Invariant Embedding Technique Research Articles

Recovery of the main fracturing direction of oil reservoirs from seismic and AVO data

In the present work, we formulate and solve an inverse problem to recover the main fracturing direction of petroleum reservoirs from seismic and AVO data. Equivalent media theory provides a starting point for modeling an equivalent anisotropic medium starting from an isotropic background with fractures. We then represent the equivalent medium in a new coordinate system whose x-axis is parallel to the reservoir's top. Using the classical theory of wave propagation and the Invariant Embedding technique (specially to deal with anisotropic media), we solve exactly the elastodynamic equations of motion to compute the reflection of P, SV, SH waves at the reservoir's top, without the need of using the so-called Zoeppritz approximation. In the inverse problem formulation, we assume that we know a priori the reservoir isotropic background elastic constants as well as the fracture system behavior, by means of its effective compliance. We then apply a metaheuristic to recover the main fracturing direction whose input data are the amplitudes of the reflected waves recorded by the geophones and the output are two Euler-angles representing the fracturing direction. We validate the methodology using noiseless synthetic data and further test its robustness using incomplete and noisy synthetic data.

An analytical–numerical formulation to modelling wave propagation in double-cased oil wells

A reliable non-destructive test to assess the integrity of double-cased wells is still an open task. In the present work, we introduce an analytical–numerical formulation for modelling the wave propagation in such structures. The formulation was developed with the aid of the invariant embedding technique and is numerically unconditionally stable even for high frequencies. Usually, the invariant embedding technique is employed in cases of solid stratified media. Here, due to the nature of the structure under investigation, we had to extend the technique to accommodate fluid layers as well. We illustrated the use and relevance of the developed formulation modelling the wave propagation in a typical double-cased pipe and identifying the frequencies and angles of incidence that most strongly interact with defects placed at the critical casing layer/cement sheath interface. Based on a robustness analysis, we further indicated what would presumably be the best inspecting field. In that sense, we hope our formulation can help in the optimum experiment design of strategies aimed at assessing the structural integrity of double-cased wells.

A Novel Unified Computational Approach to Constructing Shortest-Length or Equal Tails Confidence Intervals in Terms of Pivotal Quantities and Quantile Functions

A confidence interval is a range of values that gives the user a sense of how precisely a statistic estimates a parameter. In the present paper, a new simple computational method is proposed for simultaneous constructing and comparing confidence intervals of shortest length and equal tails in order to make efficient decisions under parametric uncertainty. This method represents a simple and computationally attractive numerical technique for finding the shortest-length and equal tails confidence intervals using pivotal quantities, which are developed from either maximum likelihood estimates or sufficient statistics. In statistics, a pivotal quantity (or pivot) is a function of observations and unobservable parameters such that the function’s probability distribution does not depend on the unknown parameters (including nuisance parameters). Finding a pivotal quantity is not discussed, but the choice a “good” pivotal quantity is essential for the resulting confidence interval to be useful. The unified computational technique yields intervals in several situations which have previously required separate analyses using more advanced techniques and tables for numerical solutions. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate nuisance parameters from the problem via the technique of invariant statistical embedding and averaging in terms of pivotal quantities (ISE&APQ). It should be noted that the well-known classical approach to constructing confidence intervals of the shortest length considers at least three versions of possible solutions and is in need of information about the form of the probability distribution of pivotal quantity in order to determine an adequate version of the correct solution. The proposed method does not need such information. It receives this information through the quantiles of the probability distribution of pivotal quantity. Therefore, the proposed method automatically recognizes an adequate version of the correct solution. To illustrate this method, numerical examples are given. In detail, the Pareto distribution is discussed.

A Novel Technique for Optimization of Statistical Decisions under Parametric Uncertainty through Invariant Statistical Embedding and Averaging in Terms of Pivotal Quantities

In the present paper, a new technique of invariant embedding of sample statistics in a decision criterion (performance index) and averaging this criterion via pivotal quantity is proposed for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty. The technique of invariant statistical embedding and averaging in terms of pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. To illustrate the proposed technique of ISE&APQ, application examples are given.

