The author is with the Department of Electrical and Electronic Engineering, Paisley. College of Technology, High Street, Paisley, Renfrewshire PA1 2BE, United.
Optimal deconvolution smoother T.J. Moir. B.Sc. Ph.D., A.M.I.E.E. Indexing terms:
Control theory, Control equipment and applications
Abstract: The optimal fixed-lag deconvolution smoothing problem is solved in the frequency domain. It is shown how the smoother may be implemented online by using an innovations moving average description, thus avoiding a two-pass algorithm, involving causal and anticausal filtering. The optimal smoother is given in polynomial matrix form, which makes the solution relatively simple to implement.
1
Elv(t)vT(tX = R
Introduction
The deconvolution problem is widely encountered in the engineering literature. The problem differs from the more usual problem of removing a signal embedded in noise with the addition of removing the effect of any distorting system. In reflection seismology the signal to be estimated is considered to be white (the primary reflectogram) and the distorting system is the seismic wavelet. Mendel [1] has given a state-space solution to the fixed-lag minimum variance deconvolution (MVD) problem. The solution involves passing the data obtained from a seismic trace through a Kalman filter and then processing (off-line) the innovations sequence with an anticausal filter. Since the MVD solution in [1] involves a Kalman filter it can be applied to time-varying and non-stationary systems. A detailed comparison has been given by Chi and Mendel [2] between various deconvolution techniques. This has shown the steady-state Kalman filter to be equivalent to the prediction error filter of Robinson [3] and the steadystate MVD filter equivalent to the least-squares inverse filterofBerkhout[4]. This paper considers the stationary optimal deconvolution problem. The solution is found in transfer function matrix form and in polynomial matrix form. It is shown how the transfer function of the optimal smoother involves the inverse of the return difference matrix of the stationary Kalman-Bucy filter. This relates to the Wiener filtering problem as previously investigated by Barrett [5] and Shaked [6]. Further, the polynomial matrix solution gives smoothed estimates as a function of a weighted sum of the innovations sequence and does not require a two pass algorithm. The deconvolution smoother can thus be implemented on-line.
(4)
T
Elrtt)v (tft = 0
(5)
seismic wavelet
|V(t)
M(t)
W(t) -»KX) primary "" y(t) reflectogram Fig. 1 System description
noise sequence * Z ( t ) seismic trace
The matrices Q and R are assumed to be symmetric and positive definite and the system is assumed to have reached a statistical steady state. An equivalent description of the process of Fig. 1 is the innovations description given by Fig. 2
Z(t) €(t)
Fig. 2
Innovations description
thus z(t)=
Fit-Mi)
(6)
where {s(i)} is the zero-mean innovations sequence with covariance matrix Ele(t)eT(t)-] = Re
(7)
Then from eqns. 6, 1 and 2 the spectral density of the seismic trace becomes v
) = W(z)QWT(z-1)
+R
(8)
and 2
System description
The system description is shown in Fig. 1. The measurement (seismic trace) vector z(t) is given by Z(t)
= y(t) + v(t)
(1) where the signal vector y(t) is generated by the weighting sequence W(t) thus: (2)
y(t)=
where y(t) e Rr, n{t) e Rq and we assume r = q. The Gaussian white noise sequences n(t) and v(t) are assumed to be zero-mean with covariances T
EW)n (t)}
=Q
(3)
Paper 4321D (C9), first received 12th April and in revised form 16th August 1985 The author is with the Department of Electrical and Electronic Engineering, Paisley College of Technology, High Street, Paisley, Renfrewshire PA1 2BE, United Kingdom
IEE PROCEEDINGS,
Vol. 133, Pt. D, No. 1, JANUARY
F(z) 4
(9)
W{z) *
(10)
The result of eqn. 8 is now well established and is first given by Arcasoy [7], where it is shown that F(z) is the return difference matrix of the steady-state Kalman-Bucy filter. Polynomial matrix description The left matrix fraction description of the system W{z) can be written as (11) and F(z) becomes F(z) = A(z-lylD(z-1)
(12) l
where the polynomial matrices A(z~ ), D(z~ ) and C(z~l) can be expressed as: A(z-l) = Ir + AlZ~l
+ A2z~2
l
+ ••• + Anaz-"°
1986
Authorized licensed use limited to: Massey University. Downloaded on February 18,2010 at 14:24:50 EST from IEEE Xplore. Restrictions apply.
(13a) 13
(13b) D2z~2
') = / ,
Dndz-nd
(13c)
The poles of det \_A{q)~], det [C(q)] and det [D(g)] are assumed to lie strictly outside the unit circle on the qplane. The equivalent polynomial matrix spectral factorisation becomes: = C(z-l)QCT(z) It is assumed that nc 3
+ A{z~l)RAT{z)
(14)
na and na = nd.
The deconvolution problem
This Section will examine the deconvolution problem in the frequency domain and derive an optimal causal fixedlag deconvolution smoothing filter. Consider Fig. 3, where v(t)
W(z)
p(t)
The mean squared error J becomes J = trace Ree(0)
(19)
1 f* = trace— I See(eJe) dd = - ? - trace [HAA* - Hj QW*]rj* — = 0 (29) 2nj J z By Cauchy's residue theorem, for eqn. 29 to be true we require the poles of T, where T= HAA* -HjQW*
Proof of Theorem I Define the estimation error covariance matrix: RJn -m) = Ele(n)eT(m)-] and the z-transform power spectral density matrix of the error: See(z)=
RJi)z-i,\z\ =
(30)
to lie outside the unit circle on the z-plane [9]. Multiply from the right by A* ~ l TA*'1
= HA-
HjQW*A*-1
= HA- {HIQW*A*~i}+
-
{HIQW*A*-1}_
and, rearranging, TA*- 1 + {HjQW*A*-l}_
= HA - {HIQW*A*~1}
+
(31) 14
IEE PROCEEDINGS, Vol. 133, Pt. D, No. 1, JANUARY 1986 Authorized licensed use limited to: Massey University. Downloaded on February 18,2010 at 14:24:50 EST from IEEE Xplore. Restrictions apply.
