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Partitioning the variance between space and time

Fubao Sun,1 Michael L. Roderick,1,2 Graham D. Farquhar,1 Wee Ho Lim,1

Yongqiang Zhang,3 Neil Bennett,4 and Stephen H. Roxburgh5

Received 22 March 2010; revised 17 May 2010; accepted 20 May 2010; published 22 June 2010.

[1] Here we decompose the spacetime variance of nearsurface air temperature using monthly observations for theglobal land surface (excluding Antarctica) from 19012000.To do that, we developed a new method for partitioningthe total spacet ime variance, here called the grandvariance, into separate spatial and temporal components.The temporal component is, in turn, further partitioned intothe variance relating to different time periods and we usemonthly data to decompose intra and inter annualcomponents of the variance. The results show that thespatial and temporal components of the variance of nearsurface air temperature have both, on average, decreased

over time primarily because of reductions in the equator

to

pole (northern) tempe ratur e gradi ent, and because in coldregions, winter is generally warming faster than summer.We also found that in most regions, the interannualvariance in nearsurface air temperature has increased.Citation: Sun, F., M. L. Roderick, G. D. Farquhar, W. H.

Lim, Y. Zhang, N. Bennett, and S. H. Roxburgh (2010), Parti-

tioning the variance between space and time, Geophys. Res. Lett.,

37, L12704, doi:10.1029/2010GL043323.

1. Introduction

[2] Regional climates are usually defined using long termaverages. Currently, there is much interest in changes in the

mean, but equally, many hydrologic [Arora and Boer, 2006;Milly et al., 2008] and ecologic [Zimmermann et al., 2009]processes are known to be sensitive to changes in theextremes of climate variables. For that reason, the varianceof nearsurface air temperature has been studied over daily[Karl et al., 1995], monthly [Parker et al., 1994; Michaelset al., 1998] and annual [Seneviratne et al., 2006; Stoufferand Weatherald, 2007; Boer, 2009] time scales. Some-times, spatial variance is used [e.g., HernndezDeckers andvon Storch, 2010], as opposed to the more familiar temporalvariance.

[3] The space and time components of the abovenotedvariances must be interrelated in some way and the aim ofthis paper is to establish those relationships. In section 2, we

present a new theoretical approach that partitions the total

spacetime variance into separate spatial and temporal com-ponents. The temporal variance is further decomposed intointra and interannual components. The new method buildson the traditional Analysis of Variance (ANOVA) technique[von Storch and Zwiers, 1999; Wilks, 2006] and extends thatby deriving analytical relationships linking the various timeand space scales. In section 3, we describe the temperaturedata. In section 4, we use the new approach to examine theclimatology of, and changes over time in, the variance of theglobal nearsurface air temperature over land from 1901 to2000.

2. Mathematical Derivation

[4] Historical databases are typically available at regulartime intervals (e.g., monthly) and on regular spatial grids.Here we consider the more general case of irregularly dis-tributed regions. As shown in Figure S1 (in the auxiliarymaterial), a twodimensional array is formed as

Z zij

mn1

where i = 1, m months (time) and j = 1, n regions (space).6

2.1. Partitioning Between Space and Time

[5] The grand variance across space and time sg2 and total

sum of squares SSg for the differences from the grand meanmg =Pm

i1Pn

j1 wj zij/(m n) are,

2

g SSg

m n 12

SSg Xmi1

Xnj1

wj zij g

23

where for n regions having area Aj, we define weights, wj =Aj/(

Pnj1 Aj/n), as the proportional area of the jth region

scaled as a proportion of the mean regional area. For thespecial case where each region has the same area, we havewj = 1.2.1.1. SpaceFirst Formulation

[6] The total sum of squares (equation (3)) can berewritten by incorporating the spatial mean of each monthms(i) =

Pnj1 wj zij/n as follows,

SSg Xmi1

Xnj1

wj zij s i s i g 2

4

1Research School of Biology, Australian National University,Canberra, ACT, Australia.

2Research School of Earth Sciences, Australian National University,Canberra, ACT, Australia.

3CSIRO Land and Water, Canberra, ACT, Australia.4Department of Engineering, Australian National University,

Canberra, ACT, Australia.5CSIRO Sustainable Agriculture Flagship and CSIRO Sustainable

Ecosystems, Canberra, ACT, Australia.

Copyright 2010 by the American Geophysical Union.00948276/10/2010GL043323

6Auxiliary materials are available in the HTML. doi:10.1029/2010GL043323.

