ADAPTIVE FILTE APPLICATIONS TO HEAVE COMPENSATION

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1 ADAPTIVE FILTE APPLICATIONS TO HEAVE COMPENSATION D.G. Lainiotis, K. Plataniotis, and C. Chardamgoas Florida Institute of Technology, Melbourne, FL and University of Patras, Patras, Greece. Abstract - The problem of heave motion compensation is addressed in this paper. A significant class of the Eainiotis partitioning approach is applied and comparisons are made with the Kalman filter based approach, with respect to their computational complexity and performance. It is shown that the linear Lainiotis filter is well suited for on-line implementation, since orders of magnitude reduction of processing time is achieved, while in the case where model parameter uncertainty exists, the adaptive Lainiotis filter has excellent performance. 1. INTRODUCTION In many sea-related problem, as seismic experiments for oil exploration, a deep towed signal source and a sensor are employed. The motion in the vertical axis of the source and the sensor (heave) affect the reflection records. The heave effects can be partially removed by using the estimates of the vertical source motions from hydrostatic pressure and motion sensors to delay or advance the pulse firing instants relative to a clock pulse transfer [ 11. It is preferable that filtering can be applied in a real-time mode during acquisition of the reflection responses. Sensors and manipulation process are subject to random noise which is dependent on the sea state, the current and the relative position of the ship with respect to the waves. The appropriate choice of the covariances matrices of the the noise is an important issue that is crucial to the success of the heave compensation strategy. The need for the design of fast, efficient and practically implementable optimal filters that provide the required estimates is apparent. A lot of studies have been reported for the solution to this problem, most of them utilizing Kalman filter-based approach 113, [21. This approach has two main drawbacks : 1. Due to the fact that the design of the Kalman filter is based on the assumption of complete knowledge of the model which describes the heave dynamics, there is a degradation in the estimate quality in the case of a mismatch between the model used to desig,n the filter and the actual model, 2. When the model is peric&c in time, somethiing very reasinable dor sea-related motion dynamics, the Kalman filter, due to its computational complexity, is inappropriate for on-line applications, and In this paper, the Lainiotir; multimodel partitioning approach [3]-[5] is proposed. The linear Lainiotis filter is used in the case where the model is (completely known, but periodicin time, reducing signficantly the processing time. The adaptive Lainiotis filter is used in order to provide adaptability in a changing environment and its performance is evaluated with respect to that of the Kalman filter. Specifically, the paper is organized as follows : In section 2 the model that describes the heave motion dynamics is given. In section 3 the linear Kalmaun and Lainiotis filters are given and discussed for the case of time-invariant and periodic models, when completely model knowledge is assumed. In section 4 the adaptive Lainiotis filter is presented and discussed for the case of partially unknown linear systems. In section 5 the simulation results are presented, and finally, conclusions are given in section PROBLEM FORMULATION The mathematical model nsed, that describes the heave motion dynamics appeared in [ l], is given by : x(t) = F x(t) t G w(t) z(t) = H x(t) + v(t) where x(t) and z(t) is the 2x1 and 1x1 state and measurement processes, respectively; (vv(t)) and (v(t)) are the 1x1 and 1x1 plant and measurement nciise random processes, respectively, which are independent, zero mean white Gaussian processes with covariances Q(t) and R(t), respectively. F is the 2x2 state transition matrix, G is the 2x1 noise matrix, and H is the 1x1 observation matrix. The initial stae vector x(0)t is independent /92 $ IEEE

