Estimating representative surface fluxes using eddy covariance leads
invariably to questions concerning inclusion or exclusion of low-frequency
flux contributions. For studies where fluxes are linked to local physical
parameters and up-scaled through numerical modelling efforts, low-frequency
contributions interfere with our ability to isolate local biogeochemical
processes of interest, as represented by turbulent fluxes. No method
currently exists to disentangle low-frequency contributions on flux
estimates. Here, we present a novel comprehensive numerical scheme to
identify and separate out low-frequency contributions to vertical turbulent
surface fluxes. For high flux rates (

The eddy covariance (EC) technique allows for direct, continuous and non-invasive tower-based ecosystem-scale estimation of surface–atmosphere scalar fluxes by simultaneous sampling of atmospheric fluctuations of wind and scalars (e.g., Baldocchi, 2008). These characteristics, along with ease of operation, have promoted the widespread application of the technique in both short-term experiments and long-term monitoring network operations (e.g., FLUXNET, CarboEurope, EuroFlux, and AmeriFlux).

Reliable flux estimation in a local environment is often complicated by a number of issues relating to the large range of fluctuation-scales which drive fluxes (Stull, 1988). Fluxes driven by high-frequency fluctuations (turbulence) are inherently local in nature, whereas fluxes driven by low-frequency fluctuations are associated with e.g., topographical forcing on the observed flow, or large-scale meteorological phenomena, including gravity waves, deep convection and large roll vortices (Lee et al., 2004). Traditionally the presence of a spectral gap (Stull, 1988) is assumed to exist between these contributions, allowing investigators to disentangle contributions simply by separating continuous observations into quasi-stationary intervals each yielding one flux estimate. However, the existence of a distinct spectral gap is unclear (Lee et al., 2004) and a growing body of work suggests that low-frequency contributions may often be non-negligible, even for relatively flat sites. Furthermore studies have shown that the low-frequency contributions are highly site-specific and characterized by significant uncertainty (Aubinet et al., 2010; Loescher et al., 2006; Yi et al., 2008). Hence, observations of atmospheric fluctuations are likely to reflect some degree of convolution between signals of local turbulent contributions and site/time-specific low-frequency contributions.

The importance of including vertical low-frequency contributions in studies is debated. For instance, some studies suggest that inclusion may improve closure in energy and carbon-balance studies (Finnigan et al., 2003; Mahrt, 1998; Sakai et al., 2001; von Randow et al., 2002), while other studies suggest otherwise (Aubinet et al., 2010). Kanda et al. (2004) demonstrated that, although the systematic bias decreased when including low-frequency contributions, the variance between flux estimates increases greatly. In other words, any single flux estimate becomes vulnerable to random low-frequency contributions, and thus increasingly difficult to interpret in terms of local surface fluxes. Moreover, it has been commented that horizontal low-frequency contributions, which are typically assumed negligible, may become significant during conditions of low turbulence intensity and gravitational flows (Yi et al., 2008) as well as during flow disturbance associated with complex topography (Zeri et al., 2010).

Accordingly, we can distinguish between two principal applications of the EC technique: (1) process-oriented studies in which fluxes are being linked to local biogeochemical processes for parametric insight into universal causal flux relationships and up-scaled through numerical modelling, and (2) long-term net ecosystem-exchange studies in which the flux estimates are understood to be site-specific, applying only for the unique conditions of a particular ecosystem. This study will focus on the former, and we will refer to the turbulence driven fluxes as locally meaningful fluxes, following Lee et al. (2004).