GEOPHYSICAL RESEARCH LETTERS, VOL. 37, L12704, doi:10.1029/2010GL043323, 2010

L12704 1 of 6
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This can be expanded and rearranged as follows,

SSg Xmi1

Xnj1

wj zij s i 2

2Xmi1

s i g Xn

j1

wj zij s i " #

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0Xmi1

s i g 2 Xn

j1

wj

" #|fflfflfflfflffl{zfflfflfflfflffl}

n

Xmi1

Xnj1

wj zij s i

2n

Xmi1

s i g

25

[7] In classical experimental analysis using the ANOVAmethod, equation (5) with wj = 1 is usually termed the totalsum of squares and is partitioned into treatment and errorcomponents [von Storch and Zwiers, 1999; Wilks, 2006].Here we express SSg in terms of variances as

SSg n 1 Xmi1

2

s i n m 1 2

t 6

where ss2(i)

Pnj1 wj(zij ms(i))

2/(n 1) is the spatialvariance of the ith month and can be displayed as a timeseries; and st

2(m) Pm

i1 (ms(i) mg)2/(m 1) is the tem-

poral variance of the spatial means, shown in Figure S1.2.1.2. TimeFirst Formulation

[8] The above analysis was based on the inclusion of thespatial mean for each month (equation (4)) in the originalexpression for SSg. That is, we considered the spatial com-ponent first. We can also derive an alternative expression forSSg by first including the temporal mean for the jth region

given by mt(j) = Pmi1 zij/m in equation (3) as follows,SSg

Xmi1

Xnj1

wj zij t j t j g 2

7

that is again expanded and rearranged as follows,

SSg Xnj1

wjXmi1

zij t j 2" #

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} m1 2t j

2Xnj1

wj t j g Xm

i1

zij t j " #

|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 m

Xnj1

wj t j g 2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n1 2s

m 1 Xnj1

wj2

t j m n 1 2

s 8

where st2(j)

Pmi1 (zij mt(j))

2/(m 1) is the temporalvariance of the jth region and can be displayed as a map;ss2(m)

Pnj1 wj(mt(j) mg)

2/(n 1) is the spatial variance ofthe temporal means, as shown in Figure S1.

2.1.3. Partitioning Schemes[9] By substituting equations (6) and (8) into equation (2),

alternative relationships that partition the grand varianceinto spatial and temporal components with area weights arederived. For the spacefirst formulation we have,

2

g

m n 1

m n 1Xm

i1

2

s i

m|fflfflfflfflffl{zfflfflfflfflffl}2s

n m 1

m n 12

t

9

and for the timefirst formulation,

2

g n m 1

m n 1

Xnj1

wj2

t j

n|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2t

m n 1

m n 12

s 10

with definitions of 2s Pm

i1 ss2(i)/m, the mean of the

spatial variances of the m months and 2t Pn

j1 wjst2(j)/n,

the mean of the temporal variances of the n regions. The

coefficients appearing in both expressions,m n1

mn 1and

n m1 mn 1 vary from

n1n 1 and m1

m 1, respectively, e.g.,

both are 0.99 when m = 100 and n = 100 and larger than 0.9when m 10 and n 10. Note that when m ) 1 and n ) 1,the sample variances calculated here will approach therespective population variances and the coefficients becomeunity (see Section S.1 in the supporting online material). Insummary, when m and n are both larger than 10, theinfluence of the sample size on the partitioning is small.

[10] The grand variance is the sum of two terms. In thespacefirst formulation (equation (9)), the first term is anaverage of the spatial variances at each time step and thesecond is a single number that represents the temporalvariance of the spatial means. In the timefirst formulation

(equation (10)), the first term is an average of the temporalvariances for each region and the second is a single numberthat represents the spatial variance of the climatology. Thechoice between equations (9) and (10) depends on thepurpose of the study. The spacefirst approach is useful forcalculating globally aggregated statistics. The timefirstapproach is more useful for regional studies and has theadded property, explained below, that the temporal com-ponent can be decomposed into the variance contributionfrom different time periods.

2.2. Partitioning the Temporal Variance in theTimeFirst Formulation

[11] For a given region, st2(j) in equation (10) is the

temporal variance of the time series. This variance willdepend on the level of temporal aggregation, e.g., months,years, etc. Similarly, the numerical values of the spatialvariance terms will also depend on the degree of spatialaggregation. Here we assume the spatial sampling units arefixed and focus on formulating the problem in terms oftemporal aggregation but note that the same underlyingapproach could be adopted for spatial aggregation.

