SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION *

BY

B. URSIN **

ABSTRACT URSIN, B., 1979, Seismic Signal Detection and Parameter Estimation, Geophysical

Prospecting 27, 1-15.

In the mathematical theory of seismic signal detection and parameter estimation given, the seismic measurements are assumed to consist of a sum of signals corrupted by additive Gaussian white noise uncorrelated to the signals. Each signal is assumed to consist of a signal pulse multiplied by a space-dependent amplitude function and with a space-dependent arrival time. The signal pulse, amplitude, and arrival time are estimated by the method of maximum likelihood.

For this signal-and-noise model, the maximum likelihood method is equivalent to the method of least squares which will be shown to correspond to using the signal energy as coherency measure. The semblance coefficient is equal to the signal energy divided by the measurement energy. For this signal model we get a more general form of the semblance coefficient which reduces to the usual expression in the case of a constant signal amplitude function.

The signal pulse, amplitude, and arrival time can be estimated by a simple iterative algorithm. The effectiveness of the algorithm on seismic field data is demonstrated.

INTRODUCTION

A seismic measurement (a group of seismic traces) is assumed to consist of a sum of signals corrupted by additive noise. Each signal (primary or multiple reflection) is assumed to consist of a signal pulse multiplied by a space-dependent amplitude function ancl with a space-dependent arrival time. The arrival times are given a parametric representation so that each arrival time function is a fixed function of the space coordinates and a parameter vector which has at least two components, the two-way travel ,time and the stacking velocity. It is assumed that the noise is white Gaussian noise with zero mean value, that the noise is uncorrelated to the signal, and that the different signals do not overlap.

The signal pulse, amplitude function, and arrival time parameter vector will be estimated by the method of maximum likelihood (Helstrom 1968).

* Paper read at the thirty-ninth meeting of the European Association of Exploration Geophysicists in Zagreb, June rg77.

** GECO A.S., Veritasveien I, N-1322 Hovik, Norway. Geophysical Prospecting 27 I

2 B. URS~N

For the assumed signal-and-noise model the logarithm of the likelihood ratio is maximized with respect to the unknown signal parameters; This results in an iterative scheme for signal detection and parameter estimation.

Some of the results given here have previously been presented at a course arranged by the Norwegian Petroleum Society (Ursin 1976).

The estimation algorithm has been successfully used in reducing the effect of multiple reflections on the seismic section (Brandsaeter and Ursin 1977). The signal pulse, amplitude, and arrival time of each multiple reflection are estimated, and the multiple reflected pulses are subtracted from the measure- ment data. The corrected traces are then further corrected for normal moveout and stacked so that the primary reflections are enhanced.

The purpose of the seismic velocity analysis is to estimate the arrival times of the reflected signals. The usual approach to this problem is to maximize a coherency measure which is computed for different values of the arrival time parameter vector (see Garotta and Michon 1967, Schneider and Backus 1968, and Taner and Koehler 1969). Neidell and Taner (1971) have given a broad discussion of the most common coherency measures.

Ursin (1977) has applied the maximum likelihood technique to the usual signal model in which the signal amplitude function is assumed to be constant. For this simple signal model the maximum likelihood detector is equivalent to a least squares detector which corresponds to using the signal energy as coherency measure. This is also equivalent to the correlation scheme proposed by Schneider and Backus (1968). The use of the semblance coefficient proposed by Taner and Koehler (1969) corresponds to a normalized least squares detector. The semblance coefficient is very similar to a filter performance parameter which is used in the design of least squares deconvolution filters (see Treitel and Robinson 1966).

The signal model used here results in a more general definition of the signal energy and of the semblance coefficient which both take into account variations in the signal amplitude.

The primary seismic pulse or signature is needed in the design of shaping filters, in the construction of synthetic seismograms, and it provides infor- mation about the seismic source and the absorption of seismic waves in the earth. White and O’Brien (1974) have discussed the problem of estimating the seismic pulse from a single trace. They assume that the signal and the noise are uncorrelated, that the reflection series is white, and that the seismic pulse is minimum delay. The present algorithm is developed under the assump- tions that the signal and the noise are uncorrelated, and that the different signal pulses do not overlap.

The performance of the algorithm on seismic field data is shown in two examples.

SEISWC SIGNAL DETWTIOH AMD PARA~MBTER tiSTI:IdATlON' 3

MAXIMUM LIKELIHOOD DETECTION AND PARAMETER ESTIMATION

The measurement data are m (k, x) =u (t, x) + w (t, x). (1)

The sensors (array elements) are located at x = XJ, j = I, 2, . . . , M, and the data are given as discrete samples at t = At, n = o, I, . . . , N.

