Introduction
The temporal variations in Earth's surface temperature are well described as
scaling on an extended range of timescales. In this parsimonious
characterisation, a parameter β describes how the fluctuation levels on
the different timescales are related to each other. The β-parameter
can be defined via the scaling of the spectral density function of the signal
by the relation
S(f)=〈|T̃(f)|2〉∼f-β,
where T̃(f) is the Fourier transform of the time record T(t) and
〈…〉 denotes an ensemble average. An alternative is to
measure the range of the variability on the longest timescales within a time
window of length Δt by
TΔt(t)=2Δt∑i=tt+Δt/2T(t)-2Δt∑i=t+Δt/2t+ΔtT(t),
and to define β via the following relation :
〈|TΔt(t)|2〉∼Δtβ-1.
In this description, the temperature fluctuations would decrease with scale
if β<1, implying that the climate fluctuations become less prominent as
we consider longer timescales, a picture which is somewhat different from
the rich long-range variability indicated by proxy reconstructions of past
climate. On the other hand, a value β>0 would imply that variability
increases with scale, a property that (if it were valid on a large range of
timescales) would lead to levels of temperature variability inconsistent
with reality. It is therefore a natural a priori working hypothesis that
Earth's typical temperature fluctuations, the climate noise, is characterised
by β≃1. Such a process is called a 1/f noise.
The 1/f description of Earth's temperature is of course an idealised model.
The reality is that the climate system consists of many components that
respond to perturbations on different characteristic timescales, and the
temperature signal can be seen as an aggregation of signals with different
timescale characteristics. Since it is difficult to recognise pronounced
timescales in the temperature records, a scaling description is both
convenient and accurate. However, we are aware that the scaling is not
perfect, and that there are structures in the climate system that deviate
from the scaling law. One example is the El Niño Southern Oscillation
(ENSO), which places larger fluctuations on the times scales of a few years
than what can be expected from a scaling model. Other examples are the
Dansgaard–Oeschger (DO) cycles in the Greenland climate during the last
glacial period, encompassing repeated and rapids shifts between a cold
stadial state and a much warmer interstadial state. The result of this
phenomenon is that the glacial climate in Greenland has much larger
millennial-scale fluctuations than what can be expected from a 1/f
description. However, as we demonstrate in this paper, the temperature
variations of both the stadial and interstadial climate states fit well with
the 1/f-scaling, telling us that the deviation from 1/f scaling in the
glacial climate arise from these regime shifting events. As we go to even
longer timescales, we also observe anomalous fluctuation levels on timescales from 104 to 105 years that can be identified with the shifting
between glacial and interglacial conditions.
One could argue that the DO cycles and the glaciation cycles are intrinsic to
the climate system and should not be treated as special events, and their
variations should be reflected in a scaling description of the climate. This
idea was forwarded by , elaborated in many later papers,
and expanded to timescales up to almost 1 Gyr in . Here
several scaling regimes are proposed, including a “break” in
the scaling law with an exponent β≈1.8 on timescales longer
than a century. A scaling model invoking two scaling regimes can account for
the millennial-scale temperature fluctuations that are produced by the DO
cycles, which are anomalous with respect to a 1/f model. However, the
estimated scaling exponent will depend on the average “density” of DO
events in the ice-core record used for the estimate, and since the events are
not uniformly distributed over time, there is no uniquely defined scaling
exponent for the last glacial period. Moreover, the scaling law would not be
useful as a climate-noise model to use as a null hypothesis for determining
the significance of particular trends and events, such as the anthropogenic
warming over the last centuries.
The main message of this paper is that the 1/f noise characterisation of
the temporal fluctuations in global mean surface temperature is very robust.
It is an accurate description for the Holocene climate, but it is also valid
under both stadial and interstadial conditions during glaciations, and during
both glacial and interglacial conditions in the quaternary climate. The 1/f
character of the climate noise provides us with robust estimates of future
natural climate variability, even in the present state of global warming. Such an
estimate would of course be invalidated by a future regime shift (a tipping
point) to a warmer climate state provoked by anthropogenic forcing. A future
observed change in the 1/f character of the noise could therefore be taken
as an early warning signal for such a shift.
