Dynamics of the Cold Surface Layer of Polythermal Storglaciären, Sweden

Home , Plucking (glaciation), Upper Staircase

Dynamics of the cold surface layer of polythermal Storglaci¨aren, Sweden

Rickard Pettersson

Department of Physical Geography and Quaternary Geology Stockholm University Stockholm 2004 c Rickard Pettersson ISSN: 1650–4992 ISBN: 91–7265–907–6 Cover: Storglaci¨aren, 7th October 2000. Layout: Rickard Pettersson using LATEX (except paper I, c American Geophysical Union) Printed by Tryckindusti Information AB, 2004 Doctoral dissertation 2004 Rickard Pettersson Department of Physical Geography and Quaternary Geology Stockholm University

Abstract. Polythermal glaciers, i.e. glaciers with a combination of ice at and below the freezing point, are widespread in arctic and subarctic environments. The polythermal structure has major implications for glacier hydrology, ice flow and glacial erosion. However, the interplay of factors governing its spatial and temporal variations such as net mass balance, ice advection and water content in the ice is poorly investigated and as yet not fully understood. This study deals with a thorough investigation of the polythermal regime on Storglaciären, northern Sweden, a small valley glacier with a cold surface layer in the ablation area. Extensive field work was performed including mapping of the cold surface layer using ground-penetrating radar, ice temperature measurements, mass balance and ice velocity measurements. Analyses of these data combined with numerical modelling were used specifically to investigate the spatial and temporal variability of the cold surface layer, the spatial distribution of the water content just below the cold surface layer transition, the effect of radar frequency on the detection of the surface layer, and the sensitivity of the cold surface layer to changes in forcing. A comparison between direct temperature measurements in boreholes and ground- penetrating surveys shows that the radar-inferred cold-temperate transition depth is within ±1 m from the melting point of ice at frequencies above ∼300 MHz. At frequencies below ∼155 MHz, the accuracy degrades because of reduced scattering efficiency that occurs when the scatterers become much smaller compared to the wavelength. The mapped spatial pattern of the englacial cold-temperate transition boundary is complex. This pattern reflects the observed spatial variation in net loss of ice at the surface by ablation and vertical advection of ice, which is suggested to provide the predominant forcing of the cold surface layer thickness pattern. This is further supported by thermomechanical modeling of the cold surface layer, which indicates high sensitivity of the cold surface layer thickness to changes in vertical advection rates. The water content is the least investigated quantity that is relevant for the thermal regime of glaciers, but also the most difficult to assess. Spatial variability of absolute water content in the temperate ice immediately below the cold surface layer on Storglaciären was determined by combining relative estimates of water content from ground-penetrating radar data with absolute determination from temperature measurements and the thermal boundary condition at the freezing front. These measurements indicate large-scale spatial variability in the water content, which seems to arise from variations in entrapment of water at the firn-ice transition. However, this variability cannot alone explain the spatial pattern in the thermal regime on Storglaciären. Repeated surveys of the cold surface layer show a 22% average thinning of the cold surface layer on Storglaciären between 1989 and 2001. Transient thermomechanical modeling results suggest that the cold surface layer adapts to new equilibrium conditions in only a few decades after a perturbation in the forcing is introduced. An increased winter air temperature since mid-1980s seems to be the cause of the observed thinning of the cold surface layer. Over the last decades, mass balance measurements indicate that the glacier has been close to a steady state. The quasi-steady state situation is also reflected in the vertical advection, which shows no significant changes during the last decades. Increased winter temperatures at the ice surface would result in a slow-down of the formation of cold ice at the base of the cold surface layer and lead to a larger imbalance between net loss of ice at the surface and freezing of temperate ice at the cold-temperate transition.

Keywords: Polythermal glaciers, cold surface layer, ground-penetrating radar, spatial and temporal variability, water content, thermomechanic modeling, Storglaci¨aren, Sweden

Dynamics of the cold surface layer of polythermal Storglaci¨aren, Sweden

Rickard Pettersson

This doctoral dissertation consists of this summary and the following papers, which are referred to by their roman numbers in the text.

I Pettersson, R., P. Jansson, and P. Holmlund, 2003. Cold surface layer thinning on Storglaci¨aren, Sweden, observed by repeated ground penetrating radar surveys. Journal of Geophysical Research, 108(F1), 6004, doi: 10.1029/2003JF000024. II Pettersson, R., P. Jansson, and H. Blatter, 2003. Spatial variability in water content at the cold-temperate transition surface of the polythermal Storglaci¨aren, Sweden. Accepted for publication in Journal of Geophysical Research.

III Pettersson, R., 2004: Frequency dependence in detection of the cold- temperate transition surface in a polythermal glacier. Submitted to Radio Science. IV Pettersson, R., P. Jansson, H. Blatter, and H. Huwald 2004. Spatial pattern and stability of the cold surface layer on Storglaci¨aren, Sweden. Manuscript.

The co-authorship of the papers reflects that they are part of a collaborative research project. In Paper I all three authors planned and outlined the study. I collected all the field data except the radar data from 1989, which was provided by Per Holmlund. I performed all data processing and analysis and wrote the manuscript after comprehensive discussions with Peter Jansson. In Paper II,the study was outlined and planned by Peter Jansson, Heinz Blatter, and me. The idea for absolute water content calibration came from Peter Jansson and Heinz Blatter. I collected and processed all the field data. I wrote the text except for the part on strain heating, which was written jointly by Heinz Blatter and me. The text was elaborated in conjunction with discussions with Peter Jansson. In Paper III, All work was done by me. Paper IV was outlined and planned by Peter Jans- son and me. I performed the fieldwork, data processing, and analysis. The model was provided by Heinz Blatter and Hendrik Huwald, but was further adapted and developed by me. I am responsible for the text in the manuscript.

Introduction the fraction of the frozen rim also affect erosion rates of the substratum, which is important for the con- Polythermal glaciers consist of both temperate ice centrations of dissolved and suspended materials in with temperature at the melting point and cold proglacial streams [Hodson et al., 1997; Hodgkins, ice with temperature below the pressure melting 1997; Humborg et al., 2002]. Hence, the extent and point throughout the year. The ablation area of such variability of the thermal regime of polythermal glaciers is often characterized by a cap of cold ice glaciers may significantly influence the glacial envi- covering a temperate core and is typically frozen to ronment. the bed along their margins and terminus in areas Little is known about the dynamics and stability where the cold ice meets the substratum. The pres- of the thermal regime of polythermal glaciers. Sev- ence and distribution of cold and temperate ice will eral studies of the polythermal structure have been influence the glacier system in a number of ways. made [e.g. Dowdeswell et al., 1984; Blatter, 1987; The existence of both cold and temperate ice in poly- Björnsson et al., 1996], but few of these have em- thermal glaciers may cause zonation in hydrological phasized the spatial distribution and the processes processes, as the presence of cold ice affects water involved in the thermodynamics of the polythermal flow through glaciers by routing or preventing wa- structure. Yet, the thermal conditions of polythermal ter movement [Jansson, 1996; Hodgkins, 1997; Skid- glaciers can have a profound effect on both glacier hy- more and Sharp, 1999; Copland, 2001]. The polyther- drology and dynamics, but also glacial geology pro- mal structure may also be responsible for isolated cesses. Therefore, it is important to gain a better rapid hydrological events, for example, when cold understanding of what influences the extent and dy- glacier margins create dams of subglacial water that namics of the polythermal structure of glaciers in or- eventually are breached, releasing a glacier outburst der to advance the understanding of both hydrolog- flood [Skidmore and Sharp, 1999] or drainage of lakes ical and dynamic processes in polythermal glaciers. dammed by polythermal ice masses [Klingbjer, 2004]. This thesis concerns investigations on the spatial The hydrology of glaciers is closely linked to glacier distribution and stability of the thermal structure of dynamics [Iken and Bindschadler, 1986; Willis, 1995; Storglaciären, northern Sweden, as well as different Jansson, 1996] and the polythermal structure can parameters providing forcing of the extent and stability of the polythermal structure, such as ice flow, lead to a distinct flow regime with unique spatial surface and englacial temperature and water content and seasonal characteristics [Iken, 1974; Andreasen, of the temperate ice. 1985; Rabus and Echelmeyer, 1997]. The unique dynamics of a polythermal glacier, together with the temperature-dependence of the ice viscosity, make Distribution of polythermal glaciers the dynamics of a polythermal ice mass a challenge to understand and model, not only because the in- Polythermal glaciers are common at high latitudes, creased gradients in strain rates and sliding need but are not restricted to these areas. They also exist to be considered, but also because of heat fluxes in other areas with continental climate, for example and phase transitions within the ice mass. [Fowler at high elevations in the European Alps [Haeberli, and Larson, 1978; Hutter, 1982; Blatter and Hutter, 1976] and in the interior of China [Maohuan, 1990]. 1991]. The degree of continentality of the prevailing climate The polythermal temperature regime of glaciers results in many different forms of polythermal struc- typically forms in continental climates, where low tures, which span from almost entirely cold glaciers winter temperatures cause the cold wave to pene- with a finite temperate near-basal layer to predomi- trate deep into the ice. The low mass turnover, typ- nantly temperate glaciers with a thin perennial sur- ical for continental glaciers, inhibits the removal of face layer with cold ice in the ablation area. Figure 1 all cold ice by melting during the melt season. As cli- shows six typical polythermal structures found in val- mate varies, it is likely that the polythermal struc- ley glaciers, which largely originate from different cli- ture is also subject to change [Blatter and Hutter, matic environments [Baranowski, 1977; Blatter and 1991; Krass, 1991; Hodgkins et al., 1999]. Changes Hutter, 1991]. However, local conditions in ice flow in the transition between areas of temperate and and variations in the local climate may also greatly cold bed conditions may be associated with glacier influence the polythermal structure, causing many surges [Hamilton and Dowdeswell, 1996; Jiskoot et intermediate types. al., 2000; Murray et al., 2000], although this may not Figures 1a and b illustrate glaciers under cold be the only controlling factor [Clarke et al., 1984; conditions, where the accumulation areas have neg- Clarke and Blake, 1991]. Changes in the extent of ligible melt and the ice formed is cold. Increased