For process-oriented studies, a number of typical approaches exist to estimate locally meaningful fluxes. These include: (1) adjusting the flux averaging time to strike an appropriate balance between adequate sampling of the turbulent flux contribution while avoiding excessive inclusion of low-frequency contributions (Sun et al., 2006); (2) ensuring horizontal homogeneous conditions within the foot print of the flux; (3) estimating vertical low-frequency contribution by performing profile measurements of fluxes on a single tower (Lee, 1998; Leuning et al., 2008) and filtering out observations reflecting excessive low-frequency influence (Novick et al., 2014); (4) filtering observations based on co-spectral similarity with theoretical co-spectra assumed to represent local flux distributions for ideal site-conditions (Hojstrup, 1981, 1982; Hunt et al., 1985; Kaimal, 1978; Kaimal et al., 1972; Moore, 1986; Moraes, 1988; Moraes and Epstein, 1987; Olesen et al., 1984); (5) estimating the ideal turbulent contribution by matching the observed co-spectral peak with that of a theoretical distribution (Sorensen and Larsen, 2010).

While each method has its merits, none is universally applicable and without its caveats. In the absence of a distinct spectral gap between contributions, separating flux contributions by adjusting the flux averaging time will inevitably fail. Moreover, given the evolving nature of the natural flow, a proposed spectral gap is likely to change in character over time, indicating that setting a fixed averaging time for an entire experiment inevitably causes some misrepresentation of fluxes. In the limit of low absolute covariance (i.e., small fluxes) a relative large variance of the co-spectra estimates complicate the comparison between observed and theoretical co-spectra. While such cases could be treated as reflecting observations approaching the detection limit of the system, and discarded accordingly, they are important for exchange studies over low-flux surfaces such as sea ice, creating a demand for a new approach here.

Ensuring co-spectral similarity requires a number of site/system-specific empirical co-spectral corrections to account for high-frequency non-white noise/dampening produced by the presence of the EC system in the observed flow as well as signal dampening in closed path systems, greatly complicating the approach (Aubinet et al., 2000; De Ligne et al., 2010; Kaimal, 1968; Massman and Ibrom, 2008; Moncrieff et al., 1997; Moore, 1986; Silverman, 1968). Matching the co-spectral peak solves the issue of excessive scaling offset mentioned above, but increases the risk of subjective analysis.

Here, we present a novel method for estimating locally meaningful
atmosphere–surface fluxes despite low-frequency influences, using a single
eddy covariance system and a numerical modelling scheme for ogive
optimization. Accordingly we call this method ogive optimization. Ogive
optimization makes no assumptions regarding optimal flux averaging time or
the presence of a spectral gap and improves the flux estimates by also
considering contributions in the very high/low frequency ranges. To evaluate
the method, we applied it on eddy covariance observations of sensible heat,
latent heat and CO

A number of typical observational situations shown in terms of co-spectra (top row) and ogives (bottom row). Shown are the turbulent fluxes (red) and low-frequency/noise/dampening components (blue).

The theory of eddy covariance is well established (e.g., Baldocchi, 2008).
Average surface fluxes of sensible heat, latent heat and CO

Flux estimates Eq. (1) may be decomposed into frequency-dependent
contributions, called co-spectra

Subsequently we may perform an ogive analysis (Desjardins et al., 1989;
Foken et al., 2006; Lee et al., 2004). The analysis requires the same basic
assumptions and involves the cumulative summation of co-spectral energy,
starting from the highest frequencies,

Figure 1 illustrates a number of observational situations showing examples of how low-frequency influence could affect our ability to capture local fluxes. In the figure, situations are shown using both co-spectral and ogive plots.

In the ideal case (Fig. 1a), turbulent and low-frequency flux contributions
are separated by a spectral gap, allowing investigators to isolate the
former simply by choosing an appropriate flux averaging time

Given the unclear existence of a spectral gap (Lee et al., 2004),
however, another more general situation is the case (Fig. 1c) of overlapping
contributions from low-frequency motions, turbulence and site and
instrument-specific non-white noise/dampening. One way to strike a balance
between adequate inclusion of the turbulent contribution and exclusion of
excessive low-frequency influence is by adjusting

Observations reflecting excessive low-frequency influence, relative to the turbulent contribution, (Fig. 1e) are typically discarded. This is because strong relative low-frequency influence results in non-negligible flux contribution to the overall estimate and further obstructs any efforts to separate contributions by adjusting the flux averaging time (Fig. 1f). The use of the term relative in this context refers to the fact that an identical problem can arise despite modest low-frequency influence when estimating fluxes in a low-flux environment. Flux estimation in such environments are often further complicated by a high ratio of co-spectral variance to actual turbulent flux contribution. This prohibits unambiguous evaluation of similarity between observed co-spectra and theoretical co-spectrum distributions, as well as proper estimation of the co-spectral peak (Sorensen and Larsen, 2010).

