iterative time limited signal restoration the purpose of this paper is to show that time limited restoration of shift invariant blurred signals can be done by means of fixed point solutions of contraction mappings under rather general conditions for the distortion operator all our results are valid for multidimensional signals an application of these results to the iterative extrapolation of band limited discrete images is shown

second order differential operators with feller wentzell type boundary conditions in this paper we study a basic generation problem concerning the second order differential operator ad2dx2 bddx c in the space c 0 1 of complex continuous functions equipped with feller wentzell type boundary conditions which originates from the work of feller w feller the parabolic differential equations and the associated semi groups of transformations ann of math 2 55 1952 468 519 we prove successfully that the operator under suitable assumptions generates a strongly continuous cosine function on c 0 1 or on a subspace of c 0 1 by means of an operator matrix analysis combined with perturbation approximation and similarity techniques

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

direct inversion of time domain electromagnetic data this note describes a simple method for converting transient electromagnetic em sounding data into profiles of conductivity versus depth based on an approximate image representation for the decaying induced ground currents the method can provide one dimensional 1 d inversion for any kind of time domain em sounding data the discussion here is limited to the case of central loop sounding in particular i apply the inversion to some time derivative central loop sounding data and demonstrate that essentially the same interpretation of the geoelectric section is obtained with the simple image method as is obtained using traditional iterative least squares fitting to layered models this simple inverse can be computed much more quickly than an iterative least squares inverse making it possible to estimate the geoelectric section concomitant with data acquisition

a linear time varying system framework for noniterative discrete time band limited signal extrapolation a technique for the analysis and design of noniterative algorithms for discrete time band limited signal extrapolation is described the approach involves modeling the extrapolation process as a linear time varying ltv system or filter together with a previously developed fourier theory for ltv systems this model provides a frequency domain transfer function representation for the extrapolation system this representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well known least squares optimal algorithms for band limited extrapolation moreover the frequency domain setting provides a conceptually attractive means for understanding the process of extrapolation itself additionally a least squares approximation methodology for designing ltv filters for band limited extrapolation is developed the design technique is shown to unify a broad class of algorithms for extrapolating discrete time data and further to provide a means for designing new and improved extrapolation algorithms

continuation techniques for a certain class of analytic functions this paper deals with the numerical continuation problem of analytic functions g z given by g z int omega k z t f t dt where k defines a bounded operator on l 2 omega it is assumed that g is known over a finite segment a of the real line where g is to be sampled our continuation techniques emerged partly from a generalization of the landweber iteration we show that a certain discrete approximation of the proposed iterative technique yields two step noniterative algorithms for solving the continuation problem we also prove convergence of these approximations to the sought function g special emphasis on the continuation problem for the case k z t e 2 pi izt is given and some related numerical examples are presented the continuation problem when the known part of g z is contaminated with some noise is addressed and some techniques for solving this problem are also provided

approximation of local convoluted semigroups abstract we extend the trotter kato theorem on c 0 semigroups to local convoluted semigroups on dual spaces and apply these results to the general banach space setting compared to known results we obtain weaker convergence assumptions on the resolvent

3d resistivity mapping of airborne em data a 3d resistivity mapping technique has been developed to provide fast estimates of resistivity distributions in airborne electromagnetic surveys this proposed 3d mapping method consists of an approximate 3d linear inverse operator and a generalized subspace solver the 3d inverse operator can be generated using any forward approximation that is linear in resistivity the generalized subspace method is an alternative to the conjugate gradient method and it reduces the original large linear system of equations to a much smaller but nonlinear one that is solved iteratively the major benefit of using generalized subspace methods is that subspace vectors can be built based upon physical principles such as skin and investigation depths since the 3d mapping is a linear inverse problem no iteration and thus no forward modeling nor sensitivity updating is needed the 3d resistivity mapping technique can be used directly to estimate 3d resistivity distribution or to provide a model update during an intermediate iteration in a nonlinear 3d inversion synthetic and field data examples indicate that the 3d mapping can provide quantitative information about the resistivity and spatial distributions of the 3d targets

