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P ?^NASA TECHNICAL MEMORANDUM NASA TM-77804

ALGORITHM FOR SPACE Tm ANALYSIS OF DATA ON GEOMAGNETIC FIELD

N. V. Kulanin

(NASA-TM-77404) ALGORITHM POE SPACE-TIME N85-176C9ANALYSIS OF DATA CN GEOMAG6E1IC FIELD(National Aeronautics and SpaceAdministration) 27 p HC A03/@I A01 CSCL 12A Unclas

G3S64 13719

Translation of: "Algorithm prostranstvenno-Vremenno AnalizaDannykh o Geomagnitnam Pole," IN: "Algorithmy i rezult'tatyoUrabotki dannykh v MTsD," (Algoritlnns and Results of DataInterpretation in WDQ , edited by V.P. Golovkov and Yu. S.Tyupkin, Soviet Geo-physical committee, USSR Academy ofSciences, Moscow, 1978, pp 1-5, 44-S4

.j;^ ^^^^a^asas

',Zza rOL,

NATIONAL AERONAUTICS AND SPACE. ADMINISTRATIONKASHINGTON, D.C. 20546 NOVEMBER 1984

tr

---- — -- _.._ J

I

STANDARD TITLE PAGE

1. Rrport Nn •^-^^-' »»_ 7. CovernmrN Aar++ton No. 3, R.cipi.M'c Cvlelop No.

NASA TM-77804 _.^ _^. Titl. and Subl lile 5. Rrpofl 0.1,

ALGORITW FOR SPACE-TIME ANALYSIS OF DATA ON6. P. ffurminp 0, 9 . . 1+0110 ^ Cod.GEOMAGNETIC FIELD

7, Aulhor(+) 8. P.rfo rminp Orpon(+vilon R.porl No,

N.V. Kulanin10, Work Unil No,

--- ---- H. Conbocl or Gfont No.NAS I-4005

.___Y. --9, Peffuninv 0,vmi+ation Nam, and Add,,,$Leo kanner Associates 13. Typr of Report and P.iiod Co.rr.dRedwood City, California 94063 Translation

12. Sponminp Ap.ney Nom. and Addre..

National Aeronautics and Space Administration 1e, 5panfor ( np A 9 rncy Cod.Washington, D.C. 20546

15. Supplem.nlary Nola

Translation of: "Algorithm prostranstvenno-Vremenno Analiza Dannykh oGeomagnitnam Pole, " IN: "Algorithy i rezult t taty obrabotki dannykh vMPsC,", (Algorithms and Results of i)ata'Interpretation in 4VDC), editedby V.P. Golovkov andYu. S. Tyupkin, Soviet Geo-physical committee, USSRAcademy of Sciences, Moscow, 1978, pp 1-5. 44-54

16. Abboot

The algorithm described, calculated for the execution of the space-timeanalysis of large volumes of diverse data quite arbitrarily distributedon the surface of the Earth and in time and realized in the form of a -program for the BESM-6'computer, is now at the stage of completion of theverification of its applicability for analyzing a large part off: the datacontained in the Veynberg catalog with the involvement of some independentresults pertaining both to epochs earlier than those contained in thiscatalog and later epochs, in the 20th century.

17, Y,cy Woad. (5eleetetl by A uthor(s)) 18. Di.ni bulion Statement

Unclassified Unlimited

19. Security Clv++i 1. (of this'r P.'11 7J. Sn emiq Cler.if. (of Ihl. pov.) 21• No. of Papo, 22. Prie.

Unclassifie^ Unclassified 18

n !'1 I

I,

C o."r

i »

ALGORITHMS AND RESULTS OF DATA INTERPRETATION IN WDC

V.P. Golovkov and Yu.S. Tyupkin

Soviet Geophysical Committee

FOREWORD

/5V,

The broad use of computers in geophysics opened up two possi-

bilities for researchers: construct complex theoretical models,

on the one hand, and interpret large volumes of data, on the other.

