**Research article**
02 Mar 2018

**Research article** | 02 Mar 2018

# Decorrelation scales for Arctic Ocean hydrography – Part I: Amerasian Basin

Hiroshi Sumata Frank Kauker Michael Karcher Benjamin Rabe Mary-Louise Timmermans Axel Behrendt Rüdiger Gerdes Ursula Schauer Koji Shimada Kyoung-Ho Cho and Takashi Kikuchi

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**Hiroshi Sumata et al.**Hiroshi Sumata Frank Kauker Michael Karcher Benjamin Rabe Mary-Louise Timmermans Axel Behrendt Rüdiger Gerdes Ursula Schauer Koji Shimada Kyoung-Ho Cho and Takashi Kikuchi

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^{1}Alfred-Wegener-Institut Helmholtz-Zentrum für Polar- und Meeresforschung, Bremerhaven, Germany^{2}Ocean Atmosphere Systems, Hamburg, Germany^{3}Yale University, Department of Geology and Geophysics, New Haven, CT, USA^{4}Jacobs University, Physics and Earth Sciences, Bremen, Germany^{5}Tokyo University of Marine Science and Technology, Tokyo, Japan^{6}Korea Polar Research Institute, Incheon, South Korea^{7}Japan Agency for Marine-Earth Science and Technology, Yokosuka, Japan

^{1}Alfred-Wegener-Institut Helmholtz-Zentrum für Polar- und Meeresforschung, Bremerhaven, Germany^{2}Ocean Atmosphere Systems, Hamburg, Germany^{3}Yale University, Department of Geology and Geophysics, New Haven, CT, USA^{4}Jacobs University, Physics and Earth Sciences, Bremen, Germany^{5}Tokyo University of Marine Science and Technology, Tokyo, Japan^{6}Korea Polar Research Institute, Incheon, South Korea^{7}Japan Agency for Marine-Earth Science and Technology, Yokosuka, Japan

**Correspondence**: Hiroshi Sumata (hiroshi.sumata@awi.de)

**Correspondence**: Hiroshi Sumata (hiroshi.sumata@awi.de)

Received: 18 Aug 2017 – Discussion started: 06 Sep 2017 – Revised: 21 Dec 2017 – Accepted: 08 Jan 2018 – Published: 02 Mar 2018

Any use of observational data for data assimilation requires adequate information of their representativeness in space and time. This is particularly important for sparse, non-synoptic data, which comprise the bulk of oceanic in situ observations in the Arctic. To quantify spatial and temporal scales of temperature and salinity variations, we estimate the autocorrelation function and associated decorrelation scales for the Amerasian Basin of the Arctic Ocean. For this purpose, we compile historical measurements from 1980 to 2015. Assuming spatial and temporal homogeneity of the decorrelation scale in the basin interior (abyssal plain area), we calculate autocorrelations as a function of spatial distance and temporal lag. The examination of the functional form of autocorrelation in each depth range reveals that the autocorrelation is well described by a Gaussian function in space and time. We derive decorrelation scales of 150–200 km in space and 100–300 days in time. These scales are directly applicable to quantify the representation error, which is essential for use of ocean in situ measurements in data assimilation. We also describe how the estimated autocorrelation function and decorrelation scale should be applied for cost function calculation in a data assimilation system.

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Any use of observational data requires assumptions, or better knowledge, about the representativeness of each measurement in space and time. This holds even more for in situ observations from data-sparse regions, such as the Arctic Ocean. Interpolation guided by the statistical properties of observed quantities can provide Arctic-wide fields, while data assimilation using comprehensive dynamical models and assimilation methods can, in addition, provide fields that are consistent with the modeled physics. Also, sampling strategies have to take the knowledge of the representativeness of point measurement into account. The temporal and spatial scales, for which a single measurement is representative, depend on local dynamics, external forcing, and the influence of lateral water–mass influxes. Here, we make an attempt to estimate those length scales and timescales in the Arctic Ocean based on observational data from the period 1980–2015. This will be achieved by estimating the autocorrelation function and decorrelation scales of temperature and salinity.

Autocorrelation functions and associated decorrelation scales are useful
measures to characterize physical phenomena occurring in the ocean (Stammer,
1997; Eden, 2007). These functions describe spatial and temporal ranges over
which ocean properties coherently vary, and the scales provide a measure of
the spatial and temporal extent of the variations. The functional form of the
autocorrelation depends on the physical properties, the considered scales
(e.g., synoptic versus mesoscale) and the area. Many studies have estimated
autocorrelation functions through analysis of in situ ocean measurements
(e.g., Meyers et al., 1991; Chu et al., 2002; Delcroix et al., 2005) and
satellite observations (e.g., Kuragano and Kamachi, 2000; Hosoda and
Kawamura, 2004; Tzorti et al., 2016). Generally, the estimated autocorrelation
functions have exponential or Gaussian form (Molinari and Festa, 2000). The
decorrelation scales are usually given by the *e*-folding scale of the
corresponding autocorrelation functions (see McLean, 2010 for a summary of
different definitions).

Estimated decorrelation scales have been applied to a variety of ocean studies. In the context of dynamical studies, the decorrelation scale is used as a measure of the scale of prevailing phenomena and used to relate dynamical processes with the observed signals (e.g., Stammer, 1997; Ito et al., 2004; Kim and Kosro, 2013). In optimal interpolation and objective mapping, the decorrelation scale gives a measure of influential radius of a point measurement; the autocorrelation function, together with the associated decorrelation scale, provides the weight of a point measurement on mean field estimates (Meyers et al., 1991; Chu et al., 1997; Davis, 1998; Wong et al., 2003; Böhme and Send, 2005). For observation network design, decorrelation scales are one guide to estimate optimal sampling intervals in space and time (Sprintall and Meyers, 1991; White, 1995; Delcroix et al., 2005).

One of the prevalent and growing applications of decorrelation scales is data assimilation. Data assimilation synthesizes observed data and modeled physics based on statistical theories. This is an effective approach to fill the gap between observation and modeling studies (Wunsch, 2006; Blayo et al., 2015). Generally, data assimilation minimizes a model–data misfit with an assessment of errors; the autocorrelation function and the decorrelation scale are necessary for these error assessments (Carton et al., 2000; Forget and Wunsch, 2007). For a model–data misfit calculation, the difference of the spatial (and temporal) scales represented by a model and by the observations should be taken into account. Physical properties simulated in general circulation models (GCMs) represent mean values over each grid cell for a certain temporal period, whereas those from in situ measurements represent values at a localized point in space and in time. The error resulting from the difference of the scales represented by these two approaches is referred to as representation error (see van Leeuwen, 2015 for a summary). The autocorrelation function and the decorrelation scales provide a direct measure of the representation error. In ocean data assimilation, an assessment of the representation error is particularly important, since it is generally an order of magnitude larger than the measurement (instrument) error (Ingleby and Huddleston, 2007).

