## 1. Introduction

Quantification of climate variability requires, from a statistical point of view, the estimation of its full spatiotemporal covariance function. Given the nonstationarity of the variability, this is not only a function of spatial distance in either direction and of time lag; it also depends on space and time. Even for a two-dimensional spatial domain (e.g., for global temperature fields), this yields already a six-dimensional covariance function. Therefore, it is often desirable to define a simple metric that concisely summarizes the covariance structure—whether it be for climate model intercomparison or validation, for the formulation of stochastic forecasting models (Lovejoy et al. 2018), or for assigning a representative spatiotemporal scale to an observational data point (Smith et al. 1994), as needed for interpolation of spatially incomplete fields (Cowtan and Way 2014) or for data assimilation into paleoclimate models (Hakim et al. 2016).

To characterize the spatial structure, a common approach is to define a measure of the effective spatial degrees of freedom (ESDOF), to be interpreted as the effective number of independent spatial samples. Thus, the size of the entire spatial domain divided by this number equals the effective spatial extent of the associated fluctuations. Inherent to any ESDOF measure is a spatial averaging operator, implicitly removing any information on spatial nonstationarity. Effectively, the entire spatial correlation structure is condensed into a single number. A variety of ESDOF measures of different complexity have been presented in the literature (Livezey and Chen 1983; Smith et al. 1994; Jones et al. 1997; Wang and Shen 1999; Bretherton et al. 1999, and references therein).

To also consider the temporal structure, the variability may be filtered in the time domain before application of an ESDOF measure. Jones et al. (1997), for example, apply their ESDOF measure to annual mean and to decadal mean temperature fields, and find a reduction of the ESDOFs by roughly a factor of 2 toward decadal scales, both for observations and models. In qualitative agreement with this are the results of North et al. (2011) and Rypdal et al. (2015), who demonstrate that in a simple diffusive energy balance model (EBM) the decorrelation length scale increases toward lower frequencies, which is consistent with fewer ESDOFs at longer time scales. The latter study confirms this result also for a climate model simulation over a range of frequencies from annual up to centennial time scales. These findings motivate us to formulate a frequency-dependent ESDOF measure as a simple summarizing metric of climate variability.

Section 2 illustrates the frequency dependence of the spatial correlation structure, using a paleoclimate model simulation. This further motivates the formulation of a frequency-dependent ESDOF measure to be introduced in section 3. Application examples are presented in section 4, followed by a discussion of the measure in section 5. The conclusions and potential future applications are itemized in section 6.

## 2. Frequency dependence of the spatial correlation structure

This section illustrates the fact that the spatial covariance structure of climate variability may indeed exhibit a notable frequency dependence. This is demonstrated, in particular, for near-surface temperature variability generated by a paleoclimate model simulation, which allows us to investigate a wide range of frequencies from monthly to millennial time scales.

Specifically, we use monthly mean 2-m temperature fields from the Transient Climate of the Last 21,000 Years (TraCE-21ka) coupled general circulation model simulation (Liu et al. 2009; He 2011) with T31 atmospheric horizontal resolution. We restrict our analysis to the preindustrial Holocene by selecting the last 7000 years under preindustrial conditions (i.e., up to 1890 CE) of the simulation, including only natural (i.e., nonanthropogenic) external forcings [for a detailed description of the forcings, see He (2011)]. Finally, the obtained time series are linearly detrended at each grid point.

From these data we compute the spatial correlation function, defined as *R*(**r**, **r**′) is the covariance between a reference location **r** and any other location **r**′ on the globe, and *σ*^{2}(**r**) = *R*(**r**, **r**) is the local variance. To demonstrate frequency dependence, correlations are estimated from the data within narrow frequency bands centered at various frequencies *ν*. These frequency-dependent correlation estimates are based on the smoothed and normalized cospectral density, as detailed in section 3d, including only information on simultaneous correlation. Figure 1 shows the spatial correlation function for three different frequencies, which correspond to millennial, multidecadal, and interannual time scales, and for three selected reference locations. These are located over the northern North Atlantic, over central Europe, and over the western tropical Pacific.

It turns out that the area that is positively correlated with the reference location (red and yellow colors), which characterizes the spatial extent of the temperature fluctuations, increases with time scale. In addition, the spatial extent varies with latitude, most strikingly at interannual time scales. In this frequency band, the largest positively correlated area is found for the tropical reference location, extending over most of the tropical belt around the globe, in contrast to the much smaller decorrelation length scales found in the extratropics. These findings are consistent with the results of Jones et al. (1997, see their Figs. 3 and 4), comparing decorrelation length scales at interannual and decadal time scales, obtained from both observations and general circulation models. Furthermore, for some reference locations, most clearly for the tropical one at interannual and multidecadal time scales, the spatial correlation function exhibits a large degree of anisotropy, which is also in agreement with the analysis of model temperature data by Jones et al. (1997, see their Fig. B1).

To quantify the spatial extent of the temperature fluctuations, Jones et al. (1997) fit an exponential decay function to the estimated correlation as a function of spatial distance. An alternative measure of the spatial extent is the global average (or integral) of the spatial correlation function. An advantage of this measure is its insensitivity to anisotropic structures. In Fig. 1, the global average (with respect to **r**′) of the spatial correlation function for a particular reference location **r**, denoted by **r**.^{1} The obtained values reconfirm the above findings regarding the dependence of the spatial extent on frequency and location. For the reference location over the northern North Atlantic (top row), for example,

In addition to this frequency dependence of the effective spatial extent, Fig. 1 also reveals an increased occurrence of anticorrelations at short time scales. In particular, at interannual time scales (right column), considerable areas are anticorrelated with the reference location (blue colors), preferably in the extratropics. Although some of those negative correlation estimates might not be statistically significant, it suggests that at those time scales and locations the correlation structure is notably influenced by teleconnections, such as the North Atlantic Oscillation, Pacific–North American pattern, or El Niño–Southern Oscillation (ENSO) phenomenon [see, e.g., Wallace and Gutzler (1981) or Feldstein and Franzke (2017) for an overview of atmospheric teleconnections]. In some cases this may even lead to a negative global average of the correlation function, an example of which is given by the reference location over central Europe at interannual time scales (Fig. 1f). In these cases the global average of the spatial correlation function has a different interpretation, as to be explained in the next section in the context of the ESDOF measures.

*ρ*(

**r**,

**r**′) are shown, with fixed

**r**and varying

**r**′, Fig. 2 (top row) shows maps of

**r**, again for the same three frequencies as before. These results confirm the previous findings, namely that (i) the effective spatial extent increases with time scale, (ii) it increases toward the tropics, particularly at interannual time scales, and (iii) anticorrelations occur preferably at these shorter time scales and in the extratropics, suggesting an involvement of teleconnections. At the top of each panel in Fig. 2 (top row) the global average of

**r**′ and the second one to

**r**. From the specified values, the fluctuations at millennial, multidecadal and interannual time scales have an average spatial extent that fits roughly 4, 10, and 25 times onto the globe, respectively.

