A Fast Optimal De convolution Algorithm
I EEE
T RAN SACT I ON S
ON
GEOS CIEN CE AN D
R EM O TE S EN S I N G
OCT OBER 1 9 9 1
VOLUME GE-
19
NUMBER 4
(ISSN 0196-2692)
A PUBLICATION OF THE IEEE GEOSCIENCE AND REMOTE SENSING SOCIETY
PAPERS
Regression Techniques for Oceanographic Parameter Retrieval Using Space-Borne Microwave Radiometry . . . . . .
......................................................................
R. Hofer and E. G. Njoku 178
Freshwater Ice Thickness Observations Using Passive Microwave Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...............................................
D.K.Hall,J.L.Fosfer,A.T.C.Chang,andA.Rango
189
Direct Determination of the Two-Dimensional Image Spectrum from Raw Synthetic Aperture Radar Data . . . . . . .
..................................................................................
P . J. Marlin 1 9 4
HF Radio Wave Transmission Over Sea Ice and Remote Sensing Possibilities . . . . . . . . . . D. A. Hi// and J. R. Wait 2 0 4
HF Ground Wave Propagation Over Mixed Land, Sea, and Sea-Ice Paths . . . . . . . . . . . . . D. A. Hi// and J. R. Wait 210
A Fast Optimal Deconvolution
Algorithm for Real Seismic Data Using Kalman Predictor Model . . . . . . . . . . . . . . .
.................................................
A. K. Mahalonabis, S. Prasad, and K. P. Mohandas 2 1 6
Application of Prediction Error Filters for the Detection of Weak Teleseismic Events . . . . . . . . . . . . . W . Stammler 2 2 2
Towards a General Theory of Induced Electrical Polarization in Geophysical Exploration . . . . . . . . . . . . J. 4. Waif 231
Quasi-Static Magnetic-Field Technique for Determining Position and Orientation . . . . . . . . . . . . . . . . . . F. H. Raab 2 3 5
Miniature Interferometer Terminals for Earth Surveying: Ambiguity and Multipath with Global Positioning System
...........................................................
C. C. Counselman and S. A. Courtwitch
244
EDlTORlAL BOARD FOR ,980
...................................................................
1981 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..Followspage
253
253
IEEE
TRANSACTlONS
ON GEOSCIENCE
A. Kovacs and R. M. Morey, “Anisotropic properties of sea ice in
the 50-150 MHz range,” J. Ceophys. Res., vol. 84, pp. 57495759, ,979.
17. L. Wentworth and M. Cohn, “Electrical properties of sea ice at
0.1 to 30 MC/S,” 3, Res. NBS, vol. 68D, pp. 681-691, 1964.
A. W. Biggs, “Geophysical exploration in polar areas with very
low frequency phase variations.“lEEE mms. Anrenn~s~oto~ulgor.,
vol. AP-16, pp. 364-365, May ,968.
J. R. Wait and D. A. Hill, “Excitation of the HF surface wave by
vertical and horizontal antennas,” Radio Sci,, vol. 14, pp. 767780, 1979.
I. R. Wait, “Theory of electromagnetic surface waves over geological conductors,” Pure Appl.
Ceophys.,
vol. 28, pp. 47-56,
1954.
D. A. Rothrock and A. S. Thorndike, “Geometric properties of
AND REMOTE
SENSING,
VOL.
GE-19. NO. 4, OCTOBER
,981
the underside of sea ice,” J. Geophys.
Ret, vol. 85, no. CT, pp.
3955-3963, July 20.1980.
I221 E. L. Feinberg. “Propagation of radio waves along the surface of
the earth,” USSR Academy of Sciences, Moscow, USSR, 1961.
[23] K. Furutsu, “Propagation of electromagnetic waves over spherical
earth across boundaries,” J. Ret N.B.S., Sec. D, vo,. 67, no. 1,
PP. 39-62, Jan. 1963.
I
David A. Hill (M’72SM’76),
page 209 of this issue.
for a photograph and biography please see
I)
James R. Wait (SM’56-F’62),
see page 209 of this issue.
for a photograph and biography please
A Fast Optimal Deconvolution Algorithm for Real
Seismic Data Using Kalman Predictor Model
A. K. MAHALANABIS,
SENIOR MEMBER, IEEE,
SURENDRA PRASAD,
Abstract-The paper is concerned with an application of P recently
proposed algorithm for minimal r&ation of stochastic dynamic systerns to tie problem of demnvolution
of reflection seismograms.
Results of real data processing are preSented in order to establish the
advantages of this algorithm in terms of computational lequixements
md accuracy of sigwl estimation vis&is the widely accepted algorithm of Robinson and Treitel.
1. lNTRODUCTlON
HE PROBLEMS of estimating the locations of the reflectmg layers and strengths of the reflection coefficients by
processing real reflection seismograms have drawn wide attention in the literature. One of the most successful approaches
developed by Robinson and Treitel (11, [2] utllizes the fol-
T.
Manuscript
received November 11,198O;revised
April 2, 1981.
A. K. Mahalanabis and S. Prasad are wit,, the Department of Electrical Engineerinp, lndian Institute of Technology, New Delhi 110 016,
India.
K. P. Mobandas is with the Department of Electrical Engineering,
Indian htit”te of Technology, New Delhi 110 016, India, on leave
from Cabcut Regional Engineering College, C&cut, Keralx 673 601,
India, under the Quality Improvement Program of the Government of
India.
MEMBER, IEEE, AND
K. P. MOHANDAS
lowing model for the seismic reflections y,:
Yk =zk +nk
(1)
where nL is the noise component of the measurements andzk
is the true reflection signal. A simple model assumed forrk by
the above named workers is
zlr =I$ aibr-i
(2)
where ni represents the reflection coefficient of the jth layer
and b, is the weighting sequence which combines the characteristics of the seismic source utilised in the exploration with
those of the reflecting layers.
tt has been shown [l] that an effective estimate of the
reflection coefficient can be obtained from the prediction
error of zk. The algorithm proposed in [l] and widely
adopted in the real-life situations, is based on the following
autoregressive form of the d-step predictor (d = 1,2, .):
0196.2892/81/0010-0216$00.75
(3)
0 1981 IEEE
Relation (3) implies that an estimate ofz, can be obtained by
using N previous values of the measurements. The coefficients
a, . a,, , a,,, needed to build the predictor are obtained
fr&;he nor& equations [I]
7r r d
1
order than the AR predictor necessary to produce a comparable deconvolution accuracy. Numerical results based on real
data collected from the Institute of Petroleum Exploration,
Dehradun, India, are presented in order to illustrate the effectiveness of the new algorithm.
