CWP-655

Visualization of 3D tensor fields derived from seismicimages

Chris Engelsma & Dave Hale

Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA

(a) (b)

Figure 1. A 3D seismic image with a traced layer displaying the tensors as ellipsoids (a), and removing that slice shows the3-dimensional structure of the tensors from that layer (b).

ABSTRACTIn image processing, tensors derived from seismic images are used as parametersin procedures such as structure-oriented smoothing. Visualizing these tensorsallows us to qualitatively assess their computation and construction. We de-scribe a computationally effective technique to render these tensors as ellipsoidglyphs.

Key words: seismic visualization ellipsoid tensor

1 INTRODUCTION

Visualizing tensor fields has always been a complicatedtask. While there are many ways in which scientists canvisualize scalar or vector fields, displaying tensor fieldsin an intuitive manner remains a challenge. In recentyears, a number of techniques have been proposed dis-cussing methods for displaying 3D tensors. In the med-ical industry, the continuity of tensor fields is empha-sized by constructing hyperstreamlines or streamtubesfor diffusion MRI tensors (Delmarcelle and Hesselink,1993; Jianu et al, 2009). In stress evaluation, the effectsof a tensor field on a given media have been visualizedthrough bending mesh volumes, simulating the effects

of a stress tensor to demonstrate anisotropic deforma-tion (Zheng and Pang, 2002). In geophysics, tensors arebeing used to help guide seismic horizon tracing (Holltet al, 2009). Recently, a number of methods have beenproposed for displaying 3D tensors.

For the purposes of image processing in explo-ration geophysics, tensors are often derived directlyfrom the images. While a method such as hyperstream-lining would give insight into the continuity of the tensorfield, it is also advantageous to render each tensor in-dividually by depicting them in an intuitive manner asFigure 1 demonstrates. These discrete representations oftensors in 3D are called “glyphs”. Glyph representationcan take many forms, and the benefits of choosing differ-

188 C. Engelsma & D. Hale

(a) (b) (c)

Figure 2. A scalar field (a), a vector field (b), and a tensor field (c). The visualization complexity increases dramatically withthe number of quantities represented at each node.

ent shapes have been explored (Kindlmann, 2004). How-ever, for the purpose of this paper, each tensor glyph isrepresented as an ellipsoid.

Tensors help increase the efficacy of image process-ing by guiding the orientation of the operation. Thisis the principle behind structure-oriented smoothing(Hale, 2009). Because these tensors are used as param-eters in different processing techniques, we must de-termine their accuracy. We therefore wish to explore amethod of visualizing these parameters in a discretizedmanner, allowing us to evaluate any arbitrary tensor.We describe an algorithm to visualize tensors derivedfrom seismic images, and demonstrate methods for eval-uating the tensor’s authenticity. By displaying tensorsas ellipsoid glyphs, this visualization method providesan intuitive and interactive method for relating the ten-sors directly back to the image. We also expedite therendering process by making our method computation-ally fast and efficient.

2 TENSOR GEOMETRY

The challenge with visualizing tensors stems from theirmultivariate nature. With scalar fields, each sample is arepresentative of one number. Vector fields follow thesame concept, but each point is now represented bythree numbers in 3D. Tensor fields introduce anotherstep in intricacy because we are now representing sixunique numbers at every point in space. A visual repre-sentation of this increasing complexity is shown in Fig-ure 2. Simultaneous visualization of six numbers extendsbeyond conventional visualization techniques unless weunderstand the geometry of the tensors.

2.1 Metric tensor field D(x)

An ideal structure-oriented procedure which honors thedominant structural features of our image (e.g. rockbedding layers and faults), requires a tensor field thataccurately represents these features. This is accom-plished by first computing the structure tensors S(x)

(van Vliet and Verbeek, 1995; Fehmers and Hocker,2003), which are smooth outer-products of image gradi-ents. The eigen-decomposition of a 3D structure tensorS(x) yields:

S = λuuuT + λvvvT + λwwwT , (1)

where the eigenvalues of λu, λv, and λw are sorted sothat

λu ≥ λv ≥ λw ≥ 0. (2)

By convention, u is defined as the eigenvector that tra-verses the direction of the largest gradient. In a 3D seis-mic image, this typically refers to the direction perpen-dicular to geologic layering. Both eigenvectors v and wtend to lie in the plane of locally planar features in theimage.

