Traveltime Inversion Research Articles

Application of Reciprocity Principle in Ultrasound Bone Tomography

Abstract Full-matrix synthetic aperture imaging (FMSA) is a well-established and widely used modality for ultrasound imaging Nevertheless, the accompanying relatively large amount of data acquisition, storage, and process results in heavy costs of computation and memory, which further hinders FMSA from real-time implementation. Moreover, the most existing FMSAs employed the hypothesis uniform sound velocity to reconstruct the soft tissues, which is not suitable for hard tissue imaging, irrespective of the high acoustic impedance contrast (i.e., the difference of sound velocity) between the bone and soft tissues. To overcome these limitations, a half-matrix ultrasound bone tomography method combining the reciprocity principle and sound velocity estimation was proposed for cortical cross-section imaging. The travel-time inversion was firstly utilized for sound velocity prediction, and then half-matrix synthetic aperture (HMSA) with ray-tracing was conducted to image the cortical cross-section. An ex-vivo bovine femur was used to validate the effectiveness of the method. Compared to the full-matrix capture method, the proposed method was demonstrated to be an accurate and cost-effective modality for cortical bone tomography.

Inversion for Inferring Solar Meridional Circulation: The Case with Constraints on Angular Momentum Transport inside the Sun

We have carried out inversions of travel times as measured by Gizon et al. to infer the internal profile of the solar meridional circulation (MC). A linear inverse problem has been solved by the regularized least-squares method with a constraint that the angular momentum (AM) transport by MC should be equatorward (HK21-type constraint). Our motivation for using this constraint is based on the result by Hotta & Kusano (hereafter HK21), where the solar equator-fast rotation was reproduced successfully without any manipulation. The inversion result indicates that the MC profile is a double-cell structure if the so-called HK21 regime, in which AM transported by MC sustains the equator-fast rotation, correctly describes the physics inside the solar convective zone. The sum of the squared residuals computed with the inferred double-cell MC profile is comparable to that computed with the single-cell MC profile obtained when we exclude the HK21-type constraint, showing that both profiles can explain the data more or less at the same level. However, we also find that adding the HK21-type constraint degrades the resolution of the averaging kernels. Although it is difficult for us to determine the large-scale morphology of the solar MC at the moment, our attempt highlights the relevance of investigating the solar MC profile from both theoretical and observational perspectives.

Characteristic reflection wavefield-based wave equation traveltime inversion

Introducing a high-precision deep-depth velocity model is essential for accurately imaging seismic data in complex areas such as deepwater and deep depth in oil and gas exploration. Conventional methods, such as reflection full-waveform inversion (RFWI), face challenges, notably the cycle-skipping issue, especially when the initial velocity model is far away from the true model. To address this, we develop a novel wave equation traveltime inversion (WETI) using the characteristic reflection wavefield (CRW) to update the low-wavenumber components of the velocity model. The CRWs are primary reflections in seismic data that contribute significantly to velocity model updates. We implement a multistep approach, including migration, characteristic reflector structure (CRS) picking, and demigration to extract the CRW. This extraction, constrained by CRS, ensures accurate matching between the observed and the simulated CRWs, even with an inaccurate initial velocity model, thus enabling an accurate traveltime shift estimation during inversion. Unlike conventional methods that use all reflection data simultaneously, CRW-WETI strategically selects fewer CRWs, enhancing inversion convexity and avoiding local minima from the cycle-skipping problem. This selective inversion begins with a limited number of CRWs to ensure stability and progressively incorporates more as the WETI process advances, eventually transitioning into conventional RFWI. Moreover, with the CRS constraint, CRW can be selected based on the target region. The target-oriented velocity inversion can also be implemented easily to update a local anomalous velocity model. Numerical examples of synthetic and field data demonstrate the effectiveness and validity of our method.

Stable neural network-based traveltime tomography using hard-constrained measurements

Traveltime tomography, or travel time inversion, has been one of the primary seismological tools for decades and has been used to understand Earth's properties and dynamic processes. An accurate, preferably flexible, eikonal solver to compute the travel time field is at the heart of the inversion process. However, most conventional eikonal solvers suffer from first-order convergence errors and difficulties dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have successfully addressed those challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multi-term nature of the loss function. To improve this, we propose a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We employ the theory of functional connections to the traveltime tomography problem, which allows for the use of a single loss term for training the PINN model. Its efficiency, stability, and flexibility in tackling various cases, including topography-dependent and 3D models, are attested through rigorous numerical tests, thus providing an efficient and stable PINN-based traveltime tomography.

