A shadowing-based inflation scheme for ensemble data assimilation
AA shadowing-based inflation scheme for ensemble dataassimilation
Thomas Bellsky
Department of Mathematics and Statistics, University of Maine, Orono, ME, USA
Lewis Mitchell ∗ School of Mathematical Sciences, University of Adelaide, Adelaide, South Australia,Australia
Abstract
Artificial ensemble inflation is a common technique in ensemble data assimi-lation, whereby the ensemble covariance is periodically increased in order toprevent deviation of the ensemble from the observations and possible ensemblecollapse. This manuscript introduces a new form of covariance inflation for en-semble data assimilation based upon shadowing ideas from dynamical systemstheory. We present results from a low order nonlinear chaotic system that sup-port using shadowing inflation, demonstrating that shadowing inflation is morerobust to parameter tuning than standard multiplicative covariance inflation,often leading to longer forecast shadowing times.
Keywords: data assimilation, shadowing, covariance inflation, chaoticdynamics, ensemble methods ∗ Corresponding author.
Telephone : +61 8 8313 5424
Facsimile : +61 8 8313 3696
Email address: [email protected] (Lewis Mitchell)
URL: http://maths.adelaide.edu.au/lewis.mitchell (Lewis Mitchell)
Preprint submitted to Physica D: Nonlinear Phenomena September 10, 2018 a r X i v : . [ phy s i c s . d a t a - a n ] S e p . Introduction Ensemble filtering methods are Monte Carlo approximations to the Bayesianupdate problem of combining a prior probability density function for a sys-tem state with a likelihood function for observational data. Such methods arewidely used across the geophysical sciences, for improving forecasts in numericalweather and climate prediction [1, 2] as well as in ocean [3, 4] and atmosphericscience [5, 6]. A key advantage of using ensemble methods is the approxima-tion of distributions by finite-size ensembles, leading to a massive computationaladvantage and the ability to represent otherwise inaccessible high-dimensionaldistributions [7]. The ensemble Kalman filter (EnKF) [8] along with its variantsand extensions such as the local ensemble transform Kalman filter (LETKF)[9] are efficient data assimilation methods which have proven highly effectiveacross a range of applications involving both state and parameter estimation[10, 11, 12, 13].Ensemble approximations come at an important cost: the use of finite-sizeensembles generically leads to sampling errors in ensemble-based Kalman filter-ing techniques, which can manifest in a number of ways. The most commonissue related to insufficient ensemble size is misspecification of error covariances.Analysis ensemble spreads are often routinely underestimated [14], or possiblyoverestimated, particularly in sparse observational grids [15, 16, 17]. Covari-ance underestimation can lead to filter divergence where the analysis ensemblebecomes overconfident in model forecasts and fails to track the true systemstates or observations, in some cases even leading to numerical instabilities inthe forecast model which ultimately catastrophically diverge to machine infinity[18, 19]. A mechanism for such filter divergence is a finite-size ensemble aligningaway from a sufficiently strong attractor, which can lead to integrating a stiffdynamical system [20].One remedy to counter the underestimation of the error covariance and pos-sible filter divergence is to artificially inflate the ensemble covariance. Thisartificial inflation can be done simply by periodically adding noise or applying2 multiplicative factor greater than one to the error covariance [21, 22]. Therealso exist adaptive and hybrid covariance inflation methods which modify theerror covariance by accounting for various features of the forecast model andensemble [23, 24, 25, 26]. Additionally, there exist hybrid methods avoidinginflation altogether when the analysis strongly dominates the prior [27]. Othermethods to prevent filter divergence involve modifying the basic ensemble fil-tering algorithms through judicious stochastic parameterization [28, 29]. Finiteensemble sizes can also lead to spurious correlations appearing in the error co-variance matrices. Such spurious correlations may be ameliorated by spatiallylocalizing the effects of observations [5].Ideally, a forecast will remain close to the true state or observations for aslong as possible and not