Polar Stratospheric Cloud Visualization: Volume ...

Polar Stratospheric Cloud Visualization: Volume Reconstruction from
Intersecting Curvilinear Cross Sections

Jessica R. Croucha, Chris Weigleb, Jonathan Gleasonc, and Yuzhong Shend

aDepartment of Computer Science, Old Dominion University;
bUniversity of Tennessee/Oak Ridge National Lab, Joint Institute for Computational Sciences;

cNASA Langley Research Center;
dDepartment of Electrical and Computer Engineering, Old Dominion University

ABSTRACT

The CALIPSO satellite launched by NASA in 2006 uses an on-board lidar instrument to measure the vertical distribution
of clouds and aerosols along the orbital path. This satellite’s dense vertical sampling of the atmosphere provides previously
unavailable information about the altitude and composition of clouds, including the polar stratospheric clouds (PSCs) that
play an important role in the annual formation of polar ozone holes. Reconstruction of cloud surfaces through interpolation
of CALIPSO data is challenging due to the sparsity of the data in the non-vertical dimensions and the complex sampling
pattern created by intersecting non-planar orbital paths. This paper presents a method for computing cloud surfaces by
reconstructing a continuous cloud surface distance field. The distance field reconstruction is performed via shape-based
interpolation of the cloud contours on each cross section using a medial axis representation of each contour. The inter-
polation algorithm employs a projection operator that is defined in terms of (latitude, longitude, altitude) coordinates, so
that projection between cross sections follows the earth’s curved atmosphere and preserves cloud altitude. This process
successfully interpolated cloud contours from CALIPSO data acquired during the 2006 polar winter and enabled three
dimensional visualization of the PSCs.

Keywords: Volume reconstruction, shape-based interpolation, shape model, polar stratospheric cloud

Polar stratospheric clouds (PSCs) form during the winter months over the earth’s polar regions and, over the Antarctic,
play a key role in the destruction of large quantities of ozone during the polar spring. Atmospheric scientists are interested
in studying Antarctic PSC formation and tracking PSC volume to better understand and predict ozone hole formation.
Of concern is that a continued increase in greenhouse gases may cause the polar stratosphere to cool, thereby allowing
PSCs to occur more frequently or persist longer - especially in the Arctic where they are currently less widespread than
over Antarctica. The study of PSCs is also motivated by the need for advances in atmospheric and climate modeling
capabilities; improvements are dependent on developing more accurate representations of cloud processes and a better
understanding of cloud-radiation interactions.

PSC formation is made possible by conditions unique to the polar winter where temperatures are cold enough to con-
dense nitric acid onto particles. Circular wind patterns over Antarctica referred to as the Polar Vortex prevent mixing of
air parcels, creating a relatively stagnant chemical system that is maintained for months. PSCs affect ozone by provid-
ing catalytic surfaces for heterogeneous chemical reactions to occur during the polar night that activate ozone-destructive
reservoirs of benign anthropogenic chlorine compounds into reactive forms. When sunlight returns in late August and
September, these reactive compounds are further transformed by ultraviolet light into chlorine free radicals which catalyti-
cally destroy ozone. Because large PSC particles also sediment and irreversibly redistribute odd nitrogen (a process known
as denitrification), ozone depletion is exacerbated by the slowing chemical cycles involved with odd nitrogen that return
chlorine to benign reservoir species.