II. PROBLEM FORMULATION
(4)
where r, = 1/L Cf=, ~,+~y~ is the autocorrelation coefficients
of lag I, (I = 0, 1, .) and N is the order of the predictor.
More recently, Mendel [3], [4] has reformulated the seismic
deconvolution problem using a state variable model for yk.
He has shown that the reflection coefficients can be taken as
the input sequence in the following Gauss-Markov model for
the vectorxk:
and that the measurements yk are related to the vector xk
through the model
yk = hTx, + nk.
(6)
The dimension of the vector xL, the matrix F, and the vectors
g and h are assumed to be known parameters of the model and
the problem of estimation of the reflection coefficients is
treated as that of estimating a white Gaussian input sequence.
The solution proposed by Mendel makes use of the fact that
though the faltered estimates of nL are identically null, the
smoothed estimates of this sequence are finite. He has also
shown that the smoothed estimates of (I~ can be obtained in
terms of the tiltered and smoothed estimates ofxk.
The algorithm of Robinson and Treitel, unfortunately,
requires prior assumptions regarding both the model structure
and the number of t?mx N in the predictor (3). The published literature indicates the need to use as many as 100 to
‘ZOO terms in this series for effective real seismic data processing. The inclusion of such a large number of terms, in turn,
increases the computational complexity. Additionally experience with time series modelling and prediction [I I], [12]
shows that better results are possible with a mixed autoregressive moving average (ARMA) type predictors. The state variable model of Mendel being equivalent to an ARMA model is
thus advantageous. Unfortunately, Mend& development
requires II p rio ri knowledge of not only the model structure
but also of the model parameters. This may make it impossible to process real data through this algorithm since the measurement of the input wavelet is not always feasible.
The aim of the present study has been to explore the possibility of using a recently proposed stochastic system identification algorithm for deconvolution of real seismic data. This
algorithm has the advantage of yielding the minimal order
Kabnan predictor model of the concerned process directly
through processing of the output autocorrelation functions of
fmite lags. It is shown that this new algorithm is also capable
of reducing the computational burden significantly since the
optimum Kalman predictor turns out to be of much lower
Consider the following Gauss-Markov model for representing
the seismic reflections:
x~+~ =Fxk +uk
(7)
yk = hTx, + nk
(6)
where uk represents a zero-mean white Gaussian vector COTspending to the productgak in (5). It is assumed that the pair
(F, hT) is completely observable. The recorded values of the
reflection sequence yk are to be processed first for estimating
the unknown parameters of the model (6), (7), before the
white noise estimation can be performed. It is well known
that [S] the number of unknown parameters in the assumed
mode (S), (6). which correspond to the elements of the matrix
F the vector g and h, the covariance ofuk and the variance of
nk, are too many to be successfully identified from the records
of Yk, The best that can be done is that of identifying the so
called Kalman predictor model equivalent of the Gauss-Markov
model (6), (7), which corresponds to the following pair of
equations:
X^*+l,k=FjZk,k-l +%
y,=hT,k,k-,
+ek
(8)
(9)
where skklk-, represents the minimum variance predicted estimate of xk based on y,,y, , ,yk-, , ek represents the
innovations sequence, and K represents the asymptotic value
of the Kalman predictor gain.
The parameters of the predictor model are the elements of
the matrix F of the vectors h and K and the variance Q of the
innovations sequence. The total number of unknown parameters is reduced significantly by using F and h in the companion forms
1
TO
(10)
and
hT=[l
O...O].
(11)
The first problem to be discussed in the paper is that of
identifying the predictor model (S), (9) by processing the real
seismic data. This requires the determination of the structural
parameter, viz., the unknown order of the vector+ and then
the estimation of the elements of F, K, and Q. Once the
model has been successfully identified, the next problem to be
discussed is that of estimating the white Gaussian reflection
sequence ilk.
Note that the problem of identification of the Kalman pre-
dictor is essentially an off-line computational problem whereas
the reflection coefficient estimation is essentially an on-line
computational problem. Also this latter problem in essence
requires us to record the innovation sequence eL which being
the error of prediction represents the desired estimate of ak.
The identification algorithm exploited in the present investigation is based on the works of Mehra [6] and Tse and
Weinert 171. The basic relation involved in the development
of this algorithm is the expression for the output autocorrelation functions, defined by
‘r =Eb+c+,Yd
where I is the chosen lag and E{.) denotes the expectation
operation. Corresponding to the predictor model (8), (9), it is
easy to check that r, satisfies the following relations:
r, = h=Ph + Q,
I=0
=,,TF’-ls ,
I>0
(12)
where s = FPh + KQ and P is the asymptotic value of the
covariance Pk defmed by
Pk =EGk,k-LZ,k-,I.
(13)
It is possible to show that P is given as the solution of the
following nonlinear equations [7] :
P = FPFT + KQK=.
(14)
Using (12) for I = 1,2,. , n, it is possible to see that the
following relation holds:
PI1
rhr
i
s.
(15)
In view of the complete observability of (F, hT), it follows
that the n X n matrix which premultiplies the vector s on the
right-hand side of (15) is nonsingular so that an explicit
expression for the vector s can be obtained. The assumed
canonical forms for hT and F further ensures that this matrix
is in fact an identity matrix so that the vectors can be computed easily using the relation
Pl
S’ I2
(1’4
rn
H
If (12) is now written for I = n + 1 and the above expression
for s is utilized along with the Cayley-Hamilton theorem, one
gets the relation [6]
~~+~=-f firi+i-,. j>l.
i=,
(17)
Choosing j = 1,2,. , n in (17), it is then possible to obtain
the following vector matrix relation:
r(n) = -R(n)f
(18)
where
R(n)= y I
.