We then compute anisotropic metric tensors D(x),using a process outlined by Hale (2009), whereby wecompute image semblances. We choose the eigenvectorsof each metric tensor D to be the same as those forthe corresponding structure tensor S. The difference be-tween D and S lie only in their eigenvalues. Specifically,the eigen-decomposition on D(x) is

D = s3uuT + s2vvT + s1wwT , (3)

where we construct eigenvalues s1, s2, and s3 such that

0 ≤ s3 ≤ s2 ≤ s1 ≤ 1. (4)

Our metric tensor D(x) is a 3 × 3 symmetric, positive-definite matrix. The largest eigenvalue s1, correspond-ing to the eigenvector w, is semblance computed withina locally linear (1D) set of voxels aligned with w. Eacheigenvalue s2, corresponding to the eigenvector v, issemblance computed within a locally planar (2D) setof voxels orthogonal to the corresponding eigenvectoru. (The plane orthogonal to u contains the eigenvec-tors v and w). Finally, each eigenvalue s3 represents

Visualization of 3D tensor fields 189

xT Ax = 1

x

Figure 3. A non-axis-aligned ellipsoid.

semblance computed for a locally spherical (3D) set ofvoxels.

2.2 Ellipsoid glyphs

We consider the definition of an ellipsoid to be

xT Ax = 1, (5)

where A is a square symmetric positive-definite matrix,and x is any point along the surface of the ellipsoidwhich satisfies this equation (note Figure 3). We canlikewise define a unit sphere in the same manner byreplacing matrix A with the identity matrix I.

Equation 2 provides a useful definition because itdescribes an ellipsoid that is not axis-aligned; the eigen-vectors of A are arbitrarily aligned in space. Consid-ering the definition of eigenvector orthonormality, wedefine the eigenvectors as the three principle axis radii,and the inverse of the square root of the eigenvalues astheir respective sizes (see Figure 4) (Strang, 2003). Thisgeometric relationship enables us to construct the ten-sors as ellipsoids; in a computer, these are illustrated asglyphs.

2.3 Geologic analogy

Ellipsoid glyph representations of metric tensors demon-strate the local orientation of the image. In particular,for a perfectly horizontal layer, we expect the ellipsoidto be oblate, because λu > λv ≈ λw. Likewise, for com-plete isotropy within an image, we expect our ellipsoidto be a sphere (λu = λv = λw). The local geologic ori-entation of the formation is also reflected, so ellipsoidsincorporate the same strike and dip qualities as theircorresponding locations in the seismic image. Becausethe intention of displaying these ellipsoids is to qualita-tively assess the how accurately the tensors have beenconstructed, we expect that they follow the bedding lay-ers in the image. This allows us to judge the veracity ofimage processing techniques guided by these tensors.

u√λu

v√λv

w√λw

AV = VDFigure 4. Eigenvalue and eigenvector relationship to thethree principle radii of an ellipsoid. Each axis within theellipsoid is equal to the eigenvector divided by the squareroot of their eigenvalues.

3 ACCELERATED RENDERING

Constructing each glyph requires computing the loca-tion of roughly one thousand vertices to be used in atriangle mesh (see Figure 5). This computation becomescostly when one begins to display a large set of ellipsoidsthroughout a 3D survey. We therefore expedite the ren-dering process. If we first compute the vertex locationsfor a unit circle, we then obtain the desired ellipsoid byapplying the appropriate matrix transformations. Thisgreatly reduces the computational cost.

We compare the equations of a unit circle,

xT x = 1, (6)

to our desired transformed ellipsoid coordinates,

yT Ay = 1. (7)

We also define the eigen-decomposition of A to be

A = VDVT , (8)

where V is a 3×3 orthogonal matrix containing theeigenvectors of A stored as column vectors, and D isa diagonal matrix storing eigenvalues λu ≥ λv ≥ λw.We now replace A in equation 8 with equation 7, andwe get

yT VDVT y = 1. (9)

Given the property of a diagonal matrix that D =

D12 D

12 , we expand equation 9 to get

yT VD12 D

12 VT y = 1. (10)

Equation 10 is the equation of an ellipsoid in terms of

190 C. Engelsma & D. Hale

(a)

(b)

Figure 5. Two glyphs: a unit sphere (a) and an ellipsoid (b).The ellipsoid was computed using matrix transformations onthe sphere’s mesh.

the eigenvectors and eigenvalues of A. From equation 6,we observe that the variable x represents the coordi-nates of a unit sphere. To transform the sphere into anellipsoid, we derive our desired coordinates y in termsof our computed coordinates x:

y = VD− 12 x. (11)

V represents a rotation matrix which realigns the princi-

ple axes of the unit sphere. The matrix D− 12 is a nonuni-

form scaling matrix containing the inverse of the squareroot of the eigenvalues. Performing equation 11 is morecomputationally efficient than explicitly computing eachvertex because this process passes 12 numbers to thegraphics card instead of recalculating one thousand co-

Figure 6. Ellipsoids selected to follow a single layer by“point-and-click” method. The strike and dip of the localformation is apparent.

ordinates for each ellipsoid. The vertex coordinates fora unit sphere are only computed once and stored.