Much of our present understanding of PSCs has been derived from remote sensing observations. Solar occultation
satellite instruments first identified PSC’s in the late 1970’s1,2 and their long multi-year record has helped established an

Further author information: (Send correspondence to J.R.C.)
J.R.C.: [email protected]
C.W.: [email protected]
J.G.: [email protected]
Y.S.: [email protected]

(a) (b)

Figure 1. (a)Illustration of CALIPSO satellite acquiring vertical aerosol profile data. Image credit: Dan Lyons, NASA. (b) CALIPSO
orbit path on June 26, 2006. Image credit: Kathy Powell, NASA.

understanding of their occurrence at temperatures < 195 − 200K as well as their spatial distribution and seasonal variation.
Polarization-sensitive airborne and ground-based lidar data have further revealed that PSC particles occur in three primary
microphysical forms - liquid ternary (H2SO4 − HNO3 − H2O) solution droplets, solid nitric acid trihydrate (NAT) crystals,
and H2O ice crystals. Both solar occultation and lidar data, however, have inherent shortcomings that limit a more detailed
understanding of PSC processes. Solar occultation data have very coarse spatial resolution (hundreds of kilometers), are
limited to 14-15 profiles per day in each hemisphere, and are restricted to occur only at the day-night terminator. Ground-
based lidar PSC data are recorded from a fixed location and are often interrupted by interference from optically thick
tropospheric clouds. Airborne lidar PSC data are both short-term and limited to along-track spatial coverage.

The Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellite mission offers an oppor-
tunity to take advantage of the most favorable aspects of the existing PSC remote sensing instruments. CALIPSO is a
joint mission between the National Aeronautics and Space Administration (NASA) and the French Space Agency, Centre
National d’Etudes Spatiales (CNES), that was launched on April 28, 2006. The primary instrument of the mission is a
polarization-sensitive dual-wavelength lidar that provides measurements of the vertical distribution of clouds and aerosols
over the Earth’s globe.3 Measurements can be made during the polar night and not just along the terminator. The high
vertical resolution of the data along the orbit, illustrated in Figure 1(a), will help to unlock features that were unresolved
by solar occultation satellite observations.

In this paper we describe a surface reconstruction technique for Antarctic PSCs, an atmospheric feature for which
CALIPSO provides a particularly rich data set due to its polar-orbiting path. The goal of this work is to construct a dense
sampling of the polar atmosphere from sparse satellite data to enable visualization of atmospheric features, particularly
polar stratospheric clouds. By applying shape-based interpolation to the sparse, intersecting, curvilinear satellite images,
we construct a volume image suitable for the application of polygonization algorithms such as Marching Cubes. The
resulting polygon meshes provide NASA scientists with the first surface reconstructions of any data from the CALIPSO
satellite.

The remainder of this paper is organized as follows: section 1 gives sampling characteristics of the CALIPSO instru-
ments and data, section 2 describes the relevant previous work, section 3 describes our algorithms for reconstructing PSCs,
section 4 shows the results of the algorithm applied real PSC data from the 2006 Antarctic winter, and section 5 presents
our conclusions and suggests future directions for this work.

1. DATA DESCRIPTION

The primary instrument aboard the CALIPSO spacecraft is the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP).
Lidar (Laser Imaging Detection and Ranging) is an imaging technology similar in principle to radar that measures the
backscatter of an emmitted laser beam to determine characteristics of its target. CALIOP is the first polarization lidar
instrument in orbit4 and provides high-resolution vertical profiles of clouds and aerosols. The laser generates pulses at 532
nm and 1064 nm wavelengths, and a 1-meter diameter telescope measures time and intensity of backscattered light of both

wavelengths along with the parallel and perpendicular polarization components at 532 nm.4 The intensity of backscatter as
well as the ratio of backscatter and extinction at the two wavelengths is used to determine the location and size of average
particles observed.5 Additionally, polarization components of the backscatter are sensitive to irregularly shaped particles
and can be used to derive the composition of atmospheric particles observed. The result is a fundamental sampling of the
atmosphere along the satellite’s orbital path.

CALIPSO acquires data from the Earth’s surface to an altitude of 40 km, thus providing measurements of the tropo-
sphere and most of the stratosphere. These are the first and second innermost layers of the Earth’s atmosphere and are
defined by their molecular composition. The troposphere and stratosphere are separated by a boundary region called the
tropopause that generally prevents mixing between the two layers. The trans-boundary profile measurements provided by
CALIPSO allow distinction of atmospheric features at different altitude layers that previously were only measurable with
in situ sampling and aircraft-based instruments.

