a4 inverse convolution filter

7/24/2019 A4 Inverse Convolution Filter

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(XOPHYSICS, VOL. SSiI, NO 1 (FEBKUAKY, 1962). p,, 4 1X, 14 Fl(;%

INVERSE CONVOLUTION FILTERS*

R.

B. RICEt

Tte difficult problem of trying to locate stratigraphic traps with the reflection seismograph would be simplified

(at least in good record areas) if it were possible to perform the inverse of the reflection process, i.e., to divide out

the reflection wavelet of which the record is composed, leaving only the impulses representing the reflection coeffi-

cients. This process has been discussed by Robinson under the title predictive decomposition, but his approach

requires that the basic composition wavelet be a one-sided, damped, minimum-phase time function. Most seismic

wavelets which we observe or are accustomed to working with (e.g. the symmetric Ricker wavelet) are not of this

class. The purpose of this paper is to discuss a digital computer approach to the problem. Finite, bounded inverse

filter functions are obtained which will reduce seismic wave forms to best approximations to the unit impulse in the

least squares sense. The degree of approximation obtained depends upon the time length of the inverse filter. Inverse

filter functions of moderate length produce approximate unit impulses whose breadths are 50 percent or less than

those of the original wavelets. Hence, these filters will increase resolution well beyond the practical limits of in-

strumental filters. Their effectiveness is more or less sensitive to variations in the peak frequency and shape of the

composition wavelet, and to interference, depending upon individual conditions. Although this sensitivity problem

can be solved to some extent through the proper design of the inverse filter, it is aggravated by the usual lack of

knowledge about the form of the composition wavelet.

INTRODUCTION

The problem of trying to locate stratigraphic

traps with the reflection seismograph is a difficult

one. One of the reasons for the difficulty is the

lark of detail or resolution on the seismic record.

Hence, the problem would be simplified (at least

in good record areas) if it were possible to perform

the inverse of the reflection process, i.e., to

divide out the reflection wavelet of which the

record is composed, leaving only the impulses

representing the reflection coefficients. From

another point of view, this process consists of

applying a filter which contracts each reflection

wavelet to a spike representing the arrival time

and amplitude of that reflection.

In his paper dealing in part with wavelet con-

traction, Ricker (195 3) discusses this problem

from an instrumental point of view. H e w as able

to design electronic filters which will reduce con-

ventional seismic wavelets to 70 or 80 percent of

their original breadth. T he app lication of such

filters to seismograms only achieves a partial

transformation back to the reflection coefficient

function, b ut a considerable improvement in reso-

lution is effected.

Robinson (1957) has treated the inverse filter-

ing problem unde r the title predictive decompo-

sition. Starting with a seismic trace or portion

thereof, he gives theore tical a nd statistical meth-

ods for computing (1) one form of the seismic

wavelet composing the trace and (2) the predic-

tion operator or inverse wavelet fo r effecting the

contraction to a sequence of spikes. His tech-

niques are based on discrete or digitized time

functions which are amenab le to treatment on

digital computers. This approach has the advan-

tage that com pletely arbitrary wavelet shapes or

impulsive responses,

which may be quite ex-

pensive if not impossible to simulate electroni-

ically, are handled with ease.

Robinson restricts his attention to cases in

which the basic composition wavelet is a one-

sided, damped, minimum-phase time function, a

mathem atical definition of which will be given

later. The reason for imposing this restriction is

that these are the only wavelets for which

bounded, one-sided inverses exist. However, most

seismic wavelets which we observe or are accus-

tomed to working with (e.g. the theoretical

Ricker wavelet) are not of this restricted class.

* Presentedat the 30th Annu al SEC Meeting, G alveston, Texas, November 9, 196 0. Manusc ript receivedby the

Editor June 9, 196 1.

t The Ohio Oil Com pany, Littleton, Colorado.

4


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inverse Convolution Filteti 6

f(t)

g(t)

-80)

?

v

-+z-/-

v-

h(t)= -6 (11= laf(r,(t-r) dr

-a,

FIG.

1. Th e problem of determining an inverse filter

function, g(t), which will transform a given wavelet,

f(t), into a negative unit impulse, --6(t).

The purpose of this paper is to present methods

for compu ting approx imate inverse filter functions

for arbitrary, non-minimum-phase seismic wave-

lets, and to discuss some of the properties of these

inverses which may affect their application to

actual seismic records.

