Slow surge of Trapridge Glacier, Yukon Territory, Canada



[1] Trapridge Glacier, a polythermal surge-type glacier located in the St. Elias Mountains, Yukon Territory, Canada, passed through a complete surge cycle between 1951 and 2005. Air photos (1951–1981) and ground-based optical surveys (1969–2005) are used to quantify the modifications in flow and geometry that occurred over this period. Yearly averaged flow records suggest that the active phase began ∼1980, and lasted until ∼2000. The average velocity in the central area of the glacier went from 16 m yr−1 in 1974 to 39 m yr−1 in 1980; it peaked at 42 m yr−1 in 1984, and remained above 25 m yr−1 until 2001. Over that interval, the flow decelerated by steps, in 4-year pulses. After a particularly vigorous acceleration in 1997–1999, the glacier gradually slowed to presurge velocities. In 2005, the flow was less than 9 m yr−1. Digital elevation models are generated by stereographic analysis of air photos for 1951, 1970, 1972, 1977, and 1981. These models are updated annually using ground-based survey data and a novel implementation of Bayesian kriging. Over the course of the surge, the front of active ice advanced 450 m and the glacier area increased by 10%, with an associated thinning of the ice. The previous surge of Trapridge Glacier, starting before 1939 and ending before 1951, led to a terminus advance of ∼1 km. Comparison of the two surges suggests that the 1930s surge started with a slow progression similar to what we observed in the 1980s and 1990s, and switched to a faster flow mode after 1941. This second phase was never attained in the recent surge, probably owing to a lack of mass.

1. Introduction

[2] Trapridge Glacier (61°14′N, 140°14′20W), Yukon Territory, Canada was identified as a surge-type glacier by Austin Post and has received two cycles of scientific study. The first, in the 1930s and 1940s, coincided with an impressive, though poorly documented surge; the second, from 1969 to the present, spans the onset and termination of a weak surge with a long active phase. The surge began between 1974 and 1977 and had effectively ceased by 2005. The peak of the surge, ∼1984, was associated with a fourfold increase in flow rate relative to the presurge and postsurge quiescence. Total advance of the active front was 450 m, giving a maximum glacier length of ∼2.75 km. This displacement was much less than that of the previous surge (>1 km).

[3] Meier and Post [1969] identified 204 surging glaciers in western North America, and outlined the following characteristics of surge behavior. (1) Surges occur repeatedly, and the duration of the cycle is generally constant for a single glacier. (2) Surges are short-lived (<1 to 6 years, generally 2–3 years) and quiescent phases are much longer (∼15 to >100 years, commonly 20–30 years). (3) The surge-phase average ice velocity can be 10–100 times that of the quiescent phase. During the surge, ice drains rapidly from an upglacier reservoir area to a downglacier receiving area, causing thickness changes of 10–100 m and horizontal displacements of <1 km to ∼10 km (10−1 or more of the glacier length). Cumulative ice displacement is 10 to 100 times greater than the total quiescent phase displacement. (4) During the quiescent phase, ice accumulates in the reservoir area and melts in the receiving area, thus gradually reversing the changes that occurred during the surge.

[4] This description of the surging phenomenon as observed in Alaska and the Yukon characterizes classic surge-type behavior and is epitomized by the well-studied 1982–1983 surge of Variegated Glacier [e.g., Kamb et al., 1985; Raymond, 1987]. The transition from slow to fast flow occurred over a period of a few months, with an hundredfold increase in ice velocity. The surge termination was very rapid (a few hours), and coincided with an outburst of turbid water and a rapid drop in the pressure of the basal water system. This rapid transition from fast to slow flow has been interpreted as resulting from the collapse of a linked-cavity drainage system and the establishment of an efficient channelized system [Kamb et al., 1985].

[5] Studies of surging glaciers in the arctic archipelago of Svalbard have established the existence of a different style of surge behavior. Dowdeswell et al. [1991] compared 8 Svalbard surge-type glaciers to 36 glaciers from other areas of the world (mainly from western North America, Iceland, the Pamirs, and the Caucasus) for which active phase duration is known. The glaciers in Svalbard have a much longer active phase, from 3 to 10 years, compared to 1–2 years for most other glaciers. In Svalbard, ice velocities during the active phase are considerably lower, and termination of the surge is gradual, with velocities decreasing over several years. The quiescent phase is also much longer, 50–500 years compared to a typical 20–30 years for most other surge-type glaciers. Comparing the surge characteristics of Variegated Glacier and Monacobreen, a Svalbard tidewater glacier that last surged in the 1990s, Murray et al. [2003] argue that the differences between the two groups can be explained by an intrinsic difference in surge mechanism. They propose that Svalbard surges are controlled by the thermal mechanism modeled by Fowler et al. [2001], while fast-surging temperate glaciers are controlled hydraulically, as modeled by Kamb et al. [1985].

[6] The recent surge of Trapridge Glacier did not follow the prototypical structure of an Alaskan surge and was even slower than most Svalbard surges. It is an unusual case, broadening the spectrum of observed surge behavior. This behavior is also atypical of the St. Elias region, an area with a high density of surging glaciers [see, e.g., Clarke, 1976, Figure 1]. The most well-known of them, for example, Steele, Hazard, Lowell, Hodgson, Disappointment, and Dusty Glaciers, are fast-surging. Exceptions have been reported, such as Backe Glacier, a small glacier SE of Trapridge formerly known as “Jackal” Glacier, which has experienced a surge of more than 6 years [Meier and Post, 1969, p. 815], an occurrence sufficiently anomalous in the region for Meier and Post [1969] to question whether it should be classified as a surge at all. This region includes both subpolar (polythermal) and temperate glaciers, and thus provides an interesting context for debate on whether temperate and polythermal glaciers have different surge mechanisms. Steele, Hazard, Backe and Trapridge Glaciers have been drilled and are known to be polythermal. “Ice worms,” which are cold sensitive and are often considered as an indicator of temperate ice [Hartzell et al., 2005], have been observed on Lowell Glacier. In addition to its unusual structure, the 1980s surge of Trapridge Glacier has the characteristic of having been observed as carefully from underneath as from the surface, and thus offers a unique context to test plausible surge models. Instrumentation of the glacier surface, interior, and base over the last 34 years yielded data sets and models for the thermal structure of the glacier [Jarvis and Clarke, 1975; Clarke and Blake, 1991], its hydrology [Stone and Clarke, 1993; Murray and Clarke, 1995; Stone and Clarke, 1996; Flowers and Clarke, 1999, 2002a, 2002b], and the hydro-mechanical coupling at the bed-ice interface [Blake et al., 1992, 1994; Fischer et al., 1999; Fischer and Clarke, 2001; Kavanaugh and Clarke, 2001]. These are the key elements necessary to understand glacier flow, and have been used to inspire, parameterize, and test surge models [e.g., Fowler et al., 2001]. However, these various pieces of the puzzle have yet to be united into a spatially resolved flow model of Trapridge Glacier. This paper attempts to set the scene for such a task by providing historical context and necessary model inputs such as flow fields and digital elevation models (DEMs).

[7] The structure of the paper is as follows. Section 2 presents the photographic and ground survey data sets that were used to constrain the ∼1930s and 1980s surge. In section 3, ground survey measurements of flow poles are analyzed to map the flow evolution of the 1980s surge. Corresponding geometrical changes are resolved using a Bayesian kriging algorithm, which allows us to generate a time-evolving DEM of the glacier (section 4). Driving stress and ice creep velocities are estimated from these DEMs and a map of the bed topography. We discuss plausible scenarios for the 1930s surge and suggest how it differed from the most recent surge. We conclude by proposing that the existence of slow and fast surges in temperate and polythermal glaciers can be explained in term of multiple switch events in the hydraulic system, and suggest how the Trapridge Glacier data sets could be used to verify that hypothesis.

