Multiscale Inversion Research Articles

An illustrated guide to: Parsimonious multi-scale full-waveform inversion

Having been a seemingly unreachable ideal for decades, 3-D full-waveform inversion applied to massive seismic datasets has become reality in recent years. Often achieving unprecedented resolution, it has provided new insight into the structure of the Earth, from the upper few metres of soil to the entire globe. Motivated by these successes, the technology is now being translated to medical ultrasound and non-destructive testing. Despite remarkable progress, the computational cost of full-waveform inversion continues to be a major concern. It limits the amount of data that can be exploited, and it largely inhibits quantitative and comprehensive uncertainty analyses. These notes complement a presentation on recent developments in full-waveform inversion that are intended to reduce computational cost and assimilate more data, thereby improving tomographic resolution. The suite of strategies includes flexible and user-friendly spectral-element simulations, the design of wavefield-adapted meshes that harness prior information on wavefield geometry, dynamic mini-batch optimisation that naturally takes advantage of data redundancies, and collaborative multi-scale updating to jointly constrain crustal and mantle structure.

WUS324: Multiscale Full Waveform Inversion Approaching Convergence Improves Waveform Fits While Imaging Seismic Structure of the Western United States

AbstractWe report a new model of radially anisotropic crustal and upper mantle structure of the western United States (WUS324) obtained from full waveform inversion of earthquake data. We ran three multiscale inversion stages beyond model WUS256 (Rodgers et al., 2022, https://doi.org/10.1029/2022jb024549) allowing them to approach convergence to fit a larger data set to a shorter minimum period of 16 s. WUS324 is based on 324 total iterations from its starting model, significantly more (16 times) than previous studies. Waveform misfit reductions are 66%–70% for both the inversion data and an independent validation data set providing confidence in the predictive power of the model. WUS324 provides much better fits and reveals shear wavespeed, vS, structure of this large region with more detail than previous waveform tomography models. We show representative images demonstrating the resolution of diverse seismic structure across this highly heterogeneous region including oceanic lithosphere, subducting slabs and continental magmatism.

Deep-learning optimization using the gradient of a custom objective function: A full-waveform inversion example study on the convolutional objective function

The integration of conventional high-performance full-waveform inversion (FWI) algorithms with deep-learning frameworks is an innovative and promising research direction with the potential to enhance and broaden the application prospects of this field significantly. Automatic differentiation with backpropagation techniques can derive the gradients of model parameters in an equivalent manner to that of adjoint methods; however, it requires a substantial amount of computer memory, particularly when considering the time-step layers. In addition, some excellent objective functions suitable for FWI are not available in deep-learning frameworks. In comparison, the adjoint method, based on effective boundary storage technology, offers greater practicality for calculating the gradients of various objective functions. Therefore, this paper develops a novel approach toward FWI that integrates deep-learning optimization with high-performance gradient computation. In particular, our method inputs model parameter gradients from a custom objective function into the deep-learning framework. Herein, we use the acoustic equation with variable density as an example to demonstrate how a convolutional objective function, along with its corresponding velocity and density gradients, can be used for optimized inversion, multiscale inversion, and deep network parameterization-based multiscale inversion within the deep-learning framework. This approach provides a paradigm for deep-learning-optimized FWI, which we apply to synthetic and field data scenarios.

Multiscale 3-D Stochastic Inversion of Frequency-Domain Airborne Electromagnetic Data

