Decreasing flow uncertainty in Bayesian inverse problems through Lagrangian drifter control
DDecreasing flow uncertainty in Bayesian inverseproblems through Lagrangian drifter control
D. McDougall † C. K. R. T. Jones ‡ November 11, 2018
Abstract
Commonplace in oceanography is the collection of ocean drifter posi-tions. Ocean drifters are devices that sit on the surface of the ocean andmove with the flow, transmitting their position via GPS to stations onland. Using drifter data, it is possible to obtain a posterior on the under-lying flow. This problem, however, is highly underdetermined. Throughcontrolling an ocean drifter, we attempt to improve our knowledge of theunderlying flow. We do this by instructing the drifter to explore parts ofthe flow currently uncharted, thereby obtaining fresh observations. Theefficacy of a control is determined by its effect on the variance of theposterior distribution. A smaller variance is interpreted as a better un-derstanding of the flow. We show a systematic reduction in variance canbe achieved by utilising controls that allow the drifter to navigate new orinteresting flow structures, a good example of which are eddies.
The context of the problem we address in this paper is that of reconstructing aflow field from Lagrangian observations. This is an identical twin experiment inwhich a true flow field is unknown but from which Lagrangian type observationsare extracted. It is assumed that little is known about the functional form ofthe flow field except that is barotropic, incompressible and either steady orwith simple known time dependence. Note that, since it is incompressible andtwo-dimensional (barotropic), the field is given by a stream function ψ ( x, y, t ).The objective is then to reconstruct an estimate of this stream function fromLagrangian observations, along with an associated uncertainty.The question addressed is whether the uncertainty of the reconstructioncan be reduced by strategic observations using Lagrangian type instruments.The measuring devices are assumed to be controllable and their position can † Institute for Computational Engineering and Sciences, University of Texas at Austin, TX,USA ‡ Mathematics Department, University of North Carolina at Chapel Hill, NC, USA a r X i v : . [ s t a t . C O ] A ug ecreasing flow uncertainty ecreasing flow uncertainty We begin by prescribing the stream function of the flow field the drifters willmove in. We will call this flow field the ‘truth’ and later we try to reconstructit from noisy observations. The truth flow we will use is an explicit solution tothe barotropic vorticity equations [44], ψ ( x, y, t ) = − cy + A sin(2 πkx ) sin(2 πy ) + εψ ( x, y, t ) , on the two dimensional torus ( x, y ) ∈ T , where the perturbation we will use isgiven by ψ ( x, y, t ) = sin(2 πx − πt ) sin(4 πy ) . The corresponding flow equation is as follows ∂ v ∂t = ε∂ t ∇ ⊥ ψ , t > ε = 0 and the underlying flowis steady. The second case is when ε (cid:54) = 0 and the time-dependent perturbationsmears the underlying flow in the x -direction. Drifters placed in the flow v willobey d x d t = v ( x , t ) + f ( x , t ) . The function f is called the control , the choice of which requires explicit diction.We consider two cases of control: a) flow-independent; and b) a posteriori. Flow-independent controls, are controls that do not systematically utilise information ecreasing flow uncertainty v . A posteriori controls harness information froma previous Bayesian update. Our soup-to-nuts methodology for assessing theefficacy for each case of control is as follows. First, drifter dynamics are obtainedby solving d x d t = v ( x , t ) , < t ≤ t K/ (2)d x d t = v ( x , t ) + f ( x ) , t K/ < t ≤ t K , (3)where v solves (1). Then observations of the drifter locations x are collectedinto an observation vector for both the controlled and uncontrolled parts y k = x ( t k ) + η k , η k ∼ N (0 , σ I ) , k = 1 , . . . , K, (cid:32) y = G ( v ) + η, η ∼ N (0 , σ I K ) , (4)where v is the initial condition of the model (1). The map G is called the forward operator and maps the object we wish to infer to the space in whichobservations are taken.Flow-independent controls f are independent of y . We will utilise two suchcontrols: a time-independent zonal control f ( ζ, f ( ζ, ζ ). The a posteriori control we execute is one that forcesdrifter paths to be non-transverse to streamlines of the underlying flow. Namely, f ( x ) = − ζ ∇ P ( ψ | y ), where ψ ( x ) = ψ ( x , ζ and the resulting drifter path on the posteriordistribution over the initial condition of the model P ( v | y , y ).Encompassing our beliefs about how the initial condition, v , should lookinto a prior probability measure, µ , it is possible to express the posterior dis-tribution in terms of the prior and the data using Bayes’s theorem. Bayes’stheorem posed in an infinite dimensional space says that the posterior prob-ability measure on v , µ y , is absolutely continuous with respect to the priorproability measure [53]. Furthermore, the Radon-Nikodym derivative betweenthem is given by the likelihood distribution of the data,d µ y d µ ( v ) = 1 Z ( y ) exp (cid:18) σ (cid:107)G ( v ) − y (cid:107) (cid:19) , where the operator G is exactly the forward operator as described in (4) and Z ( y ) is a normalising constant. We utilise a Gaussian prior measure on the flowinitial condition, µ ∼ N (0 , ( − ∆) − α ). For our purposes, we choose α = 3 sothat draws from the prior are almost surely in the Sobolev space H ( T ) [53, 10].The posterior is a high dimensional non-Gaussian distribution requiring carefulprobing by use of a suitable numerical method. The reader is referred to [53] fora full and detailed treatment of Bayesian inverse problems on function spaces.To solve the above Bayesian inverse problem, we use a Markov chain MonteCarlo (MCMC) method. MCMC methods are a class of computational tech-niques for drawing samples from a unknown target distribution. Through-out this paper, we have chosen to sample the posterior using a random walk ecreasing flow uncertainty Flow-independent controls concern the influence of an ocean drifter withoutusing knowledge of the underlying flow. They are constructed in such a way asto be independent of y . Figure 1(a) shows the variance of the horizontal component of the flow as afunction of control magnitude in the max norm, the L norm and L norm.The horizontal axis denotes the strength of the control. The vertical blackdotted line corresponds to a critical value for the magnitude. Values of ζ lessthan this correspond to controls not strong enough to force the drifter out ofthe eddy. Conversely, values bigger correspond to controls that push the drifterout of the eddy.Experiments were done for ζ = 0 , . , . , . . . , . ,
3. The case ζ = 1 .
75 wasthe first experiment in which we observed the drifter leaving the recirculationregime. The black line shows the maximum value of the variance over thedomain [0 , × [0 , . L norm and L norm, respectively. The minimum value of the variance is small enough to bedifficult to see on the plot but remains consistently small, so it has been omittedfor clarity reasons. There are some notable points to make here. Firstly, abovethe critical value (where the drifter leaves the eddy) we see that the size of thevariance decreases in all of our chosen norms. We have learned more about theflow around the truth by forcing the drifter to cross a transport boundary andenter a new flow regime. Secondly, below the critical region (where the drifter ecreasing flow uncertainty . . . . . . . ζ . . . . . × − max x,y Var( u | y ζ ) k Var( u | y ζ ) k L k Var( u | y ζ ) k L (a) The norm of the variance decreases asthe glider is forced across the transportboundary and out of the eddy. The bumpoccurs as a result of the glider exploring aslow part of the flow. . . . . . . . ζ . . . . . × − (b) The norm of the variance decreases asthe glider is forced across the transportboundary and out of the eddy. The secondbump appears because the glider re-enters atime-dependent eddy. Figure 1: Posterior variance as a function of control magnitude, ζ , for (a) thetime-independent model; and (b) the time-dependent model. . . . . . . . . . . . . . . . . . . . . × − Figure 2: Horizontal component of the posterior variance for the case ζ = 0.The black area in the lower left corresponds neatly with the region in whichobservations are taken.For small ζ , the controlled and uncontrolled paths along which we takeobservations are close. Their closeness and the size of σ creates a delicateinterplay between