Geophysical Problems Research Articles

Estimation of CO 2 saturation maps from synthetic seismic data using a deep-learning method with a multi-scale approach

In the context of Carbon Capture and Storage (CCS), monitoring the behaviour of CO 2 within subsurface reservoirs is pivotal for efficient and safe storage operations. This study proposes a new approach to enhance the accuracy of predicting CO 2 saturation maps directly from shot gathers, using a combination of deep learning (DL) and feature extraction methods. We employ a U-Net model, specifically tailored to solve regression tasks, and two-dimensional continuous wavelet transform (2D-CWT) to analyse shot gathers at different scales. We introduce a novel approach using multi-channel input data for the DL model, combining shot gathers and CWT images. The DL model was trained and tested using both single channel and multichannel input. The single channel datasets included shot gathers and CWT images at the first scale, while the multi-channel datasets integrated shot gathers with either two or four scales of CWT. We conducted a sensitivity analysis on the number of training epochs to compare the performance of the model with multichannel and single channel input. Additionally, a Transfer Learning approach was implemented to improve performance in noisy conditions by leveraging knowledge from a pre-trained model on noise-free images. Monte Carlo dropout was used to predict pixel-scale variability and provided a prediction variability with a standard deviation ranging from 0.019 to 0.194, enhancing the robustness of CO 2 saturation estimates. Our results suggest that combining feature extraction using 2D-CWT with the U-Net architecture improves the prediction performance of CO 2 saturation in a synthetic CCS model. Additionally, using multi-channel input instead of single channel is more efficient as it requires less training and prediction time to achieve comparable results. This confirms the effectiveness of 2D-CWT as a pre-processing technique, contributing valuable insights and advances in the field of data-driven geophysical inversion problems.

Geostatistical Inversion of Frequency-Domain Electromagnetic Data for Near Surface Modelling

The detailed characterization of near-surface deposits is important for both environmental and economic reasons. These shallow subsurface systems can be very complex and heterogenous due to natural dynamics and anthropogenic interferences. Modeling techniques based exclusively on direct sampling generate limited informed three-dimensional models of the near-surface. Geophysical methods provide valuable and additional information to model the spatial distribution of the near-surface for locations where direct observations are not available. From this set of methods, frequency-domain electromagnetic induction (FDEM) has been successfully applied to image complex near-surface deposits. Yet, predicting the spatial distribution of relevant subsurface properties from geophysical data, and the integration of direct observations, is not straightforward. It requires solving a challenging geophysical inversion problem. Geostatistical modelling tools have been effectively applied to couple direct observations with geophysical data such as seismic reflection. We propose an iterative geostatistical FDEM inversion method able to integrate data from direct measurements of the near-surface with surface loop-loop FDEM measurements to simultaneously predict high-resolution models of electrical conductivity and magnetic susceptibility, and their associated uncertainty. The iterative geostatistical inversion method is based on stochastic sequential simulation and co-simulation as model perturbation and update techniques. The iterative optimization is based on the local data misfit between observed and simulated FDEM data, weighted by the sensitivity of the acquisition equipment. The proposed method is first demonstrated for a synthetic landfill data set created based on real data collected at a mine tailing disposal site in Portugal, and on a real data set collected at a site with archaeological features in Knowlton, UK. The results show the ability of the proposed method to accurately predict and characterize the spatial distribution of electrical conductivity and magnetic susceptibility down to the depth of interest while reproducing the recorded FDEM data.

Cross-fade sampling: extremely efficient Bayesian inversion for a variety of geophysical problems

