Elsevier Editorial System(tm) for Computers & Graphics Manuscript Draft Manuscript Number: CAG-D-12-00026R2 Title: Surface-Based Flow Visualization Article Type: Survey Paper Keywords: Survey; Flow Visualization; Surfaces. Corresponding Author: mr matt edmunds, Corresponding Author's Institution: Swansea University First Author: matt edmunds Order of Authors: matt edmunds; Robert S Laramee; Guoning Chen; Nelson Max; Eugene Zhang; Colin Ware Abstract: With increasing computing power, it is possible to process more complex fluid simulations. However, a gap between increasing data size and our ability to visualize them still remains. Despite the great amount of progress that has been made in the field of flow visualization over the last two decades, a number of challenges remain. Whilst the visualization of 2D flow has many good solutions, the visualization of 3D flow still poses many problems. Challenges such as domain coverage, speed of computation, and perception remain key directions for further research. Flow visualization with a focus on surface-based techniques forms the basis of this literature survey, including surface construction techniques and visualization methods applied to surfaces. We detail our investigation into these algorithms with discussions of their applicability and their relative strengths and drawbacks. We review the most important challenges when considering such visualizations. The result is an up-to-date overview of the current state-of-the-art that highlights both solved and unsolved problems in this rapidly evolving branch of research.

Surface-Based Flow VisualizationSubmission1

1Submission

Abstract

With increasing computing power, it is possible to process more complex fluid simulations. However, a gap between increasingdata size and our ability to visualize them still remains. Despite the great amount of progress that has been made in the field offlow visualization over the last two decades, a number of challenges remain. Whilst the visualization of 2D flow has many goodsolutions, the visualization of 3D flow still poses many problems. Challenges such as domain coverage, speed of computation, andperception remain key directions for further research. Flow visualization with a focus on surface-based techniques forms the basisof this literature survey, including surface construction techniques and visualization methods applied to surfaces. We detail ourinvestigation into these algorithms with discussions of their applicability and their relative strengths and drawbacks. We review themost important challenges when considering such visualizations. The result is an up-to-date overview of the current state-of-the-artthat highlights both solved and unsolved problems in this rapidly evolving branch of research.

Keywords: Survey, Flow Visualization, Surfaces

1. Introduction

Flow visualization is a powerful means for exploring, ana-lyzing and communicating simulation or experimental results.Flow visualization is characterized by a range of differing tech-niques such as direct, feature, texture, and geometric-based rep-resentations [PVH∗03, LHD∗04]. Each technique has a rangeof differing accuracies and speeds [LEG∗08]. The phenomenato be studied can be sampled using regular or irregular grids,which can stem from steady-state or unsteady flow. There aremany challenges to overcome in this field of research. The topicof flow visualization with surfaces has become an increasinglyimportant field of research in recent years (see Figure 1). This isdue to the advantages that surface-based techniques offer overmore traditional curve-based methods and the maturity of 2Dflow visualization. This provides strong motivation for study-ing and categorizing the breadth and depth of surface-based re-search for flow visualization.

1.1. Sources of Flow DataThere are many different origins of vector data which can

be categorized, for example: flow simulation and flow mea-surement. Flow simulation is often used to predict real-worldconditions both as an aid to design and for the analysis of largephysical systems which may not be captured or recorded withexisting devices. This is especially true for very large and ex-pensive projects such as aircraft, ship, and car design. The cost

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Figure 1: This histogram shows the number of publications per year focusedon flow visualization with surfaces. It indicates the growing momentum andimportance of this topic.

of building physical prototypes for testing can be great. There-fore the ability to simulate the test conditions using virtual rep-resentations such as CAD (Computer Aided Design) data com-bined with computational systems such as CFD (ComputationalFluid Dynamics) is highly beneficial.

1.2. Applications

The application of flow visualization techniques to real-world problems is essential for engineers and practitioners togain an understanding of the information the data contains. Cur-rent techniques have been integrated into a wide variety of testand simulation systems, for example Fluent [ANS10]. Allow-ing the engineers to explore and evaluate data from within thesesystems in visual forms is key to effectively gain insight. Man-ually processing large amounts of numerical data is time con-suming, prone to error, and is only performed by specialists.

Preprint submitted to Computers & Graphics July 27, 2012

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Figure 2: A multi-resolution visualization using glyphs illustrating the flow atthe surface of a cooling jacket as part of an engine simulation. Color is mappedto velocity magnitude. Image courtesy of R.S.Laramee et al. [PL08].

The graphical representation and exploration of data not onlyallows for much faster analysis but also enables non-experts tounderstand the underlying phenomenon.

As datasets increase in size and complexity it becomes moreimportant to support effective exploration of their features andcharacteristics. Many of today’s applications for flow visualiza-tion are centered around fields such as aerospace, automotive,energy production, and other scientific research. Automotivedesign tends to focus on aerodynamic drag to help improve theefficiency of the vehicle, and airflow around the engine com-partments which provide much needed cooling for the variousengine/cockpit heat exchange systems. The heat exchange sys-tems themselves are also an important focus for analysis. (seeFigures 2 and 3). Aerospace also focuses on aerodynamics andengine design for aircraft, with similar such focuses for spacecraft. Electromagnetism and turbine design within the energyproduction industries are common applications of vector fieldvisualizations [RP96].

An application example within astrophysics is the visual-ization of 2008 IEEE Visualization Design Contest [IEE10];the simulation of ionization front instability. Another areaof study is the topic of acoustic flow to simulate and visual-ize such things as engine exhaust, and speaker cabinet design[Tan10]. The medical field uses visualizations to study suchphenomena as blood flow, for example, when designing heartpumps to support failing hearts. Another area of utilization isweather systems analysis by organizations such as the MET of-fice [GOV10].

1.3. Challenges

Surface-based approaches share some common problems as-sociated with flow visualization in general. Examples of thesechallenges include: large, time-dependent simulation data re-quiring the utilization of out-of-core techniques; and the han-dling of unstructured data. Surface-based methods also face

their own unique challenges which we discuss in more detailthroughout the paper.

Construction. Surface construction is a key topic for this sur-vey. Surfaces must represent an accurate approximation ofthe underlying simulation. Adequate sampling must be main-tained while reducing the extra computational overhead asso-ciated with over-sampling. Resulting meshes must also remainsmooth in the presence of various flow phenomena such as vor-tex cores, and highly divergent or convergent flow. A largeamount of effort has been put into the creation of various typesof surfaces and these form a large portion of this survey. SeeSection 2 for literature that addresses this challenge.

Occlusion. When using surfaces the problem of occlusion oc-curs frequently. This may stem from multiple surfaces that oc-clude one another, a large surface that produces self occlusion,or a combination of both. There are several approaches that canbe taken depending on the surface type to reduce this problem.A general approach is to use transparency. With integral sur-faces, i.e., surfaces to which the flow field is tangent, we havemore options. Advanced texture mapping may also be used.Additionally, integral surface seeding positions may be changedto reduce clutter. See Section 3 for literature that addresses thischallenge.

Information Content. While surfaces offer many advantages interms of perception, a basic visualization of the surface alonemay not provide sufficient information about the underlyingdata. For example a stream surface alone does not show thebehavior of inner flow contained within the surface. A reviewof the research that enhances the resulting visualization of sur-faces is also provided in this survey. See Section 3 for literaturethat addresses this challenge.

Placement and Seeding. Interactive placement is the mostcommon method currently used. There is a strong correlation

Figure 3: Evenly spaced streamlines on the boundary surface of a coolingjacket flow simulation. Image courtesy of R.S.Laramee et al. [SLCZ09].

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ClassificationConstructing Surfaces for Flow Visualization

Integral: Stream/Path Integral: Streak/Time Implicit Topological[Hul92]s, [USM96]s,

[SBH∗01]s, [GTS∗04]s,

[STWE07]t, [GKT∗08]t,

[PCY09]s, [SWS09]s,

[MLZ09]t, [PS09]s.

[YMM10]t. [SRWS10]s.

[vFWTS08a]t, [KGJ09]t,

[BFTW09]t, [MLZ10]t,

[FBTW10]t,

[vW93]s, [WJE00]s,

[Gel01]s.

[TWHS03]s, [WTHS04]s,[TSW∗05]t, [BSDW12]t

Rendering Flow on Surfaces for VisualizationDirect Geometric Texture: Static Texture Texture: Dynamic Texture

[PL08]s, [PGL∗12]s. [LMG97]s, [LMGP97]s,

[WH06]s, [SLCZ09]s,

[BWF∗10]s, [HGH∗10]t,

[EML∗11]s, [ELM∗12]s,

[ELC∗12]s.

[vW91]s, [dLvW95]s,

[FC95]t, [MKFI97],

[BSH97]s, [SK98]t,

[Wei09]t, [PZ10]s.

[LJH03]t, [vW03]t,

[LvWJH04]t, [LSH04]s,

[LWSH04]s, [WE04]t,

[LGD∗05]s, [LGSH06]s,

[BSWE06]s, [LTWH08]t.

Table 1: This table classifies surface techniques into two main categories; Constructing surfaces for flow visualization and Rendering flow on surfaces for visual-ization. The table also sub-classifies the surface construction into Integral, Implicit, and Topological, with Integral surfaces further divided between stream/pathand streak surfaces. Additional sub-classification of this section into point based, triangle based and quad based primitives are shown with color. The renderingsection is sub classified into Direct, Geometric, and Texture based techniques, with the texture based category further subdivided into static texture and dynamictexture based techniques. Additional sub-classification of this section into Parameter Space , and Image Space techniques are displayed with color. An additionalsuffix is used to represent techniques applied to steady-state (s), and time-dependent (t) vector fields. Each of the entries are ordered chronologically within eachsubcategory.

between seeding and occlusion of integral surfaces. Seeding toomany surfaces, or seeding them in such a way that they occupythe same region of the domain leads to high levels of occlu-sion. The placement of isosurfaces is a function of the selectedisovalue. Choosing optimal isovalues is an analogous problem.See Sections 2.2, 2.3, 3.1 and 3.2 for literature which studiesthese challenges.

1.4. ClassificationThe classification in this paper represents the subtopics of

flow visualization with surfaces. The classification highlightsareas which are more mature and areas which require additionalwork. Refer to Table 1.

The two main classifications deal with surface constructiontechniques e.g., the fluid flow is represented by the surface cur-vature, and surface rendering techniques e.g., fluid flow prop-erties that are visualized on the surface geometry. This classifi-cation highlights the difference between applying visualizationtechniques to surfaces, and the underlying surface construction.

Each of the two classifications are further sub-classified. Thesurface construction is classified into Integral, Implicit, andTopological techniques, while the visualization of flow on sur-faces is divided into Direct, Geometric, and Texture-based tech-niques. Topological techniques visualize flow topology explic-itly using surfaces to do so. Surface construction is also sub-classified into triangle, point, and quad-based meshing tech-niques, while the visualization of flow on surfaces is sub-

classified into parameter space-based, and image space-basedtechniques.

Another possible choice for the visualization of flow onsurfaces is classification into single chart (single parametriza-tion) and a collection of charts (atlas-based parameterization).The image space methods would be treated similar to othersimple parameterizations such as C-Space techniques. Thesub-categories are further divided into steady flow and time-dependent flow. The papers within each subcategory are or-dered chronologically. We note that this survey does not coverflat or planar surfaces or slices like those described by Laramee[Lar03].

1.5. Contributions and Summary

Surface techniques fall into two main categories: construc-tion of surfaces and visualization techniques applied to sur-faces. Surface construction techniques are a fairly well re-searched topic. The majority of techniques are variations andextensions of the original Hultquist method [Hul92]. Thesetechniques are either faster, more accurate, cater to large datadomains, or address topology. The self-occluding problems in-herent of surfaces are partially addressed by Loffelmann et al.who effectively create holes in the surface [LMGP97], Theiselet al. [TWHS03] use connectors to represent the separating sur-face, and the general approach of using translucency to alleviateocclusion e.g., see Sections 2 and 3.

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Figure 4: The classifications of construction techniques illustrates a chronological flow of work from author to author. The child-parent relationships indicates thekey ideas that are progressed. The flow of work diverges and in some cases converges as new concepts are built on top of previous ideas. The charts also show theoriginating work, and where key work is continued.

The methods of van Wijk [vW93] and Westermann et al.[WJE00] concentrate on deriving a scalar field from a vectorfield and then employing isosurface techniques to represent thedomain. Although these techniques provide good domain cov-erage, visual clutter and occlusion can result. The projection ofvector information onto a surface by Laramee et al. [LGSH06],improves performance and perception of the flow local to thatsurface, but the surface occlusion issue remains due to the na-ture of the image space-based techniques. In addition, isosur-face work in the area of unsteady flow data is limited.

A fairly common theme throughout the different surfacetypes is the application of additional visualization techniques toenhance the surfaces. Parameter and image space-based tech-niques are the main focus of these visualizations as they canprovide effective interactive solutions. Given a surface (it canbe any type, as long as it is manifold) and a vector field de-fined on it, the flow behavior can be illustrated with the de-sired dimension of visual mapping, such as 0D (hedgehogs),1D (streamlines), or 2D (textures). This enables not only thedirect display of the flow data in the Eulerian point of view, orthe visualization of the behavior of the selected particles in theLagrangian point of view, but also the complete (dense) imageof the flow behavior over the surfaces. In addition, combinedwith other conventional visualization techniques, such as colorcoding and animation, more complete flow information includ-ing both vector magnitude and orientation, as well as the timevarying characteristics, can be conveyed.

