Definition of a Multifield - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 103

We further require that the various fields have a shared embedded space-time domain
D D S D T .

= ×

D i D S and D i D T , for all i

⊂ ⊂

The subsets can also be lower-dimensional, such as domain boundaries.

10.2.4 Uniqueness of Representation

It is possible that there is not one unique representation for a field or multifield
according to the definition given above. E.g., it would be possible to shift the indices
for the components of a vector to the parameter space. Or one could even consider
a vector field already as a multifield consisting of several scalar fields.
Another example is the representation of multi-run simulations. Here two cases
can be distinguished. On the one hand, a simulation can be run repeatedly for different
(combinations of) parameters. In this case, a multifield representation is obtained
straightforwardly by making the parameter space a part of the domain.
On the other hand, stochastic ensemble simulations, as done typically in climate
research, generate a field with distributions as its values instead of scalars or vectors.
Another option would be to interpret each run as a separate field.
The decision for the specific choice of the domain and range of a field should be
guided by the semantics of the multifield and its fields.

10.3 Multifields and Related Concepts

The above definition of a multifield provides enough flexibility to capture most of
the types of compound datasets that occur in practice. In this section we identify the
types of data that can be represented by our multifield definition and describe the
actual mapping to this abstract mathematical representation.
The term multi-channel data is commonly used to describe data having multiple
variables (quantities, attributes) per point in space-time. This fits straightforwardly
into our multifield concept and is in fact a standard case of it.
The same holds, conceptually, also for multi-modal data . Practically, a differ-
ence is that in multi-modal data the single fields are typically given in different
discretization and on overlapping domains. Multi-modal data are generated in sepa-
rate processes, and data from different modalities have to be registered first, before
they can be represented as a multifield.
Multi-material simulations generate data for different materials. These can be
represented by different fields of a multifield. Domains can overlap if materials can
mix, but also at material boundaries because of discretization.
The term multi-dimensional data can be used to give a vague description of the
complexity, but by not making a distinction between the dimension of the domain