Forward Operator Research Articles

Image-domain seismic inversion by deblurring with invertible recurrent inference machines

In complex geologic settings and in the presence of sparse acquisition systems, seismic migration images manifest as nonstationary blurred versions of the unknown subsurface model. Thus, image-domain deblurring is an important step to produce interpretable and high-resolution models of the subsurface. Most deblurring methods focus on inverting seismic images for their underlying reflectivity by iterative least-squares inversion of a local Hessian approximation; this is obtained by either direct modeling of the so-called point-spread functions (PSFs) or by a migration-demigration process. In this work, we adopt a novel deep-learning (DL) framework, based on invertible recurrent inference machines (i-RIMs), which allows approaching any inverse problem as a supervised learning task informed by the known modeling operator (convolution with PSFs in our case): our algorithm can directly invert migrated images for impedance perturbation models, assisted with the prior information of a smooth velocity model and the modeling operator. Because i-RIMs are constrained by the forward operator, they implicitly learn to shape/regularize output models in a training-data-driven fashion. As such, the resulting deblurred images indicate great robustness to noise in the data and spectral deficiencies (e.g., due to limited acquisition). The key role played by the i-RIM network design and the inclusion of the forward operator in the training process is supported by several synthetic examples. Finally, using field data, we find that i-RIM-based deblurring has great potential in yielding robust, high-quality relative impedance estimates from migrated seismic images. Our approach could be of importance toward future DL-based quantitative reservoir characterization and monitoring.

Koszul duality, minimal model and L∞-structure for differential algebras with weight

A differential algebra with weight is an abstraction of both the derivation (weight zero) and the forward and backward difference operators (weight ±1). In 2010 Loday established the Koszul duality for the operad of differential algebras of weight zero. He did not treat the case of nonzero weight, noting that new techniques are needed since the operad is no longer quadratic. This paper continues Loday's work and establishes the Koszul duality in the case of nonzero weight. In the process, the minimal model and the Koszul dual homotopy cooperad of the operad governing differential algebras with weight are determined. As a consequence, a notion of homotopy differential algebras with weight is obtained and the deformation complex as well as its L∞-algebra structure for differential algebras with weight are deduced.

Joint 1DVar retrievals of tropospheric temperature and water vapor from Global Navigation Satellite System radio occultation (GNSS-RO) and microwave radiometer observations

Abstract. Global Navigation Satellite System radio occultation (GNSS-RO) and microwave radiometry (MWR) are two of the most impactful spaceborne remote sensing techniques for numerical weather prediction (NWP). These two techniques provide complementary information about atmospheric temperature and water vapor structure. GNSS-RO provides high vertical resolution measurements with cloud penetration capability, but the temperature and moisture are coupled in the GNSS-RO retrieval process and their separation requires the use of a priori information or auxiliary observations. On the other hand, the MWR measures brightness temperature (Tb) in numerous frequency bands related to the temperature and water vapor structure but is limited by poor vertical resolution (> 2 km) and precipitation. In this study, we combine these two technologies in an optimal estimation approach, 1D variation method (1DVar), to improve the characterization of the complex thermodynamic structures in the lower troposphere. This study employs both simulated and operational observations. GNSS-RO bending angle and MWR Tb observations are used as inputs to the joint retrieval, where bending can be modeled by an Abel integral and Tb can be modeled by a radiative transfer model (RTM) that takes into account atmospheric absorption, as well as surface reflection and emission. By incorporating the forward operators into the 1DVar method, the strength of both techniques can be combined to bridge individual weaknesses. Applying 1DVar to the data simulated from large eddy simulation (LES) is shown to reduce GNSS-RO temperature and water vapor retrieval biases at the lower troposphere while simultaneously capturing the fine-scale variability that MWR cannot resolve. A sensitivity analysis is also conducted to quantify the impact of the a priori information and error covariance used in different retrieval scenarios. The applicability of 1DVar joint retrieval to the actual GNSS-RO and MWR observations is also demonstrated through combining collocated COSMIC-2 and Suomi-NPP (National Polar-orbiting Partnership) measurements.

Simultaneous Reconstruction of Speed of Sound and Nonlinearity Parameter in a Paraxial Model of Vibro-Acoustography in Frequency Domain

Abstract In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton’s method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton’s method for its solution.

Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators

The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ∆h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ∆h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ∆h are also proved by demonstrating that the ∆h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ∆h bivariate Appell polynomials is provided, and symmetric identities for the ∆h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ∆h bivariate Appell polynomials. Additionally, generating relations for the ∆h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials.

Linearized waveform inversion for vertical transversely isotropic elastic media: Methodology and multi-parameter crosstalk analysis

Seismic migration and inversion are closely related techniques to portray subsurface images and identify hydrocarbon reservoirs. Seismic migration aims at obtaining structural images of subsurface geologic discontinuities. More specifically, seismic migration estimates the reflectivity function (stacked average reflectivity or pre-stack angle-dependent reflectivity) from seismic reflection data. On the other hand, seismic inversion quantitatively estimates the intrinsic rock properties of subsurface formulations. Such seismic inversion methods are applicable to detect hydrocarbon reservoirs that may exhibit lateral variations in the inverted parameters. Although there exist many differences, pre-stack seismic migration is similar with the first iteration of the general linearized seismic inversion.Usually, seismic migration and inversion techniques assume an acoustic or isotropic elastic medium. Unconventional reservoirs such as shale and tight sand formation have notable anisotropic property. We present a linearized waveform inversion (LWI) scheme for weakly anisotropic elastic media with vertical transversely isotropic (VTI) symmetry. It is based on two-way anisotropic elastic wave equation and simultaneously inverts for the localized perturbations (ΔVp0/Vp0,ΔVs0/Vs0,Δϵ,Δδ) from the long-wavelength reference model. Our proposed VTI-elastic LWI is an iterative method that requires a forward and an adjoint operator acting on vectors in each iteration. We derive the forward Born approximation operator by perturbation theory and adjoint operator via adjoint-state method. The inversion has improved the quality of the images and reduces the multi-parameter crosstalk comparing with the adjoint-based images. We have observed that the multi-parameter crosstalk problem is more prominent in the inversion images for Thomsen anisotropy parameters. Especially, the Thomsen parameter δ is the most difficult to resolve. We also analyze the multi-parameter crosstalk using scattering radiation patterns.The linearized waveform inversion for VTI-elastic media presented in this article provides quantitative information of the rock properties that has the potential to help identify hydrocarbon reservoirs.

Impact of HY‐2B SMR radiance assimilation on CMA global medium‐range weather forecasts

AbstractThe Chinese second ocean dynamic environment satellite Haiyang‐2B (HY‐2B) was successfully launched on October 25, 2018, carrying a scanning microwave radiometer (SMR) to provide information about the ocean and atmosphere. For the first time, this study investigated the impact of radiance data from the HY‐2B SMR in the four‐dimensional variational (4DVar) data assimilation system of the Global Forecast System developed by the China Meteorology Administration (CMA‐GFS). Prior to the radiance assimilation, we evaluated the data quality and developed suitable quality control (QC) procedures for the HY‐2B SMR observations. In addition, the cloud liquid water path (CLWP) and ocean wind speed (OWS) were introduced as new bias correction (BC) predictors to remove the systematic bias originating from the forward operator in the CMA‐GFS 4DVar system. The assimilation effects of SMR observations were evaluated through one‐month cycling experiments. Verification with independent observations and reanalysis data has demonstrated that assimilating SMR clear‐sky radiance can greatly reduce the analysis errors for temperature and humidity, while indirectly improving the quality of wind analysis. Furthermore, significant improvements in geopotential height forecasts were found for days 1–5 in the lower troposphere of the tropical region.

Efficient learning methods for large-scale optimal inversion design

In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $ p $ and $ q $ norms for $ {\rm L}^p-{\rm L}^q $ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.