1 RICKARD PETTERSSON

cumulation areas can extend into warmer climatic a zones, where extensive melting occurs. The meltwater percolates down into the firn pack and refreezes ELA on contact with cold firn. This releases latent heat that warms up the firn pack to temperate conditions. b Thus, both temperate and cold ice are produced and both cold and temperate ice can be present in the ablation area (Figure 1c). The cold part of the accumulation area can also be restricted to the uppermost accumulation area and most of the ice produced c becomes temperate (Figure 1d). These types of polythermal glaciers have been observed in the European Alps where there are accumulation areas at high altitude [Fisher, 1955; Haeberli, 1976; Suter, 2001] but d also on small Greenlandic ice caps [Loewe, 1966]. Figure 1e shows a glacier that is temperate with the exception of a cold surface layer in the ablation area. Polythermal glaciers of this kind are widespread on Svalbard and under more continen- e tal conditions on the eastern side of the Scandi- navian mountains [Schytt, 1968; Dowdeswell et al., 1984; Holmlund and Eriksson, 1989; Björnsson et al., 1996]. When melting starts in spring, the accumulation area quickly becomes temperate from the release f of latent heat when percolating water refreezes. In the ablation area the ice is almost impermeable to water. In spring when the ice surface is cold, the meltwater freezes in contact with the cold ice, all the released latent heat is used to warm the ice surface until it reaches the melting point. Heat conduction into the ice is limited because ice is a poor conductor. In addition, the energy available is limited because the surface temperature cannot rise above the melt- Figure 1. Schematic view of different polyther- ing point. All surplus energy supplied to the surface mal structures (modified after Blatter and Hutter, is used to melt the ice at the surface. The only way to [1991]). The gray color indicates temperate ice. Equi- reduce efficiently the cold surface layer is by surface librium line altitude (ELA) is indicated with a line. ablation. The ice will remain cold throughout the The figure is not intended as attempt to classify year if the melt rate is small. The net ablation in- polythermal glaciers, nor cover all possible types; it creases at lower altitudes, which can cause the entire should only give a sense of the range of polythermal cold layer to be removed, while still present at higher structures encountered in glaciers. elevations (Figure 1f). In this project all the work has been done on Storglaciären in northern Sweden, which has a polythermal structure as in Figure 1e. In strain heating at depth can increase the tempera- further discussions, I will focus on this type of poly- ture to the pressure melting point, causing a tem- thermal glacier, having a cold surface layer in the perate layer near the bed below an otherwise cold ice ablation area, but with a temperate accumulation mass. Such conditions have been observed on several area. glaciers in the Yukon Territory, Canada [Clarke and Goodman, 1975; Clarke et al., 1984], the Canadian Arctic [Müller, 1975; Blatter, 1987; Blatter and Kap- Polythermal conditions for glaciers in penberger, 1988; Copland and Sharp, 2001], Alaska, Scandinavian mountains USA [Rabus and Echelmeyer, 1997], and China [Mao- In the Scandinavian mountains both temperate and huan, 1990]. polythermal glaciers exist because of the strong east- On glaciers with accumulation areas that span west gradient in the climate over the mountain chain. a large elevation interval, the lower part of the ac- The Scandinavian climate is influenced by the pre-

2 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN dominant westerly winds as a consequence of the gen- 2000 eral circulation pattern. The warm Gulf stream in the north Atlantic allows warm and moist air condi- 1600 Elevation (m a. s. l) 70˚N tions to prevail far north along the western Scandina- vian coast [Liljequist, 1970]. When the air masses are 1200 forced over the Scandinavian mountain chain, most of the precipitation is lost on the western side of the 800 mountain chain. The moist and warm conditions on 400 the western side at the mountains favor temperate MMårmaglaciärenårmaglacirmaglaciärenren glaciers. On the eastern side of the mountains precip- StorglaciSStorglaciärentorglaciärenren 0 itation and temperature are generally lower and have stronger seasonal amplitude. The winter temperature 100 km is lower due to larger distance from the ocean and greater outgoing radiation because of the generally lower cloudiness east of the mountains [Liljequist, 1970]. The continental type of climate of the eastern 65˚N side of the mountains with cold winters and relatively low precipitation during the winter months, is favorable for the formation of a cold surface layer of the type in Figure 1e [Schytt, 1968; Holmlund and Eriks- 15˚E 20˚E son, 1989]. The continentality of the climate in the Scandinavian mountains increases towards the east Figure 2. Location of Storglaciären and [Angstr¨˚ om, 1974]. This climate gradient is reflected M˚armaglaciären. in the thickness of the cold surface layer. For example, Storglaciären has a average cold surface layer thickness of 30 m corresponding to 15% of its vol- ter snow cover and by the variations in air tempera- ume, while M˚armaglaciären located ∼50 km east of ture over the year [Goodrich, 1982]. The snow cover Storglaciären (Figure 2) has an average cold surface insulates the ice from the winter cold and a thick layer thickness of 65 m or 40% of its volume [Holm- snow cover tends to give higher temperatures at the lund et al., 1996]. ice-snow interface than the winter air temperatures [Hooke et al., 1983]. However, if the snow cover is >1 m thick, the ice surface temperature becomes ef- Dynamics of the cold surface layer fectively shielded from fluctuations in air temperature and remains nearly constant [Haeberli, 1973]. The thickness of the cold surface layer of a Scandina- In this case, the surface temperature is largely influ- vian polythermal glacier (Figure 1e) is a function of enced by the heat transfer from the underlying ice. the surface energy balance, net ablation at the sur- Mann and Schmidt [2003] showed that about half of face, liquid water content of the ice entering the cold the variance in surface temperature during the cold- layer, as well as the velocity field of the glacier [Blat- season could be explained by variations in snow cover ter and Hutter, 1991]. Figure 3 shows a conceptual thickness and the thermal conditions in the underly- model summarizing these factors and their contri- ing ground, thus causing a feedback mechanism on bution at a point. For simplicity, the model is in a the temperature profile in the ice. steady state with respect to both the glacier geome- The negative temperature gradient through the try and the thickness of cold surface layer. cold surface layer causes heat flow towards the glacier Given an initial state (Figure 3a), the temper- surface and the cold-temperate transition surface ature profile through the cold surface layer is gov- (CTS) to migrate downward by freezing of temperate erned by the ice surface temperature and the ver- ice at the base of the cold surface layer (Figure 3b). tical advection of ice towards the surface. During The displacement rate of the freezing front depends summer the surface temperature cannot rise above on both the amount of released latent heat when the the melting point of ice, since heat condition into temperate ice freezes and the ability to conduct the the ice is limited and all the available energy is used heat away from the freezing front, which is deter- to melt the ice. Thus, the temperature profile in mined by the temperature gradient in the cold ice the ice is largely governed by the low temperatures [Hutter et al., 1990; Blatter and Hutter, 1991]. The during winter. The ice surface temperature during amount of heat released depends on the volume of winter is determined by the thickness of the win- liquid water in the temperate ice that must freeze

3 RICKARD PETTERSSON

Atmospheric Forcing and Ice flow heat conduction a) b) c) d) e) Net mass balance

Cold ice

Deformation

CTS

Temperate ice Vertical Freezing of Horizontal advection temperate ice advection

Figure 3. Conceptual model over the cold surface layer on a Scandinavian type polythermal glacier. The cold-temperate transition surface (CTS) is drawn with an inclined base to indicate that the cold surface layer thickness may vary spatially. when the freezing front migrates. The overall change content and the temperature profile through the cold in position of the CTS is given by the sum of the surface layer. downward migration caused by freezing of temper- The conceptual model presented above depicts ate ice at the base of the cold surface layer and the the important properties providing the forcing of opposite directed ice advection. the cold surface layer thickness of a Scandinavian- During a mass balance year some ice will also be type polythermal glacier. It also shows that the in- lost at the surface, as the negative net mass balance teraction between the different components is com- in the ablation area will cause a thinning of the cold plex. However, the relative importance of the different components that provide forcing of the cold sur- surface layer (Figure 3b). The thickness of the cold face layer thickness is largely unknown, which implies surface layer is in steady state when the overall move- that the response in the cold surface layer thickness ment of the freezing front and the net loss of ice are to spatial or temporal variations in the forcing is not equal in size. fully understood. The glacier flow provides also several additional Considering the fact that the polythermal condi- factors to be considered. The horizontal velocity tions and changes in the thermal regime may have component advects ice laterally into the area that a profound effect on the glacier hydrology and dy- may change the cold surface layer thickness from re- namics as well as glacial geological processes, it is gional variations in layer thickness (Figure 3c). The therefore important to gain a better understanding importance of this factor depends on how fast the of what influences the extent and dynamics of the position of the CTS can adapt to new conditions. polythermal structure of glaciers. Vertical strain (usually extensional from the com- pressional horizontal strains typically found in the ablation area), will change the thickness further (Fig- Field site ure 3d). Storglaciären in northern Sweden (67◦55’N, 18◦35’E, If the glacier is in steady state, the vertical advec- Figure 2), is a small polythermal valley glacier. Stor- tion of ice will also equal the amount of net ablation glaciären has a long record of mass balance, ice flow, at the surface (Figure 3e). Note that it is not nec- and hydrology studies [e.g. Holmlund et al., 1996; essary that the glacier itself is in steady state for Jansson, 1996]. In addition, detailed information of the cold surface layer to be in steady state and vice the overall geometry is available from several topo- versa. This is because the stability of the cold surface graphic surface maps [Holmlund, 1996] and a bed to- layer depends on the ice advection and net ablation pographic map from radio-echo soundings and bore- at the surface, as well as on the downward migration hole drilling [Hertzfeld et al., 1993]. The polyther- of the freezing front, which is controlled by the water mal structure of the glacier was verified by Schytt

