Introduction
The depth-dependent temperature in marine sediments is controlled by the
amount of heat exchange with the water above and the deeper regions of
Earth's mantle, as well as on the thermal properties of the sediment.
When the water temperature is time-independent and there are no heat sinks
and sources, a steady state is achieved where the vertical heat flow is
constant – at least in timescales of decades.
Periodically changing water temperatures are measurable to different depths,
depending on the amplitude, period and the sediment thermal properties.
A reliable forward model to describe the sediment
temperature in the steady state or with (periodically) changing water
temperatures exists see e.g., and will be
introduced in Sect. .
Measured and modeled subsurface temperatures are analyzed for different
purposes . In climate research borehole temperature data
are inverted for the ground surface history . Here, the
surface temperature is modeled as a Fourier series and often more than one
temperature–depth profile is used. The aim is to obtain mean temperatures
over time intervals of several years and hence the
mathematical models are linear but high-dimensional. The involved inversion
schemes are adapted to these models. A more simple model is introduced later,
where these inversion schemes are not strictly necessary.
Other studies focus on the background heat flow e.g.,. Under the circumstances where the surface temperature can be
regarded as constant and no heat sinks or sources are present, the measured
subsurface temperatures show a nearly linear increase with depth. Taking
changing thermal properties of the sediment into account, the geothermal
gradient can be determined directly from temperature measurements
. To decide from these data if the steady state
geothermal gradient is prevailing, the Bullard method is applied
. If the geothermal gradient is the purpose of the
investigation, a steady state is strictly required and any disturbance
originating from surface temperature variations is regarded as noise. In the
deep sea this is usually not an issue, as the water column already filters
the surface temperature variations and the bottom water temperature is
(almost) constant . In regions with shallower water or
onshore this is a rather big problem and is often only solvable by using
temperature measurements from deep boreholes.
In this work the bottom water temperature history is of interest, but on
smaller timescales. The aim is to estimate the bottom water temperature
variations over the last year based only on one single measured profile of
depth-dependent temperature and thermal diffusivity. A parameterized function
for the bottom water temperature is introduced, which results in a non-linear
but low-dimensional operator. As only one measured profile is required, this
could help to understand the (temperature) dynamics of water basins where
continuous temperature monitoring is difficult to realize (e.g., in the Arctic
Ocean).
Besides artificial data sets, results for two measured data sets from the German North Sea and the Davis
Strait are presented. The regions where these measurements were performed are quite
different and thus show the broad field of usability of the presented
method.
Forward model setup
For the theoretical framework, the sediment is considered as
a horizontally layered half space, where no temperature change happens
in each of the horizontal directions and thus the
one-dimensional heat equation can be applied:
ρCv∂tT(x,t)=∂x(λ∂xT(x,t))+∂x(vT(x,t))+H(x,t),
where x∈Ω⊂R≥0 corresponds to the depth
below the sea floor at x=0, t>0 is the time, v is the vertical
movement of material and H an inner source. The sediment density ρ,
specific heat capacity Cv and thermal conductivity λ are typically
depth-dependent. In the example areas, the volumetric heat capacity ρCv shows only very small changes with depth and will therefore be
considered as constant see also. With the relation
κ=λρCv for the thermal diffusivity the parameters can be reduced to
only one per layer.
The equation above states that the deviation of the temperature T(x,t)
with respect to time equals the amount of diffusion (first term on the right
hand side of the equation) plus the heat transported with the material by
v (second term) and the generated heat H(x,t). In the presented
settings, the advection term ∂x(vT(x,t)) is neglected. In
regions affected by hydrothermal convection this term can make a big
difference, so data sets where the fluid flow is rather low and thus
advection has no influence have been picked.
Earth's heat source is modeled as a steady state heat flow that contributes
to the model via the lower boundary and thus there is no source
term. With these reductions the heat equation can be simplified to the
model equation:
∂tT(x,t)-∂x(κ(x)∂xT(x,t))=0∀x>0,t>0.
Thermal properties and boundary conditions
The geothermal gradient is the first derivative of the steady state solution
of the model equation with a constant homogeneous boundary value at x=0. The
steady state heat flow will be denoted with q and thus ∂xTsteady(x,t)=qλ=cg holds for all x>0,t>0.
While the bottom water temperature is constant in the steady state,
in general it is time-dependent. This deviation of the bottom water
temperature will be denoted by
Twaterf:R≥0→R, where f
is a vector of parameters. The parameters in f are to be
reconstructed.
With initial and boundary conditions, the model
Eq. () becomes a solvable initial-boundary-value
problem. In geophysical models aiming for temperature fields in the
earth, it is quite common to model the region Ω with infinite
depth .This approach is also used here,
but in the numerical implementation the region will always have
a finite depth xE which is sufficiently large.
