Code and readme of a MATLAB-based graphical user interface program for computing functionals of the geopotential

We present a novel graphical user interface program GrafLab (GRAvity Field LABoratory) for spherical harmonic synthesis (SHS) created in MATLAB®. This program allows to comfortably compute 38 various functionals of the geopotential up to ultra-high degrees and orders of spherical harmonic expansion. For the most difficult part of the SHS, namely the evaluation of the fully normalized associated Legendre functions (fnALFs), we used three different approaches according to required maximum degree: (i) the standard forward column method (up to maximum degree 1800, in some cases up to degree 2190); (ii) the modified forward column method combined with Horner's scheme (up to maximum degree 2700); (iii) the extended-range arithmetic (up to an arbitrary maximum degree). For the maximum degree 2190, the SHS with fnALFs evaluated using the extended-range arithmetic approach takes only approximately 2-3 times longer than its standard arithmetic counterpart, i.e. the standard forward column method. In the GrafLab, the functionals of the geopotential can be evaluated on a regular grid or point-wise, while the input coordinates can either be read from a data file or entered manually. For the computation on a regular grid we decided to apply the lumped coefficients approach due to significant time-efficiency of this method. Furthermore, if a full variance-covariance matrix of spherical harmonic coefficients is available, it is possible to compute the commission errors of the functionals. When computing on a regular grid, the output functionals or their commission errors may be depicted on a map using automatically selected cartographic projection.

Characteristics of the Vernagtferner mass balance for the period from 1964 to 2014

The main characteristics of the Vernagtferner mass balance are sumarized in the table below. The mass balance years from 1964/65 to 2003/2004 are listed. The table includes the total area of the glacier (basis for the calculations), the equilibrium line altitude (ELA), percentage of the accumulation area in relation to the total area (AAR) and the specific net mass balance in mm w.e. (water equivalent) per year. It becomes clear that, after a rather minor growth period in the mid 1970's, the glacier continually lost mass since the beginning of the 1980's. Besides that, a clear increase of mass balance years with extreme mass losses could be observed in the last decade. The "glacier-friendly" summer with a well-balanced mass balance in 1999 could only interrupt the series of years with extreme mass losses, but this means no change in the trend. The minor mass loss in 1999 was caused by a winter snow cover above average, which prevented the glacier from becoming snow free over large areas and thus resulted in a lower ice melt. Although real summer conditions in 2000 were mainly restricted to August and produced a snow free area only slightly larger than in 1999, there have been further ice losses. This trend of negative mass balance continued also in the years 2001 and 2002. Nevertheless, the losses are moderate because a smaller part of the glacier became ice free until autumn (appr. 50 %). The summer 2003 caused a loss of ice in a dimension never seen since the beginning of the scientific investigations. This resulted from a combination of different factors: after only a moderate winter snowcover the glacier became snow free very early. For the first time the ablation area spanned over the entire glacier (blue fields in the mass balance tables!). Only one short snowfall event interrupted the ablation period, which lasted twice as long as in the years of large losses in the 1990's. The extreme mass loss in 2003 will also influence the mass balance in the following year 2004. The graphical representation of the elevation distribution of the specific mass balance together with the absolute mass balance can be found individually for each year by choosing one of the mass balance values from the table. These diagrams also include the area-height-distribution of the glacier and the ablation area. A tabular version of the numeric values in dependence of the elevation, provided separately for the accumulation area, the ablation area and the total glacier, can be found in colums "Persistent Identifier". The tables include the results for three different parts of the glacier and for the total glacier.