Isotopic measurements in corals are used to reconstruct past sea surface temperature. These reconstructions are based on calibration regression analyses using paired measurements of modern isotopic composition and sea surface temperature. It is shown that error in these measurements of sea surface temperature can lead to substantial bias in reconstruction. Provided the variance of the measurement error is known or can be estimated, a simple correction can eliminate this bias.
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 Isotopic and other geochemical measurements in corals have been used to reconstruct past sea surface temperature [Beck et al., 1992; Fairbanks et al., 1997; Gagan et al., 2000]. As reviewed in more detail below, reconstruction is based on a model of the relationship between the isotopic measurement and sea surface temperature. The model is fit by regression using a calibration sample consisting of paired isotopic and sea surface temperature measurements. This fitting assumes that sea surface temperature is measured without error during the calibration period. This assumption is not always realistic. For example, error can arise when sea surface temperature is not measured at the location of the coral or when the pairing of sea surface temperature and isotopic measurements is imperfect. The purpose of this paper is to analyze the effect of a violation of this assumption on reconstruction. The analysis shows that measurement error can lead to a substantial bias in reconstruction. Provided a good estimate of the variance of the measurement error is available, this bias can be essentially eliminated by a simple correction.
2. The Effect of Measurement Error on Reconstruction and Its Correction
 Let Y denote an isotopic measurement in a coral and let SST denote the corresponding sea surface temperature. These quantities are assumed to be related through the simple linear regression model:
where βo and β1 are unknown regression parameters and ɛ is an error with mean 0 and unknown variance σɛ2. This error includes both error in the isotopic measurement and variability in Y unrelated to variability in SST. This model can be fit by ordinary least squares (OLS) regression using a calibration sample consisting of paired isotopic and SST measurements. Let o and 1 be the estimates of βo and β1, respectively.
 Suppose now that an isotopic measurement Yo is made outside the calibration sample. Interest centers on reconstructing the corresponding sea surface temperature SSTo. The reconstructed value found by inverting equation (1) is:
Provided that the estimates o and 1 are unbiased and that SSTo is not too far outside the SST range in the calibration sample, STo is also approximately unbiased. The alternative approach based on interchanging Y and SST in the regression equation is known to perform worse, particularly in extrapolating to conditions outside the calibration sample [e.g., Shukla, 1972].
 Returning to the calibration regression, suppose that, in addition to measurement error in Y, there is measurement error in SST. Specifically, suppose that:
where SSTmeas is measured sea surface temperature, SSTtrue is the true sea surface temperature, and η is a measurement error with mean 0 and variance ση2. It is perhaps under-appreciated that, in this case, the estimates o and 1 are biased [e.g., Fuller, 1987]. In qualitative terms, the effect of measurement error in SST is to attenuate the fitted regression model (i.e., on average, 1 is closer to 0 than β1). This leads to what might be called accentuation in SST reconstruction, with the reconstruction of a cold SST being too cold and the reconstruction of a warm SST being too warm.
 It is possible to go beyond this qualitative result. Let σSST2 be the variance of the values of SSTtrue in the calibration sample and let:
be the ratio of the variances of the true and measured values of SST in the calibration sample. It is well-known that, on average:
where is the mean of SSTtrue in the calibration sample [Fuller, 1987]. It follows upon substitution of these expressions into equation (2) that on average:
Finally, upon substituting the expression for λ in equation (5) into equation (6) and subtracting the result from the expression for SSTo, the average reconstruction bias is:
Provided that ση2 is known or can be estimated, an estimate λ is given by:
where σmeas2 is the sample variance of the measured SST values in the calibration sample. This estimate of λ can be used to construct the bias-corrected estimates of β1 and βo:
where and meas are the average of the isotopic and SST measurements in the calibration sample, respectively. Finally, these estimates can be used in turn to construct an approximately unbiased reconstruction:
3. An Illustration
 In this section, we illustrate the reconstruction bias and its correction discussed in the previous section using part of the Galapagos coral stable oxygen calibration data set of Wellington et al. . These data, which are described in detail in the auxiliary material by Wellington et al.  and are available at http://www.ngdc.noaa.gov/paleo/coral/galapagos.html, consist of 24 monthly measurements of δ18O (PDB) covering the period 1993–1994 extracted from each of two corals (designated as Ur-1 and Ur-3) of the species Pavona clavus at approximately 3 m depth in Urvina Bay, Isabela Island (0.23 S 91.14 W) in the Galapagos Islands, Ecuador.
 The two time series of δ18O are plotted in Figure 1. Figure 2 shows the corresponding values of SST measured at the site. These measurements, which we will treat as correct (i.e., as SSTtrue), have average = 23.79°C and standard deviation σSST = 1.48°C. The estimates of the parameters βo and β1 found by OLS regression of δ18O on SSTtrue for the two corals are given in Table 1.
