Local ensemble Kalman filtering in the presence of model bias

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1 Tellus (2006), 58A, Copyright C Blackwell Muksgaard, 2006 Prited i Sigapore. All rights reserved TELLUS Local esemle Kalma filterig i the presece of model ias By SEUNG-JONG BAEK 1, BRIAN R. HUNT 2, EUGENIA KALNAY 3, EDWARD OTT 1,4 ad ISTVAN SZUNYOGH 3, 1 Istitute for Research i Electroics ad Applied Physics, ad Departmet of Electrical ad Computer Egieerig, Uiversity of Marylad, College Park, MD, USA; 2 Istitute for Physical Sciece ad Techology, ad Departmet of Mathematics, Uiversity of Marylad, College Park, MD, USA; 3 Istitute for Physical Sciece ad Techology, ad Departmet of Atmospheric ad Oceaic Sciece, Uiversity of Marylad, College Park, MD, USA; 4 Departmet of Physics, Uiversity of Marylad, College Park, MD, USA (Mauscript received 6 April 2005; i fial form 5 Decemer 2005) ABSTRACT We modify the local esemle Kalma filter (LEKF) to icorporate the effect of forecast model ias. The method is ased o augmetatio of the atmospheric state y estimates of the model ias, ad we cosider differet ways of modelig (i.e. parameterizig) the model ias. We evaluate the effectiveess of the proposed augmeted state esemle Kalma filter through umerical experimets icorporatig various model iases ito the model of Lorez ad Emauel. Our results highlight the critical role played y the selectio of a good parameterizatio model for represetig the form of the possile ias i the forecast model. I particular, we fid that forecasts ca e greatly improved provided that a good model parameterizig the model ias is used to augmet the state i the Kalma filter. 1. Itroductio I may situatios the dyamics of a real process may differ from those of the est availale model of that process. We refer to this differece as model error. Model error is thought to e a key issue i weather forecastig i that the presece of model error ca lead to cause large discrepacies etwee the forecasts ad the true atmospheric states. I this coectio, we ote (i) that Kalma filters have ee cosidered for estimatig atmospheric states to e used as iitial coditios i forecast models (Ghil et al., 1981), ad, (ii) that the geeral Kalma filter methodology has log ee adapted to accout for model error (Friedlad, 1969). Recetly, the esemle techique has ee proposed as a computatioally feasile meas of applyig Kalma filterig to the very high dimesioal states iheret i gloal atmospheric models (Evese, 1994; Houtekamer ad Mitchell, 1998, 2001; Aderso, 2001; Bishop et al., 2001; Hamill et al., 2001; Whitaker ad Hamill, 2002; Ott et al., 2004a). Oe of our goals i this paper is to ivestigate the icorporatio of model error correctio i a esemle Kalma filter. Here, we cosider oly the case where the evolutio of the model error is govered y a determiistic equatio, i.e. the model error has o radom com- Correspodig author. sjaek@glue.umd.edu DOI: /j x poet. We refer to this type of error as model ias, sice whe the state of the model is descried y a proailistic variale, as is the case i data assimilatio, such errors ecome equal to the expected error i the model forecast. We will restrict our cosideratios to the example of oe particular esemle Kalma filter, the Local Esemle Kalma Filter (LEKF) proposed y Ott et al. (2004a). [This scheme has ee successfully tested (Szuyogh et al., 2005) o a reduced resolutio versio of the operatioal Gloal Forecast System (GFS) of the Natioal Ceters for Evirometal Predictio (NCEP) for the perfect forecast model sceario.] We elieve that our results i the preset paper, usig the LEKF example, may also e more geerally applicale, providig a idicatio of what to expect if model ias correctio is attempted usig other related esemle Kalma filter methods. I additio, some of our ideas may also e useful i desigig weakly costraied 4DVAR schemes (e.g. Derer, 1989; Zupaski, 1997), which also allow for model errors. The techique we propose elogs to the family of schemes usually called state space augmetatio methods (Coh, 1997). I these techiques the state vector is augmeted with the ucertai model parameters, ad the augmeted state is estimated usig the forecast model i cojuctio with oservatios. I priciple, the ucertai parameters ca occur i otherwise completely kow forecast model equatios. I such a case, the augmeted state space approach may provide a accurate 293

2 294 S. J. BAEK ET AL. estimate of the parameters eve for a highly chaotic system, as recetly demostrated o a simple model y Aa ad Hargreaves (2004). I reality, the equatios goverig the motio of the atmospheric flow are ot kow exactly, thus ucertaities also arise due to our limited kowledge of the dyamics. Also, estimatig all parameters of the forecast model equatios would ot e computatioally feasile due to the large umer of parameters. A practical approach, first suggested y Derer (1989), is to assume that the ucertaities i the forecast model ca e approximately represeted i the form of a limited umer of ulk error terms. The the task is to estimate the parameters of the ulk error terms. We recall that sice the error terms are modeled as radom vectors, the parameters to e estimated are the mea errors (model iases). The iformatio ecapsulated i the ias ca e used either to modify the forecast model equatios or to modify the aalysis scheme. Here we follow the secod approach. That is, we treat the forecast model as a lack ox, that does ot yield the true time evolutio of the atmosphere, ad we attempt to use this lack ox i cojuctio with oservatios to accout for model ias i the state estimatio. The two key compoets of the aforemetioed strategy are the selectio of a ias model that efficietly represets the ias ad the desig of a computatioal strategy that ca efficietly estimate the parameters of the ias model. The most commo assumptio is that the ias is costat or has a simple evolutio i time. It is also frequetly assumed that the ucertaities i the forecast model state ad the ias are ucorrelated. These assumptios were used to derive the ias estimatio schemes of Dee ad Da Silva (1998), Dee ad Todlig (2000), Carto et al. (2000), Marti et al. (2002) ad Bell et al. (2004). The scheme we propose allows for correlatios etwee the ucertaities of the forecast model state ad the ias. This additioal flexiility is ecessitated y the structure