A complete description of the physical and mathematical algorithms used in the generation of the data products can be found in the Algorithm Theoretical Basis Documents (ATBD).

Satellite altimeters measure the time it takes a laser pulse to travel from the source to the earth's surface and back to the receiver. The GLAS laser pulse illuminates an approximately 60 m ellipsoidal footprint. Surface elevation is determined from the subtracting the GLAS-measured range vector from the altitude vector giving the satellite orbit above the earth. The range is corrected for instrument artifacts and atmospheric delays. Tides are then applied to the resultant elevation (Zwally et al. 2002). Surface height measurements represent a mean value over the footprint area. Surface slope and roughness within a footprint increase the return pulse width, or the time interval over which the signal is returned.

Over the Greenland and Antarctic ice sheets, the return waveform resembles a simple Gaussian curve with a single peak, in the absence of forward scattering by clouds. The number of Gaussian peaks in the returns depends on the surface type. A single Gaussian peak is expected over oceans, sea ice, and ice sheets. Multiple peaks may result over land and irregular ice sheet surfaces, particularly crevasses, cliffs, and areas of ice fog and low clouds. The return signal is represented as a sum of Gaussian peaks plus a bias (Brenner et al. 2000). The modeled waveform is defined as:

where:

N_p = number of peaks in the waveform
A_m = amplitude of the m^th peak
ε = bias (noise level) of the waveform
t_m = peak position
σ_m = standard deviation of the m^th peak

The following graphs show examples of waveforms over two different surfaces. Click on the thumbnails to view larger images of sample waveforms.

	Narrow waveform for a flat, uniform surface
	Bimodal waveform, with the highest peak most likely representing a tree or building in the scene

Surface elevation

A standard parameterization is used to calculate surface elevation for ice sheets, oceans, and sea ice, using the elevation of the maximum peak and no more than two Gaussian functions with a minimum spacing of 30 ns (4.5 m) between Gaussian centers. For land elevations, the centroid of the return signal is used; a maximum of six Gaussians is allowed with 5 ns (75 cm) minimum spacing.

GLAS reports a mean elevation for each ice sheet surface measurement. When multiple peaks are present, the position of the centroid of the maximum peak determines the range. The maximum peak is used because detector saturation causes a small ringing peak that does not represent the surface. This range is corrected for atmospheric delay. The corrected range, satellite position above the ellipsoid, and off-nadir pointing angle are input for calculating surface elevation.

The increment in range between the last and first peaks allows users to recalculate the surface elevation based on the first peak. The number of peaks indicates the surface variability. The range to the centroid of the received waveform, the range from the beginning to end signals, and the skewness and kurtosis of the received waveform are also useful in evaluating surface characteristics.

Note: See the Error Sources section for important updates and corrections to surface elevation for Release 34.

Surface roughness

Researchers can measure the level of surface roughness by studying the waveform. The broader the received pulse, the greater the roughness (Brenner et al. 2000, Schutz 2002). Surface roughness and slope are interrelated. The following equation for surface roughness of a rough, flat surface assumes zero slope:

where:

Δξ = surface roughness (m)
S_P = RMS received pulse width (s)
S_l = RMS transmitted pulse width (s)
S_h = RMS width of receiver impulse response (s)
c = effective velocity of light (m/s)
E = laser energy (millijoules)

The following equation for surface slope of a flat, smooth surface assumes zero roughness:

where:

S = slope
c = effective velocity of light (m/s)
z = altimeter height above the terrain (m)
q_T = halfwidth divergence angle of the laser beam
E = laser energy (millijoules)
S_P = RMS received pulse width (s)
S_l = RMS transmitted pulse width (s)
S_h = RMS width of receiver impulse response (s)

Surface slope and roughness values are only accurate for single-pulse waveforms.

Land surface relief is the range of elevations within the footprint caused by slope and roughness. For footprints lacking vegetation or cultural features, the interpretation of the land surface elevation is similar to that of ice sheets, sea ice, and oceans. For footprints that contain vegetation or cultural features, the surface height distribution created by slope and roughness is combined with the height distribution of those features.

The primary goals of the waveform and surface elevation algorithms are as follows:

• Define and determine the range to the mean surface elevation from the waveforms.

• Define mean surface elevation by correcting the range for all known errors and applying tides.

• Identify and interpret multiple surfaces within the return.

• Determine surface roughness, reflectivity, variability, and slope based on waveform characteristics.

The GLAS algorithms were adapted from existing waveform analysis algorithms including those from the Shuttle Laser Altimeter, satellite radar altimeters, and aircraft laser altimeters over land and ice sheet surfaces. The GLAS instrument records signals from four types of surfaces: ice sheets, sea ice, land, and ocean. The science requirements differ for these different surfaces. The algorithms described in this section are the same for all surface types. Different input parameters, however, lead to differences in the final results (Brenner et al. 2000).

Because of the complexity of the Earth's surfaces, The ICESat laser altimeter waveforms are complex and need to be processed on the ground to determine the location on the waveform that represents the elevation and surface characteristics within the footprint. Based on the assumptions that the surface within the laser footprint is a combination of a smooth sloping surface and a rough (random distribution) flat surface, and the transmitted pulse has a Gaussian shape, the received waveform can be modeled as a Gaussian.