STATEMENT OF THE PROBLEM

The problem to be treated here is indicated in

Figure 1. Given som e arbitrary seismic wave

form, f(t), which is assumed to be the basic re-

flection wavelet composing a seismogram, find, by

digital means, an inverse filter function, g(t),

which will transform f(t) into a unit impulse, or

the best approximation thereto in some sense. In

Figure 1, and in what follows, the negative unit

impulse, or delta function, is used so that the

trough of the input waveform will correspond to

a trough of the output function.

It is assumed that we are dealing with linear

systems, so that the filter process is defined m ath-

ematically by the familiar convolution integral:

S

co

h(t) =

f(7)g(t - TW, (1)--m

where /z(t), the output function, is equal to the

negative delta function, -6(t), only when the

problem has an exact solution in terms of a finite,

bounded g(t). As will be seen later, given a w ave

form

f(t),

there is in general no finite, bounded

g(t) ,which will produce the exact negative delta

function for It(t). In these casesf(t) and -6(t) may

be called incompatible. The chief concern of

this study is to inves tigate the effectiveness of

titer functions, g(t), which produce the best ap-

proximations to -6(t) in some sense.

It should be mentioned that electronic circuit

theoreticians have been concerned with the in-

verse convolution filtering problem for some time

from the point of view of synthesizing electron-

ically the impulsive response of a black box

when one knows the input and output wave forms

(e.g., Ba Hli (19.54), and Kautz (1954 )). However,

they rarely deal with delta function outputs,

being more concerned with situations in which

f(t)

and h(t) are compatible than with incom-

patible ones. For compatible cases, simple

methods are available for determining g(t). Hence,

their principal problem is to obtain rational frac-

tion approximations to the system function,

i.e. the Laplace transform of g(f), which lead to

optimum realizable networks.

MATHEMATICAL BACKGROUND

In actual cases,f(t) and

g(t)

must be taken to

be zero everywhere except on some finite time

interval. Hence, relation (1) may be written as

h(t) =

S

t

_f(T)& - TW.

(1

0

There are several possible approaches to the

problem of obtaining g(t) when f(t) and h(t) are

known. One method is to take the Laplace trans-

form, or the more general LaPlace-Stieltjes

transform , of both sides which yields the well-

known result (e.g., Gardner and Barnes (1942),

P. 22g),

H(s) - F(s)

G(s),

(2)

where B(S), F(s), and G(s) are the Laplace trans-

forms of h(t),f(t), and g(t), respectively. Since our

interest is in discrete representations, the Laplace

transforms may be replaced by the Dirichlet

series representations:

G(s) = 5 Xye--u(A~)s, and

v=i

(3)

H(s) = 2 B,e-YcAt)

=I

where A,,

X,, and B, represent the areas under

the curves f(t), g(t), and h(t), respectively, for the

vth equal interval At, and where it is assume d that


3/15

6

I?. B. Rice

rl,=O for v>m, X,=0 for v>N, and B,=O for

v >)z. Letting a,, x,,

and b, represent midpoint

amplitude values of f(t), g(t), and h(t), respec-

tively, and taking At= 1, the above relations re-

duce to the approximate expressions:

WL

F(s) = c UCS,

v=l

G(s) = 5

x,e-vs,

and

(4)

v=l

H(s) = 2 b,ecYs.

v=l

FIG. 2. The exact inverse of a 50 cps Ricker wavelet

obtained by polynomial division.

Since these are polynomials in ~8, it is evident

that the unknown amplitudes, xy, can he obtained

by synthetic division of H(s) by F(s).

Another method is to represent the time func-

tions in terms of the socalled generating func-

tions of Laplace (e.g., Jordan (1947) p 21). The

generating function, F(u), of the function, f(t),

can he written as

F(u) = 5

avu,

=I

(5)

is attempted. For /z(l) equal to the negative unit

impulse, its polynomial approximation reduces to

a single term,

--u. Whe n this is divided by the

polynomial representation of the wavelet f(t),

usually the quotien t or inverse coefficients oscil-

late and increase more and mo re rapidly until

they are exceeding all reasonable bounds. This is

illustrated in Figure 2 which shows the first por-

tion of the inverse resulting for a symm etric 50

cps Ricker wavelet.

where a, has the same meaning as before; and

similarly for G(U) and H(u). It has been shown

by Piety (195 1) that, under the proper conditions,

the generating function of the convolution inte-

gral (1) can he approximated accurately by

H(u) = F(u) G(u).

(6)

Thus, the unknow n amplitudes, xy, can again be

obtained through polynomial division of H(u)

by F(u).

