a global model of the ionospheric f2 peak height based on ...€¦ · a global model of the...

Ann. Geophys., 27, 3203–3212, 2009www.ann-geophys.net/27/3203/2009/© Author(s) 2009. This work is distributed underthe Creative Commons Attribution 3.0 License.

AnnalesGeophysicae

A global model of the ionospheric F2 peak height based onEOF analysis

M.-L. Zhang, C. Liu, W. Wan, L. Liu, and B. Ning

Beijing National Observatory of Space Environment, Institute of Geology and Geophysics, Chinese Academy of Sciences,Beijing 100029, China

Received: 20 October 2008 – Revised: 26 June 2009 – Accepted: 9 July 2009 – Published: 14 August 2009

Abstract. The ionospheric F2 peak heighthmF2 is an impor-tant parameter that is much needed in ionospheric researchand practical applications. In this paper, an attempt is madeto develop a global model ofhmF2. ThehmF2 data, used toconstruct the global model, are converted from the monthlymedian hourly values of the ionospheric propagation factorM(3000)F2 observed by ionosondes/digisondes distributedglobally, based on the strong anti-correlation existed betweenhmF2 and M(3000)F2. The empirical orthogonal function(EOF) analysis method, combined with harmonic functionand regression analysis, is used to construct the model. Thetechnique used in the global modelling involves two layersof EOF analysis of the dataset. The first layer EOF anal-ysis is applied to thehmF2 dataset which decomposed thedataset into a series of orthogonal functions (EOF base func-tions) Ek and their associated EOF coefficientsPk. Thebase functionsEk represent the intrinsic characteristic vari-ations of the dataset with the modified dip latitude and lo-cal time, the coefficientsPk represents the variations of thedataset with the universal time, season as well as solar cy-cle activity levels. The second layer EOF analysis is appliedto the EOF coefficientsPk obtained in the first layer EOFanalysis. The coefficientsAk, obtained in the second layerEOF analysis, are then modelled with the harmonic func-tions representing the seasonal (annual and semi-annual) andsolar cycle variations, with their amplitudes changing withthe F10.7 index, a proxy of the solar activity level. Thus,the constructed global model incorporates the geographicallocation, diurnal, seasonal as well as solar cycle variationsof hmF2 through the combination of EOF analysis and theharmonic function expressions of the associated EOF coef-ficients. Comparisons between the model results and obser-vational data were consistent, indicating that the modelling

Correspondence to:M.-L. Zhang([email protected])

technique used is very promising when used to constructthe global model ofhmF2 and it has the potential of beingused for the global modelling/mapping of other ionosphericparameters. Statistical analysis on model-data comparisonshowed that our constructed model ofhmF2, based on theEOF expansion method, compares better with the observa-tional data than the model currently used in the InternationalReference Ionosphere (IRI) model.

Keywords. Ionosphere (Equatorial ionosphere; Mid-latitude ionosphere; Modelling and forecasting)

1 Introduction

Ionospheric modelling has been one of the leading ways tostudy the ionosphere. An ionospheric model can be either atheoretical model (also named physical model or first princi-ple model) which is derived from various laws of physics andbased on the numerical solution of the equations describingthe spatial and temporal distribution of medium parametersor an empirical (or semi-empirical) model which is derivedfrom the observational results. Empirical ionospheric mod-els are not only important for ionospheric research, but alsovery useful and much needed in a wide range of practicalapplications, such as radio and telecommunication, satellitetracking, Earth observation from space, etc.

In many empirical ionospheric models, such as the Inter-national Reference Ionosphere (IRI) (Bilitza, 1990; 2001)and NeQuick (Radicella and Leitinger, 2001; Leitinger etal., 2005) models, the calculation of the electron densityprofile usually uses the critical points such as F2, F1 andE layer peaks as anchor points, using parameters offoF2,hmF2, foF1,hmF1, foE andhmE (the critical frequencies andpeak heights of the F2, F1 and E layers, respectively) as in-puts. They are based on the global or regional “maps” of theionospheric peak parameters. Since the ionospheric electron