whether they are statistically indistinguishable or not. Ifthey are indistinguishable up to two or three standard deviations, this couldexplain the increase and then decrease of the variance below the critical value.Secondly, as ζ increases initially, the controlled path gets pushed down nearthe elliptic stagnation point of the flow (see figure 3). If this region is an areawhere the flow is smaller in magnitude than the flow along the uncontrolledpath, this is equivalent to an increase in the magnitude of the control relativeto the underlying flow. This leads to the observations becoming polluted by f .Exploring this further, we compute the mean magnitude of the flow alongthe controlled path of the drifter. More formally, we solve (3) to obtain a set of ecreasing flow uncertainty . . . . . . . . . . . . Figure 3: True glider path (black) for some positive ζ less than the critical value.Blue lines are streamlines of the true flow. Red crosses are zeros of the flow:fixed points of the passive glider equation.points { zk = z ( t k ) } Kk =1 . Then we compute the mean flow magnitude as follows (cid:104) v (cid:105) = 2 K K (cid:88) k = K/ v ( z k ) . (5)This quantity is computed for each fixed ζ the result is plotted in figure 4. Themean flow magnitude is given by the magenta line in this figure and the blackdotted line depicts the flow magnitude. Notice the first three values of ζ whichthe mean flow magnitude decreases in. This is equivalent to and increase in themagnitude of the control relative to the magnitude of the underlying flow and sothe information gain from taking observations here decreases. This correspondsnicely with the first three values of ζ in figure 1(a) that show an increase invariance. Notice also that for the other values of ζ the mean flow magnitudeshows a mostly increasing trend, consistent with a decrease in the posteriorvariance.Note that the region below the critical value correspond to control magni-tudes that are too small to push the glider out of the eddy in the unperturbedcase ε = 0. The region above the critical value corresponds to values of ζ forwhich the glider leaves the eddy, this is also in the unperturbed case. Exper-iments were done for ζ = 0 , . , . , . . . , . ,
3. In the case ε = 0, the value ζ = 1 .
75 was the first experiment in which we observed the glider leaving therecirculation regime. The black line shows the maximum value of the vari-ance over the domain [0 , × [0 , . L norm and L norm, respectively. There are some notable points to make.Firstly, below the critical magnitude (where the glider leaves the eddy in theunperturbed case) we see a sizeable reduction of posterior variance in the maxnorm as the critical magnitude is approached. To establish a connection in un-certainty quantification between the time independent and time-periodic caseis of great scientific interest and that connection has been made evident here.Note that as ζ increases and progresses further into the region above the criticalmagnitude, the posterior variance repeats the increasing/decreasing structureinduced by the eddy that we observed in the region below the critical control ecreasing flow uncertainty . . . . . . . ζ . . . . h v i Figure 4: Mean magitude of the flow along the control path (purple) against thesize of the control (black dashed line). When the gradient of the flow magnitudeis large compared with that of the control magnitude, the posterior variance issmall.magnitude. The new effects introduced into this region are purely form thetime-dependent nature of the moving eddy. The reason for their presence ismuch the same as in the time-independent case; observations trapped within aneddy regime.We have learned more about the flow around the truth by forcing the gliderinto the meandering jet flow regime. The benefits of such a control occur atexactly the same place as in the time-independent case; as the drifter leavesthe eddy in the unperturbed flow. However, extra care is required when theflow is time-dependent and the eddy moves. One cannot simply apply the samecontrol techniques as is evidenced by the extra bump in variance in the regionabove the critical magnitude. Of particular use would be extra eddy-trackinginformation to construct an a posteriori control to keep the variance small.
