Deconvolution Filter in Seismic Data Processing

石油学会誌 Sekiyu Gakkaishi, 31, (6), 439-447 (1988) 439

Deconvolution Filter in Seismic Data Processing

Toshifumi MATSUOKA

Geophysics Department, Japan Petroleum Exploration Co., Ltd.,2-17-22, Akasaka, Minato-ku, Tokyo 107

(Received December 21, 1987)

Information of importance in reflection seismic method resides in the reflectivity series. In order to

extract this information from obtained data, a deconvolution filter is the core of seismic data processing.

Since whitening deconvolution was first introduced by Robinson1), it has been used in the oil industry on

a routine basis. Whitening deconvolution is based on following assumptions: (1) A reflectivity series is

an uncorrelated random series. (2) A seismic wavelet is minimum phase. Deconvolution outputs,

however, ofter show that the minimum phase assumption on seismic wavelet is incorrect. In this paper

we deeply investigate in depth this assumption and consider several types of deconvolution methods.

Predictive deconvolution can be applied to both minimum and non-minimum phase wavelets and it is

equivalent to phase correction method2),3) for Vibroseis data, in a mathematical sense. In the case of

whitening deconvolution, to obviate the minimum phase assumption, a statistical wavelet estimation

approach,4) using property of reflectivity series, is explained.

1. Introduction

The objective of seismic surveys is to obtaininformation about subsurface geology. Seismicwaves propagate into the earth and are reflected atinterfaces between subsurface layers with differentacoustic impedances. These reflected waves aredetected by receivers set up on the surface of theearth and are digitally recorded. Various kinds ofsophisticated filtering techniques have beendeveloped in order to enhance quality of obtaineddata. Especially, deconvolution plays one of themore important roles in such enhancement. Manyauthors, therefore, have developed various tech-niques of deconvolution filtering. In this paper,basic concepts of deconvolution and recentdevelopments are discussed.

2. Wavelet Theory

From physical point of view, a seismic trace isconsidered to be an impulse response of the earth

system. Input to this system is impulse of elasticenergy generated by various methods such asdynamite and other nonexplosive sources, whileoutput contains not only the reflectivity series butalso undesirable effects such as multiples, ghost-ing reflections, and attenuation which are con-volved with reflectivity series (Fig. 1).

Since Ricker5) first studied a mechanism ofelastic wave propagation, many authors6),7),8) havetried to clarify the inherent mechanism of wavepropagation. They demonstrated theoreticallyand experimentally that a wavelet changes itsshape with time according to the law ofattenuation and dispersion.

An actual seismic trace consists of many seismicwavelets overlapped with different strengths andarrival times, and can be expressed mathematicallyby the convolution product of a reflectivity series,

{qt}, with a finite-length seismic wavelet {ws} as

xt=∑ms=0wsqt-s (1)

Fig. 1 The Impulse Response of the Earth System; the

System Consists of a Reflectivity Series and a

Wavelet

石油学会誌 Sekiyu Gakkaishi, Vol. 31, No. 6, 1988

440

Thus deconvolution, that is the inverse operationof the convolution in Eq. (1), restores thereflectivity series {qt} from the seismic trace, {xt}

(Fig. 2).There are two approaches to the deconvolution

of a seismic trace, deterministic and statisticalmethods. The deterministic approach, which iscalled wavelet processing, requires a prior knowl-edge of the seismic wavelet.9) In marineenvironments, it is sometimes possible to obtain abasic wavelet which contains ghosting reflectionsand the effects of the data recording system.10)Applying the inverse operator of this wavelet toseismic trace, we can estimate the reflectivityseries. This approach, however, can be appliedonly to marine seismic surveys.

On the other hand, statistical approach does notrequire any kind of prior information about theseismic wavelet. Therefore, it is the most commonmethod used in seismic data processing in order tosuppress multiples and to compress the wavelet.In this paper, we focus our attention on statisticaldeconvolution methods and statistical waveletestimation technique.

3. Whitening Deconvolution

In Eq. (1), although there are two unknowns,wavelet {ws} and reflectivity series {qt}, we haveonly one measurement series {xt}. Deconvolution,therefore, is always an underdetermined problemin a mathematical sense. This problem can beovercome by placing certain constraints on either

{ws} or {qt}, or on both. Different constraints leadto different deconvolution methods. Whiteningdeconvolution developed by Robinson1) requires aminimum phase assumption on seismic waveletwhich is briefly summarized blow.