From the point of view of complex variable

theory, this is the case for which F(u), with u

interpreted as a complex variable, has at least one

root inside the unit circle 1UI = 1. Then C(U) can-

not ha ve a convergent power series expansion on

and within the unit circle and hence no hounded

inverse exists. In the special case in which F(u)

has no roots on the

unit circle,

but has roots both

interior and exterior to it, G (U) has a Laurent

series expansion

The accuracy of any of these approximations

will depend upon the size of the interval used in

sampling the original functions. It is known that,

if the waveforms have a cut-off frequency of fC,

then 1/2f, is an adequa te interpolation interval.

In the absence of a realistic cut-off frequency,

an arbitrary cut-off m ay he made at that point at

which the amp litude spectrum is down, say 40 db,

or l/100 of the peak value.

G(u) =

2 gsu

s=-cc

which converges on the unit circle. Such an ex-

pans ion is called two-sided in contras t to a

one-sided power series expansion. It is possible

in this instance, if the convergence is rapid

enough, to use a reasonable number of terms of

this two-sided expansion as the inverse w avelet,

but we shall not be concerned with this approach.

In all the studies described in this paper, a sam-

pling interval of 2 ms has been used, with a cor-

responding cut-off frequency of 250 cps.

Having defined the inverse titer function in

terms of a polynomial division process, the next

step is to find o ut what happens when this division

If F(u) has all its roots within the unit circle,

then the Laure nt expansion reduces to the singu-

lar part

G(u) =

5

gsu


4/15

Inverse Convolution Filters

7

which converges on the unit circle but which is

polynomial multiplication and coefficients of like

strictly nonrealizable because it has no finite

terms on either side of the equality are equated,

starting time. Again, if the convergence is rapid

the following series of linear simultaneous equa-

enough, it is possible to use a finite numbe r of

tions in the unknown inverse amplitudes, xi, is

terms of this expans ion as the inverse filter. If,

obtained:

Gn Ul +

L&f-1

? + . . . + a1 X,

a,

rz+

. + a2 LL + a1

-rnL+

a, xm+ xm-l .xmtl + . .

in addition to roots within the unit circle, F(u)

also has one or more roots on the unit circle, then

the above expansion for C(u) will not converge on

the unit circle, and hence it cannot in any way be

used as an inverse operator.

In any of the above cases, F(u ) is non-mini-

mum-phase (at least one root inside the unit

circle), and an attempted one-sided polynomial

division of H(u) by F(zt) will produce an un-

bounded inverse.

If, on the other ha nd, F(zL) has no roots on or

within the unit circle, it is called minimum -

phase. Then G(u) does have a convergent power

series expansion for / UI _< , which can be used

for the inverse filter. This is the case treated by

Robinson (1957), as mentioned in the intro-

duction.

In sum mary then, there is usually no one-sided

bounded inverse which will transform observed or

theoretical seismic wavelets into the unit im pulse.

This problem may be examined from another

point of-view which may be more lucid and which

will form the basis for later rem arks abo ut the

solution. Aga in, if convolution is represented as

1 Filters, whether electronic or digital, are usually

termed realizable only when they perm it one to work

in real time. How ever, if the filtering is done with

respect o nominal time (e.g. the time scaleon a re-

cordedseism ogram ), hen nonrealizable filters can be

used.Such nonrealizable filters are in fact filters with

large time-delays. Usually, electronic ilters are used or

real time applicationsand digital computers or nominal

time applications,although there are num erousexcep-

tions.

zz

bm

=b

m+l (7)

a1 x,-m+1

= b,_,+l

a,x,_,,+l ...=

b,,

For h(t) equal to the negative unit impulse, bl= - 1

and

b, = 0

for v > 1. Hence, the polynomial division

process is equivalent to solving the first eq uation

for n-r, substituting this in the second equation

with bp=O, and solving for .rz, etc. Since this is a

set of infinitely many equations in infinitely many

unknowns, the solution will never terminate un-

less the values of .t obtained in the solution of the

first n--m+1 equations exactly satisfy the last

m- 1 equa tions. This is the condition for corn-

patibility mentioned earlier.

LEAST SQUARES COMPUTATION OF INVERSE FILTERS

Since, in general, there is no exact solution to

the problem, methods for obtaining approximate

solutions are called for. T here are several possibili-

ties. One may relax requirements on the wavelet

f(t), on the unit imp ulse, or on both. T hen one can

ask for x,s which will give the best appro ximation

to these modified functions in the Tchebycheff

sense (minimize the max imum deviation), in the

least squares sense (minimizing the sum of the

squares of the deviations), or in some other sense.

A number of approaches have been tried. It ap-

pears that the most useful solution is obtained in

terms of the best least squares approximation to

the unit impulse. Tha t is to say, assuming all the

x,s in equation 7 are zero for v > N, we ask for the

uniqu e set of IS which w ill minimize the sum

of the squ ares of the deviations of the calcu-

lated

b, s

from the desired ones,

bl= -

1, and

b,=O for v>l.


