Invariant Imbedding Technique Research Articles

Design and Validation of the Invariant Imbedded T-Matrix Scattering Model for Atmospheric Particles with Arbitrary Shapes

Light scattering by non-spherical particles is an important factor influencing atmospheric radiative transfer. To accurately simulate the scattering properties of non-spherical particles, the Invariant Imbedded T-matrix method (IIM T-Matrix) is developed by combining the Lorenz–Mie theory and invariant imbedding technique. In this model, the non-spherical particle is regarded as an inhomogeneous sphere and discretized into multiple spherical layers in the spherical coordinate system. The T-matrix of the inscribed sphere is firstly calculated by the Lorenz–Mie theory, and then taking it as the initial value, the T-matrix is updated layer by layer by using the invariant imbedding technique. To improve the computational efficiency, the model is further parallelized by the OpenMP technique. To verify the simulation accuracy of the IIM T-Matrix method, the results of the model are compared with those of the EBCM (Extended Boundary Condition Method) T-Matrix method, DDA (Discrete Dipole Approximation) and MRTD (Multi-Resolution Time Domain). The results show that the scattering phase matrix simulated by the IIM T-Matrix method closely agrees with that of the well-tested models, indicating that the IIM T-Matrix method is a powerful tool for the light scattering simulation of non-spherical particles. Since the IIM T-Matrix method is derived from the volume integral equation, compared to the T-Matrix method which is based on surface integral principles (i.e., “EBCM” or the “null field method”), it can be applied to the scattering calculations of particle with arbitrary shapes and inhomogeneous compositions, which can greatly expand the application scope of the T-Matrix method.

A model to predict spatial, spectral and vertical changes in the average cosine of the underwater light fields: Implications for remote sensing of shelf-sea waters

An optical model is developed using experimental data of Inherent Optical Properties (IOP) from oceanic, coastal and productive lagoon waters in order to calculate vertical and spectral profiles of the average cosine in a wide variety of waters within coastal and shelf-sea environments. The results are compared with those generated using a radiative transfer numerical model based on the invariant imbedding technique (HydroLight) with realistic depth-dependent IOPs and appropriate surface and bottom boundary conditions and the results from three existing models (Haltrin, 1998; Timofeyeva, 1971; Talaulikar et al., 2014). The average cosine predicted by the new model shows good agreement with the values obtained directly from radiative transfer calculations for a broad range of the IOPs and solar zenith conditions. Good correlations with excellent linearity with significantly low errors demonstrate a good deal of confidence of the model for accomplishing further applications. Since knowledge of spatial and temporal structures of the average cosine is of great importance to our understanding of the particle dynamics of pelagic ecosystems and coastal processes, efforts were made to apply the present model to both multispectral MODIS-Aqua imagery and hyperspectral (HICO) images acquired over the Arabian Sea and coastal lagoons of the Bay of Bengal dominated by river plumes and phytoplankton blooms. Maps of the average cosine derived from these data demonstrated significant changes in the magnitude and spectral behavior of the average cosine (from nearly featureless to strong spectral features and inflections) from different water types. Substantial changes in its spatial and spectral structures associated with highly productive waters, phytoplankton blooms and sediment plumes, as compared with open ocean areas, are well supported by the theoretical and experimental studies. The advantages of the new model in comparison with existing models are its capability to predict the vertical, spatial, spectral and temporal structures of average cosine with greater accuracy.

Invariant Imbedding and the Radiation Transfer in a Plane-Parallel Inhomogeneous Atmosphere

The invariant imbedding technique is applied to the problems of radiation transfer in a plane-parallel inhomogeneous atmosphere. All the parameters which describe the elementary event of scattering and the distribution of the energy sources are allowed to vary with depth. Mathematically, the considered standard problems of the theory are reduced to initial-value problems which are better adapted to capabilities of the modern high speed computers. The reflectance of an atmosphere is shown to play a prominent role in describing the diffusion process since all the other characteristics of the radiation field are expressed through it. Three transfer problems frequently encountered in astrophysical applications are discussed: the radiation diffusion in the source-free medium, in a medium with arbitrarily distributed energy sources, as well as the problem of finding the statistical mean quantities, characteristics of the multiple scattering in the atmosphere.