Joint inversion of induced polarization and hydraulic tomography data for hydraulic conductivity imaging

SUMMARY For accurate modelling of groundwater flow and transport processes within an aquifer, precise knowledge about hydraulic conductivity K and its small-scale heterogeneities is fundamental. Methods based on pumping tests, such as hydraulic tomography (HT), allow for retrieving reliable K-estimates, but are limited in their ability to image structural features with high resolution, since the data from time-consuming hydraulic tests are commonly sparse. In contrast, geophysical methods like induced polarization (IP) can potentially yield structural images of much higher resolution, but depend on empirical petrophysical laws that may introduce significant uncertainties to the K-estimation. Therefore, this paper presents a joint inversion procedure for both HT and IP data, which allows for combining the complementary abilities of both methods. Within this approach, a traveltime inversion is applied to the HT data, while the IP inversion is based on a full-decay time-domain forward response, as well as a reparametrization of the Cole–Cole model to invert for K directly. The joint inversion is tested on a synthetic model mimicking horizontally layered sediments, and the results are compared with the individual HT and IP inversions. It is shown that jointly inverting both data sets consistently improves the results by combining the complementary sensitivities of the two methods, and that the inversion is more robust against changes in the experimental setups. Furthermore, we illustrate how a joint inversion approach can correct biases within the petrophysical laws by including reliable K-information from hydraulic tests and still preserving the high-resolution structural information from IP. The different inversion results are compared based on the structural similarity index (SSIM), which underlines the robustness of the joint inversion compared to using the data individually. Hence, the combined application of HT and IP within field surveys and a subsequent joint inversion of both data sets may improve our understanding of hydraulically relevant subsurface structures, and thus the reliability of groundwater modelling results.

Development of an ANN-Based Technique for Inversion of Seismic Refraction Travel Times

Non-uniqueness in the inversion of seismic data can be considered the main challenge for the application of such data. Prior information, such as downhole data, can be used to control this problem. However, in most cases, prior information is not available; accordingly, geophysicists/analysts have to suppose a primary model for the observed data and then find the final adequate layered earth model through trial and error. In this study, a new technique was developed based on the artificial neural network (ANN) for the inversion of seismic refraction data in the absence of prior information. In this regard, a sequential multilayer perceptron (SMLP) was proposed, which integrates the sequential information of the model parameters to predict a reasonable layered earth model. In fact, at first, a multilayer perceptron (MLP) (First-MLP) was trained by synthetic data; then, a layered earth model, i.e., the primary model, was predicted for the observed data. Next, using the primary model, a range for each of the model parameters, i.e., thickness and P-wave velocity, for each layer was defined. Subsequently, new synthetic samples were generated based on the determined ranges. Finally, using another MLP (Second-MLP), which was trained by the new synthetic samples, the final model for the observed data was estimated. The proposed method was also tested by employing different synthetic data with and without noise. Moreover, the SMLP inversion technique was used to analyze the experimental seismic refraction dataset at a dam construction site. The results for both synthetic and experimental data confirmed the reliability of the proposed SMLP inversion technique.

Assessment of hydraulic fracture height and distribution using timelapse crosswell shear-wave data

An in-depth study of the impact of hydraulic fracturing operations on the Marcellus shale formation and overburden is presented by analyzing a unique timelapse crosswell seismic survey suitable for four-component shear wave vector rotations in order to study fracture azimuth and intensity. The borehole source used for the survey generated both oriented compressional and shear wave energy, and the seismic acquisition included perpendicular source orientations at each shot level to allow isolation of fast and slow shear modes using the Alford rotation and linear transform technique. Estimates of fast shear azimuth and slow shear time delay before and after hydraulic fracturing indicate fracture creation throughout the overburden above the Marcellus shale and a possible vertical breach of the bounding Tully limestone formation. The magnitude of the timelapse change is consistent between travel-time and slow shear lag inversion, but the absolute change is small, suggesting that the increase in fracture intensity may not be significant. These results point to operational inefficiencies during hydraulic fracturing operations that could negatively impact production and pose risks to future development. The study demonstrates the value of crosswell seismic surveys for reservoir monitoring and highlights observations that might be missed by a monitoring program that does not include timelapse seismic data and analysis of shear wave changes due to hydraulic fracturing.