(top) Maps of the global average of the spatial correlation function *D*_{ρ} and *D*_{R}, respectively, to be defined in section 3a.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

(top) Maps of the global average of the spatial correlation function *D*_{ρ} and *D*_{R}, respectively, to be defined in section 3a.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

(top) Maps of the global average of the spatial correlation function *D*_{ρ} and *D*_{R}, respectively, to be defined in section 3a.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

**r**′ and the second one to

**r**, and

*σ*

^{2}(

**r**). Because

The effect of this variance weighting can be seen in Fig. 2 (bottom row), showing maps of **r′**) of *R*(**r**, **r**′). Smallest values are preferably found in the tropics, particularly at millennial to multidecadal time scales, and largest values in the northern extratropics, which is consistent with typical spatial variance patterns of near-surface temperature. The global average of this effective spatial extent is equal to

## 3. Effective spatial degrees of freedom

In this section a simple ESDOF measure is defined and its interpretation is provided in terms of idealized spatial fields. Subsequently, estimators of slightly different variants of the measure are presented and their basic properties are specified. Finally, the measures are equipped with an explicit dependence on frequency, allowing for the estimation of ESDOF-frequency spectra, which can then serve to infer the basic space–time scaling behavior of climate variability.

### a. Definition

Various ESDOF definitions have been proposed in the literature, each of which has its specific interpretation. Accordingly, a number of ESDOF measures of different complexity have been presented, depending on the purpose of application. These range from simple variance ratios computed directly from the time-varying spatial fields [as used by, e.g., Smith et al. (1994) and Jones et al. (1997)] to ratios of higher-moment statistics, sometimes involving principal component analysis, and moment matching approaches [see Wang and Shen (1999) and Bretherton et al. (1999), and references therein, for an overview]. Here we are adopting one of the simplest ESDOF definitions that is based on a variance ratio and that is directly related to the average spatial extent of the fluctuations discussed in the previous section.

For a precise definition, consider a zero-mean random field *X*(**r**), defined on a continuous spatial domain *A*, where *A* is an area and the differential *X* (**r**) could be a global temperature field and *A* the surface area of Earth. The spatial covariance function can then be expressed as *x*(**r**).

*A*of the domain. Thus, its reciprocal can be interpreted as the effective number of independent spatial samples. Accordingly, a simple ESDOF measure can be defined as

^{2}Thus, the ESDOF measure

*D*

_{R}is given by the ratio of the spatial average of the local variance to the variance of the spatial average, and it is identical to the measure of Smith et al. (1994) [see their Eq. (7)] and of Jones et al. (1997) [see their Eq. (10)]. Alternatively, an ESDOF measure can be defined based on the averaged spatial correlation function

*D*

_{R}=

*D*

_{ρ}, where here and in the following stationarity refers to the spatial dimension (unless otherwise noted). For fields with nonstationary variance, the measure

*D*

_{R}depends on the spatial distribution of the local variance, as discussed in the previous section in the context of

*D*

_{ρ}can be obtained by standardizing the field at each location, as detailed in section 3c.

*e*-folding length scale of the correlation function. In particular, for spherical domains, where the distance between two locations

**r**and

**r′**can be expressed in terms of their angular distance

*θ*, one can define the radial covariance function

**r**. Here, the operator

**r′**with the same angular distance

*θ*relative to

**r**. With this, one can then define the normalized average radial covariance function,

**r**as previously defined. Likewise one can define the radial correlation function

*θ*> 0.

*θ*∈ [0,

*π*], then the radial functions

*D*

_{R}and

*D*

_{ρ}, respectively. Specifically, by defining the global, area-weighted mean of any radial function

*x*(

*θ*) over the interval [0,

*π*] as

*D*

_{R}and

*D*

_{ρ}are readily obtained, according to (3) and (5), respectively. Now, in order to construct an alternative ESDOF measure, one can fit a parametric correlation function, for example, an exponential decay function, to either

*π*]. Note that for random fields with stationary variance,

In the present study, we use *D*_{R} as our main ESDOF measure. We compare *D*_{R} to *D*_{ρ} to demonstrate possible effects of data standardization, and we also compare *D*_{R} to *D*_{R,fit} to illustrate how various aspects of the radial correlation structure may affect the measure *D*_{R} in different ways. Furthermore, because *D*_{R} is identical to the first ESDOF measure of Jones et al. (1997) [see their Eq. (10)], we can compare estimates of this measure between the two studies. Finally, one can loosely compare estimates of *D*_{R,fit} to the second ESDOF measure of Jones et al. (1997) [see their Eq. (14)] because it corresponds to the closely related measure *D*_{ρ,fit}. An overview of the various measures and their relation to those of Jones et al. (1997) is provided by Table 1.

List of ESDOF measures defined in the present study and their correspondence to those previously defined by Jones et al. (1997).

### b. Interpretation

#### 1) Sensitivity to the correlation structure

A few idealized example cases are presented to illustrate the interpretation of the ESDOF measures *D*_{R} and *D*_{ρ} for different spatial correlation structures. Since only the correlation structure varies between these cases, we can consider fields with stationary variance, such that *D*_{R} = *D*_{ρ}, and simply write *D* for the ESDOF measure. We distinguish between cases without and cases with anticorrelations.

As an example without anticorrelations and with stationary correlation function, consider the idealized case of a random field defined on the entire sphere, and which has a rotationally symmetric correlation function, falling off from one at the reference location to zero at the opposite point on the sphere, according to a cosine bell-shaped structure [i.e., according to cos^{2}(*θ*/2)], in which case *A*/2, which fits *D* = 2 times onto the sphere. As an example with nonstationary correlation function, consider the idealized case of coherent but mutually uncorrelated hemispheres, that is, with a correlation of one for all pairs of locations within a hemisphere, but with zero interhemispheric correlation. Again *A*/2. Likewise, one could divide the entire domain *N*_{box} equally sized and uncorrelated boxes, such that *D* = *N*_{box}.

In cases without anticorrelations, it can be useful to introduce an associated length scale measure *L*. For spherical domains, it may be defined as the radius of a spherical cap, the area of which covers that fraction of *A* that is equal to *R*_{E} is the radius of Earth, and *A* relative to the surface of the entire sphere.

