2.1. Instrument platform

The laser altimetry system is part of the SOAR airborne geophysical suite installed in a ski-equipped Twin Otter aircraft (Reference Bell, Childers, Arko, Blankenship and BrozenaBell and others, 1999; Reference Blankenship, Alley and BindschadlerBlankenship and others, 2001). The geophysical-mapping systems on board the aircraft include a gravimeter, magnetometer, laser altimeter and ice-penetrating radar. Other equipment mounted on the aircraft includes a central processing unit, time code generator, three dual-frequency GPS receivers and an inertial navigation system (INS). During most of the surveys performed by SOAR, all sensors collected data simultaneously, providing direct measurements of ice surface elevation, internal layering of the ice, bedrock geometry and composition, and basal water conditions. However, only measurements from the laser-altimeter system, which includes the auxiliary components mentioned above, are presented here. The laser altimeter measurements are made while the aircraft flies along a predetermined flight path guided by real-time GPS.

2.2. Differential GPS positioning

The accuracy of differential GPS positioning is largely dependent upon the accuracy of the base-station positioning. This is an especially important consideration for this project, because the base stations are located on a moving ice sheet. Several Ashtech Z-12 dual-frequency receivers located at different base camps were used as static base stations for laser altimetry flights. Because the base stations were moving, new positions had to be determined on a daily basis. Basestation GPS receivers collected data on each day that laser flights were conducted. SOAR provided the daily position of each base station, which was calculated using the automated GPS inferred positioning system (GIPSY) developed by the Jet Propulsion Laboratory (JPL) at the California Institute of Technology, Pasadena, CA. GIPSYreported a maximum rootmean-square (rms) error of 0.02 m for each position coordinate. Figure 2 illustrates the horizontal and vertical positions reported by GIPSY for all 24 hour surveys of one station. The daily positions are plotted and a fit is calculated to remove random positioning errors. The base-station position used for each survey flight is taken from the fitted line.

Fig. 2.

Positions reported by GIPSY (crosses) for one 1997/98 base station set up at Down-B. (a) The distribution of vertical positions for each 24 hour survey. (b) The horizontal positions for each 24 hour survey. Coordinates are based on a polar stereographic projection with the same orientation as Figures 1, 8 and 9. SOAR provided the positions and calculated the fits (black line) that are drawn through the data points.

In 1997/98, base camps were at Siple Dome and Down-B. Byrd Station and Siple Dome were used in 1999/2000 (Fig. 1). The daily GIPSY positions were used to track the movement of each of the stations. Two of these base stations, Byrd Station and Siple Dome, are located on slow-moving (12.5 and 1.5 m a⁻¹, respectively) portions of the ice sheet. The Down-B base station is located near the middle of the fast-moving (>500 m a⁻¹) Whillans Ice Stream. Laser surveys last 3–7 hours, with an average flight time of approximately 4 hours. In 4 hours, the reference GPS receivers at Down-B, Byrd Station and Siple Dome moved horizontally about 0.23, 0.006 and 0.0007 m, respectively. Consequently, errors in GPS positioning due to ice movement are only expected for the five flights that originated at the Down-B base camp. For Down-B flights, small errors (<0.23 m) in horizontal aircraft positioning are expected, though there should be almost no effect on the vertical positioning.

Once the base-station position was established, the relative position of the Ashtech or TurboRogue receiver on the aircraft was calculated using the Trimble GPSurvey processing software. A variety of base/rover combinations were processed for each flight to insure that the best possible solution was obtained. This was most important when the aircraft departed from one base and landed at another. The best solutions were consistently obtained when the base station located at the origin of each flight was used.

Reference Shi and CannonShi and Cannon (1995) show that the accuracy of differential GPS positioning on a moving aircraft can be at the 0.1 m level if tropospheric, ionospheric, precise satellite orbits (ephemerides) and multipath corrections are used during differential, carrier phase post-processing. All of these corrections were applied with GPSurvey, which reported a maximum rms of 0.2 m for the 22 flights used during the two seasons. Flights with larger positioning errors were excluded from this analysis. Reference Bell, Childers, Arko, Blankenship and BrozenaBell and others (1999) reported similar errors when using the Kinematic and Rapid Static (KARS) software to process SOAR GPS data.