The first of these possibilities outpaced the other in realization

because data reduction and data input into computers has thus far

remained the bottleneck in the overall system, where the instru-

mentation has the design-wise obsolescent analog data output.

This collection of articles contains the results of studies in

both the methodological and interpretational areas in one way or

another associated with the interpretation of large volumes of

geophysical data of the most diverse specialization. Granted all

the narrowness of the problems solved in each study, as a whole

they illustrate to some extent the possibilities that appeared

before researchers when both large volumes of data in machine-

readable form and computers with sufficiently developed standard

or specialized mathematical software were made available to them.

Naturally, these possibilities have surfaced first.of all in

the World Data Center; firstly, it has computers, secondly, a data-

base, and thirdly, since it is the data processing laboratory of

the CAPG [expansion unknown], it engages in working out methods,

algorithms, and packages of specialized geophysical programs.

In preparing this collection, we hoped that it will better

display the possibilities of the World Data Center than the center

i'Numbers in the margin indicate pagination in the foreign text.

1

catalogs and adverti.sing brochures. Additionally, all the data

used in the studies, ,just like the packages of programs, are

accessible to broad classes of users at the World Data Center in

the USSR and abroad.

V.P. Golovkov

ALGORITHM FOR SPACE-TIME ANALYSIS OF DATA ON GEOMAGNETIC /44FIELD

N.V. Kulanin

Several algorithms for space-time analyses of the geomagnetic

field have been published in the past decade, including some using

a system of basis functions differing from a system of spherical

harmonics [1], as well as algorithms of spherical harmonic analyses

admitting a sequence of spherical harmonics distinct from the canon-

ical sequence [2,37•

Given the unquestioned advantages of each of the above-men-

tioned algorithms, for the most part they impose on the original

data requirements that become burdensome when a conversion is made

to temporal integrals encompassing the ante-Gaussian epochs, where,

as a rule, data of a different type (Z,I,T ) were quite arbitrarily(and often very unsuitably) distributed in time and space.

This study proposes an algorithm for space-time analysis im-

posing virtually no special requirements either on the composition

of the initial data, which may be quite diverse (Z,11 , H ,T,or on their distribution in time and space. The

primary constraints figuring in the specific realization of the

algorithm on a computer stem exclusively from the limited possi-

bilities of the computer involved.

The algorithm uses and to a certain extent develops the ideas

advanced in the studies [4,5]. The main "traditional" aspect of

2

the algorithm can be regarded as the use in the system of basis

functions of a canonical sequence of spherical harmonics (here it

is provided, as a variant, that some groups of coefficients and

their time derivatives be recorded with a certain orientation

toward the Jukutake method [61).

Description of method. Underlying the algorithm is the re-

presentation of the geomagnetic field potential on the spherical.

surface of the Earth with radius RI at the point (t9,h) at the

instant of time t (by limitation to internal sources) in theform of a sum of the series [7]

supplemented by the representation of the secular variation of

the coefficients in the neighborhood of the central epoch

where ^ry^t)-^9ti^tJ H ^^ (t) ^^ti 1^=t

is the S-th derivative in time at the point t = t0.

Based on Eqs. (1) and (2), the expression for the components

of the geomagnetic field intensity can be written in the form

where E— X— , .I/ — or Z is the intensity component,E/CYH % &A) t - -u^ is the corresponding basis space-time function,

the k-th in the canonical sequence,9GK (M,(V^/t/ /5 ^-to <^ m +^ CIL/(il+/ _/n^^ f (CU^ %^

- nz-tn ln_t (co, o)J" ^„z) rnP"'l ^.sc^)^i z6LLiu

/45

3

here the subscript If is associated withrbs m and ,S as follows:

for r,2 = 0, ,^ . 0 -• K = I

I I 2

S,n + II I 0 Yn, + 2

(fort )I I I 5",+ 3I ..... .I .. .....2 Sim•+ 2I I 0 2 S", + 3 t

( for g e ) )

2. . .3Sm..... .. + .y...... .

and Nm .n .w( nm+2 ) ( 4,+I ).