exhibit any form of divergence, catastrophic or other-wise. Mathematical shadowing theory, developed in the context of hyperbolicsystems [30, 31], provides rigorous results guaranteeing the existence of truemodel trajectories which remain close to a given pseudo-trajectory (one whichis almost an actual trajectory of the model) for arbitrarily long times. Suchtheory has been advanced to show that numerical solutions of chaotic systemsdo indeed approximate true trajectories [32, 33]. Other studies have determinedmethods of numerical approximation for hyperbolic periodic orbits that shadowactual periodic orbits [34]. Of course, for non-hyperbolic systems it can beproven that no such shadowing trajectory exists (e.g. [35]), in which case theaim becomes to find shadowing trajectories which nonetheless remain close topseudo-trajectories with only small mismatches [36].Operationally, finding shadowing trajectories is greatly limited by model er-ror, confounding sources of error from observations, as well as the sparsity ofobservations. Nonetheless, [37] proposed a simple method, later extended in[38], for inflating ensemble forecasts in a non-hyperbolic chaotic dynamical sys-tem. Their method, which only inflates the ensemble in directions in whichuncertainty is shrinking, has had success in increasing the shadowing time of amodel forecast. While this method of course does not guarantee the existenceof shadowing trajectories in the dynamical systems sense, it has nonetheless3hown favourable numerical results in low-dimensional chaotic dynamical sys-tems, suggesting a potential application to data assimilation problems.In this article we introduce a new covariance inflation method for ensem-ble data assimilation which addresses the general problem of filter divergenceusing the shadowing-based approach from [37, 38]. Our method aims to judi-ciously inflate the ensemble only in directions in which the ensemble is growingoverconfident, with the intention of keeping the analysis ensemble close to theattractor. This is done through an algorithm that identifies the contractingensemble directions over the forecast cycle, and then inflates the ensemble onlyin these contracting directions before performing the DA analysis. We applyour shadowing inflation method within the context of the LETKF, and com-pare it with the standard multiplicative inflation method via numerical twinexperiments on a low-dimensional non-hyperbolic nonlinear chaotic system ex-hibiting dynamics akin to those in the atmosphere. As we will demonstratethrough numerical simulations, the proposed approach works well in comparisonto standard multiplicative methods, in that shadowing inflation is less sensitiveto inflation parameter tuning, reduces error and dispersion, while maintainingensemble reliability.The remainder of this paper is organized as follows: Section 2 describes en-semble data assimilation and covariance inflation, and proposes the shadowinginflation algorithm used to ameliorate issues related to covariance underestima-tion. Section 3 details the model and setup of our numerical experiments, andSection 4 presents numerical results comparing standard multiplicative inflationto our proposed shadowing inflation method. We conclude with a discussion inSection 5.
2. Ensemble data assimilation
A general ensemble Kalman filter is a Monte Carlo data assimilation tech-nique based on the Kalman filter [39]. The Kalman filter is an algorithm fordetermining a state estimate using both a model prediction and observational4ata. An ensemble Kalman filter is particularly useful, because it extends thelinear Kalman filter to nonlinear models and is computational efficient for largestate space vectors [9].To mathematically describe this type of filter, we assume we have someforecast model M that sequentially determines a model state z . The modeladvances the previous analysis state to the forecast state z f , which at time t j is: z f ( i ) j = M (cid:16) z a ( i ) j − (cid:17) . (1)Thus, M takes the previous analysis state estimate and updates it forwardin time. The index i notates a particular ensemble state, where there are k ensemble states: (cid:110) z a ( i ) j − : i = 1 , , . . . , k (cid:111) . (2)Furthermore, there exist spatial observations of the state at time t j , denotedby the vector y . Typically, the number of observations is much less than thesize of the state space dim( y ) (cid:28) dim( z ). In this formulation, it is assumed thatthere is a linear observation operator H that projects the state space to theobservation space: y j = Hz j + (cid:15) j . (3)The z j above represents the true state at time t j and the observational erroris assumed to be a Gaussian random variable (cid:15) j ∼ N (0 , R j ), where R j is thecovariance matrix for the observations. For simplicity, the time-step notation j will be dropped in the forthcoming notation.The ensemble Kalman filter [8, 40] is a reduced rank filter, where an ensem-ble of k analysis states from the previous time step Z a = { z a , z a , . . . z ak } is eachindividually advanced forward by the forecast model to determine the back-ground forecast ensemble Z f = (cid:110) z f , z f , . . . z fk (cid:111) . Then the background forecastcovariance is formulated as: P f = 1 k − Z f (cid:48) ( Z f (cid:48) ) T , (4)5here the i -th column of Z f (cid:48) is z fi − ¯ z f , with ¯ z f indicating the mean ¯ z f = k (cid:80) ki =1 z fi . Thus, the background forecast covariance (4) is not invertible sinceit is of rank k −
1, so various ensemble Kalman filter methods perform a change ofcoordinates to determine a Kalman filter update step, where transform methodsavoid computing the background forecast covariance altogether. In particular,the local ensemble transform Kalman filter (LETKF) [9] updates the covarianceanalysis as: P a = Z f (cid:48) ˜ P a (cid:16) Z f (cid:48) (cid:17) T , (5)where ˜ P a represents the analysis error covariance in a k dimensional space:˜ P a = (cid:104) ( k − I + ( HZ f (cid:48) ) T R − (cid:16) Z f (cid:48) (cid:17)(cid:105) , (6)and updates each ensemble state as: z ai = ¯ z f + Z f (cid:48) ˜ P a (cid:16) HZ f (cid:48) (cid:17) T R − (cid:16) y − Hz fi (cid:17) . (7)In this study, we use the LETKF as our data assimilation method. TheLETKF makes use of localization, where a state location is updated by onlyconsidering nearby observations. A basic technique for performing localizationis to assign some universal localization radius r , and then only update a statelocation using observations within r units of that location. A particular issue with ensemble Kalman filters is that ensemble states oftentend to the ensemble mean with small uncertainty. This can lead to the problemof ensemble collapse, where the EnKF analysis leads to an overconfident, butincorrect state, no longer shadowing the truth.Ensemble covariance inflation is a procedure to avoid underestimating uncer-tainties and ensemble collapse. These methods artificially inflate uncertaintiesin the background covariance. As discussed in the introduction, there are avariety of techniques for performing covariance inflation. One common methodis multiplicative inflation, where the background forecast covariance is inflated6y a multiplicative factor 1 + δ for δ >
0, thus P f → (1 + δ ) P f . A similar tech-nique is additive inflation, which adds noise Υ to the background covariance P f → P f + Υ . We present a new type of covariance inflation, based on ideas from [38]. Thatwork examines how long a forecast ensemble “shadows” a true solution, but thetechniques do not involve any observations or data assimilation. As discussedin the introduction, when an ensemble shadows the true solution, this can bedescribed as the spread of the ensemble (cloud of uncertainty) containing thetrue solution. Typically, one might begin with a well-distributed ensemble ofstate solutions Z ∈ R N × k made up of k ensemble members. Each ensemblemember is propagated forward by the forecast model M , and a singular valuedecomposition (SVD) is performed at each evaluation time step t j : Z j = U j S j V Tj , (8)where the resulting singular values (the diagonal components of S j ) determinethe length of the axes of ensemble spread and the singular vector U j determinesthe direction. The SVD approximates the region of uncertainty after modelpropagation, with expanding directions of uncertainty stretching the ellipse andcollapsing directions of uncertainty shrinking the ellipse, which is well describedin [37]. We remark that in reality the SVD provides a linear approximation tothe true ellipse of uncertainty created by the nonlinear evolution of the forecastdynamics, however for short forecast intervals this difference often remains small.In [38], the concept of stalking, an aggressive form of shadowing, is intro-duced. Under stalking, at each evaluation