Notation

• A_e Effective antenna aperture (m²)

• A^" Received power amplitude (W)

• β Two-way beam-width coefficient

• c Speed of light, 3 x 10⁸ m s^-1

• C_s Speed of light in snow (ms^-1)

• d_p Depth of penetration (m)

• dS Differential shell area

• dR Differential shell thickness

• dP_r Differential received power

• δ Two-way incremental ranging time

• ∈_s Relative complex dielectric constant of snow

• ∈_i Relative complex dielectric constant of ice

• f Altimeter frequency

• G Antenna gain at nadir

• γ One-way beam-width coefficient

• Г Fresnel reflection coefficient

• h Height of satellite (m)

• K Volume coefficient

• k_a Absorption coefficient of snow (m^-1)

• k_e Extinction coefficient of snow (m^-1)

• k_s Scattering coefficient of snow (m^-1)

• l Path length in snow (m)

• λ Altimeter free-space wavelength (m)

• r Transmitted power at satellite (W)

• R_m Power density incident upon snow surface (W m⁻²)

• Pt Power density transmitted across air-snow boundary (W m⁻²)

• P_FS Flat-surface impulse response

• P_r Received power at satellite (W)

• P_s Average surface-impulse response

• P_T Transmitted pulse profile

• ψ Azimuth angle

• r Radius of ice grains

• R Satellite range (m)

• ρ_s Density of snow (Mg m⁻³)

• ρ_i A Density of ice (Mg m⁻³)

• S(δ) Brown surface-scattering model

• SV(δ) Combined surface-/volume-scattering model

• SWH Ocean significant wave height (m)

• σ_s r.m.s. surface roughness (m)

• σ_vs Back-scattering cross-section per unit volume (m²m⁻³)

• t Time

• t₀ Round-trip time corresponding to intersection of altimeter pulse with a flat surface

• T Power-transmission coefficient

• θ Elevation angle

• θ_3dB Half-power beam width of antenna pattern

• V(δ) Volume-scattering model

Introduction

The ice sheets of Greenland and Antarctica constitute 10% of the surface land area of the Earth, and the accumulated ice comprises 90% of the global fresh-water reserves. Ice sheets exist as a consequence of climate and are coupled closely with the natural environment through complex exchanges of energy and mass. An understanding of ice-sheet dynamics is a fundamental requirement to achieve a realistic model of climatic and oceanographie processes. Even though the ice sheets play a critical role in global processes, they remain the most poorly explored and little-studied areas on the Earth’s surface. Research in the polar regions is hampered by the vast size of the ice sheets, their remoteness and the harshness of the polar environment (Reference Thomas,Thomas and others, 1985).

Some global-circulation models predict a world-wide temperature increase between 3.0 and 5.5 K due to doubling in the current levels of carbon dioxide and other greenhouse gases (Reference Schneider,Schneider, 1989). Also, the models show that the expected climatic warming is more strongly focused in the polar latitudes (Reference Bryan,Bryan and others, 1982; Reference Schlesinger,Schlesinger, 1984; Reference Hansen,Hansen and others, 1988). Enhanced ice-sheet melting could raise global sea levels, impact the circulation of deep ocean wateres and seriously alter the heat/momentum transfer between the ocean and the atmosphere.

Major changes in the volume of the polar ice sheets relate directly to global sea levels. Because the Greenland and Antarctic ice sheets contain a volume of ice equivalent to 70 m of sea level, volume changes as small as 1% are significant. During the last interglacial period, the sea level was 6 m higher than present-day levels. Reference MercermMercer (1978) proposed the collapse of the West Antarctic ice sheet as a possible cause of the 6 m sea-level increase. During the past century, the sea level, as recorded from tide-gauge data around the world, apparently rose by 10–20 cm. Although both thermal expansion of the oceans and ice-sheet melting contribute to sea-level rise, no more than 25% of this increase can be attributed to thermal expansion (Reference Etkins, and Epstein,Etkins and Epstein, 1982; Reference Barnett,Barnett, 1984). The apparent increase met with some skepticism because the rate of sea-level rise exhibited a large geographic variability. However, recent results, which adjust the tide-gauge data to remove contributions of glacial isostatic adjustment to local sea-level trends, show that a globally coherent increase in sea level of 2.4 ± 0.9 mm year⁻¹ is occurring (Reference Peltier, and Tushingham,Peltier and Tushingham, 1989).

