Modes of high-latitude electric field variability derived from DE-2 measurements:Empirical Orthogonal Function(EOF)analysis Tomoko Matsuo1and Arthur D.Richmond
High Altitude Obrvatory,National Center for Atmospheric Rearch,Boulder,Colorado,USA
Douglas W.Nychka
Geophysical Statistics Project,National Center for Atmospheric Rearch,Boulder,Colorado,USA
Received 14 September 2001; revid 4 January 2002; accepted 4 January 2002; published 9 April 2002.
[1]In this study we characterize dominant modes of high-latitude electric field variability as a t of two-dimensional empirical orthogonal functions(EOFs),bad on a quential non-linear regression analysis of the electric field derived from plasma drift measurements during the Dynamics Explorer-2(DE-2)satellite mission(1981–1983).Together with the mean fields,11EOFs are capable of reprenting68%of the squared electric field in the original data,leaving only a fairly random component as a residual. While such mathematically independent EOFs do not necessarily reprent physically independent modes of variability,each of the first two EOFs is actually related to
a widely known physically prominent effect on the convection patterns.Variability associated with the interplanetary magnetic field(IMF)B Y component emerges as the primary mode,and variability associated with the IMF B Z effect emerges as the condary mode.The tertiary mode reflects variability in the cusp region,but is not significantly correlated with the IMF.I NDEX T ERMS:2411Ionosphere: Electric fields(2712);2431Ionosphere:Ionosphere/magnetosphere interactions(2736);2784Magnetospheric Physics:Solar wind/ magnetosphere interactions;9820General or Miscellaneous: Techniques applicable in three or more fields
1.Introduction
[2]The term electric field variability in the high-latitude ,Codrescu et al.,1995,2000]has been ud in a very broad n meaning the residual which encompass all perturba-tions from the mean electric field:
E0¼EÀE;ð1Þwhere E0is the perturbation of the electric field E from the mean electric field E.The mean electric field E could be either the climatological electric field,which is mathematically or statisti-cally ,Foster et al.,1986;Heelis,1984;Heppner and Maynard,1987;Papitashvili et al.,1994;Ruohoniemi and Greenwald,1996;Weimer,2001],or the sample mean of a given data t.I
n this study we attempt to decompo this electric field variability at a given time t into veral dominant modes that can be reprented by a t of empirical orthogonal functions(EOFs)as E0r;t
希望英语官网ðÞ¼a1ðÞtðÞÁEOF1ðÞrðÞþÁÁÁ
þa vðÞtðÞÁEOF vðÞrðÞþÁÁÁþe0r;t
ðÞ;ð2Þwhere r reprents spatial position,a(n)(t)is the time-dependent coefficient of the n th EOF,or EOF(n)(r),and e0(r,t)now denotes the residual after subtracting the mean and the sum of weighted EOFs from the total electric field E.
[3]Virtually,an EOF is a principal component,although we don’t take the approach of the conventional principal component analysis(PCA),which is the eigenanalysis of a covariance matrix estimated from non-spar obrvations on a regular grid or at a regular ,Ramsay and Silverman,1997].Applica-tions of such a classical PCA,or so called the method of natural orthogonal components(MNOC),in the field of ionospheric physics can be ,in Faynberg[1975];Golovkov et al. [1978];and Sun et al.[1998].Instead of attempting to estimate the sample covariance matrix from our spar obrvational data,we estimate the principal components(two-dimensional EOFs) directly from one-dimensional measurements of plasma drifts al
ong satellite tracks.This approach is related to the work of James et al.[2000].