In mineral, environmental, and engineering explorations, we frequently encounter geological bodies with varied sizes, depths, and conductivity contrasts with surround rocks and try to interpret them with single survey data. The conventional three-dimensional (3-D) inversions significantly rely on the size of the grids, which should be smaller than the smallest geological target to achieve a good recovery to anomalous electric conductivity. However, this will create a large amount of unknowns to be solved and cost significant time and memory. In this paper, we present a multi-scale (MS) stochastic inversion scheme based on shearlet transform for airborne electromagnetic (AEM) data. The shearlet possesses the features of multi-direction and multi-scale, allowing it to effectively characterize the underground conductivity distribution in the transformed domain. To address the practical implementation of the method, we use a compressed sensing method in the forward modeling and sensitivity calculation, and employ a preconditioner that accounts for both the sampling rate and gradient noise to achieve a fast stochastic 3-D inversion. By gradually updating the coefficients from the coarse to fine scales, we obtain the multi-scale information on the underground electric conductivity. The synthetic data inversion shows that the proposed MS method can better recover multiple geological bodies with different sizes and depths with less time consumption. Finally, we conduct 3-D inversions of a field dataset acquired from Byneset, Norway. The results show very good agreement with the geological information.

Adaptive structure-based full-waveform inversion

In simultaneous-shot full-waveform inversion (FWI), the re-started limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm can be used to suppress crosstalk, but crosstalk cannot be completely eliminated from the inversion results. To further solve the crosstalk problem caused by the interference among individual shots in simultaneous-shot FWI, adaptive structure-based smoothing is applied to the FWI with the restarted L-BFGS algorithm. Structure-based smoothing can mitigate the crosstalk by highlighting structure boundaries. To perform structure-based smoothing, an implicit anisotropic diffusion equation is solved. We carry out a multiscale inversion strategy. Because the FWI results in the low-frequency band contain less structural information and more crosstalk, structure-based smoothing is applied to the frequency band at frequencies higher than the peak frequency to prevent negative effects from the model structure in the low-frequency band. The estimated structural information is iteratively updated during the inversion process. Furthermore, structure-based smoothing is only added to the iterations with invariant encoding to reduce oversmoothing. The invariant encoding means that the shot encoding remains unchanged between iterations. Numerical experiments with an overthrust model indicate that the proposed adaptive structure-based FWI method can effectively suppress artifacts and provide a high-resolution inversion result, even when the encoded data are contaminated with strong noise. Another advantage of the adaptive structure-based FWI method is that no seismic migration needs to be performed, which makes it more efficient for real data.

Multi-scale ground penetrating radar full waveform inversion with hybrid Tikhonov and total-variation regularization for different geometric structure

Abstract Full waveform inversion (FWI) is a high-resolution technique to estimate the parameters of dielectric permittivity (ϵ) and electrical conductivity (σ) and identify the structure of the subsurface for Ground Penetrating Radar (GPR) application. However, permittivity and conductivity parameters can be coupled in bi-parameter GPR inversion. This coupling effect leads to the crosstalk in the FWI result. To solve this problem, we propose a novel approach to use the multi-scale FWI with the hybrid regularization method, which combines Tikhonov and total-variation (TV) regularizers that simultaneously invert the ϵ and the σ parameters, which improve the inversion accuracy and reduce the crosstalk effect. The multi-scale strategy uses the Wiener filtering to process the GPR data in different frequency ranges. Then, the low frequencies signal updates the bottom part and subsequently increases the frequencies to invert for the shallow areas. The Tikhonov regularization stabilizes the reconstruction of the smoothly varying background part. In contrast, Total Variation (TV) regularization can recover the large contrasts associated with the LNAPL model. The new Tikhonov-TV (TT) regularization can mitigate the crosstalk caused by the parameter coupling effect. Numerical tests with typical GPR models demonstrate that the proposed multi-scale TT-FWI strategy can effectively eliminate the crosstalk and improve the reconstruction accuracy when the model parameters have a different structure.

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 traveltime 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 multiparameter 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 non-linear 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 used to perform multiscale 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.

Multi-Scale Acoustic Velocity Inversion Based on a Convolutional Neural Network

The full waveform inversion at this stage still has many problems in the recovery of deep background velocities. Velocity modeling based on end-to-end deep learning usually lacks a generalization capability. The proposed method is a multi-scale convolutional neural network velocity inversion (Ms-CNNVI) that incorporates a multi-scale strategy into the CNN-based velocity inversion algorithm for the first time. This approach improves the accuracy of the inversion by integrating a multi-scale strategy from low-frequency to high-frequency inversion and by incorporating a smoothing strategy in the multi-scale (MS) convolutional neural network (CNN) inversion process. Furthermore, using angle-domain reverse time migration (RTM) for dataset construction in Ms-CNNVI significantly improves the inversion efficiency. Numerical tests showcase the efficacy of the suggested approach.

