In order to have a scaling description of the climate system that is not
inherently non-stationary, the rapid shifts between stadials and
interstadials during the last glaciation (the Dansgaard-Oeschger events)
cannot be included in the scaling law. The same is true for the shifts
between the glacial and interglacial states in the Quaternary climate. When
these events are omitted from a scaling analysis the climate noise is
consistent with a

The temporal variations in Earth's surface temperature are well described as

The

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

The main message of this paper is that the

The analysis in this work is based on four data sets for temperature
fluctuations: the HadCRUT4 monthly global mean surface temperature

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

Let us denote by

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

We use two methods to analyse the scaling of temperature records. The first
is a simple periodogram estimation of the spectral power density

The wavelet variance method can be adapted to the case of unevenly sampled
data using the method described in

In Fig.

Figure

The exponent

A more complete scaling analysis can be performed if one imposes the more
restrictive assumption that the wavelet-based structure functions

By Eqs. (

In Fig.

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

A distribution is leptokurtic if it has high kurtosis compared with a normal distribution. This means that the probability density function has a high central peak and fatter tails.

on the shorter timescalesAdmittedly 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.

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

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.

As Fig. 5, but for a jump-diffusion process with memory as
described in Appendix C. The parameter value

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

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

The

Let

The characteristic function of the random variable

A Poisson jump process is defined via the Lévy exponent

Let

This paper was supported by the Norwegian Research Council (KLIMAFORSK programme) under grant no. 229754. Edited by: M. Crucifix