Correlation‐based reflection waveform inversion by one‐way wave equations

Reflection full waveform inversion can update subsurface velocity structure of the deeper part, but tends to get stuck in the local minima associated with the waveform misfit function. These local minima cause cycle skipping if the initial background velocity model is far from the true model. Since conventional reflection full waveform inversion using two‐way wave equation in time domain is computationally expensive and consumes a large amount of memory, we implement a correlation‐based reflection waveform inversion using one‐way wave equations to retrieve the background velocity. In this method, one‐way wave equations are used for the seismic wave forward modelling, migration/de‐migration and the gradient computation of objective function in frequency domain. Compared with the method using two‐way wave equation, the proposed method benefits from the lower computational cost of one‐way wave equations without significant accuracy reduction in the cases without steep dips. It also largely reduces the memory requirement by an order of magnitude than implementation using two‐way wave equation both for two‐ and three‐dimensional situations. Through numerical analysis, we also find that one‐way wave equations can better construct the low wavenumber reflection wavepath without producing high‐amplitude short‐wavelength components near the image points in the reflection full waveform inversion gradient. Synthetic test and real data application show that the proposed method efficiently updates the background velocity model.     

3D acoustic waveform inversion in the Laplace domain using an iterative solver

In this paper we propose a 3D acoustic full waveform inversion algorithm in the Laplace domain. The partial differential equation for the 3D acoustic wave equation in the Laplace domain is reformulated as a linear system of algebraic equations using the finite element method and the resulting linear system is solved by a preconditioned conjugate gradient method. The numerical solutions obtained by our modelling algorithm are verified through a comparison with the corresponding analytical solutions and the appropriate dispersion analysis. In the Laplace‐domain waveform inversion, the logarithm of the Laplace transformed wavefields mainly contains long‐wavelength information about the underlying velocity model. As a result, the algorithm smoothes a small‐scale structure but roughly identifies large‐scale features within a certain depth determined by the range of offsets and Laplace damping constants employed. Our algorithm thus provides a useful complementary process to time‐ or frequency‐domain waveform inversion, which cannot recover a large‐scale structure when low‐frequency signals are weak or absent. The algorithm is demonstrated on a synthetic example: the SEG/EAGE 3D salt‐dome model. The numerical test is limited to a Laplace‐domain synthetic data set for the inversion. In order to verify the usefulness of the inverted velocity model, we perform the 3D reverse time migration. The migration results show that our inversion results can be used as an initial model for the subsequent high‐resolution waveform inversion. Further studies are needed to perform the inversion using time‐domain synthetic data with noise or real data, thereby investigating robustness to noise.     

Seismic envelope inversion: reduction of local minima and noise resistance

Waveform inversion met severe challenge in retrieving long‐wavelength background structure. We have proposed to use envelope inversion to recover the large‐scale component of the model. Using the large‐scale background recovered by envelope inversion as new starting model, we can get much better result than the conventional full waveform inversion. By comparing the configurations of the misfit functional between the envelope inversion and the conventional waveform inversion, we show that envelope inversion can greatly reduce the local minimum problem. The combination of envelope inversion and waveform inversion can deliver more faithful and accurate final result with almost no extra computation cost compared to the conventional full waveform inversion. We also tested the noise resistance ability of envelope inversion to Gaussian noise and seismic interference noise. The results showed that envelope inversion is insensitive to Gaussian noise and, to a certain extent, insensitive to seismic interference noise. This indicates the robustness of this method and its potential use for noisy data.     