To assess the performance of the GPSurvey processing, one 1997/98 survey was also processed with the National Aeronautics and Space Administration’s (NASA) GPS Inferred Trajectories for Aircrafts and Rockets (unpublished GITAR program documentation by C. Martin in 1991). The two solutions agree well, having a maximum difference of 0.01 m in the horizontal and 0.2 m in the vertical (Fig. 3). Neglecting the take-off and landing portions of the flight, the maximum difference is 0.08 m in the vertical.

Fig. 3.

Comparison of two GPS solutions for the same flight produced using different software, GPSurvey and GITAR (J. Sonntag, EG & G, NASA Wallops Flight Facility,VA). Black line is the elevation of the aircraft during the flight. Gray line is the elevation difference found when the GPSurvey solution is subtracted from the GITAR solution.The agreement is poorest during take-off and landing of the aircraft.

2.3. Acquiring laser ranges

The Azimuth LRY 500 is a diode pumped Nd:YAG (neodymium: yttrium aluminum garnet) pulsed laser transceiver, operating in the near-infrared domain (1064 nm). Pulsed lasers measure the travel time of a laser pulse from the laser firing point (LFP) to the surface and back to the receiver. To measure the time between the transmitted and the received pulses, the Azimuth LRY 500 rangefinder uses 50% constant fraction discrimination. The timer starts at some consistent point on each transmitted pulse. Each timing event ends when the return pulse strength reaches half of its maximum amplitude.Thus the need of a “range walk” correction is eliminated. The manufacturer reports a single-pulse accuracy of 0.1 m for distances up to 1.7 km. When the beam divergence is set for 4.5 μrad, it produces a footprint with 1.5 m diameter from the 300 m nominal flight altitude. The system is capable of 1000 pulses per second, but only 500 pulses are measured here to reduce the size of the data stream. To avoid measuring surface roughness and to further reduce the size of the data stream, 64 pulses were integrated 8 times per second (Reference Blankenship, Alley and BindschadlerBlankenship and others, 2001). This results in one average elevation for every 1.5 m by 8 m area when the aircraft flies at 300 m terrain clearance.

2.4. Measuring aircraft attitude

The attitude of the aircraft, i.e. the heading, pitch and roll angles, determines the pointing direction of the laser. These angles are measured with a laser gyroscope that is part of the Litton Aero Products LTN-92 INS unit. The INS has a quoted accuracy of 0.05° in all three angles (Reference Vaughn, Bufton, Krabill and RabineVaughn and others, 1996), which translates to a <0.06 m error in calculated surface elevations when the aircraft is flying at <300 m terrain clearance and the off-nadir pointing angles are <15°. These angles are not exceeded during survey missions, so the measured attitude contributes very little to the overall error of the procedure.

2.5. Solving timing issues

Each data-collecting system operates independently and not in synchrony with the others. Universal Time Coordinated (UTC) is used to reference the system collections. Laser and INS measurements are tagged with a counter time that is corrected to UTC using information provided by the GPS time-code generator. An additional correction is required for the individual attitude parameters, because the attitude angles measured by the INS are not instantaneously available on the local digital bus. The manufacturer specifies that these time lags are as follows:110 ms in true heading, 60 ms in pitch and 50 ms in roll for the LTN-92 (Reference Vaughn, Bufton, Krabill and RabineVaughn and others, 1996). To check whether these values are correct, pitch and roll maneuvers were conducted over a flat test field, because timing offsets cause deformation of the measured surface when off-nadir angles are large. This test yielded slightly different time lags of 60 ms in roll and 85 ms in pitch. The measured time lags are preferred over the reported ones. The final step in correlating the data streams is to interpolate the INS and GPS data for the times when laser ranges have been recorded. Interpolation is necessary because the GPS data are recorded at 2 Hz, while laser measurements are recorded at 500/64 Hz, and INS measurements at 8 Hz. Different interpolation schemes were tested and provided essentially the same results. Therefore a simple linear interpolation was used.