According to a probability model of a sedimenta-tion process11), reflectivity series {qt} can berepresented by a random process such that

E[qtqt']=σ2δtt' (2)

where E is an expectation operator and δtt' is the

Kronecker delta function. A time series character-

ized by Eq. (2) is called a white noise series withvariance σ2. Seismic trace{xt}in Eq.(1), therefore,

is considered to be a weighted sum of white noiseseries, where weighting function is the seismicwavelet. Such a time series is known as a movingaverage process (MA process) of order m.

As is well known, any time series is character-

ized by its autocovariance function, r(τ),

r(τ)=E[xtxt+τ]

Substituting Eq. (1) into the above equation andusing Eq. (2), we obtain

r(τ)=E[∑ms=0wsqt-s∑ms'=0ws'qt+τ-s']

=σ2∑ms=0wsws+τ

This equation shows that autocovariance function

of seismic trace is autocovariance function of

seismic wavelet times the variance of reflectivity

series. Power spectrum of wavelet, therefore, isobtained as Fourier transform of r(τ). Unfor-

tunately, however, this function does not contain

phase information of the wavelet. This introducesan ambiguity of reconstruction of Eq. (1) whenonly autocovariance function is given. This iseasily understood from the following discussion:

Let us consider a polynomial equation of order

2m:

ZmR(Z)=Zm∑mτ=-mr(τ)Zτ (3)

To obtain wavelet from autocovariance function,factorization of ZmR(Z) into two real polynomials,W(Z) and Zm W(Z-1), is required

ZmR(Z)=W(Z)ZmW(Z-1)

where

W(Z)=∑ms=0wsZs=Πmi=1(Zi-Z)

Since autocovariance function is an even function,

r(τ)=r(-τ), it follows that if Zi is a root of Eq.(3),

then Z-1i is also a root. Accordingly, there are, at

Fig. 2 The Reflectivity Series Obtained by Applying

the Deconvolution Operator to the Seismic

Trace


441

most, 2m different ways to choose roots forconstructing W(Z).

To overcome this ambiguity, Robinson1) pro-

posed to choose those m roots of ZmR(Z) havingmodulus greater than one. The reconstructedW(Z) using these roots is a minimum phasewavelet. In other words, it is an invertible movingaverage process. Using the Z-transform, Eq. (1)can be expressed as

X(Z)=W(Z)Q(Z) (4)

where

X(Z)=∑tXtZt

and

Q(Z)=∑qtZt

From Eq. (4) Q(Z) becomes

(5)

As it is well known, since W(Z) is minimum phase,1/W(Z) can be expanded in a power series of Z as

1/W(Z)=1+a1Z+a2Z2+…

Truncating this series at n-th order gives

Q(Z)=X(Z)A(Z)

or equivalently

qt=xt+a1x-1+…+anxt-n (6)

where a=(1, a1, …… an)T is the deconvolution

operator, or the prediction error operator. A timeseries shown in Eq. (6) is known as an auto-regressive process (AR process) of order n.

We note here that minimum phase assumptionis just a mathematical requirement to solve theambiguity Different constraints, therefore, lead todifferent equations. Once the unique representa-tion of {qt} is given by convolutional form as Eq.

(6), deconvolution operator, an=(1, a1,…,an)T, may

be easily obtained from seismic trace, {xt}, byfollowing recipe.

Multiplying both sides of Eq. (6) by xt-l andtaking expectation, we obtain

E[xt-lxt]+a1E[xt-lxt-l]+…

+anE[xt-lxt-n]=E[xt-lqt]

l=0,1,2,…,n

Since ql and Xt-l, l>0 are independent of each

other

we have

r(0)+a1r(1)+…+anr(n) =σ2

r(1)+a1r(0)+…+anr(n-1)=0

r(2)+a1r(1)+…+anr(n-2)=0

r(n)+ar(n-1)+…+anr(0)=0

Omitting the first equation, the following matrix

representation of the normal equations is obtain-

ed:

(7)

Since this coefficient matrix is Toeplitz form, the

Levinson recursion can be applied to solve aboveequation. Applying obtained deconvolution

operator, a=(1, a1, a2,…, an)T, to seismic trace{xt}

gives reflectivity series which is a white noiseseries. This is why Robinson's method is called"whitening deconvolution".

