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Uncertain Transport in Unsteady Flows
Tobias Rapp * Carsten Dachsbacher † Institute for Visualization and Data AnalysisKarlsruhe Institute of Technology(a) Backward diffusion barrier strength (DBS) (b) Transport uncertainty
Figure 1: We visualize transport under uncertainties in the Red Sea ensemble dataset. The backward DBS in (a) indicates materialsurfaces that are maximally diffusive. Since the DBS assumes only small-scale stochastic deviations, we propose a complementaryvisualization of the absolute scale of uncertainties in the Lagrangian frame (b). A BSTRACT
We study uncertainty in the dynamics of time-dependent flows byidentifying barriers and enhancers to stochastic transport. This topo-logical segmentation is closely related to the theory of Lagrangiancoherent structures and is based on a recently introduced quantity,the diffusion barrier strength (DBS). The DBS is defined similarto the finite-time Lyapunov exponent (FTLE), but incorporates dif-fusion during flow integration. Height ridges of the DBS indicatestochastic transport barriers and enhancers, i.e. material surfaces thatare minimally or maximally diffusive. To apply these concepts toreal-world data, we represent uncertainty in a flow by a stochasticdifferential equation that consists of a deterministic and a stochas-tic component modeled by a Gaussian. With this formulation weidentify barriers and enhancers to stochastic transport, without per-forming expensive Monte Carlo simulation and with a computationalcomplexity comparable to FTLE. In addition, we propose a comple-mentary visualization to convey the absolute scale of uncertaintiesin the Lagrangian frame of reference. This enables us to study un-certainty in real-world datasets, for example due to small deviations,data reduction, or estimated from multiple ensemble runs.
Index Terms:
Human-centered computing—Visualization—Visu-alization application domains—Scientific visualization; * e-mail: [email protected] † e-mail: [email protected] NTRODUCTION
Although most experiments and simulations produce deterministicdata, uncertainty exists in all measured or simulated flows. Thisuncertainty might be estimated from repeated simulation runs ormeasurements, it might be introduced by data processing and re-duction, or it can be explicitly modeled. Studying uncertainty isespecially relevant in unsteady flows, where small variations in theinitial conditions can cause dramatic changes to the flow. In thispaper, we investigate uncertainties in the Lagrangian transport, i.e.the advection of a material by the flow.For deterministic flows, the Lagrangian coherent structures (LCS)identify a topological skeleton of the flow dynamics in a finite-timeinterval. Recent work has extended the definitions of coherent struc-tures to uncertain flows. The probabilistic [4] or averaged [21]transport is estimated using a Monte Carlo approach, i.e. by advect-ing a large amount of particles. While the LCS are theoretically wellestablished, this is, to our knowledge, not the case for its probabilis-tic extension.Based on recent work from Haller, Karrasch, and Kogelbauer [8,9], we employ the diffusion barrier strength (DBS) to identify trans-port barriers and enhancers to stochastic flows. These are materialsurfaces that show minimal or maximal stochastic cross flux. Byassuming only small stochastic deviations, Monte Carlo integrationis avoided and only the deterministic part of the flow has to beadvected. To this end, we first define uncertain unsteady flows asstochastic differential equations that consist of an advective com-ponent and an added stochastic component modeled as a Gaussian.The central limit theorem makes this assumption reasonable. Inthis paper, we discuss how to model uncertainty information in thisstochastic differential equation, e.g. due to data reduction, to model1 a r X i v : . [ phy s i c s . g e o - ph ] A ug small-scale deviations, or to model aggregated ensemble members.To complement the visualization of stochastic transport barriersand enhancers, which is based on the assumption of small-scaledeviations, we propose a novel visualization of the scale of uncer-tainties encountered during advection. In several experiments, weinvestigate the relationship between the stochastic transport barriersand enhancers, Lagrangian coherent structures, and its probabilisticextensions.To summarize, our contributions are:• We model stochastic flows with small deviations to Gaussianflow fields (Sect. 5),• We propose a novel visualization of transport uncertain-ties (Sect. 6),• We apply the theory of stochastic transport barriers and en-hancers to real-world data and compare it to probabilistic ex-tensions of the LCS (Sect. 7). ELATED W ORK
Uncertainty visualization has been an active research topic in thefield of visualization for more than two decades [12]. Several sur-veys [1, 16] motivate uncertainty visualization, introduce its chal-lenges, and different sources of uncertainty. In this study, we focuson the visualization of uncertain and unsteady flows. We recapitu-late the theory of Lagrangian coherent structures before we discussuncertain Lagrangian approaches.