Data, methods and results
The analysis in this work is based on four data sets for temperature
fluctuations: the HadCRUT4 monthly global mean surface temperature
in the period 1880–2011 CE (Common Era), the Moberg
Northern Hemisphere reconstruction for annual mean temperatures in the years
1–1978 CE , as well as temperature reconstructions
from the North Greenland Ice Core Project (NGRIP) and
the European Project for Ice Coring in Antarctica (EPICA)
. For the NGRIP ice core we have used 20-year means of
δ18O going back 60 kyr. For the EPICA ice core we have
temperature reconstructions going back over 300 kyr, but the data are sampled
at uneven time intervals and the time between subsequent data points becomes
very large as we go back more than 200 kyr. In addition we have used annual
data for radiative forcing in the time period 1880–2011 CE
to remove the anthropogenic component in HadCRUT4 data.
Plots of all four data records are shown in Fig. 1.
(a) The δ18O concentration in the NGRIP
ice core dating back to 60 kyr before present (BP). Here present means AD
2000 (= 2000 CE). The data are given as 20-year mean values. The time
series are split into stadial (blue) and interstadial (red) periods.
(b) The temperature reconstruction from the EPICA ice core. The
shown time series are sampled with a time resolution of roughly 200 years. The
temperature curve in the glacial periods is given in a blue colour.
(c) The Moberg reconstruction for the mean surface temperature in
the Northern Hemisphere. The data are given with annual resolution.
(d) The HadCRUT4 monthly global mean surface temperature where the
anthropogenic component has been removed using a linear-response model.
Global versus local scaling
On the face of it, it is difficult to discern scaling laws for the climate
noise on timescales longer than millennia, since we do not have
high-resolution global (or hemispheric) temperature reconstructions for time
periods longer than two kyr. The ice core data available only allow us to
reconstruct temperatures locally in Greenland and Antarctica, and we know from
the instrumental record that local and regional continental temperatures
scale differently from the global mean surface temperatures on timescales
shorter than millennial. The differences we find are that local temperature
scaling exponents βl are smaller than global temperature
exponents βg, and that the ocean temperatures scale with
higher exponents than land temperatures. Since there are strong spatial
correlations in the climate system, it is possible that all local
temperatures are scaling with a lower exponent than the global. In
this phenomenon is illustrated in an explicit
stochastic spatio-temporal model. In this model, which is fitted to
observational instrumental data, we find the relationship βg=2βl. This relationship is derived under the highly inaccurate
assumption that all local temperatures scale with the same exponent, but it
is still a useful approximation in the following sections, where we will argue that we
can use local and regional temperature records to discern the scaling of the
global mean surface temperature on timescales of 10 kyr and longer. We do
this by showing that the assumption that βg/βl>1 is valid on very long times scales leads to the
impossible result that the variance of global averages becomes larger than
the mean variance of local averages. Thus we conclude that βl
converges to βg on a sufficiently long timescale, and we
estimate an upper limit for that timescale.
Let us denote by σg and σl the standard
deviations of the global surface temperature and a local temperature
respectively, on a monthly timescale. From Eq. () it follows that
the ratio between the variances for the global and local temperatures at timescale Δt is
ρ=(σgσl)2(Δtτ)βg-βl,
where τ=1month. Unless we expect global temperatures to have
larger variations than the local temperature at timescale Δt (the
global temperature can not have a larger standard deviation than the average
standard deviation of the local temperatures), we must have ρ>1, or
equivalently,
Δt<τ(σlσg)2/(βg-βl).
