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First-arrival traveltime tomography (FATT) is used to delineate shallow velocity structures to identify static effects in oil exploration as well as to characterize the near surface for geotechnical purposes. Because FATT is generally used for land seismic data processing, it becomes necessary to consider irregular topography especially when performing wave-based tomography. However, the standard Cartesian finite-difference method cannot properly handle irregular topography. Hence, the embedded boundary method (EBM) is incorporated into the frequency-domain damped-wave equation to correctly describe irregular topography. The developed modeling algorithm is used to calculate first-arrival traveltimes and to perform FATT. The EBM-based modeling algorithm accurately describes the irregular surfaces of numerical velocity models on a regular mesh by exploiting the mirror image principle. The accuracy of the EBM-based traveltime calculation is validated by using two homogeneous velocity models with dipping and complex surfaces. The validation results demonstrate that our algorithm is unaffected by the staircase approximation. FATT is then applied to synthetic and real data to demonstrate the applicability of the developed algorithm to velocity models with complex topography. For the real data example, the inverted velocity model is used to apply static corrections. The processing results demonstrate an improvement in the continuity of seismic events, thus confirming the accuracy of the developed FATT method.

Land seismic data generally have time-static issues due to irregular topography and weathered layers at shallow depths. Unless the time static is handled appropriately, interpretation of the subsurface structures can be easily distorted. Therefore, static corrections are commonly applied to land seismic data. The near-surface velocity, which is required for static corrections, can be inferred from first-arrival traveltime tomography , which must consider the irregular topography, as the land seismic data are generally obtained in irregular topography. This paper proposes a refraction traveltime tomography technique that is applicable to an irregular topographic model. This technique uses unstructured meshes to express an irregular topography, and traveltimes calculated from the frequency-domain damped wavefields using the finite element method. The diagonal elements of the approximate Hessian matrix were adopted for preconditioning, and the principle of reciprocity was introduced to efficiently calculate the Frechet derivative. We also included regularization to resolve the ill-posed inverse problem , and used the nonlinear conjugate gradient method to solve the inverse problem. As the damped wavefields were used, there were no issues associated with artificial reflections caused by unstructured meshes. In addition, the shadow zone problem could be circumvented because this method is based on the exact wave equation, which does not require a high-frequency assumption. Furthermore, the proposed method was both robust to an initial velocity model and efficient compared to full wavefield inversions. Through synthetic and field data examples, our method was shown to successfully reconstruct shallow velocity structures. To verify our method, static corrections were roughly applied to the field data using the estimated near-surface velocity. By comparing common shot gathers and stack sections with and without static corrections, we confirmed that the proposed tomography algorithm can be used to correct the statics of land seismic data.

As full waveform inversion (FWI) requires large computation time, a variety of techniques have been suggested to reduce the computational burden. In this study, we use wavenumber-space-time domain modelling, which directly generates common mid-point (CMP) gathers, to implement the FWI algorithm. The modelling technique, which is suitable for laterally invariant velocity models, synthesises CMP gathers efficiently by using limited wavenumber components, and therefore allows reduced computation time for FWI. To consider the accuracy as well as the efficiency of FWI, the Gauss-Newton method using the approximate Hessian matrix is chosen in this study. Rather than using all of the wavenumber components, our FWI algorithm can be accelerated by using only a few components. The wavenumber components can be chosen through an analysis of the residual wavefields. To validate the usefulness of our method, we first use a one-dimensional (1D) velocity model. From the 1D model example, we note that our FWI algorithm ca...