5/15

8

R. B. Rice

The derivation of the least squares solution is

most conveniently given in terms of matrix alge-

bra. Let A represent the nXN matrix of co-

efficients, X the Nth order solution column vector,

B

the nth order column vector of right-hand ele-

ments, and

E

the nth order column vector whose

elements are the deviations of the calculated

b,s

from the desired ones. Then,

AX-B==,

(8)

and the sum of the squares of the deviations is

EE = (AX - B)AX - B)

= (XA - B(AX - B)

(9)

= XAAX - XAB - BAX + BB,

where the primes denote transposes.

To obtain the normal equations, take the par-

tial derivatives of (9) w ith respect to each of the

elements of X and set the result equal to zero. We

then obtain the normal equations,

XAAlJk + UAAX - UkAB

- BAUk = 0

for all k,

(10)

where U k is the Nth order unit column vector

with unity in the kth position. Rearrange (10) so

that

(X AA - BA)u~

= Uk-AAX + AB). (10)

Now, AA is an Nth order square matrix. Hence

XAA and (X AABA) are Nth order row

vectors. On the other hand , (--AAX+AB) is

an Nth order column vector. Thus , for any k, the

left-hand side of (10 ) is the element in the kth

pos ition of the row vector (XAA - BA) and the

right-hand side is the element of the kth position

of the column vector (-AAX+AB). Since the

equality is valid for all

k, we

must have

(X AA - BA) = - AAX + AB

or

AAX = AB

(11)

which gives

X = (ilA)-AB

(12)

and

X =

BA(AA)-I.

(13)

Substituting these quantities in (9), the least

squares error reduces to

EE = BB -

It is easily verified from

BAX.

(11)

these relations that

X satisfying AX=B is both a necessary and

sufficient condition for EE=O . As indicated

earlier, this can only happen in special situations

for finite X, X and

B,

which, in matrix language,

are those cases for which the rank of the aug-

mented matrix AB is equal to the rank of A. How-

ever, when

B,

by adding Os,

A,

and X are allowed

to become infinite, A approaches a square matrix,

and there is an exact, though useless, X which

satisfies AX=B. This is the same solution ob-

tained by the successive substitution or polyno-

mial division processes mentioned earlier.

From these heuristic considerations and m ore

detailed analyses of the form of EE for low-order

systems, the following theorem is conjectured to

be true, although a general proof is not yet at

hand:

Theorem:The least squares error EE decreases

monotonically as the number of nonzero terms in

the inverse filter func tion, X, is allowed to in-

crease.

If true, this theorem indicates that a wavelet

may be reduced to as accurate an approximation

to the unit impulse as one desires by an appropri-

ate choice of the length of the inverse filter func-

tion. This is an important consideration for appli-

cations and will be illustrated later.

Before considering some specific inverse filters

computed by the least squares method, it is of

interest to look at the form of the normal equa-

tions (11) in more detail. The matrix of coefficients

AA has the symm etric form,

010

AtA = .

,: :

where

(LYM

a,,-,,-1 . . .

uo

j

m--k

o k = c aiai+k,

i=l

k = 0, 1, . . . , 12 m,

with o&= 0 for k > m- 1. Now the (Yk re the ampli-

tudes of the autocorrelation of the original wave-

let, with

CYo =

2 Ui2

s-1


6/15


9

being the maxim um center value, 0 11 he next

value on either side of the center, etc.

The right-hand mem ber of (11) is

AB =

c

aA,

2=1

m

C

a&i+1

i=l

(16)

For the case when the negative unit impulse is

placed at the mthpoint (b,= -1, b,=Ofor vfm),

this vector reduces to

I

-a,

. I

a,-1

i,

(16

I - a2m--m

the elements of which are the negative amplitudes

of the original wavelet in reverse order down to

the aZ,+,th one.

The case of symm etric wavelets and symm etric

inverses is the one dealt w ith most frequently in

the illustrations to follow. For this case, the m

normal equations (11) can be reduced to $(m + 1)

independent equations, thereby greatly reducing

the computation time and the amount of com-

puter storage required.

ILLUSTRATIVE EXAMPLES

The above observations show that the normal

Figure 3 shows the least sq uares inverses and

equations are easily formed from the amplitudes

resulting approximations to the symm etric nega-

of the original wavelet and its autocorrelation.

tive unit impulse obtained for symm etric Ricker

Hence, with a polynomial multiplication routine

wavelets peaked at 75 and 37.5 cps, respectively

to calculate the autocorrelation, least squares in-

verse filters can be computed using a standard

program for solving systems of linear equations.

