P‐wave velocity distribution in basalt flows of the Enni Formation in the Faroe Islands from refraction seismic analysis

The main objective of this work is to establish the applicability of shallow surface‐seismic traveltime tomography in basalt‐covered areas. A densely sampled ～1300‐m long surface seismic profile, acquired as part of the SeiFaBa project in 2003 ( Japsen et al. 2006 ) at Glyvursnes in the Faroe Islands, served as the basis to evaluate the performance of the tomographic method in basalt‐covered areas. The profile is centred at a ～700‐m deep well. V_P, V_S and density logs, a zero‐offset VSP, downhole‐geophone recordings and geological mapping in the area provided good means of control. The inversion was performed with facilities of the Wide Angle Reflection/Refraction Profiling program package ( Ditmar et al. 1999 ). We tested many inversion sequences while varying the inversion parameters. Modelled traveltimes were verified by full‐waveform modelling. Typically an inversion sequence consists in several iterations that proceed until a satisfactory solution is reached. However, in the present case with high velocity contrasts in the subsurface we obtained the best result with two iterations: first obtaining a smooth starting model with small traveltime residuals by inverting with a high smoothing constraint and then inverting with the lowest possible smoothing constraint to allow the inversion to have the full benefit of the traveltime residuals. The tomogram gives usable velocity information for the near‐surface geology in the area but fails to reproduce the expected velocity distribution of the layered basalt flows. Based on the analysis of the tomogram and geological mapping in the area, a model was defined that correctly models first arrivals from both surface seismic data and downhole‐geophone data.     

Attenuation tomography: An application to gas-hydrate and free-gas detection

We estimate the quality factor (Q) from seismic reflections by using a tomographic inversion algorithm based on the frequency‐shift method. The algorithm is verified with a synthetic case and is applied to offshore data, acquired at western Svalbard, to detect the presence of bottom‐simulating reflectors (BSR) and gas hydrates. An array of 20 ocean‐bottom seismographs has been used. The combined use of traveltime and attenuation tomography provides a 3D velocity–Q cube, which can be used to map the spatial distribution of the gas‐hydrate concentration and free‐gas saturation. In general, high P‐wave velocity and quality factor indicate the presence of solid hydrates and low P‐wave velocity and quality factor correspond to free‐gas bearing sediments. The Q‐values vary between 200 and 25, with higher values (150–200) above the BSR and lower values below the BSR (25–40). These results seem to confirm that hydrates cement the grains, and attenuation decreases with increasing hydrate concentration.     

Modeling of Wide-Angle Seismic Attributes Using the Gaussian Beam Method

In this study, observed seismic attributes from shot gather 11 of the SAREX experiment are used to derive a preliminary velocity and attenuation model for the northern end of the profile in southern Alberta. Shot gather 11 was selected because of its prominent Pn arrivals and good signal to noise ratio. The 2-D Gaussian beam method was used to perform the modeling of the seismic attributes including travel times, peak envelope amplitudes and pulse instantaneous frequencies for selected phases. The preliminary model was obtained from the seismic attributes from shot gather 11 starting from prior tomographic results. The amplitudes and instantaneous frequencies were used to constrain the velocity and attenuation structure, with the amplitudes being more sensitive to the velocity gradients and the instantaneous frequencies more sensitive to the attenuation structure. The resulting velocity model has a velocity discontinuity between the upper and lower crust, and lower velocity gradients in the upper and lower crust compared to earlier studies. The attenuation model has Q _p ^-1 values between 0.011 and 0.004 in the upper crust, 0.0019 in the lower crust and a laterally variable Q _p ^-1 in the upper mantle. The Q _p ^-1 values are similar to those found in Archean terranes from other studies. Although the results from a single gather are non-unique, the initial model derived here provides a self-consistent starting point for a more complete seismic attribute inversion for the velocity and attenuation structure.     

