Full Waveform Inversion Algorithm Research Articles

Real-Time Dual-Parameter Full-Waveform Inversion of GPR Data Based on Robust Deep Learning

Summary Ground penetrating radar (GPR) is becoming an increasingly important tool for understanding the shallow electrical structures of the earth and planets due to its adaptability to harsh detection environments, efficient data acquisition and accurate detection results. GPR full-waveform can simultaneously constrain the permittivity and resistivity of the medium, providing more comprehensive geophysical information and reducing the non-uniqueness of inversion. However, given the highly non-linear inverse problem and the massive data resulted from high temporal and spatial samplings, traditional full-waveform inversion algorithms are prohibitively costly. Inspired by Google's vision semantic segmentation system, we develop a robust deep learning-guided network that integrates geology and geophysics knowledge to support the real-time translation of zero-offset GPR data into dual-parameter electrical structures. We test our proposed network using synthetic data, which demonstrates that the algorithm can provide an accurate dual-parameter electrical model from a GPR sounding in milliseconds on a common laptop PC, exhibiting high robustness and adaptability to noise interference and extreme values of model parameters. We also apply our network to field data gathered for pollutant investigation in the US. The resulting dual-parameter structure provides a more comprehensive and realistic depiction of subsurface electrical properties and reveals the migration and aging of pollutants. Our algorithm's real-time and accurate advantages are expected to further unleash the potential of GPR technology and enable it to play a more significant role in earth and planetary exploration.

A time-domain SGFD-FK hybrid method for 2D teleseismic elastic wave modeling and inversion

AbstractFull waveform inversion (FWI) has proved to be a reliable tool for high-resolution imaging of lithospheric structures at various depths down to the upper mantle. However, when the size of the model is large, the computational burden is significative and applications are restricted to low frequencies. To tackle this issue, we developed a new 2D time-domain hybrid method to simulate high-frequency teleseismic body waves propagating through a local heterogeneous elastic Earth model: The frequency-wavenumber (FK) integration method is coupled with the staggered grid finite difference method (SGFD). The FK method is used to compute the wavefield due to obliquely incident plane P and SV waves in a 1D multilayered half space that excites a local heterogeneous region. Inside this region, the velocity-stress staggered grid FD method (SGFD) is used to accurately deal with wave propagation in heterogeneous media. Spurious waves that might be generated at the boundaries of the local region are avoided using convolutional perfectly matched layers (CPML). This new hybrid method inherits the low-memory requirements of the FK method and the accuracy, efficiency and easy implementation of the SGFD. The new hybrid method is benchmarked against the analytical FK method for some canonical models and shows good agreement with analytical solutions. Subsequently, our modeling tool is incorporated into a full waveform inversion algorithm adapted for teleseismic configurations to invert the incident P wave and its coda. The inversion is carried out using a gradient approach that is efficiently implemented via the adjoint-state method. The results suggest that our hybrid method and FWI algorithm represent a valuable tool for 2D forward and inverse regional applications using teleseismic data sets.

Joint Data and Model-Driven Simultaneous Inversion of Velocity and Density

Summary Density is an important parameter for both geological research and geophysical exploration. However, for model-driven seismic inversion methods, high-fidelity density inversion is challenging due to seismic wave travel-time insensitivity to density, and crosstalk that density has with velocity. To circumvent the challenge of density inversion, some inversion methods treat density as a constant value or derive density from velocity through empirical equation. On the other hand, deep learning approaches are completely driven by data and have strong target-oriented characteristics, offering a new way to solve multi-parameter coupling problems. Nevertheless, the accuracy of the inversion results of data-driven algorithms is directly related to the amount and diversity of the training data, and thus, they lack the universality of model-driven algorithms. To achieve accurate density inversion, we propose a simultaneous inversion algorithm for velocity and density that combines the advantages of data- and model- driven approaches: A neural network model (U-T), combining the U-net and Transformer architectures, is proposed to construct nonlinear mappings between seismic data as inputs and the velocity and density as predictions. Next, the model-driven inversion algorithm uses the U-T prediction as the initial model to obtain the final accurate solution. In the model-driven module, envelope-based sparse constrained deconvolution is used to obtain full-band seismic data, while a variable dominant frequency full waveform inversion algorithm is employed to perform multi-scale inversion, ultimately leading to accurate inversion results of velocity and density. The performance of the algorithm on the Sigsbee2A and Marmousi models demonstrates its effectiveness.