요약: 탄성파 자료의 영상화 과정에서 입력자료인 속도 모델에 불연속면이 있는 경우 반사파에 의해 참반사보정(migration) 결과가 왜곡될 수 있다. 따라서 참반사보정을 위한 속도 모델은 지층 경계면에서 샘 파동장과 수신기 파동장을 구할 때 발생하는 반사파를 제거하기 위해 평활화(smoothing)하여 사용하는 것이 일반적이다. 그러나 속도 모델을 평활화할 경우 지층 경계면에서 속도 정보가 달라져 지하구조 영상이 왜곡될 가능성이 있다. 본 연구에서는 이러한 단점을최소화하기 위해 속도가 불연속인 층간의 음향 임피던스를 일정하게 만들어 샘 파동장과 수신기 파동장을 구할 때 발생하는 반사파를 줄이고자 하였다. 음향 임피던스를 일정하게 만들기 위해 속도 차이를 보상하는 가상의 밀도(fake density)를 정의하고 참반사보정에 사용하였다. 음향 임피던스가 모든 층에서 일정할 때, 반사면에서 수직 입사파의 반사계수가영이 되고 반사파가 최소화되어 참반사보정 결과를 향상시킬 수 있다. 이를 검증하기 위해 셀기반 유한차분법을 이용하여 거꿀시간 참반사보정(reverse-time migration) 알고리듬을 구현하였다. 수치예제를 통해 속도 대비가 큰 지층 경계면에서 참반사보정 영상의 품질이 향상되는 것을 확인할 수 있고, 특히 천부 지층에서 성능 개선효과가 큰 것을 관찰할 수 있다.주요어: 참반사보정, 반사파, 음향 임피던스, 가상 밀도, 셀기반 유한차분법Abstract: Migration image can be distorted due to reflected waves in the source and receiver wavefields whendiscontinuities of input velocity model exist in seismic imaging. To remove reflected waves coming from layer interfaces,it is a common practice to smooth the velocity model for migration. If the velocity model is smoothed, however, thesubsurface image can be distorted because the velocity changes around interfaces. In this paper, we attempt to minimizethe distortion by reducing reflection energy in the source and receiver wavefields through acoustic impedancehomogenization. To make acoustic impedance constant, we define fake density model and use it for migration. Whenthe acoustic impedance is constant over all layers, the reflection coefficient at normal incidence becomes zero and theminimized reflection energy results in the improvement of migration result. To verify our algorithm, we implement thereverse-time migration using cell-based finite-difference method. Through numerical examples, we can note that themigration image is improved at the layer interfaces with high velocity contrast, and it shows the marked improvementparticularly in the shallow part. Keywords: Migration, reflection energy, acoustic impedance, fake density, cell-based finite-difference method

The regularisation method simply involves adding a regularisation term to a misfit function, thereby generally stabilising ill-posed inverse problems. In this paper, a pure-gradient based full waveform inversion is carried out in the frequency domain, and smoothing regularisation is exploited in the form of preconditioning, contrary to conventional regularisations in general inverse theory. For more accurate descriptions of subsurface structures, we select adaptive regularisation rather than isotropic smoothing regularisation. The adaptive smoothing regularisation method is used to both smooth the image and differentiate between layers and boundaries. Dip information is estimated using the Hilbert transform, an adaptive smoothing filter with an edge-preserving property is designed from the estimated dip information, and full waveform inversion is then performed using the adaptive smoothing filter. The adaptive smoothing filter is periodically updated during inversion iterations to adapt to the inverted ve...

The wavenumber–space–time domain wave equation is derived from the 2D time-domain wave equation in the horizontally stratified media by taking the Fourier transform of the equation with respect to one of the spatial variables. The derived wave equation is solved by the finite-difference method and analyzed in terms of stability and dispersion. Although there are lots of wavenumbers to be solved according to the sampling theorem, the modeling can be accelerated by limiting the range of wavenumber used for the inverse Fourier transform. The proposed method generates common mid-point (CMP) gathers directly without sorting process by applying the shifting theorem. We show that our algorithm is efficient and generates the same results as ones from the 2D modeling through two numerical examples. The proposed modeling algorithm can be used for the inversion of CMP gathers or amplitude-versus-offset inversion.

In general, smoothing filters regularize functions by reducing differences between adjacent values. The smoothing filters, therefore, can regularize inverse solutions and produce more accurate subsurface structure when we apply it to full waveform inversion. If we apply a smoothing filter with a constant coefficient to subsurface image or velocity model, it will make layer interfaces and fault structures vague because it does not consider any information of geologic structures and variations of velocity. In this study, we develop a selective smoothing regularization technique, which adapts smoothing coefficients according to inversion iteration, to solve the weakness of smoothing regularization with a constant coefficient. First, we determine appropriate frequencies and analyze the corresponding wavenumber coverage. Then, we define effective maximum wavenumber as 99 percentile of wavenumber spectrum in order to choose smoothing coefficients which can effectively limit the wavenumber coverage. By adapting the chosen smoothing coefficients according to the iteration, we can implement multi-scale full waveform inversion while inverting multifrequency components simultaneously. Through the successful inversion example on a salt model with high-contrast velocity structures, we can note that our method effectively regularizes the inverse solution. We also verify that our scheme is applicable to field data through the numerical example to the synthetic data containing random noise.