In summary, the observed seismic data ismodelled by a moving average process, Eq. (1),with an assumption that the reflectivity series is awhite noise series. When seismic wavelet isminimum phase, seismic trace is expressed as an

autoregressive process as Eq. (6). Operator forwhitening deconvolution is determined by solvingnormal equations given in Eq. (7) combined withautocovariance function of the data Finally,reflectivity series can be estimated by convolvingobtained deconvolution operator with seismictrace.

4. Vibroseis Deconvolution

Although physical arguments have been pre-sented why minimum phase is a likely phasespectrum for some sources, deconvolution resultsof ter show that the minimum phase assumption isincorrect. In this section, a case of a non-minimum phase wavelet is considered

Robinson's deconvolution theorem states that ifseismic wavelet is minimum phase, deconvolutionoperator is the inverse of seismic wavelet itself,

A(Z)=1/W(Z)


442

However, since Eq. (7) contains only autocovar-iance function of the data, deconvolution operatoris, obtained from Eq. (7), even in case of a non-minimum phase wavelet. When seismic wavelet isnon-minimum phase, the obtained operator is

given by

A(Z)=1/Wm(Z)

Where Wm(Z) is a corresponding minimum phasewavelet which has exactly the same powerspectrum as the original wavelet W(Z). Byapplying this operator to seismic trace, the outputof whitening deconvolution becomes

O(Z)=1/Wm(Z)X(Z)

=W(Z)/Wm(Z)Q(Z) =P(Z)Q(Z)

where P(Z) is an all-pass filter whose powerspectrum is identity, but which changes the phasespectrum of Q(Z).

In case the seismic wavelet is non-minimum

phase, therefore, the output of whitening decon-volution has to be phase corrected. Thismotivated the phase correction method by Ristowand Jurczyk12) for Vibroseis data. The Vibroseisdata consisting of Klauder wavelet is obtained byautocorrelating vibrator sweep signal to aquireddata. Because of this operation, the Klauderwavelet is zero phase.

If Vibroseis data is obscured by multiples, theseismic trace can be written as

X(Z)=K(Z)M(Z)Q(Z)=W(Z)Q(Z)

where K(Z) and M(Z) are Z-transforms of theKlauder wavelet and the multiples, respectively.As is well known, the multiples have minimum

phase property, and, thus the deconvolutionoperator becomes

A(Z)=1/Wm(Z)=1/Km(Z)M(Z)

where Km(Z) is the minimum phase equivalent toKlauder wavelet. By using this operator, theoutput of whitening deconvolution is obtained as

O(Z)=K(Z)/Km(Z)Q(Z)

Phase correction method consists of applying

whitening deconvolution followed by convolution

with the corresponding minimum phase wavelet.

Therefore, the output of phase correction methodfinally becomes

O(Z)=K(Z)Q(Z)

Here, the undesirable effect of multiples has beeneliminated.

5. Predictive Deconvolution

Since whitening deconvolution was introducedto seismic data processing, it has been widely usedin the oil industry. Whitening deconvolution,however, cannot control resolution of the output.To overcome this problem, Peacock and Treitel13)developed a predictive deconvolution theorywhich could control output wave form.

Rewriting Eq. (6), we have

xt=b1xt-1+b2xt-2+…+bnxt-n+qt

where

bi=-ai i=1,2,… ,n

Above equation shows that the value of one stepahead, xt, is predicted by convolving past datawith the prediction operator {bs}. Prediction errorseries in this case corresponds to the white noiseseries {qt}.

Let us consider the case of prediction α+1 step

ahead

xt+α=b1'xt-1+b2'xt-2+…

+bn'xt-n+q't+α (8)

where {bs'} is the prediction operator for value of

Let us consider the case of prediction α+1 step

ahead

xt+α=b1'xt-1+b2'xt-2+…

+bn'xt-n+q't+α (8)

α+1 step ahead. Rewriting Eq.(8), we have

q't+α=xt+α-b1'xt-1-b2xt-2-…-bn'xt-n

In this case, the α step future value, xt+α, is

predicted by xt-1, xt-2, …, xt-n values. The

prediction error q't+α, therefore, is no longer a

white noise series. Thus, the output of predictionerror filter (the deconvolution filter) is not asimple spike as in whitening deconvolution.

