Inversion Objective Function Research Articles

Huber inversion-based reverse-time migration with de-primary imaging condition and curvelet-domain sparse constraint

Least-squares reverse-time migration (LSRTM) formulates reverse-time migration (RTM) in the least-squares inversion framework to obtain the optimal reflectivity image. It can generate images with more accurate amplitudes, higher resolution, and fewer artifacts than RTM. However, three problems still exist: (1) inversion can be dominated by strong events in the residual; (2) low-wavenumber artifacts in the gradient affect convergence speed and imaging results; (3) high-wavenumber noise is also amplified as iteration increases. To solve these three problems, we have improved LSRTM: firstly, we use Huber-norm as the objective function to emphasize the weak reflectors during the inversion; secondly, we adapt the de-primary imaging condition to remove the low-wavenumber artifacts above strong reflectors as well as the false high-wavenumber reflectors in the gradient; thirdly, we apply the L 1 -norm sparse constraint in the curvelet-domain as the regularization term to suppress the high-wavenumber migration noise. As the new inversion objective function contains the non-smooth L 1 -norm, we use a modified iterative soft thresholding (IST) method to update along the Polak-Ribière conjugate-gradient direction by using a preconditioned non-linear conjugate-gradient (PNCG) method. The numerical examples, especially the Sigsbee2A model, demonstrate that the Huber inversion-based RTM can generate high-quality images by mitigating migration artifacts and improving the contribution of weak reflection events.

Three-dimensional time-lapse inversion of transient electromagnetic data, with application at an Icelandic geothermal site

SUMMARY Transient electromagnetic (TEM) is an efficient non-invasive method to map electrical conductivity distribution in the subsurface. This paper presents an inversion scheme for 3-D TEM time-lapse (TL) data using a generalized minimum support (MS) norm and its application to monitoring conductivity changes over time. In particular, two challenges for TL TEM applications are addressed: (i) the survey repetition with slightly different acquisition position, that is, because systems are not installed and (ii) non-optimal data coverage above the TL anomalies, for instance, due to the presence of infrastructure that limits the acquisition layout because of coupling. To address these issues, we developed a new TEM TL inversion scheme with the following features: (1) a multimesh approach for model definition and forward computations, which allows for seamless integration of data sets with different acquisition layouts; (2) 3-D sensitivity calculation during the inversion, which allows retrieving conductivity changes in-between TEM soundings and (3) simultaneous inversion of two data sets at once, imposing TL constraints defined in terms of a generalized MS norm, which ensures compact TL changes. We assess the relevance of our implementations through a synthetic example and a field example. In the synthetic example, we study the capability of the inversion scheme to retrieve compact time-lapse changes despite slight changes in the acquisition layout and the effect of data coverage on the retrieval of TL changes. Results from the synthetic tests are used for interpreting field data, which consists of two TEM data sets collected in 2019 and 2020 at the Nesjavellir high-temperature geothermal site (Iceland) within a monitoring project of H2S sequestration. Furthermore, the field example illustrates the effect of the trade-off between data misfit and TL constraints in the inversion objective function, using the tuning settings of the generalized MS norm. Based on the results from both the synthetic and field cases, we show that our implementation of 3-D TL inversion has a robust performance for TEM monitoring.

Elastic Impedance Simultaneous Inversion for Multiple Partial Angle Stack Seismic Data with Joint Sparse Constraint

Elastic impedance (EI) inversion for partial angle stack seismic data is a key technology in seismic reservoir prediction within the oil and gas industry. EI inversion provides a consistent framework to invert partial angle stack seismic data, just as the AI inversion does for post-stack data. The commonly used EI inversion process is angle by angle. Hence, the inverted EI for different angles may be nonconforming, especially for the seismic data with a low signal-to-noise ratio. This paper proposes to simultaneously invert multiple partial angle stack seismic data to obtain EI for different angles at once. To obtain conformable EI, we used the joint sparse constraint on the reflection coefficients for different angles. Then, the objective function for simultaneous EI inversion was constructed. Next, synthetic seismic data profiles with three different angles were used to show the superiority of the proposed EI inversion method compared to the conventional method. At last, a real seismic data line was used to test the feasibility of the proposed method in practice. The inversion results of synthetic data and real data showed that it provides an effective new alternative method to estimate EI from partial stack seismic data.