2.Data Set
艾薇儿好听歌曲[4]The Dynamics Explorer-2(DE-2)data t ud in the EOF analysis includes the complete bulk ion drift velocity V,which is measured by both the Retarding Potential Analyzer(RPA)[Hanson et al.,1981]and Ion Drift Meter(IDM)[Heelis et al.,1981]on board the DE-2satellite during the whole DE-2mission(1981–1983).The ion velocity has been combined with the model magnetic field B calculated using the International Geomagnetic Reference Model(IGRF),in order to obtain the electric field according to E=ÀVÂB.The actual data fed into the non-linear regression analysis are40-s averages(about2.5°),as the focus of the EOF analysis is the electric field variability on large scales and as the resolution of the basis functions ud in the analysis is also of the order of2.5°.Finally,all the quantities are calculated in Modified magnetic Apex coordinates[Richmond,1995],and all the data ud in this study are from the northern hemisphere. [5]In addition,1-hour average interplanetary magnetic field (IMF)data in geocentric solar magnetospheric(GSM)coordinates are ud to compare with the time-dependent coefficients of the EOFs.
3.Analysis Method
[6]Each EOF(principal component)is determined in a quen-tial manner,by performing a non-linear regression analysis on the data t.For the n th EOF the two free vector parameters,a j(n)and b k(n)(e below),are alternatively determined in an iterative process in which the following cost function L(n)is minimized:
L vðÞ¼
X J
j
X I
i
Y vðÞijÀa vðÞj
X50
k¼1
jeffery
b vðÞ
k
北京艺考培训中心
X kij
"#2
吸血鬼日记第五季插曲;ð3Þ
GEOPHYSICAL RESEARCH LETTERS, VOL. 29, NO. 7, 1107, 10.1029/2001GL014077, 2002 1Also at Institute for Terrestrial and Planetary Atmospheres,State
University of New York,Stony Brook,New York,USA.
Copyright2002by the American Geophysical Union.
0094-8276/02/2001GL014077
11- 1
with constraints
X b v ðÞk
德国留学生的优势是什么2¼1;
X
b v ðÞk b v Àn ðÞk
¼0n ¼1;...;v À1;
where the vector Y ij (n )
contains the I residual obrvations (e below)at a location i on a satellite track j ,a j (n )is a weighting factor
for a track j ,b k (n )
is a regression coefficient,and the matrix element X kij is the kth basis function evaluated at a location i on a satellite track j .For the X basis functions linear combinations of the functions described in [Richmond and Kamide ,1988]are ud.A total of 50basis functions are chon using th
e 50dominant principal components of the background error covariance matrix C u which was also developed in that study.
[7]Prior to the estimation of EOFs,the mean fields (here denoted by superscript n =0)need to be determined by a linear regression that minimizes L (0)in equation (3),in which a j (0)is t to 1,the constraints on b k (0)are lifted,and Y ij (0)
are the 40-s average electric field obrvations.Once the mean fields are determined,the first EOF analysis is conducted on the residual obrvations,which are obtained by subtracting the mean fields,or Æb k (0)X kij ,from Y ij (0)
.When an EOF is found,the subquent EOF is determined in a space orthogonal to all the earlier obtained EOFs.This orthogonal space is obtained by Gram-Schmidt orthogonali-
关注英文zation.A non-linear regression (e equation (3))is conducted on the residual fields which are obtained by subtracting the previously determined EOFs multiplied by their respective a j (n )s.The n th EOF at a given location r l on the sphere,or EOF (n )(r l ),is expresd as
英语四级算分器EOF v ðÞr l ðÞ¼
X 50k ¼1
b v ðÞk X v ðÞ
kl ;
ð4Þ
In total 34,929data points from 650satellite tracks are ud in the
analysis,which are the maximum numbers of i and j in equation (3)respectively.