Early indications of a warmer climate in the polar regions are likely to show as ice thinning, increased calving from ice shelves and an inland migration of ice-shelf grounding lines (Reference Thomas,, Sanderson, and RoseThomas and others, 1979). Because ice flows along the direction of maximum regional surface slope, elevation data are used to delineate major drainage basins, ice flowlines, ice streams and grounding lines. Moreover, repeated mapping of surface elevations can be used to detect changes in the mass balance (volume growth) of the ice sheets (Zwally in Reference Nye,Nye, 1975; Reference Brooks,, Campbell,, Ramseier, Stanley and ZwallyBrooks and others, 1978). A positive mass balance is associated with ice-sheet thickening, while a negative mass balance is associated with thinning. Although most mountain glaciers are retreating, there is no consensus as to the behavior of the Greenland and Antarctic ice sheets. Localized field measurements indicate that some regions of the ice sheets are thickening while others are thinning. Systematic monitoring of surface elevations requires accurate repetitive measurements over the millions of square miles that comprise the continental ice sheets.

Even though space-borne altimeters were primarily designed for oceanographie applications, the orbits of Seasat (Reference MacArthur,MacArthur, 1978; Reference Townsend,Townsend, 1980) and Geosat (Reference MacArthur,MacArthur and others, 1987) extended to latitudes of ±1°, covering major parts of the Greenland and Antarctic ice sheets. The European Space Agency’s ERS-1 satellite altimeter currently covers latitudes up to ±82°. Reference Robin,Robin (1966) first proposed satellite altimetry for surveying the ice sheets. Over the last two decades, space-borne altimeters provided the only proven means for measuring surface elevations with the precision and spatial coverage required for meaningful ice-sheet studies (Reference Thomas,Thomas and others, 1985; Reference Thomas,Thomas, 1991). Data sets provided by the Geos-3, Seasat and Geosat satellites have been used to produce surface-elevation maps of large parts of Greenland and Antarctica with a 2 m accuracy (Reference Brooks,, Campbell,, Ramseier, Stanley and ZwallyBrooks and others, 1978; Reference Brooks, and Norcross,Brooks, 1982; Reference Zwally,, Bindschadler,, Brenner,, Martin, and ThomasZwally and others, 1983; Reference Bindschadler,, Zwally,, Major, and Brenner,Bindschadler and others, 1989). This accomplished in a few years what would have taken decades of intensive field surveying to produce. Maps of specific surface features such as ice margins, grounding lines and ice shelves have also been produced (Reference Thomas,, Martin, and ZwallyThomas and others, 1983; Reference Brooks,Brooks and Norcross, 1984; Reference Cudlip, and Mclntyre,Cudlip and Mclntyre, 1987; Reference Partington,, Cudlip,, Mclntyre and King-HelePartington and others, 1987; Reference Zwally,, Stephenson,, Bindschadler, and Thomas,Zwally and others, 1987; Reference Ridley,, Cudlip,, Mclntyre and RapleyRidley and others, 1989; Reference Stephenson, and Zwally,Stephenson and Zwally, 1989). In addition, the elevation data were used to assess the mass balance of the Jakovshavns Isbrae drainage basin in western Greenland (Reference Bindschadler,Bindschadler, 1984).

One of the most important glaciological applications of altimetery data to date was the combined analysis of the Geos-3, Seasat and Geosat data sets, spanning a period of 10 years, to assess the growth of the Greenland ice sheet. By analyzing satellite cross-over points and comparing changes in elevation, Reference Zwally,, Brenner,, Major, Martin and BindschadlerZwally and others (1989) estimated that the southern part of the Greenland ice sheet (south of 72°N) grew at an average rate of 23 cm year⁻¹ from 1978 to 1986. Reference Zwally,Zwally (1989) suggested an increase in precipitation rates caused by a warmer polar climate as a possible cause of the mass imbalance. The cross-over analysis demonstrates the ability of satellite-radar altimetry to produce results of global significance.

