Numerical models of the atmosphere, oceans, and sea ice are divided into horizontal grid cells that can range in size from a few kilometers to hundreds of kilometers. In these models, many surface-level variables are assumed to be uniform over a grid cell. Using a year of in situ data from the experiment to study the Surface Heat Budget of the Arctic Ocean (SHEBA), the authors investigate the accuracy of this assumption of gridcell uniformity for the surface-level variables pressure, air temperature, wind speed, humidity, and incoming longwave radiation. The paper bases its analysis on three statistics: the monthly average and, for each season, the spatial correlation function and the spatial bias. For five SHEBA sites, which had a maximum separation of 12 km, the analysis supports the assumption of gridcell uniformity in pressure, air temperature, wind speed, and humidity in all seasons. In winter, when the incidence of fractional cloudiness is largest, the incoming longwave radiation may not be uniform over a grid cell. In other seasons, the bimodal distribution in cloud cover—either clear skies or total cloud cover—tends to homogenize the incoming radiation at scales of 12 km and less.
Global and regional climate models and weather prediction models are based on equations of motion in which the relevant variables are continuous in space and time. In the numerical representation of these models, however, space is divided into horizontal grid cells that range in size from a few kilometers to 100 km (e.g., Collins et al. 2006; Hunke et al. 2010; Bromwich et al. 2009; Tastula et al. 2012), and the assumption is that most surface-level variables have the same value over the whole grid cell. For example, the near-surface air temperature is taken to be the same value over an entire climate model grid cell that may be 100 km on a side. We have data from the experiment to study the Surface Heat Budget of the Arctic Ocean (SHEBA; Uttal et al. 2002) to assess how appropriate this assumption of gridcell uniformity is over Arctic sea ice.
During the yearlong SHEBA deployment (October 1997–October 1998), the Atmospheric Surface Flux Group (ASFG; Andreas et al. 1999; Persson et al. 2002) maintained five sites with separations from one another of up to 12 km. By comparing simultaneous data between pairs of sites, we can evaluate the spatial variability of the data and, thereby, see whether various near-surface atmospheric variables lose uniformity with increasing separation. For this analysis, we compare the monthly averages at each site and evaluate two spatial statistics for each season: the spatial correlation function and the bias between sites as a function of separation.
The variables that the five ASFG SHEBA sites measured include near-surface barometric pressure, wind speed and direction, air temperature, and relative humidity; the four broadband radiation components, incoming and outgoing longwave and shortwave radiation; surface temperature; and the turbulent surface fluxes of momentum and sensible heat. For each site, we also ran the SHEBA bulk flux algorithm that Andreas et al. (2010a,b) developed to compute the surface fluxes of momentum and sensible and latent heat. Here, we report spatial statistics for the so-called state variables: surface-level pressure, air temperature, wind speed, relative humidity, and incoming longwave radiation. We have reported our preliminary analysis of these and some of the other variables in conference papers (Andreas and Jordan 2013; Andreas 2015).
Figure 1 shows one of the hypotheses that we want to test with the SHEBA data. The spatial correlation function of surface-level variables is known to fall off from one with increasing distance between sites. This is the behavior of correlation functions computed from turbulence data that we are familiar with (e.g., Lumley and Panofsky 1964, 14–16; Kaimal and Finnigan 1994, 33–35; Andreas and Treviño 1997; Treviño and Andreas 2008). Previously, Wilks (2006, 58–59, 67–69) and Gunst (1995) showed examples of correlation functions for meteorological variables, and Worby et al. (2008) described a spatial correlation analysis of sea ice datasets.
As Fig. 1 depicts, if the computed spatial correlation function for variable V falls off approximately exponentially (Kaimal and Finnigan 1994, p. 35), we could characterize the spatial variability with the e-folding distance Δ such that the spatial correlation function for variable V at sites A and B, which have separation D, could be represented as
This e-folding distance is commonly taken as the separation beyond which variables are considered uncorrelated.
Thorndike’s (1982) analysis of surface-level pressure, one of the variables in our dataset, sets the stage for our correlation analysis. He computed the spatial correlation function for two years of surface-level pressures measured on buoys drifting within the Arctic sea ice. His plots suggest that the e-folding distance for his pressure data is roughly 1500 km; but the buoys, evidently, had minimum separations of several hundred kilometers and maximum separations of about 2000 km. Consequently, Thorndike evaluated disturbances only for meso-α scales (e.g., Orlanski 1975). Because our SHEBA sites, on the other hand, had separations from a couple hundred meters to 12 km, our analysis is for submeso and meso-γ scales and is, therefore, relevant to grid spacings in current mesoscale models.
Our second hypothesis is that surface-level variables may differ between sea ice sites because of differences in sea ice thickness and snow depth at the sites. Ice thickness and snow depth have such a profound effect on surface temperature that variables that respond strongly to surface temperature—like emitted longwave radiation and the turbulent surface heat fluxes—may have natural variability that is not necessarily a function of site separation. The state variables, which adjust in response to the surface fluxes, may thus also differ between sites simply because of differences in ice thickness and snow depth.