4 IMPLEMENTATION METHODS

Here we discuss two methods for overlaying ellipsoidglyphs on 3D seismic data. Both methods offer differ-ent techniques to visualize tensors, and both may beused for different investigative purposes. We show twoapproaches to displaying tensor fields: a point-and-clickand an axis-aligned panel method.

4.1 Point-and-click method

Figure 6 shows ellipsoids that are selected along a givenlayer. This is performed by a succession of mouse clickswhich place an ellipsoid’s center on the sample nearestto the cursor. Note that every ellipsoid appears oblatewith varying thicknesses, and that each ellipsoid hasa dip that reflects the local orientation. Focusing onthe shape of the ellipsoids is important for determin-ing whether or not the tensor field has been correctlycomputed. Note also that the ellipsoids appear in 3Drelative to the image slice. This allows the user to ro-tate freely, preserving the location and visibility of thetensor.

In a similar way, the user can drag the cursor alongthe image and observe the changes in the ellipsoids ateach point in space. By not sticking the ellipsoids asin Figure 6, the user can watch the tensor mold to thelayers and identify discrepancies this way.

4.2 Axis-aligned panels

Placement of ellipsoids along an axis-aligned panel (seeFigure 7) shows an overall distribution of tensor clus-ters. This process involves discretizing tensors along a

Visualization of 3D tensor fields 191

Figure 7. A panel of tensor ellipsoids. Each ellipsoid represents a metric tensor, and is equally sampled along the x-axis panel.The closeup emphasizes the variation in shape as well as angle of each ellipsoid at each point in space.

3D seismic panel, allowing the user to qualitatively as-sess many tensors simultaneously. From a macroscopicviewpoint, this will enable the user to grab a broadperspective of the underlying structure. While Figure 7shows ellipsoids attached to a single panel, displaying el-lipsoids on all three axis-aligned panels is a reasonableinterpretation method as well.

The caveat of this approach is that ellipsoids willnot necessarily fall directly on a point of interest. Be-cause the ellipsoids are evenly sampled along the panel,the user is only permitted to see tensors that lie on thatsampling interval. For a more detailed survey of tensorellipsoids, the point-and-click approach is more effective.

5 CONCLUSIONS

Displaying tensor fields is an ongoing topic of researchin the field of visualization. The inherent problem withdisplaying tensors is due to the amount of informationcontained in each sample. In geophysical applications ofimage processing, tensors are derived from the seismicimages in order to design structure-oriented operations.For the purpose of quality assessment, we choose to dis-play these tensors as glyphs shaped as ellipsoids.

Constructing ellipsoids from tensors works in ourfavor, as our metric tensors fit this geometric relation-ship. By performing an eigen-decomposition of the ten-sor matrix, we obtain three orthonormal eigenvectors

and their corresponding eigenvalues, which can be rep-resented as the three principle axis directions and theircorresponding radii. Expediting the process involvesprecomputing the vertices of a unit sphere, and per-forming both a rotation and scaling matrix.

Because our tensors are derived from the seismicimage, we show that the shape of the ellipsoid relates tothe local orientation of the image. Flat layers yield disc-shaped, oblate ellipsoids, and isotropic environments aremore spherical. Ellipsoids must have the same strike anddip of the surrounding area.

We demonstrate two methods of displaying tensorfields: the first dynamically selects ellipsoids at a clickedvoxel; the second involves discretizing ellipsoids alongan axis-aligned panel. Both techniques provide intuitivevisualization of the geologic substructure, with point-and-click placement allowing for detailed investigationof a specific point.

6 ACKNOWLEDGEMENTS

We thank the Rocky Mountain Oilfield Testing Center,a facility of the U.S. Department of Energy, for provid-ing us with the data used for Figures 1, 6, and 7. Wealso thank our colleagues for their continual discussionand feedback on this topic. We also thank Diane for hervery helpful editing of this paper.

192 C. Engelsma & D. Hale

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