.
”
“‘y”
tr” r”+l . ..‘zn-.J
It is noted that for a system of order n, the matrix R(p) is
nonsingular for p Q n, but is singular for p > n. This fact can
be utilized successfully to determine the unknown order of
the given system for which output datayk are available. Making “se of the ergodic hypothesis, the output autocorrelation
sequence I, is estimated using the time average
1 =
c Yl+rYf.
I’=L+,
(19)
With L chosen sufficiently large ?, approximates r, sufficiently
closely. Choosing a large L is not difficult in seismic data
processing since a seismic trace sampled at 2 or 4 ms provides
2000-3000 samples.
In order to find the unknown value of the order n of the
Gauss-Markov model needed to tit the observed values of&
it is necessary and sufficient to test the ranks of a set of matrices R(p) of increasing dimensions. If it is found that the
determinants of all matrices are nonzem for p = 1,2, , J
and are zero for p > J, then the order n is selected asf. When
the measured data is noisy, it is observed that the determinants
of these matrices do not vanish, but the correct system order
can still be found unambiguously by looking for a large fall in
the determinant magnitude as discussed in [7].
After having determined the value ofn, the unknown coefticient vector f is estimated from (19) with r, replaced by i;. In
other words one computesf^using
the relation
f= -R(n)-’ i(n).
(20)
The numerical values of the parameters evaluated by (20) are
known to be sensitive to errors in the estimation of the autocorrelation coefficients by (19). But this will not adversely
affect the deconvolution since an effective deconvolution requires only a good waveshape and not necessarily accurate
parameter values.
The other parameters of the model, viz., the predictor gain
K, the innovations variance Q and the state variance P can be
obtained by iterating the following set of equations [7]:
P, = o
Qi=ro - h’P,h,
i=o, 1;”
Ki=(s-FP,h)Q:,
i=o,1;,.
pit1 = FPiFT + K,QiK,r,
i=o,1;.,.
(21)
This set of equations have been shown [7] to converge to
M‘&HAl,ANAB,S et a,.: AI,GOR,THM FOR REAL SElSMlC
DATA
the correct values for P, K, and Q after a limited number of
iterations for a stable F matrix. It may be pointed out that a
similar algorithm has also been developed by Son and Anderson [ 141 independently.
The simple identification algorithm discussed above has been
tested with gain corrected real seismic data supplied by the
Institute of Petroleum Exploration. A total of 500 data points
each from 24 traces of a marine seismic section have been processed. It has been observed that the system order determined
differs from trace to trace, but in most of the cases the system
order for getting good results for deconvolution was less than
five. Table I gives a summary of the results of identification.
It should be mentioned that the predictor gain and the innovations variance did not converge to steady values within a reasonable number of iterations for three traces. This appears to
be due to the poor signal quality of these traces and implies
that the present method is not guaranteed to work in all
situations.
IV. RESULTS
OF
PREDECTIVE
DECONVOLU~~ON
The identified models of Table I have been utilized for deconvolution of the actual seismic section. As mentioned
earlier, the deconvolution philosophy of Robinson and Treitel
[I], [2] extended to the KaIman predictor formulation gives
the innovations ek as the deconvolved output. Of course, the
prediction error falter in state variable form has also been discussed by Ott and Meder [lo] but their formulation is not in
a form suitable for application to real seismic data processing.
It may also be pointed out that Made1 [13] has shown that
this fdter can be very bad at low SNR’s.
The results of processing of real data are shown in Figs. 1
and 2. Since there is no question of knowing the actual values
of the reflection coefficients, in this case, the best that can be
done is to compare the results obtained using the proposed
algorithm with those of the popular algorithm of Robinson and
Treitel. Fig. l(a) shows the 24 traces plotted after applying
gain correction alone and before applying NM0 correction.
Fig. l(b) shows the deconvolved output of the classical prediction error falter of Robinson and Treitel making use of a
predictor of order 100. And Figs. 2(a) and (b) the plots of
the input data and the output of the newly proposed deconvolution algorithm. It is clear that the events of interest have
been made very clear in Fig. 2(b) compared to Figs. I(a) or
@I
Coming, next, to the computational requirements, a comparison of the two algorithms cannot be based on “operations
count.” This is so because the number of iterations required
for identifying the model in the proposed method is partially
dependent on the data quality. An overall comparison can:
however, be made on the actual CPU time on a given system.
Table II summarizes the time required to process 1000 samples
each of 48 traces constituting a seismic section on an IBM
370/145 system. It is clear that the method proposed here is
about thrice as fast as the Robinson and Treitel method.
As a further check on the superiority of the Kalman predictor over the 100 term AR predictor, whiteness tests on the
residuals (i.e., the deconvolved output) have also been performed. Figs, 3(a) and (b) show typical plots of the autocor-
relation functions of the residual for one of the traces as obtained respectively from the AR and the Kahnan predictors.
The KaIman predictor is seen to yield a more nearly white
residual than the 100 term AR predictor.
V. CONCLUD~NC REMARKS
It has been demonstrated that an optimum asymptotic Kalman predictor can be identified using the available output data
of a seismic experiment. This can be used to estimate the
white Gaussian reflection coefficients of the seismogram
mod&d by (5) and (6). The main advantages of the proposed algorithm are its applicability to real data and the comparative computational ease in identifying the best possible
model. These advantages are easily verified looking at the
results of real data processing presented in Tables I and II and
in Figs. 1,2, and 3.
As mentioned in the introductory section, the problem of
estimation of the reflection sequence has been shown by
Fig. 3. (a) Normalized autocorrclation
function of the residual of the
100 point AR predictor. Number of samples outside 1.96&E
limit = 7. (b) Normalized autocorrelation
function of the residual
of the Kalman predictor. Number of samples outside 1.96/G
limit = 4.