Coupled Inversion of Amplitudes and Traveltimes of Primaries and Multiples for Monochannel Seismic Surveys

Engineers need to know properties of shallow marine sediments to build piers, pipelines and even offshore windfarms. We present a method for estimating the density, P velocity and thickness of these sediments. The traveltime inversion of primary and multiple reflections enables their semiquantitative estimation in marine surveys when using a minimal acquisition system such as a monochannel Boomer. Picking errors, ambient noise and interfering events lead to significant errors in the estimates. Similar, albeit milder, instabilities occur when inverting the signal amplitudes to determine the reflectivity of the layer interfaces. In this paper, we introduce a coupling between the separate inversion of amplitudes and traveltimes to obtain a better Earth model. The P velocity shows up in two stable terms provided by the separate inversions: the acoustic impedance of shallow sediments (through the amplitudes) and the transit time across the sediment layer (through the traveltimes). We couple the two inversion engines by imposing a smoothness condition on velocity and density and thickness of the layer while keeping the impedance and traveltime constant. We thus exploit the ambiguity of the solution to introduce geological criteria and reduce the noise contribution. We validated the proposed method with synthetic and real data.

Hybrid Discrete Fracture Network Inversion of Hydraulic Tomography Data From a Fractured‐Porous Field Site

AbstractThe accurate characterization of hydraulic conductivity heterogeneities in an aquifer is crucial for predicting flow and transport processes correctly. Hydraulic tomography (HT) experiments are often used to infer the hydraulically relevant features, but the correct inversion of the data remains a challenging task. We implemented a discrete fracture network (DFN) inversion approach that is expanded by considering a nonzero matrix permeability. The hybrid model allows the accurate characterization of fractured‐porous sites by taking into account both matrix and fracture flow. This novel inversion algorithm is successfully applied to HT data acquired at a field site in Goettingen (Germany), and the results are compared with those of a standard travel time inversion. Furthermore, we validate the inversion results by using them as the underlying material parameters for simulating heat tracer experiments and comparing the modeled temperature responses with those of heat tracer tests actually conducted at the site. It is shown that the DFN ensemble predicts the thermal response of the experiments correctly for the two major fractures in terms of location, amplitude, and time‐dependent behavior of the temperature anomaly, as long as the stochastic nature of the results is taken into account. We conclude that considering both matrix and fracture flow in a hybrid DFN inversion approach can lead to significant improvements in flow and transport modeling at fractured‐porous sites.

Elastic Wave Equation Traveltime Inversion With Dynamic Time Warping Based on High-Speed Train Seismic Data

Elastic Wave Equation Traveltime Inversion With Dynamic Time Warping Based on High-Speed Train Seismic Data

Lithospheric underthrusting beneath the northern Eastern Cordillera plateau, Colombian Andes: Insights from analyzing tomographic inversion using local earthquake travel time and gravity data

The northern Eastern Cordillera in the Colombian Andes is a wide (~200 km) high-elevation (>2.5 km) subduction-related asymmetric plateau. This plateau is currently subjected to shortening and magmatic addition, and yet the nature of the lowermost crust and its transition to the underlying mantle remains poorly constrained. To improve our knowledge about the structure of the deep crust beneath the EC plateau, we conducted a joint inversion of travel times of local earthquakes and gravity data. Gravity contributes to up to 0.5% in dVp and dVs with no affectation of the shape or location of the discussed anomalies. Along the latitudinal profile with the highest resolution (~5.7°N), two anomalies are identified at depths of 40–60 km beneath the plateau. A western low-velocity anomaly is interpreted as crustal material underthrust eastward beneath the northwestern EC. This process is triggering the abrupt change in topography between the adjacent low-elevation basin and the orogenic plateau. Additionally, a high-velocity anomaly beneath the eastern flank of the EC is likely related to mantle metasomatism and westward underthrusting of the foreland lithosphere. This metasomatism is either related to the interaction between mafic magmas and the uppermost mantle or silica enrichment that occurred during a past episode of flat subduction. Evidence for foreland underthrusting includes relocated earthquakes at 33–42 km depth within the thrust system between the EC and the foreland, along with increased shortening in the eastern part of the plateau, an increase in exhumation rates, the eastward migration of deformation towards the foreland, and dominated thick-skinned deformation in the upper crust of the foreland with an eastward vergence of thrusts and related folds. Furthermore, within the central part of the plateau, our results show low velocities between 30 and 40 km depth, consistent with previous constraints, suggesting magmatic underplating beneath the Paipa-Iza volcanic complex.