요약: 동시 송신원 전파형 역산 기법은 계산량을 획기적으로 줄여 전파형 역산의 적용성을 높여준다. 그러나 다수의 송신원 모음 자료를 동시에 모델링하여 사용하기 때문에 관측 자료의 수진기 위치가 송신원에 따라 다른 경우, 나머지(residual) 파동장에 불필요한 값을 생성하게 되고 이는 파형역산의 수렴성을 저해하게 된다. 특히, 제한된 벌림 거리(offset)를 갖는 스트리머 방식의 탐사자료는 동시 송신원 기법을 적용하기에 가장 어려운 자료 형태이다. 이러한 문제점을 극복하기 위해 최근에 global correlation에 기반한 목적함수가 제안되었고, 시간영역 전파형 역산에 성공적으로 적용되었다.그러나 이 기법은 변형된 목적함수를 사용하기 때문에 나머지 파동장이 왜곡되고 경우에 따라 역산 결과에 부정적인 영향을 주기도 한다. 또한, 여러 가지 장점을 갖고 있는 주파수 영역 파형역산에 적용된 사례는 아직 보고된 적이 없다. 본논문에서는 이러한 나머지 파동장의 왜곡을 최소화하기 위해 global correlation 계산 시 사용하는 자료에 진폭감쇠 기법을 적용한다. 진폭감쇠를 적용한 자료는 global correlation의 특성을 최적화하여 나머지 파동장의 왜곡을 줄이고 파형역산 결과를 향상시킨다. 시간 영역에서 구한 나머지 파동장을 주파수 영역에서 역전파시킴으로써 global correlation기법을주파수 영역에서 구현한다. 스트리머 방식의 합성 탐사자료를 이용한 예제를 통해 본 논문에서 제안한 기법이 기존의global correlation 목적함수에 기반한 동시 송신원 전파형 역산보다 향상된 결과를 얻을 수 있음을 보여준다. 주요어: 전파형 역산, 동시 송신원, 주파수 영역, 스트리머 탐사 자료, global correlationAbstract: The simultaneous-source full waveform inversion improves the applicability of full waveform inversion byreducing the computational cost. Since this technique adopts simultaneous multi-source for forward modeling, unwantedevents remain in the residual seismograms when the receiver geometry of field acquisition is different from that ofnumerical modeling. As a result, these events impede the convergence of the full waveform inversion. In particular, thestreamer-type data with limited offsets is the most difficult data to apply the simultaneous-source technique. To overcomethis problem, the global-correlation-based objective function was suggested and it was successfully applied to thesimultaneous-source full waveform inversion in time domain. However, this method distorts residual wavefields due tothe modified objective function and has a negative influence on the inversion result. In addition, this method has notbeen applied to the frequency-domain simultaneous-source full waveform inversion. In this paper, we apply a time-damping function to the observed and modeled data, which are used to compute global correlation, to minimize thedistortion of residual wavefields. Since the damped wavefields optimize the performance of the global correlation, itmitigates the distortion of the residual wavefields and improves the inversion result. Our algorithm incorporates the global-correlation-based full waveform inversion into the frequency domain by back-propagating the time-domain residualwavefields in the frequency domain. Through the numerical examples using the streamer-type data, we show that ourinversion algorithm better describes the velocity structure than the conventional global correlation approach does.Keywords: full waveform inversion, simultaneous-source, frequency-domain, streamer data,2012년 6월 19일 접수; 2012년 7월 4일 수정; 2012년 7월 13일 채택; global correlation*Corresponding authorE-mail: pyunsj@inha.ac.krAddress: Department of Energy Resources Engineering, Inha University, 253 Yonghyun-dong Nam-gu, Incheon 402-751, Korea

Summary It is difficult to apply the encoded simultaneous-source full waveform inversion technique to the marine-streamer data because of acquisition geometry. The difference of acquisition geometry between the observed and modeled data produces unwanted events in the residual seismogram. To avoid these unwanted residuals, the objective function based on global correlation was suggested by several researchers. However, the objective function based on global correlation unpredictably modifies the residual seismogram, which may result in distortion of inverted velocity model. In this paper, we propose an alternative approach to avoid these distorted residual seismograms and to properly cope with the acquisition geometry problem. Our algorithm modifies the offset-limited observed data to full-offset data by combining the original observed data with the modeled data. This modification leads to undistorted residual seismogram for the simultaneous multiple shot data. As a result, we can perform the effective simultaneous-source full waveform inversion even for the marine-streamer acquisition. Through the numerical examples, we show that our inversion algorithm better describes the velocity structure than the global correlation approach does.