TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-11-1967-2017Sea ice local surface topography from single-pass satellite InSAR
measurements: a feasibility studyDierkingWolfgangwolfgang.dierking@awi.dehttps://orcid.org/0000-0002-5031-648XLangOliverBuscheThomasAlfred Wegener Institute Helmholtz Center for Polar and Marine
Research, 27570 Bremerhaven, GermanyArctic University of Norway,
9019 Tromsø, NorwayAirbus Defence and Space, 14467 Potsdam,
GermanyGerman Aerospace Center (DLR), 82234 Weßling,
GermanyWolfgang Dierking (wolfgang.dierking@awi.de)29August20171141967198514March201721March201721June20178July2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/11/1967/2017/tc-11-1967-2017.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/11/1967/2017/tc-11-1967-2017.pdf
Quantitative parameters characterizing the sea ice surface topography are
needed in geophysical investigations such as studies on atmosphere–ice
interactions or sea ice mechanics. Recently, the use of space-borne
single-pass interferometric synthetic aperture radar (InSAR) for retrieving
the ice surface topography has attracted notice among geophysicists. In this
paper the potential of InSAR measurements is examined for several satellite
configurations and radar frequencies, considering statistics of heights and
widths of ice ridges as well as possible magnitudes of ice drift. It is shown
that, theoretically, surface height variations can be retrieved with relative
errors ≤ 0.5 m. In practice, however, the sea ice drift and open water
leads may contribute significantly to the measured interferometric phase.
Another essential factor is the dependence of the achievable interferometric
baseline on the satellite orbit configurations. Possibilities to assess the
influence of different factors on the measurement accuracy are demonstrated:
signal-to-noise ratio, presence of a snow layer, and the penetration depth
into the ice. Practical examples of sea surface height retrievals from
bistatic SAR images collected during the TanDEM-X Science Phase are
presented.
Introduction
Sea ice motion on scales of tens of meters to hundreds of kilometers changes
as a function of time and space, dependent on variations of the forces
exerted on the ice by wind, ocean currents, tides, and internal ice stress.
Blocking of motion occurs along coastlines, around islands, and at other
obstacles such as icebergs. The result is either a local opening of the ice
or formation of ice ridges, rubble fields, and shear zones, leading to an
intermittent change of the ice surface topography. In this article, the
potential of interferometric synthetic aperture radar (InSAR) for measuring
sea ice surface topography is addressed.
The shape and roughness of the ice surface determines the aerodynamic
coupling between the ice and the atmospheric boundary layer (e.g., Garbrecht
et al., 2002). Changes of surface height often indicate undulations of ice
thickness, although ice depth changes do not necessarily mirror the surface
undulations. For example, the ridge keel is usually much broader than the
ridge sail, and its depth exceeds the sail height by a factor between 4 and
more than 10 (Strub-Klein and Sudom, 2012). In some cases sail and keel do
not occur concurrently (e.g., Tin and Jeffries, 2003). Nevertheless, ice
thickness can be deduced from measurements of surface height variations
using statistical approaches (Strub-Klein and Sudom, 2012; Petty et al.,
2016). (Here we use the notation “height variations” to indicate elevation
changes on the order of decimeters to a few meters, which can be retrieved
by InSAR methods. With “surface roughness” we include also undulations in
the range of millimeters to centimeters, which strongly influence the
intensity of the backscattered radar signal.) Another option for indirect
thickness retrieval is to measure the ice freeboard (the distance between
the ice surface and the local water level) employing Cryosat-2 altimeter
data and from this to calculate the ice thickness, assuming hydrostatic
equilibrium and realistic ranges of ice density and snow mass load (Rickers
et al., 2014). The determination of the ice freeboard is carried out at the
margins of ice floes adjacent to open water leads or to leads covered with
thin ice.
Topographic measurements over sea ice have been carried out by means of
helicopter-borne laser profilers (e.g., Dierking, 1995) or airborne laser
scanners (Farrell et al., 2011). The relative height error of such sensors
is on the order of 0.1 m, the footprint size between 0.1 and 2 m, and the
spatial sampling on ground ranges from 0.2 to 5 m. The largest of these
values approximately mark the upper limits that are necessary to resolve the
surface height changes of, e.g., ice ridge cross sections with sufficient
detail, considering the fact that the width of most ridges varies between
less than a meter and 40 m, with only few exceptions reaching more than 70 m
(Strub-Klein and Sudom, 2012). The laser altimeter on ICESat-2 (to be
launched in late 2017) will have a 10 m footprint and an along-track
sampling of 0.5 m (Farrell et al., 2011).
Until now, the majority of the published InSAR studies deal with data
acquired over stationary ice (called “fast ice”). The reason is that with
the spaceborne systems employed in those studies (i.e., ERS, ALOS PALSAR,
and Cosmo SkyMED), the necessary image pairs could only be acquired with
temporal gaps of tens of hours to several days. In the case of drifting ice,
such time differences are much too large for achieving the magnitude of
correlation between the two images that is necessary for a reliable
interferometric height retrieval. Hence the investigations concentrated on
indications of differential motion due to deformation processes in fast ice,
links between ice properties and interferometric coherence, and mapping of
fast ice extent. (Dammert et al., 1998; Meyer et al., 2011; Berg et al.,
2015).
The interferometric processing and height retrieval is based on the phase
difference between two radar signals received from the same ground area
element but from slightly different sensor positions. The geometric distance
between the two sensor positions is called the baseline and consists of an
along- and an across-track component (Bal, Bac). The former is
oriented parallel, the latter perpendicular to the satellite velocity
vector. The along-track baseline causes a time lag between signal 1 and
signal 2 received from a given surface element. This lag is denoted by temporal
baseline and can vary from several days (repeat-pass InSAR) to a few
microseconds (single-pass InSAR). An image showing the spatial variations of
phase differences is called an interferogram. The phase difference can only
assume values in the range from 0 to 2π, which is usually represented
by a matching color cycle in the interferogram. In worst cases,
interferograms may reveal only noise-like patches, indicating a total
decorrelation between the received signals. Contiguous patterns of recurring
color cycles called fringes represent continuously increasing or
decreasing phase differences between well-correlated signals. The
interferometric phase difference Δϕ is defined by (Madsen and
Zebker, 1998)
Δϕ=Δϕtopo(Bac)+Δϕmov(Bal)+Δϕnoise+2πn.
This equation states that the measured phase difference may contain
information about height variations of the ground surface (Δϕtopo) as well as about ground movements taking place between the
reception of signal 1 and signal 2 (Δϕmov). The component
of the across-track baseline perpendicular to the line-of-sight direction
determines the sensitivity to height variations, the length of the
along-track baseline the sensitivity to ground displacements along the
line of sight. The phase noise is caused by surface and volume scattering
effects, by radar system noise, and – in the case of repeat-track InSAR – by
atmospheric and ionospheric wave propagation delays. The last term takes
into account that multiples of 2π may have to be added to the measured
phase difference in further processing of the data (called “phase
unwrapping”). Another important parameter that is determined from the image
pair is the interferometric coherence, which represents the degree of
correlation between both images.
Optimal conditions for retrieving sea ice topography and movement are given
when two satellites fly as a tandem in close formation (“single-pass
InSAR”). The opportunity to study the potential of single-pass satellite
InSAR for mapping of sea ice topography arose during the TanDEM-X Science
Phase, which started in September 2014 and lasted for 17 months (Maurer et
al., 2016). The TanDEM-X mission (TerraSAR-X add-on for Digital Elevation
Measurements) has primarily been designed for topographic mapping of the
Earth's land masses (Krieger et al., 2007). In standard operation mode the
achievable relative accuracies are 2–4 m vertically (dependent on slope of
terrain and land cover type) and 3 m horizontally at a horizontal sampling
of 12 m (Krieger et al., 2007). This mode is optimized for topographic
mapping of the land surface but is not sufficient for retrieving height
variations of the sea ice surface. The Science Phase was initiated to
demonstrate new products and applications such as digital elevation models
with higher accuracies than in standard mode or measurements of ocean
currents. It consisted of different sub-phases, among them a large
cross-track baseline formation with mean along-track separation of zero that
was initiated in March 2015. Data takes were performed in a bistatic mode
(see below). The comparatively large baselines in this phase translated to a
very high sensitivity for object elevations on the order of decimeters. The
data that are presented in this paper were acquired during the large
cross-track baseline formation.
