Final Project Report

NASA Contract NAS5-28617

"Constraints on Geomagnetic Secular Variation

Modeling from Electromagnetism and Fluid Dynamics

of the Earth's Core"

Covering work performed between

18 October 1984 and 28 February 1986

Principal Investigator: Prof. Edward R. Benton

Campus Box 391

Dept. of Astrophysical, Planetary

Atmospheric Sciences

University of Colorado

Boulder, CO 80309

Telephone. (303) 492-7988

28 February 1986

fNASA-CB-177857) CONSIBAINIS ON GEOMAGNETIC N86-24063

SECULAB VARIATION MODELING F£CM

ELECTBOMAGNETISM AND FLUID DYNAMICS OF THE

EARTH'S COBE Final Report, 16 Oct. 198U - Onclas

28 Feb. ,1986 (Colorado Oniv.) 58 p G3/46 09117

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Introduction

The work performed under this three year project is the result of a

collaborative, but separately funded, effort between Edward R. Benton of the

University of Colorado, Ronald H. Estes of Science Applications Research, Inc.

(formerly of Businesss and Technological Systems, Inc.) and Robert A. Langel of

the NASA Goddard Space Flight Center. Theoretical work of Professor Benton

has been funded under NASA Contracts NAS5-27671 (covering the period from

18 May 1983 through 17 October 1984) and NAS5-28617 (for the period from

9 November 1984 through 28 February 1986) to the University of Colorado.

Development of geomagnetic field models and appropriate algorithms by Mr.

Estes is funded under NASA Contract NAS5-28671 to Business and

Technological Systems, Inc. Dr. Langel serves as co-investigator to this joint

project.

During the contract period now ending, work at the University of Colorado

has concentrated first on preparing a paper that describes and derives a set of

physical constraints for geomagnetic field modeling that form the theoretical

basis for this project. That work, now published, is attached as a part of this

report. The work initiated this past year is the detailed implementation of

that theory in a new set of geomagnetic field models so constructed that they

satisfy (approximately) a subset of the constraints derived in the above paper.

This latter work was presented (orally) at the Vth Scientific Assembly of IAGA
(International Association of Geomagetism and Aeronomy), Prague,

Czechoslovakia, August 1985. It has been written up for publication in Physics

of the Earth and Planetary Interiors under the co-authorship of Benton, Estes

and Langel. It gives a full, accurate, yet concise descripton of our joint

activities and progress during the past year, so it is included as the main part

of this report. The major conclusions reached are that the incorporation into

geomagnetic field models of non-linear constraints that arise from

consideration of the physics and dynamics of the earth's core, is numerically

feasible and leads to better short range predictability than do unconstrained

models

In the next time period we intend to extend the preliminary models

constructed so far by lengthening the data interval and incorporating a greater

number of constraints. We will also explore alternative representation to low

order polynomials for the time dependence of secular variation.

Other pieces of work, that were partially supported by this project include:

Benton, E.R., and Alldredge, L.R., "On the Interpretation of the Geomagnetic

Energy Spectrum," 44 page typescript submitted to Physics of the Earth and

Planetary Interiors. October 1985.

Voorhies, C.V., "Steady Flows at the Top of Earth's Core Derived from

Geomagnetic Field Models," 95 page typescript submitted to Journal of

Geophysical Research. January 1986.
Voorhies, C.V. and Backus, G.E., "Steady Flows at the Top of the Core from

Geomagnetic Field Models: The Steady Motions Theorem," Geophysical and

Astrophvsical Fluid Dynamics. 52. 163-173, 1985.
Geomagnetic Field Modeling Incorporating

Constraints from Frozen-Flux Electromagnetism

Edward R. Benton

Dept. of Astrophysical, Planetary & Atmospheric Sciences

Campus Box 391

University of Colorado

Boulder, CO 80309

Ronald H. Estes

Science Applications Research, Inc.

4400 Forbes Blvd

Lanham, MD 20706

Robert A. Langel

Code 622, Geophysics Branch

NASA/Goddard Space Flight Center

Greenbelt, MD 20771
Abstract

A spherical harmonic representation of the geomagnetic field and its

secular variation for epoch 1980, designated GSFCX9/84), is derived and

evaluated. At three epochs (1977.5, 1980.0, 1982.5) this model incorporates

conservation of magnetic flux through five selected patches of area on the

core-mantle boundary bounded by the zero contours of vertical magnetic field.

These fifteen non-linear constraints are included like data in an iterative least

squares parameter estimation procedure that starts with the recently derived

unconstrained field model designated GSFC( 12/83), Langel and Estes (1985).

Convergence is approached within three iterations.

The constrained model is evaluated by comparing its predictive capability

outside the time span of its data, in terms of residuals at magnetic

observatories, with that for the unconstrained model. The new model

demonstrates significantly improved predictability.

Next, it is established that the flux of magnetic secular variation out of

the northern (or southern) geographic hemisphere of the core-mantle boundary

is nearly conserved by a remake of the field model designated GSFC(9/80). The

GSFC(9/84) model is then examined and found to satisfy this independent linear

constraint on secular variation very well.
1. Introduction

Advances in geomagnetic field modeling typically involve a sequence of

steps. First, a new theoretical concept is put forward. Then It is tested

against data. Finally, if the tests are passed, the concept is adopted and

incorporated into standard modeling procedure. An early example of this chain

of events began with Gilbert's announced belief in 1600 that the Earth itself is

the source of the observed magnetic field. This idea was confirmed in 1839 by

Gauss, whose spherical harmonic analysis of existing data clearly showed that

the part of the geomagnetic field originating from sources outside the Earth

was insignificant compared to the main field of internal origin. As a result, it

is now standard modeling practice to ignore external fields from the outset

(actually, recent satellite data are sufficiently extensive that some modelers

have again begun to restore the very low harmonics of the external field into

their data fitting algorithms).

A modern version of this same process can be thought of as originating

when Roberts and Scott (1965) advocated that a hydromagnetic hypothesis be

invoked to assist in modeling short-term geomagnetic secular variation. Their

idea was that, during short enough time intervals, there would be Insufficient

time for magnetic diffusion to separate fluid parcels in the earth's core from

magnetic field lines. One important consequence of this "frozen-flux

hypothesis" is that magnetic flux tubes bounded by curves just below the
core-mantle boundary (CMB) which always consist of the same fluid parcels

then move laterally with the fluid and conserve their strength. Backus (1968)

discovered that the contours on which the vertical magnetic field vanishes on

the GIB, called "null-flux curves/' constitute such moving fluid boundaries of

material magnetic flux tubes. Then Hide (1978) pointed out the following way

to test the frozen-flux concept against data from seismology. Roberts and

Scott (1965) had noticed that even though the absolute magnetic flux crossing

any geocentric spherical surface of radius r > b, where b is the radius of the

CMB, can change in time; yet, the absolute flux crossing the top of a perfectly

conducting core is invariant in time. Hide (1978) proposed to evaluate the

absolute flux integral (or pole strength, after Bondi and Gold, 1950)

P(r,t) = I*/* | Br(r,e,4>,t) | sin 6ded<|>, (1)

o o

for various radii and times. Here Br is the vertical magnetic field at time t in

spherical coordinates r, 9, <j>, with e, co-latitude, <j>, east longitude. With r

decreasing from the Earth's surface, the value r = b would be that radius at

which P first becomes stationary in time.

Hide and Malin ( 1 98 1 ) used the above magnetic method for determining the

radius of the Earth's core and achieved moderate success. Voorhies and Benton

(1982), adopting a different technique and more recent data, achieved

agreement with the seismic value for b (3485 km) to within 1.8%. The average
of many further determinations by Voorhies (1984) agreees to within 06%, so

there is good evidence for believing the global aspect of the frozen-flux

hypothesis on short time scales.