Now the left-hand side of eqn. 31 is analytic inside the unit circle, while the right-hand side is analytic outside the unit circle. Hence, the two are identically zero. Thus HA-{z~lQW*A*-l}+
Now C T ( z ) D T ( z ) - 1 = P 0 + P , z + P 2 z 2 + ••• a convergent matrix power series so that
=0
{CT(z)DT(zylz-l}+
or H = {z~lQW*A*-l}+A'
(32)
Finally to ensure H is a minimum, we require the sufficient condition for optimality d3
>0
(33)
= P,(z)z-1
f o r/ ^ l
Pl(z)
= P 0+ Plz + - - + P l z
l
rj(WQW* + R)rj* — > 0 (34) z / 3 is always positive as the integral is symmetric with respect to the unit circle. This completes the proof. Remark 1: The uncausal solution to eqn. 29 gives an unrealizable smoother H,,\ *\-i = H,QW*(AA*)
(35)
which for / = 0 gives rise to the Berkhout two-sided leastsquares inverse filter [4]. Remark 2: Noting that Q is symmetric, when the measurement noise is zero (R —> 0), the spectral factors become WQ 112
(36a) (366)
ls
(46)
which expresses the smoother entirely in terms of polynomial matrices and the process and innovations covariance matrices. Eqn. 46 may be used to obtain the transfer function matrix of the optimal smoother. The solution for the process noise (primary refiectogram) estimate may also be found in the following: By using the fact that, from eqn. 6, the measurement z(t) = F(z)e(t)
(47)
then clearly (48)
from which, by multiplying eqn. 46, (49) so that the optimal smoothed (x(t) is expressed in terms of a vector moving average of order /:
then by substituting eqns. 36 into eqn. 32 we have
(50)
H
(37) R=0
(38)
which is not surprisingly the inverse of the transfer function W, delayed by / steps. 4
(45)
and a computational algorithm for determining P,(z) given in Appendix 10.1. Substituting eqn. 44 in eqn. 42 we obtain H(z; I) =
1 -— trace
(44)
where
6=0
giving / 3 > 0, or
U
(43)
Polynomial matrix optimal deconvolution smoother
The solution to the deconvolution smoothing problem given in the preceding Section has the difficulty that it requires the removal of the {.} + brackets. This Section considers a polynomial matrix solution to the problem. Polynomial matrix solutions have become popular recently, due to the work of Kucera [10], and are particularly effective in self-tuning control and filtering. We begin by using the optimal smoother of eqn. 17
and substituting for W and A with their matrix fraction equivalents. Using eqns. 8, 12 and 18 we obtain A(z) = A{z-lylD{z-l)Rl12
and the summation is from i = 1 as Po = 0 (see Appendix 10.1). For / = 1, a one-step 'smoothed' estimate is given by
from which it is seen that the estimate is simply a constant gain matrix, scaling the innovations sequence. 5
Relationship to the state-space
A stable state-space representation of the system of Fig. 1 may be defined: x(t + i)
(52)
y(t) = Nx(t)
(53)
z(t) = y(t) + v(t)
(54)
where the state vector x(t)£R", O is an n x n matrix, F is an n x q matrix and N is an r x n matrix. Then the z-transform of the system
(39) and
from eqn. 11 WT{z-1) = CT(z)AT(zy1
(40)
H(z; /) = {QCT(z)AT(z)-lAT(z)D-T(z)R-TI2z-l}
(41)
which becomes H{z; I) = Q{CT(z)DT(z)-lz-'}+ R;'D{z-'yU{z-') IEE PROCEEDINGS, Vol. 133, Pt. D, No. 1, JANUARY
(56)
represents the return difference matrix [7]. Here K is the Kalman gain matrix in the steady state given by
+
x R-ll2D(z-lylA{z-1)
F{z) = I + N(zl -)
which, for this example, becomes
The optimal smoother of eqn. 17 becomes zKT(I
(62)
-
for / = 1, a simple solution exists as z—* 0 within the {.} + brackets: #(z; 1) = QTTNTR-lF'l{z)
(63)
which is a transfer function matrix expression in terms of the inverse of the return difference matrix. Shaked [6] has shown how the return difference matrix defines the stationary Kalman-Bucy filter. Here it is interesting to see the equivalent expression as applied to deconvolution. Further, the innovations form of eqn. 63 gives fat - l/t) = QTTNTR-h(t)
(64)
which is a state-space form of eqn. 51 and is also given by Mendel [1], More generally, the optimal deconvolution smoother becomes R-^"1^"' 6
6
8
10
12
14
16
18 20
(65)
Illustrative example
To show the performance of the fixed-lag deconvolution smoother consider the simple scalar example with Q = q, R = r. Employing the polynomial smoother of eqn. 46: D(z~l) is found from the spectral factorisation of eqn. 14 1
(1 + az" )(l + az)r + q = re(l + dz-^l
\j>2f
A
" -6
8
10
12
14
16
18 20
2
4
8
10
12
14
16
18 20
+ dz)
from which d = — y + y/y2 — 1 (\d\
