1 Spatial Modelling of the Terrestrial Environment - Georeferencial

Snow Depth and Snow Water Equivalent Monitoring 39
to investigate highly local-scale or even micro-scale spatial variability. However, the actual
number of stations regularly reporting snow depth or SWE is rather less than this average
figure suggests.) To further investigate the spatial variability of snow depth or SWE, the
analysis of snow depth variograms was undertaken.
Intuitively, there is an inherent spatial dependence of snow depth or SWE in a snow
field because locations closer together tend to be more similar than those further apart (see
Tobler’s first Law of Geography). This concept can be used to encapsulate the micro-scale
of SWE or snow depth variation. ‘Spatial similarity’ of snow depth also can be present over
greater distances on account of local or regional controls on snow accumulation (in mountainous
terrain or along the track of a snow storm). In other words, spatial autocorrelation
of snow depth is probably present at different scales from micro-scales through to broad
regional scales. Quantification of the spatial dependence of a variable may be expressed by
the semi-variogram. In statistics, observations of a selected property are often modelled by
a random variable and the spatial set of random variables covering the region of interest is
known as a random function (Isaaks and Srivastava, 1989). In geostatistics, a sample of a
spatially varying property is commonly represented as a regionalized variable, that is, as a
realization of a random function. The semi-variance (γ ) may be defined as half the expected
squared difference between the random functions Z(x) and Z(x + h) at a particular lag h.
The variogram, defined as a parameter of the random function model, is then the function
that relates semi-variance to lag:
γ (h) = 1 2 E[{Z(x) − Z(x + h)}2 ] (1)
The sample variogram γ (h) can be estimated for p(h) pairs of observations or realizations,
{z (x l + h) , l = 1, 2,...,p (h)} by:
ˆγ (h) = 1 p(h)
2 p(h) ∑
{z(x l ) − z(x l + h)} 2 . (2)
l=1
As Oliver (2001) states: “[the variogram] provides an unbiased description of the scale and
pattern of spatial variation”. It is a useful tool, therefore, for analysing the scale(s) of variation
characterized by datasets of measurements of snow depth or snow water equivalent.
Variograms were calculated for measured point snow depth data (cm) for three different
datasets to investigate spatial dependencies present at different sampling scales. The
three dataset frameworks described in Figure 3.1 (the WMO, COOP and FSU snow depth
datasets) were used to explore the characteristics of snow depth spatial variability in each
dataset. Ideally, datasets containing consistent measurements of SWE should be used, but
globally, they are not available routinely so snow depth records are used. Daily samples of
snow depth from each dataset were selected for northern hemisphere mid-winters (early
February) from the respective archives. Although the same time periods are not represented
in each case, the data provide some sense of spatial variability characterized at each scale
of measurement.
The point snow depth data were projected to the Equal Area Scaleable Earth Grid (EASEgrid)
(Armstrong and Brodzik, 1995) and the variograms computed for 2 February, 2001
(WMO), 10 February, 1990 (FSU), 12 February, 1989 (COOP) and 12 February, 1994
(COOP). The WMO data cover the entire northern hemispehere while the FSU data cover