With an orbital period of approximately 99 minutes, CALIPSO makes observations over the poles about 14 times each
day.6 The CALIOP instrument produces laser pulses at a rate of 20.16 Hz as it orbits; the resulting sampling resolution
of the instrument is 30 m vertical and 333 m horizontal.4 As illustrated in Figure 1(b), the orbital path of the CALIPSO
satellite provides rich data near the poles, but no data directly over the poles or within approximately 10 degrees latitude of
the poles. CALIPSO’s orbital cross sections of the atmosphere are densely sampled vertically through the atmosphere and
along the orbital trajectory. However, the small number of orbits in one 24 hour period provides for a limited number of
corresponding samples of the atmosphere transverse to the path trajectories. Further complicating surface reconstruction,
the cross sections provided by CALIPSO are curvilinear and intersect. The nature of the CALIPSO data leads us to
investigate surface reconstruction from scattered points or sparse cross sections.

2. PREVIOUS WORK

Surface reconstruction from scattered points is a challenging problem that arises in many applications, including scientific
visualization, 3D range scanning, and medical imaging. It is a problem that has garnered much previous attention; surface
reconstruction is often pursued to facilitate three dimensional rendering and support analysis of objects’ geometry, volume,
and shape characteristics.

Surfaces can be represented either explicitly or implicitly. Explicit representations define the coordinates of a surface,
e.g., parametric surfaces and triangulated surfaces, while implicit representations define a surface as an iso-surface of a
scalar function.7–12 Both implicit and explicit surface representations can be useful in the process of surface reconstruction.
If data samples are arranged uniformly in Euclidean space, such as the ones obtained in computer tomography (CT) or
magnetic resonance imaging (MRI), the well-known Marching Cubes algorithm13 can be applied to generate a triangular
mesh that follows an isosurface. If data is non-uniformly sampled, a volume grid may be defined and a scalar function
such as the signed distance function may be used to interpolate the non-uniformly sampled data points at each vertex of the
volume grid. Once a volume grid is constructed, an isosurface can be tesselated as before.9,10,14–18 This technique can be
used, for example, to perform surface reconstruction from range images.14,19,20 A signed distance function is calculated at
each point in the volume grid as a weighted sum of the signed distances from that grid point to each range surface, and the
observed surface can be constructed by extracting an iso-surface from the volume grid.14

In some situations, including many medical applications, a volume is sampled along object cross sections. However,
the sampling resolution within cross sections and that perpendicular to cross sections may differ significantly. In the
case of freehand ultrasound, the cross sections are not parallel to each other, creating a unique type of irregularity in the
sampling geometry.21,22 Surface reconstruction utilizing a scalar function such as the distance function has been shown
to be appropriate for such applications. A simple method for distance calculation is shape-based interpolation.23,24 This
method first identifies object contours within each cross section and assigns positive distances to data points within the
object contours, negative distances to data points outside the object contours, and zero distance to those on the object
contours. A volumetric distance field can be formed by linearly interpolating the distances defined at the cross sections,
resulting in a volume suitable for iso-surface extraction.

Although shape-based interpolation works well for cross sections which are largely consistent from one cross section
to the next, it produces undesirable artifacts when the contours change shape or position too quickly across the set of
cross sections. An improvement to straightforward linear interpolation was based on the idea of interpolating along the
directions connecting the centroids of cross sections of the same object25 rather than in the directions perpendicular to

the cross sections. This approach works well for shapes that have simple structures, but it cannot handle complex shapes
that change significantly between cross sections. Other variations of centroid-based distance interpolations have been
proposed26,27 that operate under the highly restrictive assumption that each cross section contains only one contour. Clearly
this assumption cannot be satisfied for complex shapes. A somewhat more accomodating approach is the dynamic elastic
interpolation method,28 in which a series of intermediate contours between the start contour and goal contour are iteratively
generated based on the elastic matching method,29 and multiple contours of each cross section are considered separately.

