### Recent Submissions

• #### Rayleigh Wave Dispersion Spectrum Inversion Across Scales

(Surveys in Geophysics, Springer Science and Business Media LLC, 2021-10-19) [Article]
Traditional approaches of using dispersion curves for S-wave velocity reconstruction have limitations, principally, the 1D-layered model assumption and the automatic/manual picking of dispersion curves. At the same time, conventional full-waveform inversion (FWI) can easily converge to a non-global minimum when applied directly to complicated surface waves. Alternatively, the recently introduced wave equation dispersion spectrum inversion method can avoid these limitations, by applying the adjoint state method on the dispersion spectra of the observed and predicted data and utilizing the local similarity objective function to depress cycle skipping. We apply the wave equation dispersion spectrum inversion to three real datasets of different scales: tens of meters scale active-source data for estimating shallow targets, tens of kilometers scale ambient noise data for reservoir characterization and a continental-scale seismic array data for imaging the crust and uppermost mantle. We use these three open datasets from exploration to crustal scale seismology to demonstrate the effectiveness of the inversion method. The dispersion spectrum inversion method adapts well to the different-scale data without any special tuning. The main benefits of the proposed method over traditional methods are that (1) it can handle lateral variations; (2) it avoids direct picking dispersion curves; (3) it utilizes both the fundamental and higher modes of Rayleigh waves, and (4) the inversion can be solved using gradient-based local optimizations. Compared to the conventional 1D inversion, the dispersion spectrum inversion requires more computational cost since it requires solving the 2D/3D elastic wave equation in each iteration. A good match between the observed and predicted dispersion spectra also leads to a reasonably good match between the observed and predicted waveforms, though the inversion does not aim to match the waveforms.
• #### Machine Learning Enabled Traveltime Inversion Based on the Horizontal Source Location Perturbation

(GEOPHYSICS, Society of Exploration Geophysicists, 2021-10-17) [Article]
Gradient based traveltime tomography, which aims to minimize the difference between modeled and observed first arrival times, is a highly non-linear optimization problem. Stabilization of this inverse problem often requires employing regularization. While regularization helps avoid local minima solutions, it might cause low resolution tomograms because of its inherent smoothing property. On the other hand, although conventional ray-based tomography can be robust in terms of the uniqueness of the solution, it suffers from the limitations inherent in ray tracing, which limits its use in complex media. To mitigate the aforementioned drawbacks of gradient and ray-based tomography, we approach the problem in a completely novel way leveraging data-driven inversion techniques based on training deep convolutional neural networks (DCNN). Since DCNN often face challenges in detecting high level features from the relatively smooth traveltime data, we use this type of network to map horizontal changes in observed first arrival traveltimes caused by a source shift to lateral velocity variations. The relationship between them is explained by a linearized eikonal equation. Construction of the velocity models from this predicted lateral variation requires information from, for example, a vertical well-log in the area. This vertical profile is then used to build a tomogram from the output of the network. Both synthetic and field data results verify that the suggested approach estimates the velocity models reliably. Because of the limited depth penetration of first arrival traveltimes, the method is particularly favorable for near-surface applications.
• #### Ensemble Kalman filtering with colored observation noise

(Quarterly Journal of the Royal Meteorological Society, Wiley, 2021-10-15) [Article]
The Kalman filter (KF) is derived under the assumption of time-independent (white) observation noise. Although this assumption can be reasonable in many ocean and atmospheric applications, the recent increase in sensors coverage such as the launching of new constellations of satellites with global spatio-temporal coverage will provide high density of oceanic and atmospheric observations that are expected to have time-dependent (colored) error statistics. In this situation, the KF update has been shown to generally provide overconfident probability estimates, which may degrade the filter performance. Different KF-based schemes accounting for time-correlated observation noise were proposed for small systems by modeling the colored noise as a first-order autoregressive model driven by white Gaussian noise. This work introduces new ensemble Kalman filters (EnKFs) that account for colored observational noises for efficient data assimilation into large-scale oceanic and atmospheric applications. More specifically, we follow the standard and the one-step-ahead smoothing formulations of the Bayesian filtering problem with colored observational noise, modeled as an autoregressive model, to derive two (deterministic) EnKFs. We demonstrate the relevance of the colored observational noise-aware EnKFs and analyze their performances through extensive numerical experiments conducted with the Lorenz-96 model.
• #### Enhancing Fracture Network Characterization: A Data-Driven, Outcrop-Based Analysis

(Wiley, 2021-10-11) [Preprint]
The stochastic discrete fracture network (SDFN) model is a practical approach to model complex fracture systems in the subsurface. However, it is impossible to validate the correctness and quality of an SDFN model because the comprehensive subsurface structure is never known. We utilize a pixel-based fracture detection algorithm to digitize 80 published outcrop maps of different scales at different locations. The key fracture properties, including fracture lengths, orientations, intensities, topological structures, clusters and flow are then analyzed. Our findings provide significant justifications for statistical distributions used in SDFN modellings. In addition, the shortcomings of current SDFN models are discussed. We find that fracture lengths follow multiple (instead of single) power-law distributions with varying exponents. Large fractures tend to have large exponents, possibly because of a small coalescence probability. Most small-scale natural fracture networks have scattered orientations, corresponding to a small κ value (κ<3) in a von Mises--Fisher distribution. Large fracture systems collected in this research usually have more concentrated orientations with large κ values. Fracture intensities are spatially clustered at all scales. A fractal spatial density distribution, which introduces clustered fracture positions, can better capture the spatial clustering than a uniform distribution. Natural fracture networks usually have a significant proportion of T-type nodes, which is unavailable in conventional SDFN models. Thus a rule-based algorithm to mimic the fracture growth and form T-type nodes is necessary. Most outcrop maps show good topological connectivity. However, sealing patterns and stress impact must be considered to evaluate the hydraulic connectivity of fracture networks.
• #### Mathematical modeling of immune responses against sars-cov-2 using an ensemble kalman filter

(Mathematics, MDPI AG, 2021-09-30) [Article]
In this paper, a mathematical model was developed to simulate SARS-CoV-2 dynamics in infected patients. The model considers both the innate and adaptive immune responses and consists of healthy cells, infected cells, viral load, cytokines, natural killer cells, cytotoxic T-lymphocytes, B-lymphocytes, plasma cells, and antibody levels. First, a mathematical analysis was performed to discuss the model’s equilibrium points and compute the basic reproduction number. The accuracy of such mathematical models may be affected by many sources of uncertainties due to the incomplete representation of the biological process and poorly known parameters. This may strongly limit their performance and prediction skills. A state-of-the-art data assimilation technique, the ensemble Kalman filter (EnKF), was then used to enhance the model’s behavior by incorporating available data to determine the best possible estimate of the model’s state and parameters. The proposed assimilation system was applied on the real viral load datasets of six COVID-19 patients. The results demonstrate the efficiency of the proposed assimilation system in improving the model predictions by up to 40%.
• #### PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splitting

(arXiv, 2021-09-29) [Preprint]
Solving for the frequency-domain scattered wavefield via physics-informed neural network (PINN) has great potential in seismic modeling and inversion. However, when dealing with high-frequency wavefields, its accuracy and training cost limits its applications. Thus, we propose a novel implementation of PINN using frequency upscaling and neuron splitting, which allows the neural network model to grow in size as we increase the frequency while leveraging the information from the pre-trained model for lower-frequency wavefields, resulting in fast convergence to high-accuracy solutions. Numerical results show that, compared to the commonly used PINN with random initialization, the proposed PINN exhibits notable superiority in terms of convergence and accuracy and can achieve neuron based high-frequency wavefield solutions with a two-hidden-layer model.
• #### Seismic velocity modeling in the digital transformation era: a review of the role of machine learning

