quasibiennial variations in fractal dimension of solar total irradiance
TRANSCRIPT
803
ISSN 0016-7932, Geomagnetism and Aeronomy, 2007, Vol. 47, No. 6, pp. 803–809. © Pleiades Publishing, Ltd., 2007.Original Russian Text © G.S. Ivanov-Kholodny, E.I. Mogilevsky, V.E. Chertoprud, 2007, published in Geomagnetizm i Aeronomiya, 2007, Vol. 47, No. 6, pp. 848–854.
1. INTRODUCTION
An analysis of variations in the solar magnetic field,performed in [Ivanov-Kholodny et al., 2004, 2006]using moving estimates of their FD obtained with thehelp of the Higuchi [1988] technique, indicated thatsuch an approach is valid even when comparativelyshort series of observations are studied. This caused theauthors to analyze again QBVs of the solar total irradi-ance [Ivanov-Kholodny et al., 2000, 2002] using theFD estimates obtained with the help of the Higuchitechnique.
The aim of this consideration is to determinewhether FD of the solar constant is permanent and, ifFD is variable, to elucidate whether QBVs, similar tothose found in the solar and ionospheric parameters, arepresent in these variations.
2. DATA PROCESSING
As in [Ivanov-Kholodny et al., 2000, 2002], theNimbus-7 [Hoyt et al., 1992] daily average measure-ments of the solar total irradiance (
L
) in 1978–1992 andthe indices of solar activity and the ionosphere (Wolfnumbers
W
, solar radioemission flux
F
10.7
, and criticalfrequency
f
2
of the ionospheric
F
2
layer normalized tonoon [Chertoprud and Shashun’kina, 1999]) were usedas initial data. We analyzed the series of observations
(1)Xt L t( ) W t( ) F10.7 t( ) f 2 t( ), , ,{ };=
t 1 2 … 4963; t[ ], , , days,= =
obtained after filling sparse gaps in the daily data of
L
and
f
2
(by interpolation or insertion of a similar recordfrom the other part of the series) and three-day movingaveraging of all data at an interval of one day. The timeorigin
t
is November 15, 1978.
Further calculations were performed on a movingannual interval. For the specified annual interval cen-tered at instant
í
, the
x
t
=
X
t
+
T
– 183
(
t
= 1, 2, …, 365)
data were taken from sets (1). These data were used toestimate the average annual
X
values denoted as
YX
(
T
)
and
DX
(
T
)
fractal dimensions calculated according tothe Higuchi [1988] technique. In addition to the listedvalues, we calculated the
S
= [
⟨
L
2
⟩
–
⟨
L
⟩
2
]
1/2
index (here,
⟨⟩
is the symbol of the 27-day moving averaging) and itsannual moving average
YS
(
T
)
. Index
S
proposed in[Ivanov-Kholodny et al., 2000] characterizes the vari-ability of the
solar constant
on the scale of a month.