Thus, we turn to the linear analysis, in which it is re-

quired to determine the unknown parameters WK representingthe coefficients and their time derivatives

(both at the instant of time f ° Z ° ).

In order to apply Eq. (3), all the initial data must be re- /46

duced to the form of the components X , y or Z in accord-

ance with the type of data; this is done by iteration., in which

the (n + 1)-th step is determined by the formulas

y ^"*t ,Szuz_D H^"^YP7_ J/ X (n)

SOZ (4)

X^ t) T X (n)3T N

where c^^ . H I . 11 7 are the results of measuring,.D ,H J andT, respectively, X(`I) YM )ZP"LX^ y- and Z are the components

computed at the n-th iteration step, f//")= /mi}z^ ^1^")z3 '^ (x^" z+ J r(")z ; here X(^ „^^°^,' ^'^°/ are specified on thebasis of (3) in accordance with the given initial values of the

coefficients ^nw and their time derivatives at the instant

f=ea

4

l • a

We note that the use by of the fourth or fifth of the

transforms (4) is determined by the value of the polar angle B'.

In place of (3), we now can write

^KUKe,I (r4(5)z,... (5)N=1

where is the result of iteration (left-hand side of (4)) at

the/ P-th of the total number /3 of measurement points T e , where

(-C.Ac4-ep) We note that the functions (which we

assume, just like the variable Fe, , to have already been multipliedby the square roots of the measurement weightings formed on the

basis of Eqs. (4)) generally speaking are nonorthogonal in the

given domain (and all the more so on the set of points IT h .Eq. (5) is approximate even for a, fairly large 9M , since the

quantities Fr are subjected to the influence of measurement errors.

Based on these data we can only approximately determine the coeffi-

cients 44 .

One of the methods of determination is to solve the system of

conditional equations

under the condition

/47DLTM

K., a iezr

(least-squares method). The coefficients tL^^^ thus obtained con-

tain the errors e c(„ =cox -^^" (where CIr are the true values of thecoefficients that we introduced from considerations of convenience

in manipulations (see also below Eq. (15)), whose value generally

depends on the choice of the number of terms in the series in

Eq. (6) -- of the quantity R .

The question suggests itself: how can one choose the optimal

value of /Z and further, how can one refine the coefficients ^I_KF

5

within the limits of the selected spectrum such that IdCe"*1 =rnin 7

This problem is discussed below with reference to the concepts in

the study 151.

Let us turn from the system of functions &x(j) to the func-

tions ^,A) that are orthogonal on the given set of points

f 7ej , i.e., that satisfy the condition

(1 H 1-!n) —^qb 1 f t 7

)^^mI Kflrt

( I)1 e a f K f M f 0, KOM

Orthogonalization is carried out according to the Gram-Schmidt

formulas:

T KKP^p(fx(.IJ,Ir=^z,..., R

The coefficients of orthogonalization providing for Eq. (7) are

equal to

Single application of Eqs. (8) and (9) when solving large-

scale problems ( k and :V are large) may not provide for the

rigorous orthogonality of the function CP (1) owing to the accumu-

lation of truncation errors.

Here we propose an iterative process of orthogonalization

differing somewhat from the process given in 151, since the

latter, as can be shown, requires additional correction.

The next step in the iterative process is determined by the

formulas:

(9)

6

(^

(ZJ /48

Z)^ 'd ^z)(Z^^ (z^r) to

^^.I) PK ^/) i 1 ^fJ

( )

C4X

S (` 'I t' CPP_719We can show that the iterative process can lead to the at-

tainment of rigorous orthogonality, namely as o[;p ^

C,r (2) and 0-1Zp= ^KP--drptxt

But the question of the size of the domain of its convergence can,

most likely, be solved only empirically on the basis of trial

calculations for each specific case. We note only that, in addi-

tion to the excessive value of P and Al , the size of the domainof convergence is unfavorably affected by the poor causality of

the system of normal equations, which can be obtained on the basis

of system (6).

