Volume 22, Issue 5

This paper presents a novel covariation approach for seismic inversion under the assumption that the real seismic signals follow non-Gaussian α-stable distribution. To verify the correctness and effectiveness of the proposed method, two computer inversion simulation experiments on synthetic data inversion and real seismic acoustic impedance inversion are conducted using covariation, covariance and Bayesian methods. In the inversion experiment, covariation method performances better than the covariance and Bayesian, and the inverse results are very close to the true solutions. In the real seismic data inversion case, the sample quantile method was applied to estimate the characteristic exponent of a seismic trace from a nearby well before inversion. The estimated characteristic was applied in the 2D seismic impedance inversion where the moment p of the whole survey is set to be less than it. The results from the inversion of the 2D real data set are consistent with the well log interpretation.

We study boundary value problems on compact graphs with a cycle for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering coefficients of the differential equation from spectral data. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing its solution.

We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the far field operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non-negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the test function.

In this paper we study a linear inverse problem with a biological interpretation, which is modeled by a Fredholm integral equation of the first kind. When the kernel in the Fredholm equation is represented by step functions, we obtain identifiability, stability and reconstruction results. Furthermore, we provide a numerical reconstruction algorithm for the kernel, whose main feature is that a non-regular mesh has to be used to ensure the invertibility of the matrix representing the numerical discretization of the system. Finally, a second identifiability result for a polynomial approximation of degree less than nine of the kernel is also established.

So-called local a posteriori accuracy estimates for approximate solutions of ill-posed inverse problems are under investigation. We present an approach to local a posteriori estimation along with the numerical algorithm for calculating the estimators. We analyze the local optimality in order of obtained a posteriori accuracy estimates and introduce a new important notion of locally extra-optimal regularizing algorithm as a method for the solution of ill-posed problems having optimal in order local a posteriori accuracy estimates. Finally, we demonstrate the results of numerical experiments on applications of locally extra-optimal regularizing methods, constructed on the base of Tikhonov regularization, for calculating local a posteriori accuracy estimates.

In the present paper minimization problems of the information costs for numerical solving severely ill-posed problems are studied. A projection scheme of discretization which is economical in a sense of used harmonics is presented. By means of this scheme the order estimate of the minimal radius of Galerkin information is computed.