(Journal of Petroleum Exploration and Production Technology, Springer Science and Business Media LLC, 2021-09-28) [Article]
Seismic velocity modeling is a crucial step in seismic processing that enables the use of velocity information from both seismic and wells to map the depth and thickness of subsurface layers interpreted from seismic images. The velocity can be obtained in the form of normal moveout (NMO) velocity or by an inversion (optimization) process such as in full-waveform inversion (FWI). These methods have several limitations. These limitations include enormous time consumption in the case of NMO due to manual and heavy human involvement in the picking. As an optimization problem, it incurs high cost and suffers from nonlinearity issues. Researchers have proposed various machine learning (ML) techniques including unsupervised, supervised, and semi-supervised learning methods to model the velocity more efficiently. The focus of the studies is mostly to automate the NMO velocity picking, improve the convergence in FWI, and apply FWI using ML directly from the data. In the purview of the digital transformation roadmap of the petroleum industry, this paper presents a chronologic review of these studies, appraises the progress made so far, and concludes with a set of recommendations to overcome the prevailing challenges through the implementation of more advanced ML methodologies. We hope that this work will benefit experts, young professionals, and ML enthusiasts to help push forward their research efforts to achieving complete automation of the NMO velocity and further enhancing the performance of ML applications used in the FWI framework.
• #### Seasonal Simulations of Summer Aerosol Optical Depth over the Arabian Peninsula using WRF-Chem : Validation, Climatology, and Variability

(International Journal of Climatology, Wiley, 2021-09-27) [Article]
This study investigates the climatology and variability of summer Aerosol Optical Depth (AOD) over the Arabian Peninsula (AP) using a long-term high-resolution Weather Research and Forecasting model coupled with the chemistry module (WRF-Chem) simulation, available ground-based and satellite observations, and reanalysis products from 2008 to 2018. The simulated spatial distribution of the summer AOD agrees well with the satellite observations and reanalysis over the AP, with spatial correlation coefficients of 0.81/0.83/0.89 with MODIS-A/MODIS-T/MERRA-2, respectively. Higher values of summertime AOD are broadly found over the eastern AP regions and the southern Red Sea and minima over the northern Red Sea and northwest AP, consistent with observational datasets. The WRF-Chem simulation suggests that the two regions of high AOD are associated with dust advected from the Tigris–Euphrates by the northwesterly summer Shamal wind in the eastern AP and from the African Sahara via Sudan by westerly winds through the Tokar Gap for the southern AP. The high AOD over the south-central east AP is due to locally generated dust by the action of northerly winds, modulated by variations in relative humidity, vertical motion, soil moisture, and soil temperature over the desert regions. The vertical extent of this dust is primarily driven by upward motion triggered by thermal convection over the local source region. In terms of interannual variability, summer AOD exhibits significant year-to-year variations over the AP region. In particular, enhanced (reduced) AOD over the southern AP (Persian Gulf) is observed during La Niña conditions, favored by stronger (weaker) Tokar westerly (northwesterly summer Shamal) winds.
• #### Preconditioned BFGS-based Uncertainty Quantification in Elastic Full Waveform Inversion

(Geophysical Journal International, Oxford University Press (OUP), 2021-09-21) [Article]
Full Waveform Inversion (FWI) has become an essential technique for mapping geophysical subsurface structures. However, proper uncertainty quantification is often lacking in current applications. In theory, uncertainty quantification is related to the inverse Hessian (or the posterior covariance matrix). Even for common geophysical inverse problems its calculation is beyond the computational and storage capacities of the largest high-performance computing systems. In this study, we amend the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to perform uncertainty quantification for large-scale applications. For seismic inverse problems, the limited-memory BFGS (L-BFGS) method prevails as the most efficient quasi-Newton method. We aim to augment it further to obtain an approximate inverse Hessian for uncertainty quantification in FWI. To facilitate retrieval of the inverse Hessian, we combine BFGS (essentially a full-history L-BFGS) with randomized singular value decomposition to determine a low-rank approximation of the inverse Hessian. Setting the rank number equal to the number of iterations makes this solution efficient and memory-affordable even for large-scale problems. Furthermore, based on the Gauss-Newton method, we formulate different initial, diagonal Hessian matrices as preconditioners for the inverse scheme and compare their performances in elastic FWI applications. We highlight our approach with the elastic Marmousi benchmark model, demonstrating the applicability of preconditioned BFGS for large-scale FWI and uncertainty quantification.
• #### Shear wave velocity structure beneath Northeast China from joint inversion of receiver functions and Rayleigh wave group velocities: Implications for intraplate volcanism

