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How accurately do we know the temperature of the surface of the earth?

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Abstract

The earth’s near surface air temperature is important in a variety of applications including for quantifying global warming. We analyze 6 monthly series of atmospheric temperatures from 1880 to 2012, each produced with different methodologies. We first estimate the relative error by systematically determining how close the different series are to each other, the error at a given time scale is quantified by the root mean square fluctuations in the pairwise differences between the series as well as between the individual series and the average of all the available series. By examining the differences systematically from months to over a century, we find that the standard short range correlation assumption is untenable, that the differences in the series have long range statistical dependencies and that the error is roughly constant between 1 month and one century—over most of the scale range, varying between ±0.03 and ±0.05 K. The second part estimates the absolute measurement errors. First we make a stochastic model of both the true earth temperature and then of the measurement errors. The former involves a scaling (fractional Gaussian noise) natural variability term as well as a linear (anthropogenic) trend. The measurement error model involves three terms: a classical short range error, a term due to missing data and a scale reduction term due to insufficient space–time averaging. We find that at 1 month, the classical error is ≈±0.01 K, it decreases rapidly at longer times and it is dominated by the others. Up to 10–20 years, the missing data error gives the dominate contribution to the error: 15 ± 10% of the temperature variance; at scales >10 years, the scale reduction factor dominates, it increases the amplitude of the temperature anomalies by 11 ± 8% (these uncertainties quantify the series to series variations). Finally, both the model itself as well as the statistical sampling and analysis techniques are verified on stochastic simulations that show that the model well reproduces the individual series fluctuation statistics as well as the series to series fluctuation statistics. The stochastic model allows us to conclude that with 90% certainty, the absolute monthly and globally averaged temperature will lie in the range −0.109 to 0.127 °C of the measured temperature. Similarly, with 90% certainty, for a given series, the temperature change since 1880 is correctly estimated to within ±0.108 of its value.

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Acknowledgements

The author thanks R. Hébert, L. del Rio Amador and David Clarke for useful discussions. This work was unfunded, there were no conflicts of interest. The data were downloaded from the publically accessible sites to be found in the corresponding references (first paragraph, Sect. 2).

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Appendices

Appendix A: some useful properties of fractional Gaussian noise

In this appendix, we give a brief summary of some useful properties of fGn; a longer review is given in (Lovejoy et al. 2015b) and a full mathematical exposé in (Biagini et al. 2008). The standard (“s”) fGn process \(G_H^{\left( s \right)}\left( t \right)\) with parameter H, can be defined as:

$$\begin{array}{*{20}{c}} {G_H^{\left( s \right)}\left( t \right) = \frac{{{c_H}}}{{\Gamma \left( {1/2 + H} \right)}}\int\limits_{ - \infty }^t {{{\left( {t - t'} \right)}^{ - \left( {1/2 - H} \right)}}\gamma \left( {t'} \right)dt'} ;}&{ - 1 < H < 0} \end{array}$$
(30)

γ(t) is a unit Gaussian “δ correlated” white noise with <γ>= 0 and:

$$\left\langle {\gamma \left( t \right)\gamma \left( {t'} \right)} \right\rangle = \delta \left( {t - t'} \right)$$
(31)

where “δ” is the Dirac function. The constant c H is a constant chosen so as to make the expression for the statistics particularly simple, see below. It may be useful to note that fGn is related by differentiation to the more familiar Fractional Brownian motion (fBm) process. We can see by inspection of Eq. 16 that \(G_H^{\left( s \right)}\left( t \right)\) is statistically stationary and by taking ensemble averages of both sides of Eq. 16 we see that the mean vanishes: \(\left\langle {G_H^{\left( s \right)}\left( t \right)} \right\rangle = 0\). When H = −1/2, the process \(G_{ - 1/2}^{\left( s \right)}\left( t \right)\) is simply a Gaussian white noise.

Now, take the average of G H over τ; the “τ resolution anomaly fluctuation”:

$$G_{H,\tau }^{\left( s \right)}\left( t \right) = \frac{1}{\tau }\int\limits_{t - \tau }^t {G_H^{\left( s \right)}\left( {t'} \right)dt'} $$
(32)

If c H is now chosen such that:

$${c_H} = {\left( {\frac{\pi }{{2cos\left( {\pi H} \right)\Gamma \left( { - 2H - 2} \right)}}} \right)^{1/2}}$$
(33)

then we have:

$$\begin{array}{*{20}{c}} {\left\langle {G_{H,\tau }^{\left( s \right)}{{\left( t \right)}^2}} \right\rangle = {\tau ^{2H}};}&{ - 1 < H < 0} \end{array}$$
(34)

This shows that a fundamental property of fGn is that in the small scale limit (τ ≥ 0), the variance diverges and H is scaling exponent of the root mean square (RMS) value. This singular small scale behaviour is responsible for the strong power law resolution effects in fGn. Since \(\left\langle {G_H^{\left( s \right)}\left( t \right)} \right\rangle = 0\), sample functions G H(t) fluctuate about zero with successive fluctuations tending to cancel each other out; this is the hallmark of macroweather.