Coherence-enhancing anisotropic diffusion filter for 3D high-resolution reconstruction of P-wave velocity and density using full-waveform inversion: Application to a North Sea ocean bottom cable data set

Regularization is a central topic in the study of the solutions of ill-posed inverse problems. High-resolution seismic imaging using full-waveform inversion (FWI) belongs to this category of problems. Regularization through anisotropic diffusion, a technique that emerged in the field of image processing, is an interesting alternative to conventional regularization strategies. Exploiting the structural information of a given image, it has the capability to smooth this image along its main structures. The main difficulty is how to design the anisotropic diffusion operator. The concept of coherence enhancing proposed in 2D is extended in 3D and applied so as to filter and enhance the structural coherence of the model updates within an FWI algorithm. The benefits of this strategy are investigated on a 2D synthetic experiment before considering the multiparameter inversion of a 3D field data set from the North Sea up to 10 Hz. From this data, the vertical velocity and the density are simultaneously reconstructed. Compared with a conventional nonstationary Gaussian regularization strategy, the models obtained using the coherence-enhancing anisotropic diffusion strategy indicate an enhanced resolution, especially for the density model. The high-resolution reflectivity image computed from the impedance volume clearly illustrates the benefit this filtering approach can deliver in terms of structural interpretation.

High-Precision Corrosion Detection via SH1 Guided Wave Based on Full Waveform Inversion.

Corrosion detection for industrial settings is crucial for safe and efficient operations. Due to its high imaging resolution, the guided-wave full-waveform inversion tomography technique has significant potential for corrosion detection of plate metals. Limited by the long wavelengths of A0 and S0 mode waves, this method exhibits inadequate detection resolution for the earlier shallow and small corrosion defects. Based on the relatively short wavelength characteristics of the SH1 mode wave, we propose a high-precision corrosion detection method via SH1 guided wave using the full waveform inversion algorithms. By conducting finite element simulations of ultrasonic-guided waves on aluminum plates with varying corrosion defects, a comparison was made to assess the detection precision across A0, S0, and SH1 modes. The comparison results showed that, whether for regular or irregular defects, the SH1 mode wave always exhibited higher imaging accuracy than the A0 and S0 mode waves for shallow and small-sized defects. The corresponding experiments were conducted on an aluminum plate with simple or complex defects. The results of the experiments reconfirmed that the full waveform inversion method using SH1 guided wave can effectively reconstruct the shape and size of small and shallow corrosion defects within aluminum plates.

Field assessment of elastic full-waveform inversion of combined accelerometer and distributed acoustic sensing data in a vertical seismic profile configuration

Seismic data are a significant facilitator for monitoring in carbon capture and sequestration projects, providing high-resolution images of fluid migration, using, for example, full-waveform inversion (FWI). Distributed acoustic sensing (DAS), a relatively novel technology for wavefield sampling, is well suited for this type of monitoring. Using noninvasive optical fibers, DAS allows for dense spatial sampling along the entire length of the wellbore, without disrupting operations. Permanently installed in the wellbore, typically behind casing, DAS offers highly repeatable and dense sampling of the transmitted wave modes crucial to seismic monitoring of injected carbon dioxide ([Formula: see text]). However, the DAS data consist of measurements of strain along the tangent of the fiber and therefore do not transfer directly to conventional FWI algorithms. Incorporation of DAS data in their native strain (or strain-rate) form in standard FWI algorithms, requires changing the definition of the receiver sampling operator to use geometric information about the fiber to supply tangential strain measurements to the FWI residual. The theoretical developments are applied to invert field vertical seismic profile data acquired with DAS fiber and accelerometers at a [Formula: see text] sequestration site in Newell Country, Alberta. Our method incorporates DAS data and accelerometer data in one objective function and allows us to tune the relative importance we wish to place on each data set. This method also transfers to noncollocated sensors, for example, surface-deployed geophones and borehole fiber. The inverted models contain features expected from the geology of the field site, and data modeled in the inverted models compare favorably with the field data for these sensor types. The models are derived from data acquired prior to [Formula: see text] injection, representing baseline models for future time-lapse studies planned at the field research station.