Threshold probability of non-terminal type in finite horizon Markov decision processes

We consider a class of problems concerned with maximizing probabilities, given stage-wise targets, which generalizes the standard threshold probability problem in Markov decision processes. The objective function is the probability that, at all stages, the associatively combined accumulation of rewards earned up to that point takes its value in a specified stage-wise interval. It is shown that this class reduces to the case of the nonnegative-valued multiplicative criterion through an invariant imbedding technique. We derive a recursive formula for the optimal value function and an effective method for obtaining the optimal policies.

Reconstruction of relaxation modulus for viscoelastic medium with complete data

The inverse scattering problem for viscoelastic medium with wave impedance mismatch at the rear boundary is studied in this paper. The propagation operators of the viscoelastic medium are defined first and the integro-differential equations of these operators are derived via the invariant imbedding technique. For the inverse scattering problem, a new inversion procedure to reconstruct the relaxation modulus of the viscoelastic medium is developed, which utilizes a complete set of data, namely the one-side reflection data for one round trip through the viscoelastic slab. These data are complete in the sense that they can be extended to arbitrary time t in the inversion procedure. This inverse method is implemented numerically on several problems at the end of the paper. The more general case of a medium consisting of a stack of homogeneous viscoelastic medium layers is also considered.

Random amplifying medium: statistics of super-reflectance and time-delay fluctuations

A light wave incident on a random amplifying medium (RAM) may be reflected with a reflection coefficient ( r) exceeding unity. This super-reflectance and the associated time-delay will be non-self-averaging, and show sample-specific fluctuations characteristic of coherent transport through random media. The statistics of these random fluctuations involve a synergetic inter-play of disorder (localization) and coherent amplification (non-resonant distributed feedback), which is disallowed for its electronic (fermionic) counterpart. In this work, we review and report on some of our recent analytical results for these statistics for the case of a long one-dimensional (1D) RAM that physically corresponds to a single-channel optical fibre having gain and disorder, and operating in the reflection mode. Our main result is that the coherent amplification makes the probability density of the super-reflection peak for some r>1 with a long tail, while the otherwise infinite mean delay-time is rendered finite. Interestingly, our treatment of amplification provides a natural counter to monitor the time-delay for the corresponding passive (e.g., electronic) problem as well. Our approach, based on the invariant imbedding technique, is valid below the threshold for lasing, i.e., for the localization length < the gain-length.

Inverse scattering for inhomogeneous viscoelastic media

In this paper, the inverse scattering problems for the full inhomogeneous viscoelastic medium are studied via the invariant imbedding technique. Special attention is paid to the propagation operators of the viscoelastic medium and the imbedding equations for these operators are derived. For the inverse scattering problems, it is shown that the reflection data can be extended from one round trip through the iscoelastic slab to arbitrary time with the help of the propagation operators, hence the reconstruction of the relaxation modulus is sufficient to be considered only in one round trip. It is also shown that only one-side measurement reflection data are not sufficient to reconstruct the relaxation modulus and the density of the medium simultaneously. The corresponding numerical examples are presented. For the case that the relaxation modulus of the medium is modeled by two independent functions, an iterative inversion procedure is proposed to recover the relaxation modulus and the density simultaneously with the input two-side normally reflection data.

Nonlinear system identification using additive dynamic neural networks-two on-line approaches

This paper proposes a class of additive dynamic connectionist (ADC) models for identification of unknown dynamic systems. These models work in continuous time and are linear in their parameters. Also, for this kind of model two on-line learning or parameter adaptation algorithms are developed: one based on gradient techniques and sensitivity analysis of the model output trajectories versus the model parameters and the other based on variational calculus, that lead to an off-line solution and an invariant imbedding technique that converts the off-line solution to an on-line one. These learning methods are developed using matrix calculus techniques in order to implement them in an automatic manner with the help of a symbolic manipulation package. The good behavior of the class of identification models and the two learning methods is tested on two simulated plants and a data set from a real plant and compared, in this case, with a feedforward static (FFS) identifier.