The boundary conditions need to be set up to satisfy the physical
conditions, which are the stimulation from a 1-year periodic
function of the bottom water temperature and a zero-flow condition at
the lower boundary. The geothermal gradient as the solution to the
static heat equation will be added after the modeling process. Thus,
a homogeneous Neumann condition can be set at the lower boundary in
the time-dependent part and the Robin boundary condition at the
sediment–water interface. The latter describes the fact that the heat
flows out of the sediment when the sediment is warmer than the water
above and into the sediment when it is cooler.
Knowing the thermal diffusivity κ(x) of the sediment and the
parameters f∈DT of a continuous function Twaterf(t) for the bottom water temperature, this yields a full
set of equations to determine the temperature in marine sediment at every
place and time: when u(x,t) solves
Eqs. ()–(), the sediment's temperature
is given by Ttotal(x,t)=u(x,t)+cgx,
∂tu(x,t)-∂x(κ(x)∂xu(x,t))=0∀x∈Ω,t>0,κ(x)∂xu(0,t)=hρcv⋅u(0,t)-Twaterf(t)∀t>0,∂xu(xE,t)=0∀t>0ifxE<∞,limx→∞∂xu(x,t)=0∀t>0ifxE=∞,u(x,0)=u0(x)∀x∈Ω.
Here, h(t) is the heat transfer coefficient and a measure on how well the
heat energy passes the sediment–water
interface. discuss the value of this
coefficient and propose to use an average value of h=150Wm-2K for the German North and Baltic seas.
They also showed that the influence of deviations of this parameter is negligible and
therefore this constant value will be used throughout the paper.
The volumetric heat capacity ρcv is also assumed to be known from measurements.
Bottom water temperature functions
The temperature at Earth's surface is an overlay of many
sinusoidal functions with different periods in terms of Fourier
series. As the deviation of the bottom water
temperature for 1 year is the aim of the reconstruction, the following simple model with only
a 1-year period ω=2π365 is used:
Twaterf(t)=A+Bcos(ωt+φ(d)),f=(A,B,d)T,
where A and B (in ∘C) denote the average
temperature and amplitude, respectively. The annual minimum takes place
on a day d>0, which leads to a phase delay φ(d)=ω(3652-d) of the cosine function, where the definition space of
f is restricted to DT=R≥0×R≥0×[0,365].
The example data set from the German North Sea shows influences of
smaller periods besides the annual deviation, as the average depth is
only 100m . Temperature and salinity of the North
Sea are mainly governed by a general cyclonic circulation, which
renews the water in the timescale of 1 year
. The freshwater input from rivers is comparably
small. The central part of the North
Sea becomes stratified due to heating in the summer but gets
vertically mixed during winter. At the western and southern coasts,
the vertical stratification is prevented by strong tidal currents
for detail on the oceanography of the German North Sea
see. Thus, in the North Sea the simple
model for the bottom water temperature is a sufficient approximation. However, the
temperature deviation will be slightly noisy due to the shallow depths.
Also, the parameters will change slightly in different
regions, relating to the tidal currents.
Baffin Bay and Davis Strait are characterized by the northwards flowing West
Greenland Current moving temperate saline Irminger Water from
the Atlantic Ocean in the top layers and cold low-salinity Polar
Water in the bottom layers . The data set from the Davis Strait is
obtained from 1300m depth, where the seasonal influence is mostly
damped by the water column and the cold Polar Water is dominant. Thus,
the parameters A and specifically B are expected to be near
0∘C .
Bottom water temperature (top panel) as input to the forward
model operator and the solution (bottom panel) at different days of
the year. A constant thermal diffusivity of κ=8×10-7m2s-1 was used
and the geothermal gradient (black in the lower panel) was set to cg=0.03Km-1. The bottom water temperature function is a cosine with
a mean value of 8.2∘C, an amplitude of ±5.9K and the minimum on the 62 day of the year. The
colors indicate the bottom water temperature and thus the day of the
year when the temperature–depth profile is plotted. The bright
orange temperature–depth profile is the solution on the 220th day of
the year, when the bottom water temperature is near the annual
maximum of 14.1∘C.
General behavior
The general behavior of the forward model is depicted in Fig. .
A parameter set typical for the German North Sea of f=(10.5,7,45)T is
used for the bottom water deviation. For the geothermal gradient a literature
value of cg=0.03Km-1 is used. The
image shows that the sediment experiences a large temperature range over 1
year in the upper meters. As the depth increases, the covered temperature
range gets smaller.
The attenuation of the amplitude with depth is clearly visible and so
is the delay of the temperature. It can be observed that the current
temperature–depth profile contains the bottom water temperatures from
the last 3–4 months. Following the orange line indicating
the day of the highest bottom water temperature, the sediment temperature shows
a decrease with depth. At a depth of
4m the lowest temperature, as evidence of the last winter, is
reached and with further increasing depth the temperatures also
increase again. The blue line indicating the day of the coldest bottom
water temperature shows a mirror-inverted curve.
