Exact Wave Equation Research Articles

Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method

Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

Seismic response of stratified rock slopes due to incident P and SV waves using a semi-analytical approach

Based on the theory of elastodynamics and the concept of wave-field separation, a semi-analytical method for the scattering and diffraction of P-SV waves by stratified rock slopes is proposed in this study. The total wave fields are decomposed into the free field with no irregularity and the scattered wave field generated by slope topography. The former is solved directly with the aid of a dynamic stiffness matrix, and the latter is modeled by applying the inclined and horizontal fictitious distributed loads on the corresponding boundaries. Since the methodology is derived from the exact wave equations and the non-singular frequency-domain Green's functions, it can be applied to seismic dynamic response problems for layered slopes with arbitrary incident waves and layer thicknesses, and the presented solutions are high-precision and well-converged. Using a Ricker wavelet as input, the accuracy and effectiveness of the proposed method for both vertically and obliquely incident P and SV waves are fully verified. Taking three actual earthquake records including Taft wave, El Centro wave, and Loma Prieta wave as inputs, several analysis examples are presented. The parametric studies demonstrate that a significant elevation amplification effect can be seen in the ground motion of rock slopes due to the interaction of the upper slope feature with the underlying bedrock half-space. The PGA amplification factor, Fourier spectrum and standard spectral ratio of rock slopes are evidently affected by the slope height, slope angle and shear velocity of the medium. Subsequently, sensitivity analyses are performed on model slopes with varying incident wave frequencies and incident angles to illustrate the broad applicability of the proposed method. Finally, a case study is presented to evaluate the practicality of the presented procedure and highlight the benefits of performing a rapid semi-analytical solution to assess potential hazards of earthquake-induced rock slides.

Coupling between turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH approach

Context. The development of space-borne missions such as CoRoT and Kepler now provides us with numerous and precise asteroseismic measurements that allow us to put better constraints on our theoretical knowledge of the physics of stellar interiors. In order to utilise the full potential of these measurements, however, we need a better theoretical understanding of the coupling between stellar oscillations and turbulent convection. Aims. The aim of this series of papers is to build a new formalism specifically tailored to study the impact of turbulence on the global modes of oscillation in solar-like stars. In building this formalism, we circumvent some fundamental limitations inherent to the more traditional approaches, in particular the need for separate equations for turbulence and oscillations, and the reduction of the turbulent cascade to a unique length and timescale. In this first paper we derive a linear wave equation that directly and consistently contains the turbulence as an input to the model, and therefore naturally contains the information on the coupling between the turbulence and the modes through the stochasticity of the equations. Methods. We use a Lagrangian stochastic model of turbulence based on probability density function methods to describe the evolution of the properties of individual fluid particles through stochastic differential equations. We then transcribe these stochastic differential equations from a Lagrangian frame to a Eulerian frame more adapted to the analysis of stellar oscillations. We combine this method with smoothed particle hydrodynamics, where all the mean fields appearing in the Lagrangian stochastic model are estimated directly from the set of fluid particles themselves, through the use of a weighting kernel function allowing to filter the particles present in any given vicinity. The resulting stochastic differential equations on Eulerian variables are then linearised. As a first step the gas is considered to follow a polytropic relation, and the turbulence is assumed anelastic. Results. We obtain a stochastic linear wave equation governing the time evolution of the relevant wave variables, while at the same time containing the effect of turbulence. The wave equation generalises the classical, unperturbed propagation of acoustic waves in a stratified medium (which reduces to the exact deterministic wave equation in the absence of turbulence) to a form that, by construction, accounts for the impact of turbulence on the mode in a consistent way. The effect of turbulence consists of a non-homogeneous forcing term, responsible for the stochastic driving of the mode, and a stochastic perturbation to the homogeneous part of the wave equation, responsible for both the damping of the mode and the modal surface effects. Conclusions. The stochastic wave equation obtained here represents our baseline framework to properly infer properties of turbulence-oscillation coupling, and can therefore be used to constrain the properties of the turbulence itself with the help of asteroseismic observations. This will be the subject of the rest of the papers in this series.

Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity.