We can now assume that we have a system of functions cI.) K ('r}')

that are orthogonal on the given set of points Here

Eqs. (6) take on the form

K-1

d

By expressing in (11) the functions c'K in terms of GK in

accordance with (8) and using the linear independence of G K , we lt,

getR

Qi ,44^^

K (_

7

where

V/1 /i ^2`d" P" /f=1,,2,...)R;p-lz...,K1 (13)MOP

We note that the values of / it do not depend on the choice

of k. . By multipying (11) scalarly by `?o(j) , we find ([' is

the vector ( E,^ ) :

The values of CG are also not dependent on the choice of /49although system (11) is inconsistent and the equality between

E^ and the sums ^ c^aG,r(c^°E R ^x(fc) is approximate.)

In accordance with (3), let us determine the exact (true)

values of the coefficients (4 as satisfying the system of equa-tions when there is a hypothetical expansion of the initial system

of points fl ! v to `(tI .- rM, ...r if/' , as Al I`r .

K dr., GK I rc)=E Ae e=^z^..yn; ^3c1,..pA/' (15)

where A a is the error of the quantity Ee' due to an error of

measurement at the point ` r- .

Based on data of N measurements, the following system of equa-

tions can be set up

xZ GC 0, (je>= LZn 6e>=Fee e /,z, ... ,AI (16)

';, `The more or less weak dei endence of ^E7uc and CLn on R can occuri;+i ='

as a consequence of the nonlinearity of the expression for the

right-hand sides of the conditional equations (6) in terms of the

kt functions Gx (see (4)).i

8

By subtracting (16) from (15), we get

N W ^fe) ^E Q K K eKIM/

K Coe e Q e,

where A Celf I` txl /( ^^/ e -1,$, ..., h/

On the other hand,p from (16) it follows that

x^ 4 c^C,^,.GK (C^- E A C" ^ ^Ye/ (17)H.f

therefore

^AaKCibK (tc)°K^^,IC'MOi r 1Vt6e=rye(19)

and

A a,,)

, ri= 2,..., Al (19)

From (16) and (17) and referring to (8) and (13) we find

N'VC4,

where the quantities Aa.r are determined by Eq. (19)•

In most applications and, in any case, when the space-time /50

algorithm is applied to data from the ante-Gaussian epochs

n. (^ ) (21)d ;P,7N*t^K GK

for the vast majority of measurement stations.

Referring to (21) in Eqs. (18) and (19) and admitting only

a weak correlation between c(, G, (^e) and (^^) , we find,. N./

1 b (22)Oa r //CPO,R 1 1

i 9

Ifl

Ea

in any case. for K'«n/ 7

where a and aa;, are the root-mean-square errors of the quantities

E^ and a.,, . We note that 6 does not depend on e , since theweightings of the measurements were allowed for, as assumed, in the

quantities [symbols not given in text;.

Let us elucidate how the errors of the coefficients at the

functions G„ ('f) change if we convert from system (16) to the

system of conditional equations

Z GQK GK(V)^ KL a,,cP,e)c^=e^ e44 ...

(23)

According to (12) and ;20)

41,4 % ,,p.aHence, from (20) it then follows that

PI Al

Lti CL K LG K— K- r. K n F &Rol i=K u( 4 tor

K

AlI

(24)

where a z =0 ^-a SIR From (7), (19), and (21) it follows that

CA Aa'j=0(eitK);M[afAaK]-O (25)

where MIX-1 is the mathema ,,ical expectation of oc. , and (24) in

place of (25) leads to

M (_"(,6 j. )11 jZ: lp* ^ ^ 6E C a`^ z (26)

From this It follows that in order for the quantity /^ C^a K )1

to become smaller in the transition from k to 9 +X , it is nec-

essary and sufficient that the inequality

10

/^ z z N

„^'RI/,n`ni^t 4 3 'R/^,K a'R,t(^Mi l K a, ,+ t^4L,ai)(27)

be satisfied.

