IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL SMC-5, NO 1, JANUARY 1975 46 Modern Control Concepts in Hydrology NGUYEN DUONG, C BYRON WINN, AND GEAROLD R JOHNSON Abstract-Two approaches to an identification problem in hydrology are presented, based upon concepts from modern control and estimation theory The first approach treats the identification of unknown parameters in a hydrologic system subject to noisy inputs as an adaptive linear stochastic control problem; the second approach alters the model equation to account for the random part in the inputs, and then uses a nonlinear estimation scheme to estimate the unknown parameters Both approaches use state-space concepts The identification schemes are sequential and adaptive and can handle either time-invariant or time-dependent parameters They are used to identify parameters in the Prasad model of rainfall-runoff The results obtained are encouraging and confirm the results from two previous studies; the first using numerical integration of the model equation along with a trial-and-error procedure, and the second using a quasi-linearization technique The proposed approaches offer a systematic way of analyzing the rainfall-runoff process when the input data are imbedded in noise INTRODUCTION V ? UCH OF THE insight gained in other fields, lvi especially in systems engineering, is directly applicable to hydrology [1] Since modern control and estimation theory have been applied successfully to aerospace engineering problems (e.g., satellite tracking, orbit determination, space navigation, etc.) in the last two decades, and since there are many similarities between these problems and the identification of unknown parameters of hydrologic processes (i.e., the models are not known precisely, the system under study is stochastic and highly nonlinear, and there is noise in the observations), the application of this approach to the study of hydrologic systems has been investigated many of the component processes in hydrology are nonlinear [3] due to i) the time variability of watersheds due to the natural processes of weathering, erosion, climatic changes, etc., ii) the uncertainty with respect to the states and characteristics of the interior elements of the system in time, and iii) the inherent nonlinearity of the processes of mass and energy transfer that constitute the hydrologic cycle Thus hydrologic rocesses can be considered as y g yn d nonlinear dynamic distributed-parameter systems with partially known or unknown structures operating in a continuously changing environment The inputs and outputs of these systems are measurable, but the data obtained are imbedded in noise again with partially known or unknown characteristics For a detailed study of hydrologic systems the mathematical models developed should be nonlinear dynamic however, At the present distributed-parameter unavoidable invariance is oftentime, of models the assumption space because of a lack of data on parameter distribution The subdivision of large watersheds into environmental zones, where environmental conditions that affect the behavior of hydrologic systems can be assumed as uniform, and the use of a lumped-parameter model for each zone is then required to improve the modeling situation By routing the flow spacewise through all the lumped-parameter models representing the environmental zones, the total simulation of the entire watershed would represent an adistribute-paramete Lumped-parameter models of hydrologic systems can be divided into deterministic and stochastic models The Characteristics of Hydrologic Systems deterministic approach is often called parametric modeling Ahydrologic system may bedefinedasaninterconnection The choice of the model is determined by the type of in its natural ~~~problem to be solved Parametric models require input of physical elements that are related to water state The essential feature of a hydrologic system lies in its data with considerable detail in time, therefore, they model role in generating outputs (i.e., runoff, etc.) from inputs transient responses well and are most widely used for short(i.e., rainfall, snowmelt, temperature, etc.) or in interrelating term simulation or for prediction for water management inputs and outputs The stochastic nature of the inputs and purposes [1] Stochastic models have the advantage of outputs of hydrologic systems has been discussed by into account the chance dependent nature of hydroYevjevich Stochastic synthesis models are concerned Yevjevch [2] [2] Ilogic events Hydrologic processes are complex time-varying dis-wihtesmlioofherainhpbtenipuad tributed phenomena, which are controlled by an unknown output data (cross correlation models) and between sucnumber of climatic and physiographic factors The later cessiv vale ofe correlation models) descriptors tend to be static or to