Jon F. Claerbout

Toward a unified theory of reflector mapping

Schemes for seismic mapping of reflectors in the presence of an arbitrary velocity model, dipping and curved reflectors, diffractions, ghosts, surface elevation variations, and multiple reflections are reviewed and reduced to a single formula involving up and downgoing waves. The mapping formula may be implemented without undue complexity by means of difference approximations to the relativistic Schroedinger equation.

ROBUST MODELING WITH ERRATIC DATA

An attractive alternative to least‐squares data modeling techniques is the use of absolute value error criteria. Unlike the least‐squares techniques the inclusion of some infinite blunders along with the data will hardly affect the solution to an otherwise well‐posed problem. An example of this great stability is seen when an average is, determined by using the median rather than the arithmetic mean. Algorithms for absolute error minimization are often approximately as costly as least‐squares algorithms; however, unlike least‐squares, they naturally lend themselves to inequality or bounding constraints on models.

SYNTHESIS OF A LAYERED MEDIUM FROM ITS ACOUSTIC TRANSMISSION RESPONSE

A direct (noniterative) method is presented to determine an acoustic layered medium from the seismogram due to a time‐limited plane wave incident from the lower halfspace. It is shown that one side of the autocorrelation of the seismogram due to an impulsive source at depth is the seismogram due to an impulsive source on the surface. This transforms the problem to the acoustic reflections problem as solved by Kunetz. Both the deep source time function and the layering can be determined from a surface seismogram.

Velocity‐stack and slant‐stack stochastic inversion

Normal moveout (NMO) and stacking, an important step in analysis of reflection seismic data, involves summation of seismic data over paths represented by a family of hyperbolic curves. This summation process is a linear transformation and maps the data into what might be called a velocity space: a two‐dimensional set of points indexed by time and velocity. Examination of data in velocity space is used for analysis of subsurface velocities and filtering of undesired coherent events (e.g., multiples), but the filtering step is useful only if an approximate inverse to the NMO and stack operation is available. One way to effect velocity filtering is to use the operator LT (defined as NMO and stacking) and its adjoint L as a transform pair, but this leads to unacceptable filtered output. Designing a better estimated inverse to L than LT is a generalization of the inversion problem of computerized tomography: deconvolving out the point‐spread function after back projection. The inversion process is complicated ...

Acoustic daylight imaging via spectral factorization; helioseismology and reservoir monitoring

The acoustic time history of the sun’ s surface is a stochastic t x y -cube of information. Helioseismologists cross-correlate these noise traces to produce impulse response seismograms, providing the proof of concept for a long-standing geophysical conjecture. We pack the x y -mesh of time series into a single super-long one-dimensional time series. We apply Kolmogoroff spectral factorization to the super-trace, unpack, and findthe multidimensional acoustic impulse response of the sun. State-of-the-art seismic exploration recording equipment offers tens of thousands of channels, and permanent recording installations are becoming economically realistic. Helioseismology, therefore, provides a conceptual prototype for using natural noises for continuous reservoir monitoring.

Coarse grid calculations of waves in inhomogeneous media with application to delineation of complicated seismic structure

The multidimensional scalar wave equation at a single frequency is split into two equations. One controls the downgoing transmitted wave; the other controls the upcoming reflected wave. The equations are coupled, but in many reflection seismology situations the transmitted wave may be calculated without consideration of the reflected wave. The reflected wave is then calculated from the transmitted wave and the assumed velocity field. The waves are described by a modulation on up‐ or downgoing plane waves. This modulation function is calculated by difference equations on a grid. Despite complicated velocity models (steep faults, buried focus, etc.), the grid may be quite coarse if waves of interest do not propagate at large angles from the vertical. A one‐dimensional grid may be used for a two‐dimensional velocity model. With approximations, a point source emitting waves spreading in three dimensions may be included on the one‐dimensional grid. Calculation time for representative models is a few seconds. P...

Making scientific computations reproducible

To verify a research papers computational results, readers typically have to recreate them from scratch. ReDoc is a simple software filing system for authors that lets readers easily reproduce computational results using standardized rules and commands.

Velocity estimation and downward continuation by wavefront synthesis

A “wave stack” is any stack over a common shot or geophone gather in which the moveout is independent of time. It synthesizes a particular wavefront by superposition of the many spherical wavefronts of raw data. Unlike the common midpoint stack, wave stacks retain the important property of being the sampling of a wave field and, as such, permit wave‐equation treatment of formerly difficult or impossible problems. Seismic sections of field data generated by wave stacks that synthesized slanted downgoing plane waves showed a similarity in appearance to the common midpoint stacks. In signal‐to‐noise ratio they lay between the single offset section and the midpoint stack. The angle selectivity of the slanted plane‐wave stacks permitted detection of a reflector that was not visible on either the midpoint stack or the raw gathers. Simple velocity estimation in slant frame coordinates differs only in detail from standard frame coordinates. Because of the wave field character of data that have been slant plane‐wa...

Downward continuation of moveout-corrected seismograms

Earlier work developed a method of migration of seismic data based on numerical solutions of partial differential equations. The method was designed for the geometry of a single source with a line of surface receivers. Here the method is extended to the geometry of stacked sections, or what is nearly the same thing, to the geometry where a source and receiver move together along the surface as in marine profiling. The basic idea simply stated is that the best receiver line for any reflector is just at (or above) the reflector. Data received at a surface line of receivers may be extrapolated by computer to data at a hypothetical receiver line at any depth. By considering migration before stacking over offset, it is found that certain ambiguities in velocity analysis may be avoided.

Surface‐consistent residual statics estimation by stack‐power maximization

Application of incorrect static shifts will decrease the power of the common‐depth‐point stack. Conversely, static shifts can be estimated by maximizing the stack power. We tried it and it worked well enough that we recommend it in routine practice for data with low signal‐to‐noise ratio. Likewise, stack power was maximized by adjustment of surface‐consistent phase correction.