4 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN

[1968] who measured ice temperatures at a point in rect method that gives an accurate description of the the center of the ablation area down to a depth of thermal regime is to measure the temperature in situ 30 m. Ice temperatures of –2◦C at 16 m depth and using temperature sensors [e.g. Paterson, 1971; Har- –1◦C at 28 m depth were found. Hooke et al. [1983] rison, 1972 ; Müller, 1975; Hooke et al., 1983; Clarke measured negative temperatures throughout the year et al., 1984; Blatter and Kappenberger, 1988; Jania in 16 shallow boreholes distributed over the abla- et al., 1996]. However, temperature measurements tion area. Holmlund and Eriksson [1989] mapped the are point measurements. To describe the spatial pat- thermal regime with ground-penetrating radar and tern over a larger area would require a similarly large detected a cold surface layer over the entire abla- number of such measurements well distributed over tion area, with a thickness of between 20 to 60 m. In the area. This is time consuming, requires access to addition, they could prove the absence of a cold sur- many boreholes for the temperature sensors, and the face layer in the accumulation area, except in an area use of a large number of sensors. A more qualitative dominated by the accumulation of superimposed ice. assessment of the thermal regime of a glacier can be They further showed that the cold surface layer was derived from the surface drainage pattern and glacier thickest along the margins of the glacier. hydrology [e.g. Liestøl, 1988; Hodgkins, 1997; Hodson This wealth of background information makes and Ferguson, 1999]. However, this only provides an Storglaciären a good choice for further detailed stud- estimate of the thermal regime of the whole glacier. ies of the polythermal structure. In addition, the Ground-penetrating radar (GPR) also provides a nearby Tarfala Research Station provides supporting method to estimate the distribution of temperate infrastructure with technical and logistic advantages; and cold ice within an ice mass. Basically, the GPR from Tarfala the glacier is also easily accessible both is operated from the glacier surface and transmits winter and summer for fieldwork. an electromagnetic wave that travels through the ice and reflects at inhomogeneities in electrical properties within the ice or at the ice-bed interface. Some of Objectives the reflected energy travels back to the surface where the returned signal is registered by the receiver. The In this thesis, I have focused on the dynamics of the large gradient in electrical properties between cold cold surface layer of Storglaciären. The objective has ice lacking free water and temperate ice containing a been to improve our understanding of the mecha- fraction of free water creates strong scattering of the nisms controlling the thickness and stability of the GPR wave, which is easily detected by the GPR. The cold surface layer. More specifically, I have focused depth to the reflecting features is calculated from the on the following: time it takes for the reflected radar pulse to return to 1. To detect and estimate any changes in the poly- the receiver at the surface, as well as the propagation thermal structure of Storglaciären. speed of the radar signal. An overview of the electromagnetic wave propagation and reflection is given in 2. To investigate the properties controlling the Appendix A. The strong reflections from liquid water thickness of the cold surface layer, primarily the within the ice allow us to map the extent of temper- water content of the temperate ice below the ate ice. The advantage of the GPR method is that cold surface layer, which is the least investigated it is possible to cover a large area with continuous and constrained controlling factor for the poly- profiles in a short time. The main tool in investigat- thermal regime. ing the spatial variability in this study has been the 3. To study the sensitivity of the properties con- GPR technique. trolling the thickness of the cold surface layer in order to gain a better understanding of the processes involved. Detection of CTS with ground Even though the processes studied are specific for the penetrating radar type of polythermal glaciers depicted in Figure 1e, many of the mechanisms are directly applicable to The basis for GPR detection of the CTS is the reflec- other types of polythermal glaciers. tion of the GPR signal from the liquid water within the ice. However, the method only gives information about the distribution of liquid water in the ice and Fieldwork and methods not the actual temperature of the ice. To overcome this disadvantage GPR surveys can be combined There are several ways to gain information about the with direct temperature measurements in boreholes thermal regime of polythermal glaciers. The most di- by using these temperature measurements as cali-

5 RICKARD PETTERSSON

Temperature (˚C) a strong decrease in scattering eﬃciency when the –8 –4 0 wavelength of the GPR signal becomes much larger 0 than the scattering object, while the scattering ef-

10 ficiency is invariable for sizes comparable or larger Depth (m) than the wavelength [Bohren and Huffman, 1998]. 20 The results show that the CTS reflection increases 30 in depth up to a few meters at frequencies of 130– 180 MHz compared with results at higher frequen- 40 cies, but without significant loss of returned power. 50 This suggests that the scatterers in the upper part of the temperate ice are sub-decimeter in scale or ~700m much smaller than the wavelength at 130–180 MHz, but that the scatterers become comparable in size Figure 4. Comparison between temperature mea- with the wavelength farther down in the temperate surement and the CTS reflection in radargram. The ice and thus cause scattering. Furthermore, this also arrows above the radargram show the location of the indicates that even though the CTS can be detected thermistor string. at frequencies lower than ∼200 MHz it is not certain that the physical CTS is detected accurately. bration for the GPR [Björnsson et al., 1996; Jania et al., 1996; Ødeg˚ard et al., 1997; Moore et al., 1999]. Mapping of the cold surface layer In earlier studies, poor depth resolution (>3m)in the temperate measurements resulted in large uncer- In Paper I, the cold surface layer was mapped with tainties in the agreement between the radar-inferred a continuous-wave stepped-frequency radar [Hamran CTS reflection and the physical CTS. In paper I, we and Aarholt, 1993] using a frequency of 700–900 MHz compare densely spaced temperature measurements and 320–370 MHz. An overview for the mode of op- (0.5 m) across the CTS with GPR surveys (Figure 4). eration and data processing of frequency-modulated The agreement between the melting point and the radars is given in Appendix B. Figure 5 shows a typ- CTS reflection in the radargram is ±1 m, thus indi- ical GPR profile on Storglaciären. The GPR system cating that GPR surveys are a good tool for mapping was mounted on a sled towed by a snowmobile (Fig- the polythermal regime in glaciers. ure 6). The travel speed was kept as low as possible The reflection in the radargram that indicates the given the snow conditions and snowmobile function- CTS is caused by scattering from small water voids ality in order to decrease the effect of spatial aliasing present in the temperate ice beneath the cold sur- along the GPR profile. Spatial aliasing results from face layer. Watts and England [1976] showed that the insufficient spatial sampling to resolve the reflection scattering from these voids depends on frequency and with high accuracy and is closely linked to the GPR that the scattering diminishes rapidly at low frequen- signal wavelength in the ice. In order to avoid spa- cies. Ødeg˚ard et al. [1997] compared the detection of tial aliasing, traces within the GPR profile should the CTS at different GPR-frequency bands between be less than one quarter-wavelength apart [Yilmaz, 30 and 1000 MHz and could not detect any change in 2001]. In my case, the radar system usually records the detection of the CTS at frequencies above 320– 2 traces s−1, an average traveling speed of ∼1.5 m s−1 370 MHz. However, at a frequency of 30–80 MHz the givesadistanceof0.75mbetweentraces,whichis CTS reflection was lost or difficult to interpret. This larger than 5.3×10−2 morλ/4 at 700–900 MHz. To suggests that the choice of frequency may influence avoid spatial aliasing, the sampling rate must be in- the accuracy in the mapping of the CTS. creased to ∼30 traces s−1 or the traveling speed re- In paper III, I extend the work of Ødeg˚ard et al. duced to ∼0.1 m s−1. This is nearly impossible to [1997] by using surveys of the CTS at different center achieve in practice. However, the aim of the map- frequencies between 130 and 1300 MHz, combining ping of the CTS is not to resolve small-scale features this with theories of electromagnetic scattering from in the CTS, but rather to depict the larger trends small spherical objects [e.g. Bohren and Huffman, in the CTS. For this purpose, the chosen sampling 1998]. The aim is not only to investigate the fre- interval is sufficient as long as the CTS is not steeply quency dependence of the detection accuracy of the dipping [Daniels, 1996]. CTS, but also to gain information about the order of The sampling criterion used to avoid aliasing is magnitude in size of the voids causing the CTS re- also valid for three-dimensional GPR surveys, ad- flection. Electromagnetic scattering theory predicts vocating lateral spacing similar to the along profile

6 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN

0 10

20 Intensity (dB) Depth (m) 40

60 80 0 400 800 1200 1600 2000 Distance (m)