Data
The temperature reconstructing method (to be described in
Sect. ) was tested for artificial and measured data
sets containing about 20 data points in a depth of up to
4 m. The artificial data sets were generated using the forward
model and adding some white noise, while the example data sets from
the German North Sea and the Davis Strait were measured using the
FIELAX VibroHeat and HeatFlow measuring devices, respectively. The
locations of the example data sets are shown in Fig. .
Locations of the two example data sets (red). In the upper
panel, the data's location near the island of Borkum in the German
North Sea is depicted. Additionally the observation station
FINO1 is marked (green). In the lower panel, the data's
location west of the coast of Greenland, near Nuuk, is shown.
Results of a VibroHeat survey in the North Sea north of
Borkum in 2011 showing in situ temperature, thermal conductivities,
thermal diffusivity, and volumetric heat capacity as a function of
depth.
The principle for the measurements of depth-dependent thermal
parameters originates from the classical method of determining
steady state heat flow values for the oceanic crust from deep sea
sediments. Heat flow values are determined based on Fourier's law of
thermal conduction from the steady state (undisturbed) temperature
gradient and thermal conductivities. The design of deep sea
Lister-type heat flow probes follows the concept by
where a thermistor string parallel to a massive
strength member penetrates into the sediment by its own weight. A total of 22
thermistors record the sediment temperature during the whole
process. The in situ temperature is determined from the decay of the
frictional heat accompanying the penetration, while the thermal
conductivity and diffusivity values are calculated from the
temperature decay of an artificial, exactly defined, calibrated heat
pulse which heats up the sediment .
This method is normally used in deep sea soft sediments, where the
heat flow probe penetrates due to its own weight. Shallow water
sediments in the North and Baltic seas are characterized by more shear
resistant sediments such as sands, tills, and clays, where this classical
method of penetration by gravity alone is not applicable. For this
reason, a thermistor string has been combined with a standard VKG
vibrocorer . The measuring procedure follows the
classical way: the system is lowered towards the sediment, penetrates
the sediment by vibrocoring and rests in the sediment for in situ
temperature measurement and heat pulse decay recording. With this
system, a penetration depth of up to 6m is possible. The
accuracy of the thermistor strings is 2mK and the
resolution is 1mK for both systems.
The processing of the raw data from both measuring devices is handled
with the same software tool. This processing algorithm allows determining
in situ temperatures and thermal material properties with an inversion algorithm following
. Based on an assumption of thermal decay
around a cylindrical symmetric infinite line source, steady state
in situ temperatures, thermal conductivity and thermal diffusivity are
determined in an iterative inversion procedure. From thermal
conductivity and diffusivity also the volumetric thermal capacity can
be determined by ρcv=λ/κ. All thermal properties can be
obtained with less than 0.5% error.
The in situ geothermal gradient can also be determined directly
from these measurements for details see.
When the measurements are not deep enough, the determination of the in situ
geothermal gradient is not possible. The global heat flow database collects
data sets and can thus be used to find an appropriate regional approximation
. The proposed inversion method is based on routine thermal
measurements aiming for the geothermal heat flow. In the processing of these
measurements with the algorithm from , the in situ thermal
properties and the geothermal heat flow are determined. Any sedimentation
effects are dealt with in this algorithm and can thus be omitted in the
inversion schemes. Therefore, the geothermal heat flow is a known input to
the model, and approximated values are used where the in situ determination
was not possible. A simultaneous inversion of the bottom water function
parameters together with the heat flow could decrease the influence of
measurement errors; however, this will not be handled in this paper.
Figure shows depth-dependent properties from
a thermal measurement in the North Sea offshore from Borkum in June 2011. Contrary
to the general positive temperature gradient in Earth's crust,
a temperature decrease with increasing depth is observed, i.e., a negative
temperature gradient. This is due to the seasonal variation of bottom
water temperature in the North Sea. As the measurement was performed in June,
the influence of the warm summer temperatures is seen in the upper
thermistors, while the decrease towards the lower thermistors is a relict of low
winter temperatures.
The inversion scheme was performed for various data sets from the German North and
Baltic seas being measured with the VibroHeat device and data sets from
the Davis Strait and the Baffin Bay, which were measured using the
classic HeatFlow probe. Excluding some measurements from areas within
the Baltic Sea where the bottom water temperature deviation differs
too much from the simple model Eq. (), the obtained results
were all within the same range of quality and accuracy. The two data sets
presented here are chosen for demonstration of the method.
Inverse problem setup
Discretization of the forward model operator
The inverse problem is to determine the bottom water parameters f from (a
priori known) values for given geothermal gradient cg, heat
transfer coefficient h for the Robin boundary condition, and measurements
of the thermal diffusivity κ(x̃) and the temperature in the
sediment gε(x̃,t∗). Here, x̃∈Rk is the depth vector according to the k sensors of the
measuring device and t*>0 a fixed time.
