A Mapping Methodology Adapted to all Polar and Subpolar Oceans with a Stretching/Shrinking Constraint
1. Introduction
The polar and subpolar oceans of both hemispheres are crucial areas in oceanography for many reasons, such as the ability of cold polar and subpolar waters to dissolve more atmospheric carbon, the high biological productivity of these nutrient-rich waters, and the changes in dense water formation and mean sea level due to glaciers melting into the ocean. In the context of a changing climate, polar and subpolar regions have exhibited greater variability than other oceanic regions (Nakanowatari et al. 2007; Schmidtko et al. 2014; Haumann et al. 2016; Huang et al. 2017). Understanding variations of the polar ocean properties at scales ranging from seasonal to multidecadal is therefore crucial. Despite their importance, polar regions have been relatively less well sampled compared with other oceanic areas, due to limited data availability. The presence of seasonal or multiyear sea ice, difficult weather conditions, and the remote location of these regions are the main reasons for the sparse sampling there. However, over the past two decades, the density of data in many polar regions has dramatically increased owing to the deployment of profiling floats (e.g., the Argo program; Roemmich et al. 2009), instrumented seals of the Marine Mammals Exploring the Oceans Pole to Pole program (MEOP; Treasure et al. 2017), and continued ship-based measurements (e.g., the Southern Ocean Observation System; Rintoul et al. 2010). As a result, some of the polar or subpolar regions, such as the area around the Antarctic Circumpolar Current in the Southern Ocean (∼2000 profiles per 106 km2 per season between 45° and 60°S), have become moderately well sampled throughout the year. Other regions may be well sampled in summer but are more poorly sampled in winter (e.g., the Greenland Sea with ∼10 000 profiles per 106 km2 between July and September, and 3500 profiles per 106 km2 between January and March).
Mapping these ocean properties, irregularly sampled in time and space, onto a regular grid, is a very useful tool to better understand the distribution of ocean variables in polar and subpolar regions and their spatiotemporal changes. In the high-latitude regions affected by a strong Coriolis force, the weakly stratified flow tends to follow the isobaths (Pedlosky 1987), and several mapping schemes used in polar or subpolar regions consequently include a topographic constraint. Ridgway et al. (2002) used a least squares fitted scheme combined with a weighted mean function including a topographic constraint to map hydrographic data around Australia and the Australian–Antarctic Basin. Böehme and Send (2005) and Böehme et al. (2008) used a scheme based on the objective analysis of Bretherton et al. (1976), in which one of the terms of the weighting function accounted for the difference in potential vorticity between the grid point and observations. A very effective methodology, also derived from objective analysis, was developed by Schmidtko et al. (2013). This scheme introduced both a “path-following” term based on topographic constraint and latitude and a term accounting for the presence of density fronts, resulting in a particularly accurate mapping that avoided oversmoothing across the shelf and fronts. A global monthly climatology product with a resolution of 0.5°, called MIMOC, was generated through this method, and several mapping methodologies are now based on Schmidtko et al.’s (2013) scheme (e.g., Pauthenet et al. 2021; Sallée et al. 2021). The aforementioned mapping procedures are effective, but their computational requirements are heavy and their implementation is not trivial.
Shimada et al. (2017) developed a simple weighted average scheme including only two constraints: a distance constraint (r) and depth difference between the grid point and the observations to be mapped (Δh) as a topographic constraint. This topographic constraint integrated (TCI) mapping scheme has comparatively light computational requirements and its procedure is relatively tractable. The seasonal climatology produced in the Southern Ocean with this method compared favorably with MIMOC, yielding significantly lower root-mean-square (RMS) error between mapped values and observations.
However, the topographic constraint of Shimada et al. (2017) is not fully consistent in terms of the conservation of potential vorticity. Under the barotropic assumption, a relative change in potential vorticity is indeed approximately proportional to the relative change in bottom depth (see section 3), whereas the topographic constraint Δh represents an absolute change in depth.
In addition, while the implementation of the Shimada et al. (2017) mapping is straightforward, the determination of decorrelation scales is not trivial. This is also the case for most mapping methods, as underlined by Heuzé et al. (2015). Thus, a “streamlined” procedure is needed to allow nonexperts to estimate decorrelation scales with relative ease and little risk of error.
In this paper, we expand upon the methodology of Shimada et al. (2017) by introducing a water column shrinking/stretching constraint Φ = Δh/h, where h is the bottom depth at the grid point, to replace the topographic constraint Δh. In principle, our shrinking stretching constraint integrated (SSCI) method is consistent with the conservation of potential vorticity.
The purpose of our study is not to replace the well-known and tested methodologies that are based on objective analysis or to criticize their efficiency. Rather, our rationale is to provide a straightforward methodology, clearly based on a simple geophysical fluid dynamics principle, that can be used to quickly generate custom-made climatologies of good quality in all shelf and slope conditions. Importantly, we also precisely describe the procedure for determining the decorrelation scales, using the methodology of Shimada et al. (2017) as a base. This approach makes our method well adapted to all polar and subpolar regions.
We first further delineate the principle and the implementation of the method in section 3. As we found that the determination of decorrelation length scales is the most challenging part of this method, we also provide guidance on determining the decorrelation length scales in various slope and data density conditions. In section 4, we test our method by mapping temperature and salinity in the southern Sea of Okhotsk (broad and shallow shelf, sharp slope) and by comparing the results with those obtained through Shimada et al.’s (2017) scheme. To demonstrate the versatility of our method, we also undertake a similar comparison in the Southern Ocean (slanted shelf, gentle slope, deeper topography), using the latest observation data available.