*A*= 1 that is divided into two subsections of size

*a*and 1 −

*a*(with 0 ≤

*a*≤ 1), and let all locations within either subsection have a correlation of

*ρ*(

**r**,

**r**′) = 1, and all cross correlations between the subsections be equal to

*ρ*(

**r**,

**r**′) =

*r*(with −1 ≤

*r*≤ 1). Performing the two spatial averages involved in (1), or likewise (2), we obtain

*r*= 0 and

*a*= 1/2, this corresponds to the previous case with

*D*=

*N*

_{box}= 2. For

*r*< 0 and 0 <

*a*< 1, the correlation structure represents an idealized teleconnection dipole covering the domain

*a*= 1/2 yields

*D*= 2/(

*r*+ 1), which shows that for negative

*r*the ESDOF measure grows beyond

*D*= 2, and for

*r*= −1, corresponding to a “perfect teleconnection dipole” setting, it is not defined. Thus,

*D*does not behave like a conventional ESDOF measure in this case. In cases with perfectly anticorrelated subsections,

*r*= −1, the measure is rather sensitive to the spatial partitioning

*a*of the teleconnection dipole,

*D*= [

*a*− (1 −

*a*)]

^{−2}. Hence, whenever significant anticorrelations occur in the random field, then

*D*is not an ESDOF measure in the conventional sense, but a more general summarizing metric of the spatial correlation structure. The usefulness of such a metric, for applications to fields describing climate variability, is demonstrated in section 4.

Figure 3 illustrates a few of the abovementioned idealized cases with stationary variance, and how the corresponding value of *D* comes about. Specifically, a case with a stationary Matérn class correlation function is shown in Fig. 3a, a case with a nonstationary (box-shaped) correlation function, but with the same value of *D*, is shown in Fig. 3b, and an example of an idealized teleconnection dipole in Fig. 3c.

Schematic illustration of different spatial correlation structures, shown, for simplicity, for the case of a one-dimensional, continuous, and cyclic spatial domain of size *A* = 1. (a) Scenario with a Matérn class correlation function *ρ*(*x*, *x*′) that is stationary in space; i.e., it is identical for any reference location *x*. Colored dots indicate 21 example reference locations, for every third of which the correlation function is shown by a line with the same color. For *x* = 1/2 (blue symbols) the integral of *ρ*(*x*, *x*′) is illustrated by shading, and its value is specified as a number. This scenario has *D* = 7 spatial degrees of freedom. (b) As in (a), but for a nonstationary scenario with seven equally sized and uncorrelated boxes with *D* = 7. (c) Scenario of a spatial domain split into two anticorrelated sections, representing an idealized teleconnection dipole. It corresponds to the setting *a* = 2/3, *r* = −1, with *D* = 9 according to Eq. (10); with reference locations shown only for *x* < *a*.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

Schematic illustration of different spatial correlation structures, shown, for simplicity, for the case of a one-dimensional, continuous, and cyclic spatial domain of size *A* = 1. (a) Scenario with a Matérn class correlation function *ρ*(*x*, *x*′) that is stationary in space; i.e., it is identical for any reference location *x*. Colored dots indicate 21 example reference locations, for every third of which the correlation function is shown by a line with the same color. For *x* = 1/2 (blue symbols) the integral of *ρ*(*x*, *x*′) is illustrated by shading, and its value is specified as a number. This scenario has *D* = 7 spatial degrees of freedom. (b) As in (a), but for a nonstationary scenario with seven equally sized and uncorrelated boxes with *D* = 7. (c) Scenario of a spatial domain split into two anticorrelated sections, representing an idealized teleconnection dipole. It corresponds to the setting *a* = 2/3, *r* = −1, with *D* = 9 according to Eq. (10); with reference locations shown only for *x* < *a*.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

Schematic illustration of different spatial correlation structures, shown, for simplicity, for the case of a one-dimensional, continuous, and cyclic spatial domain of size *A* = 1. (a) Scenario with a Matérn class correlation function *ρ*(*x*, *x*′) that is stationary in space; i.e., it is identical for any reference location *x*. Colored dots indicate 21 example reference locations, for every third of which the correlation function is shown by a line with the same color. For *x* = 1/2 (blue symbols) the integral of *ρ*(*x*, *x*′) is illustrated by shading, and its value is specified as a number. This scenario has *D* = 7 spatial degrees of freedom. (b) As in (a), but for a nonstationary scenario with seven equally sized and uncorrelated boxes with *D* = 7. (c) Scenario of a spatial domain split into two anticorrelated sections, representing an idealized teleconnection dipole. It corresponds to the setting *a* = 2/3, *r* = −1, with *D* = 9 according to Eq. (10); with reference locations shown only for *x* < *a*.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

#### 2) Lower and upper bounds

*D*

_{R}and

*D*

_{ρ}attain their lowest possible values, denoted by min(

*D*

_{R}) and min(

*D*

_{ρ}), respectively (under variation of the spatial correlation function, given the spatial distribution of the local variance), if the variability is spatially coherent. Specifically,

*ρ*(

**r**,

**r**′) = 1 for all

**r**,

**r′**with

*σ*

^{2}(

**r**),

*σ*

^{2}(

**r**′) > 0, and

*ρ*(

**r**,

**r**′) = 1 for all

**r**,

**r′**. The lower bound of

*D*

_{R}depends on the spatial distribution of

*σ*(

**r**). For example, if

*σ*(

**r**) = 1 over one hemisphere and zero over the other, then min(

*D*

_{R}) = (1/2)/(1/4) = 2.

Corresponding upper bounds, on the other hand, do not exist. In cases with **r**, **r**′, both measures may become arbitrarily large (*D*_{ρ}, *D*_{R} → ∞) for random fields approaching continuous spatial white noise. However, we may also have *D*_{ρ}, *D*_{R} → ∞ in cases with nonzero decorrelation lengths if the random field exhibits anticorrelations, *ρ*(**r**, **r**′) < 0 for some **r**, **r**′, that is, if there is strong cancelation between positive and negative fluctuations (e.g., a perfect teleconnection dipole). This latter case occurs, for instance, if the variability of the underlying field *X* arises exclusively from redistribution within the spatial domain *D*_{ρ} and *D*_{R} are meaningful (and, thus, applicable) only if

#### 3) Sensitivity to spatial redistribution

In general, however, the presence of a process associated with redistribution does not necessarily imply an (unbounded) increase of the ESDOF measures. In particular, when the process of redistribution is combined with a nonconservative process (in the sense that it does not conserve

*η*(stationary in space and time, implying

*D*

_{R}=

*D*

_{ρ}), and is modified by local thermal damping with time scale

*τ*, and by horizontal diffusion with length scale

*λ*. The temporal evolution of this system is determined by the equation

If the white noise forcing *η* is truncated at a given wavenumber *N*_{T} (to ensure that spatial averaging over the random fluctuations leaves a nonzero residual, such that *λ* = 0), then the resulting field of *X* is associated with finite *D*_{R} and *D*_{ρ} according to the finiteness of *N*_{T}. However, if diffusion is included (*λ >* 0), then the variance of the local variability, *D*_{R} (and, thus, *D*_{ρ}) is also reduced. If, on the other hand, diffusion is replaced by another process of redistribution that is stochastically independent of the forced variability, then *D*_{R} and *D*_{ρ} in this case.