To obtain predictive deconvolution operator.the same recipe as we used for whiteningdeconvolution is adopted. Multipling both sides

of Eq. (8) by xt-l and taking expectation, thefollowing equations are obtained

(9)


443

Note that, because prediction operator {bs'} has thereverse sign of prediction error operator {as}, thereis no minus sign on the right hand side of Eq. (9).By using {bs'} obtained from Eq. (9), thedeconvolution operator for prediction distance α

is

{ft}=1,0,0, …,0,}α zeros -b1',-b2,, …,-b'n}

(10)

Although Peacock and Treitel 13) had shown

that resolution control is achieved by specifying α,

little literature exists which discusses the form ofactual output14). Recently, Matsuoka and Ulrych2,3)

studied the recursive relationship between predic-tive error operators, which can be used to inves-tigate output wave forms.

Using Z-transform, Eq. (10) becomes

Fα(Z)=1-ZαBα(Z) (11)

where

Bα(Z)=∑ni=1bi'Zi

Following considerable algebraic manipulation,Ulrych and Matsuoka3) show that Eq. (11) can beexpressed as

Fα(Z)=F1-1(Z)|αF1(Z) (12)

Here, F1(Z) is deconvolution operator with

prediction distance equal to one, and F1-1(Z)|α is

the inverse of F1(Z), truncated at the α-th term.

Predictive deconvolution operator with arbitrary

prediction distance α, can be expressed as the

product of whitening deconvolution operator

F1(Z) and inverse of F1(Z) truncated at α.

First of all, let us consider output wave form in

the case of minimum phase wavelet. The output

of predictive deconvolution with prediction

distance α is

Oα(Z)=Fα(Z)X(Z)=Fα(Z)W(Z)Q(Z)

Using Eq. (12), this equation can be written as

Oα(Z)=F1-1(Z)|αF1(Z)K(Z)Q(Z)

As already mentioned, whitening deconvolutionoperator F1(Z) is simply the inverse of waveletW(Z), whenever the wavelet is minimum phase.Consequently, the following equation is obtained

F1-1(Z)|α=W(Z)|α

Therefore, the output of predictive deconvolution

is

Oα(Z)=W(Z)|αQ(Z) (13)

Eq.(13)helps us to understand the basic concept

of predictive deconvolution. The output Oα(Z)

produced by unit gap, α=1, or whitening

deconvolution, is simply reflectivity series. In case

α is greater than one, more interesting results are

obtained. In this case, the predictive deconvolu-

tion output Oα(Z) is convolution of reflectivity

series with wavelet truncated at lag α. Peacock

and Tritel13) stated that the predictive deconvolu-

tion technique is versatile in that the choice of α

can control resolution. Above result, however,

emphasizes the importance of choosing α so that

output of deconvolution does not contain unde-sirable oscillations. Fig. 3 illustrates this result,showing the output of predictive deconvolution

applied to a minimum phase wavelet with various

prediction distances.So far, we have assumed that the wavelet is

minimum phase and that F1(Z) can be modelled by

W(Z). In fact, seismic data often contain non-minimum phase wavelets such as the Vibroseiswavelet. From Eq. (12), the output of predictivedeconvolution with prediction distance α is

Oα(Z)=F1-1(Z)|αF1(Z)W(Z)Q(Z)

where K(Z) is the Vibroseis wavelet. Since K(Z) isnot minimum phase, F1(Z) is no longer simply itsinverse. In this case, F1(Z) is the inverse of acorresponding minimum phase wavelet Km(Z).Therefore, we have

F1-1(Z)|α=Km(Z)|α

Consequently, Oα(Z) becomes

Fig. 3 Output of Predictive Deconvolution Showing

Minimum Phase Input Followed by Outputs

for Prediction Distance α=3, 6, 9, and 12

Points


444

Oα(Z)=Km(Z)|αF1(Z)K(Z)Q(Z)

