Least-squares reverse time migration of multiples

In marine seismic acquisition, surface-related multiples constitute a significant portion of the acquired data. Typically, multiples are removed during early-stage data processing because they can lead to phantom reflectors during migration that might result in erroneous geologic interpretations. However, if properly dealt with, multiples can provide valuable extra information and can complement primaries in illuminating the subsurface. Reverse time migration has limitations in imaging multiples. A computationally efficient inversion procedure is proposed which jointly maps primaries and multiples to the true reflectors and estimates the source function on the fly. As a result, high-quality, mostly artifact-free broad-band images are obtained in which the imprint of the source function is partly removed at a computationally affordable expense compared with the combined costs of wave-equation-based surface-related multiple elimination and reverse time migration. All this is achieved by including the total downgoing wavefields as areal sources in least-squares migration in combination with curvelet-domain sparsity promotion. The proposed method is applied to a shallow-water marine data set from the North Sea which contains abundant short-period surface-related multiples, and the efficacy of the method is shown in eliminating coherent imaging artifacts associated with multiples. The benefits of joint imaging of primaries and multiples are demonstrated compared with imaging these signal components separately.

Least-squares migration (LSM) of seismic data is supposed to produce images of subsurface structures with better quality than standard migration if we have an accurate migration velocity model. However, LSM suffers from data mismatch problems and migration artifacts when noise pollutes the recorded profiles. This study has developed a reweighted least-squares reverse time migration (RWLSRTM) method to overcome the problems caused by such noise. We first verify that spiky noise and free-surface multiples lead to the mismatch problems and should be eliminated from the data residual. The primary- and multiple-guided weighting matrices are then derived for RWLSRTM to reduce the noise in the data residual. The weighting matrices impose constraints on the data residual such that spiky noise and free-surface multiple reflections are reduced whereas primary reflections are preserved. The weights for spiky noise and multiple reflections are controlled by a dynamic threshold parameter decreasing with iterations for better results. Finally, we use an iteratively reweighted least-squares algorithm to minimize the weighted data residual. We conduct numerical tests using the synthetic data and compared the results of this method with the results of standard LSRTM. The results suggest that RWLSRTM is more robust than standard LSRTM when the seismic data contain spiky noise and multiple reflections. Moreover, our method not only suppresses the migration artifacts, but it also accelerates the convergence.

Reverse time migration: A prospect of seismic imaging methodology

We present a target-oriented version of least squares migration (LSM). A small target region of the model is defined, and the associated target data are identified by Kirchhoff modeling. The target data are much smaller in size than the original data. These data are migrated in the least squares sense to obtain the reflectivity, but the computational cost is significantly reduced due to a smaller data set and target size. Results with UNOCAL’s poststack marine data show the LSM images to be noticeably more focused compared to Kirchhoff migration images. In addition, use of the migration deconvolution image as a starting model improves the ability of LSM to focus seismic energy. An important implication is that target-oriented LSM might be as affordable as standard Kirchhoff migration.

The simultaneous-source shooting technique can accelerate field acquisition and improve spatial sampling but it will cause strong interferences in the recorded data and artifacts in the final image. The previously proposed structural smoothing operator can effectively attenuate artifacts for relatively simple reflection structures during least-squares inversion, but it will cause damage to complicated reflection events such as discontinuities. To preserve discontinuities in a seismic image, we apply the singular spectrum analysis (SSA) operator to attenuate artifacts during least-squares inversion. Considering that global SSA cannot deal with overcomplicated data very well, we use local SSA to remove noise and to better preserve the steeply dipping components. The local SSA operator corresponds to a local low-rank constraint applied in the inversion process. The migration operator used in the study is the reverse time migration (RTM) operator. Tests using the Marmousi model showed the superior performance of the proposed algorithm in preserving the discontinuities of seismic images.

Reverse time migration (RTM) is achieved by a forward and reverse time propagation of source and receiver wavefields respectively, followed by an imaging condition. The quality of the image from any RTM implementation depends directly on the method’s ability to separate the true reflection data from the backscattered correlation noise between the source and receiver wavefields. In this paper we demonstrate the application of an inverse scattering imaging condition that significantly reduces this backscattered correlation noise and results in a much higher quality subsurface image than is achieved by typical RTM imaging conditions.