In order to fulfil the stationarity requirement described in Sect. 2.1,
continuous observations are typically subdivided into averaging intervals.
Averaging interval time

The following is an iterative scheme for developing averaging intervals
based on basic data quality requirements. Data collected during a
field-experiment is considered continuous with end points

Using an iterative bisectional algorithm for enhanced computational speed,
combinations of

Finally, signals with very rapid evolutions such as transient signals in
dynamic systems like eddy covariance observations may undergo abrupt changes
associated with observational interference e.g., electrical interference or
instrument error. These are referred to as dropouts and discontinuities in
Vickers and Mahrt (1997). Global transforms, like the Fourier
transform, are usually not able to detect these events. In contrast, Wavelet
transforms such as the Haar transform, permit a localized evolutionary
spectral study of signals, thus allowing for detection of subtle signal
discontinuities leading to semi-permanent changes (Lee et al., 2004;
Mahrt, 1991; Vickers and Mahrt, 1997). In this study we perform the Haar
analysis for the data set

As noted, adjusting the flux averaging time will not generally allow for separation of turbulent and low-frequency flux contributions. Subtracting a running mean from observed signals, as opposed to the conventional linear detrending, allows for enhanced filtering of low-frequency contributions alone (Sakai et al., 2001; Mcmillen, 1988). Consequently some combination of data-set length (averaging time) and running mean window size might allow for filtering out of low-frequency contributions while retaining turbulent contributions. Note that both adjusting the flux averaging time and subtracting a running mean from the observed signal may, in many cases, provide sufficient separation of turbulent fluxes and low-frequency contributions. Here we apply both to arrive at a more generally applicable approach. We visualize this concept by calculating co-spectra, and corresponding ogives, for a very large number of data permutations and derive a map of the resulting ogive density pattern (Fig. 2a). The figure illustrates the density of 10 000 individual sensible heat-flux ogives based on the following data perturbations: 50 linear increments on the averaging time axis between 10 min and the maximum time available (60 min in this example) and 200 linear increments for the running mean window in the range of 1 min to half the length of the data set in question (30 min in this example). The standard 30 min linear detrended ogive is marked in red.

What is clear immediately in this particular example is the strong
consistency between individual ogive representations. This suggests that the
fluxes are very well defined for this particular period with an actual flux
around

Theoretical cospectra (red line) and equivalent ogives
(black line) are shown for two cases:

Unfortunately not all ogive density maps indicate as well defined fluxes as shown in Fig. 2. In such cases, answering the overall question of most likely flux requires the fitting, or optimization, of an ogive model to the ogive density map. With the introduction of an optimization aspect the advantage of performing the analysis for ogives, as opposed to co-spectra, becomes clear. In the limit of low absolute covariance (i.e., small fluxes), co-spectra typically become increasingly characterized by both positive and negative frequency-wise flux contributions. The co-spectral model, however, can only account for fluxes in one direction. Observed and modelled ogives, in contrast, are able to describe and account for this bidirectionality.

The basic premise in our model solution is that a region exists in the
mid-to-high frequency range of the ogive representation, which is least
impacted by instrument-specific non-white noise, dampening and low-frequency
influences. This was illustrated in Fig. 1b, d, f. While such a region is clearly
evident in Fig. 2a for

To describe the most likely flux resulting from a given ogive density
pattern, we apply the generalized co-spectral distribution model (Lee
et al., 2004)