Eikonal-equation-based characteristic reflection traveltime tomography

Automatically picking reflection traveltimes from the prestack shot gathers and positioning and updating the reflector depths during inversion are challenging in conventional eikonal-equation-based adjoint-state reflection traveltime tomography methods. To solve these problems, we develop an eikonal-equation-based adjoint-state characteristic reflection traveltime tomography (ASCRT) method. This method automatically extracts the reflection traveltimes by sequentially applying characteristic reflector picking in the depth migration section, followed by eikonal-equation-based traveltime calculation and an event-tracking algorithm. We also develop an efficient eikonal-equation-based kinematic imaging method to rapidly position and update the reflector depths. With our ASCRT method, the characteristic reflector positions and the velocity above the reflector are alternately updated. Our ASCRT method is performed with a layer-stripping strategy that sequentially uses selected characteristic reflectors to update the velocity model from shallow to deep. Compared with the wave equation-based reflection traveltime inversion methods, the efficiency of the ASCRT method is improved by several orders of magnitude, and the memory consumption is remarkably reduced. Synthetic and field data examples are presented to demonstrate the effectiveness and efficiency of our method.

Monochromatic wave equation reflection traveltime inversion

Wave equation reflection traveltime inversion (WRTI) is a fundamental method for interrogating the deep subsurface macrovelocity model. The time-domain WRTI calculates the gradient through the crosscorrelation of the source background and receiver scattered wavefields and the crosscorrelation of the source scattered and receiver background wavefields. During the backward propagation of the adjoint source, access to the source background and scattered snapshots of the entire propagation time is required. Because the source and receiver wavefields are extrapolated in opposite temporal directions, these source background and scattered wavefields are either stored in memory or reconstructed by additional wavefield extrapolation. This leads to a heavy memory burden and/or high computational cost. To address these challenges, we develop a new frequency-domain WRTI that is based on a monofrequency. By using the wavenumber coverage and gradient analyses, we prove that the monochromatic WRTI can produce comparable wavenumbers with time-domain and multifrequency-based WRTI. Our method directly stores monochromatic wavefields in memory because only one frequency is used for the gradient calculation. Therefore, our method significantly outperforms the conventional time-domain implementation in terms of memory requirement and computational cost. The accuracy of the monochromatic WRTI method is validated by gradient and result comparisons of several models with different complexities. Finally, we determine the effectiveness of our method by applying it to offshore streamer data.

Converted-wave reflection traveltime inversion with free-surface multiples for ocean-bottom-node data

Advancements in ocean-bottom-node (OBN) technologies have enabled the recording of high-quality multicomponent seismic data, which can provide crucial information about subsurface properties through elastic seismic imaging. However, imaging multicomponent data is challenging due to the requirement for P- and S-wave velocity models. Conventional processing relies on estimating the S-wave velocity model using a primary converted wave (C wave) because the pure S-wave primaries are not available if using standard air-gun sources. When a free surface is present, the C waves recorded with geophones can be contaminated by source-side P-wave multiples, especially water-layer reverberations, which are difficult to remove even using state-of-the-art wavefield decomposition methods. These C-wave multiples can cause significant mismatches between synthetic and observed data during reflection traveltime or waveform inversion if only primary P-to-S (PS) conversions are simulated. To address these issues and reliably build an S-wave velocity model for PS imaging, we develop a C-wave reflection traveltime inversion method that takes source-side free-surface multiples into account. The traveltime misfit of C-wave data is extracted using dynamic image warping to ensure a better match between the synthetic and observed data. The functional gradient of the objective function is derived using the adjoint state method. Based on sensitivity kernel analyses, a convenient gradient preconditioning strategy is developed to robustly separate the contribution of primary and multiple C waves during backpropagation. This strategy can effectively mitigate the high-wavenumber crosstalk associated with nonphysical wavepaths in the gradient and thus allow more accurate updating for the S-wave velocity model. The synthetic data and shallow-water OBN data from the East China Sea indicate that this approach can reliably recover the S-wave macrovelocity structures for PS imaging.

