RADIAL BASIS FUNCTIONS FITTING METHODS AND SMOOTH PIECEWISE ALGEBRAIC APPROXIMATION AS APPLIED TO DETERMINE POSTGLACIAL TILT IN THE CANADIAN PRAIRIES

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RADIAL BASIS FUNCTIONS FITTING METHODS AND SMOOTH
PIECEWISE ALGEBRAIC APPROXIMATION AS APPLIED TO
DETERMINE POSTGLACIAL TILT IN THE CANADIAN
PRAIRIES
Azadeh Koohzare, Petr Vanícek and Marcelo Santos
University of New Brunswick, Department of
Geodesy and Geomatics Engineering, Fredericton,
N.B., E3B 5A3, Canada Email a.koohzare_at_unb.ca
Introduction Geodetic data can be combined to
yield adjusted Vertical Crustal Movements (VCM).
However, as a result of scattered data or lack of
data in some areas, the estimates of velocity
surface models are uncertain. Central North
America is an example of such areas in the study
of postglacial rebound. Data on vertical crustal
movement in the Canadian Prairies has been sparse
and has not been included in postglacial rebound
modeling. Therefore, a more precise knowledge of
postglacial rebound tilt in this region will
assist to verify the ICE models which in turn
expand the studies of the histories of the major
lakes in the area and benefit shore
industries. In this study, different approaches
to scattered data fitting are considered to
compute the best surface of VCM that meets as
many as possible of the following goals Optimum
Approximation The best fit to the
data Stability The Computed surface should be
numerically stable. Independency The surface
should be independent of the choice of node
points. Quality The surface should be of high
visual quality(i.e., the surface should be at
least C(1)-continuous) Usability For large real
regional data sets, the surface should be
manageable.
Figures 4 and 5 show the surfaces which depict
the vertical crustal movements in the Prairies
using piecewise polynomial approximation of
degree 2 and 3. Figures 6,7a demonstrate the
contours of the vertical velocities computed
using SPAA method (polynomials of 2nd, 3rd order
for each patch) and the standard deviation of the
surface is plotted in Figure 6,7b.
Figure 4 Vertical Velocity surface (mm/yr)
computed using piecewise polynomials, Degree 2.
Figure 5 Vertical Velocity surface (mm/yr)
computed using bicubic piecewise polynomials.
Overview There are totally 2287 relevelled
segments in the area of interest in addition to
the 63 years monthly sea level records of
Churchill tide gauge. Figure 1 shows the
distribution of data in the region.
Figure 6 (a) The contours of the vertical
velocities (mm/yr) computed using piecewise
polynomials of 2nd order. (b) The standard
deviation of the surface.

Figure 1 Data distribution used in computations.
The blue shaded polygon represents the area where
the patches are selected. Continuous lines show
relevelled segments. Star indicates location of
Churchill tide gauge with the monthly mean sea
level trend of 9.5 mm/a .
Two approximations of scattered relevelling data
set in the Prairies was first computed using
radial basis function methods. By a radial
function, we mean a function for some
function In this study, we first, used the
Multiquadrics method of Hardy (1971) and radial
Cubic Spline method as two commonly used basis
function for the scattered data. Since our data
are of different types, mostly height difference
differences (gradients), special attention should
be paid to the location of nodal points in the MQ
and Cubic spline methods. The locations of nodal
points were determined by the review of leveling
profiles to find the maxima ad minima of the
relative motion for each route of relevelling.
(Holdahl, et al., 1978) Figures 2 and 3 show the
surfaces of VCM produced using MQ method and
radial C spline. As it is seen in the plots,
there are some peaks in southern Manitoba which
are in the location of nodal points and are the
artifacts of the constant shape parameter. The
surfaces are highly depends on the location of
the nodal points and this limits the methods,
when a representation of a some relative data are
sought. However, it is proved that these methods
are of highly quality for fitting the surfaces to
absolute values. (Dyn, et al., 1986)
Figure 7 (a) The contours of the vertical
velocities (mm/yr) computed using piecewise
polynomials of 3rd order. (b) The standard
deviation of the surface.

• Discussion
• A quick look at the maps of VCM compiled using
  SPAA (Figures 6a and 7a) shows that there is
  subsidence in the south part of the region. This
  is also resulted from the last studies of Carrera
  et al. 1991. Both degree of piecewise polynomials
  led to almost the same a posteriori variance
  factor ( ) and the standard deviations
  of the estimated vertical velocities for both
  cases vary between 0.5-1.5 mm/yr in the region
  where there is data.
• The tilt computed from our model is consistent
  with ICE-3G model of GIA, However, the absolute
  value for the vertical velocities (which comes
  from Churchill tide gauge) throughout the region
  has disagreement with GIA models. As a result,
  while the location of zero line is further south
  in GIA models, in our model the zero line loops
  and the suspected postglacial rebound zero line
  from our model is biased compared to GIA models.
  According to Lambert et al., 1998, the rate of
  decrease in absolute gravity values at Churchill
  and Manitoba show also disagreement in Manitoba
  with ICE3G and the standard Earth model. It was
  suggested by Lambert et al., 1998 that a thinning
  of Laurentide ice-sheet over the Prairies for the
  ICE-3G model leads to a better fit to the
  absolute gravity data. This might be a case in
  our studies, too and needs further
  investigations.
• Conclusion
• The VCM generated using the methods of MQ and
  radial splines depend largely on the node points
  and the shape parameters, and since our data is
  relative velocities, this makes some limitations
  in the use of those fitting methods. Using SPAA
  method, the solution is quite stable and doesnt
  depend on the choice of node points.
• The analysis conducted for this paper
  provides some indication into the usefulness of
  geodetic observations in evaluation of
  postglacial rebound models and provides us with
  important insight to model VCM.
• Acknowledgment We would like to thank the GEOIDE
  (GEOmatics for Informed DEcisions) Networkof
  Centres of Excellence of Canada for their
  financial support of this research.

Figure 2 Vertical Velocity surface (mm/yr)
computed using MQ method.
Figure 3 Vertical Velocity surface (mm/yr)
computed using Cubic spline method.
We, then used the method of Smooth Piecewise
Algebraic Approximation (hereafter SPAA) to fit a
surface to a scattered data in the Prairies. The
area was divided into 6 patches for the piecewise
approximation. The size of the patches is
dictated by the data distribution. Polynomial
surfaces of 2d and 3rd order were fitted to each
patch separately to obtain initial values for the
coefficients and then the simultaneous solutions
were computed after enforcing the constraints
along the boundaries between the patches to
satisfy the continuity and smoothness of the
final solutions. (For details about the method of
SPAA see Koohzare et al., 2006).
CGU Annual Scientific Meeting, Banff Alberta
May14-17, 2006