During the summer of 2004, the front area of the Jakobshavn Isbræ was monitored using a geodetic-photogrammetric survey with temporarily coincident precise observations of local ocean tides in the Disko Bay close to Ilulissat. The geodetic and photogrammetric observations were conducted at the southern margin of the glacier front. The largest observed horizontal flow velocities are in the central part of the front with values up to 45 m/d. This is a factor of 2 greater than the average velocities at the front area observed in the last century. Our new observations confirm previous estimates of an acceleration of glacier flow during the last decade. The photogrammetric survey provided flow trajectories for 4000 surface points with a time resolution of 30 min. These flow trajectories were used to compare the vertical motion of the glacier with the observed tides. The existence of a free-floating glacier tongue in 2004 was confirmed by these data. However, it occupied only a small belt, of at most a few 100 m width, in the central part of the glacier front. Horizontal motion did not appear to depend on the tidal phase, unlike some of the fast-moving ice streams of West Antarctica.
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 The Jakobshavn Isbræ (Figure 1) is one of the most dynamic glaciers of the world. Covering a catchment area of 110,000 km2, or 6.5% of the area of the Greenland ice sheet, the rate of volume lost at its front by calving has been estimated at about 35 km3/yr over the last century [Bennike et al., 2004]. More than 100 years ago, the flow velocity was determined near the glacier front by Hammer  and Engell . Aerial photographs formed the basis for velocity determinations in the 1950s and 60s [Carbonell and Bauer, 1968; Bauer et al., 1968]. Together with later theodolite measurements, collectively, these observations are consistent with a flow velocity at the floating glacier tongue of about 20–22 m/d [Lingle et al., 1981]. A sophisticated investigation of Jakobshavn Isbræ's ice dynamics was later carried out by Fastook et al. , which included ice-surface elevations, surface velocities and strain rates based on airborne photogrammetry.
 Along with the neighboring edge of the inland ice, the glacier front advanced during the Last Glacial Maximum and then retreated behind its present position during the early Holocene to mid-Holocene climate optimum [Weidick et al., 2004]. During the Little Ice Age the glacier readvanced, followed by a remarkable retreat of the ice edge of several kilometers. Generally, since 1850 the front of the floating glacier tongue has been in retreat, intermittently having achieved relatively stable positions. Somewhat smaller fluctuations in stability mark the period between 1960 and 2001 (Figure 1, bottom) [Weidick, 1992; Sohn et al., 1998; Luckman and Murray, 2005; I. Joughin et al., Fastest glacier in Greenland doubles speed, 2004, http://www.nasa.gov/vision/earth/lookingatearth/jakobshavn.html].
 In 2001 the front began to retreat once again (Figure 1, bottom), with a remarkably accelerated flow velocity [Joughin et al., 2004], reaching 13,000 m/yr (=35 m/d). In 1997 thinning of the glacier surface began, with observed rates of more than 10 m/yr at low elevations, with an increase in net ice loss from 10 km3/yr between 1997 and 2002 to about 20 km3/yr in 2002/2003 [Krabill et al., 2004].
 The causal mechanisms for this acceleration, and for the combined frontal retreat and decreasing surface heights, are not yet fully understood. It may be a consequence of a smaller buttressing force near the glacier front due to the decrease in total ice floatation area, as observed after the collapse of the Larsen ice shelf in Antarctica [De Angelis and Skvarca, 2003]. The mechanism could also have a dynamic origin [Thomas, 2004], or it may even be a response to present (and/or former) climate changes. Within this context, the interaction of the frontal zone of tidewater glaciers with the ocean, and especially with tides, has been identified as a key target for additional research [Thomas et al., 2003; Bindschadler et al., 2003].
Lingle et al.  was first to investigate the tidal flexing of the floating tongue at the Jakobshavn Isbræ and later the grounding zone was mapped by Echelmeyer et al. . The motivation for this field investigation is to measure the present-day displacement field of the frontal area of the glacier with a sufficiently robust time resolution to determine the horizontal velocity and to reveal any tidally induced motion in the vertical and horizontal directions. Since the glacier surface is completely crevassed, a survey method which did not require access to the glacier was selected. Therefore geodetic survey techniques, including tide gauge observations, were combined with precise digital photogrammetry adjacent to the glacier.