One important aspect considered is the concept of local fluxes that cannot
be observed directly. The problem may arise in the low-frequency range as
over/under estimation of covariance due to inclusion of low-frequency
contributions or the use of inadequate averaging times. Similarly, in the
high-frequency range the problem may arise in the form of under-estimation of covariance
due to inadequate sensor frequency, attenuation and distortion by both the
spatial averaging of the sensors, and the sampling and filtering of the
sensor electronics. This is illustrated in Fig. 3a. Actual flux, represented
by an ideal theoretical co-spectrum (red line) is shown alongside a
corresponding ogive (black line). In the case of insufficient observation
time (here 20 min) and observational frequency (here

Combined the corrections amount to

One intriguing consequence of including a modelling and optimization aspect is that the inevitable occurrence of overlapping data intervals does not relate linearly to interdependency of successive flux estimates, suggesting that the ogive optimization approach allows for very high temporal resolution of flux evolution at less expense in terms of flux independence.

Illustration of site locations and conditions.

To evaluate the ogive optimization method, five sites reflecting different
environments in terms of ecosystem, topography and flux strengths
(

The Abisko field-site (Fig. 4a) is located in Stordalen (68^{®}, Lymington UK) mounted on top of the mast at 2.9 m
height and an LI-7500 open path gas analyzer (LI-COR^{®},
Lincoln, NE, USA) on a boom extending towards the southwest at 2.5 m height,
with a 0.43 m horizontal offset along the boom and a slight tilt of the
instrument relative to the vertical plane to allow water dripping. Data were
logged at 10 Hz. Observations reflecting wind origins along the boom axis
were filtered out to limit flow distortion. Flux estimates were evaluated in
intervals of

The RIMI (Risø Integrated Environmental Project) site (Fig. 4b) is an active FLUXNET site (e.g., Groenendijk et
al., 2011; Stoy et al., 2013; Yi et al., 2010) located in a large, flat,
homogeneous grassland area (55^{®},
Lymington UK) and an LI-7500 open path gas analyzer (LI-COR^{®},
Lincoln, NE, USA) mounted on the same boom at heights of 2.2 and 2.1 m respectively extending from the side of a 10 m mast, with a horizontal
offset along the boom of 0.40 m. Raw data were logged at 20 Hz. Observations
reflecting wind origins along the boom axis were filtered out to limit flow
distortion. Flux estimates were evaluated in intervals of

Young Sound (Fig. 4c) is the entrance of a 7 km wide fjord in NE Greenland
characterized by thick fast sea ice within the fjord and an ice-free polynya
at the mouth of the fjord (Rysgaard et al., 2003). Continuous eddy
covariance observations were conducted at three sites within the fjord
system in the period 20 March to 27 April 2012. Two separate field-stations,
one static and one mobile, were used at three different locations (ICEI,
POLYI and DNB). ICEI (74^{®},
Elmshorn, Germany) was mounted at a height of 3.1 m and an LI-7500A open
path gas analyzer (LI-COR^{®}, Lincoln, NE, USA) was mounted at
an angle of 70^{®}, Lymington UK) was
mounted at a height of 3.7 m relative to the snow surface and an enclosed
LI-7200 gas analyzer (LI-COR^{®}, Lincoln, NE, USA) was mounted
with a 65 cm inlet tube terminating directly under the sonic anemometer. Raw
data were logged at 10 Hz. Sea-ice and snow-cover thickness was
approximately 110 and 75 cm for the ICEI and DNB sites, respectively, and
approximately 25 and 20 cm for the POLYI site. Average air temperature
increased from

During post-processing, a number of instrument-specific corrections are
needed to adjust for instrument-bias. For the sonic anemometers (Gill R2,
Gill R3, Gill Windmaster Pro and METEK USA-1) these include the following: an empirical
angle of attack correction (Nakai and Shimoyama, 2012), and
humidity and crosswind corrections (Liu et al., 2001; Schotanus et al.,
1983). We convert all observations to mixing ratios (Burba et al., 2012)
using the Webb–Pearman–Leuning correction when necessary (Sahlee et al.,
2008; Webb et al., 1980) as recommended by Ibrom et al. (2007). The need
for instrument heating corrections (Burba et al., 2008) associated with
operation of the open path LI-7500 in a cold environment (Daneborg, POLYI
and ICEI) is alleviated by using the newer LI-7500A with a “cold” setting
correcting observations down to

60 min observation of sensible heat flux recorded at
Abisko on 2 July 2012 at 9:15 p.m. LT. Shown are

In the following, we describe several typical cases observed and the associated performance of the ogive optimization method.

