Wave processes in vertically-inhomogeneous media: a new strategy for a velocity inversion

Abstract

AbsIracL lhis paper describes a closed cycle of mathematical modelling of wave propagation processes. The half-space z > 0 is assumed to be filled with a verticallyinhomogeneous medium with the wave propagation velacity =(;). A source located on lhe free surface z = 0 causes the wave process U ( z , y , z , t), described by the initial boundary value problem for the wave equation.We consider two main problems: (1) Assuming c.(z) is lmown for all z, we wish to calculate the wave field U ( z , y , z , t ) ; (2) If ~( z )

Several remarks on numerical solution of the one-dimensional coefficient inverse problem

“…As can be noticed, three graphs in the figure coincide practically. [5,50], [5,100], [5,150], [5,200], [5,300], [5,500], [5,900], [5,1900] [5,10], [5,20], [5,30], [5,40], [5,50] H. b) Behaviour of the functional depending on changing the parameter α. The set of parameters: ν 1 = {0, 10 −2 }, 100 values of f from the interval [5,50] H with a uniform step; two graphs are presented for α = 1 and α = 3 (compare with graph in Figure 5 for α = 10 −1 ) Figure 6a just as Figure 5b demonstrates a behaviour of the functional depending on increasing of the interval of time frequencies.…”

“…One of the standard methods often used in practice for numerical solution of inverse problems is the Fourier transform with respect to time variable (see, for example, [9,10,11] and papers cited in them). In this investigation we will use the Laplace transform.…”

The inverse problem of acoustics: determination of source wavelet and velocity

“…Following [2], we rewrite the initial statement (2.1)-(2.4) in the following way: Then the image of the set M under the action of the operator A will be called the set of acceptable data D (D = Λ(Μ)).…”

“…This approach often called "inversion" was described in many works (see [2,6,13,14]). The main difficulties associated with its application are well-known.…”

“…The articles [2,13] are devoted to the investigation of the first problem, and therefore we shall not consider it in detail. The main attention will be given to the solution of the second problem -simultaneous determination of the form of the sounding signal and the velocity structure of the vertically-inhomogeneous medium in the case of the point source and the receivers located at the free surface.…”