Multiple Gaussian functions are used to model waveforms resulting from multi-layer surfaces. Parameters such as mean surface elevations, pulse amplitude, pulse width, and signal noise level can be extracted through inverse model fitting. Although the centroid of the waveform can be used to represent the mean surface, this is not always preferable. Over multi-layer surfaces, it is more helpful to identify the individual layers and their associated mean elevations. Distortion of the waveform shape by non-surface characteristics such as atmospheric forward scattering or detector/amplifier saturation causes the centroid of the waveform to misrepresent the actual mean surface. These effects can be diminished by fitting a Gaussian to the return and using the centroid of the Gaussian as the mean surface elevation.

The initial estimates and covariance matrix are set to optimize the fit to the leading edge of the return for the maximum-peak Gaussian. Two Gaussians are allowed to account for the extended tail of the returns that have high forward scattering components, or two distinct surfaces in the footprint.

A standard parameterization is tuned to ice sheet-type surfaces, which normally have one major reflection point. An alternative parameterization is tuned to more complex land surfaces with multiple reflection points; the last reflection should be from the surface. The results of the algorithm are useful for discerning the nature of ice sheets and land undulations, and perhaps multiple vegetation canopies near the land surface. For smooth terrain without intervening reflections, the results of both parameterizations are identical.

For land surfaces, the algorithm characterizes the return pulse by fitting Gaussian distributions to each mode (peak) in the waveform. Surface elevation over land is derived from the centroid of the return.

Inferences of sea surface elevation and ocean wave height are determined from an average of many consecutive footprints, since the 60 m footprint resolution of GLAS covers less area than a typical ocean wave. The maximum wave height is calculated from the highest and lowest surface elevations inferred from all the pulse widths for wavelengths less than twice the footprint diameter (approximately 120 m). Generally, a sea ice footprint contains a combination of rough and smooth ice. The average elevation is typically represented by the centroid of the latest Gaussian peak in the return pulse (Brenner et al. 2000).

GLAS Release 34 Correction of Release 33 Data Issues

1. Correction to the ICESat Data Product Surface Elevation due to an Error in the Range Determination from Transmit-Pulse Reference-Point Selection (Centroid vs Gaussian)
   • It was determined that an important correction to the surface elevations on the ICESat products was not applied. The range from which the surface elevation should have been calculated is from the midpoint of the Gaussian peak on the transmit pulse to the midpoint of the Gaussian peak on the received pulse. However the location of the centroid of the transmit pulse was inadvertently used and the difference (defined as G–C) between the transmitted pulse centroid and Gaussian peak was never applied. The effect of this error of omission varies on a shot-to-shot basis and for the calibration passes over the Salar de Uyuni in Bolivia varied from ± 6 cm over the mission. Similar results were found when tabulating the track-averaged corrections (G–C) over Antarctica. Note that the (G–C) values are fairly constant within a campaign, so any elevation data adjustments made to data release 633 or older for "campaign-level" biases that were determined to an independent reference surface would eliminate the G–C error from the elevation data on an average basis.
   • The surface elevations on GLA06 (global elevation product), GLA12 (ice sheet), GLA13 (sea ice), and GLA15 (ocean) are affected. The GLA14 (land) elevations are computed using the centroid of the received pulse, as well as the centroid of the transmit pulse, and therefore the elevation calculation on GLA14 is correct as intended. However, GLA14 data users who use the range Increment for the up to six Gaussian fitted peaks may want to use the G–C correction also.
2. Dry Troposphere Correction Jitter
   • It was reported that at times the dry troposphere range correction contained several centimeters of jitter. The issue was traced to the threshold range being used to determine the dry troposphere correction in regions of rapidly changing shape of the return signal waveforms. This was normally in poor signal areas or very rough terrain. At times the threshold range relative to the standard fit would jump tens of meters and induce several centimeter changes in the dry troposphere correction. Changes were made to use the centroid range to determine the troposphere correction. The centroid closely follows the standard fit range.
   • During the investigation it was determined that the GSAS code was using only one of the 6–hour met files for the pressure inputs. Code was modified to properly interpolate between the two 6–hour met files.
3. The GLAS product high resolution DEM was determined to have issues in the Southern Latitude when the source was SRTM data. It was reported to have wrong values that showed as banding.
   • There was an interpolation error in the southern latitudes during the creation of the SRTM track files (ANC51). The code error caused the DEM values to be correct at the mid point latitude between the interpolation points, with increasing errors on either side. This resulted in the banding error seen when comparing GLAS SRTM DEM values with the source SRTM DEM. The higher the DEM gradient the larger the error so they were significant over Australia. The code was fixed and track files were re–created.
4. The order of preference for which values are used when the SRTM and CDED overlap was changed. Starting with this release the SRTM is always used in the overlap region.
5. Some parameters on GLA14 were invalid when i_elev was valid.
   • It was determined that if any standard fit for a one second was invalid, checks were not made to determine if GLA14 had valid elevations during that second. Changes were made to provide the parameters based on a valid alternate fit if the standard fit was invalid for a shot.
6. GLA12, 13, 14 and 15 atmosphere character confidence flag was always zero.
   • It was determined that the atmosphere confidence flag from GLA09 was not being placed on GLA12, 13, 14, and 15. Changes were made to correctly put the confidence flags on the products.
7. Occasional mismatch of GLA09 atmosphere characteristic flag value and the value reported on GLA06 and 14.
   • On occasion the atmosphere characteristic flag was zero on GLA06 and 14 when a non-zero value was on GLA09. It was determined that the time tolerance check for the one second GLA06 time to the GLA09 time tag did not allow for sufficient jitter between the time tags. This caused a GLA09 record not to be selected as the source for the values. Several other parameters were also found to be invalid.
8. Saturation Correction - see Saturation Correction Guidance.

For further detail, see the GSAS V6.1 Release Notes May 2014 document.