Heinert, M. and Riedel, B. (2007):
————————————————————————————————–
Parametric modelling of the geometrical ice-ocean
interaction in the Ekstroemisen grounding zone based on
short time-series
————————————————————————————————–
Geophys. J. Int. 169: 407–420. DOI: 10.1111/j.1365-246X.2007.03364.x
Technische Universität Carolo-Wilhelmina zu Braunschweig, Institut für Geodäsie und Photogrammetrie
Gaußstraße 22, D-38106 Braunschweig, e-mail: m.heinert@tu-bs.de, b.riedel@tu-bs.de
Abstract
Due to the ocean tide impact the Ekstroemisen (Ekstroem Ice Shelf, Antarctica) shows at its assumed grounding
line still vertical displacements. These can reach amplitudes of five centimetres here. The low tide amplitudes are
smaller than the corresponding high tide amplitudes. This indicates that the ice body touches the bedrock during
the low tides. Even one kilometre behind the assumed grounding line an elastic feedback of the ice body can be
observed. The maximum vertical displacements can reach still one centimetre.
These results are based on data collected during the austral summer 1997 in the grounding zone of the
Ekstroemisen. This region has been studied by using a combined sensor-field of continuous GPS (Global
positioning system), gravimeter and tiltmeter instruments to derive the elastic response of the ice body on ocean
tides. The resulting time-series which have to be analysed are biased by outliers, data gaps or complex trend
functions. In the case of very short time-series it is nearly impossible to analyse them without assuming
hypotheses.
It can be shown that standard-algorithms like correlation-functions, different Fourier-transforms or functional
regressions are not suitable to solve the parameterization of time series in the region of the grounding zone,
where several kinematic and gravimetric processes overlap. After a thorough analysis of these time series with an
average length of eight days we managed to homogenize the results of the hybrid and non-simultaneous
observations. On the base of non-linear programming we created a set of robust parameters to describe the ice
shelf motion in the grounding zone.
Key words: Tides, Spectral analysis, Fourier transform, Geodesy, Global Positioning System (GPS), Gravity
1 Introduction
fully a system or a sub-system to understand it and its
interaction with the surrounding systems. To determine
the transport of material, energy or information across
the system boundary one has to define or identify this
boundary (Riedel 2002). In order to estimate the reaction of the ice sheet on sea level change, it is neccessary
to understand the dynamical processes in the grounding
zone. The vicinity of the grounding line is called the
transition or grounding zone between the grounded ice
sheet and the ice shelf.
The continent Antarctica and its surrounding oceans
represent a sub-system in the system earth. It plays
an important role in the discussion about greenhouse
warming and related sea level rise. The system or subsystem Antarctica consists of the inland ice sheet, the
surrounding ice shelves, the atmosphere and the polar
oceans. Ice shelves play an important role for the dynamics and possibly also for the stability of the Antarctic ice sheet. Therefore it is necessary to define care-
Vaughan (1994) characterized the grounding line by
1
-7
5°
-6
-60
°
8° 20
0°
8° 30
derived by terrestrial geodetical measurements (Karsten
and Ritter 1990). In 1993-94 seismic shootings allowed
a far more accurate determination of the grounding zone
(Mayer 1996).
The tidal investigations in the grounding zone were
inspired by Holdsworth’s studies on Erebus glacier
tongue and his elastic beam approach (Holdsworth
1969, 1974). In this approach a simple elastic beam
model is used, to get an approximation of the tidal deflection in dependency on the Young modulus. In order to obtain input data for a two-dimensional finite
element model, field data have been collected at the
south-western grounding line of the Ekstroem Ice Shelf
(Riedel 2002). The main emphasis of the field campaign was placed on collecting continuous observations
of the horizontal as well as the vertical component of
the ice displacement and their lateral variations across
the grounding zone. In this study the question will be
addressed of how the ice shelf interacts geometrically
with the ocean tides. Therefore we analyse the observations on the cross-section of the grounding zone. This
will examplarily be done by using the observations that
have been obtained on the Ekstroemisen using GPS receivers, gravimeters and tiltmeters.
Due to fact that on average the time series cover only
an observation interval of eight days we can state that
the standard techniques of the time-series analysis are
not sufficient. For our purposes the only helpful standard algorithm of the time series analysis was the fast
Fourier transform. The time series were even too short
to determine the most important tides (O 1 , K1 , M2 , S2 )
qualitative. Any reliable quantification of the parameters amplitude, frequency and phase lag was impossible. All the different spectra solutions like complex fast
Fourier transform and zero padded complex fast Fourier
transform of the data or Fourier cosine transform yield
results with similar short comings (fig. 2). Only three of
the four main tides could be determined significantly.