MAHALANABlS
et 01.: ALGORlTHM FOR REAL SE,SM,C
DATA
Mende] to be equivalent to the problem of estimation of the
white noise sequence ak in (5). It should be possible to obtain
the smoothed estimates of this sequence if the unknown
parameters of the model (5)-(6) are known 131. The system
identification method discussed in the present paper yields
the optimal estimates of the matrices F and h in the canonical
form. It should be possible to use these estimates along with
an adaptive Kabnan faltering algorithm [9] in order to obtain
the covariances of gok and variance of nk, This would then
permit the results of the present paper to be extended to cover
the white noise estimation problem.
Further, Mendel and Kormylo [4] have shown that the
uncertainty associated with the reflection from oil bearing
structures can be accounted for by using the modelok = yLwL,
where yk is a binary sequence which can take values zero and
one and wk is a Gaussian sequence. It has been shown that a
combined detection estimation algorithm using the maximum
likelihood approach can be utilized for estimating nk in such
cases. The possibility of extending the method of the present
paper to the detection estimation situation is also being
investigated.
221
A. K. Mahalanabis (M’66-SM’78) was born in
Kishoreganj, Bangladesh in 1934 and has been
educated in the University of Calcutta, Calcutta,
India, where he received the D.Phil.(S.C.)
degree in 1961.
“..-.... .._..
.
c.._ I.-...” “^&ye b research h
control theory and its apptica, tions. He is currently a Professor of Electrica 1 Engineering in
the Indian Institute of Tee hnology, Delhi,
India. and he is in charge of .,~
development group in automation His corrent
research interests are mainly in the area of Stochastic Estimation and
Control with Applications in Modetiog and Identification of Physical
Systems from real data.
Dr. Mahalanabis i? a member of the Editorial Board of the Journal
Optimal Confrol Applicofions and Methods (Wiley), and is the Chairman of the International Programme Committee of the IFAC Symposium on Theory and Applications of Digital Control to be held in
New Delhi in January 1982.
.),
*
ACKNOWLEDGMENT
The authors would like to record their appreciation of the
help received from Dr. V. C. Mohan and Dr. N. D. J. Rao of
the Institute of Petroleum Exploration, Debra Dun in the form
of the data used in this work and in the form of valuable discussions and computational facilities. Also the authors wish to
acknowledge the useful comments of the anonymous reviewers
on the original version of this paper.
REFERENCES
[l] E. A. Robinson, “Predictive decomposition of seismic traces,”
Geophysicr,no.4,vol. 22, pp. 767-178, Oct. ,957.
[2] K. L. Peacock and Sven Treitel, “Predictive doconvolution,”
Geophysics,vol. 34, no. 2, pp. 155-167, Apr. 1969.
131 1. ht. Mendel, ?+‘h~te Noise Estimators for seismic data Processing in oil exploration,“IEEE Trans. Auromot. Confr., vol. K-22,
pp. 694-707, Oct. 1971.
[4] John Konnylo and 1. hf. Mendel, “On maximum likelihood estimation of reflection coefficients,” in Rot. 48th Annu. Meef.
SE0 (San Francisco, CA).
[S] E. Tse and 1. J. Anton, “Identifiability of parameters,” IEEE
?hons. Automat. Con*., vol. AC-17, Pp. 637-646, Oct. 1972.
(61 R. K. Mehra, “On line identification of linear dynamic systems,‘*
IEEE Trans. Au10mar. Con,,.,vol. AC-16, Pp. 12-21, Feb. 1971.
[7] E. Tse and H. Weinert, “Structure determination and parameter
identification of multivariable systems,” IEEE Tkns. Automaf.
Confr., vol. AC-20, pp. 603-612,Oct. 1975.
[E] 1. S. Meditch, Stochasfic Optimal Linear Esfimafion and Control. New York, McGraw-Hitl, 1969.
[P] B. D. 0. Anderson and I. B. Moore, Optimal Filfering. Englewood Cliffs, NJ: Prentice-Hall, 1979.
[IO) N. Ott and H. G. Meder, “The Kahnan t?lter as a prediction
error filter,” Geophysical Prospecfing, vol. 20, pp. 549-560.
Ill] C. E. P. Box and G. M. Jenkins, Time Series Analysis, ForecasfSan Francisco, CA: Holden Day, 1976.
iwl?nd Control.
1121 R. L. Kashyap and A. R. Rae, “Dynamic stochastic models from
empirical data,” New York: Academic Press, 1976.
1131 1. M. Mendel, “A quantitative evaluation of Ott and Meder’s prediction error filter,” Geophysical Prospecting, vol. 25, no. 4, pp.
692-698,1917.
[14] L. H. Son and B. D. 0. Anderson, “Design of Katman filters
using signal model output statistics,” plot. Insf. Elec. ,%n., vol.
120, no. 2,pP. 312-318, Feb. 1973.
Surendra Rasad (S’72-M’75) was born in New
Delhi, India, on July 10, 1948. He received
the B.Tech. degree in electronics and electrical
communication engineering from the Indian
Institute of Technology (UT.) Kharagpor,
India, in 1969, and the M.Tech. sod Ph.D.
degees io electrical engineering from the
LLT., New Delhi, India, in 1971 and 1974,
respectively.
He has been teaching
New Delhi
. at “eLLT.,
since August 1971, wnere
IS presently an
Assistant Professor. His teaching and research interests are in radar/
sonar signal processing, communications, and computers. He was a
Visiting Research Fellow at the Loughborough University of Technology, U.\., during the period from August 1976 to August 1977 where
he was mvolved in developing algorithms for adaptive array processing
for H.F. arrays. Cunently he is engaged in research in the areas of
sonar and seismic signal processing, underwater communications and
array signal processing.
*
K. P. Mohandas was born in Alleppy, Kerala
State, India, in 1946 and received the
B.Sc.(T.ngg.) degree from Kerala University in
1968 and M.Tech. degree from the Indian
Institute of Technology, Madras, in 1973, both
hg in T.K.M. College of Engineering, &loo
from 1968 to 1969 and since 1969 he is witi
the Calicut Regional Engineering College,
Calicut, Kerala State, India. He is currently
completing the requirements for the Ph.D. degree in electrical engineering at the Indian Institute of Technology, Delhi, under the Qutity
Improvement Programme of Government of India.