A significant body of literature that either tests or uses the frozen-flux

hypothesis now exists (a representative sample includes Backus and Le Mouel

1986; Benton I979a,b, 198la,b; Benton and Muth 1979; Bloxham and Gubbins

1985, 1986; Gubbins 1982, 1983, 1984; Gubbins and Bloxham, 1985; Gubbins

and Roberts 1983, Le Mouel et al., 1984; Madden and LeMouel 1982; Shure et al.,

1983; Voorhies and Backus, 1985; Voorhies and Benton 1982; Whaler 1980,

1982, 1984). The time therefore appears to be appropriate to begin

incorporating into short-term field modeling at least global and probably

regional constraints from the frozen-flux theory of the core. The qualifier

"short-term" here is because the assumption of no flux diffusion must fail on

sufficiently long time scales (as well as on too short length scales) Backus

(1968), Booker (1969). Voorhies (1984), and Bloxham and Gubbins (1985) find

evidence for possible flux diffusion in a small null-flux patch beneath the

South Atlantic (unfortunately, an area not sampled by the network of magnetic

observatories). As a result, we believe it vital to test thoroughly any field

model constrained by frozen-flux. Bloxham and Gubbins (198b) recently

introduced into a field model conservation of magnetic flux in each patch of

area on the CMB bounded by a null-flux curve. They applied their method to
observatory data at 1959.5, observatory plus POGO data at 1969.5, and MAGSAT

data at 1980. They tested the constrained model by comparing its estimated

modeling errors with the misfit between the model and its data and concluded,

tentatively , that flux was not conserved. Yet, as they note, this sort of test is

very heavily colored by the reliability of their error estimates.

In this paper we employ a similar iterative penalty method (Luenberger,

1973) to a linearized form of only that subset of flux conservation integrals

which are believed to involve well-determined null-flux curves so that the

frozen-flux assumption is appropriate. Moreover, the time span of the data

used is only the five years centered on 1980 to ensure that flux diffusion

should be minimal. We also introduce two quite different tests of the

constrained model which are independent of error estimates. In the first test

both unconstrained and constrained models are constructed from the same

starting data set. Then their ability to hindcast and to forecast the magnetic

field before and after the time interval spanning the data input are compared by

examining the temporal growth of model residuals to observatory data. The

second test involves first invoking a new fluid dynamical assumption to

simplify the vertical angular momentum balance at the top of the core (Benton,

1985). If the unsteady change in vertical absolute vorticity is produced

entirely by Coriolis and inertia! torques on a fluid parcel, then one predicts

that the total (not absolute) magnetic flux out of either the northern or
southern geographic hemisphere of the CMB is conserved by a perfectly

conducting core. Time differentiation then provides a simple linear, analytic

constraint on the zonal secular variation coefficients of odd degree. Instead of

invoking this "magnetic constraint from vorticity dynamics" into our field

models we first test it by devloping a high quality field model (a remake of the

GSFC(9/80) model) containing no frozen-flux constraints whatever. Finding

that its zeroth order, odd degree secular variation coefficients do indeed

satisfy the above linear constraint rather well, we then examine the field

model G5FC(9/84) to determine whether or not it too obeys this constraint.

In Section 2, we describe the frozen-flux constraints of interest. Then, in

Section 3 the mathematical technique for least squares minimization

incorporating non-linear constraints like additional data is introduced. The

new constrained model, GSFC(9/84), and its unconstrained counterpart are

developed in Section 4 and assessed in Section 5. Conclusions are summarized

in Section 6. The simple analytic constraint on secular variation is derived in

Appendix A and the remade GSFC(9/80) field model, designated GSFC( 10/84), is

presented in Appendix B.

2. Frozen-Flux Constraints for Geomagnetic Field Modeling

A plausible first model for the coupled electromagnetisrn and fluid

dynamics of the Earth's core can be based upon the following simplifying

assumptions:
8

1) the core fluid is inviscid, incompressible, perfectly conducting, and its

magnetic permeability is that of vacuum

2) the CMB is a smooth sphere (of radius r=b) everywhere on which

vertical fluid motion vanishes

3) unsteady change in vertical absolute vorticity of fluid adjacent to the

CMB is produced only by Coriolis and inertial torques (i.e. baroclinic,

Lorentz and viscous torques are ignored).

Under these assumptions, the vertical components of the induction equation and

the absolute vorticity equation, when evaluated at the CMB, reduce to (Benton,

1985):

3B 9B 3B

at b 30 bsine 3<|> 3r

_w_8£l=£.9u. (3)

8t b3e bsine 3<i> 3r

Here u, v, w are the vertical, southward, eastward components of motion and C

is the vertical component of absolute vorticity, i.e.

-*-»-»

£' = r • V x (v+Qxr) (A]

where v is the fluid velocity vector relative to the mantle and Q is the angular

velocity of the earth (assumed constant on the short time scales of interest

here). Benton (1985) notes the existence of four classes of conserved magnetic

or vorticity flux integrals that result from this physical model:
JJ Br dS = constant (5)

SM

JJ Br dS = constant (6)

JJ £' dS = constant (7)

5M

JJ £' dS = constant (8)

where S^ and Sv denote area patches on the CMB bounded by null-flux curves

(where Br = 0) and "null-spin curves" (where £' = 0). Here, equation (5) results

from assumptions (1) and (2) above, whereas (8) results from assumptions (2)

and (3); all three assumptions are needed to derive (6) and (7).

The total number of independent constraints that emerge from these

integrals is quite large. For example, If Nj, N2 are, respectively, the number of

distinct null-flux curves and null-spin curves on the CMB, then one obtains 2Nj

constraints from eqs. (5) and (7) and 2N2 constraints from eqs. (6) and (8)

(where it may be noted that Nj null-flux curves divide the CMB into Nj + 1

distinct "magnetic patches," 5^; but because the total magnetic flux across the

entire CMB vanishes in the absence of monopoles, the number of independent
10

magnetic constraints is just Nj; a similar result follows for the null-spin

curves, because absolute vorticity is a solenoidal vector). Benton (1985) notes

further that in all likelihood vertical absolute vorticity is strongly dominated

by planetary vorticity, 2Q, everywhere at the top of the core. Then there is

only one null-spin curve (^=1), the "absolute vorticity equator," and it nearly

coincides with the geographic equator. Two interesting consequences of this

are first that north-south fluid motion vanishes at the geographic equator in

this model

v = 0 onr = b at 6 = 3Ifor all <j>, t , (9)

2

and secondly, that intersections of the main magnetic and absolute vorticity

equators, say N^ in number, contribute a further Nj boundaries of mixed type

for area patches that move with the fluid and therefore conserve the magnetic

and absolute vorticity flux enclosed by them. Each such mixed patch is the area

bounded by a segment of the geomagnetic equator and the absolute vorticity

equator between two successive intersections of those two curves. If the

velocity potential and stream function for the surface fluid motion of the core

are each expanded in spherical harmonics to truncation level N4 (Voorhies,

1984), then eq. (9) provides 2N4*1 independent constraints on the resulting
1 1

velocity coefficients. In toto then, we have a total of NC independent

constraints, where

+ N + N+N) + 1 . (10)

For typical numerical values we have Nj =8, N2=1, N3=6, N4=8 giving Nc=47.

In this paper we have selected five constraints of the form of eq. (5) from

this larger set to incorporate into a field model. Equation (5) expresses

conservation of magnetic flux through magnetic patches on the CMB. Because

the boundary of each such integral is a null-flux curve on which Br = 0, there is

locally zero contribution to each integral just at its boundary. Nonetheless, the

boundary location is a highly non-linear function of the model parameters

(taken here as Schmidt quasi-normalized Gauss coefficients because we

assume the mantle to be an insulator). Consequently, these magnetic

constraints are non-linear and an iterative approach is adopted.

Another independent constraint, based upon (6), is useful for assessing the

constrained models. Conservation of total (not absolute) magnetic flux through

the northern geographic hemisphere of the CMB is shown in Appendix A to be

expressible as a simple, linear analytic constraint on the zonal Gauss

coefficients of odd degree (Chapman and Bartels, 1940):

9B N

_J - j2Tfjir/2_x sine^ = % Fngn°(t) = 0 (11)

2rr(a/b)3 o o 3t n=l
where FQ = 0, F, = 1, Fn+2 = - n(n+ir'(a/b)2

Fn for n = 0, 1,2.... Here,

a = 6371.2 km is the mean radius of the Earth and gn° is the zeroth order

secular variation coefficient of degree n. The first seven non-zero terms of

this constraint take the form

g,o .1(3)2 ^o * I(a)4g5o _ 5_(a)6 gy> + 35_(a}8 ggo . 63_ (a} 1 0 g* } j0

2 b 8 b 16 b 128 b 256 b

1024 b

This model therefore predicts that the present rapid decay rate of the axial

dipole moment of the Earth should show up as enhanced excitation of the

weighted odd degree zonal harmonics of secular variation.