The signal u (t, x) consists of a sequence of K reflected pulses (primary and multiple reflections)

LJ (t, x) = i uk (t, x) = ii ak (x) pk (t-T (Ok, x)), (2) k-1 k-l

where each individual signal uk (t, x) consists of a pulse pk (t) multiplied by an amplitude function @k (x) and with a variable arrival time T (ok, x). It is assumed that the arrival times can be given a parametric representation so that each arrival time function is a fixed function T (0, x) of x and a para- meter vector 8 which has at least two components, the two-way travel time and the stacking velocity. The signal pulses are assumed to be of finite duration so that pk (NAt) # 0 only for n = 0, I, . . . , L. The pulse duration LAt is assumed to be small compared to the duration of the measurement NAt. It is also assumed that the signals ak (x) pk (t-T (ok, x)) do not overlap, so that at any point in the T-x-plane there is only one signal present.

Each individual signal uk (t, x) will remain unchanged if we divide the amplitude function ak (~6) by a constant and multiply the signal pulse pk (t) by the same constant. In order to get ak (x) and pk (t) uniquely defined we need a normalization procedure. There exist a number of possibilities, and we have choosen to normalize so that a (xi) =I.

The noise w (t, x) is assumed to be white Gaussian noise with zero mean value. The covariance function of the noise is

E {w (IzAt, Q) w (KAt, xi)} = W8 (yt - K) 6 (i-j). (3)

E { } denotes expected value, and 6 (n) is given by

8 (n) = 1

I n=o onfo.

It is also assumed that the noise is uncorrelated to the signal: E {u (nht, xi) w @At, XJ)} = o. (5)

An estimate of the signal is denoted by

fik (t, X) = dk (X) $k (t-T (ik, X)) (6)

where $k (t) and dk (x) are estimates of the signal pulse and amplitude function, A and ok is an estimate of the arrival time vector.

4 B. URSIN

In order to simplify the notation we shall use the inner product (which

depends on 6,)

(u,v) = i :u(nAt+T(b /c, 4, Q) v (fiAt + T (h xt) xi) (7) ?l=o *=1

and the norm ljull = (u, u)s.

We now want to discriminate between the two hypotheses

Ha:m(t+T~lc,~),~)=~(t+T(~rc,~),~) signal absent

HI : m (t + T ($c, x), x) = uk (x) pk (t) + w (t + T (b,t, x), x) signal present

This can be done by computing the logarithm of the likelihood ratio and deciding upon HI if the log likelihood ratio exceeds a certain predetermined value. But since the arrival time functions are unknown, we compute the log likelihood ratio for a number of possible arrival times and decide that there is signal present if a local maximum of the log likelihood function exceeds a given value.

The likelihood ratio is defined as the ratio between the probability density of the measurement data assuming that a signal is present and the same probability density assuming that there is no signal present. With the assump- tions made about the signal and the noise, the likelihood ratio is

,. P (m, %t I HI) -P

A= ( = =

p (m, 6, I HO) exp

= exp i - $ (-- 2 b fik) + II fh PI).

The logarithm of the likelihood ratio is then

log A = -$f (2 (m, h) - II fikl12>,

(8)

(9)

where

and

(m, ii,) = g Zk m (aAt + T (let, xj), q) dk (q) fik (aAt). (11) j--1 n=o

SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION 5

The maximum likelihood estimates of ak (XI) and pk (~zht) satisfy the equations

- z lk m (aAt + T (&, xj), xj) fik (nAt) = o ?&=o

(12. a)

and

3 log A

36k @Ai) = z Ijk (nht) ; & (xj)2

,=1

These equations can be solved iteratively by alternately keeping dk (~j) and fik (nAt) fixed. We note that by multiplying (12. a) with dk (XJ) and summing over j or by multiplying (12. b) with $k (+zAt) and summing over n we obtain

(m h) = II Qk l12. (13)

The log likelihood ratio thus becomes

log A = $ 11 Qk 112. (14)

COHERENCY MEASURES

Using the results in the previous section we can derive several related coherency measures. The method of maximum likelihood gives the logarithmic likelihood ratio in (14), which is equal to the signal energy normalized with twice the spectral density of the noise.