Lagrangian Coherent Structures
LCS are attracting and re-pelling material surfaces that separate regions of different flow behav-ior in a finite-time interval [7]. Since these material surfaces showminimal or maximal cross flux, they control the global transportand mixing behavior [15, 22]. Due to differing views on coherency,several approaches exist to characterize LCS [6], each of whichmay lead to different results. The finite-time Lyapunov exponent(FTLE) measures the separation or attraction of infinitesimally closetracer particles and is closely related to the LCS [10], which canbe defined as height ridges of the FTLE. The extraction of LCS asheight ridges has been investigated by Sadlo and Peikert [18] andsubsequent studies [17, 20]. Although research on the identificationof LCS is still ongoing, the FTLE has been established as a powerfulvisualization of the time-dependent flow dynamics. The efficientcomputation of the FTLE has been an active research area since itrequires a dense integration of tracer particles [2, 14]. In general, tocompute the FTLE in an n -dimensional time-dependent flow v ( x , t ) ,a particle at position x ∈ R n at time t is advected to time t bysolving an ordinary differential equationd x ( t ) = v ( x ( t ) , t ) d t , x ( t ) = x . (1)The FTLE is computed from the flow map φ ( x , t , t ) , which maps aposition x at time t to a position at time t . From the spatial gradient ∇ φ , the right Cauchy-Green strain tensor is defined as: C ( x , t , t ) : = ∇ φ ( x , t , t ) (cid:62) ∇ φ ( x , t , t ) . (2)The FTLE is then computed using the largest eigenvalue λ max of thestrain tensor:FTLE ( x , t , t ) : = | t − t | log (cid:112) λ max ( C ( x , t , t )) . (3)The FTLE thus describes the average exponential stretching of aninfinitesimally close volume at time t when the flow is integrated to t . Uncertain Lagrangian Transport
In uncertain flows, particlesare advected stochastically. The flow map thus describes a distri-bution of positions where particles might be advected. Schneideret al. [21] estimate this stochastic flow map using a Monte Carloapproach. The authors then estimate the variance in the stochasticflow map, which defines the finite-time variance analysis (FTVA),a FTLE-like metric. Hummel et al. [11] discuss the comparativevisual analysis of Lagrangian transport in CFD ensembles based onthe FTVA. Guo et al. [4] propose two extensions of the FTLE: byestimating the expectation of the strain tensor and then computinga single FTLE value (FTLE-D), or by estimating a distribution ofFTLEs (D-FTLE). Both approaches depend on Monte Carlo esti-mation of the stochastic flow [5]. In this study, we present a newquantity that does not require expensive Monte Carlo estimation andis built upon a more solid theoretical foundation.
TOCHASTIC F LOWS
To visualize uncertainty in the transport in unsteady flows, we firstintroduce a stochastic flow as a deterministic flow with small stochas-tic deviations. More formally, we model an uncertain flow by astochastic differential equation (SDE), i.e. we extend the ordinarydifferential equation from Equation 1 with a stochastic componentd x ( t ) = v ( x ( t ) , t ) d t (cid:124) (cid:123)(cid:122) (cid:125) deterministic + √ sB ( x ( t ) , t ) d W ( t ) (cid:124) (cid:123)(cid:122) (cid:125) stochastic . (4)Here, W ( t ) is an n -dimensional Wiener process with disturbance √ sB ( x ( t ) , t ) . The Wiener process W consists of independent stan-dard Gaussian distributions at every time t . The notation d W ( t ) represents a random variable that is distributed with respect to astandard, multivariate Gaussian. The disturbance, which controls thescaling and anisotropy, is separated into a scaling parameter s > B ∈ R n × n . In the following, we willassume only small deviations, i.e. s is small. Numerical Integration
In general, SDEs can be solved by nu-merical integration using e.g. the Euler-Marayuma or the Runge-Kutta methods for SDEs [13]. These Markov chain Monte Carlostrategies involve sampling of the stochastic component. The nu-merical integration is thus significantly more involved compared todeterministic flows since it requires a large amount of stochasticallyintegrated particles. At the same time, it is non-trivial to decide howmany particles should be integrated. For these reasons, we want toavoid the numerical integration of stochastic flows.