3. Least Squares Parameter Estimation with Non-Linear Constraints

The Bayesian parameter estimation algorithm provides a methodology for

including a priori statistical information on the parameter space in obtaining

least squares solutions (Luenberger, 1973). Let x denote the parameter vector,

y the measurement vector and v the random noise vector with zero mean and

covariance matrix R. Then the observation equation is

y = F(x) + v . (13)

In the linear case, F(x) = Ax so that

y = Ax + v . (14)

Then if xa denotes an a priori estimate of the parameters with a priori
13

covariance matrix Qa, the estimate for the parameter state vector x is

x = (ATR~ ' A + Qg'1 )H [AV1 y + Qa~' xa] . (15)

This result may be obtained by minimizing, with respect to x, the least squares

norm J^, where

JLS(x) = (y-Ax)TR"1(y-Ax)*(x-xa)

TQa"1

(x-xa) . (16)

It is clear that the a priori information is included in the formalism as

additional data.

For the non-linear problem, an iterative- approach is required. Linearizing

about a nominal solution XQ, we have

F(x) - F(x0) = A(x0)(x-x0) * ... . (17)

A Gauss Iteration procedure yields the equation for the (n+l)st approximation

to the solution estimate where

8xn+) = (AT(xA

n) FT1

 A(xA

n) * Qg'1

 }~] [A1"^) R'1 5yn + Qg'1 [x^-xj]

and

8yn = y - A(xn) xn .

The rate of convergence of the procedure depends on how good the nominal
14

guess XQ is, and on the non-linear character of the function F(x). For highly

non-linear problems, other techniques utilizing higher order derivatives may be

required.

The implementation of constraints may be accomplished within the

estimator formalism by Lagrange multiplier techniques (see Gubblns, 1984) or

by considering the constraint equations to be data. If a non-linear constraint

equation is represented as

G(x) = g (19)

then the (n+1)st iteration relation is

5xn+1 = [AT(xA

n) FT1

 A(xn) + Qa-' + CT(xn) R^1 C(xn)l ~

]

• [AT(xn) R-1 8yn + Qg'1 (xa-xn) + CT(Jn) R^1 8gnl (20)

where

C(xn) = 1^ (21)

8x xn

The weight matrix Rc~' reflects the stiffness of the constraint, where Rc is

the "covariance" matrix of the "observed" constraint. This error measure may

be chosen to represent the estimated numerical precision lost in the computing

process, or to represent modeling errors Inherent in the constraint. This result

is obtained by minimizing the least squares norm
.A /»

JL5 = Syn

TR-> Syn * (x-Ha)T

 Q^1 (x-Ka) * SgR^ 1 Sgn , (22)

This approach has also been taken by Bloxham and Gubbins (1985).

In the absence of constraints, the least squares estimation process

converges rapidly (in usually no more than two iterations) when the time

dependence of Gauss coefficients is a power series. In that representation, the

observations of magnetic components X, Y, Z, are linear functions of the Gauss

coefficients whereas the scalar field intensity, horizontal field, declination,

and inclination are nonlinear functions of those parameters. While the

measurements will be nonlinear functions of the parameters, they are at least,

still represented in analytic form. In contrast, most of the frozen-flux

constraints are not available in analytic form. For example, the null-flux

curves on which the vertical magnetic field vanishes are extremely complex

functions of the model parameters and must be determined numerically. As a

result numerical methods have been used to compute the matrix

C

T

(xn)C(xn)

for each iteration, n = 0,1,2,..., until convergence was approached.

4. Derivation of a Field Model Constrained by Flux Conservation

To investigate the influence of frozen-flux constraints on geomagnetic

field models, we now apply the non-linear algorithm described in Section 3 to a
variation of the G5FC( 12/83) field model (Langel and Estes, 1985). That model

utilized quiet, scalar and vector MAG5AT data and annual means from a

selected set of observatories for the years 1977 through 1982. Internal main

field and secular variation Gauss coefficients were included to order and

degree 13 and 10, respectively. The degree one external harmonics were also

retained and vector biases were calculated for each observatory. Moreover, the

internal axial dipole coefficient and the external coefficients were corrected

for linear Dst variation. However, those internal coefficients which were not

determined by the data inversion to the 95% confidence level were forced to be

zero in the final solution. Because the incorporation of dynamic constraints

from the core could improve the observability of such suppressed coefficients,

the G5FC( 12/83) model has now been recomputed with all of the above

coefficients retained in the solution and with all partial derivatives included

in the normal equations matrix, A^R^A. This starting model is designated

GSFC(9/84-0). The number of obervatories (by year) used in GSFC(9/84-0) is

86 (1977.5), 91(1978.5), 90 (1979.5), 90 (1980.5), 45 (1981.5), 8 (1982.5).

The GSFC(9/84-0) field model, utilizing high quality MAGSAT data,

represents the geomagnetic field at and above the Earth's surface very

accurately at epoch 1980.0. Figure 1 displays its radial component at the

core-mantle boundary at 1980.0, using all Gauss coefficients to order and

degree 13. This model has ten distinct null-flux curves, all but two of which
17

(one near the north pole and one beneath the eastern North Pacific) also appear

in the recent stochastic inversion model at 1980 constructed by Bloxham and

Gubbins (1986). Because the high degree Gauss coefficients, which are less

well determined, multiply spherical harmonics that become more dominant at

the CMB than at the Earth's surface, such computations become subject to large

uncertainty. Benton et al. (1982) examined the effect of varying the spherical

harmonic truncation level, N, on geomagnetic properties at the CMB, finding

considerable dependence in null-flux curve location and included magnetic flux.

In this work we have therefore selected only five null-flux curves for which to

impose the constraints. Those five, labeled in Figure 1, are present in the

model of Gubbins and Bloxham (1986), and are also reasonably stable features

of the magnetic maps at the CMB prepared by Benton et al. (1979) for N ranging

from 9 to 12.

The methodology for computing magnetic flux through each null-flux curve

(whose enclosed area is referred to as a null-flux patch, Backus, 1968) is to

overlay a i ° x 1' grid and then to approximate the field within each 1° x 1° cell

by its value at the center. The required partial derivatives with respect to the

model parameters were also determined numerically for each cell by varying

the nominal Gauss coefficient value. We note that the magnetic flux through

null-flux patches depends upon both main field and secular variation

coefficients because the integrand, Bp and boundary of integration, the
18

null-flux curves, each vary with time.

For each of the five selected null-flux patches, the constraints are

imposed at epochs 1977.5, 1980.0 and 1982.5 in the form of equation (5) with

the constants set to the values obtained from the G5FC(9/84-0) model at

1980.0. We are therefore adding 15 nonlinear constraints to the G5FC(9/84-0)

data set in such a way as to force conservation of magnetic flux through those

five null-flux patches. The elements of the diagnonal weight matrix RC~^ were

set to represent an error measure of 10~5 MWb on the "observed flux," this

value being selected (by numerical experimentation) to provide a stable

inversion of the normal equations matrix.

The solution was advanced through three iterations using equation (20),

starting with the GSFC(9/84-0) model as the nominal parameter vector for

iteration number one. No a priori information was assumed for the nominal

model, so Oa ' is zero in equation (20). Each iteration was performed in two

stages. First, the MAGSAT and observatory data set published for the

GSFC( 12/83) model were processed and the normal equations matrix,

A

T

(xn)R~'A(xn) and the vector AT(xn)R~'syn were accumulated. From this

information an "unconstrained" model was generated for each iteration, they
are designated GSFC(9/84-l), GSFC(9/84-2), and GSFC(9/84-3) for iterations

one, two and three, respectively. Note that for iteration one, GSFC(9/84-l) is

identical to G5FC(9/84-0) because the latter had already converged from the

G5FC( 12/83) data set. In the second stage of each iteration, the flux and its

partial derivatives for the fifteen observation equations were computed and

the matrix C^"(xn)Rc~'c(xn) and the vector CT(xn)Rc~'sgn were formed and then

added to the first stage quantities. The "constrained" models so obtained are

denoted as GSFC(9/84-lC), G5FC(9/84-2C) and G5FC(9/84-3C) for iterations

one, two and three respectively. The final constrained model is also referred

to simply as GSFC(9/84).

5. Assessment of the Flux-Constrained Geomagnetic Field Model.

GSFC(9/84)

Table 1 has been prepared to reveal how well or poorly the various

iterative models satisfy the imposed constraints. The model name is given in

the first column, followed by the epoch (in years beyond 1900). The first three

entries of the third column give the magnetic flux (in MWb) through null-flux

patch number one at the three epochs indicated whereas the offset number

(-6.9) is the rate of change of flux (In MWb/yr) at 1980. The maximum

variation In flux (in MWb) between a single model at the three epochs Is listed

in the fourth column. The final column gives the root sum square flux rate (in

MWb/yr) for all five null-flux patches. For a perfectly constrained solution,
20

the flux values at each epoch for each single null-flux patch would coincide and

the flux rates would be zero. Table 1 shows a trend towards reduced total flux

variation for the five year data interval as the model converges. The poorest

improvement is for the extreme southern null-flux patch number 2, whose

initial 5 year maximum flux variation of 30 MWb is reduced to 19 MWb after

three iterations. The constraints are clearly strongly felt with the other four

null-flux patches. For all five patches the root sum square flux rate at 1980 is

reduced from 16.1 MWb/yr to 5.9 MWb/yr as the iterations proceed.