time step, artificial uncertainty isinserted in the shrinking directions of the ellipse. Their results determined thatthe stalking methodology often led to ensembles shadowing the truth for a longerforecast period.We adapt the idea of ensemble stalking for the purpose of ensemble inflation,which we call shadowing inflation . We remark that while shadowing also has7a) Standard z t Z (1 + δ ) Z f Z f Z a ∗ y (b) Shadowing z t Z ¯ z f I + M Z f (cid:48) Z f Z a ∗ y Figure 1: The two figures above are two dimensional cartoons illustrating standard multi-plicative inflation and shadowing inflation. In both, there exists an uncertainty for the truetrajectory z t , where the initial data’s uncertainty Z is well-distributed about the initial con-dition. In (a) the forecast model carries forward the ensemble of trajectories to a futuretime, where some overlap occurs between the actual uncertainty Z t and the final analysisuncertainty Z a , after the forecast ensemble Z f has been inflated by a factor 1 + δ and theobservation y has been assimilated. In (b) the proposed shadowing inflation scheme is illus-trated, where only the shrinking dimension in the uncertainty Z f is inflated after the forecast,leading to the analysis ensemble Z a . The shadowing inflation scheme often leads to a greateroverlap between the analysis ensemble and the true uncertainty, with subsequent forecastsachieving a longer shadowing time. t , the forecast model has determined the forecasted ensemble backgroundstate Z f ( t ) = Z f ∈ R N × k . From this forecasted ensemble state, the shadowinginflation scheme performs the following steps:1. Form the matrix: Z f (cid:48) ( t ) = Z f (cid:48) ∈ R N × k (recall the i -th column of Z f (cid:48) is z f ( i ) − ¯ z f ) at the beginning of some DA analysis time step t . Froma recent, but previous, model step t − (occurring after the last DA cycle), wealso have a forecasted ensemble background state Z f ( t − ) = Z f − , from which wesimilarly form: Z f (cid:48) ( t − ) = Z f (cid:48)− ∈ R N × k . Here, we are implicitly assuming thereare multiple forecasts steps between each DA cycle. For instance, a numericalweather forecast is typically advanced over many incremental time intervalsduring the 6 hour period between a DA cycle.2. Perform a singular value decomposition on both: Z f (cid:48) = U SV T and Z f (cid:48)− = U − S − V T − (here S consists of up to k − s i ( t )and S − consists of up to k − s i ( t − )). The lengthand direction of the ensemble ellipsoid (spread) will be determined by s i u i ,where the u i ’s are the columns of U . Due to the non-uniqueness of the SVD,the directions of the singular vectors in U − need to be matched with theircorresponding vectors in U . We do this by calculating the absolute value of thedot products between all pairs of vectors, and then selecting the set of pairswith maximal absolute values.3. Determine the columns u c of a new matrix U c by determining all i forwhich: u c = { u i : s i ( t ) < s i ( t − ) }
4. Form the inflation matrix: M = I + δU c U Tc ; (9)where δ > Z b ( t ) =¯ z f I + M Z f (cid:48) , from which the data assimilation process is continued to determinethe analysis state. 9 schematic diagram illustrating the method plus the analysis step is givenin Figure 1. This process only inflates the contracting eigendirections, andshadowing inflation only acts in the directions spanned by the analysis ensemble.It performs no inflation on the expanding eigendirections, however this couldbe easily incorporated (as could a deflation in these directions) if desired. Inthe numerical experiments which follow we found that inflating the expandingeigendirections was detrimental to the analysis.
3. Model and experimental setup
We use the Lorenz-96 model [41] as a test bed for our experiments andresults. Lorenz-96 is a conceptual model that determines a ‘weather’ state on alatitude circle: dz i dt = ( z i +1 + z i − ) z i − − z i + F. In this model, the nonlinear terms mimic advection and conserve the total en-ergy. The linear term dissipates the total energy. F is the forcing, which stronglydetermines chaotic properties. For our experiments, we take N = 40 locations onthis latitude circle. We assume the standard forcing F = 8, which correspondsto a chaoticity similar to true atmospheric dynamics [41]. The climatologicalstandard deviation for the system with these parameters is σ clim = 3 . h = 0 .