2. Setup of Field Observations and Results
2.1. Ocean Tides
 In order to provide accurate tidal information coincident with the velocity observations, a pressure gauge (Aanderaa WLR 7) was deployed north of the harbor entrance of Ilulissat. It operated successfully from 10 to 29 August 2004. Although a second pressure gauge was deployed at the field site, the distribution of dense calf ice in the Isfjord prevented its secure installation and manual readings using a levelling rod were performed as an alternative. These readings reveal an excellent agreement with the pressure gauge record at Ilulissat (Figure 2), and, consequently, it was determined that the tides recorded at the Ilulissat site are sufficiently representative for the entire frontal zone of the Jakobshavn Isbræ. The phases and amplitudes of the main tidal constituents were thus determined. These show deviations from existing tidal models that are large enough to be important to our study (Table 1 and Figure 2). The observations confirm that the total tidal variation of sea level in the Isfjord area during our field observations was on the order of 2 meters, similar to the tidal range observed by Lingle et al.,  using theodolite methods.
Table 1. Main Tidal Constituents for Ilulissat From Tide Gauge Observations in August 2004 and From the Oceanographic Model FES99a
2.2. Geodetic Reference and Theodolite Observations of Glacier Motion
 South of the glacier front and near the field camp, a GPS benchmark was monumented in bedrock and observed during our entire period of study. Additionally, existing GPS sites nearby in Ilulissat and Kangerlussuaq were occupied simultaneously, since these sites are determined precisely in the International Terrestrial Reference Frame [Dietrich et al., 2005]. The International Terrestrial Reference Frame (ITRF2000) [Altamimi et al., 2002] provided the geodetic reference frame for the analysis of all local observations, hereafter referred to as the object space. The endpoints of two geodetic baselines were also observed with GPS. In addition, electro-optical distance measurements were performed in combination with theodolite observations in order to provide bedrock reference sites that play the role of fiducial targets in the foreground of the photogrammetric survey.
 From the ends of the two baselines (length about 130 m) repeated theodolite observations (24 hours time spacing) to natural, identifiable targets at the glacier (e.g., seracs and ice towers) were performed. Many of the targets could also be identified in the digital photogrammetric images. As a result, the positions of these targets were obtained by intersection of the observed horizontal angle in the geodetic reference frame (Figure 3).
 Observations of the horizontal and vertical angles were complex. The two endpoints of the baseline were the maximum distance to locate and identify the same targets for observation. For ideal targets, this kind of theodolite measurements would easily allow submeter accuracies for position determination. In our case, however, slightly different views from the two endpoints could induce larger errors, since the natural targets may change structure when seen from another perspective. The accuracy obtained here is therefore quite difficult to quantify. We roughly estimate that the RMS for the coordinate determination is about 2–3 m, leading to errors of 3–4 m/d for the velocity determinations (Table 2).
Table 2. Coordinates and Velocities Based on Theodolite Observationsa
The target IDs correspond to the ones in Figure 4. Targets 1–11 were observed from the western basis on 20–21 August 2004; IDs 12–20 were observed from the eastern basis on 23–24 August 2004.
 One single observation set to all targets took half an hour at each point. Because of the high glacier velocity the targets move significantly during that time span. Therefore we observed first at one endpoint (horizontal angles and exact time), then at the second endpoint and finally returned to the first endpoint. Doing so, we were able to interpolate between the angles observed twice at the first endpoint and relate this exactly to the observation epoch of the second endpoint, thus eliminating the effect of glacier motion during the observation.
 In order to determine the glacial flow with sufficiently high spatial and temporal resolution, we applied digital photogrammetry with monocular image sequence analysis. For this purpose, a 4500 × 3000 pixel digital camera (Kodak DCS 14n) was installed close to the glacier front area (Figures 3 and 4, top). The camera was positioned on a hill overlooking the glacier front area. It imaged a region of the main glacier flow at a distance between 2 to 5 km with the viewing direction at right angle to the glacier flow direction. It was operated in an interval mode, taking images every 30 min, over observation periods of at least 24 hours.