Because the unconstrained solutions are optimum in the least squares

sense, the imposition of constraints must degrade the fit to the data. To

examine the amount of this degradation we consider the cost function, J^,

being minimized by the estimation procedure. With equation (22) written in the

form JLS = Syn

TR~' Syn + Sg^R^1

 8gn its first term represents the weighted

sum of the squares of the model misfits to the MAGSAT and observatory data,

now designated by Q. Table 2 presents the values of Q for the two data types

separately, as well as in toto, and for the observatory data, the Q values are

shown for each magnetic field component, X,Y,Z separately. As expected, the

inclusion of constraints increases the misfit, but by less than one percent

overall, for the final iteration. Although the solution appears to have

converged with respect to the massive set of MAGSAT data used, the situation
21

is less clear with respect to the observatory data. This probably reflects the

fact that the constraints affect the fit to secular variation (i.e. observatory

data) more strongly than they affect the fit to the short span of MAGSAT data.

For completeness, Table 2 also lists the root mean square of the residuals of

the various models to all of the MAGSAT data (in the third column) and to the

three components of the observatory data (in the fifth column). This again

shows that imposing these constraints only causes the misfit to the data (in

terms of residuals) to grow slightly.

A stringent test, both of the frozen-flux approximation and the utility of

including the resulting physical constraints into geomagnetic field models, is

provided by comparing how well or poorly unconstrained and constrained

models predict the magnetic field evolution when they are extrapolated beyond

the data interval that defined the models. The results of this test are

displayed in Figures 2-5 which are, respectively, temporal plots of the root

mean square residuals of the X,Y,Z magnetic field compoonents, and the scalar

intensity, B = (X2-

|

-Y2+Z^)1/2. Here X,Y,Z are the northward, eastward and

downward components and the residuals are the differences between the anual

means data and model values at the number of observatories indicated beneath

the dates along the abscissa. The observatory biases calculated for each model

were used in computing these statistics and the standard deviations are

plotted with a different symbol for each model. Within the resolution available
22

on these plots the points for G5FC(9/84-3) and GSFC(9/84-3C) coincide,

respectively, with those for G5FC(9/84-2) and GSFC(9/84-2C) so they are not

displayed separately. The vertical bars crossing the abscissa at 1977.5 and

1981.5 delineate the primary 4 year interval for observatory data. The dashed

vertical line at 1982.5 is to emphasize that only 8 observatories contributed to

the solution at that epoch so the statistics for 1982.5 are most indicative of

predictive errors. Because only nine observatories contributed to the standard

deviation at 1984.5 it must be considered unreliable.

Figures 2-5 generally reveal that the final constrained model is in better

agreement with observatory data when extrapolated 2 years, either forward or

backward, outside the primary data interval 1977.5-1981.5. The reduction in

root mean square residual is more pronounced in the relatively poorly predicted

vertical component, Z (Figure 4) than in the horizontal components. For

example, azat 1975.5 for constrained model G5FC(9/84-2C) is only about 75%

of that for unconstrained model G5FC( 12/83). The improvement in the Y

component (Figure 3) is not dramatic, but those residuals are uniformly smaller

anyway, than for the X and Z component. The standard deviation in field

intensity, Figure 5, is reduced by up to about 20% by imposition of the

constraints.

Because the predictability of Z is enhanced most, and that is the

component needed for extrapolation to the core-mantle boundary in studies
23

designed to evaluate the fluid motions just beneath the CMB, these findings are

viewed as encouraging. It should be noted that the improved prediction

capability has been achieved without altering the mathematical form of the

secular variation model.

A different test of these constrained models is provided by asking whether

they also satisfy other independent theoretical frozen-flux constraints,

specifically the one arising from consideration of vorticity dynamics as

discussed in Section 2 and Appendix A. Because the validity of that constraint

has not yet been firmly established, we first derived a new field model,

designated G5FC( 10/84) which is totally free of all frozen-flux constraints.

This model, essentially a remake of the G5FC(9/80) field model, is described

and documented in Appendix B.

The possibility that the secular variation coefficients of this model

satisfy the theoretical constraint given in equations (11 ) and (12) is examined

in Figure 6. The normalized flux of secular variation out of the northern

geographic hemisphere is plotted versus degree of spherical harmonic for three

epochs well within the 1960-1982 data interval used for the model. As higher

odd degree zonal harmonics are successively added, the cumulative sum in

equation (11 ) or (12) does indeed appear to approach zero. This model appears

to satisfy the constraint exactly at some epoch shortly after 1970, i.e. very

close to the centroid of the data used.
24

Having established that G5FC( 10/84) supports flux conservation through

the northern geographic hemisphere for a decade or so, we now ask whether the

other models introduced in this paper tend to obey or to violate this constraint.

Table 3 lists the axial dipole coefficient decay rate, g^0 in nT/yr, i.e. the first

term in equation (12), as well as the cumulative sum of all the terms in (12)

for six different field models, evaluated at 1980 (the center of their data

intervals).

When compared with the results in Figure 6, we find that the final

constrained model, G5FC(9/84-3C) also conserves very well the northern

hemispheric magnetic flux crossing the CMB. It might be argued that this must

follow because conservation of magnetic flux southward of the magnetic

equator in null-patch number l of Figure 1 was impressed as a constraint in

that model. However, that reasoning is not justified for two reasons. Firstly,

the southern geographic total hemispheric magnetic flux (equal and opposite to

that considered in arriving at equations 11, 12) differs, by about 9%, from the

absolute magnetic flux southward of the magnetic equator, and furthermore,

the flux in null-flux patch 2 was not strongly conserved in the three iterations

calculated herein. Thus, there is room for the G5FC(9/84-3C) model to violate

this constraint, should the data warrant it.

Readers interested in obtaining the Gauss coefficients of this model may

obtain them from R.H. Estes.
25

*

6. Conclusion

This paper has introduced into geomagnetic field modeling a selected set

of highly nonlinear constraints that arise from the frozen-flux theory of

electromagnetism for the earth's core. Forcing magnetic flux through five

null-flux patches to be driven toward the same values at each of three slightly

different epochs (1977.5, 1980.0, 1982.5) adds fifteen constraints to the

satellite and observatory data bases that define the models. The constraints

are incorporated like data in an iterative Bayesian parameter estimation

procedure that approaches convergence after three iterations.

The final constrained model, G5FC(9/84), has been tested by examining the

temporal growth of its residuals to observatory data during the two year

intervals that precede and follow the span of data that defined the model. This

stringent test was passed favorably in the sense that the constrained model

was found to be a somewhat better hindcaster and forecaster of the time

evolution of the geomagnetic field (especially the vertical component) than is

the unconstrained model used to start the iterative modeling algorithm. Also,

the imposition of the constraints caused the misfit between the model and the

satellite and observatory data to grow only slightly.

Another longer duration field model, totally unconstrained by flux

conservation, has also been developed and presented as an improvement to the

recent GSFC(9/80) field model. This model is found to have a secular variation
26

component which maintains, for a decade, nearly a null value for the flux of

geomagnetic secular variation out of the northern geographic hemisphere of the

core-mantle boundary, in accordance with a recent theoretical prediction of

Benton (1985). The new constrained model was then examined and it too

appears to satisfy this independent constraint.

We conclude that the incorporation into short duration geomagnetic field

models of constraints arising from the frozen-flux theory of the earth's core is

both feasible and of some predictive value. It remains to be determined by how

much the data interval defining the models and constraints can be lengthened

and also whether the inclusion of more constraints that are available will

sustain the above conclusion.

Acknowledgements

The work of ERB reported here has been supported, in part, at the

University of Colorado by N5F Grant EAR-7926120 and NASA Contracts

NAS5-27671 and NA55-28617 as well as at the U.S. Geological Survey, Denver,

Colorado. RHE has been supported by NASA Contracts NA55-27672 and

NA55-28671 to his former employer, Business and Technological Systems, Inc.