2. Ground and Aerial Survey Data Sets

2.1. The 1930s Surge

[8] Scientific expeditions organized by W. A. Wood from 1935 to 1941 provide early information on the evolution of Trapridge Glacier. The expedition geologist R. P. Sharp commented on the advance of the glacier: “[Trapridge Glacier], draining from the east slope of Mount Wood, advanced eastward at least 200 feet [∼60 m] between 1939 and 1941. As observed from the air in August 1949, this glacier was advanced far beyond its terminal position of 1941” [Sharp, 1951]. We have not seen the photograph from Sharp's 1949 flight, but a photogrammetric aerial survey conducted by the Canadian Government in 1951 shows the observed displacement (Figure 1c).

Figure 1.

Terminus of Trapridge Glacier in (a) 1941 and (b) 2004. We use the 2004 glacier outline as a proxy for the 1941 position, and (c) superpose it on the 1951 air photo to estimate the advance that occurred between 1941 and 1951 (1941 photo by R. P. Sharp).

[9] To estimate the displacement that occurred between 1941 and 1951, we use the 2004 terminus position as a proxy for the 1941 presurge position. Figures 1a and 1b show the terminus of Trapridge Glacier in 1941 and 2004. Despite a difference in camera position, one can see that the terminus locations were similar. In 2004, a glacier outline was obtained by walking a handheld GPS around the ice margin. Superposing this 1941 proxy on the georeferenced 1951 air photo (Figure 1c) reveals an advance of the order of ∼1 km in that interval. The crevasses on the 1951 picture do not look fresh, but rather show signs of healing, indicating that the advanced had stopped when the photo was taken. This phase of the 1930s surge lasted therefore at most 10 years, corresponding to a minimum average advance rate of ∼100 m yr−1. Of course, the advance could have occurred in a much shorter interval anytime between these two observations and the advance rate might have been much faster. As most surging glaciers in the area are characterized by fast active phases, it was assumed that most of the displacement happened over a short period of time, perhaps from 1 to 3 years. The structure of the recent surge, however, brings us to reconsider this assumption (see section 5). Its evolution was tracked over 36 years by ground survey measurements, and further geometric constraints are inferred from available air photos.

2.2. Photogrammetric DEMs, 1951–1981

[10] Photographic surveys were carried out by the Canadian government in the St. Elias Mountains between 1948 and 1981. From these, five pairs of air photos (1951, 1970, 1972, 1977, and 1981; see Figure 2) were found to have sufficient overlap and texture to enable stereographic reconstructions of Trapridge Glacier and its surroundings. From each stereo pair, two DEMs were created: a 20-m (x, y) resolution model for Trapridge Glacier, and a 100-m (x, y) resolution model of the entire Rusty Creek basin. Integraph Softcopy Workstation software was used to collect elevation data and digitize outlines of glaciers and snow patches. The difficulty of locating ground control points and their limited distribution are the main factor limiting the accuracy of the DEMs. The 12 targets used to establish the absolute orientation of the photos were only visible on the 1981 images. For all other years, the ground control points had to be located using neighboring natural features. Given the rapid morphological changes in this active landscape and the yearly variations in ice and snow cover, establishing the exact location of the targets was often difficult, thus reducing the accuracy of the absolute orientation. Vertical uncertainty of the 1970, 1972, 1977, and 1981 photos was estimated to be ±10 m, while the 1951 elevation field was deemed accurate to ±15 m. Survey measurements taken on and around the glacier in the field seasons of 1972 and 1981 allowed for a cross validation of these DEMs. The difference between the ground survey measurements (precision ∼0.1 m) and the interpolated DEM elevation was within the ±10 m uncertainty. The larger uncertainty attributed to the 1951 DEM is due to various factors. The orientation of this stereo pair was particulary problematic because of the significant morphological changes that have occurred near the ground control points between 1951 and 1981. In addition, the stereoscopic rendering of elevation was limited by a lack of parallax because the photos had too much overlap (about 80%). This further limited the accuracy with which the control points could be located, and hindered the collection of elevation data. Finally, a lack of contrast in the snow-covered upper basin on the 1951 images made it impossible to collect elevation data in that area.

Figure 2.

Georeferenced air photos of Trapridge Glacier. The origin of the frames (top left corner) is located at 533,820 m east, 687,320 m north in UTM coordinates (a) 9 July 1951: roll number, A13136; photo number, 44–45; scale, 70000; (b) 5 August 1970: roll number, A21523; photo number, 123–124; scale, 60000; (c) 11 August 1972: roll number, A22998; photo number, 66–67, scale, 80000; (d) 4 August 1977: roll number, A24762; photo number, 130–131; scale, 70000; (e) 24 August 1981: roll number, A25841; photo number, 35–36; scale, 80000. Scanning resolution for all photos: 14 μm.

2.3. Ground Survey Measurements, 1969–2005

[11] The principal ground survey station is situated on the north lateral moraine. Distances are determined using a Geodimeter 120 laser ranger, and angles are measured using a Wild T2 theodolite. Ranges are corrected for ambient temperature, and temporal drifts are accounted for by closing rounds on a reference target. By repeating measurements on fixed targets, we estimated the survey precision to be 0.1 m. Measurements of flow pole position are available for 1969–1972, 1974, and 1980–2005. Except during periods of low visibility, the flow pole array is surveyed every day of the field season. Typically, flow poles melt out in 2–4 years, and the network extent varies from year to year. The data collected from 1969 to 2005 yielded 497 uninterrupted time series, ranging from <1 to 24 years in duration.

[12] To eliminate the inescapable measurement blunders, each pole trajectory was inspected visually for consistency. Once the obvious outliers were weeded out, more subtle blunders were expunged by a numerical algorithm designed to detect the unrealistic displacement resulting from a faulty measurement. (The interested reader is referred to Frappé [2006, Appendix B] for the details of the algorithm (the thesis can be downloaded at In total, 10,609 flow pole measurements were inspected; 391 (3.7%) data points were considered suspicious and therefore rejected. Because targets are measured many times during the field season, the elimination of a measurement does not necessarily break the continuity of the time series. Of the 1416 annually averaged flow vectors calculated from the raw data, only 59 were lost after the elimination of dubious measurements. Annually averaged ice velocities are calculated using backward differencing; that is, the flow vector takes the end-point year as the nominal time position. Flow poles can also be used to locate the ice surface by subtracting the vertical distance from the prism to the glacier surface (measured at the beginning and end of each field season) from the pole position. Three longitudinal profile lines are also surveyed every year by walking a mobile target along the length of the glacier and taking measurements every 25 m or so.

3. Flow Evolution

[13] Given the frequent melt-out of poles and their tendency to disappear and reappear depending on snow conditions, the flow information extracted from these data varies greatly in coverage. This variability makes it difficult to represent changes in the structure and the intensity of the flow in a consistent manner. Sacrificing some spatial resolution for temporal continuity, we separated the glacier into five zones (Figure 3a) and considered the flow vectors in each zone as an ensemble. Figure 3b shows time series of the median value and spatial variability of the ice velocity in each zone.

Figure 3.

(a) Glacier outline and flow pole array in 1970 and 2004. The five flow zones are superposed. (b) Time series of zonally averaged ice flow. For each year of data, the median value of surface ice speed is plotted. The box indicates the upper and lower quartiles of the speed distribution, while the whiskers show the extent of the data. They extend out to the most extreme value within 1.5 times the interquartile range; any value outside this limit is plotted as a circle.

[14] Several features of the recent flow history are worth noting.

[15] 1. In the early 1970s, the thinning remnants of the receiving area known as the “apron” (zones C, D, and beyond) flowed at less than 1 m yr−1, while the ice upglacier (zones A and B) was flowing at 10–15 m yr−1. A zone of strongly emergent flow, resulting from the collision of active and inactive ice, was detected near the terminus [Collins, 1972]. It was only in 1980, after a hiatus of 6 years in the field observations, that the morphologic consequences of these contrasting flow regimes became obvious: a wave-like bulge had formed at the boundary between the two zones (Figures 2d and 2e). Insertion of thermistor strings in 1972, 1980, and 1981 established that the apron was cold-based, while the thicker ice upflow from the bulge was warm-based [Jarvis and Clarke, 1975; Clarke et al., 1984; Clarke and Blake, 1991].