The main benefits and contributions of this survey survey are:

• A review of the latest research developments in flow visu-alization with surfaces.

• The introduction of a novel classification scheme based onchallenges including; construction, rendering, occlusion,and perception. This scheme lends itself to an intuitivegrouping of papers that are naturally related to one another.

• The classification highlights both mature areas wheremany solutions have been provided and unsolved prob-lems.

• A concise overview in the area of flow visualization withsurfaces for those who are interested in the topic and wish-ing to carry out research in this area.

This report is divided into four main sections: First is a re-view of the surface construction techniques in Section 2. Thena review of flow visualization on surfaces in Section 3. Ananalysis of the different sub-classifications is conducted in Sec-tion 4 with an emphasis on the initial seed or surface place-ment/generation, perception, visual clutter and occlusion. Fi-nally the survey finishes with conclusions drawn from the anal-ysis, and proposed areas of future work in Section 5.

2. Constructing Surfaces For Flow Visualization

This section studies the different construction techniques sur-veyed. Figure 4 shows a chronological flow of work from au-thor to author. The child-parent relationships indicate the originand evolution of key ideas. The work diverges as new conceptsare built on top of previous ideas.

We start this section with a discussion about flow data and as-sociated challenges, before moving on to the sub-classificationof primitive types used for the surface mesh representations.Following this we study the range of surface construction tech-niques. These are divided into Integral, Implicit, and Topologi-cal surface construction methods.

Steady State, Time Dependent and Large Complex DataEarly work focuses on processing steady-state simulations

which represent a static flow field or single snapshots of flow intime. As the ability to process larger amounts of data increases,the research into processing this data follows. Data samplingbecomes denser, the size of the simulation domain increases,and multiple time steps are incorporated and processed. Thestructure of the data can be complex, incorporating a range ofassociated scalar quantities representing range of additional at-tributes.

Velocity data is comprised of a set of x, y, z componentsfor each sample point within the data domain. For exam-ple a 3D steady-state vector field vs(p) ∈ R3 where vs(p)

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Figure 5: Visualization of the flow field of a tornado with a point-based streamsurface. The stream surface is seeded along a straight line in the center of therespective image. Image courtesy of D.Weiskopf et al. [STWE07].

=[vx(x, y, z) vy(x, y, z) vz(x, y, z)

]for p ∈ Ω, vs ∈ R3

and Ω ⊂ R3, where Ω may be a 3D regular, structured,unstructured or irregular grid. For unsteady flow we havea time-dependent vector field vt(p) ∈ R3 where vt(p) =[vx(x, y, z, t) vy(x, y, z, t) vz(x, y, z, t)

]for p ∈ Ω, vt ∈ R3 and

Ω ⊂ R3.Our survey discusses work addressing the challenges of both

steady and time-dependent data. In Table 1 we use an ’s’ suf-fix to the citation for techniques which process steady statevector fields and a ’t’ suffix for techniques addressing time-dependent data. The trend of moving from steady-state towardtime-dependent data can be observed in the chronological clas-sification of work.

Point vs. Triangle vs. Quad-based Construction Techniques

Examining the sub-classification of surface mesh construc-tion techniques into point-, triangle-, and quad-based methodsyields some interesting insights into these approaches. Point-based surfaces are generally simpler and faster to construct asthey do not require any mesh construction computation to rep-resent a closed surface. This approach can be limiting regard-ing rendering options. Rendering the vertices as simple pointsprites, disks, or spheres is effective for dense vertex represen-tations, but gaps or inconsistencies can appear when viewingin close proximity to the surface. Lighting can also be a chal-lenge with this technique as the methods available for normalcalculation become limited and increase computation. Anotherapproach to rendering is using image-based techniques whichdon’t explicitly require geometric primitives to render closedsurfaces.

Schafhitzel et al. [STWE07] present a point based stream andpath surface algorithm where the vertices for the surface repre-sentation are generated on the GPU. With a dense output, eachof the vertices and their normals are used to render a closed sur-face as small, lit, point sprites, as in Figure 5. A texture-basedclosed surface rendering can also be achieved using LIC [CL93]performed in image space. With a similar approach, Ferstl etal. [FBTW10] present a streak surface algorithm which has

a rendering option using spherical point sprites to represent aclosed surface.

To represent a geometric mesh for rendering we must definehow the mesh is constructed. Of the two common methods fordefining a mesh, one is defined by constructing an array of ver-tices in the correct order for rendering the primitive. This ap-proach can hold redundant instances of the same vertices for agiven mesh, with a larger memory footprint and increased datatraversing the graphics bus at the cost of valuable bandwidth.In some simple surface construction implementations howeverthis method can be easier to implement.

A second approach to defining a mesh for rendering primi-tives is the utilization of an indexing array along with the arrayrepresenting the vertices of the surface. An additional indexarray specifies each vertex in the correct order to construct thegeometric primitive. In the context of surfaces the result of thisapproach is a much smaller data array, but the addition of anindex array. The index array can be compressed, dependingon the quantity of vertices, by using data types requiring lessmemory such as unsigned characters or unsigned short integersinstead of integers. This also significantly reduces the graphicsbandwidth.

Another constraint on implementations is the method usedfor constructing the data and index arrays. The most commonmesh primitive used is the triangle. This is the default for sur-face construction techniques such as isosurfaces as used by vanWijk [vW93] and Westermann et al. [WJE00]. This is a bi-product of early graphics card support for rendering triangles.

The most common approach for meshing integral surfaceswith triangles is a greedy minimal tiling strategy as describedby Hultquist [Hul92]. Other approaches include; additionalprocessing for streak surfaces where the surface topologychanges with time as in the work by Krishnan et al. [KGJ09]as shown in Figure 6, and processing of irregular grids such astetrahedra as described by Scheuermann et al. [SBH∗01].

With advancing front integration techniques, a simple ap-proach to generating a mesh is the direct use of the quad patchrepresented by the bounding streamlines and timelines. Thisapproach initially requires less computational expense, how-

Figure 6: Time surface mesh in the Ellipsoid dataset. Although the surfacehas undergone strong deformation, the mesh remains in good condition. Imagecourtesy of C.Garth et al. [KGJ09].

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Figure 7: The algorithm by McLoughlin et al. [MLZ09] handles widely di-verging flow while maintaining the desired organized advancing front. Thissurface is colored according to the underlying flow characteristics: left rotationis mapped to yellow, right rotation is mapped to orange, parallel flow is mappedto green, divergence is mapped to blue, convergence is mapped to red. Imagecourtesy of R.S.Laramee et al. [MLZ09].

ever dealing with issues of sheering quads, t-junctions and ad-ditional normal computations (one per quad corner rather thanone per triangle) can have a significant impact. Refer to Fig-ure 7 and see McLoughlin et al. [MLZ09, MLZ10] and Schnei-der et al. [SWS09]. Peikert and Sadlo [PS09] construct theirsurface geometry in an incremental manner advancing fromthe initial curve structure attempting to avoiding the issue ofquad sheering. The algorithm by van Gelder et al. [Gel01] alsolends itself to quad-based meshing due to the connectivity ofthe curvilinear grids.

2.1. Integral Construction Techniques

In this subsection we present an overview of integral sur-face construction techniques. This work is subdivided intostream/path, and streak/time surface construction algorithms.The similarity between the stream and path surface construc-tion provides a natural classification for discussion in the nextsubsection. Following the review of stream/path surfaces wethen present a study of streak surface construction. The mainchallenges addressed by the literature in this subsection are ac-curate surface construction, performance time, and continuousrepresentations.

2.1.1. Stream/Path Surface ConstructionA streamline is a curve which is tangent to the velocity field

at every point along its length. A Streamline is the trace of amassless particle from an initial location (seed point). Stream-lines show the direction fluid flow within a steady-state flowdomain. if v(p) is a three dimensional vector field, the stream-line through a point x0 is the solution I(x0, t) to the differentialequation:

ddt

I(x0, t) = v(I(x0, t))

with the initial condition I(x0, 0) = x0. A stream surface is thetrace of a one-dimensional seeding curve C through the flow.The resulting surface is everywhere tangent to the local flow. A

stream surface S is defined by:

S (s, t) := I(C(s), t)

S is the union or continuum of integral curves passing throughthe seeding curve C. S (s,−) coincides with an individual in-tegral curve, and S (−, t) coincides with individual time lines[GKT∗08].

Since there is no normal component of the velocity alongstreamlines and stream surfaces, mass cannot cross their bound-ary and therefore they are useful for separating distinct regionsof similar flow behavior. In practical applications a discretizedapproximation of the stream surface is constructed by tracingdiscretized seeding curves through the vector field using inte-gration methods such as the fourth-order Runge Kutta integra-tion scheme.

Hultquist introduces one of the first methods [Hul92] for thegeneration of stream surface approximations. This techniqueincludes strategies for controlling the density of particles acrossthe advancing front. Points can be added to the advancing frontwhen the sampling rate becomes too sparse and removed whenit becomes too dense. Neighboring pairs of streamlines are tiledwith triangles to form ribbons. These connected ribbons thenform the surface. Ribbons which encounter rapid divergenceof flow may be torn/ripped to allow the surface to flow aroundan obstacle. The separate portions of the surface are then com-puted independently. Hultquist builds on work by Belie [Bel87]and Kerlick [Ker90] who describe narrow stream ribbon meth-ods, and Schroeder et al. [SVL91] who describe a stream prim-itive called the stream polygon.

Ueng et al. [USM96] expand the stream surface work tostream ribbons, stream tubes, and streamlines on unstructuredgrids. The authors build on the stream polygons algorithmby Darmofal and Haimes [DH92], the research by Ma andSmith [MS93], and the steam ribbons of Pagendarm [PW94].This paper describes extending the techniques to unstructuredgrids by converting the physical coordinate system to a canon-ical coordinate system. The main idea is the use of a special-ized fourth-order Runge Kutta integrator which requires onlyone matrix-vector multiplication and one vector-vector additionto calculate the successive streamline vertices. This technique

Figure 8: The formation of vortices at the apex of a delta wing illustrated withthe use of a stream surface. Image courtesy of C.Garth et al. [GTS∗04].

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significantly simplifies the construction of the geometric primi-tives, reducing the computational cost and therefore improvingspeed.

Scheuermann et al. [SBH∗01] present a method of streamsurface construction on tetrahedral grids. The technique prop-agates the surface through the tetrahedral grid, one tetrahedronat a time, calculating on the fly where the surface intersects thetetrahedron. This approach enables the inclusion of topologi-cal information from the cells such as singularities. When thesurface passes through the tetrahedron, the curve segments endpoints are traced as streamlines through the next cell. For eachpoint on a streamline, a line is added connecting it to its coun-terpoint. These are then clipped against the faces of the tetrahe-dron cell and the result forms the boundary of a polygonal sur-face within the cell. This method is inherently compatible withmulti-resolution grids and handles increased grid resolution inintricate flow regions. This work is a significant improvementover the previous irregular grid work, improving surface con-struction within complicated areas, typically of more interest influid dynamics.

Garth et al. [GTS∗04] improve on the Hultquist method andshowed how to obtain surfaces with higher accuracy in areas ofintricate flow. See Figure 8. The improvements are achievedby employing arc length particle propagation and additionalcurvature-based front refinement. They also considered visu-alization options such as color mapping of vector field relatedvariables going beyond straightforward surface rendering. Anovel method to determine boundary surfaces of vortex coresand a scheme for phenomenological extraction of vortex corelines using stream surfaces is discussed and its accuracy is com-pared to one of the most established standard techniques.

Next we review the concept of a path surface. We start bydescribing a pathline. A pathline or particle trace is the tra-jectory that a massless particle takes in time dependent fluidflow. If v(x, t) is a three-dimensional vector field for x in domainΩ ∈ R3 and t in a time interval [T0,T1] the pathline I(x0, t0; t)passing through x0 at time t0 is the solution to the ordinary dif-ferential equation:

ddt

I(x0, t0; t) = v(I(x0, t0; t), t)

with the initial condition I(x0, t0; t0) = x0. A path surface is thetrajectory of a massless curve C in time-dependent fluid flow.A path surface P is defined by:

P(s, t) := I(C(s), t0; t)

Where P is the union or continuum of pathlines passing throughthe seeding curve C at time t0. S (s,−) coincides with an indi-vidual integral curve, and S (−, t) coincides with an individualtime lines [GKT∗08]. The first example of this is the work bySchafhitzel et al. [STWE07] who introduce a point-based algo-rithm for stream and path surface construction and rendering.

Schafhitzel et al. combine and build on three specific ar-eas: stream surface computation Hultquist [Hul92], renderingof point-based surfaces Zwicker et al. [ZPKG02], and texture-based flow visualization on surfaces (Weiskopf et al. [WE04]).

Figure 9: Path surface visualization of vortex shedding from an ellipsoid. Thetransparent surface consists of 508,169 triangles. Different layers identified bycolor mapping. Image courtesy of C.Garth et al. [GKT∗08]. c© IEEE/TVCG.

