مهاجرت لرزه‌ای کیرشهف با تفکیک‌پذیری بالا به روش کمترین مربعات منظم شده با نُرم-1

نویسندگان

1 دانشجوی دکتری، گروه فیزیک زمین، موسسه ژئوفیزیک دانشگاه تهران، ایران

2 استاد، گروه فیزیک زمین، موسسه ژئوفیزیک دانشگاه تهران، ایران

3 دانشیار، گروه فیزیک زمین، موسسه ژئوفیزیک دانشگاه تهران، ایران

چکیده

مهاجرت به روش کیرشهف یکی از ساده‌ترین و رایج‌ترین الگوریتم‌های مهاجرت داده‌های لرزه‌ای است. از آنجا که عملگر مهاجرت کیرشهف، الحاقی عملگر مدل‌سازی است، قادر به بازسازی درست دامنه بازتاب‌ها نبوده و تصویر نهایی مهاجرت یافته دارای وضوح کافی نخواهد بود. مهاجرت کمترین مربعات برای رفع این مشکل و بازسازی صحیح دامنه معرفی شد اما بخاطر ابعاد بزرگ ماتریس‌ها، حل مسأله به‌صورت تکراری انجام می‌شود که زمان‌بر است. اگرچه در مقایسه با حل الحاقی، حل کمترین مربعات موجب بهبود دامنه می‌شود، ولی تصویر حاصل کماکان وضوح کافی نخواهد داشت. در این مقاله با منظم سازی نرم-1 برای تزریق تنکی به جواب کمترین مربعات کیرشهف یک روش مهاجرت با تفکیک‌پذیری بالا ارائه می‌شود. در اینجا مهاجرت لرزه‌ای به‌شکل یک مسأله بهینه‌سازی با قید تنکی فرمول‌بندی و با الگوریتم شکافت عملگری برگمن حل می‌شود. از خصوصیات مطلوب این الگوریتم همگرایی بالا و حل مسائل مقید بدون نیاز به محاسبات وارون ماتریس و تنها با استفاده از عملگرهای مهاجرت و مدل‌سازی است. نتایج حاصل از داده‌های شبیه‌سازی شده عملکرد بسیار بهتر الگوریتم پیشنهادی به لحاظ تفکیک‌پذیری در قیاس با الگوریتم مرسوم مهاجرت کیرشهف را نشان می‌دهند. مهاجرت کمترین مربعات قادر به کاهش اثرات ناشی از ناقص بودن داده در تصویر مهاجرت یافته می‌باشد. لذا روش پیشنهادی نیز با افزایش کیفیت تصویر حاصل از مهاجرت کمترین مربعات، تصویری مهاجرت یافته از یک داده ناقص با تفکیک‌پذیری بالاتری تولید خواهد کرد. نتایج حاصل از اعمال روش بر روی داده مصنوعی و واقعی عملکرد مطلوب آن را نشان می‌دهد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

High resolution Kirchhoff seismic migration via 1-norm regularized least-squares

نویسندگان [English]

  • Toktam Zand 1
  • Hamid Reza Siahkoohi 2
  • Ali Gholami 3
1 Ph.D. Student, Department of Earth Physics, Institute of Geophysics, University of Tehran, Iran
2 Professor, Department of Earth Physics, Institute of Geophysics, University of Tehran, Iran
3 Associate Professor, Department of Earth Physics, Institute of Geophysics, University of Tehran, Iran
چکیده [English]

For decades Kirchhoff migration has been one of the simplest migration algorithms and also the most frequently used method of migration in industry. This is due to its relatively low computational cost and its flexibility in handling acquisition and topography irregularities. The standard seismic migration operator can be regarded as the adjoint of a seismic forward modeling operator, which acts on a set of subsurface parameters to generate the observed data. Such adjoint operators are able to provide an approximate inverse of the forward modeling operator and only recover the time of the events (Claerbout, 1992). They cannot retrieve the amplitude of reflections, thus leading to a decrease in the resolution of the final migrated image. The standard seismic migration (adjoint) operators can be modified to better approximate the inverse operators. Least-squares migration (LSM) techniques have been developed to fully inverse the forward modeling procedures by minimizing the difference between observed and modeled data in a least-squares sense. An LSM is able to reduce the (Kirchhoff) migration artifacts, enhance the resolution and retrieve seismic amplitudes. Although implementing LSM instead of conventional migration, leads to resolution enhancement. It also brings some new numerical and computational challenges which need to be addressed properly. Due to the ill-conditioned nature of the inverse operator and also incompleteness of the data, the method generates unavoidable artifacts which severely degrade the resolution of the migrated image obtained by the non-regularized LSM method. The instability of LSM methods suggests developing a regularized algorithm capable of including reasonable physical constraints. Including the seismic wavelet into the migration operator, migration will generate the earth reflectivity image which can be considered as a sparse image, so applying the sparseness constraint, e.g., via the minimization of the 1-norm of reflectivity model, can help to regularize the model and prevent it from getting noisy artifacts (Gholami and Sacchi, 2013).
In this article, based on the Bregmanized operator splitting (BOS), we propose a high resolution migration algorithm by applying sparseness constraints to the solution of least-squares Kirchhoff migration (LSKM). The Bregmanized operator splitting is employed as a solver of the generated sparsity-promoting LSKM for its simplicity, efficiency, stability and fast convergence. Independence of matrix inversion and fast convergence rate are two main properties of the proposed algorithm. Numerical results from field and synthetic seismic data show that migrated sections generated by this 1-norm regularized Kirchhoff migration method are more focused than those generated by the conventional Kirchhoff/LS migration.
Regular spatial sampling of the data at Nyquist rate is another major challenge which may not be achieved in practice due to the coarse source-receiver distributions and presence of possible gaps in the recording lines. The proposed model-based migration algorithm is able to handle the incompleteness issues and is stable in the presence of noise in the data. In this article, we tested the performance of our proposed method on synthetic data in the presence of coarse sampling and also acquisition gaps. The results confirmed that the proposed sparsity-promoting migration is able to generate accurate migrated images from incomplete and inaccurate data.

کلیدواژه‌ها [English]

  • Kirchhoff migration
  • Inverse operator
  • Least-squares
  • Bregmanized operator splitting
  • Sparsity-constrained
  • Incomplete data
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