4.Empirical Orthogonal Functions
[8]Eleven EOFs are obtained by performing the non-linear regression analysis on the DE-2data.As a measure of the amount of variability captured by each EOF,the reduction in the value of the cost function (3)made by subtracting each EOF component from the residual obrvations is shown in Figure 1.It is estimated as:
½L
n ðÞ
ÀL
n À1ðÞ
=
X J j
X I i
E 2ij
for n th EOF ;
½L ð0ÞÀ
X J j
X I i
ðY ð0Þ
ij Þ2 =
X J j
X I i
E 2ij
娘娘腔英文
for the mean ;
where E ij is the vector containing both components of the original 2-s or 4-s electric field obrvations at location i on track j ,and the total number of original obrvations is I .Note that the value of
ÆÆE ij 2
is 20%higher than the sum of the squared 40-s average
electric field,or ÆÆ(Y ij (0))2
.The information contained in each EOF decreas rapidly as the order of the EOF increas,and 68%of the squared electric field in the original data t is captured by eleven EOFs plus the mean.
[9]The first three EOFs and the mean are shown in Plate 1in the form of the electric potential.Together with the mean,they reprent 54%of the total squared electric field.The primary mode of the variability,reprented by the first EOF,is dominated by a convection vortex at the center of the polar cap,and can be interpreted partly as the effect of the IMF B Y component on the high-latitude convection pattern.This interpretation is supported by the fact that the correlation between a j (1)and the IMF B Y is higher (correlation coefficient À0.35)than the correlation of a j (1)with the IMF B Z (less than 0.05).The negative correlation indicates that a j (1)tends to be negative for a positive IMF B Y value,
making
Figure 1.The contribution to the total squared electric field,captured by the mean and the first eleven EOFs,is shown in percent.A total of 68%is explained by the mean and eleven
EOFs.Plate 1.The reprented as
electric field potential contours in a magnetic-latitude/magnetic-local-time frame with a perimeter latitude of 50°.Red contours are positive,and blue are negative.In Panel A),the mean electric potential emerges as a typical two-cell pattern.The contour interval is 5kV ,and the maximum and minimum values are indicated at the bottom left of the Panel.The EOFs in Panel B)–D)have normalized units between À1and 1,and are plotted with contour intervals of 0.2.The first EOF is dominated by a polar convection vortex,the cond EOF is dominated by a two-cell pattern,and the third EOF shows a small elongated cell-like feature around magnetic noon and 75°latitude.
11 - 2 MATSUO ET AL.: EOF ANALYSIS OF DE-2 ELECTRIC FIELD
the dusk cell larger if a j(1)ÁEOF(1)is added to the mean,and a(1) tends to be positive for a negative IMF B y value,making the dawn cell larger if a j(1)ÁEOF(1)is added to the mean.This is the familiar IMF B Y effect on the convection pattern which is demonstrated in [e.g.,Heppner and Maynard,1987].What is manifested as the condary mode of variability appears to be the variation of the convection pattern associated with the IMF B Z(correlation coef-ficient0.54).The positive correlation means that the cond EOF weakens the two-cell pattern of the mean field for a positive IMF B Z,and intensifies it for a negative IMF B Z.The intensification is accompanied by an expansion of the two-cell pattern to the lower latitudes.The features are also consistent with well-known IMF B Z effects on the high-latitude convection pattern.The third EOF, characterized by the elongated feature located at the throat(where the sunward convection from dawn and dusk sides converges in the day side),ems to be related to the variability in the cusp region. Even though the third EOF rembles the difference between potential patterns for B Y positive and negative like tho of Heppner and Maynard[1987]in the throat region,no significant correlation between a j(3)and the IMF B y is detected.
5.Discussion and Conclusion
[10]In conclusion,the EOF analysis is able to successfully decompo most of the high-latitude electric field variability into 11principal components.The dominant EOF is significantly influenced by the IMF B Y component,while the condary EOF is significantly influenced by the IMF B Z component.The tertiary EOF is associated with variability in the cusp region.The varia-bility caud by the first three EOFs can be quantified as15%, 12%,5%of the total variability of the squared electric field in the original data t.Much of the rest of the power remains in the higher order of EOFs,which appear disorganized or fairly random as suggested by the low amount of variability captured by the EOFs(e Figure1).