Multiscale Inversion of Leaf Area Index in Citrus Tree by Merging UAV LiDAR with Multispectral Remote Sensing Data

The LAI (leaf area index) is an important parameter describing the canopy structure of citrus trees and characterizing plant photosynthesis, as well as providing an important basis for selecting parameters for orchard plant protection operations. By fusing LiDAR data with multispectral data, it can make up for the lack of rich spatial features of multispectral data, thus obtaining higher LAI inversion accuracy. This study proposed a multiscale LAI inversion method for citrus orchard based on the fusion of point cloud data and multispectral data. By comparing various machine learning algorithms, the mapping relationship between the characteristic parameters in multispectral data and point cloud data and citrus LAI was established, and we established the inversion model based on this, by removing redundant features through redundancy analysis. The experiment results showed that the BP neural network performs the best at both the community scale and the individual scale. After removing redundant features, the R2, RMSE, and MAE of the BP neural network at the community scale and individual scale were 0.896, 0.112, 0.086, and 0.794, 0.408, 0.328, respectively. By adding the three-dimensional gap fraction feature to the two-dimensional vegetation index features, the R2 at community scale and individual scale increased by 4.43% and 7.29%, respectively. The conclusion of this study suggests that the fusion of point cloud and multispectral data exhibits superior accuracy in multiscale citrus LAI inversion compared to relying solely on a single data source. This study proposes a fast and efficient multiscale LAI inversion method for citrus, which provides a new idea for the orchard precise management and the precision of plant protection operation.

STFS-urban: Spatio-temporal flood simulation model for urban areas

Amidst escalating urbanization and increasing extreme climatic events, strengthening flood resilience strategies in global cities has become imperative. This study introduces an innovative spatiotemporal urban flood simulation model that seamlessly integrates diverse refined and multi-spatiotemporal scales, ranging from 7.5 to 60 min and 100–2000 m, respectively. The model comprises multi-scale radar rainfall inversion (MRI), fine-grained coupled flood simulation model (FGCFS), and transformer-CNN flood prediction (TCFP) modules. Employing the Nanjing urban area as a case study, the model's efficacy is subjected to rigorous assessment. The advantages derived from integrated refinement coupling and boundary conditions through FGCFS and TCFP are accentuated. Impressively, the results underscore the robust performance of radar rainfall inversion across most scales, revealing a correlation coefficient surpassing 0.8 and a root-mean-square error of under 5.2 mm. FGCFS achieves optimal simulated water depth changes at 7.5 min × 500 m resolution, with the Nash efficiency coefficient exceeding 0.69 (0.94 at YS observation point and 0.89 at SXM observation point), alongside percentage deviations below 12.89 (3.59 at SXM observation point and 2.42 at XJL observation point). TCFP's learning proficiency is showcased through error convergence to 0.002 m after twenty iterations, particularly suitable for resolutions below 4 m. Notably, both FGCFS and TCFP demonstrate efficient utilization of resources, enabling streamlined simulations across varying data resolutions. Consequently, our study propels a sophisticated framework harmonizing multi-scale data integration, refinement coupling, and dynamic allocation. Our work extends beyond practical solutions, offering a glimpse into the future of flood simulation modeling, and reaffirming its pivotal role within the realm of environmental research and management.