More recently, Treece et al. described a method that takes multiple contours into account without requiring separation
of contours for different objects.22 In this approach the contours on each cross section are represented by a set of disks
derived from the distance transform of the cross sections, and the direction of interpolation is allowed to vary along the
cross section. Treece et al. addressed the traditional branching and correspondence problems with the concept of region
correspondence.21 A set of maximal discs was used not only to represent cross section contours, but the associated distance
transform was also used to help determine the correspondence between disks on neighboring cross sections. External discs
(i.e., outside the contours) were used in addition to internal discs so that the placement of holes and surface concavities
could help determine region correspondence. This method yielded satisfactory surfaces for cross sections of a variety of
complex shapes.

The method presented in this paper is an adaptation of Treece et al.’s work21 to curvilinear cross sections of atmospheric
features. We adapt the work of Treece et al. in the following ways: by handling curved cross sections each of which
intersects the others in multiple locations, by constructing correspondence vectors in two directions (to the east and west
neighboring cross section) for every point on each cross section instead of only in one direction, and by leveraging the
natural polar coordinate system of our data (as well as spatial data structures) to accelerate nearest-neighbor queries.

3. METHODS

The goal of this work is to create tessellated surfaces that fit the PSC cross sections visible in CALIPSO data so that 3D
rendering is possible. This is accomplished through a multi-step process that begins with NASA-provided PSC segmenta-
tions of CALIPSO data. A compact representation of the PSC contours is computed by sampling the medial axis of each
contour with a set of maximal disks. A distance map is computed for each cross section, in which each pixel stores the
scalar distance between the pixel and the nearest point on the cloud surface. This distance field is then interpolated to fill a
voxel volume by using the set of maximal disks to guide a shape-based interpolation between cross sections. A tessellated
surface is extracted from this volume by finding the 0-valued isosurface using a Marching Cubes algorithm.

Each of the steps in this process is described in detail in the following sub-sections. Section 3.1 discusses the distance
map computation and de-noising of the cross-section images. Section 3.2 explains how the set of maximal disks for each
cross section is computed, and the correspondence relationship between disks on neighboring cross sections is discussed
in section 3.3. The method for interpolating the distance map at arbitrary points in the voxel volume is presented in section
3.4. Reconstruction is achieved by straight forward application of Marching Cubes, discussed in section 3.5, and validation
of the method using synthetic data is provided in section 3.6.

3.1 Image Processing

NASA provides a preliminary Level 2 CALIPSO Data Product that includes, for each point in a cross section, a Fea-
ture Classification Flags bitfield,5 eliminating the need for an explicit segmentation step. This bitfield labels the atmo-
spheric features as determined from the recorded CALIOP measurements.30 As illustrated in Figure 2, our target feature,
the PSC, may be classified as either cloud or stratospheric feature. In the preliminary data, all features with a base altitude
over the tropopause are classified stratospheric, and PSCs with a base altitude below the tropopause or vertically adjacent
to a cloud in the troposphere are classified cloud. We combine the two classifications into one segmentation. The bitfields
reserve room to denote subclassifications of many feature types, however this information is not present in the preliminary
data. We anticipate utilizing PSC subclassifications directly for segmentation once the algorithms for classifying features
are finalized by NASA in the near future.

The preliminary Feature Classification Flag contains a number of artifacts caused, in part, by data classification over
variable footprints. Adjacent regions of low backscatter intensity may be integrated to determine what feature, if any, is
present.5 The integration can lead to islands and holes at multiple scales as well as square-shaped boundaries unlikely to
accurately represent cloud formations. Dense clouds can also prevent the laser beam from penetrating through to lower

(a) Unfiltered (b) Filtered

Figure 2. An example of unfiltered feature masks from CALIPSO data (a). The green pixels are labeled as cloud features, and the red
pixels are labeled stratospheric features. After filtering to remove integration artifacts, we have a PSC feature mask (b).

clouds, creating data holes beneath some cloud features. Shown in Figure 2, morphological operators and median filters
are applied to the cross-sectional images of Feature Classification Flags to reduce the influence of these artifacts. These
processing steps remove small holes and improve the smoothness of the cloud segmentations, resulting in more plausible
cloud contours. However, we anticipate that the image processing steps will not be necessary when the final, improved
PSC classification data is released by NASA.