SUMMARY This paper introduces cross-fade sampling, a computationally efficient Markov Chain Monte Carlo simulation method that uses a semi-analytical approach to quickly solve Bayesian inverse problems that do not themselves have an analytical solution. Cross-fading is efficient in two ways. First, it requires fewer samples to obtain the same quality simulation of the target probability density function (PDF). Secondly, it is much faster to evaluate the posterior probability of each sample than conventional sampling methods for simulating Bayesian posterior PDFs. Conventional methods require evaluating the prior probability (which describes your a priori constraints) and data likelihood (which describes the fit between the observations and the predictions of the model) for each sample model. However, cross-fading does not require evaluating the data likelihood, meaning that ‘big data’ can be fit with zero additional computational cost. Further, the cross-fading approach can be used to calculate the marginal likelihood associated with a model design, facilitating model comparison and Bayesian model averaging. Topics covered in this paper include derivation of the cross-fade approach and how it can be used to simulate Bayesian posterior PDFs and compute the marginal likelihood, discussion of the class of problems to which cross-fading can be applied (with examples from earthquake statistics, earthquake ground motion modelling, volcanic eruption forecasting, and finite fault slip modelling), demonstration of efficiency relative to existing sampling methods and discussion of how cross-fading can be used to account for prediction errors (i.e. epistemic errors) as part of the geophysical inverse problem.

Minimum entropy constrained cooperative inversion of electrical resistivity, seismic and magnetic data

Geophysical methods are widely used to gather information about the subsurface as they are non-intrusive and comparably cheap. However, the solution to the geophysical inverse problem is inherently non-unique, which introduces considerable uncertainties. As a partial remedy to this problem, independently acquired geophysical data sets can be jointly inverted to reduce ambiguities in the resulting multi-physical subsurface images. A novel cooperative inversion approach with joint minimum entropy constraints is used to create more consistent multi-physical images with sharper boundaries with respect to the single-method inversions. Here, this approach is implemented in an open-source software and its applicability on electrical resistivity tomography (ERT), seismic refraction tomography (SRT), and magnetic data is investigated. A synthetic 2D ERT and SRT data study is used to demonstrate the approach and to investigate the influence of the governing parameters. The findings showcase the advantage of the joint minimum entropy (JME) stabilizer over separate, conventional smoothness-constrained inversions. The method is then used to analyze field data from Rockeskyller Kopf, Germany. 3D ERT and magnetic data are combined and the results confirm the expected volcanic diatreme structure with improved details. The multi-physical images of both methods are consistent in some regions, as similar boundaries are produced in the resulting models. Because of its sensitivity to hydrologic conditions in the subsurface, observations suggest that the ERT method senses different structures than the magnetic method. These structures in the ERT result do not seem to be enforced on the magnetic susceptibility distribution, showcasing the flexibility of the approach. Both investigations outline the importance of a suitable parameter and reference model selection for the performance of the approach and suggest careful parameter tests prior to the joint inversion. With proper settings, the JME inversion is a promising tool for geophysical imaging, however, this work also identifies some objectives for future studies and additional research to explore and optimize the method.

Bayesian neural network and Bayesian physics-informed neural network via variational inference for seismic petrophysical inversion

Bayesian neural network via variational inference (BNN-VI) aims to estimate probability distribution of the training parameters of the neural network, which are difficult to compute with traditional methods, by proposing a family of densities and finding the candidates that are close to the target. Variational inference is a Bayesian method that provides a faster tool than Markov chain Monte Carlo sampling algorithms. However, to provide an accurate estimation of the most likely models, BNN-VI requires sufficient training data, which is not always available in geophysical inverse problems. We propose a method that combines a Bayesian approach with a physics-informed neural networks algorithms via variational inference, namely Bayesian physics-informed neural networks via variational inference (BPINN-VI) and apply it to a geophysical inverse problem for the estimation of petrophysical properties from seismic data. The method is based on a Bayesian network approach in which the objective function is defined by the Kullback–Leibler (KL) divergence. In addition, the physical relations between data (seismic measurements) and model variables (petrophysical properties) are embedded into neural networks through a physics-dependent loss term. We tested the proposed method on a pre-stack seismic dataset from the North Sea with limited direct measurements of the target petrophysical properties from well logs. The outcome is the most likely model of petrophysical properties. Compared to BNN-VI, the BPINN-VI results show higher correlation and R2 coefficients, and lower mean square error as well as lower uncertainty and higher lateral continuity.

