Dispersive wave processing: A model-based solution

UCRL-JC-124477 PREPRINT Dispersive Wave Processing: A Model-Based Solution J. V. Candy D. C. Chambers This paper was prepared for submittal to the 13th Annual Asilomar Conference on Signals, Systems and Computers Pacific Grove, CA November 3-6, 1996 October30,1996 DISCLAIMER This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes. Dispersive Wave Processing: A ModeI-Based Solution J. V. Candy D. C. Chambers P.O. Box 808, L-437 Lawrence Livermore National Laboratory Livermore, CA 94550 Abstract Wave pmpagatwn thnmgh various media repwsents a significant pmbkm in mang appkcdions in a-cousi?icaand ekctnnnagnetks especially when h medium is dispersive. We pose a geneml dispersive wave propagationmodel that could easilIInqnwsent miang classes of dispersive waves and proceed to develop a mo&l-based Strwcture. The processor employing this underlyi genend solution to the model-based ‘ifisperstve wave estimation probkm is developed using the Bayesian maximum a-posterioti (MAP) approach which leads to the nonlinear extended Kaiman jilter (EKF) processor. 1. . Introduction Dispersive wave propagation through various media is a significant problem in many applications ranging from radar target identification where electromagnetic waves propagate through the atmosphere to discern the nature of a reflected pulse classifying an intruder as friend or foe, or in submarine detection and localization where the propagation of acoustic waves through the ever-changing dispersive ocean medium causes great concern when trying to detect the presence of a hostile target and track ita movements. Therefore, there ia a need to develop generic characterizations of dispersive waves whkh do not depend on the details of the physical system primarily because the required details such as governing equations for the system and their solutions may be imperfectly known. In this paper we will show that a model-based signal processing scheme applicable to any dispersive wave system can be developed from the basic properties of wave propagation in a dispersive medium. Our approach will be to develop a stat~space description of a dispersive wave measured by a sensor or array of sensors. For simplicity we restrict ourselves to one-dimensional waves, but the generalization to higher dimensions ia straightforward. The wave pulse is sssumed to be generated by an impulsive source at a known position and a known time. We consider the source pulse as a superposition of wave components of many frequencies. Since the system ia dispersive, each component propagatea at a different speed resulting in a spreading or dispersing of the pulse over space and time as it propagates. This spreading is described by the dispersion relation of the system which relates the frequency of each component to its wave number. We will show that a complete state-space representation of the wave can be formulated from the dispersion re lation combined with an envelope or amplitude modulation function. The dispersion relation completely describes the propagation properties of the dispersive system, while the envelope is related to the tiltia.1 conditions. Once specified, it is then possible to develop a generic model-based processing scheme for dispersive waves. The primary motivation for this approach follows the dispersive wave characterisation developed in the text by Whithsm 1]. ThB processor evolves directly from the modlfi J plane wave, internal wave techniques developed using an approximation of the dispersion relation 2]. In contrast, the generic dispersive approach re1ies exclusively on the underlying envelope and dispersion relation to develop an optimal Bayesisn processor. It ia thk model-baaed approach that we employ using a dynamic propagation model incorporated into an optimal estimation scheme to pr~ tide the neccsary internal wave enhancement. In section 1, we present the general dispersive wave representation and outline the correspondh-ig modelbased processor MBP) using the Bayesian maximum a posterion (MA L ) approach for the underlying wave estimation or equivalent signal enhancement problem. Next, employing this general solution, we apply it to the problem of internal wave estimation directly from an empirical dispersion relation in section 2, while applying it to simulated data in section 3. 