The acoustic modes of a rotating fluid-filled cavity can be used to determine the effective rotation rate of a fluid (since the resonant frequencies are modified by the flows). To be accurate, this method requires a prior knowledge of the acoustic modes in rotating fluids. Contrary to the Coriolis force, centrifugal gravity has received much less attention in the experimental context. Motivated by on-going experiments in rotating ellipsoids, we study how global rotation and buoyancy modify the acoustic modes of fluid-filled ellipsoids in isothermal (or isentropic) hydrostatic equilibrium. We go beyond the standard acoustic equation, which neglects solid-body rotation and gravity, by deriving an exact wave equation for the acoustic velocity. We then solve the wave problem using a polynomial spectral method in ellipsoids, which is compared with finite-element solutions of the primitive fluid-dynamic equations. We show that the centrifugal acceleration has measurable effects on the acoustic frequencies when MΩ≳0.3, where MΩ is the rotational Mach number defined as the ratio of the sonic and rotational time scales. Such a regime can be reached with experiments rotating at a few tens of Hz by replacing air with a highly compressible gas (e.g., SF6 or C4F8).

Magnetosonic shocklets in electron-positron-ion plasmas

The formation and propagation characteristics of the magnetosonic shocks are investigated in a three-component magnetoplasma, consisting of warm dynamical ions with hot inertialess electrons and positrons. By solving the well-known nonlinear magnetohydrodynamic equations within the framework of diagonalization method, a set of two exact nonlinear wave equations is derived, which permits for both analytical as well as numerical solutions. It is shown that coherent solitary structures are formed at time τ = 0, which transform into the magnetosonic shocklets with the passage of time (τ > 0) because of the rapid steepening of ion-fluid velocity and magnetic field perturbations. The nonlinear ponderomotive force could be the major responsible force behind the wave steepening and wave breaking in such plasmas. Furthermore, the variation of plasma β, the positron-to-electron density ratio p and the ion-to-electron temperature ratio σ significantly modifies the profiles of the ion-fluid velocity and magnetic field. The present study might be useful for understanding the nonlinear propagation of magnetosonic shocklets in interstellar medium and in laboratory plasma experiments, where an additional positron-component exist.

Bayesian Time-lapse Difference Inversion Based on the exact Zoeppritz Equations with Blockiness Constraint

Accurately inverting changes in the reservoir elastic parameters that are caused by oil and gas exploitation is of great importance in accurately describing reservoir dynamics and enhancing recovery. Previously numerous time-lapse seismic inversion methods based on the approximate formulas of exact Zoeppritz equations or wave equations have been used to estimate these changes. However the low accuracy of calculations using approximate formulas and the significant calculation effort for the wave equations seriously limits the field application of these methods. However, these limitations can be overcome by using exact Zoeppritz equations. Therefore, we study the time-lapse seismic difference inversion method using the exact Zoeppritz equations. Firstly, the forward equation of time-lapse seismic difference data is derived based on the exact Zoeppritz equations. Secondly, the objective function based on Bayesian inversion theory is constructed using this equation, with the changes in elastic parameters assumed to obey a Gaussian distribution. In order to capture the sharp time-lapse changes of elastic parameters and further enhance the resolution of the inversion results, the blockiness constraint, which follows the differentiable Laplace distribution, is added to the prior Gaussian background model. All examples of its application show that the proposed method can obtain stable and reasonable P- and S-wave velocities and density changes from the difference data. The accuracy of estimation is higher than for existing methods, which verifies the effectiveness and feasibility of the new method. It can provide high-quality seismic inversion results for dynamic detailed reservoir description and well location during development.

Enabling numerically exact local solver for waveform inversion—a low-rank approach

As in many fields, in seismic imaging, the data in the field is collected over a relatively large medium even though only a part of that medium is truly of interest. This results in significant waste in computation as a typical inversion algorithm requires many solutions of the wave equation throughout the entire domain, even if only a small part of the domain is being updated. One way to mitigate this is to use a numerically exact local wave equation solver to perform waveform inversions in an area of interest, where the idea is to compute accurate solutions of the wave equation within a subdomain of interest. Although such solvers exist, many require the computation of Green’s function matrices in the background domain. For large-scale seismic data acquisition, the computation of the Green’s function matrices is prohibitively expensive since it involves solving thousands of partial differential equations in the background model. To mitigate this, in this work, we propose to exploit the low-rank structure of the full subsurface Green’s function. Using carefully selected 2D stylized models, we first show that the full subsurface Green’s function tensor organized as a matrix exhibits the low-rank structure in a transform domain. We then propose a randomized singular value decomposition–based framework to compute the low-rank approximation of the Green’s functions, where the cost of wave equation solves depends on the rank of the underlying Green’s function matrix instead of the number of grid points at the surface of the background model and on the boundary of the local domain. Next, we validate the proposed framework by performing time-lapse waveform inversion using the 2D Marmousi model. Finally, we demonstrate a rank-minimization–based framework to compute the low-rank factorized form of the Green’s function matrices in large-scale 3D seismic data acquisition.

