Abstract. Glaciers in the northern Patagonian Andes (35– 46 ◦ S) have shown a dramatic decline in area in the last decades. However, little is known about glacier mass balance changes in this region. This study presents a geodetic mass balance estimate of Monte Tronador (41.15 ◦ S; 71.88 ◦ W) glaciers by comparing a Pléiades digital elevation model (DEM) acquired in 2012 with the Shuttle Radar Topogra- phy Mission (SRTM) X-band DEM acquired in 2000. We find a slightly negative Monte-Tronador-wide mass budget of − 0.17 m w.e. a −1 (ranging from − 0.54 to 0.14 m w.e. a −1 for individual glaciers) and a slightly negative trend in glacier extent (−0.16 % a −1 ) over the 2000–2012 period. With a few exceptions, debris-covered valley glaciers that descend be- low a bedrock cliff are losing mass at higher rates, while mountain glaciers with termini located above this cliff are closer to mass equilibrium. Climate variations over the last decades show a notable increase in warm season temper- atures in the late 1970s but limited warming afterwards. These warmer conditions combined with an overall dry- ing trend may explain the moderate ice mass loss observed at Monte Tronador. The almost balanced mass budget of mountain glaciers suggests that they are probably approach- ing a dynamic equilibrium with current (post-1977) climate, whereas the valley glaciers tongues will continue to retreat. The slightly negative overall mass budget of Monte Tronador glaciers contrasts with the highly negative mass balance esti- mates observed in the Patagonian ice fields further south.
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al. (2013) until 2010 and unpublished IES data for 2011). The difference in the geodetic mass balance results be- tween the east and west part of Drangajökull highlights how difficult it is to extrapolate mass balance records from one glacier to another, even over short distances. The results, showing ∼ 3 times more negative mass balance rate for the eastern part of Drangajökull than the western part for the en- tire period 1946–2011, are not reflected in changing spatial trends of summer temperature during the period. The sum- mer temperature measured east of Drangajökull is typically ∼ 1 ◦ C lower than revealed by measurements west of Dran- gajökull (Fig. 9c) and this is rather consistent throughout the survey period. Daily precipitation maps (1 km × 1 km cell size) in 1958–2006 deduced from ERA-40 (Uppala et al., 2005) by dynamic downscaling with linear model of oro- graphic precipitation (an update of Crochet et al. (2007) de- scribed in Jóhannesson et al., 2007) do not indicate a strong trend in winter precipitation from east to west. The modeled winter precipitation may, however, not be representative for winter accumulation due to excess lee drying in the modeled precipitation or transport of snow from east to west by snow drift; the most common wind direction on Drangajökull is from the northeast. Most of the precipitation also falls on the glacier when the wind blows from the northeast. Ongoing geodetic mass balance studies of Drangajökull on a seasonal timescale may reveal further answers.
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The 10 glaciers selected for this study all have long-term mass-balance programmes and geodetic surveys that cover (the larger part of) the period with annual measurements (Ta- ble 1, Fig. 1). Glaciers with short-term series without con- current geodetic surveys are not considered here. The glacio- logical series are continuous, except Langfjordjøkelen where glaciological measurements are lacking for 2 years (1994, 1995). The longest series is Storbreen where measurements began already in 1949; the shortest series are for Hansebreen, Austdalsbreen, and Langfjordjøkelen where measurements began in the late 1980s (Table 1). All glaciers are part of a glacier complex (thus sharing border with at least one other glacier flow unit) except for Storbreen (Andreassen et al., 2012b). The glaciers in southern Norway are located along a west–east transect, extending from a wet maritime climate, where Ålfotbreen and Hansebreen are located, to drier con- ditions in the interior, where Gråsubreen is located (Fig. 1). Engabreen and Langfjordjøkelen are located near the coast in the central and northern parts of Norway respectively and represent the glaciers with the lowest minimum and max- imum elevation respectively. The glaciers range greatly in size from 2.1 (Gråsubreen) to 46.6 km 2 (Nigardsbreen). One glacier, Austdalsbreen, calves into a regulated lake.