2 The matrices F, G, H are given by : r K(k+l) = P(k+l/k) HT (k+l) P -l(k+l/k) (12) T P (k+l/k) = H(k+l) P(k+l/k) H (k+1) + R(k+l) (13) H= - -1 l o where wo is the natural frequency of the system, and Q, is a quality factor. The discretized motion equations are : where T is the discretization = exp (IT) (5) r(k) = exp (IT) dr G (6) More details about the model and the values of the parameters, can be found in [l]. The objective is to obtain the optimal, in the mean square sense (mmse) estimate;'x(k/k) of x(k), using the noisy measurements z(k)=(z(l),... z(k)). In the next sections such estimation algorithms will be presented and discussed. 3. LINEAR FlLTERING Assuming that all the parameters of the model are known in advanced, the most common approach used is the design of the Kalman filter in order to obtain the required estimates. The mmse state estimates ^x(k/k) and the corresponding error covariance P(k/k) are given by [lo] : It can be noticed from the above equations that even in the case of time invariant models, the Kalman filter is time varying. Due to its computational complexity it is inappropriate for online applications. In a radically different approach taken by Lainiotis [3]-[51 the initial state vector is partitioned into the sum of two independent gaussian vectors, the nominal vector x and the unknown and random vector x. In other words, th% partitioning approach decomposes the original estimation problem into a simpler one, namely the one with partially known initial condition, and a parameter estimation problem pertaining to the unknown part x of the initial state. The resulting filter is the linear Lainiotisrper step partitioning filter (LLPSPF), and is given by : "x+l/k+l) = 2 (k+l/k+l) (k+ljc) P(k/k+l) 2 (k+ l/k+ I) = K (k+ 1) z(k+l) n 0,&+1) = at(k+l,k) HT(k+l) A(k+l) (18) P (k+l/k+l) = [I - K (k+1) H(k+l) ] Q(k) (19) 0 (k+l,k) = [I - K (k+l) H(k+l)] w+l,k) (20) K,(k+l) = Q(k) HT(k+l) A(k+l) P(k+l/k+l) = [I-k(k+l) H(k+l)] P(k+l/k) (8) 278

3 Comments : - Both the above filters, have the same performance for linear, Gaussian models, since they are different realizations of the same optimal mmse estimator. They only differ on the computational requirements, and thus the amount of processing time required for their implementation. - In the case where the model is time invariant, the Kalman filter is time varying, while the LLPSPF equations can be greatly simplified, namely the quantities 0, K, K become time invariant and can be computdonfy oke!?t the beginning of the filtering session[7]. - Even in the case where the model is periodic in time, the quantities of the LLPSPF referred above, need only be computed for the first period, stored and then used as needed. In this situation the LLPSPF requires remarkable less computations than the Kalman filter. This results to processing time savings which is very useful in on-line applications. Fig. 1 shows the computational requirements (in terms of normalized operations) of these filters when the number of sensors is increasing from 1 to 100, for both time invariant and periodic models. It can be easily seen the superiority of the LLPSPF in processing time savings, especially when the number of sensors is very large. 4. ADAPTIVE FILTERING It is very unrealistic to expect the filter designer to know in advance which model describes the heave motion dynamics ab any time. Thus, adaptive estimation techniques must be used in order to improve the estimation results. The adaptive estimation problem considered is specified by the following equations : where all quantities are as described previously, and 0 is an unknown finite dimensional parameter vector, which if known, would completely specify the model. Moreover, 0 is considered to be a random variable with known or assumed a-priori density p(q/o)=p(0). The processes [ w(k)) and (v(k)) are still uncorrelated when conditioned on 0, with covariances Q(k,O) and R(k,O), respectively. Given the measurement set Z(k) = [z(l), z(2),...,z(k)), the op- timal mmse estimate?(i&) of x(k) and the com:sponding error covariance P(k/k) are given by [31-[51 : where?@&;e) and P(k/k;Bi) are the 0-conditional mse state estimate and the corresponding B-conditional error covariance matrix. They are obtained from the corresponding linear filter matched to the model with parameter value 0 and initialized with initial conditions x(o/ol;0) and P(O/O;0).!2 is the sample space of 0 and p(0/k) is the a-posteriori pdf given by the following recursive Baps rule formula : where L(k/k;0) is the likelihiood ratio given by : where Pz(k/k-1;0) is the 0-conditional measurement error covariance matrix and z(k/k-1;8) is the $-conditional inn ovation sequence. Comments : - The above equations pertain to the case that tlhe pdf associated with 0 is a continuous function of 0. When this is the case, one is faced with the need for a nondenumemhle infinity of linear filters for the exact realization of the opti~nal estimator. The usual approximation performed to overcolme this difficulty is to approximate 0 :s. pdf by a finite sum, i.e., to discretize the sample space Cb. There exist, of come, cases in which the sample space is in itself naturally discrete. In this case, the integrals in (27)-(30) are replaced b~ summations running over all possible values of parameter 0. - It is comforting to know that when the true pameter value lies inside the sample space that the adaptive estimator assumes, the estimator converges to this value. TWhen the true parameter value is outside the assumed sample-space, the estimator converges to that value in the sample space that is 279