### c. Estimation

*i*= 1, …,

*N*, that is given at a finite number,

*N*, of positions

**r**

_{i}, being a discrete subset of the spatial domain

*x*

_{i}is then given by the weighted sum

*w*

_{i}are normalized according to

*D*

_{R}as

*β*

_{R}− 1, where

*β*

_{R}= 1 and

*δ*

_{R}= 0 if

*D*

_{R}= min(

*D*

_{R}). For the case of the opposite limit, that is, if

*D*

_{R}→ ∞), appendix B shows that

*β*

_{R}and

*δ*

_{R}increase monotonically, but stay below an upper bound, given by

*κ*(>4) denotes the degrees of freedom of the variance estimates. Here

*β*

_{R,max}is the expected value and

*χ*

^{2}(

*κ*, 1) distribution. These upper bounds are based on the additional assumption that

*κ*= 8.

*D*

_{ρ}from data with nonstationary variance, one may use the standardized time series

*X*

_{i}, to define the estimator of

*D*

_{ρ}as

*D*

_{ρ}≤

*N*.

*θ*, whereas the discrete locations

**r**

_{i}are generally not aligned in circles around each other. However, this issue can be circumvented by transforming the spatial fields into wavenumber space. Specifically, for the case of discrete fields covering the entire sphere, we can decompose the discrete fields into their spherical harmonic components (with total wavenumber

*n*and zonal wavenumber

*m*) and then estimate the variance

*n*th-order Legendre polynomial

^{3}and

*N*

_{T}is the truncation wavenumber (with, for example,

*N*

_{T}= 31 in the case of the TraCE-21ka paleoclimate model simulation). This approach shifts the discretization in the spatial dimension to a discretization in wavenumber space, and the spatial dimension appears now in the analytically given Legendre polynomials. With this we can write the estimator of the normalized average radial covariance function (for fields covering the entire sphere):

*D*

_{R,fit}as

### d. Frequency dependence

#### 1) Definition

In the following, the above ESDOF measures are equipped with an explicit dependence on frequency. For this purpose, the variances *σ*^{2}(**r**) and *ν*), assuming the field is stationary in time. By analogy with the relation *ν*, given by the real part of the cross-spectral density *ν* is given by

*D*

_{R}, by analogy with (4), as

*D*

_{ρ}, by analogy with (5), as

*ν*, then

*D*

_{R}(

*ν*) =

*D*

_{ρ}(

*ν*) at that frequency. Note, that the frequency-independent measure

*D*

_{R}can be recovered from the frequency-dependent version by integration, performed separately on the numerator and the denominator:

*D*

_{R}valid for some limited frequency band can be obtained by integration over that frequency range. Note, however, that

*D*

_{ρ}can similarly be recovered from

*D*

_{ρ}(

*ν*) only if

*X*(

**r**) is white noise (in time) at each location

**r**. By analogy with (11) and (12), the lower bounds of these measures are given by

**r**,

**r**′ with

**r**,

**r**′. Finally, the frequency-dependent version of the normalized average radial covariance function can be defined, by analogy with (6), as

*D*

_{R,fit}, by analogy with (8), as

#### 2) Estimation

*ν*. The relative bias and scatter of this estimator are identical to those of the frequency-independent version, given by (15) and (16), respectively, with the exception that

*κ*now denotes the degrees of freedom of the spectral density (rather than variance) estimates.

*ν*of the spatial average of the standardized

*X*

_{i}, but now standardized and averaged in the Fourier domain. Specifically, it is estimated using the standardized Fourier transforms

*ν*

_{k}that fall into the spectral smoothing window (centered at

*ν*) involved in the estimation of

*D*

_{ρ}(

*ν*) directly using all the correlations

*ρ*

_{ij}(

*ν*), that is, according to

#### 3) Illustration

*X*into spherical harmonics (see North et al. 2011; Rypdal et al. 2015), we obtain an expression for

*D*

_{R}of the North-EBM,

*n*and the truncation wavenumber

*N*

_{T}. To obtain (31) from (20), note, that, for the case of the North-EBM, (i) the power spectral density of the local variability

*n*= 0 of the same expression.

The above ESDOF spectrum (31) is shown in Fig. 4 (solid black line), for the North-EBM scaled to the dimensions of a sphere with the radius of Earth, *R*_{E}, and with the damping time scale set to *τ* = 10 years and the diffusion length scale to *λ* = 1.25 × *R*_{E}, and using a truncation wavenumber of *N*_{T} = 31 (corresponding to T31 horizontal resolution). The ESDOF spectrum is shown for a frequency range from millennial to monthly time scales. It turns out that the ESDOFs are strongly decreasing, from a few hundred at monthly time scales to just above one ESDOF at multidecadal time scales, following a power-law scaling (i.e., ~*ν*^{p}) with exponent *p* = 1, and remain constantly low at centennial and longer time scales according to a scaling exponent of *p* = 0. The break between these scaling regimes is located at a time scale of 2*πτ* ≈ 63 years for the above parameter choice. In the undamped regime *n*) are diffused away more quickly than larger-scale structures. Thus, at higher frequencies, there are more small-scale structures contributing to the total variance of the local variability, relative to the variance contribution from the global average (given by the spherical harmonic with *n* = 0, which is not affected by diffusion at all). In the damped regime *τ* represents the longest time scale of the system (such that in this regime, all spherical harmonics have the same frequency scaling with *p* = 0 in accordance with the white noise forcing).