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3204 M.-L. Zhang et al.: Global model of the ionospheric F2 peak height based on EOF analysis

density has its maximum values in the F2 layer, this layeris the most important region of the ionosphere which is pri-marily responsible for the reflection of radio waves in high-frequency communication and broadcasting. For this reason,F2 peak is the key anchor point to determine the ionosphericelectron density profiles. Therefore, the F2 layer peak heighthmF2 is one of the most important parameters in ionosphericempirical modelling. Due to the lack of observational data ofhmF2, in practice, people usually obtainedhmF2 based on itsstrong anti-correlations with the ionospheric propagation fac-tor M(3000)F2 whose value is provided usually by the CCIR(International Radio Consultative Committee) or now calledITU (International Telecommunication Union) M(3000)F2model (CCIR report, 1967). The CCIR M(3000)F2 modelpredicts M(3000)F2 based on the 12-month running aver-age sunspot number Rz12, thenhmF2 is calculated basedon this CCIR M(3000)F2 model values using Eqs. (1–6)listed in Sect. 2. However, recently some authors (Adeniyiet al., 2003; Obrou et al., 2003; Zhang et al., 2004; 2007)found that in the equatorial and low-latitude regions, the val-ues ofhmF2 converted from the CCIR M(3000)F2 modelvalue using Eqs. (1–6) have remarkable discrepancies withthe observationalhmF2. They revealed that the discrepanciesstemmed from the inaccuracy of CCIR M(3000)F2 model.Because when the measured M(3000)F2 values are used, thehmF2 converted using Eqs. (1–6) agrees very well with theobservational result derived from the manually edited tracesof ionograms using ionogram inversion programs. There-fore, there is a necessity to update the existing M(3000)F2model or construct directly a global model ofhmF2. This isan urgent main task of the IRI working group community. Tothis goal, recently, some new modeling techniques have beenproposed to model these parameters. For example, Oyeyemiet al. (2007) proposed a new modelling technique based onthe application of neural network to model the M(3000)F2parameter, whereas Liu et al. (2008) attempted to modelthe M(3000)F2 parameter based on the empirical orthogonalfunction (EOF) analysis of the observational dataset. Fur-thermore, recently Gulyaeva et al. (2008) derived a numeri-cal model ofhmF2 from the topside database of about 90 000electron density profile provided by ISIS1, ISIS2, IK19 andCosmos-1809 satellites for the period of 1969–1987. In thispaper, we pursue constructing a global model ofhmF2 usinga modelling technique which is based on the empirical or-thogonal function (EOF) decomposition of the globalhmF2dataset and the modelling of its associated EOF coefficients.In the following sections, we will first mention the datasetused for our model construction in Sect. 2, and then in Sect. 3we will describe the modelling technique used. The mod-elling results with some discussions will be shown in Sect. 4and the last section (Sect. 5) is the summary and conclusion.

2 Data and transformation equations

The fact thathmF2 is a parameter which is not easy toobtain from measurement makes it difficulty to have anobservationalhmF2 dataset with enough spatial coverageand history length of data accumulation that can be usedfor global modelling. However, it has been shown (Shi-mazaki, 1955; Wright and Mcduffie, 1960) thathmF2 isstrongly anti-correlated to the ionospheric propagation fac-tor M(3000)F2 and, fortunately, M(3000)F2 can be routinelyscaled from ionograms recorded by ionosondes/digisondesdistributed globally and its data has already been accumu-lated for a very long time. This enables us to construct adatabase ofhmF2 from the global M(3000)F2 database forour modelling study. The original empirical formula be-tweenhmF2 and M(3000)F2, i.e.,hmF2=1490/M(3000)F2-176, was derived by Shimazaki (1955). However, it wasfound, by later researchers, that a correction term1M shouldbe added and the formula now has the format (Wright andMcduffie, 1960; Bradley and Dudeney,1973; Eyfrig, 1973;Bilitza et al., 1979):

hmF2=1490

M(3000)F2+1M−176 (1)

The correction term1M in Eq. (1) accounts for the delay-effect caused by the ionizations in the E layer that is relatedto thefoF2/foE ratio. There are various expressions of1M

derived by different authors (Bradley and Dudeney, 1973;Eyfrig, 1973; Bilitza et al., 1979). The expression of1M

we used in our calculation ofhmF2 is the one derived by Bil-itza et al. (1979), which has being used in the IRI model sinceits first release in 1978 (Rawer et al., 1978):

1M =F1(R12) ·F2(R12,8)

foF2/foE−F3(R12)+F4(R12) (2)

where

F1(R12) = 0.00232·R12+0.222 (3)

F2(R12,8) = 1−R12/150·exp(−82/1600) (4)

F3(R12) = 1.2−0.0116·exp(R12/41.84) (5)

F4(R12) = 0.096·(R12−25)/150 (6)

WhereR12 is the 12-month running mean of the sunspotnumber,8 is the magnetic dip latitude which is related tothe magnetic inclinationI as2tg8=tgI .