Assessing uncertainty in refraction seismic traveltime inversion using a global inversion strategy

To analyse and invert refraction seismic travel time data, different approaches and techniques have been proposed. One common approach is to invert first‐break travel times employing local optimization approaches. However, these approaches result in a single velocity model, and it is difficult to assess the quality and to quantify uncertainties and non‐uniqueness of the found solution. To address these problems, we propose an inversion strategy relying on a global optimization approach known as particle swarm optimization. With this approach we generate an ensemble of acceptable velocity models, i.e., models explaining our data equally well. We test and evaluate our approach using synthetic seismic travel times and field data collected across a creeping hillslope in the Austrian Alps. Our synthetic study mimics a layered near‐surface environment, including a sharp velocity increase with depth and complex refractor topography. Analysing the generated ensemble of acceptable solutions using different statistical measures demonstrates that our inversion strategy is able to reconstruct the input velocity model, including reasonable, quantitative estimates of uncertainty. Our field data set is inverted, employing the same strategy, and we further compare our results with the velocity model obtained by a standard local optimization approach and the information from a nearby borehole. This comparison shows that both inversion strategies result in geologically reasonable models (in agreement with the borehole information). However, analysing the model variability of the ensemble generated using our global approach indicates that the result of the local optimization approach is part of this model ensemble. Our results show the benefit of employing a global inversion strategy to generate near‐surface velocity models from refraction seismic data sets, especially in cases where no detailed a priori information regarding subsurface structures and velocity variations is available.     

An efficient and self‐adaptive approach for Q value optimization

The time‐invariant gain‐limit‐constrained inverse Q‐filter can control the numerical instability of the inverse Q‐filter, but it often suppresses the high frequencies at later times and reduces the seismic resolution. To improve the seismic resolution and obtain high‐quality seismic data, we propose a self‐adaptive approach to optimize the Q value for the inverse Q‐filter amplitude compensation. The optimized Q value is self‐adaptive to the cutoff frequency of the effective frequency band for the seismic data, the gain limit of the inverse Q‐filter amplitude compensation, the inverse Q‐filter amplitude compensation function, and the medium quality factor. In the processing of the inverse Q‐filter amplitude compensation, the optimized Q value, corresponding gain limit, and amplitude compensation function are used simultaneously; then, the energy in the effective frequency band for the seismic data can be recovered, and the seismic resolution can be enhanced at all times. Furthermore, the small gain limit or time‐variant bandpass filter after the inverse Q‐filter amplitude compensation is considered to control the signal‐to‐noise ratio, and the time‐variant bandpass filter is based on the cutoff frequency of the effective frequency band for the seismic data. Synthetic and real data examples demonstrate that the self‐adaptive approach for Q value optimization is efficient, and the inverse Q‐filter amplitude compensation with the optimized Q value produces high‐resolution and low‐noise seismic data.     

Joint viscoacoustic waveform inversion with the separated upgoing and downgoing wavefields in zero-offset vertical seismic profile data

The attenuation of seismic waves propagating in reservoirs can be obtained accurately from the data analysis of vertical seismic profile in terms of the quality-factor Q. The common methods usually use the downgoing wavefields in vertical seismic profile data. However, the downgoing wavefields consist of more than 90% energy of the spectrum of the vertical seismic profile data, making it difficult to estimate the viscoacoustic parameters accurately. Thus, a joint viscoacoustic waveform inversion of velocity and quality-factor is proposed based on the multi-objective functions and analysis of the difference between the results inverted from the separated upgoing and downgoing wavefields. A simple separating step is accomplished by the reflectivity method to obtain the individual wavefields in vertical seismic profile data, and then a joint inversion is carried out to make full use of the information of the individual wavefields and improve the convergence of viscoacoustic full-waveform inversion. The sensitivity analysis of the different wavefields to the velocity and quality-factor shows that the upgoing and downgoing wavefields contribute differently to the viscoacoustic parameters. A numerical example validates our method can improve the accuracy of viscoacoustic parameters compared with the direct inversion using full wavefield and the separate inversion using upgoing or downgoing wavefield. The application on real field data indicates our method can recover a reliable viscoacoustic model, which helps reservoir appraisal.     

Tomographic inversion of near-surface <Emphasis Type="Italic">Q</Emphasis> factor by combining surface and cross-hole seismic surveys

The estimation of the quality factor Q plays a fundamental role in enhancing seismic resolution via absorption compensation in the near-surface layer. We present a new geometry that can be used to acquire field data by combining surface and cross-hole surveys to decrease the effect of geophone coupling on Q estimation. In this study, we drilled number of receiver holes around the source hole, each hole has different depth and each geophone is placed geophones into the bottom of each receiver hole to avoid the effect of geophone coupling with the borehole wall on Q estimation in conventional cross-hole seismic surveys. We also propose a novel tomographic inversion of the Q factor without the effect of the source signature, and examine its stability and reliability using synthetic data. We estimate the Q factors of the near-surface layer in two different frequency bands using field data acquired in the Dagang Oilfield. The results show that seismic absorption in the near-surface layer is much greater than that in the subsurface strata. Thus, it is of critical practical importance to enhance the seismic solution by compensating for near-surface absorption. In addition, we derive different Q factors from two frequency bands, which can be treated, to some extent, as evidence of a frequency-dependent Q.     