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To find if the differences between the methods only occur in Austria, or also on glaciers in other regions and climates, the results are compared to published data. Griesgletscher (Funk et al., 1997), Gulkana glacier (Cox and March, 2004), Stor- glaci¨aren (Zemp et al., 2010; Koblet et al., 2010), South Cas- cade Glacier (Krimmel, 1999), Lemon Creek glacier (Miller and Pelto, 1999), Storbreen (Andreassen, 1999) and Zongo glacier (Soruco et al., 2009) have been studied, and the geodetic and direct mass balance data is available for 26 time periods between 1940 and 2006. The average length of the periods is 18.2 yr (Table 4). The longest period lasts 57 yr, the shortest 2 yr. The temporal distribution of the data is similar to that of the Austrian data. Where only the vol- ume change without information on the mass balance or on the density of the surface layer was provided, the geode- tic mass balance was calculated assuming a mean density of 850 kg m −3 . The glaciers are located in different regions and different local climates, but the average annual geodetic mass balance is − 0.4 m w.e. a −1 and thus similar to the mean of the Austrian data ( − 0.5 m w.e. a −1 ). The mean difference is − 0.4 m w.e., the maximum difference 3.6 m w.e. and the minimum difference − 7.2 m w.e.
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The historic maps used to derive 1950 and 1960 DEMs are missing the upper section of the glacier. Historic photos ver- ify that in the mid-20th-century Sperry Glacier extended to the top of Gunsight Peak, as it does today. To enable con- sistent geodetic mass balance calculation for the entirety of Sperry Glacier, we filled in elevation change over this miss- ing section (Fig. 3) using modern 2005–2014 results. We opted for this remedy, rather than the alternative of truncat- ing 2005–2014 data, in order to be consistent with glacio- logical data, which were generated for the entire glacier sur- face including the upper section. By using modern eleva- tion change rate results to infill missing historic data, we assumed that the rate of mass change in that area near the cirque headwall was the same through the study interval (1950–2014). Given that we had no way to test the valid- ity of this assumption, and to ensure it did not fundamentally alter geodetic mass balance results, we also computed results for the truncated glacier (Tables S1 and S2 in Supplement). The difference between the truncated and infilled geodetic mass balance was ≤ 0.04 m w.e. yr −1 for both 1950–1960 and 1960–2005 (Table S2), which is less than the accounted- for geodetic mass balance error (0.05 m w.e. yr −1 for 1950– 1960, 0.12 m w.e. yr −1 for 1960–2005) (see Sect. 4.1). Infill- ing data for the missing upper section therefore does not al- ter geodetic mass balance results beyond the reported uncer- tainty bounds.
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Here, we explore the potential of using a coupled SMB– ice flow model for regional projections, by modelling the fu- ture evolution of glaciers in the European Alps with such a model. For this, we extend the Global Glacier Evolu- tion Model (GloGEM) of Huss and Hock (2015) by intro- ducing an ice flow component. We refer to this model as GloGEMflow in the following. Our approach is furthermore novel, as glacier-specific geodetic mass balance estimates are used for model calibration, and the future glacier evolution relies directly on regional climate change projections from the EURO-CORDEX (COordinated Regional climate Down- scaling EXperiment applied over Europe) ensemble (Jacob et al., 2014; Kotlarski et al., 2014). This is, to the best of our knowledge, the first regional glacier modelling study in the Alps directly making use of this high-resolution regional climate model (RCM) output. In contrast to a forcing with a general circulation model (GCM), an RCM driven by a GCM can provide information on much smaller scales, support- ing a more in-depth impact assessment and providing pro- jections with much detail and more accurate representation of localised events.
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Abstract. Area and volume changes and the average geodetic mass balance of the non-surging outlet glaciers of the southeast Vatnajökull ice cap, Iceland, during different time periods between ∼ 1890 and 2010, are derived from a multi-temporal glacier inventory. A series of digital elevation models (DEMs) ( ∼ 1890, 1904, 1936, 1945, 1989, 2002, 2010) are compiled from glacial geomorphological features, historical photographs, maps, aerial images, DGPS measurements and a lidar survey. Given the mapped basal topography, we estimate volume changes since the end of the Little Ice Age (LIA) ∼ 1890. The variable volume loss of the outlets to similar climate forcing is related to their different hypsometry, basal topography, and the presence of proglacial lakes. In the post-LIA period, the glacierized area decreased by 164 km 2 (or from 1014 to 851 km 2 ) and the glaciers had lost 10–30 % of their ∼ 1890 area by 2010 (anywhere from 3 to 36 km 2 ). The glacier surface lowered by 150–270 m near the terminus and the outlet glaciers collectively lost 60 ± 8 km 3 of ice, which is equivalent to 0.15 ± 0.02 mm of sea-level rise. The volume loss of individual glaciers was in the range of 15–50 %, corresponding to a geodetic mass balance between − 0.70 and − 0.32 m w.e. a −1 . The annual rate of mass change during the post-LIA period was most negative in 2002–2010, on average − 1.34 ± 0.12 m w.e. a −1 , which is among the most negative mass balance values recorded worldwide in the early 21st century.