Streaming prediction-error filters

Prediction-error filters (PEFs) are essential for seismic deconvolution and other geophysical estimation problems. We find that nonstationary multidimensional PEFs can be computed in a “streaming” manner, wherein the filter is updated incrementally by accepting one new data point at a time. The computational cost of estimating a streaming PEF is reduced to the cost of a single convolution. In other words, the cost of the PEF design while filtering is equivalent to the cost of applying the filter. Moreover, the nonlinear operation of finding and applying a streaming PEF is invertible at a similar cost, which enables a fast approach for missing data interpolation.

Towards a new standard for seismic moment tensor inversion containing 3-D earth structure uncertainty

SUMMARY Moment tensor (MT) inversion is a classical geophysical inverse problem that infers a force-equivalent model of a seismic source from seismological observations. Like other inverse problems, the accuracy of the inversion depends on the reliability of the forward problem simulating waveforms from the source location through an Earth structural model. Apart from errors in data, the error in forward waveform simulation, also known as theory error, is a significant source of error contributing to the misfit function between the predicted and observed waveforms. Here, we set up numerical experiments to comprehensively probe the sensitivity of the linearized MT inversion to 3-D regional earth model errors, a known predominant factor of the theory error. Using the Monte Carlo method, we estimate the empirical structural covariance matrices to characterize the waveform mismatch due to the imperfect knowledge of Earth's structure. First, although the inversion accuracy deteriorates with increasing model errors, incorporating the structural covariance matrices into the misfit function improves the accuracy of inversion results for all theorized error distributions. Secondly, we propose a slightly modified form of the structural covariance matrix, which further enhances the inversion outcome. Lastly, as the true structural errors are likely spatially correlated, we highlight the importance of adequately treating the correlation into the MT inversion because of its significant impact on inversion. Overall, as a preliminary effort in quantifying 3-D structural errors on MT inversion, this study proves the computational feasibility by means of numerical experiments and will hopefully provide a way forward for future work on this topic.

Uncertainty Quantification in Radial Anisotropy Models Based on Transdimensional Bayesian Inversion of Receiver Functions and Surface-Wave Dispersion: Case Study Sri Lanka

ABSTRACT In geophysical inference problems, quantification of data uncertainties is required to balance the data-fitting ability of the model and its complexity. The transdimensional hierarchical Bayesian approach is a powerful tool to evaluate the level of uncertainty and determine the complexity of the model by treating data errors and model dimensions as unknown. In this article, we take account of the uncertainty through the whole procedure, thus developing a two-step fully Bayesian approach with coupled uncertainty propagation to estimate the crustal isotropic and radial anisotropy (RA) model based on Rayleigh and Love dispersion as well as receiver functions (RFs). First, 2D surface-wave tomography is applied to determine period-wise ambient noise phase velocity maps and their uncertainty for Rayleigh and Love waves. Probabilistic profiles of the isotropic average VS and RA as a function of depth are then derived at station sites by inverting the local surface-wave dispersion and model errors and RFs jointly. The workflow is applied to a temporary seismic broadband array covering all of Sri Lanka. The probabilistic results enable us to effectively quantify the uncertainty of the final RA model and provide robust inferences. The shear-wave velocity results show that the range of Moho depths is between 30 and 40 km, with the thickest crust (38–40 km) beneath the central Highland Complex. Positive RA (VSH>VSV) observed in the upper crust is attributed to subhorizontal alignment of metamorphic foliation and stretched layers resulting from deformation. Negative RA (VSV>VSH) in the midcrust of central Sri Lanka may indicate the existence of melt inclusions and could result from the uplift and folding process. The positive RA in the lower crust could be caused by crustal channel flow in a collision orogeny.

Charakterystyka złoża formacji Baharyia, pola naftowe Neag-1, 2 i 3, Pustynia Zachodnia, Egipt