Before introducing the solution method to solve this inverse problem,
the forward model needs to be briefly formalized. It can be shown that
the initial-boundary problem,
Eqs. ()–(), has a unique
solution in the weak sense see e.g.,. Thus,
the forward operator F:f↦T(x,t) mapping the
parameters of the bottom water temperature to the solution of the
initial-boundary problem is well-defined. For this operator, differentiability with
respect to f can be shown if the function for the bottom water
temperature deviation Twaterf is continuously
differentiable with respect to the parameters f. The continuous
differentiability of the cosine function in Eq. () with
respect to the three parameters A, B and d is obvious.
When interested in greater timescales it is common to model Earth's
crust as a homogeneous half space, i.e., Ω has infinite depth
and the thermal diffusivity κ is constant over the region. In
this case, the solution to the initial-boundary problem is
analytical. Thus, in the numerical calculation of the temperature in
the sediment the only occurring errors are rounding errors on the
scale of the machine accuracy and no discretization errors. However,
as the sediment is modeled as a layered region with finite depth
there are two more sources of error: the discretization error and an
error resulting from the finite depth.
The discretization is realized using the method of lines. This method
and its convergence properties are broadly analyzed by
. The mesh size and time steps of the discretization
as well as the depth of the lower boundary need to be determined such
that the numerical solution is not too far away from the true
solution. In other words, the scheme determines a parameter setting such that
the relative error between the abovementioned analytical solution and
the numerical solution does not exceed the limit of 10-3 K. This limit
was chosen in reference to the accuracy of the data obtained
with the FIELAX VibroHeat device.
Having access to a numerical and nonlinear forward model
F:R3→Rk mapping the three model parameters
A, B and d to the numerically approximated temperature for time
t∗>0 at points of the depth vector x̃∈Rk,
the inverse method can be discussed based
on the observation that F possesses a gradient ∇F∈Rk×3. This matrix
had maximal rank (3) in all performed numerical examples.
Since the measured sediment temperature (and also the measured thermal
diffusivity) may suffer from measurement errors, the
temperature data are assumed to be a vector gε such that ‖g-gε‖≤ε, where g is the undisturbed
data. Furthermore, an exact parameter vector f+ is assumed to exist
such that F(f+)=g (i.e., the model is valid and represents
reality). The aim of the inversion scheme is to reconstruct f+.
The study started with a non-linear iterative Newton algorithm
as a first simple approach, which already provided good
results. Therefore, this approach is discussed first. For comparison
the iterative REGINN REGularization by INexact Newton methods; method has been adapted to the model.
The Newton algorithm
Sticking to the notation of , the ongoing iteration for the
solution of the non-linear equation F(f)=gε with
disturbed data gε is considered:
fn+1ε=fnε+snε,n∈N.
The iteration step snε is to be determined, so that the exact
solution fn+1ε=f+ is obtained.
Obviously, sn+=f+-fnε solves this
equation. The approach of the Newton algorithm is to determine a good
approximation to sn+.
As F is differentiable with derivative ∇F, the following
equation holds:
∇F(fnε)sn+=g-Ffnε-Ef+,fnε=bn.
Here, E(v,w) denotes the linearization error. This linearization
error and therefore the right side bn is not known but only
a disturbed version bnδ is. So sn+ is approximated by
solving
∇Ffnεs=bnδ.
For ill-posed problems, solving this linear equation can be quite
problematic; however, as the derivative has full rank, Gaussian elimination can be used to determine snε.
Applied to the simple model, the iteration converges to a solution of
F(f)=gε. However, as the method is not minimizing for the
exact right side bn=g-F(f+) but for a disturbed version, the
best result may not be the minimal result. Therefore, the iteration is stopped
whenever the reconstructed data are about as near to the
noisy data gε as the noisy data are away from the exact
data. Details on this discrepancy principle can be found, e.g., in
. As the value of ε is not known for
real data sets, an approximation of ε=0.005 is used.
A regularized inexact Gauss–Newton inverse method
For the inversion of the simple model, Eq. (), for the bottom water
temperature deviation, the Newton algorithm yielded stable results (as will be shown in Sect. ), such that stabilizing the inversion by using a
so-called regularization scheme was not necessary. Such schemes can
balance stability and accuracy of a solution to an inverse problem
see
and could be of importance if a more complex input model for the
bottom water temperature is used. This will not be covered in this work, but
will be discussed as a suggestion for further research in the last
section. One advantage of regularization schemes in general and of the
REGINN algorithm published by is their
proven convergence properties for noisy data or ill-conditioned
non-linear equations.