Table 1. Estimates of βo and β1 for the Two Urvina Bay Corals Ur-1 and Ur-3 Based on Regressions of δ18O on SSTtrue and on SSTmeas and Based on Correcting the Latter for Bias According to Equation (7)
Estimate of βo
Estimate of β1
 Suppose that the SST measurements at the Urvina Bay site had not been available and that measurements at another nearby site studied by Wellington et al.  at Bartolomé are used instead in calibration (i.e., as SSTmeas). The time series of SSTmeas is also shown in Figure 2. The difference SSTtrue – SSTmeas has average 0.15°C and standard deviation 0.92°C, which we will take as ση. The estimated value of λ is 0.72. The estimates of βo and β1 found by OLS regression of δ18O in the Urvina Bay corals on SSTmeas from Bartolomé are also reported in Table 1. These results are broadly consistent with the theoretical result in equation (5) - that, it should be emphasized, applies on average and not in each particular case. The bias-corrected estimates of βo and β1 given by equation (9) are also reported in Table 1.
 To illustrate the effects on SST reconstruction, Figure 3 shows SST reconstruction plots – that is, the reconstructed SST value as a function of the δ18O measurement – for each of the two Urvina Bay corals based on the three sets of parameter estimates given in Table 1. The accentuation bias due to using SSTmeas instead of SSTtrue in the calibration regressions is clear, reaching 1.5°C for δ18O values within the range of the calibration data. The use of the bias-corrected estimates essentially eliminates this bias in the first coral and reduces it substantially in the second (e.g., a maximum reconstruction bias of 0.5°C for δ18O values within the range of the calibration data).
 The purpose of this paper has been to analyze the effect on SST reconstruction from corals of measurement error in calibration SST data. Although the paper has focused on reconstruction from isotopic measurements, the same results pertain to reconstructions from any proxy for which calibration is based on ordinary least squares. The basic result, given in equation (6), is that even relatively modest measurement error can lead to a relatively large accentuation bias, particularly at and beyond the range of conditions during the calibration period. Provided the variance ση2 of the measurement error can be estimated, the bias-corrected reconstruction based on equations (8) and (9) provides a simple way to correct for this bias. The estimation of ση2 , which is discussed below, is not always easy. However, even in the absence of a good estimate, the quantification of accentuation bias presented here can be useful for bracketing the potential reconstruction bias.
 Before proceeding, a word is in order about an alternative approach. A widely known approach to fitting a simple linear regression model in the presence of measurement error in the regressor is reduced major axis (RMA) or orthogonal regression. In OLS regression, the parameters are estimated by minimizing the sum of vertical deviations between the data points and the regression line. In contrast, in RMA regression, the parameters are estimated by minimizing the sum of orthogonal deviations between the data points and the regression line. Shen and Dunbar  discussed this method in analyzing coral isotope records [see also Quinn et al., 1998; Quinn and Sampson, 2002]. In terms of the model defined by equations (1) and (3), RMA regression assumes that the variance of ɛ is equal to the variance of η. There is no reason to believe that this assumption is warranted. The method can be extended to allow for differences in these variances [see, e.g., Carroll et al., 1995]. To do so, it is necessary to have an estimate of both variances. As ɛ includes both isotopic measurement error and natural variability in Y unrelated to SST, it is not sufficient to consider only the variance of the former in this extension. More generally, finding a good estimate of the variance of ɛ is not straightforward. In particular, the estimate based on OLS regression will be biased in the presence of measurement error in SST [Carroll et al., 1995]. A final disadvantage is that, even with estimates of the variances of ɛ and η, the actual fitting can be complicated.
 Returning to the method described in this paper, the estimation of ση2 depends on the specific nature of the measurement error. When the time series of SSTtrue is short or absent altogether, one common approach is to use a spatial interpolation or gridding algorithm to estimate SSTtrue from measurements at a number of one or more nearby locations. When this is done to extend the calibration period beyond that covered by the time series of SSTtrue, then the period of overlap may be long enough to allow estimation of ση2. When this period of overlap is very short or non-existent, then estimation of ση2 must be based on a knowledge of spatial variability in SST as reflected, for example, in the spatial covariance function [Kaplan et al., 1998]. In fact, many gridding methods provide a direct estimate of the variance of the interpolation error which corresponds to ση2. It is worth pointing out that the use of a poor estimate of ση2 could lead to a bias-corrected reconstruction whose bias is actually greater than that of the uncorrected reconstruction.
 Finally, it has been assumed in this paper that the goal is to reconstruct SSTtrue (i.e., at the site of the coral). If instead the goal is to reconstruct SSTmeas, then the issue of measurement does not arise. However, this does assume a linear relationship between SSTtrue and SSTmeas and that this relationship remains stable over the period of reconstruction.
 The helpful comments of four anonymous reviewers are acknowledged with gratitude.