A necessity of decorrelation scale in ocean data assimilation also comes from the sparseness of ocean measurements. An autocorrelation function is necessary to constrain locations distant from a measurement. Li et al. (2003) pointed out that an assimilation of sparsely distributed data into an eddy-permitting model, without taking its influential radius into account, causes serious problems around the locations where the data are assimilated. Artificial eddies appear around the location of the data, since the density at the data location differs from densities at their surrounding grid points in the model. They also pointed out that the assimilated information disappears on the timescale determined by the model's local advection and diffusion. Note that this situation cannot be solved by applying advanced data assimilation techniques (e.g., 4DVar, EnKF), since the artificial eddies are dynamically consistent with the modeled physics. Autocorrelation function and decorrelation scale provide necessary information to solve such problems by imposing a spatial and temporal radius of influence of each measurement (Forget and Wunsch, 2007; Zuo et al., 2011).

Practically, autocorrelation functions are used to define an “observation operator” in data assimilation systems. The observation operator maps modeled variables onto observational points. If the operator is properly defined, a point measurement will constrain the model, not only at the location where measurements exist but also in areas distant from the measurement. An implementation of such an observation operator makes it possible to fully exploit the potential of sparsely distributed measurements, and can solve problems such as those reported by Li et al. (2003). This is of particular importance as the ocean models used for assimilation become eddy-permitting. An additional important feature of the autocorrelation function is to constrain the scale of temporally varying fluctuations. Unlike the static interpolation approaches, data assimilation provides a four-dimensional analysis field. In order to appropriately assimilate observed temporal fluctuations, the temporal scale of fluctuations should be implemented in the observation operator.

In the midlatitude and equatorial regions, there are a number of
decorrelation scale estimates (e.g., White and Meyers, 1982; Chu et al.,
1997, 2002; Deser et al., 2003; Martins et al., 2015), and these have been
applied for a variety of studies including data assimilation (see the papers
mentioned above). On the other hand, while a few studies have examined scales
of temperature and salinity variability in the Arctic Ocean (e.g., Timmermans
and Winsor, 2013; Marcinko et al., 2015), there has been no assessment of
basin-wide decorrelation scales of *T*∕*S* field to date. One reason is that
sea-ice cover greatly inhibits sea surface observation by remote sensing.
Another reason is the sparse coverage of in situ ocean measurements due to
the inaccessibility and the absence of an Argo float network (that has
provided essential data for midlatitude and Southern Ocean studies; e.g.,
McLean, 2010; Reeve et al., 2016). In the last decade, however, the
number of observational activities has been increasing significantly, with
the growing concern about the sea-ice retreat and its potential impact on
global climate (see, e.g., Ortiz et al., 2011 and references therein). In
addition to the increasing number of research cruises, autonomous observation
platforms (e.g., ice-tethered profilers – ITPs; Krishfield et al., 2008a;
Toole et al., 2011) now provide data throughout a full seasonal cycle in the
Arctic. The data acquired from these research activities enable us for the
first time to estimate basin-wide decorrelation scales for *T* and *S*
profiles in the Arctic Ocean.

The objective for the following study is to estimate the autocorrelation functions and decorrelation scales of temperature and salinity in the Arctic Ocean at different depths. Few modelling studies have focused on applications of ocean in situ measurements in the Arctic, due to the absence of comprehensive historical archives and representation error estimates. Only the climatology (PHC3.0; Steele et al., 2001) has been widely applied for model validation (e.g., Ilıcak et al., 2016). In recent years, however, assimilations of in situ measurements in the Arctic Ocean have started (Panteleev et al., 2004, 2007; Nguyen et al., 2011; Zuo et al., 2011; Sakov et al., 2012). To promote and enhance the ongoing ocean data assimilations, archiving historical measurements and estimating decorrelation scales are indispensable. To achieve the objective of the present study, we (1) compile historical observations of temperature and salinity in the Arctic Ocean, (2) construct a background mean field necessary for the decorrelation scale estimate, (3) examine the functional form of autocorrelation in temporal- and spatial-lag space, and finally (4) provide an autocorrelation function, decorrelation scales, and representation error covariance, which are directly applicable to error assessment in ocean data assimilation. Note that the estimation of the autocorrelation quantifies basin-scale variability. Smaller-scale variability (e.g., mesoscale eddies on the deformation scale; Zhao et al., 2014) remains unresolved and is an intrinsic part of the autocorrelation function. The study area is the Amerasian Basin. As will be described in Sect. 3, the second step mentioned above requires a different approach for other regions of the Arctic Ocean. The vertical depth range of the analysis is limited to between 0 to 400 m depth due to data availability.

The rest of the paper is organized as follows: Sect. 2 describes the compilation of historical data and quality-control procedures applied prior to the analysis. Section 3 describes the background temperature and salinity field construction and trend analyses. Section 4 describes examination of two-dimensional autocorrelation functions in spatial- and temporal-lag space, and provides decorrelation scale and error covariance estimates. Section 5 gives conclusions.

## 2.1 Compilation of historical data

Since there is no comprehensive in situ ocean data archive for the Arctic, we compile historical temperature and salinity measurements with the objective not only to use the data for the present decorrelation scale estimate but also to prepare an archive for future applications in model validation and data assimilation. Since the existing archived data from the Arctic Ocean are widely dispersed in various datasets with different formats, we compile these data into one archive with a standard format focusing on the Arctic and northern North Atlantic Ocean (Table 1). The original data (Table 1) were acquired from various observational platforms (e.g., research vessels, moorings, ITPs, and Argo floats) by conductivity–temperature–depth (CTD) sensors and expendable CTDs (XCTDs). The archiving effort of this study originates from the data compilation described by Rabe et al. (2011, 2014) and Somavilla et al. (2013), and is ongoing thanks to support from many oceanographers. The archived data will be available online (https://www.pangaea.de) after a profile-based thorough quality check (except those data which require additional consent from data providers). This public archive is described in Behrendt et al. (2017).

The archived information for each measurement profile includes cruise name,
station number, data type, time stamp, geographical location, bottom depth
(if available), measurement depth (pressure is converted to depth by the
method described by Saunders, 1981), temperature, salinity, data quality
information provided in the original dataset (if available), and data source
information. The spatial coverage of the archived data ranges from
45^{∘} N to the pole on the Atlantic Ocean side and from
64^{∘} N (Bering Strait) to the pole on the Pacific Ocean side.
The temporal coverage is from 1980 to 2015. Figure 1 shows an example of the
spatial distribution of the archived data (0–20 m depth range, north of
64^{∘} N) for the entire period. The archived data cover the
entire Arctic and northern North Atlantic oceans, while the biggest data
gaps are on the East Siberian Shelf and north of the Canadian Arctic Archipelago. A
basic quality check is applied to the archived data before the duplication
checks and statistical screening, described in the following subsections.
The basic quality check is composed of (1) a bathymetric test using the merged
IBCAO/ETOPO5 (Jakobsson et al., 2012) with a tolerance of 20 m, (2) a valid range test for
temperature (−2.2 ^{∘}C $<T<\mathrm{30.0}$ ^{∘}C) and
salinity (0 psu $<S<\mathrm{40.0}$ psu), and (3) a vertical stability
test. The bathymetric test is applied to remove data with inconsistent
geographic locations (i.e., either on land or indicating profile
information at depths deeper than the sea floor at their location). This
test excluded a number of erroneous profiles with position errors. The
vertical stability test is applied to remove spike data points found in CTD
and XCTD profiles. If the stability test program finds vertical density
inversions, the data points are removed from the profile. If a data point
violates one of the criteria, it is removed from the archive.

