Smooth Velocity Model Research Articles

Non-uniqueness of waveform inversions in the Laplace domain

The non-uniqueness of the solution is one of the major obstacles for a successful full waveform inversion. This paper shows that there are at least two solutions for waveform inversions in the Laplace domain: the true velocity model; the long-wavelength background velocity model. The Laplace domain inversion can recover the true velocity model if provided with an accurate initial model, but it will instead result in a smooth long-wavelength velocity model if the initial model is inaccurate and smooth. In this case, the inverted long-wavelength velocity model can still be used for either migration or as an initial velocity model for frequency domain waveform inversion.

Comparison of ray‐ and adjoint‐based sensitivity kernels for body‐wave seismic tomography

We compare finite‐frequency sensitivity kernels computed from ray‐theoretical wavefields (“banana‐doughnut” kernels) and from full waveform computations (often called “adjoint” kernels) in order to evaluate resolution, accuracy and computational cost. We focus here on body‐wave seismic tomography at regional and local scales. Our results show that: (1) for homogeneous reference media, ray‐based and adjoint kernels agree except for the expected differences in the regions close to the source and the receiver, where near‐field contributions are neglected in the ray‐based kernels; (2) for a smooth 3D background velocity model, the differences in predicted delay times for the two methods are generally well below 10 % of the delay for P waves, though as much as 20 % for S‐waves, suggesting that extra care should be taken when performing S‐wave tomography with ray‐based “banana doughnut” kernels.

Imaging permafrost velocity structure using high resolution 3D seismic tomography

A 3D seismic survey (Mallik 3D), covering [Formula: see text] in the Mackenzie Delta area of Canada’s north, was conducted by industry in 2002. Numerous lakes and marine inundation create a complex near-surface structure in the permafrost terrain. Much of the near subsurface remains frozen but significant melt zones exist particularly from perennially unfrozen water bodies. This results in an irregular distribution of permafrost ice creating a complex pattern of low and high frequency near-surface velocity variations which induce significant traveltime distortions in surface seismic data. A high resolution 3D traveltime tomography study was employed to map the permafrost velocity structure utilizing first-arrival traveltimes picked from 3D seismic shot records. Approximately 900,000 traveltime picks from 3167 shots were used in the inversion. Tomographic inversion of the first-arrival traveltimes resulted in a smooth velocity model for the upper 200 m of the subsurface. Ray coverage in the model is excellent down to 200 m providing effective control for estimating velocities through tomographic inversion. Resolution tests conducted through horizontal and vertical checkerboard tests confirm the robustness of the velocity model in detailing small scale velocity variations. Well velocities were used to validate tomographic velocities. The tomographic velocities do not show systematic correlation with well velocities. The velocity model clearly images the permafrost velocity structure in lateral and vertical directions. It is inferred from the velocity model that the permafrost structure in the near subsurface is discontinuous. Extensions of surface water bodies in depth, characterized by low P-wave velocities, are well imaged by the velocity model. Deep lakes with unfrozen water, inferred from the tomographic velocity model, correlate with areas of strong amplitude blanking and frequency attenuation observed in processed reflection seismic stack sections.

A simple smoothed velocity model of the uppermost Earth’s crust derived from joint inversion of Pg and Sg waves

The paper presents some results of seismic experiments carried out on the territory of northern Moravia and Silesia, roughly delimited by the coordinates 16°E–19°E and 49°N–51°N. The experiments were aimed at compiling a velocity model of the uppermost Earth’s crust using the database of arrival times of Pg and Sg waves recorded at a fairly large number of seismic stations, which enabled us to produce a simple 1D-layered velocity model of the region. The velocity model was computed using the traditional tomographic iterative process composed of consecutive solutions of linear equations. Based on the analysis of velocity distribution, it was found that the velocities of Pg and Sg waves increase from about 5.9 and 3.3 km/s at the surface, to about 6.1 and 3.5 km/s at a depth of 11 km, respectively.

