Wave-equation Traveltime Inversion Research Articles

Finite-frequency sensitivity kernels and hierarchical traveltime and waveform inversion of direct and reflected waves from vertical seismic profile data

Wave-equation-based traveltime or waveform inversion updates the velocity model along finite-frequency sensitivity kernels and exhibits higher accuracy than ray-based tomography. Owing to special geometry, the kernels for vertical seismic profile (VSP) data are different from those for surface seismic data. We have used the Born approximation to compute the traveltime and waveform kernels for direct and reflected waves from VSP data. Based on the property of sensitivity kernels for different information and arrivals, we develop a hierarchical inversion scheme: joint wave-equation traveltime inversion (JWETI) of direct and reflected waves, joint waveform inversion (JWI) of direct and reflected waves, and full-waveform inversion (FWI) of the reflected wave. The traveltime kernels, the waveform kernels, and the migration isochrones are used to retrieve the long-, intermediate-, and short-wavelength components of the velocity model, respectively. Numerical tests on synthetic and real walkway VSP data reveal that the JWETI stage obtains a plausible velocity macromodel, which can be treated as a starting model for subsequent JWI and FWI stages. The JWI stage is a transition between JWETI and FWI and helps to improve the structures of the deep layers of the model. The FWI stage offers some high-wavenumber contents to the inverted model. For the VSP survey, the transmitted direct waves can be used to reduce the dependence of FWI on the initial model. However, the conventional reflection-based wave-equation traveltime inversion and FWI still suffer from cycle skipping when the initial model is far away from the real model. In contrast, the hierarchical inversion scheme can yield reasonable inversion results around the borehole, even when large velocity errors are present in the starting model.

Autoencoded Elastic Wave-Equation Traveltime Inversion: Toward Reliable Near-Surface Tomogram

Due to unexpected environmental variations and poor consistencies in land data acquisition, complex near-surface seismograms are usually polluted unreasonably with a low signal-to-noise ratio (SNR). These complicated circumstances bring more challenges in identifying accurate first arrivals for the following wave-equation traveltime inversion (WT). Recently, the autoencoder (AE) is a typical unsupervised learning network whose basic principle is to compress the input seismic data for their intrinsic features in the latent space with an encoder and, thereafter, to decipher these features for seismic profiles as the output with a decoder. This process is fully automatic with high stability and is not very sensitive to data quality. In this paper, we propose an elastic WT inversion algorithm based on the AE method (AEWT) to invert the P-velocity model. Compared with the standard WT, the AEWT method automatically extracts the intrinsic features of the refractions with AE as reference data for the misfit functionals. Feature images in the latent space show similar but intensified sensitivity to the traveltimes with respect to velocity perturbations. We present one synthetic and two field data tests for comparing the proposed AEWT and the standard WT tomograms to investigate the locations of a buried fault and the depth of a buried sinkhole. All these experiments demonstrate that the proposed elastic AEWT method can reduce errors caused by low SNR and obtain a more reliable and stable P-velocity tomogram.

Monochromatic wave-equation traveltime inversion in vertical transverse isotropic media: Theory and application

Wave-equation traveltime inversion (WTI) is a promising tool for reconstructing the intrinsic anisotropy properties of the earth’s interior. However, conventional time-domain implementation of anisotropic WTI involves crosscorrelation between source- and receiver-side time-domain wavefields for gradient calculation, which requires heavy memory consumption and computational cost for storing/recomputing the source-side time-domain wavefields. To overcome this problem, we develop a novel frequency-domain anisotropic WTI method using only monochromatic wavefields during inversion. Thus, the memory requirement and computational cost are significantly reduced, which makes the monochromatic anisotropic WTI practical for real large-size applications. We focus on transversely isotropic media with a vertical symmetry axis (VTI) and aim to invert the P-wave vertical velocity [Formula: see text] and anisotropic parameter [Formula: see text]. The subspace approach is adopted for calculating multiple step lengths to balance the updates of [Formula: see text] and [Formula: see text] values in our VTI WTI method. Synthetic numerical examples based on an inclusion model and the BP 2007 model prove that the proposed monochromatic VTI WTI method can reconstruct background VTI models as accurately as VTI WTI with multiple frequencies. Then, we apply our method to an ocean-bottom seismic data set acquired at the Qiuyue field in the East China Sea. The inverted vertical velocity [Formula: see text] matches well with the long-wavelength component of the sonic well-logging velocity. Compared with the inverted isotropic velocity model, the inverted VTI model produces more focused seismic imaging reflectors throughout the section and more accurately positions the reflector depths, which is confirmed by the sonic well logging.