of our techique (see Sectio 2.4), ad is affordale due to the high computatioal efficiecy of the LEKF approach. I Sectio 2, we itroduce three differet ias models. Bias Model I is a simple additive correctio to the model forecast. Bias Model II is motivated y evisioig a situatio i which the forecast model evolutio takes place o a attractor that is shifted from the attractor for the true system evolutio. Bias Model III is essetially a comiatio of Bias Models I ad II. Sectio 3 presets the results of umerical experimets with the Lorez- 96 model (e.g. Lorez ad Emauel, 1998) for several cases of the differece etwee the forecast model evolutio ad the evolutio of the true state. Coclusios ad discussio follow i Sectio 4. A mai result is the importace of selectig a ias model that effectively parameterizes the form of possile ias i the forecast model. I particular, if the ias model ca parameterize the possile ias of the forecast model, the our results suggest that sustatial improvemet i forecasts may result. O the other had, if the parameterizatio of the model ias through the ias model does ot sufficietly capture the form of the true iases i the forecast model, the sustatial forecast improvemets were ot otaied i our umerical experimets. 2. Bias modelig ad data assimilatio The discrepacy etwee the forecast model evolutio ad the evolutio of the real atmosphere has two sources: (i) due to umerical solutio o a grid of a fiite umer of poits, the forecast model state is a fiite dimesioal represetatio of the ifiite dimesioal atmospheric fields, ad (ii) the equatios that gover the true evolutio of the atmosphere are ot kow exactly. These two sources of forecast model errors are ot idepedet, sice the errors i the forecast model formulatio are maily associated with the iheret prolems of cosiderig oly a limited umer of iteractios etwee the fiite umer of compoets of the state vector ad the imperfect represetatio of the effects of the sugrid processes o the motios at the resolved scales. Deotig the true atmospheric state at t y x t, the true atmospheric evolutio is deoted x t +1 = Ft ( x t ), (1) where F t is the (ukow) true atmospheric evolutio operator ad x t +1 is the true atmospheric state at time t +1 = t + t. Deotig the forecast model state at time t y x m, the lack ox produces a forecast model state at time t +1, x m +1 = ( ) Fm x m, (2) where F m is the forecast model evolutio operator that mimics F t. Note that the dimesios of x t ad x m are, i geeral, differet; for example, i the case of real weather forecastig the true state is ifiite dimesioal ad the forecast model state is fiite dimesioal. I what follows we will treat a sceario i which x m ad x t have the same (fiite) dimesioality, ad we heceforth assume this circumstace. With respect to the situatio of atmospheric weather forecastig, this assumptio restricts the character of the errors ad their meas (iases) that ca e addressed y our ias models. I particular, we regard our treatmet to follow as addressig oly those types of forecast model iases that ca e represeted as dyamics i the state space of the forecast model variales. Thus, we igore dyamics that occur at smaller scales tha the forecast model resolves. O the other had, if we thik of the dyamics at the uresolved scales as radom perturatios to the forecast model dyamics, our methods may e ale to correct for the mea ias due to such perturatios. Meawhile, the ucertaity i the small-scale fluctuatios is modeled as represetativeess error i the oservatio error statistics. I this sectio, we defie three ways of modelig the ias that ca arise due to forecast model error. We refer to these as Bias Model I, Bias Model II ad Bias Model III.

3 LEKF FOR MODEL BIAS 295 Fig. 1. Illustratio of Bias Model I: x t is the true state evolvig accordig to F t from the previous true state, ad x m is the forecast model state evolvig accordig to F m from the previous true state Bias Model I I geeral, it is desired to have the forecast model state as close to the true state as possile so that, assumig that the forecast model ad the true evolutio operators are the same, the forecast model state stays ear the true state after its evolutio. I practice, however, the forecast model evolutio operator differs from the true evolutio operator. As a result, eve if we evolve the forecast model state from a iitial coditio correspodig to the true state at the iitial time (e.g. x m 1 = xt 1 ), it is likely that the forecast model state departs from the true state as it evolves. I Bias Model I, we attempt to estimate t = Ft ( x t 1) F m ( x t 1), (3) i.e. the departure of the forecast model state from the true state as illustrated i Fig. 1. I order to estimate t, we must use some model of how it is related to past values of the ias i t (i < ). For example, we ca assume that the ias is costat i time. If so, the true system evolutio ca e writte i terms of the model evolutio as follows: x t = Fm( ) x t 1 + t, (4) t = t 1. (5) Though we could write this system more cocisely as x t = F m (x t 1 ) +, where is a ukow parameter vector, we write the system i terms of the augmeted state vector (x t, t )iorder to facilitate the iterative estimatio of oth x t ad t y our data assimilatio procedure. More geerally, we ca replace eq. (5) with aother model for the ias of the form, t = G ( t 1, xt 1), (6) where G is the evolutio operator for the ias correctio term. Aother alterative is to assume that the model error evolutio is a Markov process. I that case, t is represeted as t 1 multiplied y a matrix that descries the temporal covariace e- Fig. 2. Illustratio of data assimilatio with Bias Model I: Data assimilatio produces a uiased aalysis for the true state, x a, ad a aalysis for the ias correctio term, a. twee the model errors at differet spatial locatio, ad the righthad side of eq. (6) also icludes a additive radom term (e.g. Jazwiski, 1970; Daley, 1992; Zupaski, 1997). The give a estimate (x a 1, a 1 ) of the augmeted state vector at time t 1 (the aalysis from the previous data assimilatio), we take the forecast (or ackgroud ) of this vector (x, ) at time t to e x = ( ) Fm x a 1 +, (7) = G ( a 1, xa 1), (8) where we have assumed ias evolutio y eq. (6). We the perform data assimilatio usig (x, ), ad the oservatios at time t to otai the aalysis (x a, a ). This way of