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In this study, we (1) produce a six decade (1958–2018) geodetic mass balance record for three valley glaciers in the Monte San Lorenzo massif (Río Oro, Río Lácteo, and San Lorenzo Sur), using a combination of multi-sourced DEMs (based on aerial photos, SRTM, SPOT 5, and Pleiades data) on a two- decadal to sub-decadal sale, providing this way the longest glacier mass balance record so far for the Patagonian Andes, (2) provide an updated 2000–2018 geodetic mass balance assessment for other 15 glaciers in the Monte San Lorenzo, (3) derive glacier area and length changes and (4) discuss the influence of climatic trends in the region and the formation and growth of proglacial lakes at glacier terminus that may help in giving an explanation to the observed glacier changes in the intervening years. Data presented in this paper will be made available in the WGMS and/or GLIMS database.
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Typical random errors reported in the literature for glacier- wide mass balance estimates obtained with these methods are of about ± 200 mm w.e. yr −1 (Lliboutry, 1974; Braithwaite and Olesen, 1989; Cogley and Adams, 1998; Cogley, 2009). The accuracy indicated by the investigators carrying out mass balance measurements in the nine Italian glaciers range be- tween ±0.05 and ±0.30 m w.e. yr −1 (WGMS, 2015; Cartu- ran, unpublished data). Assessments based on the compari- son between the direct and the geodetic mass balance have been published for the Careser, La Mare and Lunga glaciers, indicating that the discrepancy between the two methods is lower than the lowest detectable bias (following Zemp et al., 2013), and revealing that a calibration of the direct mass bal- ance results is not required (Carturan et al., 2013a; Galos et al., 2015; Carturan, unpublished data).
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We perform a regional evaluation of the simulations by com- paring surface height changes with those measured by satel- lite altimetry in the period 2003–2008 (Moholdt et al., 2010). Note that measured height changes are also a result of glacier dynamics (although the retreat or advance of the calving front was not accounted for). The model, however, does not in- clude any glacier dynamics and the numbers are therefore not directly comparable. Still, both model and satellite data show Austfonna and northeastern Spitsbergen to be the only re- gions with positive surface height change during these years, and northwestern Spitsbergen as the region with the largest surface lowering (Table 4). The other three regions all show moderate lowering in both estimates. The model therefore seems to capture regional differences very well during this period. The mass loss from calving flux has been estimated to be 6.75 km 3 yr −1 (w.e.) for the years 2000–2006 (Blaszczyk et al., 2009), which corresponds to an additional lowering of about 0.2 m yr −1 . This suggests that the model in general simulates too much surface lowering in this period. However, the time periods are different and the estimate of Moholdt et al. (2010) does not include the effect of retreat or advance of the calving front. One must therefore still be cautious when comparing these numbers.
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In this paper we have establish many results on split geodetic number, nonsplit geodetic number, strong split geodetic number of strong product of graph and some observation on split geode- tic number, nonsplit geodetic number, strong split geodetic number of composition graphs and join of graphs.