## 2.2 Duplication check

Since data obtained from various sources are prone to duplication issues, it
is necessary to identify and remove duplicated data from the archive. A
number of past studies, which compiled large oceanographic datasets, have
suggested various automated procedures to deal with duplicate profiles
(e.g., Ingleby and Huddleston, 2007; Gronell and Wijfefels, 2008; Good et al., 2013).
In this study, we apply a simple duplication-check algorithm suitable for the present application. Since we are concerned
only with basin-scale variability in this analysis, we count profiles that
have small spatial and temporal separations as duplicates. The threshold
applied for time difference between profiles is 1 day (date coincidence) and
that applied for geographical location difference is 0.05^{∘} in
longitude and 0.01^{∘} in latitude, respectively; to account for the
effect of convergence of meridians toward the pole, a threshold of 2 km separation
is also applied. If duplication is found (i.e., both temporal and
spatial separation conditions above are met), the profiles are flagged. The
profile with the highest reliability according to the data provider's own
quality control is retained. For example, if we directly obtain data from
PIs who have already applied their own quality-control procedure, we give
the data higher priority than those from other data archives (e.g., World
Ocean Database, 2013). The final duplication-checked archive is used as input
for the statistical screening described below.

## 2.3 Statistical screening

Since the archive contains a number of data that have not been quality
controlled, we apply an additional quality-control procedure (QC) before our
analyses. Note that although we describe the QC procedure as it is applied
to the entire raw dataset in this section, we will use only data from 0 to 400 m
depth (after the QC) in the present scale analysis as mentioned in the
introduction. The QC is composed of two steps: the first step is a grid-based
screening; the second step is an area-based screening. Both steps are
based on statistics of the data samples in discretized depth ranges. We
divide the vertical profiles of temperature (*T*) and salinity (*S*) measurements
into 50 depth bins (from a 20 m interval near the sea surface to a 200 m
interval in the deep ocean; Fig. 2a). If there are more than two
measurements for a certain depth range from one profile, the measurement
values (*T* and *S*) are averaged. The statistics are calculated and applied in
each depth range separately.

First, we apply a grid-based screening. The grid-based screening takes the
difference in statistics (mean and standard deviation) in different
locations into account. We define 111 km × 111 km (corresponding to
1^{∘} × 1^{∘} at the Equator) grid cells over the
entire archive domain. The mean (*μ*) and standard deviation (*σ*)
of *T* and *S* on each grid cell and in each depth range are calculated from the
data within the surrounding 555 km × 555 km (5^{∘} × 5^{∘})
area. *T* and *S* values outside 5 times the standard
deviation (*μ*±5*σ*) on each grid cell are removed from the
archive (the procedure is repeated twice).

Second, we apply an area-based screening for the data deeper than 750 m
depth. In this step, we apply more rigorous statistics calculated from the
entire basin and shelf area. This step is necessary to remove problematic
data in data-sparse areas and data-sparse depth ranges, since the
grid-based screening cannot provide good statistics in these areas due to the
small sample size (no ITP data below 750 m). We classify the archived data
into six subdomains based on the characteristics of dynamical regimes (Nurser
and Bacon, 2014): (1) Amerasian Basin, (2) Amerasian shelf and shelf slope,
(3) Siberian Shelf and shelf slope, (4) Eurasian Basin, (5) Barents and Kara
seas including their shelf slopes, and (6) Nordic Seas (Fig. 2b). Mean and
standard deviation are calculated in individual subdomains. Then, data
outside 5 times the standard deviation (*μ*±5*σ*) are removed
(repeated twice). In this paper, we focus only on the results for the
Amerasian Basin; regions 2–5 are considered in a separate analysis.

The result of the statistical screening in the Amerasian Basin is shown in Fig. 3. The combined statistical screening successfully removes spurious data in deep depth ranges, while retaining the relatively larger variability in shallow depth ranges. After the combined statistical screening, the vertically discretized data are used for the analyses in the following section.

In this section, we describe the construction of a background mean field of
*T* and *S*, which represents the basin-wide climatology in the Amerasian Basin.
The background mean fields will be used to calculate anomaly fields
necessary for the decorrelation scale estimates. For the construction of the
background mean field, we first examine the functional form and spatial
scale of the mean field variation (Sect. 3.1). Second, we apply the derived
functional form and scale for the background mean field construction (Sect. 3.2).
The temporal linear trends of *T* and *S* are also examined to account for
the effect of a long-term temporal change of the mean field (Sect. 3.3).

## 3.1 Spatial scale of variation

To derive the scale for the background field construction, we examine the
spatial scale of variation in each depth range (the vertical layers defined
in Fig. 2b are used throughout this study to provide decorrelation scales
directly applicable to data assimilation systems using *z*-coordinate
systems). In this estimation, we assume isotropy and homogeneity of the
spatial scale of variation in a basin. These assumptions are valid if (1) planetary- and
(2) topographic-*β* effects do not dominate in a basin,
and (3) no dominant oceanic structure extends toward one specific direction.
The first and second conditions are satisfied in the high-latitude Amerasian
Basin (small planetary-*β* effect) away from marginal shelf slopes,
where a large topographic-*β* effect is expected. The third condition
is also satisfied in the deep Amerasian Basin, although not necessarily in
other sectors of the Arctic Ocean and Nordic Seas. For example, in the
Eurasian Basin, there is a prominent extension of the frontal structure along
the shelf slope associated with the warm Atlantic-water inflow (Anderson et al., 1994; Rudels et al., 2013).
The location of the front is not necessarily trapped over the shelf
slope but can be detached from the slope (Jones, 2001). Further, in the Nordic
Seas, there are meridionally extending dominant current systems, i.e., the
East Greenland Current, Norwegian Current, and West Spitsbergen
Current (Hopkins, 1991). These features require a scale examination that takes a
spatial anisotropy into account; a different approach for scale estimation
will be applied to the Eurasian Basin in a forthcoming paper. For our
purposes here, the Amerasian Basin is defined by the area where total water
depth is deeper than 1000 m. This definition excludes the area affected by
coastal currents and topographically trapped flows (associated with the
submarine Northwind Ridge, for example).

To estimate the spatial scale of variation, we introduce a structure function (Davis et al., 2008; Todd et al., 2013) with the assumption of spatial and temporal isotropy of variation,

where *x* and *t* are the spatial and temporal separations from location *x*_{0} and
time *t*_{0}, Ω is the observed property (in this case, either *T* or
*S*), and 〈 ⋅ 〉 is the averaging operator
over space and time. The structure function, *φ*_{x, t}, gives
the mean square difference between two measurements as a function of spatial
and temporal separations. It was initially introduced by Kolmogorov (1941) to provide
a statistical description of a field without specifying the mean and
variance of the field. This is an appropriate approach for the present
purpose, since we do not have a priori information regarding the statistics
of the background field. We calculate the structure function from all
available data in the Amerasian Basin (all depth bins shallower than 400 m):

where *N* is the number of available data pairs, the spatial and temporal
separations of which are *x* and *t*, and ΔΩ_{i (x,t)} is the
difference of observed values of the *i*th pair. We introduce a function *f*,
which measures the normalized root mean square difference (RMSD) of any two
measurements:

where *φ*_{bg} is defined by all the possible combinations of
available data in the basin in a certain depth range:

and *M* is the number of all available data. *φ*_{bg} is a measure
of the size of basin-wide and long-term variations; i.e., we introduce it as
the “background” mean squared difference used to normalize *φ*_{x, t}.