In Sect. 2, relevant theoretical equations are introduced that are needed
to assess whether a given SAR configuration is suitable for measuring the
sea ice surface topography. The possible performance of satellite
configurations that at present are either operational or under discussion is
investigated in Sect. 3, succeeded by preliminary results of measurements
from the TanDEM-X Science Phase (Sect. 4). While ideal conditions are
assumed in Sect. 2, real-world factors that influence interferometric
measurements over sea ice are investigated in Sect. 5. Finally, the
conclusions emphasize the major findings of this feasibility study.
Basic concepts
In this section, ideal conditions are assumed, i.e., the sea ice does not
move, all parameters appearing in the relevant equations can be accurately
determined, and the penetration depth of the radar signal into the ice is
negligible. Equivalent to the assumption of stationary ice is considering an
along-track baseline of zero, which means that the interferometric phase is
not affected by ice drift. Biasing and disturbing factors and their effect
on height retrievals are discussed in
Sect. 5. The potential to retrieve
sea ice surface topography from single-pass InSAR can be assessed by
evaluating the height of ambiguity ha (which is the height difference
related to one phase cycle, i.e., Δϕ=2π) and the
relative height error σh (Madsen and Zebker, 1998):
ha=λHtanθpBn,σh=Htanθλ2πpBnσΔϕ=ha2πσΔϕ,
where λ is wavelength, H is orbit height, θ is incidence
angle, Bn is baseline perpendicular to the line of sight (the projection
of Bac perpendicular to the slant range), and σΔϕ
is phase noise. The estimation of the latter is described later in the text.
The factor p equals 1 if one image is acquired in monostatic and the second
in bistatic mode (e.g., in a tandem, where one satellite transmits and both
receive in a synchronized operation mode) and p=2 if both images are
acquired in monostatic mode (i.e., both satellites transmit and receive
independently). A discussion of pros and cons of the monostatic and bistatic
mode can be found in Krieger et al. (2007). The incidence angle θ is
the average of the respective incidence angles at scene centers valid for
the acquisitions of image 1 and 2. A high sensitivity to topography is
achieved when the ambiguity height is small. At first sight this implies
that a large normal (perpendicular) baseline, a short wavelength, a low
orbit, and a steep incidence angle are favorable conditions for the
retrieval of topographic data. However, the normal baseline cannot be
arbitrarily large, and for the incidence angle additional
dependencies have to be taken into account, as is shown below. Satellites at
smaller orbit heights are more severely affected by atmospheric friction.
The question concerning baselines achievable in space is discussed in
Sect. 5. Another limitation is caused by the nature of the surface and
volume scattering mechanisms. The received radar signal is the coherent sum
of contributions from different scattering objects that are arbitrarily
distributed in the ground resolution cell. Because electromagnetic
interactions between single scatterers are random, the backscattering
intensity can vary significantly around the mean value (the “speckle”
effect). In the case of a satellite tandem, the radar intensities measured over
a given surface element from two different positions differ due to speckle.
This difference is spatially randomly distributed. The critical baseline
marks the total loss of correlation (i.e., the point of total decorrelation)
between the two images from which the interferogram is generated. It is
defined by (Madsen and Zebker, 1998)
Bcn=λHpΔycos2θ,
where Bcn is the critical perpendicular baseline, and Δy is the
single-look ground-range resolution. It is assumed that the surface slope
equals zero and can hence be ignored. The critical baseline is defined as
the baseline corresponding to a fringe rate of 2π per range resolution
cell. Baseline decorrelation is less severe at longer radar wavelengths,
finer spatial resolution, and larger incidence angles. Equation (4) is valid
for the case that only surface scattering but no volume scattering takes
place. It is emphasized here that two counteracting effects have to be
considered: a larger baseline on the one hand increases the height
sensitivity but on the other hand decreases the coherence. Hence, there must
be an optimal baseline that is a trade-off between those counteracting
effects. A way to calculate an optimal baseline is given below.
Relative height error normalized by ground range resolution as a
function of the ratio between normal and critical baseline, shown for
(a)γN=1 and two incidence angles and (b) for
an incidence angle of 25∘ and different values of noise levels.
The relative accuracy of the retrieved heights of sea ice deformation
structures depends on the phase noise, which is a function of the
signal-to-noise ratio (SNR) and the baseline decorrelation. The phase noise
can be expressed as a function of the interferometric coherence (Rosen et
al., 2000):
σΔϕ2=12NL1-γ2γ2.
Here, NL is the number of independent estimates (number of looks) used
to derive the phase differences, and γ is the interferometric
coherence, which in the case of a single-pass system, along-track baseline
of zero, and pure surface scattering is given by γ=γGγN (Rosen et al., 2000; Bamler and Hartl, 1998). The
first factor is the geometric baseline or surface correlation:
γG=1-BnBcnBn≤Bcn.
In the derivation of Eq. (6) it was assumed that the system transfer function is
rectangular and that no spectral shift filter is applied (see Bamler and
Hartl, 1998, Sect. 3.8). The correlation as a function of system noise is
γN=1+1SNR-1.
This equation is valid if the noise in image 1 is independent of the noise
in image 2 and both noise levels are of same magnitude.
Considering the balance between a large sensitivity to surface elevation
changes (i.e., the most favorable value of ha) and the baseline
decorrelation, the optimal baseline that minimizes the height error can be
obtained from Eq. (8), which is the result of combining Eqs. (2), (3), (4),
and (5):
σh=Δysinθcosθ22πγN-2-(1-x)2x(1-x),
where x=Bn/Bcn, and it was assumed that NL=1. By
evaluating the derivative of Eq. (8) for γN=1, it is
found that the optimal normal baseline is Bn=aBcn with
a=0.382. For γN<1, the factor a increases. It can be
determined from a cubic equation resulting from the derivative of Eq. (8). It is
a=0.483 for γN=0.5 and a=0.453 for γN=0.75. If γN=1, the baseline correlation is γG=0.618, and the phase noise σΔϕ=0.9 rad. The
former is in agreement with the “optimum correlation” derived by Rodriguez
and Martin (1992) for γN=1.
The expression sinθcosθ2-3/2π-1 ranges from 0 at
θ=0 and 90∘ to a maximum of 0.0563 at
θ=45∘. In Fig. 1a, the normalized relative height error
σh/Δy is shown as a function of Bn/Bcn for
γN=1 and incidence angles of 25 and 40∘. Relatively small height errors can be obtained over a wider interval of
Bn/Bcn ratios from 0.2 to 0.6. In Fig. 1b, the effect of system
noise is demonstrated for an incidence angle of 25∘, assuming values of
γN=0.75 and 0.5, corresponding to low SNRs of 5 and
0 dB, with the latter as limiting case. A low SNR not only affects the
achievable height accuracy and the length of the optimal normal baseline but
also narrows the interval of Bn/Bcn ratios in which the σh can be regarded as still acceptable. Note that the ratio
Bn/Bcn does not depend on p.
Ambiguity heights ha and relative height errors
σh(rounded values) for optimal baselines Bn,
determined for different satellite configurations (λ – radar
wavelength; H – orbit height; θ – radar incidence angle; Δy – ground range resolution; Bcn – critical baseline). It is assumed that p=1, NL=1,
and γN≈1 in Eqs. 2, 4, and 8.
BandLCXKuKaλ (m)0.24 0.055 0.031 0.022 0.0084 H (km)745 km 700 500 780 740 θ (∘)25402540254025402540Δy (m)4.22.74.65.02.81.93.52.38.95.8Bcn (km)5211210.213.16.713.96.012.70.851.8Bn (km)19.843.13.95.02.65.32.34.90.320.69ha (m)4.23.54.66.42.82.43.53.08.97.5σh (m)0.600.500.660.920.400.350.500.421.31.1
Noise-equivalent sigma zero (NESZ) for different satellite missions.