TOCHASTIC T RANSPORT B ARRIERS AND E NHANCERS
In this section, we introduce stochastic transport barriers and en-hancers. We focus on an intuitive introduction and refer to the workof Haller, Karrasch, and Kogelbauer for the formal derivation [8, 9].Transport barriers are inhibitors of the spread of substances in aflow, whilst transport enhancers maximize such diffusion or mixingprocesses. Remarkably, these barriers and enhancers do not dependon the actual value of the diffusivities, i.e. the scaling parameter s .They are also well-defined for deterministic flows when we considerthe case of s →
0. In this case, they present an alternative to theLagrangian coherent structures, but do not depend on any specificdefinition of coherency. The diffusion barrier strength (DBS) visual-izes the barriers and enhancers, which can be defined as ridges of theDBS, similar to the LCS that can be defined as ridges of the FTLE.The DBS is computed from a deterministic flow v and a diffusioncomponent that describes the amount and anisotropy of diffusion ateach point in space and time. First, we introduce the tensor T fromthe gradient of the flow map ∇ φ and the diffusion D ∈ R n × n as T ( x , t , t ) : = [ ∇ φ ( x , t , t )] − D ( x , t )[ ∇ φ ( x , t , t )] − T . (5)2 If the diffusion is isotropic, i.e. D ≡ I , then T ( x , t , t ) = C ( x , t , t ) − . (6)However, we have to incorporate the diffusion D ( x , t ) at every timein the interval [ t , t ] , in contrast to the FTLE that only considersthe deformation at the end of the time interval. Therefore, the time-averaged, diffusivity-weighted right Cauchy-Green strain tensor ¯ C is computed as¯ C ( x , t , t ) : = | t − t | (cid:90) t t det ( D ( x , t )) T ( x , t , t ) − d t , (7)where x is the position during integration at time t , i.e. x = φ ( x , t , t ) .Since we need only the inverse of T , we compute T ( x , t , t ) − = [ φ ( x , t , t )] (cid:124) D ( x , t ) − [ ∇ φ ( x , t , t )] (8)instead of Equation 5. Lastly, Haller et al. [8] define the DBS as thetrace of ¯ C . Since this quantity is exponential, we take the logarithmfor visualization:DBS ( x , t , t ) : = log (cid:0) tr ( ¯ C ( x , t , t )) (cid:1) . (9)Although the integral in Equation 7 might seem daunting at first, weare already performing this integration when computing the flowmap φ . Thus, to compute the DBS, we integrate the deterministicflow v and at each step evaluate T − to accumulate the diffusivity-weighted and time-averaged strain tensor ¯ C . ODELING D IFFUSION
To compute the DBS, we require a scale-independent diffusion com-ponent D . For completely deterministic flows, we set the diffusionto the identity matrix, i.e. D = I . For stochastic flows, with a scale-independent disturbance B (cf. Equation 4), the diffusion is definedas D = BB (cid:124) . (10)For uncertain and unsteady flows modeled by Gaussians, we nowdiscuss how to obtain B . The scale-independent disturbance repre-sents the anisotropy and the scaling relative to other regions of theflow. Given a Gaussian with covariance C ( x , t ) , we want to separateit into a global scaling parameter s and a disturbance B ( x , t ) .Since the disturbance should be, on average, centered around theidentity matrix I , we standardize all covariance matrices. That is,given the set of all covariance matrices C , we subtract the mean ofall variances, i.e. the diagonal elements of each covariance matrix C ∈ C . Then, we divide out the maximal standard deviation over alldimensions: B ( x , t ) = C ( x , t ) − I µ C σ max C , (11)where µ C is the mean of all variances: µ C : = E [ C , ] ... E [ C n − , n − ] (12)and σ max C is the maximum of the standard deviation of all variancesin C : σ max C : = max (cid:16)(cid:113) E [ C i , i − µ C i , i ] (cid:17) , where i = ... n − . (13)Although a different scaling than σ max C could be used since it iscanceled out in Equation 7, our definition increases the numericalstability. ISUALIZING T RANSPORT U NCERTAINTY