[12] Here we aggregate from months to years by againreorganizing the data into a 2dimensional array (Figure S1).Formally, the mmonths time series of the jth region is:[zkl]pq, where k = 1, p months per year (p = 12) and l = 1,q year (q m/p). Following the previous mathematical

SUN ET AL.: VARIANCES PARTITIONING BETWEEN SPACE AND TIME L12704L12704

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derivation but without the weights, the temporal variancest2(j) is further partitioned into separate components as,

2

t j q p 1

p q 1

Xql1

2

a l

q|fflfflfflfflffl{zfflfflfflfflffl}2a

p q 1

p q 12

e 11

2

t j p q 1

p q 1

Xpk1

2

e k

p

|fflfflfflfflffl{zfflfflfflfflffl}2e

q p 1

p q 12

a 12

with definitions of2a Pq

l1 sa2(l)/q the interannual mean

of the intraannual variances sa2(l) for the q years and 2e Pp

k1 se2(k)/p the intraannual mean of the interannual

variances se2(k) for the p months; se

2(m) is the interannualvariance for the yearly means; sa

2(m) is the intraannualvariance for the mean annual cycle (Figure S1).

2.3. Relation to Other Approaches

[13] Much previous climate related research has beenperformed using anomalies and that can be readily incor-porated within the theory presented here. For example,Westra and Sharma [2010] calculated anomalies by sub-tracting the spatial (i.e., global) mean at each month. Thevariance of those anomalies is the spatial component inequation (9). The same applies where climate data isdeseasonalized by subtracting the mean annual cycle [e.g.,Parker et al., 1994; Held and Soden, 2006]. The resultingvariance equals the interannual component of the temporalvariance in equation (12). Similar examples can be foundthroughout the literature.

3. Data and Methods

[14] Here we use the theory to examine the variance innearsurface air temperature. We use the CRU TS2.1 gridded0.5 0.5 monthly air temperature database for the global

land surface from 1901 to 2000 [Mitchell and Jones, 2005].Note that this database excludes Antarctica. We aggregatedthe database to a spatial resolution of 2.5 2.5. The cli-matology has been calculated for the 100 year period. Toexamine changes in variance over time, we divided the100yr period into 10 10 year blocks and calculated thedecadal grand variance and the components for each suc-cessive 10 year block. Trends were calculated using linearregression (ordinary least squares).

4. Results and Discussion

4.1. Twentieth Century Climatology, 19012000

[15] The 20th century climatology for the grand varianceand components is described in Table 1. In the spacefirstformulation (equation (9)), most of the grand variance(246.44C2 = 0.9997 217.53 + 0.9992 29.00) is due to thespatial variance for each month (217.53C2) (see Figure S2in the auxiliary material for the time series from which thiscomponent is computed) with a much smaller contributionfrom the temporal variance of the spatial means (29.00C2).In the timefirst formulation (equation (10)), the same grandvariance is partitioned into the temporal variance averagedover all regions (75.11C2) (see Figure S3 in the auxiliarymaterial for the spatial map of this component) and thespatial variance of the longterm mean temperature for eachregion (171.44C2). Again, the spatial component dom-

inates. That arises because the spatial differences across theglobal land surface at a given time are generally much largerthan the differences over time in a given region.

[16] As per the theory the temporal variance averagedover all regions (75.11C2) can be further partitioned usingtwo alternative schemes (equations (11) and (12)). In thefirst scheme, the intraannual variance averaged over allyears (81.36C2) is dominant when compared to the interannual variance of the annual mean (0.47C2). Similarly, inthe second scheme (equation (12)), the interannual varianceof the mean monthly temperatures (2.63C2) is muchsmaller than the intraannual variance of the mean annualcycle (79.03C2). That would be expected the differences

Table 1. Century Climatology (19012000) of the Grand Variance (C2) of Monthly Air Temperature and the Underlying Variance

Componentsa

Spatial Component Temporal Component

m(n 1)/[(m n) 1] 2s n(m 1)/[(m n) 1] st2(m)

SpaceFirst (equation (9)) 0.9997 217.53 0.9992 29.00

Temporal Component Spatial Component

n(m 1)/[(m n) 1] 2

t m(n 1)/[(m n) 1] ss2

(m)