takig forecast model error ito accout is illustrated i Fig. 2, ad is the geeral scheme used i several previous methods appearig i the literature (e.g. Dee ad Da Silva, 1998; Dee ad Todlig, 2000; Carto et al., 2000; Marti et al., 2002; Bell et al., 2004). The vector y i Fig. 2 is the oservatio of the true state at time t, which we assume to oey a model equatio of the form, y = H ( x t ) + ɛ, (9) where H is the oservatio operator mappig the true states to the oservatios. ad ɛ is the oservatioal oise. Basically, i Bias Model I it is supposed that the est forecast is produced whe the iput to the forecast model evolutio is as close to the truth as possile. Oe ca imagie prolems with this. For example, say that atmospheric alace for the forecast model is ot the same as that for the true atmosphere. The, if a very good estimate of the true state at time t is iserted ito the forecast model, the forecast model state at time t could ofte e out of alace, ad spurious gravity wave activity might e excited. I the practice of umerical weather predictio, such spurious gravity wave activity is preveted y a filterig process, called iitializatio, applied to the fields provided y the data assimilatio process (e.g. Machehauer, 1977; Baer ad Triia, 1977; Lych ad Huag, 1992; Lych, 1997). The geeral wisdom is that a well-desiged Kalma filter might elimiate the

4 296 S. J. BAEK ET AL. eed for iitializatio process. This cosideratio motivates Bias Model II Bias Model II A cosequece of the imperfect model is that the forecast model system has a differet attractor from the true system. I some cases, it might e desirale to let the forecast model state follow its ow attractor, sice pluggig a very good approximatio of the true state ito the forecast model system ca result i completely differet dyamics (like gravity wave excitatio). I additio, oe ca evisio a situatio i which forecast model dyamics ad true dyamics ecome more similar through a (a priori ukow) coordiate trasformatio. For istace, such trasformatios were rigorously derived to correct for trucatio errors i umerical solutio of the two-dimesioal Navier Stokes equatios (Margoli et al., 2003). Havig foud a similar trasformatio for the weather predictio model, we may otai a etter estimate of the true trajectory y applyig this trasformatio to a appropriate forecast model trajectory after it has ee computed tha y forcig the forecast model state to e close to the truth ad the computig its trajectory. For simplicity, we assume the trasformatio is just a shift of the forecast model state to the true state, ad we defie the ias c t at time t y c t = Ft( x 1) ( ) t F m x m 1 = F t( x 1) ( t F m x t 1 1) ct. (10) A schematic illustratio of this ias model is show i Fig. 3. The forecast model state is ot pushed to the true state. Istead, it mimics the true dyamics i a shifted locatio of the state space. Ulike the ias i Bias Model I, the ias i Bias Model II at time t depeds ot oly o F t, F m, ad x t 1, ut also o the previous ias c t 1. Noetheless, we may assume that for some choice of c t 1, the correctio term ct approximately oeys a simplified evolutio model such as c t = ct 1, or more geerally c t = G c (c t 1, xt 1 ). I terms of this model, we approximate the true system evolutio y the augmeted model system, x t = ( Fm x t 1 ) ct 1 + c t, (11) c t = G c ( c t 1, xt 1). (12) For this ias model (ad for Bias Model III to follow), our goal is ot that the aalysis state vector x a closely approximates the true state x t, ut rather that it approximates the est forecast model state x m = xt ct from which to make future forecasts. Thus, we rewrite eqs. (11) ad (12) as x m = Fm ( x m 1), (13) c t = Gc ( c t 1, xm 1), (14) where G(c, x) = G c (c, x + c). We ca the write the ackgroud augmeted state vector (x, c ) i terms of the previous aalysis (x a 1, ca 1 ) as follows: x = Fm ( x a 1), (15) c = Gc ( c a 1, xa 1), (16) I takig this approach, oe must keep i mid that the ias should e added to the forecast model state vector wheever makig comparisos to oservatios. Thus istead of eq. (9), we use the oservatio model, y = H ( x m + ct ) + ɛ (17) whe performig data assimilatio. The aalysis (x a, ca ) represets a approximatio to the augmeted state vector (x m, ct ), ad thus forecasts made usig x a as the iitial coditio should also e corrected y the approximated ias i order to etter predict the true system. Data assimilatio with Bias Model II is illustrated i Fig. 4. Fig. 3. Illustratio of Bias Model II: x t ad xm evolve accordig to their ow dyamics ut the ehavior of the forecast model is similar to the ehavior of the truth. Fig. 4. Illustratio of data assimilatio with Bias Model II i the case H(x) = x: Data assimilatio produces a aalysis of the est forecast model state, x m = xt ct, ad a aalysis for the correctio term, ct.

5 LEKF FOR MODEL BIAS 297 To the est of our kowledge, Bias Model II is a ovel approach to the effects of model errors o the accuracy of the state estimates. Hase (2002) also argued for the model attractor, ut he suggested the use of a multi-model approach as opposed to the state augmetatio we propose. Fig. 5. Illustratio of a local state cetered aout locatio m Bias Model III I Bias Model III, we comie Bias Model I ad Bias Model II. Formally, we comie the equatios descriig the previous two ias models i the followig maer. We evolve the aalysis augmeted state vector (x a 1, a 1, ca 1 ) to the ackgroud at the ext step usig the model x = ( ) Fm x a 1 + (18) = ( G x a 1, a 1, ) ca 1 (19) c = Gc ( x a 1, a 1, ca 1), (20) ad we compare the ackgroud state with oservatios accordig to eq. (17). Sice x ad c represet the est availale approximatios to x m ad ct prior to the data assimilatio at time t, the oservatio icremet we use is y H ( x + c ). (21) Notice that if G (x,, c) = 0, the this model reduces to Bias Model II, while if G c (x,, c) = 0, this model reduces to Bias Model I. I its simplest form, our model uses G (x,, c) = ad G c (x,, c) = c. However, we fid a slightly differet ias evolutio fuctio to e advatageous i some situatios (see Sectio 3.6) Augmeted local esemle Kalma filter For the purposes of all susequet discussio we heceforth take the system state at time t to e a scalar variale x,i defied o a discrete oe-dimesioal spatial domai, i = 1, 2,..., N. Thus we represet the system state as a