The stream/path surface generation is modified to run on theGPU in a highly parallel fashion. Seed points are generatedand integrated through the vector field. To maintain a roughlyeven density, integrated points along the advancing front are in-serted and removed. The surface rendering method is based onpoint set surfaces (PSS) and is extended to include stored con-nectivity information, this enables quick access to neighboringpoints. The authors’ approach to the hybrid object/image spaceLIC method displays clear line patterns which show a choice ofpath lines.

More recently, Garth et al. [GKT∗08] replaced the advanc-ing front paradigm by an incremental time line approximationscheme. See Figure 9. This allows them to keep particle inte-gration localized in time. The authors propose a decoupling ofthe surface geometry and graphical representation, and a curverefinement scheme which is used to approximate time lines,yielding accurate path surfaces in large time-dependent vectorfields.

Following recent work by Bachthaler and Weiskopf [BW08]who describe the use of tracing structures perpendicular to thevector field to generate animated LIC patterns orthogonal to theflow direction, and work by Rosanwo et al. [RPH∗09] which in-troduces dual streamline seeding based on streamlines orthog-onal to the vector field, Palmerius et al. [PCY09] introduce theconcept of perpendicular surfaces. These surfaces are perpen-dicular to the underlying vector field.

The authors describe the common properties of such surfacesand discuss the issues of non-zero helicity density, and stopconditions. The construction method starts at a predefined seedpoint propagating outwards in a clockwise spiral fashion whereeach new point is integrated perpendicular to the flow field for agiven distance. The stop criteria are: length limit, accumulatedwinding angle limit, and maximum orientation error as a resultof vector fields with non-zero helicity. Convergence or diver-gence is characterized by cone shaped surfaces. A combinationof both results are saddle shaped surfaces. Vortices distort thesurfaces by tearing them apart and producing a fan like pattern.A fast approach for generating the surfaces and stop conditionsis also described.

Extending the previous work by Garth et al [GKT∗08],Schneider et al. [SWS09] produce a more accurate and

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Figure 10: 2D manifolds visualizing the critical point of the Lorenz dynamicalsystem. Image courtesy of R.Peikert et al. [PS09].

smoother timeline interpolation using a fourth-order Hermiteinterpolation scheme when adjusting the advancing front den-sity. The Hermite interpolation between streamlines requiresthe covariant derivatives to be calculated with respect to s(streamline) and t (timeline). An additional surface accuracyerror criterion is used to dictate when coarsening or refinementtakes place. The error based refinement strategy splits a ribbonwhen the local interpolation error exceeds a given bound. Thiserror is estimated directly by seeding a new short streamlinefrom a position between neighboring streamlines at time tn − 1integrating it to time tn.

Alternatively, McLoughlin et al. [MLZ09] describe a sim-ple and fast CPU-based method for creating accurate streamand path surfaces using quad primitives. The authors propose amethod which is based on a small set of simple, local operationsperformed on quad primitives, requiring no global re-meshingstrategy. To handle divergent, convergent, and rotational flow,the sampling rate of the advancing front is updated. The quadis either divided (in areas of divergence) or collapsed (in areasof convergence). For curvature, a test of the advancing frontrotation is conducted. If true then the advancing front integra-tion step size is reduced by a factor dependent on the amount ofrotation.

Using topology for the construction and placement of flowgeometry, Peikert and Sadlo present topology relevant methodsfor constructing seeding curves to produce topologically-basedstream surfaces [PS09]. The authors build on work by Garth etal. [GTS∗04] expanding the notion of feature visualization ap-plying stream surfaces to a range of singularities and periodicorbits. The discretised offset curves constructed at the topolog-ical structures are used to initialize the stream surface propaga-tion. See Figure 10. The authors construct their stream surfacefrom quads which are divided in areas of divergence to maintaina consistent mesh topology. This is achieved by subdividing be-tween neighboring nodes with a cubic interpolant after tracingback a fixed number of steps. The nodes of the mesh retain anumber of attributes representing the flow field, which are used

for controlling the growth of the surface and texturing.The work by Yan et al. [YMM10] proposes a number of sur-

face surgery operations during integration to reveal the fractalgeometry (thin sheet rotating around and tending to the attrac-tors) of the strange attractors in 3D vector fields which pre-viously were difficult to visualize. Their method consists ofthree major steps. First a polygonal surface is advected and de-formed according to the vector field. This polygonal surfaceis initialized as some regular shape, such as a torus, which ne-glects the fractal dimension of the strange attractor. Second,due to the possible high distortion, the polygonal surface mayneed refinement. In this step, a GPU-based adaptive subdivi-sion (i.e. edge division) of mesh is applied to preserve the nec-essary features. A mesh decimation on the CPU may also beconducted to reduce the resolution of uninteresting portion ofthe surface for memory efficiency. Third, a mesh re-tiling isperformed to maintain the consistent triangulation of the thinsheet structure when approaching the attractor and to correctthe self-intersection artifacts. This method has shown its util-ity through examples with strange attractors and is expected toapply to other integral surface computations.

Another extension to steam surface algorithms inspiredby Peikert and Sadlo [PS09], is the work by Schneider etal. [SRWS10] whose algorithm detects singularities within theflow field and deals with them appropriately, rather than the cur-rent methods of continuous refinement or splitting the surface.The authors use a pre-processing step to generate the requiredtopological information. The stream surface algorithm then de-tects intersection with the separating two dimensional manifoldof a saddle point. The resulting surface will either follow a newdirection appropriate to the local vector field when encounter-ing a node saddle (See Figure 11) or split when encountering aspiral saddle.

2.1.2. Streak/Time Surface ConstructionThe main challenges addressed by the methods presented

here are computational time and maintaining a continuous dy-namic surface.

A streakline is the line joining a set of massless particlesthat have all been seeded successively over time at the samespatial location in time dependent flow. Dye steadily injected

Figure 11: A stream surface in a linear vector field with highly divergingstreamlines where the angle criterion (130 degrees) for splitting the surfacefails because the surface does not run into the saddle. The red part of the sur-face would have been left out if the angle criterion were to be used. Imagecourtesy of D.Schneider et al. [SRWS10].

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into the fluid at a fixed point extends along a streakline. If v isa three-dimensional vector field defined over a domain Ω ∈ R3

and time interval [T0,T1] to find the streakline L(x,T0,T1; s) forparticles seeded at x0, starting at time T0, as it appears at timeT1, we must solve separately the pathline equation for I(x0, t0; t)for each t0 ∈ [T0,T1], and then let L(x,T0,T1; s) = I(x0,T1 −

s; s), for s ∈ [0,T1 − T0]. If seeded at the same location in asteady state flow field streamlines, pathlines and streaklines areidentical.

A streak surface is the smooth union of streaklines fromseeding locations along a continuous curve C. A streak surfaceK is the union of all particles emanating continuously from aparameterized curve C(u) over time interval [t0, t1] and movingwith the flow from the time of seeding t. In terms of individualstreaklines it can be described as:

K(u,T0,T1; t) := L(C(u),T0,T1; t)

Introducing the first streak surface approximation, VonFunck et al. [vFWTS08a] represent smoke structures as a tri-angular mesh of fixed topology, connectivity and resolution.The transparency of each triangle is represented by α whereα = αdensity αshape αcurvature α f ade. The density componentαdensity is a representation of the smoke optical model by con-sidering the triangle primitive to be a small prism filled withsmoke. The shape component αshape of the triangle is defined asthe ratio of its shortest edge to the radius of its circumcircle andrepresents its distortion. The curvature αcurvature is representedby the local mean surface curvature. The fade α f ade is defined asan increase in transparency over time. To effectively render thetransparency at real-time frame rates a depth peeling algorithmis utilized. The authors demonstrate modifications to the algo-rithm to mimic smoke nozzles and wool tufts. This techniqueis the first step in generating streak surfaces, and addresses theocclusion issues associated with complex flow structures repre-sented by surfaces.

Focusing on performance of large, time-varying vector fields,Krishnan et al. [KGJ09] propose a method for time and streaksurface generation. Their approach enables parallelization bydecoupling the surface advection and surface refinement. Theauthors build on work by Von Funck [vFWTS08a] with ex-tensions to time surfaces while parallelising the pipeline andimproving the meshing scheme using techniques described byBridson [Bri03]. This paper describes the algorithm with re-spect to time surfaces and then explores the extension to streaksurfaces. A time surface is generated in two steps. First eachpoint of the initial surface mesh is advected over the next timeinterval. The mesh is then passed to the adaptation phase wherethree basic operations, edge split, edge flip, and edge collapseare applied to refine the surface. To prevent irreparable changesto the mesh, the integration time step is automatically chosenrequiring that no vertex moves further from its current positionthan a predetermined function of maximum velocity. Streaksurface evolution is refined using a similar approach account-ing for the new particles seeded continuously from the seedingcurve. The surface visualization uses a combination of texturemapping, lighting effects, and depth peeling for transparency.

The use of these effects helps with occlusion and depth percep-tion.

Continuing in the same theme, Burger et al. [BFTW09] de-scribe two methods of streak surface construction for the visu-alization of unsteady flow. Building on work by von Funck etal. [vFWTS08a] the authors present the first real-time approachfor adaptive streak surface integration and high quality render-ing. See Figure 12. The first approach computes a quad-basedsurface where each quad patch is independent of all others. Thisindependence enables parallel processing and rendering of eachpatch on the GPU. The refinement of these patches is performedindependently and is based on an area criterion. If the criterionthreshold is met the quad patch is split along the longest edgeand its opposite edge, forming two new independent patches.The patches are rendered directly from the vertex buffer usinga two-pass approach to fill gaps left by the refinement process.

The second approach computes a point-based interconnect-ing triangular mesh which is modified during the refinementprocess. Each advected timeline is stored in its own vertexbuffer in order, and refined every time step. The refinementprocess is completed in three passes: time line refinement, con-nectivity update, streak line refinement. When inserting pointsthe location is determined by fitting a cubic polynomial andbisecting it equally between the two diverging points. The con-nectivity of the mesh is then updated by searching the previousand next timelines for any given point’s nearest neighbor. Thethird pass computes the maximum euclidean distance betweenneighboring timelines. The complete time line is then added orremoved. The mesh is then rendered after a final pass computesthe mesh triangulation.

Extending their work on quad-based stream and path sur-faces, McLoughlin et al. [MLZ10] present a novel streak sur-face algorithm using quad primitives. The refinement of thesurface is achieved by performing local operations on a quad-by-quad basis. Quads may be split or merged to maintain suffi-cient sampling in regions of divergence and convergence. Shearflow is handled by updating the topology of the mesh to main-tain fairly regular quads. This method is designed for and im-plemented on the CPU and generally achieves interactive framerates.

Following the collection of works which define methods for

Figure 12: The visualization of a transparent streak surface rendered usingdepth peeling and generated on the GPU. Image courtesy of H.Theisel et al.[BFTW09]. c© IEEE/TVCG.

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Figure 13: Particle based surface visualization. Red particles correspond topoints on the separating surface. Green particles serve as context information.They correspond to points on time surfaces, which are released from the planarprobe at a fixed frequency. Image courtesy of H.Theisel et al. [FBTW10]. c©IEEE/TVCG.

streak surface construction, Ferstl et al. [FBTW10] introducereal-time construction and rendering of surfaces, which rep-resent Lagrangian Coherent Structures (LCS), in conjunctionwith the rendering of the streak surface particles. See Fig-ure 13. This technique interactively displays the separationsurfaces leading to new possibilities of studying complex flowphenomena. The user can interactively change the seeding pa-rameters, and visually display the separation surfaces, resultingin a visually guided exploration of separation surfaces in 3Dtime-dependent vector fields.

LCS are computed by extracting the ridges in the finite timeLyapunov exponent (FTLE) field. The paper builds on the workby Sadlo and Peikert [SP07] who describe a filtered ridge ex-traction technique based on adaptive mesh refinement. Themethod enables a substantial speed-up by avoiding the seed-ing of trajectories in regions where no ridges are present or donot satisfy the prescribed filter criteria such as a minimum finiteLyapunov exponent.

The finite-time Lyapunov exponent (FTLE) [Hal01] quanti-fies the local of separation behavior of the flow. It is used tomeasure the rate of separation of infinitesimally close flow tra-jectories. The pathline solution x = I(x0, t0; t) of Section 2.1.1for fixed times t0 and t can be considered as a flow map froma position x0 to the pathline position x where it is advected bythe flow at time t. Using the flow map, the Cauchy-Green de-formation tensor field, Ct

t0 is obtained by left-multiplying theJacobian matrix of the flow map with its transpose [Mas99]:

Ctt0 (x) =

[∂(x0, t0; t)∂(x0)

]T [∂(x0, t0; t)∂(x0)

]From this, the FTLE is computed by:

FTLEtt0 (x) =

1t − t0

ln√λmax(Ct

t0 (x))

where λmax(M) is the maximum eigenvalue of matrix M[Hal01].

FTLE requires the choice of a temporal window, the effectof a change in the time-window length has not been studiedsufficiently [PPF∗10]. A common use of FTLE is to extractLagrangian Coherent Structures (LCS). LCS are extracted froman FTLE field by ridge extraction [SP07].

