ñòð. 48 |

o(t) = g(t )i( )d = (g i) (t) : (11.57)

;1

;

Problem d: Let the Fourier transform of i(t) be given by I(!), the Fourier transform of

o(t) by O(!) and the Fourier transform of g(t) by G(!). Use (11.57) to show that

these Fourier transforms are related by:

O(!) = 2 G(!)I(!) : (11.58)

Expressions (11.57) and (11.58) are key results in the theory in linear time-invariant lters.

The rst expression states that one only needs to know the response g(t) to a single

impulse to compute the output of the lter to any input signal i(t). Equation (11.58)

has two important consequences. First, if one knows the Fourier transform G(!) of the

impulse response, one can compute the Fourier transform O(!) of the output. An inverse

Fourier transform then gives the output o(t) in the time domain.

Problem e: Show that G(!)e;i!t is the response of the system to the input signal e;i!t .

11.8. THE DEREVERBERATION FILTER 143

This means that if one knows the response of the lter to the harmonic signal e;i!t at any

frequency, one knows G(!) and the response to any input signal can be determined.

The second important consequence of (11.58) is that the output at frequency ! does

depend only at the input and impulse response at the same frequency !, but not on other

frequencies. This last property does not hold for nonlinear systems, because in that case

di erent frequency components of the input signal are mixed by the non-linearity of the

system. An example of this phenomenon is given by Snieder 55] who shows that observed

variations in the earth's climate contain frequency components that cannot be explained

by periodic variations in the orbital parameters in the earth, but which are due to the

nonlinear character of the climate response to the amount of energy received by the sun.

The fact that a lter can either be used by specifying its Fourier transform G(!) (or

equivalently the response to an harmonic input exp or by prescribing the impulse

;i!t)

response g(t) implies that a lter can be designed either in the frequency domain or in

the time domain. In section (11.8) the action of a lter is designed in the time domain.

A Fourier transform then leads to a compact description of the lter response in the

frequency domain. In section (11.9) the converse route is taken the lter is designed in

the frequency domain, and a Fourier transform is used to derive an expression for the lter

in the time domain.

As a last reminder it should be mentioned that although the theory of linear lters

is introduced here for lters that act in the time domain, the theory is of course equally

valid for lters in the spatial domain. In the case the wave number k plays the role that

the angular frequency played in this section. Since there may be more than one spatial

dimension, the theory must in that case be generalized to include higher-dimensional

spatial Fourier transforms. However, this does not change the principles involved.

11.8 The dereverberation lter

As an example of a lter that is derived in the time domain we consider here the description

of reverberations on marine seismics. Suppose a seismic survey is carried out at sea. In

such an experiment a ship tows a string of hydrophones that record the pressure variations

in the water just below the surface of the water, see gure (11.3). Since the pressure at

the surface of the water vanishes, the surface of the water totally re ects pressure waves

and the re ection coe cient for re ection at the water surface is equal to Let the

;1.

re ection coe cient for waves re ecting upwards from the water bottom be denoted by r.

Since the constrast between the water and the solid earth below is not small, this re ection

coe cient can be considerable.

Problem a: Give a physical argument why this re ection coe cients must be smaller or

equal than unity: r 1.

Since the re ection coe cient of the water bottom is not small, waves can bounce back and

forth repeatedly between the water surface and the water bottom. These reverberations

are an unwanted artifact in seismic experiments. The reason for this is that a wave that

has bounced back and forth in the water layer can be misinterpreted on a seismic section

as a re ector in the earth. For this reason one wants to eliminate these reverberations

from seismic data.

CHAPTER 11. FOURIER ANALYSIS

144

r 2 i(t-2T)

- r i(t-T)

i(t)

water

solid earth

Figure 11.3: The generation of reverberations in a marine seismic experiment.

Suppose the the wave eld recorded by the hydrophones in the absense of reverberations

is denoted by i(t). Let the time it takes for wave to travel from the water surface to the

water bottom and back be denoted by T.

Problem b: Show that the wave that has bounced back and forth once is given by i(t;

;r

T). Hint determine the amplitude of this wave from the re ection coe cients it

encounters on its path and account for the time delay due to the bouncing up and

down once in the water layer.

Problem c: Generalize this result to the wave that bounces back and forth n-times in

the water layer and show that the signal o(t) recorded by the hydrophones is given

by:

o(t) = i(t) r i(t T) + r2 i(t 2T ) +

; ; ;

or 1

Xn

o(t) = (;r) i(t nT) (11.59)

;

n=0

see gure (11.3).

The notation i(t) and o(t) that was used in the previous section is deliberately used here.

The action of the reverberation in the water layer is seen as a linear lter. The input

of the lter i(t) is the wave eld that would have been recorded if the waves would not

bounce back and forth in the water layer. The output is the wave eld that results from the

reverberations in the water layer. In a marine seismic experiment one records the wave eld

o(t) while one would like to know the signal i(t) that contains just the re ections from

below the water bottom. The process of removing the reverberations from the signal is

called \dereverberation." The aim of this section is to derive a dereverberation lter that

allows us to extract the input i(t) from the recorded output o(t).

Problem d: Can you see a way to determine i(t) from (11.59) when o(t) is given?

11.8. THE DEREVERBERATION FILTER 145

Problem e: It may not be obvious that expression (11.59) describes a linear lter of the

form (11.57) that maps the input i(t) onto the output o(t). Show that expression

(11.59) can be written in the form (11.57) with the impulse response g(t) given by:

1

X

(;r)n (t nT)

g(t) = (11.60)

;

n=0

with (t) the Dirac delta function.

Problem f: Show that g(t) is indeed the impulse response, in other words: show that

if a delta function is incident as a primary arrival at the water surface, that the

reverbarations within the water layer lead to the signal (11.60).

You probably discovered it is not simple to solve problem d. However, the problem

becomes much simpler by carrying out the analysis in the frequency domain. Let the

Fourier transforms of i(t) and o(t) as de ned by the transform (11.43) be denoted by

I(!) and O(!) respectively. It follows from expression (11.59) that one needs to nd the

Fourier transform of i(t nT).

;

Problem g: According to the de nition (11.43) the Fourier transform of i(t; ) is given

R1

by 1=2 ;1 i(t ) exp i!t dt. Use a change of the integration variable to show

;

that the Fourier transform of i(t ) is given by I(!) exp i! .

;

ñòð. 48 |