The determination of FD of the
x
t
process (
t
=1, 2,…,
N
) was reduced to two operations. The first opera-tion is the calculation of the
l
(
k
) = (
N
– 1)
⟨|
x
t
+
k
–
x
t
|⟩
k
–2
curve length
of the
x
t
time series at different shifts
k
(e.g.,
k
= 3, 4, 5, …, 27). In this operation, averaging
|
x
t
+
k
–
x
t
|
was performed in two stages: first, the averagevalue of
|
x
t
+
k
–
x
t
|
quantities on the {
t
=
m
+ (
i
– 1)
k
,
i
=1, 2, …, [(
N
–
m
)/
k
]
} set was found for each
m
= 1, 2,…,
k
; then, the arithmetic mean of these estimates wascalculated (here [
µ
] denotes the integer part of numeral
Quasibiennial Variations in Fractal Dimensionof Solar Total Irradiance
G. S. Ivanov-Kholodny, E. I. Mogilevsky, and V. E. Chertoprud
Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radiowave Propagation, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia
Received October 31, 2006; in final form, March 5, 2007
Abstract
—Based on the Nimbus-7 (1978–1992) data and the parameters of solar activity (Wolf numbers
W
,solar radioemission
F
10.7
) and the ionosphere (
f
2
index of the critical frequency of the ionospheric
F
2
layer nor-malized to noon), the fractal dimension (FD) of the variations in the solar total irradiance (
L
) has been deter-mined on the moving annual interval using the Higuchi technique. It has been established that FD estimatessubstantially vary in time. Quasibiennial variations (QBVs), which similarly manifest themselves in all consid-ered processes, are detected in these variations. It is interesting that all
fractal
QBVs are in phase with QBVsof solar irradiance (
L
) and are almost in antiphase with QBVs of initial (filtered)
W
,
F
10.7
, and
f
2
indices. Thepresence of QBVs in the solar processes and in their FD and noncoincidence of the former with the latter inphase indicate that QBVs have a two-component structure. The obtained results also indicate that an analysisof the annual FD estimates of the solar and ionospheric processes in studying variations in these processes isreliable.
PACS numbers: 96.60.Ub
DOI:
10.1134/S001679320706014X
804
GEOMAGNETISM AND AERONOMY
Vol. 47
No. 6
2007
IVANOV-KHOLODNY et
al.
µ
). The second operation in determining FD is the esti-mation of coefficient
D
in the regression equations
(2)
According to the definition (2),
D
is the averageslope of the
z(x) curve, where z = – , andx = . For ideal fractal function z(x) approaches thelinear function with increasing volume of the N sample,and the D estimates tend to one value—fractal dimen-sion—independently of the selected k ranges. Takinginto account that the fractal scheme approximatelydescribes observed processes, and the sample is limited(N = 365), one should anticipate random (due to thefinite sample volume) and regular (due to nonideal frac-tality) deviations of function z(x) from a linear function.In this case the D estimates can substantially depend onthe selected range of k. The selection of the k range isless critical for analyzing time variations in FD, whichare more resistant to the variation in this range (errorsin determining coefficient D, related to nonideal fracta-lity, will change FD variations by an almost constantvalue if deviations from ideal fractality are constant intime). The range of shifts 3 ≤ k ≤ 27 was the main quan-tity in the further calculations.
Using the monthly step of the annual interval dis-placement over the 13.5-year interval of Xt observa-tions, we obtained the indices describing the time vari-ations in FD:
(3)
where DX(m) is the moving estimate of FD of the Xt
process, and m is the time of the center of the annual
l k( )log C D k.log–=
l/ klog( )logklog
DX m( ) DL m( ) DW m( ) DF10.7 m( ), D f 2 m( ), ,{ },=
m[ ] month, m 1 2 … 151,, , ,= =
interval on which FD has been determined (m = 1 forMay 1979). FD variations (3) were compared withchanges in running average annual values YX(m) of Xt.To determine QBVs in DX(m) and YX(m) changes, wetook the transform
(4)
i.e., we used the difference filtering applied when solarand magnetic activities were analyzed in [Ivanov-Kholodny and Chertoprud, 1993]. To increase the reli-ability, we averaged the monthly values of DX(m),YX(m), ∆DX(m), and ∆X(m) over three adjacent points(the denotations are the same).
3. ANALYSIS OF QBV
All initial information for a further analysis isincluded in the DX(m), YX(m), ∆DX(m), and ∆X(m)indices calculated on the 12-year time interval betweentwo maximums of solar cyclic activity (maximums ofsolar cycles 21 and 22). Figure 1, which presents thetime variations in YL (curve 1), YF10.7 (curve 2), and DL(curve 3), characterizes the variations in the movingestimates of FD for the annual changes in the solar totalirradiance. Together with DL, the linear trend of thisquantity is presented, and the triple value of the DLstandard error (3s) (the determination of which is dis-cussed in the next section) is shown by vertical bars.The errors of YL and YF10.7 estimations, caused by theobservation errors, are not more than 0.5% of theamplitude of variations in these quantities (the differ-ence between the maximal and minimal values) and areomitted in Fig. 1 because of their smallness.