ESDOF measure *D*_{R}(*ν*) (left vertical axis) and associated length scale *L* (right vertical axis) as a function of frequency *ν*, for the North-EBM. The true value *D*_{R(ebm)}(*ν*), according to Eq. (31), is shown by the solid black line, and the expected biased value, *β*_{R}*D*_{R(ebm)}(*ν*), by the central dashed black line. The upper and lower dashed black lines mark the upper and lower 5% quantiles of a scaled-inv-*χ*^{2}[*κ*, *D*_{R(ebm)}(*ν*)] distribution, respectively. Indicated bias and scatter represent their expected upper bounds in the limit of large *D*_{R}. The red line shows

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

ESDOF measure *D*_{R}(*ν*) (left vertical axis) and associated length scale *L* (right vertical axis) as a function of frequency *ν*, for the North-EBM. The true value *D*_{R(ebm)}(*ν*), according to Eq. (31), is shown by the solid black line, and the expected biased value, *β*_{R}*D*_{R(ebm)}(*ν*), by the central dashed black line. The upper and lower dashed black lines mark the upper and lower 5% quantiles of a scaled-inv-*χ*^{2}[*κ*, *D*_{R(ebm)}(*ν*)] distribution, respectively. Indicated bias and scatter represent their expected upper bounds in the limit of large *D*_{R}. The red line shows

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

ESDOF measure *D*_{R}(*ν*) (left vertical axis) and associated length scale *L* (right vertical axis) as a function of frequency *ν*, for the North-EBM. The true value *D*_{R(ebm)}(*ν*), according to Eq. (31), is shown by the solid black line, and the expected biased value, *β*_{R}*D*_{R(ebm)}(*ν*), by the central dashed black line. The upper and lower dashed black lines mark the upper and lower 5% quantiles of a scaled-inv-*χ*^{2}[*κ*, *D*_{R(ebm)}(*ν*)] distribution, respectively. Indicated bias and scatter represent their expected upper bounds in the limit of large *D*_{R}. The red line shows

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

The expected biased estimate in the limit of large *D*_{R} is given by *κ* = 18.9 degrees of freedom, consistent with a modified Daniell spectral smoothing window with 11 weights, which we also used for *D*_{R} (at high frequencies) and their waning in the opposite limit (at low frequencies).

### e. Summary

A frequency-dependent ESDOF measure, denoted

## 4. Application to climate model temperature and precipitation fields

This section demonstrates the applicability of the previously defined ESDOF measures and their usefulness to characterize the covariance structure of climate variability. The measures are applied to 7000 years of monthly mean 2-m temperature and precipitation fields, both obtained from the TraCE-21ka paleoclimate model simulation and processed as described in section 2 for temperature.

As shown in Fig. 5a, the ESDOF estimator *p* = 1/2 across the entire frequency range, that is, from monthly up to millennial time scales. This scaling behavior is interrupted only by a small drop at ENSO time scales between 2 and 5 years and by the annual cycle (including its higher harmonics). According to this measure, there are about 220 ESDOFs at monthly and about four ESDOFs at millennial time scales. As explained in the previous section, this may indicate (i) a smaller spatial extent of the fluctuations or (ii) an increased occurrence of anticorrelations caused by teleconnections, toward the shorter time scales. As already suggested by Figs. 1 and 2, there seems to be a transition from scenario (i) at millennial to decadal time scales toward scenario (ii) at time scales shorter than decadal.

(a) ESDOF estimator *L* (right vertical axis) as a function of frequency *ν*, estimated from global 2-m temperature fields of the TraCE-21ka paleoclimate model simulation, shown by the gray line. Colored line segments indicate the same measure, estimated from the millennial (violet), centennial (blue), decadal (green), annual (red), and monthly (cyan) frequency band, with colored dots at the central frequencies. Black dots indicate the ESDOF estimator *e*-folding length scale *θ*_{e} (open dots) or over the entire interval [0, *π*] (closed dots); see text for details. The dotted brown line follows a power-law frequency scaling with exponent *p* = 0.45. (b) Estimator *e*.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

(a) ESDOF estimator *L* (right vertical axis) as a function of frequency *ν*, estimated from global 2-m temperature fields of the TraCE-21ka paleoclimate model simulation, shown by the gray line. Colored line segments indicate the same measure, estimated from the millennial (violet), centennial (blue), decadal (green), annual (red), and monthly (cyan) frequency band, with colored dots at the central frequencies. Black dots indicate the ESDOF estimator *e*-folding length scale *θ*_{e} (open dots) or over the entire interval [0, *π*] (closed dots); see text for details. The dotted brown line follows a power-law frequency scaling with exponent *p* = 0.45. (b) Estimator *e*.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

(a) ESDOF estimator *L* (right vertical axis) as a function of frequency *ν*, estimated from global 2-m temperature fields of the TraCE-21ka paleoclimate model simulation, shown by the gray line. Colored line segments indicate the same measure, estimated from the millennial (violet), centennial (blue), decadal (green), annual (red), and monthly (cyan) frequency band, with colored dots at the central frequencies. Black dots indicate the ESDOF estimator *e*-folding length scale *θ*_{e} (open dots) or over the entire interval [0, *π*] (closed dots); see text for details. The dotted brown line follows a power-law frequency scaling with exponent *p* = 0.45. (b) Estimator *e*.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

To make this more explicit, Fig. 5b shows estimates of the normalized average radial covariance function, ^{4} It turns out that from millennial to decadal time scales the *e*-folding length scale decreases [corresponding to scenario (i)] but then saturates toward the shorter time scales where changes in the spatial correlation structure are mainly associated with an increased occurrence of anticorrelations [corresponding to scenario (ii)]. This result can be expressed in terms of the ESDOF estimator *e*-folding length scale *θ*_{e}.^{5} The resulting values of *e*-folding length scale *θ*_{e} used to compute *π*] [similar to the procedure of Jones et al. (1997)],

In Fig. 6a the estimator *D*_{R}(*ν*) is a frequency-dependent version of the second ESDOF measure of Jones et al. (1997), according to their Eq. (10), we can compare our results to the ESDOF values obtained by Jones et al. (1997) (see their Table 1). Estimating *D*_{R}(*ν*) up to *ν* = (1 yr)^{−1} or *ν* = (10 yr)^{−1}, respectively, according to (22). This yields

ESDOF estimators *L* (right vertical axis) as a function of frequency *ν*, with colors as indicated in (a), estimated from the TraCE-21ka paleoclimate model simulation. ESDOF estimates are shown for 2-m temperature (T2m) and for stratiform precipitation (Precip-strat), and for global fields (global), the region between 30°S and 30°N (tropics), the northern extratropics (>30°N), and northern extratropical ocean areas (>30°N, ocean), as indicated above each panel.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

ESDOF estimators *L* (right vertical axis) as a function of frequency *ν*, with colors as indicated in (a), estimated from the TraCE-21ka paleoclimate model simulation. ESDOF estimates are shown for 2-m temperature (T2m) and for stratiform precipitation (Precip-strat), and for global fields (global), the region between 30°S and 30°N (tropics), the northern extratropics (>30°N), and northern extratropical ocean areas (>30°N, ocean), as indicated above each panel.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

ESDOF estimators *L* (right vertical axis) as a function of frequency *ν*, with colors as indicated in (a), estimated from the TraCE-21ka paleoclimate model simulation. ESDOF estimates are shown for 2-m temperature (T2m) and for stratiform precipitation (Precip-strat), and for global fields (global), the region between 30°S and 30°N (tropics), the northern extratropics (>30°N), and northern extratropical ocean areas (>30°N, ocean), as indicated above each panel.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

For all of the above estimates, the ESDOF values at decadal time scales are roughly half of the values at annual time scales. This suggests that the reduction of the global ESDOFs toward longer time scales is a robust feature across models and observations. This behavior is also consistent, at least qualitatively, with the simple North-EBM (see section 3d and Fig. 4), where the ESDOF reduction is caused solely by spatial diffusion. Although in TraCE-21ka the ESDOF scaling is flatter and it does not exhibit a scaling break, even such a scaling behavior can still be obtained by a simple diffusive EBM [the fractional EBM of Rypdal et al. (2015); not shown]. This suggests that diffusive processes might be the basic reason for the reduction of ESDOFs toward longer time scales, at least for internal (i.e., unforced) climate variability at time scales longer than those notably influenced by teleconnections. The above ESDOF spectra are characterizing the spatial covariance structure in a global mean sense. However, differences in this structure are to be expected between different regions, for example, between the extratropics (governed by quasigeostrophic scaling) and the tropics (characterized by strongly divergent circulations).