The hmF2 data, used in our present study, is calculatedfrom M(3000)F2 based on Eqs. (1–6). The use of Eqs. (1–6) is justified by many works (e.g., Adeniyi et al., 2003;Obrou et al., 2003; Zhang et al., 2004; 2007) that showedwhen the measured M(3000)F2 values are used as input,the hmF2 value obtained with Eqs. (1–6) agrees very wellwith the observational ones derived from the manually editedtraces of ionograms using ionogram inversion programs. As

Ann. Geophys., 27, 3203–3212, 2009 www.ann-geophys.net/27/3203/2009/

M.-L. Zhang et al.: Global model of the ionospheric F2 peak height based on EOF analysis 3205

for the accuracy of Eq. (1), it is estimated by Dudeney(1983) that the uncertainty ofhmF2 calculated from observedM(3000)F2 is within 4–5%, i.e., about 15–20 km. Thus, therelationship between M(3000)F2 andhmF2 can be taken astrue. In the present study, data of M(3000)F2,foF2 andfoEused to construct thehmF2 database were downloaded fromthe Space Physics Interactive Data Resource (SPIDR) web-site http://spidr.ngdc.noaa.gov/. Figure 1 shows the globaldistribution of the stations used for the presenthmF2 globalmodelling study. In the present work, we used the monthlymedian data to do the modelling. Therefore, the magneticactivity effects are ignored in the model.

3 Modelling technique description

3.1 Fundamental of EOF analysis method

The modelling technique we used to modelhmF2 is mainlybased on the empirical orthogonal function (EOF) expan-sion or decomposition of the dataset. EOF analysis methodwas actually invented by Pearson (1901). This method hasbeen widely and successfully used by meteorologists andoceanographers to analysis the spatial and temporal varia-tions of physical fields since Lorenz (1956) introduced itinto meteorology. In the ionospheric field, it was Dvinskikh(1988) who first introduced the EOF analysis into the em-pirical modelling of the ionospheric parameters. His pioneerwork was followed by other works (e.g., Singer and Tauben-heim, 1990; Bossy and Rawer, 1990; Singer and Dvinskikh,1991; Dvinskikh and Naidenova, 1991). It has been shownby many researchers that the EOF analysis is a powerfulmethod in the ionospheric data analysis and empirical mod-elling (e.g., Dvinskikh, 1988; Singer and Dvinskikh, 1991;Dvinskikh and Naidenova, 1991; Daniell et al., 1995; Marshet al., 2004; Matsuo et al., 2002, 2005; Mao et al., 2005,2008; Materassia and Mitchell, 2005; Zhao et al., 2005;Zapfe et al., 2006, Liu et al., 2008; Wan et al., 2009, andmany more).

EOF analysis involves a mathematical procedure thattransforms a dataset into a number of uncorrelated compo-nents called principal components, with any two componentsorthogonal to each other. Thus, this method sometimes isalso named Principal Component Analysis (PCA) or NaturalOrthogonal Component (NOC) algorithm. The underlyingphysical meaning of EOF analysis is that the variation of aphysical field variable is mainly controlled by some indepen-dent processes that can be separated. This method decom-poses a dataset into a series of eigen function (or base func-tions) and its associated coefficients, with the base functionsorthogonal to each other. As we know, people usually use theFourier or Spherical Harmonic analysis method to decom-pose a dataset. However, the base function set used in Fourieror Spherical Harmonic analysis is predesigned artificially. InEOF decomposition, the orthogonal base functions are not

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180−90

−60

−30

0

30

60

90

Geographic Longitude(deg)

Geo

grap

hic

Latit

ude(

deg)

Fig. 1. Global distribution of the stations used for the present mod-elling study.

artificially designed in advance, but are naturally determinedby the experimental dataset to be decomposed. Therefore,they possess the inherent characteristics of the original data,and the eigen series will converge very quickly. This makesit possible to use only a few orders of EOF components torepresent most of the variance of the original dataset. Hence,the EOF expansion has advantages in data analysis and rep-resentation. In the following, we will briefly describe thefundamentals of the EOF analysis method. For further de-tails, readers are referred to Dvinskikh (1988), Storch andZwiers (1999) and Xu and Kamide (2004).