vector x = [x,1, x,2,...,x,n ] T, where the superscript T deotes the traspose. Oce a suitale model for the ias is chose, it ca e icorporated ito the formulatio of the Kalma filter. For example, i the case of Bias Model III, the ew equatios ca e otaied y replacig the state x, y the augmeted state, v = [x,, c ] T, i the Kalma filter equatios. Here, the correctio terms, = [,1,,2,...,,N ] T, ad c = [c,1, c,2,...,c,n ] T,havethe same dimesio, N, which is typically equal to the umer of grid-poit variales i a umerical weather predictio model. By isertig the augmeted state ito the Kalma filter equatios, we assume that ψ (v ), the ackgroud proaility distriutio of the augmeted state, is Gaussia; that is, ψ ( [ ) v exp 1 ( v 2 ) T ( ) v P 1 ( v v ) ] v, (22) where v is the ackgroud mea of the augmeted state, ad P v is the ackgroud error covariace matrix for the augmeted state. The mai computatioal challege i desigig a augmeted Kalma filter is to fid a computatioally efficiet approach to estimate P v, whose dimesio icreases y N whe a ew parameter is added to the state. Oe frequetly applied approach to reduce the computatioal urde associated with the estimatio of P v is to assume that may etries of the matrix are zero, e.g. y assumig that the (o-augmeted) state ad the ias parameters are ucorrelated. We propose a differet approach, which ivolves estimatig the ackgroud mea ad the ackgroud error covariace matrix y a esemle, ad solvig the esemle Kalma filter equatios locally i grid space applyig the Local Esemle Kalma Filter (Ott et al., 2004a) to the augmeted state. The LEKF scheme estimates local states as illustrated i Fig. 5. I particular, cosiderig the LEKF procedure without model ias correctio (i.e. as i Ott et al., 2004a), for each poit m o the spatial grid, we cosider a eighorhood cosistig of the 2l + 1 poits cetered at m; these poits have locatios m l, m l + 1,..., m,..., m + l 1, m + l (e.g. l = 3 i Fig. 5). At time t = t, the LEKF does data assimilatio o local regios cetered at each grid poit usig the local state, x (m) = x,m l x,m l+1. x,m. x,m+l 1 x,m+l. (23) The gloal aalysis state (i.e. the aalysis state at t = t at each grid poit over the etire grid) is the take to e the state at the ceter of each local regio (see Ott et al., 2004a, for further discussio). I order to adapt the LEKF to correct for model ias, we augmet each local state to iclude the ias estimate of the ias model employed. For example, for Bias Model III, we form a augmeted local state, v (m) = [x (m), (m), c (m)] T, for the data assimilatio at locatio m. Similarly, for Bias Model I, v (m) = [x (m), (m)] T, ad, for Bias Model II, v (m) = [x (m), c (m)] T. Sice the augmeted local state is derived from the gloal state v, it ca e also assumed to have a Gaussia

6 298 S. J. BAEK ET AL. distriutio, ψ m ( v (m) ) exp { 1 [ v 2 (m) v (m)] T [ P (m)] 1 [ v (m) v (m)]}, (24) where v (m) is the ackgroud mea of the augmeted local state, ad P (m) is the ackgroud error covariace matrix for the augmeted local state. I this way, the dimesio of the space for data assimilatio is reduced to 2(2l + 1) for Bias Model I or II ad to 3(2l + 1) for Bias Model III. A importat property of this scheme is that it allows for (ad also requires) the estimatio of cross-correlatios etwee ucertaities i the state estimates ad ucertaities i the estimatio of the model ias terms. 3. Numerical experimets 3.1. Experimetal setup For testig our assimilatio scheme, we cosider the Lorez- 96 model (Lorez ad Emauel, 1998) which is defied y the system of differetial equatios, dx i = (x i+1 x i 2 )x i 1 x i +, i = 1,...,N, (25) where x 1 = x N 1, x 0 = x N, x N+1 = x 1 ad is a costat. The variales form a cyclic chai ad may e thought of as roughly aalogous to the values of some uspecified scalar meteorological quatity at N equally spaced sites alog a latitude circle. For compactess of otatio, we will also represet eq. (25) as dx = L(x), (26) where x = [x 1, x 2,...,x N ] T. We solve eq. (25) with a fourthorder Ruge Kutta method usig a time step of 0.05 dimesioless uits for which the system is computatioally stale. Lorez ad Emauel (1998) cosider this time step as roughly correspodig to 6 h of real atmospheric evolutio. With = 8.0 ad N = 40, Lorez ad Emauel demostrate that the system (25) results i a westward (i.e. i the directio of low idex of locatios) progressio of idividual maxima ad miima ad a eastward progressio of the ceter of activity with a domiat waveumer of 8. I additio, they also fid that the system is chaotic with 13 positive Lyapuov expoets ad a Lyapuov dimesio of Throughout our umerical experimets we use = 8.0 ad N = 40. I what follows we will assume that our forecast model dyamics is give y eq. (25) ut that the true dyamics of the system whose state we are cocered with oeys dyamics that may differ from those of our forecast model. We will cosider situatios i which the true dyamics differ from the forecast model i three ways, which we refer to as Type A truth ias, Type B truth ias, ad Type C truth ias. The dyamical ehaviors of the true system i these three cases are as follows: dx = L(x) + β (Type A), (27) dx = L(x + ζ) (Type B), (28) dx = L(x + ζ) + β (Type C), (29) where β = [β 1,β 2,...,β N ] T ad ζ = [ζ 1,ζ 2,...,ζ N ] T. Whe the true dyamics is descried y the same equatio [eq. (25)] as the forecast model, we say that the forecast model is perfect. Note that Bias Model I would e a atural choice if it were kow that the deviatio of the true dyamics from the model dyamics (26) was such that the true dyamics eloged to a family of systems of the form give y (27) (Type A truth ias). Similar statemets apply with regard to the relatio etwee Bias Model II ad Type B truth ias ad etwee Bias Model III ad Type C truth ias. With small values of β ad ζ, we cojecture that the systems (27) (29) exhiit ehaviors similar to those of system (25). I our umerical experimets, the elemets of β ad ζ vary i space (i) ad have the forms ( β i = A si 2π i 1 ), (30) N ( ζ i = B si 2π i 1 N ), i = 1,...,N, (31) where A ad B are scalar costats. The true states are geerated y itegratig oe of the three eqs. (27) (29), while the forecast model states are geerated y itegratig eq. (26). The evolutio operators, F t ad F m, are the itegratios of the aove eqs. (26) (29) from some time t to t + t where t = 0.05 ad the