Wave scattering from biased elastic dielectrics

The scattering of harmonic transverse waves is studied in a layered solid half-space which is prestressed and electrically biased. Electromechanic modes propagating normally to the stratification planes are analyzed using a wave-splitting method and a sufficient condition is derived for the existence of the solution. The invariant imbedding technique is employed to obtain a differential equation for the reflectivity matrix. This equation is exploited to formulate the inverse problem whose solution is presented in an iterative form. A useful result is then established about the convergence of the series so determined.

An invariant imbedding analysis of general wave scattering problems

The invariant imbedding technique, via the solution of a Riccati-type equation, is modified to calculate the wave fields inside and scattered from a strongly (laterally and vertically) heterogenous, anisotropic inclusion, which may be large but remains compact. The factorization underlying this approach is carried out with respect to direction of average power flow rather than the more conventional factorization with respect to local direction of propagation. The solution of the operator Riccati equation is related to the Dirichlet-to-Neumann map. The formulation is robust in the sense that it can handle a rather extreme range of modal wave speeds, and allows continuous as well as discontinuous medium variations on different (wave) length scales. It also, inherently, takes care of critical-angle phenomena. The algorithm, based on the invariant imbedding approach, yields the internal fields for a full survey of sources and receivers simultaneously. The wave field solution in the inclusion is coupled to the external field via a boundary element approach.

Surface-wave mode coupling for efficient forward modelling and inversion of body-wave phases

SUMMARY The importance of surface-wave mode coupling in the modelling of body-wave phases by surface-wave mode summation is studied by means of sensitivity kernels obtained with the Born approximation and exact solutions of the Invariant Imbedding Technique. It is shown that, independent of the character of the lateral heterogeneity, surface-wave mode coupling is required to model body-wave phase perturbations and that neglecting intermode coupling, as in the WKBJ method for surface waves, can lead to large biases. Because methods which describe surfacewave mode coupling in an exact fashion are computationally too expensive to use in inversion schemes, the Scalar Exponent Approximation (SEA) is presented, which is a computationally efficient method and takes mode coupling into account. Since, instead of Earth normal modes, surface-wave modes are used, the summation over the angular order l is carried out analytically. This means that the number of modes and mode interactions needed is significantly reduced which assures an efficient manner of modelling. It is shown that the SEA is accurate in modelling body-wave phase perturbations for geophysically realistic configurations. Because, in contrast to the WKBJ sensitivity kernels, mode coupling introduces sensitivity kernels which also depend on the position along the source-receiver path, the SEA requires a larger model parameter set in inversions. A procedure is presented which reorganizes the model parameter set and leads to a reduced set of physically relevant model parameters. The combination of the SEA and the reorganization of the model parameters can be used efficiently in large-scale 3-D inversions which incorporate the important effects of surface-wave mode coupling.

Energy loss of kilovolt electrons in thin films

The energy spectrum of electrons transmitted through thin foils is obtained by the invariant imbedding technique. Strong influence of the threshold behavior of the inelastic cross section on the multiple collision process is emphasized. Comparison with experimental data shows that the developed theory gives an adequate description of the process of the electron penetration.

A new approach to resonance line scattering in planetary atmospheres

The invariant imbedding technique is applied to the problem of resonance line scattering with frequency redistribution. By applying the principles of invariance to the resonance line scattering problem a set of non-linear, integrodifferential equations for the scattering and transmission functions are derived. Thus, an alternate approach to the resonance line scattering problem is presented. Rather than solving for the specific intensity directly the invariant imbedding equations may be solved for the scattering and transmission functions from which the specific intensity may be calculated. The technique is very general and in principle, can accommodate scattering in inhomogeneous atmospheres with arbitrary frequency redistribution and scattering phase functions. Solutions to the invariant imbedding equations in some simple cases are presented. If the scattering is isotropic and completely noncoherent and the atmosphere may be modeled as homogeneous, the scattering and transmission functions may be expressed in terms of generalized H, X and Y functions. This technique is extended to the problems of coupled spectral lines and scattering of radiation from internal sources.