2. Model-Based Dispersive Proce~r In this section we develop the underlying dispersive wave model and cast it into state-space form. Once this is accomplished, a Bayesian maximum a posteriori solution ia outliied and the resulting processor is shown to lead to the extended Kslman filter (EKF) solution 3. .The complete model-based solution ia then specilJ ed m terms of the model and algorithm showing that intuitively the d=peraion relation is the fundamental quantity that must be known a-priori. 2.1 Dispersive Stat+A3pace Propagator First, we develop the stat~space representation of a genersi dispersive wave system obtained from a simple physical characterization of a dispersive wave measured by a sensor or array of sensors. In conatrsst to the usual approach of clsssifyhg non-dispersive waves in terms of their inherent differential equations . .-. -., (hyperbolic, el- liptic, etc.), we use a solution rather than propagation equation for our dispersive prototype. Therefore, following Whitham 3], we define a generic dispersive wave as any system wL “ch admits solutions of the general form l+, t) = +, t) Sin[@(z, t)] (1) where u is the measured field and cr(z, t), 6(z, t) are the respective envelope or amplitude modulation and phsse functions. The phase ia assumed to be monotonic in x and t, and the envelope is assumed to be slowly varying compared to the phase. The phase function describes the oscillatory character of a wave, while the slowly varying envelope allows modulation of the wave without destroying its wavdike character. The local values of wave number and jiegwency can be deiined as a K(Z, t) = ~, +, M t) ~.— ~ . (2) These functions are also assumed to be slowly varying and describe the frequency modulation of a dispersive wave train. By slowly varying we mean that we can approximate the phase junction as e(z, t) = I@, t)z – (4@, t)t. (3) The combination of Eqs. 1 and 3 can be considered an asymptotic solution to some dispersive wave system. To complete the specilkation of a dispersive wave system, we define the disperswn rvlatwq u = U(IC,x, t). ThM is generally an algebraic function of K(Z, i!) but can also depend on x and t separately to represent time varying, nonuniform wave systems. Here we will write w = w(~) where the z and t dependence through the wave number function tc(~, t)and any system nonuniformity is implied. This and the envelope are the only parta of the description which are unique to the particular type of wave system under investigation. The choice of dispersion relation enables the differentiation between acoustic ra&ation, electromagnetic radiation, ocean surface waves, internal gravity waves, or sny other wave type. Thus, the dispersion relation is equivalent to the governing equations for a particular wave system [1]. Our only restriction on it in th~ paper ia that it is independent of the envelope, CY(Z,t). ThB restricts our formulation to linear dispersive waves. From Eqs. 2 and 3 it can be shown that the phase fronts of any wave travel at the phase speed defined by (4) while the points of constant wave number ~ travel at the group velocity defined by (5) These two speeds are not the saime in general and are functions of wave number K. The group velocity has the additional significance of being the energy propagation speed for the wave system, that is, the energy in the wave packet is carried at th~ velocity. As such it plays a central role in the statespace formulation of a general dispersive wave system. Now consider the problem where an impulse occurs at time t = O at the spatial origin, z = O. The impulse can be represented by the superposition of wave components with various wave numbers. A wave train is generated by the impulse and propagates aways from the origin. Each wave number component in the train propagatea with group velocity given by Eq. 5. If a sensor is placed at a distance z away from the origin, then the toad wave number, K(Z, t),observed at time t >0 ia related to z by the group velocity, z = Cg (K(z, t))t. (6) This relation ia simply a restatement of the definition of the group velocity as the speed at whkh a given wave number 6(z, t)propagates in the wave train. It is the group velocity that plays the dominant role in dispersive wave propagation. Note also