2. Data
The bathymetric data used in this study are from the General Bathymetric Chart of the Oceans (GEBCO) 2022 15″-Arc product (https://www.gebco.net/data_and_products/gridded_bathymetry_data/, which we rescaled to 1′ for the sake of faster computation), and the base for the temperature and salinity datasets is historical conductivity–temperature–depth (CTD) data from the World Ocean Database 2018 (WOD18; Boyer et al. 2018). In the Sea of Okhotsk, data from 29 profiling floats deployed as part of a joint study between Japan, Russia, and the United States and a cooperative study between Hokkaido University and the University of Washington, as well as data from the Russian R/V Khromov, have been added to the WOD2018 dataset. These data constitute a considerable addition to the WOD2018 dataset and represent most of the data available in the study region after the year 2000. Last, biologging data from seven seals and sea lions in the Sea of Okhotsk in 2015 (Nakanowatari et al. 2017) are also part of the dataset. The total number of temperature measurements is 9745 at the 26.7σθ potential density level in the southern Sea of Okhotsk (Fig. 1b). Only profiles including both temperature and salinity measurements are included in our dataset.
(a) Bathymetric map of the southern Sea of Okhotsk. The thick purple and blue arrows represent the East Sakhalin Current (ESC) and its coastal branch, respectively, and the thick orange arrow represents the pathway of the Soya Warm Current (SC). The ESC and SC respectively transport dense shelf water (DSW) and Soya Warm Current Water (SCW), which are also indicated on the map. (b) Distribution of raw temperature data in the study region at 26.7σθ. The thick solid black line represents the 200 m isobath. The mapping is carried out over the domain delimited by the purple box. (c) A relatively sharper slope and (d) a gentler slope bathymetric profile along the two solid yellow arrow lines in (a). (e),(f) The number of data points available in the southern Sea of Okhotsk on each density and depth level, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
On a depth grid (called z grid hereafter), the data at shallow depths exhibit large variability and the determination of decorrelation scales should be based on the estimates at deeper levels (e.g., 200 m and below; Shimada et al. 2017) for the larger areas including the shelf region and the deep basin. In this constraint, the z grid is not appropriate for determining the decorrelation scales in the Sea of Okhotsk. Therefore, we adopted the estimation on a sigma grid (from 26.5σθ to 27.0σθ at an interval of 0.05σθ). This density range corresponds to depths of ∼50 to ∼500 m and covers the upper layer of the Okhotsk Sea Intermediate Water (26.8σθ–27.3σθ, 200–800 m). This upper layer is the level most affected by spatial variability in the Sea of Okhotsk (e.g., Itoh et al. 2003; Ohshima et al. 2017; Mensah et al. 2019). The data density between the 26.5σθ and the 26.9σθ layers is sufficient in both shallow and deep regions, but no data exist over the Sakhalin shelf at 26.95σθ and 27.0σθ.
Once the decorrelation scales were determined from the sigma grid, the values obtained were used to map data on both the sigma grid and a z grid of 26 vertical levels (0, 10, 20, 30, 50, 75, 100, 125, 150, 200, 250 m, every 100 m between 300 and 1500, 1750, and 2000 m). The horizontal grid size was 5 km × 5 km, and the grid domain is shown in Fig. 1b. Results for the Sea of Okhotsk will be presented throughout the paper only until 500 m, as the spatial variability of properties is insignificant below this depth.
3. Principle
a. The stretching and shrinking constraint
Polar and subpolar regions are characterized by a weakly stratified water column and, owing to their high latitude, a large Coriolis parameter f. Under such conditions, the flow tends to follow the isobaths (Pedlosky 1987; Cushman-Roisin and Beckers 2011). In other words, the decorrelation scales should be much larger in the along-isobath than in the cross-isobath direction. Dynamically, this feature can be explained in terms of the barotropic potential vorticity P = f/h, where h is the bottom depth and f is the Coriolis parameter. In the weakly stratified polar and subpolar oceans, the conservation of potential vorticity is the dominant constraint. Two water masses tend to mix more easily when their relative difference in potential vorticity (ΔP/P) is smaller; thus, water properties tend to be more uniform along smaller ΔP/P. ΔP/P can be the key parameter in these oceans. Over the relatively short distances O(102) km involved in the mapping of data, differences in f are negligible compared with differences in h, and for Δh/h ≪ 1, ΔP/P can be approximated as −Δh/h. Thus, Δh/h, which represents the degree of stretching (positive Δh/h) and shrinking (negative Δh/h), should be the key quantity when considering cross-isobath decorrelation scales.
The TCI mapping scheme adopts Δh as the key parameter. However, it is not the best parameter in terms of potential vorticity conservation because similar values of Δh could have very different effects depending on the local depth. For example, Δh = +50 m corresponds to a relative change Δh/h = 0.02 for a grid point located at 2500 m depth, whereas it corresponds to Δh/h = 0.5 for a grid point located at 100 m depth (i.e., a change in potential vorticity 25 times larger than the former case).
In principle, Eq. (2), with its h denominator term, is consistent with the principle of conservation of potential vorticity and our method bears similarity to that of Böehme et al. (2008). Their method used an actual potential vorticity term (=f/h) as a constraint on their 23°-wide meridional domain. The main difference with Böehme et al. (2008) is that our mapping methodology is a simple weighted mean rather than an objective mapping (Bretherton et al. 1976); hence, we do not use the covariance matrix in our scheme. This choice is justified by Shimada et al.’s (2017) finding that in heterogeneously sampled regions, objective analysis tends to yield a noisier field than the simple weighted average scheme they implemented. Schmidtko et al. (2013) carried out a global mapping and thus accounted for both depth and latitudinal variations. Their path-following approach was based on the ratio of the depth (and latitude) of the mapped grid point over that of the surrounding grid points, and thus, it is not directly analogous to the conservation of potential vorticity.