∆DX m( ) 2DX m( ) DX m 12–( )– DX m 12+( );–=
∆X m( ) 2YX m( ) YX m 12–( )– YX m 12+( ),–=
1.9
1980
YL, W m–2
Years
1373
1985
1372
1371
DL
1.8
1.71990
60
100
140
180
220
YF10.7
1
2
33s
Fig. 1. Time variations in YL (1) YF10.7 (2), and DL (3).
GEOMAGNETISM AND AERONOMY Vol. 47 No. 6 2007
QUASIBIENNIAL VARIATIONS IN FRACTAL DIMENSION 805
An analysis of the data used to construct plot 1 indi-cates that (1) the DL value is not constant (i.e., FDs ofthe solar constant substantially vary in time); (2) thelinear trend, the randomness of which is rejected by asignificance level of 0.4%, is traced in DL variations;(3) the 11-year component is not detected in DL varia-tions, but QBVs are pronounced. In contrast to DL vari-ations, the 11-year cycle, which similarly manifestsitself in both processes (the correlation coefficientbetween YL and YF10.7 is 0.8), predominates in the vari-ations in the solar total irradiance, whereas QBVs arehardly discernible and are reliably revealed only afterfiltering [Ivanov-Kholodny et al., 2000]. Thus, funda-mental differences exist in the character of variations inirradiance L and its FD.
Additional information about variations in solarconstant FD can be obtained by comparing these varia-tions with the variations in FD of the solar activityparameters. Figure 2 presents the variations in FD ofDL, DW, and DF10.7 (curves 1–3) as an example and theDL0, DW0, and (DF10.7)0 curves approximating varia-tions in DL, DW, and DF10.7 by polynomials of the thirdorder (curves 1a–3a). The estimates of the sample val-ues ⟨DX⟩ and standard deviations σDX for these pro-cesses DX(m) are:
(5)
where the ⟨DX⟩ values with Neff ≈ 12 are indicated aserrors of σDX/(Neff)1/2 estimates.
Figure 2 and estimates (5) indicate that the averageFD values for all considered processes lie in the narrowrange (1.81 ± 0.05). In this case ⟨DL⟩ significantly (with
Dl⟨ ⟩ 1.86 0.02, σDL± 0.054,= =
DW⟨ ⟩ 1.81 0.02, σDW± 0.064,= =
DF10.7⟨ ⟩ 1.76 0.03, σDF10.7± 0.090,= =
a significance level of 3%) exceeds ⟨DF10.7⟩; i.e.,changes in the solar constant are more rugged[Mandel’brot, 2004] than variations in the solar radio-emission.
The FD variations observed in Fig. 2 pronouncedlyexceed the error level (standard deviations σDX aretwo–three times as large as errors s) and are dividedinto two components: slow (1‡–3‡) and fast QBVs. Fig-ure 2 indicates that slow variations in FD of the solarconstant substantially differ from slow variations in FDof the solar activity indices. In particular, the standarddeviation of the (DL0 – DW0) difference is σ(DL0 –DW0) = 0.056, which is commensurable with the stan-dard deviations of the DL and DW(5) variations (5).