that the resulting wave train does not have cohstant wavelength, since the whole range of wave numbers is still Dreamt. that is,A(z,t)= *. The actual sensor measurement u(t) at z is given by combining Eqs. 1 and 3, that is, u(t) = CY(Z,t; K) sin[~(t)z – ti(~)t], (7) where we have suppressed the dependence of ~ on x and allowed the envelope to be a function of IC. We will choose the wave number ~(t)at z as our state variable and develop a dynamical equation for its temporal evolution by differentiating Eq. 6 using the chain rule (8) to obtain o= d/c dc~(tt) —’t ~ d~ [1 + Cg(lc). (9) Now solving for ~ and substituting the expression for group velocity in terms of our original dispersion relation, we obtain $=-H*I[=1-1’ “0> “0) which shows how the wave number evolves dynamically (wmporsJly) as a function of the underlying dispersion..relation w(~). If we couple this expression back to the original dispersive wave solution, then we can have a general continuous-time, spatio-temporal, d~persive wave, stat-space representation with state defined by IG(t. J uppose we sample th~ wave with an array of L sensors oriented in the direction of propagation, that is, x+xt,l= l,... , L, giving L wave numbers and L initial conditions. If the entire state-space is to be initialized at the same time, care must be taken to select the initialization time to be after the leading edge of the wave has passed through the entire array. Let to be the time the leading edge passes the . ...... sensor L, the sen.mr farthest, from the origin, then xL = Cg(EL(G)) %, where KL(to) is the wave number for the maximum group velocity. The initial conditions for the other sensors in the array are obtained by solving Ze = c (nt(tO)) tO for each 4. Thus, the “spatially” yled, &spersive wave, stat~space model is given && =-+ [*]-’, [*] &hd%(6(tk+l)luk+l) =o. U =it M A (11) L. We can further diacretize th~ model temporally by sampliig t + t&, and also by replaciig the derivatives with their first difference approximations. Since we know that the dispersive medium in which the wave propagatea is uncertain, we can also characterize uncertainties with statistkal models, one of which is the well-known Gauss-Markw model 13. Performing these operations we achieve our desir al result, the dispersive wave state-space discrete, spati&emporal, Gauss-Markov model: /C~(t&+l)= tL&) Kt(tk)– K.I )Shl = CY.(t&; K&J -1 ; t?=l,..., L, (12) C [K, ~(tk+llk+l) where this expression is the corrected estimate (below) and shown in the algorithm of Table 6.63 of [3]. Thus, the model-baaed solution to this wave enhancement problem can be ~leved using the nonlinear extended Kalman filter (EKF) algorithm which is given (simply) as: tk] + V(tk), t(tk) . Dispersive Model-Based = U(tk)- ii(tk+llk) Correction where a.], CIO]are the respective nonlinear vector system an measurement functions with the corresponding state and measurement mvariancea defined by p~tk+l) and &(tk), with the system and measurement jacobians, A [~] s $# and C [K] - ~. The Subsequent development of our processor will rely on this statistical formulation for both simulation and estimation. 2.2 ., ii~(t~+lpJ= 12(tk;k.t(t~+llk)) sin [ke(tk+llk)~g – U(k[)tkj /!=l,..., L; J . . (14) = ~(tk+llk) + K(tk+l)~(tk+l), Innouatim K(tk+l) = a [~, t~] + Aw(tk), = Diierentiating the posterior density and noting that k(tk+ll~) and ‘f(tk) are both functions of the data set, SS (See Uk, We obtain that &Ap(tk+l) = ii(tk+llk+l) Refr. 3, pp. 80-81 for more details) [@k)Z~ – fd(/Cc)t&] + t@k), where Wt(tk) and u~(t~) are assumed zero mean, gauasian noise sources with respective Covarian-, &w(t&). The mer~ v~tor Gau-M~kov &Jt~), form can be found in Refr. 3 and is simply given by U(tk) (13) p Prediction: [9 1 +Awt(tk); $ [*I The minimum variance solution to this problem can be obtained by the maximizing a posterior density, leading to the so-called MAP equatio~ t,%; ZQ(t) = a (t; Icl) sin[fc@t – (4&)t], IQ(to), 1=1,..., and a set of noisy measurements, {U(tk ) }, FIND the best (minimum error variance) estimate of the wave, that is, find fi(tk). Processor Next we outline the model-based processor (MBP) baaed on the vector representation of the wave numbers and wave-field, that is, we define the vectors, U(tk) [U,(tk), ,U~(tk)]Tand /C - [~l(tk), . . . . ~~(tk)]~. Once the wave is