Not only will our analysis have implications for three-dimensional modeling with climate models and weather forecast models, it is germane to one-dimensional modeling (Brun et al. 2008)—for example, with snow models such as PIEKTUK-D (a one-dimensional model named after the Inuktituk word for blowing snow; Chung et al. 2011), with the thermodynamic code in many sea ice models (Bitz and Lipscomb 1999; Hunke et al. 2010, 2013), and with single-column models (e.g., Lipscomb 2001; Holland 2003; Morrison et al. 2005). That is, after our analysis, we will be able to speculate on the area over which a one-dimensional model’s results are valid. Likewise, our analysis will provide guidance on how far data measured on buoys drifting in sea ice can be extrapolated. Conversely, this spatial analysis can be used to decide how closely drifting buoys or other such observational platforms must be placed to provide data that cover specific regions or the entire Arctic.
Here, we divide the SHEBA data series into the four typical Arctic seasons (e.g., Walsh et al. 1985; Lindsay 1998)—autumn (September, October, and November), winter (December, January, and February), spring (March, April, and May), and summer (June, July, and August)—and calculate spatial statistics in each season. We find that near-surface variables like pressure, air temperature, and wind speed are well correlated in all seasons up to the distance limit in our data, 12 km. The correlation is so good, in fact, that we cannot evaluate the e-folding distance Δ. We can say only that Δ is much larger than 12 km. Consequently, the assumption of gridcell uniformity is appropriate for these variables—at least for grid cells of 12 km or less. And such scales are pertinent because the mesoscale Weather Research and Forecasting (WRF) Model is already being used with grid cells this small (e.g., Tastula et al. 2012). Relative humidity, which we reported on elsewhere (Andreas et al. 2002; Andreas and Jordan 2013), is also spatially homogeneous over grid cells up to 12 km.
The measured incoming longwave radiation also appears to be spatially homogeneous in autumn, spring, and summer, when totally overcast skies dominate the Arctic Ocean. In winter, on the other hand, our data hint at the possibility of spatial variability in the incoming longwave radiation because winter cloud fraction is more variable than in the other seasons.
2. SHEBA data
The SHEBA ASFG dataset that we consider here comprises observations at five levels on a 20-m tower in the main SHEBA camp (Persson et al. 2002; Grachev et al. 2005). Each tower level had a three-axis sonic anemometer/thermometer from Applied Technologies, Inc. (K-type sonic), that yielded the mean wind speed and direction and the turbulent fluxes of momentum and sensible heat by eddy covariance. Each level also had instruments from Vaisala (model HMP235) that measured mean air temperature and relative humidity.
The lower four levels on the tower were, nominally, at 2.2, 3.2, 5.1, and 8.9 m; the upper level changed from 13.8 m in winter to 18.2 m in summer (Persson et al. 2002; Brunke et al. 2006). For the temperature, humidity, and wind speed measurements analyzed here, we used data from the lowest level that reported good data. This was generally the 2.2-m level.
Near this main tower were paired up-looking and down-looking Eppley broadband shortwave and longwave radiometers. These were equipped with blowers to mitigate frost formation on the radiometer domes. We calculated surface temperature with the data from the up-looking and down-looking longwave radiometers (Andreas et al. 2010a,b).
Besides this main site, the Atmospheric Surface Flux Group maintained four remote sites that were 0.25–12 km from the tower and were off the power grid for the main SHEBA ice camp. These sites were instrumented with Flux-PAM stations from the instrument pool at the National Center for Atmospheric Research (Militzer et al. 1995; Horst et al. 1997). Along with the main tower, three Portable Automated Mesonet (PAM) sites ran for the entire SHEBA year: Atlanta, Baltimore, and Florida.
Site Cleveland was also deployed early in the experiment but was damaged by a ridging event in early February 1998 and went offline for several months for repairs. This equipment was redeployed in early April 1998 at a site called Seattle and repositioned again in early June to a site called Maui. We will refer to the data stream from this PAM station as C-S-M (i.e., Cleveland–Seattle–Maui).
The PAM sites measured the same variables that the main ASFG site did but at one level only (cf. Brunke et al. 2006; Andreas et al. 2010a,b). That is, this equipment measured wind speed and direction, air temperature, relative humidity, momentum flux, and sensible heat flux at single levels that were 2–3 m above the surface. We will henceforth refer to all of these low-level wind speed, temperature, and humidity data from both the tower and the PAM sites as the 2-m values. All data from the main tower site and these PAM sites were averaged hourly. In our subsequent analyses, we use these hourly data in our calculations.
Each PAM site measured broadband incoming and outgoing longwave and shortwave radiation, respectively, with Eppley and Kipp and Zonen radiometers and used Vaisala sensors (model HMD50Y) to measure air temperature and humidity. Because of high relative humidity throughout the year (Andreas et al. 2002), the domes of these radiometers were prone to collecting frost that compromised their measurements. And such icing was not obvious in the individual radiometer data. In March and early April 1998, however, all PAM radiometers were fitted with heaters and blowers that kept the domes virtually ice free through the end of the experiment.
For measuring wind speed and direction and the turbulent momentum and sensible heat fluxes, each PAM site was originally deployed with a Gill R2 Solent sonic anemometer/thermometer (sonics). Icing also affected these Gill sonics, but we noticed that the sonics from Applied Technologies on the main tower shed this frost better than the Gill sonics. Hence, by the end of February 1998, we had replaced the Gill sonics on all the PAM sites except Florida (which was near the main camp and easiest to maintain) with sonics from Applied Technologies.
As a second measure to prevent frost formation, we installed heaters on all the PAM sonics by the end of February 1998. Unlike the radiometers, the sonics could identify bad data. This data quality indicator was the percentage of good 10-Hz samples collected during an hour (Fig. 2). When that percentage fell below 99.5%, the heaters turned on automatically and ran until the percentage of good data was again above 99.5%. In using data from the PAM sonics, we retained for our analysis only hours with at least 20% good data (i.e., 12 good minutes during an hour; Table 1).