Multi-scale data joint inversion of minerals and porosity in altered igneous reservoirs—A case study in the South China Sea

There are abundant igneous gas reservoirs in the South China Sea with significant value of research, and lithology classification, mineral analysis and porosity inversion are important links in reservoir evaluation. However, affected by the diverse lithology, complicated mineral and widespread alteration, conventional logging lithology classification and mineral inversion become considerably difficult. At the same time, owing to the limitation of the wireline log response equation, the quantity and accuracy of minerals can hardly meet the exploration requirements of igneous formations. To overcome those issues, this study takes the South China Sea as an example, and combines multi-scale data such as micro rock slices, petrophysical experiments, wireline log and element cutting log to establish a set of joint inversion methods for minerals and porosity of altered igneous rocks. Specifically, we define the lithology and mineral characteristics through core slices and mineral data, and establish an igneous multi-mineral volumetric model. Then we determine element cutting log correction method based on core element data, and combine wireline log and corrected element cutting log to perform the lithology classification and joint inversion of minerals and porosity. However, it is always difficult to determine the elemental eigenvalues of different minerals in inversion. This paper uses multiple linear regression methods to solve this problem. Finally, an integrated inversion technique for altered igneous formations was developed. The results show that the corrected element cutting log are in good agreement with the core element data, and the mineral and porosity results obtained from the joint inversion based on the wireline log and corrected element cutting log are also in good agreement with the core data from X-ray diffraction. The results demonstrate that the inversion technique is applicable and this study provides a new direction for the mineral inversion research of altered igneous formations.

Face spoofing detection based on multi-scale color inversion dual-stream convolutional neural network

Face spoofing detection based on multi-scale color inversion dual-stream convolutional neural network

Evaluation of iron deposition in the motor CSTC loop of a Chinese family with paroxysmal kinesigenic dyskinesia using quantitative susceptibility mapping.

Previous studies have revealed structural, functional, and metabolic changes in brain regions inside the cortico-striatal-thalamo-cortical (CSTC) loop in patients with paroxysmal kinesigenic dyskinesia (PKD), whereas no quantitative susceptibility mapping (QSM)-related studies have explored brain iron deposition in these areas. A total of eight familial PKD patients and 10 of their healthy family members (normal controls) were recruited and underwent QSM on a 3T magnetic resonance imaging system. Magnetic susceptibility maps were reconstructed using a multi-scale dipole inversion algorithm. Thereafter, we specifically analyzed changes in local mean susceptibility values in cortical regions and subcortical nuclei inside the motor CSTC loop. Compared with normal controls, PKD patients had altered brain iron levels. In the cortical gray matter area involved with the motor CSTC loop, susceptibility values were generally elevated, especially in the bilateral M1 and PMv regions. In the subcortical nuclei regions involved with the motor CSTC loop, susceptibility values were generally lower, especially in the bilateral substantia nigra regions. Our results provide new evidence for the neuropathogenesis of PKD and suggest that an imbalance in brain iron levels may play a role in PKD.

A multiscale inversion algorithm for engineering inspection using GPR data

A numerical algorithm based on multiscale modeling is presented, to postprocess ground penetrating radar measurements obtained when inspecting a reinforced concrete slab, to characterize the position and shape of rebars. The method is based on minimizing the difference between the measured signal and the signal generated by a guessed location of the inclusion. Results for three specific examples are presented, including synthetic and real data. Comparison between inversion results and causative bodies is encouraging, despite the simplicity of the approach.

Characterization of Mechanical Properties of CNFs and the Assembled Microfibers Through a Multi-scale Optimization-Based Inversion Method

Characterization of Mechanical Properties of CNFs and the Assembled Microfibers Through a Multi-scale Optimization-Based Inversion Method

Integrating U-Nets Into a Multiscale Full-Waveform Inversion for Salt Body Building

Integrating U-Nets Into a Multiscale Full-Waveform Inversion for Salt Body Building