[16] 2. Sometime between 1974 and 1977, the glacier started a new surge. The median ice speed in zone A went from 16 m yr−1 in 1974 to 39 m yr−1 in 1980, while in zone B it went from 7 m yr−1 to 27 m yr−1. The bulge is already visible on the 1977 air photo (Figure 2), indicating that the surge had already started. After 1980, it took three years for the flow in zone B to reach velocities comparable to that of zone A, but thereafter the two zones behaved synchronously. The active ice front reached zone C during the 1974–1980 interval. The concomitant presence of active and stagnant ice in that zone explains the wide distribution of velocities observed from 1974 to 1986. Profile line surveys reveal that the bulge entered zone D in 1988 (Figure 7 in section 4.3). Ice velocity in that zone was not measured until 1994, when a flow pole was recorded as advancing at 32 m yr−1.

[17] 3. The upper basin reacted mildly to the onset of the surge, its median speed value going from 11 m yr−1 in 1974 to 17 m yr−1 in 1981. The change occurred in the part of the upper basin that is aligned with the lower basin; flow poles situated in the higher northern end of the upper basin move at very low rates (<3 m yr−1), and do not follow the fluctuations observed in the rest of the glacier. The apparent slowdown in 1987 is due to the fact that only poles in that area where surveyed. The data from 1994 to present are more regular in coverage, and show that the upper basin experienced fluctuations in flow similar to that of the lower basin, albeit of lesser amplitude.

[18] 4. From 1985 to 1997, the flow in the lower basin was slowing-down, following curiously regular 4-year pulses: 1 or 2 years of mild acceleration (1–2 m yr−2), followed by 2 or 3 years of steeper deceleration (3–4 m yr−2). The 1997–1999 acceleration was uncharacteristically vigorous: the velocity in zone A went from 20.5 m yr−1 to 28.5 m yr−1, implying a mean acceleration of 4 m yr−2.

[19] 5. After this last pulse, the glacier gradually slowed to presurge velocities. In 2005, the lower basin was flowing at less than 9 m yr−1.

[20] On the basis of this flow history, we can divide the 1969–2005 period in three parts: the tail end of the quiescence phase (from 1969 to sometime after 1974), the surge (starting before 1977, ending ∼1999), and the return to quiescence (1999–2005). Table 1 summarizes the flow characteristics in each zone during these three periods.

Table 1. Summary of Flow History, 1969–2005a
ZoneQuiescence 1972, m yr−1SurgeQuiescence 2005, m yr−1
Start 1980, m yr−1Peak YearPeak Speed, m yr−1Stop 1999, m yr−1
  • a

    The glacier is divided into five zones (Figure 3a), and the surface ice speed data in each zone is treated as an ensemble. Median values are given for 5 years in the time series: during the quiescent phase (1972), after the onset of the surge (1980), at the peak of the surge (different year for each zone), before the waning phase (1999), and at the end of the time series, as the glacier returns to quiescent values (2005). There were no flow poles in zone D in 2005.


4. Time-Evolving DEM

[21] We adapted a Bayesian kriging method developed by Omre [1987] to generate iteratively a DEM time series for Trapridge Glacier. To our knowledge, this is the first time this tool has been used for time-space analysis. The procedure is simple: At each time step, the field resulting from the previous iteration is taken as the prior model, and the available data are used to update the prior. The resulting estimate is passed on as prior model for the next time step, with its estimation variance used as model uncertainty. Figure 4 illustrates this procedure. Bayesian kriging theory and the procedure for parameter selection are explained in Appendix A. Bayesian kriging also allows us to include the photogrammetric DEMs in the inference process. The 1970 photo DEM is used to initiate the time series, providing a background model for the kriging of the 1969 and 1970 survey data. Similarly, the 1972 photogrammetric DEM is used as prior for the 1972 survey data, and the 1981 photo DEM is used as prior for the 1980 and 1981 models. The kriged 1981 DEM, is then used as prior for the 1982 data, and so forth. We chose to use the kriged 1981 DEM rather than the photogrammetric DEM as prior because it incorporates the ground survey measurements and is therefore more precise.

Figure 4.

One time step in the generation of the DEM time series. (a) The 1995 DEM, used as prior for the kriging of the 1996 field. Contours indicate ice thickness, in 20-m increments, and the colormap gives the field standard deviation, used as prior uncertainty. (b) Survey data from 1996, used to update the 1995 DEM. (c) Resulting 1996 ice thickness field (contours) and estimation standard deviation (colormap). Note how the upper basin has a low estimation variance, despite the fact that only a few measurements were taken in that area in 1996. A radar campaign carried out in the upper basin in 1995 provided detailed information on its topography. Because the inference of the 1996 DEM uses the 1995 DEM as prior, this information is propagated in time. Because the precision of the prior is included explicitly in the inference process, the 1996 variance field accounts for this extra information. (d) Change in elevation between 1995 and 1996. In the next time step (yr → yr + 1), the 1996 DEM will be used as prior for the kriging of the 1997 data, its standard deviation giving the prior uncertainty.

[22] Taking the previous time step as prior model might seem contradictory to the aim of detecting changes in geometry. However, this assumption of stationarity does not prevent the evolution of the model. Wherever data are available, the kriged model follows the data, and the prior has no influence on the estimation. Where no data points are available within correlation range of an estimation location, the estimate takes the value of the prior. Since the most active areas are also generally those that are the most densely surveyed, the assumption of stationarity is not problematic. In regions where no data are available, the nearest photogrammetric DEM is used as reference.

[23] The cliff-like terminus that formed after the disappearance of the apron in 1988 is difficult to reproduce using Bayesian kriging. It could only be replicated if data were available on both sides of the cliff, which is generally not the case. Only the center and north profile lines continue all the way to the terminus and explicitly mark its base. Except on rare occasions, no other measurements are made at the base of the terminus. In the absence of data, Bayesian kriging will assume a gradual transition to the prior model; in the absence of proglacial data constraining the glacier boundaries, measurements of targets next to the terminus can affect the kriged ice thickness field beyond the glacier limits. The best way to correct this effect is to impose the glacier boundaries after the interpolation, by setting all ice thickness values beyond the outline to zero. This approach was also adopted by Flowers [2000], and avoids the overspill of the glacier, replacing the numerically tapered profile by a more representative cliff-like terminus (Figure 5). The approach requires the definition of a glacier outline for each year.

Figure 5.

In the absence of data beyond the glacier outline, the kriged ice thickness decreases gradually, creating a tapered terminus. To render the cliffs that formed at the terminus in the 1990s, the field is forced to zero outside the glacier outline. The result is illustrated here for the 2004 model.

4.1. Estimation of Glacier Outlines

[24] Glacier outlines for 1951, 1970, 1972, 1977, and 1981 were digitized on the stereoscopic pairs used to generate the photogrammetric DEMs. In addition, a walk around the glacier with a handheld GPS provided an outline for 2004. For all other years, we drew nominal outlines based on the known position of the snout and on other measurements near the glacier boundary. The 1970 photogrammetric outline begins the outline time series, and the 2004 GPS outline ends it, giving two known end-points. To quantify the uncertainty of the other outlines, we drew minimal and maximal limits around the nominal outline based on hard evidence (minimal extent marked by survey measurements, known position of the snout, reference outlines marking the definite minimal (1981) and maximal (2004) limits), and on a subjective appreciation of plausible rates of advance (or retreat).