A comment on the parameter H is now in order. In treatments of fBm, it is usual to use the parameter H confined to the unit interval i.e. to characterize the scaling of the increments of fBm. However, fBm (and fGn) are very special scaling processes, and even in low intermittency regimes such as macroweather—they are at best approximate models of reality. Therefore, it is better to define H more generally as the fluctuation exponent (Eq. 9); with this definition H is also useful for more general (multifractal) scaling processes although the common interpretation of H as the “Hurst exponent” is only valid for fBm in the usual fGn literature, the parameter H is the fluctuation exponent of it’s integral, fBm, i.e. it is larger by unity that that used here.

1.1 Anomalies

An anomaly is the average deviation from the long term average and since \(\left\langle {G_H^{\left( s \right)}\left( t \right)} \right\rangle = 0\), the anomaly fluctuation over interval Δt is simply G H at resolution Δt rather than τ:

$$\begin{array}{*{20}c} {\left( {\Delta G_{{H,\tau }}^{{\left( s \right)}} \left( {\Delta t} \right)} \right)_{{anom}} = \frac{1}{{\Delta t}}\displaystyle\int\limits_{{t - \Delta t}}^{t} {G_{{H,\tau }}^{{\left( s \right)}} \left( {t^{\prime}} \right)dt^{\prime} = } \frac{1}{{\Delta t}}\displaystyle\int\limits_{{t - \Delta t}}^{t} {G_{H}^{{\left( s \right)}} \left( {t^{\prime}} \right)dt^{\prime} = G_{{H,\Delta t}}^{{\left( s \right)}} \left( t \right)} ;} & {\Delta t> \tau } \\ \end{array}$$
(35)

Hence using Eq. 34:

$$\begin{array}{*{20}{c}} {\left\langle {\left( {\Delta G_{H,\tau }^{\left( s \right)}\left( {\Delta t} \right)} \right)_{anom}^2} \right\rangle = \Delta {t^{2H}};}&{ - 1 < H < 0} \end{array}$$
(36)

1.2 Differences

In the large Δt limit we have:

$$\begin{array}{*{20}{c}} {\left\langle {\left( {\Delta G_{H,\tau }^{\left( s \right)}\left( {\Delta t} \right)} \right)_{diff}^2} \right\rangle \approx 2{\tau ^{2H}}\left( {1 - \left( {H + 1} \right)\left( {2H + 1} \right){{\left( {\frac{{\Delta t}}{\tau }} \right)}^{2H}}} \right);}&{\Delta t>> \tau } \end{array}$$
(37)

Since H < 0, the differences asymptote to the value 2τ2H (double the variance). Notice that since H < 0, the differences are not scaling with Δt.

1.3 Haar fluctuations

For the Haar fluctuation we obtain:

$$\begin{array}{*{20}{c}} {\left\langle {\left( {\Delta G_{H,\tau }^{\left( s \right)}\left( {\Delta t} \right)} \right)_{Haar}^2} \right\rangle = 4\Delta {t^{2H}}\left( {{2^{ - 2H}} - 1} \right);}&{\Delta t \geqslant 2\tau } \end{array}$$
(38)

this scales as Δt 2H and does not depend on the resolution τ (Lovejoy et al. 2015a).

Since we will use Haar fluctuations throughout, it is convenient to define the fGn G H (t) with a nonstandard normalization replacing the constant c H in Eq. 30 by H :

$${c'_H} = \frac{{{c_H}}}{{2\sqrt {{2^{ - 2H}} - 1} }}$$
(39)

With this we can define \({G_{H,\tau }} = \frac{{G_{H,\tau }^{\left( s \right)}}}{{2\sqrt {{2^{ - 2H}} - 1} }}\) so that:

$$\begin{array}{*{20}{c}} {\left\langle {\left( {\Delta {G_{H,\tau }}\left( {\Delta t} \right)} \right)_{Haar}^2} \right\rangle = \Delta {t^{2H}};}&{\Delta t \geqslant 2\tau } \end{array}.$$
(40)

Appendix B: estimating the parameters of the measurement model

In this appendix, we describe how we estimated the statistics of the amplitudes of the measurement series noises (δu, B, ε, for the scale reduction factor, missing data and conventional measurement error respectively).