The function *f* in Eq. (3) is a unitless measure of RMSD between two
measurements as a function of spatial and temporal separations. If *φ*_{x, t} ∼ *φ*_{bg}, i.e., the mean difference between
two measurements with (*x*, *t*) separation is comparable to those of
“large” distance measurement pairs, then *f*∼ 0. This indicates that
no coherent structure exists between data with (*x*, *t*) separation. If
${\mathit{\phi}}_{x,\phantom{\rule{0.125em}{0ex}}t}\ll {\mathit{\phi}}_{\text{bg}}$, i.e., the mean difference between
measurements with (*x*, *t*) separation is sufficiently small compared to
that between sufficiently distant data pairs, then *f*→1. This indicates
a strong coherence exists between the data with (*x*, *t*) separation
(ultimately, *f*=1, if the spatial and temporal separations are exactly
zero). Note that the function *f* is not an autocorrelation function,
although it has similar properties (e.g., decays from 1 to 0 for spatial and
temporal separations from zero to infinity). The function *f* measures the
scale of the coherent structure of the mean field, whereas an autocorrelation
function measures the scale of coherent variation of anomalies. A structure
function *φ* can be directly related to an autocorrelation function,
if we can define *φ* by the anomaly from the mean field (e.g., Gandin
1965; Molinari and Festa, 2000). Since we have no a priori statistical
information regarding the mean field, we cannot relate the structure function
*φ* with the autocorrelation in our case. The correspondence to the
geostatistical approach is given in Appendix A.

In order to examine the functional form of *f*, we construct data pairs from
all possible combinations of data in each depth range, classify the pairs
into 50 × 36 bins (50 bins for spatial separation with 10 km
intervals and 36 bins for temporal separation with a 10-day interval), and
calculate *f* in respective bins. For the binning, we suppose that the spatial
and temporal scales of variation are much larger than the scale used for the
binning, the validity of which is recursively confirmed by the scales
estimated. Examples of the functional form of *f* for *T* and *S* in spatial
and temporal separation space are shown in Fig. 4. Small separation gives
large *f* values, while *f*∼0 when the separation is sufficiently large. Note
that *f* decays with an increase in temporal separation in shallow depth
ranges with a timescale of approximately 90–120 days (Fig. 4a, b), while
*f* is relatively insensitive to temporal separations at depths deeper than
80 m (Fig. 4c, d), which is a manifestation of the seasonality. This
seasonality is taken into account to estimate the background mean field in
Sect. 3.2. Note that we limit our analysis here to consider only the upper
water column, from 0 to 400 m depth, as uncertainties in the uncalibrated
(“level-2”) ITP salinity data are comparable to the temporal and spatial
variability of salinity in the Amerasian Basin below 500 m (see Appendix B).

To closely examine the functional form of *f*, we calculate the temporal
(0- to 90-day) average of *f* in respective depth ranges. A survey of the
two-dimensional functional form over all depth ranges (shallower than 400 m)
revealed that 90 days is a reasonable choice to account for seasonal
variation (not shown). Figure 5 shows the 90-day averaged functional form of
*f* in different depth ranges (thin-dotted lines) and the average for all
depth ranges (0–400 m; thick-dotted black line). Although the scale of
variation varies with depth, the functional form of it be reasonably
approximated by a Gaussian function (thick-solid blue line). Note that *f*
does not come close to 1, even if the spatial separation nears 0 km, because
the present examination excludes self combination of data (i.e., $\mathrm{\Delta}\mathrm{\Omega}(\mathrm{0},\mathrm{0})=\mathrm{0}$), deals with a 0- to 90-day average, and does not resolve
mesoscale fluctuations smaller than those at 10 km scale (the spatial separation of
the bin).

The *e*-folding scales of the fitted Gaussian function for *T* and *S* are
summarized in Fig. 6. The *T* profile (dashed black line) exhibits a large
spatial scale of variation (∼ 200 km) near the sea surface, indicating
the effect of the large-scale thermal forcing at the sea surface. The *T*
profile deeper than 100 m depth is nearly constant (120–150 km). The
salinity profile (solid blue line), on the other hand, exhibits nearly
constant scale (130–150 km) from the sea surface to 400 m depth,
indicating small contributions from large-scale surface salinity fluxes at
the sea surface. We apply the *e*-folding scale of each depth level and the
Gaussian function to estimate the background mean field.

## 3.2 Background mean field

To take the seasonal variation into account, we divide the observed data into
four seasons (January–March, April–June, July–September, and
October–December), and construct the background mean *T* and *S* fields in
each season. This is supported by the fact that the temporal *e*-folding
scale is approximately 90 days in shallow layers (Fig. 4a, b) and even longer
in the deeper layers. The background field is derived by applying a spatial
Gaussian filter with an *e*-folding scale given by the spatial scale of
variation in each depth range (Fig. 6). The background field for Ω_{i} is given by

where *N* is the number of measurements, whose distance from the *i*th
measurement (Ω_{i}) is less than 3 times the *e*-folding scale (i.e.,
$\Vert {x}_{i}-{x}_{n}\Vert <\mathrm{3}L$; see below), ${W}_{n}^{\prime}$ is the normalized weighting
function for the *n*th data point, and Ω_{n} is the *n*th measurement
surrounding the *i*th measurement. The normalized weighting function ${W}_{n}^{\prime}$
is given by

where *W*_{n} is the Gaussian weighting function:

where *x*_{i} and *x*_{n} are the geographical location of Ω_{i} and
Ω_{n}, respectively, and *L*(*z*) is the *e*-folding scale of the
Gaussian filter as a function of depth (Fig. 6). An example of the derived
background field for *T* and *S* in summer is shown in Fig. 7. The field
captures a warm and fresh water mass distribution in the Canada Basin and its
smooth transition toward cold and saline water in the northeastern Amerasian
Basin. For the anomaly field calculation, we require the background field at
the locations where observational data exist. Therefore, we do not apply any
spatial and/or temporal interpolations even in data-sparse seasons (winter
and spring).

## 3.3 Temporal trend

For the present anomaly derivation, we also take the temporal trend from 1980
to 2015 into account. The trend is estimated in each
111 km × 111 km grid cell (1^{∘} × 1^{∘} at Equator
scale), in each depth range, and in each season (Mann–Kendall rank
statistics; Kendall, 1938) with a significance level of 5 %). The size of the grid
cells is chosen to be consistent with the spatial scale of variation
(Sect. 3.2). Figure 8 shows representative *T* and *S* trends in the
60–80 m depth range in summer and the corresponding average time series for
those grid cells for which the trend is statistically significant. A warming
(∼ 0.5 ^{∘} decade^{−1}) and freshening
(∼ 0.5 psu decade^{−1}) trend in the Canada Basin is evident in
this depth range. The freshening trend extends from the sea surface to 400 m
depth without a significant change in spatial pattern, whereas the *T* trend
changes sign and spatial pattern with depth. A positive trend in *T* is
observed in the depth range from 0 to 160 m over the whole analyzed time
period (i.e., through the Pacific-water/upper halocline layers, represented
by red line in Fig. 8c), while after the year 2002 a decreasing trend in *T*
is observed in the central Canada Basin in the 200–400 m depth range (lower
halocline/Atlantic-water layer, represented by blue line in Fig. 8c). A
positive trend is observed along the southern perimeter of the Canada Basin
in 250–400 m depth range (Atlantic-water layer, represented by black line
in Fig. 8c and d).