Now we provide the analogue of figure 1(a) for the bi-directional forcing function.This is shown in figure 5(a). We see similar behaviour for the variance of theposterior distribution again. Below the critical magnitude, the values of ζ forwhich the drifter is not forced hard enough to leave the recirculation regime, wesee an initial increase in the size of the posterior variance. Then we observe adecrease in posterior variance as ζ approaches a value large enough to push thedrifter out of the eddy regime, the region above the critical value.To explain the initial increase in the posterior variance below the criticalmagnitude, we calculate the mean flow magnitude just as in (5). This is shownin figure 6. We see an initial period where the mean flow along the controlledpath remains almost constant. As a consequence, the magnitude of the forcingincreases relative to the magnitude of the flow. This pollutes the observationsand leads to an increased posterior variance just as we have observed in the ecreasing flow uncertainty . . . . . . ζ . . . . . × − max x,y Var( u | y ζ ) k Var( u | y ζ ) k L k Var( u | y ζ ) k L (a) The norm of the variance decreases asthe glider is forced across the transportboundary and out of the eddy. . . . . . . ζ . . . . . × − (b) The norm of the variance decreases asthe glider is forced across the transportboundary and out of the eddy. The time-dependent part of the model pollutes thevariance once the glider leaves the eddy. Figure 5: Posterior variance as a function of control magnitude, ζ , for (a) thetime-independent model; and (b) the time-dependent model.previous section. We also see the opposite effect; the big jump in flow magnitudeat ζ = 0 . . . . . . . ζ h v i Figure 6: Mean magitude of the flow along the control path (purple) against thesize of the control (black dashed line). When the gradient of the flow magnitudeis large compared with that of the control magnitude, the posterior variance issmall.The cases of forcing explored thus far are f ( z ) = ( ζ, (cid:62) and f ( z ) = ( ζ, ζ ) (cid:62) .The main results are summarised by referring to figure 1(a) and figure 5(a).In these two cases, we see strikingly similar structure of the posterior varianceas a function of control magnitude. The initial increase in posterior variancewithin the eddy; decreasing posterior variance as the flow path of the drifterapproaches the transport boundary and small posterior variance (compared tothe case ζ = 0) once a new flow regime is being observed. Compare the values ecreasing flow uncertainty ζ for with this behaviour occurs. Notice that the values of ζ in figure 1(a)are about three times larger than those in figure 5(a). One factor at play hereis the relative magnitude of the controls in each case. For ζ = 1, the controlhas magnitude 1 in the zonal case, and magnitude √ √
2, notice that the valueof ζ for which the drifter first leaves the eddy, is ζ = √ and this is still smallerthan ζ = 1 . x -directional case. The final factor affecting the scalingis the dynamics of the system after the forcing has been applied. Controllingin only the horizontal direction will require a larger magnitude force to pushthe drifter out of the eddy than when forcing in both the x and y directionssimultaneously.An analogue for figure 1(b) for the new forcing function is shown in fig-ure 5(b). We see similar behaviour for the variance of the posterior distribution.Again, the region below the critical magnitude corresponds to values of ζ thatare not big enough to push the drifter out of the recirculation regime in the unperturbed case. Just as in figure 1(b), we see the unperturbed eddy affectingthe variance of the posterior distribution on the flow in the classic ‘bump’ fash-ion. We observe a reduction in posterior variance as ζ approaches a value largeenough to push the glider out of the eddy regime (in the case ε = 0). In theregion below the critical magnitude, the time-dependent flow effects take overand push the variance up. Again, a connection of uncertainty quantification ismade between the time-independent case and the case where the flow is per-turbed by a time-periodic disturbance, this connection lies entirely within theregion below the critical control magnitude. In section 3 we concluded that crossing a transport boundary and enteringa new flow regime has the desirable effect of reducing the posterior variance.Crossing into new flow regimes with a stationary flow can be translated totravelling transversely against the streamlines of the underlying flow. For therecirculation regime located in the bottom-left area, particles in the fluid willmove in an clockwise fashion. The gradient of the stream function will thereforepoint in towards the fixed point at z = (1 / , / z = (3 / , / f ( z ) = − ζ ∇ z ( E ( ψ | y )) , (6)for the controlled drifter model, where ψ is the stream function of the flow v . Therationale behind this choice is that, if the posterior mean stream function is agood estimator of the flow, the drifter will be forced transversely with the streamlines and escape the recirculation regime and allow us to make observations ina new flow regime.Figure 7(a) depicts the variance of the horizontal component as the strengthof the control, ζ , is varied. Note that we do not see the same behaviour as ecreasing flow uncertainty ζ for which the posterior variance oscillates, leading to a lack ofinformation gain in the knowledge of the flow. From about ζ = 0 . ζ = 0 . ζ = 0 . ζ = 0 .