The GPS data seem to have one diurnal and two semidiurnals tidal waves and the gravimetric data showed
two diurnals and one semi-diurnal tidal signal. These
time-series are extremely short to derive a model of the
tidal response.
0°
°
-30
8° 40
A0
BC
°
60
-90°
-90°
90°
12
0°
180°
D200
A003
A3
D100
A002
A2
A004
A4
GL 0
km 05
km 1
A001
A1
100
0
8° 40
8° 30
1
2
3
4
5
km
Figure 1: Map of Ekstroemisen grounding zone with
the positions of continuous working GPS receivers,
gravimeters and tiltmeters. The black square in the inlay
map shows the position of Ekstroemisen in Antarctica.
the change of surface slope from a steep, undulating surface of grounded ice to the gently sloping surface of a
floating ice shelf. It represents the limit of tidal flexure
of the ice surface and the position of the bedrock under the ice body, where ice starts to float on the ocean.
This feature is the onset of ablation processes at the
ice-ocean interface right at the grounding line. Another
characterization of the grounding zones is the transition processes of the ice from shear stress dominated
dynamics to a longitudinal stress regime. But looking
closer, the definition of a grounding line becomes comparable to that of a shore line. Its position varies and
depends on the shore structure, the ocean tides and possible sea level changes.
The Ekstroemisen (Ekstroem Ice Shelf) is one of the
smaller ice shelves in Queen Maud Land belonging to
East Antarctica. The only German overwintering station
Neumayer (70 ◦ 39’S, 08◦ 15’W) has been established up
on the northern part of the Ekstroemisen (fig. 1). The
part of the Ekstroemisen grounding zone that has been
investigated in this study lies 140 km south of the Neumayer station on the south-western rim of the ice shelf.
A first rough impression of this area has been obtained
from the topograpical cross-sections which have been
2 Data sources
The continuous geodetic monitoring in the vicinity of
the grounding zone using GPS, gravimeters and tiltmeters started on February 10 th and had to be finished
2
0.3
O1
M2
K1
(a)
S2
Figure 3: Observation periods of the GPS (black) and
gravimetric (grey) stations.
0.1
only 16 days later due to bad weather conditions (Riedel
et al. 1999). To obtain the pure geometrical response
of the ice body in the grounding zone caused by the
ocean tides, six GPS receivers were set up each with
a sampling rate of 15 seconds. One receiver acted as
the permanent base station at the site KOTA, which is
situated on a bedrock near the temporal airfield Windy
Corner in the Kottas Mountains at the eastern margin
of the Heimefrontfjella (Heinert et al. 1998). A second receiver was set up as the local reference station
at site A001 during the whole campaign (Riedel and
Heinert 1998). The daily position of the northward drifting site A001 was given by the baseline solution of
KOTA-A001. An additional kinematic processing of the
300km long baseline yields no signifcant tidal signals
at A001. The network of the remaining GPS receivers
was recording simultaneously on three respectively four
additional sites. Their tidal signal is a result of the kinematic baselines to station A001.
The map in figure 1 shows the distribution of the GPS
sites in the grounding zone. The sites A001 and A004
were placed on grounded ice while the sites A003,
D100 and D200 were set up on the ice shelf. The site
A002 marks the grounding line position, where the ice
body is assumed to start floating on the ocean. At the
sites BC, GL0 (close to A002), KM05, KM1 (close to
A004) and KM2 each one tiltmeter and one gravimeter were buried in a snow pit. The gravimeter and tiltmeter raw data were recorded in intervalls from five
to 300 seconds depending on the data logger storage.
The gravimeter output shows a combined signal of solid
earth tides and the response of the ice on the ocean tides.
The decrease of the local acceleration can be interpreted
as a Bouguer plate of water (Torge 1991) with an unknown thickness that increases under the ice body during the high tide. With a combined local free-air and
amplitude [m]
0.2
0.0
0.00
0.01
0.02
0.03
0.04
0.05
frequency [mHz]
0.06
0.07
0.08
0.3
O1
M2
K1
(b)
S2
amplitude [m]
0.2
0.1
0.0
0.00
0.01
0.02
0.03
0.04
0.05
frequency [mHz]
0.06
0.07
0.08
Figure 2: The periodicities in the spectra of the GPS
observations at the nearly undisturbed station A003 (a)
indicate an ocean tide signal, while the gravimetrical
observations at station KM2 yield a distribution of periodicities (b), which are typical for the gravimeter registrations of the tides of the solid earth. The fast Fourier
transform has been done on the original data with (line)
and without (broken line) zero padding. It can be compared with the Fourier cosine transform on the autocovariance function (dotted line).