A special case of interest is the one for w hich

the original wav elet is symm etric; m, n, an8

N( =n-m+ 1) are odd; and the negative unit

impulse is placed at the $(n+l)st point. Then,

for n>2m -1, the right-hand vector (16) will be

symmetric around the center value -a;(,+i),

with zeros at the top and bottom, and the inverse

filter function will be sym metric. If the inverse

filter is taken to be the same length as the original

wavelet (N=m, n=2m-1,

b,=-1),

there are

no zeros at the upper right- and lower left-hand

corners of AA (15). Also the right-hand mem bers

(16) consist precisely of the negative amplitudes

of the original wavelet with no additional zeros.

RICKER

LEAST SQUARES

APPROXIMATE

WAVELET INVERSE

UNIT IMPULSE

-+lk-

T

.OI

set

75

cps

T

37.5 cps

FE. 3. Least squares nversesand approximateunit impulses or 7 5 and 37.5 cps Ricker wav elets.


7/15

10

R. B. Rice

FREQUENCY cps

I%. 4. Amplitude spectra of 75 an d 37.5 cps inverses

of Figure 3.

The tails of the Ricker wavelets have been cut

off at a level of 0.002 percent of the maximum

amplitude. In each case, the number of points in

the inverse is the same as the numb er defining the

Ricker wavelet.

The resulting approximation to the negative

unit impulse for the 7 5 cps case is narrower and

has less ripple on either side of the center trough

than the one for the 3 7.5 cps case. However, it

should be noted that in each instance the center

trough of the Ricker wavelet has been reduced to

about 50 percent of its original breadth. This

rule-of-thumb holds for any frequency when the

inverse and the wavelet are of the same length.

LVote that the two inverses are not sim ilar in

shape as one might expect. The reason is that the

same interpolation interval (2ms) has been used

in both cases, so that the frequency character-

istics of the wavelets and inverses have different

relationships to the cut-off frequency (250 cps).

Figure 4 shows the amplitude spectra for the

7.5 and 37.5 cps inverse filters. The spectrum of

the 7.5 cps inverse rises smoothly to a peak at

250 cps, whereas the 37.5 cps inverse spectrum

has a local maximum at 140 cps and the major

peak at about 2 00 cps. Sate the small amount of

low-frequency content in both cases which, as will

be seen later, can be troublesome. The phase

spectra (not shown) are linear with the slope of

course depending upon the choice of zero time

The corresponding inverses for 50 and 25 cps

symm etric Ricker wavelets are exhibited in Figure

5. Again, the wavelets are reduced by the inverses

to about 5 0 percent of their o riginal breadths.

Figure 6 shows the amplitude spectra for the

50 and 2.5cps inverses. The spectrum of the 2.5cps

inverse rises smoothly to a peak frequency of

about 100 cps, then falls off erratically. On the

other hand, the 50 cps spectrum p eaks locally at

170 cps and has its m ajor peak at about 215 cps.

The theorem conjectured above is illustrated

in Figure 7 which show s the inverses and app roxi-

mate unit impulses for the 7 5 cps Ricker wavelet

RICKER LEAST SQUARES

APPROXIMATE

WAVELET

INVERSE UNIT IMPULSE

-+lk-

T

.OI set

50 cps

25

cps

lr

FIG. 5. Least squares nversesand approxima te unit impulses or 50 an d 25 cps Ricker wavelets.


8/15

Inverse Convolution Filters 11

)I=

0

50

II

FREQUENCY, cps

150 200 250

an

:

I

50 cps &

INVERSE ,

I

FIG. 6. Am plitude spectraof 50 and 25 cps nverses

of Figure 5.

75 cps RICKER

WAVELET

INVERSE

when the number of points in the inverse is in-

creased from 17 to 25, then to 41. The improve-

ment in the shape of the unit impulse approxima-

tion is quite striking although the breadth, which

is limited by the cut-off frequen cy, remains es-

sentially constant. The spectra for these inverses

are presented in Figure 8. Sate that, as the in-

verse increases in length, the spectrum exhibits

an increasingly steeper slope starting at higher

and higher frequencies.

These results indicate that the least squares in-

verse filters can be made as nearly perfect as one

desires, except for the limitations imposed on the

size of systems of normal equations that can be

solved on today s digital com puters. Ex perience

to date indicates that double-precision arithmetic

(18 or 20 decimal digits) is required for about

20th order or larger systems. Round-off error is

more troublesome than in most cases because

there are few or no zeros present in the coefficient

matrix.