Elastic properties of seafloor sediments from the modelling of amplitudes of multiple water waves recorded on the seafloor off Bear Island,North Atlantic

Wide‐angle seismic data acquired by use of air‐guns and ocean bottom seismometers (OBS) contain strong direct water arrivals and multiples, generally considered as noise and thus not included in the modelling. However, a recent study showed that standard ray‐tracing modelling of the water multiples recorded off the Bear Island, North Atlantic, provided a reliable estimate of the velocity distribution in the water layer. Here, we demonstrate that including the amplitudes in the modelling provide valuable information about the V_P contrast at the seafloor, as well as the V_P/V_S ratio and attenuation (Q_P) of the uppermost sediments. The V_P contrast at the seafloor is estimated at about 250 m/s, within a precision of approximately ±30 m/s. The V_P/V_S ratio in the uppermost sedimentary layer is modelled in the range 2.25–2.50 and the Q_P factor is estimated at 1000 for the water, 30–50 for the uppermost layer and 40–50 for the second sedimentary layer. The values obtained for the sediments suggest a lithology dominated by silty clays, with porosity below average.     

Seismic dynamic monitoring in CO<Subscript>2</Subscript> flooding based on characterization of frequency-dependent velocity factor

The phase velocity of seismic waves varies with the propagation frequency, and thus frequency-dependent phenomena appear when CO₂ gas is injected into a reservoir. By dynamically considering these phenomena with reservoir conditions it is thus feasible to extract the frequency-dependent velocity factor with the aim of monitoring changes in the reservoir both before and after CO₂ injection. In the paper, we derive a quantitative expression for the frequency-dependent factor based on the Robinson seismic convolution model. In addition, an inversion equation with a frequency-dependent velocity factor is constructed, and a procedure is implemented using the following four processing steps: decomposition of the spectrum by generalized S transform, wavelet extraction of cross-well seismic traces, spectrum equalization processing, and an extraction method for frequency-dependent velocity factor based on the damped least-square algorithm. An attenuation layered model is then established based on changes in the Q value of the viscoelastic medium, and spectra of migration profiles from forward modeling are obtained and analyzed. Frequency-dependent factors are extracted and compared, and the effectiveness of the method is then verified using a synthetic data. The frequency-dependent velocity factor is finally applied to target processing and oil displacement monitoring based on real seismic data obtained before and after CO₂ injection in the G89 well block within Shengli oilfield. Profiles and slices of the frequency-dependent factor determine its ability to indicate differences in CO₂ flooding, and the predicting results are highly consistent with those of practical investigations within the well block.     

Borehole study of compressional and shear attenuation of basalt flows penetrated by the Brugdan and William wells on the Faroes shelf

We investigated the seismic attenuation of compressional (P‐) and converted shear (S‐) waves through stacked basalt flows using short‐offset vertical seismic profile (VSP) recordings from the Brugdan (6104/21–1) and William (6005/13–1A) wells in the Faroe‐Shetland Trough. The seismic quality factors (Q) were evaluated with the classical spectral ratio method and a root‐mean‐square time‐domain amplitude technique. We found the latter method showed more robust results when analysing signals within the basalt sequence. For the Brugdan well we calculated effective Q estimates of 22–26 and 13–17 for P‐ and S‐waves, respectively, and 25–33 for P‐waves in the William well. An effective Q_S/Q_P ratio of 0.50–0.77 was found from a depth interval in the basalt flow sequence where we expect fully saturated rocks. P‐wave quality factor estimates are consistent with results from other VSP experiments in the North Atlantic Margin, while the S‐wave quality factor is one of the first estimates from a stacked basalt formation using VSP data. Synthetic modelling demonstrates that seismic attenuation for P‐ and S‐waves in the stacked basalt flow sequence is mainly caused by one‐dimensional scattering, while intrinsic absorption is small.     