Near-absence of low-frequency influence is observed leading to a
strong similarity between the ogive density pattern, the 30 min linear
detrended ogive and the modelled ogive. This is illustrated in Fig. 5a for a
case of sensible heat flux at the Abisko site. Disregarding the
high-frequency component associated with extrapolation of model results, seen
here to contribute

Cases where non-negligible low-frequency influence on the
flux estimate is observed for CO

60 min observation of CO

An example of ambivalence caused by bimodality in the ogive
density pattern is illustrated in Fig. 7, for a case of sensible heat flux
at the Abisko site. Such cases indicate that fluxes are changing within the
sampling period. The ogive optimization method is seen to capture the
turbulent flux contribution with the strongest data density. Had both modes
been of equal ogive density, the choice of mode during subjective evaluation
would be based on the quality of the model ogive optimization, and the
length of the time-series responsible for the modes. If both ogive models
were equally good, the choice would fall on the mode produced by the ogives
which consist of shorter time-series as they represent a more instantaneous
flux estimate relative to the mode produced by longer time-series. The
sampling period in question was characterized by a slightly stable
atmosphere

60 min observation of sensible heat flux recorded at Abisko on 10 July 2012 at 8:50 p.m. LT. The illustration is similar to Fig. 5.

The inadequacy of applying a fixed averaging interval for flux
estimation becomes apparent in Fig. 8, for a case of sensible heat flux at
the Daneborg site. Here, the ogive density pattern is seen to reflect a
gradual evolution in the ogive flux pattern with increasing averaging time.
The standard 30 min averaging time is seen to be too long and also to
increasingly reflect low-frequency interference (Fig. 8a). This is
consistent with an abrupt increase in atmospheric temperature (Fig. 8b) and
decrease in covariance (Fig. 8c) around 30–40 min. The ogive optimization
method identifies the appropriate flux estimate (Fig. 8a), whereas the
standard 30 min linear detrending method fails on account of
in-stationarity. In addition, the case is a perfect example of how
co-spectral evaluation of frequency-wise contributions can be misleading
(Fig. 8a, inner plot). The observational period in question was
characterized by a stable atmosphere

40 min observation of sensible heat flux recorded at Daneborg on 13 April 2012 at 3:30 p.m. LT. The illustration is similar to Fig. 5.

The inadequacy of applying a fixed averaging interval for flux
estimation becomes apparent in Fig. 8, for a case of sensible heat flux at
the Daneborg site. Here, the ogive density pattern is seen to reflect a
gradual evolution in the ogive flux pattern with increasing averaging time.
The standard 30 min averaging time is seen to be too long and also to
increasingly reflect low-frequency interference (Fig. 8a). This is
consistent with an abrupt increase in atmospheric temperature (Fig. 8b) and
decrease in covariance (Fig. 8c) around 30–40 min. The ogive optimization
method identifies the appropriate flux estimate (Fig. 8a), whereas the
standard 30 min linear detrending method fails on account of
in-stationarity. In addition, the case is a perfect example of how
co-spectral evaluation of frequency-wise contributions can be misleading
(Fig. 8a, inner plot). The observational period in question was
characterized by a stable atmosphere

Signals may be degraded for a number of reasons such as
instrument failure, electronic interference etc. Such a case is illustrated
in Fig. 9, for a case of CO

60 min observation of CO

During conditions of strong high-frequency dampening caused by
the use of a closed path instrument, the ogive optimization method
automatically shifts the high-frequency bound on optimization towards lower
frequencies to avoid influence of the dampened frequencies during
optimization. This is illustrated in Fig. 10 for a case of latent heat flux
at the ICEI site. Here the upper optimization bound is shifted back to

60 min observation of latent heat flux recorded at ICEI
on 27 March 2012 at 1:30 a.m. LT. The illustration is similar to
Fig. 5, except for the addition of a smoothed raw signals of atmospheric
H

Relative difference in percent (see Eq. 5) is shown logarithmically as a function of absolute flux estimate for all investigated sites. Also shown are the median (red line), standard deviation (light gray area) and 25–75 % percentile (dark gray area) of the relative differences. In the bottom of the figure, histograms of absolute ogive optimization flux estimate ranges are shown for each site. Numbers indicated to the left of the histograms are the respective maximum values.