change slowly in relation In stochastic simulation models, statistical measures of to the time scale of hydrologic fluctuations Observations of hdologicviablesiar used t atefture events o results in the laboratory and in the field also indicate that which probability levels are attached However, in this case long-term records, which in many instances are not h Manuscript received June 13, 1973; revised July 24, 1974 This work avial,aeneddt.siaete'aaeeso was supported by the National Aeronautics and Space Administration stochastic model in order to obtain a proper representation of their stochastic nature Stochastic simulation models under Contract NAS8-28655 N Duong iS with the Vietnamese Air Force and the University of uulyaeue o lnigproe odvlpmn Saigon, both in Saigon, Republic of South Vietnam uulyaeue o lnigproe odvlpmn C B Winn and G R Johnson are both with the Mechanical Engi- ";equally likely" long-term traces of monthly streamfiow neering Department and the University Computer Center, Colorado State University, Fort Collins, Colo 80521 'or similar smoothly varying responses ~ Lapproximtiontto ~ ~ ~ ~taking system n 47 DUONG et al.: CONTROL IN HYDROLOGY For dynamic systems that are well-characterized by finiteorder ordinary differential equations (differential systems) domain is to be preferred, the use of the state-space approach offers a great deal of convenience conceptually, notationally, and analytically The study reported in this paper deals with the applications of the state-variable approach from modern control and estimation theory to the identification of unknown parameters of nonlinear lumped-parameter response models of hydrologic systems subject to noisy input-output data The hydrologic system examined is the rainfall-runoff process STATE-SPACE APPROACH FOR IDENTIFICATION OF NONLINEAR HYDROLOGIC SYSTEMS FROM NOISY OBSERVATIONS Techniques used in the past for the determination of the instantaneous unit hydrograph and the identification of unknown parameters in a conceptual model of a hydrologic process have not been adequate This is because the input and output hydrologic data are imbedded in noise, the hydrologic processes are nonlinear, and the changing of the environmental conditions in time may affect the model output In this section a new approach for hydrological studies is investigated using state-space concepts Techniques for optimal adaptive identification of the unknown parameters and of the control inputs for the rainfall-runoff process are presented 2.0 1.5 - - - | _ ° ,A ( 10.r 25.0 _ 20.0 i 15.0 lo o l ll _ _ 5.0 o.cX- 0.00 0.05 0.10 Qp 0.20 0.15 ins./Hr Fig Plots of ao, al, and bo versus ~~dtn = 0.30 Q, for Willscreek Basin Kulandaiswamy Model Direct runoff may be considered as the result of the storage equation can now be written as transformation of rainfall excess by a basin system The physical process of this transformation is very complex, S = ao(Q)Q + al Q + dt depending mainly upon the storage effects in the basin Kulandaiswamy [4] derived the following general ex- With the continuity equation pression for the storage N ~ dnQ + , bm(Q,U) dmU dtm (1) S= , a,(Q,U) dt-S = U(t - QT) n=O 0.25 b0U (3) (4) t the rainfall-runoff process can be represented by the followwhere S iS the storage, t iS the time, N and M are and an(Q, U) and bm(Q, U) are parametric functions of the ing differential equation dU direct runoff Q and the excess rainfall U To apply (1) to the d2Q dQ (5) + Q = U -boal + A(Q) study of the rainfall-runoff process in a particular watershed, dt dt dt the values of N and M must be determined Both Q(t) and U(t) are available in the form of curves and differentia- where tion has to be done by numerical approximation techniques A(Q) = ao + Taking into consideration the nature of the curves repdQ resenting Q(t) and U(t) and the magnitude of error likely to be introduced by numerical differentiation, the values A plot of Q versus A(Q) was made for various basins, and of N= and M were adopted by Kulandaiswamy two types of regions could be differentiated The system equations for these regions are the following For this simplification, (1) reduces to 1) Nonlinear region: (2) S = a0(Q,U)Q + a1(Q,U) 2Q + b0(Q,U)U Q dQ + -odU (6 atdt2Q+(1 integers,indfertalquio Qda Plots of a0, a1, and bo versus Qp, the peak discharge, for a representative watershed are illustrated in Fig Kulandaiswamy found that a1 and bo vary from storm to storm but not show any well-defined trend in the variations, hence, he took these two parameters as constants [5] The 2) Linear region: 1t d 2Q +2dQ dt2 dt =U- odU dt (7) IEEE TRANSACTIONS ON 48 SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1975 The general nonlinear storage equation (1) proposed by to the authors was related to Prasad's work, only the Prasad Kulandaiswamy has been adopted by many hydrologists model