D. J. Verschuur

Adaptive surface‐related multiple elimination

The major amount of multiple energy in seismic data is related to the large reflectivity of the surface. A method is proposed for the elimination of all surface‐related multiples by means of a process that removes the influence of the surface reflectivity from the data. An important property of the proposed multiple elimination process is that no knowledge of the subsurface is required. On the other hand, the source signature and the surface reflectivity do need to be provided. As a consequence, the proposed process has been implemented adaptively, meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple‐free data equals the inversion residue. Results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion.

Estimation of multiple scattering by iterative inversion; Part 1, Theoretical considerations

A review has been given of the surface-related multiple problem by making use of the so-called feedback model. From the resulting equations it has been concluded that the proposed solution does not require any properties of the subsurface. However, source-detector and reflectivity properties of the surface need be specified. Those properties have been quantified in a surface operator and this operator is estimated as part of the multiple removal problem. The surface-related multiple removal algorithm has been formulated in terms of a Neumann series and in terms of an iterative equation. The Neumann formulation requires a nonlinear optimization process for the surface operator; while the iterative formulation needs a number of linear optimizations. The iterative formulation also has the advantage that it can be integrated easily with another multiple removal method. An algorithm for the removal of internal multiples has been proposed as well. This algorithm is an extension of the surface-related method. Removal of internal multiples requires knowledge of the macro velocity model between the surface and the upper boundary of the multiple generating layer. In Part II (also published in this issue) the success of the proposed algorithms has been demonstrated on numerical experiments and field data examples.

Estimation of multiple scattering by iterative inversion; Part II, Practical aspects and examples

A surface-related multiple-elimination method can be formulated as an iterative procedure: the output of one iteration step is used as input for the next iteration step (part I of this paper). In this paper (part II) it is shown that the procedure can be made very efficient if a good initial estimate of the multiple-free data set can be provided in the first iteration, and in many situations, the Radon-based multiple-elimination method may provide such an estimate. It is also shown that for each iteration, the inverse source wavelet can be accurately estimated by a linear (least-squares) inversion process. Optionally, source and detector variations and directivity effects can be included, although the examples are given without these options. The iterative multiple elimination process, together with the source wavelet estimation, are illustrated with numerical experiments as well as with field data examples. The results show that the surface-related multiple-elimination process is very effective in time gates where the moveout properties of primaries and multiples are very similar (generally deep data), as well as for situations with a complex multiple-generating system.

Estimating primaries by sparse inversion and application to near-offset data reconstruction

Accurate removal of surface-related multiples remains a challenge in many cases. To overcome typical inaccuracies in current multiple-removal techniques, we have developed a new primary-estimation method: estimation of primaries by sparse inversion (EPSI). EPSI is based on the same primary-multiple model as surface-related multiple elimination (SRME) and also requires no subsurface model. Unlike SRME, EPSI estimates the primaries as unknowns in a multidimensional inversion process rather than in a subtraction process. Furthermore, it does not depend on interpolated missing near-offset data because it can reconstruct missing data simultaneously. Sparseness plays a key role in the new primary-estimation procedure. The method was tested on 2D synthetic data.

Seismic migration of blended shot records with surface-related multiple scattering

This paper focuses on the concept of using blended data and multiple scattering directly in the migration process, meaning that the blended input data for the proposed migration algorithm includes blended surface-related multiples. It also means that both primary and multiple scattering contribute to the seismic image of the subsurface. Essential in our approach is that multiples are not included in the Greens functions but are part of the incident wavefields, utilizing the so-called double illumination property. We find that complex incident wavefields, such as blended primaries and/or blended multiples, require a reformulation of the imaging principle in order to provide broadband angle-dependent reflection properties.

Imaging of multiple reflections

Current multiple-removal algorithms in seismic processing use either differential moveout or predictability. If the differential moveout between primaries and multiples is small, prediction is the only option available. In the last decade, multidimensional prediction-error filtering by weighted convolution, such as surface-related multiple elimination (SRME), have proved to be very successful in practice. So far, multiples have been considered as noise and have been discarded after the removal process. In this paper, we argue that multiple reflections contain a wealth of information that can be used in seismic processing to improve the resolution of reservoir images beyond current capability. In the near future, one may expect that the so-called weighted-crosscorrelation (WCC) concept may offer an attractive alternative in approaching the multiple problem. WCC creates an option to avoid the adaptive subtraction process as applied in prediction-error algorithms. Moreover, it allows the transformation of multiples into primaries. The latter means that seismic imaging with primaries and multiples (nonlinear process) can be implemented by a sequence of linear processes, including the transformation of multiples into primaries and the imaging of primaries.