Velocity model building from seismic reflection data by full‐waveform inversion

Full‐waveform inversion is re‐emerging as a powerful data‐fitting procedure for quantitative seismic imaging of the subsurface from wide‐azimuth seismic data. This method is suitable to build high‐resolution velocity models provided that the targeted area is sampled by both diving waves and reflected waves. However, the conventional formulation of full‐waveform inversion prevents the reconstruction of the small wavenumber components of the velocity model when the subsurface is sampled by reflected waves only. This typically occurs as the depth becomes significant with respect to the length of the receiver array. This study first aims to highlight the limits of the conventional form of full‐waveform inversion when applied to seismic reflection data, through a simple canonical example of seismic imaging and to propose a new inversion workflow that overcomes these limitations. The governing idea is to decompose the subsurface model as a background part, which we seek to update and a singular part that corresponds to some prior knowledge of the reflectivity. Forcing this scale uncoupling in the full‐waveform inversion formalism brings out the transmitted wavepaths that connect the sources and receivers to the reflectors in the sensitivity kernel of the full‐waveform inversion, which is otherwise dominated by the migration impulse responses formed by the correlation of the downgoing direct wavefields coming from the shot and receiver positions. This transmission regime makes full‐waveform inversion amenable to the update of the long‐to‐intermediate wavelengths of the background model from the wide scattering‐angle information. However, we show that this prior knowledge of the reflectivity does not prevent the use of a suitable misfit measurement based on cross‐correlation, to avoid cycle‐skipping issues as well as a suitable inversion domain as the pseudo‐depth domain that allows us to preserve the invariant property of the zero‐offset time. This latter feature is useful to avoid updating the reflectivity information at each non‐linear iteration of the full‐waveform inversion, hence considerably reducing the computational cost of the entire workflow. Prior information of the reflectivity in the full‐waveform inversion formalism, a robust misfit function that prevents cycle‐skipping issues and a suitable inversion domain that preserves the seismic invariant are the three key ingredients that should ensure well‐posedness and computational efficiency of full‐waveform inversion algorithms for seismic reflection data.     

The natural combination of full and image‐based waveform inversion

Integrating migration velocity analysis and full waveform inversion can help reduce the high non‐linearity of the classic full waveform inversion objective function. The combination of inverting for the long and short wavelength components of the velocity model using a dual objective function that is sensitive to both components is still very expensive and have produced mixed results. We develop an approach that includes both components integrated to complement each other. We specifically utilize the image to generate reflections in our synthetic data only when the velocity model is not capable of producing such reflections. As a result, we get the migration velocity analysis working when we need it, and we mitigate its influence when the velocity model produces accurate reflections (possibly first for the low frequencies). This is achieved using a novel objective function that includes both objectives. Applications to a layered model and the Marmousi model demonstrate the main features of the approach.     

2D acoustic‐elastic coupled waveform inversion in the Laplace domain

Although waveform inversion has been intensively studied in an effort to properly delineate the Earth's structures since the early 1980s, most of the time‐ and frequency‐domain waveform inversion algorithms still have critical limitations in their applications to field data. This may be attributed to the highly non‐linear objective function and the unreliable low‐frequency components. To overcome the weaknesses of conventional waveform inversion algorithms, the acoustic Laplace‐domain waveform inversion has been proposed. The Laplace‐domain waveform inversion has been known to provide a long‐wavelength velocity model even for field data, which may be because it employs the zero‐frequency component of the damped wavefield and a well‐behaved logarithmic objective function. However, its applications have been confined to 2D acoustic media. We extend the Laplace‐domain waveform inversion algorithm to a 2D acoustic‐elastic coupled medium, which is encountered in marine exploration environments. In 2D acoustic‐elastic coupled media, the Laplace‐domain pressures behave differently from those of 2D acoustic media, although the overall features are similar to each other. The main differences are that the pressure wavefields for acoustic‐elastic coupled media show negative values even for simple geological structures unlike in acoustic media, when the Laplace damping constant is small and the water depth is shallow. The negative values may result from more complicated wave propagation in elastic media and at fluid‐solid interfaces. Our Laplace‐domain waveform inversion algorithm is also based on the finite‐element method and logarithmic wavefields. To compute gradient direction, we apply the back‐propagation technique. Under the assumption that density is fixed, P‐ and S‐wave velocity models are inverted from the pressure data. We applied our inversion algorithm to the SEG/EAGE salt model and the numerical results showed that the Laplace‐domain waveform inversion successfully recovers the long‐wavelength structures of the P‐ and S‐wave velocity models from the noise‐free data. The models inverted by the Laplace‐domain waveform inversion were able to be successfully used as initial models in the subsequent frequency‐domain waveform inversion, which is performed to describe the short‐wavelength structures of the true models.     