Figure 5. GPR profile along the centerline on Storglaciären. The profile starts just above the equilibrium line and follows the glacier centerline down to the terminus. The CTS is visible as an increase in scattering along the profile. At a distance of ∼1900 m, the cold surface layer reaches the bed. An interesting feature in the radargram is the strong convex reflections down glacier from the equilibrium line ending abruptly at the CTS reflection. These are interpreted as ice structures with higher water content. spacing of λ/4[Yilmaz, 2001]. In my case, a large set ±10 mm+1 ppm of the distance between the base of profiles were collected to give good spatial cov- station and the point to be determined. The max- erage of the glacier and the distances between pro- imum distance in our case was 5 km, thus giving files were, on average, 150 m. This distance could an accuracy of ±15 mm. The GPS stores the posi- certainly give rise to spatial aliasing even for larger tion each 10 s, thus every 20th trace is accurately trends in the CTS. To test the effect of this large positioned (at a sampling rate of 2 traces s−1). The distance between the profiles, the CTS were mapped positioning of the individual traces between the GPS using a denser set of profiles in a small area and then positions was determined by linear interpolation. Er- compared with the spatially more coarsely-mapped rors in the synchronization between the GPS and CTS. Figure 7 shows the CTS mapped with pro- GPR and the interpolation introduce additional un- files separated by ∼150 m and ∼5 m. They show certainties to the positioning of each trace, resulting very close similarities and most of the details are de- in a total positioning error of ∼1 m. The depth to the picted using the more sparsely-spaced profiles. Spar- CTS reflection in each of the profiles was interpreted ely spaced profiles will not prevent that spatial alias- and digitalized manually. Combining the position of ing from occurring, but the test shows that major the GPR traces and the digitalized depth of the CTS trends in the CTS are resolved as long there is no makes it possible to produce a map of the cold sur- dramatic change in CTS depth. face layer thickness. The positioning of the individual profiles was By nature, the collected data have an irregularly made with a global positioning system (GPS) spatial distribution, dominantly along profiles on the mounted on the snowmobile. The collected posi- glacier. Thus, the raw data needs to be interpolated tions were adjusted for uncertainties in the GPS to a grid of data to produce a digital map of the system by using differential correction. The correc- CTS. In all papers, the maps of the cold surface tion is calculated from simultaneous collected data layer thickness have been interpolated from the raw at a base station with known coordinates. In simple data set using the kriging-interpolating method [e.g. terms, the difference between the estimated position Isaaks and Srivastava, 1989]. Kriging is commonly and the known coordinates is used is used to cor- used in geology, hydrogeology and hydrology applica- rect the unknown positions. The accuracy becomes tions [Delhomme, 1979; Villeneuve et al., 1979; Jour-

7 RICKARD PETTERSSON

Figure 6. GPR equipment mounted on a snowmobile sledge. The DGPS is mounted on a pole at the back of the snowmobile. Photo: P. Jansson. nel and Huijbregts, 1984], but is also used to inter- tance to the prescribed grid node to be estimated. By polate glacier bed topography [e.g. Hertzfeld et al., summing up all weighted variables that influence the 1993; Flowers and Clarke, 1999]. Kriging is a statisti- grid node, we obtain the variable value at the spe- cally robust method because it is unbiased and does cific grid node. Thus, the value of each grid node is not alter the data set mean or change the data dis- estimated from the sum of weighted measured points tribution [Kitanidis, 1997]. It is an estimation pro- in the neighbourhood of the node. The result of the cedure to determine unknown values at prescribed kriging procedure is a regular spaced grid with in- points in a regular grid from spatially irregular val- terpolated values that can be turned into contoured ues and a semivariogram model. The semivariogram thickness maps. is a measure of how data on average changes as a The survey of the cold surface layer in the ab- function of distance from the estimation point. The lation area on Storglaciären shows a cold surface underlying principle is that, on average, two closely layer with an average thickness of 31 m, with a much spaced observations are more similar than two ob- thicker cold layer along the margins and in an area servations farther apart. The proper semivariogram close to the equilibrium line. A maximum depth of model is chosen from constructing an experimental the cold surface layer of 65 m was recorded along semivariogram from the sample points and obtaining the southern margin of the glacier. The area with the best fit of different mathematically simple semi- thinnest cold surface layer coincides with an area variogram models. It is important to find the proper where the glacier flows over a bedrock threshold and values for the different parameters in the modelled just south of the kinematic centerline. This pattern semivariogram, since different parameter sets may agrees well with an earlier survey of the cold surface have substantial impact on the spatial distribution layer performed by Holmlund and Eriksson [1989]. of the interpolated dataset [e.g. Hock and Jensen, In Paper IV, we investigate the processes involved 1999]. Based on the adopted semivariogram model, and the causes of the observed spatial pattern. The each sample point is weighted depending on the dis- results show that the cold surface layer thickness

8 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN

1991; Hutter et al., 1988]. The water content in the 30 ice entering the cold surface layer inﬂuences the freez- 30 ing rate at the base of the cold surface layer and thus the new formation of the cold surface layer. However, 10 the knowledge of the water content in temperate ice is limited, often restricted to few observations, and

Cold surface layer thickness (m) the estimated values often show a large variation [Lli- boutry and Duval, 1985; Moore et al., 1999; Murray et al., 2000]. In paper II, we investigate the spatial dis- 30 tribution of water content just below the CTS. The back-scattered energy of GPR signals at the CTS de-

30 pends on the amount of water present. However, this only provides relative values and the determinations 50 of water contents must be calibrated. We determine the absolute water content at a few calibration points

30 by using a calorimetric method based on estimating 30 the rate of freezing of temperate ice, which is directly related to the amount of liquid water [Hutter 70 et al., 1990]. The rate of freezing at the CTS is estimated from the displacement of the freezing front 50 50 N along thermistor strings installed across the cold- 100 m temperate transition. The results show that there is a distinct difference in water content in a ∼100 m wide area across the glacier with higher values north of the Figure 7. Maps of the cold surface layer using differ- glacier centerline and lower to the area south. This ent spacing between GPR profiles. The dashed lines difference may influence the spatial pattern of the indicate the position of individual GPR profiles. cold surface layer, since lower water content could result in a faster downward migration of the CTS and, hence, a thicker cold surface layer. However, com- shows close similarities with the spatial distribution paring the cold surface layer thickness map with the of net ablation and emergence velocity. The cold sur- spatial distribution of water content reveals that thin face layer becomes thin in areas with strong negative cold surface layer coincide with areas of low water net ablation and high emergence velocities. As stated content south of the centerline. Thus, we conclude earlier, the thickness of the cold surface layer depends that the spatial variations in water content are not on the balance between removal of cold ice at the sur- a limiting factor for the spatial pattern of the cold face and the freezing of temperate ice at the base of surface thickness (Paper IV). the cold surface layer. Net loss of ice at the surface is the only efficient way to thin the cold surface layer; a high negative net ablation should thus result in Thinning and sensitivity of the cold a shallow CTS depth. High emergence velocity and surface layer strong negative net mass balance causes a weak temperature gradient close to the CTS and hence limits The cold surface layer on Storglaciären was mapped the heat transport from the freezing front. In turn, already in 1989 by Holmlund and Eriksson [1989]. the lower heat flux slows the downward migration of Paper I presents a comparison between the GPR the freezing front. Thus, a high removal of cold ice at surveys of the cold surface layer on Storglaciären, the surface, in combination with reduced downward separated by 12 years. The comparison shows a con- migration of the freezing front, results in a thinner siderable thinning of the cold surface layer, although cold surface layer. the thinning is not uniform. The parameters providing forcing of the cold surface layer are winter air temperature, mass balance, ice flow, and the water Water content at the CTS content in the temperate ice (Figure 3). Variation in time of these parameters reflects changes of the cold An important forcing factor that influences the ther- surface layer thickness. mal regime of polythermal glaciers is the liquid wa- Depending on the response time for a perturba- ter content in the temperate ice [Blatter and Hutter, tion in the forcing, the observed thinning could be

9 RICKARD PETTERSSON Difference in CTS depth (m) Local acc. area

0 1976–1988 2

0 –1 –5

4 Net balance (m w.eq) 3 –10 2 N 500 m 1 0 1989–2001 –1

Figure 8. Comparison of GPR surveys of the cold 2 –2 surface layer made between 1999 and 2001. The hatched pattern indicates where the thinning is 1 0 –1 –2 within the uncertainty limits of the comparison. The large diﬀerences near the margins are probably arti- facts due to poorly overlapping data and diﬃculties in the interpolation of the data. the result of a relatively recent change or part of

a long-term trend in the forcing of the cold surface Difference (m w.eq) layer. Figure 8 shows a comparison of surveys of the 1 0.5 cold surface layer between 1999 and 2001, using the same methodology as in paper I. Although there are 0 0 large areas lying within the uncertainty limits of the 0 comparison, the results show that the thinning of the cold surface layer is an ongoing process. In paper IV, –1 we use a simple one-dimensional thermomechanical N model of the cold surface layer to estimate the re- 1000 m sponse time for the cold surface layer to reach a new steady state following a perturbation in the forcing of Figure 9. Average mass balance for the periods the thickness. The results indicate that the response 1976–1988 and 1989–2001 for Storglaciären (Data time is on a decadal scale. Thus, the perturbation from Tarfala Research Station). The average differ- in the forcing that caused the thinning probably oc- ence is 0.15 m w.eq., thus indicating that the glacier curred during the last decades. is close to steady state. The cause of the thinning of the cold surface layer is investigated in paper IV by examining data available from the last decades on mass balance, ice minus area in 2001 (Figure 10). However, for ma- flow, and air temperature. The mass balance mea- jority of the emergence velocity along the centerline surements indicate that the glacier has been close shows no large differences in estimated emergence ve- to steady state during the last decades [Holmlund, locity between these periods along most of the cen- 1988; Holmlund and Jansson, 1999]. Figure 9 shows terline (P. Jansson et al., manuscript in prep., 2004). the average net mass balance for the periods 1976– This infers that the thinning of the cold surface layer 1988 and 1989–2001 as well as the difference between may not be the consequence of a considerable change the periods. The differences in net balance and abla- in flow regime of the glacier. Figure 11 shows the tion are small and thus cannot explain the observed monthly mean temperature from Tarfala Research thinning of the cold surface layer. Measurements of Station, which is located <3 km from Storglaciären, the ice velocity along the centerline on Storglaciären for the period 1965–2002. The recorded tempera- made in 1965–1967 [Schytt, 1968], 1983–1985 [Hooke ture show a slight decrease in summer temperatures et al., 1989] and data from 2001 (Paper II and IV) throughout the whole period, but a significant in- indicate an increase in emergence velocity in the ter- crease in winter temperatures of about 1◦C occur-