states are availale at every discrete time t = t 0 + t, where t 0 is the time at which a experimet egis ad is a positive iteger. We assume that the oservatios are availale at every time t for 0 ad the state variales themselves are directly oserved. Thus the oservatio operator i eqs. (9) ad (17) is the idetity operator [i.e. H (x) = x]. We also assume that the oservatioal oise ɛ has zero-expected value ad is ucorrelated, white ad Gaussia with variace σ 2. Thus the local oservatio error covariace matrix is a diagoal matrix whose compoets are σ 2. Correspodigly, we geerate our simulated oservatios (9) y addig ucorrelated Gaussia radom umers with variace σ 2 to the true state variales x t i ad form a local oservatio y (m) = [y,m l,...,y,m+l ] T. Throughout our umerical experimets, we take σ 2 = Data assimilatios are doe at every itegratio time t. The aalysis error is defied as e a = xa xt, (32) for Bias Model I, ad e a = xa + ca xt, (33)

7 LEKF FOR MODEL BIAS 299 for Bias Model II ad Bias Model III, where x a is the esemle mea of the aalysis ad where c a is the esemle mea of the estimate of the Type II ias. We use the root-mea-square (rms) of the aalysis error, RMS { } e a 1 = N N ( ) e a 2,,i (34) i=1 to assess the quality of the aalysis at a give time, ad the time mea of the rms error over a log time iterval T, e a = 1 T 0 +T = 0 +1 RMS { e a }, T 1, (35) to measure the overall performace of the assimilatio scheme. Here, 0 is the time we allow for the aalysis to coverge to the true state. To improve the aalyses i our umerical experimets, we employ variace iflatio, ˆP a (m) ˆP a (m) + μ k I k, (36) where ˆP a (m) is the local aalysis error covariace matrix defied i the iteral coordiate system (Ott et al., 2004a) whose asis is the set of eigevectors of the local ackgroud error covariace matrix P (m), μ is a iflatio coefficiet, ad = Trace{ ˆP a (m)}. This particular form of variace iflatio was proposed i Ott et al. (2004a) where it is referred to as ehaced variace iflatio. Ehaced variace iflatio has the effect of ehacig the estimated proaility of error i directios that formally show oly very small error proaility. [This modificatio of ˆP a (m) also modifies the esemle perturatios through the square root filter; see Ott et al. (2004a).] The geeral purpose of employig a variace iflatio is to correct for the loss of variace i the esemle due to o-liearities ad samplig errors. Most importatly, variace iflatio ca also stailize the Kalma filter i the presece of model errors, as it was show i Ott et al. (2004) for the Loraz-96 model ad i Whitaker et al. (2004) for the NCEP GFS model preparig a historical reaalysis data set. Sice variace iflatio schemes are computatioally less expesive tha the state augmetatio method, we hope to see that the techique we propose here lead to larger improvemets i the accuracy of the state estimates tha what ca e achieved y simply tuig the variace iflatio coefficiet μ. For the dimesio of local states used i the LEKF, we select 13 (i.e. l = 6) which is kow to e a good choice for the Lorez model (25) with = 8 ad N = 40 (Ott et al., 2004a). Hece, the augmeted local states have 26 dimesios for the states used i Bias Model I or Bias Model II, ad 39 dimesios for the states used i Bias Model III. I our umerical experimets, we choose the umer of esemle memers to e the same as the dimesio of the augmeted local states so that the local ackgroud error covariace matrix has full rak. This choice meas that the esemle size is 13 whe ias is ot estimated i the assimilatio, 26 whe Bias Model I or II is used, ad 39 whe Bias Model III is used. Thus we take ito accout the added dimesioality of the augmeted local states, aticipatig that this icreased dimesioality ecessitates correspodigly icreased esemle size i order to properly represet it. This icreased esemle size is part of the added computatioal cost that is paid i order to correct for model ias. I practice, for give computer resources, the eed for a large esemle may thus ecessitate cosideratio of eefit trade-offs amog esemle size, local domai size, model resolutio, etc. Fially, for ias evolutio [eqs. (8), (16), (19) ad (20)] we use G (x,, c) = ad G c (x,, c) = c, util Sectio 3.6 where we cosider differet evolutio Perfect forecast model We first test our ias models for the case of a perfect forecast model, i.e. for the case whe the true values of β ad ζ i eqs. (27) (29) are 0. I this case, the evolutio operators for the true state ad the forecast model state are idetical, F t = F m. I order to geerate the true states for this ru, we first itegrate eq. (25) for 10 4 time steps from a radom iitial coditio, allowig the system to approach its attractor. After this, we perform data assimilatio at every time step. The iitial esemle memers for the first data assimilatio are geerated y addig idepedet, zero mea, ormally distriuted radom umers of variace 1.3 to the true state at every spatial poit i. Before otaiig the rms time mea of the aalysis error (35), we ru data assimilatios to allow covergece. Past this time (deoted 0 ), it is foud that the rms aalysis error reaches a statistically steady state i which it fluctuates aout a temporally costat mea value which we deote e a. The data plotted i Fig. 6 show the time-averaged rms aalysis error (35) as a fuctio of the iflatio coefficiet μ (36). Here, the rms aalysis error is averaged over T = time steps. For the case i which o state augmetatio is employed i the assimilatio (data plotted as symols) the est performace is otaied ear μ = for which e a = The other three curves i Fig. 6 show the aalysis errors for the cases whe the same oservatios are assimilated with usig the three differet ias models of Sectio 2.4 i the state estimatio. I these cases, we try to estimate a ias that is zero i reality. The estimated ias terms ted to fluctuate aout zero, resultig i a slight ( 8%) icrease of the error. The aove results from examiig this case provide a stadard agaist which we ca compare results that we will susequetly otai for situatios with error i the forecast model. I Fig. 6, we see that the miima of e a appear at a lower iflatio coefficiet whe the states are augmeted y a estimate of the ias, (i.e. for Bias Models I ad III). I order to see why this occurs we cosider the local perturatios for the augmeted

