The HadSST3 gridded fields provide several critical components, the temperature anomaly, the number of observations, and several estimates of the uncertainty (Kennedy et al., 2011a, b). The grid cell uncertainties and observation counts allow one to treat some grid cells as having greater confidence than others. Unlike land surface station data, where each monthly average represents many temperature observations, the ocean observation counts are a true measure of the number of instantaneous SST measurements.

Analogous to Rohde et al. (2013a), the core of the interpolation approach is to generate a kriging-based field using an assumed distance-based correlation function. As with Rohde et al. (2013a), a correlation-based approach is used rather than the more common covariance-based approach to simplify the computational considerations and should be adequate as long as the variance changes relatively slowly with changes in position. A review of both the HadSST data and climate model outputs suggested that the temperature-to-distance correlation function could be modeled effectively via the same spherical correlation function approach used for land surface temperatures: 1R(d)=R01-ddmax21+d2dmax,d<dmaxR(d)=0,d≥dmax. The empirically estimated distance parameter dmax was found to have a value of 2680 km based on the spatial variance of the HadSST monthly averages. This is similar to, though somewhat smaller than, the 3310 km scale adopted in the land surface temperature study (Rohde et al., 2013a). By contrast, the local correlation parameter R0=0.47 was estimated to be much lower in the oceans (compared to 0.86 on land). This is due to two factors. Firstly, ocean observations are individual measurements whereas land observations reflect monthly averages. Secondly, the typical monthly fluctuations in the oceanic environment are much smaller than on land, causing a reduced signal-to-noise ratio. The estimation of R0 was based on a comparison of the variance in HadSST grid cells with a single measurement to those with > 100 observations. The latter condition provides a proxy for cells where the random portion of measurement and sampling uncertainty could plausibly be neglected.

Figure 1 shows an empirically estimated average correlation versus distance between HadSST grid cells. This shows the empirical length scale, though a larger intercept is used (∼ 0.75), reflecting the fact that the average HadSST grid cell incorporates many observations. The lower value for R0 represents the typical relationship between a single measurement and the monthly average.

This treatment, using a single scale length for the whole ocean, simplifies the analysis; however, it does ignore some of the real variations across the oceans. For example, in regions with boundary currents, upwelling–downwelling, or complex ocean-to-land geographies, the scale length of monthly average temperature variations may be smaller than suggested here. In practice, the 5 ∘ × 5∘ gridding of HadSST already precludes a detailed analysis of most small features. The interpolation presented here primarily serves to improve the representation by smoothing over noise and filling gaps, but it will not necessarily capture the smallest features.

The distance correlation function gives rise to a kriging formulation. 2T(x,t)=θt+∑jK(xj,x,t)(SST(xj,t)-θt) 3K(x1,x,t)⋮K(xN,x,t)=D(x1,t)R(x1-x2)⋯R(x1-xN)R(x2-x1)D(x2,t)⋮⋱⋮R(xN-x1)⋯D(xN-1,t)R(xN-1-xN)R(xN-xN-1)D(xN,t)-1R(x1-x)⋮R(xN-x) 4D(xj,t)=1+(Neff(xj,t)-1)R0Neff(xj,t) 5Neff(xj,t)=max⁡sm2(σm(xj,t))2,1.

Here t is the current month, T(x,t) is the interpolated temperature at a general location x, SST(xj,t) is the HadSST anomaly value in the grid cell centered at location xj, σm(xj,t) is the measurement uncertainty associated with location xj, and sm is the average measurement uncertainty of a single measurement. Neff(xj,t) is then an effective number of independent measurements associated with the grid cell. Though HadSST provides the true number of observations per cell, N(xj,t), we found that Neff(xj,t), which incorporates the measurement uncertainty, appeared to give superior results than simply relying on the reported number of observations. The incorporation of Neff(xj,t) into the determination of the kriging coefficients K has the effect of giving greater weight to grid cells with less uncertainty. For integer values of Neff(xj,t), the formulation of D(xj,t) is mathematically equivalent to having xj appear Neff(xj,t) independent times in the correlation matrix. Note also that any empty HadSST grid cells at time t are omitted from the matrix formulation for K.