From the facts discussed here, we can conclude

that predictive deconvolution for Vibroseis data is

equivalent to applying whitening deconvolution

and subsequently convolving with the correspond-ing minimum phase wavelet truncated at α term

Km(Z)|α. Therefore, in case of long prediction

distance, α is large, predictive deconvolution and

phase correction method12) discussed in the

previous chapter, are mathematically identical.This enables us to explain empirical observation

to the effect that predictive deconvolution can

eliminate multiples in Vibroseis data.15)

6. Bispectrum Deconvolution

In order to obviate the minimum phaseassumption for seismic wavelet, a statisticalmethod to reconstruct phase spectrum of {ws} from

{xt} is considered in this section.Investigating actual well-log data, Hosken16)

has found that probability distribution of refec-tion coefficients is non-Gaussian. This studymotivated Matsuoka and Ulrych4) to developalgorithms for estimation of phase spectrum of theseismic wavelet. Because of the non-Gaussiancharacter of reflectivity series, the seismic trace hashigher order spectra which contain phase informa-tion of the wavelet.

The third order spectrum, or the bispectrum, isdefined by double Fourier transform of the thirdorder cumulant

B(ω1,ω2)=∬∞-∞C(τ1,τ2)

exp(-i(ω1τ1+ω2τ2))dτ1dτ2

where

c(τ1,τ2)=E[xtxt+τ1xt+τ2]

Since reflectivity series is uncorrelated, the

bispectrum of seismic trace takes a simple from,

(14)

where

W(ω)=∑sωsexp (-iωs)

and

γ3q=E[q3t]

Let us write

W(ω)=H(ω)exp(iφ(ω))

and

Bw(ω1,ω2)=G(ω1,ω2)exp(iψ(ω1,ω2))

where Bw(ω1, ω2)is bispectrum of the wavelet

having following relationship

B(ω1,ω2)=γ3qBw(ω1, ω2) (15)

Using Eq. (14) and (15), we obtain

ψ(ω1,ω2)=φ(ω1)+φ(ω2)-φ(ω1+ω2)

(16)

Above equation is the governing equation to

determine phase spectrum of the wavelet, φ(ω),

from estimated phase of the bispectrum, ψ(ω1,ω2).

Methods have been suggested to solve Eq.(12)

by Brillinger17) (BR), Lii and Rosenblatt18) (LR),

and Matsuoka and Ulrych4)(MU). The BR and LR

algorithms reconstruct φ(ω) from ψ (ω1,ω2)

recursively whereas the Matsuoka and Ulrych

(MU) method does so in a non-recursive least-squares fashion. Details of the algorithms aresummarized below:

(i) BR AlgorithmBrillinger originally suggested a recursive

equation,

φ(ω)={2∫ω0φ(λ)dλ-∫ω0ψ(λ,ω-λ)dλ}/ω

which was modified by Matsuoka and Ulrych4) fordigital computations resulting in an expression

(17)

where

S(n)=∑ni=0ψ(i,n-i)

and n=0 corresponds to ω=0 and n=N to ω=π.

To obtain φ(n) from the known S(n), two initial

values, namely, φ(0) and φ(1) are required. The

value of φ(0) may be zero or ±π and is determined

from B(0, 0). φ(1) is obtained by the following

equation

where

φ(N)=±kπ, k=0,1,2,… (18)

Since an addition of kπ to φ(N) is equivalent to a

pure time delay, φ(N) can be set to zero. Com-

bining with these initial values, Eq.(17)can be

solved recursively.


445

(ii) LR Algorithm

For digital computation, let ω1=i, ω2=j. Then

Eq.(16) becomes

ψ(i,j)=φ(i)+φ(j)-φ(i+j)

Letting j=1 and summing the above equation

with respect to i from 0 to n-1, we obtain

∑n-1i=0ψ(i,1)=φ(0)+nφ(1)-φ(n)

or

φ(n)=-∑n-1i=0ψ(i,1)+φ(0)+nφ(1)

Hence, with φ(0)(=0) and φ(1) which is obtained

from condition of φ(N)=0, as in the BR algorithm,

we can determine fi(n). Note that the LR

algorithm is essentially limited to use of the

bispectrum values along the line ψ(i,1) or ψ(1,j),

i,j=0,1,…,N. This is in contrast to the BR and MU

algorithms which determine the phase spectrum

using all bispectrum values.