Meteorological data suggest dramatic warming has occurred in the Tibetan Plateau since the late 1980s and that the mag- nitude of warming is much greater than that in the low- elevation regions (Kang et al., 2010). This warming has re- sulted in a continuous negative mass balance (or mass loss) of glaciers during the last decade, ranging from the Himalayas to the north of the Tibetan Plateau except for the north- western Tibetan Plateau (e.g., Yao et al., 2012; Bolch et al., 2012; Gardelle et al., 2012; Neckel et al., 2014). In recent years, the altitude of the equilibrium line for some of the observed glaciers has risen beyond the highest elevations of the glaciers; that is, there is no more net accumulation area and subsequently the entire glacier is becoming an ab- lation area (Yao et al., 2012). Although glacier mass balance varies depending on climate change and geographical condi- tions as shown on the Tibetan Plateau (e.g., Yao et al., 2012; Bolch et al., 2012), our 3 H and Hg ice core records confirm that the upper glacier areas (e.g., about 5750–6000 m a.s.l.) have rapidly been transforming into ablation areas in recent decades. In particular, extensive ablation has caused sub- stantial mass loss of the Nyainqêntanglha and Geladaindong glaciers since at least the 1950s in the southern part and the 1980s in the central part of the Tibetan Plateau, respectively. We suggest that the glaciers on the southern to the central Tibetan Plateau might be melting faster than previous data have shown (Liu et al., 2006; Jacob et al., 2012; Gardner et al., 2013). Ice losses on such a large scale and at such
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( ∼ 0.12 m) at the end of the ice season. A strong loss of la- tent heat flux during the entire period generated some 0.23 m of lake ice sublimation at the ice surface. The observed air– ice interface evolution (Fig. 2) revealed the integrated im- pacts of surface sublimation and melting (during the late sea- son), which could not be instrumentally delineated from each other. By regrouping the modeled ice mass balance compo- nents, we can calculate the evolution of the ice surface (i.e., surface sublimation + melting) and ice bottom, and compare them with the measurements (Fig. 4b). Although the modeled ice bottom depth is 4.2 cm larger than the measured one (Ta- ble 2), the HIGHTSI model captured the general evolution very well at both the ice surface and bottom. The modeled to- tal ice thickness (i.e., Depth B −Depth S ) is in good agreement with the observations (Fig. 4c). However, during day 460, the ice melting was stopped due to a snowfall event. This short- term pause was not revealed by the model since the snow thickness was assumed to be zero.
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The surface water- groundwater interaction computed using the oxygen isotope mass balance and conductivity mass balance method is presented in Table.3 and graphically presented in Fig.3 and 4. The percentage contribution of base flow determined by the oxygen isotope mass balance method showed a minimum contribution of 17% in the upstream to a maximum contribution of 43% in the downstream during the pre monsoon season. During the post monsoon season, the base flow was 38%, 27% and 15% for the downstream, midstream and upstream respectively. The results show that the base flow of groundwater contributes significantly in the maintenance of the river flow during both pre monsoon and post monsoon seasons. The ground water contribution to the river was also observed to be pronounced in the lower reaches and was relatively higher in the pre monsoon season.
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The standard distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. Although this concept has been known for a very long time, it is only in recent decades that received considerable attention as a subject of its own. In 1990 Buckley and Harary wrote the book Distance in graphs. In 2004 Handbook of graph theory edited by Groes and Yellen contains a section devoted exclusively to distance in graphs. A number of results on distance come from the fact that two vertices u and v are adjacent if and only if d(u,v)=1 and two distinct vertices u and v are non adjacent if and only if d(u,v) ≥ 2. Hence any concept whose definition relies on the adjacency or non adjacency of two vertices in a graph can be restated in term of distance. In this paper we find the geodetic polynomials of some graphs, also we have found the detour geodetic polynomial of some graphs.
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We used a windowing approach to account for uncertainty in the modern ice volume. All runs that fall within a cer- tain distance of the modern ice volume in 2005 AD (dashed line in Fig. 3) are kept, whereas the others are discarded. We investigate window widths of ± 20 %, 10 %, 5 %, and 2.5 % of the modern ice volume. These estimates of uncertainty in modern ice volume are somewhat ad hoc, but as we show later, they have little influence on our projection uncertainty. This method of confronting model results with data (Hilborn and Mangel, 1997) is highly simplified; we dis- cuss alternative approaches in Sect. 5.5. Total ice volume is a reasonable comparison metric because we are interested in future sea level change, and because inferences about the past state of the ice sheet are usually stated in terms of vol- ume changes relative to the present (e.g. Alley et al., 2010, their Fig. 13). Other metrics for comparing simulated and observed ice sheets include the ice-covered area, maximum ice thickness (Ritz et al., 1997; Stone et al., 2010), the root mean squared ice thickness (e.g. Greve and Otsu, 2007), the distribution of ice surface velocities (e.g. Aschwanden et al., 2009), and the partitioning of mass loss between sur- face melting and calving (Robinson et al., 2011). Using an aggregate measure such as total ice volume helps avoid non- trivial statistical issues with autocorrelation, in which adja- cent residuals between observations and model predictions are not independent of one another (e.g. Bloomfield and Ny- chka, 1992).