θt is a free parameter at each time t and effectively represents the global ocean-average temperature anomaly. Its value is found iteratively by insisting that the spatial average of T(x,t)-θt=0.

It is instructive to note that this kriging formulation has the property that T(xj,t)→SST(xj,t) in the limit that Neff(xj,t)→∞, but will ordinarily produce a temperature estimate based on a weighted average of multiple HasSST grid points in the case that Neff(xj,t) is small or moderate. The latter property can be useful in suppressing noise at grid locations with high uncertainty and/or very few measurements.

It is also important to recognize that though the correlation function R(d) has a very long tail, this does not mean that average necessarily extends over a large area. In general, the kriging coefficients K(xj,x,t) constructed in this way will heavily favor the nearest several data points. As long as nearby data are available, little weight will be given to distant grid cells. However, the long tail of the correlation function means that the kriging will attempt to fill large holes using distant data if no nearby data are available.

An absolute value field was also created by applying a similar interpolation to the HadSST climatology. 6C(x,m)=P(x,m)+∑j(KB(xj,x,m)(SSTCLIM(xj,m)-P(x,m))) C(x,m) is the interpolated climatology for month m, SSTCLIM(xj,m) is the reported climatology, and KB(xj,x,m) is a set of kriging parameters, which are the same as K(xj,x,m) except that R0 and D(xj,t) are both replaced with 1, effectively treating the SSTCLIM(xj,m) as if it has no uncertainty. P(xm) is a background prediction function dependent only on the month and the latitude of x. It is described as a piecewise cubic spline with 11 knots as free parameters equally spaced in the cosine of latitude. These free parameters are chosen to minimize the spatial average of C(x,m)-P(x,m). By construction, C(xj,m)=SSTCLIM(xj,m) for all xj values, and this construction merely provides a way of interpolating between grid cell centers.

In addition to the above description, a physical cutoff was applied to the absolute temperature C(x,m)+T(x,t) at a fixed minimum temperature of -1.8 ∘C, which is the freezing temperature of seawater. If the interpolation would suggest a value lower than this, T(x,t) was adjusted accordingly to maintain the minimum value of -1.8 ∘C. Such adjustments are rare.

Finally, one last interpolation is performed using an assumption of temporal persistence. Unlike land temperature anomalies, where the temporal correlation is often only a couple weeks, ocean temperature anomalies typically have a temporal correlation measured in months. This can be exploited to estimate ocean temperatures based on adjacent months when no other information is available.

Analogous to Rohde et al. (2013a), a diagnostic criterion can be constructed V(x,t)=∑jK(xj,x,t). Because of the nature of the kriging coefficients, V(x,t)→1 in the presence of dense data and V(x,t)→0 if there are no HadSST data in the neighborhood of x.

The final estimate of the SST, including a temporal persistence adjustment for regions of low V(x,t), is then 7Tfinal(x,t)=T(x,t)+(1-V(x,t))V(x,t+1)T(x,t+1)+V(x,t-1)T(x,t-1)V(x,t+1)+V(x,t-1)-θt. Here, t-1 and t+1 refer to the temperature field 1 month earlier and 1 month later, respectively. This adjustment allows for a modest reduction in uncertainty at early times when data are temporally sparse.

As described, this analysis is agnostic about the resolution used to sample the final temperature field. In practice, we generally use the same 15 984-element equal-area grid as Rohde et al. (2013a) to calculate Tfinal(x,t), though with non-ocean elements masked out.

The ocean-average uncertainty in our ocean reconstruction is estimated following essentially the same model as adopted by HadSST3. HadSST3 estimates the total reconstruction uncertainty as the combination of measurement uncertainty, coverage uncertainty, and bias uncertainty (Kennedy et al., 2011a, b). Bias uncertainty, σbias, which reflects biases created due to variations over time in the ways that SST has been measured, is brought forward essentially unchanged by our analysis process (Fig. 2). Due to its slowly varying nature, this uncertainty remains the most important limitation of the detection of long-term averages.