A Novel Intelligent Inversion Method of Hydrogeological Parameters Based on the Disturbance-Inspired Equilibrium Optimizer

Accurate and quick acquisition of hydrogeological parameters is the critical issue for groundwater numerical simulation and sustainability of the water sources. A novel intelligent inversion method of hydrogeological parameter, based on the global optimization algorithm called the disturbance-inspired equilibrium optimizer (DIEO), is developed. Firstly, the mathematical model and the framework of DIEO are reported. Several types of mathematical benchmark functions are used to test the performance of the DIEO. Furthermore, the intelligent inversion of hydrogeological parameters of pumping tests is transformed into the global optimization problem, which can be solved by meta-heuristic algorithms. The objective function for hydrogeological parameter inversion is constructed, and the novel inversion method based on DIEO is finally proposed. To further validate the competitiveness and efficiency of the proposed intelligent inversion method, three types of case studies are carried out. The results show that the proposed intelligent inversion method is reliable for obtaining the hydrogeological parameters accurately and quickly, providing a reference for the inversion of parameters in other fields.

AVA inversion of pre-stack seismic data with cross-gradient constraints

The ill-posed feature is one basic attribute of geophysical inversion methods. As an example of geophysical inverse problem, AVA (Amplitude variation with incident angle) inversion of pre-stack seismic data is susceptible to noise and uncertainty in the acquisition. To get stable and accurate inversion results, the regularization constraints on model parameter need to be added into the objective function of AVA inversion. In AVA inversion, the most commonly used regularization includes sparsity constraint (e.g. L1-norm regularization, Cauchy regularization) and a priori model parameters constraint, and so forth. However, the existing AVA inversion methods do not consider the structural similarity of different model parameters. All of the different model parameters represent the same underground geological structure, so they should have similar structure. This paper adopts the cross gradient to measure the structural similarity of different model parameters. Next, the cross gradients of different model parameters are added into the objective function of AVA inversion as a regularization term to implement structural similarity constraint. Results of the model numerical tests and real seismic data indicate that the AVA inversion with cross-gradient constraint has higher stability compared to the AVA inversion without structural similarity constraint, especially for the density inversion results.

3-D Joint Inversion of Gravity and Magnetic Data Using Data-Space and Truncated Gauss–Newton Methods

Gravity and magnetic inversion are important methods for comprehensive quantitative interpretation of data obtained in, e.g., mineral, oil and gas, and geothermal exploration. At present, the 3-D joint inversion technology of gravity and magnetic data is facing challenges from large-scale data exploration applications. In this letter, a new algorithm for 3-D joint inversion of gravity and magnetic data with high accuracy and low computational cost is presented. We use the geometric trellis method to perform fast forward calculations and then introduce the sparse constraint and adaptive sensitivity matrix into the model constraint terms. The inexact structural resemblance method is then used to add the cross-gradient constraint penalty term to the objective function. Finally, an algorithm (DS-TGN) combining data-space (DS) and truncated Gauss–Newton (TGN) methods is used to solve the joint inversion objective function. Numerical experiments with synthetic data show that the proposed algorithm can significantly reduce the computational cost and obtain high accuracy density and magnetization models with structural resemblance and sharp boundaries. We also apply the DS-TGN algorithm to data obtained in the area of Greater Khingan in northwestern Heilongjiang, China. The underground density and magnetization distribution results provide a high-resolution geological model for the detection of skarn-type deposits.

Poststack Impedance Inversion With Geological Structure-Guided Total Variation Constraint

Impedance inversion is an effective method to estimate the properties of subsurface model from poststack seismic data. However, owing to the low quality of seismic data and other reasons, impedance inversion methods usually suffer from spatial discontinuities and low resolution. In order to overcome these problems, we have developed an impedance inversion method based on geological structure-guided total variation (GSGTV) constraint. Different from traditional TV constraint, GSGTV considers the spatial distribution of geological structures instead of imposing gradient constraints on impedance only in fixed directions (horizontal and vertical). Therefore, GSGTV not only retains the advantage of traditional TV that can preserve the edge of layer, but also overcomes the shortcomings of traditional TV that is only suitable for inverting block structures. This ensures that GSGTV can simultaneously improve the resolution and spatial continuity of the inversion result. The realization of impedance inversion method based on GSGTV constraint requires three steps. First, estimate the local structural orientation of the subsurface from seismic data. Then, construct GSGTV regularization term based on the local structural orientation. Finally, establish the inversion objective function and solve the function by the split-Bregman iterative algorithm. We compared the proposed method and the impedance inversion method based on traditional TV constraint with synthetic and real seismic data. The inversion results confirm that our method can improve the resolution and spatial continuity.