An integration was achieved between different bore holes and laboratory measured data using several petrophysical parameters of the Baharyia Formation encountered in Neag-1,2&3 oil fields. It illustrates the key control factors affecting the Baharyia reservoir quality. The obtained petrophysical relationships could be used widely in both exploration geophysics and hydrocarbon reservoir production. It provides and demonstrates solutions for both geological and geophysical engineering problems. The measured porosity and permeability are ranging from 2.5 to 32 % and 0.005 to 874 mD respectively. The influence of diagenesis on both reservoir porosity and permeability has been investigated. Pore filling minerals has been classified into four classes by XRD- analysis technique. A reliable regression equation was reached between reservoir permeability and mineral pore fillings. Several relationships among rock permeability, porosity and density obtained from open hole logs were recognized. The pore throat distribution has been laboratory measured by use of MICP technique for some selected samples. The calculated reservoir storage and flow capacity indicate four major fluid flow types which are controlled by the variations in reservoir pore space framework. Formation resistivity factor – porosity relation was accomplished under reservoir conditions, while the Archie’s 2nd equation was outlined. The Archie’s parameters (a, m &n) were calculated for shaly and clean sandstones of the Baharyia Formation. Both cation exchange capacity (CEC), Mounce potential (MP) and mercury injection capillary pressure (MICP) were measured to distinguish reservoir facies.

Gradient-based joint inversion of point-source moment tensor and station-specific time-shifts

SUMMARY The misalignment of the observation and predicted waveforms in regional moment tensor inversion is mainly due to seismic models’ incomplete representation of the Earth's heterogeneities. Current moment tensor inversion techniques, allowing station-specific time-shifts to account for the model error, are computationally expensive. Here, we propose a gradient-based method to jointly invert moment-tensor parameters, centroid depth and unknown station-specific time-shifts utilizing the modern functionalities in deep learning frameworks. A $L_2^2$ misfit function between predicted synthetic and time-shifted observed seismograms is defined in the spectral domain, which is differentiable to all unknowns. The inverse problem is solved by minimizing the misfit function with a gradient descent algorithm. The method's feasibility, robustness and scalability are demonstrated using synthetic experiments and real earthquake data in the Long Valley Caldera, California. This work presents an example of fresh opportunities to apply advanced computational infrastructures developed in deep learning to geophysical problems.

Combining “Deep Learning” and Physically Constrained Neural Networks to Derive Complex Glaciological Change Processes from Modern High-Resolution Satellite Imagery: Application of the GEOCLASS-Image System to Create VarioCNN for Glacier Surges

The objectives of this paper are to investigate the trade-offs between a physically constrained neural network and a deep, convolutional neural network and to design a combined ML approach (“VarioCNN”). Our solution is provided in the framework of a cyberinfrastructure that includes a +newly designed ML software, GEOCLASS-image (v1.0), modern high-resolution satellite image data sets (Maxar WorldView data), and instructions/descriptions that may facilitate solving similar spatial classification problems. Combining the advantages of the physically-driven connectionist-geostatistical classification method with those of an efficient CNN, VarioCNN provides a means for rapid and efficient extraction of complex geophysical information from submeter resolution satellite imagery. A retraining loop overcomes the difficulties of creating a labeled training data set. Computational analyses and developments are centered on a specific, but generalizable, geophysical problem: The classification of crevasse types that form during the surge of a glacier system. A surge is a glacial catastrophe, an acceleration of a glacier to typically 100–200 times its normal velocity. GEOCLASS-image is applied to study the current (2016-2024) surge in the Negribreen Glacier System, Svalbard. The geophysical result is a description of the structural evolution and expansion of the surge, based on crevasse types that capture ice deformation in six simplified classes.

A robust numerical method based on a deep-learning operator for solving the 2D acoustic wave equation

The numerical solution of a wave equation plays a crucial role in computational geophysics problems, which forms the foundation of inverse problems and directly impacts the high-precision imaging results of earth models. However, common numerical methods often lead to significant computational and storage requirements. Due to the heavy reliance on forward-modeling methods in inversion techniques, particularly full-waveform inversion, enhancing the computational efficiency and reducing the storage demands of traditional numerical methods have become key issues in computational geophysics. We develop the deep Lax-Wendroff correction (DeepLWC) method, a deep-learning-based numerical format for solving 2D hyperbolic wave equations. DeepLWC combines the advantages of traditional numerical schemes with a deep neural network. We provide a detailed comparison of this method with the representative traditional Lax-Wendroff correction method. Our numerical results indicate that the DeepLWC significantly improves the calculation speed (by more than 10 times) and reduces the space needed for storage by more than 10,000 times compared with traditional numerical methods. In contrast to the more popular physics-informed neural network method, DeepLWC maximizes the advantages of traditional mathematical methods in solving partial differential equations and uses a new sampling approach, leading to improved accuracy and faster computations. It is particularly worth pointing out that DeepLWC introduces a novel research paradigm for numerical equation solving, which can be combined with various traditional numerical methods, enabling acceleration and reduction in the storage requirements of conventional approaches.