On the timescale of months, the fluctuation levels of local continental
temperatures is about two orders of magnitude larger than the fluctuation
level for the global mean temperature. If we also use βg=1
and βl=1/2 we obtain the condition Δt<105 months∼10 kyr, i.e. on timescales longer than 10 kyr the ratio
βg/βl can no longer be larger than unity. A
similar estimate can be obtained from the NGRIP ice core data. In the
Holocene the 20-year resolution temperature reconstructions from Greenland
has a standard deviation which is about five times greater than the 20-year
moving average of the Moberg reconstruction for the Northern hemisphere.
Applying the same argument restricts the timescale for which Greenland
scaling exponent is smaller than the global scaling exponent to approximately
10 kyr.
(a) The δ18O concentration in the NGRIP
ice core. The data are given as 20-year mean values. Two different parts of
the times series are shown. The blue curve represents the
δ18O concentration in a time period starting approximately
50 kyr before present (BP) and has a duration of approximately 8500 years.
As in Fig. , present means AD 2000 (= 2000 CE). The black curve
represents the δ18O concentration in a long stadial period
that started about 22 kyrs BP and has a duration of approximately 8500
years. (b) The wavelet scaling functions estimated from the two
parts of the NGRIP data set. The blue points are the estimates from the part
of the NGRIP ice core that is shown as a blue curve in (a), and
which contains DO cycles. The black points are the estimates from the
part of the NGRIP ice core that is shown as a black curve in (a),
and which does not contain any DO cycles.
Based on the reasoning above, we expect scaling of the ice core data to be
similar to the global scaling on sufficiently long timescales. In the
remainder of this paper we demonstrate that the scaling in the ice core data
on timescales up to hundreds of kyr is similar to the 1/f scaling we
observe in global temperature up to a few millennia. This suggests that the
1/f scaling on very long timescales in ice core data is a reflection of
the scaling in global temperatures on these scales.
Methods for estimation of scaling
We use two methods to analyse the scaling of temperature records. The first
is a simple periodogram estimation of the spectral power density S(f). This
estimator can also be applied to data with uneven time sampling using the
Lomb-Scargle method . The other method is to take the
wavelet transform of the temperature data:
WΔt(t)=1Δt∫T(t′)ψ(t-t′Δt)dt′
and construct the mean square of the wavelet coefficients: the wavelet
variance. This is a standard technique for estimating the scaling exponent
β , and it is known that
〈|WΔt(t)|2〉∼Δtβ.
We choose to use the so-called Haar wavelet
ψ(t)=1t∈[0,1/2)-1t∈[1/2,1)0otherwise,
and the integral in Eq. () is computed as a sum. With this wavelet we have the relation
WΔt(t)=TΔt(t)Δt,
between the wavelet transform and the Haar fluctuation of Eq. ().
The power spectral density and the wavelet variance are equivalent
representations of the second order statistics of the time record, one in
frequency domain and the other in time domain, and Eqs. () and
() show that they are characterised by the same exponent β if
there is scaling of the second moment. By the Wiener-Khinchin theorem, these
second-order moments are also equivalent to the autocorrelation function.
Hence, scaling in the second-order statistics plays a special role,
irrespective of the scaling or non-scaling of other moments.
(a) For each time series considered in this paper we show
double-logarithmic plots of the wavelet fluctuation 〈|W(t,Δt)|2〉 as a function of the timescale Δt. The green triangles
and the green circles represent the HadCRUT4 monthly global mean surface
temperatures with and without the anthropogenic component respectively. The
black circles are the analysis of the Moberg Northern Hemisphere
reconstruction. The analysis of the 20-year mean NGRIP data is shown as the
blue diamonds, the purple triangles and the red diamonds. The blue diamonds
show the results of the analysis of the entire data set dating back to
60 kyrs BP. The red diamonds are the results of the analysis preformed on
the stadial periods only, and the purple triangles are the results of the
analysis of the interstadial periods only. The results for the EPICA ice core
data are shown as the orange stars and the black crosses. The orange stars
are obtained by analysis of the entire data set dating back 200 kyrs, and
the black crosses are obtained by only analysing the two most recent
glaciations. The two solid lines have slopes β=1 and β=1.8.