3
Bouguer gradient of 0.04308 mGal · m −1 it is possible
to get geometrical height variations based on the approaches of Thiel (1960) and Doake (1992). The pure
geometrical information of the tiltmeters give us the
possibility to separate sites with pure earth tides from
those which have got mixed tidal signals in the gravimetrical data. As a result only the site KM2 showed
unbiased earth tides, as no tilt signal exists there. Furthermore, the tilt information is an indicator for the deformation intensity as we assume that a tilt change only
can be found in a region where the tidal energy is dissipated. Accordingly, we found no tilt at the site BC
which has a minimum distance of five kilometres from
any grounded ice (figure 1). An overview of the periods
of observation at the different sites is given in figure 3.
than the quantile of the Student distribution
α
P t̂ ≤ tf ;1− α2 = 1 −
2
with a high-significance threshold α = 0.01. The quantile tf ;1−α of this moving average can be computed using the degree of freedom f
f=
Outliers, noise, data gaps as well as linear and nonlinear trends distort our data more or less seriously. One
has to face the problem that these biasing effects cannot be treated independently of each other: the outliers
and the noise falsify the trend estimation, a linear filter algorithm requires data without gaps. Also, outliers
and noise are not easily to reduce, if a signal cannot be
filtered. Consequently, we screened, analysed and modelled our data in a recursive way and developed our own
adapted algorithms.
At first, we eliminated the extreme outliers in every
data segment. Therefore we compared a single value x t
of the time series with the result of a moving average x̄ t
with half-sided cosine weight function
b
τ 1 π
xt−τ 1 + cos
b − 1 τ =1
b
b−1
(b − 1)2
· sx t =
.
b
2·b
A detected single outlier can be substituted by the value
of the moving average. It is useful to add a random value
derived from a stationary Gaussian process with the local standard deviation s xt . However, this algorithm has
to be stopped if more than five outliers in sequence
have been detected. Otherwise this algorithm can create a random walk process (Schlittgen and Streitberg
1997). If there are more than five outliers then this sequence should be treated as a data gap. The treatment
of data gaps is a serious problem with respect to analyses, which require equidistant data. A quite easy and
sufficient way is given by Melchior’s method (Kobarg
1988) on the base of Labrouste’s combination of ordinates (Lecolazet 1956). For time series of tides this
interpolation combines highly positively correlated values of the access intervals to fill the gaps. Typically,
the highly correlated intervals of tide observations are
shifted by 24, 48 and 72 hours.
3 Data screening
x̄t =
(3)
(1)
x̃t
=
−
with the width b = 10, which is equivalent with two
and half minutes given by our sampling rate. Accordingly, we can test whether x t and x̄t are statistically
equal (null hypothesis) or x t differs significantly (alternative hypothesis). The estimation of the Student ratio t̂
can be written as
b−1
|xt − x̄t |
= |xt − x̄t | (2)
t̂ =
b
sx t
(xt−τ − x̄t )2
+
1
(xt−72h + xt+72h )
20
3
(xt−48h + xt+48h )
10
3
(xt−24h + xt+24h )
4
(4)
In distorted time series it can happen that such an invariable interpolation tries to use values of the access
intervals, in which data are missing as well. Consequently, we tried to change the source intervals. One
possible solution is the use of access intervals within
time intervals of high negative correlations with respect
to the gap. In those cases we used a modified Melchior
τ =1
with the empirical local standard deviation s xt . The null
hypothesis can be accepted if the estimated t̂ is smaller
4
algorithm with heuristically determined weights.
x̃t
=
+
access intervals
+
(a)
−
45
46
47
48
filling interval
49
50
51
52
45
46
47
filling interval
48
49
53
(b)
50
51
52
x̃t =
53
45
46
47
t
50
51
52
uτ · xt±(I+τ ) with I ∈ N
(6)
The question is, how to determine the unknown weight
vector u. In an undistorted interval of the time series
it is possible to build differences between the known
values xt and modelled values x̃ t , whereby one starts
with a random set of weights. To get an optimal set of
weights the mean squares
2
(x̃t (u, I, b) − xt ) =
vt2
(7)
(c)
weight function
filling interval
48
49
b
τ =1
days of the year
access intervals
(5)
To be as flexible as possible we developed a correlative Melchior algorithm (fig. 4). By using the YuleWalker equation (Schlittgen and Streitberg 1997) it is
possible to determine the weights of an auto-regressive
process AR[p] of the order p from well-known autocorrelations. Unfortunately, this combination uses values that are directly neighboured to those we are looking for. That means for data gap, which should be filled,
that it can only consist of one single value. Let a data
gap have the size of n values. Then we need each the
number of b values of x t before and behind a data gap
to interpolate one value x̃ t of the gap. So, the first element u1 · xt±(I+1) of the combination has to be I ≥ n
values away from the gap at time t.