05 simulatesa 6 hour Earth weather forecast [41]. We discretize this model on a h/
10 time-step, performing DA updates at multiples of h . When performing shadowinginflation, we determine the expanding (and contracting) directions of uncer-tainty by examining the singular value decomposition at the assimilation timestep t a and the previous model step t a − h/ t = 22), create N obs equally-spaced, fixed, synthetic observations by adding Gaussian noise with error co-variance R = 0 . I to every observed spatial location, and allow the system tospin up for 10 days from a random initial condition before performing experi-ments. We perform 100 separate simulations, where the majority of our results10rovide the median of the RMS error and the corresponding interquartile rangeof RMS error over the 100 simulations. We focus on the median and interquar-tile range of our sample of simulations here due to the skewed nature of thesample – a small number of initial condition and observation sets lead to filterdivergence with both forms of inflation, which dominate the calculation of meanRMS errors and obscure the trends observed. We use k = 20 ensemble membersin all experiments, randomly initialized by adding Gaussian noise with errorcovariance equal in size to that of the observations to the truth at t = 0. Inall experiments we set the localization radius r = 5, which we found to producethe best performance in the LETKF with global multiplicative inflation, and set N obs = 8. We vary our constant inflation parameter, in Equation 9, anywherefrom δ = 0 .
005 to δ = 0 . .
005 to 1 .
4. Results
In Figure 2 we show example analyses made by the standard multiplicativeinflation and shadowing inflation methods with a shadowing inflation parameterof δ = 0 .
02 (and multiplicative inflation factor of 1 . z and z coordinates shown),where the spread around the truth appears relatively large. On the other hand,11 −
50 100 150 200 − Analysis cycle
TruthShadowing inflationMultiplicative inflation
Figure 2: Example analysis trajectories for standard (red, solid) and shadowing (blue, dashed)methods. Top: observed component. Bottom: unobserved component. Black, solid line showsthe truth. − − − − z Standard − − − − − Shadowing
EnsembleForecast meanTruth
Figure 3: Example ensemble trajectories for standard (left) and shadowing (right) inflationschemes. The grey trajectories show forecasts generated by individual ensemble members aftereach analysis, while the red dashed line shows the mean of these ensemble forecasts. The solidblack line shows the true trajectory being estimated. the shadowing inflation scheme avoids this overinflation by only inflating theensemble in contracting directions, leading to an ensemble which is more tightlyclustered around the truth throughout. This suggests that shadowing inflationis most beneficial near the edges of the model attractor, similar to results in[38] for the forecasting problem. We might also expect shadowing inflation tooutperform the standard method near the stable manifold of a saddle, wheretrajectories in the analysis ensemble might diverge dramatically due to beingfalsely initialized on both sides of the manifold. Note also that while the anal-ysis mean for each ensemble is not visibly dissimilar in our example (Figure 2,analysis cycles 120–140), there is a substantial difference in how well each en-semble represents the truth. Indeed, in Figure 3 the forecast mean (red dashedline) in this region is reasonably close to the truth and not dissimilar betweenthe two inflation methods, however the ensemble spread is noticeably worse forthe global multiplicative inflation.Figure 2 suggests that the shadowing inflation scheme is most beneficial inthe unobserved subspace. To explore this we plot the median and interquartile13MS errors at each DA step over the entire 110 day simulation (here δ = 0 . . δ . This figure varies the shadowing inflation factor δ from0 .
005 to 0 .
1, and equivalently varies the fixed multiplicative inflation factor 1+ δ from 1 .
005 to 1 .
1. Figures 5a-c respectively plot the median and interquartileRMS errors of the multiplicative inflation scheme (solid, red) and the shadowinginflation scheme (dashed, blue) for a) all locations (log scale), b) only observedlocations, c) only unobserved locations (log scale). The optimal multiplicativeinflation parameter is near δ = 0 .
01 and the optimal shadowing inflation param-eter is near δ = 0 .
05. We see that the shadowing inflation scheme is much morerobust to the predetermined inflation factor than the multiplicative scheme,which increases in error for increasing values of δ past 0 . δ = 0 .
02, Figure 6b is for an inflation factor of δ = 0 .
05 (where this forecast begins at the last day of the simulation plottedin Figure 4a), and Figure 6c is for an inflation factor of δ = 0 .