Choosing Appropriate Regularization Parameters by Splitting Data into Training and Validation Sets—Application in Global Surface-Wave Tomography

Abstract Many linear(ized) geophysical inverse problems cannot be solved without regularization. Finding the regularization parameter that best balances the model complexity and data misfit is often a key step in the inversion problem. Traditionally, this is done by first plotting the measure of model complexity versus data misfit for different values of regularization parameter, which manifests as an L-shaped curve, and then choosing the regularization parameter corresponding to the corner point on the L-curve. For this approach, the difference in units between model complexity and data misfit must be considered, otherwise the result will be strongly affected by the scaling between these two quantities. Inspired by the machine learning literature, we here propose an extension to the traditional L-curve method. We first split the raw dataset into training and validation sets, obtain a solution by performing inversion on the training set only, and calculate data misfits on the validation set. We demonstrate the efficacy of this approach with a toy example and with two synthetic datasets. In realistic global surface-wave tomography studies where sampling of the Earth is nonuniform, we devise a procedure to generate a validation dataset with sampling as uniform as possible. We then show that the regularization parameter can be determined using this validation set, and this determination is apparently robust to the ratio of data split between training and validation sets. For both synthetic tests and realistic inversions, we find that our procedure can produce a minimal point that can be easily identified on the misfit curves calculated on the validation sets, and avoids the nuances encountered in the traditional L-curve analysis.

A depth-variant wavelet extraction method based on the orthogonal matching pursuit for direct inversion

The results of pre-stack depth migration reflect subsurface structures directly. However, it is an urgent geophysical problem that how to deal with this non-stationarity exhibited by depth-domain seismic data and achieve high precision reservoir prediction. To address this issue, we develop a depth-variant wavelet extraction method based on the orthogonal matching pursuit and attain direct inversion of the depth-domain acoustic impedance using the depth-variant wavelets. First, the parameters of source wavelets estimated by complex seismic trace analysis are utilized to construct the overcomplete dictionary. Secondly, the overcomplete dictionary is used to decompose the depth-domain seismic traces by orthogonal matching pursuit. Thirdly, we extract the depth-variant Ricker wavelets from the decomposed optimal atoms. Test of examples demonstrates that the proposed depth-variant wavelet method has high accuracy. The synthetic seismogram obtained by the proposed method has high Pearson correlation coefficients of 99%, which verifies that our method is feasible and accurate. Next, a depth-domain impedance trend constraint is introduced into the objective function of the conventional basis pursuit inversion to enhance the lateral continuity. Finally, the objective function with impedance trend constraint is solved by basis pursuit for depth-domain acoustic impedance. We test the direct depth-domain acoustic impedance inversion method on the synthetic and field data, which indicates that our method has high accuracy and robustness. The mean relative error of its inversion result is only 0.9% for the noise-free example and about 5% for the strong noisy example. Therefore, our method can be effectively applied to the seismic-well calibration and seismic impedance inversion in the depth domain. And it has powerful potential in reservoir characterization, fluid prediction and attribute extraction in the depth domain. Additionally, the energy distribution features of the Ricker wavelet are derived and analyzed, which can guide the application of the Ricker wavelet in seismic signal analysis.