Summary Recent work on autofocusing with the Marchenko equation has shown how the Green's function for a virtual source in the subsurface can be obtained from reflection data. The response to the virtual source is the Green's function from the location of the virtual source to the surface. The Green's function is retrieved using only the reflection response of the medium and an estimate of the first arrival at the surface from the virtual source. Current techniques, however, only include primaries and internal multiples. Therefore, all surface-related multiples must be removed from the reflection response prior to Green's function retrieval. Here, we extend the Marchenko equation to retrieve the Green's function that includes primaries, internal multiples, and free-surface multiples. In other words, we retrieve the Green's function in the presence of a free surface. We use the associated Green's function for imaging the subsurface. The information needed for the retrieval are the reflection response at the surface and an estimate of the first arrival at the surface from the virtual source. The reflection response, in this case, includes the free-surface multiples; this makes it possible to include these multiples in the imaging operator and it obviates the need for surface-related multiple elimination.

Recent work on autofocusing with the Marchenko equation has shown how the Green's function for a virtual source in the subsurface can be obtained from reflection data. The response to the virtual source is the Green's function from the location of the virtual source to the surface. The Green's function is retrieved using only the reflection response of the medium and an estimate of the first arrival at the surface from the virtual source. Current techniques, however, only include primaries and internal multiples. Therefore, all surface-related multiples must be removed from the reflection response prior to the Green's function retrieval. Here, we present a new scheme that includes primaries, internal multiples, and free-surface multiples. In other words, we retrieve the Green's function in the presence of the free surface. The information needed for the retrieval are the reflection response at the acquisition surface and an estimate of the first arrival at the surface from the virtual source. The reflection response, in this case, includes the free-surface multiples; this makes it possible to include these multiples in the imaging operator and it obviates the need for surface-related multiple elimination.

Reverse time migration (RTM) is a technique used to obtain high-resolution images of underground reflectors; however, this method is computationally intensive when dealing with large amounts of seismic data. Multi-source RTM can significantly reduce the computational cost by processing multiple shots simultaneously. However, multi-source-based methods frequently result in crosstalk artifacts in the migrated images, causing serious interference in the imaging signals. Plane-wave migration, as a mainstream multi-source method, can yield migrated images with plane waves in different angles by implementing phase encoding of the source and receiver wavefields; however, this method frequently requires a trade-off between computational efficiency and imaging quality. We propose a method based on deep learning for removing crosstalk artifacts and enhancing the image quality of plane-wave migration images. We designed a convolutional neural network that accepts an input of seven plane-wave images at different angles and outputs a clear and enhanced image. We built over 500 1024×256 velocity models, and employed each of them using plane-wave migration to produce raw images at 0°, ±10°, ±20°, and ±30° as input of the network. Labels are high-resolution images computed from the corresponding reflectivity models by convolving with a Ricker wavelet. Random sub-images with a size of 512×128 were used for training the network. Numerical examples demonstrated the effectiveness of the trained network in crosstalk removal and imaging enhancement. The proposed method is superior to both the conventional RTM and plane-wave RTM (PWRTM) in imaging resolution. Moreover, the proposed method requires only seven migrations, significantly improving the computational efficiency. In the numerical examples, the processing time required by our method was approximately 1.6% and 10% of that required by RTM and PWRTM, respectively.

Recent work on retrieving the Green’s function with the Marchenko equation shows how these functions for a virtual source in the subsurface can be obtained from reflection data. The response to the virtual source is the Green’s function from the location of the virtual source to the surface. The Green’s function is retrieved using only the reflection response of the medium and an estimate of the first arrival at the surface from the virtual source. Current techniques, however, only include primaries and internal multiples. Therefore, all surface-related multiples must be removed from the reflection response prior to Green’s function retrieval. We have extended the Marchenko equation to retrieve the Green’s function that includes primaries, internal multiples, and free-surface multiples. In other words, we have retrieved the Green’s function in the presence of a free surface. The information needed for the retrieval is the same as the current techniques, with the only difference being that the reflection response now also includes free-surface multiples. The inclusion of these multiples makes it possible to include them in the imaging operator, and it obviates the need for surface-related multiple elimination. This type of imaging with Green’s functions is called Marchenko imaging.