Our SSCI scheme basically follows the TCI scheme but uses a constraint that is more dynamically consistent with the potential vorticity conservation. For this reason, the SSCI scheme yields better mapped water properties than the TCI in topographic environments typical of subpolar and polar marginal seas—specifically, a broad shelf connected to the deep basin via a sharp continental slope. This improvement will be demonstrated in section 4 through a case study of the Sea of Okhotsk.
b. Determination of the decorrelation scales
The procedure to determine the decorrelation scales is based on the estimation of the 2D correlation of temperature as a function of along-slope distance and cross-slope depth difference following Shimada et al. (2017). This procedure involves the following steps: 1) determining the isobath direction, 2) building correlograms from temperature data at selected grid points, 3) fitting an autocorrelation function to the correlogram to estimate the decorrelation scales at every grid point, and 4) vertical and horizontal averaging of the decorrelation scales.
We incorporate several modifications into this methodology. Estimating the decorrelation scales is a nontrivial task that depends on several variables, as described in the following text. The value of each of these variables needs to be adjusted depending on the topographic conditions and spatial variability of the water properties, and we provide guidance on such adjustments. The section below thus provides a thorough, step-by-step description of the methodology to determine decorrelation scales, which should be valid for most conditions. All MATLAB routines necessary to perform the following procedure are available on a repository (see link in the acknowledgments section).
1) Isobath direction
At each sigma level, the temperature data and the grid are originally defined in an (x, y) orthogonal coordinate system. However, the coordinates of the temperature data used to estimate the decorrelation scales Lr and LΦ of Eq. (2) need to be translated into an along–across-isobath coordinate system (r, Φ), which is also orthogonal. Thus, the first step of the procedure consists of determining the depth h and isobath direction at each grid point from the GEBCO bathymetric dataset.
2) Grid definition for building a correlogram
The next step consists of classifying temperature anomaly data in the previously defined (r, Φ) coordinate system. The data are averaged into boxes of length δr (Fig. 2d) and width δΦ (or δh, Fig. 2e), from which a correlogram can be estimated (Fig. 3). A correlogram is a graphic quantifying the correlation of temperatures in a two-dimensional grid. The two dimensions here are the along-isobath distance r (x axis), and either the stretching parameter Φ (SSCI, y axis, Figs. 3a–c) or the depth difference Δh (TCI).
Building a correlogram from raw temperature data, part I. (a) Raw temperature data within a distance dmax + 10% (=247 km) around the grid point indicated by the black triangle. (b) Along-isobath distance r and (c) depth difference Δh (in m) and shrinking/stretching parameter (unitless) of the observation points referenced to the grid point. (d) The close-up view around the grid point, located midslope at 1700 m depth. The red box has a length δr of 75 km and a width of 10 km. The black arrow indicates the isobath direction at this grid point. (e) The bathymetric section in the cross-isobath direction including the grid point of (d). At middepth, with a slope α of 310 m (10 km)−1, Eq. (4a) yields n = 11, and Eq. (4b) yields δh = 310 m, which corresponds to a horizontal distance of 10 km. Each gray line indicates the limit of the n boxes.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
Building a correlogram from raw temperature data, part II. (a) Temperature classified as a function of r (x axis) and Δh or Φ (y axis). The grid domain and boxes’ size are determined from the steps described in Fig. 2, and boxes with no data are greyed. The parameters dmax = 225 km and Δhmax = 1700 m (or Φmax = 1) are the maximum values along the x and y axes, respectively, and each box has a length δr = 75 km and a width δh = 310 m (or δΦ = 0.18). (b) Correlogram of temperature estimated from the data in (a), and (c) autocorrelation function (ACF) fitted onto the correlogram. The correlation length scale Lr (=75 km) and LΦ (=0.72) or LΔh (=1240 m) were determined by fitting the e-folding ellipse onto the ACF in (c). The ellipse is also shown in (b).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
First, observations located within a maximum distance dmax and a maximum depth difference Δhmax (or stretching parameter Φmax) of the grid point are selected, and then classified into m × n boxes of length δr and width δΦ (or δh). The determination of the four parameters (dmax, Φmax, δr, δΦ) or (dmax, Δhmax, δr, δh) is crucial to properly estimate the decorrelation scales. Here we provide guidance to determine the value of each of these terms. To obtain the decorrelation scales, we strongly suggest building the correlogram from temperature data since these data are the most widely available and their accuracy is higher than that of salinity or other seawater properties.
The most straightforward term to be determined is Φmax or Δhmax, which is simply the maximum (in absolute value) stretching or depth difference between the grid point and the observations. In the case shown in Fig. 2, the grid point is located at a depth of 1700 m and the maximum bottom depth of all the observation points is 3400 m. Thus, Δhmax = 1700 m and Φmax = 1.00.
The distance or radius dmax and the horizontal box length δr are determined empirically via a sensitivity test because the decorrelation scales are sensitive to these two parameters. Details of the sensitivity test procedure and its results are provided in the appendix. From the sensitivity test results, values dmax = 225 km and δr = 75 km were chosen.
The value of all parameters necessary to determine the correlation scales in the southern Sea of Okhotsk (and the Southern Ocean) are listed in Table 1.
List, definition, and values of the parameters necessary to determine the decorrelation scales for the SSCI and TCI mapping schemes.
3) Building of correlogram
Following Shimada et al. (2017), two quality indicators are estimated for the correlograms of each of these grid points:
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The signal-over-noise ratio (SN ratio):where 1 is the correlation coefficient at 0 lag in the correlogram (the center of the correlogram in Fig. 3b) and μ0 is the value that the correlation coefficient at 0 lag would take if it was obtained by interpolation of the correlation coefficients from the surrounding boxes. An SN ratio value of at least 1 is desirable.
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The variability correlation coefficient: The 0-lag cross-correlation coefficient between the correlogram (Fig. 3b) and the ACF [Fig. 3c, Eq. (5)]. The variability correlation coefficient indicates how well the model defined by the ACF fits the actual correlogram, and a value of 0.8 is considered as the good-quality threshold.
