Convolution Procedure Research Articles

Picosecond time-resolved cars in isotopically mixed crystals of benzene

Time-resolved coherent anti-Stokes Raman experiments are reported for the benzene crystal 991 cm −1 mode and in mixed C 6H 6/C 6D 6 crystals. Addition of C 6D 6 results in an increase in the rate at which the excitation is trapped. This is explained by the recent theory of Velsko and Hochstrasser which predicts that the trapping rate should undergo a maximum at some isotopic impurity concentration, consistent with observations from time and frequency domain studies. A proper convolution procedure for CARS experiments in the time domain is presented.

Optical absorption and magnetic circular dichroism of Pb2+centres in KBr: lineshape calculation

The lineshapes of the absorption and magnetic circular dichroism spectra of KBr:Pb2+ have been calculated at several temperatures in the range 20-250K. The Hamiltonian of the system, which includes the linear electron-phonon interaction was diagonalised directly. Integration over the coordinates which describe the interaction with modes of Eg and T2g symmetry was carried out by the gaussian quadrature method. The interaction with the A1g modes was accomplished by the usual convolution procedure. The conventional semiclassical approximation has been modified to take into account the difference in the effective frequencies of the cubic and non-cubic vibrational modes. Comparison of the calculated and experimental spectra show good agreement up to about 120K. The discrepancies at higher temperatures are considered to be due primarily to higher terms in the electron-phonon interaction; a conclusion that is supported by a moments analysis of the experimental spectra.

Optical absorption spectrum of KI:Sn2+. II. Theoretical calculation of the line shape

The line shape of the optical absorption spectrum of KI:Sn2+ has been calculated at different temperatures in the range 10–300 K. The eigenvalues and eigenfunctions corresponding to the electronic states of the (a1g) (t1u) configuration of the impurity ion have been obtained within that configuration by considering a Hamiltonian which includes exchange, spin-orbit and electron-lattice interactions. The 12×12 Hamiltonian matrix was diagonalized directly, thereby dispensing with the need for a perturbation approach. Interactions of the electrons of the impurity ion with lattice vibrations of A1g, Eg, and T2g symmetry were accounted for. Integration over the interaction mode coordinates Q2, Q3,...,Q6 was carried out by the Gaussian quadrupole method and the interaction with the A1g mode achieved by the usual convolution procedure. Within the framework of the conventional theory which assumes only cubic centers and takes into account only the linear terms in the electron-lattice interaction, reasonable agreement between theory and experiment could not be obtained. Therefore, centers with tetragonal symmetry were also considered by including a term in the Hamiltonian which represents interaction with the static crystal field of appropriate symmetry due to a nearby cation vacancy. Satisfactory results were obtained only when the coexistence of cubic and tetragonal centers in the approximate ratio 3:1 was assumed. Further improvement was achieved by taking into account the quadratic terms in the electron-lattice interaction with the A1g mode. The residual discrepancy is attributed, among other factors, to the neglect of quadratic terms in the interaction with the Eg and T2g modes.

Wave field extrapolation techniques in seismic migration, a tutorial

The objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration, i.e., the Kirchhoff‐summation approach, the plane‐wave method (k–f method), and the finite‐difference technique. Particular emphasis is given to the relationship between the different methods. By formulating the problem in the space‐frequency domain (x, y, ω‐domain), a systems approach can be adopted which results in simple and concise expressions. These expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure. In the situation of lateral velocity variations, the (de)convolution procedure becomes space‐variant. The space‐frequency domain is most suitable for recursive depth migration. In addition, frequency dependent properties such as absorption, dispersion, and spatial bandwidth can be handled easily. It is shown that all extrapolation methods are based on two equations: Taylor series and wave equation. In the Kirchhoff‐summation approach all terms of the Taylor series are summed to an exact analytical expression—the Kirchhoff‐integral for plane surfaces. It formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications. The Fourier‐transformed version of the Kirchhoff‐integral is used in the plane wave method (k–f method). This actually means that spatial (de)convolution in the x, y, ω‐domain is translated into multiplication in the [Formula: see text], [Formula: see text], ω‐domain. Of course, this is not allowed if the extrapolation operators are space‐variant. In explicit finite‐difference techniques a truncated version of the Taylor series is used with some optimum adjustments of the coefficients. For only one or two terms in the Taylor series, a spatial low‐pass filter must be applied to compensate for the amplitude errors at high tilt angles. Explicit methods are simple and most suitable for three‐dimensional (3-D) applications. In implicit finite‐difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator. Properly designed implicit schemes do not show amplitude errors and, therefore, amplitude correction filters need not be applied. In comparison with explicit schemes, implicit schemes are more sensitive to improper boundary conditions at both ends of the data file. It is shown that the forward seismic model can be elegantly described by a matrix equation, using separate operators for downward and upward traveling waves. Using this model, inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to conpensate for the upward propagation effects.