will be used in the investigation of the performance in the simulation of the rainfall-runoff process by lumped- of the proposed identification schemes parameter response models, but the approach used in the determination of the model parameters has been criticized Reformulation of Prasad Model in State Space by Eagleson [6] Kulandaiswamy used characteristics of the Equation (9) can be written as surface runoff hydrograph at peak discharge (dQ/dt = 0), q d2Q (I KNQN-1 dQ_ I Q + on the falling limb (U = 0), and on the rising limb up to the U K2! dt VK2! dt2 \K2! and plots of to get various end of rainfall excess ao, a,, (10) bo versus Qp and Q versus A(Q); then from these plots the evaluation The c determined m, were values of a1, c1, bo, and The estimation of the unknown parameters K1, K2, of ao from a single discharge (the peak discharge) and a1,bo and N can be accomplished by applying a Kalman filtering from a portion of the surface runoff hydrograph should be algorithm to the augmented two-dimensional state vector g g method that can evaluate the model replaced by some other r model Defining the following the observation and coefficients over the full range of observed discharges transformations X=Q X2=Q X3=K1 Prasad Model X,= X2 X =Kl X4=- X5=N K2 A simplification of the preceding model by retaining (11) only two terms of the general nonlinear storage equation was proposed by Prasad [7]; in this case the storage and the assumption that the model coefficients are time invariant, (10) can be written in the following form equation is jXi r (8) IX2 X3x4x5X5 S = KQN + K2dQdt KQ K where K1, K2, and N are the parameters to be estimated In his study, Prasad assumed that K1, K2, and Nare constant for a particular hydrograph Using the continuity equation, the following differential equation for the rainfall-runoff process is obtained K2 d2Q + QN21 dQ KINQNl dQ + t Q U U (9) X2 + X4(U -X) (12) 0| |X3| = 40 LXJ L or, in abbreviated notation, J X(t) = f[X(t),R(t)] (13) Equation (13) is the model equation in state space Let Y(t) denote the measured runoff that is imbedded in noise; one then has X Comparing the Prasad model for nonlinear storage ((8)) with the Kulandaiswamy model defined by (3), one can recognize 1X21 that ao(Q) and a, have been taken as K'NQN-l and K2, respectively, and bo = (14) Y(t) = [I 0 0| X3 |+ D In the Prasad model, the time-invariant coefficients X4 Kl, K2, and N were originally evaluated by a trial-andXs error method, which is computationally inefficient and t requires the knowledge of the initial conditions with or,i n abbreviated notation, (15) Y(t) = h[X(t)] + v(t) sufficient accuracy These coefficients were later computed by Labadie [8] using quasi-linearization, which represented where v(t) represents the noise term a significant improvement over the trial-and-error proThe formulation of the estimation schemes for estimating cedure but which has two main inherent weaknesses: p i) initial approximations must be within, or at least close for the study of the rainfall-runoff process implemented or the convex the solution, optimal to, region surrounding were from the storm of April 10, 1953, over used data convergence is not convergence does notThe iS not not attained; attained; iii)i.ifif 'convergencedoes convergence River Basin above Catlin, Vermilion of the Fork South ~~~~~~~~the ^ * t is it initial of set approximations, result for a particular selected were data These in order to compare the to determine systematicallyabetter set ofIllinois no possible to present state-space methods with the results from earlier tesults frm aprox inti results from these initial annroximations These approaches for estimating the model coefficients studies using the same data set All results presented were ar sutbeol.o eemnitcmdl n r o obtained from computer programs written in Extended suitable for the analysis of real input-output data where the Fortran for use on the Control Data Corporation 6400 values to be used in the model are imbedded in partially digital computer at Colorado State University known or unknown noise The two methods proposed herein are very useful in solving parameter identification Adaptive Control Approach An adaptive control algorithm for the estimation of the problems for this case Since the Prasad model for rainfallrunoffis typically nonlinear, and the data set made available state variables and the unknown parameters of a timeo a a 49 DUONG et al.: CONTROL IN HYDROLOGY varying system with noisy data has been developed by Duong [9] It is suitable for those cases where the errorcovariance matrix of the input disturbance is unknown Basically, it consists of modeling the hydrologic system as an optimal stochastic control problem, applying the separation principle [10], using a Gauss-Markov process to model the unknown parameters, and using an adaptive filtering scheme to estimate the observation error covariance matrix R and the process error covariance matrix Q Survey papers in this area were presented by Sage and Husa [11], Weiss [12], and Mehra [13] The matrices F(t) and G(t) in the linearized expression of (12) have the following forms: 0 E1 -X3X4X5X O0 O F(t)- ' -X2X4X5XX5?' 