주파수-파수 영역에서의 Curvelet 변환 기반 탄성파 트레이스 내삽

Abstract

In recent years, as the importance of seismic trace interpolation techniques has been increased, many studies have been performed to reconstruct the missing traces from the irregularly under-sampled seismic data. Among these trace interpolation techniques, a projection onto convex set (POCS) algorithm has gained popularity for its robustness and easy implementation. However, the conventional POCS algorithm using Fourier transform has the limitation in interpolating the seismic data containing curved events. The curvelet transform-based POCS algorithm in t-x domain, which was proposed by Yang et al. (2012) in order to overcome this limitation, can reconstruct the curved events effectively, but it has trouble in interpolating the weak events when they are interpolated with strong events together. To solve this problem, I introduced a new curvelet transform-based POCS algorithm which applies curvelet transform to the 2D Fourier transformed data in the f-k domain instead of the data in the t-x domain for each iteration of POCS algorithm. For verifying the efficiency of the introduced method, the proposed method was applied to the data generated by Marmousi2 model and Hess VTI synthetic data. The results clearly demonstrate that the presented algorithm is superior to the curvelet transform-based POCS in t-x domain especially for reconstructing the events with relatively weak amplitudes in the data containing the events with various amplitudes. In order to analyze strong and weak points of the proposed method in detail, comparative analyses of curvelet transform-based POCS in t-x domain and in f-k domain were performed. According to these comparative analyses, when the events have similar amplitudes, both methods interpolate the missing parts of the events well regardless of the moveout of the event. However, when the data have events with different amplitudes and similar moveouts, relatively weak events are better reconstructed by the curvelet transform-based POCS in f-k domain than in t-x domain. However, if the events have significantly different moveouts as well as different amplitudes, even the proposed method has trouble in filling the missing part of the relatively weak events. Further experiments show that to obtain the reasonable result for this case, the amplitude of relatively weak event should be at least 30% of that of strong event.|최근 탄성파 탐사 자료 내삽의 중요성이 증가하면서 불규칙하게 샘플링된 자료의 복구에 대한 많은 연구가 활발히 진행되어 왔다. 탄성파 탐사 내삽 방법 중 POCS (Projection onto convex set) 내삽 기법은 사용이 간편하고 적용이 용이하다는 장점이 있어 많은 관심을 받아 왔다. 그러나, 기존의 Fourier 변환 기반 POCS 내삽 기법의 경우 곡선형을 가지는 다차원 탄성파 트레이스의 복원에 한계를 지닌다. 이를 극복하기 위하여 Yang et al. (2012)은 curvelet 변환을 적용한 POCS 내삽 기법을 제안하였고, 곡선형 경향을 가지는 이벤트의 복구에 탁월함을 확인하였다. 하지만, curvelet 변환 기반 POCS 기법은 진폭차이가 큰 신호들을 동시에 복구할 경우 상대적으로 진폭이 약한 신호의 복구에 한계점을 지닌다. 따라서 이 연구에서는 이러한 문제를 개선하기 위해 시간-공간 영역의 자료를 curvelet 변환하여 적용하는 기존의 POCS 방법과는 다르게 주파수-파수 영역으로 변환된 자료를 curvelet 변환하여 적용하는 새로운 curvelet 변환 기반 POCS 기법을 제안하였다. 제안된 방법의 효율성을 확인하기 위하여 강한 진폭과 약한 진폭을 가지는 이벤트들이 포함된Marmousi2 자료와 Hess VTI이방성 자료를 이용하여 수치 모형 실험을 실시하였다. 그 결과, 이 연구에서 제안된 방법이 기존의 curvelet 변환 기반 POCS 기법에 비하여 상대적으로 약한 진폭을 가진 이벤트들의 복원에도 탁월한 복구결과를 가져오는 것을 확인하였다. 또한, 제안된 방법의 장∙단점을 자세히 분석하기 위하여 두 가지 이벤트를 가지는 자료를 통하여 기존의 curvelet 변환 기반 POCS 기법과의 비교 분석을 실시하였다. 그 결과, 두 가지 이벤트의 진폭이 유사한 경우 도착 시간 차의 차이에 상관 없이 두 방법 모두 탁월한 복구 결과를 보이나, 강한 진폭과 약한 진폭을 동시에 가지는 탄성파 탐사 자료에서 각각 이벤트의 도착 시간 차가 유사한 경우는 제안된 방법이 복구에 탁월한 효과를 보이는 것을 확인하였다. 그러나, 도착 시간 차가 비교적 큰 경우에는 제안된 방법에서도 복구의 한계를 보였다. 이 경우에 합리적인 복구 결과를 얻기 위한 최적의 이벤트 간 진폭 비를 조사하였는데, 약한 이벤트의 진폭이 강한 이벤트 진폭의 30% 이상의 크기를 가지면 합리적인 복구결과를 가져오는 것을 확인하였다.; In recent years, as the importance of seismic trace interpolation techniques has been increased, many studies have been performed to reconstruct the missing traces from the irregularly under-sampled seismic data. Among these trace interpolation techniques, a projection onto convex set (POCS) algorithm has gained popularity for its robustness and easy implementation. However, the conventional POCS algorithm using Fourier transform has the limitation in interpolating the seismic data containing curved events. The curvelet transform-based POCS algorithm in t-x domain, which was proposed by Yang et al. (2012) in order to overcome this limitation, can reconstruct the curved events effectively, but it has trouble in interpolating the weak events when they are interpolated with strong events together. To solve this problem, I introduced a new curvelet transform-based POCS algorithm which applies curvelet transform to the 2D Fourier transformed data in the f-k domain instead of the data in the t-x domain for each iteration of POCS algorithm. For verifying the efficiency of the introduced method, the proposed method was applied to the data generated by Marmousi2 model and Hess VTI synthetic data. The results clearly demonstrate that the presented algorithm is superior to the curvelet transform-based POCS in t-x domain especially for reconstructing the events with relatively weak amplitudes in the data containing the events with various amplitudes. In order to analyze strong and weak points of the proposed method in detail, comparative analyses of curvelet transform-based POCS in t-x domain and in f-k domain were performed. According to these comparative analyses, when the events have similar amplitudes, both methods interpolate the missing parts of the events well regardless of the moveout of the event. However, when the data have events with different amplitudes and similar moveouts, relatively weak events are better reconstructed by the curvelet transform-based POCS in f-k domain than in t-x domain. However, if the events have significantly different moveouts as well as different amplitudes, even the proposed method has trouble in filling the missing part of the relatively weak events. Further experiments show that to obtain the reasonable result for this case, the amplitude of relatively weak event should be at least 30% of that of strong event.