By J.K. Costain, C. Coruh

The cloth during this quantity offers the fundamental idea essential to comprehend the foundations in the back of imaging the subsurface of the Earth utilizing mirrored image and refraction seismology. For mirrored image seismology, the tip product is a ''record section'' from a suite of ''wiggly traces'' which are recorded within the box from which information regarding the homes of subsurface constitution and rock should be derived. For the main half, the foundations of imaging are an identical whatever the intensity to the objective; a similar mathematical history is important for focusing on a shallow water desk as for investigating the bottom of the earth's continental ''crust'' at a intensity of 30-50 km.

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More commonly referred to as a Discrete Fourier Series (DFS) or simply a Fourier Series. 3. Discrete-Time Fourier Transform (DTFT). Time is discrete, frequency is continuous. 4. Discrete Fourier Transform (DFT). Both time and frequency are discrete. The introduction of the "fast Fourier transform" (FFT) version of the CHAPTER 3. 1: Notation for the Fourier transform (FT) Analysis Synthesis /•+OO F{y) = I f(t)e~i2*vtdt f(t) = / J — oo FHe+i2"'^ J — oo /(£) is continuous, aperiodic, and can be complex oo < t < oo F(i^) is complex, continuous, and aperiodic OO < V < OO oo < t < oo OO < V < 00 DFT has revolutionized geophysics and other fields that deal with large data sets.

The value of z is the same for 9 = 0° and 0 = 360° so there are n distinct roots as k ranges from 0 to n — 1. , z2^-l. 3. When the phase 6 of the phasor is equal to zero, then z = r = 1. Therefore, in the general polar form with k an integer, I _ ei(0+2k*) _ ei(2fc7r) and the root rxln can be given as ll/n _ ei{2k*/n) or l 1 / n = cos (2kn/n) + i sin {2kn/n) forfc= 0,1,2, • • • n - 1. From this result we see that the n roots of unity are located around the unit circle in the complex plane. 4 Logarithm of a Complex Number In general if z = eu then UJ = lnz CHAPTER 2.

A discrete signal is one that is defined only for integer values of the independent variable(s); the dependent variable could take on a continuous range of values. For a digital signal, however, the dependent variable must also be discrete. Digital signals are thus considered to be a subset of discrete signals. Not only is the time quantitized, but so also is the amplitude. There are four types of Fourier transforms in the nomenclature of conventional signal processing. They are 1. Fourier Transform (FT).