55, we showfigure 4. This figure presents the true path of the drifter for ζ = 0 . , . . . , . ζ = 0 . ζ = 0 .
55. Notice that as ζ increases, the true path forms a kinkand forms a trajectory close to the zero of the flow at ( x, y ) = (7 / , / ζ = 0 .
52 and ζ = 0 .
55, where it is approximately 3 × − . In the case of the bi-directionalcontrol, where the relative size of the flow increases for the values of ζ thatgive a reduction in variance , it occurs between ζ = 0 .
25 and ζ = 0 .
625 whereit is approximately 1 . × − . This is about an order of magnitude bigger,crystallising the tradeoff between polluting the observed data versus exploring‘interesting’ parts of the flow. If the posterior mean is a good estimator ofthe underlying flow, utilising a control of this nature is beneficial if the driftersnavigates close to a hyperbolic fixed point of the passive drifter model equation.The first thing to note is that we do not see the same behaviour as we dofor the two na¨ıve controls chosen in section 3. Nor do we see similar structureswhen compared with figure 7(a). For each value of ζ , it is the case that the truepath navigates to the time-dependent eddy surrounding the zero of the flow atthe point ( x, y ) = (3 / , / ζ = 0 .
21 and ζ = 0 .
27 because the truepath is navigating towards one of the hyperbolic fixed points of the eddy. Anovel connection is established between the behaviour of these two controls inboth the time-independent case and the time-periodic case. ecreasing flow uncertainty .
30 0 .
35 0 .
40 0 .
45 0 .
50 0 . ζ . . . . . × − max x,y Var( u | y ζ ) k Var( u | y ζ ) k L k Var( u | y ζ ) k L (a) The norm of the variance decreases asthe glider is forced towards the a saddlepoint in the flow. No clear gain is madeotherwise. .
25 0 .
30 0 .
35 0 .
40 0 .
45 0 .
50 0 . ζ . . . . . × − (b) No clear gain is made in the case of thetime-dependent model. Figure 7: Posterior variance as a function of control magnitude, ζ , for the aposteriori control in the case of: (a) a time-independent model; and (b) a time-dependent model. .
30 0 .
35 0 .
40 0 .
45 0 .
50 0 . ζ . . . . . h v i Figure 8: Mean magitude of the flow along the control path (purple) againstthe size of the control (black dashed line). Though the gradient of the flowmagnitude is small compared with that of the control magnitude, the posteriorvariance decreases because the net gain in flow knowledge by observing near asaddle point outweighs the net loss by the control polluting the observations. . . . . . . . . . . . . Figure 9: The true drifter paths for each value of ζ for the experiments shown infigure 7(a). The pink path corresponds to the magntidue ζ = 0 . ζ = 0 .
