Fast Forward Algorithm Research Articles

Fictitious Point Technique Based on Finite-Difference Method for 2.5D Direct-Current Resistivity Forward Problem

With the widespread application of the direct-current resistivity method, searching for accurate and fast-forward algorithms has become the focus of research for geophysicists and engineers. Three-dimensional forward modeling can be the best way to identify geo-electrical anomalies but are hampered by computational limitations because of the large amount of data. A practical compromise, or even alternative, is represented by 2.5D modeling characterized using a 3D source in a 2D medium. Thus, we develop a 2.5D direct-current resistivity forward modeling algorithm. The algorithm incorporates the finite-difference approximation and fictitious point technique that can improve the efficiency and accuracy of numerical simulation. Firstly, from the boundary value problem of the electric potential generated by the point source, the discrete expressions of the governing equation are derived from the finite-difference approach. The numerical solutions of the discrete electric potential are calculated after the approximate treatment of the boundary conditions with a finite-difference method based on a fictitious point scheme. Secondly, through the simulation of a homogeneous half-space model and a one-dimensional model, and compared with the analytical results, the correctness and stability of the finite-difference forward algorithm are verified. Lastly, through the numerical simulation for a two-dimensional model, 2.5D direct-current sounding responses are summarized, which can provide a qualitative interpretation of field data.

Faster quantum sampling of Markov chains in nonregular graphs with fewer qubits

Sampling from the stationary distribution is one of the fundamental tasks of Markov chain-based algorithms and has important applications in machine learning, combinatorial optimization and network science. For the quantum case, qsampling from Markov chains can be constructed as preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. In this paper, a new qsampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks and works without any limit compared with existing results. In detail, we build a qsampling algorithm that not only accelerates non-regular graphs but also keeps the speed-up of existing quantum algorithms for regular graphs. In non-regular graphs, the invocation of the quantum fast-forward algorithm accelerates existing state-of-the-art qsampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms we reduce log n, where n is the number of graph vertices. In regular graphs, our result matches other quantum algorithms, and our reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared to the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm is the first algorithm to achieve complete quadratic acceleration (without log factor) over the classical case without any limit. To enlarge success probability amplitude amplification is introduced. We construct a new reflection on stationary state with fewer ancilla qubits and think it may have independent application.

A fast forward algorithm of azimuthal gamma imaging logging while drilling

Logging while drilling (LWD) technology can evaluate the formation interface and stratigraphic properties around the borehole in real time, so as to adjust the drilling trajectory and effectively improve the reservoir encounter rate. Fast forward algorithm can be used with LWD, by comparing the fast forward results with the actual gamma LWD results, it can distinguish the formation interface more accurately. Fast forward algorithm also can provide a theoretical basis for geosteering and azimuthal gamma logging interpretation. In this paper, based on the spatial distribution law of gamma rays and The Monte Carlo N-particle code (MCNP)-driven spatial sensitivity function, a 3D fast forward method of gamma ray LWD is proposed. Compared with the traditional algorithms, the new method can realize the fast simulations of gamma ray logging in four sectors, which enables fast azimuthal imaging. The comparison shows that whether in vertical wells or high-angle wells, the results of the proposed method are in high agreement with the MCNP fine simulations. In addition, the new method improves the calculation speed by tens of thousands of times with comparable accuracy. Finally, the gamma logging-while-drilling data of a highly deviated well in a field case verifies the accuracy and applicability of the method. The algorithm can meet the requirements of LWD for fast forward modeling of azimuthal natural gamma ray logging, which is expected to be applied to geosteering and logging interpretation of high angle and horizontal wells.

Intelligent Planning of Tourist Routes Based on Cloud Computing and Marching Algorithm

In order to improve the effect of modern tourism route planning, this article combines the cloud computing advancement algorithm to construct a tourism route intelligent planning system and combines the current people's tourism needs to carry out intelligent tourism route planning. Based on the fast forward algorithm of the program function equation, this article uses the fast forward algorithm to solve the discrete nonlinear program function equation and directly obtain the cloud computing data. In addition, the self-intersection phenomenon of the wave front must be considered when describing the characteristics of the cloud computing information field by using the process of the equivalent cloud computing information front expanding to the surrounding. Finally, this article constructs the intelligent system structure. The research results show that the intelligent travel route planning system based on cloud computing and marching algorithm proposed in this article can play an important role in modern smart tourism.