4) Estimation of the decorrelation scales
Maps of vertically averaged values and vertical profiles of horizontally averaged values for Lr, LΦ, and LΔh are displayed in Figs. 4a–c and Figs. 4d–g, respectively. At each vertical level, outliers for each variable were previously filtered out by removing the values exceeding the level’s average ±3 standard deviations. The maps reveal large length scales (Lr, Fig. 4a) and small scales of LΦ (Fig. 4b) and LΔh (Fig. 4c) along the Sakhalin slope; that is, temperatures are well correlated along the isobaths and change abruptly across the isobath. Off the Hokkaido shelf, Lr is moderate whereas LΦ and LΔh are large, indicating that the depth constraint is relatively weak there. Off the Kuril shelf, both the Lr and LΦ (LΔh) are small, indicating poor along- and across-slope temperature correlations.
Maps of vertically averaged decorrelation scales (a) Lr, (b) LΦ, and (c) LΔh. The black dashed lines represent the 100 and 200 m isobaths, and the solid black lines the 500, 1000, 2000, and 3000 m isobaths. (d) Vertical profiles of horizontally averaged Lr, (e) LΦ (red line and dots) and LΔh (black line and dots), (f) variability correlation coefficient, and (g) SN ratio. The error bars represent ±1 standard deviation. In (d), the gray bars represent the number of estimates at each depth level. The thicker lines in (d)–(g) highlight the vertical levels chosen to estimate the average value of decorrelation scales [marked by triangles in (d) and (e)] used to map the data. The red and black lines represent the values obtained for the SSCI and the TCI methods, respectively. If values are identical, only the red line is shown, as in (d) and (f).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
In the deep Kuril Basin, LΔh is heterogeneous, varying between 500 and 2500 m, whereas, in shallower regions, such as east of Sakhalin Island, LΔh barely exceeds 200 m (Fig. 4c). However, the SSCI yields low values below 0.75 for LΦ in most of the domain, especially in the regions where the bottom depth exceeds 500 m (Fig. 4b). As a result, the LΦ field is more homogeneous than the LΔh field and has a higher SN ratio (Fig. 4g), which exemplifies the fundamental difference between the TCI and SSCI methods.
The difference in Lr, LΦ, and LΔh values between vertical levels is small for all parameters (Figs. 4d,e), although the standard deviations are high due to the spatial variability of the decorrelation scales seen in Figs. 4a–c. The final value adopted to map the data is the average of (horizontally averaged) Lr and LΔh/LΦ for all levels where the variability correlation coefficient exceeds 0.8 (Fig. 4f) and the SN ratio exceeds 1 (Fig. 4g). These conditions are fulfilled in the 26.5–26.9σθ range, yielding average values Lr = 97 km, LΔh = 932 m, and LΦ = 0.56. We used a spatially averaged value for the decorrelation scales for the whole domain because Lr, LΦ, and LΔh can only be obtained in limited areas (Figs. 4a–c) and cannot be spatially interpolated and extrapolated to the whole domain. In addition, the decorrelation scale values depend on local data variability and density, and a spatial average prevents resorting to subjective choices.
4. Application
a. Application to the Sea of Okhotsk
In the southern Sea of Okhotsk, two broad and shallow shelves exist off the coast of the Hokkaido and Sakhalin Islands. The Hokkaido shelf extends northeastward from the coast, and the distance between the coastline and the 200 m isobath is mostly greater than 100 km, except on the eastern side of Hokkaido Island (Figs. 1a,b). The Sakhalin shelf extends eastward from the coastline, with a width varying between 50 km (north of 49°N) and 200 km in the Terpenia Bay (between 47° and 49°N). It is narrowest (about 20 km in width) at the southern tip of Sakhalin Island (Fig. 1a). The south of the Sakhalin shelf and the Hokkaido shelf are connected to the Kuril Basin (depth > 3000 m) by a sharp continental slope (Figs. 1a,b).
The East Sakhalin Current flows along and over the Sakhalin shelf, transporting cold and fresh waters, with modified characteristics of dense shelf water (DSW; Itoh et al. 2003; Mensah et al. 2019). The Soya Warm Current flows over the Hokkaido shelf, transporting warm and saline Soya Warm Current Water (SCW; Itoh et al. 2003; Ohshima et al. 2017). The water mass prevalent in the Kuril Basin is the Okhotsk Sea Intermediate Water, which is colder and fresher than SCW and warmer and saltier than DSW. Thus, within a small area, three water masses with different characteristics exist, with the largest changes of characteristics occurring across the Hokkaido shelf, where the warmest and saltiest SCW and the freshest and coldest DSW-influenced waters are separated by a sharp gradient (Fig. 1b). This area represents a particularly challenging environment to map.
The temperature and salinity data in the Sea of Okhotsk were mapped bimonthly with three different methods: 1) a classic isodirectional weighted mean [using only the distance constraint in Eq. (1)] with Lr = 97 km; 2) the TCI method with Lr = 97 km, LΔh = 932 m; and 3) the SSCI method with Lr = 97 km, LΦ = 0.56. The different mappings were carried out using their respective weighting function, with observations being selected within a search radius equaling 1.5 times Lr around each grid point. The search radius was extended to 1.75 times and 2 times Lr if less than five observations were present within the previous radius. If less than five points existed within 2 times the Lr search radius around the grid point, no value was assigned to that grid point. The distribution of raw data for each bimonthly period at 26.7σθ is shown in online supplementary Fig. S1.
1) Horizontal distribution of properties and error
Mapped May–June temperature data in the southern Sea of Okhotsk at 26.7σθ obtained with (a) isodirectional weighted mean, (b) TCI mapping scheme, and (c) SSCI mapping scheme. (d)–(f) The mapping error between observed data and mapped data with each of the respective methodologies. (g) Difference between the error associated with the SSCI mapping and the isodirectional mapping scheme. (h) Difference between the SSCI error and the TCI error. The blue (red) color indicates that the SSCI error is smaller (larger). (i) Standard deviation of the May–June temperature data, estimated within 33 km and 0.19Φ of each grid point. The numbers in (d)–(f) and (i) respectively represent the horizontal average of the mapping error and standard deviation for the whole mapping domain (see Fig. 1b).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
Bimonthly horizontally averaged mapping error (in °C) at 26.7σθ (sigma grid) and 50 m depth (depth grid) for each of the three mapping methodologies evaluated in this study and standard deviation (Std) of the temperature field at these levels. SSCI values are in boldface.