Data-Driven Seismic Impedance Inversion Based on Multi-Scale Strategy

Seismic impedance inversion is one of the most commonly used techniques for reservoir characterization. High accuracy and high resolution seismic impedance is a prerequisite for subsequent reservoir interpretation. The data-driven approach offers the opportunity for accurate impedance prediction by establishing a nonlinear mapping between seismic data and impedance. However, existing data-driven methods take the raw seismic data directly as input, making it difficult for the network to learn high frequency weak signal information and resulting in low resolution inversion results. In order to mitigate the above issues, a data-driven seismic impedance inversion method based on multi-scale strategy is proposed. The method first obtains seismic data at different scales using frequency division techniques and do normalization on the extracted multi-scale data to ensure the consistency of the seismic signal energy in different frequency bands. The multi-scale seismic data will then be fed into the network, which helps the network to learn the high frequency information features more easily, and ultimately obtain higher resolution inversion results. We use the most commonly used convolutional neural network (CNN) as an example to demonstrate that the proposed multi-scale data-driven seismic impedance inversion method can improve the resolution of the inversion results. In addition, since the above seismic impedance inversion method is executed trace-by-trace, the f-x prediction filtering technique is introduced to improve the lateral continuity of the inversion results and obtain more geologically reliable impedance profiles. The validity of the proposed method is verified in the application of synthetic model data as well as an actual data set.

Shallow geological structures building by joint waveform inversion based on curvelet decomposition

Shallow geological imaging structures often play an essential role in the field of environmental and engineering geophysics. Early arrival waveform inversion (EWI) is high-resolution method to imaging near-surface structures. We proposed a joint waveform and envelope inversion method based on curvelet transform (CJWEI) to build shallow geological structures. By taking advantage of their complementary sensitivities to near-surface structure, we can constrain both low- and high-wavenumber components of structures, as well as mitigating the cycle-skipping problem. Curvelet transform was used to decompose seismic data into different scales and provide a multi-scale inversion strategy to further reduce non-uniqueness of the inversion. With synthetic test, we demonstrate that our method can constrain the anomalies and hidden layers in the shallow structure more efficiently as well as robust in terms of noise.

Source-Independent Waveform Inversion Method for Ground Penetrating Radar Based on Envelope Objective Function

For the full waveform inversion, it is necessary to provide an accurate source wavelet for forwarding modeling in the iteration. The source wavelet estimation method based on deconvolution technology can solve this problem to some extent, but we find that the estimated source wavelet is not accurate and needs to be manually corrected repeatedly in the iteration. This process is highly operator-intensive, and the update process is time-consuming and increases the potential for errors. We propose a source-independent waveform inversion (SIEWI) scheme for cross-hole GPR data, and use the envelope objective function combined with this method to effectively reduce the nonlinearity of inversion. The residual field used by SIEWI to construct the gradient inherits the characteristics of the envelope wavefield. Compared with full waveform inversion (FWI), SIEWI is more robust and less sensitive to frequency components and inaccurate source wavelet. To avoid cycle jumping, the multi-scale strategy effectively utilizes the properties of convolutional wavefields. In one iteration, the wavefield is decomposed into multiple frequency bands through multiple convolutions in the time domain to construct a multi-scale inversion strategy that preferentially inverts low-frequency information.

Curvelet-Based Joint Waveform and Envelope Inversion of Early-Arrival Imaging Shallow Geological Structure

Abstract Near-surface imaging structures often plays a significant role in the field of environmental and engineering geophysics. Early-arrival waveform inversion (EWI) is state-of-the-art method to imaging near-surface structures due to its high resolution. However, the method faces with cycle-skipping issue which might lead to an unexpected local minimum. Envelope inversion (EI) could deal with this issue which contributes to the ultralow-frequency information extracted from the envelope but has a low resolution. We have developed a curvelet-based joint waveform and envelope inversion (CJWEI) method for inverting imaging near-surface velocity structures. By inverting two types of data, we are able to recover the low- and high-wavenumber structures and mitigate the cycle-skipping problem. Curvelet transform was used to decompose seismic data into different scales and provide a multiscale inversion strategy to further reduce non-uniqueness of waveform inversion efficiently. With synthetic test and real data application, we demonstrate that our method can constrain the anomalies and hidden layers in the shallow structure more efficiently as well as is robust in terms of noise. The proposed multiscale joint inversion offers a computational efficiency and high precision to imaging fine-scale shallow underground structures.