The warming and freshening trend in the Pacific-water layer has already been reported by many studies (e.g., Proshutinsky et al., 2009; Jackson et al., 2010; Giles et al., 2012; Timmermans et al., 2014). The cooling trend in the central Canada Basin and the warming trend along its southern perimeter are a consequence of deepening of the warm Atlantic water in the central basin and concurrent upwelling of warm Atlantic water at the boundaries, a manifestation of an intensification of the anticyclonic Beaufort Gyre in recent years (e.g., McLaughlin et al., 2009; Karcher et al., 2012; Zhong and Zhao, 2014). Although similar trends can be found in other seasons (from winter to spring), they are not statistically significant.

The temporal trend in each location is used to define a time-varying
background field. Since the temporal distribution of the archived data is not
spatially uniform, the representative time (i.e., the time that the temporal
mean value represents) of the background field ${\stackrel{\mathrm{\u203e}}{\mathrm{\Omega}}}_{i}$ varies
with space. The representative time is used as a tie point (offset) to
connect the mean and trend. Taking the effect of the representative time into
account, the time-varying background field for Ω_{i} is defined by

where *a*(*x*) is the temporal trend at location *x*, *t* is the time,
*t*_{rep}(*x*) is the representative time of the background mean field
${\stackrel{\mathrm{\u203e}}{\mathrm{\Omega}}}_{i}$ at location *x*. We calculate the representative time
in each 111 km × 111 km area by the average of measurement times
of all the data contained in the corresponding area and apply it to define
the time-varying background field (see the Supplement). For the area where no
trend can be deduced, we apply a constant background field,
${\stackrel{\mathrm{\u0303}}{\mathrm{\Omega}}}_{i}={\stackrel{\mathrm{\u203e}}{\mathrm{\Omega}}}_{i}$.

## 4.1 Autocorrelation function

Decorrelation scales used in oceanographic studies are generally defined by
an *e*-folding scale of an autocorrelation function, which has a Gaussian or
exponential functional form (Molinari and Festa, 2000). Practically, the autocorrelation
functions are obtained from a series of autocorrelations estimated by
differently lagged points (e.g., White and Meyers, 1982; Meyers et al., 1991). An autocorrelation for
Δ** l** lag is given by

where cov(**Ω**_{l}, **Ω**_{l+Δl}) is an autocovariance
between two data series **Ω**_{l} and **Ω**_{l+Δl}, the
temporal and/or spatial lag between which is Δ** l**, and
var(

**Ω**

_{l}) and var(

**Ω**

_{l+Δl}) are the variances of

**Ω**

_{l}and

**Ω**

_{l+Δl}, respectively. We assume isotropy and homogeneity of the autocorrelation in the Amerasian Basin, supported by the weak planetary-

*β*effect in polar regions and the homogeneity of the Rossby radius in the Amerasian Basin (Nurser and Bacon, 2014; Zao et al., 2014). These assumptions enable us to calculate the autocorrelation from data series, which are composed of data pairs having the same temporal and spatial lag Δ

**but come from different locations in the basin and from different times (e.g., Sprintall and Meyers, 1991; Chu et al., 1997, 2002), i.e.,**

*l*
where *N* is the number of data pairs, the spatial and temporal lags between
which are Δ** l**, ${\mathrm{\Omega}}_{n}^{\prime}$ is the anomaly value of the

*n*th data, ${{\widehat{\mathrm{\Omega}}}^{\prime}}_{n}$ is the anomaly value of the paired data which locates Δ

**-lagged point from ${\mathrm{\Omega}}_{n}^{\prime}$.**

*l*The anomaly dataset **Ω**^{′} is defined by subtracting the time-varying
background field $\stackrel{\mathrm{\u0303}}{\mathbf{\Omega}}$ from the observed data **Ω**. Each
anomaly datum of the set is paired with the other anomalies to construct a set
of anomaly data pairs, which consists of all possible combinations of two
anomaly data. The data pairs are classified into discretized bins, according
to the spatial and temporal lags of the paired data (50 spatial bins with a
10 km interval and 73 temporal bins with a 5-day interval; i.e., the
examination window is 500 km lag × 365-day lag). The spatial and
temporal sizes of the bin are designed to capture the functional form of the
autocorrelation relevant for basin-scale data assimilation (i.e., the
functional form of the autocorrelation describing mesoscale fluctuations are
not examined in this analysis). Each bin has a sufficient number of data
pairs to calculate an autocorrelation (*N*>*O*(10^{3}); see
Fig. 9a). Figure 9b, c show examples of the autocorrelation functions for *T*
and *S* in the 40–60 m depth range. There is a clear decrease of
autocorrelation with increasing spatial and temporal lags, although with some
variability about this relationship.

Temporal and spatial averages of the autocorrelation are calculated to
identify its functional form by fitting a suitable empirical function.
Figure 10a and b show the temporal average of the spatial autocorrelation
functions of *T* and *S* for different depth ranges. To account for the
effect of differences of temporal autocorrelation scales in different depth
ranges, we define the temporal average by a 0- to 30-day lag in shallow levels
(0–140 m depth range) and by a 0- to 60-day lag in deeper levels (below
140 m). The functions generally show their highest values at zero-spatial
lag, with decreasing values as the spatial lag increases. Some functions
exhibit a second peak around a spatial lag of 200–300 km. We examine the
relation between the second peaks and associated background mean field of *T*
and *S* in different depth ranges, and find that the peaks derive from the
circular *T* and/or *S* structure of the Beaufort Gyre (see Appendix C).
Since the Beaufort Gyre is characterized by bowl-shaped isosurfaces of *T*
and *S* associated with surface downward Ekman pumping, coherent variation of
the isosurfaces gives rise to the second peak. To eliminate the effect of the
second peak for our scale estimate, we use the autocorrelation functions just
for a spatial lag of 0–150 km to compute a fitting function. We tested
exponential and Gaussian functions for the fitting and found that the
Gaussian function is generally suitable to represent the
observationally derived spatial autocorrelations (Fig. 10c, d).

The temporal autocorrelation is also examined by taking spatial-lag averages
(0–20 km) of the two-dimensional autocorrelations of *T* and *S*. Figure 11a,
b show the averaged temporal autocorrelation functions in various depth
ranges. The functions show their highest values at zero-temporal lag and a
reduction towards large temporal lags, whereas the functions from many depth
ranges clearly exhibit an annual cycle. Since the seasonal variability of the
background field is already taken into account (Sect. 3.2), the annual cycle
found in the temporal autocorrelations indicates the effect of persistent
atmospheric forcing, the timescale of which is longer than 1 year (e.g.,
Arctic Oscillation, Thompson and Wallace, 1998; North Atlantic Oscillation,
Hurrell, 1995; Wallace, 2000), and/or spin-up/-down process of gyre-scale
circulation, the timescale of which is estimated as 3–4 years (Yoshizawa et
al., 2015). To remove the effect of the annual cycle found in Fig. 11a, b, we
use the autocorrelation functions from 0 to 200 days of temporal lag to find a
fitting function for the temporal autocorrelation. We again tested
exponential and Gaussian forms for the fitting, and found that the Gaussian
functions are suitable to represent the form of the temporal autocorrelation
functions (Fig. 11c, d).