Let Yij =Y (ti , xj ) represent the value of a variable (e.g.,hmF2) at the spatial pointxj (e.g., longitude and latitude)at time ti , i=1, 2, . . . ,m andj=1, 2, . . . ,n. ThenYm×n isa matrix ofm rows andn columns. Suppose there are to-tally r independent physical processes affecting the variationof the variable, thenY can be decomposed into a series ofbase functionEk(xj ),(j = 1,2,...,n) and its associated co-efficientsAk(ti),(i = 1,2,...,m).

Y =

r∑k=1

Y k(t,x) =

r∑k=1

Ak(t)EkT (x) (7)

WhereEk= [ek

1,ek2,...,e

kn]

T andAk= [ak

1,ak2,...,ak

m]T . The

superscriptT over a matrix means the transpose of the ma-trix. The base functionsEk(k = 1,2,...,r) are uncorrelated,that is, they are orthogonal over space:

EkT El=

n∑j=1

ekj e

lj =

{1 k = l

0 k 6= l(8)

ekj andak

i are obtained by minimizing

δ =

m∑i=1

n∑j=1

[Yij −

r∑k=1

aki e

kj

]2

(9)

www.ann-geophys.net/27/3203/2009/ Ann. Geophys., 27, 3203–3212, 2009

http://spidr.ngdc.noaa.gov/


Fig. 2. A sample map of the standard deviation obtained when doingthe kriging interpolation.

with respect toeki andek

j . In fact, Ek(k = 1,2,...,r) is the

orthogonal eigenvectors of the symmetric matrixS = Y T Y ,which can be obtained by solving the equation

SEk= λkE

k (10)

Whereλk is the eigen value. OnceEk is found, the associ-ated coefficientAk can be computed by

Ak= YEk (11)

It can be proved that

δ =

m∑i=1

n∑j=1

[Yij −

r∑k=1

aki e

kj

]2

=

m∑i=1

λi −

r∑i=1

λi (12)

The percentage variance of the dataset captured by the firstr

components isr∑

i=1λi

/m∑

i=1λi ×100%.

3.2 Data preprocessing and decomposition

Before we can apply the EOF decomposition to the dataset ofhmF2, we need to do some data preprocessing work. Sincethe time periods with available data are different for differ-ent stations, we must normalize the data of each station tothe same time period. In this study, we choose the years of1975–1985 covering both ascending and descending phasesof the solar cycle activities as the time period used for globalmodelling. This time period was chosen because data duringthis period are available for most of the stations used. To fillthe missing data for any station during this period, we pre-processed the data as follows. First, for each single station,with its observational values of M(3000)F2,foF2 andfoE asinput, M(3000)F2 is converted intohmF2 using Eqs. (1–6).WhenfoE is not available, the value obtained from IRI modelis used as input. With the convertedhmF2 dataset, single sta-tion model ofhmF2 is constructed for each individual stationusing the modelling technique described in Liu et al. (2008).

Then, the missing data (if any), for any chosen station dur-ing the time period of 1975–1985, are filled with the singlestation model values.

After normalizing all the chosen stations’ data to the timeperiod of 1975–1985 as described above, data at evenly dis-tributed grids (5◦ ×5◦ in a latitudinal range of 85◦ S–85◦ Nand longitudinal range of 0–360◦ E) were then obtained byinterpolation using Kriging method. To test the validity ofthe Kriging method in our data preprocessing, the accuracyof Kriging maps is estimated. Figure 2 is a sample but typ-ical map of the standard deviation obtained when doing theKriging interpolation. In Fig. 2, the black points representstations whose data is used in interpolation. As can be seenfrom the figure, the standard deviations of the Kriging maps,at the large areas where observations are sparse, are less than10 km, which is a very general result in our data preprocess-ing. This means the accuracy of the Kriging maps obtainedin our data preprocessing is quite acceptable. The preparedgridded data are the dataset to which the EOF decompositionis going to be applied.