8 300 S. J. BAEK ET AL Data assimilatio with Type A truth ias Fig. 6. Time-averaged rms aalysis error, e a, versus variace iflatio coefficiet, μ, for the perfect forecast model experimet: With a perfect forecast model, ay attempt to estimate ad correct for a ias results i slightly higher aalysis error. local ackgroud, δv (m) = v (m) v (m), (37) where {v (m)} are the esemle memers of the augmeted local ackgroud. These perturatios are used to estimate the local ackgroud error covariace matrix (Ott et al., 2004a) P (m) = V (m) [ V (m)] T, (38) where V (m) = k 1 2 [ δv (1) (m) δv (2) (m) δv (k+1) (m) ] (39) I this experimet, we perform data assimilatio usig the three augmeted local states as descried i Sectio 2.4 ad a uaugmeted state whe the true state is evolved usig eq. (27) with A = 0.2 = 1.6 i eq. (30) correspodig to Type A truth ias. Agai the forecast model state is evolved usig eq. (25). We ca approximate the ias t give y eq. (3) as follows. Recall that F t ad F m are the time t maps of the true dyamics [eq. (27)] ad the forecast model [eq. (26)] ad that eq. (3) is ased o the assumptio that x m (t 1 ) = x t (t 1 ). Takig the differece etwee the true equatio (27) ad the model equatio (26), d (xt x m ) = L(x t ) + β L(x m ) β, (41) the, itegratig eq. (41) for the time iterval t 1 t t with the iitial coditio x t (t 1 ) = x m (t 1 ) = x t 1, to otai t = x t (t ) x m (t ), yields t x t x m β = β t. (42) t 1 Usig eqs. (3) ad (42) we otai t β t. For the situatio i this sectio, t = 0.05, ad we have take β to e costat i time, β i ( = A si 2π i 1 ) N ( = 1.6 si 2π i 1 ). (43) N Time-averaged rms aalysis errors for each case are show i Fig. 7. I the case where the ias is ot estimated, the error is aroud at μ 0.7, which is still lower tha the rms error of the oisy oservatios. If we, however, augmet the state usig Bias Model I the error is reduced dramatically, ad slightly more if we augmet the state usig Bias Model III, yieldig e a = ad 0.061, respectively, at μ 10 5 [Fig. 7()]. If, however, we augmet the state usig Bias Model II, the the ad k + 1 is the umer of the local esemle memers. We rewrite eq. (37) usig eq. (7) as δv (m) = [ δx δ ] [ (m) = (m) δ x (m) + δ δ (m) ] (m), (40) where {δx (m)} are perturatios for the local ackgroud, x = Fm (x a j) 1 ), ad {δ( (m)} are perturatios for the local predictio for Bias Model I. I our experimets we oserve that {δ (m)} are oly weakly correlated with each other ad almost ucorrelated with {δ x (m)}. Effectively, therefore, ucorrelated radom vectors are added to the state perturatios {δ x (m)} i eq. (40). I cosequece, P (m) (38) is effectively iflated mostly o the diagoal compoets y the amout of the variace of {δ (m)}. (This effective iflatio created y usig eq. (7) to otai the ackgroud will also e oserved whe the forecast model is ot perfect as show i the followig susectios.) Fig. 7. Time-averaged rms aalysis error, e a, versus μ for the case of Type A truth ias: Note that () shows the same results as (a) for Bias Model I ad Bias Model III ut for a differet vertical scale.

9 LEKF FOR MODEL BIAS 301 Fig. 8. The average ias estimate of locatio i is show as. The approximate true ias β i t is show as the solid curve. rms error is aroud at μ 0.3, worse tha what is otaied whe o ias estimatio is employed. Here, we see agai that usig the uiased ackgroud (7) for data assimilatio effectively iflates the local ackgroud error covariace matrix, ad a smaller variace iflatio yields the lowest aalysis error, e a. The good results otaied whe the state is augmeted usig either Bias Model I or Bias Model III might reasoaly e ascried to the fact that estimatio of t ca e regarded as correctig for precisely the form of truth ias that is preset whe the truth evolves y eq. (27). I Fig. 8, we plot a, the time average of the esemle mea of the ias estimate, a = 1 T 0 +T = 0 +1 a, (44) a = 1 k+1 a( j), (45) k + 1 i=1 where T = 2000 ad 0 = , for the experimet with the state augmeted usig Bias Model I at μ = 10 5 (where the rms error is miimum i Fig. 7). We see that a agrees well with the approximatio to t give y (42) ad (43) (show as the solid curve). Also, although the shape is ot show here, for the case that the state is augmeted usig Bias Model III, a agai agrees well with (42) ad (43). We ow examie the aalysis errors usig Bias Model I ad III. I Fig. 9, we plot the aalysis error (32) averaged over 5000 time steps, e a = 1 T 0 +T = 0 +1 e a, (46) (here, T = 5000 ad 0 = ) for the case that o ias estimatio is performed i data assimilatio ( ), the case that the estimatio is performed usig Bias Model I i the assimilatio ( ), ad the case that the ias estimatio is performed usig Bias Model III ( ). The variace iflatio coefficiets are μ = 1.0, 10 5, ad 10 5, respectively, at which the errors, e a, are miimum for each case. I order to uderstad why Bias Model III does etter tha Bias Model I for this case, recall that oth models determie the ackgroud state x at time t y x = Fm (x a 1 ) + where Fig. 9. Time-average of the aalysis error as a fuctio of locatio: The Type A truth ias is corrected est whe we perform the assimilatio usig Bias Model III. is approximately costat i time [eqs. (7) ad (18)]. However, Bias Model I tries to make x close to xt, whereas Bias Model III tries to make x + c close to xt, where c is also approximately costat i time. Suppose that has coverged to the timeidepedet approximatio β t of t [see eq. (42)], that c has coverged to a costat vector c, ad that the aalysis at time t 1 is perfect: x a 1 + c = xt. Cosider the model trajectory x m (t) of eq. (26) with x m (t 1 ) = x a 1 ad the true trajectory x t (t) of eq. (27) with x t (t 1 ) = x t 1. The xt = xt (t ) ad x = xm (t ) +β t, ad the ackgroud is most accurate if x + c = xt. As i eq. (41), x t ( x + c) = x t (t ) ( x m (t ) + β t + c) = x t (t 1 ) ( x m (t 1 ) + β t + c ) t + [L(x t (t)) + β L(x m (t))] t 1 = x t 1 ( x a 1 + β t + c) = + t t t 1 [L(x t (t)) L(x m (t))] + β t t 1 [L(x t (t)) L(x m (t))]. (47) Thus, we desire that the average value of L (x t (t)) L(x m (t)) to e as small as possile over the iterval t 1 t t. Usig c = β t/2 makes this average zero to first order, ad is thus superior to usig c = 0, which correspods to Bias Model I. (see Fig. 10.) This is cofirmed i Fig. 11, from which oe oserves c a β t/2 i the experimet with Bias Model III.