In the first approximation we assume that the quantity

AG a when there is a change in R. undergoes random

fluctuations about the mean value equal to zero. Then the re-

quirement of a decrease, on the average, in the quantity

MLA air ) .°1 when there is an increase in R- leads to the

condition

G(. R e L (28)

We will assume the quantity a R tj to be "random" with respect

to the fixed quantity CL t # A , i.e., we assume that

1 F -d.Thus, Eqs. (19) and (21) allow us to consider the "distribu-

tion" of the quantity CL K,, to be norm (here we assume that /3 is

large enough); from (28) it follows that

a ' < Is' = sM— CL,g .jf a- R ,. I

where is the minimum admissible probability of inequality

(28) being satisfied, and X is determined from the equation

^b'A (13"-S)-L xls =P

where 'P *(Z ) is an integral error function.

/51

11

Thus, for example, S (0,93) - 0,4, S (0,85) - 0,5.

In contrast to the study [5], by solving the problem of mini-

mization /IC(a al,')'] , we provide for the second approximation.

Here we assumeN

where R. is the largest value of K for which inequality (28)

is still valid. We also introduce the notation:

rryy (R

i ^yY&)l i

dZ //qR hY o I ^a!A F/

Then inequality (27) can be written in the form

Z `+S S Z -6 14 70 (29)

By solving the corresponding quadratic equation, we determine

Z4c - -,J ± ,5 +ba (30)

Analogously to the solution of inequality (28), we assume

/52

the quantities Z' and S" to be fixed, and [symbol not given]

and [symbol not given] to be random, such that

Mrs ^-^; r^cs^=,s;.^c^^=3z ,.^^,s1.-^^ c 31)

Thus, the solution of inequality (29) can be reduced to esti-

mating the probability of the random variable 7, entering the do-

main outside the random interval ( kfo Zz ). We note that Z and

Y --as linear functions of the coefficients CZ - are normally

distributed and at the same time are uncorrelated, since they

12

contain the coefficients aj with different subscripts, leading to

the conclusion that Z and $ are independent. By applying the

theorems of addition and multipllcaticn of probabilities [8] andwith .reference to (30) and (31), let us find the desired probabi-

lity in the form

t

"°z s a (32)

fifCI- ^5 IRVaAs a result:,, by referring to (12) and (25), we get the values

of the coefficients CK,

refined in the sense of an approximation

to dK

K y fawN• C=K Ci^be ai

(33)

[subscript in left-hand side reads "refined"]

and also

^,p vro 1N. ' L--X. UC of Q.( •

where

I when n /( yy or when M->.;c^u

0 in the remaining cases

Here K4 is the boundary value of the subscript K determining

the transition to the optimization regime defined by Eqs. (32), (33),

is the minimum probability that inequality (29) will be satis-

fied. Since the refinement of ceK was carried out successively

RM-if times (but not more than k.-A,;6 times), i.e., repeatedly, F'

it is best to set 0,5, '1

We note that by forming t^^,^ with the aid of the algorithm,

1 by using Eqs. (12) and (13) we actually replace ZR Mrra4m with the

dtK in Eq. (12), which means that in the transition across the

13

value of (Eq. (33) from C,.r-0 a replacement in Eq. (12) /53of the form (4. ) 1 --(CeK* )1 at'. , , which ultimately requiresthe substitution in Eq. (32) of the form ,^^► 5^-/31.ca^

The possibilities of the procedure of refining the coeffi-

cients W,, 4, by Eqs. (32) and (33) are now being inves-

tigated with test models.