When the analysis is restricted to the tropics (defined here as the region between 30°S and 30°N), shown in Fig. 6d, the scaling is different. Specifically, the scaling exponent *p* is near zero at time scales longer than those of ENSO variability, with about 5 to 10 ESDOFs, depending on whether *A* of the spatial domain. Whereas for global fields, *D* = 1 corresponds to *L* = 2000 km (the entire sphere), for the tropical region, for example, covering half of the sphere, *D* = 1 corresponds to *L* = 10 000 km (one hemisphere).

By contrast, in the northern extratropics, shown in Fig. 6b, a slight opposite scaling break is found near *ν* = (10 yr)^{−1}. Thus, the frequency dependence of the ESDOFs is stronger at decadal to millennial and weaker at annual to monthly time scales. In addition, there is a narrow peak with fewer ESDOFs at *ν* = (13 yr)^{−1}, which, however, is due to an unrealistic feature of this model simulation, similar to what was found in other simulations with this model, although with different forcing and resolution, as reported by Danabasoglu (2008). This feature is associated with unrealistically large variability confined to the northern North Atlantic. It is this spatial confinement of the variance that causes the upward peak in the lower bound ^{6} When the analysis includes both hemispheres, however, the annual cycle signature is more complex because of interhemispheric anticorrelations (Figs. 6a,d).

The frequency scaling of the ESDOF measures may be quite different when other variables are investigated. As an example, Fig. 6c shows

## 5. Discussion

As shown in section 3, the lower bound of the ESDOF measure *D*_{R}, defined by (4), depends on the spatial distribution of the variance [see (11)]. More specifically, the measure attains a value still greater than one, even if all locations within the spatial domain are perfectly correlated, which may seem to be counterintuitive, as one may expect *D*_{R} = 1 in this case. This suggests defining the corrected measure *D*_{R} in close alignment with the measure *D*_{ρ}, which is independent of the distribution of variance. Given that the estimation of *D*_{R,c} is computationally less expensive than estimation of *D*_{ρ} (especially in the spectral domain, where standardization has to be performed at each frequency), one may prefer *D*_{R,c} over *D*_{ρ}. However, Fig. 6f shows that there are cases where the correction would rather create a mismatch with *D*_{ρ}. This implies that, in general, it is better to use *D*_{R} together with its lower bound, or, alternatively, the measure *D*_{ρ}.

Nonetheless, it is useful to realize that, while *D*_{R} represents a scale-oriented ESDOF measure, *D*_{R,c} can be interpreted as a number-oriented measure. To understand this distinction, consider the idealized case of *N*_{box} (= *N*_{box,1} + *N*_{box,0}) equally sized and uncorrelated boxes, with *N*_{box,1} boxes having variance one and *N*_{box,0} boxes having variance zero. Then, from the definition of the measures, we obtain *D*_{R} = *N*_{box}, which is the reciprocal of the (fractional) spatial *scale* of the box-shaped fluctuations, whereas we have *D*_{R,c} = *N*_{box,1}, which is the *number* of the fluctuations.

Although, strictly speaking, this interpretation breaks down for smooth spatial variance distributions (as is the case for climate variability), it helps to differentiate our measure *D*_{R} from the ESDOF measure *X*_{i} by the squared anomalies *X*_{i} are normally distributed), we obtain *D*_{R} is a scale-oriented ESDOF measure based on raw anomalies,

Using squared anomalies has the advantage that anticorrelations do not counterintuitively increase the ESDOFs, such that a perfect teleconnection dipole is associated with one ESDOF, rather than diverging ESDOFs (*D*_{R} → ∞, as in our case). Hence, one may want to define a frequency-dependent version of ^{7} Although squaring in the Fourier domain or in the cross-spectral density domain does actually turn anticorrelations into positive correlations, it also turns phase lags into negative correlations, again leading to a counterintuitive increase of the resulting ESDOF measure. Alternatively, one may square the real parts of the cross-spectral densities, but this yields a strong positive bias of the variance of the global average and, thus, a strong negative bias of the ESDOF measure, as it also occurs for *κ* ≫ *D*, is not applicable to our case, where *κ* is small as it represents the spectral degrees of freedom, such that we may easily have *κ* < *D*. Thus, it would be necessary to provide an analytically derived bias correction valid also for small *κ*.

## 6. Conclusions

A frequency-dependent ESDOF measure is introduced that can serve as a simple and easy to compute summarizing metric of the covariance structure of a time-varying field.

The underlying ESDOF measure is based on the variance of the spatial average of an anomaly field. The measure exploits the fact that the variance of the spatial average depends on the spatial correlation, as for largely uncorrelated anomalies the spatial average leaves only a small residual, whereas the residual is large for a spatially coherent field.

An interpretation of the measure is provided for two settings: (i) for fields dominated by positive correlations, the measure characterizes the average spatial extent of the fluctuations; and (ii) for fields dominated by anticorrelated structures (due to, for example, teleconnections), it characterizes the relative spatial partitioning of the associated patterns into the respective anticorrelated spatial subsections.

Two variants of the underlying measure are defined: with and without standardization of anomalies, the latter variant being identical, in its frequency-independent version, to the ESDOF measure used by Smith et al. (1994) and Jones et al. (1997).

Both measures are equipped with an explicit dependence on frequency by replacing variances by power spectral densities, which allows for displaying ESDOF spectra across frequencies.

Estimators for both measures are provided, and their bias and scatter are characterized.

It is shown how the average radial covariance function can be estimated from gridded fields to construct alternative frequency-dependent ESDOF measures based on other aspects of the spatial covariance function than its integral.

These frequency-dependent ESDOF estimators are applied to temperature and to precipitation fields obtained from a paleoclimate model simulation. From the results we can conclude the following:

Our frequency-dependent ESDOF measures are successful at providing concise information on the covariance structure of climate variability and, in particular, at detecting distinct space–time scaling regimes in the frequency domain.