A comparison between Laplace domain and frequency domain methods for inverting seismic waveforms

We compared the ability of full waveform inversion to recover background velocity models from data containing no low-frequency information using the frequency and Laplace domains. Low-frequency information is crucial for recovering background velocity when using frequency-domain waveform inversions. However, the dearth of low-frequency information in field data makes frequency-domain inversion impractical without accurate starting velocity models. Instead, by performing waveform inversion in the Laplace domain, one can recover a smooth velocity model that can be used for either migration or for subsequent frequency-domain inversion as an accurate initial velocity model. The Laplace-transformed wavefield can be thought of as the zero-frequency component of a damped wavefield over a range of damping constants. In this paper, we compare results obtained from both frequency- and Laplace-domain inversions and confirm that the Laplace-domain inversion can be used to recover background velocity from real data without low-frequency information. We also demonstrate that the Laplace-domain inversion can provide the frequency-domain inversion with smooth initial velocity models for better inversion results.

Two-dimensional Laplace-domain waveform inversion using adaptive meshes: an experience of the 2004 BP velocity-analysis benchmark data set

SUMMARY To delineate the Earth's structure precisely by time- or frequency-domain waveform inversions, an appropriate initial model that contains long-wavelength structures of the true model is essential. A Laplace-domain waveform inversion, which can provide a smooth velocity model that is equivalent to a long-wavelength velocity model from scratch, was recently proposed. The Laplace-domain waveform inversion can be used for obtaining a good initial model for full-waveform inversions. To improve the applicability of the 2-D Laplace-domain waveform inversion to field data, we adopted the adaptive finite element not only to deal with sources and receivers at shallow depths but also to reduce the computational burden using larger elements below the sources and receivers. For the boundary condition on the bounded domain for the Laplace-domain modelling and inversion, the unique property of the Laplace-domain wavefield, which was damped rapidly from a source, was taken into account. The computational domain was extended further from a source and then the Dirichlet boundary condition was applied to the outer edges of the extended model. With these two adaptations, our Laplace-domain waveform inversion could then be tested on a very large 2004 BP velocity-analysis benchmark data set. Numerical tests of the BP model, which contains three salt domes of high-velocity contrast, were conducted on a self-made synthetic data set by Laplace-domain modelling and a time-domain original data set. For the two data sets, a Laplace-domain waveform inversion using the adaptive mesh successfully provided a smooth velocity model that contained long-wavelength structures of the true velocity model. The smooth velocity model obtained by the Laplace-domain waveform inversion was used as a good initial model for the frequency-domain waveform inversion of the original BP data set. The resulting frequency-domain inversion recovered almost every feature of the salt domes except for some subsalt structures.

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

ABSTRACTElastic 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.

Testing robust inversion strategies for three-dimensional Moho topography based on CELEBRATION 2000 data

SUMMARY In this paper, we present results of the 3-D tomographic modelling of the crustal structure and Moho topography applied to data recorded in SE Poland during the CELEBRATION 2000 seismic experiment. The target area covers ca. 500 km × 500 km and represents a complex geological setting from old Precambrian platform (East European Craton, EEC), through the crustal blocks (terranes) that form the Trans-European Suture Zone, to the young Alpine orogen—the Carpathians. We test two different inversion strategies using two different algorithms: (i) coupled inversion of Pg and PmP arrivals to constrain both the crustal velocities and the Moho depths; (ii) decoupled inversion of PmP arrivals only using previously obtained smooth 3-D crustal velocity model. The coupled inversion of 11 700 Pg and 3100 PmP arrivals results in a much smoother crustal velocity field than the one previously obtained by inversion of first arrivals only. Also, the obtained Moho structure is much smoother than the Moho map compiled from the existing 2-D models. Decoupled inversion of the PmP reflections provides Moho structure comparable in resolution to the compiled map. Synthetic tests indicate that with our data set we are able to resolve larger than 100-km-size Moho structures. The modelled Moho is shallowest in the area of the Upper Silesian Block (ca. 32 km), then it is deepening by ca. 10 km over 100-km-wide zone along the margin of the EEC and finally it reaches up to 48 km depth in the area of the EEC. Our favoured strategy for modelling large refraction/wide-angle reflection data set consists of derivation of a smooth crustal velocity model by a first-arrival tomography and then complementing this model by a decoupled inversion of PmP reflections in order to constrain minimum-structure Moho topography.