1. In Figure 6a,14 igure 4: The median and interquartile RMS errors for the multiplicative inflation scheme(red, solid) and shadowing inflation scheme (blue, dashed) are plotted over the entire simula-tion time (with a shadowing inflation parameter δ = 0 .
05 and multiplicative inflation 1 . σ clim = 3 .
63 and observa-tional error standard deviation is √ . ≈ . igure 5: Top three figures above plot the median and interquartile RMS errors of the mul-tiplicative inflation scheme (solid, red) and the shadowing inflation scheme (dashed, blue),varying the inflation factor. (a) is a log scale of the RMS error of all locations, (b) is the RMSerror of observed locations, and (c) is a log scale of the RMS error of unobserved locations.Here, the localization radius is r = 5 and there are 8 fixed observations. δ = 0 .
02. The convex shape of the dia-grams indicates that both methods are overdispersive, however the shadowinginflation method is significantly less underconfident (across both observed andunobserved variables) than the standard multiplicative inflation method.
5. Discussion
This work has introduced a new shadowing-based inflation method for en-semble data assimilation. We have tested this shadowing inflation scheme nu-merically on a low-dimensional nonlinear system exhibiting chaotic dynamicsreminiscent of those in the atmosphere. Comparing shadowing inflation withstandard global multiplicative covariance inflation, we have found that shad-owing inflation outperforms the standard multiplicative inflation over a rangeof inflation values exhibiting a relative insensitivity to parameter tuning, oftenleading to longer forecast shadowing times, and maintaining ensemble reliability.All experiments for the present work were performed using a perfect model– an obvious area for further exploration is the case of model error, whereensemble inflation must compensate for structural deficiencies in the forecastmodel. That shadowing inflation tends to perform best at the extremities of theattractor (as shown in Figure 3) suggests that it might be a useful method insituations involving model error, as it is near these extremes, or near a saddlepoint in the stable manifold, where model error should have a large detrimentaleffect. Future work will involve coupling shadowing inflation methods with17 igure 6: This figure plots the median and interquartile RMS errors of all locations for themultiplicative inflation scheme (solid, red) and the shadowing inflation scheme (dashed, blue)for a ten day forecast at the end of the 110 day DA simulation. (a) is for a shadowing inflationfactor of δ = 0 .
02, (b) is for a shadowing inflation factor of δ = 0 .
05, and (c) is for a shadowinginflation factor of δ = 0 .
10 (and a corresponding multiplicative inflation factor of 1 + δ in allcases). Here, the localization radius is r = 5 and there are 8 fixed observations.
10 15 2000.020.040.060.080.1 (a) all variables
Standard inflationShadowing inflation5 10 15 2000.020.040.060.080.1 P r obab ili t y (b) observed (c) unobserved Figure 7: Rank histograms, N obs = 8, r = 5, δ = 0 . s i , a more attractor-based schememay show further improvements. Note that such methods would necessarily beconstrained to low-dimensional systems where the shape of the attractor canbe reasonably estimated. The shadowing inflation factor may also be relatedto the observational density. The use of uniformly-spaced observations leads tolarge unobserved gaps for some values of N obs (e.g. N obs = 7); exploring thedependence of optimal inflation level on N obs will require us to use a differentobservational setup to that employed here.For simplicity we formulated the method here involving an extra SVD ofthe full forecast ensemble deviation matrix, which would be computationallyimpractical in large systems. Furthermore, analysis ensembles tend to be un-derdispersive in high-dimensional systems, requiring large inflation factors –care must therefore be taken when performing inflation with such systems. Sig-nificant computational savings could be made in high-dimensional systems byperforming the shadowing inflation within the ensemble space. However, how tobest implement this decomposition, particularly while ensuring fidelity and suf-ficient smoothness of the interpolated fields, remains a question for exploration.Future work will formulate a shadowing inflation scheme within the ensemblesubspace and compare the accuracy and computational cost when making thismodification. Acknowledgments
The authors acknowledge and thank the Mathematics and Climate ResearchNetwork, in particular National Science Foundation support DMS-0940271 and20MS-0940314.
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