Deep crustal structure and tectonic implications of the Nansha Trough from wide-angle reflection and refraction seismic data

Deep crustal structure of the Nansha Trough (NT) plays a crucial role in understanding its crustal nature and magmatic activity. This study used the data from a wide-angle seismic profile (NDL1), which passes through the Jinghong seamount (JHSM) and Yinqing seamount (YQSM) along the trough. By using 2D forward ray-tracing and joint refraction and reflection travel-time inversion methods, we obtained the velocity structure of the NT. The model results show a thinned continental crust with a thickness of ∼9–18 km in the east of the trough, which is variable drastically below the seamounts. The upper crust slightly varies at a thickness of ∼5 km except at the seamounts, while the lower crust shows lateral variations in thickness and velocity, accompanied by high velocities below the JHSM due to the magmatic addition. Velocity models of the NDL1 also reveal crustal differences between the JHSM and YQSM. Combined with the magnetic anomaly and multichannel seismic data, we infer that the seamounts formed from multiple stages of magmatism after the spreading of the South China Sea and the NT are still active. High velocities in the lower crust result from the mantle upwelling and decompression melting of flexural responses of the NT, which may provide a genesis clue for the formation of the submarine seamounts.

Slab tearing and lithospheric structures in Luzon island, Philippines: constraints from P- and S-wave local earthquake tomography

Luzon Island is a complex setting of seismicity and magmatism caused by the subduction of the South China Sea lithosphere and the presence of a major strike-slip fault system, the Philippine Fault. Previous studies of the structure of this subduction zone have suggested that a ridge subduction system resulted in a slab tearing along the ridge. On the other hand, the Philippine Fault plays an important role in understanding how major strike-slip faults deform and displace at a continental scale. To constrain the lithospheric geological structure in the area and refine the slab tearing model, we performed a P- and S-wave seismic tomography travel time inversion using local earthquakes. The dataset has been combined from seismic phases reported by the International Seismological Centre and new pickings from six broadband seismic stations in northern Luzon. The three-dimensional P- and S-wave velocity models in Luzon Island were analyzed by applying the LOTOS package with a one-dimensional velocity model obtained from the VELEST program. Our tomographic images indicate contrasting velocity structures across the Philippine Fault to a depth of 60 km. Therefore, we suggest that the Philippine Fault might be a lithospheric structure that displaces both the crust and the upper mantle. The results also indicate regions of low-velocity slab windows from a depth of 40 km, which are interpreted as the sites of slab tearing. Compared with focal mechanisms and earthquake occurrence in this region, we propose that slab tearing extends from the fossil ridge and creates regional kinematic perturbations. The tearing produces shallow upwelling magma to stay in the chambers beneath the crust, which is in contrast to the magmatic system observed in other regions.

Crosscorrelation-based dynamic time warping and its application in wave equation reflection traveltime inversion

Crosscorrelation and dynamic time warping (DTW) are ubiquitous in time-shift estimation. However, the small-shift limitation of crosscorrelation and the instability and high sensitivity to noise of DTW seriously hinder their applications in complex situations. In this study, we develop a method called crosscorrelation-based DTW (CDTW) to address these issues. Our method constructs error matrices by local crosscorrelation instead of Euclidean distance to minimize the sensitivity to noise. The new error matrices are calculated in local windows and contain local structure similarity information of the two input signals. It improves the stability of the algorithm and makes the CDTW method less sensitive to noise and amplitude modulation. Our method estimates the time-varying shift using dynamic programming as the conventional DTW after the new error matrix is formulated. Numerical tests on pairs of signals and seismic images prove that our method can accurately estimate time shifts in cases of time-varying amplitude modulation and strong random noise contamination. Finally, we apply the CDTW method to wave equation reflection traveltime inversion (WRTI) and develop a CDTW-based WRTI method. Synthetic and field applications prove that this method can construct good background velocity models with the reliable reflection traveltime shifts produced by the CDTW method.