(b) As in (a), but instead of the wavelet fluctuation
function we show the spectral density function S(f). The two solid lines
have slopes -β with β=1 and β=1.8.
The wavelet variance method can be adapted to the case of unevenly sampled
data using the method described in . In the present
work, we obtain very similar results using the periodogram and the wavelet
variance estimators. Claims have been made that higher-order statistics in
the form of a multifractal characterisation are an essential part of the
statistical description of these data (e.g. , Chapter
11). For this reason we include a brief analysis of higher moments of the
data in Sect. 2.4, and discuss their significance in Sect. 2.5.
Results of second-order analysis
In Fig. we show the wavelet fluctuation 〈|W(t,Δt)|2〉 estimated for two different segments of the NGRIP data. Both time
series have the same number of data points and both represent time intervals
of 8500 years. The differences between the two time series is that one
contains DO cycles, whereas the other does not. The estimated wavelet
fluctuations and the spectral density scale very differently for the two time
series, and this motivates us to separate stadial and interstadial conditions
when we analyse the scaling in NGRIP data. This separation is shown in
Fig. a, where the red curve represents the δ18O
concentration in interstadial periods and the blue curve represents the
δ18O concentration in stadial periods. We have followed
in defining the dates for the onsets of the
interstadials and we have defined the start dates for the stadial periods to
be just after the rapid temperature decrease that typically follows the slow
cooling in the interstadial periods. In Fig. we show the spectral
density function and the wavelet scaling function for the stadial data (red
diamonds) and the interstadial periods (purple triangles), which both display
an approximate 1/f scaling, but where the fluctuation variance in the
stadial data is larger than in the interstadial data. These results are
different from what is obtained when considering the NGRIP data (during the
last glaciation) as a single time series (shown as blue diamonds). If we were
to define a single scaling exponent for the whole time series, then we would
obtain an estimate β≈1.4.
Figure shows that the scaling of the stadial and interstadial
NGRIP data are similar to the scaling of global temperatures on shorter timescales during the Holocene. We have included an analysis of the instrumental
temperature record both with (green triangles) and without the anthropogenic
component (green disks). The anthropogenic component can be removed by
subtracting the response to the anthropogenic forcing in a simple linear
response model of the type considered in . We have also
included an analysis of the Moberg Northern Hemisphere reconstruction (black
squares), and we observe that the composite scaling wavelet variance function
and the composite spectral density function obtained by combining the
instrumental data with the Moberg reconstruction, is consistent with a 1/f
model on timescales from months to centuries. Since the NGRIP data also
show 1/f scaling, and since we believe that the scaling of the NGRIP data
is a reflection of global scaling on timescales longer than a millennium, it
is illustrative to adjust the fluctuation levels of the NGRIP data so that
its Holocene part has a standard deviation close to that of the standard
deviation of the 20-year means of the Moberg reconstruction in the same time
period. This means that we use the adjusted NGRIP data as a proxy for global
temperature on millennial scales. The effect of this adjustment is only a
vertical shift of the wavelet scaling function and the spectral density
functions in the double-logarithmic plots, so that it becomes easier to
compare the scaling of the NGRIP data with the Moberg reconstruction and the
instrumental data. We do not apply any adjustments of the fluctuation levels
of the stadial and interstadial periods relative to each other. The same
adjustment is applied to the EPICA data, and here we also consider the
scaling of the glacials and interglacials separately as shown in
Fig. b. The scaling estimated from the EPICA data for glacial
periods (black crosses in Fig. ) follows closely the scaling of the
NGRIP data analysed as a single time series (blue diamonds). This shows that
the glacial climates have similar characteristics in Greenland and in
Antarctica. Careful examination of the figure shows that the fluctuations grow
slightly faster with the scale Δt in the NGRIP time series than for
the glacial periods of the EPICA time series. This is expected since the
regime shifting events in Antarctica associated with the DO cycles are much
less pronounced than in Greenland . In the EPICA data
we cannot estimate a scaling exponent for the dynamics in periods without
regime shifts, but our results for the EPICA data are consistent with a
description of the climate as a 1/f climate noise plus regime shifts. If we
analyse the EPICA data without omitting the interglacials, then the
fluctuations increase even faster with the scale Δt (orange stars in
Fig. ). This effect is completely analogous to the effect of
shifting between the stadial and interstadial conditions during glaciations.
