ation 
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S. 
       
  
Y 
AN > 
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i 
  
  
  
  
Figure 1.2- PTV - Exemple 1: velocity field in a 
turbulent channel flow (1 second of 
flow data with about 500 simultaneous 
velocity vectors, 2D-projection) 
  
Figure 1.3- PTV - Example 2: velocity field gene- 
rated in a aquarium (0.5 seconds of 
flow data with about 800 simultaneous 
velocity vectors, 2D-pro jection) 
Another example of 3D data acquisition is the 
application of 3D laser induced fluorescence 
(3D-LIF) to examinations of mixing processes (Dahm 
et al, 1990). "Unlike PTV, LIF "is based on 
continuous visualization of flow structures by 
fluorescent material, which emits light of a 
certain wavelength 1l, when animated by a laser 
with a different wavélength 1 . By scanning an 
observation volume with a laser lightsheet in 
depth and recording images of the illuminated 
slices with high-speed cameras, 3D fluorescence 
concentration data can be acquires  quasi- 
simultaneously. From these data e.g. concentration 
gradient vector fields and scalar energy 
dissipation fields can be derived, which contain 
information about the efficiency of mixing 
processes. 
2. THE METHOD 
(This paragraph contains an exposition of the 
deterministic and stochastic approaches, with 
special regard to 3D problems; for a view over 2D 
problems and time series and for more information 
See: Sansó/Tcherning, 1983; Ammanati et al., 1983; 
Sansó/Schuh, 1987; de Haan/Mussio, 1989; Barzaghi/ 
Crippa, 1990). 
2.1 Covariance estimation and covariance function 
modelling 
The collocation method requires appropriate models 
to interpolate the empirical autocovariance and 
crosscovariance function of the signal, obtained 
from the residuals of linear interpolations. 
This model function is used (in addition to the 
model found by linear interpolation) to predict 
the value of the studied quantities. 
An hypothesis was made: the residuals can be seen 
as realizations of a continuous, isotropic, and 
normal stochastic process which is stationary of 
2nd order with mean zero and covariance function 
of the kind: 
C(P,,P,) - CCIP, = PII) 
With X(P ) the n observations at the different 
points P,...,P,...P the estimate of the 
3 1 ex n 3 
empirical autocovariance function at the space 
interval po is calculated from: 
UJ 
n i 
a > (1) 
[U) zi V —ÓZÓETT Vv 
7 n i=L ‘1 ER 3=1 3 
i 
where fo, V P, > pi I PT PII = pon 
and v= X -X k = 1,n 
and the estimates of empirical  crosscovariance 
function at the space interval r are computed 
from: 
m1) 
n i 
1 1 
unl. ) (1) 
Yoel? ] ^ y Nom 
i=1 i 2) j=1 3 
  
where JA, V P, > p71) I P = Pll =< pm 
E VP sr p - Js r0 
j j qub à