days of the year
access intervals
1
(xt−30h + xt+30h )
10
1
(xt−24h + xt+24h )
2
1
(xt−18h + xt+18h )
10
1
(xt−6h + xt+6h )
5
t
have to be minimized. The most comfortable way to
solve this minimization problem is to use one of the
various non-linear optimization algorithms (see section
5 and appx.). We recommend that the interval of the
time series, which is used for the weight determination,
should have the number of 3·(I+2b) more or less undistorted observations. If one starts to fill the smaller gaps
first, it should be possible to fill all the gaps in a time series by using these three methods. To sum up one may
say that Melchior’s method and its first adaptation is
quite useful for tidal signals, but only the second adaptation can be used for other signals without the typical
tidal character as well.
53
days of the year
Figure 4: Data filling: (a) Melchior’s method, (b) an
adapted Melchior method and (c) an autocorrelative
Melchior method with the symmetrical weight function.
5
4 Data analysis
For various questions we need only special frequency
ranges. Therefore, we filtered the time series by using
a linear high- or low pass filter. As weight function we
used the so-called Gaussian core
g (τ ) =
τ2
1
√ · e− Θ2
Θ π
(8)
with the form parameter
√
∆t
2
and τ =
,
Θ=
e
b
Figure 5: Example of the convergence of the minimization procedure using the steepest descent (left) or the
quasi-Newton method (right).
whereby b determines the width of the filter. The related
filter gain function is given by
H (ν) = e−π
2 2
ν Θ2
.
The phase lag is given by
(9)
ϕi = arctan
The separation between high and low frequencies
was defined by the filter width of b =
ˆ 15 min. A statistical test yields the result whether the high frequent noise
follows the normal distribution. Accordingly, the use of
all statistical standard methods, which depend on normal distribution, gives no cause for concern. For further
investigations it was, of cause, necessary to determine
the trend of station motion from the time series. On one
hand this result represents a station velocity and on the
other hand the search for periodicities becomes possible
after the linear trend reduction. The minimization of the
simple linear regression
(xt − (β0 + β1 t))2
(10)
bi
ai
.
(12)
The periodical answer of an ice body on the periodicities to the ocean tides can be different from the
theoretical ocean tides. Thus, this regression concept is
not sufficient. Consequently, for a motion model, that
should be free of hypotheses, the frequencies have to
be treated as unknown parameters as well. Therefore,
we created a model in which all the free parameters u i
are estimated in the same recursion. The model refers to
both the linear trend given by its zero value x 0 and the
related gradient m and the periodicities which can be
described by their amplitude A, the frequency ν and the
phase lag ϕ. In case that the model has to fit a gravimetrical time series the parameter e determines the effect
of the tides of the solid earth:
τ
was sufficient for these purposes.
Φt (u) = x0 + m · t + e · E(t)
n
+
Ai · sin (2πνi · t + ϕi )
5 Parametric modelling of iceocean interaction
(13)
i=1
with
For the modelling of ocean tides it is quite common to
use a regression of the parameters amplitudes and linearized phase lags
ai · cos (2πνi t) + bi · sin (2πνi t) , (11)
Φt (νν ) =
uT = [x0 m e A1 ν1 ϕ1 . . . An νn ϕn ]
(14)
and with n = 4 representing the four main tides
O1 , K1 , M2 and S2 .
At first, the geometrical variation has to be deduced
from the variation of the local gravimetrical signal.
This signal still contains the tides of the solid earth,
which had to be estimated together with the ocean
tides. Therefore, we used the global earth tide model
i
whereby the frequencies ν i of the tides are assumed to
be known:
νi = νO1 , νK1 , νM2 , νS2 , ....
6
7
5
1
3
0
1
amplitude [m]
distance to the grounding line [km]
9
-1
-3
36
38
40
42
44
46
48
50
52
time [day of the year]
54
56
58
60
Figure 6: The quite short ocean tide registration (left) along the profile crossing the grounding line yield highly
distorted time series, which hardly can be interpreted. After a thorough data screening and modelling (right) the
time series show typical ocean tide oscillations.