Removal of internal multiples with the common-focus-point (CFP) approach. Part 1: Explanation of the theory

Removal of surface and internal multiples can be formulated by removing the influence of downward-scattering boundaries and downward-scattering layers. The involved algorithms can be applied in a model-driven or a data-driven way. A unified description is proposed that relates both types of algorithms based on wave theory. The algorithm for the removal of surface multiples shows that muted shot records play the role of multichannel prediction filters. The algorithm for the removal of internal multiples shows that muted CFP gathers play the role of multichannel prediction filters. The internal multiple removal algorithm is illustrated with numerical examples. The conclusion is that the layer-related version of the algorithm has significant practical advantages.

Estimation of primaries and near-offset reconstruction by sparse inversion: Marine data applications

ost wave-equation-based multiple removal algorithms are based on prediction and subtraction of multiples. Especially for shallow water, the prediction strongly relies on a correct interpolation of the missing near offsets. The subtraction of predicted multiples from the data can easily lead to the distortion of primaries if primaries and multiples overlap. Recently, a new approach for surface-related multiple removal was proposed: the estimation of primaries by sparse inversion (EPSI), which is based on a full waveform inversion approach. EPSI is based on the same primary-multiple model as surface-related multiple elimination (SRME) and does not require a subsurface model. In contrast to SRME, EPSI estimates the primaries as unknowns in a multidimensional inversion process rather than a subtraction process.The multidimensional primary impulse responses are parameterized by band-limited spikes, which are estimated such that they, along with their corresponding multiples, match the input data. An interesting aspect of the EPSI method is that it produces a residual, which is the part of the input data not explained by primaries and multiples. This residual can be analyzed and may provide useful information on the primary estimation process. Furthermore, it has been demonstrated that EPSI is also capable of reconstructing the missing near offsets from the multiples. The proposed method is applied to a field data set with moderate water depth, where it is demonstrated that the results are comparable with SRME. This data set is used to illustrate the residual. For a shallow-water field data set, it is shown that EPSI gives a better result than the standard SRME result caused by EPSIs capability to reconstruct the missing near offsets.

Surface‐related multiple elimination on land seismic data—Strategies via case studies

Three processing strategies for the estimation and subsequent elimination of surface-related multiple energy on land seismic data are presented. They can be applied in a prestack mode (to shot and common-midpoint gathers) or in a poststack mode. The algorithm for the multiple attenuation is based on wave theoretical principles in which the data are used as a prediction operator. The estimated multiples are then adaptively subtracted from the input data to obtain primary-only data. A processing step prior to applying multiple elimination is an important component of these methodologies, particularly in the prestack analysis. Its aim is to regularize the data, improve the S/N ratio, and balance the seismic amplitudes. This results in smooth prediction operators. The effectiveness of these schemes in suppressing multiples is demonstrated with a number of case studies involving processing land seismic data.

Adaptive curvelet-domain primary-multiple separation

In many exploration areas, successful separation of primaries and multiples greatly determines the quality of seismic imaging. Despite major advances made by surface-related multiple elimination (SRME), amplitude errors in the predicted multiples remain a problem. When these errors vary for each type of multiple in different ways (as a function of offset, time, and dip), they pose a serious challenge for conventional least-squares matching and for the recently introduced separation by curvelet-domain thresholding. We propose a data-adaptive method that corrects amplitude errors, which vary smoothly as a function of location, scale (frequency band), and angle. With this method, the amplitudes can be corrected by an elementwise curvelet-domain scaling of the predicted multiples. We show that this scaling leads to successful estimation of primaries, despite amplitude, sign, timing, and phase errors in the predicted multiples. Our results on synthetic and real data show distinct improvements over conventional least-squares matching in terms of better suppression of multiple energy and high-frequency clutter and better recovery of estimated primaries.