Satellite missionNESZReferenceTandem-L-32 to -27 dBKrieger et al. (2010)Sentinel-1-22 dB (the actual value(https://sentinel.esa.int/web/sentinel/user-guides/sentinel-1-sar)depends on range position)TerraSAR-X/TanDEM-X-19 to -26 dBTerraSAR-X Ground Segment Basic ProductSpecification Document, TX-GS-DD-3302 (2008)Ku-band concept-24 to -29 dBLopez-Dekker et al. (2014)SIGNAL-13 dBInternal documentResults
Besides the TanDEM-X mission, satellite configurations operated at other
frequency bands are taken into account: (a) Tandem-L (Krieger et al., 2010);
(b) a C-band tandem with wavelength and orbit altitude of Sentinel-1
(Torres et al., 2012) considering the possibility of adding a passive
companion satellite for interferometric measurements; (c) a
Ku-band tandem based on a concept presented by Lopez-Dekker et
al. (2014) for a single platform meant for measuring ocean currents; and (d)
a scenario for Ka band that is adopted from a proposal for an ESA
Earth Explorer mission (Börner et al., 2010).
Wavelengths and orbit altitudes listed in Table 1 are taken from the
references given above. In this study, we selected incidence angles of 25 and
40∘ for all five mission scenarios. The slant range resolution
Δρ (Δy=Δρ/sinθl, where
θl is the local incidence angle) for TSX is 1.2 m for the
High-Resolution Spotlight (SL) and single-polarized Stripmap (SM) imaging
modes (TerraSAR-X Ground Segment, Basic Product Specification Document,
http://sss.terrasar-x.dlr.de/). For Sentinel-1 Stripmap mode, the slant
range resolution is 2 m at θl=25.6∘ and 3.3 m at
θl=41∘ (Aulard-Macler, 2012). For Tandem-L we
used a bandwidth of 85 MHz (Krieger et al., 2010) for calculating Δy. Tandem-L orbit parameters are under discussion (status end 2016: altitude
745 km). For the Ku-band mission we assumed a bandwidth of
100 MHz instead of 10 MHz as used by Lopez-Dekker et al. (2014). The
bandwidth for SIGNAL (Ka band) is assumed to be 40 MHz, based on
discussions on the mission concept. The normal baselines given in the table
are 0.382×Bcn, with the critical baseline from Eq. (4).
Height of ambiguity and relative height error are calculated using Eqs. (2)
and (3), respectively. The small ambiguity heights for TanDEM-X may
require phase unwrapping if the actual ridge height results in a phase
difference > 2π. In a fringe corresponding to 10 m height
difference, a ridge of 2 m height extends only over one fifth of the fringe
width, which may be not sufficient for the retrieval of the ice surface
relief, in particular if the phase noise is high. The mean maximum heights of
first-year ice ridges reported for different areas of the Arctic range from
1.1 to 3.3 m (Strub-Klein and Sudom, 2012), and the average heights across
the transverse section of a ridge are even smaller (see Sect. 5). Except for
SIGNAL, the achievable minimum relative height errors (under ideal
conditions) are hence on the limit that is necessary for a meaningful
retrieval of a rough sea ice surface topography. From the investigated
configurations, the lowest error is achieved with TanDEM-X, the largest for
SIGNAL. If the bandwidth of the Ka-band mission is increased,
e.g., to 100 MHz, the relative height error is similar to the
Ku-band values (0.49 and 0.42, respectively).
Ranges of backscattering coefficients σ0 for different sea
ice types, with examples. Numbers given in italics are for HH polarization.
Ice typeRadar band polarization/incidence angleσ0 rangeLocation/sensor/referenceLead iceC-23 to -13 dBBeaufort SeaFirst-yearVV-25 to -11 dBERS-1Multi-year20–26∘-13 to -8 dBKwok and Cunningham (1994)Young iceC-18 to -11 dBBarents Sea, Svalbard Storfjord, Fram StraitVV (HH)-30 to-15 dBAirborne SAR (ESAR, DLR)First-year30–45∘-14 to -8 dBDierking (2010)-16 to-10 dBBrash-14 to -6 dB-15 to-8 dBRidges-7 to -2 dB-8 to-4 dBYoung iceL-27 to -14 dBBarents Sea, Svalbard Storfjord, Fram StraitVV (HH)-29 to-12 dBAirborne SAR (ESAR, DLR)First-year30–45∘-23 to -16 dBDierking (2010)-23 to-16 dBBrash-14 to -10 dB-14 to-8 dBRidges-10 to -6 dB-10 to-6 dBNo distinctionKu-16 to -2 dBEntire ArcticVV + HH mergedScatterometer20-60∘Ezraty and Cavanié (1999)
Effect of a low signal-to-noise ratio (SNR) on the relative height
error σh.
The values of Table 1 are valid if the correlation related to system noise,
γN, is still close to 1, i.e., the signal-to-noise ratio of the
measurements is larger than about 17 dB (corresponding to γN=0.98). However, the noise-equivalent sigma zero (NESZ) for satellite
SARs is relatively high and hence the SNR for thin and smooth level ice low.
For the satellite missions used as examples in this study,
NESZ values are given in Table 2. Examples of
radar backscattering coefficients σ0 typical for different ice
classes and conditions are listed in Table 3. Dierking (2013) compared C-
and X-band images of sea ice in the Beaufort Sea and found that their
intensity variations was highly correlated for level and deformed ice except
for nilas (thin ice) forming in an area of open water. In about 85 % of the
investigated cases, the X-band intensities varied between -19 dB and
-8 dB and were 3–5 dB higher than at C band. This compares well with
measurements reported by Tucker et al. (1991) (see also overview presented in
Dierking, 2012). Publications of radar measurements over sea ice at
Ku and Ka bands are sparse. Dierking (2012)
summarized the results of field and laboratory studies. In the incidence
angle range from 25 to 40∘, values between -15 and +7 dB are
reported at Ku band, with nilas revealing the lowest and
multi-year ice the largest intensities. Corresponding numbers for
Ka band are -12 and +10 dB.
The ranges of backscattering intensities and noise levels for the different
radar systems considered here indicate that the SNR typically varies between
0 and 25 dB under real conditions. Smooth thin level ice (without frost
flower coverage, snow crusts, and air inclusions in the ice volume) reveals
relatively low backscattering intensities; hence SNRs are between 0 and 10 dB.
For thicker level ice (surface roughness scales from millimeters to
centimeters) and ice with a considerable fraction of air inclusions, the SNR
ranges from 5 to 15 dB in most cases. Deformation structures in the ice
cover (e.g., ridges, brash ice), imaged at higher spatial resolutions below
about 50 m, may reveal spots of very large intensities originating from ice
blocks and fragments with their surface oriented normally towards the radar.
Stronger multiple and volume scattering arises from piles of ice blocks. The
difference between the backscattering coefficient and the NESZ will hence be
roughly 15–25 dB. For the three given ice classes, the corresponding values
of γN are 0.5–0.91 (thin level ice), 0.76–0.96 (thicker level
ice), and 0.96–0.99 (deformation structures).
In Table 4, the effect of low backscattering intensities on the achievable
height accuracy is demonstrated, using SNR = 10 and 5 dB. For the
former case one obtains γN=0.91, Bn/Bcn=0.418,
γG=0.582, and σΔϕ=1.13 rad and for
the latter case γN=0.75, Bn/Bcn=0.454,
γG=0.546, and σΔϕ=1.55 rad. The
given signal-to-noise ratios are typical for new ice and smooth first-year
level ice at lower radar frequencies. For SNR = 10 dB, the relative height
error is larger by a factor between 1.1 and 1.2 and for SNR = 5 dB by a
factor from 1.4 to 1.5.
A distinct surface topography and height variations of 1 m and more are
usually observed in areas of first-year ice (thickness > 0.3 m) and
multi-year sea ice (thickness > 2 m). Considering the ridge height
statistics provided by Strub-Klein and Sudom (2012) the relative height error
σh should optimally be less or equal to 0.5 m for
meaningful height retrievals. Ridges and extended deformation zones reveal
large backscattering intensities at C and L band (Dierking and Dall,
2007). Hence, the SNR is high, whereas the SNR over level ice
areas is relatively low and the relative height error correspondingly high.
If one, for example, assumes that ice ridges, which appear as narrow high intensity
zones in a SAR image, are distributed in areas of level ice with lower
backscattering intensities, the SNR and hence the achievable height accuracy
may vary significantly within short distances. This effect is less severe at
higher radar frequencies (X, Ku, and Ka bands),
since the intensity contrasts between deformed and level ice are considerably
lower. The reason is that the relative backscattering from level ice is
stronger since its surface appears rougher to the shorter radar wavelengths.