## 4.2 Decorrelation scale

The spatial and temporal decorrelation scales of *T* and *S* are derived from the
*e*-folding scales of the fitted spatial and temporal autocorrelation
functions in the respective depth ranges. The spatial autocorrelation
function is represented by the Gaussian form,

where *A*_{s} is the autocorrelation at zero-spatial lag, *x* is a
spatial lag, and *d*_{s} is the spatial decorrelation scale. The
temporal autocorrelation function has the same formula but exchanges
*A*_{s} for *A*_{t}, *x* for *t*, and *d*_{s} for
*d*_{t}, where *A*_{t}, *t*, and *d*_{t} are the
autocorrelation at zero-temporal lag, temporal lag, and temporal
decorrelation scale, respectively. The autocorrelation at zero-temporal and
-spatial lag (*A*_{s} and *A*_{t}) represents the effect of
unresolved fluctuations, which have a scale smaller than the resolution of
the present analysis at 10 km resolution in space and 5-day resolution in
time (1–*A*_{s} represents the magnitude of unresolved fluctuations
relative to the basin-scale fluctuations). The effect of mesoscale eddies
with the scale of the deformation radius (order of 10 km horizontally) is
described by this parameter.

Figure 12 summarizes the vertical profiles of the spatial and temporal
decorrelation scales (*d*_{s} and *d*_{t}) of *T* and *S* with
the associated parameters for zero-lag autocorrelations (*A*_{s} and
*A*_{t}). The zero-lag autocorrelations (Fig. 12a, c) show smaller
values (0.6–0.7) in the upper 100 m depth range, indicating active
mesoscale processes (e.g., eddy activity observed in the Pacific-water layer; e.g., Zhao et al., 2014).
The zero-lag autocorrelations for spatial
(Fig. 12a) and temporal lags (Fig. 12c) exhibit similar profiles, confirming
the appropriateness of the spatial and temporal averages used for the
functional form examinations. The vertical profiles of the decorrelation
scale (Fig. 12b, d) indicate an influence of the sea surface boundary
condition at shallow levels. The spatial decorrelation scale near the sea
surface (∼ 200 km) is larger than it is in deeper layers
(∼ 150 km), as a consequence of the direct influence of the atmosphere
and sea ice, the spatial scale of which is larger than the scale of intrinsic
ocean processes. The temporal decorrelation scale near the surface
(100–150 days), on the other hand, is shorter than that of the deeper layers
(200–300 days), possibly due to the effect of short-timescale variation of
the atmospheric field and associated sea-ice motion. It is interesting to
note that the scales of the mean field and of the variance are very similar
(e.g., compare Figs. 6 and 12b). We currently have no explanation for this
feature but assume that it is a peculiarity based on the dynamics of the
analyzed basin. In forthcoming papers, we plan to analyze the scales in the
Eurasian basin and over the Arctic shelf slope and will revisit this
question.

Note that the *T* and *S* profiles exhibit similar vertical profiles in the
depth range shallower than 250 m, while discrepancies stand out in levels
deeper than 250 m (Fig. 12b, d). This may be due to small calibration errors
associated with our use of ITP level-2 (i.e., not the fully calibrated
level-3) data (see Krishfield et al., 2008b; Johnson et al., 2007). In order
to incorporate as many data as possible, we have included all available ITP
level-2 data, where level-3 data are not yet available. This strategy is
beneficial for scale estimation of temperature (ITP level-2 temperature data
have the same accuracy as level-3 data, within ±0.001 ^{∘}C) in the
entire depth range and salinity shallower than 250 m depth. On the other
hand, since salinity variability decreases with depth (Fig. 3b), the
uncalibrated ITP level-2 salinity data may yield non-negligible spurious
variation at levels deeper than 250 m, which may deteriorate the accuracy of
the scale estimates for salinity in this depth range.

## 4.3 Error covariance

The autocorrelation function derived in Sect. 4.1 can be related to an error
covariance by Eq. (9). Since the variance in Eq. (9) used to normalize the
covariance does not depend on spatial and/or temporal separation in
principle (see the assumption in Sect. 4.1), it can be represented by a
variance calculated from all the data in the Amerasian Basin. Therefore, the
error covariance associated with the representation error is given by a
function of spatial and temporal separations, *x* and *t*:

where *ρ*(*x*, *t*) is the autocorrelation function, and
var_{bg} is the background mean variance defined by

The vertical profiles of var_{bg} for *T* and *S* are shown in
Fig. 13. The background mean variance clearly reflects the vertical
stratification in the Amerasian Basin (e.g., McLaughlin et al., 2004; Shimada
et al., 2005), with highest variance in the depth ranges of vertical extrema
in the profile. The temperature profile exhibits two minima (in the mixed
layer and around 130 m depth) and two maxima (approximately in 70 and
250 m; Fig. 3a). These shallow extrema are associated with the
seasonally, spatially, and interannually varying near-surface temperature
maximum (see, e.g., McPhee et al., 1998), and Pacific summer water layers
(see, e.g., Timmermans et al., 2014). The deep minimum corresponds to the Pacific
winter water layer plus variations in the deeper Atlantic water (see, e.g.,
Shimada et al., 2005; Fig. 2). The vertical profile of salinity variance also
exhibits good correspondence with salinity stratification and its variation
(Fig. 3b), with smallest variance (approximately 120 m depth)
corresponding to weakest salinity stratification and largest (around 180 m)
corresponding to the stratification boundary between the upper and lower
halocline. The derived covariance is also necessary to complete the model–
observation misfit calculation, as summarized in the following section.

We examined spatial and temporal scales of *T* and *S* anomalies from the
mean fields in the Amerasian Basin. To provide scales describing the
anomalies, we examined the autocorrelation of *T* and *S* measurements and
calculated spatial and temporal decorrelation scales. Historical *T* and *S*
measurements in the Arctic and northern North Atlantic oceans were compiled
for this study and for future applications to Arctic Ocean data
assimilations. The resulting quality-controlled archive was used to construct
a background mean field, from which anomaly fields were derived. By assuming
spatial and temporal homogeneity of the autocorrelation function in the basin
interior, we calculated autocorrelations as a function of spatial and
temporal lags. The examination revealed that the autocorrelation function can
be well described by a Gaussian function in space and time. The spatial and
temporal decorrelation scales were estimated to be 150–200 km in space and
100–300 days in time (*e*-folding scales of the autocorrelation function).
The spatial decorrelation scale is relatively large near the sea surface,
while the temporal scale is relatively small near the surface. Mesoscale
fluctuations, with scales smaller than 10 km and shorter than 5 days, are
represented by the zero-lag autocorrelation. The zero-lag autocorrelation
should be re-examined in future work to describe the autocorrelation smaller
than the Rossby radius by fully exploiting ITP data.