2.3.1. Subpixel-Accuracy Image Sequence Analysis
 The image motion between two patches g1, g2 of two consecutive images I1, I2 is determined by cross correlation using the normalized cross correlation coefficient ρ:
dimension of the image patches (typically 40 × 40 pixels),
average intensity of image patches g1, g2.
 The maximum of the cross correlation coefficient over a predefined two-dimensional search area provides the X-Y image movement with pixel accuracy. Fitting a parabola through the neighborhood of the maximum, subpixel accuracy can be reached by analyzing the vertex of the parabola. Degradation in precision related to the movement of shadows between consecutive images were compensated for by forward-oriented search regions and continuity constraints for the image motion between more than two consecutive image patches. The algorithm can be implemented rather efficiently and uses fractions of a millisecond of computation time per patch.
 Using the method, a total of 4000 image patches were tracked through a slightly greater than 24-hour sequence consisting of 53 subsequent images, started on 20 August 2004, 19:30 UTC. Camera motion during image acquisition was determined by tracking four reference targets (Figure 4, top) through the image sequences using the cross correlation technique. Glacier point trajectories were corrected for, up to a level of 5 pixels of camera motion. Figure 4 (bottom) shows a subset of 24-hour flow trajectories in the central front of the observed region, overlaid on a section of one digital image of the sequence. The average length of the trajectories is 140 pixels with a vertical motion of 5 pixels, translating into a forward motion of up to 45 m/d and a vertical motion amplitude of up to 1.40 m.
 In addition to the 4000 image patches also the trajectories of the 500 surface targets described in the next paragraph were determined in the same way.
2.3.2. Georeferencing of Trajectories
 The trajectories determined by cross correlation must be scaled and the observed glacier points have to be transformed from image space into the object space. Since no reference targets could be attached to the glacier surface itself, geodetic observations to natural targets (section 2.2) were used in conjunction with the fiducial markers in the foreground in order to reference single points in image space to the object space. This procedure was followed for 7 images that were acquired for a photogrammetric stereo analysis, as well as for one image of the image sequence. Slight perturbations of the view in the photos allowed estimation of camera positions and orientations for all images by means of a bundle adjustment. In the next step, well-textured surface targets across the entire glacier which could be identified in all seven images were transformed photogrammetrically from image to object space. Additional points were added around/along critical areas, such as peaks, edges or recognizable undulations which would otherwise cause discontinuities in the coordinates. The coordinates of more than 500 surface targets and their associated variances form a dense mesh across the observed glacier surface. Consequently, both their image coordinates, X and Y, and their coordinates x, y and z in object space are retrieved. A minimum curvature bicubic spline interpolation was performed for each coordinate direction (x, y, z) to fill any unsampled space. A transformation of any arbitrary image point into the object space (ITRF) is therewith realized.
2.4.1. Fitting of the Trajectories and Parameter Estimation
 All trajectories (Xi(tj), Yi(tj)), where i stands for point number and j for epoch number, were scaled by their distance obtained from the georeferencing procedure and used to perform a least square fit. For this purpose, a reference curve ((tj), (tj)) was created. Its variability in the vertical was in accordance with the observed tide (Figure 2) and the horizontal change of coordinates corresponded to a reference velocity of 40 m/d. As unknown parameters for the fit, two translations tX, tY (in X and Y), a rotation α (describing a deviation of the trajectory from the horizontal direction), a scale factor in Y (describing the damping of the tidal signal) and a scale factor in X (for scaling the reference velocity of 40 m/d) were estimated. The resulting observation equations for the least squares fit are
The RMS value of the fit as well as the adjusted parameters and their RMS values are obtained for each trajectory.