TimeFirst (equation (10)) 0.9992 75.11 0.9997 171.44

Intraannual Component Inter annual Component

q(p 1)/[(p q) 1] 2a p(q 1)/[(p q) 1] se2(m)

IntraFirst (equation (11)) 0.9174 81.36 0.9908 0.47

Interannual Component Intra annual Component

p(q 1)/[(p q) 1] 2e q(p 1)/[(p q) 1] s2a(m)

InterFirst (equation (12)) 0.9908 2.63 0.9174 79.03

aThe coefficients are based on the 100year block: m = 1200 months (q = 100 years and p = 12 months per year) and n = 3399 gridboxes. Partitioningthe grand variance: sg

2 = 246.44. Partitioning the temporal variance: 2t = 75.11.


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between years are much smaller than variations within ayear.

4.2. Decadal Trends, 19012000

[17] The decadal grand variance generally declined overtime (Figures 1a and 2a) and was approximately comple-mentary with the global mean nearsurface air temperature

(Figures 1b and S4). The grand variance decreased from250.59C2 in the first decade to 240.56C2 in the last decadeof the 20th century (Figure 1b). The average rate of changeis 8.57C2 per century and the corresponding standarddeviation sg decreased at 0.27C per century. (Note: Thevariance change can be related to the change in the standard

deviation byd2g

dt= 2sg

dgdt

.)[18] The overall trend has been broken down into the

respective components (Table 2). (Trends of the correspondingpopulation variances in Table S1.) The spatial components,in either the space (equation (9) and Figure 1c) or timefirst(equation (10) and Figure 1f) approaches generally dominatethe overall decline in the grand variance (Figure 1a) while

the temporal components also decline (Figures 1d and 1e).As in the climatology, the trends in the grand variances areconserved in the partitioned spatial and temporal compo-nents (Table 2). Of interest here is the trend in the temporalvariances for individual regions (Figure 1e). This componentis calculated as an average over space and the underlyinggridboxlevel data can be displayed as a map (Figure 2a)that is more readily interpreted. In general, the climatolog-

ical variance in colder regions (e.g., Northern China, Russia,Greenland) is larger (Figure S3) and the trends in the tem-poral variance in those regions are also larger (Figure 2a)than in warm tropical regions. The large declines in thetemporal variance in the abovenoted cold regions aregenerally the result of winter temperatures increasing fasterthan summer temperatures [e.g., Stine et al., 2009]. Simi-larly, the decline in the spatial variance (Figures 1c and 1f) isdue to reductions in the equatortopole (northern) temper-ature gradient [e.g., Karl et al., 1995].

[19] The trend in the temporal variance, regardless of thescheme used (equations (11) or (12)), is dominated by theintraannual component (Figures 1g and 1j) that decreased

Figure 1. Trends in the decadal (a) grand variance sg2

, and (b) grand mean mg of monthly air temperature over global landsurface over the 10 decades from 1901 to 2000. The underlying spacetime variance components per (c and d) the spacefirst formulation (equation (9)) with (e and f) the timefirst formulation (equation (10)). The temporal variance (Figure 1e) isfurther partitioned into intra and interannual components in (g and h) equation (11) and (i and j) equation (12). The firstblock indicates 19011910, and so on.


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Figure 2. Trends in the decadal temporal variance and the intra and interannual components for 19012000. (a) Temporalvariance, st

2(j) and components as follows: (b) mean of the intraannual variances 2a (equation (11)), (c) interannual variance

se2(m) of the yearly means (equation (11)), (d) mean of the interannual variances 2e (equation (12)), (e) intraannual variance

sa2(m) of the mean annual cycle (equation (12)).

Table 2. Trends of the Decadal Grand Variance (C2per Century) of Monthly Air Temperature and the Underlying Variance Components

for the Global Land Surface (19012000)a

m(n 1)/[(m n) 1] d2s /dt n(m 1)/[(m n) 1] dst2(m)/dt

SpaceFirst (equation (9)) 0.9997 7.12 0.9917 1.46

n(m 1)/[(m n) 1] d2t /dt m(n 1)/[(m n) 1] dss2(m)/dt

TimeFirst (equation (10)) 0.9917 3.58 0.9997 5.01

q(p 1)/[(p q) 1] d2a/dt p(q 1)/[(p q) 1] dse2(m)/dt

IntraFirst (equation (11)) 0.9244 4.08 0.9076 0.21

p(q 1)/[(p q) 1] d2e /dt q(p 1)/[(p q) 1] dsa2(m)/dt

InterFirst (equation (12)) 0.9076 0.82 0.9244 4.68

aTrends are calculated based on 10year blocks: m = 120 (q = 10, p = 12) and n = 3399. Trend of the grand variance: dsg2/dt = 8.57. Trend of the

temporal variance: d2t /dt = 3.58.