A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains

In the frequency domain, gradient-based local-optimization methods of waveform inversions have been unsuccessful at inverting subsurface parameters without an accurate starting model. Such methods could not correct automatically for poor starting models because multiple local minima made it difficult to approach the true global minimum. In this study, we compared the behavior of objective functions in the frequency and Laplace domains. Wavefields in the Laplace domain correspond to the zero-frequency component of a damped wavefield; thus, the Laplace-domain waveform inversion can image smooth velocity models. Objective functions in the Laplace-domain inversion have a smoother surface and fewer local minima than in the frequency-domain inversion. We applied the waveform inversion to a 2D slice of the acoustic SEG/EAGE salt model in the Laplace domain and recovered smooth velocity models from inaccurate initial velocity conditions. We also successfully imaged velocities of the salt, SEG overthrust, and Institut Francais du Petrole Marmousi models with the frequency-domain inversion method by using the inverted velocity model of the Laplace-domain inversion as the initial model.

Regularization in slope tomography

Seismic imaging in depth is limited by the accuracy of velocity model estimation. Slope tomography uses the slowness components and traveltimes of picked reflection or diffraction events for velocity model building. The unavoidable data incompleteness requires additional information to assure stability to inversion. One natural constraint for ray-based tomography is a smooth velocity model. We propose a new, reflection-angle-based kind of smoothness constraint as regularization in slope tomography and have compared its effects to three other, more conventional constraints. The effects of these constraints were evaluated through angle-domain common-image gathers, computed with wave-equation migration using the estimated velocity model. We found that the smoothness constraints have a distinct effect on the velocity model but a weaker effect on the migrated data. In numerical tests on synthetic data, the new constraint leads to geologically more consistent models.

First-arrival traveltimes inversion based on a minimal number of parameters in shallow cross-well GPR tomography

Ground-penetrating radar (GPR) is a powerful tool to characterize the subsurface electromagnetic properties, and therefore related quantities such as the water-content. In this paper, we present a new algorithm for cross-well tomographic inversion of GPR first-arrival traveltimes. It is based on a blocky description of the propagating medium, and only a limited number of parameters are inverted. This methodology has been developed for media with significant contrast of electrical permittivity, for instance near-surface sedimentary media, where the occurrence of head waves is frequent. In a first step, the distribution of the electromagnetic wave velocity is determined between two wells. In a second step, we address the question of the layers continuity for several successive tomographic panels. This methodology allows us to obtain a 2D velocity profile of the medium between successive pairs of wells. The inversion scheme takes into account both the direct and refracted waves generated at interfaces with strong velocity contrast. It was tested on synthetic data for a single pair of wells, and on real data, first for one tomographic panel, then for a set of three successive tomographic panels. For comparison, inversion was also conducted with a classical LSQR algorithm, which delivers a smooth velocity model, and does not take into account head waves. As expected, the velocities obtained with our method are more contrasted, and free from the head waves generated artefacts.

Viscoelastic true-amplitude prestack reverse-time depth migration

A new true-amplitude prestack elastic depth-migration algorithm includes compensation for transmission and anelastic attenuation losses in an isotropic medium. Geometric spreading and its compensation are incorporated by extrapolating up- and downgoing waves using a full two-way wave equation. Intrinsic attenuation is simulated and compensated for using composite memory variables derived from standard linear solid relaxation mechanisms. Zoeppritz equations and their approximations are used to compute and analyze the angle-dependent reflection/transmission coefficients; converted energy is included at each interface. Transmission losses for compressional waves are compensated, based on estimation of angle-dependent elastic reflectivity using a two-pass recursive procedure. The image condition is the ratio of the compressional receiver/source wavefield amplitudes. Application to synthetic data from a dipping-layer model and a salt model accurately extracts P-velocity, S-velocity, density, and P-wave impedance beneath the target reflector, even under a salt overhang. Factors not explicitly considered include building of the smooth background velocity and attenuation models, estimates of the source time function, directivity and coupling, multipathing arrivals, and effects of attenuation and anisotropy on the reflection/transmission coefficients.