0,03
(b)
(a)
0,3
S2
0,2
K1
O1
0,1
0,0
0,02
S2
0,0
2,0
4,0
6,0
8,0
10,0
-2,0
distance to assumed grounding line [km]
O1
tilt phase lag [rad]
phase lag [rad]
6,0
(c)
5,0
4,0
S2
3,0
K1
2,0
M2
1,0
-2,0
(d)
0,5
O1
5,0
4,0
S2
3,0
K1
2,0
M2
1,0
0,0
2,0
4,0
6,0
8,0
-2,0
10,0
gravimetrical pre-factor [-]
distance to assumed grounding line [km]
local tilt [mrad]
-1,5
-1,0
-0,5
0,0
distance to assumed grounding line [km]
0,0
0,0
0,5
O1
0,01
0,00
-2,0
6,0
K1
M2
M2
amplitude [m]
amplitude [m]
0,4
(e)
0,4
0,3
M2
0,2
S2
K1
O1
0,1
0,0
-1,1
0,0
2,0
4,0
6,0
8,0
distance to assumed grounding line [km]
10,0
(f)
-1,2
-1,3
-1,4
-2,0
0,0
2,0
4,0
6,0
8,0
10,0
-2
distance to assumed grounding line [km]
0
2
4
6
8
10
distance to assumed grounding line [km]
Figure 7: In subfigure a) the amplitudes of the shelf ice oscillation due to ocean tides modelled from combined
GPS and gravimeter observations are shown and these amplitudes are zoomed around the grounding line’s vicinity
in (b). The phase lags of identical stations have got small differences dependent on the sensor type: GPS and
gravimeters (c) or tiltmeters (d). The measured (black line and dots) and modelled (grey line, no dots) local tilt
due to the ocean tides is an indicator of the regions of ice body deformation (e). The gravimetrical pre-factors
(f) are necessary to fit the tides of the rigid earth derived from the global model ETGTAB to the real gravimeter
observations (13).
7
7.5
7.5
(a)
2.5
amplitude [cm]
amplitude [cm]
5.0
(b)
2.0
0
-2.5
0.1
0.0
-0.1
-5.0
-2.0
-7.5
42
43
44
45
46
47 48 49 50
time [day of the year]
51
52
53
43
54
46
47
3.0
O1
M2
K1
O1
(c)
S2
2.0
1.0
0.01
0.02
0.03
0.04
0.05
frequency [mHz]
0.06
M2
K1
amplitude [cm]
amplitude [cm]
45
time [day of the year]
3.0
0.0
0.00
44
2.0
1.0
0.0
0.00
0.07
(d)
S2
0.01
0.02
0.03
0.04
0.05
frequency [mHz]
0.06
0.07
Figure 8: Vertical station motions and their spectra due to the tides: measured (black) and modelled (grey) tidal
motion at (a) station A002 (assumed grounding line) indicates a grounding of the ice shelf during the low tide.
The related spectra (c) proof the non-harmonic tides by showing side peaks. The tidal motion of the station A004
(b) can only be described by an upward folded tide model. The related spectra (d) show a frequency doubling that
indicates an elastic feedback motion at this station situated one kilometre behind the assumed grounding line.
ETGTAB (Timmen and Wenzel 1995). The function
E(t) represents this earth tide model in our equation
(eq. 13). To fit this model to the gravimetrical data, we
defined the pre-factor e. This factor changes locally,
what may be caused by ocean loading effects (fig. 7).
To get the parameter vector u we had to minimize the
residuals vt between the model Φ t and real data l(t) by
using the least mean squares.
2
ft (u) =
(Φt (u) − lt )) =
vt2 .
(15)
case of systematic errors. Due to these restrictions of the
Gauss-Newton methods, we decided to use the methods
of non-linear programming.
The simplest recursion that is used for non-linear programming is given by the gradient method which is
also known as method of steepest descent or as NewtonRaphson method (see appx.).
Starting with random elements of the parameter vector u0 , the minimization follows the steepest descent of
the solution function ∇f (u 0 ) with step length α and
the parameter vector is updated by
Most of the Gauss-Newton adjustment methods including the regression require good parameter approximations and huge matrixes. Furthermore, these methods
are based on the estimation function of the least mean
squares. Accordingly, it is impossible to change the estimation function to, e.g., the least median of squares
making it quite difficult to get robust solutions in the
uk+1 = uk − αk · ∇f (uk ) .
t
t
(16)
The determination of the step length is a particular problem. Most of the algorithms estimate the step length
quite conservatively, which slows down the convergence (fig. 5 left). In addition the Newton method uses
a local Hessian matrix H. This matrix contains the
8
derivatives of the second order. The inversion of this
matrix determines an optimal step size (see appx.) of
the iteration (fig. 5 right).
uk+1
amplitude and phase lag stability. This turned out as to
be quite difficult for the data around the grounding zone,
where the magnitudes of the different effects change
rather rapidly in space and time.
