![](bg9.png)
All of these expressions adopted methods that rep-
resent the shape of an object by connecting primitive
segments when tracing its borderline. On the other
hand, Cohn took a different, approach to represent
a concave object (Cohn, 1995). He regarded differ-
ences in the closure and the object itself as regions
and represented the spatial relations of these regions.
Kumokawa et al. also proposed a different represen-
tation for a concave shape using closure (Kumokawa
and Takahashi, 2008).
A study by Kulik et al. applied QSR to landscape
silhouettes (Kulik and Egenhofer, 2003). They pro-
posed a description language for the shape of an open
line. They defined several primitives comprising two
consecutive vectors depending on relative lengths and
angles; regarded the borderline of a silhouette of a
landscape as a pattern of connections between these
primitives; and deduced landscape features, including
mountain, valley, and plateau. They also proposed a
transformation from the refined level to the abstract
level. The differences between Kulik’s method and
ours are: first, he used straight lines as primitives,
whereas we use curves; second, his target silhou-
ette was always in the vertical direction, whereas our
method can be applied to rotated forms; third, he nei-
ther formalized the method nor discussed the validity
of the representation, whereas we both define the va-
lidity of the representation and prove one-to-one rela-
tion with the model.
In addition, whereas all extant studies treated the
essentially one-dimensional data of a borderline, we
treated the two-dimensional data of a stratum consist-
ing of multiple regions.
8 CONCLUSIONS
We have discussed qualitative representation and rea-
soning for strata.
We developed a model for local data from a typ-
ical fold, and proposed its representation in the form
of a pair of sequences of symbols that stand for the
configuration of a layer and the shapes of the border-
lines between layers. This representation is suitable
to show the main features of strata: one layer extends
in one direction if there is no fault, and the relations of
interconnections between layers are unchanged even
if the width of a layer, shape, or axis of a fold changes.
We defined the required validity of the representa-
tion, and then showed that the valid representation and
that of the model have a one-to-one relation. More-
over, we defined several operations on the represen-
tation, and showed that they preserve its validity. We
also showed that global data can be generated by con-
necting local data with the same configuration. This
enables derivation of relations among multiple local
data collected in different locations or at different
times. Our main contribution is to show symbolic
treatment of strata and provide a basis for logically
explaining the process of landscape generation.
In future studies, we intend to identify sets of rep-
resentations obtained from repetitive application of
connections of local data. We are also considering the
formalization needed to explain the strata-generation
process, as well as a qualitative simulation for possi-
ble future morphological changes.
ACKNOWLEDGEMENTS
This research is supported by JSPS Kakenhi
JP21K12020. The authors would like to thank Mo-
tohiro Tsuboi for giving useful advice from the field
of geology.
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