Anisotropic elastic full‐waveform inversion of walkaway vertical seismic profiling data from the Arabian Gulf

Borehole seismic addresses the need for high‐resolution images and elastic parameters of the subsurface. Full‐waveform inversion of vertical seismic profile data is a promising technology with the potential to recover quantitative information about elastic properties of the medium. Full‐waveform inversion has the capability to process the entire wavefield and to address the wave propagation effects contained in the borehole data—multi‐component measurements; anisotropic effects; compressional and shear waves; and transmitted, converted, and reflected waves and multiples. Full‐waveform inversion, therefore, has the potential to provide a more accurate result compared with conventional processing methods. We present a feasibility study with results of the application of high‐frequency (up to 60 Hz) anisotropic elastic full‐waveform inversion to a walkaway vertical seismic profile data from the Arabian Gulf. Full‐waveform inversion has reproduced the majority of the wave events and recovered a geologically plausible layered model with physically meaningful values of the medium.     

Elastic full waveform inversion of multi‐component OBC seismic data

Elastic full waveform inversion of seismic reflection data represents a data‐driven form of analysis leading to quantification of sub‐surface parameters in depth. In previous studies attention has been given to P‐wave data recorded in the marine environment, using either acoustic or elastic inversion schemes. In this paper we exploit both P‐waves and mode‐converted S‐waves in the marine environment in the inversion for both P‐ and S‐wave velocities by using wide‐angle, multi‐component, ocean‐bottom cable seismic data. An elastic waveform inversion scheme operating in the time domain was used, allowing accurate modelling of the full wavefield, including the elastic amplitude variation with offset response of reflected arrivals and mode‐converted events. A series of one‐ and two‐dimensional synthetic examples are presented, demonstrating the ability to invert for and thereby to quantify both P‐ and S‐wave velocities for different velocity models. In particular, for more realistic low velocity models, including a typically soft seabed, an effective strategy for inversion is proposed to exploit both P‐ and mode‐converted PS‐waves. Whilst P‐wave events are exploited for inversion for P‐wave velocity, examples show the contribution of both P‐ and PS‐waves to the successful recovery of S‐wave velocity.     

Acoustic full waveform inversion of synthetic land and marine data in the Laplace domain

Elastic waves, such as Rayleigh and mode‐converted waves, together with amplitude versus offset variations, serve as noise in full waveform inversion using the acoustic approximation. Heavy preprocessing must be applied to remove elastic effects to invert land or marine data using the acoustic inversion method in the time or frequency domains. Full waveform inversion using the elastic wave equation should be one alternative; however, multi‐parameter inversion is expensive and sensitive to the starting velocity model. We implement full acoustic waveform inversion of synthetic land and marine data in the Laplace domain with minimum preprocessing (i.e., muting) to remove elastic effects. The damping in the Laplace transform can be thought of as an automatic time windowing. Numerical examples show that Laplace‐domain acoustic inversion can yield correct smooth velocity models even with the noise originating from elastic waves. This offers the opportunity to develop an accurate smooth starting model for subsequent inversion in the frequency domain.     

Research Note: Insights into the data dependency on anisotropy: an inversion prospective

While velocity contrasts are responsible for most of the events recorded in our data, the long wavelength behavior of the velocity model is responsible for the geometrical shape of these events. For isotropic acoustic materials, the wave dependency on the long (wave propagation) and short (scattering) wavelength velocity components is stationary with the propagation angle. On the other hand, in representing a transversely isotropic with a vertical symmetry axis medium with the normal moveout velocity, the anellepticity parameter η, the vertical scaling parameter δ, and the sensitivity of waves vary with the polar angle for both the long and short wavelength features of the anisotropic dimensionless medium parameters (δ and η). For horizontal reflectors at reasonable depths, the long wavelength features of the η model is reasonably constrained by the long offsets, whereas the short wavelength features produce very week reflections at even reasonable offsets. Thus, for surface acquired seismic data, we could mainly invert for smooth η responsible for the geometrical shape of reflections. On the other hand, while the δ long wavelength components mildly affects the recorded data, its short wavelength variations can produce reflections at even zero offset, with a behavior pattern synonymous to density. The lack of the long wavelength δ information will mildly effect focusing but will cause misplacement of events in depth. With low enough frequencies (very low), we may be able to recover the long wavelength δ using full waveform inversion. However, unlike velocity, the frequencies needed for that should be ultra‐low to produce long‐wavelength scattering‐based model information as δ perturbations do not exert scattering at large offsets. For a combination given by the horizontal velocity, η, and ε, the diving wave influence of η is absorbed by the horizontal velocity, severely limiting the η influence on the data and full waveform inversion. As a result, with a good smooth η estimation, for example, from tomography, we can focus the full waveform inversion to invert for only the horizontal velocity and maybe ε as a parameter to fit the amplitude. This is possibly the most practical parametrization for inversion of surface seismic data in transversely isotropic with vertical symmetry axis media.     