10 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN

10 ) 2 –1 2001 1983 0

–10 Temperature (˚C) –2 Emergence velocity (m a –20 1970 1980 1990 2000 0 1000 2000 Time (a) Distance from terminus (m) Figure 11. Monthly mean temperature at Tarfala Figure 10. The emergence velocity along the cen- Research Station. The dashed line and dotted line terline of Storglaci¨aren. The dashed line and solid show the 5-year running average of the summer line show the emergence velocity estimated from ice temperature (June–August) and winter temperature velocity measurements made in 1983 [Hooke et al., (October–April), respectively. Solid line shows the 1989] and 2001, respectively. 5-year running average of the annual mean temperature (Data from Tarfala Research Station). ring in the mid 1980s [Grudd and Schneider, 1996].

This increase in air temperature, possible in com- 2 bination with slight increase in the insulating snow with temperate bed conditions by about 0.1 km is cover thickness may have reduced the cooling effect probably negligible for the overall erosional capacity on the ice and may have altered the temperature of Storglaciären. distribution in the cold surface layer. A weaker tem- The small reduction in the frozen rim on Stor- perature gradient of the cold surface layer would lead glaciären, when compared to the relatively large to a slowing down of the freezing rate at the CTS. change in the cold surface layer can be explained by Thus, the observed thinning of the cold surface layer the bedrock topography beneath the glacier. Stor- on Storglaciären may have been caused by a stronger glaciären occupies a valley with steep sidewalls (Fig- imbalance between the freezing at the base of the ure 12a) and has a relatively thin cold surface layer. cold surface layer and the in general unchanged net A thinning of the cold surface layer would lead to rel- loss of cold ice at the surface. atively small changes in the contact area between the cold surface layer and the substratum. For a glacier having a cold surface layer of the same thickness An implication of the cold surface but occupying a valley with shallower angle walls, layer thinning a similar decrease in the thickness of the cold surface layer would result in larger areas becoming tem- The erosive capacity of a glacier depends on the perate (Figure 12b). Assuming the same change in temperature conditions at the bed. Under temperate cold surface layer but a larger total thickness of the conditions, the presence of meltwater favours sliding, cold surface layer would lead to increased areas that which enhances abrasion and plucking. Under cold become temperate, since the slope of sidewalls de- conditions, some sliding and erosion can occur, but creases towards the valley bottom (Figure 12c). the effectiveness is reduced considerable [Echelmeyer M˚armaglaciären, about 50 km northeast of Stor- and Zhongxiang, 1987; Cuffey et al., 2000]. The ob- glaciären (Figure 2), occupies a wider valley and served thinning of the cold surface layer on Stor- has a thicker cold surface layer than Storglaciären glaciären over the last 12 years has led to a decrease [Holmlund et al., 1996]. Since the thinning of the in size of the area exhibiting cold bed conditions at cold surface layer on Storglaciären is suggested to the margins of the glacier (Paper I, Figure 5), which be caused by an increased winter temperature, we could lead to an increased erosion of the substratum. could speculate that this increase in air temperature However, the areas that have become temperate are has occurred on a regional scale and that a simi- small, about 0.1 km2. On Storglaciären, about 85%, lar decrease of the cold surface layer has occurred or 2.7 km2 of the total area, has temperate bed condi- on M˚armaglaciären. Such decrease in cold surface tions [Holmlund et al., 1996]. An increase in the areas layer thickness would probably result in much larger

11 RICKARD PETTERSSON

a) b) c)

⌬a

Figure 12. Conceptual view of changes in extent of the bottom frozen margins of a polythermal glacier when the cold surface layer decreases in thickness. The change is small for a glacier occupying a narrow valley (Figure 12a) compared with when the glacier occupies a wider valley (Figure 12b). The change of the frozen rim becomes even larger if the cold surface layer is thick (Figure 12c). basal areas of M˚armaglaciären becoming temperate with the thinnest cold surface layer exhibit very small and thus the impact on the glacier system and the changes. erosional capacity of the glacier would also be much The stability of the cold surface layer depends on larger. the net ice ablation at the surface and the downward migration of CTS, which are controlled by the temperature gradient through the cold surface layer and Summary of papers the water content in the temperate ice arriving at CTS. Annual mass balance measurements show no The following is a short summary of each paper in evidence of any substantial increased net ablation the thesis. The papers are arranged in chronological between the survey dates that could explain the ob- order. served thinning. However, small increase in average winter air-temperature, a limiting factor for the tem- Paper I perature gradient through the cold surface layer, may Pettersson, R., P. Jansson, and P. Holmlund, provide a partial explanation. The weaker tempera- 2003. Cold surface layer thinning on Storglac- ture gradient reduces the transport of latent heat iären, Sweden, observed by repeated ground from the CTS, thus slowing down its downward mi- penetrating radar surveys. Journal of Geophys- gration. ical Research, 108 (F1), 6004, doi: 10.1029/ 2003JF000024. Paper II In this study, we compare two GPR surveys of the Pettersson, R., P. Jansson, and H. Blatter, 2004. cold surface layer on Storglaciären in order to in- Spatial variability in water content at the cold- vestigate the possibility of detecting changes in the temperate transition surface of the polythermal Storglaciären, Sweden. Accepted for pub- thickness of the cold surface layer. The first survey lication in Journal of Geophysical Research. was made in 1989 and the second in 2001 and covers a large part of the ablation area. We also compare This paper presents a method for estimating the spa- GPR profiles at two center frequencies with detailed tial distribution of absolute water content in the tem- temperature measurements in bore holes and show perate ice just beneath the cold surface layer. This that the CTS can be detected and mapped with an estimate is made by combining estimates of relative accuracy of about ∼1 m at both frequencies. A com- water content from GPR backscatter strength with parison between these two GPR surveys of the cold absolute determinations from temperature measure- surface layer shows a substantial and complex thin- ments and the boundary condition at the freezing ning of the cold layer. An overall decrease of 8.3 m or front. The estimated distribution and the absolute 22% of average thickness of the CTS depth is much values of the water content provide insight about the larger than uncertainties in CTS depth determina- origin of the liquid water within the temperate ice. tions. The largest thinning seems to have occurred The relative water content along the CTS is esti- in areas with thick cold surface layer, while areas mated from 18 densely space GPR profiles in an area

12 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN of about 100×600 m located close to the equilibrium temperate ice indicating that the mean radii of the line on Storglaciären. Three of the profiles crossed voids causing the cold-temperate transition would be boreholes equipped with thermistors for estimating on a decimeter scale if they were spherical in shape. values of absolute water content, which provide cal- The spatial variation in CTS-depth difference be- ibration points for the relative values. The results tween 155 MHz and higher frequencies may rise from show a water content of 0.80%, 0.75% and 0.58% at spatial variations in void size or number density of three calibration points and a mean water content scatterers. of 0.8% with a standard deviation of ±0.26% for the extrapolated water content. The spatial distribution Paper IV of the water content shows a distinct pattern with Pettersson, R., P. Jansson, H. Blatter, and low and higher water content on either side of the H. Huwald 2004. Spatial pattern and stability glacier centerline, respectively. of the cold surface layer on Storglaciären, Swe- The water content in temperate ice may have four den. Manuscript. possible origins; strain heating due to deformation of the ice; hydrostatic pressure changes; input of water In this paper, we have investigated the sensitivity at the surface; and water originating from the firn- and the causes to an observed thinning of the cold ice transition in the accumulation area. A rough es- surface layer on Storglaciären, northern Sweden, be- timate of the strain heating indicates that it would tween 1989 and 2001 using a one-dimensional ther- be insufficient to explain the observed water content. momechanic model and measurements of ice surface Hydrostatic pressure changes could contribute to the temperatures, vertical velocities, and net balance dis- water content only if large differences in hydrostatic tributed over the ablation area. Spatial resemblance pressure exist between the point of ice formation and between pattern of the cold surface layer, net balance the point of water content observation. Furthermore, and emergence velocity together with high sensitivity on polythermal glaciers the water input from the sur- to different emergence velocities in the model suggest face in the ablation area is estimated to be small due that the net ablation and vertical ice advection is the to the subzero temperatures at the surface. We hy- dominant forcing for the spatial pattern of the cold pothesize that the differing water content is due to surface layer. Transient modeling results suggest that that ice on either side of the centerline originates the cold surface layer adapts to a new equilibrium af- from different cirques, thus implying spatial varia- ter a perturbation in the forcing on a few decades. tions in the entrapment of water in the firn-ice tran- No significant change in ice flow or mass balance, sition process. both measured on the glacier, has been observed at Storglaciären the last decades. Instead, an increase ◦ Paper III of 1 C in winter air temperature since the mid-1980s is probably the cause for the observed thinning of the Pettersson, R., 2004. Frequency dependence in detection of the cold-temperate transition surface cold surface layer. Increased winter temperatures at in a polythermal glacier. Submitted to Radio the ice surface would result in a slow-down in forma- Science. tion of cold ice at the base of the cold surface layer and lead to a larger imbalance between net loss of In this study, I investigate the properties of the wa- ice at the surface and freezing of temperate ice at ter inclusions that cause the CTS reflection by con- the cold-temperate transition. The model results in- ducting repeated GPR surveys at different frequen- dicate that the cold surface layer thins more in areas cies. The theory of the scattering of electromagnetic with lower emergence velocity at an increased ice sur- waves from small particles indicates a strong decrease face temperature, which partly explains the observed in scattering efficiency from scatterers that are much complex thinning pattern of the cold surface layer. smaller than the wavelength of the incident wave. Hence, by changing the wavelength of the GPR signal and studying changes in scattering from the CTS Conclusion can yield an order of magnitude estimation of the scattering void sizes. My comparison of radar surveys GPR provides an excellent tool for mapping the of the cold-temperate transition at different frequen- distribution of cold and temperate ice in polyther- cies shows an increase in depth to the cold-temperate mal glaciers. Comparing direct temperature mea- transition reflection at a center frequency of 155 MHz surements in boreholes and GPR surveys of the CTS compared to the CTS depth detected at frequencies suggests that the radar-inferred CTS is within ±1m above 345 MHz. This is interpreted as a reduction of from the melting point of ice at frequencies 320– scattering efficiency from the uppermost part of the 370 MHz. At frequencies below 155 MHz center-