The REGINN algorithm is introduced and analyzed by and
. The algorithm is an inexact Newton method,
i.e., it consists of an outer Newton iteration and an inner regularization
iteration which determines the Newton iteration step. Thus, the outer
iteration follows the same idea as the abovementioned simple Newton method,
Eq. (), where the iteration step snε
solves Eq. (), and which is stopped with the discrepancy
principle. The determination of the iteration steps can be implemented with
any regularization scheme and the analysis is done for a general formulation.
Statistics for the inversion of artificial data. The uppermost part
contains the inversion results for artificial data with 0.1% white
noise added, the middle part artificial data with 1.0% noise. The
lowermost part shows inversion results to artificial data with 0.1%
white noise added after the seasonal forcing already contained noise of about
8%.
Data
Artificial with 0.1% noise
exact parameters
A
B
d
8.2
5.9
62
Newton results
A
B
d
mean error
max error
mean
8.20
5.89
61.97
0.31 %
0.65 %
variances
0.00
0.00
0.05
REGINN result
A
B
d
mean error
max error
mean
8.20
5.90
61.96
0.21 %
0.59 %
variances
0.00
0.00
0.03
Data
Artificial with 1.0% noise
exact parameters
A
B
d
8.2
5.9
62
Newton results
A
B
d
mean error
max error
mean
8.22
5.96
61.84
2.88 %
7.29 %
variances
0.05
0.25
2.90
REGINN result
A
B
d
mean error
max error
mean
8.36
6.26
63.45
2.63 %
5.68 %
variances
0.02
0.13
1.64
Data
Artificial with 1.0% noise and 8.0% water noise
exact parameters
A
B
d
8.2
5.9
62
Newton results
A
B
d
mean error
max error
mean
8.25
5.90
62.88
6.86 %
17.54 %
variances
0.29
2.22
22.72
REGINN result
A
B
d
mean error
max error
mean
8.29
5.89
62.27
8.69 %
14.92 %
variances
0.99
10.07
29.51
For this work, the algorithm published by , where the inner
iteration is a conjugate gradient (CG) method, was adapted. As the algorithm
has two nested iterations, the outer (Newton) iteration will be referred to
with superscript n and the inner (CG) iteration with superscript m.
The idea of the algorithm starts again with Eq. (),
∇Ffnεs=bnδ,
for the Newton iteration step snε and tackles this
linear system using the CG method.
The CG method is designed to minimize the residuum ‖bnδ-∇F(fnε)sm‖ in every iteration step m=1,2,3,… in enlarging subspaces. As the residuum at the end of
every iteration step is the minimum in the corresponding subspace, the
method is the most efficient method possible. For more information on
the CG method see e.g., and . The inner
iteration is stopped when the linearized residuum fulfills a certain
accuracy estimation for details on the choice of this
estimation see. The outer iteration is again stopped
with the discrepancy principle. Thus, the
reconstructions from the two algorithms can be directly compared.
Results
Artificial data for measured thermal diffusivity
In this section, the sensitivity of the algorithms in a layered half-space
setting is analyzed. Therefore, the measured thermal diffusivity and the
depth-vector of the sensors are used together with a random day of the year
t∗ and a random vector of the seasonal forcing parameters to produce
artificial data. With added zero-mean Gaussian noise, disturbed data sets
gε were obtained. For comparability of the results, the same
water parameters fExact=(8.2,5.9,62)T and literature values for
cg and h are used for all the experiments presented here.
As the derivative is not a square matrix, any theoretical analysis of
the convergence behavior of the Newton or REGINN algorithm
does not apply. To cope with this lack of theoretical information on
the general behavior of the introduced inverse problem, the algorithms
were executed repeatedly for randomly disturbed artificial data sets, produced from
the same parameters, and the variances were calculated. Thus, a bound for the emerging reconstruction
error depending on the noise level in the data can be given. The results are
shown in Table .
For data with a noise level of 0.1 % (see Table ) the
reconstruction error has the same magnitude for both algorithms. As the
initial guess is chosen randomly from the definition space and the
variances are very small, it can be stated that both algorithms work
stably and that the initial guess has no influence.
For a noise level of 1.0 % in the data, a maximum reconstruction error
of about 8 % is obtained, while the mean value of all reconstructions is
still close to the exact parameter values with a mean error of only 3 %
(see Table ) with both algorithms. This suggests that the overall
result of the inversion can be improved by executing the algorithm several
times for different (randomly chosen) intial guess every time and using the
resulting mean value.
The overall mean parameter values foverall=(8.29,6.11,62.6)T yield a relative reconstruction error of
≈1.1 %.
As the penetration depth of the first sensor is not exactly known, the
algorithm was also tested for vertically shifted data. Here, the distance
between two thermistors, e.g., 13.5cm for the Borkum data, was
added to the depth values. With, again, 20 repetitions of each algorithm, the
mean values and the variances only differed from the non-shifted data in the
third decimal place.