This means that the differences of the SNRs between deformed and level ice
are not as large as at the lower frequencies, which is an advantage for
topographic mapping.
Examples
The bistatic formation during the TanDEM-X Science Phase started on
12 March 2015. All in all, over 40 bistatic image pairs were acquired around
the city of Barrow (since December 2016 renamed to Utqiaġvik), off the
Coast of Alaska, USA, predominantly with large interferometric baselines. In
the area of interest, a station is located that is equipped with sea ice
radar and a webcam, both of which acquire imagery regularly. For our study two
TanDEM-X image pairs were selected to generate preliminary maps of sea ice
surface topography by applying a standard SAR-interferometric approach. The
main processing steps included spectral filtering of the input images,
interferogram generation and flat earth removal, interferogram filtering, and
phase-to-height conversion.
Webcam (a) and sea ice radar image (b) obtained
from the Geographic Information Network of Alaska, University of Alaska
Fairbanks, taken 29 March 2015
(http://feeder.gina.alaska.edu/radar-uaf-barrow-seaice-images).
The first example shown in Fig. 2 was generated from data that were gathered
on 29 March 2015, close to the coastline of Barrow. For the bistatic mode p
equals 1, the incidence angle θ was 27.3∘, the normal
baseline Bn amounted to 1113 m, and the length of the along-track
baseline was 138 m. With an orbit height of H=514 km and a radar
wavelength λ=0.031 m, one obtains ha=7.4 m for the
height of ambiguity (Eq. 2). The critical normal baseline Bcn is
8072 m (Eq. 4), and the relative error σh varies between
0.66 m for a signal-to-noise-ratio of 10 and 0.51 for
SNR = 100 (Eq. 8, SNR given as linear value). The area from which the
elevation profile depicted in Fig. 2c was retrieved was landfast ice, and
thus
we can neglect contributions of ice movement to the interferometric phase
caused by the along-track component of the baseline (this issue is addressed
in the next section). The profile reveals single prominent ridges with
realistic heights. Unfortunately, coincident data of surface topography
obtained by other sensors (e.g., laser profiler or scanner) are not available
for this area and day. The general characteristic of the ice surface
structure obtained from SAR compares well with the structures that can be
recognized in the webcam image of the Barrow station and in the sea ice radar
image. Both are shown in Fig. 3.
We emphasize the fact that the backscattered radar intensity is not
necessarily directly linked to the ice surface topography. In the extreme
case, ice with a perfectly smooth surface may reveal strong volume
backscattering, if air bubbles are present in the ice volume and the
penetration depth of the radar waves is large. Surface undulations on the
scale of meters are not directly mirrored in the variations of the
backscattered signals, since the radar response is more sensitive to the
size and tilt of single surface elements such as ice blocks and their
orientation relative to the radar look direction but not sensitive to
elevation changes typical for a sea ice surface. One major source of the
backscattered radar signal is the small-scale ice surface roughness with
amplitudes and wavelengths in the range of millimeters to centimeters.
An empirical estimation of the relative height error is derived by evaluation
of the local height statistics within a representative area with a relatively
flat and homogeneous sea ice surface (red polygon in Fig. 2a). The area was
located close to the coastline, several kilometers northeast of Barrow. The
standard deviation of the surface height for this sample area is 0.12 m,
calculated from the retrieved DEM, which has a spatial resolution of 12 m.
The one-look resolution of the data was 2.5 m in ground range and 6.6 m in
azimuth direction. Assuming that the standard deviation is caused by noise
effects and neglecting the correlation between adjacent pixels, the number of
looks in the height map is approximately 8.7, and the theoretical relative
height error according to Eq. (8) is between 0.51/8.7 to
0.66/8.7 m, i.e., 0.17 and 0.22 m. The empirical evaluation of a
local height statistics hence compares reasonably with the theoretical
derivation in Sect. 2.
A second example from an area located southwest of Barrow can be found in
Fig. 4. The data were acquired on 20 March 2015 with a normal baseline of
833 m, an along-track baseline of 42 m, and an incidence angle of
37.2∘. The height of ambiguity is 14.5 m. The amplitude image
(Fig. 4b) reveals that the profile – when starting at point B and moving to
the left – crosses a zone of landfast ice (dark grey belt with bright
structures), a coastal polynya, i.e., an open water area with indications of
wind-driven Langmuir circulation (dark grey area with bright stripes), a
narrow zone of thin ice (dark grey zone), and pack ice (bright grey) in which
open water leads (dark areas) are embedded. For the retrieved elevation
difference between the landfast ice (distance from 11 000 to 13 000 in
Fig. 4c) with elevations between -3 and -2 m and the drifting pack ice
(2500 to 6000) with elevations around zero, we did not find an explanation.
As Eq. (1) reveals, ice movements along the radar line of sight cause
additional phase differences of the backscattered signal. However, the drift
speed of the pack ice calculated from the observed height difference is too
large to be realistic. The open water area (distance from 7000 to 11 000 m)
and the lead (1100 to 2500 m) crossed by the profile A–B appear as rugged ice
terrain in the height map with heights between two and almost eight meters.
We suppose that these apparent height changes are in effect caused by the
influence of surface currents in the open water areas. The along-track
baseline of 42 m corresponds to a temporal baseline of 6 ms. This time
interval is shorter than the decorrelation time of a water surface, which
ranges from about 8 to 10 ms at X band (Romeiser and Thompson, 2000).
Hence, the requirement for a measurable phase difference is fulfilled. At
L band, for example, the decorrelation time is larger by a factor of 10
(Romeiser and Thompson, 2000), which means that it is possible to measure
phase differences at even larger temporal baselines. The interferometric
phase of open water areas is in general proportional to the mean surface
current parallel to the radar look direction and contains also contributions
associated with the velocity of small wind-induced ripple waves and with the
surface currents due to the orbital motion generated by longer ocean waves
(Romeiser and Thompson, 2000). In the special situation shown in Fig. 4b, the
open water areas bounded by the light blue lines reveal alternating dark and
bright strips in the SAR image. This pattern is typical for Langmuir
circulation, in which streaks of ice nearly parallel to the wind direction
appear on the water surface (Leibovich, 1983). The streaks are visible
manifestations of the convergence zones between counter-rotating vortices that
are present in the near-surface water layer, with their axes of rotation
parallel to the wind. The surface current is composed of a component parallel
to the streaks and a component perpendicular to them. The former is largest
in the zones of convergence and smaller in the zones of divergence. The
latter changes direction between neighboring vortices (Leibovich, 1983). The
large “height” variations in the open water areas of Fig. 4 may hence be
caused by this complex current pattern and possible wave–current
interactions. Since the streaks of ice are located in the zones of
convergence, their surface is rough (on scales of centimeters), and the
backscattered radar intensity is high. Because of lacking complementary data
the analysis of Fig. 4 remains on a qualitative level. Nevertheless, the
example demonstrates the need to systematically study the influence of open
water surface currents and ice drift on the retrieval of sea ice topography.
Note that at longer temporal baselines the interferometric coherence measured
over water is very low. In such cases a coherence map can be used to detect
the presence of openings in the sea ice cover. Not only areas of open water
contaminate the height retrieval from InSAR measurements but also spatial
variations of the ice drift and rotation of the pack ice. This is discussed
in detail in the following section.
DiscussionInfluence of sea ice motion
Since most parts of the sea ice cover are in steady motion, along-track
baselines cause additional phase shifts that affect the retrieval of
topographic heights (see Eq. 1). In addition, the movement of the ice
between the acquisitions of image 1 and image 2 leads to decorrelation
effects due to speckle. In the case of single-pass InSAR with very small
along-track baselines (otherwise surface height retrievals will be severely
hampered, see below), the effect of temporal decorrelation can be neglected
for sea ice. The interferometric phase Δϕmov for a
baseline Bal in along-track direction (corresponding to the along-track
distance of the positions from which the images are acquired) is (Madsen
and Zebker, 1998)
Δϕmov=-2πpuLOSBalvλ,
where v is ground velocity of the SAR platform, uLOS is line-of-sight or radial object
velocity, λ is radar wavelength, Bal is along-track baseline,
and p is explained above. The radial velocity can assume positive and
negative values, dependent on the direction of the movement (towards or away
from the radar). It is determined from the sea ice drift velocity u by
uLOS=usinθcosφ, where θ is the incidence angle
and φ is the azimuth angle between the direction of the ice drift and
its across-track component. Here it is assumed that the vertical component
of the ice displacement is zero. Whether this assumption is justified is
discussed below. For a given phase difference the corresponding along-track baseline
depends linearly on λ and v and decreases with increasing
uLOS. Typical average sea ice drift velocities range mostly from 0 to
0.35 m s-1 (1.26 km h-1) (Rampal et al., 2009), but for instantaneous radial
velocities Kræmer et al. (2015) found values up to 0.6 m s-1 from analyses
of the Doppler shift of SAR signals.