0 0 01 E2 E3 0O0 ?0 19 176 168 18 160 17 16 152 144 1536 v1v 128 14 13 120 l 12 11 104 #096 1.20~~~~observation 10 1.18 0 1.16 0 0_ 11 1.14 E2 X5U-X - E3 =-Xx - -_0 1.10 1)Xj-2-X4 1.08 X2X3X5X' X #observation 1.12 where -X2X3X4X5(X5 # I (16) = Sof 1.06 -l -X2X3X4XX5-X log (XI) 1.04 and 1.02 B(t) = [O X4 0 0]T 1.00 An error term e(t), used to model such effects as unknown dynamics and truncation errors, was chosen to be a zeromean white noise process with covariance matrix ~~~~~~pdated O 01 l = | 0] # observation Fig Estimation of time-invariant parameters by linear adaptive control approach O O | as soon as new values of the estimate were obtained The values of the time-invariant parameters in the model converged relatively quickly to their optimal estimates lfter only ten iterations, as shown in Fig The resulting optimal estimates are The observation noise covariance matrix has the form KI = 19.99 R =0.0001 K2 = 0.16 N = 1.18 R was taken to be much smaller than because the observed Defining the same coefficients as those used by Labadie outputs of the rainfall-runoff system were relatively noisefree compared to the inputs; also, there are errors in the [8] yields model equations due to the incomplete knowledge of the N = 1.18 16 A 3.77 Al KIN K2 K2 nature of the system For this linear stochastic control problem, the statetransition matrix (D must be computed carefully to avoid There values are not much different from those obtained introducing further errors into the model equations, there- by Prasad using a trial-and-error procedure and numerical fore, in the study second-order terms were also taken into integration of the nonlinear equation Prasad obtained the the computation of D ,Since the state variables of the rain- values 3.79, 0.076, and 1.27, for At, A2, and N, respectively fall-runoff system vary relatively slowly with time, the Labadie, using a quasi-linearization technique, obtained interval between two consecutive observations (one hour) is 4.473, 0.0943, and 1.27, respectively The differences are subdivided into only ten subintervals to avoid excessive mainly due to the noise terms introduced into the model computational requirements, and the value of 'D is computed equations to make them more realistic and to conform with the nature of the rainfall-runoff data using standard approximation formulas Values of the estimated surface runoff compared with The control gains were computed first, based on nominal values of the state Later, values of the control gains were observed runoff are shown in Fig with and without = 50 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1975 .01 0030 ~ 0027 A_ o o 0024 with control without control observed runoff | 0021 0018 -.0015s 1/ 0012 C; > m ° ooosl j r_ m.\ 0009 | control of the input disturbance From these results, the following remarks can be made i) Adaptive control of the inputs is important for the analysis of the rainfall-runoff process Without controlling the rainfall data, the system identification results are poor, therefore, the estimation of the surface runoff from these noisy inputs is unacceptable ii) Controlling the inputs requires little additional computer time For the particular data set under study, with the same initial conditions mentioned previously, only six seconds were needed in addition to the computer time used by the Kalman filtering scheme, including the adaptive estimation of the error-covariance matrices R and Q iii) Even with input-control procedures, the approximation of a nonlinear system by linearized equations will not provide good results if the system under study is highly nonlinear The effect of adaptive estimation of the model-error covariance matrix can be seen from Fig In this test case, the Q matrix was 0006 0003 10 15 20 e adaptive est of Q 0030 O - 0027 0024 0021 ie 0err observed runoff 0 01 25 # observati,on L Fig Estimate of direct runoff with and without control err .O 100 i _ Without adaptive estimation of the Q_ matrix at each state, the filter started to diverge at the twentieth observation Finally, the performance of the adaptive control algorithm with and without rectification of the nominal state at each stage is shown in Fig for comparison As expected, rectification improves the filter performance Nonlinear Estimation Approach If the input noise characteristics are known, a nonlinear estimation approach may be used to estimate the state and unknown parameters in a hydrologic system This approach is simpler and provides faster convergence than