2.3 Interpolation Algorithms
2.3.1 Inverse Distance Weighting (IDW)
Inverse Distance Weighting (IDW) is one of the most widely used interpolation techniques, proposed by Shepard in 1968 [46, 47]. This technique is categorised as a deterministic method and is based on the assumption that the value to be interpolated is likely to be more similar to the nearer observed values than to those at a greater distance. This technique is expressed in the following:
ˆ f(x,y) = N
∑
i=1 λ(dsi)i · vi (2.1)12 Literature Review λ(dsi)i= d−us si ∑Ni=1d −us si (2.2) dsi= q (xi−x)2+ (y i−y)2 (2.3)
where ˆf(x,y)indicates that we are estimating the value at locationx,y;λ(dsi)iis the weighting mechanism;dsiis the spatial distance (2-dimensional Euclidean distance) between the point to be interpolated (x,yin this case) and the known data point (vi);us is a user-defined parameter
that is used to adjust the diminishing strength in relation to increased spatial distance; and,N is the total number of known points. If the configurationus=2 is applied, this IDW becomes
the Inverse Distance Squared technique [48]. Yet, the parameterus need not always be to
two, and can be adjusted to improve performance [49]. The complexity of conventional IDW isO(N)which can be seen from Equation 2.1.
The so-called ‘zero distance problem’ has been discussed by de Mesnard [50]. For the case where location to be interpolated is exactly the same as one of the reference points (i.e. dsi =0), Shepard [46] does not interpolate that particular location because we already have full knowledge at that point – thediscretecase. Unfortunately, such implementations may not be realistic when the mean within a particular area (suburb, city, state, country) is being estimated (i.e.dsi→0) and, for these situations, utilising thecontinuouscase is more satisfactory.
Improved IDW
IDW is based on the notion that nearer data points will have more influence compared to those further away, and so including every data point throughout the map to interpolate a single point is unnecessary. This is because, as the distance is further away, the particular point has very little influence on the original value. An illustration of the improved version of IDW can be seen in Figure 2.1, and the parameterR can be optimised to improve the quality of IDW [49]. The improved IDW comes with two major advantages in terms of computational efficiency: (a) The processing time does not increase as the number of known points increases; (b) We can further improve the performance by applying the kd-tree data structure algorithm, reducing the computational time from O(N) to O(logN) [51]. This feature is crucial because the basic computational time required for the STI algorithm is O(T×N), instead ofO(N), where T is the number of user-defined window lengths to be included in the interpolation process.
2.3 Interpolation Algorithms 13
Fig. 2.1 A demonstration of the improved IDW. The dots are the sample data points, and the ‘×’ is the point location to be interpolated. Ris a user-defined radius parameter that indicates the farthest distance to be included from the point×. On this case, only a total of six sample points will have an influence on the interpolation process. However, no empirical approach has been developed to obtain the optimal value for the parameterR.
STI - Extension approach
The extension approach to the IDW’s STI method (2-D Space and 1-D Time problem) was described by Li and co-authors in 2014 [51]:
ˆ f(t,x,y) = stend
∑
tstart N∑
i=1 λ(dsti)st,i·vt,i (2.4) λ(dsti)st,i= d−ust sti ∑Ni d−stiust (2.5) dsti= q (xi−x)2+ (y i−y)2+c2(ti−t)2 (2.6)wheredstiis the spatio-temporal distance between the measured(xi,yi,cti)and unmeasured (x,y,ct) location point; ust is the spatio-temporal diminishing strength as distancedsti in- creases; and, cis the user-defined temporal factor that converts the time unit to a spatial distance unit; However, there is still no empirical information on how to justify choice of the temporal factor (c), and a naive choice does not yield optimal results [51].
14 Literature Review
The extension of this approach to the 3-D space and 1-D time problem can be expressed in a slight variation based on Equation 2.6, so that it becomes:
dsti=
q
(xi−x)2+ (y
i−y)2+ (zi−z)2+c2(ti−t)2 (2.7)
wherezis the 3rd spatial dimension in which always seen as the altitude (surface height).
STI - Reduction approach (ST Product Method)
This method was proposed by Liet al. and is constructed in the following way [45]:
ˆ f(x,y,t) = N
∑
i=1 λ(dsi)s,i · fˆ(t) (2.8) ˆ f(t) = ti2−t ti2−ti1vi1+ t−ti1 ti2−ti1vi2 (2.9) where the spatial weightingλ(dsi)s,iis equivalent to Equation 2.2; ˆf(t)is the estimated value at timet;ti1andti2are the first (previous) and second (next) time indices, and similarly,vi1 andvi2are the first (previous) and second (next) value at corresponding timet.It is important to note that such an algorithm (Equation 2.9) relies on the assumption that values at the same location (vi1andvi2) but at different times (ti1andti2) are provided. Nevertheless, one of the assumptions of this dissertation is that the data are not necessarily collected in a finely-gridded manner (in both the spatial and temporal dimensions). Due to the fact that this algorithm does not meet the assumptions of this thesis, it will not be considered and applied to the simulation in this work.