2.2. Implicit Construction Techniques

This section reviews a set of surface construction techniqueswhich are described by solving some function of the underlyingflow field. The motivation for this type of technique includeavoiding compound error associated with integration schemesand meshing challenges resulting from convergent, divergentand shear flow. The work in this area is limited to stream surfacerepresentations in steady-state velocity data.

van Wijk [vW93] introduces a method for the global repre-sentation of the stream surfaces as implicit surfaces f (x) = C.Once f is defined at the boundary, the definition is then ex-tended to the interior of the domain by specifying that it is con-stant on streamlines. C can be varied to efficiently generate afamily of stream surfaces. The originating curves are definedat the boundary by the value of f . This method greatly dif-fers from the advancing front methods introduced by Hultquist[Hul92]. Two methods are presented to derive f ; The first isbased on solving the convection equations, and the second isbased on backward tracing trajectories from grid points. The3D stream function defines a scalar field from which tradi-tional isosurface extraction techniques are then used to createthe stream surfaces.

Taking this concept a step further, Westermann et al.[WJE00] present a technique for converting a vector field toa scalar level set representation. See Figure 14. An analysisof the subsequent distorted level-set representation of time sur-faces is conducted before combining geometrical and topolog-ical considerations to derive a multiscale representation. Thisis implemented with the automatic placement of a sparse setof graphical primitives, depicting homogeneous streams withinthe fields. The final step is to visualize the scalar field with iso-surfaces. The advantage of this technique is full domain cov-erage as the van Wijk [vW93] method constructs surfaces onlywhere intersections with the boundaries occur.

With a different approach van Gelder. [Gel01] introduce asemi-global method which does not suffer from the compounderror from integral surface generation or computational over-head and error seen in the global approaches. Stream surfacesare constructed on 3D curvilinear grids which satisfy the con-

Figure 14: Dense flow fields are first converted into a scalar field, and thendisplayed and analyzed by means of level-sets in this field. Image courtesy ofR.Westermann et al. [WJE00].

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straints of a region expressed as integrals, instead of solvinga local ordinary differential equation. The constraints are ex-pressed as a series of solvable quadratic minimization prob-lems. The solution exploits the fact that the matrix of eachquadratic form is tridiagonal and symmetric. The author de-scribes the transformation of the curvilinear grid into parameterspace to simplify the stream surface construction problem.

2.3. Topological Surface Construction TechniquesThe challenge of topology based methods is to separate or

segment the flow into areas of similar behavior. As part ofthis process singularities and separatrices are extracted from theflow field. In steady-state flow the separatrices are stream sur-faces. The topological structures can also useful for supportingother flow visualization methods, and is the inspiration for tech-niques is this section.

Theisel et al. [TWHS03] present an approach for construct-ing saddle connectors in place of separating stream surfaces is asignificant effort to help alleviate occlusion. A saddle connec-tor is a streamline which connects two saddle points. Build-ing on work by Theisel and Seidel [TS03] the authors applysaddle connectors in three dimensions. This work is extendedby Weinkauf et al. [WTHS04] who introduce the concept ofseparating surfaces originating from boundary switch curves.A boundary switch curve is a curve generated at the domainboundary where inflow changes to outflow or vice versa e.g.,flow is parallel with the boundary surface. This is achieved byjoining saddle points to boundary switch curves, or betweeneach other, using a type of streamline called boundary switchconnectors. The idea of using streamline connectors in placeof separating surfaces reduces visual clutter as can be seen inFigure 15.

Inspired by Theisel and Seidel’s work on tracking featuresin 2D [TS03], Theisel et al. [TSW∗05] introduce a method forvisualizing the propagation of vortex core lines over time. Thecontextual surfaces are shown emanating from the vortex corelines in Figure 16. Two 4D vector fields are computed which

Figure 15: Topological skeleton showing saddle connectors, singularities andboundary switch connectors. Image courtesy of H.Theisel et al. [WTHS04].

Figure 16: Time surfaces shown as contextual information emanating from thevisualized vortex core lines. Image courtesy of H.Theisel et al. [TSW∗05].

act as feature flow fields such that their integration surfaces (e.g.stream surfaces) provide the vortex core structures. The featureflow field is equivalent to the parallel vector (PV) approachesby Peikert and Roth [PR99]. In addition, this work describes amethod to extract and classify local bifurcations of vortex corelines in space-time through the tracking and analysis of PV linesin the feature flow field.

Avoiding the integration of hyperbolic trajectories by replac-ing them with intersections of LCS while utilizing LIC to revealthe tangential dynamics, Bachthaler et al. [BSDW12] stack 2Dvector fields according to time to generate a 3D space-time vec-tor field. The LCS ridge structures are computed from an FTLEscalar field generated in three dimensions. The hyperbolic tra-jectories are mapped to saturation. Visualizing the LCS dynam-ics the authors apply the method described by Weiskopf et al.[WE04]. To address the problem of occlusion in the space-timevisualization of the LCS, the paper describes restricting the vi-sualization to bands around the LCS intersection curves. Theauthors adopt the concept of hyperbolic trajectories and space-time streak manifolds.

3. Rendering Flow on Surfaces for Visualization

This section presents a survey of techniques that enhance therendering and visualization of surfaces used for flow visualiza-tion. Figure 17 show a chronological flow of techniques demon-strating the child-parent relationships and key ideas that are pro-gressed. The charts also show the originating work, and wherekey work is continued.

We start this section with a discussion about the conceptualdifferences of Parameter Space and Image Space techniques.We then examine the rendering techniques. The techniques inthis section are classified into Direct, Geometric and Texture-based. The texture-based subsection is further divided intostatic and dynamic type textures.

Parameter Space and Image Space TechniquesOne approach to applying texture properties on surfaces is

via the use of a parameterization. Applying textures to surfaces

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Figure 17: The classifications of rendering techniques show a chronological flow of work from author to author. The child-parent relationships indicates the keyideas that are progressed. The flow of work diverges and in some cases converge as new concepts are built on top of previous ideas. The charts also show theoriginating work, and where key work is continued.

becomes particularly suitable when the whole surface can beparameterized globally in two dimensions as shown by Forsselland Cohen [FC95]. The drawbacks with this approach includechallenges such as distorted textures as a result of the mappingbetween object space and parameter space. A global param-eterization for many types of surface is not available such asisosurfaces generated from marching cubes algorithms.

A more recent approach is the use of image space techniquesto accelerate computation. The general approach is to projectthe surface geometry to image space and then apply a seriesof image space techniques. These techniques can range fromadvecting dense noise textures [LJH03] to rendering attributesfrom the underlying data to the surface such as streamlines[SLCZ09], or illustrating various perceptual attributes of thesurface such as silhouette edge highlighting [HGH∗10].

3.1. Direct Rendering Techniques

Direct visualization techniques are the most primitive meth-ods of flow visualization. Typical examples involve placing anarrow glyph at each sample point in the domain to representthe vector data or mapping to some scalar attribute of the localvector field. Direct techniques are simpler to implement andenable direct investigation of the flow field. However, thesetechniques may suffer from visual complexity and imagery that

Figure 18: The combination of velocity-range glyphs (Disk glyph) and stream-let tubes (Arrow glyph) is applied to provide both detailed (the range glyph)and summary (the streamlet) information of the vector field direction. Imagecourtesy of R.S.Laramee et al. [PGL∗12].

lacks in visual coherency. They also suffer from serious occlu-sion problems when applied to 3D data sets. This idea providesmotivation for the work classified in this section.

Extending the direct visualization paradigm, combining itwith clustering techniques and utilizing image space methods,Peng and Laramee introduce a glyph placement technique per-formed in image-space which visualizes boundary flow [PL08].This concept builds on work by Laramee et al. [LvWJH04] byprojecting the visible vector field from object-space to image-space, then generating evenly-spaced glyphs on a regular grid.The glyphs are a clustered approximation of the underlying vec-tor field mesh resolution. Calculated on the fly, this algorithm isefficient and can handle large unstructured, adaptive resolutionmeshes.

Following their previous image space work [PL08], Peng etal. [PGL∗12] present a novel, robust, automatic vector fieldclustering algorithm that produces intuitive images of vec-tor fields on large, unstructured, adaptive resolution boundarymeshes from CFD. Their bottom-up, hierarchical approach isthe first to combine the properties of the underlying vector fieldand mesh into a unified error-driven representation. See Fig-ure 18. Clusters are generated automatically, no surface pa-rameterization is required, and large meshes are processed ef-ficiently. Users can interactively control the level of detail byadjusting a range of clustering distance measures. This workalso introduces novel visualizations of clusters inspired by sta-tistical methods.

3.2. Geometric Illustration Techniques

In this section we study a range of geometric illustration tech-niques. The purpose of illustrative techniques is to enhance theoften complex surface representations commonly found in flowdata. A general solution to this problem is to use transparency.With surfaces we have additional options. Surface primitiveshave well defined normals thus they can offer perceptual advan-tages including: lighting and shading which provide intuitivedepth cues, the ability to texture map, image space techniquessuch as silhouette edge highlighting and the placement of addi-tional geometry on the surface.

Loffelmann et al. [LMGP97] introduce methods for plac-ing arrow images on a stream surface using a regular tiling,and offsetting portions of the surface, leaving arrow shapedholes which alleviate occlusion. The authors are inspired by

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Figure 19: Stream arrows offset from a stream surface. The portion offsetleaves holes, reducing occlusion by the surface. Image courtesy of H.Hauser etal. [LMG97].

Shaw’s [AS92] artistic approach to the use of stream surfaces tohelp with occlusion within the dynamical systems. The authorsextend this work as their method provides unsatisfactory resultsin areas of divergence and convergence [LMG97]. The arrowscould become too small or too large to provide quality streamsurface visualizations. The proposed novel enhancement is ahierarchical approach which stores a stack of scaled stream ar-rows as textures applying the suitably scaled arrow to the sur-faces maintaining consistent proportionality. See Figure 19.

Weiskopf and Hauser [WH06] explore a GPU based shadingmethod for the evaluation and visualization of surface shapes.Cycle shading and hatched cycle shading methods extend theidea of natural surface highlights in a regular repeating pattern.This technique can be used where Phong illumination is ap-propriate as curvature or mesh connectivity information is notrequired. To reduce depth induced aliasing an approach similarto Mip Mapping is introduced. The effectiveness of cycle shad-ing for the assessment of surface quality is demonstrated by auser study.

Image-based streamline seeding on surfaces by Spencer etal. [SLCZ09] utilize the perceptual properties of surfaces bydisplaying the local vector field as as evenly-spaced stream-lines [JL97], while maintaining lighting, shading and other use-ful depth cues. This image-based approach is efficient, handlescomplex data formats, and is fast. The evenly spaced stream-lines are produced by scanning the image, checking the z depthof the fragments (where the fragments z depth is obtained fromthe depth buffer), and initializing the seeding where the frag-ment z depth is non zero (i.e., the fragment is part of some ob-ject in the field of view) and no other streamline is closer thanthe user specified minimum distance.

The illustration of stream surfaces is explored by Born etal. [BWF∗10] who are inspired by traditional flow illustrationsdrawn by Dallmann, Abraham and Shaw in the early 1980’s.The authors describe techniques for silhouettes and feature

lines, halftoning, illustrative surface streamlines, cuts, and slabsto visually describe the surface shape. User interactive explo-ration with these techniques allows insight into the inner flowstructures of the data. Implemented on the GPU, this work re-quires no pre-processing for the final visualizations.

Further exploration of illustration techniques is performed byHummel et al. [HGH∗10] who present a novel application ofnon-photorealistic rendering methods to the visualization of in-tegral surfaces. See Figure 20. The paper examines how trans-parency and texturing techniques can be applied to surface ge-ometry to enhancing the users perception of shape and direc-tion. They describe angle, normal variation, window and sil-houette transparencies with adaptive pattern texturing. The au-thors present this work in a combined rendering pipeline imple-mented on the GPU. This maintains interactivity and removesthe need for expensive surface processing to generate visualiza-tions.

Defining seeding curves at the domain boundaries from iso-lines generated from a scalar field, Edmunds et al. [EML∗11]automatically seed and generate stream surfaces integratedthrough the 3D flow. The scalar field is derived from the exittrajectory of the flow from the domain capturing the topologi-cal constructs of the flow at the boundary. See Figure 21. Thiswork also discusses strategies for resolving occlusion resultingfrom seeding multiple surfaces.

Using a vector field clustering strategy combined with a de-rived curvature field to define stream surface seeding locationand seeding curve contour, Edmunds et al. [ELM∗12] use arange of illustrative techniques to enhance the visualization.The illustrations, as with surface placement, specifically targetproviding the required visual information for the engineer toaccurately interpret the flow characteristics. Figure 22 demon-strates the focus and context approach to the surface placementand visualizes the applied illustrative techniques.

Figure 20: Illustrative techniques applied to a stream surface. This techniqueshows windowed transparency and two sided surface coloring. Image courtesyof C.Garth et al. [HGH∗10].

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Figure 21: A set of streamsurfaces seeded automatically on a tornado simula-tion. The image shows surfaces with edge highlighting improving the percep-tion and allows insight into the behavior of the inner flow structures. Imagecourtesy of M.Edmunds et al. [EML∗11].

