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Seismic Data Processing and Interpretation 1. Introduction The purpose of seismic processing is to manipulate the acquired data into an image that can be used to infer the sub-surface structure. Only minimal processing would be required if we had a perfect acquisition system. Processing consists of the application of a series of computer routines to the acquired data guided by the hand of the processing geophysicist. The interpreter should be involved at all stages to check that processing decisions do not radically alter the interpretability of the results in a detrimental manner. Processing routines generally fall into one of the following categories: * enhancing signal at the expense of noise * providing velocity information * collapsing diffractions and placing dipping events in their true subsurface locations (migration) * increasing resolution. 2. Processing Steps There are number of steps involved from seismic data acquisition to interpretation of subsurface structure. Some of the common steps are summarized below: Acquisition Static Correction Processing Velocity Analysis NMO/DMO Stacking Migration (Time/Depth, Kirchhof’s, f-k domain ) Interpretation Seismic data to subsurface geology In order to work with above steps (or to work more with seismic data), a number of signal processing operations are needed to accomplish the job. Some of them are: i) Sampling data, ii) Mute, iii) Amplitude recovery/ corrections, iv) Filtering, v) Deconvolution, v) f-k analysis etc. Some signal processing tools are explained in section three. 1 2.1 Data acquisition Shot gather: 2 Multiple shotpoints: If more than one shot location is used, reflections arising from the same point on the interface will be detected at different geophones. The common point of reflection is known as the common midpoint (CMP). CMP: 3 CMP gather: The CMP gather lies in the heart of seismic processing for two main reasons: i) the variation of travel time with offset, the moveout will depend only on the velocity of the subsurface layers (horizontal uniform layers) and the subsurface velocity can be derived. ii) The reflected seismic energy is usually very weak. It is imperative to increase the signal to noise ratio of most data. 2.1.1 Static Correction Static corrections are applied to seismic data to compensate for the effects of variations in elevation, weathering thickness, weathering velocity, or reference to a datum. The objective is tio determine the reflection arrival times which would have been observed in all measurements had been made on a flat plane with no weathering or low velocity material present. These corrections are base don uplole data, refraction first break or event smoothing. 4 2.2.1 Stacking (Velocity Analysis, NMO/DMO) This section is an approach of iterative process of applying NMO, DMO and standard velocity analysis. DMO improves the quality of the stack and the usefulness of the stacking velocity field. A variety of methods are available (constant velocity stacks, constant velocity gathers, semblance) which work to different extents with different data types. NMO and DMO are used the final velocity field after convergence. 2.2.2 Migration Migration will lead to the final product, either as depth or time. We can apply migration using velocities based on our velocity analysis if they are good enough, by testing a range of different velocities to determine which collapse diffractions correctly, or by using other information. Care is required to produce a generally smooth velocity field. A seismic section before and after migration is shown below for example. 5 2.2.3 Interpretation This is the final section, one can say finished product of the seismic processing steps. Subsurface geologies are generally derived from this unit. 6 3. Signal Processing Tools 3.1 Basic Fourier Theory The Fourier series expansion of f is: where, for any non-negative integer n: is the nth harmonic (in radians) of the function f, are the even Fourier coefficients of f, and are the odd Fourier coefficients of f. Equivalently, in exponential form, where: i is the imaginary unit, and in accordance with Euler's formula. Example 1: Simple Fourier series Let f be periodic of period 2π, with f(x) = x for x from −π to π. Note that this function is a periodic version of the identity function. Plot of a periodic identity function - a sawtooth wave. Animated plot of the first five successive partial Fourier series We will compute the Fourier coefficients for this function. 7 Notice that an are 0 because the are odd functions. Hence the Fourier series for this function is: 3.2 Fouriere Transform In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. A useful analogy is the relationship between a series of pure notes (the frequency components) and a musical chord (the function itself). In mathematical physics, the Fourier transform of a signal can be thought of as that signal in the "frequency domain." This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function. (See also fractional Fourier transform and linear canonical transform for generalizations.) 8 Suppose is a complex-valued Lebesgue integrable function. The Fourier transform to the frequency domain, , is given by the function: , for every real number When the independent variable t represents time (with SI unit of seconds), the transform variable ω represents angular frequency (in radians per second). Other notations for this same function are: and . The function is complex-valued in general. ( represents the imaginary unit.) If is defined as above, and is sufficiently smooth, then it can be reconstructed by the inverse transform: , for every real number The interpretation of is aided by expressing it in polar coordinate form, , where: the amplitude the phase Then the inverse transform can be written: which is a recombination of all the frequency components of . Each component is a complex sinusoid of the form whose amplitude is proportional to and whose initial phase angle (at t = 0) is . 3.2.1 Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, to solve partial differential equations, and to perform other operations such as convolutions. The DFT can be computed efficiently in practice using a fast Fourier transform (FFT) algorithm. The sequence of N complex numbers x0, ..., xN−1 is transformed into the sequence of N complex numbers X0, ..., XN−1 by the DFT according to the formula: where e is the base of the natural logarithm, is the imaginary unit (i2 = − 1), and π is Pi. The transform is sometimes denoted by the symbol , as in or . The inverse discrete Fourier transform (IDFT) is given by 9 Note that the normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1/N. A normalization of for both the DFT and IDFT makes the transforms unitary, which has some theoretical advantages, but it is often more practical in numerical computation to perform the scaling all at once as above (and a unit scaling can be convenient in other ways). (The convention of a negative sign in the exponent is often convenient because it means that Xk is the amplitude of a "positive frequency" 2πk / N. Equivalently, the DFT is often thought of as a matched filter: when looking for a frequency of +1, one correlates the incoming signal with a frequency of −1.) In the following discussion the terms "sequence" and "vector" will be considered interchangeable. 