Introduction (ps.gz 23K) , pdf, , (src 23K)

Karrenbach SEP Introduction intro Over time, seismic land data acquisition evolved from the recording of only the vertical component of the seismic displacement field to multi-component recordings. An increasing number of surveys for detailed qualitative subsurface-parameter studies uses multi-component recordings. A three-component source, together with a three-component receiver, enables recording of the total elastic wave field. It is, theoretically, the most complete elastic measurement one can obtain from the subsurface for a given surface acquisition area. This chapter explains why a tensor wave field can be constructed from individual vector wave fields. The construction can be done successfully only when the vector source and receiver behavior does not vary throughout the data set. Using reciprocity principles for vector wave fields, I design an equalization scheme that removes non-reciprocal inconsistencies in the data. To test this method I use finite-difference modeling scheme that allows modeling of linear wave propagation as well as nonlinear wave propagation with ...

Spatial variations in field data
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var9c This chapter illustrates strong spatial variability in a multi-component surface seismic data set. One of the simplest methods for analyzing variability is looking at the total or partial energy in a seismic trace. This value is a good indicator of spatial variability throughout the entire data set. I show, using a nine-component land and a conventional marine data set, that the variability in the seismic land data is greater than for a marine dataset, because wave propagation on land is complicated by the presence of near surface heterogeneities, which are virtually absent in marine data. Such near-surface properties dominate the transfer of energy to and from the earth for multi-component sources and receivers. A nine-component land dataset This section analyses a nine-component data set that was recorded with equal and constant source-receiver spacing in a split spread geometry. I applied a despiking algorithm to remove unreasonable samples ...

Solving for the symmetry
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solving In this chapter I derive a source equalization criterion from a general wave equation and show its relation to other processing techniques. The seismic inverse problem can be cast in many different forms: some purely deterministic, some stochastic, still others in an optimization frame work. In the general elastic wave equation is the matrix of elastic parameters (stiffness), the density and is a vector wave field. The source is activated at location . and its transpose are spatial derivative matrices. I present a more detailed description and implementation of those operators in Chapter genmod . The seismic inverse problem in all generality would have to find ...

Reciprocity of tensor wavefields
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A general method of producing numerical code that preserves reciprocity would be very valuable. Do you see how to do this ? reciprocity This chapter consists of two parts: one shows the validity of the reciprocity principle for anisotropic heterogeneous media using a properly designed numerical modeling algorithm; the second examines a nine-component seismic land data set for reciprocity and finds only a few reciprocal events. Such an observation leads to the conclusion that even if the geometry of the experiment was quite reciprocal, the multi-component sources and receivers did not behave consistently reciprocal. Reciprocity principles in wave propagation problems are ...

Finite-difference modeling of tensor fields
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genmod In this chapter I describe numerical algorithms based primarily on finite-differences that model wave propagation in arbitrary complex media. Spatial complexity is not restricted, and for seismic problems arbitrary stress-strain relationship can be employed. The numerical implementation is modular and written in Ratfor90 (Ratfor + Fortran90 or High Performance Fortran) such that complex methods can be easily composed by using simple building blocks. The scheme used in model experiments described in this thesis allows us , in particular, to model nonlinear source effects and wave propagation in anisotropic media. I use this algorithm for modeling seismic wave propagation, although any problem that is representable as a second order tensorial partial differential equation can be directly modeled. In particular, it allows us to look at effects introduced by the medium without having to switch methods depending on the medium type. ...

Accuracy of reflection coefficients
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Accuracy of reflection coefficients The question if finite-difference modeling is adequate for amplitude versus offset analysis (AVO) AVO has long been a subject of discussion. Obviously, if temporal and spatial discretization is small, it can approximate continuous derivatives well, and finite-differences give the correct result. However, the question remains: does it give correct responses for discretizations that are desirable and commonly used? i.e., as large time steps that are desired for usual seismogram discretizations and as large spatial steps that can be tolerated by the model description. A definite answer cannot be given directly, because it depends on several factors: how a staggered grid is employed, how boundary conditions are treated, and what kind of time update is used. I chose to test whether FD modeling of reflection amplitudes is adequate ...