The estimated function and the scales, together with the associated error covariance, are directly applicable to model–observation misfit calculation in data assimilation systems, which intend to assimilate a spatially and temporally varying field. A cost function measuring the model–observation misfit is given by

where ** d** is the data vector,

**is the model vector,**

*m**H*is the observation operator, and

**R**is the observation error covariance matrix. The current study gives the descriptive form of

*H*and

**R**. An observation operator,

*H*, which takes spatial and temporal representativeness of each measurement into account, is given as follows:

where *i* refers to the *i*th in situ measurement, *j* refers to the modeled
variable at the *j*th model grid point, *ρ* is the autocorrelation
between (*x*, *t*)-distant locations, *x*_{ij} and *t*_{ij} are the spatial
and temporal separations between the *i*th measurement and the *j*th model
grid point. The operator *H*_{i}(** m**) maps the model field

**to the**

*m**i*th measurement location (in space and time), in accordance with the influence of the measurement. We can describe the autocorrelation function

*ρ*by the results shown in Sect. 4.1 and 4.2 in the following formula:

where *A* is the autocorrelation between zero-lag locations (*x*<10 km and
*t*<5 days) representing the contributions from unresolved-scale
fluctuations (Fig. 12a); *d*_{s} and *d*_{t} are the spatial
and temporal decorrelation scales (Fig. 12b, d), respectively. This formula
provides the representation error of a point measurement at (*x*,
*t*)-distant locations. Note that the current formula enables us to quantify
errors of modeled *T* and *S* not only at the location where the measurements
exist but also at the locations distant from the measurements. The present
study also provides error covariance matrix **R** associated with the
representation error. The representation error covariance between the *i*th
and the *i*^{′}th measurements is

where $\mathit{\rho}({x}_{i{i}^{\prime}}$, ${t}_{i{i}^{\prime}})$ is the autocorrelation between *i*th and
*i*^{′}th measurements, the spatial and temporal separations between which are
given by ${x}_{i{i}^{\prime}}$ and ${t}_{i{i}^{\prime}}$, and var_{bg} is the background error
variance given as a function of depth (Fig. 13). As summarized here, the
current study provides a full descriptive formula to exploit ocean in situ
measurements in the Amerasian Basin for a model–observation misfit
calculation.

The present scale estimates pose a requirement from a basin-scale data assimilation on a sampling strategy. Static interpolation approaches (e.g., optimal interpolation (Gandin, 1965; Reynolds and Smith, 1994), objective mapping (Wong et al., 2003; Böhme and Send, 2005; Böhme et al., 2008), and data-interpolating variational analyses (Troupin et al., 2010, 2012; Korablev, 2014) exploit statistical information of data to derive a mean analysis field. Data assimilation approaches, in addition, exploit modeled physics and provide temporally and spatially varying four-dimensional analysis fields. The former approaches need a scale representing the mean field, while the latter, in addition, needs spatial and temporal scales representing the anomaly field to fully exploit the information embedded in in situ data. For Arctic Ocean studies, statistical interpolation has been using decorrelation scales of 300–500 km (Steele et al., 2001; Proshutinsky et al., 2009; Rabe et al., 2011, 2014), while the present study suggests the necessity of a smaller measurement interval (150–200 km in space and 100–300 days in time) to describe the anomaly field by a basin-scale data assimilation.

Further studies are necessary to interpret the decorrelation scale of *S* and
*T* in the context of ocean dynamics and relate it to the hydrographic
features in the Amerasian Basin. The scale of ocean variability is governed
by external forcings and by various physical processes in the ocean. The
local dynamic response to local external forcing (i.e., vertical normal mode
in response to basin-scale wind stress curl; Pedlosky, 1987; Olbers et al.,
2012) is one very likely mechanism to explain the shape of the vertical
profile of the scale. Near the sea surface, the decorrelation scales should be
examined in relation to the scale of atmosphere and sea-ice variability
(Walsh, 1978; Walsh and Chapman, 1990), and the dynamical processes governing
the mixed layer (Peralta-Ferriz and Woodgate, 2015). The effect of remote
forcing is another important issue to be examined. Advection of anomalous
water masses introduces scales governed by mechanisms outside of the basin
and/or shelf–basin interaction, such as the inflow of anomalous Pacific water
into the deep basin (Steele et al., 2004; Itoh et al., 2012), its
modification processes on the shelf (Pickart et al., 2005, Woodgate et al.,
2005), the advection of anomalous Atlantic water (McLaughlin et al., 2009;
Karcher et al., 2012), or variations of freshwater supply due to river runoff
(Lammers et al., 2001). In this study, we employed level surfaces, as we focus
on the applicability of the decorrelation scales for model validation and
data assimilation (many models use the so called *z*-coordinate system). For
future studies which aim at a dynamical interpretation of the decorrelation
scales, an analysis in isopycnal coordinates would be a logical next step.
Autocorrelation and decorrelation scale estimates for other parts of the
Arctic Ocean (i.e., the Eurasian Basin and over the shelf slopes) will be
presented in forthcoming papers.

The data in the Amerasian Basin were collected and made
available by the following research programs: Arctic Switchyard project
(http://www.ldeo.columbia.edu/Switchyard), Baufort Gyre Exploration Program
based at the Woods Hole Oceanographic Institution
(http://www.whoi.edu/beaufortgyre) in collaboration with researchers from
Fisheries and Oceans Canada at the Institute of Ocean Sciences, the second
and third Chinese National Arctic Research Expeditions (Shi, 2009a, b),
Ice-Tethered Profiler Program (Toole at al., 2011; Krishfield et al., 2008)
based at the Woods Hole Oceanographic Institution (http://www.whoi/edu/itp),
JAMSTEC Compact Arctic Drifter (J-CAD) measurements by the North Pole
Environmental Observatory Project led by University of Washington in
collaboration with researchers from Japan Agency for Marine-Earth Science and
Technology (JAMSTEC) (Kikuchi et al., 2004), the KPDC
(http://kpdc.kopri.re.kr) data archived from the project titled “K-AOOS”
(Korea Polar Research Institute, PM17040) funded by the Ministry of Oceans
and Fisheries, South Korea, LOMROG 2007 Oden cruise (Bjork and Gothenburg
University, 2012), Nansen and Amundsen Basins Observational System
(NABOS/CAOBS) based at the University of Alaska Fairbanks
(http://nabos.iarc.uaf.edu/index.php), North Pole Environmental Observatory
(NPEO) (Morison et al., 2011), RV *Mirai* cruises operated by JAMSTEC
(http://www.godac.jamstec.go.jp/darwin/), Submarine Arctic Science Program
(SCICEX) (SCICEX Science Advisory Committee, 2009, updated 2014), the UNCLOS 2011 program
by Fisheries and Oceans Canada at the Institute of Ocean Science in
collaboration with JAMSTEC (Guéguen et al., 2015), and World Ocean
Database 2013 (Boyer et al., 2013).

Since data analysis software based on geostatistical approaches (e.g.,
iSATiS, SURFER) is used in oceanographic studies in recent years, it is
useful for providing a summary of the relation between the current approach
and geostatistical approaches. The spatial scale of variation estimated in
Sect. 3.1 is a different notation of the variogram concept used in
geostatistics. In the present formula, we normalize the variance by the sill
of the variogram, and a root-squared value is considered. This is because a
variogram deals with a variance (i.e., spatial scale of the squared
difference between two measurements), while we intend to quantify the spatial
scale of difference between two measurements. We also defined the function by
the value subtracted from 1, in order to obtain a function decaying to zero
at infinity. This is done for mathematical convenience in order to obtain a
Gaussian-like function. This is preferable for the framework of the best
linear unbiased estimator (BLUE), which is constituting the basis of data
assimilation theories. Since the spatial scale of variation originates from
the same concept as variograms, it can be related to the terminology used in
geostatistical approaches. The function *f* (i.e., normalized
root mean square difference) at zero separation (Fig. 5) is

where *N*_{g} is a nugget of the semivariogram plot. The estimated scale (the
spatial scale of variation) describes the square root of the scale described
by a variogram, although it is not easy to find an exact correspondence,
since empirical functions describing the two functions may differ. If we
directly translate the function *f* into a semivariance used to plot a
semivariogram, our formulation corresponds to an empirical semivariance
with the following form:

where *A* is the function *f* value at zero separation, which is related to
the nugget in Eq. (A1). Since we modeled the function *f* by a Gaussian
formula, we cannot define the “range” in the corresponding semivariogram
(the range goes to infinity in a Gaussian formula). After obtaining a
background mean field by using the spatial scale of variation, we do not have
to rely on geostatistical approaches any longer, since we can directly
calculate the autocorrelation by variance and autocovariance (Eq. 9).