4.2. Bed Topography

[25] To obtain an ice thickness field from the DEMs, a map of the bed topography is necessary. A first map of the bed of Trapridge Glacier was produced by Flowers and Clarke [1999] after an extensive radar survey campaign carried out in 1994–1997. This map covers the area that was glaciated at the time, but it does not extend to the 1951 limits. To quantify the volume of ice accumulated in the receiving area, this bed map was extended using information gathered from the 1981 photogrammetric DEM. In 1981, most of the apron had melted away, and the once ice-covered bed was exposed. The preservation in proglacial sediments of fragile subglacial features such as flutes and crevasse-fill ridges is evidence that erosion of the bed after its deglaciation had been minimal. Thus the 1981 proglacial field, as captured on the photogrammetric DEM, can be used as a proxy for the glacier bed in 1951. We kriged a new bed topography map, using both the radar survey data and the portions of the 1981 that showed exposed ground. The covariance parameters were obtained by an approximate maximum likelihood algorithm [Pardo-Igúzquiza and Dowd, 1997; Pardo-Igúzquiza, 1998], and the statistical model was tested by analysis of orthonormal residuals [Kitanidis, 1991]. The standard deviation of the estimate, which can be interpreted as a measure of uncertainty, varies between 1 and 10 m over the ice-covered area, with most cells having a standard deviation close to 5 m. This bed topography map allows us to use equivalently either ice thickness or ice surface topography as the field to interpolate. To eliminate the downward slope trend, we kriged ice thickness rather than ice surface topography.

4.3. Geometrical Evolution

[26] On the basis of the survey and photogrammetric data available, digital elevation models of Trapridge Glacier were generated for 1951, 1970–1972, 1974, 1977, and 1980–2005. To each model is associated an ice thickness field h(x), a glacier outline, and an estimation variance field σ2(x) (Figure 6).

Figure 6.

(a) Ice thickness field and (b) standard deviation field. The collection of points in the higher reaches of the upper basin was not possible in 1951 because of a lack of texture on the air photos. The standard deviation can be used as a measure of uncertainty on the ice surface topography. It does not include the uncertainty on the bed topography.

[27] The 1970s were characterized by the rapid disappearance of the ice accumulated in the receiving area of the 1930s surge (Figures 2b2e). From 1980 to 1988, the bulge that formed at the onset of the surge propagated downglacier at an average speed of 30 m yr−1, traveling 240 m (Figure 7). In 1988, it reached the limits of the downwasting apron and formed the terminus. The active front continued to advance until 2002, at an average rate of 14 m yr−1 (200 m displacement), then practically stopped, moving only 10 m between 2002 and 2005.

Figure 7.

The bulge is drawn at 3-year intervals from 1981 to 2005. The 2002 and 2005 profiles are nearly identical, and barely distinguishable at this scale. Notice the progressive thinning of the terminus front. The 1987 profile lacks data between the bulge and the tip of the apron. In the absence of data, kriging interpolates a smooth return to the prior model, which explains the unrealistic thickness of the apron for that year. The flow zones defined in Figure 3 are indicated in bold capital letters.

[28] Table 2 summarizes the overall changes in glacier area, thickness, and volume. The kriging standard deviation σ(x) is interpreted as estimation uncertainty. While this is not strictly speaking a valid interpretation of this statistical quantity (for a discussion on this topic, see Delhomme [1979]), it does give some sense of the influence of the data on the interpolation. We believe, however, that because of the method used for the selection of the covariance parameters, the estimation standard deviation tends to underestimate our knowledge of the field (see Appendix A).

Table 2. Summary of Geometrical and Dynamical Changes, 1951–2005
Total area, km23.08 ± 0.082.34 ± 0.211.97 ± 0.192.09 ± 0.082.11 ± 0.05
Total volume, km3N/A0.13 ± 0.030.13 ± 0.020.13 ± 0.020.13 ± 0.03
Mean ice thickness, m6272776563
Median slope, degrees1111111010
Median driving stress, kPa1011181219895
Median creep velocity, m yr−
Lower basin
Total area, km22.12 ± 0.071.39 ± 0.201.02 ± 0.181.14 ± 0.071.16 ± 0.05
Total volume, km30.10 ± 0.030.05 ± 0.020.05 ± 0.010.06 ± 0.010.06 ± 0.01
Mean ice thickness, m5357675653
Median driving stress, kPa881031218884
Median creep velocity, m yr−
Upper basin (total area = 0.94 km2 (fixed))
Total volume, km3N/A0.08 ± 0.010.08 ± 0.010.07 ± 0.020.07 ± 0.02
Mean ice thickness, m8185857576
Median driving stress, kPa145136119114113
Median creep velocity, m yr−

[29] The integration of this uncertainty over the glacier area amounts to 15–23% of the total ice volume. The glacier outline, different for each year, is used as integration bounds. The effect of its uncertainty on the total ice volume calculation can be quantified by integrating the ice volume contained between the maximal and minimal outlines (see section 4.1). As it happens, the relative volume uncertainty associated to the definition of the glacier outlines is small (<4%). Because survey measurements do not cover the entire glacier area (especially in the early years), much of the uncertainty on the total ice volume follows from the uncertainty of the background models, and is thus limited by the accuracy of the photogrammetric DEMs. The fact that the kriging variance and weights account for the uncertainty of the background model is a characteristic trait of Bayesian kriging, and insures that the uncertainty on the photogrammetric DEMs is accounted for.

[30] The approximate conservation of total ice volume seems to indicate that the advance of the glacier is not a result of a positive mass balance, but rather is due to a redistribution of mass. While the model precision is insufficient to completely exclude a mass balance effect, the study of ice distribution along the centerline, where the DEMs are most precise, clearly indicates a mass transfer from the upper basin to the lower areas over the course of the surge. The spreading of the glacier from 1980 to 2002, yielding an net increase of 10% in area, was accompanied by an equivalent reduction in ice thickness. Both the upper and lower basin thinned by ∼10 m (Table 2). This thinning can be observed in the evolution of the profile (Figure 7), in ice thickness maps (Figure 6), and in the mean ice thickness time series (Figure 8a). Most of the spreading and thinning occurred between 1991 and 1996; over that interval, the terminus was advancing at an average rate of 21 m yr−1. Part of this sudden thinning is due to a delayed detection of the upper basin drawdown. Only a dozen flow poles give regular information on the ice level in the upper basin, and their coverage is very limited; the first thorough survey of the upper basin occurred during radar sounding campaign in 1995. Subtracting the 1995 DEM from the 1994 DEM show an average thinning of ∼10 m in that region. The hypothesis of a delay in detection would explain why the DEMs show an apparent 10% increase in total ice volume starting gradually from 1985 and ending suddenly in 1995. Because the drawdown in the upper basin was not detected before the 1995 radar campaign, the transfer of mass from the upper basin to the lower basin that started in the 1980s yields an apparent net increase in total ice volume. The increase is within the uncertainty bounds, and we therefore cannot ascertain its significance, however, the sudden return to normal ice volume after 1995 seems a clear indication that the increase is an artifact.

Figure 8.

Time series of mean ice thickness, median slope, and median stress. To eliminate the effect of the apron on the statistics, only cells with an ice thickness >20 m were considered.

4.4. Driving Stress and Creep Velocity

[31] The horizontal components of the driving stress depend on the ice thickness and surface slope and can be inferred from the DEMs and bed topography map. The k-component of the driving stress τ is

equation image

where ρI is the density of ice, g the gravitational acceleration, z the ice surface elevation, and xi denotes one of the two horizontal coordinates. Figure 8 shows time series of the mean ice thickness, median surface slope, and median driving stress for the lower basin of the glacier. To eliminate the effect of the apron on the statistics, only locations with an ice thickness greater than 20 m were considered. Figure 8 shows that the thinning of the glacier is the main control on the evolution of the driving stress, which also decreased rapidly between 1992 and 1996. The contact zone between active and stagnant ice, and later between the terminus cliff and the proglacial field, is marked by an arc of high stress. Upstream from this perturbation, there is a zone of relaxation, with low surface slope and low driving stress. This front can be detected as early as 1972, and becomes clearly defined in 1980. In the years preceding the onset of the surge (1969–1974), the mean ice thickness in the active area increased by ∼15 m, and the median driving stress by ∼20 kPa.