For TanDEM-X, the ground velocity is 7 km s-1. In the following
discussion we determine the along-track baseline which at a given
line-of-sight velocity causes a phase shift corresponding to a given relative
height error. If the height error is set to σh=0.5 m
(which represents a still acceptable accuracy, see above) and the height of
ambiguity to 5 m (representing one fringe), the corresponding phase
difference amounts to 36∘ or 0.2π rad. With p=2 and λ=0.031 m, a phase shift Δϕmov=0.2π rad is
obtained at Bal=339 m (56 m), if u=0.05 m s-1
(0.3 m s-1), φ= 0∘, and θ= 40∘,
which gives uLOS= 0.032 m s-1 (0.193 m s-1). The
baseline length Bal doubles if p= 1. In units of time, the
temporal baselines are Bal/v=0.05 and 0.008 s and thus extremely
short. The phase noise also has to be considered here, which gives a relative
velocity error of σuLOS=vλσΔΦ/(2πpBal). If the SNR is low and the baseline decorrelation not
negligible, critical phase differences due to surface motion are reached at
even shorter baselines. In the examples presented below, we assume that the
SNR ≥ 15 dB.
Along-track baseline Bal multiplied with the system
factor Cnsys versus the sea ice line-of-sight velocity for
different ratios between relative height error and ambiguity height. See text
for further explanations.
Critical along-track spatial and temporal baselines (Bal) causing phase
shifts corresponding to a height change of 0.5 m at a height of ambiguity of
5 m, calculated for small and large sea ice drift with the sensor
configurations from Table 1
(v is platform velocity over ground; uLOS is line of sight velocity of the ice).
Assumptions: p=1, phase difference per
fringe Δϕmov/2π=36∘. Because of lacking
information we used a velocity over ground of 7 km s-1 for Tandem-L.
In the following discussion we define the “system-coefficient” as
Cnsys=|p/vλ| and investigate the product
CnsysBal as a function of the magnitude of the
line-of-sight velocity for different ratios Δϕmov/2π. The result is shown in Fig. 5. From Eq. (3), the ratio Δϕmov/2π is related to σh/ha.
For a given height of ambiguity, the corresponding height error is
σh=CnsysBaluLOSha.
Here we interpret the height error as a height change that would occur if the
phase shift Δϕ were not caused by ice motion but by ice topography.
For TanDEM-X data, for example, Cnsys is =p/217 (s m-2). If
uLOS=0.032 m s-1 and the ratio ϕal/2π=0.1, one obtains CnsysBal=3.125 s m-1 (solid
curve in Fig. 5); i.e., Bal=678 m if p=1. With a height of
ambiguity of 5 m, the corresponding height change, Δh,
is 0.5 m. If the sea ice drifts in across-track direction, a line-of-sight
velocity of 0.032 m s-1 corresponds to a ground velocity of
0.075 m s-1 at θ= 25∘ and 0.05 m s-1 at θ=40∘. For the long-dashed curve in Fig. 5 (σh/ha=0.025), the product CnsysBal equals
0.78 at uLOS=0.032 m s-1. If we again set the height
change Δh=σh=0.5 m, the height of ambiguity is
20 m, and Bal=169 m for p=1. The sensitivity to ice motion
increases at larger values of ha if σh is fixed
to a constant value. Differently expressed, this means that even at short
along-track baselines the effect of ice drift cannot be neglected when the
height of ambiguity increases. In Table 5, “worst case” critical along-track baselines are listed for p=1 that cause phase shifts corresponding
to a relative height error of 0.5 m at an ambiguity height of 5 m. For p=2, they are even shorter by a factor of 2. Table 5 reveals that extreme
instantaneous line-of-sight velocities, such as reported by Kræmer et
al. (2015), cause significant phase shifts already at very short baselines, in
particular at higher radar frequencies. For line-of-sight velocities larger
than 0.2 m s-1, the change of the critical along-track baseline is
only small (see Fig. 5). Since according to Eq. (1),
surface topography and movement affect the measured interferometric phase
simultaneously, it is necessary to obtain independent data of the
line-of-sight velocity. This can in principle be achieved by estimating the
Doppler centroid from the unfocussed SAR data as described in Kræmer et
al. (2015). For this method, however, a sufficient number of neighboring
pixels has to be averaged, resulting in spatial resolutions on the order of
hundreds of meters to a few kilometers. Whether this approach for retrieving
the line-of-sight velocity independent from the InSAR measurements is useful in practice for estimating the interferometric phase shift due to ice motion needs to be investigated in detail in another study.
So far, it was assumed that the motion of the ice is rectilinear, i.e.,
along a straight line, during the time interval between the acquisitions of
images 1 and 2. However, rotational motion of single ice floes about
their vertical axis also causes phase shifts and leads to an additional
decorrelation effect (Zebker and Villasenor, 1992; Scheiber et al., 2011).
The maximum magnitudes of floe rotation rates vary between 0.02∘ per hour
in the central Arctic with its compact ice cover and 2∘ per hour in the
marginal ice zone, where the ice concentration is low (Leppäranta,
2011). In the marginal ice zone, rates of even more than 100∘ per day
were noted at rare occasions, which may have been caused by ocean eddies.
Considering the temporal baselines given in Table 5, the expected rotation
angles that occur during single-pass along-track InSAR data takes vary
between 10-9 and 10-3∘. From interferograms derived from
TanDEM-X ScanSAR images acquired at the NE coast of Greenland, Scheiber et
al. (2011) retrieved floe rotations up to 0.005∘ for a time interval of
2.6 s between the two image acquisitions. This demonstrates the very high
sensitivity to rotational movements, which is also valid for rotations
around a horizontal axis. The rotational phase shift depends on the azimuth
position x relative to the center of rotation and the rotation angle
φrot:
Δϕrot=4πλxsinφrotsin(θ),
where λ and θ are explained above. The rotation angle is
measured relative to the azimuth direction (Scheiber et al., 2011). Due to
the rotation, the scattering elements in a resolution cell change their
position, which causes decorrelation of the received radar signals. Total
decorrelation occurs at a rotation angle of λ/(2Δxsinθ), where Δx is the azimuth resolution; i.e., at higher
radar frequencies and larger incidence angles the “critical” rotation
angle is smaller. For TanDEM-X ScanSAR and Stripmap mode, the respective
angles are ±0.086 and ±0.25∘ for θ=30∘
(Scheiber et al., 2011).
In the analysis above it was assumed that the vertical ice motion is zero. As
Mahoney et al. (2016) demonstrated, already small vertical displacements of a
few millimeters (as observed, for example, when infra-gravity waves propagate in sea
ice covered areas) may cause significant phase shifts. However, in their
investigation the temporal baseline was 10 s. They reported wave amplitudes
between 1.2 and 1.8 mm with periods between 30 and 50 s. For topographic
mapping, very short temporal baselines are required, at X band, e.g.,
optimally less than 0.5 s and even much less if the line-of-sight velocities
are high (see Table 5). Hence, vertical displacements caused by infra-gravity
waves in the central ice pack can be neglected, whereas in the marginal ice
zone surface wave amplitudes can be much greater. A sudden deformation event
due to pressure or shear forces in the ice, resulting in a vertical shift of
smaller ice areas, must also be considered since it may cause
non-negligible phase shifts. But since such processes are momentary events,
the probability that the related movement is directly measured is very low.
Influence of penetration depth and horizontal resolution
Another important question that needs to be investigated is whether the sea
ice surface height retrieved from the interferometric data represents the
actual height. Unfortunately, this is not the case. One has to consider two
effects. (a) Over rugged sea ice terrain, the retrieved value is an
effective height determined by the spatial resolution of the interferogram.