Although the tiltmeter signals cannot be used directly,
they were quite important for interpretation purposes.
For example at the station KM2 there are no typical
ocean tide signals. Consequently, we know from this
tiltmeter time series that we have to expect a nearly
undistorted solid earth tide signal on the neighbouring gravimeter. The amplitude and tilt information of
all stations clearly show the ice body deformation in
the observed area between the assumed grounding line
and the station Base camp (BC). The station BC situated about eight kilometres north of the grounding
line shows the highest ocean tide impact with a maximum amplitude of about 60 centimetres. This is 80%
of the prediction from the regional Weddell Sea Model
(Robertson et al. 1998). The amplitudes of the main
tides M2 , S2 , K1 and O1 obviously decrease towards
the assumed grounding line (fig. 7).
The periodicities in the time series of the stations above
and behind the assumed grounding line do not disappear. Consequently, the spectrum of the data at the station A002-GL0 still shows the typical ocean tide peaks.
Though, the side peaks indicate that the motion is not
harmonical any more (fig. 8c). The model of vertical
motion of station A002-GL0 (fig. 8a) does not fit during the low tide. This suggests that the ice body touches
the rocks of the sea floor. On the one hand the existence
of tide signals at station A002-GL0 implies that the
grounding line as it was determined from seismic shootings is placed more south and on the other hand the nonharmonic motions support once more Vaughan’s theory
that we have to assume a wider grounding zone.
The amplitudes do not reach zero even a kilometre behind the assumed grounding line (fig. 7). Rather, there
exists a frequency doubling in the spectra of the station
A004-KM1 (fig. 8d). The model of the vertical motion
of station A004-KM1 (fig. 8b) fits only under the condition that during the low tide the convex parts of the wave
is folded upwards. That means that the ice shelf lifts up
as well during the high tide as during the low tide and
touches down in between. This is a significant evidence
for an elastic feedback at this station (fig. 9a-c).
This elastic feedback is obvious in the phase plot
(fig. 7 d) as well. The phase lag of every periodicity changes significantly at the stations above and behind the formerly assumed grounding line, whereas this
change is dependent on the tide’s frequency. The diur-
−1
= uk − ∇2 f (uk )
· ∇f (uk )
= uk − H−1
k · ∇f (uk )
(17)
The use of the frequently inverted Hessian matrix
causes a lot of numerical problems, which can result in
the termination of the recursion before convergence is
reached. To avoid these numerous inversions, a group of
so-called Quasi-Newton methods use approximations
of the inverted Hessian matrix H̃−1
k (see appx.).
uk+1 = uk − α · H̃−1
k · ∇f (uk )
(18)
Because of the approximation of the Hessian matrix
the update of the parameter vector requests an optimized step length α. This 1-dimensional line search can
be solved by a separate optimization with e.g. an onedimensional steepest gradient recursion.
6 Results
It has once more to be emphasized that all the modelling
efforts that have been done in our study, were necessary to combine the data of different sensors installed
at different places during different observation intervals
(fig. 6). This incoherence of the sensor field is caused
by the logistical and environmental situation of the investigated area. Nevertheless, we need the information
of how this ice body interacts with the ocean tides at all
places at one time.
It is straight forward to model the GPS data first. They
represent directly the geometrical height variations due
to the ocean tides. On the other hand we have to be
careful with the gravimetrical data. The solid earth
tides reach maximum amplitudes of about 15 centimetres around the Ekstroemisen grounding zone. At the
stations KM1-A004 and GL0-A002 where both GPS
and gravimeter data were available the comparison between a GPS signal and the related gravimetrical signal showed that the GPS signal is much more reliable,
even if the gravimetric signal had been corrected. Accordingly, we modelled the ocean tides using the GPS
time series first. In the second step, we tried to explain
the gravimetrical signals by fitting all known effects to
the model.The other gravimetrical time series at KM2,
KM05 and BC were modelled with respect to regional
9
(a)
inland ice
ice shelf
water
KM1-A004
bedrock
GL0-A002
(b)
ice shelf
inland ice
water
bedrock
(c)
inland ice
water
ice shelf
bedrock
ary moves during spring tides more than one kilometre
landwards and back twice a day. The ice body at the
formerly assumed grounding line – represented by the
data at the site A002-GL0 – rests several hours on the
bedrock instead of reacting unbiased on the low ocean
tides.