(Wiley, 2021-09-17) [Preprint]
A high-resolution 3-D crustal and upper-mantle shear-wave velocity model of Northeast China is established by joint inversion of receiver functions and Rayleigh wave group velocities. The teleseismic data for obtaining receiver functions are collected from 107 CEA permanent sites and 118 NECESSArray portable stations. Rayleigh wave dispersion measurements are extracted from an independent tomographic study. Our model exhibits unprecedented detail in S-velocity structure. Particularly, we discover a low S-velocity belt at 7.5-12.5 km depth covering entire Northeast China (except the Songliao basin), which is attributed to a combination of anomalous temperature, partial melts and fluid-filled faults related to Cenozoic volcanism. Localized crustal fast S-velocity anomaly under the Songliao basin is imaged and interpreted as late-Mesozoic mafic intrusions. In the upper mantle, our model confirms the presence of low velocity zones below the Changbai mountains and Lesser Xing’an mountain range, which agree with models invoking sub-lithospheric mantle upwellings. We observe a positive S-velocity anomaly at 50-90 km depth under the Songliao basin, which may represent a depleted and more refractory lithosphere inducing the absence of Cenozoic volcanism. Additionally, the average lithosphere-asthenosphere boundary depth increases from 50-70 km under the Changbai mountains to 100 km below the Songliao basin, and exceeds 125 km beneath the Greater Xing’an mountain range in the west. Furthermore, compared with other Precambrian lithospheres, Northeast China likely has a rather warm crust (~480-970 °C) and a slightly warm uppermost mantle (~1200 °C), probably associated with active volcanism. The Songliao basin possesses a moderately warm uppermost mantle (~1080 °C).
• #### Fracture Permeability Estimation Under Complex Physics: A Data-Driven Model Using Machine Learning

(SPE, 2021-09-15) [Conference Paper]
Abstract The permeability of fractures, including natural and hydraulic, are essential parameters for the modeling of fluid flow in conventional and unconventional fractured reservoirs. However, traditional analytical cubic law (CL-based) models used to estimate fracture permeability show unsatisfactory performance when dealing with different dynamic complexities of fractures. This work presents a data-driven, physics-included model based on machine learning as an alternative to traditional methods. The workflow for the development of the data-driven model includes four steps. Step 1: Identify uncertain parameters and perform Latin Hypercube Sampling (LHS). We first identify the uncertain parameters which affect the fracture permeability. We then generate training samples using LHS. Step 2: Perform training simulations and collect inputs and outputs. In this step, high-resolution simulations with parallel computing for the Navier-Stokes equations (NSEs) are run for each of the training samples. We then collect the inputs and outputs from the simulations. Step 3: Construct an optimized data-driven surrogate model. A data-driven model based on machine learning is then built to model the nonlinear mapping between the inputs and outputs collected from Step 2. Herein, Artificial Neural Network (ANN) coupling with Bayesian optimization algorithm is implemented to obtain the optimized surrogate model. Step 4: Validate the proposed data-driven model. In this step, we conduct blind validation on the proposed model with high-fidelity simulations. We further test the developed surrogate model with newly generated fracture cases with a broad range of roughness and tortuosity under different Reynolds numbers. We then compare its performance to the reference NSEs solutions. Results show that the developed data-driven model delivers good accuracy exceeding 90% for all training, validation, and test samples. This work introduces an integrated workflow for developing a data-driven, physics-included model using machine learning to estimate fracture permeability under complex physics (e.g., inertial effect). To our knowledge, this technique is introduced for the first time for the upscaling of rock fractures. The proposed model offers an efficient and accurate alternative to the traditional upscaling methods that can be readily implemented in reservoir characterization and modeling workflows.
• #### CO2 Leakage Rate Forecasting Using Optimized Deep Learning