The difference in flux estimates of the standard 30 min linear detrending
approach and the ogive optimization method is associated with both the
inclusion/exclusion of low-frequency contributions, the inadequacy of the
fixed averaging interval and the extrapolation of modelled ogives into
un-observable high/low frequencies. The relative flux difference

As Fig. 11, but here the relative difference is shown as
a function of atmospheric stability

As hypothesized in Sect. 2.2, the average relative flux difference is seen
to be very high for small absolute flux estimates, peaking at

Number of flux estimates from the conventional method
(

Depending on perspective and the character of observed fluxes at a particular
site the described thresholds may either serve as an indicator of a lower
limit to local-scale flux resolvability by the standard 30 min linear
detrending approach, or as an argument for the application of enhanced flux
estimation techniques such as the presented method. For the presented
observations the consequences are illustrated by the histograms of the
different sites (Fig. 11). Although the location of the flux threshold is a
bit unclear for latent heat flux, estimation of locally meaningful fluxes at
the three sea-ice sites Daneborg, POLYI and ICEI is essentially impossible
without accounting for low-frequency contributions. The same applies for
sensible heat flux at the Abisko lake site, latent heat flux at the grassland
site RIMI and CO

The relative flux difference was furthermore investigated in terms of
atmospheric stability (Fig. 12). Though variation in

For many flux estimates the vertical wind speed signal or the scalar signal
are non-stationary to the point of prohibiting a flux estimation using
traditional methodology. Hence the ogive optimization method may also provide
a greater number of flux estimates. This is shown in Table 1 to generally be
true for the Abisko and Daneborg sites, both of which characterized by
degraded signal quality at times. Sites RIMI and POLYI are inconclusive in
this respect and the conventional method appears superior in the case of
ICEI. The latter may be related to the very low fluxes observed for this site
(Fig. 11) suggesting the presence of a detection limit for the ogive
optimization method when using the particular instrument setup at ICEI
(LI-7200 enclosed gas analyzer) within the respective ranges

Low-frequency shifts in flux direction were found to be common in this study. To our knowledge such occurrences are not described by any existing theoretical framework, indicating a puzzling caveat to current theory. The occurrences challenge the notion that fluxes should be of same sign regardless of incident eddy scales. One explanation might be that vertical low-frequency contributions represent only one part of a net low-frequency contribution and hence is balanced by a horizontal component. Indeed the horizontal low-frequency component has been shown to be significant during certain conditions (Yi et al., 2008; Zeri et al., 2010), despite typically being assumed negligible. The finding indicates that further investigation of the interplay between low-frequency contributions, and their influence on turbulent flux estimates, is necessary.

The presented ogive optimization method has been shown to
successfully separate local from non-local flux contributions. In addition,
it enhances flux estimation by both investigation of a large range of
averaging times and running mean detrending, and extrapolation of optimized
ogive model results. The method makes no assumptions concerning appropriate
averaging time or the presence of a spectral gap, does not require the
application of transfer functions and allows for very high temporal
resolution of flux evolution. For high flux rates (

The study suggests favourable application of the ogive optimization method in most environments, particularly in environments characterized by small fluxes such as over sea ice. Overall, the notion of a dynamic and generally non-negligible overlap of low-frequency and turbulent flux contributions is confirmed.