Deep generative networks for multivariate fullstack seismic data inversion using inverse autoregressive flows

The simultaneous prediction of the subsurface distribution of facies and acoustic impedance (IP) from fullstack seismic data requires solving an inverse problem and is fundamental in natural resources exploration, carbon capture and storage, and environmental risk management. In recent years, deep generative models (DGM), such as variational autoencoders (VAE) and generative adversarial networks (GAN), were proposed to reproduce complex facies patterns honoring prior geological information. Variational Bayesian inference using inverse autoregressive flows (IAF) can be performed to infer the solution to a geophysical inverse problem from the encoded latent space of such pre-trained DGM. Successful applications of such approach on crosshole ground-penetrating radar synthetic data inversion demonstrated that the technique's accuracy is comparable to that of Markov chain Monte Carlo (MCMC) inference methods, while significantly reducing the computational cost. Nonetheless, these application examples did not account for the spatial uncertainty affecting the facies-dependent continuous physical property, from which the geophysical data are calculated. This uncertainty can significantly affect the inversion accuracy and its applicability to real data. In this work, specific VAE and GAN architectures are proposed to simultaneously predict facies and co-located IP, while accounting for their spatial uncertainties. The two types of generative networks are used in Bayesian inversion with IAF for the inversion of seismic data. The results are found to reproduce the statistics of the training images and solve the seismic inversion problem accurately, comparably to MCMC inversion. Furthermore, advantages and limitations of the two DGMs are evaluated by comparing the results obtained.

A matching pursuit approach to the geophysical inverse problem of seismic travel time tomography under the ray theory approximation

Summary Seismic travel time tomography is a geophysical imaging method to infer the 3-D interior structure of the solid Earth. Most commonly formulated as a linearized inverse problem, it maps differences between observed and expected wave travel times to interior regions where waves propagate faster or slower than the expected average. The Earth’s interior is typically parametrized by a single kind of localized basis function. Here we present an alternative approach that uses matching pursuits on large dictionaries of basis functions. Within the past decade the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed. They combine global and local trial functions. An approximation is built in a so-called best basis, chosen iteratively from an intentionally overcomplete set or dictionary. In each iteration, the choice for the next best basis element reduces the Tikhonov–Phillips functional. This is in contrast to classical methods that use either global or local basis functions. The LIPMPs have proven their applicability in inverse problems like the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging. Here, we remodel the Learning Regularized Functional Matching Pursuit (LRFMP), which is one of the LIPMPs, for travel time tomography in a ray theoretical setting. In particular, we introduce the operator, some possible trial functions and the regularization. We show a numerical proof of concept for artificial travel time delays obtained from a contrived model for velocity differences. The corresponding code is available online.

GeoTaichi: A Taichi-powered high-performance numerical simulator for multiscale geophysical problems

This study introduces GeoTaichi, an open-source high-performance numerical simulator designed for addressing multiscale geophysical problems. By leveraging the power of the Taichi parallel language, GeoTaichi maximizes the utilization of modern computer resources on multicore CPU and GPU architectures. It offers robust and reliable modules for the discrete element method (DEM), material point method (MPM), and coupled material point-discrete element method (MPDEM). These modules enable efficient solving of large-scale problems while being implemented in pure Python. The design philosophy of GeoTaichi focuses on creating a framework that is readable, extensible, and user-friendly. This paper highlights the coupling procedure of MPDEM, the code structures, and the most important features of GeoTaichi. Rigorous benchmark tests have been conducted to verify the validity and robustness of GeoTaichi. Additionally, the performance of GeoTaichi is compared with similar software tools in the field, underscoring a notable improvement in both computational efficiency and memory savings when compared to existing alternatives. Program summaryProgram title: GeoTaichiCPC Library link to program files:https://doi.org/10.17632/858bmcf7j6.1Developer's repository link:https://github.com/Yihao-Shi/GeoTaichiLicensing provisions: GNU General Public License v3.0Programming language: PythonNature of problem: The simulations of large-deformation geophysical flows and their interaction with structures play a crucial role in the field of geophysics. To address the complexities of these nonlinear problems, the discrete element method (DEM), material point method (MPM), and their coupling (MPDEM) have proven to be highly suitable numerical schemes. However, these schemes impose substantial computational demands, necessitating the development of an efficient framework that can harness modern computer resources on multicore CPU and GPU architectures.Solution method: The open-source code GeoTaichi implements the DEM, MPM, and coupled MPDEM, encompassing a range of constitutive models and contact laws for different geologic materials. The clump particle model is also introduced in DEM to solve granular mechanics involving complex-shaped particles. One significant advantage of GeoTaichi is its utilization of the Taichi parallel language, which is designed to be user-friendly and easily extensible for customized applications.

