A new role for adaptive filters in Marchenko equation-based methods for the attenuation of internal multiples
AA new role for adaptive filters in Marchenkoequation-based methods for the attenuationof internal multiples
Myrna Staring, Lele Zhang, Jan Thorbecke and Kees Wapenaar
Summary
We have seen many developments in Marchenko equation-based methods for internalmultiple attenuation in the past years. Starting from a wave-equation based method thatrequired a smooth velocity model, there are now Marchenko equation-based methods thatdo not require any model information or user-input. In principle, these methods accuratelypredict internal multiples. Therefore, the role of the adaptive filter has changed for thesemethods. Rather than needing an aggressive adaptive filter to compensate for inaccurateinternal multiple predictions, only a conservative adaptive filter is needed to compensate forminor amplitude and/or phase errors in the internal multiple predictions caused by imper-fect acquisition and preprocessing of the input data. We demonstate that a conservativeadaptive filter can be used to improve the attenuation of internal multiples when applying aMarchenko multiple elimination (MME) method to a 2D line of streamer data. In addition,we suggest that an adaptive filter can be used as a feedback mechanism to improve thepreprocessing of the input data. 1 a r X i v : . [ phy s i c s . g e o - ph ] M a r ntroduction. The use of an adaptive filter for the attenuation of internal multiples isa strongly debated topic. On the one hand, it can be argued that an adaptive filter candamage primary reflections. On the other hand, it is well-known that the field data usedfor internal multiple prediction have imperfections due to acquisition and preprocessing(e.g., inaccurate deconvolution of the source wavelet), thereby making it very challenging topredict internal multiples with the correct amplitude and phase from field data. Therefore,not using an adaptive filter might result in an incomplete attenuation of the internal multiplesand consequently an incorrect interpretation of the target area.In recent years, there have been many developments related to Marchenko equation-basedmethods for internal multiple attenuation [4]. Some more conventionally used internal multi-ple attenuation methods, for example a method proposed by [1], strongly rely on an adaptivefilter to attenuate the internal multiples in field data (the method in principle predicts in-ternal multiples with incorrect amplitudes, uses a layer stripping approach that causes errorpropagation from shallow to deep, and usually does not remove the source wavelet). In con-trast, Marchenko equation-based methods in principle predict internal multiples with thecorrect amplitude and phase, and thus typically only require a conservative adaptive filterto correct for imperfections due to acquisition and preprocessing. As a result, it is no longera debate on whether to use an aggressive adaptive filter or no adaptive filter, but on howan adaptive filter can provide a helping hand in ironing out the last details.In this paper, we will look at an example of the application of an adaptive Marchenkoequation-based method (the Marchenko multiple elimination method) on field data. We willshow how only a conservative adaptive filter is needed to improve attenuation of the internalmultiples in the data, and how it can also be used as a feedback mechanism.
Theory.
The basis of all Marchenko equation-based methods are the coupled Marchenkoequations [3]. By solving these equations iteratively, we retrieve directionally decomposedfocusing functions and Green’s functions. These wavefields can then be used for differentpurposes, for example redatuming, internal multiple prediction or homogeneous Green’sfunction retrieval [4]. In this paper, we use the projected downgoing focusing function v + as introduced by [2]: v + = ∞ (cid:88) k =0 ( θ t t R (cid:63) θ t t R ) k δ. (1)By convolving this function once more with reflection response R , we obtain the2archenko multiple elimination method (MME) [2, 5]: R t = Rv + = Rδ + α ∞ (cid:88) k =1 R ( θ t t R (cid:63) θ t t R ) k δ. (2)Internal multiple predictions are obtained by evaluating this series for every timestep t = t − (cid:15) (where (cid:15) represents the band-limitation in the data) and only saving the singlesample at timestep t . When storing all individual timesteps together, we can add themto the input reflection response to obtain reflection response R t without internal multiples.A conservative adaptive filter α can be used to adjust the internal multiple predictionswhen necessary. Although this method is computationally more expensive compared toother Marchenko equation-based methods, it does not require any model information oruser-input and is thus completely data-driven. Example. [6] have shown that this method can attenuate internal multiples in streamerdata acquired by Equinor in the Norwegian Sea, but perhaps we can do better by using aconservative adaptive filter α . The preprocessing of the dataset included 3D to 2D conver-sion, near offset reconstruction, interpolation to 25 m source and receiver spacing, waveletdeconvolution and the attenuation of surface-related multiples.Figure 1a shows an image obtained by one-way wave equation migration of the 2D pre-processed reflection response, while Figure 1b shows the resulting reflection response afteradding 6 terms of the series in Equation 2 without adaptive filter (the result presented by[6]). Instead of simply adding the internal multiple predictions, the result in Figure 1c wasobtained using a conservative adaptive filter α (filter length 3, windows of 200 dt by 50dx). The arrows at numbers 1 and 4 show a more complete attenuation of what we be-lieve to be internal multiples. The ellipses and the arrow at numbers 2, 3 and 5 show howprimary reflections become better visible. In addition, we observed that the adaptive filtermainly changed the amplitudes of the internal multiple predictions, thereby indicating thatthe input data was not optimally scaled. Based on this observation, we can go back to ourpreprocessing workflow and optimize the scaling. Conclusions.
As the prediction of internal multiples is becoming more accurate, thereis no longer a need for aggressive adaptive filters. Instead, conservative adaptive filterscan be used to attenuate internal multiples in field data more completely. In addition, aconservative adaptive filter can be used as a feedback tool to see whether amplitude and3
IG. 1: Images of the reflection response a) before internal multiple attenuation, b) after internalmultiple attenuation without adaptive filter and c) after internal multiple attenuation using aconservative adaptive filter. phase of the data were correctly preserved during preprocessing. Still, adaptive filters needto be applied with much care, and a suitable domain for subtraction needs to be chosen forevery dataset.
Acknowledgements.
We would like to thank Eric Verschuur and Equinor for providingthe field data used in this paper. This research was performed in the framework of the project’Marchenko imaging and monitoring of geophysical reflection data’, which is part of theDutch Open Technology Programme with project number 13939 and is financially supportedby NWO Domain Applied and Engineering Sciences. The research of K. Wapenaar hasreceived funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreement no. 742703). [1] Jakubowicz, H. [1998] In:
SEG Expanded Abstracts , 1527–1530.[2] van der Neut, J. and Wapenaar, K. [2016]
Geophysics , (5), T265–T284.[3] Wapenaar, K., Broggini, F., Slob, E., and Snieder, R. [2013] Physical Review Letters , ,084301.[4] Wapenaar, K., Staring, M., Brackenhoff, J., Zhang, L., Thorbecke, J. and Slob, E. [2020] In: AGE Conference and Exhibition .[5] Zhang, L. and Staring, M. [2018]
Journal of Applied Geophysics , , 429–433.[6] Zhang, L. and Slob, E. [2020] Geophysics , (2), S65–S70.(2), S65–S70.