[32] Neglecting transverse and longitudinal stresses, we use the shallow ice approximation to estimate creep velocities from the DEMs. (Note that in this text we use “creep velocity” as a short form for “vertical shear velocity,” thus excluding the other possible directions of creep.) While this approximation is more appropriate for ice sheets than for valley glaciers, it gives an upper bound on the amount of flow that can be explained by ice deformation. Where longitudinal and transverse stresses are significant, it will generally overestimate vertical shear velocities.

[33] Nye [1965] calculated numerical solutions for the deformation of a glacier in a channel of rectangular, semicircular, or parabolic cross section with uniform sliding rates and different half-width to thickness ratios (W). For each run he tabulated multiplicative “shape factors” f to correct the centerline basal stress predicted by the shallow ice approximation. The cross section of Trapridge Glacier is somewhere between a rectangle and a parabola, with a half-width to thickness ratio W ≈ 500 m/60 m ≈ 8.3. The closest shape factors calculated by Nye are equation image = 0.884 for a rectangle with a half-width to thickness ratio W = 3, and equation image = 0.806 for a parabola with W = 4. The effect of transverse stresses decreases rapidly as the half-width to thickness ratio is increased; with a half-width to thickness ratio of 8.3, we would expect the stress caused by the valley walls to be negligible in most of the central area of Trapridge Glacier if, as supposed in Nye's simulations, sliding at the base is uniform.

[34] Important longitudinal or transverse stresses occur wherever there are variations in surface slope or in basal sliding. At glacier scale, stresses due to small-scale slope variations cancel each other and only the average slope has an impact on the flow. To eliminate these high-frequency variations, we used a two-dimensional median filter, with a square window of side = 2πh〉 ≈ 440 m (22 cells). Wherever the surface slope is important, longitudinal stresses will reduce creep in the steep section and increase it in the upstream area, thus the shallow ice approximation will overestimate creep at the bulge front and underestimate creep above the front. Numerical simulations by Schoof [2006] have shown that ice creep over a plastic till is strongly affected by sliding rates at the base. Activation of sliding at the center of the glacier is the result of reduced friction in this area, and vertical shearing decreases as the balance of forces must be assumed by transverse stresses rather than vertical stresses. The central lubricated area has then a plug-flow velocity structure, while the margins have large transverse strain rates [see Schoof, 2006, Figure 4b]. At the boundary between sliding and nonsliding ice, increased friction at the bed will increase vertical shearing. In these limited areas, the shallow ice approximation may underestimate the creep velocity, while it will overestimate it in the lubricated areas. Other sources of stress, such as protuberances in the bed, breaks in bed slope, throttling of the flow at the intersection of the upper and lower basin can also inhibit or enhance the creep rate.

[35] Under the shallow ice approximation (or lamellar flow model), the vertical gradient in ice velocity u(z) is

equation image

where A(T) is a temperature-dependent viscosity parameter, and n is the exponent of Glen's flow law [Van der Veen, 1999, p. 104]. A value of n = 3 is typical. Velocity at any depth can be obtained by integrating equation (2). The temperature of Trapridge Glacier varies with depth [Jarvis and Clarke, 1975]. To obtain a flow parameter that accounts for the polythermal profile, we simplify the thermal structure of the glacier to a self-similar copy of a unique profile equation image(z/h), stretched or contracted to fit any ice thickness. On the basis of this assumption, the velocity at the glacier surface (z = h) is

equation image

where us is the sliding velocity, and AT is the vertically integrated flow parameter,

equation image

We use loosely the term “sliding” to describe any motion occurring at the base. Mathematically, it represents the boundary condition of the velocity field, u(z = 0) = us. Physically, it can be explained by sliding of the ice over the bed, deformation of the bed, or the cumulative effect of both. The temperature dependence of the flow parameter A follows the Arrhenius relation [see Paterson 1994, p. 86]. We took the thermal profile measured in hole 11C (1988) from Clarke and Blake [1991, Figure 9] as the standard profile and integrated equation (4) to obtain the corrected flow parameter AT = 5.75 × 10−15 s−1 kPa−3. Given that most of the deformation occurs in the basal ice, which for most of Trapridge Glacier is at the melting point, it is not surprising that this value is close to that tabulated for temperate glaciers, A(0°C) = 6.8 × 10−15 s−1 kPa−3, [Paterson 1994, p. 96].

[36] The difference between the calculated creep velocity and the observed flow gives a minimum value for the motion occurring at the bed. Subglacial measurement of basal ice displacement relative to the bed using slidometers has shown that 50–90% of the surface motion could be attributed to actual sliding, depending on the location [Blake et al., 1994]. The remaining displacement must be explained by creep and sediment deformation. Creep velocity as predicted by the shallow ice approximation is parallel to the surface gradient; because this is not necessarily the case for the observed velocity, the sliding vectors inferred by subtraction may not be aligned with the flow. Figure 9 maps the observed surface speed, the modeled creep speed, and the inferred sliding speed. The information obtained from flow poles is averaged over cells of 100 m × 100 m. Where the predicted creep velocity is larger than the observed flow, the sliding vector points in the direction opposed to the flow. This is an obvious indication that, in those locations at least, longitudinal or transverse stresses are significant and the shallow ice approximation is not valid. In Figure 9, the cells where calculated creep exceeds the surface velocity are marked with a crossbar to avoid misinterpreting them as zones of fast sliding.

Figure 9.

(top) Observed surface speed, (middle) modeled creep speed, and (bottom) inferred sliding speed. The information on flow deduced from flow poles is averaged over cells of 100 m × 100 m. Only vector magnitudes are shown. The creep velocity is parallel to the surface topography gradient; since this is not necessarily the case for the observed velocity, the deduced sliding vectors may not be aligned with the flow. Where the predicted creep velocity is larger than the observed flow, the sliding vector points in the direction opposed to the flow. These cells are marked by crossed squares.

[37] Figure 10 shows the contribution of sliding and creep to the observed surface velocity in each flow zone from 1981 to 2005. Before the onset of the surge, creep and sliding each accounted for approximately half of the motion of the active ice (zones A and B, 1970–1974). In the apron (zone C), 100% of the flow is due to ice deformation. This result is consistent with the independent observation that the apron was cold-based, and thus could not slide. After the onset of the surge, >80% of the motion occurs at the base. As expected, the acceleration of the flow between 1974 and 1980 is due to the activation of sliding. In the first years of the surge, creep decreased in zone A and increased in zone B because of the propagation of the bulge from one zone to the next. Because of the steep slope, the creep velocity predicted by the lamellar flow model at the bulge is quite large. This is certainly an overestimate; in reality, longitudinal stresses induced by the rapid change in surface elevation would reduce this deformation rate. However, the thickening that also accompanied the advance of the active ice front certainly contributed to a higher creep rate. The creep acceleration observed in zone B and C is probably overestimated, but real. Flow in zone C went from being dominated by creep (prior to 1980) to being principally due to sliding (sometime after 1986). In the intervening period, some flow poles in that zone were in active ice, while others were on the apron, which explains the mixed statistics. After the increase corresponding to the passage of the bulge, the creep in each zone of the lower basin decreased steadily as the glacier thinned and flattened. The 4-year pulses observed at the surface are unquestionably due to varying basal conditions.

Figure 10.

Zonally averaged velocities, 1969–2005. The creep and sliding contribution to the observed flow are separated. The zones are defined in Figure 3a.