Migration velocity analysis and waveform inversion

Least‐squares inversion of seismic reflection waveform data can reconstruct remarkably detailed models of subsurface structure and take into account essentially any physics of seismic wave propagation that can be modelled. However, the waveform inversion objective has many spurious local minima, hence convergence of descent methods (mandatory because of problem size) to useful Earth models requires accurate initial estimates of long‐scale velocity structure. Migration velocity analysis, on the other hand, is capable of correcting substantially erroneous initial estimates of velocity at long scales. Migration velocity analysis is based on prestack depth migration, which is in turn based on linearized acoustic modelling (Born or single‐scattering approximation). Two major variants of prestack depth migration, using binning of surface data and Claerbout's survey‐sinking concept respectively, are in widespread use. Each type of prestack migration produces an image volume depending on redundant parameters and supplies a condition on the image volume, which expresses consistency between data and velocity model and is hence a basis for velocity analysis. The survey‐sinking (depth‐oriented) approach to prestack migration is less subject to kinematic artefacts than is the binning‐based (surface‐oriented) approach. Because kinematic artefacts strongly violate the consistency or semblance conditions, this observation suggests that velocity analysis based on depth‐oriented prestack migration may be more appropriate in kinematically complex areas. Appropriate choice of objective (differential semblance) turns either form of migration velocity analysis into an optimization problem, for which Newton‐like methods exhibit little tendency to stagnate at nonglobal minima. The extended modelling concept links migration velocity analysis to the apparently unrelated waveform inversion approach to estimation of Earth structure: from this point of view, migration velocity analysis is a solution method for the linearized waveform inversion problem. Extended modelling also provides a basis for a nonlinear generalization of migration velocity analysis. Preliminary numerical evidence suggests a new approach to nonlinear waveform inversion, which may combine the global convergence of velocity analysis with the physical fidelity of model‐based data fitting.     

Full waveform inversion of SH‐ and Love‐wave data in near‐surface prospecting

We develop a two‐dimensional full waveform inversion approach for the simultaneous determination of S‐wave velocity and density models from SH ‐ and Love‐wave data. We illustrate the advantages of the SH/Love full waveform inversion with a simple synthetic example and demonstrate the method's applicability to a near‐surface dataset, recorded in the village ?achtice in Northwestern Slovakia. Goal of the survey was to map remains of historical building foundations in a highly heterogeneous subsurface. The seismic survey comprises two parallel SH‐profiles with maximum offsets of 24 m and covers a frequency range from 5 Hz to 80 Hz with high signal‐to‐noise ratio well suited for full waveform inversion. Using the Wiechert–Herglotz method, we determined a one‐dimensional gradient velocity model as a starting model for full waveform inversion. The two‐dimensional waveform inversion approach uses the global correlation norm as objective function in combination with a sequential inversion of low‐pass filtered field data. This mitigates the non‐linearity of the multi‐parameter inverse problem. Test computations show that the influence of visco‐elastic effects on the waveform inversion result is rather small. Further tests using a mono‐parameter shear modulus inversion reveal that the inversion of the density model has no significant impact on the final data fit. The final full waveform inversion S‐wave velocity and density models show a prominent low‐velocity weathering layer. Below this layer, the subsurface is highly heterogeneous. Minimum anomaly sizes correspond to approximately half of the dominant Love‐wavelength. The results demonstrate the ability of two‐dimensional SH waveform inversion to image shallow small‐scale soil structure. However, they do not show any evidence of foundation walls.     