A note on multifractal processes
The exponent β is well-defined as long as the power spectral density
function S(f) is a power law in f, or equivalently if the wavelet
variance 〈|W(t,Δt)|2〉 is a power law in Δt. As
mentioned in Sect. 2.3, if well-defined, the β exponent is related to
the temporal correlations in the signal via simple formulas. In fact, for a
(zero-mean) stationary process T(t) with -1<β<1 we have 〈T(t)T(t+Δt)〉∼(β+1)βΔtβ-1 and for (a
zero-mean) process with stationary increments and 1<β<3 we have
〈ΔT(t)ΔT(t+Δt)〉∼(β-1)(β-2)Δtβ-3, where ΔT(t) is the increment
process of T(t) . Thus, the results presented so far in
this paper do not rely on any assumptions of self-similar or multifractal
scaling. It is only assumed that the second-order moments 〈|W(t,Δt)|2〉 are well approximated by power-laws over an
extended range of timescales.
A more complete scaling analysis can be performed if one imposes the more
restrictive assumption that the wavelet-based structure functions 〈|W(t,Δt)|q〉 are power-laws in Δt, not only for q=2,
but for an interval of q values. The time record can be classified as
multifractal only if this is true. It is then possible to define a scaling
function η(q) via the relation
〈|WΔt(t)|q〉∼Δtη(q).
defines a scaling function ξ(Δt) from the
moment 〈|TΔt(t)|q〉. From Eq. () we observe
that their scaling function is related to ours by η(q)=ξ(q)+q/2.
By Eqs. () and () we observe that η(2)=β. If
T(t) is self-similar (or if T(t) is the increment process of a
self-similar process in the case β<1) we have η(q)=βq/2,
but in general, the η(q) may be concave (it bends down). Processes that
exhibit power-law structure functions and strictly concave scaling functions
can be characterised as multifractal intermittent. A monofractal is a special
case of the multifractal class. For a monofractal (monoscaling) process, the
scaling function is linear in q.
(a) The estimated wavelet-based structure functions
〈|W(t,Δt)|q〉 for the HadCRUT4 monthly global mean
surface temperature where the anthropogenic component has been removed using
a linear-response model. The lines show the fitted power-law functions cqΔtτ(q). The q values are q=0.1,1.0,1.5,…,4.0.
(b) The scaling function τ(q) obtained from the fitted
power laws in (a). The line is a linear fit to the estimated scaling
function, and the slope of this line is β/2 with β=0.88.
(c–d) As (a) and (b) but in this case for the
Moberg Northern Hemisphere reconstruction. (e–f) As (a)
and (b) but for the interstadial periods in the NGRIP record.
(g–h) As (a) and (b) but for the stadial periods
in the NGRIP record. (i–j) As (a) and (b) but for
the ice age periods in the EPICA record. (k–l) As (a) and
(b) but for the NGRIP record including both stadial and interstadial
periods. The red curve in (l) is the scaling function estimated from
the longest timescales, the blue curve is the scaling function estimated
from the shortest timescales, and the green curve is the scaling function
estimated using all the available timescales.
