New theoretical and practical aspects of electromagnetic soundings at low induction numbers

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1441-1451 ◽  
Author(s):  
Enrique Gómez‐Treviño ◽  
Francisco J. Esparza ◽  
Sóstenes Méndez‐Delgado
Keyword(s):  
Differential Equation ◽  
Imaginary Part ◽  
Field Data ◽  
Series Representation ◽  
Field Measurements ◽  
Magnetic Dipoles ◽  
The Real ◽  
Novel Approach ◽  
Conductivity Profile ◽  
Hertz Potential

This paper presents a theoretical yet practical study of electromagnetic (EM) soundings at low induction numbers for vertical and horizontal magnetic dipoles. The physical model is a heterogeneous half‐space with arbitrary vertical conductivity variations. The study comprises a novel approach for solving forward problems, analytical formulas for inversion, and a practical algorithm for recovering conductivity variations from field measurements. The basis of the theoretical approach is a series representation of the EM field in terms of ascending powers of frequency. At low induction numbers only two terms are required. When substituted into Maxwell's equations, one term in the series can be obtained in terms of the other. Furthermore, if the electrical conductivity varies only with depth, the imaginary part of the field can be obtained from its real part through a differential equation. The real part, which corresponds to zero frequency, plays the role of a distributed source for the frequency‐dependent imaginary part. In the case of vertical magnetic dipoles, the approach applies directly to the real and imaginary components of the magnetic field, while for horizontal dipoles one must use the Hertz potential, but the procedure is exactly the same. In each case this leads to a statement of the forward problem as the solution of a real differential equation. The solutions are integral expressions valid for arbitrary conductivity profiles. Assuming that these expressions represent integral equations for conductivity, analytical inverse formulas are derived for both vertical and horizontal dipoles. These formulas ensure a unique recovery of the conductivity profile under ideal conditions. An algorithm based on linear programming offers a variety of practical advantages for the inversion of field data. Numerical experiments and applications to field data illustrate the performance of the algorithm.

scholarly journals On inversion in the case of the fundamentally finite integrable Vekua complex differential equation

2020 ◽  
Vol 108 (122) ◽  
pp. 13-22
Author(s):  
Milos Canak ◽  
Miloljub Albijanic
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
General Solution ◽  
Boundary Value Problems ◽  
Imaginary Part ◽  
Boundary Value ◽  
The Real ◽  
Complex Differential Equation ◽  
The Core ◽  
Vekua Equation

The class of so called fundamentally finite integrable Vekua CDE is defined using the fixed point of the inversion and where one solution is equal to the coefficient of the equation. Then the different manifestations of inversion in relation to the general solution, an arbitrary analytical function inside and the core of the coefficient are examined. It shows that all the major problems of the Vekua equation theories, including boundary value problems can be interpreted and solved using the principle of inversion. The main significance of the fundamentally finite integrable Vekua equation is that the real and imaginary part of the solution can be separated, which in many mechanical and technique problems have certain physical meanings.

scholarly journals SORNETTE–IDE MODEL FOR MARKETS: TRADER EXPECTATIONS AS IMAGINARY PART

2002 ◽  
Vol 13 (04) ◽  
pp. 551-553 ◽  
Author(s):  
CHRISTIAN SCHULZE
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Imaginary Part ◽  
Complex Variable ◽  
Strong Increase ◽  
Periodic Oscillations ◽  
The Real

A nonlinear differential equation of Sornette–Ide type with noise, for a complex variable, yields endogenous crashes, preceded by roughly log-periodic oscillations in the real part, and a strong increase in the imaginary part. The latter is interpreted as the trader expectation.