Deep-blur: Blind identification and deblurring with convolutional neural networks

Abstract We propose a neural network architecture and a training procedure to estimate blurring operators and deblur images from a single degraded image. Our key assumption is that the forward operators can be parameterized by a low-dimensional vector. The models we consider include a description of the point spread function with Zernike polynomials in the pupil plane or product-convolution expansions, which incorporate space-varying operators. Numerical experiments show that the proposed method can accurately and robustly recover the blur parameters even for large noise levels. For a convolution model, the average signal-to-noise ratio of the recovered point spread function ranges from 13 dB in the noiseless regime to 8 dB in the high-noise regime. In comparison, the tested alternatives yield negative values. This operator estimate can then be used as an input for an unrolled neural network to deblur the image. Quantitative experiments on synthetic data demonstrate that this method outperforms other commonly used methods both perceptually and in terms of SSIM. The algorithm can process a 512 $ \times $ 512 image under a second on a consumer graphics card and does not require any human interaction once the operator parameterization has been set up.1

Improving the representation of the atmospheric boundary layer by direct assimilation of ground‐based microwave radiometer observations

AbstractIn a joint effort, MeteoSwiss and Deutscher Wetterdienst (DWD) address the need for improving the initial state of the atmospheric boundary layer (ABL) by exploiting ground‐based profiling observations that aim to fill the existing observational gap in the ABL. We implemented brightness‐temperature observations from ground‐based microwave radiometers (MWRs) in our data assimilation systems using a local ensemble transform Kalman filter (LETKF) with Radiative Transfer for TIROS Operational Vertical Sounder, ground‐based (RTTOV‐gb) as a forward operator. We were able to obtain a positive impact on the brightness temperature first guess and analysis, as well as a slight impact on the ABL humidity, using two MWRs at MeteoSwiss. These results led to a subsequent operational implementation of the observing system at MeteoSwiss. Furthermore, we performed an extensive set of assimilation experiments at DWD to investigate further various aspects such as the vertical localisation of selected single channels. We obtained a positive impact on the 6‐hr forecast of ABL temperature and humidity by assimilating two channels employing a dynamical localisation based on the sensitivity functions of RTTOV‐gb but also with a static localisation in a single‐channel setup. Our experiments indicate the importance of vertical localisation when using more than one channel, although reliable improvements are challenging to obtain without a larger number of observations for both assimilation and verification.

Identifiability of the parameters contained in a cyclic cohesive zone model for CFRP-to-steel bonded joints

The application of externally bonded CFRP reinforcements has shown its effectiveness in reducing crack growth and extending fatigue life of steel elements subjected to cyclic loadings. Failure usually occurs due to cohesive debonding within the adhesive interface and therefore the interfacial behavior is crucial in guaranteeing the effectiveness of the bonded system. The adoption of cohesive zone models represents a valid approach for the description of interfaces between two adherends (i.e. CFRP and steel substrate), providing a useful tool for the analytical and numerical investigation of CFRP-to-steel bonded systems. An important issue generally associated with the use of a cohesive model is the correct calibration of its parameters, necessary to guarantee a reliable use of the model. Therefore, this work proposes a robust inverse analysis procedure to investigate the identifiability of the parameters governing the fatigue behavior of an exponential cyclic cohesive zone model. Single-lap direct shear tests are considered for the numerical investigation of the interfacial bond behavior. The input data chosen for the inverse algorithm are the axial strain measurements in a discrete number of points along the bonded interface and the values of relative displacement between the two adherends measured at peak and valley of each cycle at the specimen loaded end (i.e. the global slip). Virtual data perturbed by different levels of noise are used and a meta-model reduction technique is adopted to reduce the computational cost of the forward operator and to solve the inverse problem in a stochastic context through a Monte Carlo like procedure.

Adaptive minimax optimality in statistical inverse problems via SOLIT—Sharp Optimal Lepskiĭ-Inspired Tuning

We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form fˆα=qαT∗TT∗Y , where Y is the available data, T the forward operator, qαα∈A an ordered filter, and α > 0 a regularization parameter. Whenever such a method is used in practice, α has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible α in the sense that mean squared error (MSE) E[∥fˆα−f†∥2] w.r.t. the true solution f† is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an a posteriori parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on Y and the noise level σ as well as the operator T and the filter qαα∈A and does not require any problem-dependent tuning of further parameters. We prove an oracle inequality for the corresponding MSE in a general setting and derive the rates of convergence in different scenarios. By a careful analysis we show that no other a posteriori parameter choice rule can yield a better performance in terms of the order of the convergence rate of the MSE. In particular, our results reveal that the typical understanding of Lepskiĭ-type methods in inverse problems leading to a loss of a log factor is wrong. In addition, the empirical performance of SOLIT is examined in simulations.