3.3. Texture-Based Techniques

In this subsection we present an overview of texture-basedvisualization algorithms. This work is subdivided into StaticTexture and Dynamic Texture techniques. Dense texture-basedtechniques exploit textures to display a representation of theflow and avoid the seeding challenges. The general approachuses a texture with a filtered noise pattern which is smeared andstretched according to the local velocity field. Texture-basedapproaches provide dense visualization results, show intricatedetail, and capture the characteristics of the flow even in com-plex areas of flow such as vortices, sources, and sinks.

With standard spot noise, a texture [vW91] can be character-ized by a scalar function f (x). A spot noise texture is definedas:

f (x) =∑

aih(x − xi)

in which h(x) is called the spot function. It is a function ev-erywhere zero except for an area that is small compared to thetexture size. ai is a random scaling factor with a zero mean,xi is a random position. In non-mathematical terms: spots ofrandom intensity are drawn and blended together on randompositions on a plane.

3.3.1. Static TextureThe concept of the static texture in the context of dense

texture-based methods refers to the non-animation of the flowin the final visualization. Although the images are static in na-ture the algorithms can be applied to both steady-state or time-dependent flow data. One of the early examples of this classi-fication of techniques is presented by de Leeuw and van Wijk,who first use the basic textured spot noise principle for visual-izing vector fields on surfaces and steady flow [dLvW95]. Theauthors build on work by van Wijk [vW91] extending it in fourmain areas; using a parameterized stream surface to deform thespot polygon adapting it to the shape of the local velocity field,

using a negative Gaussian high-pass filter to remove the low fre-quency components of the spot noise textures, using the graph-ics hardware to conduct a series of transformations of the ma-trix stack, normally used for the viewing pipeline, for improvedinteractive animation, and synthesizing spot noise on highly ir-regular grids by pre-distorting the spots in texture space.

Extending the original LIC method [CL93] to the visualiza-tion of flows on curvilinear surfaces, Forssell and Cohen [FC95]introduce a technique to visualize vector magnitude (velocity)as variable speed flow animation. In order to extend LIC to aparameterized surface the surface geometry and related vectorfield information is mapped to parameter space. The LIC im-age in parameter space is computed and mapped back onto thephysical surface through an inverse mapping. The new visu-alization takes into account the vector magnitudes by varyingthe speed of the phase shift in the animation, and handles un-steady flow by conducting convolution along the pathlines in-stead of streamlines. However, the direct convolution followingthe pathlines of the particles can lead to visual ambiguity sincethe forward and backward pathlines through a pixel need notmatch.

Mao et al. [MKFI97] extend the original LIC method by ap-plying it to surfaces represented by arbitrary grids in 3D. For-mer LIC methods targeted at surfaces were restricted to struc-tured grids [For94], [FC95], [SJM96]. Also, mapping a com-puted 2D LIC texture to a curvilinear grid may introduce distor-tions in the texture. The authors propose solutions to overcomethese limitations. The principle behind their algorithm relies onsolid texturing [Pea85]. The convolution of a 3D white noiseimage, with filter kernels defined along the local streamlines, is

Figure 22: This visualization demonstrates the feature-centered approach tostream surface placement. The vortex shedding is represented by the streamsurfaces along with the contextual information of the complete flow field. Im-age courtesy of M.Edmunds et al. [ELM∗12]. This is a direct numerical NavierStokes simulation by Simone Camarri and Maria-Vittoria Salvetti (Universityof Pisa), Marcelo Buffoni (Politecnico of Torino), and Angelo Iollo (Universityof Bordeaux I) [CSBI05] which is publicly available [Int]. We use a uniformlyresampled version which has been provided by Tino Weinkauf and used in vonFunck et al. for smoke visualizations [vFWTS08b].

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performed only at visible ray-surface intersections.Battke et al. [BSH97] introduce a fast LIC technique for sur-

faces in 3D space. Instead of using 2D global parameterization,which is limited to curvilinear surfaces, the authors propose a3D local parameterization scheme. In this way multiple inter-connected surfaces can be handled. An initial tessellation ofthe surface is conducted and local euclidean texture coordinatesare defined for each triangle. LIC textures are then computedby projecting a locally scaled vector field onto each planar trian-gle. To ensure a smooth transition across the tessellated surface,streamlines are followed across neighboring triangles, whichresults in a smooth textured surface. The textures are packedefficiently into texture memory by arranging similar patches inrows and then proceeding using a greedy algorithm. The resultis a smooth interactive visualization of LIC on surfaces in 3Dflow.

Shen et al. [SK98] further improve [FC95] by eliminatingthe artifacts caused by the ambiguity of backward and forwardconvolution, directly following pathlines of the flow. This en-ables the visualization of unsteady flow on surfaces. In orderto achieve this a time accurate scattering scheme, compared tothe gathering scheme in the original LIC, is used to model thetexture advection. More specifically, every pixel in the imagespace scatters its color value to its neighbors following the path-line of the particle at the center of the pixel in a small time step.The resulting color for each pixel is the weighted sum of all thecontributions from its neighbors by considering their ages. Inorder to maintain the temporal coherence, this method uses theprevious result as input for the next iteration. To improve theperformance a parallel framework is also discussed.

Weiskopf [Wei09] introduces iterative twofold convolutionas an efficient high-quality two-stage filtering method for dense

Figure 23: A side view of the surface of a 221K polygonal intake port mesh.The visualization shows ISA applied to the flow simulation data. Color ismapped to velocity. Image courtesy of R.S.Laramee et al. [LvWJH04].

texture-based vector field visualization. The first stage appliesa Lagrangian-particle-tracing-based user-specified compact fil-ter kernel. The second stage applies iterative alpha blendingfor large-scale exponential filtering. A discussion of samplingrates demonstrates this order of convolution operations facili-tates large integration step sizes. Twofold convolution can beapplied to steady and unsteady vector fields, dye and noise ad-vection, and surfaces. This work has the potential to be in-corporated in existing GPU-based 3D vector field visualizationmethods.

Advancing techniques which illustrate flow on surfaces,Palacios et al. [PZ10] present an algorithm for the visualiza-tion of N way rotationally symmetric fields (N-RoSy) on 2Dplanes and surfaces. The basic idea is to decompose the N-RoSy field into multiple distinct vector fields. Together thesedecomposed vector fields capture all N directions at each point.The LIC method is then adopted to compute a flow image foreach vector field. These flow images are blended to obtain thefinal result. A simple probability model based on the correctionof normally distributed random variables is applied to compen-sate for the loss of contrast caused by the blending of images.This algorithm is easily extended to surfaces with an additionaltransformation to combat the artifacts caused by the perspectivedifference of the N-direction vectors on the tangent plane andthe view plane, respectively.

3.3.2. Dynamic TextureDynamic textures in the context of dense texture-based meth-

ods refers to the animation of the flow in the final visualization.The technique generally known as texture advection is appliedto steady state and time dependent flow data. The main chal-lenges of this literature addressed here are computational timeand visualization of unsteady flow.

Laramee et al. [LJH03] and van Wijk [vW03] present newmethods for the synthesis of dense textures on surfaces, bring-ing the concept into the realms of interactivity. These tech-niques follow previous work in this area in which van Wijkpresents a method for producing animated textures for the vi-sualization of 2D vector fields [vW02] along with methods byJobard et al. [JEH01]. This technique uses a different orderof visualization operations than traditional work. The surfacegeometry and related vector field information are projected toimage space where the texturing is then applied.

Following this work, Laramee et al. [LvWJH04] perform acomparative analysis of two techniques: Image Space Advec-tion (ISA) [LJH03], and Image Based Flow Visualization forSurfaces (IBFVS) [vW03]. The authors discuss the best ap-plication of each technique, explaining that IBFVS is a goodchoice where the pixel to polygon ratio in image space is high,and the ISA technique is better for larger meshes in which manypolygons cover a single pixel or where there are many occludedpolygons, see Figure 23.

Laramee et al. [LSH04] then extend current texture advectiontechniques to present a novel hybrid method in which a densetexture-based flow representation is applied directly to isosur-faces. The authors build on work by van Wijk et al. [vW03] and

15

Laramee et al. [LJH03] whose methods are suitable for the vi-sualization of unsteady flow on surfaces. This paper addressesissues associated with applying texture advection to isosurfaceswhere there may be a component of the velocity that is normalto the surface, the perceptual challenges such as occlusion, andissues related to re-sampling of the 3D vector field to 2D image-space. The authors use a normal mask to dim areas of flowwhich have strong cross flow components at the isosurface, anduse a clipping plane to remove sub sets of the geometry to helpreduce occlusion.

The proposed texture advection technique by Weiskopf andErtl [WE04] depends on Lagrangian particle tracing, which issimultaneously computed in object space and in image space.This approach builds on previous texture advection work, in-troducing frame-to-frame coherence when the camera positionis changed. The input noise is modeled as a 3D texture which isscaled appropriately in object space. The authors propose dif-ferent color schemes to improve the visualization of shape andflow. This technique is implemented on the GPU and supportsinteractive visualization.

Studying the application of image-based techniques,Laramee et al. [LWSH04] visualize the flow characteristics ofthe cycle of an engine cylinder, focusing on swirl and tumble.A variety of techniques are demonstrated including iso-surfacesenhanced with an image-based technique. As another applica-tion of visualization of automotive CFD simulations, Larameeet al. [LGD∗05] analyze fluid flow within an engine coolingjacket. Again, a variety of visualization methods are demon-strated including surface-based methods such as stream sur-faces and isosurfaces.

Presenting a new extension to previous work by Larameeet al. [LSH04] and Garth et al. [GTS∗04], Laramee etal. [LGSH06] propose a hybrid method where a dense texture-based flow representation is applied directly to stream surfaces.This conveys features of the flow that otherwise would not beseen using stream surfaces alone. Texture advection applied tostream surfaces avoids issues inherent with isosurfaces such asflow normal to the surface.

Modifying the texture advection approach by [WE04] torun on a highly parallelized GPU cluster, Bachthaler et al.[BSWE06] introduce an algorithm which is scalable accordingto the number of the cluster nodes. The authors employ a sort-first strategy with image space decomposition for LIC workloaddistribution. A sort-last approach with an object-space parti-tioning of the vector field increases the amount of availableGPU memory. This work addresses the challenges of mem-ory accesses locality caused by particle tracing, dynamic loadbalancing to support view changes, and combining image-spaceand object-space decomposition. For future work, parallel ren-dering could be extended to the projection of the surface geom-etry itself in order to visualize extremely large surface meshes.

Li et al. [LTWH08] propose a global texture advectionand synthesis method for flow visualization on surfaces. Itsolves the problem of inconsistent texture correspondence be-tween visible and invisible parts of surfaces in image-basedapproaches. In order to achieve such global continuous tex-turing of flow, surfaces are firstly segmented into patches and

parameterized. These patches are then overlapped by a smallextent with adjacent ones and packed into a uniform texturespace. Next, a 2D dense texture based flow visualization tech-nique [LHD∗04] such as Graphics Processing Unit Line In-tegral Convolution (GPULIC) or Unsteady Flow Advection-Convolution (UFAC) is employed to synthesize the flow tex-ture in texture space. The overlapping patches guarantee tex-ture continuity across their discontinuous borders. The syn-thetic flow texture is mapped to the surface, compositing theoverlapping regions to avoid artifacts in texture patterns causedby inconsistent partial particle traces.

4. Discussion and Future Challenges

No individual technique provides optimal results for all prob-lems or phenomena, and many of the techniques overlap in theirapproach. The best technique depends on several factors suchas the purpose of the visualisation; presentation, detailed anal-ysis, or exploration, and the interest of the analyst or engineerstudying the data. This survey provides a study of a variety oftechniques and approaches to cater for most eventualities whenstudying 3D vector field simulations. However, there are sometopics of study which could potentially benefit from further ex-amination, experiment, and verification. We summarise anddiscuss these challenges. Some of these challenges are moregeneral, while others are specific to this survey.

• Large, Unstructured, Time-Dependent CFD Data.

• Visualizing Error and Uncertainty.

• Perceptual Challenges.

• Information Content.

• Implicit/Topological/Direct Techniques for UnsteadyFlow.

• Interactive Construction and Rendering of Time surfaces.

• Human Centered Evaluation of Flow Visualization Tech-niques.

Large, Unstructured, Time-Dependent Grids:. A significant ef-fort has gone into the study of processing large, unstructureddata [USM96] [SBH∗01] while more recently the focus hasshifted towards studying highly parallel GPU based imple-mentations along with utilization of scalable GPU based tech-nologies [BSWE06]. With the introduction of advanced GPUtechnologies a re-examination of techniques with a focus par-allelization has occurred with the introduction of new chal-lenges. With modifications required to particle tracing meth-ods [STWE07], restrictions regarding triangulation techniques[BFTW09], and scalability are all important topics for furtherparallel/GPU based research.

16

Visualizing Error and Uncertainty:. The numerical integrationof particle trajectories sampled using a piecewise interpolationof the underlying data introduces an accumulative error. Thiserror is often overlooked or misunderstood [Hul92] describedin section 1.3. Another source of error is the sampling densityof the advancing front. If an insufficient density is maintainedthen the surface may fold or become excessively coarse pro-ducing inaccurate surface representations. The difficulty in de-termining the correct splitting conditions, allowing the surfacetrajectory to split around some boundary or critical point, is alsoa potential source of inaccurate representations of the flow. Dis-tortion of the meshing techniques as described by McLoughlinat al. [MLZ10], [MLZ09], regarding distortion of non-planarquads, distortion of triangles in highly diverging flow regions orareas of high shear strain between adjacent time lines accord-ing to Berger et al. [BFTW09], and the visual representationof the point strategy by Schafhitzel et al. [STWE07], are areasof future work which should be examined. The current visu-alizations computed with these error-prone methods could leadto misleading information. Therefore, a possible future workpath is to develop proper techniques to visualize these differenttypes of uncertainty.