In Fig. we present a crude multifractal analysis of the data sets
considered in this paper using q values in the range from 0.1 to 4. For
the Holocene we find linear scaling functions for both the instrumental
record and the Moberg Northern Hemisphere reconstruction, and in the NGRIP
data we find linear scaling functions for the stadial periods and the
interstadial periods when these are analysed separately, although, as we have
already seen, there is a deviation from the 1/f scaling in the stadial
periods for timescales shorter than about 200 years. If the NGRIP record is
analysed with both stadial and interstadial stages included, then it is not
clear how to define the scaling function since the shifts between the two
types of stages cause a “break” in the power-law scaling of the
wavelet-based structure functions. If we define η(q) using the timescales shorter than 2 kyr we obtain a linear scaling function corresponding
to β=1.14, and if we use the timescales longer than 4 kyr we obtain a
linear scaling function corresponding to β=1.78. In neither case do we
obtain a strictly concave scaling function. A linear scaling function is also
obtained if we disregard the “break” in the scaling and fit power laws
using all the available timescales. In this case the scaling function
corresponds to β=1.26. For the periods of the EPICA record that
corresponds to ice ages, we find wavelet-based structure functions that are
closer to power-laws than what is observed in the NGRIP record. This is
expected since the abrupt transitions between cold and warm periods is much
less pronounced in Antarctica than in Greenland. The scaling function for the
ice-age periods in the EPICA data is linear and corresponds to β=1.18.
The results discussed above show that from this analysis there is no evidence
of multifractal intermittency in the temperature records analysed in this
paper. This is not very surprising and could be suspected by direct
inspection of the data record. The trained observer would use the fact that
if η(q) is strictly concave, then the kurtosis of WΔt(t),
〈|WΔt(t)|4〉〈|WΔt(t)|2〉2∼Δtη(4)-2η(2),
is decreasing as a power-law function of Δt, and is therefore
leptokurtic on the shorter timescales Δt. Multifractal intermittency in addition implies that the
amplitudes of the random fluctuations are clustered in time, on all timescales, as observed in intermittent turbulence or financial time series (see
e.g. ). These are not prominent features in
the time series analysed in this paper. For the NGRIP data, the
δ18O ratio slightly deviates from a normal distribution as a
result of the DO events, but this is not well described by a multifractal
model since that would require the wavelet-based structure functions to be
power-laws in Δt. In fact, what we show in this paper is that the
effect of DO events is to break the scaling, rather than to produce
multifractal scaling.
Admittedly this multifractal analysis is a crude first-order
characterisation. Our crude analysis suggests that the records analysed are
most reasonably modelled as monofractal. However, to establish this with
confidence we need to perform statistical hypothesis testing. The strategy
for such testing must consist of two elements. First, we have to test whether
we can reject the hypothesis that the observed records are realisations of a
multifractal (with monofractal as a special case) stochastic process. If this
hypothesis can be rejected, there is no point in discussing whether the
process is multifractal or monofractal. If we cannot reject the multifractal
hypothesis, we must test if we can reject that this multifractal is a
monofractal. The outcome of these tests depends on the lengths of the
observed records, since rejection of the various null hypotheses depends on
the statistical uncertainty associated with realisations of the null models.
Monte Carlo simulations of these null models is the simplest tool to
establish these uncertainties. In a forthcoming paper we will perform this
rigorous testing of the multifractal hypothesis for the data analysed in the
present paper, in addition to a wide selection of forcing data and climate
model data. The results presented here should therefore be taken as
preliminary.
A note on non-fractal processes that scale in the second moment
In this paper we have focused on scaling in second-order statistics, or more
precisely, on modelling the temperature records as stochastic processes that
exhibit scaling of the second moment, but not necessarily of other moments.
Reviewer Shaun Lovejoy strongly opposes this approach. He considers it as a
return to old quasi-Gaussian ideas that disregards the developments of
multifractal formalism and multiplicative cascades, and in his last referee
comment he raises doubts about the existence of processes that exhibit
scaling in the second moment, but not in other moments. Here we will not only
demonstrate the existence of such processes, but explain that the serious
fallacy of Lovejoy's approach is that he fails to distinguish between
multifractal noises and non-Gaussian noises that cannot be modelled within the
multiplicative cascade paradigm. Examples of the latter is the large class of
Lévy noises. In less technical terms, the issue is that a multifractal
noise may consist of uncorrelated random variables (e.g. their signs may be
uncorrelated), but they will never be independent (e.g. their squares will
be correlated). A Lévy noise, on the other hand, consists of independent
random variables, which implies that all powers of the variables will be
uncorrelated. Empirical multifractal analysis methods typically fail to
distinguish between these different classes of processes because they
implicitly assume a multifractal model. Often, the distinction is not easy to
make, because a non-Gaussian Lévy noise may have a bursty (intermittent)
appearance, and analysis must be designed to separate multifractal clustering
(correlation in higher powers) from intermittency of non-Gaussian independent
variables. Long-range memory in the process does not make the distinction
less relevant. Such processes may easily be produced from those discussed
above by convolving the zero-memory processes with a memory response kernel.