The first step in our modelling is to do the EOF analysison the gridded dataset prepared as described above. Two lay-ers EOF decomposition are applied. In the first layer, theprepared gridded dataset are decomposed into the EOF basefunctionsEk(µ, LT), which represent the variation ofhmF2with the modified dip latitude (short:Modip)µ and the localtime (LT), and the associated EOF coefficientsPk (UT,m),which represent the variations ofhmF2 with the universaltime (UT) and seasonal as well as solar cycle variations (de-noted all by the variablem) as the following

hmF2(µ,LT,UT,m)=

N∑k=1

Ek(µ,LT) ·Pk(UT,m) (13)

Here the coordinate(µ, LT) instead of geographical lati-tude/longitude or magnetic latitude/longitude is used since itwas found (Rawer, 1963) that features of the ionospheric pa-rameters, in particular those of the F2 layer, are mostly wellorganized under the coordinate (µ, LT) which is also wellconfirmed by recent research works (e.g., Azpilicueta et al.,2006) and by our experience when we tried to decomposethe dataset using other coordinates such as the geographi-cal latitude/longitude one. This is due to the fact that theionosphere is controlled by both the orientation of the Earth’srotation axis and the configuration of the geomagnetic field.Therefore, its variation depends on both the geographical andgeomagnetic latitudes, which is embedded in the modipµ

defined astgµ = I/

cos1/2ϕ, whereI is the magnetic incli-nation andϕ the geographical latitude. The ionosphere hasalso longitudinal variation. In our modelling, the longitudi-nal variation is embedded in theLT (of Ek) andUT (of Pk)variations sinceLT=UT+Longitude/15.

In the second layer of EOF decomposition, the coefficientsPk(UT,m) obtained in the first layer EOF analysis are de-composed once again into a series of base functionF

jk (UT)



LT

E1

Mod

ip(0 )

0 6 12 18 24−80

−40

0

40

80

0.017

0.018

0.019

0.02

0.021

0.022

0.023

LT

E2

Mod

ip(0 )

0 6 12 18 24−80

−40

0

40

80

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

LT

E3

Mod

ip(0 )

0 6 12 18 24−80

−40

0

40

80

−0.04

−0.02

0

0.02

0.04

LT

E4

Mod

ip(0 )

0 6 12 18 24−80

−40

0

40

80

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Fig. 3. Distribution of the first four orders of the base functions (E1–E4) obtained with the 1st layer EOF decomposition ofhmF2.

representing the variation with the universal time (UT) andthe associated coefficientsAj

k(m) representing the seasonalas well as solar cycle variations

Pk(UT,m)=

N1∑j=1

Fjk (UT) ·A

jk(m) (14)

3.3 Modelling the associated EOF coefficientsAjk(m)

The second step in our modelling is to model the EOF coef-ficientsA

jk(m) with the following harmonic functions repre-

senting the seasonal (annual and semi-annual) and solar cyclevariations. The reason justifying the usage of the followingharmonic functions with their amplitudes changing with thesolar flux index F10.7 to model the coefficientsAj

k(m) will begiven in Sect. 4 when we discuss the results obtained by theEOF decompositions.

Ajk(m) = B

j

k1(m)+Bj

k2(m)+Bj

k3(m) (15)

Bj

k1(m) = cj

k1+dj

k1F10.7(m) (16)

Bj

k2(m) = (cj

k2+dj

k2F10.7(m))cos2πm

12

+(sj

k2+ tj

k2F10.7(m))sin2πm

12(17)

Bj

k3(m) = (cj

k3+dj

k3F10.7(m))cos2πm

6

+(sj

k3+ tj

k3F10.7(m))sin2πm

6(18)

wherem is the month representing the seasonal variation, andF10.7 index is a proxy most commonly used to represent thesolar activity levels which was originally called the Coving-ton index (Covington, 1948). It is the solar radio flux densitymeasured at a wavelength of 10.7 cm and expressed in unitsof 10−22 Watts/m2/Hertz. For more details on the proxy so-lar radio flux, please refer to Tobiska (2001) and referencestherein. With these equations, the coefficientsc

j

k1, dj

k1, cj

k2,

dj

k2, sj

k2, tj

k2, cj

k3, dj

k3, sj

k3, tj

k3 are determined by the least-square fitting approaches.