T RAN SACT I ON S
ON
GEOS CIEN CE AN D
R EM O TE S EN S I N G
OCT OBER 1 9 9 1
VOLUME GE-
19
NUMBER 4
(ISSN 0196-2692)
A PUBLICATION OF THE IEEE GEOSCIENCE AND REMOTE SENSING SOCIETY
PAPERS
Regression Techniques for Oceanographic Parameter Retrieval Using Space-Borne Microwave Radiometry . . . . . .
......................................................................
R. Hofer and E. G. Njoku 178
Freshwater Ice Thickness Observations Using Passive Microwave Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...............................................
D.K.Hall,J.L.Fosfer,A.T.C.Chang,andA.Rango
189
Direct Determination of the Two-Dimensional Image Spectrum from Raw Synthetic Aperture Radar Data . . . . . . .
..................................................................................
P . J. Marlin 1 9 4
HF Radio Wave Transmission Over Sea Ice and Remote Sensing Possibilities . . . . . . . . . . D. A. Hi// and J. R. Wait 2 0 4
HF Ground Wave Propagation Over Mixed Land, Sea, and Sea-Ice Paths . . . . . . . . . . . . . D. A. Hi// and J. R. Wait 210
A Fast Optimal Deconvolution
Algorithm for Real Seismic Data Using Kalman Predictor Model . . . . . . . . . . . . . . .
.................................................
A. K. Mahalonabis, S. Prasad, and K. P. Mohandas 2 1 6
Application of Prediction Error Filters for the Detection of Weak Teleseismic Events . . . . . . . . . . . . . W . Stammler 2 2 2
Towards a General Theory of Induced Electrical Polarization in Geophysical Exploration . . . . . . . . . . . . J. 4. Waif 231
Quasi-Static Magnetic-Field Technique for Determining Position and Orientation . . . . . . . . . . . . . . . . . . F. H. Raab 2 3 5
Miniature Interferometer Terminals for Earth Surveying: Ambiguity and Multipath with Global Positioning System
...........................................................
C. C. Counselman and S. A. Courtwitch
244
EDlTORlAL BOARD FOR ,980
...................................................................
1981 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..Followspage
253
253
IEEE
TRANSACTlONS
ON GEOSCIENCE
A. Kovacs and R. M. Morey, “Anisotropic properties of sea ice in
the 50-150 MHz range,” J. Ceophys. Res., vol. 84, pp. 57495759, ,979.
17. L. Wentworth and M. Cohn, “Electrical properties of sea ice at
0.1 to 30 MC/S,” 3, Res. NBS, vol. 68D, pp. 681-691, 1964.
A. W. Biggs, “Geophysical exploration in polar areas with very
low frequency phase variations.“lEEE mms. Anrenn~s~oto~ulgor.,
vol. AP-16, pp. 364-365, May ,968.
J. R. Wait and D. A. Hill, “Excitation of the HF surface wave by
vertical and horizontal antennas,” Radio Sci,, vol. 14, pp. 767780, 1979.
I. R. Wait, “Theory of electromagnetic surface waves over geological conductors,” Pure Appl.
Ceophys.,
vol. 28, pp. 47-56,
1954.
D. A. Rothrock and A. S. Thorndike, “Geometric properties of
AND REMOTE
SENSING,
VOL.
GE-19. NO. 4, OCTOBER
,981
the underside of sea ice,” J. Geophys.
Ret, vol. 85, no. CT, pp.
3955-3963, July 20.1980.
I221 E. L. Feinberg. “Propagation of radio waves along the surface of
the earth,” USSR Academy of Sciences, Moscow, USSR, 1961.
[23] K. Furutsu, “Propagation of electromagnetic waves over spherical
earth across boundaries,” J. Ret N.B.S., Sec. D, vo,. 67, no. 1,
PP. 39-62, Jan. 1963.
I
David A. Hill (M’72SM’76),
page 209 of this issue.
for a photograph and biography please see
I)
James R. Wait (SM’56-F’62),
see page 209 of this issue.
for a photograph and biography please
A Fast Optimal Deconvolution Algorithm for Real
Seismic Data Using Kalman Predictor Model
A. K. MAHALANABIS,
SENIOR MEMBER, IEEE,
SURENDRA PRASAD,
Abstract-The paper is concerned with an application of P recently
proposed algorithm for minimal r&ation of stochastic dynamic systerns to tie problem of demnvolution
of reflection seismograms.
Results of real data processing are preSented in order to establish the
advantages of this algorithm in terms of computational lequixements
md accuracy of sigwl estimation vis&is the widely accepted algorithm of Robinson and Treitel.
1. lNTRODUCTlON
HE PROBLEMS of estimating the locations of the reflectmg layers and strengths of the reflection coefficients by
processing real reflection seismograms have drawn wide attention in the literature. One of the most successful approaches
developed by Robinson and Treitel (11, [2] utllizes the fol-
T.
Manuscript
received November 11,198O;revised
April 2, 1981.
A. K. Mahalanabis and S. Prasad are wit,, the Department of Electrical Engineerinp, lndian Institute of Technology, New Delhi 110 016,
India.
K. P. Mobandas is with the Department of Electrical Engineering,
Indian htit”te of Technology, New Delhi 110 016, India, on leave
from Cabcut Regional Engineering College, C&cut, Keralx 673 601,
India, under the Quality Improvement Program of the Government of
India.
MEMBER, IEEE, AND
K. P. MOHANDAS
lowing model for the seismic reflections y,:
Yk =zk +nk
(1)
where nL is the noise component of the measurements andzk
is the true reflection signal. A simple model assumed forrk by
the above named workers is
zlr =I$ aibr-i
(2)
where ni represents the reflection coefficient of the jth layer
and b, is the weighting sequence which combines the characteristics of the seismic source utilised in the exploration with
those of the reflecting layers.
tt has been shown [l] that an effective estimate of the
reflection coefficient can be obtained from the prediction
error of zk. The algorithm proposed in [l] and widely
adopted in the real-life situations, is based on the following
autoregressive form of the d-step predictor (d = 1,2, .):
0196.2892/81/0010-0216$00.75
(3)
0 1981 IEEE
Relation (3) implies that an estimate ofz, can be obtained by
using N previous values of the measurements. The coefficients
a, . a,, , a,,, needed to build the predictor are obtained
fr&;he nor& equations [I]
7r r d
1
order than the AR predictor necessary to produce a comparable deconvolution accuracy. Numerical results based on real
data collected from the Institute of Petroleum Exploration,
Dehradun, India, are presented in order to illustrate the effectiveness of the new algorithm.