Artificial data with noisy bottom water temperature
In Sect. 5.1, artificial data for an undisturbed bottom water temperature
function was produced. However, from the water data available via MARNET
(2014) for the German North Sea it is known that this undisturbed function is
very unlikely to be accurate. In this section, the influence of noise in the
bottom water temperature on the accuracy of the reconstruction shall be
investigated.
Different from the white noise added to the data itself, here shorter
periods of the cosine series are used to approximate the occurring
errors as accurately as possible. Thus, this section will give an insight
into the sensitivity of the model with respect to the three main
parameters, even if the original forcing was more complicated and
the measured data contains errors.
For generating artificial data in this section, the seasonal
forcing was expanded to the first 52 summands of the Fourier series for
1-year periodic functions:
Twaterf(t):=A+∑i=152Bicos(iωt+φ(di)),f=(A,B1,…,B52,d1,…,d52)T.
This cutoff Fourier series approximates periods from 1 year to 1
week.
The results of this experiment are presented in the last section of
Table . A distinct increase in the reconstruction error can be
observed for a noise level in the data of 1 % from a mean error of
3 % for an undisturbed bottom water temperature function to a mean error
of 7 % here. Although the variances increase, the mean reconstruction
result is still a very good approximation to the exact water parameters. The
reconstruction error of a single reconstruction may be greater than 15 %,
while already 20 repeated executions can decrease the reconstruction error
notably. For both algorithms, the differences between the average results and
the exact parameter values are smaller than 0.1K in the average
temperature and the amplitude and smaller than 1 day in the day of the annual
minimum. The main uncertainty occurs from the determination of the day of the
annual minimum.
From the inversions done here, it can be concluded that the three
main parameters get reconstructed quite well even if the real
bottom water temperature function is more complicated than the simple
model in Eq. ().
Inversion of the data set near Borkum. In the upper panel,
the recorded bottom water temperatures at the BSH station FINO1 are
depicted for the years 2010 (green) and 2011 (blue). Additionally,
the cosine functions as results of the inversion schemes are
plotted: the mean result of the Newton algorithm, the REGINN result,
and the overall mean. In the lower panels, the measured temperature
(left) and thermal diffusivity (right) are depicted. The resulting
temperature–depth profiles from the modeling with the inversion
results are plotted together with the measured temperatures in the
left panel.
Example: Borkum
Data sets from three different locations in the German North and Baltic seas
were studied, but only one example will be presented in detail. The location
of the thermal measurement and the nearest MARNET station
see are depicted in Fig. . Along with two
other examples, this data set is broadly discussed by .
The quality of the reconstruction results in terms of variances was the same
in all studied examples. The data set north of the island of Borkum was
chosen for the following reasons: the distance between the data set location
and the nearest MARNET station FINO1 is smaller than with the other
examples and the recorded water temperatures showed the smallest differences
from the chosen bottom water temperature function Eq. ().
Additionally, the time series of the water temperatures was available and
allowed analyzing the applicability of this model equation. As the
measurement is only 3m deep, the in situ geothermal gradient could
not be obtained from the measurements. Thus, a typical value of cg=0.03Km-1 for this area was used.
The results of the inversion are listed in Table . In
comparison to the parameter vector used in , f̃Borkum=(10.4,6.9,41)T, the overall mean
parameters fit quite well. The Newton algorithm
provides smaller values than the REGINN algorithm but also
with smaller variances in the average temperature and
amplitude. The day of the annual minimum resulting from the Newton
algorithm seems very unlikely and also has a larger variance.
Averaging all reconstructions with both algorithms,
the values fit to the educated guess. The variance on the day of the
annual minimum is here quite large, because it was
reconstructed differently by the two algorithms. The value corresponds
to a SD (standard deviation) of ≈15 days.
Results of the inversion of the data set from Borkum. The Newton
algorithm gives an unlikely estimate for the day of the annual minimum but
the overall mean parameters fit the guess from within the
desired bounds.
Data
Borkum
Newton results
A
B
d
mean
9.79
5.66
26.62
variances
0.00
0.02
2.63
REGINN result
A
B
d
mean
11.95
9.15
55.53
variances
0.03
0.09
1.71
overall results
A
B
d
mean
10.87
7.40
41.07
variances
1.20
3.14
213.78
Bottom water temperature data in this area were available from the
FINO1 station. Inverting the water data with
the same algorithm, the parameter vectors f2010=(10.0,7.9,56)T for the data from 2010 and f2011=(10.4,7.1,55)T for 2011, respectively, were obtained. In this particular case,
the average temperature and the amplitude differ less than
1K between 2 years and also the day of the annual minimum
remains nearly the same. This leads to the assumption that the simple
model for the bottom water temperature fits the natural conditions in
this area quite well.
Comparing these parameter vectors to the ones obtained from the
inversion (Table ), it is obvious that the Newton algorithm
provided too small values, while the REGINN algorithm
yielded too large values. The overall mean fits best, but the day of
the annual minimum was reconstructed better with the REGINN
algorithm alone.