Wave-equation traveltime slope inversion by combining finite difference and cross-correlation methods

As an essential step in seismic inversion, velocity macromodel building provides a low-wavenumber background velocity model for seismic imaging and full waveform inversion. Defined as the first-order partial derivative of traveltime with regard to spatial coordinates, slope information has been directly or indirectly applied in ray-based geophysical inversion to retrieve models containing low and intermediate wavenumbers. In this study, to use slopes of locally coherent events in background velocity model building, we extend the slope inversion from ray theory to wave equation, which we name wave-equation traveltime slope inversion (WSI). Unlike traditional ray-based slope inversion, which relies on tedious picking of continuous horizons, we compute slope conveniently and robustly through cross-correlation and finite difference methods. Furthermore, we present the formulation of wave-equation traveltime slope inversion under the framework of wave-equation traveltime inversion. Traveltime and slopes are distinct data attributes that play different roles in reconstructing low-wavenumber background velocity models. Compared with the classic traveltime inversion, the wave-equation traveltime slope inversion is more sensitive to local velocity changes. Numerical examples demonstrate that the proposed method improves the horizontal resolution in velocity model building, which especially shows excellent potential in obtaining a higher lateral resolution of reconstruction for steeply dipping structures than traditional wave-equation traveltime inversion. What's more, this study makes the joint inversion of travel time and slope based on the wave equation possible, which can theoretically bring a higher resolution stabilized inversion.

Influence of shear velocity on elastic full-waveform inversion: Gulf of Mexico case study using multicomponent ocean-bottom node data

Elastic full-waveform inversion (FWI) is superior to acoustic FWI due to its ability to simulate complex mode conversions in fast-varying elastic media. Using elastic FWI may become important when building velocity models in areas of large complex salt bodies that we typically see in the Gulf of Mexico. Accounting for elastic effects in FWI can reduce the artifacts that are caused by using an acoustic approximation. We often rely on petrophysics relations to define an initial shear velocity model, which also could be challenging. We demonstrate how to take advantage of the reflected converted waves from the top of the salt interface recorded in horizontal geophones to improve the estimation of the background shear velocity. Then, we illustrate the impact of the initial background shear velocity in the elastic inversion of the long-offset low-frequency hydrophone data. The workflow is based on a combination of wave-equation traveltime inversion, to cope with the cycle skipping, and FWI, to obtain higher-resolution results.

Processing the Artificial Edge-Effects for Finite-Difference Frequency-Domain in Viscoelastic Anisotropic Formations

Real sedimentary media can usually be characterized as transverse isotropy. To reveal wave propagation in the true models and improve the accuracy of migrations and evaluations, we investigated the algorithm of wavefield simulations in an anisotropic viscoelastic medium. The finite difference in the frequency domain (FDFD) has several advantages compared with that in the time domain, e.g., implementing multiple sources, multi-scaled inversion, and introducing attenuation. However, medium anisotropy will lead to the complexity of the wavefield in the calculation. The damping profile of the conventional absorption boundary is only defined in one single direction, which produces instability when the wavefields of strong anisotropy are reflected on that truncated boundary. We applied the multi-axis perfectly matched layer (M-PML) to the wavefield simulations in anisotropic viscoelastic media to overcome this issue, which defines the damping profiles along different axes. In the numerical examples, we simulated seismic wave propagation in three viscous anisotropic media and focused on the wave attenuation in the absorbing layers using time domain snapshots. The M-PML was more effective for wave absorption compared to the conventional perfectly matched layer (PML). In strongly anisotropic media, the PML became unstable, and prominent reflections appeared at truncated boundaries. In contrast, the M-PML remained stable and efficient in the same model. Finally, the modeling of the stratified cross-well model showed the applicability of this proposed algorithm to heterogeneous viscous anisotropic media. The numerical algorithm can analyze wave propagation in viscoelastic anisotropic media. It also provides a reliable forward operator for waveform inversion, wave equation travel-time inversion, and seismic migration in anisotropic viscoelastic media.

Frequency-domain wave-equation traveltime inversion with a monofrequency component

Wave-equation traveltime inversion (WTI) is a useful tool for background velocity model building. It is generally formulated and implemented in the time domain, in which the gradient is calculated by temporally crosscorrelating the source- and receiver-side wavefields. The time-domain source-side snapshots are either stored in memory or are reconstructed through back propagation. The memory requirements and computational cost of WTI are thus prohibitively expensive, especially for 3D applications. To partially alleviate this problem, we provide an implementation of WTI in the frequency domain with a monofrequency component. Because only one frequency is used, it is affordable to directly store the source- and receiver-side wavefields in memory. There is no need for wavefield reconstruction during gradient calculation. In such a way, we have dramatically reduced the memory requirements and computational cost compared with the traditional time-domain WTI realization. For practical implementation, the frequency-domain wavefield is calculated by time-domain finite-difference forward modeling and is transformed to the frequency domain by an on-the-fly discrete Fourier transform. Numerical examples on a simple lateral periodic velocity model and the Marmousi model demonstrate that our method can obtain accurate background velocity models comparable with those from time-domain WTI and frequency-domain WTI with multiple frequencies. A field data set test indicates that our method obtains a background velocity model that well predicts the seismic wave traveltime.