The authors are grateful to the funding agencies and to the above institutions

where this work was completed..
27

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32

Figure Captions

Figure 1: Contour map of the downward vertical magnetic field on the Earth's

core-mantle boundary at epoch 1980.0 according to the G5FCX9/84-0)

field model. Small numerals are field values (in gauss) for each

contour. Large numerals (1 through 5) are labels for null-flux patches.

Figure 2: Variation of a^ with time (unweighted) for observatory annual means

including model anomalies.

Figure 3: Variation of av with time (unweighted) for observatory annual

means including model anomalies.

Figure 4: Variation of tfz with time (unweighted) for observatory annual means

including model anomalies.

Figure 5: Variation of OB with time (unweighted) for observatory annual means

including model anomalies.

Figure 6: Normalized flux of secular variation through the northern geographic

hemisphere of the core-mantle boundary

3B

(—1 [2Tr [7r/2_I- sineded<t>, in nT/ur)

2TT(a/b)3 o o at

as a function of spherical harmonic degree according to the

G5FC( 10/84) field model.
TABLE 1. SOLUTION FLUX AND FLUX RATE THROUGH NULL CURVES

MWb AND MWb/YR

NULL

CURVE

MODEL EPOCH

GSFC 9/84-0

Flux 77.5

80.0

82.5

Flux Rate 80.0

GSFC 9/84-1C

Flux 77.5

80.0

82.5

Flux Rate 80.0

GSFC 9/84-2

Flux 77.5

80.0

82.5

Flux Rate 80.0

GSFC 9/84-2C

Flux 77.5

80.0

82.5

Flux Rate 80.0

GSFC 9/84-3

Flux 77.5

80.0

82.5

Flux Rate 80.0

GSFC 9/84-3C

Flux 77.5

80.0

82.5

Flux Rate 80.0

M/

VARI/

PATCH 1 \

-17454

-17449

-17488

\X MAX MAX M/

\TION VARIATION VARIATION VARIA

i PATCH 2 A PATCH 3 A PATCH 4 £

1410 156 -37

1386 131 -56

1380 107 -77

IX MAX RSS

tTION VARIATION of Flux

^ PATCH 5 A Rate

53

40

32

-6.9 39 -5.7 30 -10.0 49 -8.0 40 -3.8 21 16.1

-17456

-17443

-17472

1408 133 -56

1391 127 -52

1405 133 -56

41

38

40

-2.4 29 -0.1 17 -0.1 6 -0.2 4 -0.1 .3 2.4

-17448

-17440

-17452

1408 142 -38

1392 128 -55

1388 119 -74

-

54

40

28

-0.6 12 -3.8 20 -4.6 23 -7.4 36 -5.4 26 10.9

-17453

-17444

-17437

1400 125 -47

1390 128 -54

1396 133 -62

42

42

44

5.5 16 -1.0 10 1.8 8 -3.2 15 0.5 2 6.7

-17448

-17439

-17451

1408 141 -38

1392 129 -55

1388 118 -74

54

40

27

-0.7 12 -3.8 20 -4.7 23 -7.4 36 -5.5 27 11.0

-17436

-17438

-17450

1405 127 -61

1389 130 -56

1386 136 -54

40

41

41

-4.0 14 -3.7 19 1.9 9 1.3 7 0.2 1 5.9
TABLE 2. COST FUNCTION EVALUATION

MODEL MAGSAT (54.813 OBSERVATIONS) OBSERVATORY (972 OBSERVATIONS) TOTAL Q

0 RMS OF ALL DATA RESIDUALS (nT)

GSFC 984-0 22,332 6.38

GSFC 984-2 20,916 6.17

GSFC 984-2C 21,065 6.20

GSFC 984-3 20,916 6.17

GSFC 984-3C 21,060 6.19

0 RMS OF X, Y. Z RESIDUALS (nT)

5891 14.3

4.6

5.9

5605 14.1

4.6

5.4

5799 14.3

4.5

5.7

4239 11.2

4.5

6.2

4309 11.0

4.6

6.6

28,223

26,521

26,864

25,155

25,369
TABLE 3. COMPLIANCE OF VARIOUS FIELD MODELS WITH LINEAR ANALYTIC

CONSTRAINT ON SECULAR VARIATION'COEFFICIENTS

Model Name

GSFC(12/83)

GSFC<9/84-0)

GSFC(9/84-2)

GSFC(9/84-2C)

GSFC(9/84-3)

GSFC(9/84-3C)

Secular Variation

of the Axial Dipole

Coefficient g^CnT/yr)

26.5

26.3

24.8

24.9

24.8

24.9

Normalized Flux of

Secular Variation Out of

Northern Hemisphere

of the CMB, in nT/yr

-1.9

-6.2

-2.0

6.0

-2.0

0.6
ORIGINAL PAGE IS

POOR QUALITY

TABLE 4. GAUSS COEFFF1CIENTS FOR GSFC (10/84) AT EPOCH 1980. UNITS IN nT

Nun Nun Nun 8>nm Nun

1

I

2

2

2

J

3

3

3

4

4

4

6

4

5

5

5

5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

7

7

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

9

10

10

10

10

10

10

10

10

10

10

10

11

11

11

11

11

11

11

11

11

11

11

11

12

12

12

12

12

12

12

12

12

12

12

12

12

1J

1J

1)

1}

15

U

15

13

15

13

11

15

U

1}

0

1

0

1

2

0

1

2

5

0

1

2

5

4

0

1

2

J

4

5

0

1

2

3

4

5

6

0

1

2

J

4

5

6

7

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

11

0

1

2

3

4

5

6

7

8

9

10

11

12

0

1

2

3

4

5

6

7

8

9

10

11

12

13

-29992 10

-1955.70

-1996.60

3027. 40

1662 50

1281.10

-2180.70

1250.80

832.63

937.47

782 39

397 85

-418.89

198.36

-218.15

357.57

260.98

-74 14

-162.00

-48.10

47 89

65. 5J

42.03

-192 12

3.72

13 92

-107.61

71 46

-58.70

1.44

20.52

-12 73

0.67

10.64

-1 67

18.50

6.72

-0.02

-10.98

-7.05

4 19

2.86

6.35

-1 50

4.96

10.51

1.1!

-12.34

9 28

-3.61

-1 21

6 66

1 37

-5 00

-3.58

-3.93

2 37

-5.21

-1.74

4.74

3.21

0.87

2.16

2.83

-0.05

2.45

-0.89

-1.96

2.01

-0.11

-0.65

-0.21

1.47

1.75

-0.62

1.98

3.41

-1.82

-0.21

-0.02

-0.07

0.61

0 66

-0 75

-0.19

0.42

-0.55

-0 03

0 51

0 19

-0 20

-0 44

0 00

-0.71

-0.10

0 90

-0 54

0.35

-0.42

0.27

-0.16

0.25

0.04

0.41

0.00

5603 90

0.00

-2129 00

-199.46

0.00

-334 93

271.21

-252.73

0.00

212.53

-256.42

53 10

-297 10

0.00

45 72

149.98

-150 56

-77.67

92.06

0.00

-14.69

93.40

70.86

-43 21

-2 19

17.28

0 00

-83 07

-27.34

-4 79

16 20

17.87

-23 12

-9 79

0 00

6 97

-17.58

4 25

-22.27

9.26

16 13

-13 20

-14.90

0 00

-21.00

16.01

8.81

-4.94

-6.60

9.02

9.87

-5.68

2.20

0.00

0.94

0.57

2.58

5.63

-4.42

-0.45

-1.35

3.66

-0.57

-6 26

0.00

0.71

2.26

-1.19

-3.02

0.88

0.10

-2.45

-0.62

-1.55

-1.38

0.77

0.00

0.07

0.73

2.47

-1.39

0.11

0.15

-0.29

0.06

-0.04

-1.40

0.43

0.41

0.00

-0.28

0.44

1.62

-0.25

-0.48

-0.11

0.94

0.14

0.76

-0.01

-0.08

0.07

-0.41

20

10

-17

4

5

3

-4

-0

2

-2

-1

-9

-Z

-5

-0

-0

-0

-4

-0

0

0

-0.

3

1

0

1.1

0

-0

0.

0.

0.

0

-0

-0

0

0.

0.

0.

-0.

-0.

0.

-0

-0.

-0

0.

-0.

-0.

-0.

-0.

0.

0.

-0.

-0.

-0.

-0.

0.

-0.

0.

-0.

-0.

0.

0

-0

-0.

0.

0

-0.

-0.

0

-0

0

-0.

-0.

0.

-0.

0.

-0.

-0.

0

0.

-0

0

-0.

0

0.

-0.

-0.

