1 1 1 1 THREE-DIMENSIONAL FILTERING: Environmental soundings image enhancement Jon F. Cl rbout Cecil and Ida Green Professor of Geophysics Stanford University Book on Web at http://sepwww.stanford.edu/sep/prof email: jon@sep.stanford.edu FREEWARE, COPYRIGHT, AND PUBLIC LICENSE This disk contains freeware from many authors. Freeware is software you can copy and give away. But it is restricted in other ways. Please see author's copyrights and ``public licenses'' along with their programs. I do not certify that all of the software on this disk is freely copyable (although I believe it to be). ...
Basic operators and adjoints
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A great many of the calculations we do in science and engineering are really matrix multiplication in disguise. The first goal of this chapter is to unmask the disguise by showing many examples. Second, we see how the adjoint operator (matrix transpose) back projects information from data to the underlying model. Geophysical modeling calculations generally use linear operators that predict data from models. Our usual task is to find the inverse of these calculations, i.e., to find models (or make maps) from the data. Logically, the adjoint is the first step and a part of all subsequent steps in this inversion process. Surprisingly, in practice the adjoint sometimes does a better job than the inverse! This is because the adjoint operator tolerates imperfections in the data and does not demand that the data provide full information. Using the methods of this chapter, ...
Model fitting by least squares
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The first level of computer use in science and engineering is `` modeling ." Beginning from physical principles and design ideas, the computer mimics nature. After this, the worker looks at the result and thinks a while, then alters the modeling program and tries again. The next, deeper level of computer use is that the computer itself examines the results of modeling and reruns the modeling job. This deeper level is variously called `` fitting " or `` inversion ." We inspect the conjugate gradient (CG) method of fitting and write a subroutine for it that will be used in most of the examples in this monograph. HOW TO DIVIDE NOISY SIGNALS A single parameter inversion problem arises in Fourier analysis, where we seek a ``best answer'' at each frequency, then combine all the frequencies to get a best signal. Thus emerges a wide family of interesting and useful applications. ...
Inverse interpolation and nasty noise
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Here we look at many simple applications of the theory in chapters ajt and lsq . I wish I could postpone irritating practical details like irregularly located data and ill-mannered noises, but these difficulties arise so commonly that we must begin with them or find ourselves handicapped in most genuine applications until we do. As we dig into the topics of interpolation and noises, we soon find a wealth of important applications, fascinating theory, and some conceptual gaps I cannot yet cross. Here we first look at missing data, then irregularly spaced data, then nasty noises, and finally, anticipating higher-dimensional, computer-intensive problems, and we examine some preconditioning strategies. MISSING DATA IN ONE DIMENSION A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy. Specifying the filter chooses the interpolation philosophy. Generally the filter is a roughening filter. ...
Time-series analysis
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1 1 The many applications of least squares to the one-dimensional convolution operator constitute the subject known as `` time-series analysis .'' The autoregression filter, also known as the prediction-error filter ( PEF ), gathers statistics for us, not the autocorrelation or the spectrum directly but it gathers them indirectly as the inverse of the amplitude spectrum of its input. The PEF plays the role of the so-called ``inverse-covariance matrix'' in statistical estimation theory. Given the PEF, we use it to find missing portions of signals. Time domain versus frequency domain In the simplest applications, solutions can be most easily found in the frequency domain. When complications arise, it is better to use the time domain, directly applying the ...
Aliasing and spatial prediction
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In chapter tsa , using the principle of minimum energy we found filters given signals, we found signals given filters, and, given enough constraints, we found both at the same time. Mysteriously, one of the requirements is that the filter be a one-sided autoregression filter, i.e., the prediction-error filter (PEF). Since the output of this filter is white, the filter itself ``learns'' the spectrum of the data where the data is given, so taking this filter to areas where data is not given, we are able to estimate missing data of the appropriate spectrum which seamlessly melds the known region to the unknown. In this chapter, we extend these ideas to two dimensions. We first see that simple two-dimensional filters manipulate the spectrum of dips present in a data plane. A first goal is to uncover the form of the mysterious whitening filter in two dimensions. ...
Nonstationarity: patching
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There are many reasons for cutting data planes or image planes into pieces (patches), operating on the pieces, and then putting them back together again as depicted in Figure rayab2D . The earth's dip varies with lateral location and depth. The dip spectrum and spatial spectrum thus also varies. The dip itself is the essence of almost all earth mapping, and its spectrum plays an important role (inverse covariance matrix) in estimating any earth properties. In statistical estimation theory, the word to describe changing statistical properties is ``nonstationary''. rayab2D width=3.00in,height=1.5in ray Left is space of inputs and outputs. Right is during analysis. We begin this chapter with basic patching concepts along with supporting utility code. As in chapter md , I exhibit two-dimensional subroutines only. Three-dimensional code is an easy extension but to reduce clutter, I do not display it. It is in the CD-ROM library. The language of this chapter, ...
Plane waves in three dimensions
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In this chapter we seek a deeper understanding of plane wave s in three dimensions where the examples and theory typically refer to functions of time and two space coordinates , or to 3-D migration images where the coordinate is depth or traveltime depth. As in chapter pch we need to decompose data volumes into subcubes as in Figure rayab3D . rayab3D width=3.00in ray Left is space of inputs and outputs. Right is during analysis. In this chapter we will see that the wave model implies the 3-D whitener is not a cube filter but two planar filters. The wave model allows us to determine the scale factor of a signal, even where signals fluctuate in strength because of interference. Finally, we examine the local-monoplane concept that uses the superposition principle to distinguish a sedimentary model cube from a data cube . THE LEVELER: A VOLUME OR TWO PLANES? ...
RATional FORtran == Ratfor
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1 1 'Ratfor' Bare-bones 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. I believe Ratfor is the best available expository language for mathematical algorithms. Ratfor was invented by the people who invented C. Ratfor programs are converted to Fortran with the Ratfor preprocessor . Since the preprocessor is publicly available, Ratfor is practically as universal as Fortran. Kernighan, B.W. and Plauger, P.J., 1976, Software Tools: Addison-Wesley. Ratfor was invented at AT which makes it available directly or through many computer vendors. The original Ratfor transforms Ratfor code to Fortran 66. See http://sepwww.stanford.edu/sep/prof for a public-domain Ratfor translator to Fortran 77. You will not really need the Ratfor preprocessor or any precise definitions if you already know Fortran or almost any other computer language, because then the Ratfor language will be easy to understand. ...