13 RICKARD PETTERSSON frequency, the accuracy degrades because of reduced surface layer thickness at different vertical advection scattering efficiency from the CTS. From the theory rates. of electromagnetic scattering from spherical objects, A comparison of the two detailed surveys of the it is possible to show that this reduction in scattering cold surface layer thickness on Storglaciären shows a efficiency is a result of the water voids that cause the considerable thinning of the cold surface layer dur- CTS reflection becoming very small compared to the ing the 12 years period spanning the surveys. An GPR wavelength. This suggests that the scatterer average decrease of 8.3 m has occurred over the sur- causing the CTS reflection would be between 0.1 and veyed area, which corresponds to a decrease of 22% of 0.05 m assuming that the scatterers are spherical in the average thickness. However, the thinning has not shape. However, the voids in glacier ice may have been uniform over the studied area. Areas with both other shapes, which can influence the scattering effi- considerable thinning, with a maximum of about ciency considerably. Thus, we cannot unequivocally 20 m are recorded, as well as areas with no significant establish the exact size of the scatterers at the CTS, thinning. The areas with significant thinning of the but indicate the order of magnitude of <0.1 m for cold surface layer coincide with areas of low verti- the mean size of the scatterers causing the CTS re- cal advection, although the agreement is not perfect. flection. Modeling results of the cold surface layer thickness A controlling factor for the thickness of the cold forcing indicates that a perturbation in surface tem- surface layer is the water content beneath the cold perature or water content would give a considerably surface layer. This is the least investigated and con- larger change in CTS depth at low vertical advection strained quantity that is relevant for the thermal compared to higher advection. This suggests that ar- regime of glaciers, but also the most difficult to as- eas with low advection would thin more as compared sess. Spatial variations of absolute water content in with areas with high advection, thus explaining much the temperate ice immediately below the cold surface of the observed thinning pattern. layer are determined by combining relative estimates Sensitivity studies of the cold surface layer, using of water content from GPR data with absolute de- a one-dimensional thermomechanical model, suggest terminations from temperature measurement across that the perturbation in the forcing of the cold sur- the CTS and the thermal boundary condition at the face layer causing the thinning of the cold surface freezing front. The results show a distinct difference layer must have occurred during the last decades. An in absolute water content on either side of the kine- examination of ice velocity measurements and mass matic centerline of the glacier. This difference be- balance on Storglaciären during the last decades does tween either side of the centerline is suggested to not indicate any large changes. It is possible that be caused by ice originating in different cirques with changes in water content of the temperate ice ar- different amounts of entrapped water at the ice-firn riving at the CTS could have caused the thinning transition. The entrapment of water at the ice-firn of the cold surface layer, although evidence for such transition is considered the most important source changes is not available. An increase in winter air of liquid water in temperate ice below the cold sur- temperatures of about 1◦C recorded at Tarfala Re- face layer in the upper ablation area of Storglaciären. search Station since the mid-1980s indicates that the Contributions from other sources, such as pressure cold surface layer thinning can be the result of in- changes and strain heating are small, while any con- creased winter surface temperature over the glacier. tribution from percolating surface water probably is non-existent. The cold surface layer of Storglaciären, mapped Acknowledgements in 2001, has an average thickness of 31 m, with a complex spatial pattern. The spatial distribution of I am grateful to my supervisor Per Holmlund for in- winter surface temperature and water content of the troducing me to glaciology, for all the help and sup- temperate ice arriving at the CTS cannot explain port whenever I needed it, and for critically review- the variability in cold surface layer thickness. A thin ing my manuscripts. I appreciate the freedom you cold surface layer is found in areas with a higher have given me during my work, but at the same time negative net mass balance and high emergence ve- keeping me on track. I also want to thank my other locities. Large negative net mass balance thins the supervisors Peter Jansson, Regine Hock, and Jens- cold surface layer efficiently and high vertical advec- Ove Näslund. Peter for being an excellent supervisor tion affects the heat transport from the freezing front and friend, always open-minded and willing to dis- and hence new formation of cold ice. This is further cuss scientific and practical topics. Regine and Jens- supported by thermomechanical modeling of the cold Ove for constructive criticism on my manuscripts and surface layer, which shows large variations of the cold productive discussions. I am also deeply in debt to

14 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN all who have helped me with fieldwork, sharing data, Clarke, G.K.C., and R.H. Goodman, Radio echo sound- reading manuscripts or just moral support. Although ings and ice-temperature measurements in a surge- type glacier, J. Glaciol., 14 (70), 71–78, 1975. you are not mentioned by name, I am very grateful Copland, L., Polythermal glacier hydrology and ice flow for all your help and support. dynamics, Ph.D. thesis, Univ. of Alberta, Edmonton, I would like to thank my Ann-Marie for putting 2001. up with me during the final stage of preparation of Copland, L., and M. Sharp, Mapping thermal and hydrological conditions beneath a polythermal glacier with the thesis. Your encouragement and support when radio-echo sounding, J. Glaciol., 47 (157), 232–242, nothing seemed to work was outstanding. Finally, I 2001. am grateful to my parents who have always believed Cuffey, K.M., H. Conway, A.M. Gades, B. Hallet, R. in me and encouraged my curiosity. Without their Lorrain, J.P. Severinghaus, E.J. Steig, B. Vaughn, and J.W.C. White, Entrainment at cold glacier beds, support, this thesis had not come true. Is it possible Geol., 28 (4), 351–354, 2000. to have better parents? I doubt it. Daniels, D.J., Surface penetrating radar, 300 pp., The Financial support for the first three years of institution of Electrical Engingeers (IEE), London, this study was received from Climate Impacts Re- 1988. search Centre, supported by EU structural funds. Daniels, D.J., Surface-penetrating radar, IEE. Elec. Communicat. eng. J.,Aug., 165–181, 1996. The remaining part was funded by the Department Daniels, D.J., D.J. Gunton, and H.F. Scott, Introduction of Physical Geography and Quaternary Geology of to subsurface radar, IEE Proc., 125, F(4), 278–320, Stockholm University. 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Angstr¨˚ om, A., Sveriges klimat, 157 pp., Generalstabens and Technol., Trondheim, 2000. litografiska anstalt, Stockholm, 1974. Faulconbridge, I., Radar fundamentals, 276 pp., Argos Baranowski, S., The subpolar glaciers of Spitsbergen seen Press, Canberra, 2002. against the climate of this region, pp. 110, Acta uni- Fisher, J.E., Internal temperatures of a cold glacier and versitatis Wratislaviensis, 1977. conclusions therefrom, J. Glaciol, 2 (18), 583–591, Björnsson, H., Y. Gjessing, S.-E. Hamran, J.O. Hagen, 1955. O. Liestøl, F. Pálsson, and B. Erlingsson, The ther- Flowers, G.E., and G.K.C. Clarke, Surface and bed mal regime of sub-polar glaciers mapped by multi- topography of Trapridge glacier, Yukon territory, frequency radio-echo sounding, J. Glaciol., 42 (140), Canada: Digital elevation models and derived hy- 23–32, 1996. draulic geometry, J. Glaciol., 45 (149), 165–174, 1999. Blatter, H., On the thermal regime of an arctic valley Fowler, A.C., and D.A. 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Smith, Antennas in matter, 868 Holmlund, P., Maps of Storglaciären and their use in pp., MIT Press, Cambridge, Massachusetts, 1981. glacier monitoring studies (incl. 2 maps of the glaciers Kitanidis, P.K., Introduction to geostatistics, 249 pp., in the Tarfala valley at the scale 1:10 000), Geogr. Cambridge Univ. Press, Cambridge, 1997. Ann., 78A(2–3), 193–196, 1996. Klingbjer, P., Recurring jökulhlaups in Sälka, northern Holmlund, P., and M. Eriksson, The cold surface layer on Sweden, Geogr. Ann.,in press, 2004. Storglaciären, Geogr. Ann., 71A(3–4), 241–244, 1989. Krass, M.S., Formation of the thermal regime of subpolar Holmlund, P., and P. Jansson, The Tarfala mass balance glaciers under climate change, IAHS Publ., 208, 515– programme, Geogr. Ann., 81A(4), 621–631, 1999. 525, 1991. Holmlund, P., W. Karlén, and H. 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DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN

Appendix A: GPR wave propagation σe = σ + ε ω (A.3) and attenuation σe = σ − ωε , (A.4)

The fundamental theory behind GPR techniques is where the prime indicates the real part and the the propagation of electromagnetic waves through bis the imaginary part. ω is the angular frequency a dielectric medium or a mixture of different di- (ω =2πf) of the applied electric field. For the mag- electric media. A dielectric medium is a material netic field is there a corresponding property to the that can be polarized by an external electric field, permittivity called the magnetic permeability.The but returns to its unpolarized state once the elec- magnetic permeability is a measure of ability of a tric field is removed. There are several different po- material to magnetize and also has a complex form larization mechanisms causing the polarization in µ µ iµ. a dielectric medium. In ice, the principal polariza- = + (A.5) tion arises from rotational polarizations of the water In many situations the permittivity and permeability molecule, atomic polarization related to the stretch- are expressed as relative permittivity and permeabil- ing of the bonds between atoms and electronic po- ity in vacuum larization caused by a shift of the electron cloud around the atoms relative the nucleus [King and ε εr = (A.6) Smith, 1981]. In an ideal system, the polarization ε0 µ would occur instantaneously, but most natural ma- µr = , (A.7) terial requires a finite time for charge transport or µ0 dipole rearrangement. This introduces inertia to the where ε0 and µ0 are the permittivity and permeabil- polarization, which is different for the different polar- ity in vacuum, respectively. εr and µr are often called ization mechanisms. In an alternating electric field, dielectric and dimagnetic constants of the medium the different polarization mechanisms are reinforced [Daniels et al., 1988]. when the alternating field is in resonance with the polarization rate. Thus, the degree of polarization Electromagnetic propagation for a medium is dependent on frequency because the different polarization mechanisms become dominant The propagation characteristics of an electromag- at different frequencies of the applied field [Marcelo netic field through an isotropic homogeneous dielec- and Finn, 1992]. The polarization is also tempera- tric medium can be derived from Maxwell’s equa- ture dependent. This dependence is due to the re- tions for electromagnetic fields [e.g. Rojansky, 1979]. semblance between thermal motion of the molecules The wave equation in frequency domain for a time- and the polarization [Johari and Charette, 1975]. harmonic electromagnetic field, expressed in vector A measure of the ability of a specific medium form is to polarize is called its permittivity. The permittiv- 2 2 ity affects the wave propagation speed at which the ∇ E − k E =0 (A.8) wave propagates through the dielectric medium. In ∇2H − k2H =0, (A.9) contrast to a dielectric medium, a conducting material allows charges to move freely and the con- where E and H are the electrical and magnetic field ductivity of the conductor causes attenuation of the strength and k is the wave number or the propaga- 2 2 electromagnetic wave by dissipation of energy in the tion constant given by k = ω µε. Since the perme- 2 medium [Marcelo and Finn, 1992]. Most natural ma- ability and permittivity are complex properties, k 2 terials have both dielectric and conductive properties can be expressed as k = iωµ(σ + iωε). The con- to some degree. Mathematically, permittivity and tinued derivation of the propagation characteristics conductivity are generally divided into a real part is similar for both the magnetic and electric fields and imaginary part, where the imaginary part of the since they are related [Rojansky, 1979]. Therefore, permittivity represents losses due to dampening of the derivation for magnetic field equations will be molecule vibrations and is indistinguishable from the omitted hereafter. conductive losses. Thus, it is common to use effective Equation A.8 can be simplified by assuming a permittivity and conductivity [Daniels et al., 1988], plane wave that has only spatial dependence in the z-axis with linear polarization along the x-axis, then σ ε ε Equation A.8 becomes e = + ω (A.1) σ ∂2 ε ε E − k2 . e = + ω (A.2) ∂z2 E =0 (A.10)

19 RICKARD PETTERSSON

Assuming sinusoidal time dependence, the solution be approximated with for Equation 7 for the electric field in the z-direction 1 √ 2 is 1 1 + −kz − kz v ≈ µε 1+1 = √ . (A.17) E = E0 e + E0 e , (A.11) 2 µεe where the first term on the left side is the wave trav- Sometimes it is more convenient to express the phase eling in the positive direction, while the second term velocity in terms of dielectric constants is the wave traveling in the opposite direction. The c wave number can be divided into a real and imagi- v = √ , (A.18) nary part and expressed as k = α + iβ. Substituting εrµr this into Equation A.11 and assuming that the wave propagates only in the positive z-direction gives where c is the speed of light in vacuum. In a non- magnetic material µr = 1, the propagation velocity −αz −jβz E = E0e e , (A.12) depends only on the relative dielectric constant for the medium. Equation A.18 gives the propagation where α is related to the attenuation factor of the velocity in low-loss dielectric homogeneous materi- electromagnetic wave and β is the phase factor. This als, but natural materials often consist of a mixture is useful because it shows how the electric field prop- of several dielectric media. The propagation veloc- agates and attenuates in the dielectric medium and ity then depends on the bulk permittivity, which can can be used to derive the propagation velocity and be estimated by an empirical mixing model [Nguyen, attenuation of the electromagnetic wave in the me- 1999]. Temperate ice can be considered as a binary dia. mixture of liquid water and ice and the propagation The attenuation and phase factor are found by velocity in temperate ice becomes dependent on the substituting and solving the Equations k = α + iβ fraction of free water in the ice. This has importance and k2 = iωµ(σ + iωε) , which yields for accurately converting the travel-time of GPR signals into depth estimates and other GPR processing ⎡ ⎛ ⎞⎤ 1 procedures [Murray et al., 2000]. 2 2 1 σ + ωε α = ω ⎣ µε ⎝ 1+ − 1⎠⎦ (A.13) 2 ωε − σ Attenuation

⎡ ⎛ ⎞⎤ 1 The attenuation rate of the electromagnetic wave 2 2 1 σ + ωε due to conductive and dielectric losses is expressed β = ω ⎣ µε ⎝ 1+ +1⎠⎦ , (A.14) 2 ωε − σ in Equation A.13. The attenuation rate is given in Np m−1 and from Equation A.11 we can see that an attenuation of 1 Np m−1 means that the wave decays α β −1 where is given in nepers (Np) per meter and in to a factor e of its original value in one meter. The radians per meter. In Equation A.13 and A.14, the attenuation A, can also be expressed in dB m−1 term σ + ωε −α δ A =20loge . (A.19) ωε − σ =tan (A.15) is called loss tangent and is the ratio between con- Equation A.13 also shows that apart from conduc- ductivity loss (heat loss) and energy storage in the tive and dielectric losses expressed as the loss tan- medium due to the conductivity currents and dis- gent, the attenuation depends on the frequency of placement (polarization) currents that occur when the electromagnetic wave, with attenuation increas- the electric ﬁeld propagates through the medium. ing with frequency. In a low loss medium (tan δ 1) the attenuation can be approximated with A.1. Propagation velocity 1 √ The propagation velocity v, of an electromagnetic α ≈ ω µε tan δ. (A.20) 2 wave is given by In this case, the attenuation increases proportionally − 1 ω 1 2 with frequency [Eide, 2000]. v = = µε 1+tan2 δ +1 . (A.16) β 2 In cold glacier ice, conductive and dielectric losses are small, resulting in a low attenuation, and the In a medium with low conductivity and dielectric loss penetration depth can be considerable [Bogorodsky we have tan δ 1 and the propagation velocity can et al., 1985]. However, in temperate ice the liquid