Bimonthly horizontally averaged mapping error at 26.7σθ (sigma grid) and 50 m depth (depth grid) for salinity and standard deviation (Std) of the salinity field at these levels. SSCI values are in boldface.
The largest difference between the SSCI (Fig. 5c) and the isodirectional mapping (Fig. 5a) exists off the coast of Hokkaido, where the Soya Warm Current flows (Fig. 1a). There, the isodirectional mapping yielded a gradual temperature decrease toward the offshore, from 2°C within the 50 m isobath to about 1°C at the 500 m isobath. In contrast, the SSCI scheme yielded a clear front between the SCW, whose temperature exceeds 4°C (Fig. 5c), and the DSW-influenced waters with temperatures less than 1°C. The considerably lower mapping error yielded by the SSCI scheme (Fig. 5f) suggests that the frontal structure produced by SSCI is a more accurate representation of the actual temperature field. The difference between the two mapping errors (Fig. 5g) highlights the better quality of the SSCI mapping off the coast of Hokkaido, as well as in most of the other coastal (e.g., off Sakhalin and the Kuril Islands) and slope regions. However, the isodirectional mapping yields a smaller error in the south of Sakhalin Island. This is due to the considerably fewer data available in this area versus the immediate surrounding (supplementary Fig. S1c). In such cases, a smoother mapped field may be advantageous. Last, the mapping errors are comparable between the 100 and 200 m isobaths near Hokkaido (Fig. 5g), where the temporal variability is very high as indicated by the large standard deviation in this region (Fig. 5i).
Overall, the SSCI error for May–June was about 40% smaller than that of the isodirectional method for temperature (Table 2) and 33% for salinity (Table 3), which justifies the use of our scheme. The SSCI mapping consistently yielded smaller errors than both the TCI and the isodirectional mapping for most periods, with a relatively smaller error reduction in September–October and November–December on the sigma grid (Tables 2 and 3), which may be due to the more homogeneous temperature and salinity fields and the less sharp fronts existing in these periods (Figs. 5e,f).
The differences and similarities between the TCI and the SSCI methods highlight the advantage of the SSCI scheme. Off the Hokkaido coast, the TCI mapping resulted in the oversmoothing of the temperature field (Fig. 5b), similar to the isodirectional mapping. Indeed, on this shallow and broad shelf, the high value of LΔh (932 m), Eq. (1) yielded large weights for observations located deep into the continental slope, hence the strong smoothing. In contrast, the value of LΦ (0.56) corresponds to a depth difference of only 112 m for a grid point located at 200 m depth. This allowed Eq. (2) to yield a sharp front by giving large weights only for observations at shallow depths, on the shelf.
Off the Kuril Islands, the SSCI and TCI mappings exhibited similar temperature values (Figs. 5b,c). In this area, no shelf exists, and the bottom topography is characterized by a sharp slope from the surface to about 3000 m depth (Fig. 1d). For a grid point located at 1500 m, LΦ = 0.56 corresponds to a depth difference of 840 m, very close to the value of LΔh. This explains the closeness between the results of the two methods at this place.
2) Vertical distribution of error
Figure 6 presents vertical profiles of horizontally and temporally averaged mapping error and error difference between the different mapping methods for both the sigma grid (Figs. 6a,b) and z grid (Figs. 6c,d). On the sigma grid, SSCI was superior to the other two methods at all levels. For the z grid (Figs. 6c,d), SSCI performed better than the isodirectional mapping from the surface down to 250 m and the TCI mapping down to 125 m depth. At best, the error at 30 m was reduced by nearly 40% versus the isodirectional mapping, and 30% versus the TCI mapping (Figs. 6c,d). The better quality of the SSCI mapping at shallow depths versus offshore is also exemplified in the cross section of z-gridded θ (Figs. 7a–c), mapping error (Figs. 7d–f), and error difference (Figs. 7g–h).
Vertical profiles of horizontally averaged error associated with isodirectional, TCI, and SSCI mapping schemes for (a) sigma-grid and (c) z-grid mapping. (b),(d) Error difference profiles (in the sigma grid and z grid, respectively) between SSCI and isodirectional mapping (dashed gray curve) and between SSCI and TCI mapping (solid black curve). A negative difference indicates a smaller SSCI error.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
Cross section off the coast of Hokkaido (see Fig. 1a for the location of the section) of May–June (a)–(c) potential temperature mapped using isodirectional, TCI, and SSCI mapping schemes, respectively. (d)–(f) RMS error associated with each of the respective mappings. Difference between the RMS error associated with the SSCI scheme and that associated with (g) the isodirectional mapping scheme and (h) the TCI mapping. The blue (red) color indicates a smaller (larger) SSCI error. (i) Standard deviation of the potential temperature. Note the difference in vertical scale for the data above and below 100 m depth.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
Similar to the horizontal distribution of sigma-gridded θ (Fig. 5), the SSCI mapping yielded a much sharper front (Fig. 7c) and smaller mapping error close to the coast and over the shelf break (Fig. 7f) than the other two mapping methodologies (Figs. 7a,b,d,e,g,h). Below the upper ∼250 m, the three mapping methodologies were nearly equivalent (Figs. 7c,d) due to the small spatial variability of the temperature field at these depths (Figs. 7a–c,i). Therefore, we conclude that the SSCI mapping exhibited the highest improvement in the most crucial shelf regions (Figs. 5–7), whereas it was nearly equivalent to the other mapping methodologies in areas with little spatial variability. This conclusion is also valid for salinity (supplementary Figs. S2–S4).
b. Comparison of the TCI and SSCI mapping schemes in the Southern Ocean
The typical topographic cross section of the Southern Ocean (Figs. 8c,d) differs greatly from that of the southern Sea of Okhotsk (Figs. 1c,d). In the previous section, we demonstrated that SSCI performed better than TCI both qualitatively (Figs. 5 and 7) and quantitatively (Fig. 6, Tables 2 and 3) in a topographic environment characterized by a broad and shallow shelf connected to the oceanic basin via a sharp continental slope. In this section, we compare the performance of these two methodologies in the deeper and gently sloped topography typical of the Southern Ocean.