Woods Hole Oceanographic Institution provides ITP temperature and salinity
data at different levels of processing; here, we use both level-3 (final
processed data) and uncalibrated level-2 data when level-3 data are not
available (see Krishfield et al., 2008b). Profile-by-profile conductivity
calibration (not applied to the level-2 data) accounts for conductivity
sensor drift. The calibration method applied to level-3 data is to adjust the
potential conductivity of each profile to the value derived from
bottle-calibrated CTD stations on the deep 0.4 ^{∘}C potential
temperature surface (Krishfield et al., 2008b).

As a measure of the uncertainty of the uncalibrated ITP level-2 data, we
calculate deviations of the ITP level-2 data from the background mean field
(Sect. 3.2). We assume that the standard deviations of the background field
derived from all data represent the natural variability of *T* and *S* in each
depth level. If the standard deviation from ITP level-2 data is larger than
the natural variability, we can conclude that the ITP level-2 data have an
error (bias) expressed by the excess of the standard deviation. Figure B1
depicts vertical profiles of the standard deviations of *T* and *S*
calculated from all data, from ITP level-2 data only, and from all data
except ITP level-2 data. The *T* profiles exhibit smaller standard deviation
of ITP level-2 data than the natural variability throughout the entire water
column. On the other hand, the *S* profile shows that the standard deviation
of ITP level-2 data is larger than the natural variability below 250 m
depth, and it is almost double as large below 500 m depth. Since the spatial
scale estimated in Sect. 3.1 and the decorrelation scale estimated in
Sect. 4.2 would be deteriorated by erroneous sensor drifts, we limit our
analyses from the sea surface to 400 m depth.

To understand the source of the second peaks found around the 200–300 km lag in
the spatial autocorrelation functions, we examine their relation to the
background mean fields. The second peaks in the autocorrelation functions are
always found where the corresponding *T* and/or *S* fields exhibit the
classic circular structure associated with the anticyclonic Beaufort Gyre.
Figure C2 shows examples of the background mean fields and corresponding
autocorrelation functions for various depth ranges. The upper two panels (Fig. C2a
and c) exhibit a clear circular spatial pattern in the Canada Basin, while
the lower two panels (Fig. C2e and g) do not. The corresponding spatial
autocorrelation functions show clear second peaks around 240 km lag
corresponding to the presence of the circular pattern (Fig. C2b and d), while
they show no such peak where the circular pattern is not present (Fig. C2f
and h).

The coincidence between the second peak and the circular structure of the
Beaufort Gyre indicates that the peak captures a coherent variation of
isothermal (isohaline) depth. We employ level depth surfaces for the present
analysis; bowl-shaped isosurfaces of *T* and *S* in the Canada Basin exhibit
a circular structure on level surfaces. Due to this structure, the same
isothermal (isohaline) surface appears on a level surface as it encircles the
center of the Beaufort Gyre (Fig. C2a, c). The second peak captures a
relatively high autocorrelation between the measurements, both of which
belong to nearly the same isothermal (isohaline) surface but are separated by a
certain distance in accordance with the circular pattern. A consideration of
mechanisms governing the decorrelation scale further supports this
interpretation. The basin-scale dynamical response of the ocean to external
forcing is manifested as vertical displacements of isopycnal surfaces (with
given *T* and *S* properties), resulting in coherent variations of these
depth surfaces. For follow-on studies to the present one, it is desirable to
calculate autocorrelation functions and decorrelation scales in a way that
takes such coherent large-scale dynamic features into account. This could be
achieved by analyzing anomalies of the isohaline/isothermal depth from
their mean state. In the case of the Beaufort Gyre, we expect the
autocorrelation functions for the variation of the isohaline/isothermal depth
to have larger spatial scales than those for *T* and *S* estimated on level
surfaces. As an approximate measure of the decorrelation scales for
isohaline/isothermal depth anomalies, we fit a Gaussian function using the
value at the zero-lag correlation and the second peak obtained from the level
surface analysis (Fig. C1), resulting in roughly 200–400 km. The largest
scales we find in the 200–350 m depth range for the isothermal depths and
in the 150–400 m depth range for the isohaline depths correspond to the
depths of strong vertical gradients of *T* and *S*. For a sound analysis, a
variation of isosurface should be quantified by a variation of isosurface
depth. In such an analysis, for example, salinity is no longer a variable to
be examined, but depth of constant salinity surface, i.e., ${\left.Z(x,y,t)\right|}_{S\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\text{constant}}$, is the variable to be examined.

The supplement related to this article is available online at: https://doi.org/10.5194/os-14-161-2018-supplement.

The authors declare that they have no conflict of interest.

The authors sincerely appreciate three anonymous reviewers for their
thorough reviews and criticism on the first and second manuscripts, and two
anonymous reviewers for the constructive comments on the third manuscript.
Funding by the Helmholtz Climate Initiative REKLIM (Regional Climate Change),
a joint research project of the Helmholtz Association of German research
centers (HGF), is gratefully acknowledged. This work has partly been supported
by European Commission as part of FP7 project Ice, Climate, and Economics –
Arctic Research on Change (ICE-ARC, project no. 603887). We also would like to
express our gratitude towards the German Federal Ministry of Education and
Research (BMBF) for the support of the project “RACE II – Regional Atlantic
Circulation and Global Change” (03F0729E) and various observational efforts
listed in Table 1. The GFD-DENNOU library
(http://dennou.gaia.h.kyoto-u.ac.jp/arch/dcl/) and Ocean Data View
(Schlitzer, 2015) were used to draw the figures.

The article processing
charges for this open-access

publication were covered by a
Research

Centre of the Helmholtz Association.

Edited by: Neil Wells

Reviewed by: two anonymous referees

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- Abstract
- Introduction
- Data
- Construction of the background mean field
- Decorrelation scale
- Conclusions
- Data availability
- Appendix A: Correspondence to the geostatistical approach
- Appendix B: Error estimates of ITP level-2 data
- Appendix C: Examination of the second peak in spatial autocorrelation functions
- Competing interests
- Acknowledgements
- References
- Supplement

- Abstract
- Introduction
- Data
- Construction of the background mean field
- Decorrelation scale
- Conclusions
- Data availability
- Appendix A: Correspondence to the geostatistical approach
- Appendix B: Error estimates of ITP level-2 data
- Appendix C: Examination of the second peak in spatial autocorrelation functions
- Competing interests
- Acknowledgements
- References
- Supplement