In the upper panel of Fig. , the FINO1 data from 2010 and
2011 and, additionally, the cosine functions of the mean
inversion results are shown. In the lower panel the measured thermal
diffusivity (right) and the measured and modeled
temperature–depth profiles are depicted. The
temperature interval resulting from the Newton results is too small,
while the REGINN result has too high temperatures in the
second half of the year. From the three cosine functions, the one
resulting from the overall mean fits the recorded temperatures best.
For the temperature–depth profile on the day of the measurement this
does not hold. The model with the overall mean result has too high
temperatures. The REGINN result fits the measured temperatures better
but only to a depth of 2 m, while the Newton results fit better below 2 m depth.
The not-so-optimal fit of the overall mean results can be
due to the uncertainty of the reconstruction of the day of the annual
minimum. By shifting the cosine of the overall mean results about 1
week (such that d≈48), the cosine fits the recorded
temperatures at FINO1 better and the temperature–depth profile
fits the measurements better as well.
Results of the inversion of the data set west of Greenland. Both
algorithms give similar reconstruction values with small variances.
Data
Greenland
Newton results
A
B
d
mean
3.18
0.16
71.51
variances
0.00
0.00
2.15
REGINN result
A
B
d
mean
3.18
0.16
74.25
variances
0.00
0.00
0.93
overall results
A
B
d
mean
3.18
0.16
72.88
variances
0.00
0.00
3.43
Inversion of the data set west of Nuuk. In the upper panel,
the cosine functions as results of the inversion schemes are
plotted: the mean result of the Newton algorithm in dashed line, the
REGINN result in dashed line with dots and the overall mean in
a straight line. In the lower panels the measured temperature (left)
and thermal diffusivity (right) are depicted. The resulting
temperature–depth profiles from the modeling with the inversion
results (the line styles are the same as above) are plotted together
with the measured temperatures in the bottom left panel.
Example: Greenland
The second example data set was measured on a cruise in 2006 in the waters of
the Davis Strait and Baffin Bay, west of Greenland, the location is shown in
the lower panel of Fig. . The in situ geothermal heat flow of
cg=0.0303Km-1 was determined from these
measurements.
The measurement is located at the southern ridge of the Davis Strait,
at the passage to the Labrador Sea. The water depth is about
1300m and thus the bottom water temperature deviation is expected
to be rather small.
The reconstruction results are shown in Table . Both
algorithms reconstructed similar values. As continuous
measurements of the bottom water temperature deviation are not easy in
these parts of the Arctic Ocean, there are no measurements to compare
these values to. However, the obtained temperature interval seems
plausible.
In Fig. , the cosine functions with the reconstructed
bottom water function parameters are depicted (upper panel). In the
lower panels the measured thermal diffusivity (right) and sediment
temperature (left) as well as the modeled temperature are shown. The sediment shows a wide range
of thermal diffusivity values. The
reconstructed bottom water temperature deviations do not differ much,
nor do the modeled sediment temperatures. They all fit the measured
temperatures quite well. Looking at the low variances
(Table ), this indicates a stable method and high
applicability.
Discussion
The aim of this work was to provide a method that obtains the parameters of
a function modeling the annual bottom water temperature variation from one
instantaneous measured profile of depth-dependent sediment temperature and
thermal diffusivity. Before the obtained reconstruction results are
discussed, the desired accuracy in geophysical usage needs to be determined.
Desired accuracy of the reconstructed parameters
In comparison to the measured water temperatures
see it is obvious that the mathematical model,
Eq. (), neglects all periods smaller than 1 year.
The average temperature A varied about 0.5K between
the years 2010 and 2011 and the amplitude about
0.8K at the station FINO1 in the German North Sea.
Similar results can be obtained for other
stations in this region. Thus, an accuracy better than 1K for the
parameters A and B in the reconstruction is sufficient for the German North Sea.
However, for the usage of the presented method in other areas (like the Arctic Ocean)
the accuracy level needs to be based on
the relative error – at least for the two temperature-related
parameters: reconstruction of a parameter of the order of 0.1K
with an accuracy of ±1K is not useful. In the
experiments with artificial data sets, a relative error of magnitude
around or slightly less than the relative data error was achieved. The
above stated differences in the recorded bottom water temperature at
the station FINO1 yield a relative change of about 5%
in the average temperature and 10% in the amplitude.
The day of the annual minimum only changed about 1 day at
FINO1, but was most difficult to reconstruct in all the
experiments seen in Sect. . This is clearly due to the
fact that the cosine function has a small first derivative around the
extrema. Thus, the function Eq. () does not change a lot in
the weeks around the annual minimum, e.g., it remains over 3 weeks
in the lowest 1 % of the covered temperature interval.
Considering all this, the reconstruction of the
parameters should be better than A±5%, B±10% and d±10 days.