Finally, low-frequency shifts in flux direction were found to be common in this study. To our knowledge such occurrences are not described by any existing theoretical framework. Based on studies indicating non-negligible horizontal low-frequency contributions during certain conditions (Yi et al., 2008; Zeri et al., 2010) we hypothesize a more intricate balancing interplay between vertical and horizontal low-frequency flux contributions which, if confirmed, suggests the need for more sophisticated eddy covariance system arrays if low-frequency contributions are to be accurately included (i.e., for site-specific studies). If exclusion of low-frequency contributions is desired (i.e., for universal-process-oriented studies), the presented method should be unaffected by these questions.

As described in Sect. 2.4.2 our goal is to tune the four final model
parameters

A random guess of parameters is made within a set of reasonable bounds.
The speed and accuracy of any optimization method involving pre-set bounds
depend greatly on the reasonable choice of these bounds. Here we set

Think of optimizing a model ogive to an ogive density map as choosing a
path between two points in the Pyrenees for which you travel at the highest
possible average altitude, all the while being constrained to a certain type
of path (the ogive form and the associated parameters). In more technical
terms optimizing the four parameters of the model ogive may be thought of as
locating the point in a parameter-wise four-dimensional probability space,
for which the net ogive density reached along the path is the highest. In
this context we seek a global, as opposed to local, solution within the
probability space formed by the four parameters. Based on the initial random
guess, a local solution is determined using the MATLAB function fminsearchbnd
(available through the Mathworks^{®} file exchange) which is a
Nelder–Mead polytope direct search optimization algorithm. The algorithm is
fast for problems of low dimensionality such as ours, but not certain to
converge to a global solution. The goal is to perform a rough, but fast,
improvement of the random guess to limit processing time for the next step,
which is far more computationally expensive.

Based on the local optimization of parameters produced by fminsearchbnd,
a global solution is determined using the Differential Evolution (DE)
algorithm (Storn and Price, 1997). Differential evolution is a simple
and reliable evolutionary population-based search technique, which has been
successfully applied on a wide range of problems in a variety of scientific
fields (Mallipeddi et al., 2011). Inspired by Darwinian
evolutionary theory it optimizes a problem by iteratively improving a
population of NP candidate solutions (agents) based on random candidate
mutation (motion) and survivability within the probability space of a
multivariate problem. Mutations are governed by predefined mathematical
relations, called strategies, which depend on crossover probability

Often optimizing a smaller subset of the problem is an advantage,
particularly during low-frequency interference which persists despite
data perturbation in the mass ogive phase. One such case is shown in Fig. 13.
Optimizing in subsets is achieved by subdividing the problem into 18
frequency interval weights in the range 0 to 1, signifying 0 to 100 %
influence of a given part of the density map on the optimization output (Fig. 13a, black lines). Corresponding solutions for the 18 frequency interval
weights are shown in Fig. 13b (green lines). All solutions based on frequency
intervals with lower bound before or after the ogive density peak

60 min observation of latent heat flux recorded at
Abisko on 12 July 2012 at 8:50 p.m. LT.

The executable code of our procedure, ogive optimization, will be made available and can be acquired by e-mailing the corresponding author (jasi@envs.au.dk or lls@bios.au.dk). The program is coded in MATLAB and is optimized for use with the parallel computing toolbox.

The study received financial support from the Arctic Research Centre, Aarhus University, the DEFROST project of the Nordic Centre of Excellence program “Interaction between Climate Change and the Cryosphere”, the collaborative research project “Changing Permafrost in the Arctic and its Global Effects in the 21st century” (PAGE21), the Canada Excellence Research Chair program, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the ArcticNet Canadian network of centres of excellence. Additionally, this work is a contribution to the Arctic Science Partnership (ASP). The authors wish to thank a number of people who assisted with the Daneborg experiment; David Barber, Bruce Johnson, Kunuk Lennert, Ivali Lennert, Egon Randa Frandsen, Jens Ehn, Karl Attard and Dorte Søgaard. We furthermore wish to thank the Abisko Scientific Research station for providing infrastructure and technical help in the field. Lastly, the EU project, CarboEurope and especially CarboEurope PI, Ebba Dellwik, DTU, Denmark is acknowledged for the use of the flux data from the CarboEurope site Ll.Valby. Edited by: S. M. Noe