Splitting the wave operator
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Splitting the wave operator Karrenbach Splitting the wave operator When one applies the elastic wave equation operator for spatially varying media, several first order derivatives have to be calculated in the process. The wave equation can formally be split into two components so that derivatives are taken with respect to medium parameters and with respect to the propagating wave field. Splitting the wave equation operator allows one to adapt derivative operators to the physical quantities to be differentiated. In particular the adaption can be guided by special properties of the quantity. Practical problems can arise since the derivated quantities ...

Nonlinear tensor wave propagation modeling
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nonlinear The interaction of a seismic source device with the free surface of the earth makes the radiation characteristic different from that of the equivalent body point source . Usual seismic sources are built to produce enough energy to penetrate a few kilometers into the earth. There is no guarantee that when the source is activated, the surface material behavior will stay within a region that can be appropriately described using infinitesimal stress strain relationships. In this section I use finite-differences to model radiation characteristics and wave propagation when the equations of motion are no longer linear. I extend a nonlinear scalar equation that is normally used in one dimension into a three component, three-dimensional setting.

Appendix: Maximizing self-similarity of convolutional
operators central
(ps.gz 82K), pdf, (src 103K)

Many geophysical problems involve the repeated application of identical operators to some data. A few example can be found in recursive schemes, recursive wave extrapolations, or multi-grid methods. For numerical implementation an approximation to an operator is usually used, which deviates from the true desired behavior of the operator. What happens, then, if the approximate operator is repetitively applied multiple times ? The answer lies in the existence of a Central Limit Theorem for that particular operator. Muir and Dellinger Dellinger.sep.48.261 describe a way for recursive NMO application in the paper How to Beat the Central Limit Theorem . I illuminate this idea further and contrast it with other optimal design criteria for approximate operators. Repeated convolution with identical operators A smoothing filter, as shown in Figure triangle , for example, can be represented as a convolutional operation with a three-point filter of the form (0.5,1,0.5). ...

Multi-component Data Equalization
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source equalization Karrenbach SEP--73 srceq try P-P dataset Oseberg, compare to crosstalk In this chapter I attempt to correct the previously-discussed data set that showed extreme energy variations (Chapter var9c ) and largely non-reciprocal behavior (Chapter reciprocity ) so that the data become more reciprocal. In Chapter solving I defined the problem theoretically and gave a numerical analysis of the solution structure by treating it as a generalized inverse problem. In this chapter I apply it to a small synthetic transverse isotropic (TI) data set that is based on the Marmousi structural model and show that the equalization gives results as theory would predict. When I apply it to the nine-component data, the equalization improves data symmetry for strong coherent events. The optimization problem In chapter solving I described the solution structure to the problem of finding the symmetric data in an optimal way. In this chapter I set up a least-squares optimization problem in ...

Well logs as tensorial quantities
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tensorlog Do the Haskell comparison In this chapter I discuss my application of tensor fields to an area where generally only scalar fields have been used to improve the agreement between parameters measured in different types of experiments. The transformation of measured scalar quantities in well logs to tensors allows for an improved description of the medium. The primary example in the chapter demonstrates the prediction of surface seismic velocities from well logs. Observations based on measurements taken on different scales may not agree. However, an equivalent-medium approach can link measurements carried out at different scales. I have applied high-frequency (Dix) and low-frequency (Schoenberg ) averages to a well log. This study compares the results of a conventional velocity analysis of surface seismic data. In every physical experiment of finite size and duration carried out in nature, the measurement takes place over a certain scale. ...

Programming utilities
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programming RatFor == RATional FORtran Up to now Fortran is our most universal computer language for computational physics. For general programming, however, it has been surpassed by C . `` Ratfor '' is Fortran with C-like syntax and combines uncluttered, free-form programming with the power of clearly displaying mathematical formulas and algorithms. Ratfor accepts constructs that are possible in any of the Fortran dialects (Fortran IV, 66, 77, 90 and HPF). As illustrated in chapter genmod , I used Ratfor90 to implement an entire tool box of wave equation modules. My choice of Ratfor90 is dictated by the need for computational efficiency of parallel computers (thus Fortran90 or HPF) and the wish to actually include the code of algorithms in this thesis. The Ratfor preprocessor is now, usually included in the Fortran compiler environment, but a public domain version is also available. General ratfor language features The following paragraphs are reprinted from Jon Claerbout's ...

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