2.6. Computing the laser footprint position

The laser range recorded during the flight is a slant range to the surface. To compute the position of the laser footprint in a global, geographic coordinate system, the laser range, aircraft position and aircraft attitude are combined according to the scheme described in Reference LindenbergerLindenberger (1993),Reference Vaughn, Bufton, Krabill and RabineVaughn and others (1996) and Reference HoftonHofton and others (2000). We developed FORTRAN computer programs to filter and combine the different data streams so that the position of the illuminated laser spot on the snow surface could be reliably determined. Figure 4 shows individual measurements from each instrument and the calculated ice surface elevation for one laser flight.

Fig. 4.

Plots of the data recorded by each instrument for a laser survey over Siple Dome. (a) Aircraft elevation obtained with GPS. (b, c) Attitude of the aircraft recorded by the INS. (d) Range to the snow surface recorded by the laser altimeter. (e) Resulting elevation profile of the ice-sheet surface.

Equation (1) includes a series of coordinate transformations starting at a local reference system centered at the LFP (indicated by subscript L) and ending in the World Geodetic System 1984 (WGS84) Cartesian reference frame (indicated by subscript W). Rotations are indicated by R.

(1)

The laser ranger is mounted inside the aircraft pointing basically nadir. The ranger measures the distance between the LFP and the snow surface (S) from points along the flight trajectory . The laser range vector is first corrected for range bias and then transformed into a local aircraft reference system (indicated subscript A) centered at the GPS antenna phase center. The transformation is carried out by rotating the laser range vector by the laser mounting bias (ΔR _LS ), which accounts for the misalignment between the laser and aircraft reference frames. The offset vector between the LFP and the GPS antenna is then added .

The next rotation uses the INS mounting biases (ΔR _INS) to transform the laser vector from the local aircraft reference system into the local INS reference frame with axes defined by the roll, pitch and heading axes of the INS. The INS mounting biases account for the angular differences between the aircraft body system and the roll, pitch and yaw axes. This step can also account for the difference between local vertical and the direction perpendicular to the geodetic ellipsoid (WGS84).

The last rotation of this series transforms the laser vector from the INS reference frame to the local-level reference frame using the attitude of the aircraft (R _INS). The local-level reference frame is an Earth-tangential reference system centered at the GPS antenna. The z axis is perpendicular to the WGS84 ellipsoid and points downward. The x axis lies along the intersection of the local GPS meridian and a plane parallel with the tangent plane to the ellipsoid (i.e. points north). The y axis completes the righthand reference system and is positive to the east.

Finally the expression of the laser vector from the GPS antenna to the laser spot on the snow surface is transformed into the WGS84 global Cartesian system. Rotations for the latitude and longitude of the aircraft are performed to align the local-level reference frame axes with the WGS84 axes. Latitude and longitude values are recorded by the GPS on board the aircraft.The position of the laser footprint is computed by adding the laser vector in theWGS84 system to the vector recorded by the GPS on board the aircraft .

3.1. Ground calibration

The offset vector between the LFP and GPS antenna phase center was measured by traditional surveying methods using a theodolite and tape measure. Several symmetrical “hard” points on the aircraft (joists, flap indicators and seats) were used to define the aircraft reference system. One axis runs the length of the aircraft between symmetric points. A second axis is perpendicular to the first axis and is aligned with the aircraft wings. The third axis is vertical and perpendicular to the other two. The offset vector of [−1.378 m, 0.411 m, 1.696 m] was measured in 1996. The alignments of the INS and laser ranger were measured by electronic level to estimate the instrument mounting biases. Since the angular measurements over short distance can result in large errors, these mounting bias estimates have been refined by in-flight calibration.