(iii) MU Algorithm

Beginning with Eq.(16) once again we write

ψ(i,j)=φ(i)+φ(j)-φ(i+j)

We form all the possible equations excepting ψ(i,

0) i=0,1,…,N, since this term is always equal to

φ(0), which is assumed to be known. Some

examples of these equations are

ψ(1,1) =2φ(1)-φ(2)

ψ(1,2)=φ(1)+φ(2)-φ(3)

ψ(1,N-1)=φ(1)+φ(N-1)-φ(N)

ψ(2,2) =2φ(2)+φ(4)

ψ(2,3) =φ(2)+φ(3)-φ(5)

ψ(N/2,N/2)=2φ(N/2)-φ(N) ifNeven

Considering Eq.(18), φ(N)can be assumed to be

zero. The following matrix equation, therefore, is

obtained

Aφ=Ψ

where

Φ=(φ(1),φ(2),…,φ(N-1))T

Ψ=(ψ(1,1),ψ(1,2),…,ψ(1,N-1),ψ(2,2),…ψ(N/2,N/2))T

and A is a sparse coefficients matrix

A=

[2 -1 0 0 0 … 0

1 1 1 0 0 … 0

1 0 1 -1 0 … 0

1 0 0 0 0 … 1

0 2 0 -1 0 … 0]

The size of matrix A depends on whether N is even

or odd, and is

(N/2)×(N-1)for N even

{(N-1)/(N+1)}/4×(N-1) for N odd

The unknown phase vector Φ is determined using

the least-squares solution

Φ=(ATA)-1ATΨ

Let us emphasize that the MU algorithm is a

nonrecursive method which utilizes all bispectrum

values available.

Matsuoka and Ulrych4) compared these three

algorithms using two synthetic examples, a simplefull-band wavelet and a band-limited wavelet.

They concluded that BR and LR algorithms work

only for the full-band wavelet. On the other hand,

MU algorithm can estimate phase spectrum, not

only for full-band wavelet but also for band-

limited wavelets, owing to the application of least

squares. This approach and its applications such

as telecomunication systems19) have been studied

by many authors.20),21)

7. Discussions

Deconvolution is an universal step in the

processing of seismic data. Whitening deconvolu-tion introduced by Robinson has had a profound

impact both on resolution and on signal-to-noise

ratio of seismic sections, and, extension of this

approach to predictive deconvolution has been of

great importance in producing multiple freeseismic data.

The success of the predictive deconvolution

technique has motivated its applications, even in

case the seismic wavelet is known not to be

minimum phase. An instance is Vibroseis

processing which requires a phase correction forthe non-minimum phase contribution. In this

respect, recent work by Matsuoka and Ulrych2,3)

has considerably extended the application of pre-

dictive deconvolution to Vibroseis data. These

authors22) have investigated the use of unusually

large values of ridge regression in the solution ofnormal equations and have obviated the need of

phase correction. Extension of the technique to


446

band limited ridge regression shows even greater

promise.The restrictions of minimum phase assumption

has motivated many researchers to investigate

deconvolution techniques which obviate this

assumption. One such approach is by means ofthe bispectrum. This technique shows promise

and much future work needs to be done, such asthe investigation of stable methods for computa-

tion of the bispectrum for short data sets and

consideration on effects of the non-whiteness of

the reflectivity series.

References

1) Robinson, E. A., Ph. D. Dissertation, MIT, (1954); also inGeophysics, 32, 418 (1967).

2) Matsuoka, T., Ulrych, T. J., 55th Annual InternationalSEG Meeting, 558 (1985).

3) Ulrych, T. J., Matsuoka, T., Butsuri-Tansa, 40, 274 (1987).4) Matsuoka, T., Ulrych, T. J., Proc. IEEE, 72, 1403 (1984).5) Ricker, N., Geophysics, 18, 10 (1953).6) Futterman, W. L., J. Geophys. Res., 67, 5279 (1962).7) Knopoff, L., Rev. Geophys., 2, 625 (1964).