(SPE, 2021-09-15) [Conference Paper]
Abstract Geologic CO2 Sequestration (GCS) is a promising engineering technology to reduce global greenhouse emissions. Real-time forecasting of CO2 leakage rates is an essential aspect of large-scale GCS deployment. This work introduces a data-driven, physics-featuring surrogate model based on deep-learning technique for CO2 leakage rate forecasting. The workflow for the development of data-driven, physics-featuring surrogate model includes three steps: 1) Datasets Generation: We first identify uncertainty parameters that affect the objective of interests (i.e., CO2 leakage rates). For the identified uncertainty parameters, various realizations are then generated based on Latin Hypercube Sampling (LHS). High-fidelity simulations based on a two-phase black-oil solver within MRST are performed to generate the objective functions. Datasets including inputs (i.e., the uncertainty parameters) and outputs (CO2 leakage rates) are collected. 2) Surrogate Development: In this step, a time-series surrogate model using long short-term memory (LSTM) is constructed to map the nonlinear relationship between these uncertainty parameters as inputs and CO2 leakage rates as outputs. We perform Bayesian optimization to automate the tuning of hyperparameters and network architecture instead of the traditional trial-error tuning process. 3) Uncertainty Analysis: This step aims to perform Monte Carlo (MC) simulations using the successfully trained surrogate model to explore uncertainty propagation. The sampled realizations are collected in the form of distributions from which the probabilistic forecast of percentiles, P10, P50, and P50, are evaluated. We propose a data-driven, physics-featuring surrogate model based on LSTM for CO2 leakage rate forecasting. We demonstrate its performance in terms of accuracy and efficiency by comparing it with ground-truth solutions. The proposed deep-learning workflow shows promising potential and could be readily implemented in commercial-scale GCS for real-time monitoring applications.
• #### The potential of self-supervised networks for random noise suppression in seismic data

(arXiv, 2021-09-15) [Preprint]
Noise suppression is an essential step in any seismic processing workflow. A portion of this noise, particularly in land datasets, presents itself as random noise. In recent years, neural networks have been successfully used to denoise seismic data in a supervised fashion. However, supervised learning always comes with the often unachievable requirement of having noisy-clean data pairs for training. Using blind-spot networks, we redefine the denoising task as a self-supervised procedure where the network uses the surrounding noisy samples to estimate the noise-free value of a central sample. Based on the assumption that noise is statistically independent between samples, the network struggles to predict the noise component of the sample due to its randomnicity, whilst the signal component is accurately predicted due to its spatio-temporal coherency. Illustrated on synthetic examples, the blind-spot network is shown to be an efficient denoiser of seismic data contaminated by random noise with minimal damage to the signal; therefore, providing improvements in both the image domain and down-the-line tasks, such as inversion. To conclude the study, the suggested approach is applied to field data and the results are compared with two commonly used random denoising techniques: FX-deconvolution and Curvelet transform. By demonstrating that blind-spot networks are an efficient suppressor of random noise, we believe this is just the beginning of utilising self-supervised learning in seismic applications.
• #### The potential of self-supervised networks for random noise suppression in seismic data