We also introduced screening for icing when evaluating the quality of the PAM radiometer data to retain for analysis. The PAM data files contain two metrics for evaluating icing (www.eol.ucar.edu/isf/projects/sheba/rad.isff.html) that we will henceforth refer to as radiometer metrics. Beside the ASFG radiometers near the main tower, which were well maintained and had efficient blowers, the Atmospheric Radiation Measurement Program (ARM) also maintained a suite of radiometers in the main SHEBA camp. The two PAM radiometer metrics compare simultaneous measurements of incoming longwave radiation at each PAM site with the incoming longwave radiation measured at the tower and ARM sites (i.e., PAM − tower and PAM − ARM; Fig. 2). When frost was present on the dome of an up-looking PAM longwave radiometer, that radiometer essentially sensed the near-surface air temperature, not the sky temperature. As a result, a frosted PAM radiometer would generally yield higher values of incoming longwave radiation than would the clean tower or ARM up-looking radiometers.
Figure 2 shows hourly time series of these two radiometer metrics for PAM site Atlanta. During the first four months of the experiment, before the PAM radiometers had effective blowers and heaters, both the tower and ARM radiometer metrics were often large positive numbers. This is the signature of frosted PAM radiometers. During this same period, the Atlanta sonic corroborates this diagnosis of icing events by showing corresponding periods of few good sonic samples (Fig. 2). In our analysis, we conservatively rejected because of presumed icing any radiative fluxes for which the PAM − tower radiometer metric was greater than 20 W m−2. If the tower radiometer metric was unavailable, we tested the PAM − ARM radiometer metric for the same limit.
Figure 2 also shows a few values of radiometer metrics that have inexplicably large negative values. As a precaution, we also rejected PAM radiometer data when the PAM − tower radiometer metric was less than −20 W m−2. Again, if the tower metric was unavailable, we tested the PAM − ARM metric for the same limit.
Table 1 summarizes this screening for sonic and radiometer icing and notes which variables are affected by the screening.
Figure 3 shows the connections among the sites and emphasizes our naming convention. For calculating spatial statistics (described shortly), we look at simultaneous values of the same variable from paired sites. Because we have five sites and are analyzing pairs, for each hour we can have up to 10 different distances to use in computing the spatial statistics for any of 15 variables.
Each of the five ASFG locations had a GPS that reported hourly latitude Θ and longitude Φ. If (ΘA, ΦA) represents the latitude and longitude at site A and if (ΘB, ΦB) represents the latitude and longitude at site B, we calculated the distance D between the two sites as (e.g., Weaver and Mirouze 2013)
Here, R is the radius of curvature of the earth (6372.80 km), and
is the arc length, where ΔΘ = ΘA − ΘB and ΔΦ = ΦA − ΦB.
Figure 4 shows a histogram of the hours of unique data available for each distance. Because the ice was in continual motion, distances between sites were always changing; but the vast majority of separations in our dataset are 6 km and less. Most of the separations beyond 6 km occurred in September 1998, when Baltimore drifted rapidly away from the other sites. That is, the largest separations arise from September data when Baltimore is paired with any of the other four sites.
3. SHEBA sites
Although several other papers have documented the snow and ice characteristics of the SHEBA sites and the overall physical features of the SHEBA area (e.g., Uttal et al. 2002; Persson et al. 2002; Sturm et al. 2002; Perovich et al. 2003), for completeness, we briefly describe the five ASFG sites.
At the beginning of the experiment, in October 1997, the snow cover was only a few centimeters deep at all sites. Snow collected episodically through the winter and typically reached a maximum average depth of 0.3–0.5 m near 15 May 1998. As suggested by this range, the snow depth varied quite a bit horizontally and depended on the topography of the underlying sea ice (e.g., Sturm et al. 2002).
The snow began melting rapidly in early June and disappeared between 14 June and 4 July 1998, depending on the site, to expose bare sea ice. The ice surface continued melting at all sites into early August. Snow began falling and accumulating around 1 September, and all sites had a few centimeters of snow when we discontinued measurements in late September 1998.
The sea ice experienced a similar annual cycle. At all sites, the ice grew on its underside through the winter. At roughly the time when the snow cover disappeared at a site, the ice began melting there from below. The ice generally melted at its surface and on its bottom faster than it formed during the winter such that, through the SHEBA year, the sea ice in the vicinity of the SHEBA camp thinned.
All sites were on ponded ice during the summer, nominally from 10 June until 10 August 1998. The areal coverage of melt ponds in the vicinity of the SHEBA camp reached a maximum of 22% around 1 August 1998 (Perovich et al. 2002; Andreas et al. 2010a).
The ASFG tower site and the PAM site at Florida, which was nearby, were placed on a smooth, multiyear floe that was 2 m thick at the time of deployment. Florida began reporting on 22 October 1997; on 31 October 1997, the tower site was the last ASFG equipment to come online. This ice floe thickened to 2.5–2.8 m through the winter and thinned to about 1.5 m by the end of the deployment.
On 11 October 1997, PAM site Atlanta was also deployed on a smooth, multiyear floe. This floe, however, was only 1.5 m thick at the time. Through the winter, the ice grew to over 2 m thick by early July 1998. It then thinned rapidly from top and bottom melting such that the ice was less than a meter thick when we dismantled Atlanta on 30 September 1998.