[38] Some field evidence suggests that the creep contribution to flow might be even lower than estimated above. In 1990, a hole that had been instrumented the previous year was redrilled by tracking the cable connecting the sensor to its data logger. The geometry of the hole, which was located in the center of the lower basin, was mapped using a magnetic inclinometer. Comparison with the inclinometry map traced before the installation of the sensor showed that the total lateral deformation over that year was less than the error on the inclinometry measurements (∼0.3 m) [Blake, 1992]. The yearly creep displacement predicted by our model at this location is 3.3 m. As suggested by Schoof [2006], the fact that there is less vertical shearing than predicted by the shallow ice approximation can be explained by the presence of sliding at the bed. The reduction of friction resulting from the failure of the till yields lesser vertical shearing and greater transverse stresses, in effect creating a zone of plug-flow in the central area of the glacier. This sliding-induced plug-flow is probably what we observe in the lower basin of Trapridge Glacier.

[39] In the upper basin, which is considerably thicker (70–110 m), ice deformation plays a significant role. Averaged over the entire upper basin, the median creep velocity went from ∼8 m yr−1 in 1970, peaked at 15 m yr−1 in 1989, and dropped rapidly to roughly 7 m yr−1 after the survey of the upper basin in 1995. As was discussed previously, it is likely that the thickness of the upper basin (and therefore the creep velocity) was overestimated in the years 1985 to 1995. Irrespective of the creep estimate, the flow in the upper basin is much slower than in the lower basin, indicating lower sliding velocities. These lower sliding rates are probably due to a lack of water at the base. Only a small quantity of surface meltwater is produced each summer in the upper basin, and most of it seems to be channeled to the lower basin before it can reach the bed.

5. Discussion

[40] The fourfold increase in flow that accompanied the surge does not match the 10- to 100-fold increase taken by Meier and Post [1969] as characteristic of glacier surges. Despite the reduced acceleration, the surge was sustained long enough to yield a significant advance of the active ice front (∼20% of glacier length). Few surges of that duration have been observed, and one wonders whether this is due to their rarity, or perhaps to an observational or conceptual bias. Fast surges are more likely to be noticed and reported: conspicuous features such as sheared margins and extreme crevassing signal their activity, and field observations over a single season can be sufficient to reveal geometric change. When data on surge duration are lacking, surging glaciers are generally assumed to be fast-surging.

[41] This was certainly the case at Trapridge Glacier. The comment of Sharp [1951] (see section 2.1) indicates that the 1930s surge may have proceeded in two stages: a period of slow advance, initiated sometime before 1939, and a period of faster advance initiated sometime after 1941. Because of a predisposition to think of Trapridge Glacier as fast-surging, it has generally been thought that this second phase had been short, with most of the displacement revealed by the 1951 air photo occurring over a small period of time (Figure 11, segment BCD). The unusual density of crevasses visible on the 1951 photograph supported this assumption. The first phase of advance was seen as a presurge buildup rather than a part of the surge, a misinterpretation that was repeated in the 1980s when the wave-like bulge developed. In light of the recent surge, we revisit the assumption that the 1940s advance was short, suggesting that it might have been longer, possibly spanning most of the 1941–1951 interval (Figure 11, segment BCD). Segments BCD and BCD give two limiting cases, both yielding the 1 km displacement expected. While it is probable that the difference between a tenfold and a hundredfold increase in strain rates will have an impact on the density of crevasses, we do not think the degree of crevassing observed in 1951 is enough evidence to discriminate between the BCD and BCD scenarios. Since τequation image1/3, the stresses induced in the fast advance scenario (BCD) are only about twice greater that those of scenario BCD. Stress concentration by topographic features plays a crucial role in crevasse formation, but predicting this localized response is nontrivial. Lacking other constraints on the surge evolution, we consider any variation between these two limiting cases as admissible.

Figure 11.

Sketch of the 1930s and 1980s surge sequence. For the 1980s surge, we use the ice velocity measured in the central area of the glacier (zone A and B on Figure 3). For the 1930s surge, the ice velocity is constrained by the advance rate deduced from the three observations available (1939–1941, advance of ∼60 m [Sharp, 1951]; 1941–1951, advance of ∼1 km (Figure 1)). The disparity between these advance rates suggests that the 1930s surge may have proceeded in two phases: first, a slow advance resembling the 1980s surge (AB), second, a fast advance yielding most of the displacement (segments BCD, BCD, or any intermediate case). This hypothesis assumes that some threshold condition was reached sometime after 1941, bringing the glacier to a faster flow mode. One could argue that this assumption is not strictly necessary to explain the data, and that the difference in advance rates could be simply due to the fact that the surge was ramping up when the Wood expeditions visited the site (segment AB). The pace of the 1940s advance is uncertain; while it is clear that there was a significant acceleration, we do not know how sudden it was and how long it lasted. The truth must lie between scenario BCD, a fast Alaskan-like advance, and BCD, a more Svalbard-like advance.

[42] The progress of the surge prior to 1941 is also open for interpretation. We know there was an advance of 60 m between 1939 and 1941, and that in 1941 the terminus had reached a position similar to that it occupied ∼2000, at the end of the 1980s surge. This suggested the hypothesis that the 1939–1941 advance was part of a longer period of slow advance, similar in nature to the 1980s surge (Figure 11, segment AB). According to this hypothesis, sometime after 1941, a shift in the subglacial environment switched the glacier into a faster sliding mode. It seems likely that this catastrophic release followed the overriding of the bedrock rises that give Trapridge Glacier its distinctive trilobate terminus (Figure 1). One could argue, however, that the evidence for a surge in two phases is not compelling, and that the lower advance rates could have coincided with the raising edge of a surge that had just started (Figure 11, segment AB). This interpretation has the appeal of simplicity, but requires that the glacier was already far advanced in its quiescent state, and already had a steep terminus. Lacking information prior to 1939, it is impossible to settle between these two hypothesis, and we cannot constraint the duration of the initiation period.

[43] The terminus advance having almost completely stopped, it is now clear that the glacier will not enter a phase of advance comparable to that of the 1940s. The average terminus advance rate between 1988 and 2002 was half of that observed by Sharp [1951] between 1939 and 1941, and the glacier advanced less than 10 m between 2002 and 2005. In the same period, ice velocities have decreased to below presurge values (<10 m yr−1). From ground surveys, we know the ice cliff was ∼40 m high in 2005, while it was 45–60 m high in 1941 according to the observations of Sharp [1951]. Although imprecise, comparison of the terminus width-to-height ratio on the 1941 and 2004 photographs support these figures, suggesting that the glacier was ∼25% thicker in 1941. Could a reduction in accumulation due to the late 20th century warming observed in the St. Elias Mountains be responsible for the shutting down of Trapridge Glacier surges? What changes in subglacial conditions explain the different advance rates between the two surges? The pace of the 1980s surge contrasts not only with the previous surge, but also with the behavior of the best-known surging glaciers in the St. Elias, Steele, Hazard, Hodgson, and Lowell, which have all experienced fast surges. The lumped thermal model of Fowler et al. [2001] can simulate both short and long oscillations, and could apply to the first three glaciers, which are (or, in the case of Hodgson Glacier, thought to be) polythermal. The two surges of Trapridge Glacier would be an interesting case to test this model. In work by Fowler et al. [2001], the bed vestigial roughness (equation image, equation (3.27)) controls the ice velocity, while the thermal transition time (δ, equation (3.29)) determines how fast surge activation occurs. Both parameters scale inversely with ice thickness, either explicitly (for equation image), or through the basal shear stress (for δ). The model could therefore explain how a reduction in accumulation rates during the quiescence period would yield longer and slower surges. Inferring what mass loss would be required to explain the difference in pace between the two Trapridge surges and assessing the plausibility of the change would be a way to test this model.