Altimeter Principles and Techniques

The satellite-radar altimeter measures the precise time it takes a radiated pulse to travel to the surface and back again. If the orbital position of the satellite is known relative to a reference surface, then the measured time, converted to range, is used to derive the elevation of the reflecting surface. In addition to measuring range, the altimeter records a mean “wave form” consisting of an average of return-pulse echoes (typically 100) and estimates other geophysical parameters such as ocean wave height and surface wind speed. Since the Geos-3, Seasat and Geosat satellites were developed primarily for oceanographic applications, all on-board estimation of geophysical parameters uses a model that relates the mean return wave form to back-scattering from the ocean surface. A diagram of the altimeter pulse interaction with a flat surface and the corresponding mean return wave form is shown in Figure 1. As the incident pulse strikes the surface, a circular region that increases linearly with time is illuminated. Correspondingly, a linear increase in the leading edge of the return wave form occurs. After the trailing edge of the radar pulse intersects the surface, the region back-scattering energy to the satellite comes from an expanding annulus of constant area. At this point, the return wave-form amplitude reaches a maximum and begins to trail off due to the reduction of off-vertical scattering by the altimeter’s antenna pattern. For a rough ocean surface, the leading edge of the return pulse is “stretched” because scattering from the wave crests precedes scattering from the wave troughs as the pulse wave front progresses downward. Thus, the width of the leading edge of the return pulse is related to the height of ocean waves.

Fig. 1. The interaction of a radar-altimeter pulse with a horizontal and planar surf ace, from its initial intersection (to), through the intersection of the back of the pulse shell with the surface (t₀), to the stage where the pulse begins to be attenuated by the antenna beam (t₂). The return is from the surface only (Reference Ridley, and Partington,Ridley and Partington, 1988).

The radar-altimeter model relates the interaction of the return wave form to sea-surface state. Since the back-scatter area seen by the altimeter is restricted to a fraction of a degree around vertical, the ocean surface can be approximated by a horizontal planar surface with a large number of scattering facets distributed randomly about the mean sea surface. Reference Moore, and Williams,Moore and Williams (1957) showed that the mean altimeter return wave form as a function of time is described by the convolution of two terms,

where P_r(t) is the received power at the satellite. P_T(t) is the transmitted-pulse profile after passing through the altimeter receiver, and P_s(t) is a term involving the distribution of scatterers, their back-scattering properties and the antenna gain. Reference Barrick, and DerrBarrick (1972) used this convolutional form and obtained a double integral describing the altimeter return wave form assuming a Gaussian distribution of scattering facets. Reference Brown,Brown (1977) approximated the transmitted pulse shape, the altimeter’s antenna pattern and the range distribution of scatterers with Gaussian functions to generate an analytical solution for the mean ocean return.

The mean return wave form given by the Brown surface-scattering model is

The flat-surface impulse response, P_FS( δ)i is given by

where c is the speed of light, h is the height of the satellite and δ is the two-way incremental ranging time

The one-way beam-width coefficient, γ is

where θ₃dB is the half-power beam width of the altimeter’s antenna pattern. The parameter σ_c is determined from

where σ3_P is related to the altimeter’s transmitted pulse width, T, by

and σ is the r.m.s. surface roughness.

All on-board estimation of oceanic geophysical parameters uses the ocean surface-scattering model (e.g. Reference Hancock,, Forsythe, and LorellHancock and others, 1980; Reference Hayne,Hayne, 1981). Mean ocean-return wave forms from the Brown model appear in Figure 2, where the family of curves results from varying the r.m.s. surface roughness. For smallσ_s, the return wave forms rise sharply to the plateau region and then begin to trail off due to the reduction of antenna gain. As σ_s increases, the slope of the leading edge becomes shallower until no fall-off in the plateau region of the wave form is apparent. Over the ocean, the wave-form’s leading-edge width is used to estimate significant wave height (SWH) where

The satellite range to the mean sea surface is measured by tracking the half-power position on the leading edge of the wave form. Since the reflectivity of the ocean surface is nearly constant, the amplitude of the return wave form is used to estimate the surface wind speed. As surface winds increase, the ocean surface becomes rougher and fewer facets will contribute specularly reflected energy, causing a decrease in the overall wave-form amplitude. Over the last 15 years, various forms of the surface-scattering model have been used to study sea levels (e.g. Reference Millerm and MilbertMiller and others, 1986; Reference Cheney,, Miller,, Agreen, Doyle and DouglasCheney and others, 1989), and wave-height/wind-speed measurements were validated through comparison with in situ data (e.g. Reference Fedor,Fedor and others, 1979; Reference Brown,, Stanley, and Roy,Brown and others, 1981; Reference Fedor, and Brown,Fedor and Brown, 1982; Reference Dobson,Dobson, 1987; Reference Monaldo,Monaldo, 1988).