For the signal-and-noise model that we are considering the method of maximum likelihood is equivalent to the method of least squares, since the error function

I = 11 m-ilk 112 (15)

is minimized. By using (13) we obtain for the least squares estimate

1 = II m II2 - II& 112. (16)

We see that the method of least squares is equivalent to maximizing the signal energy

E, = //fikl12. (17)

6 B. URSIN

To use the signal energy as a coherency measure was proposed by Taner and Koehler (1969). They also proposed to use the semblance coefficient S which is equal to the signal energy normalized with the energy of the measure- ment data, that is

EU II Qk II2 ‘=E, = /lrnj12 (18)

For the assumed signal-and-noise model the semblance coefficient is

i! dk (xl)2 i & @At)2 s = icI i=,’ 130

(19)

With a (xj) = I, j = I, 2, , . ., M in (12. b) and (19) we obtain L M

1

n 2

C 2 m (%At + T (Ok, q), XJ)

s = m-o I=‘, -> M E I; m(aAt+T(“ek,xj),xj)2

n=o ,=I which is the usual definition of the semblance coefficient,

(20)

The semblance coefficient may also be defined in a different way. From eq. (16) and (18) we see that the semblance coefficient is equal to

s-$. (21) m

This is very similar to a filter performance measure that is used in least squares filter design (see Robinson and Treitel 1966). We note that the sem- blance coefficient is normalized so that o < S < I.

ESTIMATION ALGORITHM

We shall outline how the derived results can be used in an algorithm for estimating the signal parameters. We have seen that for a given estimate of the arrival time parameter vector Ok the optimal estimates of the signal pulse and amplitude must satisfy the non-linear set of equations (12. a) and (12. b).

In order to obtain improved estimates of the arrival times we shall let the estimated arrival time of each signal be given by

h A

Tk (xj) = T (ok, xj) + ATk (xj), (22) where ATk (xj) is a correction term (residual normal moveout). These cor- rection terms are computed so that the correlation between the measurement

SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION 7

and the estimate of the signal pulse is maximum. This procedure is used in Helstrom (1968, p, 274) to estimate the arrival time of a known signal of un- known amplitude in white noise.

In order to start the computations we need initial values of the estimates of the signal parameters. We shall use a constant amplitude function &g (x!) = I,

j=I, . ..) M and assume that an estimate of the arrival time parameter vector is available from a velocity analysis.

There are two ways to terminate the algorithm. The computations are stopped if the number of iterations exceeds a given number or if the change in the coherency measure used is below a given value. We are going to use the semblance coefficient, but any of the coherency measures discussed in the previous section can be used. The semblance coefficient for signal number k computed at iteration step number i is denoted by Sk.

We have the following steps in the estimation algorithm (we let the super- script i denote estimated value at step number i) :

Initiation

Let So, (x3) = I and i;‘“k (xj) =T (6,, x3) n where 8, is obtained from a velocity analysis. Compute the initial estimates of the signal pulse

fii (nAt) = 5 z m (~,ht+$~ (xi),xj) (23) 1-x

and the semblance coefficient

M i fi”, (nAt)” s; = M c-0

ZZ i m(nAt + !$ (x,), x3.)” i-1 ?b-0

(24)

Algorithm

First compute the cross correlation functions

c; (7, Xj) = i $“,- l (nAt) m (nAt +?$-I (xj) + 7, xj) . n=o

(25

The correction to the arrival time estimate is AT; (x,) = 7 for which ci (T, xj) is maximum, and we obtain

f; (x,) = i”,-’ (xj) + AT: (xj) . (26)

The new estimate of the signal amplitude function is

c: (AT6 (xj), xj) ‘% (“) = c; (AT: (x,), x1) (27)

8 B. URSIN

This is eq. (12. a) except that we have normalized so that C$ (x1) = I. The new estimate of the signal pulse is found by using (12. b), which gives

The new value of the semblance coefficient is then

Termination

The computations are terminated if the number of iterations i exceeds a given number or if the change A.$ =Si - Si-’ in the semblance coefficient is less than a given value.

The estimate of the signal amplitude at xj is obtained by correlating the measurement at xj (that is, trace number j) with the estimate of the signal pulse. The estimate of the signal pulse is computed by an optimal weighted stack of the measurement data (Robinson 1970). This is the least squares estimate of the signal pulse, given the signal amplitude function & (xj) (see Lb-sin 1974).

APPLICATION TO FIELD DATA

The application of the estimation algorithm to field data will be shown in two examples, one in which the waterbottom reflections are estimated, and another in which several primary and multiple reflections are estimated.

In both examples a pulse-shaping filter has been applied to the data. In the filter design, the energy of the desired output pulse is set equal to the energy of the estimate of the seismic pulse. The estimated pulses are given in relative units, and the estimated amplitude functions are normalized so that the amplitude of the near trace is equal to one. In plotting the shot-point records an RMS scaling procedure has been used.

Waterbottom reflections

Fig. I shows a shot point record before and after the application of a pulse- shaping filter. The filter was computed from an estimate of the first water- bottom multiple reflection, and the desired output of the filter is a delayed zero-phase pulse. In fig. 2 we see the filtered data before and after the removal of the waterbottom multiples. The estimated pulse and amplitude function of the waterbottom reflections are shown in fig. 3.

SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION 9

Fig. I. Shot-point data before and after the application of a pulse-shaping filter.