3.4 Model construction

Our model constructions are done in a reversal procedurebased on the obtained EOF base functionsEk andF

jk as well

as the coefficientscj

k1, dj

k1, cj

k2, dj

k2, sj

k2, tj

k2, cj

k3, dj

k3, sj

k3, tj

k3obtained. Specifically, to calculate the model value ofhmF2at a given geographical location, one first uses Eqs. (15–18),with the given monthm and the corresponding solar flux in-dexF10.7(m), to calculate the modelled second layer EOF co-efficientsAj

k . Then with the calculated modelled coefficients



UT

P1

Yea

r

0 6 12 18 241975

1978

1981

1984

1.4

1.5

1.6

1.7

1.8

x 104

UT

P2

Yea

r

0 6 12 18 241975

1978

1981

1984

−1500

−1000

−500

0

500

1000

UT

P3

Yea

r

0 6 12 18 241975

1978

1981

1984

−1000

−500

0

500

1000

UT

P4

Yea

r

0 6 12 18 241975

1978

1981

1984

−1000

−500

0

500

Fig. 4. Distribution of the coefficients (P1–P4) corresponding to the first four orders of base functions (E1–E4) shown in Fig. 3 obtainedwith the first layer EOF decomposition ofhmF2.

Ajk and the base functionsF j

k obtained in the second layerEOF decomposition, the modelled first layer EOF coeffi-cientsPk are calculated with Eq. (14). At last, with the calcu-lated modelledPk and the base functionsEk obtained in thefirst layer EOF decomposition, the modelled value ofhmF2is calculated using Eq. (13).

4 Results and discussion

Figure 3 shows the contour plots of the first four orders of thebase functionsEk, obtained with the first layer EOF decom-position of hmF2 dataset, versus the modified dip latitude(µ) and the local time (LT). It can be seen that apparently,the base functionsEk we obtained showed some typical fea-tures that have been observed in many data. For example,the distribution of the first order of base functionE1 mani-fests mainly a typical phenomenon related to the equatorialionization anomaly. As is well known, the equatorial ioniza-tion anomaly is a phenomenon characterized by a structurewith two crests of ionization (best represented byfoF2) atabout±17◦ dip latitude on each side of the magnetic equatorand a trough in between. It is formed as a consequence of theso called “fountain” effect. The influence of this equatorialfountain effects on the F2 peak heighthmF2 is that it will

produce a latitudinal distribution ofhmF2 with higher valuenear equatorial and low latitudes but with lower value out-side. This latitudinal distribution structure ofhmF2 is exactlywhat we see in the distribution of the first order of the basefunction E1. The second base functionE2 mainly reflectsthe north-south asymmetry which is closely related to theseasonal change of the solar zenith angle. Because we willsee, in Fig. 4, the associated EOF coefficientP2 correspond-ing to this base function shows mainly an annual variationpattern. As for the third base functionE3, the most remark-able feature to notice is the evening enhancement ofhmF2 inthe equatorial and low-latitude region. This feature is appar-ently a result of the regeneration/enhancement of the fountaineffect during the post-sunset hours due to the evening pre-reversal enhancement (PRE) of the F2 region plasma’s elec-tromagneticE ×B drift caused by the enhanced zonal elec-tric field near the magnetic equator (Fejer et al., 1995, andreferences therein). Another feature worth noticing is the en-hancement/abatement ofE3 in the north/south auroral zones.This feature can be explained by combining it with the pat-tern ofP3 shown in Fig. 4.P3 is positive/negative during thenorth winter/summer seasons. Therefore, the contribution ofthe componentE3×P3 is positive/negative during the northwinter/summer seasons. This is apparently a phenomenacaused by the difference of sunlit hours in the North/South



0 4 8 12 16 20 24−0.5

0

0.5

F1j

0 4 8 12 16 20 24−0.5

0

0.5

F2j

0 4 8 12 16 20 24−0.5

0

0.5

F3j

0 4 8 12 16 20 24−0.5

0

0.5

F4j

UT

1976 1979 1982 198502468

x 104

A1j

1976 1979 1982 1985−4000−2000

020004000

A2j

1976 1979 1982 1985

−2000

0

2000A

3j

1976 1979 1982 1985

−2000

0

2000

A4j

Year

j=1 j=2 j=3 j=1 j=2 j=3 Modeled

Fig. 5. Distribution of the base functions (Fj1 –F

j4 ) (left panels) and

their corresponding coefficients (Aj1–A

j4) (right panels) obtained

with the second layer EOF decomposition ofhmF2.