the adaptive control approach but does require knowledge of the input - noise characteristics - Optimal estimation in the nonlinear case involves the solution of an infinite-dimensional process, as shown by 0015 Kushner [14] Since the computational aspects of the truly a 0012 L' toptimum nonlinear filter are prohibitive, several approaches 0012 i l | to suboptimal filtering have been proposed in the past few C; X 00 years (Friedland and Bernstein [15], Schwartz and Stear [16], Athans et al [17], Sage and Melsa [18]) These 0006 algorithms can be roughly subdivided into the so-called first-order filters and higher-order filters with increasing 0003and computational requirements In hydrology Z l §complexity ~~because the estimate of the state of a system is usually 105 j ! q obsrvation | not required to be highly accurate only the extended (first# bsrato order) Kalman filter iS considered here z ~~~~Toimprove the performance of the extended Kalman one can pursue the technique derived by Denham ~~~~~~~~~~~filter, Fig Effect of adaptive estimation of model-error covariance and Pines [19] to reduce the effect of the measurement function (h) nonlinearity, which occurs very often in matrix, hydrology when the output data from a hydrologic system are imbedded in noise This technique is a local iteration 0018 || - - - \ i ' DUONG et al.: CONTROL IN HYDROLOGY 51 algorithm based on relinearization about the new estimate This is described in detail in Duong [9] and the application to the Prasad equation is described The continuity equation may be written as dS = U dt Q + w(t) - =_( K = Wb(t - T) = K2 ~QN-1dQt + 0027 - 0024 - 0021 - 0018 - 0009 0006 Let Xl = Q, X2 = Q, X3 = K1, X4 = 1/K2, X5 = N; then one obtains the following state equation: 0003 X3 X4 w X2 l - - ] x4 + W (19) 20 176 19 168 181 160 ~~~~~17 152 The obervatin eqution i the sme as[~jW 19) in abbreviated notation, X = f(X(t),U*) + g((X(t),w(t)) (20) The observation equation is the same as in the previous section The matrix of partial derivatives is ° X2 E1 F(t)= 0 _0 20 obse1 rvation 25 [_ 5_0 or, 15 10 Fig Effect of rectification of nominal state X2 + X4(U* - XI) =0 - (18) V5-1 |X4X3->X5X fixed normal state ak 0015 ) (U*-Q)00 U(01 +( XI with rectification , O-o o observed runoff Combining this equation with the nonlinear storage equation, one obtains dQ ~ - 0030 (17) where U* denotes the actual rainfall input data and w(t) represents the input noise which is assumed, as usual, to be a zero-mean white noise process with covariance matrix E[w(t)w(T)'] -XX4Xsl S- 0 0 -X2)XIX5XX5-' 0 E E 0 0 0 where 13 120 128 0 0_ 112 ~~~~~~~~.104 11 X5-1 10 10 15 20 25 # observation 1.20 1.18 1.16 ~~~~~~~~~~~~~~~~~~~~~.096 10 15 20 25 # observation t 1.14 1.12 -1)XIJ52 X4 E2= U-X X2X3X5Xl5-' E3= -X2X3X4Xj 5-(1( + X5 log (X1)) - 144 14 12 (21) E= -X2X3X4X5(X5 14 16 1.10 1.00 06 Assume that the observation and input noise error 04 covariance matrices have the following values, respectively, 1.02 R = 0.001 1.00 W-=0.01 10 15 20 25 and the given initial conditions are the same as in the pre # observation parameters by iterated extended ViOUS eape exampe.Asshow ssonlin Fg Kalman filter l.66,the h ptimu plu valus ausoofFig Estimation of time-invariant IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1975 52 recently begun finding their way into civil systems, but it is a anticipated that the major developments will be made in the civil systems area during the next decade This paper has presented the application of optimal estimation theory to identification problems in hydrological systems The methods presented here offer the following advantages A 01 0030 a fixed fie err o adaptive estimation oferr .0027 observed runoff over previously used techniques: i) the state-space formulation of the problem provides a useful alternate procedure 0024 >,|/ g for parameter identification of the process under study; ii) the state-space formulation provides a systematic method 0021 r y \ I f of analyzing the rainfall-runoff process; iii) the identification schemes presented are sequential and adaptive and can 0018 0018 13x handle data sets that may be imbedded in noise with unknown characteristics; iv) the techniques may be applied 0015 to stationary or nonstationary parameters; v) the com.0012 putational requirements are quite insignificant 4.009 I The adaptive control method may be used to investigate streamflow prediction for small watersheds using only the { X0 a 0006 measured runoff at the mouth of the watershed In this case, K< a 'dthe adaptive control method can be used to estimate the / J 0003 unknown precipitation inputs from measured runoff, and good short-term streamflow prediction can be obtained by the state and the estimation-error covariance S 10 1S 20 j 25 | propagating # observation matrix forward in time It is anticipated that the techniques