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at a similar rate to the overall temporal variance (Figure 1e).This can be confirmed by the maps in Figures 2a, 2b, and 2e.Of interest here is that irrespective of the partitioningscheme adopted, the interannual component of the tempo-ral variance is increasing, although the magnitude of thetrends are relatively small (Figures 1h and 1i). The map(Figures 2c and 2d) shows that the interannual variance isincreasing in most regions with decreases prominent in only

a few regions, e.g., US, eastern Australia, northern Chinaand the Himalayan plateau. That is consistent with earlierfindings of a generally small increase in the interannualcomponent of the temporal variance in most land regions[Parker et al., 1994]. The anomalous decrease in the USreported here is also consistent with earlier findings [Karlet al., 1995].

[20] In previous regional studies of changes in the tem-poral variance of air temperature, the decrease at subannualtime scales (e.g., daily, monthly) in much of the northernhemisphere [Karl et al., 1995; Michaels et al., 1998] alongwith an increase over longer periods (e.g., interannual)throughout much of Europe [Seneviratne et al., 2006] haveboth been attributed to increasing greenhouse gas con-

centrations. That can potentially explain the globally aver-aged trends (Figures 1g1j). However, that explanationcannot be universal because the sign (+/) of the trend in theinter and intraannual components of the temporal varianceis not always opposite. For example, throughout much of thetropics both the inter and intraannual variances haveincreased (Figure 2).

5. Conclusions

[21] Based on the long established concept of ANOVA,we have developed a new method of partitioning variancebetween spatial and temporal components. The grand (ortotal spacetime) variance can be split using two different

partitioning schemes that we call the space

and time

firstapproaches. In the spacefirst scheme, the grand variance isequal to the sum of two components: the average of thespatial variances at each time step and the temporal varianceof the spatial means. This scheme is useful for calculatingglobally aggregated statistics. In the timefirst scheme, thegrand variance is equal to the sum of two components: theaverage of the temporal variances from all regions andthe spatial variance of the temporal means. This scheme ismore useful for regional studies. A further advantage of thetimefirst scheme is that the resulting variance of the tem-poral component can be further partitioned into componentsrelating to different time periods. Here we used monthlydata to decompose the intra and interannual variance.

[22] Using observations of monthly temperature over theglobal land surface from 1901 to 2000 we calculated theclimatology and examined trends in the variance compo-nents over time. The climatology showed the expectedresult: the grand variance is dominated by the spatial com-ponent since the differences in air temperature betweenequatortopole are generally larger than differencesbetween seasons in a given region. In turn, the temporalcomponent is dominated by the intraannual component,i.e., in a given region, the temperature difference within ayear is larger than the difference between years.

[23] The trend analysis showed declines in the decadalgrand variance from 1901 to 2000. Using the timefirst

approach, the total decline was more or less equally parti-tioned between the spatial and temporal components of thevariance (Table 2). The decline in the spatial component isgenerally due to the northern pole warming faster than theequator. The decline in the temporal component arises froma similar phenomenon: winter temperatures are increasingfaster than summer temperatures, particularly in cold regions(e.g., Northern China, Russia, Greenland) that dominate the

overall temporal trends (Figure 2a). However, in many partsof the tropics, the temporal component of variance hasincreased slightly (Figure 2a). The temporal component wasfurther partitioned into inter and intraannual componentswith the latter dominating trends. Interestingly, changes inthe interannual variance (Figures 2c and 2d), whilst ofsmall magnitude, are generally increasing in most regionsand should be a high priority for further investigation.

[24] Acknowledgments. This work was supported by the AustralianResearch Council (DP0879763). We thank two reviewers for commentsthat improved the manuscript.

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Boer, G. J. (2009), Changes in interannual variability and decadal potentialpr ed ic ta bi li ty un de r gl ob al wa rm in g, J. Clim., 22 , 30983109,doi:10.1175/2008JCLI2835.1.

Held, I., and B. Soden (2006), Robust responses of the hydrological cycleto global warming, J. Clim., 19, 56865699.