Experimental and theoretical investigations of parameters controlling line profiles in electron-beam lithography

An experimental and theoretical investigation of the more important parameters which affect the developed profile shape in electron-beam lithography is described. The theoretical approach is based on a Monte Carlo method of simulating electron scattering in the substrate to calculate the energy dissipation in the electron resist layer for a scanned electron beam. The current distribution in the beam is taken into account with a separate convolution procedure. The developed profile shape is obtained with a threshold solubility model which predicts a threshold energy density of 6.58 × 10 21 eV/cm 3 . A development simulation using the string method is used to predict the profile shape when the development process becomes a significant factor at a resist thickness of about 0.7μm. Finally the proximity effect is investigated by means of adjacent line experiments and compared with the predictions of the threshold solubility model.

Perturbational study of a model of chemisorption including dynamic image effects

Recent calculations of the effects of dynamic screening on the valence spectra of adsorbates, based on the rigid Fermi sea approximation, are criticized as inadequate. A new perturbational scheme which takes into account the interplay between one-electron scattering and many-body screening processes is presented and used for a model calculation of the spectra of occupied adsorbate levels. The results for the shape and the shift of the first plasmon satellite obtained in the adiabatic limit are shown to be different form those obtained by the convolution procedure of the rigid Fermi sea approximation.

Perturbational study of a model of chemisorption including dynamic image effects

Recent calculations of the effects of dynamic screening on the valence spectra of adsorbates, based on the rigid Fermi sea approximation, are criticized as inadequate. A new perturbational scheme which takes into account the interplay between one-electron scattering and many-body screening processes is presented and used for a model calculation of the spectra of occupied adsorbate levels. The results for the shape and the shift of the first plasmon satellite obtained in the adiabatic limit are shown to be different from those obtained by the convolution procedure of the rigid Fermi sea approximation.

Method of Gaussian quadrature in the calculation of optical absorption and magnetic circular dichroism spectra of s2 ions in alkali halide crystals: application to KBr:In+

The problem of calculating the lineshape function for optical absorption and magnetic circular dichroism due to ionic impurities with the ns2 outer electron configuration, incorporated substitutionally in alkali halide crystals, has been reformulated. The complete energy matrix has been diagonalized directly. Integration over the interaction mode coordinates of Eg and T2g symmetry has been carried out numerically using Gaussian quadrature formulae; the interaction with the A1g mode has been taken into account by the usual convolution procedure. The method has been applied to KBr:In+. The calculated lineshape functions for optical absorption at temperatures ranging from 4 to 300 K and, for MCD at 5 K, are in good agreement with the experimentally determined lineshapes. Moreover, the theory accounts very well for the observed variation of the effective g tensor for the A band with temperature. The calculated values for the moments of the absorption and MCD lineshape functions are in reasonably satisfactory agreement with those deduced from the observed spectra.

Efficient convolution kernels for computerized tomography.

Three concepts are presented: 1) Extended kernels: The Ramachandran-Lakshminarayanan convolution kernel has one zero between each non-zero value in the spatial domain. By extending the kernel in Fourier space, it is shown that each extension leads to two additional zeros in the spatial domain of the convolution kernel, thus decreasing the required number of multiplications necessary for convolution. Any known kernel can be extended and in the limit of extension a simple backprojection reconstruction is obtained. 2) Binary kernels: A technique for generating binary approximations to any convolution kernel is described. Excluding the central element, all other elements of the kernel are approximated by an even power of two thus, multiplications are replaced by shift operations in the convolution procedure. 3) Recursive convolution: It is shown how additions can be saved by using a recursive formulation which generates new elements in the convolution procedure utilizing only a few summation steps. Results from both extended kernels and their binary approximations are described for simulated phantoms and ultrasound data obtained from breast scans of patients.