 Furthermore, a time shift Δt between the trajectory and the reference curve was determined. It turned out that the tidal response of the glacier showed a significant delay. Since such a time shift Δt can only be estimated for trajectories with a tidal amplitude its determination was not implemented directly into the algorithm according to equations (2) and (3). Instead, we shifted the reference curve by small steps of Δt: (tj + Δt), (tj + Δt) and looked for the RMS value of the fit when applying the least squares procedure according to equations (2) and (3). For all trajectories with a tidal signal the fit showed a significant dependence on Δt (Figure 5). The RMS values became a minimum for a Δt which corresponds to a delay of the tidal motion of the glacier of 35–45 min in comparison to the reference curve.
 Stacked histograms of the fit residuals for different distances are shown in Figure 6. Observations taken at night, with reduced brightness during twilight, show larger scatter. Therefore these were downweighted by a factor of 3 in the adjustment. Typical examples for the fit of a floating location and a grounded location are shown in Figure 7 (top and middle). Not all trajectories provided a reasonable fit. The example shown in Figure 7 (bottom) reveals a jump in the trajectory which was probably caused by a sudden motion pulse. Thus about 2% of the trajectories were identified and culled from the analysis.
2.4.2. Correction Due to Flow Direction
 All flow velocities determined in the image plane, as described in the previous paragraph, represent the projection of the real flow vector onto the image plane. The setup of the photogrammetric survey was designed such that the view direction is nearly perpendicular to the ice flow direction. To correct for perturbations from this ideal geometry, clearly visible flow line patterns were digitized from a georeferenced Landsat TM image. This provided the azimuthal angle of glacier flow direction for the entire area of investigation. The correction term for the resulting velocities in the image plane was always smaller than 5%.
2.4.3. Error Budget
 The glacier point trajectory error budget consists of three components: one introduced by image measurement precision, the second by the fit of the trajectories and the third by image-to-object space transformation effects. Image measurement precision can be obtained by an analysis of the steepness of the vertex of the parabolic fit into the cross correlation coefficients. The precision is on the order of 0.02–0.04 pixels for well defined targets, such as the image foreground reference targets. The glacier points themselves delivered a precision of the order of 0.1–0.3 pixels. This degradation can be explained by the rough surface of the glacier, surface topography changes in the image patches and remaining effects of moving shadows. Over a 24 hour illumination period, the glacier surface changes (moving shadows) and the RMS values of the fit to the reference trajectory (section 2.4.1) are a more general and useful measure of accuracy. Here we found an additional degradiation of the image measurements expressed by the RMS values for the residuals. The RMS values (see Figure 6) change from 0.20 m in a distance of 2200 m to 0.42 m in a distance of 3500 m. While the daylight observations are slightly more accurate, twilight observations show a degradation in accuracy on the order of 50 to 100% compared to daylight observations.
 The error introduced by trajectory georeferencing acts effectively as a scale error. This error is dependent upon the precision obtained in the adjustment of the terrestrial survey network, as well as on the local glacier surface model approximation (section 2.3.2). For the roughly 500 targets of the bundle adjustment, the error in scale (equivalent to an error in the distance from camera position) grows approximately with the square of the distance. It is on the order of 0.5% in the foreground (2 km distance) and about 2% in the background (4 km distance). For the roughly 4000 trajectories, the effects of bicubic spline interpolation of coordinates (section 2.3.2) must be added to the error budget. Comparison of noninterpolated target points located on the glacier surface to a subset of the 4000 interpolations indicates quite satisfactory results, up to distances of 3.5 km. In the far field the differences became larger indicating that the interpolation would lead to distance errors of up to 10%.
 Therefore in object space we present a sparse distribution of surface points covering the complete area of survey which belong to the group of targets in the bundle adjustment. In addition, we show a dense pattern of analyzed trajectories where the maximum distance is restricted to about 3.5 km. For this area, we are confident that the error in geolocation is not larger than 200 m.
 Combining the RMS values of the unknowns obtained by the fit (see Figure 7, top and middle plots) and the foreground and background scaling uncertainties of 0.5% and 2%, respectively, leads to an error estimate for the corresponding velocity determinations of about 0.5 m/d and 2.0 m/d.
