Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commerc...

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Inversion of seismic reflection data in the acoustic approximation

Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton--Jacobi equations Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms They are optimal in the sense that the computational complexity of the algorithms is O(N log N), where N is the total number of points in the domain The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings; provide details of the computational schemes, including higher order versions; and demonstrate the techniques in a collection of different areas

Fast Marching Methods

Multifold ground coverage by seismic techniques such as the common reflection point method provides a multiplicity of wave travel path information which allows direct determination of root‐mean‐square velocities associated with such paths. Hyperbolic searches for semblance among appropriately gathered arrays of traces form the basis upon which velocities are estimated. Measured semblances are presented as a velocity spectral display. Interpretation of this information can give velocities with meaningful accuracy for primary as well as multiple events. In addition, the velocity data can help correctly label events. This paper outlines the fundamental principles for calculating velocity spectra displays. Examples are included which demonstrate the depth and detail of geological information which may be obtained from the interpretation of such displays.

Velocity spectra-digital computer derivation and applications of velocity functions

The focusing of synthetic-aperture-radar (SAR) data using migration techniques quite similar to those used in geophysics is treated. The algorithm presented works in the omega -k/sub x/ domain. Because time delays can be easily accommodated with phase shifts that increase linearly with omega , range migration poses no problem. The algorithm is described in plane geometry first, where range migration and phase history can be exactly matched. The effects of the sphericity of the Earth, of the Earth's rotation, and of the satellite trajectory curvature are taken into account, showing that the theoretically achievable spatial resolution is well within the requirements of present day and near future SAR missions. Terrestrial swaths as wide as 100 km can be focused simultaneously with no serious degradation. The algorithm has been tested with synthetic data, with Seasat-A data, and with airplane data (NASA-AIR). The experimental results fully support the theoretical analysis. >

SAR data focusing using seismic migration techniques

Computer migration of seismic data emerged in the late 1960s as a natural outgrowth of manual migration techniques based on wavefront charts and diffraction curves. Summation (integration) along a diffraction hyperbola was recognized as a way to automate the familiar point‐to‐point coordinate transformation performed by interpreters in mapping reflections from the x, t (traveltime) domain into the x, z (depth domain). We will discuss the mathematical formulation of migration as a solution to the scalar wave equation in which surface seismic observations are the known boundary values. Solution of this boundary value problem follows standard techniques, and the migrated image is expressed as a surface integral over the known seismic observations when areal or 3-D overage exists. If only 2-D seismic coverage is available, wave equation migration is still possible by assuming the subsurface and hence surface recorded data do not vary perpendicular to the seismic profile. With this assumption, the surface inte...

/pdf/integral-formulation-for-migration-in-two-and-three-bjtokkohw7.pdf

Integral formulation for migration in two and three dimensions

Generalized linear inversion, sometimes known as model perturbation, nonlinear regression, or inverse modeling, is applied to synthetic and real seismic data sets with the objective of obtaining an impedance profile as a function of time. The impedances solved for are parameterized in a manner that describes the unknown earth using fewer variables than previous seismic generalized linear inversion techniques. In this application only single traces of common‐midpoint (CMP) processed data will be inverted. The method of generalized linear inversion (GLI) presented here is designed to improve on the shortcomings of recursive inversion with respect to relative and absolute scale of the impedance results, resolution of impedance boundaries, and distortion from residual wavelet effects. In obtaining these goals other advantageous aspects of GLI were discovered. For example, it is insensitive to noise in many cases, and it will allow an interpreter to fix the impedance of any number of known lithologies in an in...

Generalized linear inversion of reflection seismic data

The subject matter of this review paper pertains to developments during the past several years in the area of reflection seismic data processing and analysis. While this subject area is extensive in both its breadth and scope, one indisputable fact emerges: the computer is now more pervasive than ever. Processing areas which were computer intensive, such as signal enhancement, are now even more so; and those formerly exclusive domains of man, such as seismic interpretation, are beginning to feel the encroachment of the large number crunchers. What the future holds is anyone’s guess, but it is quite probable that man and computer will share the throne if the interactive seismic processing systems on the drawing boards come to pass. For the present and recent past, however, the most intensively developed areas of seismic data processing and analysis include 1) computer extraction of processing parameters such as stacking velocity and statics, 2) automated detection and tracking of reflections in multidimens...

Developments in seismic data processing and analysis (1968-1970)

As a part of the Vela Uniform program seismograms were recorded between December 19, 1961, and April 1, 1962, from instruments placed on the ocean bottom off the California coast in water depths up to 1.2 km. Three components of velocity and water pressure were recorded digitally. During the test period a total of five underground nuclear blasts originating from the Nevada test site and several small unidentified seismic events were recorded by the ocean-bottom seismometer. Recordings of pressure and vertical velocity in the frequency range of 0.5 to 5 cps for three nuclear blasts were of sufficient quality for detailed analysis. Spectral analyses were performed on several of these nuclear blast seismograms and the noise data immediately preceding them. These were compared with similar analyses of the same events obtained from a land station in the southern California area. The results indicated that signal-to-noise ratio is slightly poorer on the ocean bottom in 1.2 km of water than on a land station at the same range. The absolute noise level appears to be higher at the deep water sites relative to the land station; however, the ocean-bottom noise level is near the continental ‘average’ of Brune and Oliver for the samples investigated. Finally, the simultaneous recording of pressure and velocity appears to afford a distinct advantage in the identification and classification of arrivals recorded on the ocean bottom because of the P-V relationship that exists for the various propagation modes.

Ocean‐bottom seismic measurements off the California coast

A total of 500 hours of usable ocean‐bottom seismic data recorded on pressure and three components of velocity has been collected in three geographically separate areas of the Pacific Ocean at depths to 20,000 ft. These data are presently being analyzed to determine the extent to which monitoring seismic motion on the ocean floor can assist Project VELA UNIFORM goals of detection and identification of underground and underwater nuclear blasts. Analysis of three earthquakes and ambient noise recorded simultaneously on the ocean bottom and land reveals: 1. Ocean‐bottom signal‐to‐noise ratios are equal to or less than those seen at a comparative land station; 2. Ocean‐bottom signal and noise levels are higher than those obtained at the land station; and 3. Ocean‐bottom ambient noise power spectra increase in level towards the microseismic 6‐ to 8‐sec peak as do the land data. No strong directional ocean‐bottom noise components have been observed. Simultaneous recording of pressure and particle velocity affor...

William A. Schneider

Papers

Integral formulation for migration in two and three dimensions

Generalized linear inversion of reflection seismic data

Developments in seismic data processing and analysis (1968-1970)

Ocean‐bottom seismic measurements off the California coast

Collection and analysis of Pacific ocean-bottom seismic data