II. PROBLEM FORMULATION
(4)
where r, = 1/L Cf=, ~,+~y~ is the autocorrelation coefficients
of lag I, (I = 0, 1, .) and N is the order of the predictor.
More recently, Mendel [3], [4] has reformulated the seismic
deconvolution problem using a state variable model for yk.
He has shown that the reflection coefficients can be taken as
the input sequence in the following Gauss-Markov model for
the vectorxk:
and that the measurements yk are related to the vector xk
through the model
yk = hTx, + nk.
(6)
The dimension of the vector xL, the matrix F, and the vectors
g and h are assumed to be known parameters of the model and
the problem of estimation of the reflection coefficients is
treated as that of estimating a white Gaussian input sequence.
The solution proposed by Mendel makes use of the fact that
though the faltered estimates of nL are identically null, the
smoothed estimates of this sequence are finite. He has also
shown that the smoothed estimates of (I~ can be obtained in
terms of the tiltered and smoothed estimates ofxk.
The algorithm of Robinson and Treitel, unfortunately,
requires prior assumptions regarding both the model structure
and the number of t?mx N in the predictor (3). The published literature indicates the need to use as many as 100 to
‘ZOO terms in this series for effective real seismic data processing. The inclusion of such a large number of terms, in turn,
increases the computational complexity. Additionally experience with time series modelling and prediction [I I], [12]
shows that better results are possible with a mixed autoregressive moving average (ARMA) type predictors. The state variable model of Mendel being equivalent to an ARMA model is
thus advantageous. Unfortunately, Mend& development
requires II p rio ri knowledge of not only the model structure
but also of the model parameters. This may make it impossible to process real data through this algorithm since the measurement of the input wavelet is not always feasible.
The aim of the present study has been to explore the possibility of using a recently proposed stochastic system identification algorithm for deconvolution of real seismic data. This
algorithm has the advantage of yielding the minimal order
Kabnan predictor model of the concerned process directly
through processing of the output autocorrelation functions of
fmite lags. It is shown that this new algorithm is also capable
of reducing the computational burden significantly since the
optimum Kalman predictor turns out to be of much lower
Consider the following Gauss-Markov model for representing
the seismic reflections:
x~+~ =Fxk +uk
(7)
yk = hTx, + nk
(6)
where uk represents a zero-mean white Gaussian vector COTspending to the productgak in (5). It is assumed that the pair
(F, hT) is completely observable. The recorded values of the
reflection sequence yk are to be processed first for estimating
the unknown parameters of the model (6), (7), before the
white noise estimation can be performed. It is well known
that [S] the number of unknown parameters in the assumed
mode (S), (6). which correspond to the elements of the matrix
F the vector g and h, the covariance ofuk and the variance of
nk, are too many to be successfully identified from the records
of Yk, The best that can be done is that of identifying the so
called Kalman predictor model equivalent of the Gauss-Markov
model (6), (7), which corresponds to the following pair of
equations:
X^*+l,k=FjZk,k-l +%
y,=hT,k,k-,
+ek
(8)
(9)
where skklk-, represents the minimum variance predicted estimate of xk based on y,,y, , ,yk-, , ek represents the
innovations sequence, and K represents the asymptotic value
of the Kalman predictor gain.
The parameters of the predictor model are the elements of
the matrix F of the vectors h and K and the variance Q of the
innovations sequence. The total number of unknown parameters is reduced significantly by using F and h in the companion forms
1
TO
(10)
and
hT=[l
O...O].
(11)
The first problem to be discussed in the paper is that of
identifying the predictor model (S), (9) by processing the real
seismic data. This requires the determination of the structural
parameter, viz., the unknown order of the vector+ and then
the estimation of the elements of F, K, and Q. Once the
model has been successfully identified, the next problem to be
discussed is that of estimating the white Gaussian reflection
sequence ilk.
Note that the problem of identification of the Kalman pre-
dictor is essentially an off-line computational problem whereas
the reflection coefficient estimation is essentially an on-line
computational problem. Also this latter problem in essence
requires us to record the innovation sequence eL which being
the error of prediction represents the desired estimate of ak.
The identification algorithm exploited in the present investigation is based on the works of Mehra [6] and Tse and
Weinert 171. The basic relation involved in the development
of this algorithm is the expression for the output autocorrelation functions, defined by
‘r =Eb+c+,Yd
where I is the chosen lag and E{.) denotes the expectation
operation. Corresponding to the predictor model (8), (9), it is
easy to check that r, satisfies the following relations:
r, = h=Ph + Q,
I=0
=,,TF’-ls ,
I>0
(12)
where s = FPh + KQ and P is the asymptotic value of the
covariance Pk defmed by
Pk =EGk,k-LZ,k-,I.
(13)
It is possible to show that P is given as the solution of the
following nonlinear equations [7] :
P = FPFT + KQK=.
(14)
Using (12) for I = 1,2,. , n, it is possible to see that the
following relation holds:
PI1
rhr
i
s.
(15)
In view of the complete observability of (F, hT), it follows
that the n X n matrix which premultiplies the vector s on the
right-hand side of (15) is nonsingular so that an explicit
expression for the vector s can be obtained. The assumed
canonical forms for hT and F further ensures that this matrix
is in fact an identity matrix so that the vectors can be computed easily using the relation
Pl
S’ I2
(1’4
rn
H
If (12) is now written for I = n + 1 and the above expression
for s is utilized along with the Cayley-Hamilton theorem, one
gets the relation [6]
~~+~=-f firi+i-,. j>l.
i=,
(17)
Choosing j = 1,2,. , n in (17), it is then possible to obtain
the following vector matrix relation:
r(n) = -R(n)f
(18)
where
R(n)= y I
.