Hemispheres during winter/summer seasons. Therefore, as isdemonstrated here, the EOF decomposition analysis method,to some extent, is able to separate the variance of a datasetinto components caused by sources due to different physi-cal processes or mechanisms. This demonstrated that EOFanalysis is a powerful and advantage method to analyze andorganize the ionospheric data.

Figure 4 shows the distribution of the first four orders ofthe associated coefficientsPk obtained in the first layer EOFdecomposition ofhmF2 dataset. They correspond to the firstfour base functions shown in Fig. 3. It can be seen thatP1shows a variation pattern with a very strong dependence onthe solar cycle activity. It also shows some seasonal varia-tions. The coefficients (P2–P4) corresponding to the otherorders of base functions show most obviously the annual andsemi-annual variations, besides, they also show dependenceon the solar activity levels.

Figure 5 shows the results obtained in the second layerEOF analysis, that is, the EOF decomposition ofPk. Theleft panels are the distributions of the base functionsF

jk ob-

tained, the right panels are for the corresponding associatedcoefficientsAj

k . The different curves are for the first three

components (j=1, 2, 3). ForAjk , the results modelled with

Eqs. (15–18) are also plotted (thin green curves overlayingon each corresponding decomposed ones). As we can seefrom the right panels, the second layer EOF coefficientsA

jk

mainly contains the annual and semiannual variation com-ponents and their amplitudes also change with the solar cy-cle activity levels. This justifies the modelling of the coeffi-cientsA

jk with the harmonic functions representing the sea-

sonal (annual and semi-annual) variations as well as their so-

100 150 200 250 300 350 400 450 500100

200

300

400

500

Year= 1976

R= 0.9392

Mod

el h

mF

2(km

)

200 250 300 350 400 450 500 550 600200

300

400

500

600

Year= 1981R= 0.94538

Observed hmF2(km)

Mod

el h

mF

2(km

)

Fig. 6. Scatter plots of model value versus observational data ofhmF2.

lar cycle activity dependence represented by the F10.7 indexas expressed by Eqs. (15–18). As can be seen from the fig-ure, the modelled coefficientsAj

k replicated the original onesvery well.

As an example for showing the validity of the constructedglobal model ofhmF2, Fig. 6 shows the scatter plots of themodel value versus observational data ofhmF2. Please payattention that the observation presented here is the true obser-vational data from all individual stations available, the krig-ing and single stations’ screen points are not included here.The upper panel is for the case of the low solar activity year(1976) and the lower panel for the case of the high solar ac-tivity year (1981). It can be seen that the model values calcu-lated with our constructed model based on the EOF decom-position and the observational data show a high linearity andthe correlation coefficientsR between the model values andthe observational data are very high (R is 0.939 and 0.945 forthe years of 1976 and 1981, respectively). The high linear-ity and correlation coefficients between the modelledhmF2values and the observational data imply that the constructedmodel is able to reproduce the observational data quite well.

Figure 7a–b shows some sample plots demonstrating thecomparison between the observational data ofhmF2 and themodel values given by our EOF basedhmF2 model as wellas those given by the IRI model for the low solar activityyear 1965 (Fig. 7a) and the high solar activity year 1970(Fig. 7b) for 10 stations representative of high, middle andlow latitudes from both North and South Hemispheres. Theirgeographical coordinates are listed in Table 1. In Fig. 7a–b, from top to bottom, stations are ordered from north high,middle and low latitudes to the south low, middle and highlatitudes. In these plots, the thick green curves represent ourEOFhmF2 model results; the thick black curves represent the



250

LY164

IRI EOF_hmF2 EOF_M3000F2 OBS

250

YA462

250WK545

250

350YG431

250

MA720

hmF

2(km

)

250

350

450 SI301

250

350 HU91K

250MU43K

250HO54K

Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.

200

300 PSJ5J

300

LY164

IRI EOF_hmF2 EOF_M3000F2 OBS

250

350 YA462

250

350 WK545

250

350YG431

250

350 MA720

hmF

2(km

)

300

400

500SI301

300

400HU91K

250

350MU43K

250

350 HO54K

Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.

250

350 PSJ5J

Fig. 7. Sample plots of model values and observational data ofhmF2 for the years of(a) 1965 and(b) 1970. From top to bottom panels,stations are ordered from north high, middle and low latitudes to the south low, middle and high latitudes.