Risk and resilience-based optimal post-disruption restoration for critical infrastructures under uncertainty

Post-disruption restoration of critical infrastructures (CIs) often faces uncertainties associated with the required repair tasks and the related transportation network. However, such challenges are often overlooked in most studies on the improvement of CI resilience. In this paper, two-stage risk-averse and risk-neutral stochastic optimization models are proposed to schedule repair activities for a disrupted CI network with the objective of maximizing system resilience. Both models are developed based on a scenario-based optimization technique that accounts for the uncertainties of the repair time and the travel time spent on the underlying transportation network. Given the large number of uncertainty realizations associated with post-disruption restoration tasks, an improved fast forward algorithm based on a wait-and-see solution methodology is provided to reduce the number of chosen scenarios, which results in the desired probabilistic performance metrics. To assess the risks associated with post-disruption scheduling plans, a conditional value-at-risk (CVaR) metric is incorporated into the optimization models through a scenario reduction algorithm. The proposed restoration framework is applied to the French RTE electric power network with a DC power flow procedure, and the results demonstrate the added value of using the stochastic optimization models incorporating the travel times related to repair activities. It is essential that risk-averse decision-making under uncertainty largely impacts the optimum schedule and the expected resilience, especially in the worst-case scenarios.

A fast forward algorithm for three-dimensional gravity gradient tensor using the compact difference schemes

We propose a fast and accurate 3D large-scale forward modeling method to compute the gravity gradient tensor, in terms of a fourth-order compact finite difference scheme and the extrapolation cascadic multigrid (EXCMG) method. Firstly, the 19-point fourth-order compact finite difference scheme with unequal mesh sizes in different coordinate directions is used to discretize the governing 3D Poisson equation. The resulted symmetric positive definite linear systems are solved by the EXCMG method. Then, the second-order derivatives (gravity gradient tensor) are calculated by solving a series of tridiagonal linear systems resulting from the fourth-order compact finite difference discretization. Finally, numerical result for a cubic model with positive density is used to verify the accuracy of our presented method. It shows that our method can give nearly fourth-order accurate approximations to gravitational potential and gravity gradient tensor. The accuracy is much higher and more efficient than traditional finite difference methods.

Electromagnetic Logging Response in Multilayered Formation With Arbitrary Uniaxially Electrical Anisotropy

Transversely isotropic (TI) formation is the most popular anisotropic electrical model for computing the electromagnetic (EM) field in multilayered media. However, the TI formation considers only electric parameters such as resistivity along the horizontal and vertical directions, which are not always available in real-life situations. The realistic electric parameter distribution mainly depends on the sedimentary formation environment and could be in arbitrary directions. This article developed a fast-forward algorithm used to calculate the EM field in multilayered media with arbitrary uniaxially electrical anisotropy. Formation resistivity, anisotropic dip, and anisotropic azimuth are adopted to characterize the arbitrary uniaxial anisotropic resistivity media. Half-space Fresnel coefficients are defined and used to analyze the EM reflection and transmission characteristics between different anisotropic media. Generalized reflection and transmission matrices are derived to express the EM field in arbitrary layers. To validate the proposed algorithm, we modeled the traditional logging while drilling (LWD) and triaxial induction tool responses under different conditions. The numerical results agree with the finite element results, demonstrating the feasibility, robustness, and stability of the algorithm.

Fast inversion of array laterolog measurements in an axisymmetric medium

The array laterolog is an important tool for complex formation logging evaluation due to its high resolution and large detection depth. However, its logging responses are seriously affected by leakage events due to the surrounding rock and by mud invasion. These factors must be considered when inverting array lateral logging data, so that the inversion results reflect the true formation conditions as much as possible. The difficulties encountered in the inversion of array lateral logging data are: too many inversion parameters cause the calculation of the Jacobian matrix to be difficult and the time required to select the initial inversion values due to the slow forward-modeling speed. In this paper, we develop a fast processing method for array laterolog data. First, it is important to clearly define the main controlling factors for the array laterolog response, such as thickness, the surrounding rock, and invasion. Second, based on a depth-window technique, processing the array laterolog data for the entire well is transformed into multiple 2D inversions of the layers using a series of continuous depth windows. For each formation in a depth window, combined with the 1D equivalent fast-forward algorithm, rapid extraction of the radial resistivity profile of the formation is achieved. Finally, the 1D inversion result is used as the initial state to further eliminate the influence of surrounding rocks and layer thicknesses on the apparent resistivity response. Numerical simulation results show that the factors affecting the response of the array laterolog are the invasion properties, the layer thicknesses, and the surrounding rocks; the windowing technique greatly reduces the number of inversion parameters needed and improves the inversion speed. A real application of the method shows that 2D inversion can rapidly reconstruct the actual resistivity distribution and improve the accuracy of reservoir saturation calculations.