Using long-range transmissions in the Beaufort Gyre to test the sound-speed equation at high pressure and low temperature.

An ocean acoustic tomography array with a radius of 150 km was deployed in the central Beaufort Gyre during 2016-2017 for the Canada Basin Acoustic Propagation Experiment. Five 250-Hz transceivers were deployed in a pentagon, with a sixth transceiver at the center. A long vertical receiving array was located northwest of the central mooring. Travel-time anomalies for refracted-surface-reflected acoustic ray paths were calculated relative to travel times computed for a range-dependent sound-speed field from in situ temperature and salinity observations. Travel-time inversions for the three-dimensional sound-speed field consistent with the uncertainties in travel time [∼2 ms root mean square (rms)], receiver and source positions (∼ 3 m rms), and sound speed calculated from conductivity-temperature-depth casts could not be obtained without introducing a deep sound-speed bias (below 1000 m). Because of the precise nature of the travel-time observations with low mesoscale and internal wave variability, the conclusion is that the internationally accepted sound-speed equation (TEOS-10) gives values at high pressure (greater than 1000 m) and low temperature (less than 0 °C) that are too high by 0.14-0.16 m s-1.

Wide‐spectrum reconstruction of a velocity model based on the wave equation reflection inversion method and its application

AbstractMost seismic data used for conventional exploration lack far‐offset signals and effective low‐frequency components. This makes it difficult for conventional full waveform inversion methods to recover the low‐wavenumber components of mid‐ and deep‐layer models. To resolve the bottleneck of ray‐theoretical reflection traveltime tomography, wave equations, such as reflection inversion methods, are receiving increasing attention. We reconstructed a wide‐spectrum velocity model based on the multiscale wave‐equation reflection inversion method for model decomposition. First, using the dynamic image warping method to obtain reflection traveltime residuals, the wave‐equation reflection traveltime inversion was used to recover the low‐wavenumber components of the background model, which also solved the cycle‐skipping problem. Second, the medium‐wavenumber component of the model is then supplemented by the reflection waveform inversion, while the reflectivity model is obtained relying on the least‐squares reverse time migration method. Based on this method, the background and perturbation models are updated alternately and iteratively. At the same time, the stratum structural tensor information was extracted using the perturbed model image to construct a preconditioned operator for the stratum structural constraint, suppress unreasonably high‐wavenumber components in the generalized gradient and improve the geological consistency of the inversion results. Testing of the model using the Sigsbee2b model and seismic field data from the East China Sea showed that, compared with the conventional reflection traveltime tomography method based on prestack depth migration, the wave‐equation reflection inversion strategy with well‐matched information from traveltimes to waveforms significantly improved the accuracy of the middle and deep velocity modelling and enhanced the imaging quality of the whole body.

Locating volcanic tremor using azimuth coherence of cross-correlation

Accurate localization of volcanic tremors can provide important information about volcanic activities. However, the seismic waveform of volcanic tremor is complicated, resulting in that its location cannot be accurately determined from traditional traveltime inversion methods due to the unidentifiable first arrivals. The cross-correlation-based back-projection method, which stacked the back-projected spatial images utilizing the cross-correlation data from different station pairs, is considered as one of the most effective waveform stacking location methods for the volcanic tremor. In this method, the obtained coherence from different station pairs can improve the signal-to-noise ratio of projected energy distribution, and then constrain the potential source location. Nevertheless, not all back-projected images of station pairs are helpful to enhance the positioning effect in practice because the resolution of the energy peak in the back-projection method is affected by the selection of used station pairs. In this study, we optimized the back-projection method by establishing some criteria for the selection of the station pairs, including the distance between two stations in each pair, and the calculated azimuth angle, the cover area, and the central location between each two of all station pairs. In addition, the phase information in cross-correlation data has been also used in our method to limit the energy divergence in the resulting map. Taking the Katla volcano as a real example, we demonstrate that the proposed second-order azimuth coherent location method can increase the accuracy of locating volcanic tremors compared to the traditional methods, resulting in an improvement of approximately 70 % in location resolution.