Much more surprising seems to be the uplift of the
ice body during the low tides at site A004-KM1. This
site one kilometre behind the grounding line was assumed to be situated on grounded ice. Both the low and
the high tides induce a maximum uplift of one centimetre here. We interpret the uplift during a low tide as
an elastic feedback of the ice body. Two assumptions
of Holdsworth’s models cannot be verified for the Ekstroemisen: the ice body is not vertically fixed at the
grounding line and the grounding line is not horizontal
invariable with respect to the bedrock either.
Vaughan’s grounding line definition using a stationary bedrock-water boundary at the ice shelf bottom as
well has to be modified at first in the very case of the situation at Ekstroemisen. We assume outgoing from this
case that comparable results can be found for other regions with a shallow bedrock topography as well. Consequently, this example shows that the system boundary
between the inland ice and the ice shelf can often be instationary in space and time. The impact of this instationarity is significant and systematic. Accordingly, the
variability of the system boundary has to be taken into
account for the determination of the mass balance of an
ice shelf.
Acknowledgments
Figure 9: Schematic cross-sections through the ice shelf
at the grounding line during (a) the mean tidal level, (b)
the high tide and (c) the low tide as suggested by our
results.
Our reviewers have greatly contributed to the quality of
the manuscript. We are most grateful to our reviewer D.
Vaughan and the editor H. Schmeling who have given
us encouraging support.
nal waves cross over between station GL0-A002 and
KM05, while the semi-diurnals change over a longer
distance.
7 Conclusions
The Ekstroemisen grounding line is not a clear line that
strictly separates a grounded from a floating ice body.
Furthermore the presented results show that this bound10
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Appendix
a Taylor-polynomial of the second order (compare eq.
19).
Derivation of the method of the steepest descent
f (uk+1 ) =
The method of the steepest descent can be deduced from
the Taylor-polynomial (Bartsch 1999).
+
∞
1 ∂ i f (uk )
i
f (uk+1 ) =
(uk+1 − uk )
iu
i!
∂
k
i=0
To determine a steepest local descent, the first partial
derivative as a gradient is sufficient. Accordingly, only
the polynomial of the first order is necessary for the further analysis.
grad(f (uk )) = ∇f (uk ) =
∂f (uk )
∂uk
(20)
(21)
The stationary vector u opt marks a local optimum of
the solution function f (u opt ). Consequently, the polynomial has to be solved for u k+1 :
uk+1 = uk + ∆f (xk ) · ∇f (uk )
(22)
The difference of the values of the function ∆f (x k )
can be interpreted as the step length α. Correspondingly, the general recursion of the steepest descend
method can be expressed as follows (Rardin 1998).
uk+1 = uk ± αk ∇f (uk )
∂ 2 f (uk )
∆u2k+1
∂ 2 uk
(24)
∆uk+1 = uk+1 − uk
(19)
k = 1 . . . m.
∂f (uk )
(uk+1 − uk )
∂uk
∂f (uk )
∆uk+1
∂uk
with
with
f (uk+1 ) = f (uk ) +
f (uk ) +
(23)
The search of an optimal step length α requires a separate algorithm. Mostly, the step length is estimated quite
conservativly to avoid that too long steps miss the next
local optimum. Because of this conservative step length
estimation the convergence of this recursion needs a lot
of steps.
Derivation of the Newton method
To force a faster convergence of the algorithm it is
necessary to find an optimal step length instead of the
small linear steps α that are used for the steepest descend method. Consequently, it is possible to start with
At first the step length along the solution function has
to be defined 1 .
1
∆f uk = ∇f (uk ) ∆uk+1 + ∇2 f (uk ) ∆u2k+1 (25)
2
The derivation with respect to ∆u k allows determining
the impact of the step length of the parameter vector u k
on the solution function.
∂∆f (uk )
= ∇f (uk ) + ∇2 f (uk ) ∆uk+1
∂∆uk
(26)
We are looking for the stationary vector. Accordingly,
the derivative of the substitution function has to have
the value zero (Rardin 1998). A further progress along
the gradient would lead to an increasing distance to the
next local extremum.
∂∆f (uk ) !
=0
∂∆uk+1
(27)
The substitution function in eq. (24) can be recomposed. Remember that ∆u k in eq. (7) was defined as
the difference of the two places in the parameter space
k + 1 and k.