A New Technique of Invariant Statistical Embedding and Averaging Via Pivotal Quantities for Intelligent Constructing Efficient Statistical Decisions under Parametric Uncertainty

In the present paper, a new technique of invariant embedding of sample statistics in a decision criterion (performance index) and averaging this criterion via pivotal quantities is proposed for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. The aim of the present paper is to show how the technique of ISE&APQ may be employed in the particular case of optimization, estimation, or improvement of statistical decisions under parametric uncertainty. To illustrate the proposed technique of ISE&APQ, application examples are given.

A Novel Intelligent Technique of Invariant Statistical Embedding and Averaging via Pivotal Quantities for Optimization or Improvement of Statistical Decision Rules under Parametric Uncertainty

In the present paper, for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a decision criterion and averaging this criterion over pivots’ probability distributions is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. The aim of the present paper is to show how the technique of ISE&APQ may be employed in the particular case of optimization, estimation, or improvement of statistical decisions under parametric uncertainty. To illustrate the proposed technique of ISE&APQ, illustrative examples of intelligent constructing exact statistical tolerance limits for prediction of future outcomes coming from log-location-scale distributions under parametric uncertainty are given

Non-steady state transport of charge carriers. An approach based on invariant embedding method

In this work, we report on a model that describes the microscopic electrical transport as a transmission problem using the invariant embedding technique. Analytical expressions for the transport coefficients under non-steady-state conditions are derived allowing us to calculate carrier concentration and time-dependent conductivity. Employing measurable magnitudes, our theoretical results allow us to determine defect concentrations, carrier generation rates, cross sections of recombination, and capture by traps. This model can be employed to study the conduction processes of semiconductors and test their band and defect structure. In particular, time-dependent photoconductivity measurements of a ZnO microwire have been well fitted using our model indicating a relevant role of intrinsic point defects in this material.

Transient radiative-conductive heat transfer modeling in constructional semitransparent silica ceramics

The article deals with transient radiative – conductive heat transfer investigation in constructional silica ceramics with 10% of total porosity in 2D slab geometry. An empirical model, the so-called effective thermal conductivity (ETC) approximation, and rigorous radiative heat transfer (RCHT) problem solution are compared. Effective thermal conductivity coefficients were determined by both quasi-stationary and non-stationary sample heating and corresponding conductive heat transfer inverse problem solution. Rigorous RCHT calculations were based on radiative transfer equation solution. Spectral directional-hemispherical reflectances were used for scattering and absorption coefficients determination by an identification technique. Identification was based on direct problem solution by invariant embedding technique and parametric optimization. Mie theory and independent scattering approximation with regard to scatterers (pores) size distribution were used for mean scattering cosine estimation. It was shown that in the case of non-stationary temperature regime with rapid heating and cooling stages ETC models fail to predict through the thickness temperature distribution with absolute error less than 502013100 K. The absolute value of this error is strongly affected by the heating schedule. Moreover, calculations revealed several features of the heat flux through the unheated boundary that could not be taken into considerations by effective thermal conductivity approaches.

Identifying the ultrasonic inspecting fields that most strongly interact with adhesive bonding defects

We propose a systematic modelling procedure to identify the harmonic ultrasonic incident fields that most strongly interact with adhesive bonding defects in submerged multilayer plates. In principle, these are also the optimal choices for the interrogating fields in inverse methods for interfacial stiffness estimation. We model the adhesive bonding with the aid of the spring boundary conditions and model the bonding defects by a reduction in the spring constants. As part of the proposed procedure, we use the invariant embedding technique to compute the reflection coefficient at the top of the immersed laminates. The main advantage of this technique is its unconditional numerical stability even for evanescent waves at high frequencies. The developed algorithm is equally well suited to treating isotropic as well as anisotropic layers. We illustrate the application of the proposed procedure simulating the inspection of a three-layer isotropic plate and a 16-ply anisotropic composite laminate used in the aeronautical industry. In the simulations, we identify the frequencies and angles of incidence that will most strongly interact with bonding defects.