HernndezDeckers, D., and J. S. von Storch (2010), Energetics responsesto increases in greenhouse gas concentration, J. Clim., in press.

Karl, T. R., R. W. Knight, and N. Plummer (1995), Trends in high frequency climate variability in the twentieth century, Nature, 377,217220.

Michaels, P. J., et al. (1998), Analysis of trends in the variability of dailyand monthly historical temperature measurements, Clim. Res., 10, 2733.

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Mitchell,T. D.,and P. D. Jones (2005), Animprovedmethod of constructinga database of monthly climate observations and associated highresolutiongrids, Int. J. Climatol., 25, 693712, doi:10.1002/joc.1181.

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N. Bennett, Department of Engineering, Australian National University,Canberra, ACT 2601, Australia.

G. D. Farquhar, W. H. Lim, M. L. Roderick, and F. Sun, ResearchSchool of Biology, Australian National University, GPO Box 475,Canberra, ACT 0200, Australia. ([email protected])

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GEOPHYSICAL RESEARCH LETTERS, VOL. ???, XXXX, DOI:10.1029/,

Partitioning the variance between space and time

Fubao Sun,1 Michael L. Roderick,12 Graham D. Farquhar,1 Wee Ho Lim,1

Yongqiang Zhang,3 Neil Bennett,4 and Stephen H. Roxburgh5

Supporting online materials

S.1: Sample vs p opulation variances

Throughout the main text, the sample variance wasused and that has resulted in the (n 1), (m 1) and(mn) 1 terms (and for p, q as well) appearing in thevarious coefficients.

When m >> 1 and n >> 1, the sample variance willapproach the population variance. If population vari-ances are instead chosen, the -1 part of those termswould disappear throughout. For example, both coeffi-

cients m(n

1)(mn)1 and n(m

1)(mn)1 become mn

mn = 1. On thatbasis, the equivalent population expressions for Eqns 9and 10 are,

2g =

2s +

2t () (S1)

2g =

2t +

2s() (S2)

where 2g , 2s ,

2t (),

2t and

2s() are the corresponding

population variances. Similarly, Eqns 11 and 12 become,

2t (j) =

2a +

2e() (S3)

2t (j) = 2e +

2a() (S4)

where 2t (j), 2a,

2e(),

2e , and

2a() are the correspond-

ing population variances.The trends based on population statistics are summa-

rized in Table S1 and can be compared with sample statis-tics in Table 2 (main text). As demonstrated by thatcomparison, for large sample sizes there is little practicaldifference between sample and population level statistics.

Copyright 2010 by the American Geophysical Union.0094-8276/10/$5.00

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X - 2 SUN ET AL.: VARIANCES PARTITIONING BETWEEN SPACE AND TIME

Table S1. Same as Table 2 (main text) but for the popula-tion variance (Unit: C2 per century).

Trend of the grand variance: d2g/dt = 8.57

Space-First d2s/dt d2t ()/dt

Eqn S1 -7.12 -1.46

Time-First d2t

/dt d2s()/dtEqn S2 -3.55 -5.01

Trend of the temporal variance: d2t (j)/dt = 3.55

Intra-First d2a/dt d2e()/dt

Eqn S3 -3.74 0.19

Inter-First d2e/dt d2a()/dt

Eqn S4 0.74 -4.29

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Figure S1. Scheme for analyzing the space-time variance. a) Block of the space-time dataset; b) Partitioning the grandvariance 2g ; c) Partitioning the temporal variance

2t (j) of the jth region. Note: p q m

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X - 4 SUN ET AL.: VARIANCES PARTITIONING BETWEEN SPACE AND TIME

Time series of the spatial variances of each month in Eqn 9

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

0

100

200

300

400

500

Variance:

C2

Figure S2. Time series of the spatial variance 2s(i) (inEqn 9) for each month from January 1901 to December

2000. Note: 2s in Table 1 is the average of this timeseries

0.05 10-1 10-1/2 1 101/2 10 103/2 102 105/2 522 C2

Figure S3. Climatology (1901-2000) of the temporal

variance 2t (j) (Eqns 11 and 12). Note: 2t in Table 1 is

the average of this map

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-1.3 -0.4 0 0.4 0.8 1.2 1.6 2.0 2.9 C Century-1

Figure S4. Trend (1901-2000) in the decadal mean airtemperature t(j) (Eqns 11 and 12).

sun 2010 gl 043323

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