character.ked by the underlying Gauss-Markov representation, then diqwrsiue waue estimation problem can be specified by GIVEN the approximate Gauss-Markov model (above) characterized by the dispersive wave stat+space model (16) k(tk+llk+l) = it(tk+llk) + K(t&(tk) Gain (17) K(tk) = p(tk+~,k)cT(tk)R:l(t~) Here the predicted and corrected covariances are given in Table 6.63. Fkom the Table we see that in order to construct the optimal dispersive wave modelbased processor, we must not only specify the required initial conditions, but alaa the respective system and measurement jscobians: ~ and ~. For our general solution, we see that C [K/+, tk] = ~(tk;K4)SiIl [6/(t~)Q– . . ....ti(Ke)tk] , t = 1. . . . . L. The jacobians then follow easily as A [/$f(t~), tk] = 1-$(1 -*),4 c [Kt(tk), tk] = *sin [Kt(tk)22 – =1>...,L; ti(fi)tk] Thii completes the section on dispersive wave estinw tion, next we consider the application of this processor for internal waves. 3. Internal Wave Processor When operating in a stratified environment with reladients any excitation that disatively sharp density turbs the pycnocline rdensity profile) will generate intenud waves [4]. To apply our processor to the internal wave enhanment problem, we first recall the original dispersive wave system of Eqns. 1 and 3 and apply this structure to the internal wave dynamics where u represents the measured velocity field and cr(z, t) and 6(z, t) are the respective envelope and phase to be specified by the internal wave structure. We define the internal wave dispersion relatwn by w = (JO(K)+ fc(z, t)v. (18) where we have included the additive veloci~ term to account for the effects of a doppler shift created by the ambient current. That is, in the original formulation we have replaced the position variable by z ~ z — vt which leads to the above equation. For uJ~), we use a d~persion model based on some empirical results, the Barber approximation (5], for internal wave dispersion and group velocity. Thus, we have U*(K) = coK(t) 1 + $@(t) (19) with CO ia the initial phase velocity and NO is the maximum of the Brunt-V&is&la frequency profile. It ia also possible to derive the following approximation to the amplitude modulation function as dtd = ~ [CJIC)]3’2 sin[4K,tdTtd (20) where A ia a constant amplitude governing the overall envelope gain, TW ia a temporal window width and cJ~) is the phase speed defined below. We use the simulator [6] to synthesize internal wave dynamics corresponding to an internal wave field experiment performed in Loch Lknhe, Scotland “m 1994 [7,8]. The simulation (shown in Figures la) was performed baaed on the SNR defined by: SNR := u2/%~, where u is the energy in the true image (sealed to unit variance) and &V is the measurement noise variance extracted from experimental data. The spatiotemporal velocity field includes the entire range of 361 temporal samples at At = 5sec representing a propagation time of approximately 1/2 hour and spatially we assume a line array of 30 sensors (enough to illustrate the wave structure) spaced at Ax = 4rn representing w aperture of 120 meters. For this ... internal wave sim.. .. ulation, we choose a peak Brunt-V-ala frequency of NO = 0.137r/sec and a long wave (~= O) phase speed of CO = 0.34rn/s with the ambient current (doppler The data was contaminated shift) of v = –0.lcm/sec. with additive gaussian noise at a - 23dB SNR. Here we observe the effect of additive gaussian noise in obscuring (visually) the internal wave dynamics of the synthesized velocity field. We see the noisy internal wave spatio-temporal signals synthesized, while the enhanced wave estimates are shown in lb. The results appear quite good, however, they must be analyzed from the statistical perspective to actually assess the overall performance of the processor. With this information in hand, we specify the required dispersive wave stak space model as: f@&+l) = lcl(t~) + *+ a!