E XTRAPOLATION OF DISCONTINUOUS DATA

All elevation change measurements in Papers II-IV are discontinuous point data. Since the goal is to estimate glacier volume change and mass balance, unsampled glacier areas need to be considered. If there is a homogeneous spatial distribution of data, the mean elevation change can be used as an estimate of the area-averaged change which is converted to volume change by multiplying with the total glacier area. Otherwise, it is necessary to apply spatial extrapolation techniques to account for the uneven distribution of data. In the following two sections, we discuss how spatial interpolation and hypsometric averaging can be used to determine glacier volume change from discontinuous elevation change data. The conversion from volume change to mass balance is discussed in Sect. 4.12.

4.6.1.Spatial interpolation

Spatial interpolation is the procedure of estimating values at unsampled sites within an area covered by scattered observations. Here, we use the term to describe the process of interpolating a continuous surface from discontinuous data. It can for example be used to generate DEMs from contour maps (Paper II) or to generate continuous elevation change surfaces from scattered elevation change data (Fig. 18). In the latter case, volume change is estimated by summing all glacier elevation change values and multiplying them with the cell size at the ground (Eq. 22). Spatial interpolation requires a good spatial distribution of data in order to avoid interpolation artifacts. The ICESat data sets in Papers I-IV and Fig. 18 are limited to a few tens of profiles which are too sparse to obtain reliable interpolated surfaces.

For that reason, it was necessary to complement the ICESat data with differential SAR interferometry in order to generate a new DEM of Austfonna (Paper I).

The choice of interpolation technique depends on the characteristics of the input data (e.g. points/contours, data density and data uncertainty) and the desired properties of the interpolated surface (e.g. accuracy and smoothness). Kääb (2008) used spline, kriging and inverse-distance-weighting (IDW) algorithms to interpolate continuous elevation change surfaces from elevation change data along contour lines on Edgeøya. He found that the three interpolation methods yielded similar volume change estimates. Nuth et al. (2007) used an

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iterative finite-difference interpolation technique (Hutchinson, 1989) for the same purpose in southern Spitsbergen. This technique was also used to generate DEMs from topographic maps in Paper III. A thorough discussion about DEM generation from contour lines can be found in Wise (2000). Kääb (2008) found that it was more accurate to interpolate an elevation change surface from elevation change data along contours than to interpolate a DEM from the contours and then difference it with respect to the other DEM. Paper III acknowledges that elevation comparisons against an interpolated DEM yield a better spatial distribution of elevation change data than comparisons against contours.

Fig. 18. Interpolated elevation change surface from calculated elevation changes at crossover points (circles) between 2003-2008 ICESat laser altimetry (not shown) and 1983 airborne radio-echo sounding (dotted lines). A minimum curvature spline was used for the spatial interpolation. Note that interpolation artifacts exist in areas with few crossover points and that the results might be biased due to pressure-altitude errors in the 1983 survey (Sect. 5.5.1).

57 4.6.2.Hypsometric averaging

Due to climate and dynamic factors, there is often a relation between glacier elevation changes and elevation (Papers II-IV). This trend is difficult to account for by spatial interpolation if there is not a good spatial data coverage over most elevation bands (Fig. 18).

In such cases, it is better to parameterize elevation changes ο݄ as a function of elevation z and multiply the function ο݄ሺݖሻ with the glacier hypsometry ܣሺݖሻ over the glacier elevations to obtain estimates of volume change οܸ:

ο ൌ න ο݄ሺݖሻܣሺݖሻ݄݀ (23)

This is analogous to the hypsometric method in Eq. 2 if the parameterization is done for a fixed number of elevation bins. Paper II uses Eq. 23 over a semi-continuous hypsometry of Austfonna at a height resolution of 1 m (Paper II: Fig. 4), while Papers III-IV uses 50 m elevation bins to calculate regional volume changes. In the case of Austfonna, the choice of hypsometric resolution had no significant impact on the final volume change numbers.

Several different ο݄ሺݖሻ parameterizations are in use for hypsometric averaging. The most common approach is to calculate average elevation changes within elevation bins and assume that these average changes are representative for the entire area within the corresponding elevation bin (e.g. Arendt et al., 2002). A laser scanning study in the Canadian Arctic used the median elevation change in each elevation bin since it is less sensitive to outliers (Abdalati et al., 2004). An alternative method is to fit higher order polynomial functions to the relation between elevation change ο݄ and elevation ݖ:

ο݄ሺݖሻ ൌ ܽ_଴൅ ෍ ܽ_௡ݖ^௡

௡

ଵ

(24) where ݊ is the order of the polynomial fit. Kääb (2008) chose the polynomial order by increasing it iteratively until the improvement of the RMS to a higher order was below a certain limit. A similar approach was followed in Paper III although subjective judgments were needed in some regions to avoid runaway tails at the edges of the data. Papers II and IV use third order polynomial fits in all regions since the ݎ^ଶcoefficient of determination and the RMS error of the polynomail fits were typically stablilizing after adding the third order coefficient. All these studies found that the resulting volume change would not differ much between different orders of polynomial fits and the mean/median elevation bin methods. An advantage with the polynomial method is that it is smooth and continuous over all glacier elevations, providing elevation change estimates also in elevation bins with no data.

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