By design, the diffusion barrier strength ignores the absolute scaleof stochasticity, i.e. the amount of uncertainty of the transport. How-ever, this quantity is still relevant, especially if the amount of stochas-tic deviations varies strongly in the flow. To this end, we propose avisualization that complements the DBS by directly conveying thescale of stochastic deviations.Although it is possible to directly visualize the time-dependentvariance of a Gaussian flow field, we are interested in the uncer-tainty of the transport, which is inherently defined in a Lagrangianframe. We propose to measure the uncertainty encountered duringthe integration of a tracer particle. In other words, this visualizesthe transport of uncertainty in the flow. Moreover, this enables us tointegrate only the deterministic part of the stochastic flow and avoidstochastic numerical integration.First, we discuss how to measure the uncertainty of a Gaussianflow with covariance C ( x , t ) at a single point in time and space. Sincewe are not interested in the variance along individual dimensions,we employ the generalized variance [23,24] defined as | det ( C ( x , t )) | .Intuitively, this measures the multidimensional scatter of a Gaussian.To enable comparisons in different dimensions, we standardize thisquantity by taking the n -th root in n -dimensional space. Lastly, weaverage this measure over time during the material transport: σ T ( x , t , t ) : = | t − t | (cid:90) t t | det ( C ( x , t )) | n d t , (14)where x = φ ( x , t , t ) . ESULTS
In this section, we visualize the uncertain transport in a syntheticand a real-world dataset. Additional results, datasets, as well asexemplary source code can be found in the supplementary material.
This two-dimensional synthetic and time-dependent vector field de-scribes two counter-rotating gyres. It is commonly used for thevalidation of FTLE and LCS. In the supplementary material, we de-scribe the definition of the Double Gyre flow and perform additionalexperiments. In the following, we study the time interval [ , ] andintegrate forward in time.In Fig. 2, we employ a Gaussian error model estimated duringdata reduction to a space-time grid of size [ × × ] . In(a), we have stochastically advected a large amount of randomlydistributed particles to visualize separating manifolds in the flow.The DBS shown in (b) clearly corresponds to these structures. Theuncertainty of the transport is visualized in (c) and indicates a highuncertainty in the midst of both gyres. The DBS is low in theseregions. For reference, we illustrate the FTLE of the mean flow in(d), which does not consider the stochastic component of the flow.The FTLE indicates the presence of several smaller features aroundthe two gyres that are not depicted in the density visualization in (a)or the DBS (b) and are located in regions of high uncertainty (c).The mean D-FTLE from Guo et al. [4] shown in (e) indicates alarger amount of structures. The center of the left gyre even showsadditional structures that are not present in the FTLE or the advectedparticles (a). The transport uncertainty indicates a high uncertaintyin this area (c). However, the variance of the D-FTLE (f) is only highnear the central barrier of the flow. At the same time, the presenceof this barrier is far from uncertain. A high variance in the D-FTLEthus does not necessarily imply uncertainty of the transport barriers. This dataset from the SciVis contest 2020 is an ensemble simulationof the circulation dynamics in the Red Sea [19]. Eddies in the oceanplay a major role in the transport of energy and particles. Uncertainty3 (a) Density of advected particles (b) DBS (c) Transport uncertainty (d) FTLE (e) D-FTLE (mean) (f) D-FTLE (variance)