[44] Irrespective of its capacity to reproduce the various surge rates of Trapridge Glacier and other polythermal glaciers, this model cannot explain the surges of Lowell Glacier and the other temperate glaciers in the region. The clustered distribution of surge-type glaciers around the world and within regional basins suggests that glaciers within clusters share dynamical characteristics that make them respond to key environmental controls in a similar manner. It seems implausible that two distinct surge mechanisms would be present in the same region. Models of surge mechanism should be able to explain the entire range of intensity and duration observed in a cluster, while reflecting the inherent commonality that explain the concentration of surges in that area. The Trapridge Glacier data set might provide an opportunity to inspire and constrain such a model. The year-round recording of subglacial properties at Trapridge Glacier started in 1989 and will end in 2007. The onset of the surge was missed, but its termination occurred when the full instrument array was deployed. Before the survey data analysis presented in this paper was carried out, records of subglacial water pressure were scrutinized in search of events coinciding with the slowdown observed after 2002. No major alteration of the drainage system were detected beyond the seasonal scale (see Flowers and Clarke [2002b] for a description of the fall and spring transitions). The slow pace of the surge, the 4-year pulses, and the long termination period suggest that the single-switch model might not be the best hypothesis for this surge. Instead, we intend to revisit the subglacial data records and search for multiple switching events, the cumulative effect of which could explain the surge evolution. This hydraulically controlled model could explain the existence of fast and slow surges in term of the frequency and intensity of switch-events and would have the advantage of being applicable to both temperate and polythermal glaciers.

[45] After such a dramatic decrease in surge vigor, one wonders if Trapridge Glacier has reached a point of no return, and may never surge again. This would not be unprecedented: Dowdeswell et al. [1995] presented evidence that the warming marking the end of the Little Ice Age in Svalbard had caused some glaciers to lose their surging property. Hoinkes [1969] reported on the surges of Vernagtferner in the Ötztal Alps, which are also been thought to have stopped after 1895 because of climate warming. Nearby, Rusty Glacier (formerly “Fox” Glacier) was identified by Austin Post as surge-type but now has a strongly negative mass balance and seems incapable of recharging its reservoir area. We think it is too early to settle on the future of Trapridge Glacier. The predicted continuation of the current warming trends does not inspire much optimism for an improved mass balance. The weakness of the surge has an ambiguous effect on the prospect of a new surge. On one hand, it will have limited the upper basin drawdown, thus facilitating its recharge for another surge. On the other, the weakness of the surge entails that the flux out of the reservoir has not reduced significantly now that the surge is over. Given that the amount of ice in the upper basin does not seem to have increased in the last few years, it is unlikely to increase rapidly now that the surge is over.

6. Conclusion

[46] Air photos from the 1970s and 1980s and ground surveys measurements of flow poles and ice surface enable us to detect the onset (between 1974 and 1977), propagation (1977–1999), and termination (1999–2005) of a new surge of Trapridge Glacier. Maximum surge velocities (∼40 m yr−1) were four times the typical quiescent flow rate (∼10 m yr−1). The active phase lasted over 20 years, with ice flow on a slowing trend marked by regular 4-year pulses. A wave-like bulge formed at the boundary of active and inactive ice and propagated downglacier at an average speed of 30 m yr−1, incorporating the downwasting remnants of the cold-based ice accumulated during the previous surge. The bulge reached the terminus in 1988, 240 m downglacier from its 1980 position. Between 1988 and 2002, the cliff-like terminus advanced 200 m, (average rate of 14 m yr−1) leading to a 10% increase in glacier area. Flow rates rapidly decreased in 1999 after two years of acceleration. This last pulse was of greater amplitude than the four that preceded it. In 2005, the ice was flowing at presurge velocities.

[47] An algorithm based on Bayesian kriging was developed to merge photogrammetric DEMs and yearly ground survey measurements to generate a time-evolving DEM of this surge. This is a new technique that can be used whenever a field with spatially variable uncertainty must be updated from sparse data. The field uncertainty is taken into account when interpolation weights are attributed, and is reflected in the variance of the resulting field. The algorithm is numerically efficient and can be implemented by modifying standard ordinary kriging routines. From this DEM time series and a map of the bed topography, total ice volume, driving stress, and vertical shear velocity were estimated. Creep velocity accounts for a negligible portion of the flow observed in the lower basin; sliding or bed deformation is responsible for most of the motion in that area. In the upper basin, where the ice is thicker and flows more slowly, creep could account for as much as half the recorded motion. Total ice volume remained approximately constant (within uncertainty bounds) over the duration of the surge. The glacier advance and spread was accompanied by a drawdown of the ice, first in the upper basin, then in the upper part of the lower basin.

[48] The previous surge was much more dramatic in extent and possibly in pace, but observations are too scarce to constrain precisely its duration and structure. A terminus advance of ∼1 km occurred sometime between 1941 and 1951, preceded by a period of slower progression of unknown duration. Similarity between the terminus in 1941 and at the end of the current surge suggests that the 1930s surge started by a slow progression similar to the recent surge, and switched to a faster flow mode sometime after 1941. It has been proposed that this second phase was much more similar in style to an Alaskan-type advance, but the lack of observation between 1941 and 1951 prevents us from testing this assumption.

[49] The St. Elias region contains both temperate and polythermal surging glaciers. While most of these are thought to have experienced fast surges, in many cases the surging status is inferred from morphological features or repeated observations separated by long time intervals. The number of glaciers experiencing slow surges in the St. Elias could therefore be greater than expected. While it is clear that the thermal regime of a glacier will have an impact on its surge behavior, we think this effect is collateral to the surging phenomena rather than essential. To reflect the underlying dynamical unity suggested by the clustering of surge-type glaciers, models for surge mechanism should be able to explain the entire spectrum of surge speed and structure observed within a geographical cluster, irrespective of the thermal regime of the glaciers.

Appendix A:: Generation of Time-Evolving DEM Using Bayesian Kriging

[50] Three types of data were used to constrain the Trapridge Glacier digital elevation models (DEMs): ground survey of profile lines, ground survey of flow pole positions, and aerial photography. The description of geometrical changes is complicated by the fact that none of the three data sets is consistent in time. Flow poles, which follow the ice they are drilled in, can only offer a Lagrangian perspective on ice thickness variations. The coverage of the array also varies significantly over the span of the data set. Profile lines should in principal offer an Eulerian description of topographic changes since they are oriented relative to the surrounding landscape, but in practice their location varies slightly each year. As for air photos, they are available only for five of the 32 years of the time series. To complicate the matter further, the three data sets have different sampling density and accuracy. Profile lines are densely sampled but cover limited ground, while the flow pole array is sparser but cover a larger area of the glacier. Photogrammetric measurements are regularly sampled, but are much less precise. Some form of interpolation is therefore needed to integrate these three data sources and construct a consistent time series. Because survey measurements are mostly taken in the central part of the glacier, extrapolation of these data toward the margins is also necessary.

[51] Ordinary kriging is a standard geostatistical method for the interpolation of data [e.g., Kitanidis, 1997; Olea, 1999]. Estimation at an unsampled location is expressed as a linear combination of the surrounding data points. Given certain stationarity assumptions, the weights are optimized to minimize the estimation error variance. This automatically adjusts the weights such that areas of dense sampling do not dominate the estimation. This declustering property of kriging is particularly useful when dealing with unevenly sampled data. However, ordinary kriging does not account for data uncertainty, and cannot distinguish between two data points with different precision. Extrapolation of data using ordinary kriging is also problematic.

[52] Bayesian kriging [Omre, 1987], offer solutions to most of these limitations. It allows photogrammetric DEMs to be used as prior models to support interpolations and extrapolations based on the more precise survey data. Where survey data are available, the kriged model follows this more accurate source of information, and the prior has no influence on the estimation. Where no survey data points are available within correlation range of an estimation location, the background model offers a plausible extrapolation value. In either case, the variance of the kriged model represents the availability of data and the confidence attributed to the prior estimate. By using one year's results as background model for the interpolation of the next year's data, this approach also allows the propagation of information in time. In that way, information gained during intensive survey campaigns can be used to guide the inference for years where data is more limited.