Waveform inversion in the Laplace domain

SUMMARY For the last 30 yr, since Tarantola’s pioneering theoretical study of waveform inversion, geophysicists and applied mathematicians have utilized a waveform inversion to delineate the earth’s structures. However, successful applications of waveform inversion to real data are nominal. The failures are mainly caused by the high non-linearity of the waveform inversion and the real data containing insufficient low-frequency components. We propose a waveform inversion algorithm that is robust and is not sensitive to the initial model by exploiting the wavefield in the Laplace domain and the adjoint property of the wave equation. The wavefield in the Laplace domain is equivalent to the zero frequency component of the damped wavefield. Therefore, the inversion of Poisson’s equation in electrical prospecting can be viewed as a waveform inversion problem, exploiting the zero frequency component of an undamped wavefield. Since our inversion algorithm in the Laplace domain is strongly associated with the nature of DC inversion for Poisson’s equation in electrical prospecting, our algorithm can generate a velocity model that is equivalent to a long-wavelength velocity model (a smooth velocity model). Through numerical tests, we found that the Laplace-transformed wavefields for optimally low Laplace damping constants are critically needed in our algorithm as the low-frequency components are necessary in the waveform inversion of the frequency domain. We note that the objective function by l2 norm of the logarithmic wavefield in the Laplace domain behaves as if it has no local minimum points in low and high Laplace damping constants. Moreover, we note that the forward modelling in the Laplace domain could be accurately calculated by using a coarser grid of several hundreds of metres than that of the frequency domain. Numerical tests of the salt dome model with high-velocity contrast demonstrate the robustness of our algorithm to both synthetic and real data. To apply our algorithm to real data more successfully, we need to improve the accuracy of the Laplace transformation for the wavefields in the time domain. However, our waveform inversion in the Laplace domain can be successfully applied to real data since our algorithm has more advantages in forward modelling and inversion than those in the frequency domain.

2-D traveltime and waveform inversion for improved seismic imaging: Naga Thrust and Fold Belt, India

SUMMARY Exploration along the Naga Thrust and Fold Belt in the Assam province of Northeast India encounters geological as well as logistic challenges. Drilling for hydrocarbons, traditionally guided by surface manifestations of the Naga thrust fault, faces additional challenges in the northeast where the thrust fault gradually deepens leaving subtle surface expressions. In such an area, multichannel 2-D seismic data were collected along a line perpendicular to the trend of the thrust belt. The data have a moderate signal-to-noise ratio and suffer from ground roll and other acquisition-related noise. In addition to data quality, the complex geology of the thrust belt limits the ability of conventional seismic processing to yield a reliable velocity model which in turn leads to poor subsurface image. In this paper, we demonstrate the application of traveltime and waveform inversion as supplements to conventional seismic imaging and interpretation processes. Both traveltime and waveform inversion utilize the first arrivals that are typically discarded during conventional seismic processing. As a first step, a smooth velocity model with long wavelength characteristics of the subsurface is estimated through inversion of the first-arrival traveltimes. This velocity model is then used to obtain a Kirchhoff pre-stack depthmigrated image which in turn is used for the interpretation of the fault. Waveform inversion is applied to the central part of the seismic line to a depth of ∼1 km where the quality of the migrated image is poor. Waveform inversion is performed in the frequency domain over a series of iterations, proceeding from low to high frequency (11‐19 Hz) using the velocity model from traveltime inversion as the starting model. In the end, the pre-stack depth-migrated image and the waveform inversion model are jointly interpreted. This study demonstrates that a combination of traveltime and waveform inversion with Kirchhoff pre-stack depth migration is a promising approach for the interpretation of geological structures in a thrust belt.