3.2 Maximal Disk Representation

Multiple algorithms exist for operating on segmented images to efficiently compute the distance field31,32 and the medial
axis,32–35 and for pruning the medial axis while guaranteeing complete object coverage.36 Because the distance field
captures the full shape of the segmented objects, a sparser maximal disk representation is sufficient for our purposes. We
employ a three step, greedy algorithm for pruning the medial axis. In step one, all disks in the full medial axis are marked
undecided. In step two, the undecided disk with largest radius is marked in the maximal disk set, then all undecided disks
that overlap the in disk by more than half their area are marked out. Step two is repeated until no unmarked disks remain.
In step three, all disks marked in are collected to form the maximal disk representation. Figure 3 illustrates a maximal disk
representation of the segmented image from Figure 2.

3.3 Disk Correspondences

To interpolate the distance map between two adjacent cross sections, SA and SB, each disk on cross section SA must be
assigned a corresponding location on cross section SB, and vice versa. Disk correspondences are computed through a
procedure based on the work of Treece, et al.37 Before defining correspondence vectors, disk projection, neighborhood
disk sets, and distance map lookup operations are required.

The projection function, P(x, S), maps an arbitrary point, x, onto a data cross section S. For this application, x is
specified in terms of (latitude, longitude, altitude) coordinates. t is a scalar parameter defined on S that equals 0 at one end
of the cross section and increases along its length; S(t) represents the (latitude, longitude, altitude) coordinates of points
along the base of S. x can be projected onto S by computing the parameter value tm where the latitude of S(tm) is closest to
the latitude of x, and then replacing x’s longitude with the longitude at S(tm).

tm = min |S(t )l at − xlat | (1)
(2)
t

P(x, S) = xlat , S(tm)long, xalt

Intuitively, the projection P(x, S) moves x along a curve of constant latitude and constant altitude to the point where it
intersects cross section S. If multiple points exist at which Slat equals xlat , then the one at which Slong is nearest xlong is

Figure 3. The skeleton of the segmented image from Figure 2 is shown in green. The maximal disk representation of the same segmented
images is shown in red.

Figure 4. A continuously sampled medial axis on cross section SB, shown with projected disk da and closest boundary point x.

selected. This method of projecting points onto cross sections works reliably for curvilinear cross sections and projects
points through the curved space of the Earth’s atmosphere. Since PSCs form only within specific ranges of altitudes and
latitudes, projecting cloud contours across different altitudes and latitudes would map PSC contours into regions where
PSCs cannot occur. This motivated the development of the projection method used here that preserves altitude and latitude
and projects across longitude. This projection method is application-specific and differs from the earlier work by Treece,
et al., which assumed planar input data in a Cartesian space.

N (x, S) is a neighborhood function that returns the set of disks on cross section S that are in the vicinity of projected
point P(x, S). A neighborhood is defined as a region of a cross section that has at least ten times the diameter of the largest

disk found on the cross section. In our implementation, disks are stored in a spatially-organized data structure to facilitate
fast creation of neighborhood disk sets. I (x, S) is a function that returns the two-dimensional image map coordinates for
P(x, S). M(I (x, S)) represents the value of the distance map defined at the projection of x onto S.

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−→ = I (db, SA) − I (da, SA) (3)
lab (4)
∑[db∈N (da,SB)] (wb · l−→ab)
→−v (da, SB) = ∑[db∈N (da,SB)] wb

−→
The weights, wb, applied the vectors lab are inversely proportional to the square of two disk correspondence error
metrics, ε(da, db) and ε(db, da).