Figure 2: The Double Gyre dataset with uncertainty estimated during data reduction to a grid of size [ × × ] . (a) Temperature t (b) Temperature t Figure 3: The mean temperature distribution in t (a) and t (b)illustrates the diffusion of temperature in the Red Sea and is closelyaligned with the stochastic enhancers and barriers characterized bythe DBS.is estimated from 50 ensemble members created from perturbed ini-tial conditions. Here, we estimate the mean and covariance fromthe individual members and analyze the resulting uncertain flow.Since the depth of the dataset is irregularly spaced, we have resam-pled it to a grid of size 500 × ×
150 with 60 time steps. To aidthe understanding of the dataset, we have added topography andbathymetry [3] to our visualizations.In Fig. 1 (a) the backward DBS over a time interval of 182 hoursis shown and indicates enhancers to stochastic transport. In Fig. 3 (a)and (b) we visualize the diffusion of the temperature over time nearthe surface. Note that this diffusion corresponds to the enhancersindicated by the DBS. The transport uncertainty shown in Fig. 1 (b)indicates a high uncertainty in the gulf of Aden, the lower right partof the dataset. This suggests that we should investigate this region inmore detail, for example by looking at individual ensemble members.Note that the DBS visualizes only the transport of the aggregatedstochastic flow, but does not consider individual ensemble members.
All of our evaluations were performed using GPU acceleration on anNVIDIA Quadro RTX 8000 with CUDA. To integrate deterministicflows, a fourth-order Runge-Kutta scheme is used. Stochastic flowsare integrated using the Euler-Maruyama method with a constantnumber of 100 Monte Carlo runs.Performance measurements for different datasets are shown in Table 1: Performance measurements of our datasets.
Dataset Resolution FTLE DBS FTLE-DDouble Gyre 1024 ×
512 4 . . ×
150 2739ms 5193ms 2 , , × . . , ×
750 110 . . , Table 1. The DBS requires evaluating Equation 8 during each inte-gration step. Computing the DBS thus takes two to four times longerthan the FTLE. In comparison, methods that depend on stochasticintegration increase the runtime by several orders of magnitude.
Our results show that the DBS is significantly faster to compute thanprobabilistic extensions of the FTLE since no stochastic integrationis performed. In our experiments, the DBS closely aligns with thedensity of stochastically advected particles, whilst many featuresfrom the FTLE and its probabilistic extensions are not visible. In fact,probabilistic extensions of the FTLE show features of possible real-izations of an uncertain flow. In contrast, the DBS indicates featuresthat exist in an inherently stochastic flow. Although both approacheshave merit, it makes the probabilistic D-FTLE hard to interpret. Atfirst glance, the variance of the D-FTLE might suggest uncertain-ties of the transport barriers and enhancers, however, this is not thecase. Indeed, none of the approaches convey the actual amount ofuncertainty encountered during integration. Our visualization of thetransport uncertainties efficiently illustrates this uncertainty in theLagrangian frame, thus providing additional insights.
ONCLUSION
In this study, we introduce the theory of stochastic transport barri-ers and enhancers to the visualization community and discuss itsapplication to Gaussian flow fields. The diffusion barrier strength,a quantity similar to the FTLE, visualizes the transport behavior inuncertain and unsteady flows. Compared to probabilistic extensionsof the FTLE, the computation is significantly more efficient sinceno stochastic integration is performed. Moreover, the visualizationis considerably simplified since the results are not probabilistic oraveraged. By design, the DBS does not consider the absolute scaleof deviations, which would require stochastic integration. This as-sumption is problematic if regions of strong uncertainties exists inthe flow. To this end, we propose a complementary visualization ofthe transport uncertainties that measures generalized variance in theLagrangian frame of reference.4 R EFERENCES [1] G.-P. Bonneau, H.-C. Hege, C. R. Johnson, M. M. Oliveira, K. Pot-ter, P. Rheingans, and T. Schultz.
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