Joint wave-equation traveltime inversion of diving/direct and reflected waves for P- and S-wave velocity macromodel building

Full-waveform inversion (FWI) suffers from the local minima problem and requires a sufficiently accurate starting model to converge to the correct solution. Wave-equation traveltime inversion (WETI) is an effective tool to retrieve the long-wavelength components of the velocity model. We have developed a joint diving/direct and reflected wave WETI (JDRWETI) method to build P- and S-wave velocity macromodels. We estimate the traveltime shifts of seismic events (diving/direct waves and PP- and PS-reflections) through the dynamic warping scheme and construct a misfit function using the time shifts of diving/direct and reflected waves. We derive the adjoint wave equations and the gradients with respect to the background models based on the joint misfit function. We apply the kernel decomposition scheme to extract the kernel of the diving/direct wave and the tomography kernels of PP- and PS-reflections. For an explosive source, the kernels of the diving/direct wave and PP-reflections and the kernel of the PS-reflections are used to compute the P- and S-wave gradients of the background models, respectively. We implement JDRWETI by a two-stage inversion workflow: First, we invert the P- and S-wave velocity models using the P-wave gradients, and then we improve the S-wave velocity model using the S-wave gradients. Numerical tests on synthetic and field data sets reveal that the JDRWETI method successfully recovers the long-wavelength components of P- and S-wave velocity models, which can be used for an initial model for the subsequent elastic FWI. Moreover, the JDRWETI method prevails over the existing reflection WETI method and the cascaded diving/direct and reflected wave WETI method, especially when large velocity errors are present in the shallow part of the starting models. The JDRWETI method with the two-stage inversion workflow can give rise to reasonable inversion results even for the model with different P- and S-wave velocity structures.

Frequency-dependent reflection wave-equation traveltime inversion from walkaway vertical seismic profile data

To accurately image the geologic structures from walkaway vertical seismic profile (VSP) data, it is necessary to estimate the subsurface velocity field with high resolution and enhanced illumination of deep reservoirs. Because full-waveform inversion (FWI) suffers from cycle skipping when the initial model is far from the true model, we have adopted frequency-dependent reflection wave-equation traveltime inversion (FRWT) to generate the background velocity model for VSP migration. The upgoing reflection data are separated from the original shot gathers, and dynamic warping is used to evaluate the traveltime differences between the observed data and the calculated data. Different frequency bands of the data are inverted in sequence to reconstruct the velocity model with higher resolution. We also implement wavefield decomposition on the gradient field to extract the contributions of reflection components and improve the updated model. The inverted results obtained from FRWT can be used as the initial model for conventional FWI or the velocity model for reverse time migration. The experiments on synthetic data and field data demonstrate that our approach can effectively recover the background velocity model from walkaway VSP data.

Adaptive traveltime inversion

We have developed a method to obtain a misfit function for robust waveform inversion. In this method, called adaptive traveltime inversion (ATI), a matching filter that matches predicted data to measured data is computed. If the velocity model is relatively accurate, the resulting matching filter is close to a Dirac delta function. Its traveltime shift, which characterizes the defocusing of the matching filter, is computed by minimization of the crosscorrelation between a penalty function such as [Formula: see text] and the matching filter. ATI is constructed by minimization of the least-squares errors of the calculated traveltime shift. Further analysis indicates that the resulting traveltime shift corresponds to a first-order moment, the mean value of the resulting matching filter distribution. We extend ATI to a more general misfit function formula by computing different order moment of the resulting matching filter distribution. Choosing the penalty function in adaptive waveform inversion (AWI) as [Formula: see text], the misfit function of AWI is the second-order moment, the variance of the resulting matching filter distribution with zero mean. Because our ATI method is based on a global comparison using deconvolution, such as AWI, it can resolve the “cycle skipping” issue. We evaluate our ATI misfit function and compare it with state-of-the-art options such as least-squares inversion (L2 norm), wave-equation traveltime inversion, and AWI using schematic examples before moving to more complex examples, such as the Marmousi model. For the Marmousi model, starting with a 1D [Formula: see text] model, with data without low frequencies (no energy below 3 Hz), a meaningful estimation of the P-wave velocity model is recovered. Our ATI misfit function (first-order moment) indicates comparable performance with the AWI misfit function (the second-order moment). We also include a real data example from the Gulf of Mexico to demonstrate the effectiveness of our method.