55. The posterior variance decays as ζ approaches0 . ecreasing flow uncertainty To summarise, we have measured the performance of two na¨ıve control meth-ods, and one a posteriori control method, both in a time-independent and time-dependent flow. We have done so by observing their influence on the posteriorvariance in the mean flow direction. Section 3 addresses the na¨ıve controlsand section 4 the a posteriori control. Each control is designed to push oceandrifters into uncharted flow regimes. The three cases of control we employ hereare a purely zonal control; a control of equal magnitude in both the x and y directions; and the gradient of the posterior mean constructed using a posterioriinformation from a previous Bayesian update. In the time-independent flow, weshow a sizeable reduction of the posterior variance in the mean flow directionfor these three cases of control. We also see that on comparing the posteriorvariance for the zonal and bi-directional controls, similar structures arise whenviewed as a function of control magnitude, which dictates when the drifter leavesthe eddy and is the main influence on the posterior information. In the caseof the a posteriori control in the time-independent flow, the drifter leaves theeddy for all the values of control magnitude we have chosen. Here we observethe variance reduction occurring when the true drifter path approaches a hy-perbolic fixed point on the transport barrier of the eddy in the upper-right ofthe domain. This is evidence that oceanic transport barriers heavily influenceposterior information and sets up a novel geometric correspondence betweenthe flow structure and the posterior variance. Using the na¨ıve controls in thetime-dependent flow, we show robustness of posterior variance as a functionof the perturbation parameter. When the control magnitude is such that thedrifter leaves the eddy in the unperturbed flow, we see reduction in the posteriorvariance on the initial condition for the time-periodic flow. When employinga time-dependent a posteriori control, we see no overall net gain in posteriorvariance over the uncontrolled case. For our particular flow and drifter initialcondition, it is the case that the uncontrolled drifter path explores a hyperbolicfixed point of an eddy in the time-dependent flow more effectively than the con-trolled path. This reiterates the efficacy of control strategies and their influenceon the path along which observations are made.There are a number of ways in which this work could be generalised in orderto obtain a deeper understanding of the effects controlled ocean drifters have onflow uncertainty. For example, (i) the study of non-periodic model dynamics;(ii) the use of information from the posterior variance ; (iii) more elaboratecontrol strategies. Many other generalisations are also possible. Non-periodicmodels are more dynamically consistent with regards to their approximationof larger ocean models. We have seen the application of posterior knowledgein the construction of a control, though only through use of the mean. Thevariance of the underlying flow could be used in a similar fashion, perhaps tocontrol ocean drifters towards an area of large variance. This could have asimilar affect on the posterior distribution as the method of controlling a drifterinto a new, unexplored flow regime. Moreover, controls could be constructedto better reflect reality. Ocean gliders have a limited amount of battery power. ecreasing flow uncertainty Author McDougall would like to acknowledge the work of John Hunter (1968–2012), who led the development of an open-source and freely available plottinglibrary, matplotlib, capable of producing publication-quality graphics [23]. Allthe figures in this publication were produced with matplotlib.
References [1] J L Anderson. A Local Least Squares Framework for Ensemble Filtering.
Monthly Weather Review , 131(4):634–642, April 2003.[2] A Apte, M Hairer, A M Stuart, and J Voss. Sampling the posterior: Anapproach to non-Gaussian data assimilation.
Physica D: Nonlinear Phe-nomena , 230(1-2):50–64, June 2007.[3] A Apte, C K R T Jones, and A M Stuart. A Bayesian approach to La-grangian data assimilation.
Tellus A , 60(2):336–347, March 2008.[4] A Apte, C K R T Jones, A M Stuart, and J Voss. Data assimilation: Math-ematical and statistical perspectives.
International Journal for NumericalMethods in Fluids , 56:1033–1046, 2008.[5] Y F Atchad´e. An Adaptive Version for the Metropolis Adjusted LangevinAlgorithm with a Truncated Drift.
Methodology and Computing in AppliedProbability , 8(2):235–254, August 2006.[6] Y F Atchad´e and J S Rosenthal. On adaptive Markov chain Monte Carloalgorithms.
Bernoulli , 11(5):815–828, October 2005.[7] R W Barbieri and P S Schopf. Oceanographic applications of the Kalmanfilter. 1982.[8] L Bengtsson. 4-dimensional assimilation of meteorological observations.
