Continuous monitoring of oceanic bottom water temperatures is a complicated task, even in relatively easy-to-access basins like the North or Baltic seas. Here, a method to determine annual bottom water temperature variations from inverse modeling of instantaneous measurements of temperatures and sediment thermal properties is presented. This concept is similar to climate reconstructions over several thousand years from deep borehole data. However, in contrast, the presented method aims at reconstructing the recent temperature history of the last year from sediment thermal properties and temperatures from only a few meters depth. For solving the heat equation, a commonly used forward model is introduced and analyzed: knowing the bottom water temperature variations for the preceding years and the thermal properties of the sediments, the forward model determines the sediment temperature field. The bottom water temperature variation is modeled as an annual cosine defined by the mean temperature, the amplitude and a phase shift. As the forward model operator is non-linear but low-dimensional, common inversion schemes such as the Newton algorithm can be utilized. The algorithms are tested for artificial data with different noise levels and for two measured data sets: from the North Sea and from the Davis Strait. Both algorithms used show stable and satisfying results with reconstruction errors in the same magnitude as the initial data error. In particular, the artificial data sets are reproduced with accuracy within the bounds of the artificial noise level. Furthermore, the results for the measured North Sea data show small variances and resemble the bottom water temperature variations recorded from a nearby monitoring site with relative errors smaller than 1 % in all parameters.

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

Measured and modeled subsurface temperatures are analyzed for different
purposes

Other studies focus on the background heat flow

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.

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:

The equation above states that the deviation of the temperature

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:

The geothermal gradient is the first derivative of the steady state solution
of the model equation with a constant homogeneous boundary value at

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

With initial and boundary conditions, the model
Eq. (

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

Here,

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

The example data set from the German North Sea shows influences of
smaller periods besides the annual deviation, as the average depth is
only

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

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

The general behavior of the forward model is depicted in Fig.

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

The temperature reconstructing method (to be described in
Sect.

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

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 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

The in situ geothermal gradient can also be determined directly
from these measurements

Figure

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 inverse problem is to determine the bottom water parameters

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. (

When interested in greater timescales it is common to model Earth's
crust as a homogeneous half space, i.e.,

The discretization is realized using the method of lines. This method
and its convergence properties are broadly analyzed by

Having access to a numerical and nonlinear forward model

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

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

Sticking to the notation of

The iteration step

As

Here,

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

Applied to the simple model, the iteration converges to a solution of

For the inversion of the simple model, Eq. (

The REGINN algorithm is introduced and analyzed by

Statistics for the inversion of artificial data. The uppermost part
contains the inversion results for artificial data with

For this work, the algorithm published by

The idea of the algorithm starts again with Eq. (

The CG method is designed to minimize the residuum

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

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

For a noise level of

The overall mean parameter values

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.,

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:

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

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.

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

The results of the inversion are listed in Table

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

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

Bottom water temperature data in this area were available from the
FINO1 station. Inverting the water data with
the same algorithm, the parameter vectors

Comparing these parameter vectors to the ones obtained from the
inversion (Table

In the upper panel of Fig.

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

Results of the inversion of the data set west of Greenland. Both algorithms give similar reconstruction values with small variances.

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.

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 measurement is located at the southern ridge of the Davis Strait,
at the passage to the Labrador Sea. The water depth is about

The reconstruction results are shown in Table

In Fig.

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.

In comparison to the measured water temperatures

The average temperature

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

The day of the annual minimum only changed about

Considering all this, the reconstruction of the
parameters should be better than

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

As the function of the bottom water temperature was expanded to a cutoff
Fourier series, the variances increased to

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

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

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.

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

Before reconstructing the bottom water temperatures from real data sets, one
should carefully consider if the simple model, Eq. (

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.

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

The presented method yields stable results for artificial data with an accuracy within the bounds of the artificial noise level. The experiments with real data sets are very promising. The SDs are small for all data sets and the results matched the measured bottom water temperature variations from the monitoring stations (where available).

These results may be of major interest for oceanographers because they can provide oceanographic information for regions where long-term monitoring is not possible or too expensive. However, before applying the method to other regions, the validity of the simple model for the bottom water temperature needs to be carefully discussed.

This work would not have been possible without the kind permission of NSW (Norddeutsche Seekabelwerke GmbH) and GEUS (De Nationale Geologiske Undersøgelser for Danmark og Grønland) to use the data sets measured on cruises with partnership of FIELAX.

The authors thank M. A. Morales Maqueda and an anonymous reviewer for their very helpful comments. The article processing charges for this open-access publication were covered by the University of Bremen. Edited by: J. M. Huthnance