The idea is to use second order structure functions (Sect. 3), however from structure functions we can only estimate the squared quantities (δu 2, B 2, ε2). We therefore used an easily verifiable result, valid for a Gaussian random variable x:

$$\begin{array}{*{20}{c}} {\mu _x^{} = \pm {{\left( {\mu _{{x^2}}^2 - \frac{{\sigma _{{x^2}}^2}}{2}} \right)}^{1/4}}} \\ {\sigma _x^{} = {{\left( {\mu _{{x^2}}^2 - \mu _x^2} \right)}^{1/2}}} \end{array}$$
(41)

where \(\mu _x^{}\), \(\sigma _x^{}\) are respectively the means and standard deviations of x and \(\mu _{{x^2}}^{}\), \(\sigma _{{x^2}}^{}\) of x 2. Finally, the sign of \(\mu _x^{}\) is not determined. In the case of B, ε, this is unimportant since they are multiplied by sign symmetric random functions so that without loss of generality we can we take µ B > 0, µε > 0, but for δu, there is an ambiguity. However, since presumably the series are insufficiently averaged, we expect δu > 0 so that below, we use the plus sign.

The error in the squared fluctuation variance at each scale Δt is therefore:

$$\begin{gathered} S_i^2\left( {\Delta t} \right) - S_{}^2\left( {\Delta t} \right) = \delta u_i^2S_{}^2\left( {\Delta t} \right) + \sigma _T^2B_i^2\Delta {t^{2H}} + \sigma _T^2\varepsilon _i^2\Delta {t^{ - 1}} \\ = \sigma _T^2\varepsilon _i^2\Delta {t^{ - 1}} + \sigma _T^2\left( {\delta u_i^2 + B_i^2} \right)\Delta {t^{2H}} + {A^2}\delta u_i^2\Delta {t^2} \\ \end{gathered} $$
(42)

where St) is the ensemble averaged true earth structure function (see Eq. 25). Since at large Δt the Δt 2 term is dominant, regression of this equation against Δt 2 can conveniently be used to estimate \({\mu _{\delta u}} = 0.114\) and \({\sigma _{\delta u}} = 0.077.\) However the other terms are smaller and to obtain robust estimates it is advantageous to consider the pairwise differences as in Figs. 2, 3. Since there are six series, we have 6 × 5/2 = 15 pairs, giving us substantially more statistics with which to estimate the missing data and error amplitudes B i , ε i of the ith series (here, the index i runs from 1 to 6). Therefore, consider the differences between the ith and jth series of measurements:

$$\delta {T_{ij}}\left( t \right) = {\sigma _T}\delta {u_{ij}}G_H^{\left( 0 \right)}\left( t \right) + A\delta {u_{ij}}t + {\sigma _T}{B_{ij}}G_H^{\left( {ij} \right)}\left( t \right) + {\sigma _T}{\varepsilon _{ij}}G_{ - 1/2}^{\left( {ij} \right)}\left( t \right)$$
(43)

where \(\delta u_{ij}^2 = \delta u_i^2 + \delta u_j^2\) and we have used the mathematical result:

$$\begin{gathered} \begin{array}{*{20}{c}} {{B_{ij}}G_H^{\left( {ij} \right)}\left( t \right)\mathop = \limits^d {B_i}G_H^{\left( i \right)}\left( t \right) - {B_j}G_H^{\left( j \right)}\left( t \right);}&{B_{ij}^2 = B_i^2 + B_j^2} \end{array} \hfill \\ \begin{array}{*{20}{c}} {{\varepsilon _{ij}}G_{ - 1/2}^{\left( {ij} \right)}\left( t \right)\mathop = \limits^d {\varepsilon _i}G_{ - 1/2}^{\left( i \right)}\left( t \right) - {\varepsilon _j}G_{ - 1/2}^{\left( j \right)}\left( t \right);}&{\varepsilon _{ij}^2 = \varepsilon _i^2 + \varepsilon _j^2} \end{array} \hfill \\ \end{gathered} $$
(44)

where “\(\mathop = \limits^d \)” indicates equality in probability distributions (so that \(G_H^{\left( {ij} \right)}\left( t \right)\mathop = \limits^d G_H^{\left( i \right)}\left( t \right)\mathop = \limits^d G_H^{\left( j \right)}\left( t \right)\)). These results follow since sums and differences of independent Gaussian variables are also Gaussian and their variances add.