%’ ‘k‘to; Thus, the MAP estimator can now be constructed using the formulation of the previous section and substituting the internal wave model and above noted jacobiana. a model-based signal proWe used SSPACKYC; cessing toolbox available in MATLAB to design the processor [9,10]. The model-based wave enhancer ia able to extract the internal wave signatures quite effectively even though the embedded dispersive wave model is just an approximation to the actual wave dynamics. We show the spati-temporid interpretation (and display) of the noisy and enhanced signals in Figure 1. To confirm the processor performance that we observed visually, we perform individual whiteness tests on each of the temporal sensor outputs. We use the corresponding sensor innovation indicating the difference between the noisy and predcted measurement along with the bounds predicted by the processor. Here 95% of the innovationa should lie within the bounda indicating a reasonable processor. Performing these tests on all of the sensor innovation outputs reveala that each irdkidually satisfied these atatistkal tests indicating a zero mean/white innovations and a near optimal processa. Thus, these simulations show that the GaussMarkov formulation enables us to capture various uncertainties of internal waves as well as its associated statistics in a completely consistent framework. To further assess the feasibility of th~ approach, we ran the enhancer on other synthetic data sets at OdB,– 10dB, and –13dB with similar results, that is, the model-based approach enabled a near-optimal Bayeaian solution with all sensor innovation sequences statistically testing as zero-mean/white. We are currently applying this technique to carefully controlled experimental measurements and the preliminary results have led us to continue to puVue this approach. [5] B. C. Barber, “On the dispersion relation for trapped internal waves? J. Fluid Mechanics., 252, 31-49, 1993. ‘w 25 [6] M. Milder, Internal Waves Radiated by a Moving Source.- Analytic Simulation, RDA Report, RDATR-2702-007, 1974. 10 [7] D. D. Mantrom, “Loch Linnhe ’94: Test Operations Description and On-Site Analysis US Activ1994. ities: LLNL Report.,UCRL-ID-119197, Is 10 [8] H. F. Robey and D. L. Ravizza, “Loch Lmnhe Emeriment 1994: Baclwround Stratification and Sh&ir Measurements Pa 1: Profile Summary and Dispersion Relationa”, LLNL Report.,UCRL-ID119352, 1994. s 1 Time [9] J. V. Candy and P. M. Candy, “SSPACK-PC: A model-based signal processing package on personal computers; DSP Applic., 2,(3), 33-42, 1993. (SCC) F@we 1: Equivalent Model-Based Internal Wave Spati*Temporal Enhancement (a) Synthesized wave (-23 dB SNR). (b) Enhanced interred wave. 4. Summary In this paper we have developed a generic dispersive wave processor. Starting with a general solution to the propagation of waves in a dispersive medium, we developed the approximate (due to nonlinear systems) maximum a-posterior (MAP) solution using a Bayesian formulation of the wave field enhancement problem. The results are significant in that all that ia required ia the envelope or equivalently amplitude modulation function and diapersion relation to completely specify the underlying wave system. It is in fact the particular dwperaion relation that enables the dMerentiation between acoustic and electromagnetic rsdation, ocean surface wavea and internal gravity waves or seismic waves or any wave system for that matter. The generahty of this result enables the specification of a particular wave system by ita underlying envelope and dispersion and then applying the algorithm to obtain the MAP solution. [10] MA TLAB. Boston: The Math Works Inc., 1993. [11] J. V. Candy and . D. H. Chambers. “Model-based enhancement of internal wave images.” to appear, Special Issue on Imaging in the Ocean, IEEE J. Oceanic Engr., Jan. 1997. Acknowledgements We would like to acknowledge the motivation and support of Dr. R. Twogood, Pro am Leader LLNL Imaging & Detection Program anr Mr. James Br~, Project Leader for the Radar Ocean Imaging Project. Tbia work was performed under the auspices of the Department of Energy by the Lawrence Llvermore National Laboratory under contract W-7405-Eng-48. ., ., References [1] G. B. Whitham, Linear and Nonlinear New York Wiley, 1974. Waves, [2] J. V. Candy and D. H. Chambers. “Internal wave processing: a model-based approach.” IEEE J. Oceanic Engr., Vol. 21, (l), 37-52, 1996. [3] J. V. Candy, Signal Processing: The Model-Based Approach. New YorkMcGraw-Hill, 1986. [4] J. R. Apel, Principles of Ocean Physics, York: Academic Press, 1987. New ...... Technical Information Department • Lawrence Livermore National Laboratory University of California • Livermore, California 94551