Perceptual Challenges:. Rendering too many surfaces causesperceptual problems such as occlusion and visual complexity.Garth et al. [GTS∗04]showed the placement of stream surfacesis important in the reduction of visual clutter. See section 2 formore on this. Effective and efficient placement strategies notonly for static, but for dynamic data are areas of work which re-quire significant study. The topological constructs available insteady flow can easily disappear in dynamic flow. Therefore, aconsideration of the time period would likely be necessary. Thework on topology aware seeding is an excellent starting pointalong this direction [PS09, SRWS10], and an initial study ofstream surface placement by Edmunds et al. [EML∗11] and[ELC∗12] attempts to seed surfaces at the domain boundarybased on characteristics of the flow exiting the domain. Theresults show limited success as the technique only works whereflow exits the boundary. Further work studying the placementof surfaces to best represent the characteristics of the flow isthe feature centered approach by Edmunds et al. [ELM∗12].One possible next goal for this area of research to develop aknowledge-assisted seeding strategy for better extracting moreinformative integral surfaces.

Born et al. [BWF∗10] and Hummel et al. [HGH∗10] extendthe use of image space and GPU technologies to significantlyenhance the geometric structure of surfaces, while represent-ing attributes of the vector field to improve the perception offlow characteristics. Extending these algorithms to dynamicsurfaces such as streak and time surfaces is an area of workyet to be studied in depth. Visualizing flow characteristics onthe constantly changing surface geometry, which is not neces-sarily tangential with the underlying vector field, is one suchchallenge.

Information Content:. This future work direction and thosethat follow are more specific to this survey. Mapping glyphs

to surfaces to allow users to annotate some feature of the sur-face/flow, or allowing the user to switch between different vi-sual or attribute cues are interesting directions of future work[BWF∗10]. Palacios et al. [PZ10] wish to investigate efficientcontrast adjustment when the input images are not gray-scaleand have different hues. One example of this is to visualizeboth the major and minor eigenvector fields of a second-ordertensor. Also of interest to the authors are new decompositionstrategies that will lead to fewer images to blend, thus increas-ing the interactivity.

A largely unexplored area of further research is the effectivevisualization of multivariate data attributes normally associatedwith engineering simulations. This includes not only the vi-sualization of the attributes themselves, but also the effectiveplacement of the surfaces to best represent the information ofinterest.

Implicit/Topological/Direct Techniques for Unsteady Flow:.The work by van Wijk [vW93], Westermann et al. [WJE00],and van Gelder [Gel01] focuses on implicitly visualizing somescalar function of the underlying flow field. Section 2.2 pro-vides more detail. This approach avoids the numerical integra-tion error, refinement schemes, and user defined thresholds andparameters. However, these techniques do raise new challengessuch as higher dimensionality e.g., temporal data, and resolvingareas of highly turbulent flow are problems which still remainunsolved.

The work by Theisel et al. [TWHS03] and Weinkauf et al.[WTHS04] use topological constructs to generate visualizationsof separatrices, and singularities within the domain. Thesetechniques are applied only to steady-state flow data. Theiselet al. [TSW∗05] make the step into temporal data. Ferstl etal. [FBTW10] describe fuzzy ridge structures undergoing fre-quent topology changes with the FTLE in turbulent areas offlow, which may cause visual clutter, providing scope for futurework.

The direct methods by Peng et al. [PL08] [PGL∗12] dealwith unstructured meshes, with challenges arising from boththe re-sampling performance time and perceptual issues. Futurework includes the investigation of different measures for thederivation of mesh resolution, and there is great scope to extendthis work to temporal data.

Interactive Construction and Rendering of Time surfaces:.There has been little work studying the challenge of time sur-faces in time-dependent flow with the exception of Krishnan etal. [KGJ09] (Section 2.1.2). This work handles the test caseswell, maintaining a well-formed mesh. A study of extendingthis work to scalable parallel environments, demonstrated on awider range of flow types e.g., highly rotational flow, could beconducted with a focus on representation error compared to theground truth case.

Human Centered Evaluation of Flow Visualization Techniques:.Laidlaw et al. [LKJ∗05] conducted an extensive user studyof two-dimensional visualization techniques discussing theirrelative merits for visualizing particular characteristics of the

17

flow. Further two-dimensional user studies have been con-ducted more recently by Liu et al. [LCS∗12].These workswould not necessarily translate directly to three dimensions.

One of the key areas of future work which generally has re-ceived little attention is a thorough study of three-dimensionalvisualization techniques, outlining which technique is best un-der a given circumstance for providing the required visual in-formation. The work by Forsberg et al. [FCL09] is a step inthe right direction performing a user study of three-dimensionalflow visualization examining line and tube representations ofintegral curves with both monoscopic and stereoscopic view-ing.

This type of study performed for surface-based flow visu-alization examining the effective construction, placement, andrendering of surfaces to best handle, and effectively representcharacteristics of a given 3D flow field is an important possiblefuture work direction.

5. Conclusions and Summary

Despite the great amount of progress that has been made inthe field of surface-based flow visualization over the last twodecades, a number of challenges remain. Challenges such assurface placement, speed of computation, perception, and eval-uation remain key topics for further research.

We have introduced a novel classification scheme based onchallenges including construction, rendering, and data dimen-sionality. This scheme lends itself to an intuitive grouping ofpapers that are naturally related to each other. Our classifica-tion highlights both unsolved problems and mature areas wheremany solutions have been provided.

The result is an up-to-date overview of the current state ofthe art in this rapidly evolving branch of research.

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*Graphical Abstract (for review)

Highlights

• We detail our investigation into these algorithms with discussions of their applicability and their relative strengths and drawbacks.

• We review the most important challenges when considering such visualizations.• The result is an up-to-date overview of the current state of the art that highlights

both solved and unsolved problems in this rapidly evolving branch of research.

*Highlights (for review)

Dear Reviewers,

Please receive the revised version of our paper “Surface-Based Flow Visualisation” for the next review. We thank the reviewers for their thorough and insightful feedback. We have done our best to address the reviewers comments and concerns and we are sure that the result is an improved submission. Please find our response to the reviews enclosed starting with a summary of changes.

We look forward to hearing again from you soon.

Sincerely,

The authors.

Summary of Changes

A new future challenges section has been written to address the main concerns of reviewers #3 and #5. This new section includes a more insightful and expanded conclusion of future work paths.

We have addressed all the reviewers concerns in the revised paper. The original reviews have been included in this letter and next to each of the requirements a response has been added (in green) to highlight the content and where this has been modified in the paper.

We have done our best to correct all grammatical errors pointed out in the feedback, and added all the citations asked for by the reviewers. We have rephrased the text that is not well formulated. In what follows, we detail our response to the concerns raised by the reviews.

Reviewer #3:The authors present a survey paper on surface-based flow visualization. The paper thoroughly summarizes important recent research in this area. The main strength and contribution of the paper is presenting to the community a summary of work in this area that would be useful to researchers beginning work in this area.

The main weakness of the paper is there is very little original interpretation of the body of work discussed in the paper. The discussion and future challenges section is just one column long for a 14+ page paper. The future challenges identified are rather general (e.g., uncertainty vis, better seeding/feature extraction, user study evaluations, scaling to big data) and have been identified many times before as generally important problems for visualization, for example, see Chris Johnson's Visualization Viewpoints article on top visualization research problems from several years ago. A much more detailed analysis and interpretation of the body of work would be needed to strengthen the paper.

Response: We have included a new future work section which provides greater detail of the conclusions drawn from the reviewed material. This includes new insights, and more specific future work directions derived directly from our taxonomy of the literature provided by Table 1(which has been updated).

Two classifications of the papers discussed are provided, one for surface construction and one for surface rendering. The organization is natural and makes sense. However, there is almost no discussion of what new insights can be learned from these classifications. Ideally, the paper would include some discussion of how viewing the related work described in the taxonomy / classification here helped the authors come to some new conclusions about the most profitable / under-explored areas for future work.

Response: Our discussion of future work has been expanded based partly on Table 1 and partly from inspiring discussions with professor Colin Ware of the University of New Hampshire.

Overall, the writing is easy to follow. The discussion section at the end begins with more or less a summary of the previous 12 pages or so, it might help the reader to move this to the beginning as an introduction. To make room for more original interpretation of the work, some of the introductory material, e.g., the discussion of why flow visualization is important in many applications, could be shortened significantly or removed for this audience.

Response: We moved this summary to the introduction in Section 1.5. The discussion of why flow visualization is important has also been shortened.

*List of Corrections

Reviewer #4:Although there are a couple of (mostly recent) survey papers on different subtopics of flow visualization, there is no competing prior publication in the sub area of surface-based flow visualization. Therefore, this paper fills a gap in the literature. With the nice Fig. 1, the authors make the case for a survey paper in this sub area. Clearly, there is the need for a survey in this growing field, and this survey is timely.

In general, I find it hard to assess novelty and contribution for a survey paper. What I like about this paper is its contribution in the sense of organizing the literature (in particular, see the classification in Table 1). I also like that the relevant contents of the previous papers are explained in sufficient detail so that the reader can grasp the main ideas of those papers. Another strength is the nice presentation in the form of text and figures; a few minor grammatical errors will be easily removed by the journals copy editing process.

With all these strengths, I think that this paper should be accepted for publication. However, there are a couple of points that should be addressed in a revision:

A) Although the classification into parameter space and image space methods (see caption of Table 1, beginning of Section 3) is O.K. in general, there are a few issues. Basically, image space is just one particular choice of parametrization. Another categorization (with other advantages and disadvantages) would rather distinguish between a single chart (single parametrization) and a collection of charts (atlas-based). Then, the image space methods would be treated similar to other simple parametrizations (say C-space techniques). Related to this issue of categorization: ref [WE04] isn't really parameter space; the relevant parametrization of the space is driven by image space; the quantities that are not part of image space are then in full 3D space (not in any 2D manifold explicitly).

Response: We added a discussion of other possible categorization methods to Section 1.4.

B) There are a couple of papers that should be added to this survey. Please update Fig. 1 accordingly.

In the context of rendering of flow on surfaces, the following papers are missing and should be discussed:

1. H. Battke, D. Stalling, and H.-C. Hege: Fast Line Integral Convolution for arbitrary surfaces in 3D. In H.-C. Hege and K. Polthier, editors, Visualization and Mathematics, Springer, 1997: 181-195

2. X. Mao, M. Kikukawa, N. Fujita, and A. Imamiya: Line Integral Convolution for 3D surfaces. EG Workshop on Visualization 1997: 57-70

3. D. Weiskopf, H. Hauser: Cycle shading for the assessment and visualization of shape in one and two codimensions. Graphics Interface 2006: 219-226

4. S. Bachthaler, M. Strengert, D. Weiskopf, T. Ertl: Parallel Texture-Based Vector Field Visualization on Curved Surfaces Using GPU Cluster Computers. EGPGV 2006: 75-82

5. D. Weiskopf. Iterative Twofold Line Integral Convolution for Texture-based Vector Field Visualization. In T. Möller, B. Hamann, R. Russell (Eds.), Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, Springer, 2009: 191-211

In the context of topological surface construction techniques, the following space-time surface construction and visualization method could be included:

1. S. Bachthaler, F. Sadlo, C. Dachsbacher, D. Weiskopf: Space-time Visualization of Dynamics in Lagrangian Coherent Structures of Time-dependent 2D Vector Fields. Proceedings of IVAPP 2012

When discussing the concept of perpendicular surfaces (see p. 7, ref. [PCY09]), the following related references should be included (both use perpendicular tracing structures):

1. S. Bachthaler, D. Weiskopf: Animation of Orthogonal Texture Patterns for Vector Field Visualization. IEEE Trans. Vis. Comput. Graph. 14(4): 741-755 (2008)

2. O. Rosanwo, C. Petz, S. Prohaska, H.-C. Hege, I. Hotz: Dual streamline seeding. PacificVis 2009: 9-16

When describing ridge surfaces on p. 9: "The paper builds on recent work on the extraction of Lagrangian coherent ridges in the finite time Lyapunov exponent (FTLE) field at each time step. The separation surfaces represent these ridges, which are tracked in time", the following related paper that describes the actual

extraction of ridge surfaces for flow visualization should be discussed:

1 F. Sadlo, R. Peikert: Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction. IEEE Trans. Vis. Comput. Graph. 13(6): 1456-1463 (2007)

Response: We have added the additional papers to this survey and updated Figure 1 and Table 1 accordingly.

C) Further details:

Credits should be added to the captions for images taken from other papers.

Response: Credits have been added to the captions of the images.

Bibliographic data should be checked for completeness.

Response: Bibliographic data has been checked for completeness.

"Velocity data is comprised of scalar values ." (p. 4): This statement is wrong. The velocity vector is comprised of components, but these aren't scalars. Scalars observe certain transformation properties (and vectors do others): they remain constant under those transformations, which is not the case for the components of a vector.