INVERSION METHOD IN THE SLANT STACK DOMAIN USING AMPLITUDES OF REFLECTION ARRIVALS*

Almost all ray-tracing methods ignore the analysis of the amplitudes of seismic arrivals and therefore utilize only half of the available information. We propose a method which is a combination of ray-tracing imaging and transformation of the amplitudes of wide-aperture data. Seismic data in the conventional X-T domain are first transformed to the domain of intercept time τ and ray parameter p to recover the plane wave response. The next step is the derivation of a series of plane wave reflection coefficients, which are mapped as a function of τ and p. The reflection coefficients R(τ, p) for two arbitrarily chosen traces can then be used in our inversion method to derive a slowness-depth and a density-depth profile. It is shown that the inclusion of amplitudes of seismic arrivals (in this method, we consider the acoustic case) makes the inverse method highly stable and accurate. In a horizontally stratified medium one can recover separate profiles of velocity and density. Since this method utilizes large-offset data, it can be used for separate recovery of velocity and density to a greater depth.     

An attenuation model for distant earthquakes

Large magnitude earthquakes generated at source–site distances exceeding 100km are typified by low‐frequency (long‐period) seismic waves. Such induced ground shaking can be disproportionately destructive due to its high displacement, and possibly high velocity, shaking characteristics. Distant earthquakes represent a potentially significant safety hazard in certain low and moderate seismic regions where seismic activity is governed by major distant sources as opposed to nearby (regional) background sources. Examples are parts of the Indian sub‐continent, Eastern China and Indo‐China. The majority of ground motion attenuation relationships currently available for applications in active seismic regions may not be suitable for handling long‐distance attenuation, since the significance of distant earthquakes is mainly confined to certain low to moderate seismicity regions. Thus, the effects of distant earthquakes are often not accurately represented by conventional empirical models which were typically developed from curve‐fitting earthquake strong‐motion data from active seismic regions. Numerous well‐known existing attenuation relationships are evaluated in this paper, to highlight their limitations in long‐distance applications. In contrast, basic seismological parameters such as the Quality factor (Q‐factor) could provide a far more accurate representation for the distant attenuation behaviour of a region, but such information is seldom used by engineers in any direct manner. The aim of this paper is to develop a set of relationships that provide a convenient link between the seismological Q‐factor (amongst other factors) and response spectrum attenuation. The use of Q as an input parameter to the proposed model enables valuable local seismological information to be incorporated directly into response spectrum predictions. The application of this new modelling approach is demonstrated by examples based on the Chi‐Chi earthquake (Taiwan and South China), Gujarat earthquake (Northwest India), Nisqually earthquake (region surrounding Seattle) and Sumatran‐fault earthquake (recorded in Singapore). Field recordings have been obtained from these events for comparison with the proposed model. The accuracy of the stochastic simulations and the regression analysis have been confirmed by comparisons between the model calculations and the actual field observations. It is emphasized that obtaining representative estimates for Q for input into the model is equally important.Thus, this paper forms part of the long‐term objective of the authors to develop more effective communications across the engineering and seismological disciplines. Copyright © 2003 John Wiley & Sons, Ltd.     

Near‐field seismic effects in a homogeneous medium and their removal in vertical seismic profile attenuation estimates

To better understand (and correct for) the factors affecting the estimation of attenuation (Q), we simulate subsurface wave propagation with the Weyl/Sommerfeld integral. The complete spherical wavefield emanating from a P‐wave point source surrounded by a homogeneous, isotropic and attenuative medium is thus computed. In a resulting synthetic vertical seismic profile, we observe near‐field and far‐field responses and a 90° phase rotation between them. Depth dependence of the magnitude spectra in these two depth regions is distinctly different. The logarithm of the magnitude spectra shows a linear dependence on frequency in the far‐field but not in those depth regions where the near‐field becomes significant. Near‐field effects are one possible explanation for large positive and even negative Q‐factors in the shallow section that may be estimated from real vertical seismic profile data when applying the spectral ratio method. We outline a near‐field compensation technique that can reduce errors in the resultant Q estimates.     

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.     

