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International Archives of the Photogrammetry, Remote 
  
between —l and 2 m at Dutch standard sea-level (NAP) are 
considered as beach areas. Such areas are derived from digital 
elevation models (DEMs). Similarly, non-vegetated and dry 
zones are derived from Landsat TM imagery. Non-vegetated 
zones are selected as areas with negative NDVI values; dry 
zones are selected as areas with a wetness index lower than 
zero. The delimitation of the object beach should satisfy the 
constraints for altitude, non-vegetated and dry zones (Vasseur 
et. al., subm.). 
The description of the spatio-temporal ontology depends on two 
factors: the spatial variation of the attributes within a 
compartment and its changes in time. The spatial variation is 
inherited in several attributes, as altitude, vegetation index and 
wetness index. In the definitions for beach nourishment, the 
attributes are vaguely described in contents and geometry. 
Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
Beach compartments can be described by a membership of dry, 
non-vegetated beaches (Van de Vlag et al., 2004). Additionally, 
each attribute has a different timescale. Hence, different 
temporal scale issues need to be incorporated. Beach volumes 
derived from altitude can be described on yearly trends. The 
vegetation index has monthly fluctuations, while wetness index 
is characterized by tide fluctuations on a daily scale. 
The spatial variation of the attributes can be modeled using 
fuzzy logic, whereby beach compartments suitable for beach 
nourishment are determined by its membership to dry, non- 
vegetated beaches. Hence, a compartment is bound by two 
static. compartment. boundaries (CL.geo) and by two vague 
boundaries: the sea-beach boundary (BS.geo(#)) and the beach- 
dune boundary (BD.geo(r)). These boundaries are illustrated in 
figure 2, lower image. 
  
ndvi index | 
00501 02 ed 
vegetated 
02 07-005 
non-vegetated 
  
ndvi elevation wetness index | 
membership values membership values membership values 
1 4 1 
    
CL 
elevations 
> 
wetness index 
42° 75 + 376° 19^ 
wet 
md(i,t) 
> 
  
fuzzy boundary 
beach - sea (BS) 
  
  
  
  
    
  
  
fuzzy boundary 
beach - dune (BD) 
  
  
  
  
BD 
Figure 2: Compartment, boundaries and their various fuzzy membership functions. The lower image visualizes 
with two adjacent crisp boundaries (CL) and two fuzzy boundaries (BS) and (BD). 
The sand volume within the fuzzy compartmental method can 
be calculated, using: 
np 
C.vol(t) 2 ps x X m(i, t) x e(1, t) 
(1) 
where m(i,f), membership value of location (i) in compartment 
Cat time ¢. It is calculated as: 
wil) = min {mb(i,t), md (i, 1), mv(i,1)} 
Where mb(i,r) is the membership function of the beach object, 
Wf) that of dry object and mw(i,f) that of a non-vegetated 
Object in which pixel ; occurs at time / (see figure 2). 
Membership functions are compiled as triangular functions and 
Ve semantic based. The mb(i,f) equals 1 if altitude ranges from 
to 1 m amsl (i.c. above mean sea level). It increases linearly 
liom 0 to 1 between —1.1 to 0 m amsl and decrease linearly 
liom 1 to 0 between 1 and 3 m amsl, and it equals 0 elsewhere. 
1191 
[compartment boundary ( CL)| 
  
  
  
a compartment (C), 
The md(i,t) equals 1 if wetness index is less than -3, and 
decrease linearly from 1 to 0 for the wetness index moving from 
-3 to 3 m amsl, and it equals 0 elsewhere. Finally, the nv(i.7) 
equals 1 if the ndvi value is less than -0.05, it equals 0 if the 
ndvi is larger than 0.05 and it decrease linearly from 1 to 0 in 
between. 
To include temporal uncertainty into the beach nourishment 
processes, we consider daily fluctuations for the wetness index, 
monthly fluctuations for the vegetation index and yearly 
fluctuations for altitude. These assumptions are based on 
observation methods and applied on the most appropriate time 
scale for these attributes. However, weather influences are 
neglected as these are complicated to observe and difficult to 
model due to several time dimensions. 
Temporal membership functions are introduced, which have the 
highest values when the most reliable data can be collected. For 
vegetation, the v(/) equals 1 between 1 June and | August, it 
equals 0.5 between 1 November. until 1 March, and it is linear 
in between. Similarly nd(f) equals 1 during flood time and