APPROXIMATE

UNIT IMPULSE

FIG. 7. Approximate unit impulsesproducedby 75 cps nversesof increasing engths.


9/15

12

R. 8. Rice

I Or

.9c

; .7

?

7 6

2

a .5

SPECTRA OF 75cps

INVERSES

I

/I

II

1,

17pt II

25pt 1;

41 pt

:

:I

II

II

:

I

. .._f

0

50

100

150

200

2;o

FREQUENCY cps

FIG. 8. Am plitude spectraof 7 5 cps nversesof Figure 7

Next, it is of interest to see how effective these

inverse filters are in transform ing synthetic seis-

mograms back to the reflection coefficient func-

tions from which they were derived. The results

will be indicative of possibilities under ideal con-

dictions; i.e., assu ming that the basic composition

wavelet is known, is invariant with respect to

time and that there is no interference present.

Trace (c) of Figure 9 represents a reflection co-

efficient function obtained from an actual con-

tinuous velocity log on a Nebraska well, assuming

constant density. Trace (a) is the synthetic seis-

mogram obtained by convolving the 75 cps sym-

metric R icker wavelet of Figure 3 with T race (c),

and T race (b) is the result of applying the 75 cps

inverse of Figure 3 to Trace (a). The detailed

agreement between Trac es (b) and (c) is excellent.

If one were able to do this well on an actual sies-

mogram , certainly there wou ld be much less dif-

ficulty in making detailed, accurate stratigraphic

and lithologic interpretations from seismic records

However, in practice there are many complicating

factors which will be discussed below.

.Ol SEC.

(a) -Trace (c) x 75 cps RICKER

WAVELET

(b) - Trace (a) x INVERSE OF

75 cps RICKER WAVELET

(c)-REFLECTION COEFFICIEN

FUNCTION

(d)-Trace (e) x INVERSE OF

37.5 cps RICKER WAVELET

(e)-

Trace (c) x 37.5 cps.

RICKER WAVELET

FIG.

9. Results of ap plying 7.5and 37.5 cps inversesof Figure 3 to synthetic seismograms

computed rom a C VL on a Nebraska well.


10/15


13

Trace (e) of Figure 9 is the synthetic seismo- ties (Figure 6). For example, the 10 cps compo-

gram obtained from the same reflection coefficient nent for the 50 cps inverse is five times the 50 cps

function (c) using the 37.5 cps symm etric Ricker component. Hence, as the peak frequency of the

wavelet of Figure 3 , and Trace (d) is the result

Ricker wavelet is decreased and the am ount of

of filtering this trace with the 37 .5 cps inverse.

high-frequency content above 100 cps becomes

In this case, some of the detail has been lost, but insignificant, the low-frequency portion of the in-

the agreement w ith Trac e (c) is still fairly good. verse spectrum becomes dominant.

In Figure 10, similar results, based on the same

reflection coefficient (Trace (c)), are shown for the

cases of the 5 0 and 25 cps Ricker wavelets and

their inverses of Figure 5. The 50 cps inverse pro-

duces quite good agreement (Trace (b)) with the

original reflection coefficient function, but the 25

cps inverse does not restore m uch of the detail.

Better results can be obtained in any case by using

a longer inverse.

On the other hand, if the peak frequency of the

wavelet is greater than tha t on which the inverse

is based, as in the two cases on the right side of

Figure 11, the additional high frequencies are

amplified too much.

In the application of the inverse filtering tech-

nique to actual seismograms, there are a number

of factors which m ay significantly affect the re-

sults. These include variations in the frequency

and character of the composition wavelet, and

the presence of interference. Some of these effects

have been investigated synthetically to obtain

preliminary estimates of their significance and to

evaluate partially the possibilities of overcoming

the problems which they generate.

The over-all results indicate that the effect of

variations in frequency of the order of + 10 per-

cent is not serious. If the inverse is based on a peak

frequency which is too high, the resolution will

suffer proportionately. If the peak frequency is

too low, the amount of ripple will increase.

E$ect of Variations in Wave let Shape

For the purpose of illustrations, an arbitrary

distinction will be drawn between variations in

frequency which preserve wavelet form and

changes in wavelet shape which leave the peak

frequency un altered. This distinction may be dif-

ficult to find in practice where variable earth fdter-

ing corresponding to different reflection times,

differential effects due to weathering changes, and

variations in shooting conditions will usually af-

fect both peak frequency and wavelet shape.

However, the assump tion of linearity permits

these additive effects to be studied individua lly.

Next consider variations in wavelet shape cor-

responding to chang es in the form of the ampli-

tude and/or phase spectra of the composition

wavelet which leave the peak frequency unaltered

Obviously, there is no end to the numbe r of dif-

ferent kinds of variations that could be considered

However, space permits the illustration of two

types.