From a physical viewpoint, it is very important to distinguish between these
two classes of stochastic processes. The multifractal processes are based on
a turbulent cascade paradigm and the dynamical description is fundamentally
nonlinear. The Lévy noises, and their long-memory cousins, may arise from
non-Gaussian, independent fluctuations on the short timescales, e.g. jumps
with randomly distributed waiting times.
We distinguish between a Lévy noise T(t) and a Lévy
process X(t). The latter is a continuous-time stochastic process
with stationary, identical and independently distributed (i.i.d.) increments,
i.e. for any τ the increments X(t+τ)-X(t) have a well-defined
distribution which is independent of t. The discrete-time process
T(t)=X(t+1)-X(t), where t is the set of nonnegative integers, is a
Lévy noise. Theoretical results on the fluctuation statistics on varying
timescales of Lévy noises are most conveniently obtained by means of the
standard structure functions of the underlying Lévy process (rather than
the wavelet structure function defined in Sect. 2.4), i.e. we define
Sq(Δt)≡〈|X(t+Δt)-X(t)|q〉.
For a process which belongs to the multifractal class we have
Sq(Δt)∼Δtζ(q),
where the scaling function ζ(q) is related to η(q) for the wavelet moments and ξ(q) for the Haar fluctuation by ζ(q)=η(q)+q/2=ξ(q)+q.
In Appendix A we show that for a Lévy process the following relations hold for the second and fourth moments;
S2(Δt)=〈Y2〉Δt,
S4(Δt)=3〈Y2〉2Δt2+13kurt[Y]-1Δt,
where Y≡X(1) and kurt[Y]≡〈Y4〉/〈Y2〉2 is the kurtosis (flatness) of Y. For a Gaussian process
kurt[Y]=3, and hence S4(Δt)∝Δt2. In this case
Sq(Δt)∝Δtq/2, X(t) is a Wiener process, and T(t)
is a Gaussian white noise. For a non-Gaussian Lévy noise, Eq. ()
provides the key to distinguish it from multifractal noise. For Δt∼kurt[Y]/3-1 there is a break in the scaling of S4(Δt).
In fact, for Δt≪kurt[Y]/3-1 moments higher than q=2 will
scale more or less like Δt1 (i.e. ζ(q)→1 for large
q), while for Δt≫kurt/3-1 they will scale like Δtq/2. The latter corresponds to the scaling of a Gaussian white noise,
which is quite obvious, since the random variables are independent and the
central limit theorem implies that the fluctuations are Gaussian on the long
timescales. On the other hand, on the short timescales when the
fluctuations are still non-Gaussian, the scaling function ζ(q) bends
over to become flat for large q, which is just the behaviour we find for
multifractals. Hence, by leaving out the scales Δt>kurt[Y]/3-1 from the analysis we will be led to the conclusion that
the non-Gaussian Lévy process is multifractal. The trace-moment analysis
employed by Shaun Lovejoy is designed to
conceal the scaling behaviour on these scales and is not suitable as a model
selection test to distinguish multifractals from non-Gaussian Lévy noises
or their long-memory derivatives.
