Derived Fields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 141

27 ◦ C) are
susceptible to storms. Filtering out regions with lower temperatures and restricting
the analysis to the months from June to November helps locate storm tracks. Regions
shown in blue in Fig. 14.7 e have been filtered out. The red regions match closely with
the storm tracks shown in Fig. 14.7 d. Even though the west coast of South America
has trade winds, storms are particularly absent due to lower temperatures. The storm
prevalent regions in the Indian, Atlantic, and Pacific oceans have high values of the
comparison measure.

>

Storm tracks . The regions over the ocean with warm temperatures (

14.4 Decomposition and Componentization

In this section we examine a different situation in which multiple fields can arise as
the components of a decomposed field.

14.4.1 Hodge Decomposition

A classical example of this is the Hodge-Helmholtz decomposition [ 18 , 25 ]ofa
vector field V as follows:

=

V c + V d +

V V h (14.1)

where V c is curl-free ( × V c = 0), V d is divergence-free ( · V d = 0), and V h is

harmonic ( 0).

Such a decomposition can have applications in many scientific and engineering
domains such as fluid simulation and modeling, electromagnetism, weather predic-
tion, engine design, scientific visualization, and computer graphics. In these appli-
cations, one often needs to analyze an input vector field such as the velocity of fluid
particles and the direction of the magnetic field. One of the most important aspects of
a vector field is singularities , which are points in the domain that satisfy V

· V h = 0 and × V h =

( p ) = 0.

A singularity can be classified by its Jacobian (gradient tensor) as follows [ 6 ]:

1. source: both eigenvalues of the Jacobian are positive.
2. sink: both eigenvalues are negative.
3. center: both eigenvalues are imaginary numbers.
4. saddle: one of the eigenvalues is positive and the other negative.

Through the decomposition, the sources, sinks, and some saddles can be captured
by the curl-free component, while the centers and some other saddles are captured by
the divergence-free component. The harmonic component is often seemingly feature-
less in the planar case. However, on hyperbolic surfaces, the harmonic component
can capture the saddles that arise as a result of surface topology. For example, any