ε(da, db) = |lab| − |M(I (da, SA)) − M(I (db, SA))| (5)

ε(db, da) = |lab| − |M(I (db, SB)) − M(I (da, SB))| (6)
(7)
= 11
wb +
ε(da, db) + 1 ε(db, da) + 1

Note in the equation above that the length of l−→ab is computed using two dimensional image coordinates consistent with the
lengths recorded in the distance field; also note that the distance fields contain positive distances inside a cloud contour and
negative distances outside a contour.

The significance of the ε metric is most apparent when a continuous sampling of the medial axis of the cross section

contours is considered. In the continuous case, if the point on SB’s cloud contour closest to da is denoted x, as shown in
Fig. 4, then there always exists a disk centered at db whose boundary touches x. For this disk, |lab| = |M(I (db, SB)) −

M(I (da, SB))|, thus ε(db, da) = 0. For other disks on SB, ε(db, da) is proportional to the distance between the boundary of
the disk centered at db and the center of P(da, SB).

In practice, the medial axis of the cloud contours is discretely sampled and db is not generally contained within the
sample set. Although sampling error contributes to larger ε values, the metric remains useful. For any given da, ε(db, da)
is smallest for the db sample that represents the portion of the cloud boundary on which x is located. A correspondence
vector calculation is illustrated in Fig. 5.

For each cross section two sets of correspondence vectors are computed, one that maps disks onto the adjacent cross
section to the east, and the other that maps disks onto the adjacent cross section to the west. Because cross sections

intersect, the adjacency relationships are not constant over the whole length of a cross section. Thus, identification of the
eastern neighbor and western neighbor is a decision made on a per disk basis.

Once a correspondence vector is assigned to every disk on a cross section, a correspondence vector for any point on
the cross section can be computed as a weighted sum of the disk correspondence vectors. Thus on data slice SA, the
correspondence vector for a point x is defined as follows.

→−v (x, SA) = ∑[da∈N (x,SA)] (wa · →− ) (8)
la

∑[da∈N (x,SA)] wa

where
→−l a = I (da, SA) − I (x, SA)

1
wa = |la| − |M(I (da, SA)) − M(I (x, SA), SA)| + 1

3.4 Shape-Based Distance Map Interpolation

The first step toward computing the distance field value at an arbitrary point x is the identification of the data cross sections
immediately to the east and to the west of the point. Denoting these slices as Seast and Swest , the distance map values can be
interpolated between four points, two of which are located on the eastern cross section and two of which are located on the
western cross section. Figure 6 illustrates the relationship between the four points. The first two points are found simply
by projecting x onto the east and west cross sections.

peast = P(x, Seast ), pwest = P(x, Swest ) (9)

Weights αeast and αwest represent the percentage of the total distance between Seast and Swest that lies to the east and west

of x. |x − pwest | |x − peast |
peast − pwest |peast − pwest |
αeast = | | , αwest = (10)

The second pair of points are located by computing the point correspondence vectors for peast and pwest , weighting the
vectors according to the α distances and adding the result to peast and pwest .

peast = peast + αeast (→−v (peast ,Seast )), pwest = pwest + αwest (→−v (pwest ,Swest )) (11)

Finally, the distance field at x is computed by interpolating the four distance map values.

M(x) = αe2ast M(I (peast , Seast )) + αeast αwest M(I (peast , Seast )) (12)
+αeast αwest M(I (pwest , Swest )) + αw2est M(I (pwest , Swest ))

The result of applying this distance interpolation method to CALIPSO data can be seen in Figure 6, where a slice through
an interpolated volume is shown with superimposed cloud contours.

(a) (b) (c)

Figure 5. (a) Portion of a distance map cross-section, with disks derived from the data shown in red. Disks projected from an adjacent
strip are shown in green. (b) Corresponding portion of the adjacent cross-section, shown with maximal disks in green. The red disks are
projected from the adjacent cross-section shown on the left. (c) A correspondence vector has been computed for the fifth red disk in the
distance map on the left. lab vectors for this disk are shown with dotted black lines, and the disk correspondence vector is shown with a
solid blue line.

