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which on average are contained within the s -planes, tend to scatter more within the
average s -plane than in the direction perpendicular it.
Agterberg ( 1974 ) proposed to subject each of the three direction cosines of B-axes
in quartzphyllites belonging to the basement of the Dolomites separately to polyno-
mial trend surface analysis followed by normalization at any point on the map of the
study area in order to obtain the unit vector field. Trend surface analysis is based on the
assumption that 2-D polynomials can be used for describing trends while the residuals
(differences between original and trend values) are randomly distributed. This method
does not account for possible spatial autocorrelation of the residuals but the estimated
trends would be unbiased if the residuals satisfy a second-order stationary process ( cf .
Watson 1971 ). The method of separately fitting polynomial functions to direction
cosines of unit vectors also was used by Parker and Denham ( 1979 ). These authors
used cubic smoothing splines instead of ordinary polynomial functions.
An important paper on interpolation and smoothing of directed and undirected
linear data was published by Watson ( 1985 ). Several new algorithms were proposed
by this author using functions according to which the influence of measurements
decreases with distance from the points at which interpolation with or without
smoothing is to be applied along a line or within a map area. This type of approach
also is useful for 3-D applications using drill-hole data to extend the geological map
downwards from the topographic surface (Michael Hillier, Geological Survey of
Canada, personal information).
8.2.2 Arnisdale Gneiss Example
In the previous section it was shown that Hercynian schistosity in the San Stefano
area was reactivated to form the core of an Alpine anticline that can also be seen in
the Permotriassic rocks overlying the crystalline basement. This style of folding is
analogous to that described by Ramsay ( 1960 , 1967 ) for gneisses of the Moine
Series at Arnisdale, western Highlands of Scotland. An example is shown in
Fig. 8.10 where the pattern is for the azimuth of deformed B-lineations. Axial
traces (intersections of axial planes with the topographic surface) for the late folds
are indicated. A schematic explanation for this style of folding is given in Fig. 8.11 .
During the late folding, the foliation, which is according to parallel original
surfaces r , s , t and u , did not change in attitude. The axial plane of the late folding
(ab) makes a small angle with the original surface. The a -direction is subvertical
and is contained in the original surfaces.
Patterns such the one shown in Fig. 8.10 can become almost completely
obscured when the surfaces, on which the lineation was developed were themselves
variably oriented or when the amount of compressive strain accompanying the
original folding was variable in space. In such situations, one may never be able to
reconstruct a pattern of deformation in an accurate manner. However, by develop-
ing the vector mean for lineation from observations in different outcrops, trends can
be established providing a broad outline of the geometry of the folds and the
underlying genetic process.
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