Fig. 2. Ocean surface-scattering altimeter wave forms from the Brown model for σ₈ = 0.2–2.0 m.

The on-board algorithm that estimates the satellite range tracks the half-power point on the leading edge of the return wave form (Fig. 2). The return wave form is digitized into equally spaced range “gates”, where the center gate should coincide with the half-power point on the mean wave form. The design of the tracking algorithm is based upon the assumption that the range to the surface changes slowly and predictably with time. This is valid for the ocean surface, and the algorithm positions the center gate at the half-power point very accurately. However, over the ice sheets the half-power point deviates substantially from the center gate because of larger topographic variations. Since the telemetered range corresponds to the center gate, further processing on the ground is required to correct the range estimate for ice-sheet wave forms. This procedure is called “repacking”. The retracking correction measures the distance from the center gate to the leading edge of the return wave form. All ice-sheet altimeter data must be retracked to produce accurate elevation measurements (Reference Martin,, Zwally, Brenner and BindschadlerMartin and others, 1983; Reference Zwally,, Brenner,, Major, Martin and BindschadlerZwally and others, 1990), and one must apply further adjustments to correct for elevation errors caused by local surface slopes (Reference Brenner,, Bindschadler,, Thomas, and Zwally,Brenner and others, 1983).

The retracking algorithm used to correct range measurements over the ice sheet uses a modified form of Brown’s analytical surface-scattering model (Reference Zwally,, Brenner,, Major, Martin and BindschadlerZwally and others, 1990). The modified model includes a parameter that controls the trailing-edge slope of the return wave form to account for pointing-angle errors and ice-sheet slopes. In addition, a double-ramped form of the model is used to retrack complex wave forms that occur when two distinct and nearly equidistant surfaces contribute back-scattered power within the sampled range of the altimeter. This modified version of the Brown model retracked all Seasat and Geosat ice-sheet wave forms to obtain the corrected surface elevations now archived at both the World Data Center for Glaciology in Boulder, Colorado, and at NASA/Goddard Space Flight Center in Greenbelt, Maryland.

Recent work by Reference Ridley, and Partington,Ridley and Partington (1988) demonstrated that return wave forms from the ice sheets differed substantially from those predicted by the surface-scattering model in many cases. They proposed that volume-scattering from beneath the ice-sheet surface contributed substantially to the return power, and they developed a model based upon the numerical evaluation of an integral to describe the return wave forms. Based upon a qualitative analysis of averaged return wave forms, Reference Partington,, Zwally,, Ridley and RapleyPartington and others (1989) showed that the shape of the altimeter wave forms from Greenland corresponded roughly to surface-scattering in the low latitudes and volume-scattering in the higher latitudes. They observed only subtle variations in the shape of the averaged wave forms from the Antarctic plateau regions. Moreoever, Reference Davis, and Poznyak,Davis and Poznyak (in press) showed that 10 GHz radar signals can penetrate many meters below the ice-sheet surface of East Antarctica, and that the amount of penetration varied with surface elevation. Thus, sub-surface volume-scattering can contribute significant amounts of return energy at frequencies used by satellite altimeters.

The presence of volume-scattering suggests altimeter wave forms may contain important geophysical information about the surface conditions of the ice sheet. In this paper, we develop an analytical model that describes altimeter return wave forms from the continental ice sheets. Using Gaussian approximations for the altimeter’s antenna pattern and transmitted pulse shape, we derive an analytical solution for the volume-scattering power from beneath the ice-sheet surface. Because a closed-form solution is obtained, we combine the volume-scattering model with the Brown surface-scattering model to develop a retracking algorithm that can be applied directly to individual altimeter return wave forms (Reference Davis,Davis, in press). The term “individual wave forms” refers to wave forms telemetered from the satellite at a rate of 10 s^-1, which are themselves an average of 100 return-pulse echoes.