On 12 October 1997, we deployed the Baltimore PAM site on first-year sea ice in a refrozen polynya. This polynya was about 400 m in the north–south direction by 150 m in the east–west direction; the PAM site was near the southern edge. Multiyear, hummocky ice, some of which was over 3 m thick, surrounded the polynya. Ice in the polynya itself was 0.4 m thick at the time of deployment, thickened to about 1.4 m by mid-May 1998, then thinned to about 1 m by the time we decommissioned Baltimore on 21 September 1998.
We deployed site Cleveland on 15 October 1997 in a rubble field that extended a few hundred meters in all directions. We know of no measurements of the ice thickness here but presume that the sea ice was at least 3 m thick. A ridging event in early February damaged the equipment at this PAM site, and it was taken out of service on 6 February 1998.
After repairs, the station was redeployed at site Seattle on 16 April 1998. The station itself was placed on a refrozen melt pond of 50–60-m radius, but the surrounding ice beginning 100–200 m from the station in all directions was hummocky. In early June, this site became untenable because of ice motion, and the equipment was moved to new site Maui on 10 June. During the two-month record from Seattle, the ice at the PAM site was 1–1.5 m thick.
Maui was on a multiyear floe with gently rolling hummocks. There are no known thickness measurements from the site, but we presume that the sea ice was at least 2 m thick at the time of deployment. Maui was an active site with frequent leads and melt ponds near it until freeze-up started around 1 September. We decommissioned Maui on 20 September 1998, when ice motion upset the PAM tripod and jeopardized the equipment.
4. Quantifying the spatial variability
a. Monthly averages
As context for our later calculations, we computed the monthly averages of surface-level pressure, 2-m air temperature, 2-m wind speed, and incoming longwave radiation (Fig. 5). In each panel, the monthly average is plotted at the middle of the month. For these and all subsequent plots, we invoked the screening for frost summarized in Table 1.
In each panel of Fig. 5, the averages from April 1998 through the end of the experiment in September are very close. This is the first clue that the spatial variability in state variables over the SHEBA site from late spring through early autumn was not severe. In each panel, the error bars are ±2 standard deviation in the monthly average.
Extenuating circumstances explain most of the obvious differences in averages among sites during the first six months of the experiment. Before computing the averages in Fig. 5, we screened the data from the PAM sites for cases of sensor icing (see Table 1) and ignored hours for which our quality metrics suggested icing. Figure 6 shows the number of hours of good data for each month from each site that went into computing the averages depicted in Fig. 5.
As Table 1 shows, we did three types of screening. The barometric pressure, air temperature, and relative humidity sensors were unaffected by icing. The bottom panel in Fig. 6 shows data returns from the pressure sensors and thus represents this class of icing-resistant instruments. Likewise, the middle panel (Fig. 6) shows data returns for wind speed and is therefore also relevant to wind direction and the turbulent fluxes. Finally, the top panel (Fig. 6) shows returns from the up-looking longwave radiometers. This panel is relevant to every variable that relies on a radiation measurement: surface temperature and both incoming and outgoing longwave and shortwave radiation.
Data returns from the PAM sonic anemometers became much more reliable in March 1998, after we had installed heaters on all the sonics. Likewise, the PAM radiometers got more reliable in March when we fitted the radiometers with efficient heaters and blowers. The tower instruments were the last ones to come online; that is the reason for the small number of observations from the tower in October 1997. Similarly, in September 1998, Baltimore and Maui were decommissioned around 20 September while the other sites remained in operation longer.
By comparing the counts of good data in Fig. 6 with the averages in Fig. 5, we can explain some of the obvious discrepancies among the monthly averages. For example, the outlying pressure in October 1997 from the tower (Fig. 5, top-left panel) resulted because the tower sensor, which came online late, did not sample some of the higher pressures from earlier in October. Similarly, the C-S-M PAM station was mostly out of service in March 1998; its March pressure thus does not come from the same range of air masses as for the other four sites.
Comparing averages of the 2-m wind speed (Fig. 5) with the monthly retrieval rates (Fig. 6) reveals features of the instrument icing. When the tower sonics were fully operational, starting in November 1997 until March 1998 when the PAM sonics all got heating, the monthly averaged wind speed from the tower is markedly lower than for the PAM sites. Figure 6 shows that there were many more hours of good data from the tower sonics than from the PAM sonics during this period. Therefore, from Fig. 5 it is obvious that the tower wind speeds represent typical fall and winter conditions. Meanwhile, the PAM sonics provided good data only when the winds were high enough to preclude icing conditions or to blow any collected frost off the sonics.
b. Spatial correlation function
To compute the spatial correlation function for variable V in each season, we start by computing the covariance (Cov) for variable V between sites A and B (see Fig. 3) when they have separation D, where D represents the averaging interval [D−, D+) and D+ − D− = 0.5 km. That is, we compute this covariance as
The term PA, for example, is the position of site A, represented as (ΘA, ΦA); ND is the number of good A and B pairs for variable V that are in distance interval [D−, D+) for a given season. Subscript i is the index for a specific hour in the dataset. Furthermore,
is the average of these same data, where X denotes either site A or B.
To calculate the spatial correlation function, we also need the standard deviations σV for sites A and B for variable V for the given season when the separation is in the distance interval [D−, D+). These come from the variances as
where subscript X again denotes either site A or B.