[11]A question that remains to be answered is the nsitivity of the EOF analysis to a particular data t and a choice of the mean electric field model.For instance,if the mean electric fields E in equation(1)are determined by the empirical model of Weimer [2001],34%of the total squared electric field can be explained by the mean electric fields,in comparison with22%by the sample mean electric field(e Figure1).Thus the model of Weimer [2001]is capable of capturing some of the variability in the EOFs. Regardless of the choice of mean electric field model,it is evident that a significant amount of power is left as the electric field variability.It reinforces the importance of the electric field vari-ability in the estimation of the Joule heating in the thermosphere [Codrescu et al.,1995,2000].
[12]Future studies should be carried out using a larger data t possibly from the Super Dual Auroral Radar Network(Super-DARN)or DMSP satellites.Such comprehensive data ts will provide not only better grounds to specify generic EOFs but also a possibility to study asonal(dipole tilt)and solar cycle variations of EOFs and even EOFs associated with substorms.As demon-strated by Sun et al.[1998],a principal component analysis using ionospheric equivalent current data can successfully decompo the two dominant modes of variability associated with the DP2 and DP1current systems during substorms.In fact,the primary mode of variability associated with the DP2variations features two-cell vortices and exhibits high correlation with the IMF B Z,as reported by Nishida[1968],similarly to the electric field variability captured by the cond EOF of this study.An EOF that is similar to the variability in the DP1current system is not detected in this study,mainly becau the obrvations during substorms are out-weighed by the rest of the obrvations.Less than10%of the total DE-2satellite pass are identified as being measured during substorms[Weimer,1999].The relative roles played by electric field variability and conductivity variability in producing the variability associated with the DP1current system during sub-storms is an interesting question and can also be examined by further studies using more extensive data ts.
[13]Regarding the first EOF,it appears to have some relation to the DPY equivalent current system pr
opod by Friis-Christenn and Wilhjelm[1975],which is described as a single-cell current related to the IMF B Y.In addition,the DPY current system has intensified eastward and westward electrojets located around80°invariant latitude in the cusp region for the IMF B Y positive and negative cas,respectively[Friis-Christenn et al.,1985],which we would have expected to have some relation to our third EOF, despite the puzzling fact that the correlation between a j(3)and the IMF B Y is negligible.
[14]In Ridley et al.[2000],the relationship of the high-latitude electric potential to the IMF was characterized bad on a multi-variate linear regression analysis of the IMF B Y and B Z data and the electric potential obtained from the data assimilation procedure of Richmond and Kamide[1988].Variability in the electric potential associated with the IMF B Y component is shown in Plates1c–1d and variability associated with the IMF B Z component is shown in Plates1a–1b in Ridley et al.[2000],and they are related to the first EOF and the cond EOF of this study,respectively.
[15]Adapted to the dominant modes of the variability in the system,the EOFs are ideal basis functions for data assimilation procedures.Becau the EOF takes account of the spatial coher-ence of the variability on large scales,even a data void area can be well constrained by using EOFs as ba
sis functions.The u of EOFs can reduce the required number of basis functions and thus can be computationally more efficient in terms of reconstructing large scale features in data assimilation procedures.
[16]Acknowledgments.Tomoko Matsuo is supported by the NCAR Advanced Study Program.The authors are grateful to Kristine Henl for her efforts in pre-processing the DE-2data.The DE-2data were obtained from the NASA Data Archive and Distribution Service(NDADS)mass storage system at the National Space Science Data Center(NSSDC).The solar wind data were obtained from the NSSDC OMNIWeb at NASA.The NCAR is sponsored by the National Science Foundation.
产后康复师培训
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T.Matsuo,A.D.Richmond,and D.W.Nychka,NCAR,P.O.Box3000, Boulder,CO80307,USA.(tmatsuo@ucar.edu;richmond@ucar.edu; nychka@ucar.edu)
11- 4 MATSUO ET AL.: EOF ANALYSIS OF DE-2 ELECTRIC FIELD
