internal scatterings together with single scattering. This iterative process does not allow for separate imaging of different orders of multiples. It also requires muting of anticausal events after crosscorrelation, but this can be automated. Slob et al. (2014) take a different route, in that they decompose the focusing function into upand downgoing parts. The Marchenko method of inverse scattering is considered applicable to acoustic fields; therefore, the Marchenko-type methods such as those mentioned above should give correct amplitudes. In this study, we do not rely on the Marchenko equation. Other developments in imaging multiple scattering are given by Malcolm et al. (2011), who image prismatic waves and other higher order internal scatterings. In their work, the scatterers are separated into regions according to illumination decomposition. This decomposition splits each term of the Born scattering series into further terms, some of which are ignored. The ignored terms are the ones for which the wavefront sets of velocity perturbations have no points in common. In contrast, we propose to directly work with the Born series; i.e., we do not make any assumptions about the geometry of velocity perturbations. Moreover, our method differs from that described by Malcolm et al. (2009) in the implementation algorithm. Malcolm et al. (2009) project the full Manuscript received by the Editor 29 July 2013; revised manuscript received 28 January 2014; published online 5 August 2014. King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. E-mail: mohammad.zuberi@kaust.edu. sa; tariq.alkhalifah@kaust.edu. sa. © 2014 Society of Exploration Geophysicists. All rights reserved. S207 GEOPHYSICS, VOL. 79, NO. 5 (SEPTEMBER-OCTOBER 2014); P. S207–S216, 13 FIGS. 10.1190/GEO2013-0287.1 D ow nl oa de d 04 /2 1/ 16 to 1 09 . 1 71 . 1 37 . 2 10 . R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry . s eg . o rg / data into different orders of internal multiples by optimizing the L norm of the difference between the full data and the desired order of internal multiples, whereas in our method we do not use any optimization. In this study, we propose a generalized internal multiple imaging (GIMI) process, which would image any order of internal scattering, separately. We also prove that the conventional reverse-time migration (RTM) images first-order multiple scattering (the double bounce), but relative to the single scattering it tends to be weaker in amplitude. In this GIMI process, we rely solely on the background Green’s function based on the conventional smooth velocity model. The additional computational cost is just a crosscorrelation of the surface seismic data with the back-propagated data. The number of crosscorrelations required is one less than the order of the term in the Born scattering series (or equal to the order of internal multiple) that we intend to image. Although the theoretical derivation given here for GIMI applies to terms of all orders of the Born scattering series, numerical examples of only the first two terms (single-scattering and first-order internal multiples) are shown in the examples section. We also discuss the limitations of GIMI, namely, crosstalk and weak scatterers. The crosstalk is the result of the additional crosscorrelations with the surface data and, thus, can be mitigated by a simple procedure. A small velocity perturbation assumption is, however, necessary for the convergence of the Born scattering series. GIMI could be implemented either by the wavefield extrapolation method or more efficiently by the integral method, which involves only crosscorrelations and summations of the surface data (such as Kirchhoff migration). In this study, we use the integral method for imaging. When using the integral method, we do not have to worry about the scatterer strength (especially when we are ignoring the amplitudes) because all the scatterings are contained in the data. A nonconvergent Born series would mean that we would have internal multiples in the recorded data that are higher in amplitude than the primaries, but this is never the case. Even if internal multiples in the data have higher amplitudes than primaries, GIMI would image them, but the crosstalk noise will be very high and the simple procedure described in this study would not work. Therefore, as long as the data have internal multiples lower in amplitude than primary reflections, the crosstalk suppression procedure described in this study would work. For the first-order internal multiple imaging (second-order term in the Born scattering series), GIMI can be described as a three-step procedure (Zuberi and Alkhalifah, 2014). The first step consists of the back propagation of the recorded surface data using the background Green’s function. In the second step, we crosscorrelate the back-propagated data with the surface data. The final step consists of crosscorrelating the result of second step with the background Green’s function from the surface to the image point. This resultant image contains enhanced first-order internal multiple contribution, whereas the single-scattering contribution has been suppressed. In other words, we have selected the first-order internal multiple for imaging. We can add this image to the result from conventional single-scattering imaging, e.g., Kirchhoff migration, to give the total image. The generalized method is obtained by repeating the second step of the three-stage procedure. In the next section, wewill present the theoretical background of the proposed method. THEORY In the following derivation, we will focus on the phase aspect of imaging ignoring the amplitude. Throughout the derivation, the location of an image point in the subsurface is described by the Cartesian coordinate vector X, whereas x with and without numerical subscripts, e.g., x0, are points on the surface (the recording surface). Vectors with primes as superscripts, e.g., x 0, will be in the volume enclosed by the surface of integration. All the derivations are in the frequency domain including the resultant images represented by I. Subscripts on I represent the order of the imaged term in the Born scattering series; e.g., I2 represents the image corresponding to second-order term in the Born series or first-order internal multiples. After imaging, a summation over the frequency is implied because we adhere to the zero-time imaging condition. Although direct arrivals can be useful in imaging under certain circumstances (Zuberi and Alkhalifah, 2013), here, we shall ignore them. Surface multiples can also add to the quality of the image (double illumination discussed by Schuster et al., 2004), and they can also be used to fill in the missing data (Shan and Guitton, 2004). Nonetheless, surface multiples still cannot illuminate areas that are not illuminated by single scattering. This can be seen by considering a vertical reflector. Awave coming from the surface and incident on a vertical reflector at any angle cannot go back to the surface unless it bounces off another interface. Therefore, we shall assume that free-surface multiples have been eliminated from the data and ignore them in our study. Conventional imaging (with smooth velocity) The scattered f