Sediment transport under dry weather conditions in a small sewer system

1998 ◽  
Vol 37 (1) ◽  
pp. 155-162
Author(s):  
Flemming Schlütter ◽  
Kjeld Schaarup-Jensen
Keyword(s):  
Sediment Transport ◽  
Field Data ◽  
Field Measurements ◽  
Weather Conditions ◽  
Sewer System ◽  
Transport Models ◽  
Sewer Systems ◽  
Combined Sewer ◽  
Initial Results

Increased knowledge of the processes which govern the transport of solids in sewers is necessary in order to develop more reliable and applicable sediment transport models for sewer systems. Proper validation of these are essential. For that purpose thorough field measurements are imperative. This paper renders initial results obtained in an ongoing case study of a Danish combined sewer system in Frejlev, a small town southwest of Aalborg, Denmark. Field data are presented concerning estimation of the sediment transport during dry weather. Finally, considerations on how to approach numerical modelling is made based on numerical simulations using MOUSE TRAP (DHI 1993).

scholarly journals A novel approach to the computation of one-loop three- and four-point functions: I. The real mass case

2019 ◽  
Vol 2019 (11) ◽  
Author(s):  
J Ph Guillet ◽  
E Pilon ◽  
Y Shimizu ◽  
M S Zidi
Keyword(s):  
Building Blocks ◽  
The Real ◽  
Alternative Method ◽  
Novel Approach ◽  
Reduction Methods ◽  
Real Mass ◽  
Novel Method ◽  
Algebraic Reduction ◽  
The One ◽  
Mass Case

Abstract This article is the first of a series of three presenting an alternative method of computing the one-loop scalar integrals. This novel method enjoys a couple of interesting features as compared with the method closely following ’t Hooft and Veltman adopted previously. It directly proceeds in terms of the quantities driving algebraic reduction methods. It applies to the three-point functions and, in a similar way, to the four-point functions. It also extends to complex masses without much complication. Lastly, it extends to kinematics more general than that of the physical, e.g., collider processes relevant at one loop. This last feature may be useful when considering the application of this method beyond one loop using generalized one-loop integrals as building blocks.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

Geophysics ◽  
2021 ◽  
pp. 1-53
Author(s):  
Jiangtao Hu ◽  
Jianliang Qian ◽  
Jian Song ◽  
Min Ouyang ◽  
Junxing Cao ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Ray Tracing ◽  
Seismic Waves ◽  
Real Space ◽  
Advection Equation ◽  
Partial Differential ◽  
The Real ◽  
Eikonal Function ◽  
Complex Valued

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we propose a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. The proposed methods can be useful for migration and tomography in attenuating media.

Richardson's Iteration With Dynamic Parameters and the SIP Incomplete Factorization for the Solution of Linear Systems of Equations

1981 ◽  
Vol 21 (06) ◽  
pp. 699-708
Author(s):  
Paul E. Saylor
Keyword(s):  
Linear Systems ◽  
Field Data ◽  
Symmetric Matrix ◽  
Simple Problem ◽  
Test Problems ◽  
Physical Parameters ◽  
Incomplete Factorization ◽  
The Real ◽  
Hydrocarbon Reservoir ◽  
Linear Algebraic Equations

Abstract Reservoir simulation yields a system of linear algebraic equations, Ap=q, that may be solved by Richardson's iterative method, p(k+1)=p(k)+tkr(k), where r(k)=q-Ap(k) is the residual and t0, . . . tk are acceleration parameters. The incomplete factorization, Ka, of the strongly implicit procedure (SIP) yields an improvement of Richardson's method, p(k+1)=p(k)+tkKa−1r(k). Parameter a originates from SIP. The product of the L and U factors produced by SIP gives Ka=LU. The best values of the tk acceleration parameters may be computed dynamically by an efficient algorithm; the best value of a must be found by trial and error, which is not hard for only one value. The advantages of the method are (1) it always converges, (2) with the exception of the a parameter, parameters are computed dynamically, and (3) convergence is efficient for test problems characterized by heterogeneities and transmissibilities varying over 10 orders of magnitude. The test problems originate from field data and were suggested by industry personnel as particularly difficult. Dynamic computation of parameters is also a feature of the conjugate gradient method, but the iteration described here does not require A to be symmetric. Matrix Ka−1 A must be such that the real part of each eigenvalue is nonnegative, or the real part of each is nonpositive, but not both positive and negative. It is in this sense that the method always converges. This condition is satisfied by many simulator-generated matrices. The method also may be applied to matrices arising from the simulation of other processes, such as chemical flooding. Introduction The solution of a linear algebraic system, Ap=q, is a basic, costly step in the numerical simulation of a hydrocarbon reservoir. Many current solution methods are impractical for large linear systems arising from three-dimensional simulations or from reservoirs characterized by widely varying and discontinuous physical parameters. An iterative solution is described with these two main advantages:it is efficient for difficult problems andthe selection of iteration parameters is straightforward. The method is Richardson's method applied to a preconditioned linear system. Matrix A may be symmetric or nonsymmetric. In the simulation of multiphase flow, it is usually nonsymmetric. Convergence behavior is shown for four examples. Two of these, Examples 3 and 4, were provided by an industry laboratory (Exxon Production Research Co.), and were suggested by personnel as especially difficult to solve; SIP failed to converge and only the diagonal method1 was effective. Convergence of Richardson's method is compared with the diagonal method using data from a laboratory run. The other two examples are: Example 1, a matrix not difficult to solve, generated from field data, and Example 2, a variant of a difficult matrix described by Stone.2 The easy matrix of Example 1 is included to show the performance of Richardson's method (with preconditioning) on a simple problem.