−3 −3
−2 −2
−1 −1
0 0
1 1
2 2
3 3
x 104
x 104 x 104
−7.5
−7

(a) (b)

Figure 6. (a) Illustration of distance field interpolation. (b) A constant-altitude slice through an interpolated distance field for CALIPSO
data is shown, together with green outlines of the cross sections and blue cloud contours.

3.5 Volume Reconstruction

We sample the distance field on a regular grid in geographic coordinates (latitude, longitude, altitude), find the isocontour
by Marching Cubes,13 then transform the vertices of the isosurface into Euclidean coordinates for display. This approach
compresses the sampling of the distance field near the poles. As the PSCs primarily reside near the poles, this is an
acceptable sampling artifact. In the future, we plan to explore adaptive sampling and higher-order polytopes to sample more
densely in regions further away from the poles while maintaining the advantages of sampling in geographic coordinates.

3.6 Validation

This work presents the first surface reconstructions of PSCs from CALIPSO satellite data. It is not currently possible to
directly verify that these surfaces are valid. Instead, validation is achieved by reconstructing the surface of synthetically
generated data and comparing the reconstructed surface to the ideal surface. The synthetic data are modeled as the union of
a set of spheres. If the ith sphere is centered at ci and has radius ri, then the following function returns the distance between
a point p and the synthetic cloud surface.

u(p) = min (px − cix )2 + (py − ciy )2 + (pz − ciz )2 − ri (13)

i

For the validation experiment a data volume is constructed from 250 randomly distributed spheres. Depicted in Figure 7a,
we sample the distance function on a 241 × 201 × 141 grid and apply Marching Cubes to produce a reference surface.

We compute multiple reconstructed distance fields by our algorithm, varying the number of CALIPSO-like cross sec-
tions through the synthetic data volume and the resolution of the reconstructed distance field. We then apply Marching
Cubes to reconstruct the surface from each distance field. Figures 7b-f show several such surfaces. Note that Figure 7c also
illustrates the cross sections used to sample the reference implicit function, each cross section in green and the contours
within the cross sections in blue.

We compute the RMS error of a reconstructed surface from the implicit function value at each vertex of the surface
mesh. Table 1 presents the value of the error metric for the reference surface and for several surfaces computed by our
method. As the table and Figure 7 show, our reconstructions improve as either the number of cross sections increases or
the distance field resolution increases.

4. RESULTS

Figure 8 shows the results of PSC cloud reconstruction performed using one day of CALIPSO data every two weeks for
late in the polar winter of 2006. These reconstructions show the growth of the PSCs leading up to the September 24, 2006
record ozone hole. Each reconstruction was performed using the 14 atmospheric cross sections generated by the 14 passes
CALIPSO made through the polar region on each of the reconstruction dates. The input data was processed as described in
section 3 to produce a PSC distance field on each of the cross sections. The PSC distance field was then interpolated to fill
a 60 × 60 × 60 voxel grid that covered the polar atmosphere from a latitude of 60◦ S down to the south pole, with altitudes
ranging from 8 to 30 km.

(a) reference (b) 6 cross sections, 80x80x80 grid

(c) 12 cross sections, 80x80x80 grid (d) 25 cross sections, 80x80x80 grid

(e) 50 cross sections, 80x80x80 grid (f) 100 cross sections, 80x80x80 grid

Figure 7. Random sphere surfaces. A relatively high resolution Marching Cubes surface computed directly from the implicit function is
compared to our algorithm. For our algorithm, varying numbers of cross sections and distance field resolutions were employed. Note
that c) also shows the spacing of the cross sections.