A fast and accurate numerical implementation of the envelope model for laser–plasma dynamics

In laser-driven, plasma wakefield acceleration regimes (LWFA), when relevant scale lengths of the laser envelope and of the driven plasma waves are well separated from the wavelength and frequency of the laser fast oscillating component, a reduced physical model (usually referred to as the envelope model), has been introduced, allowing to formulate the laser–plasma equations in terms of laser cycle-averaged dynamical variables. As a main consequence, physical regimes where this reduced model applies, can be investigated with significant savings of computational resources still assuring comparable accuracy, with respect to standard Particle-In-Cell (PIC) models where all relevant space–time scales have to be resolved.Here we propose a computational framework characterized by two previously unexplored numerical implementations of the envelope model. The first one is based on explicit second order leapfrog integration of the exact wave equation for laser pulse propagation in a laboratory coordinate system in 3D cartesian geometry, replacing the usually quoted representation in an Eulerian frame moving at the speed of light. Since the laser and driven wakefield wave equations in a laboratory frame are advection dominated, we introduce a proper modification of finite differences approximating longitudinal space derivatives, to minimize dispersive numerical errors coming from the discretized advection operators. The proposed implementation, avoiding semi-implicit procedures otherwise required when dealing with a comoving frame, assures significant saving in computational time and ease of implementation for parallel platforms. The associated equation of motion for plasma particles has been integrated, as in standard PIC codes, using the Boris pusher, properly extended to take into account the specific form of the Lorentz force in the envelope model.As a second contribution, a novel numerical implementation of the plasma dynamics equations in the cold-fluid approximation, is presented. The scheme is based on the second-order one-step Adams–Bashforth time integrator coupled to upwind non-oscillatory WENO reconstruction for discretized space derivatives. The proposed integration scheme for the Eulerian fluid equations is equivalent to a leapfrog scheme with an added higher order dissipative truncation errors. It can be used either as a much faster, yet of comparable accuracy, alternative to the PIC representation of plasma particle motion, or even in a hybrid fluid–particle combination when kinetic effects and particle injection and acceleration in a wakefield have to be investigated.

Refraction traveltime tomography based on damped wave equation for irregular topographic model

Land seismic data generally have time-static issues due to irregular topography and weathered layers at shallow depths. Unless the time static is handled appropriately, interpretation of the subsurface structures can be easily distorted. Therefore, static corrections are commonly applied to land seismic data. The near-surface velocity, which is required for static corrections, can be inferred from first-arrival traveltime tomography, which must consider the irregular topography, as the land seismic data are generally obtained in irregular topography.This paper proposes a refraction traveltime tomography technique that is applicable to an irregular topographic model. This technique uses unstructured meshes to express an irregular topography, and traveltimes calculated from the frequency-domain damped wavefields using the finite element method. The diagonal elements of the approximate Hessian matrix were adopted for preconditioning, and the principle of reciprocity was introduced to efficiently calculate the Fréchet derivative. We also included regularization to resolve the ill-posed inverse problem, and used the nonlinear conjugate gradient method to solve the inverse problem.As the damped wavefields were used, there were no issues associated with artificial reflections caused by unstructured meshes. In addition, the shadow zone problem could be circumvented because this method is based on the exact wave equation, which does not require a high-frequency assumption. Furthermore, the proposed method was both robust to an initial velocity model and efficient compared to full wavefield inversions. Through synthetic and field data examples, our method was shown to successfully reconstruct shallow velocity structures. To verify our method, static corrections were roughly applied to the field data using the estimated near-surface velocity. By comparing common shot gathers and stack sections with and without static corrections, we confirmed that the proposed tomography algorithm can be used to correct the statics of land seismic data.

Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind

In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe-α|x-ct|. In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon).