The R^4 notation (twice on p. 4) is wrong because the vectors v_t are from R^3 (only their parameters have 4 elements).

"depth pealing" (p. 8) -> "depth peeling"

Paragraphs 2-5 of Section 3.3.2 should be better integrated and shortened. After all, these papers describe very similar things, and some of the contents have repetitions in the description.

The abbreviations "GPULIC" and "UFAC" are not introduced (p. 14).

Response: All the above corrections have been incorporated into the survey.

Reviewer #5:The paper provides a comprehensive survey of surface-based flow visualization techniques. While surveying a complete and comprehensive body of work, the paper has a number of problems that reduce its appeal for the target audience:

1. The fundamental mathematical concepts are not introduced in the manuscript. Definitions should be given for integral curves and surfaces, such that readers unfamiliar with these topics (or coming from other areas) are able to understand these fundamental concepts exactly.

Response: Definitions of integral curves and surfaces have been added in Section 2.1.1 and 2.1.2.

2. While the paper achieves an enumeration of relevant work, little added value is provided over a straightforward description of algorithms with a very brief discussion of relative merits. Specific problems solved by methods are pointed out ad hoc, and no clear guidelines emerge as to what method should be chosen for a specific scenario. (In my opinion, this should be among the primary contributions of a survey.) Here, the authors should identify common thematic elements (e.g. "countering integral surface distortion"), describe and illustrate them in abstract form, and then outline how specific methods incorporate these themes.

To give but one of many examples, critical points pose a difficulty to stream surface construction, and several works specifically address this aspect of stream surface computation. However, they do not play a significant role for time or streak surfaces due to their transient nature -- for readers that are not already intimately familiar with the topic, this will be entirely lost.

Response: We have identified and described common thematic elements and common challenges and added them to the introduction of each respective section of the survey. Specific methods then address these topics, respectively.

Similarly, the paper does not discuss in detail the applicability of different surface types. E.g. under what circumstances is it sensible to employ time surfaces for visualization?

Response: As we have not found much literature on the topic of time surfaces – we have not discussed their applicability in detail.

1. I find the "novel" classification scheme rather natural on one hand (stationary vs. time, explicit vs. implicit representation); on the other hand, it is misleading: implicit/topological stream surfaces should not be in a different category than stream surfaces, in my opinion, since both types of schemes are aimed at solving the same problem. Where in Table 1 are time surfaces represented?

Response: We have updated the classification. There was one paper that was miss classified. Topological methods refer to papers that visualize vector field topology explicitly and use surfaces to do so. Table 1 is updated to include the section where time surfaces are classified.

2. A similar problem exists with the secondary aspect of how to visualize flow on surfaces. E.g. under what conditions would one use a static method on a dynamic surface? The pieces to this puzzle are all contained in the text, but I would prefer that the authors explicitly address these questions, to guide readers through the body of work.

Response: We have updated the introductions of each section to highlight the common themes and challenges addressed by the literature that follows.

Overall, I think the paper should be rewritten significantly to remedy these problems and provided additional value over enumerating and briefly discussing individual contributions. I would propose to limit the scope slightly by focusing on integral surfaces alone, which should make for enough material overall if a meaningful discussion is added. In its current state, I propose to reject the paper.

Response: We have revised the paper significantly guided by the reviewers feedback. We are sure the result is an improved submission. Although the reviewer has made a good suggestion, the other two reviewers are pleased with the current scope, thus we have retained it.

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Package natbib Warning: Citation `laramee:texture' on page 14 undefined on inpu t line 75. Package natbib Warning: Citation `laramee:investigating' on page 14 undefined o n input line 76. Package natbib Warning: Citation `weiskopf:hybrid' on page 14 undefined on inpu t line 77. Package natbib Warning: Citation `laramee:visualAnalysis' on page 14 undefined on input line 78. Package natbib Warning: Citation `laramee:textureAdvection' on page 14 undefine d on input line 79. Package natbib Warning: Citation `bachthaler:parallel' on page 14 undefined on input line 80. Package natbib Warning: Citation `li:flow' on page 14 undefined on input line 8 1. ) (./figures.tex ! LaTeX Error: Cannot determine size of graphic in figure1.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.3 ...egraphics[width=1.0\linewidth]figure1.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure1.jpg Graphic file (type eps) <figure1.jpg> ! LaTeX Error: Cannot determine size of graphic in figure2.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ...

l.11 ...graphics[width=0.8\linewidth]figure2.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure2.jpg Graphic file (type eps) <figure2.jpg> Package natbib Warning: Citation `peng:vector' on page 14 undefined on input li ne 14. Package natbib Warning: Citation `peng:vector' on page 14 undefined on input li ne 14. ! LaTeX Error: Cannot determine size of graphic in figure3.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.19 ...raphics[width=0.95\linewidth]figure3.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure3.jpg Graphic file (type eps) <figure3.jpg> Package natbib Warning: Citation `spencer:evenly' on page 14 undefined on input line 21. Package natbib Warning: Citation `spencer:evenly' on page 14 undefined on input line 21. ! LaTeX Error: Cannot determine size of graphic in figure4.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.26 ...th=1.0\linewidth,height=50mm]figure4.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure4.jpg Graphic file (type eps)

<figure4.jpg> ! LaTeX Error: Cannot determine size of graphic in figure5.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.37 ...raphics[width=0.74\linewidth]figure5.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure5.jpg Graphic file (type eps) <figure5.jpg> Package natbib Warning: Citation `schafhitzel:point' on page 14 undefined on in put line 40. Package natbib Warning: Citation `schafhitzel:point' on page 14 undefined on in put line 40. ! LaTeX Error: Cannot determine size of graphic in figure6.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.45 ...graphics[width=0.8\linewidth]figure6.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure6.jpg Graphic file (type eps) <figure6.jpg> Package natbib Warning: Citation `krishnan:time' on page 14 undefined on input line 48. Package natbib Warning: Citation `krishnan:time' on page 14 undefined on input line 48. ! LaTeX Error: Cannot determine size of graphic in figure7.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation.

Type H <return> for immediate help. ... l.53 ...graphics[width=0.7\linewidth]figure7.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure7.jpg Graphic file (type eps) <figure7.jpg> Package natbib Warning: Citation `mcloughlin:easy' on page 14 undefined on inpu t line 59. Package natbib Warning: Citation `mcloughlin:easy' on page 14 undefined on inpu t line 59. Package natbib Warning: Citation `mcloughlin:easy' on page 14 undefined on inpu t line 59. Package natbib Warning: Citation `mcloughlin:easy' on page 14 undefined on inpu t line 59. ! LaTeX Error: Cannot determine size of graphic in figure8.jpg (no BoundingBox) . See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.64 ...raphics[width=0.85\linewidth]figure8.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure8.jpg Graphic file (type eps) <figure8.jpg> Package natbib Warning: Citation `garth:surface' on page 14 undefined on input line 66. Package natbib Warning: Citation `garth:surface' on page 14 undefined on input line 66. ! LaTeX Error: Cannot determine size of graphic in figure9.jpg (no BoundingBox) .

See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.71 ...graphics[width=1.0\linewidth]figure9.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure9.jpg Graphic file (type eps) <figure9.jpg> Package natbib Warning: Citation `garth:generation' on page 14 undefined on inp ut line 74. Package natbib Warning: Citation `garth:generation' on page 14 undefined on inp ut line 74. ! LaTeX Error: Cannot determine size of graphic in figure10.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.79 ...h=0.8\linewidth,height=65mm]figure10.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure10.jpg Graphic file (type eps) <figure10.jpg> Package natbib Warning: Citation `peikert:topologically' on page 14 undefined o n input line 81. Package natbib Warning: Citation `peikert:topologically' on page 14 undefined o n input line 81. ! LaTeX Error: Cannot determine size of graphic in figure11.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.86 ...=1.0\linewidth, height=37mm]figure11.jpg

Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure11.jpg Graphic file (type eps) <figure11.jpg> Package natbib Warning: Citation `schneider:topology' on page 14 undefined on i nput line 91. Package natbib Warning: Citation `schneider:topology' on page 14 undefined on i nput line 91. ! LaTeX Error: Cannot determine size of graphic in figure12.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.96 ...aphics[width=0.98\linewidth]figure12.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure12.jpg Graphic file (type eps) <figure12.jpg> Package natbib Warning: Citation `burger:interactive' on page 14 undefined on i nput line 98. Package natbib Warning: Citation `burger:interactive' on page 14 undefined on i nput line 98. ! LaTeX Error: Cannot determine size of graphic in figure13.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.103 ...aphics[width=0.8\linewidth]figure13.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure13.jpg Graphic file (type eps) <figure13.jpg>

Package natbib Warning: Citation `ferstl:isss' on page 14 undefined on input li ne 107. Package natbib Warning: Citation `ferstl:isss' on page 14 undefined on input li ne 107. ! LaTeX Error: Cannot determine size of graphic in figure14.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.112 ...phics[width=0.85\linewidth]figure14.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure14.jpg Graphic file (type eps) <figure14.jpg> Package natbib Warning: Citation `westermann:levelset' on page 14 undefined on input line 115. Package natbib Warning: Citation `westermann:levelset' on page 14 undefined on input line 115. ! LaTeX Error: Cannot determine size of graphic in figure15.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.120 ...th=1\linewidth,height=65mm]figure15.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure15.jpg Graphic file (type eps) <figure15.jpg> Package natbib Warning: Citation `weinkauf:boundary' on page 14 undefined on in put line 122. Package natbib Warning: Citation `weinkauf:boundary' on page 14 undefined on in

put line 122. ! LaTeX Error: Cannot determine size of graphic in figure16.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.127 ...phics[width=0.95\linewidth]figure16.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure16.jpg Graphic file (type eps) <figure16.jpg> Package natbib Warning: Citation `theisel:extraction' on page 14 undefined on i nput line 129. Package natbib Warning: Citation `theisel:extraction' on page 14 undefined on i nput line 129. ! LaTeX Error: Cannot determine size of graphic in figure17.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.134 ...th=1\linewidth,height=34mm]figure17.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure17.jpg Graphic file (type eps) <figure17.jpg> ! LaTeX Error: Too many unprocessed floats. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.143 \beginfigure[!b] You've lost some text. Try typing <return> to proceed. If that doesn't work, type X <return> to quit.

! Undefined control sequence. \@float@Hx ...ltovf \fi \global \setbox \@currbox \color@vbox \normalcolor \... l.143 \beginfigure[!b] The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.143 \beginfigure[!b] A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Cannot determine size of graphic in figure18.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.145 ...0.77\linewidth,height=55mm]figure18.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure18.jpg Graphic file (type eps) <figure18.jpg> Package natbib Warning: Citation `peng:clustering' on page 14 undefined on inpu t line 149. Package natbib Warning: Citation `peng:clustering' on page 14 undefined on inpu t line 149. ! Undefined control sequence. \@largefloatcheck ->\ifdim \ht \@currbox >\textheight \@tempdima -\textheigh... l.150 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue,

and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> > l.150 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. <argument> \@currbox l.150 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Argument of \@xnext has an extra . <inserted text> \par l.150 \endfigure I've run across a `' that doesn't seem to match anything. For example, `\def\a#1...' and `\a' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `' was spurious, just type `2' and it will go away. Runaway argument? \@currbox \@currlist \ifnum \count \@currbox >\z@ \advance \@pageht \@pagedp \E TC. ! Paragraph ended before \@xnext was complete. <to be read again> \par l.150 \endfigure I suspect you've forgotten a `', causing me to apply this control sequence to too much text. How can we recover? My plan is to forget the whole thing and hope for the best. ! Extra \else. \end@float ...pdima \penalty \@floatpenalty \else \vadjust \penalty -\@Miv ... l.150 \endfigure I'm ignoring this; it doesn't match any \if.