Cross sections Grid resolution RMS error Cross sections Grid resolution RMS error
25 20x20x20 3.1618 6 80x80x80 5.1172
25 40x40x40 2.8749 80x80x80 4.1208
25 60x60x60 2.7711 12 80x80x80 2.6698
25 80x80x80 2.6698 25 80x80x80 1.6474
25 2.6655 50 80x80x80 1.3148
100x100x100 0.6770 100 0.6770
Marching Cubes 241x201x141 Marching Cubes 241x201x141

Table 1. Random sphere reconstruction error metrics. A relatively high resolution Marching Cubes reconstruction is compared to our
algorithm. The number of cross sections and the resolution of the distance field reconstruction were varied for our algorithm. The RMS
error of the surfaces reconstructed from our algorithm approaches that of Marching Cubes applied directly to the implicit function.

(a) June 20 (b) July 3

(c) July 17 (d) July 31

Figure 8. Antarctic polar stratospheric cloud surfaces at approximately two-week intervals for polar winter 2006. The growth of the
PSCs is apparent from the reconstructed surfaces.

Each reconstruction took approximately 7 minutes on a 3.2 GHz PC and was performed using a Matlab implementa-
tion. An optimized, compiled implementation using more sophisticated spatial data structures would produce results more
quickly. However, since the aim of this application is post-processing and visualization of data that is collected over a
period of 24 hours or longer, there is no immediate need for a real-time implementation.

4.1 Limitations

The main limitation of this work is applicability to general scattered data or sparse contour reconstruction. The CALIPSO
satellite produces data with idiosyncratic sampling that is a poor match to more general volume reconstruction techniques.
The reconstruction technique proposed herein is applicable to CALIPSO data over the full satellite orbit and may be
applicable to data from other satellites following the same orbital path as CALIPSO.

However, there may be interesting atmospheric phenomena that are stable or slow to change. These may allow or even
require combining CALIPSO data from multiple days, at which point the included orbit paths will exhibit intersections
well away from the poles. For long term phenomena it may be necessary to change the projection function to consider
more than just the near-east and near-west orbits.

5. CONCLUSIONS AND FUTURE WORK

We have presented a method for reconstructing surface representations of polar stratospheric clouds interpolated from
segmented images in curvillinear cross sections. These surfaces enable atmospheric scientists to visualize the formation
and dispersion of PSCs throughout the year, but especially during the polar winter when the growth of PSCs may be used
to predict ozone depletion over Antarctica. The surfaces can also be used to compute statistics about the cloud formations,
such as cloud volume. Our method is the first to enable the reconstruction of surfaces from the curvilinear cross sections
provided by CALIPSO.

Revised classification data, including a PSC composition feature mask, is planned for a late 2007 release from the
NASA CALIPSO Science Team. We will run our method on the improved classification data to generate multiple surfaces
that indicate sub-features within the initial volume. The new data will provide more reliable and scientifically useful cloud

volumes which we will use to improve the accuracy of our results. Atmospheric scientists expect that the finalized PSC
classifications and corresponding volume estimations will enable improved prediction of ozone loss in the Antarctic spring.

The finalized CALIPSO data will also contain uncertainty information based on the size of the integration footprint
necessary to classify a region of the cross section. This additional information about classifications poses the standard,
but largely unsolved, problem in uncertainty visualization: how best to enable the simultaneous selection and display of
surfaces at multiple levels of certainty in a manner conducive to the underlying science.

Future work should address the limitations discussed in Section 4.1, improvements enabled by the availability of
improved input data, and the utilization of other data sources. Other data sources exist to fill the gap in CALIPSO’s
sampling at and near the Earth’s poles, such as modeled temperature values. Combining these other sources with CALIPSO
data, either in combination with other scattered point data interpolation algorithms or with modifications to our own, would
enable surface reconstruction over the poles.

6. ACKNOWLEDGMENT

The authors thank Chip Trepte for bringing this application to our attention and NASA’s Science Mission Directorate for
the CALISPO mission. These data were obtained from the NASA Langley Research Center Atmospheric Sciences Data
Center.

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