Efficient convolution kernels for computerized tomography

Three concepts are presented: 1) Extended kernels: The Ramachandran-Lakshminarayanan convolution kernel has one zero between each non-zero value in the spatial domain. By extending the kernel in Fourier space, it is shown that each extension leads to two additional zeros in the spatial domain of the convolution kernel, thus decreasing the required number of multiplications necessary for convolution. Any known kernel can be extended and in the limit of extension a simple backprojection reconstruction is obtained. 2) Binary kernels: A technique for generating binary approximations to any convolution kernel is described. Excluding the central element, all other elements of the kernel are approximated by an even power of two; thus, multiplications are replaced by shift operations in the convolution procedure. 3) Recursive convolution: It is shown how additions can be saved by using a recursive formulation which generates new elements in the convolution procedure utilizing only a few summation steps. Results from both extended kernels and their binary approximations are described for simulated phantoms and ultrasound data obtained from breast scans of patients.

Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering

A linear digital filtering algorithm is presented for rapid and accurate numerical evaluation of Hankel transform integrals of orders 0 and 1 containing related complex kernel functions. The kernel for Hankel transforms is defined as the non‐Bessel function factor of the integrand. Related transforms are defined as transforms, of either order 0 or 1, whose kernel functions are related to one another by simple algebraic relationships. Previously saved kernel evaluations are used in the algorithm to obtain rapidly either order transform following an initial convolution operation. Each order filter is designed with identical abscissas over a large range so that an adaptive convolution procedure can be applied to a large class of kernels. Different order Hankel transforms with related kernels are often found in electromagnetic (EM) applications. Because of the general nature of this algorithm, the need to design new filters should not be necessary for most applications. Accuracy of the filters is comparable to that of single‐precision numerical quadrature methods, provided well‐behaved kernels and moderate values of the transform argument are used. Filtering errors of less than 0.005 percent are demonstrated numerically using known analytical Hankel transform pairs. The digital filter accuracy is also illustrated by comparison with other published filters for computing the apparent resistivity for a Schlumberger array over a horizontally layered earth model. The algorithm is written in Fortran IV and is listed in the Appendix along with a test driver program. Detailed comments are included to define sufficiently all calling parameter requirements.

A convolution approximation for Van Hove correlation function in liquids

The relative motions of particles in a simple liquid may be described as one-dimensional radial diffusion in an effective potential of the mean force. This effective potential may be obtained by an iteration procedure with the Vineyard approximation as the initial step. Using a harmonic approximation for the potential, the “distinct” part of the Van Hove time-dependent correlation function is then the convolution of a modified radial distribution function with a modified self-correlation function. The former describes the average positions of particles whereas the latter describes the density profile around the average position. For liquid argon, this modified convolution procedure appears to give results in satisfactory agreement with molecular dynamics.

MIGRATION IN TERMS OF SPATIAL DECONVOLUTION*

AbstractThe relationship between two finite‐difference schemes (15° and 40°) and the Kirchhoff summation approach is discussed by using closed form solutions of Claerbout's approximate versions of the wave equation. Forward extrapolation is presented as a spatial convolution procedure for each frequency component. It is shown that downward extrapolation can be considered as a wavelet deconvolution procedure, the spatial wavelet being given by the wave theory. Using this concept, a three‐dimensional model for seismic data is proposed.The advantages of downward extrapolation in the space‐frequency domain are discussed.Finally, it is derived that spatial sampling imposes an upper limit on the aperture and a lower limit on the extrapolation step.

STEEP DIP FINITE‐DIFFERENCE MIGRATION*

AbstractA method is presented to derive approximate versions of the wave equation which allow finite‐difference migration for very steep dips (> 50°). It is shown that for conventional finite‐difference schemes, in addition to the dip limitation, the maximum acceptable frequency should be specified. A finite‐difference migration technique is proposed in the frequency domain. It is derived that finite‐difference wave field extrapolation in the frequency domain consists of a space‐variant convolution procedure for each frequency component, the space‐variance being defined by the lateral variation in the velocity.Finally it is shown that with finite‐difference migration, particle velocity data can be easily obtained from pressure data.