presented could be _ applied to large watersheds by dividing the large watershed Fig Estimate of direct runoff with and without adaptive estimation into environmental zones and using runoff observations of model-error covariance matrix.of moe-rorcvracemto identify the unknown parameters Routing models for adjacent zones could be incorporated into the system and the use of the state variable approach would e the model parameters converge a little faster than the case ~~~~~~~~~~~~equations, lead to a matrix equation representing the response model for the entire watershed Then, using total measured runoff, estimates are all unknown parameters in the various environmental zones K1 = 19.998 -= 0.162 K2 I N= 1.182 One would expect these results to be better than those obtained previously, since in this case a nonlinear filter has been used, therefore, model-error has been reduced This fact is verified by a better comparison of the estimated outflow to the measured outflow as presented in Fig Since the estimates of model coefficients converge to stable values, one may conclude that these coefficients are constant or can be approximated by time-invariant parameters for a particular hydrograph The effect of adaptive estimation of the model-error covariance matrix was also tested in this case Using the same set pf initial conditions as previously used, the divergence of the filter was less rapid than in the adaptive control approach The result is also shown in Fig -* and in the routing models could be estimated simultaneously [1] [2] [3] [4] [5] [6] [7] [8] REFERENCES D R Dawdy, "Mathematic modeling in hydrology," The Progress of Hydrology, in Proc Ist lnt Sem for Hydrology Professors, V Yevjevich, "The structure of inputs and outputs of hydrologic systems," in Proc 1st Bilateral United States-Japan Sem Hydrology, Jan 1971 J Amorcho and G T Orlob, "Nonlinear analysis of hydrologic systems," Water contribution 40, 1961.Resources Center, Univ Calif., Berkeley, V C Kulandaiswamy, "A basic study of the rainfall excess surface relationship in a basin system," Ph.D dissertation, Univ rOfunoff of Illinois, Urbana, 1964 V C Kulandaiswamy and C V Subramanian, "A nonlinear approach to runoff studies," presented at the Int Hydrology Symp., Fort Collins, Colo., Sept 6-8, 1967 P S Eagleson, General Report, presented at Tech Session 2, Int Hydrology Fort Collins, 1967 hydrologic Col., ofSept.a 6-8, R "Nonlinear simulation Prasad,Symp., regional system," Ph.D dissertation, Univ of Illinois, Urbana, 1967 J Labadie, "Optimal identification of nonlinear hydrologic system response models by quasi-linearization," M.S thesis, of California, Los Angeles, 1968 ~~~~~~~~~~~~~~~~~~~Univ [9] N Duong, Dep control concepts in hydrology," Ph.D ; ~~~~CONCLUSIONS Mechanical Engineering, Colorado State Univ., ; ~~~~~~~~~~~~~~~~~~dissertation,"Modern The methods of systems analysis have been applied Fort Collins, 1973 H W Sorenson, "Controllability and observability of linear, extensively to military, aerospace, and industrial systems ~~~~~~~~~[10] stochastic, time-discrete control systems," Advan Contr Syst., - in the past The development of these methods and the applications began with the military requirements in the Second World War and then were expanded significantly # Into the aerospace and industrial fields They have JUSt Theory Appl., vol 6, pp 75-158, 1968 A P Sage and G W Husa, "Algorithms for sequential adaptive ~~~~~[11]estimation of prior statistics," in Proc 8th IEEE Symp Adaptive Processes, 1969 I M Weiss, "A survey of discrete Kalman-Bucy filtering with ~~~~~~~~~~~~~[12] unknown noise covariances," presented at the AIAA Guidance, 53 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL SMC-5, NO 1, JANUARY 1975 [13] [14] [15] [16] Control, and Flight Mechanics Conf, Santa Barbara, Calif., 70-p 995, 1970 R K Mehra, "Approaches to adaptive, filtering," IEEE Trans Automat Contr (short papers), vol AC-17, pp 693-698, Oct 1972 H J Kushner, "Dynamical equations for optimum nonlinear filtering," J Differential Equations, vol 3, pp 179-190, 1967 B Friedland and I Bernstein, "Estimation of the state of a nonlinear process in the presence of non-Gaussian noise and disturbances," J Franklin Inst vol 281, pp 455-480, 1966 L Schwartz and E B Stear, "A computational comparison of several nonlinear filters," IEEE Trans Automat Contr vol AC-13, pp 83-86, Feb 1968 [17] M Athans, R P Wishner, and A Bertolini, "Suboptimal state estimation for continuous-time nonlinear system from discrete noisy measurements," IEEh Trans Automat Contr., vol AC-13, pp 504-514, Oct 1968 [18] A P Sage and J L Melsa, Estimation Theory with Application to Communications and Control New York: McGraw-Hill, 1971 [19] W F Denham and S Pines, "Sequential estimation when measurement function nonlinearity is comparable to error," AIAA J., vol 4, pp 1071-1076, 