3.2.2 Z-transform In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. The Z-transform and advanced Z-transform were introduced (under the Z-transform name) by E. I. Jury in 1958 in Sampled-Data Control Systems (John Wiley & Sons). The idea contained within the Z-transform was previously known as the "generating function method". The (unilateral) Z-transform is to discrete time domain signals what the one-sided Laplace transform is to continuous time domain signals. Definition The Z-transform, like many other integral transforms, can be defined as either a one- sided or two-sided transform. Bilateral Z-Transform The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as where n is an integer and z is, in general, a complex number: z = Aejφ where A is the magnitude of z, and φ is the angular frequency (in radians per sample). 3.3 Convolution In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval. 10 Definition The convolution of and is written . It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform: By change of variables, replacing by , it is sometimes written as: The integration range depends on the domain on which the functions are defined. While the symbol is used above, it need not represent the time domain. In the case of a finite integration range, and are often considered to extend periodically in both directions, so that the term does not imply a range violation. This use of periodic domains is sometimes called a cyclic, circular or periodic convolution. Of course, extension with zeros is also possible. Using zero-extended or infinite domains is sometimes called a linear convolution, especially in the discrete case below. If and are two independent random variables with probability distributions and , respectively, then the probability distribution of the sum is given by the convolution . For discrete functions, one can use a discrete version of the convolution. It is given by When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, in this sense (using extension with zeros as mentioned above). Generalizing the above cases, the convolution can be defined for any two integrable functions defined on a locally compact topological group (see convolutions on groups below). A different generalization is the convolution of distributions. Commutativity Associativity Distributivity Associativity with scalar multiplication for any real (or complex) number . Differentiation rule where denotes the derivative of f or, in the discrete case, the difference operator . 11 Convolution theorem The convolution theorem states that where denotes the Fourier transform of . Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform and Mellin transform. See also less trivial Titchmarsh convolution theorem. Applications Convolution and related operations are found in many applications of engineering and mathematics. 1. In statistics, as noted above, a weighted moving average is a convolution. also the probability distribution of the sum of two independent random variables is the convolution of each of their distributions. 2. In optics, many kinds of "blur" are described by convolutions. A shadow (e.g. the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the shape of the iris diaphragm. The photographic term for this is bokeh. 3. Similarly, in digital image processing, convolutional filtering plays an important role in many important algorithms in edge detection and related processes. 4. In linear acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it. 4. In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse response of a real room on a digital audio signal (see previous and next point for additional information). 5. In electrical engineering and other disciplines, the output (response) of a (stationary, or time- or space-invariant) linear system is the convolution of the input (excitation) with the system's response to an impulse or Dirac delta function. See LTI system theory and digital signal processing. 6. In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse. 7. In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. This is the fundamental problem term in the Navier Stokes Equations relating to the Clay Institute of Mathematics Millennium Problem and the associated million dollar prize. 3.4 Cross-correlation In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the "covariance" of a random vector X, which is understood to be the matrix of covariances between the scalar components of X. In signal processing, the cross-correlation (or sometimes "cross-covariance") is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, is 12 sometimes called the sliding dot product, and has applications in pattern recognition and cryptanalysis. For discrete functions fi and gi the cross-correlation is defined as where the sum is over the appropriate values of the integer j and a superscript asterisk indicates the complex conjugate. For continuous functions f (x) and g (x) the cross- correlation is defined as where the integral is over the appropriate values of t. The cross-correlation is similar in nature to the convolution of two functions. Whereas convolution involves reversing a signal, then shifting it and multiplying by another signal, correlation only involves shifting it and multiplying (no reversing). In an Autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero. If X and Y are two independent random variables with probability distributions f and g, respectively, then the probability distribution of the difference − X + Y is given by the cross-correlation f g. In contrast, the convolution f * g gives the probability distribution of the sum X + Y 3.5 Autocorrelation A plot showing 100 random numbers with a "hidden" sine function, and an autocorrelation of the series on the bottom. Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. Informally, it is a measure of 13 how well a signal matches a time-shifted version of itself, as a function of the amount of time shift. More precisely, it is the cross-correlation of a signal with itself. Autocorrelation is useful for finding repeating patterns in a signal, such as determining the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. Given a signal f(t), the continuous autocorrelation Rff(τ) is most often defined as the continuous cross-correlation integral of f(t) with itself, at lag τ. where represents the complex conjugate and * represents convolution. For a real function, . The discrete autocorrelation R at lag j for a discrete signal xn is The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For wide-sense-stationary random processes, the autocorrelations are defined as: For processes that are not stationary, these will also be functions of t, or n. For processes that are also ergodic, the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to These definitions have the advantage that they give sensible well-defined single- parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes. 14 References: 1. Hatton, L., Worthington, M. H. and Makin, J., Seismic Data Processing Theory and Practice. Blackwell Scientific Publications, ISBN 0-632-01374-5, 1986. 2. Bracewell, R. N, The Fourier Transform and its Applications. McGraw-Hill, New York, 1983. 3. Telford, W.M., Gelbert, L.P., Sherff, R.E. and Keys, D.A., Applied Geophysics. Cambridge University Press, 1990. 4. Yilmaz, O., Seismic Data Processing. Society of Exploration Geophysicists, USA, 1989. N.B. for more information please contact to: Dr. S. M. Rahman Dept. of Applied Physics & Electronic Engineering University of Rajshahi, Bangladesh. Email: smrahman@ru.ac.bd 15