-0

0

-0.

0.

-0.

0.

-0

-0.

-0.

0.

0.

0.

-0.

-0.

-0.

0.

833

.772

.773

.084

.624

.362

.301

658

.110

.186

.110

012

509

.399

.748

649

926

333

151

547

986

082

141

906

432

025

060

052

369

.047

446

959

403

298

161

480

097

193

022

243

085

417

348

741

324

113

006

002

142

345

042

276

145

219

116

099

012

038

027

106

101

049

195

028

104

017

016

004

155

050

025

019

O il

026

056

015

186

021

014

076

010

014

013

048

033

032

043

024

009

029

050

017

087

014

033

007

048

022

013

046

006

102

012

073

0 000

-15 973

0 000

-16.132

-26 477

0 000

-0 126

1 935

-7 061

0 000

3 099

1 395

4 228

-2.017

0 000

2 406

0.410

-0.722

0 978

0 961

0 000

0.281

-1 176

-0 410

-0 170

0.532

1.213

0 000

-1 352

-0.016

0 223

0 613

-0 342

0 091

0 403

0 000

-0 145

-0 292

-0 085

-0.391

0 354

-0 381

-0 539

0.357

0 000

0.113

0.090

0.252

-0.127

-0.222

-0.057

-0.049

-0.386

0.088

0 000

-0 094

-0.021

0.07S

-0.011

-0.047

-0.045

0.007

-0 007

-0.112

-0.324

0.000

-0.031

-0.002

0.067

-0.042

0 048

0 049

-0 107

-0 006

0.070

-0.066

0.009

0 000

-0.046

0.007

0 003

-0.084

-0.014

-0.023

0.011

-0.041

-0 026

0.022

-0.090

-0 048

0 000

0.010

-0.020

0.047

-0 015

-0.032

0.019

0 056

0.020

-0.010

0 015

-0.033

-0 052

-0.210

-0

-0

1

0

0

1

1

0

0

-0

0

-0

0

-0

-0

-0

-0

-0.

0

-0.

0.

-0.

0.

-0

0.

0

0

,

.6521

.1948

.0521

3539

1224

.2284

.1791

.7282

.8154

1665

1730

.7660

0530

5842

.1243

0660

2199

.2173

0072

.0342

.1104

0590

1273

0544

0476

0341

0955

0

-0

0

-2.

-0

0.

-0.

-0.

-0

0

-0.

0.

0.

-0.

0.

0.

-0.

0.

-0.

0

0.

0.

-0.

-0.

0.

0

-0.

.0000

.4694

.0000

.2172

.8740

.0000

.9699

.1680

6155

.0000

.3269

.0755

5049

.2564

.0000

.0666

1301

1276

0420

0314

0000

0760

0642

0772

0655

0048

0403

-0

-0

0

0

-0

0

0

0

0

-0

0

-0

0

-0

07241

02893

08398

01536

01135

.12058

09956

.08857

05543

.01000

.02415

05721

02582

06017

0 00000

0 03138

0.00000

-0.22115

-0 03275

0.00000

-0.07104

-0.02257

-0.07826

0.00000

-0.04263

0.00602

0 05363

-0.04315
TABLE 5. NUMBER OF OBSERVATORIES USED IN GSFC (10/84) BY YEAR

YEAR

1960.5

1961.5

1962.5

1963.5

1964.5

1965.5

1966.5

1967.5

1968.5

1969.5

1970.5

NUMBER OF OBSERVATORIES

137

146

145

143

158

164

166

163

165

164

164

YEAR

1971.5

1972.5

1973.5

1974.5

1975.5

1976.5

1977.5

1978.5

1979.5

1980.5

1981.5

1982.5

NUMBER OF OBSERVATORIES

157

154

156

156

153

149

145

141

139

118

73

17
TABLE 6

OBSERVATORIES AND BIAS SOLUTION

Observatory Latitude

ABISKO

ACACIAS

ADAK

ADDIS ABABA

AGINCOURT

ALERT

ALIBAG

ALIBAG II

ALMA ATA

ALMERIA

AMATSIA

AMBERLEY II

ANNAMALAINAGAR

APIA III

AQUILA

ARGENTINE ISLND

ARTI

BAKER LAKE

BAKER LAKE II

BAKER LAKE III

BANGUI

BANGUI II

BANGUI III

BARROW II

BARROW III

BEIJING

BELSK

BEREZNAYKI

BERE2NAYKI II

BINZA

BJORNOYA

BOULDER

BUDKOV

BYRD II

CAMBRIDGE BAY

CANBERRA

CASEY

CASTELLACCIO

CASTLE ROCK

CHA PA

CHAMBON FORET

CHELYUSKIN II

COIMBRA

COLLEGE

DALLAS

DIKSON II

DOMBAS II

DOURBES

DRUZHNAYA

DUMONT DURVILLE

DUSHETI

DYMER

EBRD

EIGHTS

ESKDALEMUIR

FORT CHURCHILL

FREDERICKSBURG

FUQUENE

FURSTNFELDBRUCK

GNANGARA

GODHAVN

GODHAVN II

GORNOTAYEZHNAYA

GREAT WHALE R

GROCKA

GUAM

(Deg)

68.36

-35.01

51.87

9.03

43.78

82.50

18.64

18.64

43.25

36.85

31.55

-43.15

11.37

-13.81

42.38

-65.24

56.43

64.33

64.33

64.33

4.44

4.44

4.44

71.30

71.32

40.04

51.84

49.82

49.82

-4.27

74.50

40.14

49.08

-80.02

69.20

-35.32

-66.28

44.43

37.24

22.35

48.02

77 .72

40.22

64.86

32.99

73.54

62.07

50.16

80.62

-66.66

42.09

50.72

40.82

-75.23

55.32

58.77

38.21

5.47

48.16

-31.78

69.24

69.24

43.68

55.27

44.63

13.58

Longitude

(Deg)

18

-57

-176

38

-79

-62

72

72

76

-2

34

172

79

-171

13

-64

58

-96

-96

-96

18

18

18

-156

-156

116

20

73

73

15

19

-105

14

-119

-105

149

110

8

-122

103

2

104

-8

-147

-96

80

9

4

58

140

44

30

0

-77

-3

-94

-77

-73

11

115

-53

-53

132

-77

20

144

.82

.69

.64

.76

.27

.50

.87

.87

.92

.46

.92

.72

.68

.77

.32

.26

.57

.03

.03

.03

.57

.57

.57

.75

.62

.17

.79

.08

.08

.37

.20

.24

.01

.52

.00

.36

.53

.93

.13

.83

.26

.28

.42

.84

.75

.56

.12

.99

.05

.01

.71

.30

.49

.17

.20

.10

.37

.74

.28

.95

.52

.52

.17

.78

.77

.87

Altitude X Bias

(Km)

0.37

0.01

0.0

2.44

0.17

0.05

0.0

0.0

1.29

0.06

0.0

0.03

0.0

0.0

0.62

0.0

0.28

0.04

0.04

0.04

0.38

0.38

0.38

0.0

0.0

0.06

0.17

0.0

0.0

0.0

0.07

1.64

0.49

1.51

0.01

0.84

0.0

0.34

0.13

0.0

0.14

0.0

0.09

0.08

0.20

0.01

0.65

0.20

0.01

0.03

0.97

0.09

0.04

0.44

0.24

0.03

0.06

2.54

0.56

0.05

0.0

0.0

0.29

0.02

0.23

0.14

(nT)

38

-8

-462

495

-15

9

-91

-187

162

2

124

13

209

-33

12

72

126

170

246

176

-127

-63

-133

54

34

607

120

-390

-356

-80

-98

-0

-25

-8

108

9

915

24

-78

-51

-54

-32

22

-4

- 56

-87

-68

20

72

-164

-205

-3

29

103

18

-113

84

128

-5

2

228

288

4

280

-17

146

.0

.9

.3

.7

.5

.0

.8

.0

.8

.9

.2

.0

.9

.3

.5

.6

.9

.0

.3

.8

.2

.3

.5

.1

.3

.1

.4

.5

.6

.7

.8

.8

.6

.2

.9

.9

.0

.8

.6

.5

.3

.6

.8

.2

.0

.4

.4

.0

.1

.4

.4

.3

.9

.0

.9

.2

.4

.5

.3

.9

.8

.7

.5

.0

.0

.8

Y Bias

(nT)