Simple Approach to Electromagnetic Scattering by Small Radially Inhomogeneous Spheres

In this paper, we use the invariant embedding technique to solve the problem of plane electromagnetic (EM) wave scattering by a nonuniform object with spherically symmetric distributions of dielectric and magnetic susceptibilities. Assuming that the size of the object is smaller or comparable with the vacuum wavelength, we consider a simple and effective approach to restore the EM fields inside the object and the scattering and absorption cross sections. The method is based on exact solutions of Maxwell’s equations and provides no limitations on the wavelengths inside the object. Therefore, it can be used to study complex cases, including those related to local amplification and absorption of waves in inhomogeneous resonant media.

Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models

Predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. In order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. In particular, this technique can be used for optimization of inventory management problems. A practical example is given.

Novel technique for medium permittivity profile reconstruction using THz pulsed spectroscopy

A new method for permittivity profile reconstruction based on terahertz time-domain spectroscopy signal processing is presented. Reconstruction is accomplished in two steps. First, the sample pulse function is reconstructed using sample time-domain reflection data. Low and high frequency noise filtering and the interpolation of the pulse function at low frequencies are then applied. Second, an invariant embedding technique is used to calculate the dielectric permittivity profile based on the sample pulse function. Method was tested experimentally on samples with known permittivity profiles. The algorithm is stable to additive Gaussian white noise as shown using mathematical modeling based on the finite-difference time-domain technique (FDTD). Possible applications of this permittivity profile reconstruction technique are discussed.

Optimization under uncertainty with applications to personnel management problems in tourism

A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. In the present paper, for improvement or optimization of statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a performance index is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, an invariant embedding technique is independent of the choice of priors. It allows one to eliminate unknown parameters from the problem and to find the best invariant decision rule, which has smaller risk than any of the well-known decision rules. In order to illustrate the application of the proposed technique for constructing optimal statistical decisions under parametric uncertainty, we discuss the following personnel management problem in tourism. A certain company provides interpreter-guides for tourists. Some of the interpreter-guides are permanent ones working on a monthly basis at a daily guaranteed salary. The problem is to determine how many permanent interpreter-guides should the company employ so that their overall costs will be minimal? We restrict attention to families of underlying distributions invariant under location and/or scale changes. A numerical example is given.

Invariant embedding technique for medium permittivity profile reconstruction using terahertz time-domain spectroscopy

Reconstructing the dielectric permittivity profile (depth dependence of sample dielectric permittivity) is an important inverse problem. We present a new method for permittivity profile reconstruction based on terahertz time-domain spectroscopy signal processing. Reconstruction is accomplished in two steps. First, the sample pulse function is reconstructed using sample time-domain reflection data. Low- and high-frequency noise filtering and the interpolation of the pulse function at low frequencies are then applied. Second, an invariant embedding technique is used to calculate the dielectric permittivity profile based on the sample pulse function. Samples with known permittivity profiles have been studied experimentally using this procedure in order to verify this algorithm. This algorithm is stable to additive Gaussian white noise as shown using mathematical modeling based on the finite-difference time-domain technique. Possible applications of this permittivity profile reconstruction technique are discussed.

Solving the Problems of Inspection Planning under Parametric Uncertainty of Underlying Models

Certain fatigued structures must be inspected in order to detect fatigue damages that would otherwise not be apparent. A technique for obtaining optimal inspection strategies is proposed for situations where it is difficult to quantify the costs associated with inspections and undetected failure. For fatigued structures for which failures (fatigue damages) are only detected at the time of inspection, it is important to be able to determine the optimal times of inspection. Fewer inspections will lead to lower fatigue reliability of the structure upon demand, and frequent inspection will lead to higher cost. When there is a fatigue reliability requirement, the problem is usually to develop an inspection strategy that meets the reliability requirements. It is assumed that only the functional form of the underlying invariant distribution of time-to-failure is specified, but some or all of its parameters are unspecified. The invariant embedding technique proposed in this paper allows one to construct an optimal inspection strategy under parametric uncertainty. This strategy represents a sequence of inspection times satisfying the specific criterion, which takes into account the predetermined value of the conditional fatigue reliability of the structure. The numerical examples are given.