In contrast to slow variations, fast variations in FDof the solar constant are distinctly reproduced in thevariations in FD of the solar activity parameters. Afterlinear filtering (4), the similarity of FDs in the consid-ered processes becomes evident. Figure 3 shows the fil-tered variations in FD of ∆DL, ∆DW, ∆DF10.7, and ∆Df2(curves 1–4). The filtered variations in the solar totalirradiance ∆L (curve 5) is presented for comparison.Vertical lines mark the instants of extremums on the ∆Lcurve. To be comparable, all variations are presented inthe standardized form, i.e., after the transformation
(6)
An analysis of the daily L data (including the stan-dard deviation of L within a day) indicates that the qual-ity of L observations and the reliability of FD determi-nation decrease at the beginning and end of the obser-vation period. Therefore, when comparing ∆DXvariations, it is reasonable to consider a shorter period:from the beginning of 1981 to the end of 1989. This
∆DX st ∆DX ∆DX⟨ ⟩–( )/σ∆DX or=
∆L st ∆L ∆L⟨ ⟩–( )/σ∆L.=
1.51980
D
Years
1.9
19901985
3s3
1.8
1.7
1.6
3s2
3s1
1a
2a
3a
1
2
3
Fig. 2. Variations in FD of DL (1), DW (2), and DF10.7 (3). Vertical bars show the triple standard errors of these values. The curvesapproximating the variations in DL, DW, and DF10.7 by polynomials of the third order are also presented (1a)–(3a).
806
GEOMAGNETISM AND AERONOMY Vol. 47 No. 6 2007
IVANOV-KHOLODNY et al.
period is subsequently used in a correlation analysis.The standard error st is not more than 0.5 for ∆DL andis less than 0.3 for the other quantities.
The variations shown in Fig. 3 are closely related toone another. The average value of ten pair correlationcoefficients ri, k (i ≠ k are the numbers of the plots inFig. 3) is 0.70 ± 0.05. The relation of plot i to theremaining plots is characterized by the average ri valueof the correlation coefficients ri, k: r1 = 0.64 ± 0.09, r2 =0.78 ± 0.06, r3 = 0.75 ± 0.06, r4 = 0.58 ± 0.13, and r5 =0.72 ± 0.04. The smallest correlation coefficient has theionospheric index ∆Df2, which seems natural: iono-
spheric disturbances are superimposed on the solareffect.
An analysis of the data used to construct Fig 3 indi-cates that (1) fast variations in FD of the solar constantare distinctly reproduced in the variations in FD of theparameters of solar activity (and the ionosphere); (2)QBVs of the solar constant (curve 5) and its FD (curve1) are almost synchronous. Synchronism between thevariations in the index itself and its FD is the specificproperty of the process, which is realized not always(see Fig. 4).
The variations in FD of ∆DL (curve 1) and in the ∆W(2), ∆F10.7 (3), ∆f2 (4), and ∆S (5) values are compared
1981
st
1
1985
0–1
10
–1
10
–1
1983 1987 1989 1991
1
2
3
4
5
10–1
10–1
st
Fig. 3. Filtered variations in FD of ∆DL (1), ∆DW (2), ∆DF10.7 (3), and ∆Df2 (4). The variations in ∆L (5) are shown for comparison.The variations are standardized.
1981
st
1
1985
0–1
10
–1
10
–1
1983 1987 1989 1991
10–1
10–1
1
2
3
4
5
st
Fig. 4. The variations in ∆DL (1) as compared to the variations in ∆W (2), ∆F10.7 (3), ∆f2 (4), and ∆S (5). The variations are stan-dardized.
GEOMAGNETISM AND AERONOMY Vol. 47 No. 6 2007
QUASIBIENNIAL VARIATIONS IN FRACTAL DIMENSION 807
in Fig. 4. To be comparable, all variations are presentedin the standardized form, i.e., after the transformationsimilar to (6). The standard errors st are not more than0.5 and 0.3 for ∆DL and the other quantities, respec-tively (see also the estimates in [Ivanov-Kholodny etal., 2000, 2000a, 2002]). Figure 4 demonstrates that theQBV extremums on the ∆W, ∆F10.7, ∆f2, ∆S curvesappear synchronously and the positions of the extrem-ums on the ∆DL curve and on the remaining plots donot coincide. Taking into account a similarity in the∆DL, ∆DW, ∆DF10.7, and ∆Df2 variations (Fig. 3), wecan conclude that QBVs in the parameters of solaractivity (and the ionosphere) are asynchronous in con-trast to the situation with the solar total irradiance L.