Imaging 3-D crustal P-wave velocity structure of western Yunnan with bulletin data

Western Yunnan is a region with intensive tectonic activity and serious earthquake risk. It is of significant importance to study three dimensional crustal structure of this region to understand the tectonic setting and disaster mechanism. Densification and digitalization of seismic networks in this region provides an opportunity to study the velocity structure with bulletin data. In this study, we collect P-wave data of 10 403 regional earthquakes recorded by 79 seismic stations from January 2008 to December 2010. In addition to first arrivals data (Pg with epicentral distance less than 200 km and Pn), the Pg (or P) data with epicentral distance more than 200 km are also considered as later direct arrivals in the tomographic inversion. We also compare the quantity and the quality of the seismic data before 2010 and after 2010. The test results show that adding the follow-up Pg phase can effectively improve the inversion ability of crustal imaging, and quantity and the data quality are significantly improved since 2010. The tomographic results show that: (1) The Honghe fault zone, which is the major fault systems in this region, may cut through the entire crust, and the velocity contrasts between two sides at lower crust beneath the Honghe fault are estimated at higher than 10%, while the velocity difference below Nujiang fault zone extends only in the upper crust; (2) Most of the earthquakes in the region occurred at the interface of high-velocity media and low-velocity media, i.e., the areas with high velocity gradient, which has been validated in other areas.     

Three‐term inversion of prestack seismic data using a weighted l2, 1 mixed norm

We present a new inversion method to estimate, from prestack seismic data, blocky P‐ and S‐wave velocity and density images and the associated sparse reflectivity levels. The method uses the three‐term Aki and Richards approximation to linearise the seismic inversion problem. To this end, we adopt a weighted mixed l_{2, 1}‐norm that promotes structured forms of sparsity, thus leading to blocky solutions in time. In addition, our algorithm incorporates a covariance or scale matrix to simultaneously constrain P‐ and S‐wave velocities and density. This a priori information is obtained by nearby well‐log data. We also include a term containing a low‐frequency background model. The l_{2, 1} mixed norm leads to a convex objective function that can be minimised using proximal algorithms. In particular, we use the fast iterative shrinkage‐thresholding algorithm. A key advantage of this algorithm is that it only requires matrix–vector multiplications and no direct matrix inversion. The latter makes our algorithm numerically stable, easy to apply, and economical in terms of computational cost. Tests on synthetic and field data show that the proposed method, contrarily to conventional l₂‐ or l₁‐norm regularised solutions, is able to provide consistent blocky and/or sparse estimators of P‐ and S‐wave velocities and density from a noisy and limited number of observations.     

Offset‐variable density improves acoustic full‐waveform inversion: a shallow marine case study

We have previously applied three‐dimensional acoustic, anisotropic, full‐waveform inversion to a shallow‐water, wide‐angle, ocean‐bottom‐cable dataset to obtain a high‐resolution velocity model. This velocity model produced an improved match between synthetic and field data, better flattening of common‐image gathers, a closer fit to well logs, and an improvement in the pre‐stack depth‐migrated image. Nevertheless, close examination reveals that there is a systematic mismatch between the observed and predicted data from this full‐waveform inversion model, with the predicted data being consistently delayed in time. We demonstrate that this mismatch cannot be produced by systematic errors in the starting model, by errors in the assumed source wavelet, by incomplete convergence, or by the use of an insufficiently fine finite‐difference mesh. Throughout these tests, the mismatch is remarkably robust with the significant exception that we do not see an analogous mismatch when inverting synthetic acoustic data. We suspect therefore that the mismatch arises because of inadequacies in the physics that are used during inversion. For ocean‐bottom‐cable data in shallow water at low frequency, apparent observed arrival times, in wide‐angle turning‐ray data, result from the characteristics of the detailed interference pattern between primary refractions, surface ghosts, and a large suite of wide‐angle multiple reflected and/or multiple refracted arrivals. In these circumstances, the dynamics of individual arrivals can strongly influence the apparent arrival times of the resultant compound waveforms. In acoustic full‐waveform inversion, we do not normally know the density of the seabed, and we do not properly account for finite shear velocity, finite attenuation, and fine‐scale anisotropy variation, all of which can influence the relative amplitudes of different interfering arrivals, which in their turn influence the apparent kinematics. Here, we demonstrate that the introduction of a non‐physical offset‐variable water density during acoustic full‐waveform inversion of this ocean‐bottom‐cable field dataset can compensate efficiently and heuristically for these inaccuracies. This approach improves the travel‐time match and consequently increases both the accuracy and resolution of the final velocity model that is obtained using purely acoustic full‐waveform inversion at minimal additional cost.     