Prestack Seismic Inversion via Nonconvex L1-2 Regularization

Using seismic data, logging information, geological interpretation data, and petrophysical data, it is possible to estimate the stratigraphic texture and elastic parameters of a study area via a seismic inversion. As such, a seismic inversion is an indispensable tool in the field of oil and gas exploration and development. However, due to unknown natural factors, seismic inversions are often ill-conditioned problems. One way to work around this unknowable information is to determine the solution to the seismic inversion using regularization methods after adding further a priori constraints. In this study, the nonconvex L1−2 regularization method is innovatively applied to the three-parameter prestack amplitude variation angle (AVA) inversion. A forward model is first derived based on the Fatti approximate formula and then low-frequency models for P impedance, S impedance, and density are established using logging and horizon data. In the Bayesian inversion framework, we derive the objective function of the prestack AVA inversion. To further improve the accuracy and stability of the inversion results, we remove the correlations between the elastic parameters that act as initial constraints in the inversion. Then, the objective function is solved by the nonconvex L1−2 regularization method. Finally, we validate our inversion method by applying it to synthetic and observational data sets. The results show that our nonconvex L1−2 regularization seismic inversion method yields results that are highly accurate, laterally continuous, and can be used to identify and locate reservoir formation boundaries. Overall, our method will be a useful tool in future work focused on predicting the location of reservoirs.

Viscoelastic Fluid Factor Inversion and Application in Luojia Oilfield Based on Broadband Impedance

Based on the physical quantity of log data, the accurate identification of oil- and gas-bearing properties may be caused by the prestack inversion of fluid prediction, which will affect the success rate of exploration and development. Prestack data contain more information of amplitude and frequency. Using the frequency-dependent viscoelastic impedance equation and Bayesian inversion framework, the objective function of frequency-dependent elastic impedance inversion can be established to realize the frequency-dependent impedance inversion at different angles. According to the elastic impedance equation of the frequency-varying viscoelastic fluid factor, the relationship between elastic impedance and the frequency-dependent viscoelastic fluid factor is established, and the prestack seismic inversion method of the frequency-dependent viscoelastic fluid factor is studied. However, one of the important factors easily neglected is that we have been using logging data to establish fluid-sensitive parameters and the lithophysical version for fluid identification, so there are differences between logging and seismic frequency bands for fluid identification. The indicator factors with higher sensitivity to fluid can be selected by laboratory measurements. This article applies this method on Luojia oilfield data and verifies this method with log interpretation results, based on the sample of rock physics obtained in a low-frequency rock physics experiment; the technique of dispersion and fluid-sensitive parameters is studied, and the fluid prediction technology of a multifrequency band rock physics template is adopted, which can build the relationship between rock physical elastic parameters and fluid properties by the multifrequency broadband impedance method.

Vector analysis of inverse objective function for electrochemical impedance spectroscopy of big capacity lead-acid battery

The important emergency power supply needs a large capacity battery. The batteries are long-term float charging. Electrochemical impedance spectroscopy (EIS) is important for remaining useful capacity warnings. This paper establishes the EIS inverse calculation technique that is centered in the vector objective function and linear interpolation search algorithm. This technique simplifies the initial value preparation and the gradient descent direction search for the inverse calculation of EIS. The root-mean-square errors of online embedded programming are smaller than the offline results of numerical software for EIS. The vector objective function targets the rotation angle of the initial polarization impedance vector to be zero during the EIS inverse calculation. This equivalently replaces the regular minimization of root-mean-square error. The linear interpolation algorithm simplifies the gradient descent direction search to a one-dimensional search for the double-layer capacitance of the initial polarization impedance. This algorithm replaces the conventional graphic method and evolutionary algorithm. The proposed algorithm is verified by the microcontroller at Valve-regulated lead-acid (VRLA) batteries (4200 Ah). The result of the algorithm is close to special numerical software for EIS. This algorithm improves the performance of an embedded management board.