4. DISCUSSION
1. FD estimates (at k0 ≤ k ≤ K0) serve at least twopurposes. First, a FD absolute value numerically char-acterizes roughness of the studied series on scalesk0−K0; second, relative variations in FDs obtained onthe moving interval with a length of Q � K0 containinformation about variability of time series parameters(roughness) beginning from scale ~Q. Taking intoaccount that a fractal scheme only approximatelydescribes a studied process, it is natural to anticipatethat an absolute FD value can substantially depend on aselected range of shifts k, whereas relative variations inFD are resistant to such a selection. A similar situationis sometimes encountered in physical measurements,when absolute values of a measured quantity are lessreliable than variations in this quantity.
Figure 5 to a certain degree illustrates the aforesaid.This figure presents the DL estimates and standardizedfiltered variations (∆DL) for two ranges of k. The aver-
age estimates (⟨DL⟩) are 1.87 ± 0.02 for 3 ≤ k ≤ 27 and1.69 ± 0.02 for 1 ≤ k ≤ 27; i.e., a change in the k rangesubstantially affects the FD absolute value. However,the standardized filtered variations (∆DL) for two con-sidered ranges of k are almost identical. Thus, it is pos-sible to analyze FD variations even if descriptions of aprocess pronouncedly differ from of an ideal fractalscheme, when it becomes difficult to estimate a FDabsolute value. We should note that a sharp decrease inthe ⟨DL⟩ value in going from 3 ≤ k ≤ 27 to 1 ≤ k ≤ 27 isrelated to the 3-day moving averaging of the initial pro-cess, as a result of which the values of the l(1) and l(2)quantities, entering into Eqs. (2), become substantiallysmaller.
2. The standard error (sD) of coefficient D, obtainedduring a regression analysis of Eqs. (2), can be used todetermine the error (s) of DX. If DX does not substan-tially change during a year, s ≈ sD. If such changes areconsiderable, the sD value gives increased s. Taking intoaccount this circumstance, we took as s the average ofthe smallest ten sD values over the entire series of obser-vations when we determined the DX error.
3. Previously, an analysis of the variations in thesolar total irradiance [Ivanov-Kholodny et al., 2000,2002] revealed QBVs in two parameters characterizingthis irradiance: in the irradiance (L) and in the standardirradiance deviation (S) within a month. QBVs in indexS are close in form and phase to QBVs in the indices ofsolar activity and the ionosphere (see Fig. 4). QBVs inirradiance itself (L) are almost in antiphase with theseQBVs.
At present, QBVs have been found out in FDs of theirradiance L and solar activity indices (see Fig. 3). Itturned out that all these QBVs are similar to QBVs inthe irradiance L. Thus, all QBVs in the processes con-
1981
DL
19851983 1987 1989 1991
1
2
3
4
2.0
1.8
1.6
210
–1–2
∆DLst
Fig. 5. Comparison of the DL estimates and the standardized filtered variations (∆DL) for two k ranges. The estimates (1) and(3) were obtained for 3 ≤ k ≤ 27; the estimates (2) and (4), for 1 ≤ k ≤ 27.
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GEOMAGNETISM AND AERONOMY Vol. 47 No. 6 2007
IVANOV-KHOLODNY et al.
sidered above belong to one of two types, similar toQBVs in L or S. It is interesting that QBVs in all indices(except L) and their FDs belong to different types. OnlyQBVs in the solar total irradiance (L) and in its FD weresimilar, and this is the distinctive feature of the irradi-ance as compared to the remaining parameters. Thepresence of QBVs in the solar processes and in theirFDs and noncoincidence of the former with the latter inphase indicate that QBVs have a two-componentstructure.