Feasibility study for an anisotropic full waveform inversion of cross‐well seismic data

Anisotropy is often observed due to the thin layering or aligned micro‐structures, like small fractures. At the scale of cross‐well tomography, the anisotropic effects cannot be neglected. In this paper, we propose a method of full‐wave inversion for transversely isotropic media and we test its robustness against structured noisy data. Optimization inversion techniques based on a least‐square formalism are used. In this framework, analytical expressions of the misfit function gradient, based on the adjoint technique in the time domain, allow one to solve the inverse problem with a high number of parameters and for a completely heterogeneous medium. The wave propagation equation for transversely isotropic media with vertical symmetry axis is solved using the finite difference method on the cylindrical system of coordinates. This system allows one to model the 3D propagation in a 2D medium with a revolution symmetry. In case of approximately horizontal layering, this approximation is sufficient. The full‐wave inversion method is applied to a crosswell synthetic 2‐component (radial and vertical) dataset generated using a 2D model with three different anisotropic regions. Complex noise has been added to these synthetic observed data. This noise is Gaussian and has the same amplitude f?k spectrum as the data. Part of the noise is localized as a coda of arrivals, the other part is not localized. Five parameter fields are estimated, (vertical) P‐wave velocity, (vertical) S‐wave velocity, volumetric mass and the Thomsen anisotropic parameters epsilon and delta. Horizontal exponential correlations have been used. The results show that the full‐wave inversion of cross‐well data is relatively robust for high‐level noise even for second‐order parameters such as Thomsen epsilon and delta anisotropic parameters.     

A nonlinear method for multiparameter inversion of pre‐stack seismic data based on anisotropic Markov random field

Multiparameter inversion for pre‐stack seismic data plays a significant role in quantitative estimation of subsurface petrophysical properties. However, it remains a complicated problem due to the non‐unique results and unstable nature of the processing; the pre‐stack seismic inversion problem is ill‐posed and band‐limited. Combining the full Zoeppritz equation and additional assumptions with edge‐preserving regularisation can help to alleviate these problems. To achieve this, we developed an inversion method by constructing a new objective function that includes edge‐preserving regularisation and soft constraints based on anisotropic Markov random fields and is intended especially for layered formations. We applied a fast simulated annealing algorithm to solve the nonlinear optimisation problem. The method directly obtains reflectivity R_PP values using the full Zoeppritz equation instead of its approximations and effectively controls the stability of the multiparameter inversion by assuming a sectionally constant S‐ and P‐wave velocity ratio and using the generalised Gardner equation. We substituted the inverted parameters, i.e., the P‐wave velocity, the fitting deviation of S‐wave velocity, and the density were inverted instead of the P‐wave velocity, the S‐wave velocity, and the density, and the generalised Gardner equation was applied as a constraint. Test results on two‐dimensional synthetic data indicated that our substitution obtained improved results for multiparameter inversion. The inverted results could be improved by utilising high‐order anisotropic Markov random field neighbourhoods at early stages and low‐order anisotropic Markov random field neighbourhoods in the later stages. Moreover, for layered formations, using a large horizontal weighting coefficient can preserve the lateral continuity of layers, and using a small vertical weighting coefficient allows for large longitudinal gradients of the interlayers. The inverted results of the field data revealed more detailed information about the layers and matched the logging curves at the wells acceptably over most parts of the curves.     

Double parameterized regularization inversion method for migration velocity analysis in transversely isotropic media with a vertical symmetry axis

Simultaneous estimation of velocity gradients and anisotropic parameters from seismic reflection data is one of the main challenges in transversely isotropic media with a vertical symmetry axis migration velocity analysis. In migration velocity analysis, we usually construct the objective function using the l₂ norm along with a linear conjugate gradient scheme to solve the inversion problem. Nevertheless, for seismic data this inversion scheme is not stable and may not converge in finite time. In order to ensure the uniform convergence of parameter inversion and improve the efficiency of migration velocity analysis, this paper develops a double parameterized regularization model and gives the corresponding algorithms. The model is based on the combination of the l₂ norm and the non‐smooth l₁ norm. For solving such an inversion problem, the quasi‐Newton method is utilized to make the iterative process stable, which can ensure the positive definiteness of the Hessian matrix. Numerical simulation indicates that this method allows fast convergence to the true model and simultaneously generates inversion results with a higher accuracy. Therefore, our proposed method is very promising for practical migration velocity analysis in anisotropic media.