Efficient source wavefield reconstruction for full-waveform inversion

Full-waveform inversion (FWI) is a useful tool for estimating the properties of subsurface media, but it has the disadvantage of incurring considerable computing costs. Therefore, we develop a method for reducing the computing memory requirement in the time domain of FWI that uses a new approach with the discrete cosine transform (DCT) and the Nyquist sampling theorem. Our method enables accurate reconstruction of a wavefield with several optimal DCT coefficients derived by considering the maximum wavenumber component of the virtual source wavefield and the maximum frequency of the source wavelet. Consequently, virtual source wavefields can be stored in a computer’s memory rather than on a disk. Our FWI algorithm is compared with existing compression methods that use DCT and wavelet transform in a numerical example. The results of this comparison demonstrate that our algorithm provides significantly improved computational efficiency.

Simultaneous waveform inversion of seismic-while-drilling data for P-wave velocity, density, and source parameters

Full-waveform inversion (FWI), as an optimization-based approach to estimating subsurface models, is limited by incomplete acquisition and illumination of the subsurface. The incorporation of additional data from new and independent raypaths should be expected to result in significant increase in the accuracy of FWI models. In principle, seismic-while-drilling (SWD) technology can supply these additional raypaths; however, it introduces a new suite of unknowns, namely precise source locations (i.e., drilling path), source signature, and radiation characteristics. A new FWI algorithm is formulated in which the source radiation patterns and positions join the velocity and density values of the grid cells as unknowns to be determined. Several numerical inversion experiments are then conducted with different source settings using a synthetic model. The SWD sources are supplemented by explosive sources and multicomponent receivers at the surface, simulating a conventional surface acquisition geometry. The subsurface model and SWD source properties are recovered and analyzed. The analysis is suggestive that SWD involvement can enhance the accuracy of FWI models, with varying degrees of enhancement depending on factors such as trajectory inclination, source density, and drill path extension. The impact of SWD-FWI over standard FWI is reduced when low-frequency data are missing, but improvements over the models constructed with no subsurface sources remain. This formulation permits general source information, such as position and moment tensor components, to be independently obtained. This inversion scheme may lead to a range of potential applications for which medium properties and source information are required.

Forge ahead in acquisition and model building via full-waveform inversion: Gulf of Mexico case study

Full-waveform inversion (FWI) has proven itself an essential tool for velocity model building using seismic data. In recent years, the geophysical community has made substantial progress in developing FWI to overcome some long-standing limitations, such as handling cycle skipping, better using reflection energy, including more physics in the inversion algorithms, and increasing inverted frequencies to achieve higher resolution. FWI does not only target the larger and midscale model updates, more and more FWI examples were presented with increased resolution which allows for extraction of FWI derived reflectivity. Interrogating the FWI kernel with given geology and acquisition geometry can provide critical information on how the acquisition design should be optimized to provide FWI a better opportunity to update the velocity model at target depth, especially in the deep part of the model. When the acquisition geometry is different in terms of offsets, azimuthal coverage, and the minimum frequency in the recording, it can be analyzed to design workflows to enable FWI to optimally update the model parameters. Obviously, recent ocean-bottom node acquisitions that record long offsets, full azimuth, and low frequencies have made FWI shine in the seismic industry. Modifying and reshaping a complex salt geometry is one of the ultimate goals of FWI. In addition to the salt boundary being an issue, there is a potential cycle-skipping problem associated with uncertainties of large salt bodies missing or misplaced even though the frequencies used to start FWI are becoming lower due to the advancement in seismic acquisition and FWI algorithms. Furthermore, if the FWI-predicted data are simulated with an acoustic engine, it could pose amplitude and phase discrepancies at high-velocity contrast interfaces. Elastic FWI alone has been proposed as a means of overcoming these challenges associated with salt and regions with high-velocity contrast. We determine the FWI progress of the past few decades with the latest examples from different acquisition geometries.

Full Waveform Inversion for NDT using ultrasonic linear arrays

Ultrasonic (UT) imaging is a widespread technique for nondestructive testing (NDT). The state-of-the-art UT image reconstruction algorithms are based on delay-and-sum (DAS) operations, which assume constant acoustic velocity across the tested objects and also neglects nonlinear effects such as diffractions and multiple reflections. These assumptions limit the reconstruction capabilities of DAS-based algorithms, especially for complex objects composed by several materials. In seismology, Full Waveform Inversion (FWI) methods have been used to obtain subsurface properties based on scattered and transmitted sound waves - which is a very similar problem to ultrasonic imaging in NDT - using the full wave information, showing promising results. In this paper, we present the application of FWI in NDT using simulated data representing acquisitions with a common linear array transducer. We review the theoretical formulation of FWI and discuss some difficulties that arise when it is applied to NDT. Image reconstruction is performed using both a state-of-the-art DAS algorithm, namely the Total Focusing Method, and FWI. The implementation targets a GPU platform, using the CUDA API. This leverages the highly parallelizable nature of the simulations and the FWI algorithm. The obtained results show that FWI can show more internal structures than TFM, with less information about the specimen. To foster further development, all source codes are provided.