Table 1. Geographical coordinates of the stations labeled inFig. 7a–b.

No Code GLAT(◦ N) GLON(◦ E)

1 LY164 64.7 18.82 YA462 62.0 129.63 WK545 45.4 141.74 YG431 31.2 130.65 MA720 20.8 203.56 SI301 1.3 103.87 HU91K −12.0 284.78 MU43K −32.0 116.49 HO54K −42.9 147.310 PSJ5J −51.7 302.2

IRI model results, whereas the blue points are the observa-tional data. As an additional comparison, results provided bythe EOF-based M(3000)F2 model (Liu et al., 2008) are alsoshown (thin red curves). They are obtained by converting themodelled M(3000)F2 tohmF2 using Eqs. (1–6). As can beseen from these plots, the EOF model results (both ofhmF2

and M(3000)F2) in general reproduced quite well the vari-ation behaviour of the observational data. Compared withthe results given by the IRI model, the EOF model resultscompare obviously better than the IRI model results with theobservational data. As for the EOF model results, it can beseen from Fig. 7a–b that, in general, the results based on theM(3000)F2 model are very similar to those provided by thehmF2 model. However, a careful inspection of the details ofthe plots indicates that thehmF2 model results are somewhatimproved over the M(3000)F2 model results.

To estimate the accuracy of the model, we made a statis-tical analysis on differences between the model values andthe observational data by calculating the root-mean-squared-error (RMSE) of the model:

RMSE=

√√√√ 1

Np

Np∑i=1

(hmF2model −hmF2obs)2 (19)

WhereNp is the total number of the data points.The calculatedRMSEsfor our EOF-basedhmF2 model are

11.8 km for the low solar activity year (1965) and 13.4 km forthe high solar activity year (1970), respectively. Those forthe results based on the EOF M(3000)F2 model are 13.3 km



and 14.3 km for the years 1965 and 1970, respectively. Asa comparison, the RMSEs for the IRI model results are18.9 km and 22.6 km for the year 1965 and 1970, respec-tively. Therefore, in general, our constructed global modelof hmF2 based on the EOF decomposition has a higher ac-curacy than the model currently used in IRI. The accuracy ofthe EOF-basedhmF2 model is also slightly higher than thatof the global EOF-based M(3000)F2 model (in terms of theconvertedhmF2).

5 Summary and conclusion

In the present study, an attempt was made to construct theglobal model of the ionospheric F2 peak height parameterhmF2 based on the EOF decomposition of the dataset and themodelling of the associated EOF coefficients with harmonicfunctions representing annual and semi-annual seasonal vari-ations. Solar cycle dependence is also taken into account inthe model by including the changes of the harmonic ampli-tudes with the solar irradiance flux index F10.7. Comparisonsbetween the model predictions and the observational data arein agreement. Statistical analysis on the differences betweenmodel values and observational data showed the constructedmodel of hmF2 based on the EOF expansion agrees betterwith the observational data than the model currently usedin IRI. The accuracy of thehmF2 model developed in thisstudy is slightly higher than that of the global EOF-basedM(3000)F2 model (in terms of convertedhmF2) previouslydeveloped by our team (Liu et al., 2008). From the point ofview of practical application, thehmF2 model is more prefer-able than the M(3000)F2 global model, since in many appli-cations it is the parameterhmF2, rather than M(3000)F2, thatis really required. It is also due to the fact that the conver-sion from M(3000)F2 tohmF2 involves thefoF2/foE ratio.It would be much more convenient to obtainhmF2 directlyfrom a reliablehmF2 global model than converting tohmF2from an M(3000)F2 model.

Acknowledgements.This research was supported by the NationalNatural Science Foundation of China (40890164, 40774092), theNational Important Basic Research Project (2006CB806306) andthe China Meteorological Administration Grant (GYHY20070613).The M(3000)F2,foF2, foE and F10.7 index data used for the mod-elling study were downloaded from the SPIDR web sitehttp://spidr.ngdc.noaa.gov/. The author Man-Lian Zhang gratefully acknowl-edges the support of K. C. Wong Education Foundation, HongKong. We would also like to thank the two referees for their sug-gestions to improve the manuscript of this paper.

Topical Editor M. Pinnock thanks Shunrong Zhang and anotheranonymous referee for their help in evaluating this paper.

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