Chronopotentiometry of systems involving follow-up chemical reactions Application of convolution and finite difference procedures

Distortion of polarization curves by double layer charging severely limits the standard application of chronopotentiometry to mechanism and rate constant analysis of systems involving fast follow-up chemical reactions. A procedure is proposed to overcome these difficulties in the case of irreversible electrochemical-chemical processes (pure kinetic conditions). It starts with the extraction of the faradaic component, if, of the imposed current and the computation of its convolution integral, I, with the function (πt)−1/2. The if-E and I-E are then simultaneously treated by a logarithmic analysis the form of which depends upon the expected mechanism. The efficiency of the method in mechanism and rate determination is illustrated experimentally on the example of the electrohydrodimerization of p-methylbenzylidene malononitrile for which discrimination between 32 possible limiting mechanisms has to be carried out. The same reaction is also used to test another type of analysis to be applied to partially reversible systems. Data processing then combines finite difference approach and convolution. It amounts to fitting the formal kinetics with the experimental data automatically in a small size computer avoiding tedious manual fitting with a priori calculated working curves. It is shown that the rate constants can be thus determined with a good accuracy even from single-step experiments.

On the determination of fluorine concentration profiles in magnesium alloy scales using the 19F(p, αγ) 16O reaction

Application of the 19F(p, αγ) 16O reaction by gamma ray counting was selected for the determination of in depth fluorine concentration profiles in the scales of Magnox reactor canning material. Because of lack of required accuracy in reaction cross-section data, the excitation function was measured in the 0.8–2.0 MeV proton energy region by the detection of 6–7 MeV gamma rays in the forward direction. It is reported that for the present investigation, a combined use of the 1.283, 1.348 and 1.375 MeV cross-section resonances is ideal for depths of up to 5 μm. A convolution procedure was used to extract fluorine concentration profiles from gamma yield curves by taking into account the reaction excitation function and the energy loss and straggling of the incident proton beam.

Chronopotentiometry of systems involving follow-up chemical reactions: Application of convolution and finite difference procedures

Distortion of polarization curves by double layer charging severely limits the standard application of chronopotentiometry to mechanism and rate constant analysis of systems involving fast follow-up chemical reactions. A procedure is proposed to overcome these difficulties in the case of irreversible electrochemical-chemical processes (pure kinetic conditions). It starts with the extraction of the faradaic component, i f, of the imposed current and the computation of its convolution integral, I, with the function ( πt) −1/2. The i f- E and I-E are then simultaneously treated by a logarithmic analysis the form of which depends upon the expected mechanism. The efficiency of the method in mechanism and rate determination is illustrated experimentally on the example of the electrohydrodimerization of p-methylbenzylidene malononitrile for which discrimination between 32 possible limiting mechanisms has to be carried out. The same reaction is also used to test another type of analysis to be applied to partially reversible systems. Data processing then combines finite difference approach and convolution. It amounts to fitting the formal kinetics with the experimental data automatically in a small size computer avoiding tedious manual fitting with a priori calculated working curves. It is shown that the rate constants can be thus determined with a good accuracy even from single-step experiments.

Convolution and deconvolution in the synthesis and analysis of staircase voltammograms

The comparison of staircase voltammograms with linear scan voltammograms is complicated by the former's response being dependent on the measurement time. A method for the prediction and removal of that dependence is given, presenting the data in a form similar to that of semi-differential analysis. An efficient algorithm for simulating the response of a reversible system to repetitive stepwise excitation is developed utilizing the fast Fourier transform convolution procedure. A brief comparison of several convolution procedures is presented.

Application of convolution procedures to chronopotentiometry

Application of convolution procedures to chronopotentiometry

Application of convolution procedures to chronopotentiometry

Convolution procedures are used to extract the faradaic information from chronopotentiometric data, in conditions where significant distortion by double layer charging occurs. The faradaic component of the imposed current is obtained, after measurement of the double layer capacitance, by differentiation of the initial chronopotentiogram. Convolution of this current with the function (π t) −1/2 leads to a potential-convoluted current relationship freed from the effect of double layer charging. The kinetic characterization of the system using a combined analysis of this relationship and that relating the faradaic current to the electrode potential is discussed for the various types of reaction mechanism. The efficiency of the proposed procedure is tested on the galvanostatic reduction of fluorenone in DMF.