On cusped solitary waves in finite water depth

It is well-known that the Camassa–Holm (CH) equation admits both of the peaked and cusped solitary waves in shallow water. However, it was an open question whether or not the exact wave equations can admit them in finite water depth. Besides, it was traditionally believed that cusped solitary waves, whose 1st-derivative tends to infinity at crest, are essentially different from peaked solitary ones with finite 1st-derivative. Currently, based on the symmetry and the exact water wave equations, Liao [1] proposed a unified wave model (UWM) for progressive gravity waves in finite water depth. The UWM admits not only all traditional smooth progressive waves but also the peaked solitary waves in finite water depth: in other words, the peaked solitary progressive waves are consistent with the traditional smooth ones. In this paper, in the frame of the linearized UWM, we give, for the first time, some explicit expressions of cusped solitary waves in finite water depth, and besides reveal a close relationship between the cusped and peaked solitary waves: a cusped solitary wave is consist of an infinite number of peaked solitary ones with the same phase speed, so that it can be regarded as a special peaked solitary wave. This also well explains why and how a cuspon has an infinite 1st-derivative at crest. Besides, it is found that, when wave height is small enough, the effect of nonlinearity is negligible for the interaction of peaked waves so that these explicit expressions are good enough approximations of peaked/cusped solitary waves in finite water depth. In addition, like peaked solitary waves, the vertical velocity of a cusped solitary wave in finite water depth is also discontinuous at crest (x=0), and especially its phase speed has nothing to do with wave height, too. All of these would deepen and enrich our understandings about the cusped solitary waves.

Steady-state resonance of multiple wave interactions in deep water

AbstractThe steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.

Pretend model of traveling wave solution of two-dimensional K-dV equation

Traveling wave resolution of Korteweg-de Vries (K-dV) solitary and numerical estimation of analytic solutions have been studied in this paper for imaginary concept. Pretend model of traveling wave deals with giant waves or series of waves created by an undersea earthquake, volcanic eruption or landslide. The concept of traveling wave is frequently used by mariners and in coastal, ocean and naval engineering. We have found some exact traveling wave solutions with relevant physical parameters using new auxiliary equation method introduced by Pang et al. (Appl. Math. Mech-Engl. Ed 31(7):929–936, 2010). We have solved the imaginary part of exact traveling wave equations analytically, and numerical results of time-dependent wave solutions have been presented graphically. This procedure has a potential to be used in more complex system for other types of K-dV equations.

A Method for Diagnosing the Sources of Infrasound in Convective Storm Simulations

AbstractThis paper presents a convenient method for diagnosing the sources of infrasound in a numerical simulation of a convective storm. The method is based on an exact acoustic wave equation for the perturbation Exner function Π′. One notable source term (Suu) in the Π′ equation is commonly associated with adiabatic vortex fluctuations, whereas another (Sm) is directly connected to the heat and mass generated or removed during phase transitions of moisture. Scale estimates suggest that other potential sources are usually unimportant. Simple numerical simulations of a disturbed vortex and evaporating cloud droplets are carried out to illustrate the infrasound of Suu and Sm. Moreover, the diagnostic method is applied to a towering cumulonimbus simulation that incorporates multiple categories of ice, liquid, and mixed-phase hydrometeors. The sensitivity of Sm to the modeling of the hail-to-rain category conversion is briefly addressed.

Acoustic shock wave propagation in a heterogeneous medium: A numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.

ELECTROMAGNETIC PULSE PROPAGATION OVER NONUNIFORM EARTH SURFACE: NUMERICAL SIMULATION

Parabolic equation method proposed by Leontowich and Fock [1,2] is an efficient simulation approach to VHF propagation over the earth surface. Deep physical analysis and advanced mathematical methods [3,4] turned Leontovich’s PE into a universal tool of diffraction theory. Its applications go far beyond the initial problem circle – e. g. [5-8]. The key role in this development played the decisive turn to straightforward numerical techniques pioneered by Malyuzhinets and Tappert [9,10]. In radio wave propagation, PE was used first to derive explicit analytical formulae for the EM field strength in model environments. A simplification has been reached by introducing the impedance boundary condition (BC) [11]. Taking into account tropospheric refraction ducts required the use of sophisticated asymptotic methods [12]. Further development (almost exclusively towards numerical implementation) was aimed at refined PE modifications [13-15], account for irregular terrain [16], introducing artificial transparent boundaries [17,18] and nonlocal BC to describe rough interfaces [19]. A non-stationary PE counterpart and a finite-difference (FD) scheme for its solution have been proposed by Claerbout and applied to seismic problems [13]. Afterwards, this “time-domain parabolic equation” (TDPE) was used to calculate acoustic propagation in ocean [20]. At the same time, little attempts of using TDPE to simulate EM pulse propagation in realistic environments are known. In this paper, we consider computational aspects of EM pulse propagation along the nonuniform earth surface. For ultrawide-band pulses without carrier, TDPE results directly from the exact wave equation written in a narrow vicinity of the wave front. To solve it by finite differences, we introduce a time-domain analog of the impedance BC and a nonlocal BC of transparency reducing the open computational domain to a strip of finite width. Numerical examples demonstrate the influence of soil conductivity on the received pulse waveform which can be used in remote sensing.