Fig. z. Filtered shotpoint data before and after the removal of the waterbottom multiples.

IO B. URSIN

Fig. ga. l’rimary reflection (the water bottom).

Fig. 3b. First multiple reflection.

Fig. 3c. Second multiple reflection. Fig. 3d. Third multiple reflection.

Fig. 3e. Fourth multiple reflection

ilMElMSl DlSTINCEiKM,

Fig. 3f. Fifth multiple reflection.

Fig. 3g. Sixth multiple reflection. Fig. 3h. Seventh multiple reflection.

: : 4.2. j ,;,. ,I, IOC,. 0 0. i ,:, : J

TIMEIHSI OISTMCEIKMI

Fig. 3i. Eighth multiple reflection.

SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION II

Composite reflections

Fig. 4 shows a shot point record before and after removing some of the multiple reflections. The corresponding velocity analyses are shown in fig. 5 and 6. The estimated pulse and amplitude function of three waterbottom reflections are shown in fig. 7.

Fig. 4. Shotpoint data before and after the removal of multiple reflections.

The estimated primary reflection at 0.6 seconds is shown in fig. 8. a, and the estimated peg-leg multiple at 1.0 seconds is shown in fig. 8. b. Fig. 8. c shows the estimated primary at 1.05 seconds. In the last part of the estimated pulse we see a peg-leg multiple (because of the reversed polarity).

Fig. g. a shows the estimate of the composite reflection at 1.4 s, and fig. g. b shows the associated peg-leg multiple at 1.8 s. In fig. 9. c we see the estimate of the primary reflection at 2.0 s.

12 B. URSIN

--2---i- 1. Y-1 -_~*--- - _c---

:.. I ---->c=d -4 -./- ._---- ri

Fig. 5. Velocity analysis of field data.

SEISMIC SIGNAL DETECTION AND PARAMETER ESTIMATION

I,0

I'S

Fig. 6. Velocity analysis of data after multiple removal.

Fig. 7a. The waterbottom primary Fig. 7b. First waterbottom multiple reflection. reflection.

Fig. 7c. Second waterbottom multiple reflection.

Fig. 8a. Second primary reflection. Fig. 8b. Peg-leg multiple of the second primary.

Fig. SC. Third primary reflection.

Fig. ga. Fourth primary reflection. Fig. gb. Peg-leg multiple of fourth primary.

Fig. gc. Fifth primary reflection.

SEISMIC SIGNAL DBTECTION AND PARAMETER ESTIMATION I.5

CONCLUSION

We have shown that the algorithm gives a satisfactory result when applied to non-overlapping events.

The algorithm has been used with good results in estimating the seismic pulse for use in the design of pulse-shaping filters and in estimating long- period multiple reflections.

The examples show that the algorithm is useful in detailed stratigraphic studies where it is important to know the amplitude, pulse shape and travel time of each event.

ACKNOWLEDGEMENT

This work has received financial support from the Norwegian Petroleum Directorate.

I would like to thank Mr. Helge Brandsaeter for his comments and for writing the computer programs which have been used in the computations.

REFERENCES

BRANDSAXTER, H., and URSIN, B., 1977, Adaptive long-period multiple attenuation, paper presented at the 39th Meeting of the E.A.E.G., Zagreb.

GAROTTA, R., and MICHON, D., 1967, Continuous analysis of the velocity function and of the moveout corrections, Geophysical Prospecting 15, 584-597.

HELSTROM, C. W., 1968, Statistical theory of signal detection, 2nd edition, Pergamon Press, Oxford.

NEIDELL, N. S., and TANER, M. T., 1971, Semblance and other coherency measures, Geophysics 36, 482-497.

ROBINSON, J. C., 1970, Statistically optimal stacking of seismic data, Geophysics 35, 436-446.

SCHNEIDER, W. A., and BACKUS, M. M., 1968, Dynamic correlation analysis, Geophysics 33, 105-126.

TANER, M. T., and KOEHLER, F., 1969, Velocity spectra-digital computer derivations and applications of velocity functions, Geophysics 34, 859-881.

TREITEL, S., and ROBINSON, E. A., 1966, The design of high-resolution digital filters, IEEE Trans. on Geoscience Electronics GE-4, 25-38.

URSIN, B., 1974, Optimal stacking-deconvolution filters, paper presented at the 36th Meeting of the E.A.E.G., Madrid.

URSIN, B., 1977, Seismic velocity estimation, Geophysical Prospecting 25, 658-666. URSIN, B., 1976, Signal detection and parameter estimation, paper presented at the

NPF course: Processing of Marine Seismic Data, Gol. WHITE, R. E., and O’BRIEN, P. N. S., 1974, Estimation of the primary seismic pulse,

Geophysical Prospecting 22, 627-651.