Figure 12 shows the approximate unit impulses

obtained by applying the 17-point, 75 cps inverse

of Figure 3 to symm etric 75 cps Ricker wavelets

with 17, 13, and 11 nonzero values, respectively.

It will be noted that this increase in the sharp-

ness of cut-off of the tails of the w avelet has no

adverse effect whatever. This is a desirable prop-

erty of the least squares compu tation technique

which would not hold for some other inverse filter

calculation methods.

I?ffect of Variations in Peak Frequency

Figure 11 shows the approximate unit impulses

resulting from the application of the 50 cps in-

verse of Figure 5 to symm etric Ricker wavelets

having peak frequencies of 44, 37.5, 25, 56.25, and

75 cps, respectively, as compared with the 50 cps

case at the top of the figure. The results on the

left for decreasing frequencies indicate a broaden-

ing effect. In fact, the approximate unit impulse

for the 2 .5 cps case is about 30 percent wider than

the 2.5 cps Ricker wavelet itself. The reason is

that the inverse amplifies the low frequencies in

the wavelet relative to the intermediate frequen-

In Figure 13, the traces on the right a re the

result of convolving the slightly asymmetric

wavelets on the left with the inverses (Figures

3 and 5 ) for the corresponding symm etric Ricker

wavelets. The asymmetric wavelets have been

obtained by adding a saw-tooth function to the

symm etric Ricker wavelets with the amplitude of

positive and negative peaks of the saw-tooth

function being about 4 percent of the maximum

trough amplitude of the Ricker wavelet. Note

that the effect of the asymm etry is very slight in

the 75 cps case, but that it increases with de-

creasing frequency, becoming completely over-

riding in the 2 5 cps case. This is explained by the


11/15

14

R. B. Rice

.Ol SEC.

-ww-

A

(a) -Trace (c) x 50

cps

RICKER

WAVELET

(b)-Trace (a) x INVERS E OF


(c)-REFLECTION coEFFiciENT

FUNCTION

id)-Trace (e) x INVER SE OF


(e)-Trace (c) x 25 cps RICKER

WAVELET

FIG. 10. Results of applying 50 and 25 cps inverses of Figure 5 to synthetic seismograms

computed from a CVL on a Nebraska well.

x 44 cps R W

4 f-

x56. 5cpsRdb

x37.5 cps RW-

V

FIG. 11. Convolutions of 50 cps inverse of Figure 5 with Ricker wavelets of different peak frequencies.


12/15

75 cps RICKER

WAVELET

17-Point

T

I3-Poin

I:

I I-Point

v

I

1;

15

APPROXIMATE

.Oi SEC.

UNIT IMPULSE

Y I--


INVERSE

u

-T--

FIG. 1 2. Effect of approximateunit impulse for 75 cps caseof cutting off tails of Rick er wave let

fact that the saw-tooth function added in each

case contains a larger proportion of high-frequency

components than the Ricker wavelets. These com-

ponents are not significant in the extreme high-

frequency range where the p eak of the 7.5 cps in-

verse spectrum occurs, but are increasingly differ-

entially amplified by the lower frequency inverses.

Most types of variations in wavelet shape would

not add such high-frequency components and

hence would not be so troublesome. If the varied

forms of the wavelet are known, the inverse

filter can be designed to circum vent the difficulties

as much as possible, although the amoun t of reso-

lution effected may have to be comprised. The

biggest problem, however, is that of determining

the form of the composition wavelet. The stand-

ard technique based on the power spectrum of the

autocorrelation of the assume d stationary time

series (e.g., Robinson (195 7)) gives no information

about the phase spectrum of the wavelet, which,

of course, strongly influences the shape. Hence,

reliance mu st be placed on theoretical or empirical

knowledge other than that contained in the seis-

mogram.

Eject of Random Noise

Finally, a few synthetic studies have been

carried out to investigate the effect of random in-

terference on the inverse filtering process. The

digital computer was used to generate random

noise spikes with an assigned density and maxi-

mum amplitude in a prescribed interval. The up-

per left-hand Trace I of Figure 14 shows such a set

of random noise spikes superimposed on the latter

portion of the 50 cps synthetic seismogram

(Trace (a)) of Figure 10. The maximum amplitude


13/15

16

Ft. B.