10 302 S. J. BAEK ET AL. Fig. 10. Figure (a) depicts the case c = 0, correspodig Bias Model I, while figure () depicts the case c β/2 i Bias Model III. Fig. 11. The average ias estimate of locatio i is show as +. The the value of β i t/2 is show as the solid curve Data assimilatio with Type B truth ias I this experimet, we simulate a ias y evolvig the true state with eq. (28) with temporally costat ζ ad estimate it with three differet augmetatio methods as doe i Sectio 3.3. To otai the true value of c t, defied y (10), we use dc t = dxt dxm = L(x t + ζ) L(x m ) = L(x t + ζ) L(x t c t ). (48) A trivial solutio to eq. (48) is c t = ζ whose ith elemet is give y ( c t,i = B si 2π i 1 ) ( = 1.6 si 2π i 1 ), (49) N N where we take B = 0.2 = 1.6 i this experimet. The time-averaged rms aalysis errors for each case are show i Fig. 12. We see that, whe we augmet the state usig Bias Model II i the assimilatio, we ca correct for the ias. The miimum rms error for this assimilatio is aout ad occurs ear μ = The miimum rms error for the assimilatio with the augmeted state usig the Bias Model III is aout ad occurs at a lower μ value (as expected) of aout μ = Without ias correctio, the miimum rms error is ad occurs ear μ = 4.0, similar to what is otaied usig Bias Model I ( e a ear μ = 0.5). Also, due to the variace iflatio effect of Bias Model I, the est assimilatio result Fig. 12. Time-averaged rms aalysis error, e a, versus μ for the case of Type B truth ias. () has a differet vertical scale from (a) for trasparecy. with the augmeted state usig the Bias Model III occurs at a lower value of variace iflatio tha the assimilatio with the augmeted state usig Bias Model II. For the same reaso, the rms error, e a, for the case of assimilatio with the augmeted state usig Bias Model I has lower values i the regio of variace iflatio, 10 5 μ 1.0, tha for the case of assimilatio with o state augmetatio. We show c a i Fig. 13 for the case i which the state is augmeted usig Bias Model II at μ = It is see that the Fig. 13. Bias estimate for the Bias Model II is show as. The true ias (49) is show as the solid curve.

11 LEKF FOR MODEL BIAS Settlig time Fig. 14. Time-averaged rms aalysis error, e a, versus μ for the case i which the truth has Type C truth ias: I order to correct for the iases, the augmeted state used i the assimilatios must cotai oth the ad c ias estimates, Here, we agai use a differet vertical scale i () from (a) for trasparecy. result (plotted as ) agrees very well with eq. (49) (plotted as the solid lie). We otai the same result for c a whe the state is augmeted usig Bias Model III at μ = Data assimilatio with Type C truth ias Now, we comie the two iases i the truth [see eq. (29)], ad estimate them with three augmeted states as doe i previous sectios. I this case, we ca regard the Type A truth ias as added to a system that already has Type B truth ias. Hece, a differetial equatio for t ca e writte as Eve though the state augmeted LEKF ca correct for various iases i the true state, for the truth iases cosidered here it requires loger settlig time for the forecast model state to coverge toward the true state as compared to the LEKF without state augmetatio. We regard the time it takes for the rms aalysis error (34) to settle ear its time-averaged rms aalysis error (35) as the settlig time. With a perfect forecast model as i Sectio 3.2, the settlig time is aroud 50 time steps with the k + 1 = 13 esemle memers we are usig for the regular (i.e. without state augmetatio) LEKF assimilatio scheme. The settlig time, however, ecomes etwee 100 ad 200 time steps whe either Bias Model I or Bias Model II are used i the assimilatio to respectively correct for Type A or B truth ias. The logest settlig time, which is ear time steps, appears whe assimilatios are doe usig Bias Model III to correct for Type B ad Type C truth iases (whe usig Bias Model III to correct for Type A truth ias, we foud settlig times that were geerally elow 500). However, it turs out that we ca easily correct this prolem y usig a priori iformatio o the ias which could e otaied y lookig at the iovatio, the differece etwee the forecast ad the oservatio, d = H ( x ) y, (52) for the case whe o ias estimatio is performed. We plot the time-average of the iovatio i Fig. 15 for the case that o ias estimatios are performed eve though Type A, B or C truth ias is preset i the truth. Sice we take the oservatios to e uiased, we ca thik of the time averaged iovatio as the forecast ias. We see that these averages are large ad vary slowly i space. d t = L(x t + ζ) + β L(x t c t ), (50) where ζ ad β are costat i time i the preset experimet. A solutio to eq. (50) is d t = β if c t = ζ; (51) that is, the idividual true ias is the same as if the system has oly oe ias. The quatities β, ζ, are thus agai give y eqs. (43) ad (49). Figure 14 shows the resultig time-averaged rms aalysis error, e a, for each estimatio method. The est result is otaied usig Bias Model III. This might e aticipated sice augmetatio y oe ias estimate aloe caot satisfy the solutio (51). The miimum rms error is aroud ad occurs ear ; this is the same as i the previous experimets. The other assimilatio methods yields e a at μ 4.0 without state augmetatio, e a at μ 0.5 usig Bias Model I, ad e a at μ 0.3 usig Bias Model II. Fig. 15. Time-averaged iovatio for the case that o ias estimatio is performed with various iases i the truth.

12 304 S. J. BAEK ET AL. I the previous experimets, the iitial esemle variace for the ias estimate is 0.1. By icreasig the iitial variace to 1.0 (somewhat larger tha the spread i the time-averaged iovatios), we ca dramatically decrease the logest settlig time from time steps to 800 time steps, while for the case that the settlig time is already small (etwee 100 ad 200 time steps) o sigificat chage i the settlig time is oserved. We ca decrease the settlig time further if we exploit the fact that the iases vary slowly i space. To icorporate this added kowledge ito our data assimilatio scheme we ow use a diffusio process for the time evolutio, eqs. (8) ad (16), of the iases, +1,i = (1 2α ) a,i + α a,i 1 + α a,i+1, (53) c +1,i = (1 2α c )c a,i + α cc a,i 1 + α cc a,i+1, (54) where α ad α c are diffusio coefficiets. By itroducig diffusio i this way, rapid spatial variatio of the ias estimates is damped, leadig to smooth spatial variatio cosistet with the actual case, eqs. (30) ad (31), ad the evidece of Fig. 15. Our experimets show that there is a modest improvemet i the settlig time i the case that there is oe type of ias i the truth (truth ias A or B) ad that it is corrected usig the correspodig ias model (Bias Model I or II, respectively); the settlig time is decreased from etwee 100 ad 200 time steps to etwee 80 ad 130 time steps whe α (or α c ) is icreased from zero to α = 0.01 (or α c = 0.01). Whe we cosider the case of Type C truth ias ad augmet the state usig Bias Model III, o-zero α diffusio (with α c = 0) ca achieve a large decrease i the settlig time, from 800 time steps to 300 time steps. All of the decreases i settlig time we have descried come without sigificat icrease i the time-averaged rms aalysis error e a. We fid that usig diffusive evolutio