Multimodal surface wave inversion with automatic differentiation

SUMMARY Investigating subsurface shear wave velocity (vs) structures using surface wave dispersion data involves minimizing a misfit function that is commonly solved through gradient-based optimization. Sensitivity kernels for model updates are commonly estimated using numerical differentiation, variational methods or implicit functions which however, may involve numerical instability and computational challenges when dealing with complex velocity models and large data sets. In this study, we propose a novel surface wave inversion framework in which error-free gradients are calculated by automatic differentiation (AD) and forward modelling is implemented by convenient computational graphs in the state-of-the-art deep learning framework. The AD-based inversion approach is first validated using two synthetic data sets. Then, the subsurface structures at three distinct locations, namely the Great Plains and the Long Beach in the US and Tong Zhou in China, are also derived using this method with seismic ambient noise data, which show nice consistency with those obtained using traditional methods. With the significantly improved computational efficiency, a great number of initial models can be inverted simultaneously to mitigate the impact of local minima and to estimate the uncertainty in the invert models. We have developed a new surface wave inversion package named ADsurf based on automatic differentiation and computational graphs in the deep learning framework, and its computational efficiency is also compared with the traditional finite-difference-based gradient estimation approach. While a great number of intriguing studies on the geophysical inverse problems have been conducted recently using deep learning for end-to-end mapping, the use of AD provided in the in the deep learning frameworks to assist and expedite the gradient computations are still underexploited in geophysics. Thus, it is expected that various geophysical inverse problems in many different areas beyond the surface wave inversion can also be tackled with this new paradigm in the future.

A weakly inhomogeneous vibrating membrane and the solotone effect in two dimensions

Abstract The phrase solotone effect refers to a persistent irregular pattern within a set of eigenvalues. This effect arises when the underlying differential equations contain discontinuities. Models of the Earth contain such discontinuities and research into solotones was originally motivated by vibration problems in geophysics. More recently, solotone effects have been discovered within heat conduction, and within quantum wells. However, at present, only one-dimensional arrangements have been shown to give rise to a solotone effect. In this paper, we compute the eigenvalues of an inhomogeneous vibrating rectangular membrane. The resulting eigenspectra, when taken as a whole, fail to display a solotone effect. Nevertheless, we show that irregular patterns do exist within selected subsets of eigenvalues. Our analysis therefore provides the first example of a solotone effect in a spatial dimension greater than one. Furthermore, this discovery could be used to extend the solotone inverse method already implemented in one dimension.

Surface loading on a self-gravitating, linear viscoelastic Earth: moving beyond Maxwell

SUMMARY Constitutive laws are a necessary ingredient in calculations of glacial isostatic adjustment (GIA) or other surface loading problems (e.g. loading by ocean tides). An idealized constitutive law governed by the Maxwell viscoelastic model is widely used but increasing attention is being directed towards more intricate constitutive laws that, in particular, include transient rheology. In this context, transient rheology collectively refers to dissipative mechanisms activated in addition to creep modelled by the Maxwell viscoelastic model. Consideration of such viscoelastic models in GIA is in its infancy and to encourage their wider use, we present constitutive laws for several experimentally derived transient rheologies and outline a flexible method in which to incorporate them into geophysical problems, such as the viscoelastic deformation of the Earth induced by surface loading. To further motivate this need, we demonstrate, via the Love number collocation method, how predictions of crustal displacement depart significantly between Earth models that adopt only Maxwell viscoelasticity and those with transient rheology. Throughout this paper, we highlight the differences in terminology and emphases between the rock mechanics, seismology and GIA communities, which have perhaps contributed towards the relative scarcity in integrating this broader and more realistic class of constitutive laws within GIA. We focus on transient rheology since the associated deformation has been demonstrated to operate on timescales that range from hours to decades. With ice mass loss enhanced at similar timescales as a consequence of anthropogenically caused climate change, the ability to model GIA with more accurate constitutive laws is an important tool to investigate such problems.