For global near-surface temperature fields, the ESDOFs exhibit a distinct reduction from monthly to millennial time scales, consistent with (i) the results of Jones et al. (1997) for annual to decadal time scales, and with (ii) a simple diffusive EBM, suggesting that horizontal diffusion is a basic reason for the reduction of the ESDOFs toward longer time scales.

For stratiform precipitation, as an example of a nondiffusive climate variable, the ESDOFs do not exhibit any distinct frequency scaling between monthly and millennial time scales.

Numerous potential applications exist for these frequency-dependent ESDOF measures. For example, as a reasonable next step, ESDOF spectra may be estimated from observational data, as this may significantly contribute to our understanding of climate variability by characterizing its space–time structure across wide ranges of frequencies. A challenge in this context will be the treatment of data gaps in space and time, in particular, when using paleoclimate data obtained from proxy records. In this latter case, frequency-dependent ESDOF estimates may provide useful information regarding the representative spatial scale of local paleoclimate reconstructions and how that scale varies with time scale. Data assimilation approaches in paleoclimate modeling (e.g., Hakim et al. 2016) may profit from this type of information. Another possible application is the comparison of models with observations, in order to validate or improve either of them. Laepple and Huybers (2014) detected large discrepancies in sea surface temperatures at long time scales for local, but not for large-scale variability, which implies different covariance structures in models versus observations. ESDOF spectra may also be used to identify climate variables and frequency bands for which the ESDOFs are independent of frequency, as in such cases the variability may possibly be described by statistical space–time factorization (provided the spatial statistics are stationary across frequencies), implying simplifications for prediction (Lovejoy et al. 2018). Finally, our frequency-dependent ESDOF measures may help to quantify the impact of external forcings on the space–time structure of climate variability, as various types of forcing may reveal different characteristic ESDOF fingerprints.

## Acknowledgments

This is a contribution to the SPACE ERC project; this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement 716092). The work profited from discussions at the Climate Variability Across Scales (CVAS) working group of the Past Global Changes (PAGES) program. We also acknowledge discussions with Igor Kröner on the properties of the estimators, and the helpful comments of three anonymous reviewers.

## APPENDIX A

### Length Scale Measure *L*

*R*is given by

*A*

_{tot}= 4

*πR*

^{2}, and the area of a spherical cap by

*A*

_{cap}= 2

*πR*

^{2}(1 − cos

*θ*), where

*θ*is half the opening angle of the cap, as seen from the center of the sphere. Then from (5), and requiring that

*L*in section 3b, we have

*D*denotes

*D*

_{R}or

*D*

_{ρ}. The radius

*L*of the cap along its curved surface is then given by

*D*(i.e., if

*D*≫ 2

*f*

_{A}),

*L*can be approximated by

## APPENDIX B

### Bias and Scatter of ESDOF Estimators

Analytic expressions for *D*_{R} are derived, for the case of a standard normally distributed random field with stationary variance and equal spatial weighting, that is, for *w*_{i} = 1/*N* for all *i*, and with all possible pairs of locations having the same correlation *ρ*_{ij} (for *i* ≠ *j*). This latter assumption requires that *ρ*_{min} < *ρ*_{ij} ≤ 1, with *ρ*_{min} = −1/(*N* − 1). We obtain *D*_{R} = 1 if *ρ*_{ij} = 1 and *D*_{R} → ∞ as *ρ*_{ij} → *ρ*_{min}.

*χ*

^{2}variable, that

*χ*

^{2}(

*κ*, 1) distribution. To investigate the behavior of the covariance term in (B1) in the limit of large

*D*

_{R}, note, that

*ρ*

_{ij}implies

*D*

_{R}= 1, this yields a correlation of one. For increasing

*D*

_{R}> 1, the correlation decreases monotonically, and approaches zero as

*D*

_{R}→ ∞. From this we imply that the covariance term in (B1) vanishes as

*D*

_{R}

*→*∞. Hence, from (B2),

*D*

_{R}= 1, the numerator and the denominator of

*β*

_{R}= 1. From a set of numerical simulations (see Fig. B1a) we infer that

*β*

_{R}(solid black line) increases monotonically with

*D*

_{R}and, thus, approaches its limit

*β*

_{R}, as stated by (15) in section 3c.

Estimates of (a) *β*_{R} and (b) *D*_{R} and of (c) *β*_{ρ} and (d) *D*_{ρ} for different values of *N* (solid colored lines and dots), obtained from numerical simulations according to the case described at the beginning of appendix B. Estimates from simulations with *N* = *D*_{R}, *D*_{ρ} are connected by solid black lines with open dots. The dashed black line in (a) indicates the upper bound *β*_{R,max}, defined by (15), and in (b) the upper bound *κ* = 22.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

Estimates of (a) *β*_{R} and (b) *D*_{R} and of (c) *β*_{ρ} and (d) *D*_{ρ} for different values of *N* (solid colored lines and dots), obtained from numerical simulations according to the case described at the beginning of appendix B. Estimates from simulations with *N* = *D*_{R}, *D*_{ρ} are connected by solid black lines with open dots. The dashed black line in (a) indicates the upper bound *β*_{R,max}, defined by (15), and in (b) the upper bound *κ* = 22.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

Estimates of (a) *β*_{R} and (b) *D*_{R} and of (c) *β*_{ρ} and (d) *D*_{ρ} for different values of *N* (solid colored lines and dots), obtained from numerical simulations according to the case described at the beginning of appendix B. Estimates from simulations with *N* = *D*_{R}, *D*_{ρ} are connected by solid black lines with open dots. The dashed black line in (a) indicates the upper bound *β*_{R,max}, defined by (15), and in (b) the upper bound *κ* = 22.

Citation: Journal of Climate 34, 18; 10.1175/JCLI-D-20-0228.1

*D*

_{R}, we can write

*D*

_{R}→ ∞ implies

*ρ*

_{ij}→

*ρ*

_{min}, we find from (B3), using the expression for

*χ*

^{2}variable, we have

*χ*

^{2}(

*κ*, 1) distribution. Then, using

*N*. In the opposite limit, that is, if

*D*

_{R}= 1, the scatter vanishes,

*D*

_{R}and, thus, approaches its limit (B8) (dashed colored lines) from below, for any given

*N*. Hence, the limit (B8) represents an upper bound to

*N*. If, additionally, we let

*N*→ ∞, then this limit reduces to

*D*

_{R}, the limit

*N*as long as

For the estimator *β*_{ρ} and *δ*_{ρ} from the numerical simulations. Although the spatial stationarity of the simulated random fields implies *D*_{R} = *D*_{ρ}, it does not imply *D*_{ρ} = *N* (solid black line in Fig. B1c), *β*_{ρ} approaches the same upper bound *β*_{R,max} (dashed black line) as *D*_{ρ} → ∞. For *D*_{ρ} < *N*, however, *β*_{ρ} can slightly exceed *β*_{R,max} for large values of *N*. Thus, the upper bound *β*_{R,max} is only approximately applicable to *D*_{ρ} > *N*, the behavior of *β*_{ρ} < 1), in particular, for small values of *N*. These differences arise essentially from the fact that the correlation estimator, implicit to *D*_{ρ} = *N* and *D*_{ρ} < *N* (Fig. B1d), whereas for *D*_{ρ} > *N*, the scatter of *β*_{R,max} and *δ*_{R,max} apply reasonably well also to *D*_{ρ} ≤ *N*.