A WebGPU-based acoustic wave simulator for ultrasound NDT

Ultrasound NDT is a standard industrial technique for health monitoring and flaw identification in several structures, such as subsea oil pipelines. Its primary purpose is to form a representative image of the internal structure of the inspected object, which provides, in many cases, early warning of structure discontinuities, making it possible to reduce repair costs and mitigate risks. One of the imaging methods that has been receiving increasing attention in the ultrasound and seismology communities is the Full Waveform Inversion (FWI), which estimates the velocity model for the inspection area by confronting the acquired information obtained by the NDT system with theoretical simulated data. Since FWI algorithms rely on successive ultrasonic wave simulations, it is essential to ensure that each simulation is optimized and contributes little to the total running time. Acoustic wave simulations can be optimized using a graphical processing unit (GPU), which provides hardware acceleration through parallelism, where each point of the simulated area can be computed simultaneously. Among all available GPU APIs, WebGPU is the next-generation standard graphics Web API that exposes modern computer graphics capabilities by giving the user low-level, general-purpose access to the GPUs. This API is designed to efficiently map to native GPU APIs, which enables applications to run on different GPUs, making implementations more scalable and interchangeable. Motivated by this flexibility, we explore the acceleration capabilities of the WebGPU API by implementing a WebGPU-based acoustic wave simulator for ultrasound NDT.

Compensating attenuation effects in full-waveform inversion with dissipation-dispersion decoupling

Seismic attenuation poses challenges to velocity model building from acoustic/elastic full-waveform inversion (FWI). In particular, when constructing the FWI gradient for velocity inversion, recent studies have indicated that the high-attenuation structure could distort the gradient by damping the amplitudes and shifting the kinematic phases, resulting in an imbalanced update and thus unreliable velocity model. These Q effects are particularly significant in the reflection acquisition geometry due to the “double-damping” issue. Here, we develop a Q-compensated FWI algorithm for constructing a Q-free FWI gradient. By using a recently developed viscoacoustic wave propagator, this compensation can be done conveniently by keeping the dispersion term and flipping the dissipation term in the wave equation when we simulate the forward and time-reversed adjoint wavefields. The resultant gradient obtained by interacting these two wavefields has correct kinematics and Q-free amplitudes. This Q-compensated FWI can balance the update between Q- and no- Q-areas, which we determine using synthetic examples. In addition, we illustrate how to take advantage of the dissipation-dispersion decoupling to determine the anomalous Q value in the Q model building workflow via Q-compensated reverse time migration.

Ultrasound computed tomography based on full waveform inversion with source directivity calibration

Ultrasound computed tomography based on full waveform inversion has the potential to provide high-resolution images of human tissues in a quantitative manner. A successful ultrasound computed tomography system requires the decent knowledge of acquisition array, including the spatial position and the directivity of each transducer, to meet the high-level demand of clinical applications. The conventional full waveform inversion algorithm assumes a point source with the omni-directional emission. Such assumption does not hold when the directivity of emitting transducer is not negligible. For a practical implementation, an efficient and accurate self-checking evaluation of directivity is crucial prior to the reconstruction of images. We propose to measure the directivity of each emitting transducer using the full-matrix captured data obtained with a water-immersed and target-free experiment. We introduce the weighted virtual point-source array to act as the proxy of emitting transducer during the numerical simulation. The weights of different points in the virtual array can be calculated from the observed data using the gradient-based local optimization method. Although the full waveform imaging method relies on the finite-difference solver of wave equation, such directivity estimation benefits from the introduction of analytical solver. The trick significantly reduces the numerical cost, enabling an automatic directivity self-check at boot. We verify the feasibility, efficiency, and accuracy of the virtual array method through simulated and experimental tests. For the experimental test, we also illustrate that full waveform inversion with directivity calibration can reduce the artifacts introduced by the conventional point source assumption, improving the quality of reconstructed images..