 If we take the velocities obtained by theodolite observations (Table 2) and compare them with the velocities at the same location as determined from the photogrammetric trajectories, the RMS value of the differences becomes 3.1 m/d. This value is consistent with our error considerations determined above and with those for the theodolite observation (section 2.2).
3. Interpretation and Discussion
3.1. Horizontal Velocity Field of the Front Area
 The flow velocities obtained for the front area are first shown in image space (Figure 8, top) with their corresponding RMS values (Figure 8, bottom). For the object space, Figure 9 (top) shows the velocities for the targets of the bundle adjustment. Figure 9 (bottom), which is restricted to a maximum distance of 3.5 km, shows the results of the dense pattern of trajectories. The scatter of the data points is small. It is possible in addition, that the glacier motion itself has small but finite spatial scatter in its motion, but our approach cannot distinguish between errors and real scatter without additional information. We observe an increase in flow velocity from the southern margin of the glacier to the central flow line. A cross section of the flow velocity has been obtained from Figure 9 (top) and Figure 10 (top). There is also an increase along the flow direction toward the calving front. As it will be shown later, the area of maximum velocity corresponds to the region of maximum slope. The maximum velocity values are larger than 45 m/d.
 Considering the retreat of the glacier front during the last few years, and comparing the velocities at the same locations with those determined by Joughin et al. , we can unequivocally state that we have observed an additional glacier acceleration. We extracted from Joughin et al. [2004,Figures 1 and 2] the velocities observed in previous years at the central location of our cross section, which is 15 km behind the 1992 calving front. The velocities over time for this location are shown in Figure 10 (bottom). The increase in velocity as obtained by Joughin et al.  up to 2003 continued in 2004. For our location the velocity shows an increase of about 35% between 2003 and 2004, which corresponds to a velocity of 16 km/yr. Over the last decade the magnitude of the glacier flow has more than doubled, which helps to explain the increasing thinning rate of the glacier that has now reached a rate of 15 m/yr [Krabill et al., 2004]. It is interesting to note that such an increase of flow velocities has also been observed for other glaciers in Greenland [Rignot et al., 2004; Rignot and Kanagaratnam, 2006].
3.2. Vertical Tidal Motion of the Front Area
 The tidal motion results are shown in Figure 11 (top) in the image space with the corresponding RMS values (Figure 11, bottom). For the object space, we again have the sparsely distributed data points from the bundle adjustment (Figure 12, top) and the densely covered area of the flow trajectories up to 3.5 km distance from the camera (Figure 12, bottom). We see the full range of free-floating tidal motion only for a small portion of the glacier front, with decreasing amplitudes observed both upstream and toward the southern margin. We infer a maximum width of the tidal zone of about 0.6 km. Further east the glacier shows no tidal motion. It is interesting to note that we see three stages of front position even in the summer of 2004. The first stage is from the ASTER image in the background of Figure 12 from 27 June 2004. The second stage is recorded by our survey conducted on 20 August 2004, and the third is the front position on 9 September 2004 extracted from the geocoded Landsat image shown in Figure 1 (bottom). In 2005, the glacier front retreated fully behind our estimated grounding line as well as behind the one mapped by Echelmeyer et al., 1991] (Figure 13). Echelmeyer et al. [1991, p. 370] state that “…the actual ’line’ of ungrounding is very difficult to identify (theoretically and observationally) on such a heavily crevassed glacier.” Their site L9, which “…was considered to be the center of the grounding zone based on an abrupt change in slope from L7 to L9” showed no tidal uplift (in contrast to L7, about 2 km downstream) in short-term theodolite surveys. An insight into the slope of the area may also be deduced from the dip angles α of our flow trajectories (Figure 14), which show a steeper part close to the glacier front and a rather gentle slope in the central part. This pattern and its location agree quite well to the elevation gradient determined by Fastook et al. [1995, Figure 6]. We conclude that the steeper part toward the fjord clearly marks the transition zone from grounding to floating, which is consistent with the increased damping of the tidal signal there (Figure 12). The gentle slope of the area following upstream could either reflect the geometry of the bedrock or, more probably, of a glacial sediment layer located between the moving ice and the solid rock.