A least-squares correlation-based full traveltime inversion for shallow subsurface velocity reconstruction

Accurate estimation of shallow subsurface velocity models with complex topography is crucial for statics corrections and imaging deep structures. The correlation-based wave-equation traveltime inversion (CWTI) method is suitable for estimating such shallow subsurface velocity structures. However, the CWTI objective function suffers an inherent resolution-loss problem because the traveltime weighted crosscorrelation misfit does not fall to zero even when the model is perfectly matched. Furthermore, the Born-approximation-based CWTI gradient cannot provide an effective model update during each iteration. To overcome these problems, we have developed a least-squares correlation-based full traveltime inversion (LCFTI) method, in which the least-squares correlation-based objective function was designed to minimize the traveltime weighted difference between the autocorrelation and the crosscorrelation. By incorporating the autocorrelation, LCFTI indicates better convergence and higher resolution than CWTI. The LCFTI model updates are derived using the Rytov approximation to avoid incorrect model updates by emphasizing phase matching. Furthermore, to accurately simulate wave propagation, we use the spectral-element method as the modeling engine, in which the mesh of the complex topography is flexibly represented. Synthetic and field data examples are performed to demonstrate the effectiveness of the proposed method in shallow subsurface velocity reconstruction.

Wave-equation traveltime inversion with multifrequency bands: Synthetic and land data examples

We have developed a wave-equation traveltime inversion method with multifrequency bands to invert for the shallow or intermediate subsurface velocity distribution. Similar to the classical wave-equation traveltime inversion, this method searches for the velocity model that minimizes the squared sum of the traveltime residuals using source wavelets with progressively higher peak frequencies. Wave-equation traveltime inversion can partially avoid the cycle-skipping problem by recovering the low-wavenumber parts of the velocity model. However, we also use the frequency information hidden in the traveltimes to obtain a more highly resolved tomogram. Therefore, we use different frequency bands when calculating the Fréchet derivatives so that tomograms with better resolution can be reconstructed. Results are validated by the zero-offset gathers from the raw data associated with moderate geometric irregularities. The improved wave-equation traveltime method is robust and merely needs a rough estimate of the starting model. Numerical tests on the synthetic and field data sets validate the above claims.

Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors

A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes the squared sum of the traveltime residuals. Even though, wave-equation traveltime inversion can partly avoid the cycle skipping problem, a good initial velocity model is required for the inversion to converge to a reasonable tomogram with different attenuation profiles. When Q model is far away from the real model, the final tomogram is very sensitive to the starting velocity model. Nevertheless, a minor or moderate perturbation of the Q model from the true one does not strongly affect the inversion if the low wavenumber information of the initial velocity model is mostly correct. These claims are validated with numerical tests on both the synthetic and field data sets.

Towards full waveform ambient noise inversion

We develop a method for the joint inversion of noise correlation functions for the distribution of noise sources and for Earth structure. The forward problem is free of assumptions required to equate noise correlations with Green functions and allows us to compute inter-station correlations for arbitrary distributions of noise sources in space and time. Using adjoint techniques, we design an iterative inversion scheme for noise sources and Earth structure based on waveform and energy differences as misfit functional. Starting from an initial model from a wave equation traveltime inversion, we recover the target velocity model with high accuracy. A key prerequisite is a good inference of the noise source distribution.

Skeletonized wave-equation inversion in vertical symmetry axis media without too much math

We have developed a tutorial for skeletonized inversion of pseudo-acoustic anisotropic vertical symmetry axis (VTI) data. We first invert for the anisotropic models using wave-equation traveltime inversion. Here, the skeletonized data are the traveltimes of transmitted and/or reflected arrivals that lead to simpler misfit functions and more robust convergence compared with full-waveform inversion. This provides a good starting model for waveform inversion. The effectiveness of this procedure is illustrated with synthetic data examples and a marine data set recorded in the Gulf of Mexico.

Wave-equation traveltime inversion: Comparison of three numerical optimization methods

Compared to traditional traveltime inversion methods, no traveltime picking, high-frequency assumption or ray tracing is necessary in wave-equation traveltime inversion (WT). Another merit of WT is that it is insensitive to the starting model. Although WT offers less detailed parts of velocity model than the full waveform inversion (FWI), it can provide a good initial velocity model for FWI. In this paper, the steepest descent, conjugate gradient and limited-memory BFGS (L-BFGS) optimization methods are used in the implementation of acoustic WT. We use synthetic crosswell data for testing and the numerical results show that L-BFGS has a faster rate of convergence and offers a reconstructed velocity model with better resolution compared to the other two gradient methods.