Estimation of S-wave velocity from wire-line logs for organic-rich rocks

Existing shear-wave velocity prediction methods for organic-rich rocks have not simultaneously taken the effects of organic matter (kerogen) distribution and pore shape on the velocity into consideration. To solve this problem, the current paper proposes a shear-wave velocity prediction method for organic-rich rocks based on the Kuster–Toksöz (KT) rock physics model and chaotic quantum particle swarm optimisation (CQPSO) algorithm. In this method, the organic-rich rocks are considered equivalent to a mixture of minerals, kerogen particles and fluid-containing pores, in which kerogen particles and pores are both equivalent to ellipsoid-shaped inclusions. The effect of kerogen distribution and pore shape on velocity is described according to the change in the aspect ratio of ellipsoids. We used the error between the predicted and measured P-wave velocities to establish the inverse objective function, and introduced the CQPSO algorithm to calculate the equivalent kerogen particle and the equivalent pore aspect ratios. Based on this, the shear-wave velocity is predicted using inverted parameters. Compared with three existing single-parameter adaptive methods, the prediction results show that the proposed method is superior and can be satisfactorily implemented for laboratory measurement and on logging data.

Reflectivity decomposition: Theory and application

Sparse reflectivity inversion of processed reflection seismic data is intended to produce reflection coefficients that represent boundaries between geologic layers. However, the objective function for sparse inversion is usually dominated by large reflection coefficients, which may result in unstable inversion for weak events, especially those interfering with strong reflections. We have determined that any seismogram can be decomposed according to the characteristics of the inverted reflection coefficients that can be sorted and subset by magnitude, sign, and sequence, and new seismic traces can be created from only reflection coefficients that pass the sorting criteria. We call this process reflectivity decomposition. For example, original inverted reflection coefficients can be decomposed by magnitude, large ones removed, the remaining reflection coefficients reconvolved with the wavelet, and this residual reinverted, thereby stabilizing inversions for the remaining weak events. As compared with inverting an original seismic trace, subtle impedance variations occurring in the vicinity of nearby strong reflections can be better revealed and characterized when only the events caused by small reflection coefficients are passed and reinverted. When we apply reflectivity decomposition to a 3D seismic data set in the Midland Basin, seismic inversion for weak events is stabilized such that previously obscured porous intervals in the original inversion can be detected and mapped, with a good correlation to the actual well logs.

Research on Pseudo-2D Joint Inversion of TEM and CSAMT Based on Well Log Constraint

The transient electromagnetic method (TEM) and controlled-source audio-frequency magnetotellurics method (CSAMT) are commonly used in detecting water abundance of rock formation and faults in coal mines. However, these methods show low accuracy, given the multiplicity of their inversion results, especially for areas with minor differences in lithology and electrical property. To improve the accuracy of electromagnetic exploration, a pseudo-2D joint inversion is performed. The objective function of this pseudo-2D joint inversion is established, and the joint inversion process is constrained by resistivity logging data. Afterward, the symmetric successive over-relaxation (SSOR) is used to realize the pseudo-2D joint inversion calculation of TEM and CSAMT with well log constraint. The effectiveness of joint inversion is verified by combining synthetic and field data. Results show that the pseudo-2D joint inversion results of TEM and CSAMT with well log constraint correspond to the actual geological situation. Compared with either TEM or CSAMT, joint inversion has a significantly better capability of reflecting water abundance in rock formation and faults.

Deformation early-warning index for heightened gravity dam during impoundment period

The mechanical parameters of materials in a dam body and dam foundation tend to change when dams are reinforced in aging processes. It is important to use an early-warning index to reflect the safety status of dams, particularly of heightened projects in the impoundment period. Herein, a new method for monitoring the safety status of heightened dams is proposed based on the deformation monitoring data of a dam structure, a statistical model, and finite-element numerical simulation. First, a fast optimization inversion method for estimation of dam mechanical parameters was developed, which used the water pressure component extracted from a statistical model, an improved inversion objective function, and a genetic optimization iterative algorithm. Then, a finite element model of a heightened concrete gravity dam was established, and the deformation behavior of the dam with rising water levels in the impoundment period was simulated. Subsequently, mechanical parameters of aged dam parts were calculated using the fast optimization inversion method with simulated deformation and the water pressure deformation component obtained by the statistical model under the same conditions of water pressure change. Finally, a new early-warning index of dam deformation was constructed by means of the forward-simulated deformation and other components of the statistical model. The early-warning index is useful for forecasting dam deformation under different water levels, especially high water levels.