an automatic technique for presentation of coincident loop impulse response transient electromagnetic data coincident loop tem sounding data are often presented by plotting the half space apparent conductivity as a function of delay time a new algorithm generates an improved presentation that plots the apparent conductivity as a function of depth the resulting data may be further processed to sharpen or spike the smoothly varying apparent conductivity depth curves in an attempt to better represent the rapid changes in conductivity that often exist in the earth the algorithm described involves an approximation but is simple easy to use and computationally efficient a layered conductivity structure is assumed so the algorithm is best for areas where the geology is approximately horizontal however the algorithm can also be used to identify anomalous features that are not infinite horizontal layers the spiked conductivity models derived from synthetic data are consistent with the original layered earth models and show a greater resolution than the apparent conductivity depth curves and sometimes ampl

review on airborne electromagnetic inverse theory and applications airborne electromagnetic aem data processing and inversion has progressed from the early conductivity depth imaging to 1d inversion for a layered earth model and most recently to 2d 3d inversions the new processes have used methods for searching for solutions applying constraints and focusing images such as occam s laterally constrained and holistic inversions for airborne data imaging and 1d inversion some algorithms are fast whereas others have control over data misfit and the vertical or lateral smoothness beyond 1d algorithms 2d 3d inversions are based on currently used methods such as isolated conductor models which are good for highly conductive orebodies in resistive background and techniques that discretized regions that are used for more complex structures and backgrounds in the latter case the computations are slow so research is focusing on time efficient computer algorithms for solving the equations such as gauss newton quasi newton and nonlinear conjugate gradient algorithms for the electromagnetic em problem the solution can be obtained in a more practical time frame if material that has minimal impact is ignored two approaches can be used to speedup the em inversion process the moving footprint and direct solver methods we hope our work will to some extent help stimulate and focus the research in aem inversion

approximations of laplace transforms and integrated semigroups abstract let f m m n be a sequence of functions from 0 to a banach space e we give a new and essential condition on f m which is weaker than the usual local lipschitz continuity condition ensuring that the convergence of f m is equivalent to the convergence of their laplace transforms this enables us to establish new approximation theorems for r times integrated semigroups on e for all r 0 as a consequence an open problem for the convergence of integrated semigroups on the whole space e is solved in essence moreover we present an application to nonhomogeneous cauchy problems

on the convergence and approximation of integrated semigroups abstract we extend the well known theorem on convergence and approximation of c 0 semigroups due to h f trotter and t kato to the context of integrated semigroups the result is obtained as a consequence of an integrated version of the trotter kato theorem based on the laplace transform an application to the complete second order equation is given

conductivity depth imaging of airborne electromagnetic step response data an adaptation of the macnae lamontagne method allows transform of airborne step response electromagnetic em data to a conductivity depth image the algorithm is based on a nonlinear transformation of the amplitude of the measured response at each delay time to an apparent mirror image depth using matrix algebra the set of mirror image depth delay time data pairs can then be used to derive a conductivity section data can be efficiently processed on a personal computer at rates faster or comparable to the rate required for collection stable conductivity fitting as a function of depth is obtained by damping the matrix inversion by specification of the first and second derivative smoothness weights of the fitted conductivity depth sounding damping parameters may be either fixed or varied along the profile their choice can be constrained by geologic control stability of the process is enhanced by accounting for the transmitter and receiver tilts the mirror image depth delay time data can also be use

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

convoluted c cosine functions and semigroups relations with ultradistribution and hyperfunction sines abstract convoluted c cosine functions and semigroups in a banach space setting extending the classes of fractionally integrated c cosine functions and semigroups are systematically analyzed structural properties of such operator families are obtained relations between convoluted c cosine functions and analytic convoluted c semigroups introduced and investigated in this paper are given through the convoluted version of the abstract weierstrass formula which is also proved in the paper ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups several examples of operators generating convoluted cosine functions analytic convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors as an application it is proved that the polyharmonic operator 2 n n n acting on l 2 0 with appropriate boundary conditions generates an exponentially bounded k n convoluted cosine function and consequently an exponentially bounded analytic k n 1 convoluted semigroup of angle 2 for suitable exponentially bounded kernels k n and k n 1