20 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN √ water influences the overall dielectric loss and the the medium becomes n = εr. However, the electric attenuation increases, especially at higher frequen- field is perpendicular to its propagation direction and cies (>1 GHz) due to relaxation effects in the water a different polarization of the electromagnetic wave [Eide, 2000]. Yet, the attenuation is reasonably low results in a different incident geometry and differ- allowing an acceptable penetration depth. The poor ent reflection coefficients. To solve this, the electric penetration often encountered on temperate glaciers field can be decomposed into two components, one not only depends on the increased attenuation, but perpendicular (horizontal polarization) and one par- also on scattering of the electromagnetic wave from allel (vertical polarization) to the plane of incidence water inclusions in the ice [Watts and England, 1976]. [Eide, 2000]. Snell’s law for an electromagnetic wave becomes Reflection and scattering n2 cos θi − n1 cos θr Γ = (A.24) Each time an electromagnetic wave passes an inter- n2 cos θr + n1 cos θi face between two different dielectric media some en- n2 cos θi − n1 cos θr Γ⊥ = , (A.25) ergy is reflected, while the remaining part of the n2 cos θi + n1 cos θr wave energy is refracted into the second medium. Some loss occurs each time an electromagnetic waves where θi and θr are the incident angle and the refrac- passes through a boundary until all energy is ab- tion angle of the electromagnetic wave, respectively. sorbed. The energy losses occurring at each interface At an angle of θi = θr = 0, Equation A.24 and A.25 between dielectric media cause degradation in pen- reduce to √ √ etration depth, but are also the fundamental cause εr1 − εr2 η2 − η1 for the ability to detect targets with GPR. Γ=√ √ = . (A.26) εr1 + εr2 η2 + η1 The degree of reflection that occurs at an interface between dielectric media is determined by the Equation A.26 shows that the fraction of reflected en- intrinsic impedance of the medium, which is the to- ergy increases with increased dielectric contrast be- tal resistance for the electromagnetic wave to prop- tween two media. The dielectric properties between agate through the medium. The intrinsic impedance water and ice are large and this is the reason why is a material property and describes the ratio be- water within the ice is easily detected with GPR. tween the electric and magnetic field strengths and The relation above for reflection and refraction is is determined by valid for an interface between two dielectric media 1 that is planar and much larger than the wavelength −iωµ µ σ − 2 η = = 1 − . (A.21) of the GPR wave. When the physical dimension of σ + iωε ε iωε the interface becomes small and irregular in shape, as compared with the wavelength of the electromag- For a dielectric medium where σ = 0, resulting in netic wave, the degree of scattering is often given σ/iωε = 0 and Equation A.21 reduces to by the radar cross-section. The radar cross-section µ is the projected area of target that will intercept η . = ε (A.22) the GPR-wave and reflect the signal back to the receiver [Faulconbridge, 2002]. The radar cross-section The fraction of the incoming energy reflected at an can be either larger or smaller than the geometrical interface between two dielectric media illuminated area depending on the shape and scattering proper- by a normal incidence electromagnetic wave can be ties of the target [Daniels, 1988]. Hence, the dielec- described by tric contrast between the scattering object and the η2 − η1 Γ= , (A.23) surrounding medium is important. The radar cross- η2 + η1 section decreases rapidly when the scattering object where Γ is the reflection coefficient and the subscripts becomes very small compared with the wavelength denote the two different dielectric media. When the of the GPR-wave [Bohren and Huffman, 1998]. incident electromagnetic wave has an arbitrary in- The large dielectric gradient between ice and liq- cidence angle, the reflection coefficient can be de- uid water in temperate ice causes strong scattering scribed by Snell’s law for optic refraction and re- and is the ultimate reason for the degraded penetra- flection [Daniels, 1988]. In this case, the index of tion capacity in temperate ice [Watts and England, refraction is an important quantity. The refraction 1976]. Success when sounding temperate glacier ice index is defined as the ratio of the speed of light in with a low frequency signal depends on if the targets vacuum to that at the investigated medium. In non- are very small when compared with the wavelength, magnetic media (µr = 1), the refraction index for resulting in reduced scattering.

DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN

Appendix B: GPR design and

processing Received Transmitted The primary objective of using GPR is often to estimate depth and extent of electrical inhomogeneities in the investigated medium. The depth to the reflecting feature is determined from the time lapse for the transmitted energy to return to the receiver and the Frequency propagation velocity of the electromagnetic energy in the medium. The spatial extent is estimated by making repeated measurements with a spatial offset, then combining the measurements into a spatial pic- Time ture of the extent of the reflections. However, the technical solutions to achieve this in a favorable way Figure B.1. Frequency-modulated continuous-wave may vary, depending on the properties of the target, radar waveform. The characteristics of the wave form predicted depth, the resolution sought and logistic controllable by the operator are the bandwidth and considerations. the modulation frequency. Stepped frequency wave- The most common type of GPR is the impulse GPR, which generates a train of short electromag- form is simply a discrete version of this waveform. netic pulses by rapid discharge of stored energy that is transmitted by a transmitting antenna. Any reflected and scattered pulse energy is registered by the receiver antenna, redirected to a receiver device For multiple targets, the beat frequency will be a and the time from the transmitted pulse to reach summation of several signals and the solution be- the receiver is determined. The receiver digitizes the comes more complicated. In practical applications, returned waveform and produce appropriate output this is solved by performing an inverse Fourier trans- for storage and display. form on the resulting beat frequency, which pro- Frequency-modulated continuous-wave (FMCW) duces a synthesized time-domain response equivalent radar is used as an alternative to impulse GPR in ap- with the received waveform from an impulse GPR plications that require high-resolution and higher dy- [Daniels, 1988]. namic range [Hamran et al., 1995; Noon et al., 1998]. The range resolution of an FMCW radar is dic- Impulse GPR uses the pulse itself as a timing mark tated by the ability of the radar to discriminate the to measure the time for the transmitted pulse to beat frequencies. Two closely spaced targets have reach the receiver. Another form of providing a time short delay time between their relative frequency re- marker is to use frequency modulation of a transmit- sponses. The short delay time result in small beat ted continuous wave. The time delay for the returned frequencies, which might be below the cut-off limit waveform and hence the target range is determined in the receiver and the two targets may not be re- from the frequency difference between the transmit- solved as two targets [Faulconbridge, 2002]. Increas- ted and the received frequency spectrum, which is ing the bandwidth of the transmitted signal produces often termed the beat frequency [Langman, 2002]. a larger beat frequency that can be detected. There- Figure B.1 illustrates conceptually a saw tooth fore, closely spaced targets with the same short time frequency modulated-continuous wave with the delay between their relative frequency responses can bandwidth ∆f, and the modulation rate τr,asa be resolved as two objects (Figure B.2). This is sim- function of time. In simple terms, the gradient of ilar to impulse radars, where the range resolution is the modulated waveform can be determined in two determined by the ability to separate two returned ways, by either the bandwidth and modulation rate echoes. A long pulse length (i.e. narrow bandwidth) or the beat frequency and the delay time for the would make the necessary time difference between returned way form. Since the gradient determined the two pulses longer to be able to separate the two in both ways is equal and the characteristics of the echoes. If the time differences are too small between transmitted waveform are known, the delay time t, the returned echoes, the reflections will mix and the for the returned waveform can be solved [Faulcon- bridge, 2002] response from the two targets cannot be resolved. Decreasing the pulse length (i.e. increasing the band- ∆f fb fbτr width) makes it easier to distinguish between re- = ⇒ τ = . (B.1) τr τ ∆f sponses from two closely spaced targets. The range

23 RICKARD PETTERSSON

a) b) Frequency Beat frequency

Time Time

Figure B.2. Increased range resolution by increasing the transmitted bandwidth. The solid line is the transmitted signal and the dashed line is the received signal. Increasing the bandwidth leads to larger beat frequency this is more likely to be detected in the receiver. resolutioninbothcasescanbeexpressedas A type of FMCW radar is the stepped frequency vτ v continuous wave (SFCW) radar. The SFCW radar ∆R = = , (B.2) 2 2B transmits a continuous wave, but stepping through a number of discrete frequencies producing a stair- v τ where is the propagation velocity, is the pulse cased version of Figure B.1, instead of transmitting B width and is bandwidth (the reciprocal of the en- a waveform with continuous frequency modulation. velope of the pulse length). In fact, the SFCW waveform can simply be seen as a The range of unambiguous returns is a timing discrete version of the FMCW waveform and the dig- issue. The FMCW waveform is transmitted repet- itized beat frequency from a SFCW radar contains itively, and identical beat frequencies can be pro- exactly the same information as the beat frequency duced for each cycle, even though the actual range to from an FMCW radar [Eide, 2000]. the target can be quite different. Hence, the FMCW One important difference from the FMCW mod- radar provides the same range, since it uses the beat ulation is that the transmitted frequency spectrum frequency to calculate the range. In the time domain, consists of a line spectrum with a frequency separa- this is analogous to the radar needing sufficient time tion determined by the bandwidth and the number for the pulses to travel to the most distant target and of transmitted frequencies. This leads to an altered back again before transmitting the next pulse. If the maximum unambiguous range, determined by returned energy from the most distant target reaches the receiver after the second pulse is transmitted, the v Ra = , (B.4) receiver interprets the distant return to be a shallow 2δf return of the second pulse. The limiting factor is the δf modulation rate and increasing the modulation rate where is the frequency separation between the dis- will increase the unambiguous range crete frequencies transmitted by the SFCW radar. In practice, the SFCW radar consists of a net- vτr Ra = . (B.3) work analyzer capable of transmitting a sequence of 2 frequencies evenly distributed within a bandwidth Note that the ambiguous range is not equivalent to [Hamran et al., 1995]. The bandwidth and number the maximum penetration depth of the radar, which of frequencies is user-definable. The return signal is is determined from the radiated power, transfer func- digitized and mixed with a replica of the transmit- tion between the antenna and probing media and ted waveform producing the beat frequency. After the attenuation and scattering in the ground (i.e. the low-pass filtering, the data is stored as a complex ar- radar equation). ray representing the beat frequency in a stair-cased

24 DYNAMICS OF THE COLD SURFACE LAYER OF POLYTHERMAL STORGLACIAREN,¨ SWEDEN waveform, where each step corresponds to the re- tions [Noon and Stickley, 1998]. To suppress the side- sponse measured at each frequency. The amplitude lobes, data can be weighted with a window function and phase of the waveform is a representation of before applying the inverse Fourier transformation. the reflection coefficient of the target, while the fre- The choice of window function depends on whether quency of the waveform is proportional to the delay a high resolution or a high dynamic range is desired time [Eide, 2001]. in the inverted response [Hamran and Aarholt, 1993]. An inverse discrete Fourier transform converts For high resolution, a square window is used while the data into the time domain. Often, the trans- for high dynamic range a Hanning window is most mitted bandwidth of the waveform is well within suitable [Eide, 2000]. The transformed data is often the frequency response of the transmitting and re- in good condition and no further processing is neces- ceiving antennas. This band-limiting function causes sary. However, the data is converted to a logarithmic an abrupt transition when converted to the time scale because of is high dynamic range. The data is domain, which produces range side-lobes that re- finally displayed using either color coding or a gray duce range resolution and cause ambiguous reflec- scale.