World Meteorological Organization , 1975.[9] A Beskos, G O Roberts, and A M Stuart. Optimal scalings for localMetropolis-Hastings chains on nonproduct targets in high dimensions.
TheAnnals of Applied Probability , 19(3):863–898, June 2009. ecreasing flow uncertainty
Gaussian Measures . American Mathematical Society, 1998.[11] E F Carter. Assimilation of Lagrangian data into a numerical model.
Dy-namics of Atmospheres and Oceans , 13(3-4):335–348, 1989.[12] S L Cotter, M Dashti, J C Robinson, and A M Stuart. Bayesian inverseproblems for functions and applications to fluid mechanics.
Inverse Prob-lems , 25(11):115008, November 2009.[13] S L Cotter, M Dashti, and A M Stuart. Approximation of Bayesian inverseproblems for PDEs.
SIAM Journal of Numerical Analysis , 48(1):322–345,2010.[14] S L Cotter, M Dashti, and A M Stuart. Variational data assimilation usingtargetted random walks.
International Journal for Numerical Methods inFluids , 68:403–421, 2011.[15] S L Cotter, G O Roberts, A M Stuart, and D White. MCMC Methods forfunctions: Modifying old algorithms to make them faster. 2012.[16] P Courtier, J-N Th´epaut, and A Hollingsworth. A strategy for operationalimplementation of 4D-Var, using an incremental approach.
Quarterly Jour-nal of the Royal Meteorological Society , 120(519):1367–1387, 1994.[17] A Doucet, N de Freitas, and N Gordon.
Sequential Monte Carlo MethodsIn Practice . 2001.[18] G Evensen.
Data Assimilation: The Ensemble Kalman Filter . Springer,2006.[19] W K Hastings. Monte Carlo sampling methods using Markov chains andtheir applications.
Biometrika , 57(1):97–109, 1970.[20] R Herbei and I McKeague. Hybrid Samplers for Ill-Posed Inverse Problems.
Scandinavian Journal of Statistics , 36(4):839—-853, 2009.[21] R Herbei, I W McKeague, and K G Speer. Gyres and Jets: Inversion ofTracer Data for Ocean Circulation Structure.
Journal of Physical Oceanog-raphy , 38(6):1180–1202, June 2008.[22] P L Houtekamer and H L Mitchell. Data Assimilation Using an EnsembleKalman Filter Technique.
Monthly Weather Review , 126:796–811, 1998.[23] J. D. Hunter. Matplotlib: A 2D Graphics Environment.
Computing inScience and Engineering , 9(3):90–95, 2007.[24] J Kaipio and E Somersalo. Statistical inverse problems: Discretization,model reduction and inverse crimes.
Journal of Computational and AppliedMathematics , 198(2):493–504, January 2007. ecreasing flow uncertainty
Inverse problems , 16(5):1487, 2000.[26] R E Kalman. A New Approach to Linear Filtering and Prediction Prob-lems.
Journal of Basic Engineering , 82(Series D):35–45, 1960.[27] R E Kalman and R S Bucy. New results in linear filtering and predictiontheory.
Journal of Basic Engineering , 83:95–107, 1961.[28] E Kalnay.
Atmospheric modeling, data assimilation and predictability .Cambridge University Press, 2002.[29] L Kuznetsov, K Ide, and C K R T Jones. A Method for Assimilation ofLagrangian Data.
Monthly Weather Review , 131:2247–2260, 2003.[30] A S Lawless, S Gratton, and N K Nichols. An investigation of incremental4D-Var using non-tangent linear models. Technical report, 2005.[31] A S Lawless, S Gratton, and N K Nichols. Approximate iterative methodsfor variational data assimilation. Technical Report April, 2005.[32] F-X Le Dimet and O Talagrand. Variational algorithms for analysis andassimilation of meteorological observations: theoretical aspects.