(b) The radar waves penetrate into the ice and snow, which means that
the horizon of maximum backscatter does not necessarily correspond to the
true ice (or snow) surface. In addition, the effect of volume decorrelation
has to be considered (see below).
Strub-Klein and Sudom (2012) present numbers for the maximum height of ridge
sails and for the average height of each sail (in both cases they list values
for maximum, minimum, and mean). The average height of a single sail (in the
following denoted as average sail height) is the mean of the heights measured
over a ridge cross section. Considering typical horizontal resolutions of
topographic maps derived from InSAR data, the retrieved interferometric
height will closely correspond to the average sail height. To be more
specific, the individual widths of ridge sails reported by Strub-Klein and
Sudom (2012) range from 1.8 to 73.2 m with mean values between 9.6 and
17.5 m for different locations in the Arctic and 7.4 for the Baltic Sea. The
corresponding average sail heights were between 0.1 and 3.3 m (minimum and
maximum from all individual ridges), with mean values from 0.3 to 1.6 m for
the different Arctic locations and 0.3 m for the Baltic. If the sail width
is larger than the spatial resolution of the SAR image, the ridge cross
section is represented by more than one height value. Dependent on the shape
of the ridge and on how many resolution cells cover its width, the elevation
derived from the resolution cell covering the sail maximum and possibly also
from adjacent cells may be higher than the average sail height, whereas the
elevations of the cells on the lower flanks of the ridges are smaller. If the
sail width is smaller than the spatial resolution of the interferogram, the
retrieved height will be smaller than the average sail height. Since ridges
of low height reveal small widths, the height error of the interferometric
retrieval may be too large to determine any useful value. Referring to Fig. 6
in the article by Strub-Klein and Sudom (2012), most ridges with widths
> 10 m are between 1.5 and 8 m high (maximum values). The statistics
presented by Strub-Klein and Sudom (2012) are based on measurements of
first-year ice ridges. Multi-year ridges (i.e., ridges which survived at least
one melting period) are more rounded, and the degree of consolidation
(bonding between single ice blocks) is higher. Because of the lack of a
multi-year ice ridge statistics we assume that their ridge height and width distributions are similar to the ones of first-year ice.
Freeboard values retrieved from radar altimeter data are typically lower
than 0.5 m, but north of Greenland's coast higher values may occur (Ricker
et al., 2014). With the relative height errors listed in Table 1, the
estimation of ice freeboard at the edges of leads is at the limit of the
achievable accuracy when using InSAR data.
Volume correlation γVol as a function
of the ratio d/ha_Vol (Eq. 13).
Critical penetration limits dc for first-year (FY) and
multi-year (MY) ice, determined from Eq. (13) for the satellite
constellations shown in Table 1 (p=1). For comparison, typical
penetration depths d for FY and MY ice are given (see text). For d>dc, the volume correlation is lower than 0.95. Also shown are
heights of ambiguity without (ha) and with (ha_Vol)
volume correction according to Eq. (12), together with the minimum ice
thickness Dmin=3.5d that is required for Eq. (13) to be valid.
For L band, Eq. (14) was applied, with D=0.5 m for FY and D=1.5 m
for MY ice (θ is the incidence angle).
The penetration depth d of radar waves (in terms of power) into ice depends
on ice salinity, temperature, volume structure, and radar frequency. If a
snow cover is present, its properties have also to be taken into account. Note
that we refer to the one-way penetration depth, which is d=κ-1,
if the extinction coefficient κ is constant with depth. It depends
on both the real and imaginary part of the dielectric constant. If the latter
is close to zero, the penetration depth approaches infinity. For saline
first-year ice, d decreases if the ice temperature and salinity increase.
Since the salinity of multi-year ice is low, variations of the penetration
depth are dominated by temperature changes. Under freezing conditions and if
volume scattering is negligible, the penetration depths at X band are about
1–7 cm into first-year ice and 5–30 cm into multi-year ice. The
corresponding values for C band are roughly twice as large. For
Ku band (Ka band), penetration depths range between 3
and 17 cm (2–8 cm) for multi-year ice and between 0.5 and 5 cm
(0.3–2 cm) for first-year ice. All numbers were taken from Lewis et
al. (1987). At L band, the penetration depths are 0.3–1 m for first-year
ice and 1–3 m for multi-year ice (Ulaby et al., 1986, Appendix E). In Shokr
and Sinha (2015, Table 8.11), the following values are given for first-year
ice with a snow cover of 13 cm and multi-year ice with 20 cm overlaid snow:
L band of 49/160 cm, C band of 7.0/32.0 cm, X band of 4.0/20.0 cm,
Ku band of 3.3/18.4 cm (interpolated value), Ka
band of
1.0/9.0 cm. Since ice blocks in ridges are often desalinated, the
effective penetration depth into the ridged ice is larger than into the
adjacent level ice, which reduces the apparent ridge height relative to the
level ice surface retrieved from the interferogram. In the following we
quantify the effect of the penetration depth.
Volume decorrelation
In Sect. 1, we defined the interferometric coherence as γ=γGγN, assuming that volume scattering can be
neglected. In the case of low-salinity sea ice, this is not always so,
which means that the coherence includes a volume component: γ=γGγNγVol. The effect of volume
decorrelation can be estimated based on Eq. (11), which was derived by
Weber Hoen and Zebker (2000):
γVol=11+πpBncosθHλtanθϵ′d′2,
with Bn, p, H, and θ defined above. Here, |γVol| is the correlation coefficient, ε′ is the real
part of the dielectric constant of the ice, d′ is the penetration length
along the propagation direction of the refracted wave at which the one-way
power falls to e-1, and λ is the radar wavelength in free
space. Note that Weber Hoen and Zeber derived Eq. (11) for the case of
repeat-pass interferometry, i.e., p=2. The penetration depth d (along
the vertical) is d=d′cosθr, where θr is
the refraction angle. The correlation decreases if the penetration depth into
the ice increases. In the following discussion, a radar resolution cell
corresponds to a volume element. Equation (11) is derived under the
assumptions that (a) the scattering medium is homogeneously lossy, (b) the
radar cross section of the scattering elements varies only as a function of
depth, (c) the volume is characterized by an exponential extinction, and
(d) the layers at depths >d do not contribute to the
backscattered signal. In the derivation a non-weighted, ideal radar transfer
function is used (Weber Hoen and Zebker, 2000). Based on the study by Dall
(2007), we modified Eq. (11). From Snell's law we obtain
cosθr=(1-sin2θ/ε′)-1/2. If the radar waves penetrate into the volume, the
height of ambiguity changes according to
ha_Vol=haε′-sin2θε′cosθ=hacϵθ.
(Note that there is a printing error in this equation in the paper by Dall,
2007.) Equation (11) then simplifies to
γVol=11+πdha_Vol2.
Equation (13) represents the absolute value of Eq. (9) in Dall (2007),
except that Dall uses the two-way penetration depth d2=d/2. Note that
ha and hence |γVol| are functions of p.
According to Eq. (13), the volume correlation depends on the ratio
between the penetration depth and the volume-corrected height of ambiguity.
However, Eq. (13) is only valid if the ice thickness exceeds the
penetration depth by a factor of 2.5; otherwise the volume correlation
additionally depends on the ratios D/ha_Vol and D/d (Dall,
2007). The correlation coefficient as a function of the ratio
d/ha_Vol is shown in Fig. 6. For the dielectric constant of
sea ice, results of measurements are presented in Hallikainen and
Winebrenner (1992, their Figs. 3.5 and 3.6) for different ice types,
depending on salinity and temperature. Those measurements were carried out
in the frequency ranges 4–5 GHz and 10–16 GHz for salinities between
0.2 and 0.5 ppt and temperatures between -50 and
-0.2 ∘C. In the first frequency interval, the real part of the
dielectric constant assumes values between 2.9 and 4.3 and in the second one
between 2.5 and 4.2. Values for multi-year ice are between 2.5 and 3.1 and for
first-year ice between 2.9 and 4.2. For the discussion of examples presented
in Table 6, we assume ε′=2.8 for multi-year ice and
ε′=3.5 for first-year ice, yielding c2.8,25=0.6380,
c3.5,25=0.5745, c2.8,40=0.7203, and c3.5,40=0.6553 for
the coefficient cεθ in Eq. (12). A value of
d/ha_Vol= 0.1 corresponds to |γVol|≈0.95 (Eq. 13).