This ρV,AB has the same properties as usual correlation coefficients. It ranges from −1.00 to +1.00. If ND = 1, it is undefined because both variances are zero. If ND = 2, ρV,AB is exactly +1.00 or −1.00: two points define a straight line, which has perfect positive or negative correlation.
which has an approximately normal distribution with mean zero and standard deviation
is therefore a test of the null hypothesis—that variable V is uncorrelated between sites A and B. For example, if we test at the 95% confidence level, we would reject the null hypothesis if |z| > 1.960. If we find |z| > 4.417, we can reject the null hypothesis at the 99.998% confidence level.
In our subsequent plots of the spatial correlation function, we show that almost all values of ρV,AB(D) are 0.90 or greater. Table 2 shows values of z for typical ρV,AB values that we computed and for several values of ND. From (8) and (9), it is also possible, using standard techniques, to calculate a 95% confidence interval for the true value of the correlation coefficient given the measurement ρV,AB(D). Table 2 also shows these 95% confidence intervals implied for the selected ρV,AB(D) and ND combinations.
From Table 2, we see that z is a very large number when we find ρV,AB(D) to be 0.90 or larger. Consequently, for virtually all the ρV,AB values that we show later, we reject the null hypothesis with greater than 99.998% confidence. That is, variable V is always highly correlated between sites A and B for separations of 12 km and less. Moreover, Table 2 shows that the confidence interval always narrows with increasing ND. As a result, to focus on just the best data in the following correlation plots, we eliminated any ρV,AB(D) values computed from fewer than 15 pairs.
A key point to remember for interpreting these computations is that the summation in (4) includes only terms for which good measurements of variable V were available simultaneously at both sites A and B. And we use only these same data in (5) and (6). This protocol is unlike how we computed the monthly averages in the previous section, where we used all good data at a single site to compute its monthly average.
c. Bias statistics
Another way to evaluate the spatial variability of surface-level variables over sea ice is to calculate the bias between sites for a given variable as a function of separation. In the same notation as above, the bias for variable V between sites A and B in a given season is
As before, PA is the position of site A, and PB is the position of site B when the separation between these sites, D, is in the interval [D−, D+). Again, ND is the number of simultaneous values of variable V at the two sites that have separation D during the season.
If paired sites are not affected by local features such as differences in ice thickness or snow depth, we expect, contrary to the correlation function, that the bias statistic will increase with increasing separation. Quite simply, we expect sites to become increasingly dissimilar as their separation increases.
The bias calculated as (11) follows a Student’s t distribution. In our current notation, the t statistic is
which has 2ND − 2 degrees of freedom. We notice in (12) that
where again X can be either A or B, is usually identified as the standard deviation of the mean . We denote this quantity as pV,X, however, because it is also an indicator of the precision with which variable V can be measured. As such, pV,X accounts for the fundamental precision of the instrument but also for the random variability in the atmospheric V field. Fairall et al. (1996), Persson et al. (2002), Andreas et al. (2010a,b), and the online report of the SHEBA Flux-PAM project (www.eol.ucar.edu/isf/projects/sheba/) reported pV values from long-term measurements with the same instruments and with instruments similar to those used during SHEBA. We summarize in Table 1 pV values synthesized from these sources.
where, from Table 1, we assume pV,A = pV,B ≡ pV. The quantity on the right here is now an indicator of the confidence we have that our calculated BV,AB(D) values do or do not indicate similarity between sites A and B. In other words, if , the bias is less than the precision in the measurements of variable V, and we must assume the sites are similar. On the other hand, if , we have instrumental evidence that the sites are dissimilar.
If, as with the correlation function, we require at least ND = 15 samples for our upcoming plots and want to test BV,AB(D) at the 95% confidence level, t = 2.048. If ND is 100 or more, the t distribution is indistinguishable from a normal distribution; in which case, t is again 1.960 for a 95% confidence test. The upshot is that, at the 95% confidence level and for the sample sizes that we consider, t is always near 2. Hence, the right side of (14) is always approximately .
Table 1 also shows these values. Later, when we discuss the bias plots, we will interpret only values of |BV,AB(D)| greater than as indicating that variable V was different at sites A and B.
Although (13) quantifies measurement precision, it provides no insights into how to treat calibration biases in the SHEBA instruments. When we calculated the bias from (11), we saw that some of the values differed consistently from zero for no physical reason. We hypothesized that this behavior resulted because the instruments at the sites were not well intercalibrated. In other words, a nonzero bias that has no obvious physical explanation and no trend with separation likely is evidence that the two instruments had different calibration offsets.
For both the correlation coefficients and these bias calculations, we eliminated from our plots any values that resulted from fewer than 15 paired observations. Remember, all the statistics that we will discuss are seasonal calculations; each set of paired sites could thus include over 2000 paired observations of any variable in any season (Fig. 6). Because of the screening for frost, however, some pairs early in the experiment had far fewer samples.
Although the SHEBA dataset includes simultaneous observations or calculations of 15 surface-level variables, including measured and estimated turbulent surface fluxes, we focus here only on what are often called state variables—those that define the atmospheric state. Alternatively, documentation for the Community Ice Code (CICE; Hunke et al. 2013, their Table 1) identifies the variables that we discuss—namely, pressure, air temperature, wind speed, relative humidity, and incoming longwave radiation—as quantities provided by the atmospheric model through a “flux coupler” to the sea ice model. The sea ice model, in turn, computes the variables that we will consider elsewhere—surface temperature and the turbulent surface fluxes—and passes these back through the flux coupler to the atmospheric model.
