Mathematics - Fourier Transforms And Waves

Mathematics - Fourier Transforms And Waves

(Parte 1 de 7)


Jon F. Clærbout

Cecil and Ida Green Professor of Geophysics Stanford University



1 Convolution and Spectra 1


2 Discrete Fourier transform 17

3.3 A matlab program for downward continuation36


Why Geophysics uses Fourier Analysis

When earth material properties are constant in any of the cartesian variables then it is useful to Fourier transform (FT) that variable.

In seismology, the earth does not change with time (the ocean does!) so for the earth, we can generally gain by Fourier transforming the time axis thereby converting time-dependent differential equations (hard) to algebraic equations (easier) in frequency (temporal frequency).

In seismology, the earth generally changes rather strongly with depth, so we cannot usefully Fourier transform the depth axis and we are stuck with differential equations in . On the other hand, we can model a layered earth where each layer has material properties that are constant in . Then we get analytic solutions in layers and we need to patch them together.

Thirty years ago, computers were so weak that we always Fourier transformed the and coordinates. That meant that their analyses were limited to earth models in which velocity was horizontally layered. Today we still often Fourier transform but not , so we reduce the partial differential equations of physics to ordinary differential equations (ODEs). A big advantage of knowing FT theory is that it enables us to visualize physical behavior without us needing to use a computer.

The Fourier transform variables are called frequencies. For each axis we have a corresponding frequency . The ’s are spatial frequencies, is the temporal frequency.

The frequency is inverse to the wavelength. Question: A seismic wave from the fast earth goes into the slow ocean. The temporal frequency stays the same. What happens to the spatial frequency (inverse spatial wavelength)?

In a layered earth, the horizonal spatial frequency is a constant function of depth. We will find this to be Snell’s law.

In a spherical coordinate system or a cylindrical coordinate system, Fourier transforms are useless but they are closely related to “spherical harmonic functions” and Bessel transformations which play a role similar to FT.

Our goal for these four lectures is to develop Fourier transform insights and use them to take observations made on the earth’s surface and “downward continue” them, to extrap- i CONTENTS olate them into the earth. This is a central tool in earth imaging.

0.0.1 Impulse response and ODEs

When Fourier transforms are applicable, it means the “earth response” now is the same as the earth response later. Switching our point of view from time to space, the applicability of Fourier transformation means that the “impulse response” here is the same as the impulse response there. An impulse is a column vector full of zeros with somewhere a one, say (where the prime means transpose the row into a column.) An impulse response is a column from the matrix

The impulse response is the that comes out when the input is an impulse. In a typical application, the matrix would be about and not the simple example that I show you above. Notice that each column in the matrix contains the same waveform

. This waveform is called the “impulse response”. The collection of impulse responses in Equation (0.1) defines the the convolution operation.

Not only do the columns of the matrix contain the same impulse response, but each row likewise contains the same thing, and that thing is the backwards impulse response

. Suppose were numerically equal to . Then equation (0.1) would be like the differential equation . Equation (0.1) would be a finitedifference representation of a differential equation. Two important ideas are equivalent; either they are both true or they are both false:

1. The columns of the matrix all hold the same impulse response. 2. The differential equation has constant coefficients.

The story gets more complicated when we look at the boundaries, the top and bottom few equations. We’l postpone that.

0.0.2 Z transforms

There is another way to think about equation (0.1) which is even more basic. It does not involve physics, differential equations, or impulse responses; it merely involves polynomials.


(That takes me back to middle school.) Let us define three polynomials.

Are you able to multiply ? If you are, then you can examine the coefficient of . You will discover that it is exactly the fifth row of equation (0.1)! Actually it is the sixth row because we started from zero. For each power of in we get one of the rows in equation (0.1). Convolution is defined to be the operation on polynomial coefficients when we multiply polynomials.

0.0.3 Frequency

The numerical value of doesn’t matter. It could have any numerical value. We haven’t needed to have any particular value. It happens that real values of lead to what are called Laplace transforms and complex values of lead to Fourier transforms.

Let us test some numerical values of . Taking we notice the earliest coefficient in each of the polynomials is strongly emphasized in creating the numerical value of the polynomial, i.e., . Likewise taking , the latest value is strongly emphasized. This undesirable weighting of early or late is avoided if we use the Fourier approach and use numerical values of that fulfill the condition . Other than that forces us to use complex values of , but there are plenty of those.

Recall the complex plane where the real axis is horizontal and the imaginary axis is vertical. For Fourier transforms, we are interested in complex numerical values of which have unit magnitude, namely, . Examples are , or .

The numerical value gives what is called the zero frequency. Evaluating , finds the zero-frequency component of . The value gives what is called the “Nyquist frequency”.

. The Nyquist frequency is the highest frequency that we can represent with sampled time functions. If our signal were then all the terms in would add together with the same polarity so that signal has a strong frequency component at the Nyquist frequency.

How about frequencies inbetween zero and Nyquist? These require us to use complex numbers. Consider , where . The signal could be segregated into its real and imaginary parts. The real part is . Its wavelength is twice as long as that of the Nyquist frequency so its frequency is exactly half. The values for used by Fourier transform are .

Now we will steal parts of Jon Claerbout’s books, “Earth Soundings Analysis, Process- iv CONTENTS ing versus Inversion” and “Basic Earth Imaging” which are freely available on the WWW1 .

To speed you along though, I trim down those chapters to their most important parts.


Chapter 1 Convolution and Spectra

Time and space are ordinarily thought of as continuous, but for the purposes of computer analysis we must discretize these axes. This is also called “sampling” or “digitizing.” You might worry that discretization is a practical evil that muddies all later theoretical analysis. Actually, physical concepts have representations that are exact in the world of discrete mathematics.

1.1 SAMPLEDDATAANDZ-TRANSFORMS Consider the idealized and simplified signal in Figure 1.1. To analyze such an observed

Figure 1.1: A continuous signal sampled at uniform time intervals. cs-triv1 [ER]

signal in a computer, it is necessary to approximate it in some way by a list of numbers. The usual way to do this is to evaluate or observe at a uniform spacing of points in time, call this discretized signal . For Figure 1.1, such a discrete approximation to the continuous function could be denoted by the vector

Naturally, if time points were closer together, the approximation would be more accurate. What we have done, then, is represent a signal by an abstract -dimensional vector.

Another way to represent a signal is as a polynomial, where the coefficients of the polynomial represent the value of at successive times. For example,


This polynomial is called a “ -transform.” What is the meaning of here? should not take on some numerical value; it is instead the unit-delay operator. For example, the coefficients of are plotted in Figure 1.2. Figure 1.2 shows

Figure 1.2: The coefficients of are the shifted version of the coefficients of . cs-triv2 [ER]

(Parte 1 de 7)