Sensitivity analysis and application of time‐lapse full‐waveform inversion: synthetic testing and field data example from the North Sea,Norway

Time‐lapse refraction can provide complementary seismic solutions for monitoring subtle subsurface changes that are challenging for conventional P‐wave reflection methods. The utilization of refraction time lapse has lagged behind in the past partly due to the lack of robust techniques that allow extracting easy‐to‐interpret reservoir information. However, with the recent emergence of the full‐waveform inversion technique as a more standard tool, we find it to be a promising platform for incorporating head waves and diving waves into the time‐lapse framework. Here we investigate the sensitivity of 2D acoustic, time‐domain, full‐waveform inversion for monitoring a shallow, weak velocity change (?30 m/s, or ?1.6%). The sensitivity tests are designed to address questions related to the feasibility and accuracy of full‐waveform inversion results for monitoring the field case of an underground gas blowout that occurred in the North Sea. The blowout caused the gas to migrate both vertically and horizontally into several shallow sand layers. Some of the shallow gas anomalies were not clearly detected by conventional 4D reflection methods (i.e., time shifts and amplitude difference) due to low 4D signal‐to‐noise ratio and weak velocity change. On the other hand, full‐waveform inversion sensitivity analysis showed that it is possible to detect the weak velocity change with the non‐optimal seismic input. Detectability was qualitative with variable degrees of accuracy depending on different inversion parameters. We inverted, the real 2D seismic data from the North Sea with a greater emphasis on refracted and diving waves’ energy (i.e., most of the reflected energy was removed for the shallow zone of interest after removing traces with offset less than 300 m). The full‐waveform inversion results provided more superior detectability compared with the conventional 4D stacked reflection difference method for a weak shallow gas anomaly (320 m deep).     

不依赖子波、基于包络的FWI初始模型建立方法研究

地震全波形反演(FWI)从理论走向实际面临着诸多难题,其中之一就是需要一个较高精度的初始模型,另一个难题就是需要一个较为精确的震源子波,初始模型和震源子波的准确程度严重影响着全波形反演的最终结果.为此,本文提出了不依赖子波、基于包络的FWI初始模型建立的方法,建立了相应的目标函数,推导出了反演的梯度,给出了伴随震源的表达式,理论上分析了不依赖子波FWI的可行性.在数值试验中,讨论了参考道的选取方式,通过分析归一化目标函数收敛速率,认为近偏移距参考道优于远偏移距参考道,在地震数据含干扰噪音时,平均道作为参考道要优于最小偏移距参考道.通过包络、包络对数、包络平方三种目标函数反演结果的比较,发现包络对数目标函数对深层的反演效果最好.通过不同子波的试验进一步验证了本方法的正确性.     

Multistep inversion workflow for 3D long‐offset damped elastic waves in the Fourier domain

We present a new workflow for imaging damped three‐dimensional elastic wavefields in the Fourier domain. The workflow employs a multiscale imaging approach, in which offset lengths are laddered, where frequency content and damping of the data are changed cyclically. Thus, the inversion process is launched using short‐offset and low‐frequency data to recover the long spatial wavelength of the image at a shallow depth. Increasing frequency and offset length leads to the recovery of the fine‐scale features of the model at greater depths. For the fixed offset, we employ (in the imaging process) a few discrete frequencies with a set of Laplace damping parameters. The forward problem is solved with a finite‐difference frequency‐domain method based on a massively parallel iterative solver. The inversion code is based upon the solution of a least squares optimisation problem and is solved using a nonlinear gradient method. It is fully parallelised for distributed memory computational platforms. Our full‐waveform inversion workflow is applied to the 3D Marmousi‐2 and SEG/EAGE Salt models with long‐offset data. The maximum inverted frequencies are 6 Hz for the Marmousi model and 2 Hz for the SEG/EAGE Salt model. The detailed structures are imaged successfully up to the depth approximately equal to one‐third of the maximum offset length at a resolution consistent with the inverted frequencies.     