Achieved accuracy of the reconstructed parameters
For the real data sets, a noise level of 1 % was assumed. While different
noise levels were considered in the experiments with artificial data, only
the obtained information on data with 1 % noise is relevant for the
applicability on real data. As seen in Table , SDs of A±0.2K, B±0.5K and d±2.5 were obtained for both
algorithms. In relative errors this equals A±2 and B±8%.
Thus, the desired accuracy was achieved with only 20 repeated inversions. By
taking the overall mean of all reconstructions from both algorithms, the
reconstruction error could be further decreased. Here, the variances and SDs
increased, but the obtained results were closer to the exact parameter
values.
As the function of the bottom water temperature was expanded to a cutoff
Fourier series, the variances increased to A±0.5K, B±1.5K and d±4.8 days (see Table ) or, in relative
errors, A±6 and B±24%. Here, the amplitude was less accurate
than desired. Still, the mean of the reconstructed values was accurate within
the desired interval.
As the variances were smaller for less noisy data in both experiments, it can be
concluded that both algorithms yield stable results for data
with a noise level of ≤1 % and for a bottom water temperature
function which varies from a plain cosine by up to
8 % (see Table ). For higher noise levels in the data or
the forcing function, the methods still converged but were not
accurate enough.
Applicability to real data
The experiments with artificial data suggested a stable method whose
accuracy could be increased by executing the algorithms repeatedly and using
a mean value of results from both algorithms. The reconstruction error did
not increase too much, when the function for the bottom water temperature was
changed to a cutoff Fourier series; the main parameters were still reconstructed sufficiently accurately, i.e., the relative errors
were smaller than 5 % for the mean temperature and 10 % for the amplitude and the
day of the annual minimum was reconstructed within 10 days.
Using real data sets, the general
form of the bottom water temperature deviation in the area of interest
needs to be studied carefully. As mentioned above some areas in the Baltic Sea cannot be
modeled with our simple model. The data sets from the North Sea, as
the one introduced in Sect. yielded
stable results, but with rather large variances. However, the results
match with the recorded water temperatures. The differences of the
results from the two inversion algorithms to the graphically obtained
parameter values used from are within the
desired error bounds. Here, the greatest differences occurred in the day
of the annual minimum but, as already discussed in Sect. , it is not possible
to determine it more precisely while A and B are only accurate to
1K.
Lastly, the results from the inversion of the data set west of Greenland have
the smallest variances, proposing reliable values. Other surveys on the
temperature (and salinity) of Baffin Bay and Davis Strait gave similar
temperature values see e.g.,. As there are currently
no long-term measurements, the results of this method are of scientific value,
if one trusts them. It is important to note that matching
the recorded temperatures is not the same as being exact:
the recorded temperatures are also measurements with errors and they
were not recorded at exactly the same locations as where the data sets were
taken.
Before reconstructing the bottom water temperatures from real data sets, one
should carefully consider if the simple model, Eq. (), is applicable
for said area. If it is already known that the sea is layered and not well
mixed or if there are strong currents along the sea floor, this method will
not give reasonable results. If one finds unreasonable results, this should
be interpreted as a strong hint that the simple model and hence the method
are not applicable. The algorithms differ in their approach to solve the
linearized system and therefore obtain different results if the data are not
exact or the function for the bottom water temperature is not of the simple
form as Eq. (). These differences in the results of the algorithms are
useful to gain an insight into the uncertainty of the results that is linked
to measurement noise and modeling errors. Thus, the method uses the overall
mean values from both algorithms and the variances can be used as an
indicator for the applicability of the simple model for the bottom water
temperature function.
Future work
A major point with the reconstruction of real data sets was that
the simple model for the bottom water temperature does not fit to all areas. Hence,
a main topic in further research will be the
generalization of the bottom water temperature model. The
implementation of the inversion algorithms can be easily adapted to
reconstruct the parameter vector of other periodic functions. The
Fourier series, introduced in Sect. , was
a reasonable start. The more coefficients of the Fourier series to be
reconstructed, the more sensors are needed to get a full-rank
derivative. For regions with more noise in the bottom water
temperature deviation, such as the Baltic Sea, the smaller periods
could possibly generate more realistic data and thus improve the
reconstruction results for such data sets.
A piece-wise constant function as in the large-scale climate history
reconstruction may also be used. Such a model is then possibly
capable of reconstructing aperiodic events in the most recent water
temperature changes. This will be of great interest for the Baltic
Sea (e.g., to identify inflows from the North Sea over the Danish
Straits) or the Arctic Sea (e.g., to indicate cold water discharge due
to iceberg calving events). Simultaneous inversion of the background
heat flow and the bottom water
temperature should also be considered. The iterative Newton algorithm is
possibly not suitable for these higher dimensional problems, but other algorithms and
approaches from climate history reconstruction can be built upon .