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the terminus was ﬂoating (Rivera et al., 2012a). Manual delineation of the glacier front between 2011 and 2017 using Landsat optical data (Fig. 7) indicates that Jorge Montt retreated by an additional ~2.5 km, likely through enhanced calving following retreat into deeper water (Rivera et al., 2012a). Given an average glacier freeboard height of 22 m above sea level at the terminus (Rivera et al., 2012a, 2012b) and width of 1.15 km, the glacier frontal retreat amounts to a mass loss rate of 0.07 Gt a −1 (~3% of the catchment's loss due to thinning, Table 1) between 2011 and 2017. This value is however likely underestimated since the slope of the glacier surface, and thus upglacier thickening, has not been considered. Due to the un- certainty associated with this calculation, and that at least part of the glacier terminus was ﬂ oating in 2010/11 (Rivera et al., 2012a) and likely during our observational period, we report this loss separately in Table 1 and do not include it in our total estimate of glacier contribution to sea level rise. Between 2011 and 2017, Jorge Montt shows the highest mass loss per unit area, 4 times above the average for the SPI as a whole (Table 1). Its absolute rate of mass loss (2.20 ± 0.38 Gt a −1 ) is comparable to what reported by Jaber (2016) for the period 2011–2014 (2.59 Gt a −1 , uncertainty not re- ported), which increased by 50% compared to the 1.72 Gt a −1 (uncertainty not reported) rate of mass loss between 2000 and 2011 (Jaber, 2016).
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We have taken advantage of the local routine for pa- tients that undergo liver transplantation to study the fate of infused exogenous albumin in conjunction with major abdominal surgery. Albumin was shifted from blood, presumably to the interstitial space, at 37 ± 17 g at the end of surgery. By postoperative day 3 a total of 48 ± 33 g albumin was lost from plasma. The institutional routine of substantial intravenous albumin administra- tion during and after liver transplantation until POD 3 reached a total of 138 ± 55 g, which is of the same mag- nitude as the measured total intravascular albumin mass. Although the cumulative perioperative albumin shift was somewhat larger than the 24 ± 17 g reported after the end of surgical procedures without albumin supply , it is not clear if, or to what degree, the exogenous albu- min contributed to this difference in albumin shift. Fluid overload has been suggested to cause capillary leakage,
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beginning with geodetic surveys at about decadal intervals. (iii) Such geodetic surveys should use sensors optimized for snow and ice surveys, be carried out towards the end of the ablation season (i.e. with minimal snow cover), and cover the entire glacier system as well as surrounding stable ter- rain (for uncertainty assessments). (iv) As a rule of thumb, an absolute difference between the glaciological and geode- tic balances that exceeds the annual random error estimated for the glaciological balance (e.g. > 0.30 m w.e. a −1 for our sample) indicates that reanalysing is urgently needed. (v) Mass balance series longer than 20 yr should be reanalysed in any case. (vi) Every mass balance series should be clearly flagged in publications and databases with its reanalysing sta- tus. (vii) More research is needed to better understand and quantify the potential error sources and related systematic and random errors (cf. Sects. 2.2–2.4). Important issues are the influence of the interpolation method on the glaciological balance; the density conversion of the geodetic balance; and the quantification of the internal balance components, espe- cially for polythermal and cold glaciers.
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Temperature (°C) 25,0 25,0 - 25,0 -27,7 -32,7 25,0 13.0 -1,48 Pressure (bara) 1,00 4,01 - 15,30 15,00 1,03 9,95 6,74 4,01 Flow (ton/h) 100 100 - 100 100 28,7 31,8 30,3 28,7 Table A7. Mass balance Case 3, 15 bara liquefaction pressure, Stream Numbering from Figure 1.
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