(arXiv, 2021-09-15) [Preprint]
Noise suppression is an essential step in any seismic processing workflow. A portion of this noise, particularly in land datasets, presents itself as random noise. In recent years, neural networks have been successfully used to denoise seismic data in a supervised fashion. However, supervised learning always comes with the often unachievable requirement of having noisy-clean data pairs for training. Using blind-spot networks, we redefine the denoising task as a self-supervised procedure where the network uses the surrounding noisy samples to estimate the noise-free value of a central sample. Based on the assumption that noise is statistically independent between samples, the network struggles to predict the noise component of the sample due to its randomnicity, whilst the signal component is accurately predicted due to its spatio-temporal coherency. Illustrated on synthetic examples, the blind-spot network is shown to be an efficient denoiser of seismic data contaminated by random noise with minimal damage to the signal; therefore, providing improvements in both the image domain and down-the-line tasks, such as inversion. To conclude the study, the suggested approach is applied to field data and the results are compared with two commonly used random denoising techniques: FX-deconvolution and Curvelet transform. By demonstrating that blind-spot networks are an efficient suppressor of random noise, we believe this is just the beginning of utilising self-supervised learning in seismic applications.
• #### Cycle-skipping mitigation using misfit measurements based on differentiable dynamic time warping

(arXiv, 2021-09-09) [Preprint]
The dynamic time warping (DTW) distance has been used as a misfit function for wave-equation inversion to mitigate the local minima issue. However, the original DTW distance is not smooth; therefore it can yield a strong discontinuity in the adjoint source. Such a weakness does not help nonlinear inverse problems converge to a plausible minimum by any means. We therefore introduce for the first time in geophysics the smooth DTW distance, which has demonstrated its performance in time series classification, clustering, and prediction as the loss function. The fundamental idea of such a distance is to replace the $\min$ operator with its smooth relaxation. Then it becomes possible to define the analytic derivative of DTW distance. The new misfit function is entitled to the differentiable DTW distance. Moreover, considering that the warping path is an indicator of the traveltime difference between the observed and synthetic trace, a penalization term is constructed based on the warping path such that the misfit accumulation by the penalized differentiable DTW distance is weighted in favor of the traveltime difference. Numerical examples demonstrate the advantage of the penalized differentiable DTW misfit function over the conventional non-differentiable one.
• #### Dual-band generative learning for low-frequency extrapolation in seismic land data

(Society of Exploration Geophysicists, 2021-09-01) [Conference Paper]
The presence of low-frequency energy in seismic data can help mitigate cycle-skipping problems in full-waveform inversion. Unfortunately, the generation and recording of low-frequency signals in seismic exploration remains a non-trivial task. Extrapolation of missing low-frequency content in field data might be addressed in a data-driven framework. In particular, deep learning models trained on synthetic data could be used for inference on the field data. Such an implementation of switching application domains remains challenging. We, therefore, propose the concept of generative dual-band learning to facilitate the knowledge transfer between synthetic and field seismic data applications of low-frequency data extrapolation. We first explain the two-step procedure for training a generative adversarial network (GAN) that extrapolates low frequencies. Then, we describe the workflow for synthetic dataset generation. Finally, we explore the feasibility of the dual-band learning concept on real near-surface land data acquired in Saudi Arabia. The presence of low-frequency energy in seismic data can help mitigate cycle-skipping problems in full-waveform inversion. Unfortunately, the generation and recording of low-frequency signals in seismic exploration remains a non-trivial task. Extrapolation of missing low-frequency content in field data might be addressed in a data-driven framework. In particular, deep learning models trained on synthetic data could be used for inference on the field data. Such an implementation of switching application domains remains challenging. We, therefore, propose the concept of generative dual-band learning to facilitate the knowledge transfer between synthetic and field seismic data applications of low-frequency data extrapolation. We first explain the two-step procedure for training a generative adversarial network (GAN) that extrapolates low frequencies. Then, we describe the workflow for synthetic dataset generation. Finally, we explore the feasibility of the dual-band learning concept on real near-surface land data acquired in Saudi Arabia.
• #### High-dimensional wavefield solutions based on neural network functions