The Water Transferred of Meandering River in Cellular Model

2012 ◽  
Vol 452-453 ◽  
pp. 842-845
Author(s):  
Min Zhang ◽  
Rui Xun Lai
Keyword(s):  
Flow Direction ◽  
Field Measurements ◽  
Treatment Method ◽  
Special Treatment ◽  
Cellular Model ◽  
Channel Processes ◽  
Flow Routing ◽  
The Real ◽  
Meandering River ◽  
Routing Scheme

We have known that many of the main features of river can be captured in a relatively simple cellular computer model. Here we examine some of the detailed characteristics of this model. We show a new tangential angle and special treatment method in curvature channel in a simple cellular model. The water distribute will be determined by the tangential angle which represent the flow direction. And the same tangential angle cellular in curvature reach will be transport in one group in filially. The results show it can make the flow routing conformed to the real river, and solve the question of curve bending coefficient is too large. The results of the routing scheme are compared with field measurements of cross-section, with the predictions of a more close to the real. It’s indicate that the curvature flow routing scheme outlined here is able to overcome some of the limitations of previous simple cellular automata models and may be suitable for use in curvature reach of river modeling water and sediment transport and channel change in complex fluvial environments. As such this research represents a small and ongoing contribution to the field of numerical simulation of curvature channel processes.

scholarly journals An estimation of effects of memory and learning experience on the eoq model with price dependent demand

2021 ◽  
Vol 55 (5) ◽  
pp. 2991-3020
Author(s):  
Mostafijur Rahaman ◽  
Sankar Prasad Mondal ◽  
Shariful Alam
Keyword(s):  
Differential Equation ◽  
Decision Making ◽  
Fractional Differential Equation ◽  
Lot Sizing ◽  
Learning Experience ◽  
Memory And Learning ◽  
Eoq Model ◽  
Novel Approach ◽  
Fractional Differential ◽  
Fuzzy Fractional Differential Equation

In this article, an economic order quantity model has been studied in view of joint impacts of the memory and learning due to experiences on the decision-making process where demand is considered as price dependant function. The senses of memory and experience-based learning are accounted by the fractional calculus and dense fuzzy lock set respectively. Here, the physical scenario is mathematically captured and presented in terms of fuzzy fractional differential equation. The α-cut defuzzification technique is used for dealing with the crisp representative of the objective function. The main credit of this article is the introduction of a smart decision-making technique incorporating some advanced components like memory, self-learning and scopes for alternative decisions to be accessed simultaneously. Besides the dynamics of the EOQ model under uncertainty is described in terms of fuzzy fractional differential equation which directs toward a novel approach for dealing with the lot-sizing problem. From the comparison of the numerical results of different scenarios (as particular cases of the proposed model), it is perceived that strong memory and learning experiences with appropriate keys in the hand of the decision maker can boost up the profitability of the retailing process.

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