Enhancing salt model resolution and subsalt imaging with elastic FWI

Full-waveform inversion (FWI) has become the centerpiece of velocity model building (VMB) in seismic data processing in recent years. It has proven capable of significantly improving the velocity model and, thus, the migration image for different acquisition types and geologic settings, including complex environments such as salt. With the advent of FWI imaging, the scope of FWI applications has extended further from VMB into the imaging landscape. However, current FWI applications in the industry prevalently employ the acoustic approximation. One common problem of acoustic FWI (A-FWI) is the apparent salt halos at the salt-sediment interface in the resulting FWI velocity and FWI image, the presence of which hinders direct interpretation and imaging focusing around salt bodies. With synthetic and field data examples, we demonstrate that this salt halo is caused mainly by the large mismatch between the elastic recorded data and the acoustic modeled data, particularly at middle to long offsets. To overcome limitations imposed by acoustic assumptions, we developed an elastic FWI (E-FWI) algorithm that combines an elastic modeling engine with the time-lag cost function, which we call elastic time-lag FWI (E-TLFWI). With a more accurate modeling engine, E-TLFWI significantly reduces the salt halo observed in its acoustic counterpart. However, we also observe that the images migrated using the A-FWI and E-FWI velocity models remain similar overall, with some slight improvements around and beneath salt boundaries, particularly near steep salt flanks, as a result of the reduced salt halo. By contrast, FWI images derived from E-TLFWI show considerable benefits over those from acoustic time-lag FWI, such as improved event focusing, better structural continuity, and higher signal-to-noise ratio. The sharpened salt boundaries and enhanced quality of the FWI images reveal the significant value of E-FWI and provide the justification for its greatly increased cost.

Application of full waveform inversion algorithm in Laplace–Fourier domain for high-contrast ultrasonic bone quantitative imaging

Background and objectivesFull waveform inversion (FWI) has been widely applied for the reconstruction of underground medium parameters in seismic communities and has made a great success. It is also a promising way to image hard tissues such as bones by ultrasonic FWI algorithm. However, the ultrasonic FWI methods for bone parameters imaging reported in literature so far are limited to the time domain and/or Fourier domain, and can only achieve quantitative imaging with acoustic velocity of bone less than 3000 m/s. Because the acoustic velocity of actual cortical bones can be as high as 4200 m/s, it is still a challenge for FWI to achieve higher parameter contrast bone imaging. MethodsHere, we proposed an ultrasonic FWI algorithm in Laplace–Fourier domain (LFDFWI) for high-contrast bone quantitative imaging. Compared to Time domain and Fourier domain, the LFDFWI algorithm is more appropriate for dealing with the presence of high contrast between bone tissues, reducing the possibility of inversion falling into a local minimum, and obtaining better inversion results. We adapted the seismic FWI algorithm to make it suitable for high-frequency ultrasonic sources and small-sized bone parameter imaging. ResultsWe conducted a series of bone models to evaluate the effectiveness of the proposed algorithm, including four kinds of bone model derived from micro computed tomography (Micro-CT) image of rat. We evaluated the experimental results based on visual analysis, error analysis and structural similarity (SSIM). The numerical simulation results showed that, when acoustic approximation is used, the proposed method can obtain accurate high-contrast images of the velocity and density parameters of bone structure, the mean relative error (MRE) in the region of interest (ROI) were all less than 2%, and the SSIM is up to 98%; when the viscoelastic approximation is used, this method can also obtain the desired high-contrast bone parameter distribution, with MRE less than 4% and SSIM higher than 74%, both of which are better than FDFWI in Fourier domain (FDFWI). ConclusionThe results demonstrated that the proposed FWI algorithm can obtain high resolution bone parameter models close to the Micro-CT image, which proves its clinical application potential.

Near-Borehole Formation Acoustic Logging Imaging: A Full Waveform Inversion Algorithm in Cylindrical Coordinates

Near-Borehole Formation Acoustic Logging Imaging: A Full Waveform Inversion Algorithm in Cylindrical Coordinates

An Auto-Calibration Algorithm for Hybrid Guided Wave Tomography Based on Full Waveform Inversion