1966 A Multilevel Approach to Planning for Capacity Expansion in Water Resource Systems W SCOTT NAINIS AND YACOV Y HAIMES, MEMBER, IEEE Abstract-Supply and demand models are developed as aids in course, to the different and often conflicting uses of water planning future water resource expenditures The supply model yields least-cost schedules of projects to meet assumed levels of demand The demand model, conversely, allocates available water supplies within a regional input-output economy to most efficiently utilize the available resource As more and more water is made available at various future points in time, the extra amount will be used for less important and productive uses However, as additional water supply is planned, the costs of supplying the resources become excessively high, since less efficient supply projects must be built For these reasons, the supply and demand models are placed in a hierarchical framework whereby the supply/demand projections are adjusted so that the additional costs of supply match the additional economic value of the supply This requires a formal extension of classical benefit-cost analysis, which is referred to as projects such as fresh water withdrawal, flood control, * r power, re on,cand oflw augmenonr fo prollutio control The objectives of water resource projects are typically more general, a good example being those stated by the Water Resources Council [36]: enhancement of 1) national efficiency, 2) environmental equality, 3) regional andl-being develpment, Single project cost-benefit analysis ignores the interaction between projects as they are constructed over time When planning for a long time horizon, all feasible project sites must be considered within the planning phase Single dynamic benefit-cost analysis I INTRODUCTION IIT.[ATER resource projects are often large in magnitude, VV both physically and monetarily Time is also an important factor in water resource planning; the time period between the conception and full utilization of a large water resource project has been estimated as 30 years Standard practice in water resource project analysis has been to formulate a cost-benefit ratio or a net benefit for an individual project based upon a given economic period (life-time), discount rate, and assumptions on quantification of direct and indirect costs and benefits of that project [16] More recently the concepts of multipurpose projects and multiobjective planning have been proposed for the standard Of project selection [22] The multiple purposes refer, of project analysis simply cannot yield insight into long term Manuscript received August 15, 1973; revised April 16, 1974 This Contract GI-34026 A revised version of this paper was presented at the International Conference on Cybernetics and Society, Washington, W S Nainis is with the Operations Research Section, Arthur D Little, Inc., Cambridge, Mass Y Y Haimes iS with the Department of Systems Engineering, Water Commission findings [21] both stress the fact that long range economic effects of water resource development and geographic effects due to water transfer are an important consideration In response to these characteristics, a work was supported in part by the National Science Foundation under Case Western Reserve University, Cleveland, Ohio water resource planning for a region that contains multiple river basins In the process of long range planning for the development of water resource systems to meet projected water needs, one should not ignore the "circular" effect of water on development [35] By this is meant that the availability of additional water supply (and other water resource related outputs) will encourage increased, and very likely marginal, usage of water The ultimate resolution of the circular effect could be a general economic equilibrium analysis as discussed in [32] The general equilibrium approach is seen as computationally infeasible, for the present, and requires excessive data This paper investigates methodologies for economically based planning models in water resource investment The Water Resources Planning Act [36] and the recent National planning model has been developed to consider investment in water resource supply projects and their economic ... obtained introducing further errors into the model equations, there- by Prasad using a trial-and-error procedure and numerical fore, in the study second-order terms were also taken into integration... knowledge of the input - noise characteristics - Optimal estimation in the nonlinear case involves the solution of an infinite-dimensional process, as shown by 0015 Kushner [14] Since the computational... al.: CONTROL IN HYDROLOGY 51 algorithm based on relinearization about the new estimate This is described in detail in Duong [9] and the application to the Prasad equation is described The continuity