 A logical deduction then may be, that the glacial sediments act as a lubricating layer between the glacier base and the bedrock. The lubricating layer can adjust to the subglacial topography, providing enhanced lubrication at the base of the ice stream. As discussed by Thomas , such demise by enhanced lubrication has important consequences for a host of positive feedback mechanisms that could severely affect any longer-term prognosis of the health and stability of the Greenland ice sheet as a whole.
 Another interesting result is the delayed tidal response w.r.t. the ocean tide. The delay is on the order of 35–45 min (section 2.4.1, Figure 5). Our tide pole records (Figure 2) close to the glacier front were in excellent agreement with the tidal phase recorded in Ilulissat. A possible explanation is that a time-dependent flow of a sediment-water mixture must be established beneath the glacier inducing a retardation in the tidal response. Tides tend to modify the force balance of the glacier basal environment. It is interesting to note in this context, that it has been shown for ice stream C, West Antarctica, that tidal forcing travels upstream as an attenuating wave at approximately 1.6 m s−1 [Anandakrishnan and Alley, 1997].
3.3. Tidal Effects on the Horizontal Motion
 An interesting question concerns the ocean tides and their influence on the horizontal glacier flow [Thomas et al., 2003]. In West Antarctica such an in-phase velocity behavior has been well documented [Bindschadler et al., 2003]. For the Jakobshavn Isbræ, Echelmeyer et al., 1991, p. 368] state “At shorter timescales, however, the speed does fluctuate, at least on the lower glacier where fluctuations on the order of 35% have been observed on tidal frequencies.” We performed a correlation analysis of the rate of horizontal displacement and the tidal phase for a representative number of flow trajectories. For free-floating parts, for floating areas with damped tidal motion, as well as for grounded portions of the glacier, we find no significant change in horizontal displacement rate that depends on the tidal phase. This means, for our limited area and time of observation we cannot confirm the statement of Echelmeyer et al. .
 Analysis of field data taken during the summer of 2004 demonstrate that the carefully processed digital image sequences are of considerable value for understanding the kinematics of fast-moving polar glaciers. The method offers the advantages of a very high temporal and spatial resolution, combined with flexible and cost-effective data acquisition. Automatic image sequence processing by correlation techniques turned out to be quite challenging because of the extreme glacier surface topography. However, careful handling of area-based image matching tools provided excellent quantitative determination of the velocity field. Obvious disadvantages of the method are the limited area coverage and the clear topographic dependence. In this sense, terrestrial photogrammetry may provide a useful complement to a variety of airborne or satellite data acquisitions. For the frontal area of the Jakobshavn Isbræ, we obtained a dense velocity field map and this provides a detailed picture of the spatial variation of the glacier flow. For the retreating front, we observed a maximum flow velocity of 45 m/d, which is equivalent to more than 16 km/yr. This is a factor of two increase with respect to the 20th century average value. The acceleration began during the last decade [Joughin et al., 2004]. For the 1-year period 2003–2004 we have determined that a further increase of about 35% in velocity has occurred. In addition to the increased glacier velocity, the results clearly show a tidally induced height variation of the glacier tongue, while there is no evidence of tidally induced short-term fluctuations in horizontal flow. We mapped the remaining portion of the floating glacier by analyzing these tidally related vertical displacements. Upstream of the 2004 front position on 20 August 2004, a zone of about 0.6 km in width could be identified by its decreasing vertical tidal deformation. However, from 2004 to 2005, the glacier front retreat has continued, and the front position of the Jakobshavn Isbræ is now behind the area which showed the vertical tidal displacements in 2004.
 The research was funded by the German Research Foundation (DFG) under grant DI 473/21-1. The Danish Polar Center and the Greenland Home Rule government provided the research permit. Furthermore, we thank Air Alpha A/S, Ilulissat, for logistic support. The helpful comments of the Associate Editor Tavi Murray and two anonymous reviewers are gratefully acknowledged. We thank Erik Ivins for carefully reading the manuscript and for many valuable suggestions.