By retracking individual wave forms, quantitative estimates of the geographic and seasonal variations in the near-surface properties of the ice sheet were determined (Reference Davis, and Zwally,Davis and Zwally, 1993). These correlate well with elevation, latitude, mean annual temperature, etc. Furthermore, by comparing retracking locations on individual wave forms, the combined model can be used to assess the accuracy of surface elevations produced by various retracking algorithms. Seasonal variations in the amount of signal penetration could produce erroneous estimates of ice-sheet elevation change, especially with algorithms assuming only surface-scattering contributions to the return power. This needs to be evaluated to allow accurate detection of changes in the mass balance of the ice sheets by temporal studies of satellite cross-overs. This is extremely important for current and future altimeter missions that cover higher latitudes, where greater signal penetration is expected to occur.

Altimeter Wave-Form Model

For the Seasat and Geosat altimeters, each return wave form is an average of 100 return pulse echoes. The return wave form, elevation estimate and other geophysical parameters are telemetered at a rate of 10 s⁻¹, which corresponds to a satellite traverse distance of 662 m. The altimeter wave form is sampled within a range window that has a width of approximately 28 m. Sub-surface volume-scattering contributes energy to the return wave form for depths up to approximately half the width of the range window. In geneneral, the material found within the top 15 m of the ice-sheet surface is a mixture of snow and firn. Firn is a term used for snow that has achieved a greater density ( > 0.4 Mg m⁻³) through recrystallization (Reference Paterson,Paterson, 1981). For simplicity, we use the term snow in this paper to describe the near-surface material, while recognizing that the density of this material is typically greater than that attributed to snow.

The geometry for an altimeter pulse that penetrates the ice-sheet surface is shown in Figure 3. The power density incident upon the snow surface is

where P is the transmitted power, G is the antenna gain toward nadir and R is the satellite range to the surface. The incident power is either scattered back from the surface toward the satellite or transmitted across the air/ snow boundary. The power transmitted across the air/ snow boundary can then be scattered back to the satellite by the ice crystals within the snow volume. When both surface-and volume-scattering are present, the surface-scattering strength is proportional to the relative complex dielectric constant, ∈_s, at the surface, while the volume-scattering strength is proportional to the dielectric discontinuities inside the medium and the density of the inhomogeneities (Reference Ulaby,, Moore, and FungUlaby and others, 1982, p. 846).

Fig. 3. Geometry for satellite-altimeter pulse that penetrates the ice-sheet surface producing volume scatter.

The signal strength after passing the air/snow boundary is

where T is the power-transmission coefficient at the surface and is related to the Fresnel reflection coefficient, Г, by

where

and ∉_s is the relative dielectric constant of snow. Typical values for the real and imaginary parts of the relative dielectric constant, Fresnel reflection coefficient and power-transmission coefficient for dry snow, wet snow, ice and ocean water are shown in Table 1. For dry snow, wet snow and ice, the power-transmission coefficient is greater than 90%. However, when only a small amount of water is contained in the snow, the imaginary part of the relative dielectric constant increases to a value that is hundreds of times larger than that for dry snow. For large values of the imaginary dielectric constant, the incident energy is severely attenuated because of high absorption losses. Thus, for wet snow, one expects surface-scattering to contribute most of the return power, while for dry snow it is clear that significant contributions from sub-surface volume-scattering is likely.

Table 1. Properties of snow, ice and ocean water at 13.5 GHz

As the energy from the expanding spherical shell penetrates the snow surface, it is attenuated by the snow volume. The reduction in power density is defined by the equation

where dP_t is the incremental attenuation of the incident power P_t, l is the path length in snow and K_e is the extinction coefficient of snow. The extinction coefficient describes the absorption and scattering losses within the snow volume. Separating variables, we obtain

or

and so

This equation is commonly applied in radar-meteorology problems to describe the attenuation of the radar signal by clouds, rain, snow and hail (Reference Battan,Battan, 1973; Reference Ulaby,, Moore, and FungUlaby and others, 1981).