If the finite thickness of sea ice is taken into account, the volume
decorrelation is a function of the three parameters ambiguity height
ha_Vol (which characterizes the radar system),
penetration depth d (which depends on salinity, temperature, and the
fraction, size, and shape of air bubbles in the ice), and ice thickness D.
The radar waves do not penetrate into the saline water below the ice. Hence
we can apply Eq. (8) given in the paper by Dall (2007). Evaluating the
magnitude of his expression, we obtain
γVol=1-Aexp-2Dd2+πdha_Vol2-1+Bexp-2Dd21+πdha_Vol21-exp-2Dd
with
A=cos2πDha_Vol-πdha_Volsin2πDha_Vol,B=cos2πDha_Vol+ha_Volπdsin2πDha_Vol.
The coefficients A and B give rise to a resonance phenomenon that, due
to the multiplication with exp(-2D/d), only occurs when the penetration
depth is larger or of same magnitude as the ice thickness. Note that
scattering from the ice–water interface is not considered here. In the case
of low ice salinity, measurable scattering contributions from the ice–water
boundary may occur as was demonstrated by Dierking et al. (1999) for Baltic
Sea ice. The development of a model including the interface scattering,
however, is beyond the scope of this study.
The comparison of the critical penetration limits and the penetration depths
listed in Table 6 reveals that volume decorrelation can be neglected at
Ka and Ku band for both first- and multi-year ice.
Equation (13) is not applicable for thin young ice (thickness < 5 cm) but
in this case topographic undulations usually cannot be reliably retrieved
considering the achievable height accuracies. At X band, volume
decorrelation has to be taken into account for low-salinity multi-year ice,
and Eq. (13) is still applicable for a larger range of the ice thickness. For
the Ka-, Ku-, and X-band mission scenarios shown in
Tables 1 and 6, the ratio d/ha_Vol≪1. This means that
according to Dall (2007) the elevation bias (relative height error) due to
volume effects equals half the one-way penetration depth. If we focus on
first-year ice with D>0.5 m (note that even on ice with D≈0.2 m ridge sails may be as high as 3 m in some cases; see Fig. 15 in
Strub-Klein and Sudom, 2012), and assume that the salinity of thinner
second-year ice is larger than for multi-year ice (which means a smaller
penetration depth in the first case), useful estimates of the critical
penetration depth according to Eq. (13) can still be obtained at C band.
The ratio d/ha_Vol is ≪ 1, which means that also in this
case the elevation bias approximately corresponds to 0.5d, that is at
maximum to about 0.3 m for low-salinity multi-year ice (see penetration
depths listed above). The situation at L band is more complicated, since
the ice thickness and penetration depths are of similar magnitude. Hence,
Eq. (14) is applied to provide estimates for the critical penetration depths,
which reveal that volume decorrelation has to be considered at L band. The
elevation bias depends both on thickness and penetration depth. The
derivation of a corresponding relationship is not carried out here.
Dall (2007) only considers cases with d/D and ha_Vol/D
approaching infinity for which he obtains a bias of 0.5D.
The determination of volume decorrelation and elevation bias requires that
the penetration depth and hence ice salinity, temperature, and – of minor
importance – volume fraction of scattering elements have to be obtained
parallel to the SAR data acquisitions which is not possible in practise. A
reliable determination of the elevation bias is difficult since it depends on
the ratios between ice thickness, penetration depth, and the volume-corrected
height of ambiguity, which change between ice types and depend on
meteorological conditions (e.g., melting and freezing conditions). Optimal
measurement conditions with relatively small penetration depths are given if
the ice temperature is close to the freezing point but still too low for
melt onset. Classification maps separating multi-year, first-year, and thin
ice obtained from the SAR intensity images are helpful for judging the
reliability of the estimated height error. The separability of ice classes,
however, depends on radar frequency and polarization. We note that the
application of Eqs. (13) and (14) requires that volume scattering is not
negligible (see Dall, 2004, Eq. 2). Volume scattering may be very low under
certain conditions, in particular at L band. The presence of snow on the
ice (see below) complicates the situation further. Tomographic radar
measurements on sea ice such as reported by Yitayew et al. (2016) may provide
useful insights regarding these issues.
Influence of snow
In real-world situations, snow layers are present on the ice. Dry snow is
almost transparent at larger radar wavelengths (penetration depth, e.g., 30 m
at a wavelength of 7.5 cm) and still highly penetrable at smaller
wavelengths (1.5 m at 1 cm); see Ulaby et al. (1982, Fig. 11.25).
Scattering from the snow surface is negligible in most cases. In the snow,
the radar wavelength decreases, and the incidence angle at the snow–ice
interface is smaller than at the air–ice interface. Hence, the results given
above for a snow-free ice surface have to be adjusted accordingly. Effects
are, among others, that ambiguity height and critical baseline decrease (Eqs. 2
and 4). In general, snow thickness is larger in areas of deformed ice. From
measurements at different sites in the Arctic, Sturm et al. (2006)
reported snow thickness variations between a few centimeters and up to 80 cm
with mean values between 9 and 21 cm. The snow thickness may vary
considerably on relatively short spatial scales due to redistribution
by wind. In the intervening smooth ice, the snow layer may be thicker
than on top of the ridges but less thick than at their lee sides. The snow
density, which determines the dielectric constant of dry snow, may also vary.
If snow thickness and density over the ridge and the adjacent level ice are
different, the total topographic phase difference includes a contribution
from the different paths along which the radar waves propagate.
In the case of strong surface winds and longer temporal baselines, the snow drift
may reduce the interferometric coherence. This effect is more pronounced at
higher radar frequencies. At Ku and Ka bands, the
major scattering horizon may not be identical with the snow–ice surface but
be located higher up in the snow layer (e.g., Willat et al., 2010, 2011).
However, taking into account a realistic height error for retrievals from
InSAR measurements, the rise of the scattering horizon is negligible. In
moist snow, penetration depths decrease significantly. If the volume moisture
content is 1 % (5 %), the depths for the given wavelengths are 70 cm
(20 cm) and 5 cm (1 cm) (Ulaby et al., 1982, Fig. 11.25.) During the
melting season, the radar signal is backscattered from the wet snow or – if
no snow is present – from the wet ice surface. One could argue that height
retrievals from images measured over melting ice provide the “real” surface
that determines the aerodynamic drag. However, topographic data are also
required from the winter season, and temporal variations and trends of the
ice surface height need to be known for estimating the ice mass balance. It
must also be considered that the backscattered intensity changes
seasonally; e.g., in the case of multi-year ice, the volume scattering
contribution is suppressed under melting conditions, and thus the total
backscattering and the SNR decrease. The backscattered intensity may
increase if superimposed ice is formed on top of first-year ice, but it may
also decrease if meltwater smoothes the small-scale surface roughness
(Onstott et al., 1987).
Normalized path difference Δs/h as a function of snow
density ρds. For explanations, see text.
If a homogeneous layer of snow with thickness h is assumed, the difference of
the paths without (s) and with snow (s′) are Δs=s-s′=h(cosθ-1-cosθr-1). The refraction angle is
sinθr=sinθε′ds-1/2. The
real part of the dielectric constant for snow is related to its density
ρds by ε′ds=1+1.9ρds
for ρds≤ 0.5 g cm-3 and
0.51+2.88ρds for ρds>0.5 g cm-3
(Hallikainen and Winebrenner, 1992). The normalized path difference Δs/h is shown in Fig. 7. If, e.g., the ridge is snow-free and the snow layer
on the neighboring level ice with ρds=0.6 g cm-3 is
40 cm thick, one obtains differences of 1.5, 3.7, and 11.2 cm at incidence
angles of 20, 30, and 45∘, respectively. Considering that these values
are considerably smaller than the “acceptable” relative height error of
0.5 m, the influence of a dry snow layer can be neglected in most cases.
This is a valuable result since snow density and thickness data valid for the
time of SAR image acquisitions are usually not available.