Impedance technique for modeling electromagnetic wave propagation in anisotropic and gyrotropic media

A convenient and physically unambiguous interpretation of the invariant embedding technique is discussed in application to problems of electromagnetic wave propagation in complex media characterized by anisotropic and gyrotropic dielectric or magnetic response.

Free-surface multiple attenuation using inverse data processing in the coupled plane-wave domain

Free-surface multiples contain a large amount of energy in the seismic data because of large reflectivity of the free surface. We propose a method for free-surface multiple attenuation by a simple muting in the inverse coupled plane-wave domain. Our method is based on inverse data processing and the well-known 2D invariant embedding technique. If the lateral variation in subsurface structure is smooth, the data are well compressed in the 2D coupled plane-wave domain, reducing computation costs and stabilizing the inversion procedure. Surface multiples and primaries are well separated in the inverse coupled plane-wave domain, and multiples can be eliminated by simple muting, which does not damage the primary energy. To reduce artifacts, wraparound, and noise introduced by the frequency-domain data matrix inversion, horizontal and vertical tapers are applied. A least-squares matrix inversion method is chosen to stabilize the inversion. Synthetic data examples show that plane-wave inverse data processing is stable and successful in attenuating free-surface multiples.

Modelling of ground penetrating radar data in stratified media using the reflectivity technique

Horizontally layered media are often encountered in shallow exploration geophysics. Ground penetrating radar (GPR) data in these environments can be modelled by techniques that are more efficient than finite difference (FD) or finite element (FE) schemes because the lateral homogeneity of the media allows us to reduce the dependence on the horizontal spatial variables through Fourier transforms on these coordinates. We adapt and implement the invariant embedding or reflectivity technique used to model elastic waves in layered media to model GPR data. The results obtained with the reflectivity and FDTD modelling techniques are in excellent agreement and the effects of the air–soil interface on the radiation pattern are correctly taken into account by the reflectivity technique. Comparison with real wide-angle GPR data shows that the reflectivity technique can satisfactorily reproduce the real GPR data. These results and the computationally efficient characteristics of the reflectivity technique (compared to FD or FE) demonstrate its usefulness in interpretation and possible model-based inversion schemes of GPR data in stratified media.

Recovering acoustic reflectivity using Dirichlet-to-Neumann maps and left- and right-operating adjoint propagators

SUMMARY Constructing an image of the Earth subsurface from acoustic wave reflections has previously been described as a recursive downward redatuming of sources and receivers. Most of the methods that have been presented involve reflectivity and propagators associated with one-way wavefield components. In this paper, we consider the reflectivity relation between two-way wavefield components, each a solution of a Helmholtz equation. To construct forward and inverse propagators, and a reflection operator, the invariant-embedding technique is followed, using Dirichlet-to-Neumann maps. Employing bilinear and sesquilinear forms, the forward- and inverse-scattering problems, respectively, are treated analogously. Through these mathematical constructs, the relationship between a causality radiation condition and symmetry, with respect to a bilinear form, is associated with the requirement of an anticausality radiation condition with respect to a sesquilinear form. Using reciprocity, sources and receivers are redatumed recursively to the reflector, employing left- and right-operating adjoint propagators. The exposition of the proposed method is formal, that is numerical applications are not derived. The key to applications lies in the explicit representation, characterization and approximation of the relevant operators (symbols) and fundamental solutions (path integrals). Existing constructive work which could be applied to the proposed method are referred to in the text.

Factorization of second-order elliptic boundary value problems by dynamic programming

We present a method to factorize a second-order boundary value problem in a cylindrical domain in a system of uncoupled first-order initial value problems, together with a nonlinear Riccati-type equation for functional operators. This uncoupling is obtained by a space invariant embedding technique along the axis of the cylinder. This method can be viewed as an infinite-dimensional generalization of the block Gauss LU factorization.