The back-scattered power from the differential snow volume propagates back through the snow volume and is transmitted across the snow/air interface once more. The differential power just below the surface is

where dS and dR are the differential shell area and thickness, σ_vs is the back-scattering cross-section per unit volume and the factor of two in the exponential accounts for the two-way attenuation of the signal through the snow. The differential received power at the altimeter is

where A_e is the effective aperture of the antenna

g²(θ) is a function that describes the two-way angular variation of antenna gain and the 4πR² term accounts for the spreading loss of the re-radiated power. For spherical coordinates, the differential area is

and the differential thickness is

where c_s is the speed of light in snow. From the geometry (Fig. 3), the path length in snow is a function of both angle and time

where t₀ is the round-trip time corresponding to the initial intersection of the pulse shell with the flat surface to

Using a Gaussian approximation for the antenna pattern, we have

where β is a constant that determines the two-way antenna beam width and is given by

The Gaussian approximation is valid out to the point on the antenna pattern for which there is no appreciable contribution to the back-scattered power (Reference Brown,Brown, 1977).

Substituting Equations (9), (10), (17)–(20) and (22) into Equation (16), we obtain

where A is a term containing all constants and is given by

Assuming no azimuthal variations in the snow properties, the integration of Equation (24) in ψ yields a factor of 2π. At any time t > t₀, the angle from nadir to the edge of the shell volume on the surface is a function of time and is given by

so that the integration of Equation (24) in ψ becomes

This equation is simplified considerably by using the small-angle approximations

For a flat surface, the maximum angle that contributes energy to the sampled return wave forms for the Seasat and Geosat altimeters is≈0.4°, making these approximations extremely accurate. The equation is further simplified by converting to the two-way incremental ranging time, 6, given by Equation (4). Making these substitutions into Equation (27), we obtain

where δ′ is a dummy variable for δ Re-arranging terms in the exponential, we obtain

Integrating Equation (30), we obtain

where

The first exponential term in Equation (31) describes the effect of the altimeter’s antenna pattern on the return power, while the second exponential term describes the effect of the signal attenuation caused by the propagation of the altimeter pulse through the snow. Note that dP_r in Equation (31) is a function of time (range) and only applies for ranges beneath the ice-sheet surface (δ > 0), where volume-scattering contributes to the received power. To obtain the total volume-scattered power as a function of time, we convolve the impulse response dP_r/dδ in Equation (31) with the transmitted pulse profile (see Equation (1)), thus

For typical space-borne altimeters, the width of the transmitted pulse, T, is less than 10 ns. The transmitted-pulse shape is essentially a quadratic shape, but, for small pulse widths, can be adequately represented by a Gaussian function (Reference Brown,Brown, 1977). Thus,

where β_r is a constant that determines the 3 dB width of the transmitted pulse. Multiplying and regrouping the terms in the exponentials, we obtain an integral of the form

where

and

This is a tabulated integral (Reference Gradshteyn, and Ryzhik,Gradshteyn and Ryzhik, 1980, p. 307) and the final result for the received power due to volume-scattering is

where A“ is the amplitude in watts given by

The closed-form analytical solution given by Equation (37) describes the average return power due to subsurface volume-scattering as a function of time. The constants in the argument of the first exponential function are determined by design characteristics of the altimeter, i.e. satellite height, antenna beam width and transmitted pulse width. In the second exponential function, the argument is controled by the geophysical paramters K_e and C_s, as well as by the width of the transmitted pulse. The most significant simplifying assumption in our derivation of the volume-scattering model is that the snowpack is homogeneous within the volume sampled by the altimeter. This allows the attenuation of the signal through the snow to be described by a constant extinction coefficient, and it permits the contribution of the differential shell volumes to the received power to be described by a constant back-scattering cross-section per unit volume. The volume-scattering model is a first-order approximation to the physical situation where the snow density and grain-sizes are known to increase with depth under typical ice-sheet conditions. The advantage of our approach is that we obtain a closed-form solution and can combine it with the Brown surface-scattering model. The combined model can be used to retrack individual altimeter wave forms over the ice sheets to obtain quantitative estimates of the near-surface geophysical properties of the ice sheet (Reference Davis, and Zwally,Davis and Zwally, 1993).