## REFERENCES

Bretherton, C. S., M. Widmann, V. P. Dymnikov, J. M. Wallace, and I. Bladé, 1999: The effective number of spatial degrees of freedom of a time-varying field.

,*J. Climate***12**, 1990–2009, https://doi.org/10.1175/1520-0442(1999)012<1990:TENOSD>2.0.CO;2.Cowtan, K., and R. G. Way, 2014: Coverage bias in the HadCRUT4 temperature series and its impact on recent temperature trends.

,*Quart. J. Roy. Meteor. Soc.***140**, 1935–1944, https://doi.org/10.1002/qj.2297.Danabasoglu, G., 2008: On multidecadal variability of the Atlantic meridional overturning circulation in the Community Climate System Model version 3.

,*J. Climate***21**, 5524–5544, https://doi.org/10.1175/2008JCLI2019.1.Feldstein, S. B., and C. L. E. Franzke, 2017: Atmospheric teleconnection patterns.

*Nonlinear and Stochastic Climate Dynamics*, C. L. E. Franzke, and T. J. O’Kane, Eds., Cambridge University Press, 54–104, https://doi.org/10.1017/9781316339251.004.Hakim, G. J., J. Emile-Geay, E. J. Steig, D. Noone, D. M. Anderson, R. Tardif, N. Steiger, and W. A. Perkins, 2016: The last millennium climate reanalysis project: Framework and first results.

,*J. Geophys. Res.***121**, 6745–6764, https://doi.org/10.1002/2016JD024751.He, F., 2011: Simulating transient climate evolution of the last deglaciation with CCSM3. Ph.D. thesis, University of Wisconsin–Madison, 171 pp.

Jones, P. D., T. J. Osborn, and K. R. Briffa, 1997: Estimating sampling errors in large-scale temperature averages.

,*J. Climate***10**, 2548–2568, https://doi.org/10.1175/1520-0442(1997)010<2548:ESEILS>2.0.CO;2.Laepple, T., and P. Huybers, 2014: Ocean surface temperature variability: Large model–data differences at decadal and longer periods.

,*Proc. Natl. Acad. Sci.***111**, 16 682–16 687, https://doi.org/10.1073/pnas.1412077111.Liu, Z., and Coauthors, 2009: Transient simulation of last deglaciation with a new mechanism for Bølling-Allerød warming.

,*Science***325**, 310–314, https://doi.org/10.1126/science.1171041.Livezey, R. E., and W. Y. Chen, 1983: Statistical field significance and its determination by Monte Carlo techniques.

,*Mon. Wea. Rev.***111**, 46–59, https://doi.org/10.1175/1520-0493(1983)111<0046:SFSAID>2.0.CO;2.Lovejoy, S., L. del Rio Amador, and R. Hébert, 2018: Harnessing butterflies: Theory and practice of the Stochastic Seasonal to Interannual Prediction System (StocSIPS).

*Nonlinear Advances in Geosciences*, A. A. Tsonis, Ed., Springer, 305–355, https://doi.org/10.1007/978-3-319-58895-7_17.North, G. R., J. Wang, and M. G. Genton, 2011: Correlation models for temperature fields.

,*J. Climate***24**, 5850–5862, https://doi.org/10.1175/2011JCLI4199.1.Rypdal, K., M. Rypdal, and H.-B. Fredriksen, 2015: Spatiotemporal long-range persistence in Earth’s temperature field: Analysis of stochastic-diffusive energy balance models.

,*J. Climate***28**, 8379–8395, https://doi.org/10.1175/JCLI-D-15-0183.1.Smith, T. M., R. W. Reynolds, and C. F. Ropelewski, 1994: Optimal averaging of seasonal sea surface temperatures and associated confidence intervals (1860–1989).

,*J. Climate***7**, 949–964, https://doi.org/10.1175/1520-0442(1994)007<0949:OAOSSS>2.0.CO;2.Wallace, J. M., and D. S. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter.

,*Mon. Wea. Rev.***109**, 784–812, https://doi.org/10.1175/1520-0493(1981)109<0784:TITGHF>2.0.CO;2.Wang, X., and S. S. Shen, 1999: Estimation of spatial degrees of freedom of a climate field.

,*J. Climate***12**, 1280–1291, https://doi.org/10.1175/1520-0442(1999)012<1280:EOSDOF>2.0.CO;2.

^{1}

This interpretation holds because the average **r** would be perfectly correlated with that area fraction and fully uncorrelated with the remaining area (keeping the integral of the spatial correlation function constant; see section 3b for further interpretation).

^{2}

Note that, for a discrete spatial field, the two averages involved in

^{3}

The Legendre polynomials are an orthogonal system if expressed as *θ*.

^{4}

The normalized average radial covariance function for a frequency band bounded by *ν*_{1} and *ν*_{2} is obtained by integrating the numerator and denominator of (25) over that frequency interval. The five frequency bands are bounded by *ν*_{1} = (7000 yr)^{−1}, *ν*_{2} = (1000 yr)^{−1} for the millennial, by *ν*_{1} = (1000 yr)^{−1}, *ν*_{2} = (100 yr)^{−1} for the centennial, by *ν*_{1} = (100 yr)^{−1}, *ν*_{2} = (10 yr)^{−1} for the decadal, by *ν*_{1} = (10 yr)^{−1}, *ν*_{2} = (14 months)^{−1} for the annual, and by *ν*_{1} = (10 months)^{−1}, *ν*_{2} = (2 months)^{−1} for the monthly band, where the annual cycle has been omitted. Narrow frequency bands (of 41 discrete frequencies) centered at the higher harmonics of the annual cycle have been omitted from the monthly band.

^{5}

Note that, given *θ*_{e}, the rhs of (26) that involves an integral over *θ* (through the operator *M*[⋅] defined in section 3a) is written out explicitly by Jones et al. (1997) [see their Eq. (14)].

^{6}

Accordingly,

^{7}

Note that squaring a harmonic oscillation doubles its frequency, and further distortions occur for discrete time signals.