4 "closer" to the true value, in the sense of Kullback's information measure minimization [9]. - It is well known that the adaptive estimation problem constitutes a class of nonlinear estimator problems. Lainiotis' partitioning approach decomposes this nonlinear problem into a linear nonadaptive part, cosisting of a bank of linear filters, each filter matched to an admissible value of 6, and a nonlinear part., consisting of the a-posteriori pdf's p(6/k), that incorporates the adaptive, learning, or system identifying nature of the adaptive estimator. - An important feature of the partitioning realization of the optimal estimator is its natural decoupled structure. Indeed all the filters needed to implement the adaptive estimator can be independently realized. This fact has the following great advantages [SI : - these filters can be implemented using parallel processing machines, saving enormous computational time - the overall realization is robust with respect to failure of any of the parallel processors 5. SIMULATION RESULTS Since the plant noise (w(k)) is dependent on the sea state and the current, it is reasonable to consider its covariance matrix Q(k) as unknown. During the fist simulation experiment Q(k) was assumed unknown but constant while in the second experiment Q(k) was assumed unknown and periodic in time. The Kalman filter was designed and tested for the matched and the unmatched case. The Adaptive Lainiotis Filter (ALF) was designed with two linear filters, each one matched to a specific value of Q. Two different realizations of the ALF were tested. The first was this described by (27)-(30) in the previous section. The second realization, instead of averaging the 6-conditional states estimates with respect to the a-posteriori probabilities p(o/k), simply selects the 0-conditional state estimate with the highest a-posteriori probability. This is reffered to as the decisiondirected ALF (ALF-DD). In order to asses the performance of the above filters, the mean square error, averaged over 50 Monte Carlo runs was used : mc MSE = [ x(k) - &/k) 3 i= 1 The simulation results are shown in figs From these graphs the following conclusions can be made : - Mismodeled dynamics play an important role in the overall filter performance.the performance of the Kalman filter corresponding to the matched cases is better than that corresponding to the mismatched ones, while the performance of the ALF is bounded above by that of the matched Kalman filter and below by that of the mismatched Kalman filter. - Both in the time invariant and time varying situation, the ALF perfoms better and correctly identifies the model in a limited number of steps. - The decisiondirected ALF identifies the correct model faster than the regular ALF. 4. CONCLUSIONS The problem of heave motion estimation was considered in this paper. The need for on-line implementation of the filtering algorithm lead us to propose the LLPSPF that has great advantages in processing time for both the time invariant model and periodic case. Since the pameters of the model are not all known in advance, the ALF was used, and its performance was compared with that of the Kalman fiter. It is shown via simulation, that the ALF identifies the correct model in limited number of steps and its performance is far better than that of the Kalman filter. REFERENCES [ 11 Ferial El-Hawary, "Compensation for source heave by use of a Kalman filter", IEEE J. of Oceanic Eng., OE-7, n. 2, [2] Ferial El-Hawary, "Pattern recognition for marine seismic explorations", in Automated pattern analysis in petroleum explorations, eds. I. Palaz, S.K. Sengupta, Springer- Verlag, [3] D.G. Lainiotis, "Optimal adaptive estimation : Structure and parameter adaptation", IEEE Trans. on AC, v.16, , [4] D.G. Lainiotis, "Partitioned estimation algorithms, I : Nonlinear estimation", J. Information Sciences, 7, , 2 280

5 1974. [51 D.G. Lainiotis, "Partitioning: A unifying frameworks for adaptive systems, I : Estimation", Roc. of the [6] D.G. Lainiotis et al., "Real time ship motion estimation using Lainiotis filters", IFAC Workshop on Expert systems and signal processing in marine automation, Denmark, [7] D.G. Lainiotis, S.K. Katsikas, "Linear and nonlinear Lainiotis filters : A survey and comparative evaluation", IFAC Workshop on Expert systems and signal processing in marine automation, Denmark, [8] S.K. Katsikas, S.D. Likothanassis, D.G. Lainiotis, "On the parallel implementations of the linear Kalman and Lainiotis filters and their efficiency", Signal Processing, 25, , [91 R.M. Hawkes, RJ. Moore, "Performance of Bayesian parameter estimators for linear signal models", IEEE Trans. on Automatic Control, AC-21, ,1976. [lo] R.E. Kalman, "A new approach to linear filtering and prediction problems", Trans. ASME J Bas. Engng., Ser D, 35-45,

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