Wave-Equation Reflection Tomography: Annihilators and Sensitivity Kernals

In seismic tomography, the finite frequency content of broad-band data leads to interference effects in the process of medium reconstruction, which are ignored in traditional ray theoretical implementations. Various ways of looking at these effects in the framework of transmission tomography can be found in the literature. Here,we consider inverse scattering of bodywaves to develop a method of wave-equation reflection tomography with broad-band waveform data— which in exploration seismics is identified as a method of wave-equation migration velocity analysis. In the transition from transmission to reflection tomography the usual cross correlation between modelled and observed waveforms of a particular phase arrival is replaced by the action of operators (annihilators) to the observed broad-bandwave fields. Using the generalized screen expansion for one-way wave propagation, we develop the Fréchet (or sensitivity) kernel, and show how it can be evaluated with an adjoint state method. We cast the reflection tomography into an optimization procedure; the kernel appears in the gradient of this procedure.We include a numerical example of evaluating the kernel in a modified Marmousi model, which illustrates the complex dependency of the kernel on frequency band and, hence, scale. In heterogeneous media the kernels reflect proper wave dynamics and do not reveal a self-similar dependence on frequency: low-frequency wave components sample preferentially the smoother parts of the model, whereas the high-frequency data are—as expected—more sensitive to the stronger heterogeneity.We develop the concept for acoustic waves but there are no inherent limitations for the extension to the fully elastic case.TOTAL (Firm)National Science Foundation (U.S.) (grant EAR-0409816

One-Way Wave Propagators For Velocity Analysis On Curvilinear Coordinates

Due to present computational limitations, migration by the one-way wave equation remains an integral tool in seismic exploration. For the realistic interpretation of common image point gathers, it is necessary that migration be free from artifacts from caustics and turning waves. In order to permit situations where turning waves occur, we perform our migration on specially chosen curvilinear coordinates where waves do not travel horizontally. We present an implementation of the curvilinear one-way wave equation using a rational approximation and discuss its application in migration velocity analysis, as well as transmission and reflection tomography

Source-Indexed Migration Velocity Analysis with Global Passive Data

The reverse-time migration of global seismic data generated by free-surface multiples is regularly used to constrain the crustal structure, but its accuracy is to a large extent determined by the accuracy of the 3-D background velocity model used for wave propagation. To this improve the velocity model and hence the accuracy of the migrated image, we wish to apply the technique of migration velocity analysis (MVA) to global passive data. Applications of MVA in the active setting typically focus on o ffset- or angle-gather annihilation, a process that takes advantage of data redundancy to form an extended image, and then applies an annihilation operator to determine the success of image formation. Due to the nature of regional-scale passive seismic arrays, it is unlikely that the data in most of these studies will be su cient to form an extended image volume for use in annihilation-based MVA. In order to make use of the sparse and irregular array design of these arrays, we turn towards a shot-pro le moveout scheme for migration velocity analysis introduced by Xie and Yang (2008). In the place of extended image annihilation, we determine the success of the migration velocity model by using a weighted image correlation power norm. We compare pairs of images formed by migrating each teleseismic source by image cross-correlation in the depth direction. We look for a suitable background model by penalizing the amount of correlation power away from zero depth shift. The total weighted correlation power between source-pro le images is then used as the error function and optimized via conjugate gradient. We present the method and a proof-of-concept with 2-D synthetic data

Reverse-time migration-based reflection tomography using teleseismic free surface multiples

Converted and multiply reflected phases from teleseismic events are routinely used to create structural images of the crust–mantle boundary (Moho) and the elasticity contrasts within the crust and upper mantle. The accuracy of these images is to a large extent determined by the background velocity model used to propagate these phases to depth. In order to improve estimates of 3-D velocity variations and, hence, improve imaging, we develop a method of reverse-time migration-based reflection tomography for use with wavefields from teleseismic earthquakes recorded at broad-band seismograph arrays. Reflection tomography makes use of data redundancy—that is, the ability to generate numerous structural images of the subsurface with different parts of the wavefield. In exploration seismology (where it is known as migration velocity analysis) reflection tomography typically involves the generation of an extended image (e.g. offset- or angle-gathers), and the fitness of the background model is evaluated through the application of image-domain annihilators. In regional-scale passive source seismology, however, annihilation-based methods are inadequate because the sparse and irregular distribution of teleseismic sources is not likely to produce illumination over a sufficient range of angles. To overcome this problem we turn towards a source-indexed moveout scheme. Instead of extended image annihilation, we determine the success of the tomographic velocity model by cross correlating images produced with multiply scattered waves from different teleseismic sources. The optimal velocity model is the one that minimizes correlation power between windowed images away from zero depth shift. We base our inversion scheme on the seismic adjoint method and a conjugate gradient solver. For each image pair, the update direction is determined by correlations between downgoing wavefields with upgoing adjoint wavefields for both images. The sensitivity kernels used in this method is similar to those found in other forms of adjoint tomography, but their shapes are controlled by the spatial distribution of the error function. We present the method and a proof-of-concept with 2-D synthetic data

Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of H\"{o}lder stability of the inverse maps, which ties naturally to the analysis of inverse problems

Transient Analysis of a Line-Focus Transducer Probing a Liquid/Solid Interface

The use of a line-focus ultrasonic transducer in a vertical scanning reflection acoustic microscope system is well known for quantitative materials characterization [1]. The technique relies on the measurement of the reflected radio frequency tone burst echo amplitude, V, as a fonction of amount of defocus, z, and analysis of the interference minima of the V(z) curve to obtain various interface wave speeds. The technique uses well developed theory [2,3,4] representing fixed frequency ultrasound generated and detected by a cylindrical lens in the frequency domain. We have developed a large aperture lensless line-focus transducer which is highly efficient and has a bandwidth wide enough to allow the generation and detection of narrow transient pulses [5]. From this transducer placed in water near a solid sample, the resulting echo waveforms have multiple features which can be interpreted as the arrival of a specularly reflected axial ray and leaky surface waves. Using this transducer, we have developed a time-resolved and polarization-sensitive testing technique for materials characterization [6]. The objective of this paper is to provide a theoretical basis for interpretation and analysis of these time domain waveforms

Characteristics of Injuries in the Logging Industry of Louisiana, USA: 1986 to 1998

Characterizing injuries and their trends will allow safety managers to concentrate their resources on the areas of safety that will be most effective in the workplace. Injuries reported to the Louisiana Office of Workers' Compensation Administration for 1986 to 1998 were characterized according to the part of the body affected, the nature of the injury, the source of the injury, and the type of accident for the timber harvesting industry. Many of the injuries in the logging sector were sprains / strains to the knees. Injuries resulting from falling onto structures and surfaces were common and rising. Although the number of accidents in each category is generally decreasing, some trends should be of concern. There was no significant linear trend in overall accident rates since 1991. While the proportion of cuts and lacerations declined, the proportion of fractures increased. This coincided with a time period when logging operations in Louisiana experienced rapid mechanization and insurance companies started enforcing the use of personal protective equipment. The proportion of transportation accidents rose more than any other category. Some suggestions on focusing and improving current safety programs are given. The need for continued and improved training of managers and employees seems to be most critical

Parameter identification in a semilinear hyperbolic system

We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigte the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings