Water reverberation filter

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Problem 9.7a

Verify equation (9.7d) for the inverse filter for water reverberation by convolving it with equation (9.7a) for the water reverberation.

Background

Multiple reflections within a water layer can cause severe reverberation (ringing) when the reflection coefficient at the sea floor is sufficiently high. We take the reflection coefficients at the top and bottom of the water layer as $ -1 $ and $ R $, and $ n\Delta $ as the round-trip traveltime in the layer, $ n $ being an integer and $ \Delta $ the sampling interval. Bounces can occur before or after a wave is reflected at deeper reflectors. We assign unit amplitude to a primary reflection with traveltime $ t $. A reflection from the same horizon that has suffered one bounce in the water layer has traveltime $ t+n\Delta $ and amplitude $ -R $. Since the bounce may take place either before or after reflection at depth, there are two waves arriving at $ t+n\Delta $, the combined amplitude being $ -2R $. There are three waves that suffer two bounces: two bounces before the deep reflection, two after, and one before plus one after, so the multiple that arrives at $ t+2n\Delta $ has amplitude $ +3R^{2} $. Reflections suffering three bounces arrive at $ t+3n\Delta $ with amplitude $ -4R^{3} $, and so on. The impulse response (see Sheriff and Geldart, 1955, 279) of the water layer is thus the infinite series


$ {\begin{aligned}w_{t}=[1,\;-2R,\;3R^{2},\;-4R^{3},\;5R^{4},\ldots \ldots .]\quad \mathrm {when} \ n=1;\end{aligned}} $ (9.7a)


$ {\begin{aligned}{\mbox{or}}\qquad \qquad w_{t}=[1,\;0,\;-2R,\;0,\;3R^{2},\;0,\;-4R^{3},\ldots .]\qquad \mathrm {when} \ n=2;\end{aligned}} $ (9.7b)

and so on for other values of $ n $.

A filter (see problem 7.11) that removes the effect of another filter is an inverse filter, and the inverse filter in this case is $ i_{t} $; it satisfies the equation


$ {\begin{aligned}w_{t}*i_{t}=\delta _{t};\end{aligned}} $ (9.7c)


$ {\begin{aligned}i_{t}&=[1,\;2R,\;R^{2},\;0,\;0,\;.\;.\;.]\qquad \mathrm {when} \ n=1\end{aligned}} $ (9.7d)


$ {\begin{aligned}&=[1,\;0,\;2R,\;0,\;R^{2},\;0,\;.\;.\;.]\quad \mathrm {when} \ n=2.\end{aligned}} $ (9.7e)

[See Sheriff and Geldart, 1995, section 15.5.3 for a discussion of $ z $-transforms.]

Solution

For $ n=1 $, $ i_{t}=\left[1,\;2R,\;R^{2}\right] $, and $ w_{t}=[1,-2R,3R^{2},-4R^{3},5R^{4},\ldots ] $. We wish to verify that $ w_{t}*i_{t}=\delta _{t} $. Using the procedure described in problem 9.2, we replace each element of $ i_{t} $ by a scaled version of $ w_{t} $; thus

1, –$ 2R $, +$ 3R^{2} $, –$ 4R^{3} $, +$ 5R^{4},\ldots $
+$ 2R $, –$ 4R^{2} $, +$ 6R^{3} $, –$ 8R^{4},\ldots $
+$ R^{2} $, –$ 2R^{3} $, +$ 3R^{4},\ldots $
= 1, 0, 0, 0, 0, … $ =\delta _{t} $.

We can proceed in the same way for other values of $ n $.

Problem 9.7b

The spectrum of the water-layer filter is shown in Figure 9.7a for $ n=1 $; the large peaks occur at the “singing frequency” $ nf_{1} $, where $ n $ is any odd number. Sketch the amplitude spectrum of the inverse filter.

Solution

In the time domain, $ w_{i}*i_{t}=\delta _{t} $; in the frequency domain this becomes $ W(f)\times I(f)=1 $. Thus the amplitude spectrum of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): I(f) is the inverse of that of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): W(f) , that is, for a given frequency,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} |I(f)|=1/|W(f)|, \end{align}

as shown in Figure 9.7b.

Figure 9.7a.  Spectrum of water-layer filter; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_1 = V_{W}/4h, h = water depth, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R = 0.5 .
Figure 9.7b.  Inverse filter spectrum (dashed).

Problem 9.7c

Verify your sketch of the water-reverberation inverse filter by transforming


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left[1, \; 2R,\; R^{2}\right]\leftrightarrow [1+2Re^{-{\hbox {j}}2\pi f{\Delta}} + R^{2} e^{-{\hbox {j}}4\pi f{\Delta}}, \end{align} (9.7f)

and calculating the value of the spectrum for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R=0.5 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f=0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/2\Delta , and $ 1/\Delta $.

Solution

We start with the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z -transform of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i_{t} , that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): I(f)=(1+2Rz+R^{2} z^{2}) . This is a complex quantity and we multiply by the conjugate complex (see Sheriff and Geldart, 1995, section 15.1.5) to get the magnitude squared. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=e^{\hbox {j}2\pi\Delta} , the conjugate complex is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): e^{+\hbox {j}2\pi\Delta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): zz^{-1} =1 ; thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} |I(f)|^{2} &=(1+2Rz+R^{2} z^{2})(1+2Rz^{-1} +R^{2} z^{-2})\\ &=(1+Rz)^{2} (1+Rz^{-1})^{2} =[(1+Rz)(1+Rz^{-1})]^{2},\end{align} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mbox {so} \qquad \qquad |I(f)|=(1+Rz)(1+Rz^{-1})=[1+R^{2} +R(e^{+\hbox {j}2\pi f\Delta} +e^{-{\hbox {j}}2\pi f\Delta})]. \end{align}

Substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R=0.5 and using Euler’s formula (Sheriff and Geldart, 1995, problem 15.12a), we get

$ {\begin{aligned}|I(f)|=[1.25+\cos \,(2\pi f\Delta )].\end{aligned}} $

Substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f=0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/2\Delta , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/\Delta , we get the following values

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} |I(f)|=2.25,\ 0.25,\ 2.25,\ \quad\mathrm{that\ is}, 9/4,\ 1/4,\ 9/4. \end{align}

The peaks and troughs in Figure 9.7a have the relative values 9, 1, 9; since these values are normalized, the agreement is exact.

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