Therefore the fluctuations in the differences are:

$$\delta \Delta {T_{ij}}\left( {\Delta t} \right) = {\sigma _T}\delta {u_{ij}}\Delta G_H^{\left( 0 \right)}\left( {\Delta t} \right) + A\delta {u_{ij}}\Delta t + {\sigma _T}{B_{ij}}\Delta G_H^{\left( {ij} \right)}\left( {\Delta t} \right) + {\sigma _T}{\varepsilon _{ij}}\Delta G_{ - 1/2}^{\left( {ij} \right)}\left( {\Delta t} \right)$$
(45)

With this, squaring and averaging, we obtain for the corresponding squared structure function:

$$S_{ij}^2\left( {\Delta t} \right) = \overline {\delta \Delta {T_{ij}}{{\left( {\Delta t} \right)}^2}} = \sigma _T^2\varepsilon _{ij}^2\Delta {t^{ - 1}} + \sigma _T^2\left( {\delta u_{ij}^2 + B_{ij}^2} \right)\Delta {t^{2H}} + {A^2}\delta u_{ij}^2\Delta {t^2}$$
(46)

We can now estimate the parameters by regression of \(S_{ij}^2\left( {\Delta t} \right)\) on the fifteen i, j pairs of difference structure functions against \(\Delta {t^{ - 1}}\), \(\Delta {t^{2H}}\) (with H = −0.1) and \(\Delta {t^2}\). To make the problem numerically more robust, we used the fact that the trend A was estimated earlier from regressions on the individual series T i (t). Similarly, for each of the six S i t)2 functions, we estimated the trends A 2δu i 2; using the estimates for A this leads to estimates of μδu , σδu , δu ij 2 = δu i 2 + δu j 2. These trends were then removed to obtain the (quadratically) detrended difference structure function \(S_{ij,\det }^2\left( {\Delta t} \right) = \sigma _T^2\varepsilon _{ij}^2\Delta {t^{ - 1}} + \sigma _T^2\left( {\delta u_{ij}^2 + B_{ij}^2} \right)\Delta {t^{2H}}\); when regressed against \(\Delta {t^{ - 1}}\), \(\Delta {t^{2H}}\), these gave robust estimates of the prefactors \(\sigma _T^2\varepsilon _{ij}^2\) and \(\sigma _T^2\left( {\delta u_{ij}^2 + B_{ij}^2} \right)\). Combined with the trend based estimates of δu ij 2, we thus obtain 15 estimates for each of the random variables, \(\varepsilon _{ij}^2\), \(B_{ij}^2\). If we assume that the parameters are independent identically distributed random variables then Eq. 38 shows that:

$$\begin{array}{*{20}{c}} {B_{ij}^2\mathop = \limits^d 2B_i^2\mathop = \limits^d 2B_j^2} \\ {\varepsilon _{ij}^2\mathop = \limits^d 2\varepsilon _i^2\mathop = \limits^d 2\varepsilon _j^2} \end{array}$$
(47)

Therefore, we use the estimates of \(\varepsilon _{ij}^2\), \(B_{ij}^2\) to obtain estimates of the statistics of \(\varepsilon _i^2\), \(B_i^2\), and then from Eq. 35, by assuming the variables are Gaussian, we obtain estimates for the means and standard deviations of \(\varepsilon _i^{}\), \(B_i^{}\). For completeness, we give the means and standard deviations of δu i , obtained from S i (Δt) as explained earlier.

$$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\mu _{\delta u}^{} = 0.114;}&{\sigma _{\delta u}^{} = 0.077} \end{array}} \\ {\begin{array}{*{20}{c}} {\mu _B^{} = 0.347;}&{\sigma _B^{} = 0.175} \end{array}} \\ {\begin{array}{*{20}{c}} {\mu _\varepsilon ^{} = 0.132;}&{\sigma _\varepsilon ^{} = 0.062} \end{array}} \end{array}$$
(48)

(due to the ambiguity in the sign, we did not take the square root of Eq. 41 to more directly yield B i , ε i ). Since the different random variables are somewhat correlated, using the above equation yields the “effective” values needed for the simulations below. For completeness, recall that we have already estimated H = −0.1, A = (5.83 ± 0.073) ×10−4 K/month and σ T  = 0.142 ± 0.01 K (Eqs. 20, 21).

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Lovejoy, S. How accurately do we know the temperature of the surface of the earth?. Clim Dyn 49, 4089–4106 (2017). https://doi.org/10.1007/s00382-017-3561-9

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