The coverage uncertainty, σcoverage, is the uncertainty in the large-scale average arising due to incomplete sampling of the spatial field. As with HadSST3, our estimate of the coverage uncertainty is constructed by sampling a known field, applying our interpolation procedure, and seeing how well we reproduce the underlying average of the known field. Following HadSST3, we used the SST fields provided by HadISST v2 as our target. The HadISST fields are spatially complete, observation-based historical reconstructions of SST and sea ice concentration (Titchner and Rayner, 2014). To estimate the coverage uncertainty associated with a specified HadSST sampling field, we mask every month of the HadISST dataset using that sampling field, interpolate the remaining data, and measure the error in the interpolated average relative to the true ocean average of the whole HadISST field. The deviations in the ocean average are then collected across all HadISST months, and the uncertainty for that coverage mask is reported as the root-mean-square average of the deviations. Using this technique, which is directly analogous to the HadSST3 coverage assessment technique, we estimate that the application of our interpolation approach typically reduces the coverage uncertainty by 20 %–40 % (Fig. 2).

Lastly, we consider the impact of our interpolation on the measurement and sampling uncertainty. Measurement uncertainty essentially captures the errors in individual observations, while sampling uncertainty reflects the fact that water temperatures can vary on timescales shorter than a month and spatial scales smaller than a grid box. Though interpolation does not change the underlying uncertainty associated with individual measurements, by adjusting the weight of individual observations in the overall average, we affect the way that individual measurement errors propagate into the global average. In particular, in the presence of sparse data, limited measurements may be extrapolated over a large area. In some circumstances, this can cause the effective uncertainty in the global average due to these uncertainties to increase. In essence, the interpolation may trade improvements in coverage uncertainty against a greater impact for measurement uncertainty. This largely limits our ability to reduce the overall uncertainty by interpolation.

The impact of measurement uncertainty on a large-scale average depends on the error correlation. If the measurement uncertainties were uncorrelated, then the error would generally be expected to decline with the square root of the number of measurements. In actuality, the measurement uncertainties are frequently correlated. In most cases, single ships report many measurements per month. Each of those measurements can have both random errors and a potential for systematic bias. For a single ship, we cannot expect this bias component of a measurement error to be reduced by increasing the number of observations. In their analysis HadSST3 models the entire error correlation matrix to understand the effect of measurement errors on the global average uncertainty.

For HadSST3, the error correlation matrices were not published. As a result, it is not possible to exactly determine the effect of our interpolation procedure on the measurement uncertainty. However, we can make a reasonable estimate. Since HadSST3 releases both the per-grid-cell measurement uncertainties and the global average measurement uncertainty, we can compare the expected measurement uncertainty treating all grid cells as independent to what is actually observed by HadSST3 using the whole error correlation matrix (Kennedy et al., 2011b). 8σuncorrelated=Σj(A(xj)σm(xj,t))2, where A(xj) is the fraction of the Earth's oceans represented by grid cell xj and σuncorrelated is the measurement uncertainty resulting from assuming that the measurement errors in individual grid cells are uncorrelated with other grid cells.

We find that the measurement uncertainty reported by HadSST3 in the ocean average is typically ∼ 2.1 times larger than σuncorrelated, with some variation over time.

We use this estimate as a benchmark to approximate the effect of error correlation on our analysis of measurement uncertainty. 9σinterpolated, measurement=σHadSST, measurementσuncorrelatedΣj(K‾(xj,t)σm(xj,t))2 10K‾(xj,t)=∫∫K(xj,x,t)dx/∫∫1dx Here the double integral denotes the integral over the surface of the ocean. Thus K‾(xj,t) is effectively the weight of the xj grid point in the global average.