! You can't use `\vadjust' in vertical mode. \end@float ...nalty \@floatpenalty \else \vadjust \penalty -\@Miv \vbox \... l.150 \endfigure Sorry, but I'm not programmed to handle this case; I'll just pretend that you didn't ask for it. If you're in the wrong mode, you might be able to return to the right one by typing `I' or `I$' or `I\par'. ! Argument of \@xnext has an extra . <inserted text> \par l.150 \endfigure I've run across a `' that doesn't seem to match anything. For example, `\def\a#1...' and `\a' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `' was spurious, just type `2' and it will go away. Runaway argument? \@currbox \@currlist \ifnum \count \@currbox >\z@ \advance \@pageht \@pagedp \E TC. ! Paragraph ended before \@xnext was complete. <to be read again> \par l.150 \endfigure I suspect you've forgotten a `', causing me to apply this control sequence to too much text. How can we recover? My plan is to forget the whole thing and hope for the best. ! LaTeX Error: Too many unprocessed floats. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.152 \beginfigure[!t] You've lost some text. Try typing <return> to proceed. If that doesn't work, type X <return> to quit. ! Undefined control sequence. \@float@Hx ...ltovf \fi \global \setbox \@currbox \color@vbox \normalcolor \... l.152 \beginfigure[!t]

The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.152 \beginfigure[!t] A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Cannot determine size of graphic in figure19.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.154 ...=0.9\linewidth,height=70mm]figure19.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure19.jpg Graphic file (type eps) <figure19.jpg> Package natbib Warning: Citation `loffelmann:hierarchical' on page 14 undefined on input line 156. Package natbib Warning: Citation `loffelmann:hierarchical' on page 14 undefined on input line 156. ! Undefined control sequence. \@largefloatcheck ->\ifdim \ht \@currbox >\textheight \@tempdima -\textheigh... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> >

l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. <argument> \@currbox l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \@elt l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Missing = inserted for \ifnum. <to be read again> \@elt l.157 \endfigure I was expecting to see `<', `=', or `>'. Didn't. ! Missing number, treated as zero. <to be read again> \@elt l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: This may be a LaTeX bug. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.157 \endfigure Call for help

! Undefined control sequence. \@addmarginpar ...a <\z@ \global \setbox \@marbox \box \@currbox \fi \@tempd... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \box l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Missing number, treated as zero. <to be read again> \@elt l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...advance \@tempdima \ht \@marbox \ifdim \@tempdima >\z@ \@l... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). <to be read again>

\global l.157 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...vance \@mparbottom \dp \@marbox \global \advance \@mparbot... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...dvance \@tempdima -\ht \@marbox \global \setbox \@marbox \... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...marbox \global \setbox \@marbox

\vbox \vskip \@tempdima \... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...\vskip \@tempdima \box \@marbox \global \ht \@marbox \z@ ... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ... \@marbox \global \ht \@marbox \z@ \global \dp \@marbox \... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global

l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). <to be read again> \global l.157 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...marbox \z@ \global \dp \@marbox \z@ \kern -\@pagedp \noint... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \kern l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). <to be read again> \kern l.157 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...arginparwidth \fi \box \@marbox

\hss \nointerlineskip \hb... l.157 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \hss l.157 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Too many unprocessed floats. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.159 \beginfigure[!b] You've lost some text. Try typing <return> to proceed. If that doesn't work, type X <return> to quit. ! Undefined control sequence. \@float@Hx ...ltovf \fi \global \setbox \@currbox \color@vbox \normalcolor \... l.159 \beginfigure[!b] The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.159 \beginfigure[!b] A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Cannot determine size of graphic in figure20.jpg (no BoundingBox

). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.161 ...0.95\linewidth,height=70mm]figure20.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure20.jpg Graphic file (type eps) <figure20.jpg> Package natbib Warning: Citation `hummel:iris' on page 14 undefined on input li ne 164. Package natbib Warning: Citation `hummel:iris' on page 14 undefined on input li ne 164. ! Undefined control sequence. \@largefloatcheck ->\ifdim \ht \@currbox >\textheight \@tempdima -\textheigh... l.165 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> > l.165 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. <argument> \@currbox l.165 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined.

! Argument of \@xnext has an extra . <inserted text> \par l.165 \endfigure I've run across a `' that doesn't seem to match anything. For example, `\def\a#1...' and `\a' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `' was spurious, just type `2' and it will go away. Runaway argument? \@currbox \@currlist \ifnum \count \@currbox >\z@ \advance \@pageht \@pagedp \E TC. ! Paragraph ended before \@xnext was complete. <to be read again> \par l.165 \endfigure I suspect you've forgotten a `', causing me to apply this control sequence to too much text. How can we recover? My plan is to forget the whole thing and hope for the best. ! Extra \else. \end@float ...pdima \penalty \@floatpenalty \else \vadjust \penalty -\@Miv ... l.165 \endfigure I'm ignoring this; it doesn't match any \if. ! You can't use `\vadjust' in vertical mode. \end@float ...nalty \@floatpenalty \else \vadjust \penalty -\@Miv \vbox \... l.165 \endfigure Sorry, but I'm not programmed to handle this case; I'll just pretend that you didn't ask for it. If you're in the wrong mode, you might be able to return to the right one by typing `I' or `I$' or `I\par'. ! Argument of \@xnext has an extra . <inserted text> \par l.165 \endfigure I've run across a `' that doesn't seem to match anything. For example, `\def\a#1...' and `\a' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `' was spurious, just type `2' and it will go away.

Runaway argument? \@currbox \@currlist \ifnum \count \@currbox >\z@ \advance \@pageht \@pagedp \E TC. ! Paragraph ended before \@xnext was complete. <to be read again> \par l.165 \endfigure I suspect you've forgotten a `', causing me to apply this control sequence to too much text. How can we recover? My plan is to forget the whole thing and hope for the best. ! LaTeX Error: Too many unprocessed floats. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.167 \beginfigure[!t] You've lost some text. Try typing <return> to proceed. If that doesn't work, type X <return> to quit. ! Undefined control sequence. \@float@Hx ...ltovf \fi \global \setbox \@currbox \color@vbox \normalcolor \... l.167 \beginfigure[!t] The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.167 \beginfigure[!t] A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Cannot determine size of graphic in figure21.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ...

l.169 ...aphics[width=1.0\linewidth]figure21.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure21.jpg Graphic file (type eps) <figure21.jpg> Package natbib Warning: Citation `edmunds:automatic' on page 14 undefined on in put line 173. Package natbib Warning: Citation `edmunds:automatic' on page 14 undefined on in put line 173. ! Undefined control sequence. \@largefloatcheck ->\ifdim \ht \@currbox >\textheight \@tempdima -\textheigh... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> > l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. <argument> \@currbox l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \@elt l.174 \endfigure

A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Missing = inserted for \ifnum. <to be read again> \@elt l.174 \endfigure I was expecting to see `<', `=', or `>'. Didn't. ! Missing number, treated as zero. <to be read again> \@elt l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: This may be a LaTeX bug. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.174 \endfigure Call for help ! Undefined control sequence. \@addmarginpar ...a <\z@ \global \setbox \@marbox \box \@currbox \fi \@tempd... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \box l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Missing number, treated as zero.

<to be read again> \@elt l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...advance \@tempdima \ht \@marbox \ifdim \@tempdima >\z@ \@l... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \protect l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) LaTeX Warning: Marginpar on page 14 moved. ! Undefined control sequence. \@addmarginpar ...vance \@mparbottom \dp \@marbox \global \advance \@mparbot... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence.

\@addmarginpar ...dvance \@tempdima -\ht \@marbox \global \setbox \@marbox \... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...marbox \global \setbox \@marbox \vbox \vskip \@tempdima \... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...\vskip \@tempdima \box \@marbox \global \ht \@marbox \z@ ... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again>

l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ... \@marbox \global \ht \@marbox \z@ \global \dp \@marbox \... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). <to be read again> \global l.174 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...marbox \z@ \global \dp \@marbox \z@ \kern -\@pagedp \noint... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again>

\kern l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). <to be read again> \kern l.174 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...arginparwidth \fi \box \@marbox \hss \nointerlineskip \hb... l.174 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \hss l.174 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Too many unprocessed floats. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.176 \beginfigure[!b] You've lost some text. Try typing <return> to proceed. If that doesn't work, type X <return> to quit. ! Undefined control sequence.

\@float@Hx ...ltovf \fi \global \setbox \@currbox \color@vbox \normalcolor \... l.176 \beginfigure[!b] The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.176 \beginfigure[!b] A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Cannot determine size of graphic in figure22.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.178 ...aphics[width=1.0\linewidth]figure22.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure22.jpg Graphic file (type eps) <figure22.jpg> Package natbib Warning: Citation `edmunds:feature' on page 14 undefined on inpu t line 189. Package natbib Warning: Citation `camarri05' on page 14 undefined on input line 189. Package natbib Warning: Citation `iCFDDatabase' on page 14 undefined on input l ine 189. Package natbib Warning: Citation `vonfunck:smoke' on page 14 undefined on input line 189. Package natbib Warning: Citation `edmunds:feature' on page 14 undefined on inpu

t line 189. Package natbib Warning: Citation `camarri05' on page 14 undefined on input line 189. Package natbib Warning: Citation `iCFDDatabase' on page 14 undefined on input l ine 189. Package natbib Warning: Citation `vonfunck:smoke' on page 14 undefined on input line 189. ! Undefined control sequence. \@largefloatcheck ->\ifdim \ht \@currbox >\textheight \@tempdima -\textheigh... l.190 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> > l.190 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. <argument> \@currbox l.190 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Argument of \@xnext has an extra . <inserted text> \par l.190 \endfigure I've run across a `' that doesn't seem to match anything. For example, `\def\a#1...' and `\a' would produce

this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `' was spurious, just type `2' and it will go away. Runaway argument? \@currbox \@currlist \ifnum \count \@currbox >\z@ \advance \@pageht \@pagedp \E TC. ! Paragraph ended before \@xnext was complete. <to be read again> \par l.190 \endfigure I suspect you've forgotten a `', causing me to apply this control sequence to too much text. How can we recover? My plan is to forget the whole thing and hope for the best. ! Extra \else. \end@float ...pdima \penalty \@floatpenalty \else \vadjust \penalty -\@Miv ... l.190 \endfigure I'm ignoring this; it doesn't match any \if. ! You can't use `\vadjust' in vertical mode. \end@float ...nalty \@floatpenalty \else \vadjust \penalty -\@Miv \vbox \... l.190 \endfigure Sorry, but I'm not programmed to handle this case; I'll just pretend that you didn't ask for it. If you're in the wrong mode, you might be able to return to the right one by typing `I' or `I$' or `I\par'. ! Argument of \@xnext has an extra . <inserted text> \par l.190 \endfigure I've run across a `' that doesn't seem to match anything. For example, `\def\a#1...' and `\a' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `' was spurious, just type `2' and it will go away. Runaway argument? \@currbox \@currlist \ifnum \count \@currbox >\z@ \advance \@pageht \@pagedp \E TC. ! Paragraph ended before \@xnext was complete. <to be read again> \par

l.190 \endfigure I suspect you've forgotten a `', causing me to apply this control sequence to too much text. How can we recover? My plan is to forget the whole thing and hope for the best. ! LaTeX Error: Too many unprocessed floats. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.192 \beginfigure[!b] You've lost some text. Try typing <return> to proceed. If that doesn't work, type X <return> to quit. ! Undefined control sequence. \@float@Hx ...ltovf \fi \global \setbox \@currbox \color@vbox \normalcolor \... l.192 \beginfigure[!b] The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.192 \beginfigure[!b] A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: Cannot determine size of graphic in figure23.jpg (no BoundingBox ). See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.194 ...aphics[width=0.9\linewidth]figure23.jpg Try typing <return> to proceed. If that doesn't work, type X <return> to quit. File: figure23.jpg Graphic file (type eps)

<figure23.jpg> Package natbib Warning: Citation `laramee:isa' on page 14 undefined on input li ne 197. Package natbib Warning: Citation `laramee:isa' on page 14 undefined on input li ne 197. ! Undefined control sequence. \@largefloatcheck ->\ifdim \ht \@currbox >\textheight \@tempdima -\textheigh... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> > l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. <argument> \@currbox l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \@elt l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Missing = inserted for \ifnum. <to be read again> \@elt

l.198 \endfigure I was expecting to see `<', `=', or `>'. Didn't. ! Missing number, treated as zero. <to be read again> \@elt l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! LaTeX Error: This may be a LaTeX bug. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.198 \endfigure Call for help ! Undefined control sequence. \@addmarginpar ...a <\z@ \global \setbox \@marbox \box \@currbox \fi \@tempd... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \box l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Missing number, treated as zero. <to be read again> \@elt l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.)

! Undefined control sequence. \@addmarginpar ...advance \@tempdima \ht \@marbox \ifdim \@tempdima >\z@ \@l... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \protect l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) LaTeX Warning: Marginpar on page 14 moved. ! Undefined control sequence. \@addmarginpar ...vance \@mparbottom \dp \@marbox \global \advance \@mparbot... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...dvance \@tempdima -\ht \@marbox \global \setbox \@marbox \... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct

spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...marbox \global \setbox \@marbox \vbox \vskip \@tempdima \... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \vbox l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Undefined control sequence. \@addmarginpar ...\vskip \@tempdima \box \@marbox \global \ht \@marbox \z@ ... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.)

! Undefined control sequence. \@addmarginpar ... \@marbox \global \ht \@marbox \z@ \global \dp \@marbox \... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \global l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). <to be read again> \global l.198 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...marbox \z@ \global \dp \@marbox \z@ \kern -\@pagedp \noint... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \kern l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted). <to be read again> \kern l.198 \endfigure Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \@addmarginpar ...arginparwidth \fi \box \@marbox \hss \nointerlineskip \hb... l.198 \endfigure The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. <to be read again> \hss l.198 \endfigure A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ) Package natbib Warning: There were undefined citations. [14] [15] (./manuscript.aux) LaTeX Warning: There were undefined references. ) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifx on line 190 was incomplete) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifnum on line 190 was incomplete) (\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifx on line 165 was incomplete)

(\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifnum on line 165 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) (\end occurred when \ifx on line 150 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) (\end occurred when \ifnum on line 150 was incomplete) Here is how much of TeX's memory you used: 7284 strings out of 493849 99222 string characters out of 3152231 216318 words of memory out of 3000000 10367 multiletter control sequences out of 15000+200000 35031 words of font info for 104 fonts, out of 3000000 for 9000 714 hyphenation exceptions out of 8191 38i,12n,37p,4477b,359s stack positions out of 5000i,500n,10000p,200000b,50000s Output written on manuscript.dvi (15 pages, 180192 bytes).