o ias estimates actually decreases e a i some cases. Figure 16 shows the time asymptotic aalysis errors as a fuctio of α for the case of Type C truth ias ad Bias Model III assimilatio. We see that a small amout of diffusio, i additio to shorteig the settlig time, also improves the time asymptotic performace of the assimilatio. This improvemet is ot see i the experimets with Bias Model I ad II. Fially, we ote that if diffusio [eqs. (53) ad (54)] is added i the perfect forecast model case (Sectio 3.2), all three cases of augmetatio have the same values of rms aalysis error as that of the uaugmeted case. That is, curves correspodig to the four cases i Fig. 6 have the same miimum values with appropriate amouts of diffusio. Evidetly diffused evolutio of the ias estimate allows the estimate to coverge to the truth faster, ad also reduces the rms aalysis error of the state augmeted estimates A simple state-depedet model error I this sectio, we itroduce a simple model error, γ x 2 i, which is proportioal to the square of the state variale. That is, the Fig. 16. Time-averaged rms aalysis error, e a, versus μ with various α ad α c = 0 for the case of Type C truth ias corrected usig Bias Model III: Small diffusio improves the performace of the assimilatio up to α = dyamic equatio for the true system is as follows: dx i = (x i+1 x i 2 )x i 1 x i γ x 2 i +, i = 1,...,N, (55) where γ = 0.05 ad = Here, is icreased to maitai chaotic ehavior of the true dyamics (itroducig γ x i 2 without chagig from its previous value of = 8.0 results i domiace of time periodic ehavior of the true dyamics). For the forecast model, we use eq. (25) with = Through the umerical experimet, we otai the time average γ x i at each poit i while γ x i 2 itself has large temporal fluctuatios ragig from 0 to aroud 10. The task we udertake here is to successfully estimate the time mea effect of the model error, γ x i 2, with our ias estimatio schemes. Usig aalysis similar to that i eqs. (41) ad (42), we otai t t (x ) i γ (x t i )2 t 2 γ i t, (56) t 1 ad hece i t for each locatio i. I Fig. 17, we plot the rms aalysis errors of the umerical experimets. Without ias estimatio the miimum rms aalysis error e a is otaied at μ 1.0. Amog the three ias models, Bias Model I produces the est result, e a at μ I terms of the differece with the error i the perfect model case ( e a 0.057), this represets a 12% improvemet toward the perfect model performace. If we employ diffusive evolutio (with α = 0.05), we ca further decrease the rms error to otai e a at μ 0.09 ad 18% improvemet toward the perfect model performace. I oth cases (with ad without diffusio), we otai i a 0.05, which is a good

13 LEKF FOR MODEL BIAS 305 To evaluate the performace of the proposed ias models for use i augmeted esemle Kalma filterig, we carried out experimets with the Lorez-96 model. While we used the origial model equatio of Lorez ad Emauel (1998) to evolve the model state, we employed altered versios of Lorez ad Emauel s equatio to geerate sets of time-series of the true states. Each alteratio of the equatio correspoded to distictly differet types of model iases. The mai results of these umerical experimets are the followig. Fig. 17. Time-averaged rms aalysis error, e a, versus μ with the simple state-depedet model error: With Bias Model I ad diffusio process, we ca improve the performace i terms of the rms aalysis error. estimate of i t, with smaller spatial variatios whe we use diffusio. We also fid that the performace of the assimilatio with Bias Model I is ot sesitive to the selectio of i the model equatio while the performace of the assimilatio without ias estimatio is sesitive to the selectio of. We ca also achieve the same performace with Bias Model III with appropriate diffusio (α = 0.05, α c = 0.2), ut we caot achieve it with Bias Model II. We cojecture that the reaso is ecause the form of the ias i eq. (55) is closer to Type A truth ias tha Type B. 4. Coclusios ad discussio I this paper, we cosidered three ias models for use i state space augmetatio strategies to mitigate the effects of model iases o forecasts. (i) Bias Model I is ased o the assumptio that the est ackgroud iformatio is otaied whe the iitial coditio of the short-term forecast that provides the ackgroud (the aalysis at the previous assimilatio time) is as close to the truth as possile. (ii) Bias Model II is ased o the assumptio that there exists a trasformatio from orits o the attractor of the forecast model to orits o the attractor of the true system. (iii) Bias Model III comies Bias Model I ad Bias Model II. While Bias Model I was cosidered y others i earlier papers for schemes other tha the LEKF, Bias Model II ad Bias Model III are (to the est of our kowledge) first itroduced i the preset paper. (i) The effectiveess of the differet ias models strogly depeds o the actual form of the true model ias. I our umerical experimets it was foud that whe the ias model was suited to the ias of the forecast model i modelig the true dyamics, the good results were otaied. However, whe this was ot the case, the results were ot improved y the model ias correctio scheme. This suggests that serious cosideratio of the choice of the ias model may e crucial i otaiig a successful scheme for model ias mitigatio. (ii) For the ias models we cosidered, Bias Model III performed as well or etter tha the other ias models i terms of average aalysis error, at the expese of requirig a larger esemle ad i some cases icreasig the settlig time. I most cases, the iclusio of parameters that were ot preset i the model ias did ot yield improved performace. However, i the case of Type A truth ias, Bias Model III did outperform Bias Model I, due to the fact that the model ias was added to a cotiuous time forecast model, while the ias correctio was applied at discrete times (see Sectio 3.3). (iii) We foud (Sectio 3.6) that the model ias correctio scheme took may more iteratio steps to coverge tha i the case i which o model iases are preset. The settlig time strogly depeds o the actual model ias, ad o the ias model employed. As a result of the possiility of log settlig times, oe might aticipate that use of these model correctio schemes may ecome prolematic. We ca, however, dramatically reduce the settlig time y icrease of the iitial esemle spread of the ias estimates. Moreover, whe the model ias is slowly varyig i space, we demostrated that choosig a diffusive evolutio of the model ias ca also reduce the settlig time. (iv) I a case with state-depedet, ad thus time-varyig, additive model error (Sectio 3.7), Bias Model I estimated a model ias that was close to the time average of the model error. I this sese, it foud the est estimate of the model error withi our costat-i-time parameterizatio. The improvemet i performace compared to o ias estimatio was modest ut sigificat. Fially, we ote that state space augmetatio is ot the oly way to accout for the effect of model errors i the state estimatio process. As we metioed earlier, variace iflatio (oth additive ad multiplicative) ca improve the resiliece of Kalma filter schemes to the effects of model errors. Promisig results were achieved y employig additive variace iflatio schemes