Plane wave shot PSDM and controlled illumination techniques

The Pre-Stack Depth Migration (PSDM) method based on wavefield continuation is the most reliable method for imaging complex structure in the subsurface, although there are large computational costs and poorly adaptive geometry. Plane wave shot migration is another method to perform exact wave equation prestack imaging with high computational efficiency and without the migration aperture problem. Moreover, wavefield energy can be compensated at the target zone by controlled illumination. In this paper, plane wave shot PSDM was implemented by the control of the plane down-going wavefield and selection of number and range of the raypaths in order to optimize the imaging effect. In addition, controlled illumination techniques are applied to enhance the imaging precision of interesting areas at different depths. Numerical calculation indicates that plane wave shot imaging is a rapid and efficient method with less computational cost and easy parallel computation compared to the single-square-root operator imaging for common shot gathers and doublesquare-root operator imaging for common midpoint gathers.

Electromagnetic fields in curved spacetimes

We consider the evolution of electromagnetic fields in curved spacetimes and calculate the exact wave equations for the associated electric and magnetic components. Our analysis is fully covariant, applies to a general spacetime and isolates all the sources that affect the propagation of these waves. Among others, we explicitly show how the different components of the gravitational field act as driving sources of electromagnetic disturbances. When applied to perturbed Friedmann–Robertson–Walker cosmologies, our results argue for a superadiabatic-type amplification of large-scale cosmological magnetic fields in Friedmann models with open spatial curvature.

Teukolsky Master Equation: de Rham Wave Equation for Gravitational and Electromagnetic Fields in Vacuum

A new version of the Teukolksy Master Equation, describing any massless field of different spin $s=1/2,1,3/2,2$ in the Kerr black hole, is presented here in the form of a wave equation containing additional curvature terms. These results suggest a relation between curvature perturbation theory in general relativity and the exact wave equations satisfied by the Weyl and the Maxwell tensors, known in the literature as the de Rham-Lichnerowicz Laplacian equations. We discuss these Laplacians both in the Newman-Penrose formalism and in the Geroch-Held-Penrose variant for an arbitrary vacuum spacetime. Perturbative expansion of these wave equations results in a recursive scheme valid for higher orders. This approach, apart from the obvious implications for the gravitational and electromagnetic wave propagation on a curved spacetime, explains and extends the results in the literature for perturbative analysis by clarifying their true origins in the exact theory.

Fast, nonperturbative, transmission inverse scattering, quantitative imaging with laboratory and simulated acoustic data

Inverse scattering provides an ultimate method for acoustic imaging for medical and other fields of application (because it is quantitative and is self-correcting for refraction, absorption, and diffraction artifacts). E. Wolf showed that, with plane wave incident excitation, spatial distributions of inhomogeneous acoustic speed of sound and absorption within bodies could be reconstructed under the Born approximation by use of a simple back propagation algorithm operating in spatial frequency space (k-space). This approach, as extended to include the Rytov approximation and/or to real-space, is known as ‘‘diffraction tomography’’ (DT). However, DT fails at frequencies above 1 MHz, for objects with the size and contrast of the human female breast. We have extended DT by replacing its approximate wave equations with an exact wave equation. We present for the first time images of laboratory objects (agar cylinders, excised tissue, etc.) made by inverting transmission data. Present algorithms are at the boundary of real time operations or for 3D imaging with PC level computer networks. We discuss problems characteristic of inverse scattering such as multiple minima solutions and the need for receiver and transmitter element beam patterns. Input of these images for computer aided diagnosis is discussed.