A1. Bayesian Kriging Theory

[53] As is customary in geostatistics, the data z(xi) are considered as spatial samplings of one given realization of a random field {Z(x, ω); x ∈ &#55349;&#56479;, ω ∈ Ω}, where x is a space coordinate vector (deterministic), &#55349;&#56479; is the spatial domain, and ω is a random variable associated with one realization in the sample space Ω. The random field does not correspond to a natural random process, but is solely an operational concept. Each realization Z(x, ωi) is a deterministic function of space; averages are taken over the population of possible fields Ω, not over the spatial domain &#55349;&#56479;. The underlying random variable ω is commonly omitted in the notation; it is customary to use capitals to denote random functions Z(x, ω) ≡ Z(x), and lower case to denote the particular realization sampled by the data z(x, ωi) ≡ z(x).

[54] Assuming a certain statistical model for this random function, the best linear estimation of the field at an unsampled location z(x0) can be calculated. The various forms of kriging differ in the constraints they impose on the statistical model. Ordinary kriging, for example, requires the random function to be second-order stationary: its mean E [Z(x)] is unknown but spatially constant, and its covariance depends only on the distance h between two locations Cov [Z(x), Z(x′)] = C(h). In Bayesian kriging, the expected behavior of the random function is supposed known a priori, with a degree of certitude that can vary spatially. To include formally this qualified guess, another random variable is introduced: {M(x, ϕ); x ∈ &#55349;&#56479;, ϕ ∈ Φ}, where Φ is the sample space, and ϕ the underlying random variable. Again, the ϕ dependence will be omitted to simplify notation. The two first moments of M(x) are defined by the user

equation image
equation image

where μM(x) is the prior model and CM(x, x′) is its covariance function. The prior covariance depends on a location-specific variance field and on a translation-symmetric correlation function: CM(x, x′) = σM(x)σM(x′)ρM(xx′). The random field Z(x) is assumed to be second-order stationary conditional to this prior model, that is

equation image
equation image

Given equations (A1) and (A2), the moments of the target field Z(x) are

equation image
equation image

The estimator for the value of the field at an unsampled location x0 is a linear combination of the random field Z(x) evaluated at the N sampled locations xi,

equation image

Those familiar with kriging might notice that this estimator is identical in form to the simple kriging estimator. In simple kriging, however, the expected value of the random function is supposed to be known exactly. In Bayesian kriging, the expected value is known within a certain uncertainty, represented by the variance of the prior. The covariance of the field (equation (A4)) has two parts, a translation-symmetric component representing the natural variability of the field around the prior, and a location-specific component accounting for the prior uncertainty. This second term is absent in simple kriging. Bayesian kriging is a general theory that includes as limiting cases simple kriging, for which the expected value of the field is exactly known, and universal kriging, for which the expected field is unknown, but supposed to follow certain functional form (generally a low-order polynomial) [Omre and Halvorsen, 1989].

[55] Given the statistical model defined by equations (A1)(A6), the weights λi's that minimize the estimation error variance can be found by linear optimization. The optimal weights obey the following system of equation [Omre, 1987]

equation image

and the optimal estimation error variance is

equation image

[56] When no data are available within one correlation range of a given estimation site, all the kriging weights λi are set to zero, and the estimate (A7) takes the value of the prior. The estimation variance (A9) is then equal to the prior variance at this site CM(x0, x0) plus the conditional variance CZM(0). The conditional variance can thus be understood as a measure of the information decay rate: every year for which no data are available within correlation range of a site, the variance of the model at this site increases by CZM(0).

[57] As we are ultimately interested in the calculation of ice volumes and stresses, we extended the common use of block kriging to the point-estimation method outlined above. To obtain estimates that are representative of the entire grid-cell rather than only of a point location, the covariance functions are integrated over the cell area. The derivation of the block-based form of Bayesian kriging from its point-based form is identical to that for ordinary kriging, which can be found in standard geostatistics textbooks [e.g., Kitanidis, 1997; Olea, 1999]. The interested reader can also see Appendix C of Frappé [2006] for a derivation of the Bayesian case. The Bayesian kriging algorithm was programmed in Matlab, built on an ordinary kriging routine published by Marcotte [1991]. The scripts are available on request.

A2. Selection of Covariance Parameters

[58] The assumption of second-order stationarity around the prior (equations (A3) and (A4)) defines the general structure of the statistical model. To complete it, analytical expressions for the conditional covariance CZM(xx′), and the prior covariance CM(x, x′) must be determined. Covariance functions are chosen within a limited set of nonnegative elementary models (gaussian, spherical, exponential, etc.), to insure the positivity of the variance [Kitanidis 1997, p. 54]. For each elementary model, three parameters need to be defined: the sill of the function (giving the field variance), the correlation range, and the microscale variability (also known as the nugget effect).

[59] When using Bayesian kriging to generate a time series, the prior and the target field are the same physical field, and their spatial characteristics are expected to be similar. The variance field, describing how much is known about the field, will vary each year on the basis of the data available; the correlation function, however, should remain the same because it depends primarily on the physical properties of the field. Assuming the spatial structure of the covariance to be constant in time reduces the number of parameters to estimate from 93 (3 per year) to 3: the conditional variance CZM(0), describing the rate of decay of information, the correlation range, and the microscale variability.

[60] In most applications of kriging, the covariance parameters are selected by fitting an experimental covariance function (or, equivalently, an experimental variogram), calculated from the data. While Omre [1987] does suggest an estimator for the conditional variogram γZM(xx′), we found this estimator to be problematic. It would often yield negative values, despite the fact that, by definition, a variogram should be positive. We used these experimental variograms to get a general sense of the spatial structure of the field, and then calculated several time series based on various sets of parameters. For each run, orthonormal residuals were calculated, and their statistical behavior compared.

[61] Orthonormal residuals analysis is an advanced cross-validation technique that allows the objective testing of a statistical model. Residuals are differences between observations and model predictions. They can be calculated by predicting the value of the field at a sampled location while ignoring the corresponding data point. Orthonormal residuals are calculated in a similar fashion, except that the cross validation follows a specific sequential procedure [Kitanidis, 1991]. Just as the estimator equation image(x0) is a random function until the actual values of the data points are substituted in the linear combination equation imageλiZ(xi), the residuals can be considered as random variables, and their statistical properties can be tested. If the statistical model is correct, the experimental residuals should have a mean of 0, a variance of 1, and be uncorrelated [Kitanidis, 1991]. Using these statistics, the performance of competing models can be objectively compared, and the best model can be chosen. After testing more than 30 combination of parameters and elementary models, we adopted a spherical model with a range of 500 m and a sill of 100 m2. No microscale variability was detected. With this set of parameters, DEMs for all of the 31 time steps yielded orthonormal residuals with a mean sufficiently close to zero. For 24 models, however, the variance of the orthonormal residuals was far enough from unity to justify the rejection of the model. For 22 out of those 24 years, the orthonormal residual variance was too small. This would indicate that the value assigned to the conditional variance CZM(0) was overestimated, and that the model can predict the data with more precision than supposed. However, a smaller conditional variance value yielded orthonormal residuals that were biased. This bias probably results from the thinning trend present in the ice thickness field for most of the 1980–2005 period. To compensate for this failure of the stationarity assumption, the conditional variance must be inflated and, since residuals are normalized by the estimation variance, this narrows the distribution of the orthonormal residuals. As a consequence, we believe that the estimation standard deviation tends to underestimate our knowledge of the target field.


[62] We gratefully acknowledge the 28 surveyors that stood on the windy moraine to collect this data set over the years. The photogrammetric analysis was carried out in collaboration with Eric Saczuk of the British Columbia Institute of Technology. Christian Schoof provided insightful comments on an early version of the manuscript. We thank Tavi Murray, Tim James, and an anonymous reviewer for their constructive criticism. This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and supported by the Arctic Institute of North America. T.-P. F.-S. was supported by a NSERC PGA scholarship. We thank our friends at the Kluane Lake Research Station for their logistical support.