Distribution of natural trends in long-term correlated records: A scaling approach

We estimate the exceedance probability $W(x,\alpha ;L)$ that, in a long-term correlated Gaussian-distributed (sub) record of length $L$ characterized by a fluctuation exponent $\alpha$ between 0.5 and 1.5, a relative increase $\Delta /{\sigma }_t$ of size larger than $x$ occurs, where $\Delta$ is the total observed increase measured by linear regression and ${\sigma }_t$ is the standard deviation around the regression line. We consider $L$ between 500 and 2000, which is the typical length scale of monthly local and reconstructed annual global temperature records. We use scaling theory to obtain an analytical expression for $W(x,\alpha ;L)$. From this expression, we can determine analytically, for a given confidence probability $Q$, the boundaries $\pm x_Q(\alpha ,L)$ of the confidence interval. In the presence of an external linear trend, the total observed increase is the sum of the natural and the external increase. An observed relative increase $\Delta /{\sigma }_t$ is considered unnatural when it is above $x_Q(\alpha ,L)$. In this case, the size of the external relative increase is bounded by $\Delta /{\sigma }_t\pm x_Q(\alpha ,L)$. We apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends.

Figure 1

Visualization of the considered quantities $\Delta$, ${\sigma }_t$, and $\alpha$ for a data set $T_i$ in a window of length 100. $\Delta$ is the total increase of the data $T_i$ in the time window and is measured by a linear regression line. ${\sigma }_t$ is the standard deviation around this regression line. The fluctuation exponent $\alpha$ characterizes the memory of the data $T_i$ in this time window. The relevant quantity in which we are interested is the dimensionless fraction $\Delta /{\sigma }_t$ when a certain value for $\alpha$ is measured.

Figure 2

Sketch of a probability density function $P\left(x\right)$. The integral over the white area defines the confidence probability $Q$. The lower and upper bounds $-x_Q$ and $x_Q$ define the confidence interval $[-x_Q,x_Q]$. If an event $x^{*}$ is outside this interval, it is considered unnatural. In this case, the minimum external relative trend is $x^{*}-x_Q$.

Figure 3

Visualisation of the data generation procedure. For each global $\alpha$ value, we generated several long data sets (long lines), which we divided into smaller subsequences of size $L$. In each of these subsequences, we measured the local $\alpha$ value by DFA2. Next we consider only those subsequences where the measured $\alpha$ value is in a certain range; for example, $\alpha =0.60\pm 0.01$ (marked subsequences). In all these sequences we measure the relative trend $x_i={\Delta }_i/{\sigma }_{t,i}$ as described in Fig. 1 to obtain the histogram $h\left(x\right)$ which, after normalization, yields the desired PDF $P(x,\alpha ;L)$.

Figure 4

(a) Probability density function $P(x,\alpha ;L)$ of the relative temperature increase $x=\Delta /{\sigma }_t$ of synthetic long-term correlated records in a time window $L=1000$. The circles represent the probability density function for $\alpha =1.0$ and the triangles are for $\alpha =0.7$. The dashed lines are Gaussians, which fit the data best for small arguments, while the straight lines are exponentials, which fit the data best for large arguments. (b)–(d) Cumulative probability $W(x,\alpha ;L)$ for $L=500$, 1000, and 2000 for fluctuation exponents $\alpha$ between 0.5 and 1.5 (from left to right).

Figure 5

Inverse slopes $x^{*}(\alpha ,L)$ of the exponential tails of $W(x,\alpha ;L)$ as a function of $\alpha$ for $L=500$, 750, 1000, 1500, and 2000 (circles from top to bottom, shifted downward for clarity by a factor of 1, 2, 4, 8, and 16, respectively). The values can be approximated by power laws (straight lines), where the $L$ dependence of the prefactor $C_L$ and of the exponent ${\delta }_L$ are shown in the insets (circles). Both parameters can be approximated by logarithmic functions (straight lines).

Figure 6

Cumulative probability $W(y,L)$ of the rescaled parameter $y=x/x^{*}(\alpha ,L)$ for $L=500$ (black), $L=1000$ (red), and $L=2000$ (yellow) for $\alpha$ between 0.5 and 1.2 (full lines). The curves for $\alpha =1.5$ are shown as dotted lines. The curves for $L=1000$ and 2000 have been shifted upward by a factor 2 and 4, respectively, for clarity. It can be seen clearly that the curves scale only for $\alpha ⩽1.2$. In the inset it is shown that $W(y,L)$ still has an $L$ dependence. $W(y,L)$ from Eq. (19), where $w_L$ has been obtained numerically, is shown as (blue) circles and (green) triangles. The agreement between the full lines and the symbols is very good.

Figure 7

$w_L^2$ as a function of $L$. The error bars are estimated by the error of $n_L{A}_L$ together with Eqs. (16) and (17). Within the error bars, $w_L^2$ can be approximated by $w_L^2=-6.32+1.41\ln \left(L\right)$.

Figure 8

Boundaries $x_Q(\alpha ,L)$ from Eq. (21) of the confidence interval as a function of $L$ for (a) $Q=95\%$ and (b) $Q=99\%$ for $\alpha$ between 0.5 and 1.2 (full lines) and between 1.3 and 1.5 (dashed lines) (from bottom to top). For comparison the numerically estimated values for $x_Q(\alpha ,L)$ are also shown (full circles). One can see clearly that Eq. (21) is only a very good approximation for $\alpha ⩽1.2$ or $L\approx 500$. For larger $\alpha$ values and $L>500$ one has to interpolate the numerical values.