3.2. In-flight calibration

Previously used calibration techniques require surveys of large, smooth, flat surfaces (e.g. lakes, ocean, runways) to resolve the misalignment between the INS and the laser (e.g. Reference Krabill, Thomas, Martin, Swift and FrederickKrabill and others, 1995; Reference HoftonHofton and others, 2000). Since flat, horizontal surfaces are not found in the interior of the West Antarctic ice sheet (WAIS) a new calibration algorithm has been developed, which facilitates calibration over arbitrary, natural surfaces (Reference Filin and CsathóFilin and Csathó, 2002). The core of the procedure is an analytical approach that combines Equation (1) with an analytical approximation of the surface. The calibration parameters are determined by least-squares adjustment.

To calibrate the system, skiways next to the Siple Dome and Byrd Station base camps were surveyed using both snowmobile-mounted GPS and laser altimetry (Fig. 5). For a more convenient analysis of the results, the data were converted into a local coordinate system. Conversion to a local frame consists of a constant translation of the origin, usually into a surveyed point within the site. In this case, the antenna phase center of the base-station GPS is used. The x axis of the local system points to true north, the z axis points toward the center of the Earth, and the y axis completes a righthand reference system.

Fig. 5.

Laser-altimetry calibration surveys and snowmobile GPS surveys at (a) Siple Dome in December 1997 and (b) Byrd Station in December 1999.

Analysis of the recoverability of the systematic errors shows that not all systematic errors can be recovered for a nadir-looking, laser profiling system (Reference FilinFilin, 2001). The effect of a heading bias is absorbed in the pitch and roll biases, and the INS and laser mounting biases cannot be separated. Therefore we combine the INS and laser mounting biases into laser-system pitch and roll mounting biases. The recovered biases also include the range bias of the laser. Pitch and roll maneuvers were performed over the calibration site to obtain a more robust solution for the mounting biases.

The difference between the ice-sheet surface reconstructed from non-calibrated and calibrated laser data and the reference surface at Byrd Station is shown in Figure 6. After the system calibration the mean of the residual is zero and its standard deviation is 0.06 m. The estimated instrument biases are −1.6° pitch, −0.1° roll and 0.12 m in range during the 1997/98 field season and −1.9° pitch, −0.2° roll and 0.6 m in range for the 1999/2000 season. In both seasons, the pitch and roll mounting biases determined by in-flight calibration agree well with the combined laser and INS mounting biases measured on the ground. The large range bias in the second season might be caused by problems related to the saturation of the return signal. Changes in atmospheric conditions such as temperature and humidity could also contribute to this error, because they affect the speed and refraction of light, which reduces measured laser ranges through the atmosphere (Reference Marini and MurrayMarini and Murray, 1973). Some users of airborne laser altimetry use an atmospheric correction, based on the measured temperature and humidity at the LFP and at the target (Reference Vaughn, Bufton, Krabill and RabineVaughn and others, 1996; Reference Ridgway, Minster, Williams, Bufton and KrabillRidgway and others, 1997). In Antarctica, however, there are too few meteorological stations located on the ice surface to effectively utilize this technique. Instead, any effects that the atmosphere may have on the laser range are considered in the range bias along with other possible range errors. Once determined, the range bias is assumed to be unchanging throughout a season and is added to each range measurement from that season.

Fig. 6.

Two laser profiles derived from the same survey along skiway S2 at Siple Dome before and after the removal of instrument range and mounting biases. Gray line is the reference snow surface derived from snowmobile-mounted GPS surveys.

3.3. Testing calibration values

After correction of range and angular biases, the SOAR 319 system performed well over short distances (∼10 km) from the base stations at Siple Dome. Repeat flights over Siple Dome’s skiway S1 show a 2.8 cm bias with a rms of 10.3 cm. The bias is attributed to surface slope, because the repeat flight-lines were usually about 25 m apart. The rms difference may be a true measurement of surface roughness caused by sastrugi. Sub-decimeter accuracy was also found when laser-derived elevations were compared to elevations measured with the snowmobile surveys (Fig. 7).

Fig. 7.

Surfaces derived from repeat laser altimetry (black lines) and snowmobile-mounted GPS (grey lines) along skiway S1 at Siple Dome.