(a) Bathymetric map and (b) distribution of raw temperature data at 200 m depth between November and April in the Southern Ocean. Data were mapped within the area inside the purple box. Cases of a relatively (c) gentler and (d) sharper slope bathymetric profiles, along the two solid yellow lines in (a).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
For the sake of consistency with the Shimada et al. (2017) study, we used a similar latitudinal extent, horizontal grid (1° longitude × 0.5°latitude) and vertical z grid (see section 2) as in that study, and compared summer climatologies (January to March). Decorrelation scales were estimated at all levels between 200 and 2000 m using all data available between November and May (Fig. 8b).
The raw dataset from which the decorrelation scales were evaluated and the mapping carried out has been augmented with the biologging (instrumented seals) CTD data from the MEOP database (Treasure et al. 2017) and Argo floats. With these additional data, the temperature data include 204 406 measurements at 200 m (Fig. 8b), for a density of 0.5 profiles per 10 km × 10 km at this depth. The values of the correlogram parameters (dmax, Φmax, δr, δΦ, δh, n) for the Southern Ocean are listed in Table 1. Following the procedures described in section 3b and the appendix, the decorrelation scales are Lr = 485 km, LΔh = 635 m, and LΦ = 0.27. By comparison, Shimada et al. (2017) found Lr = 346 km and LΔh = 584 m, which is mainly due to the shorter dmax (500 km versus 660 km in our case) and δr (100 km versus 220 km in our case) used in their study (see details in the appendix).
Similar to the Sea of Okhotsk, the error between the mapped data and the raw data was calculated for both the TCI and SSCI mappings. Figure 9 presents the vertical profiles of horizontally averaged mapping error estimated within a distance of 161 km and a stretching parameter of 0.09 (one-third of Lr and LΦ) around each grid point (Fig. 9a), and the error difference between the two methods (Fig. 9b). The results of Fig. 9 demonstrate that the two methods are, overall, quantitatively equivalent, with a depth-average error difference of 0.002°C. TCI, however, performed better than SSCI between 75 and 500 m, with a maximum error difference of +0.009°C (9% smaller than the SSCI error) at 200 m.
(a) Vertical profiles of horizontally averaged RMS error associated with TCI and SSCI mapping schemes on the z-grid mapping. (b) Error difference between SSCI and TCI mapping. A positive difference indicates a larger SSCI error.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
These differences are considered negligible, which is supported by the similarity in the horizontal distribution of θ and mapping error for both methods (Figs. 10 and 11). Even at 150 m, where the error difference between SSCI and TCI is maximum (Fig. 9b), the distributions of potential temperature obtained with the two methods display similar properties: cold waters (below 0°C) on the shelf and continental slope, warmer Circumpolar Deep Water farther offshore, and a front separating these two water masses (the Antarctic Slope Front) at the midslope (Figs. 10a,c). In this frontal region, the spatial variability is large, as indicated by the substantial (∼1.25°C) standard deviation of the raw data (Fig. 10f). Such variability of the Antarctic Slope Front may be due to mesoscale eddies generated by baroclinic instabilities (Stewart and Thompson 2016). Nevertheless, the error associated with both mappings is very low near the front (Figs. 10b,d). At 900 m depth (Fig. 11), the temperature field has a smaller spatial variability, and the two mappings also yield very similar results (Figs. 11a,c) and levels of error (Fig. 9b). Thus, in contrast to the southern Sea of Okhotsk, both mappings retain the essential characteristics of the water masses distribution and temperature gradients.
Mapped summer temperature data in the Southern Ocean at 150 m estimated with (a) TCI mapping scheme and (c) SSCI mapping scheme. (b),(d) The mapping error between observed data and mapped data with each of the respective methodologies. (e) Difference between the error associated with the SSCI mapping and the TCI mapping. The blue (red) color indicates that the SSCI error is smaller (larger). (f) The standard deviation of the summer temperature data, estimated within a distance of 161 km and a stretching parameter of 0.09 of each grid point. The number indicated in (b) and (d) represents the horizontal average of the mapping error for the whole domain.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
As in Fig. 10, but for the 900-m-depth level. Note that the color scale in (e) is similar to Shimada et al.’s (2017) Fig. 6b for comparison purposes.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
The insignificance of the discrepancies between TCI and SSCI in the Southern Ocean is also evident compared to the errors of MIMOC and the World Ocean Circulation Experiment (WOCE) relative to TCI. For example, between 200 and 1900 m, the MIMOC error was larger than that of TCI by 0.05°C (Fig. 11c of Shimada et al. 2017), whereas the error difference between TCI and SSCI is only 0.001°C on average in this depth range, with the TCI having the smaller error (Fig. 10b). The distribution of error difference between TCI and SSCI (Figs. 10e and 11e) also displays negligible values compared with the error difference between TCI, MIMOC, and WOCE (Fig. 6 of Shimada et al. 2017). Therefore, and even though regional differences exist (Figs. 10e and 11e), we conclude that SSCI is equivalent to TCI in the Southern Ocean. As some water masses have seldom been sampled in the Southern Ocean until now (e.g., ice shelf water or dense shelf water), SSCI might yield better results when more of these water masses, and the fronts associated with them, will be observed.