56.9

2.7

17.0

5.9

166.4

30.8

449.0

453.5

33.2

21.7

37.1

-6.8

-82.4

211.2

39.5

-76.2

-265.9

-50.8

-136.8

-40.1

-65.7

1072.6

-44.9

-58.5

-62.2

-224.9

145.4

8.5

-198.6

-185.8

47.7

56.0

-16.9

40.5

-89.7

41.5

-311.7

-197.1

-20.3

-115.3

-24.4

-91.1

-15.1

-52.0

17.8

-144.5

-81.5

-21.4

-676.0

-383.0

-2.6

81.6

-0.1

203.2

-52.4

41.8

-71.1

-76.2

-8.7

-126.1

-272.6

-310.8

-6.9

107.0

-50.0

90.4

Z Bias

(nT)

34.6

12.8

-96.7

117.0

-138.3

-142.2

672.2

592.4

-181.8

3.3

275. 2

89.2

-99.5

-919.1

-6.6

494.5

448.0

-99.4

-102.7

-87.8

171.6

201.2

210.3

69.3

-55.7

406.7

301.4

314.4

323.1

-50.3

29.5

-166.3

-32.5

-137.7

144.8

93.4

-678.6

-120.0

-28.9

-175.8

101.6

-81.0

21.8

-109.0

-80.9

-255.6

-242.4

78.7

1160.5

-2906.7

-102.5

106.0

-25.9

109.7

-57.9

-255.2

135.9

80.0

5.4

150.8

481.4

740.2

-58.6

-56.5

-60.7

94.1

Begin

Year

1960

1964

1964

1960

1960

1961

1960

1978

1963

1960

1976

1960

1960

1960

1960

1960

1973

1960

1969

1975

1960

1966

1973

1960

1963

1960

1960

1965

1977

1960

1960

1964

1967

1962

1972

1979

1978

1960

1970

1960

1960

1960

1960

1960

1964

1960

1960

1960

1960

1960

1960

1964

1960

1963

1960

1964

1960

1960

1960

1960

1960

1976

1960

1966

1960

1960

.5

.5

.8

.5

.5

.9

.5

.5

.4

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.3

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.6

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

.5

End

Year

1980.5

1978.5

1966.0

1980.5

1969.1

1980.5

1977.5

1980.5

1981.5

1980.5

1981.5

1977.5

1980.5

1980.5

1981.5

1982.5

1981.5

1968.5

1974.5

1980.5

1965.5

1972.5

1982.5

1962.5

1980.5

1980.5

1980.5

1976.5

1980.5

1973.5

1981.5

1980.5

1978.5

1968.3

1980.5

1981.5

1981.5

1969.5

1974.5

1978.5

1980.5

1979.5

1980.5

1982.5

1974.5

1980.5

1980.5

1981.5

1979.5

1981.5

1981.5

1979.5

1979.5

1965.4

1981.5

1980.5

1982.5

1979.5

1981.5

1981.5

1975.5

1979.0

1981.5

1980.5

1976.5

1982.5
TABLE 6 (Cont.)

Observatory Latitude

(Deg)

HALLETT STATION

HALLEY BAY

HARTLAND

HATIZYO

HAVANA

HEL

HERMANUS

HOLLANDIA

HONGKONG

HONOLULU IV

HUANCAYO

HURBANOVO

HYDERABAD

IBADAN

ISLA DE PASCUA

ISTANBL KNDILLI

JAIPUR

JULIANEHAAB II

KAKIOKA

KANOYA

KANOZAN

KELES

KERGUELEN

KIEV

KIRUNA

KLYUCHI

KODAIKANAL

KOROR

KRASNAYA PAKHRA

KSARA

LA PAZ

LA QUIACA II

LEIRVOGUR

LERWICK

LHASA

LOGRONO

LOPARSKOYE

LOVO

LUANDA BELAS

LUNPING

LVOV

LWIRO

M BOUR

MACQUARIE ISLND

MAGADAN

MAJURO

MANHAY

MAPUTO

MARION ISLAND

MAURITIUS II

MAWSON

MEANOOK

MEMAMBETSU

MIRNYY

MISALLAT

MIZUSAWA

MOCA

MOLODEZHNAYA

MONTE CAPELLINO

MOULD BAY

MUNTINLUPA

NAGYCENK

NAIROBI

NARSSARSSUAQ

NEWPORT

NIEMEGK

NOVO KAZALINSK

NOVOLAZAREVSKAY

NURMIJARVI

-72

-75

50

33

22

54

-34

-2

22

21

-12

47

17

7

-27

41

26

60

36

31

35

41

-49

50

67

55

10

7

55

33

-16

-22

64

60

29

42

68

59

-8

25

49

-2

14

-54

60

7

50

-25

-46

-20

-67

54

43

-66

29

39

3

-67

44

76

14

47

«• 1

61

48

52

45

-70

60

.32

.52

.99

.12

.97

.61

.43

.57

.36

.32

.04

.87

.41

.43

.17

.06

.92

.72

.23

.42

.25

.42

.35

.72

.83

.03

.23

.34

.48

.82

.54

.10

.18

.13

.70

.46

.25

.35

.92

.00

.90

.25

.39

.50

.12

.08

.30

.92

.88

.09

.60

.62

.91

.55

.51

.01

.34

.67

.55

.20

.38

.63

.33

.10

.26

.07

.80

.77

.51

Longitude

(Deg)

170

-26

-4

139

-82

18

19

140

114

-158

-75

18

78

3

-109

29

75

-46

140

130

139

69

70

30

20

82

77

134

37

35

-68

-65

-21

-1

91

-2

33

17

13

121

23

28

-16

158

151

171

5

32

37

57

62

-113

144

93

30

141

8

45

8

-119

121

16

36

-45

-117

12

62

11

24

.22

.68

.43

.80

.14

.82

.23

.51

.22

.00

.34

.19

.55

.90

.42

.06

.80

.03

.19

.88

.96

.21

.20

.30

.42

.90

.46

.50

.31

.89

.10

.60

.70

.18

.15

.50

.08

.83

.17

.17

.75

.80

.96

.95

.02

.38

.68

.58

.85

.55

.88

.33

.19

.02

.89

.08

.66

.85

.95

.40

.01

.72

.82

.20

.12

.68

.10

.83

.65

Altitude X Bias

(Km)

0.0

0.02

0.09

0.0

0.0

0.0

0.02

0.22

0.02

0.0

3.31

0.11

0.49

0.29

0.0

0.12

0.0

0.45

0.02

0.10

0.34

0.44

0.04

0.09

0.38

0.0

2.32

0.0

0.19

0.91

0.42

3.44

0.0

0.08

3.65

0.44

0.19

0.02

0.05

0.09

0.39

1.67

0.0

0.0

0.0

0.0

0.43

0.04

0.0

0.05

0.0

0.67

0.03

0.01

0.11

0.11

1.34

0.0

0.69

0.14

0.06

0.15

1.67

0.0

0.77

0.07

0.0

0.45

0.10

(nT)

110.1

19.3

-26.9

-83.4

72.0

56.7

21.4

-196.3

-66.9

-157.5

94.9

21.6

330.8

104.1

-15.4

202.3

188.6

121.5

5.4

4.5

-38.8

-179.8

225.7

-62.6

-809.0

199.0

-528.0

70.8

135.9

-23.5

39.2

47.0

-274.5

-113.7

193.8

21.4

115.2

60.2

285.2

31.4

152.2

263.3

136.7

279.5

-1356.0

-337.6

20.4

356.5

-849.2

500.5

32.8

105.2

-224.6

-114.5

-49.9

-117.2

-68.5

-19.6

-27.2

-19.1

-52.7

10.7

60.8

-352.2

-28.8

-17.3

-90.5

-282.2

299.3

Y Bias

(nT)

-110.8

380.7

4.8

-1031.7

162.2

-162.0

13.6

-61.3

39.0

93.7

47.5

-19.6

70.1

-325.1

584.9

118.6

-403.8

-177.3

16.3

62.7

41.8

-53.8

191.3

188.2

-1819.1

-87.5

269.9

135.3

-18.5

54.7

97.6

-4.3

593.3

166.7

29.9

3.5

333.6

-4.4

-35.0

44.8

121.4

80.8

17.9

4.2

352.3

50.5

-13.8

35.5

664.1

-221.7

21.4

6.9

147.4

48.5

67.6

49.4

-27.1

-101.8

-62.5

5.4

-51.8

-13.8

52.6

265.6

112.8

-3.8

-185.3

62.1

-107.8

Z Bias

(nT)

-177.

-34.

52.

361.

-24.

-89.

20.

-370.

-33.

-337.

15.

-46.

469.

116.

-498.

4.

-32.

351.

-80.