.
”
“‘y”
tr” r”+l . ..‘zn-.J
It is noted that for a system of order n, the matrix R(p) is
nonsingular for p Q n, but is singular for p > n. This fact can
be utilized successfully to determine the unknown order of
the given system for which output datayk are available. Making “se of the ergodic hypothesis, the output autocorrelation
sequence I, is estimated using the time average
1 =
c Yl+rYf.
I’=L+,
(19)
With L chosen sufficiently large ?, approximates r, sufficiently
closely. Choosing a large L is not difficult in seismic data
processing since a seismic trace sampled at 2 or 4 ms provides
2000-3000 samples.
In order to find the unknown value of the order n of the
Gauss-Markov model needed to tit the observed values of&
it is necessary and sufficient to test the ranks of a set of matrices R(p) of increasing dimensions. If it is found that the
determinants of all matrices are nonzem for p = 1,2, , J
and are zero for p > J, then the order n is selected asf. When
the measured data is noisy, it is observed that the determinants
of these matrices do not vanish, but the correct system order
can still be found unambiguously by looking for a large fall in
the determinant magnitude as discussed in [7].
After having determined the value ofn, the unknown coefticient vector f is estimated from (19) with r, replaced by i;. In
other words one computesf^using
the relation
f= -R(n)-’ i(n).
(20)
The numerical values of the parameters evaluated by (20) are
known to be sensitive to errors in the estimation of the autocorrelation coefficients by (19). But this will not adversely
affect the deconvolution since an effective deconvolution requires only a good waveshape and not necessarily accurate
parameter values.
The other parameters of the model, viz., the predictor gain
K, the innovations variance Q and the state variance P can be
obtained by iterating the following set of equations [7]:
P, = o
Qi=ro - h’P,h,
i=o, 1;”
Ki=(s-FP,h)Q:,
i=o,1;,.
pit1 = FPiFT + K,QiK,r,
i=o,1;.,.
(21)
This set of equations have been shown [7] to converge to
M‘&HAl,ANAB,S et a,.: AI,GOR,THM FOR REAL SElSMlC
DATA
the correct values for P, K, and Q after a limited number of
iterations for a stable F matrix. It may be pointed out that a
similar algorithm has also been developed by Son and Anderson [ 141 independently.
The simple identification algorithm discussed above has been
tested with gain corrected real seismic data supplied by the
Institute of Petroleum Exploration. A total of 500 data points
each from 24 traces of a marine seismic section have been processed. It has been observed that the system order determined
differs from trace to trace, but in most of the cases the system
order for getting good results for deconvolution was less than
five. Table I gives a summary of the results of identification.
It should be mentioned that the predictor gain and the innovations variance did not converge to steady values within a reasonable number of iterations for three traces. This appears to
be due to the poor signal quality of these traces and implies
that the present method is not guaranteed to work in all
situations.
IV. RESULTS
OF
PREDECTIVE
DECONVOLU~~ON
The identified models of Table I have been utilized for deconvolution of the actual seismic section. As mentioned
earlier, the deconvolution philosophy of Robinson and Treitel
[I], [2] extended to the KaIman predictor formulation gives
the innovations ek as the deconvolved output. Of course, the
prediction error falter in state variable form has also been discussed by Ott and Meder [lo] but their formulation is not in
a form suitable for application to real seismic data processing.
It may also be pointed out that Made1 [13] has shown that
this fdter can be very bad at low SNR’s.
The results of processing of real data are shown in Figs. 1
and 2. Since there is no question of knowing the actual values
of the reflection coefficients, in this case, the best that can be
done is to compare the results obtained using the proposed
algorithm with those of the popular algorithm of Robinson and
Treitel. Fig. l(a) shows the 24 traces plotted after applying
gain correction alone and before applying NM0 correction.
Fig. l(b) shows the deconvolved output of the classical prediction error falter of Robinson and Treitel making use of a
predictor of order 100. And Figs. 2(a) and (b) the plots of
the input data and the output of the newly proposed deconvolution algorithm. It is clear that the events of interest have
been made very clear in Fig. 2(b) compared to Figs. I(a) or
@I
Coming, next, to the computational requirements, a comparison of the two algorithms cannot be based on “operations
count.” This is so because the number of iterations required
for identifying the model in the proposed method is partially
dependent on the data quality. An overall comparison can:
however, be made on the actual CPU time on a given system.
Table II summarizes the time required to process 1000 samples
each of 48 traces constituting a seismic section on an IBM
370/145 system. It is clear that the method proposed here is
about thrice as fast as the Robinson and Treitel method.
As a further check on the superiority of the Kalman predictor over the 100 term AR predictor, whiteness tests on the
residuals (i.e., the deconvolved output) have also been performed. Figs, 3(a) and (b) show typical plots of the autocor-
relation functions of the residual for one of the traces as obtained respectively from the AR and the Kahnan predictors.
The KaIman predictor is seen to yield a more nearly white
residual than the 100 term AR predictor.
V. CONCLUD~NC REMARKS
It has been demonstrated that an optimum asymptotic Kalman predictor can be identified using the available output data
of a seismic experiment. This can be used to estimate the
white Gaussian reflection coefficients of the seismogram
mod&d by (5) and (6). The main advantages of the proposed algorithm are its applicability to real data and the comparative computational ease in identifying the best possible
model. These advantages are easily verified looking at the
results of real data processing presented in Tables I and II and
in Figs. 1,2, and 3.
As mentioned in the introductory section, the problem of
estimation of the reflection sequence has been shown by
Fig. 3. (a) Normalized autocorrclation
function of the residual of the
100 point AR predictor. Number of samples outside 1.96&E
limit = 7. (b) Normalized autocorrelation
function of the residual
of the Kalman predictor. Number of samples outside 1.96/G
limit = 4.