4. For the X process, which is the sum of severalcomponents of the Xi processes with slightly differentstatistical parameters, the character of the relationbetween the variations in the average X value and FD(strong correlation, anticorrelation, correlations at atime shift between X and FD, absence of correlation,etc.) can be related to changes in the contributions ofthe Xi components to the X process. This is illustratedby the following simplest scheme
(7)
where Xi processes are similar to flicker noise withpower spectrum indices αi (for definiteness, α1 < α2),and slowly varying amplitudes ai characterize the con-tribution of the X1 and X2 components. If the relativecontribution of the X1 and X2 components remainsunchanged (the a1/a2 ratio is constant), FD will be con-stant. If the X1 component increases (i.e., a1 increasesand a2 remains unchanged), the average X value and FDwill increase simultaneously. If the X2 componentincreases (i.e., a2 increases at a constant a1 value), theaverage X value will increase, and FD will decrease.Thus, under certain conditions, slow variations in the ai
amplitudes result in interrelated variations in the aver-age value and FD of the X process. Consequently, dif-ferent forms of interrelation between the variations inthe moving average and FD of the X process are possi-ble within the scope of the scheme (7).
As the other integral (over the surface) solar param-eters, the solar total irradiance (L) includes severalcomponents that belong to different spatial scales, andvariations in these components differ in statisticalparameters. The observed relation between QBVs inthe solar total irradiance (L) and its FD (the correlationcoefficient between ∆L and ∆DL is 0.81 ± 0.08) can beinterpreted in the scheme (7) as an indication for thepresence of the most powerful QBVs in the irradiancecomponent that belongs to the smallest spatial scale andhas the smallest α (the first component in the (7)scheme).
5. The helioseismological observations performedat the SOHO cosmic laboratory made it possible toreveal a complex layered structure of the solar magne-toplasma, including the magnetic field generation in theuppermost layer of the Sun. The presence of this struc-ture changes the kinematic and temperature regimes ofthe Sun at different depths and can principally change
X a1X1 a2X2, ai 0, Xi 0,≥>+=
the solar constant. However, a relatively slow waveradiation transfer (and related temperature regime)does not change the average solar constant at differentdepths. That is why the measurements of the solar con-stant, previously performed on spacecraft, are stillactual. At present, real data on instability of previousmeasurements of QBVs in the solar constant areabsent.
The studies of different random processes in thenature (see, e.g. [Mandel’brot, 2004]) indicate that ran-dom processes sometimes radically change their mainprocesses. It is still unclear whether variations in thesolar constant are among such processes. Therefore, itis desirable to search and analyze other data on the vari-ability of the solar total irradiance.
5. CONCLUSIONS
An analysis of the variations in the solar magneticfield, performed by Ivanov-Kholodny et al. [2004,2006] using the moving estimates of FDs of these vari-ations, indicated that such an approach can be success-fully used to study even comparatively short series ofobservations. This caused us to analyze again QBVs ofthe solar total irradiance [Ivanov-Kholodny et al., 2000,2002] using the FD estimates.
The FDs of the variations in the solar total irradiance(L) according to the Nimbus-7 data (1978–1992) and ofthe parameters of solar activity (Wolf numbers W, solarradioemission F10.7) and the ionosphere (f2 index for theF2 ionospheric layer) were calculated on the movingannual interval. It was established that the moving esti-mates of FD substantially change in time. QBVs, whichare similar for all considered processes, are observed inthese changes. It is interesting that all fractal QBVs arein phase with QBVs of the solar irradiance itself (L) andare almost in antiphase with QBVs of the initial (fil-tered) W, F10.7, and f2 indices. The presence of QBVs inthe solar processes and in their FDs and the noncoinci-dence of the former with the latter in phase indicate thatQBVs have a two-component structure. The obtainedresults also indicate that an analysis of moving annualestimates of FDs of the solar and ionospheric processesis reliable when variations in these processes arestudied.
ACKNOWLEDGMENTS
This work was supported by the Russian Foundationfor Basic Research, project no. 04-02-16374).
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QUASIBIENNIAL VARIATIONS IN FRACTAL DIMENSION 809
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