A fast forward algorithm for three-dimensional magnetic anomaly on undulating terrain

This paper presents a fast and high-precision three-dimensional magnetic anomaly forward algorithm on undulating terrain. Firslty, The forward algorithm for three-dimensional magnetic anomalies on a horizontal observation surface is developed with the Block-Toeplitz-Toeplitz-Block (BTTB) matrix. In the proposed algorithm, the cuboid elements are used to divide the underground space. And the position of the observation point is set as the projection of the central point of the cuboid element onto the observation surface. With the symmetry of the BTTB matrix, the coefficient matrix is compressed and stored to reduces the memory requirements during the computation process. In order to deal with the coefficient matrix and parameter vector computations, the matrix cross-multiplication computation is transformed into a point-multiplication computation with the fast Fourier transform (FFT) method. Then, to solve the problem of undulating terrain, a hierarchical interpolation method is proposed in forward algorithm, in which the undulating terrain is divided into several horizontal observation surfaces. Subsequently, the magnetic field on the undulating terrain is interpolated. Some 3D models with horizontal observation surface and undulating terrain are tested to verify the proposed algorithm. The results show that the forward algorithm has the advantages of fast computation speed, high precision, and low memory requirement. It is suitable for large-scale (ten millions of nodes) three-dimensional magnetic forward modeling.

A fast forward algorithm for real-time geosteering of azimuthal gamma-ray logging

Geosteering is an effective method to increase the reservoir drilling rate in horizontal wells. Based on the features of an azimuthal gamma-ray logging tool and strata spatial location, a fast forward calculation method of azimuthal gamma-ray logging is deduced by using the natural gamma ray distribution equation in formation. The response characteristics of azimuthal gamma-ray logging while drilling in the layered formation models with different thickness and position are simulated and summarized by using the method. The result indicates that the method calculates quickly, and when the tool nears a boundary, the method can be used to identify the boundary and determine the distance from the logging tool to the boundary in time. Additionally, the formation parameters of the algorithm in the field can be determined after a simple method is proposed based on the information of an offset well. Therefore, the forward method can be used for geosteering in the field. A field example validates that the forward method can be used to determine the distance from the azimuthal gamma-ray logging tool to the boundary for geosteering in real-time.

Fast linearly independent arithmetic expansions

The concept of Linearly Independent arithmetic (LIA) transforms and expansions is introduced in this paper. The recursive ways of generating forward and inverse fast transforms for LIA are presented. The paper describes basic properties and lists those LIA transforms which have convenient fast forward algorithms and easily defined inverse transforms. In addition, those transforms which require horizontal or vertical permutations to have fast transform are also discussed. The computational advantages and usefulness of new expansions based on LIA logic in comparison to known arithmetic expansions are discussed.

Inversion techniques applied to resistivity inverse problems

Reviews various inversion techniques applied to resistivity inverse problems, and proposes a fairly different approach. The advantages and disadvantages of various resistivity inversion methods are briefly summarized. A good inversion method must simultaneously minimize the effects of data error and model parameter errors. The necessary requirements for inversion of any geophysical data are a fast forward algorithm for calculating theoretical data for initial model parameters, and a technique for calculating derivatives of the data with respect to the model parameters. Unfortunately, the second requirement is not readily available for two-dimensional (2D) or three-dimensional (3D) inverse resistivity problems. A few workers have examined the problem of resistivity inversion, but no claim has yet been made for an optimum inverse method for 2D or 3D resistivity data. Most resistivity inversion methods are also restricted to surface electrode arrays. A multi-dimensional resistivity inversion method using perturbation analysis and reciprocity is proposed. The theoretical basis for this method is briefly developed. This approach is an efficient way of calculating the derivatives of data with respect to the model parameters because the inverse formulation thereby developed is a linearized form of the general non-linear problem.