As a comparative variable with nearly perfect behavior in the context of our analysis, we show in Figs. 7 and 8 the spatial correlation functions and spatial bias for barometric pressure. Pressure gradients are generally weak over Arctic sea ice (e.g., Brown 1981; Thorndike 1982); hence, the correlation coefficient is virtually one at all separations in Fig. 7. In this and all subsequent figures, the legend identifies the color scheme for the plotted points; and, in all cases, the first site listed is A, and the second site is B [see, e.g., (11)].
The bias between paired sites in Fig. 8 is almost always less than 0.2 mb (1 mb = 1 hPa), well below the limit for significant bias (Table 1). And that bias has no trend with separation between sites. Figure 8 does, however, suggest some instrument biases. Remember, in this and all other plots, the large separations in the top-left panel (Fig. 8) resulted because site Baltimore drifted away from the other sites in September 1998. By looking at Baltimore paired with the tower, Atlanta, and Florida in June, July, August, and September, we see consistent nonzero biases that have no physical explanation. The Baltimore barometer seemed to be reading high compared to the other three instruments, though it agrees pretty well with the C-S-M barometer.
b. Air temperature
Figure 9 is the time series of 2-m air temperature T2 from each of the five SHEBA sites for the entire experiment. Temperatures start well below freezing in October 1997, fall to near −40°C in December and January, rise slowly, and hover around 0°C for the three months of summer. We also see breaks in the series during the first six months, as we mentioned earlier, especially in the Cleveland–Seattle–Maui series.
From casual inspection, we see in Fig. 9 that the air temperatures appear well correlated across the sites: all difference traces cluster within 1°–2°C of zero. Figure 10, which shows the spatial correlation functions for these data, confirms this result. With the exception of two insignificant outliers, the spatial correlation functions for air temperature are above 0.95 in all seasons and for all separations. No season shows the decay in correlation with separation that we hypothesized in Fig. 1.
Although the temperature bias between sites is erratic in Fig. 11, almost all values are within the (±0.6°C) limit that indicates insignificant bias. We do, nevertheless, see at least one obvious case of instrumental bias. The red circles representing the tower–Baltimore pair always show a negative bias (also Fig. 9). Such consistent behavior in every season seems unlikely to have resulted from physical processes. More likely, either the tower was biased low, the Baltimore temperature was biased high, or both.
c. Wind speed
Another state variable, the surface-level wind speed U2 (nominally at 2 m), is not as well correlated as air temperature but generally produces correlation coefficients above 0.9 (Fig. 12). These values, again, indicate significant correlation between sites (Table 2). And these correlation coefficients have no tendency to decrease with increasing distance between paired sites.
Likewise, the spatial bias for wind speed in Fig. 13 hovers around zero in all seasons and for all separations, and all but two bias values are within ±0.6 m s−1 of zero, evidence of insignificant bias (Table 1). The bias is most erratic in autumn and winter when we have fewer samples from the PAM sonics (Fig. 6) and, therefore, have larger uncertainties in the statistics.
d. Relative humidity
In the SHEBA dataset, surface-level relative humidity, another state variable, behaves much more erratically than air temperature and wind speed. Andreas and Jordan (2013) showed the spatial correlation functions for the SHEBA surface-level relative humidity data and concluded that inconsistent instrument response among the SHEBA relative humidity sensors—which were different at the PAM sites than on the tower (Andreas et al. 2002)—explains this poor spatial coherence. After all, Andreas et al. (2002) had already demonstrated that, when figured with respect to saturation over ice, relative humidity measurements over polar sea ice all hover around 100% in all seasons.
Consequently, there is no physical reason why relative humidity should be markedly different from site to site. Rather, the seemingly erratic behavior in our plots of the statistics for relative humidity is an artifact of instrument issues (Andreas et al. 2002; Andreas and Jordan 2013). Hence, we do not show these plots but simply conclude, following Andreas et al. (2002), that relative humidity is another state variable that is well correlated for long distances over sea ice, as air temperature and wind speed are.
e. Broadband radiation
As an example of the spatial variability in a broadband radiation variable, we discuss only the incoming longwave QL↓. Because of the high emissivity of sea ice and the dominance of totally cloudy skies (discussed shortly), the emitted longwave radiation QL↑ closely follows QL↓ (e.g., Intrieri et al. 2002a). With regard to shortwave radiation, it was nonexistent or poorly measured because of low sun angle from October 1997 through February 1998. Furthermore, because clouds have similar influence on incoming longwave and incoming shortwave radiation, we can infer the spatial variability of the shortwave radiation from the incoming longwave radiation.
Figure 14 shows the seasonal spatial correlation functions computed from the incoming longwave radiation. In autumn and winter, the correlation tends to be more erratic than in spring and summer. Correlations, nevertheless, are generally above 0.95 and therefore imply significant correlation between sites.
The correlations for large separation in the top-left panel (Fig. 14) are especially noteworthy because all of these come from pairings involving Baltimore in September 1998, long after we had remedied the problem of radiometer icing. Therefore, the correlation values of 0.90–0.99 should represent autumn. At other times in autumn and winter, the correlations may be erratic because of the small numbers of samples from the PAM sites (Fig. 6).
By early April 1998, however, we had remedied the problem of radiometer icing. Therefore, we are confident in the correlation functions for spring and summer, all of which have values above 0.96.