New methods for boundary value problems

Nonlinear two-point boundary value problems abound in the theory of the urine concentrating mechanism. Systems of order 50 are commonly considered. Various approaches to these problems have been employed including quasilinearization. Three difficulties have been noted with quasilinearization: 1) Large Jacobian matrices need to be evaluated at each integration step. 2) A large amount of memory is needed to store the temporary particular and complementary solution values. 3) The differential equations may be stiff and therefore difficult to integrate.A recent conceptual advance has made possible the automatic and exact evaluation of the gradient functions. This advance makes coding of the derivatives less prone to error and enables the derivatives to be evaluated efficiently. The memory constraint is overcome by storing the temporary function values on the disk. Since access to these values is sequential, the amount of time spent storing and retrieving the values from disk is minimal. The difficulty in integrating the differential equations can be solved by using an invariant imbedding technique, which has proven useful in the past for solving stiff counterflow problems.

Invariant imbedding for block tridiagonal systems

The invariant imbedding technique is applied to the block tridiagonal systems in a general setting using, when necessary, the generalized Moore-Penrose inverse of a matrix. Three examples are given, and the existence and the stability of solutions are discussed.

Classical noise and nonlinear effects in the ideal avalanche diode

2014 A general formulation of the avalanche statistics in an ideal diode is presented, in an effort to extend the theory to the nonlinear regime. All known results, including the time dependency, are recovered with this procedure. It unifies the various treatments in a consistent way and enables an analysis of nonlinear phenomena. Where the injection transient and different drift velocities become important, the invariant imbedding technique is used. The limitations of the continuity equations as a starting point for a statistical treatment are discussed. An example is given where the classical methods (Langevin, G.-R. theorem, a-c perturbations, Fourier analysis) fail in the nonlinear case. Revue Phys. Appl. 15 (1980) 1585-1597 NOVEMBRE 1980, 1

On the numerical solution of a differential nonlinear eigenvalue problem on an infinite range

In this paper we describe the solution of a rather intractable differential eigenvalue problem arising from a stability problem in fluid dynamics. Techniques used for solving the problem include finite differences, a variety of shooting methods and Riccati transformations (invariant imbedding). A description is given of the difficulties encountered. The invariant imbedding technique was found to be the most efficient method for obtaining the solution accurately. Though our results are for one difficult problem, we anticipate that our conclusions will be widely applicable to problems of similar type.

A Cauchy problem for Milne's integral equation with a pincherle-goursat kernel

In a series of our preceding papers (cf. Bellman and Ueno, 1971a, 1971b, 1972a, 1972b, 1972c; Ueno 1972), with the aid of invariant imbedding technique, we transformed the two point boundary value problem for integral equations into the initial-value problem. In this paper, extending the procedure to the Milne's integral equation, whose kernel is expanded into a double Fourier cosine series, we show how to find a Cauchy system for the required solution of Milne's integral equation and the Fredholm resolvent. The obtained integro-differential equations are reduced to large system of ordinary differential equations with known initial conditions. Then, they are suitable for the numerical computation of the solution by the modern high-speed computer.

Invariant Imbedding and Unsteady Ground-Water Flow

The invariant imbedding technique is used to find the unknown slope of the water table near the surface reservoir in a problem of unsteady ground-water flow into a surface reservoir in a large unconfined aquifer. The original nonlinear two-point boundary-value problem is transformed into an initial-value problem and an exact solution is obtained by the technique of quasilinearization. The invariant imbedding method involves no trial-and-error procedure and is easily programmed. This technique provides a useful tool in solving field problems and it becomes extremely powerful in solving a family of problems in which a set of unknown initial conditions are required to be determined from the corresponding given set of boundary conditions.

A Comparison of Numerical Methods for Solving Optimal Control Problems

Linear systems with quadratic performance criteria typically lead to two-point boundary value problems. The numerical solution of such problems is often difficult because of roundoff and truncation errors. In this paper numerical methods for digital computer solutions are considered for a system described by a given second-order linear differential equation and the results are compared. The results show that where accurate analytic solutions cannot be obtained, the invariant imbedding technique yields more accurate results, for a given integration grid interval, than the other methods considered.