Summary Reverse-time migration (RTM) with the conventional cross-correlation imaging condition suffers from low-frequency artifacts that result from backscattered energy in the background velocity models. This problem translates to least-squares reverse-time migration (LS-RTM), where these artifacts slow down the convergence, as many of the initial iterations are spent on removing them. In RTM, this problem has been successfully addressed by the introduction of the so-called inverse scattering imaging condition, which naturally removes these artifacts. In this work, we derive the corresponding linearized forward operator of the inverse scattering imaging operator and incorporate this forward/adjoint operator pair into a sparsity-promoting (SPLS-RTM) workflow. We demonstrate on a challenging salt model, that LS-RTM with the inverse scattering imaging condition is far less prone to low-frequency artifacts than the conventional cross-correlation imaging condition, improves the convergence and does not require any type of additional image filters within the inversion. Through source subsampling and sparsity promotion, we reduce the computational cost in terms of PDE solves to a number comparable to conventional RTM, making our workflow applicable to large-scale problems.

To obtain a physical understanding of gradient-based descent methods in full-waveform inversion (FWI), we find a connection between the FWI gradient and the image provided by reverse time migration (RTM). The gradient uses the residual data as a virtual source, and RTM uses the observed data directly as the boundary condition, so the FWI gradient is similar to a time integration of the RTM image using the residual data, which physically converts the phase of the reflectivity to the phase of the velocity. Therefore, gradient-based FWI can be connected to the classical reflectivity-to-velocity/impedance inversion (RVI). We have developed a new FWI scheme that provides a self-contained and physically intuitive derivation, which naturally establishes a connection among the amplitude-preserved RTM, the Zoeppritz equations (amplitude variation with angle inversion), and RVI, and combines them into a single framework to produce a preconditioned inversion formula. In this scheme, the relative velocity update is a phase-modified and deconvolved RTM image obtained with the residual data. Consistent with the deconvolution, the multiscale approach applies a gradually widening low-pass frequency filter to the deconvolved wavelet at early iterations, and then it uses the unfiltered deconvolved wavelet for the final iterations. Our numerical testing determined that the new method makes a significant improvement to the quality of the inversion result.

Summary Seismic images provided by Reverse Time Migration can be contaminated by artefacts associated with the migration of multiples. Multiples can corrupt seismic images producing both false positives, i.e. by focusing energy at unphysical interfaces, and false negatives, i.e. by destructively interfering with primaries. Multiple prediction / primary synthesis methods are usually designed to operate on point source gathers, and can therefore be computationally demanding when large problems are considered. Here, a new scheme is presented for fully data-driven retrieval of primary responses to plane-wave sources. The proposed scheme, based on convolutions and cross-correlations of the reflection response with itself, extends a recently devised point-sources primary retrieval method for to plane-wave source data. As a result, the presented algorithm allows fully data-driven synthesis of primary reflections associated with plane-wave source data. Once primary plane-wave responses are estimated, they are used for multiple-free imaging via standard reverse time migration.

Considerable effort has been devoted to the migration of multicomponent data in elastic media with wavefield separation techniques being the most successful. Most of this work has been carried out in the time domain. In this paper, we formulate a multicomponent migration technique in the frequency domain. Reverse-time migration can be viewed as the zero-lag cross-correlation between virtual source and back-propagated wavefields. Cross-correlating the Helmholtz decomposed wavefields rather than directly correlating the vector displacement fields results in sharper, more interpretable images, contaminated by fewer crosstalk artefacts. The end products are separate P and S wave (and if desired, PS and SP) migration images. We test our migration algorithm on synthetic seismic data generated using the SEG/EAGE salt-dome, Overthrust and Marmousi-2 models. We correctly image the location and shape of the target zone for oil exploration using these data sets. Furthermore, we demonstrate that our new migration technique provides good images even when the initial velocity model is only approximate.

Acceleration of stable TTI P-wave reverse-time migration with GPUs