Thus, on a model utilizing the quasiuniform filling of the

surface of a sphere with "measurement stations" with the super-

imposition on the data synthesized at these stations, pertaining

to the same epoch (1965), of the error typical of the measurements

in the 16th through 18th centuries, we obtained X(b = 25 when

RM = 35•

Let us briefly describe the order of execution of the algo-

rithm: orthogonalization (including repeated); iteration in accor-

dance with Eqs. (4); computation of the coefficients.C4,, and of

their errors; determination of /L „e ; when 7k (where R is the

last overestimation of the possible length of the series (3));

expansion of the list of basis functions and repetition of the

preceding steps; selection of the initial data in accordance with

the ratio of nonclosure to the root-mean-square error; and repeti-

tion of therecedin steps (singly); and fin ding r '" tP g P g Y); g L d o r^o4N, 1

The algorithm described, calculated for the execution of the

space-time analysis of large volumes of diverse data quite arbi-

trarily distributed on the surface of the Earth and in time and

realized in the form of a program for the BESM-6 computer, is now

at the stage of completion of the verification of its applicabi-

lity for analyzing a large part of the data contained inthe Veyn-

berg catalog [9] with the involvement of some independent resultspertaining both to epochs earlie- than those contained in this

catalog and later epochs, in the , century.

14

L1

In addition, the algorithm has been applied to an analysis of

paleomagnetic data of the Bryunes epoch [10].

In conclusion, the author expresses his gratitude to N.P.

Ben t kova, V.P. Golovkov, and T.N. Cherevko for valuable comments

and discussion.

The author 1s also indebtedness to S.I. Braginskiy, under

whose supervision he began developing the algorithm described

above.

15

I1

REFERENCES

1. Pushkov, A.N., E.B. Faynberg, M.V. Fiskin, and T.A. Chernova,"Space-time analysis of main geomagnetic field by method ofdecomposition into natural components," In the collection:Pro stranstvenno-vremennava struktura renmarrnitnnon nnlvn

1976. tructure scow,

2. Bazarzhanov, A.D. and G.I. Kolomiytseva, "Improvement in theanalytic representation of the secular variation," Geomag-netizm i aeronomiya 7/5, 868 (1967).

3. Kolomiytseva, G.T. and A.N. Pushkov, "Analytic model of maingeomagnetic field for an interval of 2000 years," in thecollection: Prostranstvenno-vremenna a struktura eoma nit-nogo polya, Moscow, Nauka, 197 .

Cain, J.C., W.E. Daniels, S.J. Hendricks, and D.C. Jensen,"An evaluation of the main geomagnetic field, 1940-1962,"J. Geophys. Res. 70/15, (1965).

5. Mishin, V.M. and A.D.Bazarzhanov, "Selection of spectrum ofLegendre polynomials approximating the observed [text miss-ing] -field," in the collection: Geoma nitny e issledovani a[Geomagnetic Studies], Moscow, Nauka, No. , 1966.

6. Jukutake, T., "Spherical harmonic analysis of the Earth's mag-netic field for the 17-t'i and 18-th centuries," J. of Geom.and Geoelectr. 23/1, 11 (1971).

7. Yanovskiy, B.M., Zemnoy magnetizm [Terrestrial Magnetism],Leningrad, Izdatel'stvo Leningrad State University,1, 86 (1964).

8. Ventsel', Ye.S., Te^ori a vy2royatnostey [Probability Theory],Moscow, Nauka, P. 37, 1969.

Veynberg, B.P. and V.P. Shibayev, "Catalog of results of mag-netic determinations on the globe from 1520 to 1940," Ser-iva rukopisnykh trudov In-ta zemnogo magnetizma [Seriesof Manuscript Works of the Instituto of Terrestrial Magne-tism], Leningrad, 1941, Issue 5, Vols. 1-3 (with rights ofmanuscript).

10. Ben'kova, N.P., V.P. Golovkov, N.V. Kulanin, and T.N. Cherevko,"Analytic models of the geomagnetic field in the Bryunesepoch," in present collection.

16

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