(Society of Exploration Geophysicists, 2021-09-01) [Conference Paper]
Waveﬁeld solutions are critical for applications ranging from imaging to full waveform inversion. These wave-ﬁelds are often large, especially for 3D media, and multiple p o int sources, like Green’s functions. A recently introduced framework based on neu ra l networks admit-ting functional solutions to partial differential equations(PDEs) o↵ers the opportunity to solve the Helmholtz equation with a neural network (NN) model. The input to such an NN is a location in space and the output are the real and imaginary parts of the scattered waveﬁeldat that location, thus, acting like a function. The net-work is trained on random input points in space and a variance of the Helmh o lt z equation for the scatteredwaveﬁeld is used as the loss function to update the network parameters. In spite of the methods ﬂexibility, like handling irregular surfaces and complex media, and its potential for velocity model building, the cost of training the network far exceeds that of numerical solutions. Relying on the network’s ability to learn waveﬁeld features, we extend the dimension of this NN function to learn the waveﬁeld for many sources and frequencies, simultaneously. We show, in this preliminary study, that reasonable waveﬁeld solutions can be predicted using smaller networks. This includes waveﬁelds for frequencies not within the training range. The new NN function has the potential to eﬃciently represent the waveﬁeld as a function of location in space, as well as source location and frequency.
• #### Self-supervised learning for random noise suppression in seismic data

(Society of Exploration Geophysicists, 2021-09-01) [Conference Paper]
A staunch companion to seismic signals, noise consistently hinders processing and interpretation of seismic data. Borrowing ideas from the field of computer vision, we propose the use of self-supervised deep learning for the task of random noise suppression. These techniques require no clean training data and therefore remove any requirement of pre-cleaning of field data or the generation of realistic synthetic datasets for training purposes. Through the use of blind-spot networks, we show that self-supervised Noise2Void (N2V) procedure can be adapted to the seismic context, and trained solely on noisy data. An initial validation performed on a synthetic dataset corrupted by additive, white, Gaussian noise confirms that N2V can be trained to accurately separate the correlated seismic signal from the uncorrelated noise. Furthermore, when correlated and random noise are both present in the data, whilst the model cannot remove the majority of the correlated noise, a portion of it is suppressed alongside the random noise. Finally, the network is validated on a field dataset that is heavily contaminated with strong random noise caused by the surface conditions. The N2V denoising approach is shown to drastically reduce the random noise in the data. Through these examples, we have validated the effectiveness of blind-spot networks on highly oscillating signals, such as seismic data. This pave the way for the application of other self-supervised procedures to seismic data that go beyond the assumption of statistically independent noise.
• #### Misfit functions based on differentiable dynamic time warping for waveform inversion

(Society of Exploration Geophysicists, 2021-09-01) [Conference Paper]
Misfit functions based on differentiable dynamic time warping (DTW) have demonstrated an excellent performance in various sequence-related tasks. We introduce this concept in the context of waveform inversion and discuss a fast algorithm to calculate the first and second derivatives of such misfits. The DTW distance is originally calculated by solving a series of min sorting problems, thus it is not differentiable. The fundamental idea of differentiable DTW is replacing the min operator with its smooth relaxation. It is then straightforward to solve the derivatives following the differentiation rules, which results in both the differentiable misfit measurements and warping path. This path weights the traveltime mismatch more than its amplitude counterpart. We can construct a penalization term based on this warping path. The penalized misfit function is adaptable to measure traveltime and amplitude separately. Numerical examples show the flexibility and the performance of the proposed misfit function in mitigating local minima issues and retrieving the long-wavelength model features
• #### Tomographic deconvolution of reflection tomograms

(Society of Exploration Geophysicists, 2021-09-01) [Conference Paper]
We present a tomographic deconvolution procedure for highresolution imaging of velocity anomalies between reflecting interfaces. The key idea is to first invert reflection or transmission traveltimes for the background velocity model. A convolutional neural network (CNN) model is then trained to estimate the inverse to the blurred tomogram consisting of small scatterers in the background velocity model. We call this CNN a tomographic deconvolution operator because it deconvolves the blurring artifacts in traveltime slowness tomograms. This procedure is similar to that of migration deconvolution which deconvolves the migration butterfly artifacts in migration images. Results with synthetic examples show the effectiveness of this procedure in significantly sharpening the tomographic images of small scatterers.