optimal control of fractional diffusion equation in this paper we apply the classical control theory to a fractional diffusion equation in a bounded domain the fractional time derivative is considered in a riemann liouville sense we first study the existence and the uniqueness of the solution of the fractional diffusion equation in a hilbert space then we show that the considered optimal control problem has a unique solution interpreting the euler lagrange first order optimality condition with an adjoint problem defined by means of right fractional caputo derivative we obtain an optimality system for the optimal control

the usefulness of multicomponent time domain airborne electromagnetic measurements time domain airborne electromagnetic aem systems historically measure the inline horizontal x component new versions of the electromagnetic systems are designed to collect two additional components the vertical z component and the lateral horizontal y component to provide greater diagnostic information in areas where the geology is near horizontal the z component response provides greater signal to noise particularly at late delay times this allows the conductivity to be determined to greater depth in a layered environment the symmetry implies that the y component will be zero hence a nonzero y component will indicate a lateral inhomogeneity the three components can be combined to give the energy envelope of the response over a vertical plate the response profile of this envelope has a single positive peak and no side lobes the shape of the energy envelope is dependent on the flight direction but less so than the shape of the x component response profile in the interpretation of

hille yosida type theorems for o ω t n times integrated semigroups abstract for an arbitrary continuous increasing function t 0 c we introduce the concept and characterize the properties of n times integrated laplace transforms of o t vector valued functions based on these properties we have established hille yosida type theorems for o t n times integrated semigroups an explanation example is given

discrete and continuous band limited signal extrapolation this paper has two main purposes first some new algorithms for the extrapolation of multidimensional band limited sequences are presented these algorithms extend those given in 1 in two ways 1 we do not impose any restrictions on either the shape of the region containing the set of samples of the two dimensional signal to be extrapolated or on the shape of its passband zone 2 we obtain a class of sequences which are band limited extrapolations of the given data including the minimum norm solution of 1 as a special case the second objective of out paper is to relate the discrete extrapolation problem to the continuous signal extrapolation problem 2 specifically we prove that the solution obtained by using our approach for the discrete extrapolation problem tends to the solution of the continuous problem when the sampling rate used in the known part of the continuous signal to be extrapolated approaches infinity

the hille yosida and trotter kato theorems for integrated semigroups abstract we completely characterize the generators of non degenerate exponentially bounded n times integrated semigroups we deduce some perturbation results we also give a proof of the trotter kato theorem inspired from the classical one

imaging quasi layered conductive structures by simple processing of transient electromagnetic data an imaged conductivity section of a layered earth can be obtained by simple transformation of step response electromagnetic data measured in the quasi static zone this method of data transformation is presented as an alternative to conventional apparent conductivity transformations at each delay time the variation of the step response as a function of geometry transmitter and receiver location is transformed to an equivalent reference depth h which can be related to the depth of electromagnetic field diffusion the behavior of h as a function of delay time is nearly independent of the source receiver geometry the slowness dt dh divided by the magnetic permeability is almost exactly proportional to the cumulative conductance measured from the surface down to a depth h thus we can estimate an apparent conductivity which we call the imaged conductivity at depth to be d2t 0dh2 the cost of this transformation is a fraction of the cost of conventional data inversion and it does not require an a

convergence of iterative nonexpansive signal reconstruction algorithms iterative algorithms for signal reconstruction from partial time and frequency domain knowledge have proven useful in a number of application areas in this paper a general convergence proof applicable to a general class of such iterative reconstruction algorithms is presented the proof relies on the concept of a nonexpansive mapping in both the time and frequency domains two examples studied in detail are time limited extrapolation equivalently band limited extrapolation and phase only signal reconstruction the proof of convergence for the phase only iteration is a new result obtained by this method of proof the generality of the approach allows the incorporation of nonlinear constraints such as time or space domain positivity or minimum and maximum value constraints finally the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application