Seismic demigration/migration in the curvelet domain

Curvelets can represent local plane waves. They efficiently decompose seismic images and possibly imaging operators. We study how curvelets are distorted after demigration followed by migration in a different velocity model. We show that for small local velocity perturbations, the demigration/migration is reduced to a simple morphing of the initial curvelet. The derivation of the expected curvature of the curvelets shows that it is easier to sparsify the demigration/migration operator than the migration operator. An application on a 2D synthetic data set, generated in a smooth heterogeneous velocity model and with a complex reflectivity, demonstrates the usefulness of curvelets to predict what a migrated image would become in a locally different velocity model without the need for remigrating the full input data set. Curvelets are thus well suited to study the sensitivity of a prestack depth-migrated image with respect to the heterogeneous velocity model used for migration.

Practical Evaluation of P- and S-Wave Separation Via Elastic Wavefield Decomposition

The practical ability of elastic wavefield decomposition (EWD) to yield purer compressional (P) and shear (S) seismic images than conventional scalar component analysis of multi-component seismic data is examined. This vector-processing technique takes advantage of the P- and S-wave separation properties of the divergence and curl operators. Practical implementation of EWD requires information about the seismic wavefield at depth. This is achieved via downward continuation of the seismic data using a time-domain, finite-difference approach. Synthetic data experiments demonstrate that the robustness of EWD is dependent on the accuracy and smoothness of the velocity model used during the downward continuation stage of the algorithm. Velocity errors of up to 10% can be tolerated, after which significant artefacts appear in the separated records. A smooth velocity model will avoid contamination by spurious reflection events. P/S separation can still be effective where a constant-velocity model is used for data suffering from statics associated with lateral inhomogeneities in the near surface. Moderate noise contamination does not seem to significantly impact on the wavefield separation results. In fact, the downward continuation process appears to suppress random noise. Application of EWD to a real, two-component dataset from the Bowen Basin, Australia, appears to enhance the relative strength and coherency of the P- and S-wave reflection events in the extracted P and S records, respectively. Following application of EWD, the corresponding converted-wave stack exhibits better coherency and fault definition than the converted-wave stack produced from the raw horizontal-component data.

Scattering coefficients in the mantle revealed from the seismogram envelope analysis based on the multiple isotropic scattering model

At longer periods, scattered ScS waves sometimes dominate over coda waves at large lapse times. Examining recordings of seismic envelopes at 9 IRIS seismic stations of regional earthquakes with focal depths deeper than 150 km in periods from 1 to 20 s for a wide lapse time range up to 2000 s, we found significant frequency dependence. The coda decay gradient at short periods is steeper than that at longer periods; however, the change of coda gradient associated with the ScS arrival becomes distinct as the period becomes longer. In particular, a clear offset of coda amplitude appears in central Asia for 10 and 15 s period bands. The multiple isotropic scattering process of S-waves in the heterogeneous mantle can be simply simulated by using the Monte Carlo simulation method based on the radiative transfer theory in scattering media. Assuming a two-plane-layer attenuation structure and smoothed velocity model of the PREM, we estimated the average total scattering coefficients of Swaves such as 7.52 � 10 � 4 ~1.32 � 10 � 3 km � 1 and 2.08 � 10 � 4 ~6.23 � 10 � 4 km � 1 at 4 s, and 4.51 � 10 � 4 ~7.37 � 10 � 4 km � 1 and 2.80 � 10 � 5 ~2.71 � 10 � 4 km � 1 at 10 s, for the lithosphere and the upper mantle and for the lower mantle, respectively. Our results indicate that scattering occurs mostly in the lithosphere and the upper mantle and support that medium heterogeneity spreads over the whole mantle though its scattering power is small. Strong scattering occurs beneath central Asia and Papua New Guinea, whereas the scattering beneath Italy and regions of east Russia is much weaker. The numerical calculation enables us to confirm that much stronger scattering than intrinsic attenuation causes the offset behavior with coda decay gradient change after the ScS arrival for 4 and 10 s period bands in some regions. D 2005 Elsevier B.V. All rights reserved.