Data-driven multichannel poststack seismic impedance inversion via patch-ordering regularization

Seismic acoustic impedance inversion plays an important role in reservoir prediction. However, single-trace inversion methods often suffer from spatial discontinuities and instability due to poor-quality seismic records with spatially variable signal-to-noise ratios or missing traces. The specified hyperparameters for seismic inversion cannot be suitable to all seismic traces and subsurface structures. In addition, conventional multichannel inversion imposes lateral continuity with a prespecified mathematical model. However, the inversion results constrained with specified lateral regularization are inferior when the subsurface situations violate the hypothesis. A data-driven multichannel acoustic impedance inversion method with patch-ordering regularization is introduced, in which the spatial correlation of seismic reflection is used. The method decomposes the seismic profile into patches and constructs the patch-ordering matrix based on the similarity among seismic patches to record the impedance structural extension. So the patch-ordering matrix can record the spatial extension of the acoustic impedance. Then, a simple regularization with difference operators of varying weights can reduce the random noise presented in the inverted impedance profile, stabilize the inversion result, and enhance the spatial continuity of the layer extension. The objective function for multichannel poststack seismic impedance inversion can be constructed by integrating the observed seismic record and the spatial continuity in the form of patch-ordering regularization, and it can be solved effectively with the limited-memory BFGS algorithm. The synthetic and field data tests illustrate the improvement of accuracy and lateral continuity of inverted results with our method, compared to conventional model-based inversion results.

A New Approach for Blind Nonlinear Acoustic Impedance Inversion

We propose a blind nonlinear acoustic impedance inversion method. The seismic wavelet is first extracted through the Euclid deconvolution method from multichannel seismic data. Then, the acoustic impedance is inverted based on the exact nonlinear forward operator. The conventional Euclid deconvolution can theoretically estimate the reflectivity without special prior assumptions, but the method is extremely inefficient and unstable in the case of a large amount of data. We optimize the method and propose a frequency-domain algorithm to improve its efficiency. Conventional impedance inversions are almost implemented based on the linearized approximate formula, but the inversion errors will increase sharply when there is a strong reflection interface. We build the inversion objective function by the accurate nonlinear formula to improve accuracy. The total variation (TV) regularization and low-frequency components of well-logging curves are added to the objective function to make the inversion result have a block structure and converge to the absolute impedance. The nonlinear optimization problem is finally solved by the Levenberg–Marquardt (LM) algorithm. The results of synthetic data and field data verify that our method has high accuracy and good practicability.

Inversion of reservoir fluid mobility from frequency-dependent seismic data — A case study of gas-bearing reservoirs

Reservoir fluid mobility is by definition the ratio of rock permeability to fluid viscosity. This attribute can be applied to reservoir physical property and permeability evaluation. Thus far, the only means of obtaining the reservoir fluid mobility over a large range of exploration areas is based on the extraction method. However, the location of high fluid mobility obtained by the extraction method is close to the reservoir interface. To obtain fluid mobility in the middle of the reservoir, an approximate inversion method of reservoir fluid mobility from frequency-dependent seismic data is proposed. First, we have calculated the reservoir fluid mobility coefficient using well data according to the relationship of fluid parameters. Then, we establish an inversion equation based on the low-frequency reflection coefficient and the reservoir fluid mobility. Taking the reservoir fluid mobility coefficient calculated from well data as an a priori constraint, the low-frequency model is subsequently constructed and applied with the inversion equation to obtain an inversion objective function. Next, the inversion equation is solved by the basis pursuit algorithm. Finally, the proposed reservoir fluid mobility inversion method is applied to synthetic and real data of gas-bearing reservoirs. The real data processing results show that the proposed reservoir fluid mobility inversion method can estimate the fluid mobility in the actual position of the reservoir more effectively.