Rice

CONVOLUTIONS

OF WAVELETS ON LEFT WITH

INVERSES OF SYMMETRIC RICKER

WAVELETS OF SAME FREQUENCY

75 cps

MODIFIED

RICKER

WAVELETS

50 cps

FIG. 13 . Convolutionsof inversesof Figures 3 and 5 with asymm etric Ricker wavelets.

of these spikes is about 50 percent of the maxi-

mum trace amplitude. When this noise function is

convolved with 25, 50 , and 7.5 cps symm etric

Ricker wavelets, respectively, and added to the

original trace, the three lower traces on the left

are obtained. The traces on the right are the re-

sult of applying the 50 cps inverse filter of Figure

5 to the left-hand traces.

In compa ring the second trace on the right

with the upper right-hand trace, it is observed

that the details have been fairly well restored in-

the presence of the 2 5 cps noise, but that there is

a low-frequency component remaining. For the

case when the interference is made the sam e fre-

quency as the signal (third set of traces), the in-

verse filter cannot distinguish between the signal

and the noise. Hence, the interference affects the

results in direct p roportion to its density and

amplitude. In the last set of traces, it is evident

that the 75 cps noise has a disastrous effect on the

filtering process. The amplitudes on the right-

hand trace have been reduced by a factor of ten

to keep them within reasonable bounds, and there

is no resemblance to the top trace.

These examples indicate that random inter-

ference may not be too troublesome if it is of sub-

stantially lower frequency than the reflections,

although the very low frequencies will be amplified

by the inverse filter to some extent as mentioned

earlier in the discussion of Figure 11. The require-

ment is that the interference must not have sig-

nificant frequency content beyond the point at

which the spectrum of the inverse filter starts to

rise rapidly. If the cut-off frequency can be made


14/15


17

SYNTHETIC

LEFT-HAND TRACES

SEISMOGRAMS

X50 cps INVERSE FILTER

TRACE I

TRACE I

cps

NOISE

TRACE I

cps NOISE

TRACE I

cps

NOISE

FIG.

14.

Effect of rando m noise of d ifferent frequenc ies n resultsof app lying 50 cps nverse of

Figure 5 to 50 cps synthetic seismogram.

high enough and the inverse filter long enough,

this requirem ent can usually be fulfilled. If this is

not accomplished, the results, a s in the lower

right-hand trace, will be self-explanato ry.

It should be noted that significant reading er-

rors introduced in digitizing seismograms will

have the same effect as high-frequency inter-

ference. Hence, it is necessary to develop some

procedure for digitizing traces with sufficient ac-

curacy. Commercial oscillographic readers are

usually employed.

SUMMARY AND CONCLUSIONS

In summ ary, it has been demonstrated that it

is possible to compute inverse filter functions by

digital techniques which, under ideal conditions,

will increase the resolution of reflection effects on

seismograms well beyond the limits that are prac-

tical with instrume ntal filters. It has been shown

that the effectiveness of these filters is more or less

sensitive to variations in the peak frequency and

shape of the wavelet composing the seismogram,

and to interference. This sensitivity can be over-

come to some extent by d esigning the filter so that

it will not amplify unwanted frequencies. In m any

instances, the amount of resolution effected will

have to be comprom ised with the sensitivity prob-

lem, which is aggravated by the usua l lack of

knowledge about the form of the composition

wavelet. Nevertheless, the method appears to

hold enough promise as a new seismic interpreta-

tion tool in the search for stratigraphic traps to

warrant further study of its application to field

records.


15/15

18

R. B. Rice

ACKNOWLEDGMENTS

Gardner, XI. F., and Barnes, J. L., 1942, Transients in

The author is greatlv indebted to his assistant,

linear systems, v. 1: New York, John Wiley and Sons,

Inc.

Mr. R. L. Massey, for his help in most phases of

Jordan, Charles, 1947, Calculus of finite differences,

the work on this paper. He would also like to

2nd edition: New York, Chelsea Publishing Co.

Kautz, W. H., 1954, Transient synthesis in the time

thank Dr. E. A. Robinson for his helpful reading

domain: Transactions of the IRE Professional Group

of the manuscript.

on Circuit Theory, v. CT-l, n. 3, p. 29-38.

Piety, R. G., 1951, A linear operational calculus of em-

pirical functions: Phillips Petroleum Co. Research

REFERENCES

Division Report 10612.51R.

Ba Hli, F., 1954, .4 general method for time domain net-

Ricker, Norman, 1953, Wavelet contraction, wavelet

work synthesis: Transactions of the IRE Profes-

expansion, and the control of seismic resolution: Geo-

sional Group on Circuit Theory, v. CT-l, n. 3, p.

physics, v. 18, p. 7699792.

Robinson, E. A., 1957, Predictive decomposition of

21-28.

seismic traces: Geophysics, v. 22, p. 767-778.

a4 inverse convolution filter

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