In Fig. 5 we present an analysis of a synthetic jump-diffusion process, which
belongs to the class of Lévy noises. The details of this process are
explained in Appendix B. The second-order structure function is a power law
(a straight-line in the log-log plot), but the other structure functions are
not. If a scaling function is produced by fitting a straight line to the
structure functions on the long timescales, and computing the slopes, we
find the scaling function of a white Gaussian noise (the red line in
Fig. 5d). If the same is done on the short timescales, the estimated scaling
function is concave as one would expect for a multifractal (the blue curve).
In Fig. 6, we show the same for a jump-diffusion process with memory,
produced by convolving the Lévy noise with a memory kernel. Hence, the
difficulties related to distinguishing multifractals from other types of
non-Gaussian processes are not something that is limited to processes of
independent random variables.
(a) The increments of a jump-diffusion process shown in
(b). This is a non-Gaussian independent noise process.
(b) A realisation of a jump-diffusion process, and the cumulative
sum of the signal in (a). This process is the sum of a Brownian
motion and a Poisson jump process as described in Appendix B. The jump
distribution is Gaussian with a standard deviation that is 10 times greater
than the standard deviation of the increments of the Brownian motion.
(c) Sq(Δt for q=1,2,3 for the jump-diffusion process as
computed from a large ensemble of realisations of the process.
(d) Scaling function ζ(q) estimated from structure functions
like those in (c). The red line is estimated by computing the slope
of the structure-function curves on the longest timescale (Δt=500).
The blue curve is estimated from the slopes at the shortest timescale
((Δt=1). The black curve by estimating the slope of the straight
line drawn between the end points of the structure-function curves.
As Fig. 5, but for a jump-diffusion process with memory as
described in Appendix C. The parameter value β=0.4 is used.
Discussion and concluding remarks
Accurate characterization of the climate noise is essential for the detection
and evaluation of anthropogenic climate change. For instance, when we apply
standard statistical methods for estimating the significance of a temperature
trend, the result depends crucially on the so-called error model, i.e. the
model for the climate noise that is used as a null hypothesis. There is
strong evidence that the temperature fluctuations are better described by
scaling models than by so-called red-noise models (or AR(1)-type models).
However, simply characterising the climate noise as scaling does not specify
an error model. The exponent in the scaling law (the β parameter) must
also be determined, and it is usually determined from the same signal as we
are testing for trends. If we do that without detrending we risk estimating
a too high β for the error model, which yields a trend test with weak
statistical power, i.e. we may fail to detect a trend even if it is
present. It is possible to improve the statistical power in a logically
consistent way by detrending prior to estimating β, but the approach is
often (incorrectly) criticised for being circular, since β should be
estimated under the assumption that the error model (null hypothesis) is
true. However, since de-trending only has a small effect if the null
hypothesis is true, de-trending is valid under both the null hypothesis and
the alternative hypothesis.
Another approach, which is the motivation for this paper, is to characterise
the scaling of the climate noise from pre-industrial temperature records. If
we are to use the scaling exponent estimated from pre-industrial records to
demonstrate the anomalous climate event associated anthropogenic influence,
we must be confident that the temperature scaling does not change
significantly over time. We must also be confident that the scaling is
robust, in the sense that it is not too sensitive to moderate changes in the
climate state. The results presented in this paper suggest that, unless the
climate system experiences dramatic regime shifting events, we can be
confident that the natural fluctuations in global surface temperature is
approximated by 1/f-type scaling on a large range of timescales. This
result makes it easy to determine, on any timescale, if the observed
increase in global mean surface temperature is inconsistent with the natural
variability, and by how much.
The 1/f scaling described here is the same scaling observed in numerous
publications by Shaun Lovejoy and Daniel Schertzer, e.g.
. The important difference in
our interpretations is that they believe this scaling is limited to the scale
range from a few months to a century (the “macroweather regime”). Our
analysis suggests that this scaling is an expression of Nature's internal
“humming” in Quaternary surface temperature variability on all scales up to
hundreds of kyr. A natural conclusion drawn from this interpretation is that
description of the observed deviations from this scaling caused by DO events
and glacial/interglacial transitions should be sought in dynamical-stochastic
models rather than in general scaling laws .