Refraction traveltime tomography using damped monochromatic wavefield

For complicated earth models, wave-equation–based refraction-traveltime tomography is more accurate than ray-based tomography but requires more computational effort. Most of the computational effort in traveltime tomography comes from computing traveltimes and their Fréchet derivatives, which for ray-based methods can be computed directly. However, in most wave-equation traveltime-tomography algorithms, the steepest descent direction of the objective function is computed by the backprojection algorithm, without computing a Fréchet derivative directly. We propose a new wave-based refraction-traveltime–tomography procedure that computes Fréchet derivatives directly and efficiently. Our method involves solving a damped-wave equation using a frequency-domain, finite-element modeling algorithm at a single frequency and invoking the reciprocity theorem. A damping factor, which is commonly used to suppress wraparound effects in frequency-domain modeling, plays the role of suppressing multievent wavefields. By limiting the wavefield to a single first arrival, we are able to extract the first-arrival traveltime from the phase term without applying a time window. Computing the partial derivative of the damped wave-equation solution using the reciprocity theorem enables us to compute the Fréchet derivative of amplitude, as well as that of traveltime, with respect to subsurface parameters. Using the Marmousi-2 model, we demonstrate numerically that refraction traveltime tomography with large-offset data can be used to provide the smooth initial velocity model necessary for prestack depth migration.

The common focal point and common reflection surface methodologies: subsets or complements?

ABSTRACTThe common focal point (CFP) method and the common reflection surface (CRS) stack method are compared. The CRS method is a fast, highly automated procedure that provides high S/N ratio simulation of zero‐offset (ZO) images by combining, per image point, the reflection energy of an arc segment that is tangential to the reflector. It uses smooth parametrized two‐way stacking operators, based on a data‐driven triplet of attributes in 2D (eight parameters in 3D). As a spin‐off, the attributes can be used for several applications, such as the determination of the geometrical spreading factor, multiple prediction, and tomographic inversion into a smooth background velocity model. The CFP method aims at decomposing two‐way seismic reflection data into two full‐aperture one‐way propagation operators. By applying an iterative updating procedure in a half‐migrated domain, it provides non‐smooth focusing operators for prestack imaging using only the energy from one focal point at the reflector. The data‐driven operators inhibit all propagation effects of the overburden. The CFP method provides several spin‐offs, amongst which is the CFP matrix related to one focal point, which displays the reflection amplitudes as measured at the surface for each source–receiver pair. The CFP matrix can be used to determine the specular reflection source–receiver pairs and the Fresnel zone at the surface for reflection in one single focal point. Other spin‐offs are the prediction of internal multiples, the determination of reflectivity effects, velocity‐independent redatuming and tomographic inversion to obtain a velocity–depth model. The CFP method is less fast and less automated than the CRS method. From a pointwise comparison of features it is concluded that one method is not a subset of the other, but that both methods can be regarded as being to some extent complementary.

Fast migration/inversion with multivalued ray fields: Part 2—Applications to the 3D SEG/EAGE salt model

Three‐dimensional prestack depth migration is the convenient approach for seismic imaging in the case of strong lateral variations of the velocity. Because of computing limitations, it has been limited to single‐arrival kinematic Kirchhoff migration until recently. This approach fails in the case of complex media characterized by multiarrival traveltimes. We present numerical strategies for extending in three dimensions first‐arrival kinematic Kirchhoff migration to multiarrival quantitative ray‐based migration (preserved amplitude migration). We rely on wavefront construction in a smooth velocity model to compute the multivalued traveltime and amplitude maps, and the CPU efficiency of migration itself is ensured by efficient and robust interpolation or extrapolation strategies. We present an application to the synthetic 3D SEG/EAGE salt model. Taking into account multiarrivals clearly improves subsalt imaging at the price of quite limited computing costs (a 20% increase in our case, with respect to a preserved‐amplitude single‐arrival migration).