−∇f (uk ) = ∇2 f (uk ) (uk+1 − uk )
(28)
To yield the recursion of the Newton method with its
so-called Newton step the equation has to be solved for
uk+1 .
−1
uk+1 = uk ± ∇2 f (uk )
∇f (uk )
(29)
The comparison of this expression with the recursion of
the steepest descend method makes clear that instead of
an undefined step length α k (23) we can find the Newton step from the vector product of the gradient and the
inverse Hessian matrix H −1 (u), whereby this matrix is
defined as
H(uk ) = ∇2 f (uk ) =
∂ 2 f (uk )
.
∂uk,i ∂uk,j
(30)
1 The resulting substitution function is only an approximation with
all its problems (Mautz 2001).
12
Pseudo-inverse Hessian matrix approaches The BFGS approach after Rardin (1998) has got a different appearance:
for the Quasi-Newton method
The steadily repeated inversion of the Hessian matrix
forms a serious problem of singularity. The combination of numerical derivatives in an approximately linear
surrounding of the solution function, which one find far
away from local extrema, will cause matrix instabilities
and singularity. The algorithm will stop before having
converged. Therefore, this algorithm has to be modified. Typically, the recursion uses a pseudo-inverse approach of the Hessian matrix. The recursion starts because of missing information with an identity matrix
as the pseudo inverse of the state k = 0 (Grundmann
2002).
The DFP2 approach after Bazaraa and Shetty (1979)
for the pseudo-inverse Hessian matrix
H̃−1
k+1
can be expressed as
−1
H̃−1
k+1 = H̃k −
λk+1 ]T · H̃−1
[λ
k · λ k+1
=
H̃−1
k −
+
T
−1
H̃−1
k λ k+1 H̃k λ k+1
λ k+1 ]T H̃−1
[λ
k λ k+1
T
[∆uk+1 ] λ k+1
T
[∆uk+1 ] λ k+1 · h · hT
h=
T
[∆uk+1 ] λ k+1
−
H̃−1
k λ k+1
λk+1 ]T H̃−1
[λ
k λ k+1
with
3 Broyden,
Fletcher, Powell
Fletcher, Goldfarb, Shanno
∆uk+1λ k+1 H̃−1
k
.
∆uk+1λ k+1
ρk+1 =
∆uk+1 [∆uk+1 ]T
∆uk+1λ k+1
(33)
(34)
Recursion of the Quasi-Newton algorithm
Φ0 ← f (u0 )
∇f (u0 ) ←
(32)
∂f (u0 )
∂u0
H̃−1
0 ←I
2nd step: determination of the minimization direction
vector.
∆uk+1 ← H̃−1
k ∇f (uk )
3rd step: 1-dimensional minimization of the value of the
solution function along the minimization direction vector. This step is only required if the pseudo-inverse Hessian matrix H̃−1
k is used and can be solved as an own
steepest gradient minimization:
λ k+1 = λ k+1 = ∇f (uk+1 ) − ∇f (uk )
2 Davidson,
+
1st step: the decision has to be done whether the recursion reached a stationary point or not. If the convergence process falls below the self-determined threshold
, the recursion will stop.
< (recursion stops)
∇f (u0 )
≥ (recursion continues)
∆uk+1 [∆uk+1 ]
∆uk+1
H̃−1
k λ k+1 [∆uk+1 ]
∆uk+1λ k+1
with
. (31)
whereby
−
u0 ← rnd
T
+
=
T
The BFGS3 approach after Grundmann (2002) is a
polynomial extension of the DFP approach,
H̃−1
k+1
λk+1 H̃−1
k λk+1
1+
· ρ k+1
∆uk+1λ k+1
H̃−1
k
Pre-step: before the start of the recursion, the parameter vector can be chosen by random, the related output
of the solution function and its first derivatives can be
computed and finally, the pseudo-inverse of the Hessian
matrix can be defined as the positive identity matrix.
−1
˜ 2 f (uk+1 )
= ∇
T
−1
H̃−1
k · λ k+1 · H̃k · λ k+1
H̃−1
k+1
f (uk+1 ) ← minα (f (uk + α · ∆uk+1 ))
iff
13
−1
H̃−1
k = Hk .
(35)
4th step: update of the parameter vector with and optimized step length α and computation of the local gradient.
uk+1 ← uk + α · ∆uk+1
∇f (uk+1 ) ←
∂f (uk+1 )
∂uk+1
5th step: computation of the updated inverse Hessian
matrix following eq. (32) or (33).
−1
H̃
H̃−1
←
BFGS
k+1
k
6th step: the recursion can be restarted at the first step
after updating the epoch parameter.
k ←k+1
14