The similarity between the two mappings is due to the combination of a gentle and continuous slope, larger depths, and the limited amount of broad shelf regions in the Southern Ocean. First, the gentler slope induces longer cross-slope correlation scales and, hence, a smoother temperature field. At 2000 m depth, with an average slope of 80 m (10 km)–1 and LΔh = 635 m, the e-folding value of the topographic term
We also considered the possibility that the similar performances of SSCI and TCI in the Southern Ocean were due to the low data density in this ocean. The error was reevaluated for both mapping schemes, separately for regions with low (1 profile or less per 10 km2) and high (2 profiles or more per 10 km2) data density. Results show no significant differences between the high- and low-data density regions (supplementary Fig. S5). This is consistent with the fact that most grid points in the Southern Ocean have a data density greater than one observation per one decorrelation scale, a threshold above which data density is not expected to affect the mapping error (appendix D of Shimada et al. 2017). For the Sea of Okhotsk, climatologies were generated from degraded datasets, with, respectively, only 1/5 and 1/10 of the original data randomly selected before mapping. In this case, while the mapping error for both schemes increases by an average of about 30% and 75% with the 1/5- and 1/10-degraded datasets, respectively, the SSCI still performs consistently better than the TCI mapping (supplementary Fig. S6). For these reasons, it is more likely that the Southern Ocean’s topographic setup is the cause for the similar performances of SSCI and TCI in this region.
5. Conclusions
This study introduced a mapping methodology based on the potential vorticity principle to map physical variables in polar and subpolar regions with greater precision. We introduced a water column shrinking/stretching constraint (SSC) to account for the fact that the flow of weakly stratified waters in high-latitude regions tends to follow the bathymetric contours. Together with a distance constraint, this SSC term was used in the weighted mean mapping scheme defined in Eq. (2). Based on the TCI scheme proposed by Shimada et al. (2017), our SSCI method was devised as an accurate, physically sound, computationally efficient, tractable mapping scheme. In addition, the SSCI mapping scheme follows the constraint of potential vorticity conservation more precisely than the TCI [see Eq. (2) versus Eq. (1)]. The determination of decorrelation scales is a nontrivial task, as pointed out by Heuzé et al. (2015), which may be discouraging to users wishing to implement their custom-made mapping. We attempted to address this difficulty by detailing a step-by-step standard procedure for the determination of decorrelation scales (section 3b) and providing all the codes necessary to perform this task (see link in the acknowledgments section).
In this study, we demonstrated in principle (section 3a) and practice (section 4a) that SSCI outperformed TCI in regions with broad and shallow shelves connected to the deep basin via a steep slope (i.e., a topographic environment typical of marginal seas). The SSCI mapping scheme was particularly effective in areas where temperature (or any other physical variable) fronts exist, as exemplified in the case study of the Sea of Okhotsk, where the Kuril Basin is essentially occupied by Okhotsk Sea Intermediate Water, whereas the Sakhalin and Hokkaido shelves are mainly occupied by colder and fresher DSW, and warmer and saltier SCW, respectively. Such a physical setting is not particular to the Sea of Okhotsk, as water masses on the continental shelves of most polar and subpolar regions differ greatly from those off the continental slope and oceanic basin. Similar physical configurations and diversity of water masses exist in, for example, the Baffin Bay, the Labrador Sea, or the Beaufort Sea. Therefore, we conclude that SSCI should be a good mapping solution to apply in other subpolar and polar marginal seas. In addition, we also evaluated SSCI against TCI in the Southern Ocean where the bottom topography is generally slanted even in shallow regions. In this case, we found that the two mapping methodologies were overall equivalent. We conclude that SSCI and TCI are equivalent in regions with gently sloped and deeper topography, whereas SSCI is more accurate than TCI in regions of broad shelves connected to the oceanic basin via a sharp continental slope.
The analysis in this paper did not address issues related to cabbeling (i.e., mapping of neighboring data located along the same isopycnal but with different temperatures and salinities, which may result in the mapped density being artificially higher) or density inversions. Users wishing to map temperature, salinity, and density over several vertical levels can still apply a cabbeling-correcting method such as that in Schmidtko et al. (2013) and implement a correction for the density inversion. We omitted these points because this paper is intended to present a methodology that is not restricted to the mapping of temperature, salinity, and density. Mapping of various chemical properties or surface-only properties, such as sea surface height (Mizobata et al. 2020), is also among the applications of SSCI mapping.
The SSCI mapping scheme presented in this paper is one of a few alternatives that can be adopted to map ocean properties in polar and subpolar regions. We believe it is an effective and efficient method and hope that the decorrelation scales estimation—a crucial but nontrivial step of geographical mapping—has been clarified by our proposed standardization.
Acknowledgments.
This work is supported by Grants-in-Aid for Scientific Research 22K1409402, 17H01157, and 20H05707 from the Ministry of Education, Science, Sports, and Culture of Japan. We thank Y. Mitani (Kyoto University) for providing us with the biologging data from seven instrumented seals in the Sea of Okhotsk. We extend our thanks to the two anonymous reviewers whose constructive criticism greatly contributed to the improvement of this paper. Data analyses were conducted using the Pan-Okhotsk Information System of Hokkaido University and all manuscript figures were drawn using MATLAB version R22b (MathWorks, Inc., https://uk.mathworks.com/products/matlab.html). The color scale in Figs. 7g and 7h was generated using Mirko Hrovat’s (2022) RedBlue Colormap Generator with Zero as White or Black (https://www.mathworks.com/matlabcentral/fileexchange/74791-redblue-colormap-generator-with-zero-as-white-or-black), MATLAB Central File Exchange.
Data availability statement.