8) Wuenschel, P. C., Geophysics, 30, 539 (1965).9) Stone, D., Proc. IEEE, 72, 1394 (1984).

10) Ashida, Y., Toba T., Butsuri-Tanko, 31, 46 (1978).11) Schwarzacher, V., "Sedimentation Models and Quantita-

tive Stratigraphy", Elsevier Scientific Publishing Co.,

(1975).12) Ristow, D., Jurczyk, D., Geophys. Prosp., 23, 363 (1975).13) Peacock, K. I., Treitel, S., Geophysics, 34, 155 (1969).14) Silvia, M. A., Robinson, E. A., "Deconvolution of

Geophysical Time Series in the Exploration for Oil andGas", Elsevier Scientific Publishing Co., (1979).

15) Pollet, A., Lowrie, L., Matthews, J. 52nd AnnualInternational SEG Meeting, (1982).

16) Hosken, J. W. J., 50th Annual International SEGMeeting, (1980).

17) Brilliger, D. R., Biometrika, 64, 509 (1977).18) Lii, K. S., Rosenblatt, M., Ann. Statist., 10, 1195 (1982).19) Bellini, S., Rocca, F., "Digital Communications",

Biglieri, E., Prati, G. Ed., 251 (1986), Elsevier SciencePublishers.

20) Nikias, C. L., Raghuveer, M. R., Proc. IEEE, 75 (1987), in

press.21) Pan, R., Nikias, C. L., IEEE Tr. ASSP, ASSP-35, 895

(1986).22) Matsuoka, T., Ulrych, T. J., 56th Annual International

SEG Meeting, 509 (1986).


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要旨

地震探査データ処理におけるデコンボリューションフィルターの理論的考察

松岡俊文

石油資源開発(株)物理探鉱部, 107東京都港区赤坂2-17-22

石油探査に用いられるいろいろな物理探査法のうち, 反射地

震探査法は堆積層の地質について高い精度の情報が得られるた

めに現在最も広く用いられている。地表近くで人工的に作られ

た地震波は, 地下の地層境界面で反射し, その一部は再び地表

までもどって来る。地表の受振器によって観測されるのは, こ

れらの反射して来た波である。

おのおのの境界面における波の反射の挙動を表す反射係数

は, 各地層の持つ物理的な性質によって一意に決まる。これら

の境界面は深度方向に対して密に存在しているとみなせる。こ

のためこれらの反射係数は全体として一つの係数列をなす。こ

れを反射時系列と呼ぶ。この反射時系列を求める事によって,

地下の境界面を同定する事が可能となる。ところが地表で観測

されたデータは以下に示す様に地下の反射係数列そのものでは

ない。

人工的に作り出された地震波は, 一般的には孤立したパルス

状の波ではない。さらに地震波が伝播するにつれて, 地層の持

つ非弾性的な性質によって波形は変形する。このためわれわれ

が持っているデータは, システム理論によれば反射時系列とこ

の様な基本波形とのコンボリューションと見なされる。デコン

ボリューションフィルターの目的は, 観測された時系列から地

震波の波形の影響を取り除き, 地下の反射係数列を推定するこ

とである。ここで問題となるのは, この基本波形を単独で直接

観測する事は, 常に可能とは限らない。このため取得されたデー

タは一つの時系列にもかかわらず, 二つの未知量, 即ち基本波

形と反射係数列を含むことになってしまう。そのためロビンソ

ンは幾つかの物理的な仮定をもうけてデコンボリューションの

問題を初めて解いた。

ロビンソンによって開発されたデコンボリューションの手法

は, その後多くの研究者たちによって改良が加えられて来た。

本論文においては, ロビンソンによって与えられた基本波形が

最小位相特性を有するという仮定に対して, 詳細な理論的考察

を加える。そのためにまずホワイトニングデコンボリューショ

ンに対する理論的枠組みを明らかにする。この議論を踏えて,

予測型デコンボリューションは最小位相でない基本波形に対し

ても適応可能であることを示す。さらに反射係数列の統計的性

質に着目した基本波形の位相特性の推定法を示す。

Keywords

Deconvolution, Minimum phase, Reflection seismic method, Seismic data processing


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Deconvolution Filter in Seismic Data Processing