The total uncertainty in the ocean average is then found by assuming the components are independent. 11σbias2+σcoverage2+σinterpolated, measurement2 Over nearly all time periods, we find that interpolation does reduce the uncertainty associated with missing coverage. In the early period, the interpolation results in an appreciable reduction in total uncertainty. However, the total uncertainty in the global average is little changed in the recent period. This is because the bias and measurement uncertainties play a dominant role in the recent period, and the impact of these uncertainties on the global average is little changed as a result of the interpolation. However, even if the ocean-average uncertainty is not changed during the recent period, the interpolation may still aid in the interpretation of local- to regional-scale features.

The combined field is constructed by merging the Berkeley Earth land surface temperature with the interpolated SST field described above. Two versions are considered that differ only in their treatment of sea ice, using either the land air temperature (LAT) or the SST field to estimate the temperature anomaly at sea ice locations. From 1850 to near present, the sea ice locations are estimated using the ice concentration fields in HadISST v2 (Titchner and Rayner, 2014).

To combine LAT and SST data, both data sets are expressed on the same grid. To simplify the combination at cells that are part land and part ocean, we have taken to adding in the spatial climatology and doing the combination in absolute temperatures.

In the case where sea ice areas are represented by SST, the combination is straightforward: 12Tcombined(x,t)=L(x)TLAT(x,t)+(1-L(x))TSST(x,t), where L(x) is the fraction of the grid cell at location x that is land, and TLAT and TSST are respectively the LAT as estimated by Rohde et al. (2013a) and the interpolated SST as described above.

In the case where sea ice regions are treated as land, 13Tcombined(x,t)=L*(x,t)TLAT(x,t)+(1-L*(x,t))TSST(x,t), 14L*(x,t)=L(x)+(1-L(x))I(x,t), where I(x,t) is the ice fraction at location x at time t as reported by HadISST v2 (Titchner and Rayner, 2014). For this purpose, HadISST is also regridded onto the same grid as LAT and SST. As HadISST is frequently delayed by a few months compared to other climate data, it is necessary to supplement this data set when producing near-real-time estimates. For this purpose, the Sea Ice Index of the National Snow and Ice Data Center (Fetterer et al., 2017) is used for months that are not yet available in HadISST. The modern ice distribution in both HadISST and the Sea Ice Index are based on satellite observations; however, we found that the Sea Ice Index tended to have systematically more partial melting than HadISST. To maintain consistency, a distribution transform was applied to the sea ice fractions provided in the Sea Ice Index based on comparing the 2014–2018 ice fields in each dataset.

It is useful to note that regardless of whether one is using SST or LAT to estimate temperatures in association with sea ice, most such estimates involve a considerable extrapolation. In the case of LAT, for example, conditions over sea ice in the Arctic will usually be extrapolated from Greenland, Canada, Scandinavia, and Russia. Similarly, in the Antarctic, coastal stations will be extrapolated outward over the ice. By contrast, when using SST, one extrapolates from rare SST measurements that may be far removed from the sea ice edge. Or, in the case that analysis of the sea ice regions is excluded entirely, averaging methods are effectively substituting the ocean or global average temperature anomaly.

It is our belief that the anomaly field generated by extrapolating air temperatures over sea ice locations is a more sensible approach to characterizing climate change at the poles. The air temperature changes over the sea ice can be quite large even while the water temperatures underneath are not changing at all. In particular, over the last decades Arctic air has shown a very large warming trend during the winter.

Regardless of the approach used, the spatial climatology can then be calculated and removed (differing from the original only in cells with a mix of land and water/sea ice). Then the long-term trend in the climate can be computed using the spatial average of the anomaly fields.

Uncertainties for the combined record are calculated by assuming the uncertainties in LAT and SST time series are independent and can be combined in proportion to the relative area of land and ocean. In the case that LAT is used over sea ice, the uncertainties for both LAT and SST have to be slightly recalculated by assuming that the time-varying mask L*(x,t) is applied the relevant spatial averages in the uncertainty estimations described in Rohde et al. (2013a) and in the SST section above. Doing this adjustment causes a slight increase in LAT uncertainty (due to the extrapolation over sea ice) and a similar small decrease in SST uncertainty.