The bias plot for the incoming longwave radiation in Fig. 15 corroborates these observations. Again, the data seem to be more erratic in the winter (December–February) but are also erratic in spring. Still, the large majority of bias values are within 5 W m−2 of zero and therefore fall within the limit of ±7 W m−2 (Table 1) that implies insignificant bias. Moreover, as with Fig. 14, none of the data in Fig. 15 show any tendency for the similarity among sites to degrade with increasing separation.
We nevertheless have physical reasons to expect incoming longwave and incoming shortwave radiation to vary among sites for the separations represented in our dataset. Figure 16 shows histograms of hourly cloud fraction observed in the main SHEBA camp by ground-based remote sensing instruments operated by NOAA’s Earth System Research Laboratory (Intrieri et al. 2002a,b; Shupe et al. 2006). Polar cloud fraction is bimodal: it is typically either near zero or near 100% (e.g., Makshtas et al. 1999).
But according to Fig. 16, that behavior varies seasonally. In autumn, spring, and summer, total overcast dominates the record. Such ubiquitous cloud cover would spatially homogenize the incoming longwave and incoming shortwave radiation. In winter, in contrast, clear skies occurred frequently, and fractional cloud cover occurred more often than in the other seasons. While clear skies would again encourage a spatially homogeneous distribution of incoming radiation, the fractional cloud cover could produce diversity among the SHEBA sites. In other words, because both incoming longwave and incoming shortwave radiation respond quickly and dramatically to changes in cloud cover (e.g., Andreas et al. 2008), we expect to see the range of cloud fractions in winter produce uncorrelated responses in incoming longwave radiation in that season especially.
In Figs. 14 and 15, winter is the season with the most erratic behavior in the spatial correlation functions and in the bias. While this behavior is compatible with the higher incidence of fractional cloud cover in this season, neither the correlation nor the bias statistics warrant an unequivocal conclusion of dissimilarity between sites. We suspect that the hourly averaged radiation data that we used in our analysis smoothed out the variability in winter cloud fraction at each site and thereby tended to again homogenize the sites. For shorter time scales, the variability in cloud fraction could produce differences in incoming radiation between sites. In other seasons, the cloud data and our analysis of spatial variability suggest that the incoming longwave and shortwave radiation are uniform over grid cells up to 12 km.
Our need to correlate simultaneous data at every hour from all five of the sites that the SHEBA Atmospheric Surface Flux Group maintained brought to the fore some quality-control issues that may have been glossed over in earlier SHEBA publications. Namely, frost forming on the sonic anemometers/thermometers and broadband radiometers at the Flux-PAM sites, especially, degraded their measurements during the first five months of the experiment. For the sonics, evidence of the frost was missing data: if enough frost collected on the sonic transducers, the wind signals disappeared. The radiometers, on the other hand, kept reporting, even with badly frosted domes. Our way of identifying such corrupted data was to compare measurements from PAM radiometers with either the tower or ARM radiometers, which were well maintained and had efficient frost prevention from the beginning of the experiment. Section 2, Table 1, and Fig. 2 summarize the screening we instituted to eliminate unreliable measurements. Others who may use the PAM data need to be aware of these instrument issues.
Figures 5 and 6 document an interesting phenomenon associated with the instrument icing: before heaters were installed, the PAM sonics were less prone to collecting frost when the 2-m wind speed was above about 4 m s−1. That is, while the frost-resistant sonics from Applied Technologies on the 20-m tower yielded monthly averaged winds of 4 m s−1 through the first five months of SHEBA, the PAM sonics gave much higher averages (Fig. 5). We conclude that, because the PAM sonics were often not reporting in light winds because of frost formation, their average measurements were biased high. There are two plausible explanations for this behavior. Either winds above 4 m s−1 prevented the feathery frost from collecting on the sonics or these higher winds created enough mixing that the typically drier air aloft mixed down to the surface to create conditions that were unfavorable for frost to form. Regardless, potential users of the PAM data need to be aware of this bias in the wind speed data from the PAM stations.
We could place no instruments over open water, like leads or polynyas, although our instruments did adequately sample over melt ponds during the summer. Conceivably, the 2-m air temperature could be higher over open ocean water than over compact sea ice, at least in winter. Hence, significant differences in surface-level air temperature could exist over a model grid cell. But observations of Arctic leads suggest that, even in winter, the heating from the water surface does not reach very far above that surface: the heat is mostly blown downwind (cf. Glendening and Burk 1992). Andreas et al. (1979) showed temperature profiles measured upwind and downwind of wintertime leads with widths up to 85 m. In all of these, the downwind temperature at 2 m was within a few tenths of a degree Celsius of the temperature of the upwind air over compact sea ice despite the fact that the leads were 20°–25°C warmer than the upwind ice. Most of this temperature drop occurs within 10–20 cm of the surface. Makshtas (1991, 34–35) made the same observations over a polynya that was up to 119 m wide.
In other words, the 2-m air temperature can be assumed homogeneous over grid cells up to 12 km, even if the grid cell contains a small fraction of open water within the sea ice. The higher surface temperature and lower albedo of the open water must, nevertheless, still be accounted for in the radiative fluxes and in the turbulent heat fluxes within the grid cell.
The surface-level wind speed is similarly well mixed 2 m above the surface in all seasons whether the surface is water or ice.