Nonlinear AVA Inversion Based on a Novel Quadratic Approximation for Fluid Discrimination

Fluid discrimination plays an important role in reservoir exploration and development. At present, the fluid factors used for fluid discrimination are estimated by linear AVA inversion methods based on the linear approximations of the Zoeppritz equations. However, the Zoeppritz equations show that the relationship between prestack AVA reflection coefficients and reservoir parameters is highly nonlinear. Therefore, inversion methods based on linear approximations will seriously influence the nonuniqueness and uncertainty of inversion results. In this paper, a nonlinear inversion based on the quadratic approximation is carried out to reduce the nonuniqueness and uncertainty of fluid factor. Firstly, in order to directly invert the fluid factor, a novel quadratic approximation in terms of the fluid factor (ρf), shear modulus, and density on both sides of the reflection interface is derived based on poroelasticity theory. Then, a nonlinear inversion objective function is constructed using the novel quadratic approximation in a Bayesian framework, and the Gauss-Newton method is adopted to minimize this objective function. The synthetic data example shows that the new method can obtain reasonable fluid factor inversion results even in low SNR (signal-to-noise ratio) case. Finally, the proposed method is also applied to field data which shows that it can effectively discriminate reservoir fluids.

Multiscale frequency-domain seismic inversion for fracture weakness

Amplitude variation with offset and azimuth (AVOAz) inversion is an effective tool to invert the azimuthal seismic reflection data for fracture weaknesses, which play an important role in seismic characterization of fractures. Conventional AVOAz inversion is implemented in the time domain, which simultaneously uses full-frequency components of the seismic data causing an unstable and inaccurate inversion result in the case of noisy seismic data. Here we propose a novel multiscale frequency-domain seismic inversion approach to improve resolution and accuracy of fracture weakness estimates. Based on new derived frequency-domain expression of azimuthal seismic amplitude difference for a horizontal transversely isotropic (HTI) medium, we build the kernel function incorporated with the Fourier operator, which avoids the Fourier inverse transform in the frequency-domain inversion. Low frequency information in seismic data is often missing or contaminated with noises, thus we establish an inversion objective function combining low frequency regularization term and Cauchy sparse regularization term in the Bayesian framework. We adopt the multiscale strategy and perform inversion successively from low to high frequency group utilizing a limited number of frequency components with a high signal-to-noise (S/N) ratio. In addition, we compare the multiscale frequency-domain inversion approach, the frequency-domain simultaneous inversion approach, and the time-domain inversion approach using synthetic data and field data. Synthetic example demonstrates that the cross-correlation coefficients (CCs) between the true fracture weaknesses and the inversion results obtained by multiscale frequency-domain inversion approach are above 0.83 even with a S/N of 2. Compared to the time-domain inversion approach and the frequency-domain simultaneous inversion approach, the inversion results with multiscale frequency-domain inversion approach have higher resolution and accuracy. Field data example further validates the feasibility and superiority of the proposed multiscale frequency-domain inversion approach.

Dense Recurrent Neural Networks for Accelerated MRI: History-Cognizant Unrolling of Optimization Algorithms.

Inverse problems for accelerated MRI typically incorporate domain-specific knowledge about the forward encoding operator in a regularized reconstruction framework. Recently physics-driven deep learning (DL) methods have been proposed to use neural networks for data-driven regularization. These methods unroll iterative optimization algorithms to solve the inverse problem objective function, by alternating between domain-specific data consistency and data-driven regularization via neural networks. The whole unrolled network is then trained end-to-end to learn the parameters of the network. Due to simplicity of data consistency updates with gradient descent steps, proximal gradient descent (PGD) is a common approach to unroll physics-driven DL reconstruction methods. However, PGD methods have slow convergence rates, necessitating a higher number of unrolled iterations, leading to memory issues in training and slower reconstruction times in testing. Inspired by efficient variants of PGD methods that use a history of the previous iterates, we propose a history-cognizant unrolling of the optimization algorithm with dense connections across iterations for improved performance. In our approach, the gradient descent steps are calculated at a trainable combination of the outputs of all the previous regularization units. We also apply this idea to unrolling variable splitting methods with quadratic relaxation. Our results in reconstruction of the fastMRI knee dataset show that the proposed history-cognizant approach reduces residual aliasing artifacts compared to its conventional unrolled counterpart without requiring extra computational power or increasing reconstruction time.