MAHALANABlS
et 01.: ALGORlTHM FOR REAL SE,SM,C
DATA
Mende] to be equivalent to the problem of estimation of the
white noise sequence ak in (5). It should be possible to obtain
the smoothed estimates of this sequence if the unknown
parameters of the model (5)-(6) are known 131. The system
identification method discussed in the present paper yields
the optimal estimates of the matrices F and h in the canonical
form. It should be possible to use these estimates along with
an adaptive Kabnan faltering algorithm [9] in order to obtain
the covariances of gok and variance of nk, This would then
permit the results of the present paper to be extended to cover
the white noise estimation problem.
Further, Mendel and Kormylo [4] have shown that the
uncertainty associated with the reflection from oil bearing
structures can be accounted for by using the modelok = yLwL,
where yk is a binary sequence which can take values zero and
one and wk is a Gaussian sequence. It has been shown that a
combined detection estimation algorithm using the maximum
likelihood approach can be utilized for estimating nk in such
cases. The possibility of extending the method of the present
paper to the detection estimation situation is also being
investigated.
221
A. K. Mahalanabis (M’66-SM’78) was born in
Kishoreganj, Bangladesh in 1934 and has been
educated in the University of Calcutta, Calcutta,
India, where he received the D.Phil.(S.C.)
degree in 1961.
“..-.... .._..
.
c.._ I.-...” “^&ye b research h
control theory and its apptica, tions. He is currently a Professor of Electrica 1 Engineering in
the Indian Institute of Tee hnology, Delhi,
India. and he is in charge of .,~
development group in automation His corrent
research interests are mainly in the area of Stochastic Estimation and
Control with Applications in Modetiog and Identification of Physical
Systems from real data.
Dr. Mahalanabis i? a member of the Editorial Board of the Journal
Optimal Confrol Applicofions and Methods (Wiley), and is the Chairman of the International Programme Committee of the IFAC Symposium on Theory and Applications of Digital Control to be held in
New Delhi in January 1982.
.),
*
ACKNOWLEDGMENT
The authors would like to record their appreciation of the
help received from Dr. V. C. Mohan and Dr. N. D. J. Rao of
the Institute of Petroleum Exploration, Debra Dun in the form
of the data used in this work and in the form of valuable discussions and computational facilities. Also the authors wish to
acknowledge the useful comments of the anonymous reviewers
on the original version of this paper.
REFERENCES
[l] E. A. Robinson, “Predictive decomposition of seismic traces,”
Geophysicr,no.4,vol. 22, pp. 767-178, Oct. ,957.
[2] K. L. Peacock and Sven Treitel, “Predictive doconvolution,”
Geophysics,vol. 34, no. 2, pp. 155-167, Apr. 1969.
131 1. ht. Mendel, ?+‘h~te Noise Estimators for seismic data Processing in oil exploration,“IEEE Trans. Auromot. Confr., vol. K-22,
pp. 694-707, Oct. 1971.
[4] John Konnylo and 1. hf. Mendel, “On maximum likelihood estimation of reflection coefficients,” in Rot. 48th Annu. Meef.
SE0 (San Francisco, CA).
[S] E. Tse and 1. J. Anton, “Identifiability of parameters,” IEEE
?hons. Automat. Con*., vol. AC-17, Pp. 637-646, Oct. 1972.
(61 R. K. Mehra, “On line identification of linear dynamic systems,‘*
IEEE Trans. Au10mar. Con,,.,vol. AC-16, Pp. 12-21, Feb. 1971.
[7] E. Tse and H. Weinert, “Structure determination and parameter
identification of multivariable systems,” IEEE Tkns. Automaf.
Confr., vol. AC-20, pp. 603-612,Oct. 1975.
[E] 1. S. Meditch, Stochasfic Optimal Linear Esfimafion and Control. New York, McGraw-Hitl, 1969.
[P] B. D. 0. Anderson and I. B. Moore, Optimal Filfering. Englewood Cliffs, NJ: Prentice-Hall, 1979.
[IO) N. Ott and H. G. Meder, “The Kahnan t?lter as a prediction
error filter,” Geophysical Prospecfing, vol. 20, pp. 549-560.
Ill] C. E. P. Box and G. M. Jenkins, Time Series Analysis, ForecasfSan Francisco, CA: Holden Day, 1976.
iwl?nd Control.
1121 R. L. Kashyap and A. R. Rae, “Dynamic stochastic models from
empirical data,” New York: Academic Press, 1976.
1131 1. M. Mendel, “A quantitative evaluation of Ott and Meder’s prediction error filter,” Geophysical Prospecting, vol. 25, no. 4, pp.
692-698,1917.
[14] L. H. Son and B. D. 0. Anderson, “Design of Katman filters
using signal model output statistics,” plot. Insf. Elec. ,%n., vol.
120, no. 2,pP. 312-318, Feb. 1973.
Surendra Rasad (S’72-M’75) was born in New
Delhi, India, on July 10, 1948. He received
the B.Tech. degree in electronics and electrical
communication engineering from the Indian
Institute of Technology (UT.) Kharagpor,
India, in 1969, and the M.Tech. sod Ph.D.
degees io electrical engineering from the
LLT., New Delhi, India, in 1971 and 1974,
respectively.
He has been teaching
New Delhi
. at “eLLT.,
since August 1971, wnere
IS presently an
Assistant Professor. His teaching and research interests are in radar/
sonar signal processing, communications, and computers. He was a
Visiting Research Fellow at the Loughborough University of Technology, U.\., during the period from August 1976 to August 1977 where
he was mvolved in developing algorithms for adaptive array processing
for H.F. arrays. Cunently he is engaged in research in the areas of
sonar and seismic signal processing, underwater communications and
array signal processing.
*
K. P. Mohandas was born in Alleppy, Kerala
State, India, in 1946 and received the
B.Sc.(T.ngg.) degree from Kerala University in
1968 and M.Tech. degree from the Indian
Institute of Technology, Madras, in 1973, both
hg in T.K.M. College of Engineering, &loo
from 1968 to 1969 and since 1969 he is witi
the Calicut Regional Engineering College,
Calicut, Kerala State, India. He is currently
completing the requirements for the Ph.D. degree in electrical engineering at the Indian Institute of Technology, Delhi, under the Qutity
Improvement Programme of Government of India.