Combining with the dispersion characteristics of ultrasonic guided waves, full waveform inversion method shows great application potential in the quantificational detections of defects in plate- and pipe- like structures. Owing to the inversion efficiency problem, the forward modeling is generally based on the acoustic equation approximation, and a reference signal unaffected by the defects needs to be artificially selected to correct the approximation forward simulation results. This study presents an auto-calibration (without artificial selection of the reference signal) and high-precision imaging method based on the combination of the full waveform inversion and ray tomography algorithm. The ray tomography results are not only used to automatically select the reference signal based on an auto-calibration process, but also used as the macro initial model for the full inversion method, which decreases the possibility of losing in local minimum values during the inversion process and enhances the robustness of the inversion method. Therefore, compared with the classical guided wave tomography method based on full waveform inversion, relatively high-frequency transducers can be used to acquire high-frequency signals, and thus, the imaging accuracy could be effectively improved. Simulation and experiment results have verified that the global relative error of the auto-calibration method is smaller than the classical method. The good imaging results of irregular complex defects confirmed the effectiveness and applicability of the new method.

Wasserstein Distance-Based Full-Waveform Inversion With a Regularizer Powered by Learned Gradient

Full-waveform inversion (FWI) is a powerful technique for building high-quality subsurface geological structures. It is known to suffer from local minima problems when a good starting model is lost. To obtain a desirable solution, regularization constraints are needed to impose suitable priors, in particular for salt models. Recent studies have allowed cooperating the denoising operator as a specific prior with optimization algorithms to effectively solve various image reconstruction tasks. Inspired by the promising performance of regularization by denoising (RED), we propose a flexible unsupervised learning FWI framework, in which the regularizer is an extensive variant of RED that powered by learned gradient for addressing salt models inversion. To further mitigate local minima issues, the Wasserstein distance induced by optimal transport theory with a new preprocessing transformation is applied as a measure in data domain. Integrating the physical constraints featured by the wave equation and the model priors featured by the deep convolutional neural network, our method is able to produce measurement consistent and high-resolution results. We experimentally compare the proposed method with traditional total variation regularized FWI and RED regularized FWI on the well-known geological models. The numerical results demonstrate the effectiveness of our method for handling the inversion task of high-contrast media in the presence of a free-surface case. Moreover, our framework is adaptive to restrict the solutions by leveraging the regularizer with learned gradient, whose training does not need any geological images, thus paving the way to develop FWI algorithms with state-of-the-art deep learning techniques in interdisciplinary research.

Feasibility of seismic time-lapse monitoring of CO2 with rock physics parametrized full waveform inversion

SUMMARY Carbon capture and storage is an important technology for greenhouse gas mitigation. Monitoring of CO2 storage should, in addition to locating the plume, provide quantitative information on CO2 saturation. We propose a full waveform inversion (FWI) algorithm for the prediction of the spatial distribution of CO2 saturation from time-lapse seismic data. The methodology is based on the application of a rock-physics parametrized FWI scheme that allows for direct updating of reservoir properties. We derive porosity and lithology parameters from baseline data and use them as input to predict CO2 saturation from monitor data. The method is tested on synthetic time-lapse data generated for the Johansen formation model. Practical issues associated with field data applications, such as acquisition limitations, construction of the initial model, noise and uncertainty in the rock physics model, are taken into account in the simulation. The results demonstrate the robustness of our approach for reconstructing baseline and monitor models. We also illustrate the potential of the approach as compared to conventional two-step inversion algorithms, in which an elastic FWI prediction of velocities and density is followed by rock physics inversion.

A consistent multiparameter Bayesian full waveform inversion scheme for imaging heterogeneous isotropic elastic media

SUMMARY The inversion of complete seismic waveforms offers new perspectives to better constrain the elastic properties of Earth’s interior. However, models of density and seismic velocities obtained from full waveform inversions are generally characterized by very different and uneven spatial resolutions. Because the 3-D structure of the Earth represents small deviations from average reference Earth models, the absolute values of density, VP and VS in the Earth are strongly correlated. Here, we exploit this strong correlation between model parameters as a priori information introduced into a new full waveform inversion algorithm, by considering a non-diagonal 3-D model covariance matrix in which the spatial correlations of elastic properties are described with an exponential covariance function. The inverse of such a model covariance matrix is easy to compute, and we thus have all the ingredients to construct a consistent Bayesian full waveform inversion scheme. We show that taking into account the correlations between density and seismic velocities can lead to dramatic improvements on the reconstructed models of density, seismic velocities and VP/VS ratio. This new imaging approach opens new perspectives for refining tomographic images of density and seismic velocities in the lithosphere and upper mantle on a regional scale by full waveform inversion of teleseismic body waves.