Double-scale supervised inversion with a data-driven forward model for low-frequency impedance recovery

Low-frequency information is important in reducing the nonuniqueness of absolute impedance inversion and for quantitative seismic interpretation. In traditional model-driven impedance inversion methods, low-frequency impedance background is from an initial model and is almost unchanged during the inversion process. Moreover, the inversion results are limited by the quality of the modeled seismic data and the extracted wavelet. To alleviate these issues, we investigate a double-scale supervised impedance inversion method based on the gated recurrent encoder-decoder network (GREDN). We first train the decoder network of GREDN called the forward operator, which can map impedance to seismic data. We then implement the well-trained decoder as a constraint to train the encoder network of GREDN called the inverse operator. Besides matching the output of the encoder with broadband pseudo-well impedance labels, data generated by inputting the encoder output into the known decoder match the observed narrowband seismic data. Both the broadband impedance information and the already-trained decoder largely limit the solution space of the encoder. Finally, after training, only the derived optimal encoder is applied to unseen seismic traces to yield broadband impedance volumes. The proposed approach is fully data-driven and does not involve the initial model, seismic wavelet and model-driven operator. Tests on the Marmousi model illustrate that the proposed double-scale supervised impedance inversion method can effectively recover low-frequency components of the impedance model, and demonstrate that low frequencies of the predicted impedance originate from well logs. Furthermore, we apply the strategy of combining the double-scale supervised impedance inversion method with a model-driven impedance inversion method to process field seismic data. Tests on a field data set show that the predicted impedance results not only reveal a classical tectonic sedimentation history, but also match the corresponding results measured at the locations of two wells.

Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application

Integral transformations are essential for solving complex problems in business, engineering, natural sciences, computers, optical science, and modern mathematics. In this paper, we apply a general integral transform, called the Jafari transform, for solving a system of ordinary differential equations. After applying the Jafari transform, ordinary differential equations are converted to a simple system of algebraic equations that can be solved easily. Then, by using the inverse operator of the Jafari transform, we can solve the main system of ordinary differential equations. Jafari transform belongs to the class of Laplace transform and is considered a generalization to integral transforms such as Laplace, Elzaki, Sumudu, G\_transforms, Aboodh, Pourreza, etc. Jafari transform does not need a large computational work as the previous integral transforms. For the Jafari transform, we have studied some valuable properties and theories that have not been studied before. Such as the linearity property, scaling property, first and second shift properties, the transformation of periodic functions, Heaviside function, and the transformation of Dirac’s delta function, and so on. There is a mathematical model that describes the cell population dynamics in the colonic crypt and colorectal cancer. We have applied the Jafari transform for solving this model.

Two Forms of An Inverse Operator to the Generalized Bessel Potential

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor–Delsarte series.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

Differential-difference method with approximation of the inverse operator

The problem of finding an approximate solution of a nonlinear equation with operator decomposition is considered. For equations of this type, a nonlinear operator can be represented as the sum of two operators – differentiable and nondifferentiable. For numerical solving such an equation, a differential-difference method, which contains the sum of the derivative of the differentiable part and the divided difference of the nondifferentiable part of the nonlinear operator, is proposed. Also, the proposed iterative process does not require finding the inverse operator. Instead of inverting the operator, its one-step approximation is used. The analysis of the local convergence of the method under the Lipschitz condition for the first-order divided differences and the bounded second derivative is carried out and the order of convergence is established.

A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems

We propose a class of Newton-like methods with increasing convergence order for approximating the solutions of systems of nonlinear equations. Novelty of the methods is that in each step the order of convergence is increased by an amount of three at the cost of only one additional function evaluation. Another important feature is the single use of an inverse operator in each iteration, which makes the schemes attractive and computationally more efficient. Theoretical results regarding convergence and computational efficiency are verified through numerical examples including those arising from boundary value problems and integral equations. By way of comparison, it is shown that the present methods are more efficient than their existing counterparts, particularly when applied to solve the large systems of equations.

On the generalization of moyal equation for an arbitrary linear quantization

For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.

Convolutional neural networks with SegNet architecture applied to three-dimensional tomography of subsurface electrical resistivity. CNN-3D-ERT

Summary In general, the inverse problem of electrical resistivity tomography is treated using a deterministic algorithm to find a model of subsurface resistivity that can numerically match the apparent resistivity data acquired at the ground surface and has a smooth distribution that has been introduced as prior information. In this paper, we propose a new deep-learning algorithm for processing the 3D reconstruction of electrical resistivity tomography (ERT). This approach relies on the approximation of the inverse operator considered as a non-linear function linking the section of apparent resistivity as input and the underground distribution of electrical resistivity as output. This approximation is performed with a large amount of known data to obtain an accurate generalization of the inverse operator by identifying during the learning process a set of parameters assigned to the neural networks. To train the network, the subsurface resistivity models are theoretically generated by a geostatistical anisotropic Gaussian generator, and their corresponding apparent resistivity by solving numerically 3D Poisson's equation. These data are formed in a way to have the same size and trained on the convolutional neural networks with Segnet architecture containing a 3-level encoder and decoder network ending with a regression layer. The encoders including the convolutional, max-pooling and nonlinear activation operations, are sequentially performed to extract the main features of input data in lower resolution maps. On the other side, the decoders are dedicated to upsampling operations in concatenating with feature maps transferred from encoders to compensate the loss of resolution. The tool has been successfully validated on different synthetic cases and with particular attention to how data quality in terms of resolution and noise affect the effectiveness of the approach.