The MATLAB codes used to estimate the decorrelation scales (section 3b and appendix) are archived in Zenodo, a public, community-supported repository, at https://doi.org/10.5281/zenodo.8170649. These codes are publicly available, with no restrictions. The General Bathymetric Chart of the Oceans (GEBCO) 2022 15″-Arc product is available from the following link: https://www.gebco.net/data_and_products/gridded_bathymetry_data/. The marine mammal data were collected and made freely available by the International MEOP Consortium and the national programs that contribute to it (http://www.meop.net). Argo float data were collected and made freely available by the International Argo Program and the national programs that contribute to it (https://argo.ucsd.edu, https://www.ocean-ops.org). The Argo Program is part of the Global Ocean Observing System. Argo float data were downloaded together with other historical CTD data as part of the World Ocean Database 2018, a National Centers for Environmental Information standard product. The profiling float data in the Sea of Okhotsk from the joint University of Washington–Hokkaido University project will be released in 2024.
APPENDIX
Determination of dmax and δR by a Sensitivity Test
Both small-scale phenomena (e.g., water masses front, mesoscale eddies) and large-scale processes (e.g., gyre circulation) occur in the oceanic areas to be mapped by oceanographers, which may affect the estimate of the decorrelation scales. Specifically, when building a correlogram, large values for dmax (the radius within which data are selected), and δr (the correlogram’s horizontal box length) are likely to yield longer decorrelation scales since they highlight larger-scale phenomena. The opposite is true for small values of dmax and δr. This raises a problem because good mapping requires the selection of data within a large area to ensure good summarization of the data field, while retaining small-scale details, e.g., a water mass front. It is thus necessary to conduct a sensitivity study to find which range of dmax and δr produce satisfactory results, and which specific value of Lr and LΦ is the most adapted to the user’s purpose of mapping. Here, we conduct a sensitivity study of Lr and LΦ on dmax and δr and will evaluate the quality of mapping based on four parameters: the SN ratio, the variability correlation coefficient [section 3b(3)], the number of grid points evaluated, and the reduction in error versus the isodirectional mapping.
Figure A1 presents the results of the sensitivity test for each of the aforementioned variables, and displays a few remarkable features:
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The variability coefficient (Fig. A1a) is greater than the 0.8 quality threshold for most of the dmax and δr range, except for low values of δr (<∼30 km) or dmax (<125 km).
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The SN ratio (Fig. A1b) increases with dmax and generally exceeds the 1 quality threshold for dmax values equal to or greater than 200 km.
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The number of grid points evaluated (Fig. A1c) is largest for high δr (>60 km) and relatively low dmax (<300 km). Considering that a higher number of grid points evaluated should yield a more reliable estimate of decorrelation scales, we arbitrarily set a quality threshold of 0.33 NTotal, where NTotal is the number of grid points satisfying the 250 m (100 km)−1 slope condition. The threshold of 0.33 is highly subjective and can be adjusted depending on the number of grid points evaluated in a given environment.
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Lr increases (Fig. A1d) with both dmax and δr, while LΦ (Fig. A1e) increases only with dmax.
Results of the sensitivity test: (a) Variability correlation coefficient, (b) SN ratio, (c) number of grid points evaluated, (d) Lr, (e) LΦ, and (f) percentage of error decrease relative to the case of isodirectional mapping, as functions of dmax and δr. The white contour in (d)–(f) represents the range of dmax and δr fulfilling the three quality conditions: variability correlation coefficient > 0.8, SN ratio > 1, N > 0.33 NTotal, with NTotal being the total number of grid points where the slope is greater than 250 m (100 km)−1, and N is the number of grid points where correlograms could be evaluated from the raw data. The black circle locates the pair of dmax and δr that we selected for this study.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
We define the range of dmax and δr where the three quality conditions are satisfied as the optimal range (white contour in Figs. A1d–f) within which the Lr–LΦ pair should be chosen. Within this range, the error associated with the SSCI mapping is smaller by 20%–25% relative to the isodirectional mapping (Fig. A1f). We ultimately chose the pair dmax = 225 km and δr = 75 km for the following reasons: 1) both the variability coefficient, SN ratio, number of grid points, and error decrease are relatively large compared to neighboring points, and 2) the corresponding values of Lr (104 km) and LΦ (0.5) at this level are relatively low, which should allow us to reach a compromise between obtaining sufficient smoothing to produce comprehensive climatologies, and keeping a large amount of details in smaller-scale features. With the values of dmax and δr now defined, the decorrelation scales can be estimated at every density levels and their final values adopted following section 3b(4).
A similar sensitivity test was carried out for the Southern Ocean at the 1000-m-depth level and its results are shown in Fig. A2. Following the same principle as in the Sea of Okhotsk, we chose dmax = 660 km and δr = 220 km. The decorrelation scales are then estimated at all depth levels, and the vertically averaged value yields Lr = 485 km, LΦ = 0.27, and LΔh = 635 m.
As in Fig. A1, but for the Southern Ocean.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-22-0143.1
The chosen values of dmax and δr for the sensitivity experiment are mostly larger in the Southern Ocean than in the Sea of Okhotsk, owing to the larger distance scale and wider topographic features in the Southern Ocean. The most striking difference with the results in the Sea of Okhotsk consists in the higher values of SN ratio and variability correlation coefficient. This is likely due to the smaller temporal variability of the temperature field in the Southern Ocean (Fig. 10f versus Fig. 5i). Also, note that for equal values of dmax and δr, the resulting values of Lr and LΦ differ greatly between the Sea of Okhotsk and the Southern Ocean. For example, at dmax = 500 km and δr = 100 km, the decorrelation scales in the Sea of Okhotsk would be ∼120 km and ∼0.8 (Fig. A1), versus ∼310 km and 0.18 for the Southern Ocean (Fig. A2). This demonstrates that despite the sensitivity of the decorrelation scale estimation to dmax and δr, the intrinsic properties of the local topography play a major role in the magnitude of the decorrelation scales.
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