Full waveform inversion using oriented time‐domain imaging method for vertical transverse isotropic media

Full waveform inversion for reflection events is limited by its linearised update requirements given by a process equivalent to migration. Unless the background velocity model is reasonably accurate, the resulting gradient can have an inaccurate update direction leading the inversion to converge what we refer to as local minima of the objective function. In our approach, we consider mild lateral variation in the model and, thus, use a gradient given by the oriented time‐domain imaging method. Specifically, we apply the oriented time‐domain imaging on the data residual to obtain the geometrical features of the velocity perturbation. After updating the model in the time domain, we convert the perturbation from the time domain to depth using the average velocity. Considering density is constant, we can expand the conventional 1D impedance inversion method to two‐dimensional or three‐dimensional velocity inversion within the process of full waveform inversion. This method is not only capable of inverting for velocity, but it is also capable of retrieving anisotropic parameters relying on linearised representations of the reflection response. To eliminate the crosstalk artifacts between different parameters, we utilise what we consider being an optimal parametrisation for this step. To do so, we extend the prestack time‐domain migration image in incident angle dimension to incorporate angular dependence needed by the multiparameter inversion. For simple models, this approach provides an efficient and stable way to do full waveform inversion or modified seismic inversion and makes the anisotropic inversion more practicable. The proposed method still needs kinematically accurate initial models since it only recovers the high‐wavenumber part as conventional full waveform inversion method does. Results on synthetic data of isotropic and anisotropic cases illustrate the benefits and limitations of this method.     

Frequency‐domain acoustic‐elastic coupled waveform inversion using the Gauss‐Newton conjugate gradient method

We developed a frequency‐domain acoustic‐elastic coupled waveform inversion based on the Gauss‐Newton conjugate gradient method. Despite the use of a high‐performance computer system and a state‐of‐the‐art parallel computation algorithm, it remained computationally prohibitive to calculate the approximate Hessian explicitly for a large‐scale inverse problem. Therefore, we adopted the conjugate gradient least‐squares algorithm, which is frequently used for geophysical inverse problems, to implement the Gauss‐Newton method so that the approximate Hessian is calculated implicitly. Thus, there was no need to store the Hessian matrix. By simultaneously back‐propagating multi‐components consisting of the pressure and displacements, we could efficiently extract information on the subsurface structures. To verify our algorithm, we applied it to synthetic data sets generated from the Marmousi‐2 model and the modified SEG/EAGE salt model. We also extended our algorithm to the ocean‐bottom cable environment and verified it using ocean‐bottom cable data generated from the Marmousi‐2 model. With the assumption of a hard seafloor, we recovered both the P‐wave velocity of complicated subsurface structures as well as the S‐wave velocity. Although the inversion of the S‐wave velocity is not feasible for the high Poisson's ratios used to simulate a soft seafloor, several strategies exist to treat this problem. Our example using multi‐component data showed some promise in mitigating the soft seafloor effect. However, this issue still remains open.     

基于精确震源函数的解调包络多尺度全波形反演

本文提出解调包络方法来重构地震记录中缺失的低频信号,同时该方法能够降低全波形反演的非线性程度;提出伴随状态震源函数反演方法来得到精确的震源函数,并推导了梯度计算公式;解调包络方法结合低通滤波技术,实现了从低频到高频的多尺度反演策略,有效缓解了全波形反演的周波跳跃问题.数值算例证明了解调包络、伴随状态震源函数反演方法和低通滤波多尺度反演策略的可行性及优越性.震源函数反演精度测试结果表明:即使观测记录在缺失低频信息的情况下,也能反演得到精确的震源函数.缺失低频测试和抗噪能力测试结果表明:即使地震数据中缺失9Hz以下的低频信号或者信噪比极低的情况下,利用反演得到的精确震源函数进行解调包络多尺度全波形反演,同样可以得到高精度的全波形反演结果.与Hilbert包络全波形反演对比结果表明:解调包络在重构低频和降低伴随震源主频方面具有一定优势.     

频率多尺度全波形速度反演

以二维声波方程为模型,在时间域深入研究了全波形速度反演.全波形反演要解一个非线性的最小二乘问题,是一个极小化模拟数据与已知数据之间残量的过程.针对全波形反演易陷入局部极值的困难,本文提出了基于不同尺度的频率数据的"逐级反演"策略,即先基于低频尺度的波场信息进行反演,得出一个合理的初始模型,然后再利用其他不同尺度频率的波场进行反演,并且用前一尺度的迭代反演结果作为下一尺度反演的初始模型,这样逐级进行反演.文中详细阐述和推导了理论方法及公式,包括有限差分正演模拟、速度模型修正、梯度计算和算法描述,并以Marmousi复杂构造模型为例,进行了MPI并行全波形反演数值计算,得到了较好的反演结果,验证了方法的有效性和稳健性.