Tellus A ,(38A):97–110, 1986.[33] W Lee, D McDougall, and A M Stuart. Kalman filtering and smoothing forlinear wave equations with model error.
Inverse Problems , 27(9):095008,September 2011.[34] P J V Leeuwen. Nonlinear data assimilation in geosciences: an extremelyefficient particle filter.
Quarterly Journal of the Royal Meteorological Soci-ety , 136(653):1991–1999, October 2010.[35] J M Lewis and J C Derber. The use of adjoint equations to solve a varia-tional adjustment problem with advective constraints.
Tellus A , (37A):309–322, 1985.[36] A C Lorenc. Analysis methods for numerical weather prediction.
QuarterlyJournal of the Royal Meteorological Society , 112:1177–1194, 1986.[37] A C Lorenc, S P Ballard, R S Bell, N B Ingleby, P L F Andrews, D MBarker, J R Bray, A M Clayton, T Dalby, D Li, T J Payne, and F WSaunders. The Met. Office global three-dimensional variational data as-similation scheme.
Quarterly Journal of the Royal Meteorological Society ,126(570):2991–3012, October 2000.[38] I W McKeague, G Nicholls, K Speer, and R Herbei. Statistical inversionof South Atlantic circulation in an abyssal neutral density layer.
Journalof Marine Research , 63(4):683–704, July 2005. ecreasing flow uncertainty
The Journalof Chemical Physics , 21(6):1087, 1953.[40] A M Michalak. A method for enforcing parameter nonnegativity inBayesian inverse problems with an application to contaminant source iden-tification.
Water Resources Research , 39(2):1–14, 2003.[41] R N Miller. Toward the Application of the Kalman Filter to Regional OpenOcean Modeling.
Journal of Physical Oceanography , 16:72–86, 1986.[42] K Mosegaard and A Tarantola. Monte Carlo sampling of solutions to inverseproblems.
Journal of Geophysical Research , 100(B7):12431—-12447, 1995.[43] D F Parrish and S E Cohn. A Kalman filter for a two-dimensional shallow-water model. In , pages 1–8, 1985.[44] R. T. Pierrehumbert. Chaotic Mixing of Tracer and Vorticity by ModulatedTraveling Rossby Waves.
Geophysical and Astrophysical Fluid Dynamics ,58:285–320, 1991.[45] A A Robel, M Susan Lozier, S F Gary, G L Shillinger, H Bailey, andS J Bograd. Projecting uncertainty onto marine megafauna trajectories.
Deep Sea Research Part I: Oceanographic Research Papers , 58(12):915–921,September 2011.[46] G O Roberts. Weak convergence and optimal scaling of random walkMetropolis Algorithms.
Annals of Applied Probability , 7(1):110–120, 1997.[47] G O Roberts and J S Rosenthal. Optimal scaling of discrete approximationsto Langevin diffusions.
Journal of the Royal Statistical Society: Series B(Statistical Methodology) , 60(1):255–268, February 1998.[48] G O Roberts and J S Rosenthal. Optimal scaling for various Metropolis-Hastings algorithms.
Statistical Science , 16(4):351–367, November 2001.[49] A R Robinson and D B Haidvogel. Dynamical Forceast Experiments witha Baroptropic Open Ocean Model.
Journal of Physical Oceanography ,10:1928, 1981.[50] A R Robinson and W G Leslie. Estimation and Prediction of Oceanic EddyFields.
Progress in Oceanography , 14:485–510, 1985.[51] D L Rudnick, R E Davis, C C Eriksen, D M Fratantoni, and M J Perry. Un-derwater Gliders for Ocean Research.
Marine Technology Society Journal ,38(2):73–84, June 2004.[52] H W Sorenson.
Kalman filtering: theory and application . IEEE, 1960. ecreasing flow uncertainty
Acta Numerica ,pages 1–107. 2010.[54] O Talagrand and P Courtier. Variational assimilation of meteorological ob-servations with the adjoint vorticity equation. I: Theory.