Figure 4 shows a family of curves generated from the volume-scattering model. For small values of the extinction coefficient, k_e, the wave forms rise slowly because an ever-increasing snow volume contributes to the return power as the signal penetrates many meters beneath the ice-sheet surface. For larger values of k_e, the signal penetration is reduced and the wave forms rise rapidly because most of the snow volume contributing to the return power is located near the ice-sheet surface. The depth of penetration, d_p, is defined as the depth where the incident power density is reduced in magnitude to 1/e. For an extinction coefficient that varies with depth z, d_p is defined by

while for a constant extinction coefficient this reduces to

In the limit, as the extinction coefficient approaches infinity, the depth of penetration approaches zero and only surface-scattering contributes to the back-scattered power. Note that, for the wave forms with large values of k_e, the fall-off in the plateau region due to the antenna pattern is present and the wave forms are similar to those generated by the Brown surface-scattering model (Fig. 2).

Fig. 4. Family of curves generated from the volume-scattering model for k_e – 0.1–0.5 m^−l.

The volume-scattering model is combined with the Brown surface-scattering model in the following form

where S(δ) is the Brown model (Equation (2)), V(δ) is the volume-scattering model (Equation (37)) and Κ is the volume-scattering coefficient. The volume-scattering coefficient describes the magnitude of the volume-scattered power relative to the surface-scattered power. The inclusion of the surface-scattering model is necessary for two reasons. First, because a flat surface is assumed in the derivation of the volume-scattering model, the effects of surface roughness must be taken into account to obtain realistic altimeter return wave forms. As surface roughness increases, the width of the wave-form’s leading edge increases, just as in the case of surface-scattering over the ocean. Secondly, the surface area that contributes back-scattered power increases linearly up to the point when the trailing edge of the pulse intersects the surface, where the region back-scattering energy to the satellite becomes an expanding annulus of constant area. However, the volume that contributes to the back-scattered power increases linearly (approximately) throughout the entire range sampled by the altimeter. When the amount of surface-and volume-scattering is nearly equal, an initial sharp increase in return power occurs. This corresponds to the linear increase in surface area and sampled volume. However, once the surface area becomes constant, the volume that contributes back-scattered power still increases so the return power continues to increase but at a reduced rate. Thus, when the combined model is fitted to altimeter wave forms, the parameter K provides an indication of the relative contribution of the volume-scattered power.

Applications

Although the primary purpose of this paper is to present the above concepts and formulas, in this section we fit the combined model to Geosat-altimeter data to demonstrate the variations in the shape of the return wave forms that occur over the Greenland and Antarctic ice sheets. In addition, we use the combined model to simulate the effect of different antenna beam widths and operating frequencies on the shape of the return wave forms over the ice sheets. The results show that two-frequency altimeters can provide information to separate the different absorption and scattering losses from the extinction coefficient.

Summary and Discussion

A theoretical model describing the volume-scattered power for ice-sheet altimetry is derived assuming Gaussian distributions for the antenna pattern and transmitted pulse shape. The volume-scattering model is combined with the Brown surface-scattering model and applied to averaged return wave forms from the Greenland and Antarctic ice sheets. The results show that the return wave forms are characterized by varying amounts of surface-and volume-scattering. The volume coefficient, K, and extinction coefficient, k_e, are used to classify the return wave forms into one of three categories: surface-scattering, volume-scattering or a combination of the two. In general, surface-scattering is predominant in the low-altitude regions of the Greenland ice sheet, while varying combinations of surface-and volume-scattering occur at higher altitudes on the Greenland ice sheet. In the plateau region of East Antarctica, volume-scattering is the greatest contributor to the received power.

The presence of volume-scattering from the ice sheets suggests that important geophysical information concerning the near-surface conditions of the ice sheet can be derived from altimeter return wave forms. Because we obtain a closed-form analytical solution, the combined surface-/volume-scattering model can be used to retrack individual altimeter wave forms over the ice sheets. By retracking individual altimeter wave forms, quantitative estimates of the geographic and seasonal variations in the extinction coefficient of snow were determined and found to correlate well with elevation and latitude (Reference Davis, and Zwally,Davis and Zwally, 1993). The retracking algorithm used to extract surface elevations from Seasat and Geosat is based upon the Brown surface-scattering model (Reference Martin,, Zwally, Brenner and BindschadlerMartin and others, 1983). Hence, it is important to determine the effect of sub-surface volume-scattering on the precision of these elevation measurements, so that repeated altimeter mapping of surface elevations can accurately detect changes in the mass balance of the ice sheets.