Dirichlet systems with discrete relativistic operator

AbstractWe are concerned with Dirichlet systems involving the relativistic discrete operator Here, for , we denote by the forward difference operator. Besides a “universal” existence result for a system with a general nonlinearity, we obtain multiplicity of solutions for systems with parameterized nonlinearities. Our approaches mainly rely on Brouwer degree arguments and critical point theory for convex, lower semicontinuous perturbations of ‐functionals.

Impacts of radar forward operator on convective-scale data assimilation and short-term forecasts

The current study utilizes the operational data assimilation system of Deutscher Wetterdienst (DWD) to investigate the impacts of improved radar forward operator. First of all, it is shown that for experiments in which both conventional and radar data are assimilated and the latent heat nudging (LHN) is applied, the one with improved operator (i.e., with improved Mie-scattering scheme and accounting for beam broadening effect and etc.) exhibits neutral impacts on short-term forecasts. However, in subsequent experiments, in which conventional data are not assimilated and the LHN is switched off, the one with improved operator not just reduces representation error during data assimilation cycles but also enhances the short-term forecast skills. In addition, it is found that the subsequent experiments result in much shorter observation error correlation length scales for radar reflectivity data, indicating that the application of the LHN or assimilation of conventional data may increase the length scales.

Active control of electromagnetic fields in layered media

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely ill-posed optimization problem (i.e. suffering from possible non-existence or non-uniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green's function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an ill-posed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.

Staining algorithm for least-squares reverse time migration

The staining algorithm-based reverse time migration (RTM) technique has shown promise in improving imaging results for certain complex structures, such as subsalt and steeply dipping layers, by incorporating the complex domain. However, RTM suffers from low-resolution images as it utilizes the adjoint of the forward modeling operator rather than its inverse operator. To address this limitation, we propose a new approach called stained least-squares RTM (ST-LSRTM) that enhances the gradient through the integration of the staining algorithm. In ST-LSRTM, we first develop the wave equation and Born approximation, transitioning from the real to the complex domains. To simplify the linear perturbation operator, we constrain the magnitude of the imaginary velocity. Next, we compute the gradient in the complex domain. The enhanced gradient is obtained by replacing the traditional gradient with the normalized stained gradient within the stained region. Finally, we construct a misfit function based on the L2 norm and calculate the step size and gradient direction using the conjugate gradient method. Model tests demonstrate that ST-LSRTM achieves high-resolution imaging results even when the stained area is imprecise. Comparisons of waveform curves and normalized residual curves confirm that ST-LSRTM approaches true reflection coefficient model more closely compared to traditional LSRTM.

Sensitivity of cloud-phase distribution to cloud microphysics and thermodynamics in simulated deep convective clouds and SEVIRI retrievals

Abstract. The formation of ice in clouds is an important process in mixed-phase clouds, and the radiative properties and dynamical developments of clouds strongly depend on their partitioning between the liquid and ice phases. In this study, we investigated the sensitivities of the cloud phase to the ice-nucleating particle (INP) concentration and thermodynamics. Moreover, passive satellite retrieval algorithms and cloud products were evaluated to identify whether they could detect cloud microphysical and thermodynamical perturbations. Experiments were conducted using the ICOsahedral Nonhydrostatic (ICON) model at the convection-permitting resolution of about 1.2 km on a domain covering significant parts of central Europe, and they were compared to two different retrieval products based on Spinning Enhanced Visible and InfraRed Imager (SEVIRI) measurements. We selected a day with multiple isolated deep convective clouds, reaching a homogeneous freezing temperature at the cloud top. The simulated cloud liquid pixel fractions were found to decrease with increasing INP concentration, both within clouds and at the cloud top. The decrease in the cloud liquid pixel fraction was not monotonic and was stronger in high-INP cases. Cloud-top glaciation temperatures shifted toward warmer temperatures with an increasing INP concentration by as much as 8 ∘C. Moreover, the impact of the INP concentration on cloud-phase partitioning was more pronounced at the cloud top than within the cloud. Furthermore, initial and lateral boundary temperature fields were perturbed with increasing and decreasing temperature increments from 0 to ±3 and ±5 K between 3 and 12 km, respectively. Perturbing the initial thermodynamic state was also found to systematically affect the cloud-phase distribution. However, the simulated cloud-top liquid pixel fraction, diagnosed using radiative transfer simulations as input to a satellite forward operator and two different satellite remote-sensing retrieval algorithms, deviated from one of the satellite products regardless of perturbations in the INP concentration or the initial thermodynamic state for warmer subzero temperatures while agreeing with the other retrieval scheme much better, in particular for the high-INP and high-CAPE (convective available potential energy) scenarios. Perturbing the initial thermodynamic state, which artificially increases the instability of the mid- and upper-troposphere, brought the simulated cloud-top liquid pixel fraction closer to the satellite observations, especially in the warmer mixed-phase temperature range.