Achievable baselines
InSAR techniques can successfully be applied for drifting sea ice only if
image pairs are acquired with small temporal gaps on the order of
milliseconds to seconds and baselines smaller than the critical limit
determined by Eq. (4). This means that data from satellite configurations
such as TanDEM-X are required. The two satellites of the TanDEM-X mission
(TerraSAR-X and TanDEM-X) fly in a helix formation, which combines an
out-of-plane (horizontal) orbital displacement due to different ascending
nodes with a radial (vertical) separation due to different eccentricity
vectors (Krieger et al., 2007). The ascending node is the intersection of the
equatorial plane and the satellite orbit on the leg from the Southern to the
Northern Hemisphere. The TanDEM-X satellite is controlled with respect to
TerraSAR-X (Maurer et al., 2016). The maximum baseline varies along the
orbit; its length is expressed as a function of the geographical latitude (AO
TanDEM-X Science Phase manual, https://tandemx-science.dlr.de/).
Furthermore, the effective baseline is larger at smaller (steeper) incidence
angles. During the TanDEM-X Science Phase, the largest cross-track baselines
amounted to 3000 m and were achieved over the Equator (AO TanDEM-X Science
Phase manual, https://tandemx-science.dlr.de/). For a given latitude,
the baseline length can be changed by varying the eccentricity vector of the
orbit. The helix parameters are usually kept constant for certain periods to
minimize fuel consumption. For a satellite tandem we can conclude that (a) it is
in principle possible to achieve the cross-track baselines necessary for
mapping height variations of the sea ice surface; (b) the sensitivity to
surface height variations is not constant but varies as a function of
latitude; and (c) optimal conditions for measurements of the sea ice surface
topography in a given region are only possible during limited temporal
intervals because of satellite operation requirements. An important issue is
the magnitude of the along-track baseline Bal as we discussed
above. For TanDEM-X, the uncertainty of estimates of Bal amounts
to ±200 m. As a rule of thumb, Bal is twice as
large as Bn in a bistatic configuration. For TanDEM-X, it is at its
maximum at the Equator and approaches zero at the poles (Gerhard Krieger,
personal communication, December 2016).
In the literature, satellite constellations consisting of more than two
receiver microsatellites (and a satellite with an active SAR ahead or behind)
have been discussed (e.g., Krieger et al., 2003; Moreira et al., 2002). The
general advantage of configurations consisting of N>2 receiver satellites
is that the variations of the across-track baseline lengths as a function of
latitude can be minimized by picking the most suitable transmitter–receiver combination. The interferometric cartwheel consists of
satellites flying in close formation on slightly different elliptical orbits.
The orbit parameters are selected such that the formation of receiver
satellites seem to move on an ellipse centered on the orbit of the active
satellite. However, with the cartwheel, across- and along-track baselines
cannot be optimized at the same time. This is a disadvantage for retrieving
the surface relief of drifting sea ice (see above). An alternative is the
cross-track pendulum (with the TanDEM-X helix as a special case). Here, the
receiver satellites are all moving with equal velocities along circular
orbits in different orbital planes with slightly different ascending nodes
and/or inclinations. With this configuration, across-track baselines of any
desired length can be formed. If three receiver satellites are used and the
respective maximum baseline is selected, the variations are limited between
87 and 100 % of the achievable maximum. At the same time, the along-track
baselines can be set independently and kept constant (Krieger et al., 2003,
Fig. 2). However, very short along-track baselines increase the risk of
collisions at crossing points of the orbit planes. Because of the secular
drift of the ascending nodes (due to the nonspherical shape of the Earth) and
the different inclinations, the cross-track pendulum formation is not stable.
For maintaining the orbits, additional fuel is required. It is beyond the
scope of this study to propose an optimal satellite formation for the
retrieval of sea surface height undulations. But it is noted that this is a
necessary requirement for planning future satellite missions suitable for
determining surface topography on meter and sub-meter scale.
Other factors
When estimating the achievable accuracies of height retrievals, possible
errors in the determination of (a) length and angle of the normal baseline,
(b) local incidence angle, and (c) orbit altitude have to be considered and
assessed routinely in InSAR processing. The respective influence of these
errors on the final results is not made subject of this study, since the
intention was to discuss specific conditions related to the retrieval of the
topography of fast and drifting sea ice. Other factors that need to be taken
into account in InSAR processing are the accuracies of co-registration of the
two images used for generating the interferogram, filtering steps for
reducing the phase noise, flat-plane phase removal, and phase unwrapping
(Richards, 2007). Phase unwrapping, however, may only be required for large
sea ice ridges and low heights of ambiguity. The simplest approach for phase
noise reduction in the interferograms is achieved by averaging
neighboring pixels, thus increasing the number of looks, NL,
which reduces the relative height error (see Eqs. 3 and 5) but at the same
time worsens the spatial resolution, which possibly decreases the retrieved
apparent ridge heights.
Conclusions
In this paper we analyzed the application of interferometric SAR for
retrieving the surface topography of sea ice, assuming different satellite
missions with radar frequencies ranging from Ka to L band. As
a basis for judging the feasibility we used statistics of ridge heights and
widths reported in the literature. Optimal across-track baselines for
achieving the lowest possible height error vary from 40 km at L band
(incidence angle 40∘) to 320 m at Ka band (at
25∘). Relative height errors smaller than 0.5 m are achievable for
large signal-to-noise ratios (SNR < 15 dB). In particular undeformed
thin ice (without frost flower coverage) and smooth level ice reveal a low
SNR. For an SNR of 10 dB, the relative height error increases by a factor of
1.1–1.2; for SNR = 5 dB the factor is 1.4–1.5. In the case of drifting
pack ice, the influence of the ice motion on the interferometric phase must
be considered unless the line-of-sight ice velocity uLOS equals zero.
For uLOS=0.18 km h-1, along-track baselines from 3400 m
at L band to 110 m at Ka band cause phase shifts corresponding
to a relative height error of 0.5 m. If uLOS= 2.2 km h-1, which represents large wind speeds, the respective
numbers are 280 to 10 m. Wind-driven surface currents on open water areas
within the ice cover may also generate a phase shift. Hence, such areas
should be masked in the topographic map. Effects of volume decorrelation in
ice and snow are negligible at Ka and Ku band and of
minor importance at X band because the radar penetration depths are
relatively small at these frequencies. At C and L band, an increase of
the height error due to volume decorrelation has to be considered in
particular for low-salinity ice with large penetration depths. If a dry snow
layer is present on the ice, the radar wavelength at the snow–ice interface
is shorter than in air and the incidence angle is steeper, changing the
magnitude of the optimal across-track baseline. In the case of melting
conditions, radar penetration depths into the snow are reduced and approach
zero at larger snow moisture content. With the recent TanDEM-X mission, a
change of the default orbital parameters is required to achieve the necessary
across-track baselines over the polar regions. The cross-track pendulum
satellite configuration with more than two satellites can be more easily
optimized for measurements of sea ice topography than the cartwheel. The
availability of additional information in the process of retrieving sea ice
topography would be of advantage. For example, to judge the influence of sea
ice motion on the height retrieval, the line-of-sight velocity should be
determined simultaneously with the interferometric phase from the Doppler
shift of the radar signal caused by the ice movement. Another valuable
information is an ice chart showing the spatial distribution of different ice
types, derived from the SAR intensity images used for generating the
interferogram, possibly extended by images acquired at different
polarizations and/or frequencies.
The TanDEM-X data were obtained under a scientific license
(see Acknowledgements) for an approved proposal submitted in the framework
of the TanDEM-X Science Phase AO. They are not publicly
accessible.
WD worked on linking the theoretical basis for the retrieval of surface
topography with the specific conditions that need to be considered for sea
ice. OL processed the TanDEM-X data and contributed the figures for Sect. 4.
TB helped with clarifying technical aspects concerning processing of TanDEM-X
data and baseline variations along the orbit. WD wrote major parts of the
paper, and the co-authors contributed to the improvement of the text.
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors express their thankfulness to our colleagues Jørgen Dall from
the Technical University of Denmark and Gerhard Krieger from the German
Aerospace Center for useful discussions and hints on volume decorrelation and
orbit configurations. We are also grateful for the constructive comments of
Andrew Mahoney and an anonymous referee. The TanDEM-X data were provided by
the German Aerospace Center (DLR) under a scientific license, project
XTI_OCEA6971.The article processing charges
for this open-access publication were covered by a Research
Centre of the Helmholtz Association. Edited by: Lars Kaleschke Reviewed by: Andrew
Mahoney and one anonymous referee
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