Volatility of unevenly sampled fractional Brownian motion: An application to ice core records

The analysis of many natural time series and especially those related to ice core records often suffers from uneven sampling intervals. For fractional Brownian motion, we show that standard estimates of the volatility can be strongly biased due to uneven sampling. Taking these limitations into account, we study high-resolution records of temperature proxies obtained from Antarctic ice cores. We find that the volatility properties reveal a strong nonlinear component in the temperature time series for time scales of 5–200 kyr extending earlier results. These findings suggest in particular that temperature increments over these time scales appear in clusters of big and small increments—a big (positive or negative) change is most likely followed by a big (positive or negative) change and a small change is most likely followed by a small change.

Figure 1

(Color online) Time intervals $\Delta t_i=t_{i+1}-t_i$ between subsequent data points as a function of time $t_i$ for the Vostok ice core. The data set contains 3311 points and covers 422 kyr.

Figure 2

(Color online) Rescaled DFA1 for a single realization of FBM with $H=0.8$ (dashed line) and the derived $B_{0.8}^{V,R}\left(t_k\right)$ for different resolutions $R$ (solid lines): $R=25,50,100,200,400\text{yr}$, from top to bottom. The expected asymptotic behavior is a flat curve as highlighted by the dotted line.

Figure 3

(Color online) Rescaled DFA1 for the volatility series of a single realization of FBM with $H=0.8$ (dashed line) and of the derived $B_{0.8}^{V,R}\left(t_k\right)$ for different resolutions $R$ (solid lines): $R=25,50,100,200,400\text{yr}$, from top to bottom. The expected asymptotic behavior is a flat curve as highlighted by the dotted line which corresponds to ${\alpha }_V=2H-1=0.6$. The curves for $B_{0.8}^{V,R}\left(t_k\right)$ are shifted vertically for clarity.

Figure 4

(Color online) Local exponents ${\alpha }_V$ of the FBM volatility series with $H=0.3$, 0.5, 0.8, and 0.95 for different resolutions $R$. One-$\sigma$ error bars are given. The lines with symbols correspond to the case of the Vostok filter $\left[B_H^{R,V}\left(t_k\right)\right]$, while the lines without symbols correspond to the directly and evenly sampled case $\left[B_H^R\left(t_k\right)\right]$. There are clear differences between the two cases for resolutions finer than 500 yr. Note the different scales for the two rows.

Figure 5

(Color online) Local exponents ${\alpha }_V$ estimated from the modified volatility series as defined in Eqs. (6, 9) for different values of $H$ (lines with symbols). One-$\sigma$ error bars are given. The (orange) lines without symbols show the local exponents ${\alpha }_V$ for $B_H^{100}\left(t_k\right)$ ($H=0.95,0.8,0.5$ from top to bottom) for comparison because of their similar length.

Figure 6

(Color online) Local exponents ${\alpha }_V$ of the Vostok series shown as open symbols for resolutions $R=100\text{yr}$ (circles), 200 yr (squares), 500 yr (diamonds), and 1000 yr (triangles) obtained by DFA3. Closed symbols with error bars correspond to estimates obtained from 100 surrogates—see text for details. Inset: rescaled fluctuation functions of the Vostok series for the different $R$. Global fitting gives ${\alpha }_V=0.63,0.61,0.70,0.85$ for increasing $R$. For the surrogates, we find ${\alpha }_v=0.51\pm 0.03,0.50\pm 0.03,0.53\pm 0.04,0.55\pm 0.05$ for increasing $R$.

Figure 7

(Color online) Local exponents $\alpha$ of the Vostok series for different resolutions $R$ obtained by DFA3. Inset: respective rescaled fluctuation functions. Global fitting gives $\alpha =0.54,0.59,0.72,0.72$ for increasing $R$.

Figure 8

(Color online) Local exponents ${\alpha }_V$ of the modified volatility series of the Vostok data using $H=0.5$ (solid lines) and 0.6 (dotted lines) in Eq. (9) for different DFA orders. The inset shows the respective rescaled fluctuation function for $H=0.5$ indicating that ${\alpha }_V\approx 0.6$ globally.

Figure 9

(Color online) Rescaled fluctuation function of the EDC volatility series for different resolutions $R$ obtained by DFA3. Global fitting gives ${\alpha }_V=0.53,0.58,0.70,0.76,0.74$ for increasing $R$. For the surrogates (shown in Fig. 10), we find ${\alpha }_v=0.50\pm 0.02,0.50\pm 0.02,0.52\pm 0.03,0.53\pm 0.04,0.52\pm 0.04$ for increasing $R$.

Figure 10

(Color online) Local exponents ${\alpha }_V$ of the EDC series (open symbols) for different resolutions $R$ corresponding to the fluctuation functions shown in Fig. 9. The curves with closed symbols and error bars correspond to estimates obtained from 100 surrogates—see text for details.