A neural-network-based method for generating synthetic 1.6 µm near-infrared satellite images

Abstract. In combination with observations from visible satellite channels, near-infrared channels can provide valuable additional cloud information, e.g. on cloud phase and particle sizes, which is also complementary to the information content of thermal infrared channels. Exploiting near-infrared channels for operational data assimilation and model evaluation requires a sufficiently fast and accurate forward operator. This study presents an extension to the method for fast satellite image synthesis (MFASIS) that allows for simulating reflectances of the 1.6 µm near-infrared channel based on a computationally efficient neural network with the same accuracy that has already been achieved for visible channels. For this purpose, it is important to better represent vertical variations in effective cloud particle radii, as well as mixed-phase clouds and molecular absorption in the idealized profiles used to train the neural network. A new approach employing a two-layer model of water, ice and mixed-phase clouds is described, and the relative importance of the different input parameters characterizing the idealized profiles is analysed. A comprehensive data set sampled from Integrated Forecasting System (IFS) forecasts together with different parameterizations of the effective water and ice particle radii is used for the development and evaluation of the method. Further evaluation uses a month of ICOsahedral Non-hydrostatic development based on version 2.6.1 (ICON-D2) hindcasts with effective radii directly determined by the two-moment microphysics scheme of the model. In all cases, the mean absolute reflectance error achieved is about 0.01 or smaller, which is an order of magnitude smaller than typical differences between reflectance observations and corresponding model values. The errors related to the imperfect training of the neural networks present only a small contribution to the total error, and evaluating the networks takes less than a microsecond per column on standard CPUs. The method is also applicable for many other visible and near-infrared channels with weak water vapour sensitivity.

Relaxation approach for learning neural network regularizers for a class of identification problems

The present paper deals with the data-driven design of regularizers in the form of artificial neural networks, for solving certain inverse problems formulated as optimal control problems. These regularizers aim at improving accuracy, wellposedness or compensating uncertainties for a given class of optimal control problems (inner-problems). Parameterized as neural networks, their weights are chosen in order to reduce a misfit between data and observations of the state solution of the inner- optimal control problems. Learning these weights constitutes the outer-problem. Based on necessary first-order optimality conditions for the inner-problems, a relaxation approach is proposed in order to implement efficient solving of these inner-problems, namely the forward operator of the outer-problem. Optimality conditions are derived for the latter, and are implemented in numerical illustrations dealing with the inverse conductivity problem. The numerical tests show the feasibility of the relaxation approach, first for rediscovering standard L 2-regularizers, and next for designing regularizers that compensate unknown noise on the observed state of the inner-problem.

Multiparameter least‐squares reverse time migration using the viscoacoustic‐wave equation

AbstractIn viscoacoustic least‐squares reverse time migration methods, the reflectivity image associated with the Q factor is negligible, inverting only the velocity (v) parameter or v‐related variables such as squared slowness or bulk modulus. However, the Q factor influences the amplitude and phase of the seismic data, especially in basins containing gas reservoirs or storing . Therefore, the Q factor and its associated parameters must be considered in the context of viscoacoustic least‐squares reverse time migration. Thus, we propose a multiparameter viscoacoustic least‐squares reverse time migration procedure, which obtains the inverse of bulk modulus (κ) and the Q magnitude (τ) simultaneously. We derive and implement the multiparameter forward and adjoint pair Born operators and the gradient formulas concerning κ and τ parameters. Then, we apply these derivations in our proposed multiparameter approach, which can produce images with better balanced amplitudes and more resolution than conventional reverse time migration images.