Any statistical analysis is useful only if an associated metric is also reported by which to judge the significance of the original statistic. For example, in our case, we computed the spatial correlation function but also provided the value of the correlation that allows rejecting the null hypothesis. We also computed 95% confidence limits for typical values of our correlation function. Only by comparing with these significance values could we conclude that all the state variables were highly and significantly correlated for all separations in our dataset.
Likewise, we derived a precision metric for our computed bias values from Student’s t distribution. Table 1 lists the instrument precision and this associated bias limit for each of the five state variables that we consider. The bias plots, Figs. 8, 11, 13, and 15, are pretty erratic. Only with the guidance provided by this statistically derived bias limit could we conclude that this erratic behavior is random and, thus, not significant evidence of differences between sites.
Our analysis of five surface-level meteorological variables obtained from one year of SHEBA data supports the assumption of uniformity in these variables over model grid cells up to 12 km across. The variables that we analyzed were barometric pressure and surface-level (i.e., ~2 m) air temperature, wind speed, relative humidity, and incoming longwave radiation. The wind speed and longwave radiation data from the four Flux-PAM sites required careful screening to eliminate data made unreliable by frost forming on the instruments during the first five months of the experiment. Future users of the SHEBA data need to be aware of these complications.
We computed three metrics to judge the spatial variability of these variables: the monthly average and seasonal values of the spatial correlation function (7) and the spatial bias (11). Furthermore, for interpreting each of these statistics, we computed associated confidence limits or, in the case of the bias, a limiting instrument precision.
Especially in spring, summer, and early autumn, when we had the most complete data returns, the monthly averages hinted at the uniformity in conditions. In plots of the seasonal metrics as functions of separation between sites in each of the four Arctic seasons—autumn, winter, spring, and summer—we saw no obvious degradation in any of the functions as the separation between sites increased. Again, if spatial variability were significant in these variables, we would expect the spatial correlation to decrease with increasing separation, as in Fig. 1, and the spatial bias to increase with separation. But the calculated correlations and the confidence limits we placed on those values show that all state variables are highly correlated in all seasons and for all separations. All seasons included results for separations up to 6 km; the calculations for autumn had separations up to 12 km.
Clearly, the e-folding distance Δ (Fig. 1) is much larger than 12 km. Our results thus complement those from Thorndike (1982). Considering only surface-level pressure and measurements made at the meso-α scale, he found Δ to be about 1500 km. Our measurements of five state variables, including pressure, confirm that there are no unexpected perturbations at the submeso and meso-γ scales that degrade correlation at these smaller scales.
Our calculations of the spatial bias for these state variables likewise reveal no significant biases between sites. We base this conclusion on the limiting instrument precision for each variable, as listed in Table 1. In other words, only a few insignificant points in our bias plots were outside limits set by how precisely we made our SHEBA measurements.
Admittedly, our selections for the five SHEBA sites could have affected our conclusions. We deployed four of the original five sites on fairly thick ice. Only Baltimore started on thinner ice, a refrozen polynya that had ice 0.4 m thick in autumn 1997. We saw no effect of ice thickness in the metrics for pressure, air temperature, wind speed, relative humidity, or incoming longwave radiation. Other variables in our dataset that are especially sensitive to surface temperature may show consequences of ice thickness and snow depth. We will consider these variables elsewhere.
Our analysis did not reveal any consistent spatial variability in the incoming longwave radiation in autumn, spring, and summer. These seasons featured total cloud cover at least 90% of the time and clear skies most of the other times. Both regimes produce homogeneity in the incoming longwave radiation. In contrast, in winter, total cloud cover occurred 50% of the time in the SHEBA dataset; clear skies occurred 33% of the time; and fractional cloud cover between these limits occurred 17% of the time. This high incidence of fractional cloud cover could produce heterogeneity in the incoming radiation in winter, and our plots of the spatial correlation function and bias in Figs. 14 and 15, respectively, do show slightly higher variability in winter. This behavior is suggestive of heterogeneity in the incoming longwave radiation in winter, but our hourly averaging tended to smooth any short-term variability in incoming radiation in response to changing clouds. Hence, we cannot unequivocally confirm this hypothesis.
In summary, our analysis justifies the assumption that, over the central Arctic ice pack, the variables surface-level pressure, air temperature, wind speed, and humidity are uniform over model grid cells up to 12 km.
As we close, some readers may view this analysis as weakly relevant since Arctic sea ice is thinner in winter and open water is more prevalent in summer now than during the SHEBA year. But remember, climate simulations, for instance, require a spinup of several centuries (e.g., Collins et al. 2006; Hurrell et al. 2013; Stevens et al. 2013) before they can accurately simulate current conditions or predict future climate. Our analysis is certainly pertinent to these several hundred years before present.
We thank our colleagues in the SHEBA Atmospheric Surface Flux Group, who helped us collect, process, and interpret the SHEBA data: Chris Fairall, Andrey Grachev, Peter Guest, Tom Horst, and Ola Persson. We thank personnel with the Earth Observing Laboratory at the National Center for Atmospheric Research: Tony Delany, Tom Horst, Gordon Maclean, John Militzer, and Steve Semmer. Their determination and professionalism in solving our icing problems made this study possible. We also thank Matt Shupe, Janet Intrieri, and Taneil Uttal for the synthesized cloud-fraction data that went into Fig. 16. We thank John Weatherly, three anonymous reviewers, and editor John Walsh for reviews that helped us improve the manuscript. Lastly, we thank Emily Moynihan of BlytheVisual for creating Fig. 3 and the U.S. National Science Foundation, which supported this work with Award ARC 10-19322.