-33.

-54.

-39.

661.

119.

-39.

-18.

-47.

151.

222.

-87.

246.

12.

-493.

30.

-43.

34.

-496.

-3.

172.

51.

144.

30.

48.

305.

1251.

-113.

169.

-142.

-1371.

-433.

201.

-148.

67.

-438.

100.

-190.

349.

-216.

-569.

-41.

33.

-54.

-33.

555.

-127.

-82.

-5.

176.

94.

9

2

3

1

8

7

7

2

5

5

4

5

2

7

9

5

4

8

4

3

4

0

2

2

5

8

6

7

3

7

6

7

3

1

9

8

9

3

8

8

8

4

9

0

2

7

3

8

5

4

4

0

3

7

4

9

7

5

6

0

3

5

3

9

9

5

1

2

3

Begin

Year

1960.8

1960.5

1960.5

1967.7

1965.5

1960.5

1960.5

1960.5

1974.5

1961.5

1960.5

1960.5

1965.5

1960.5

1961.8

1960.5

1979.5

1960.6

1960.5

1960.5

1961.5

1960.5

1960.5

1960.5

1965.5

1967.5

1960.5

1961.5

1960.5

1960.5

1974.5

1960.5

1960.5

1960.5

1960.5

1960.5

1961.5

1960.5

1960.5

1965.8

1960.5

1960.5

1960.5

1960.5

1960.5

1964.8

1960.5

1960.5

1973.7

1960.5

1960.5

1960.5

1960.5

1960.5

1960.6

1969.5

1960.5

1965.5

1960.5

1962.8

1960.5

1961.5

1964.5

1968.9

1966.6

1960.5

1974.5

1961.5

1960.5

End

Year

1962.5

1975.5

1981.5

1982.5

1978.5

1979.5

1981.5

1962.3

1978.5

1982.5

1981.5

1979.5

1980.5

1962.5

1968.4

1979.5

1981.5

1964.6

1981.5

1981.5

1978.5

1963.5

1981.5

1963.5

1981.5

1981.5

1980.5

1966.2

1979.5

1970.5

1976.5

1982.5

1981.5

1981.5

1974.5

1976.5

1980.4

1980.5

1982.5

1981.5

1981.5

1970.5

1982.5

1981.5

1966.5

1966.1

1973.5

1982.5

1977.5

1965.5

1981.5

1980.5

1981.5

1980.5

1974.5

1978.5

1971.5

1979.5

1962.5

1980.5

1981.5

1981.5

1978.5

1979.0

1981.5

1982.5

1981.5

1979.5

1982.5
TABLE 6 (Cont.)

Observatory Latitude

(Deg)

OTTAI-JA

PAMATAI

PAMATAI II

PANAGYURISHTE

PARAMARIBO

PARATUNKA

PATRONY

PILAR

PLATEAU

PLESHENITZI

PODKAM TUNGUSKA

PORT MORESBY

PORT-ALFRED

PRUHONICE

QUETTA

REGENSBERG

RESOLUTE BAY

ROBURENT

ROBURENT II

ROI BAUDOUIN

RUDE SKOV

SABHAWALA

SAN FERNANDO

SAN JOSE LAS LA

SAN JUAN

SAN JUAN II

SAN MIGUEL III

SANAE

SCOTT BASE

SHESHAN

SHILLONG

SIMOSATO

SITKA

SODANKYLA

SOUTH GEORGIA

SOUTH POLE

SREDNIKAN IV

ST JOHN S

STEKOLINIY

STEPANOVKA

STONYHURST

SURLARI

SW1DER

SYOWA BASE

TAMANRASSET

TANANARIVE

TANGERANG

TATUOCA

TEHRAN

TENERIFE

TEOLOYUCAN

THULE II

TIHANY

TIKSI

TOLEDO

TOMSK

TOOLANGI

TRELEW

TRIVANDRUM

TRQMSO

TSUMEB

TUCSON

UELEN

UJJAIN

ULAN BATOR

VALENTIA

VANNOVSKAYA

VASSOURAS

45

-17

-17

42

5

52

52

-31

-79

54

61

-9

-46

49

30

47

74

44

44

-70

55

30

36

23

18

18

37

-70

-77

31

25

33

57

67

-54

-89

62

47

60

46

53

44

52

-69

22

-18

-6

-1

35

28

19

77

46

71

39

56

-37

-43

8

69

-19

32

66

23

47

51

37

-22

.40

.57

.57

.51

.81

.90

.17

.67

.25

.50

.60

.41

.43

.99

.19

.48

.70

.30

.30

.43

.84

.36

.46

.02

.38

.11

.77

.30

.85

.10

.57

.57

.06

.61

.28

.99

.44

.59

.12

.78

.85

.68

.12

.01

.79

.92

.17

.21

.74

.48

.75

.48

.90

.58

.88

.47

.53

.25

.48

.66

.22

.25

.16

.18

.85

.93

.95

.40

Longitude

(Deg)

-75

-i-'i9

-149

24

-55

158

104

-63

40

27

90

147

51

14

66

8

-94

7

7

24

12

77

-6

-82

-66

-66

-25

-2

166

121

91

135

-135

27

-36

-13

152

-52

151

30

-2

26

21

39

5

47

106

-48

51

-16

-99

-69

17

129

-4

84

145

-65

76

18

17

-110

-169

75

107

-10

58

-43

.55

.57

.57

.18

.22

.43

.45

.88

.50

.88

.00

.15

.87

.55

.95

.44

.90

.89

.89

.30

.46

.80

.21

.65

.12

.15

.65

.37

.78

.19

.88

.94

.32

.05

.48

.32

.31

.68

.02

.88

.47

.25

.25

.59

.53

.55

.63

.51

.38

.28

.18

.17

.89

.00

.05

.93

.47

.31

.95

.95

.70

.83

.84

.78

.05

.25

.11

.65

Altitude X Bias

(Km)

0

0

0

0

0

0

0

0

3

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

0

0

0

0

0

0

0

0

1

1

0

0

1

0

2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.75

.18

.08

.55

.0

.10

.53

.33

.61

.19

.0

.07

.0

.32

.74

.59

.02

.81

.81

.13

.04

.49

.02

.0

.09

.39

.17

.04

.01

.09

.0

.05

.02

.17

.0

.79

.60

.0

.0

.13

.11

.08

.09

.0

.37

.37

.01

.0

.36

.30

.27

.05

.18

.03

.50

.19

.45

.02

.29

.11

.08

.76

.0

.0

.0

.0

.56

.45

(nT)

154.1

-578.7

-642.9

-172.0

31.0

-329.5

30.9

18.6

78.6

291.9

66.6

25.8

-669.9

-24.5

3.4

26.3

41.5

95.2

104.1

-98.9

44.2

4.4

79.9

64.9

130.6

-35.6

719.3

-37.4

-2260.9

-220.3

-108.2

-41.0

9.5

-126.7

-74.6

-1334.5

110.3

56.3

-266.0

-101.0

23.7

28.2

-303.6

-9.4

79.9

413.5

-28.7

51.9

-75.8

-444.2

-50.0

-51.7

2.4

-79.1

15.8

12.4

12.9

121.4

251.4

98.8

62.9

-50.2

-83.0

-232.2

-30.4

145.4

199.1

102.4

Y Bias

(nT)

-148.0

-1093.3

-721.0

-169.7

62.6

230.1

49.6

3.9

-18.1

155.0

58.5

48.7

1218.2

22.7

49.4

31.8

29.0

-18.5

56.5

-3.3

-9.7

-54.7

39.2

19.9

43.8

197.2

432.4

-31.2

-927.9

81.4

-77.1

45.9

-9.4

-117.0

-359.0

-3483.0

12.1

14.0

-732.7

-705.1

16.5

-32.1

-105.5

-70.8

-205.6

5.0

10.5

-104.3

16.3

106.2

-9.8

97.7

-12.6

-151.5

2.6

-59.1

-12.6

22.2

182.8

-404.1

-80.1

-65.5

33.8

190.0

-9.5

-56.5

97.2

-67.4

2 Bias

(nT)

186.1

376.0

-124.0

-187.9

-38.2

255.6

-85.4

-16.4

-27.1

-130.1

-307.9

270.5

158.8

-82.6

-60.9

-35.4

75.6

112.9

70.5

-24.3

-46.7

19.5

-60.2

-10.9

175.5

163.4

1713.7

21.2

-3779.3

230.8

-373.9

17.5

-78.2

-598.6

98.1

71.2

95.6

1.1

49.1

62.0

-70.4

-61.6

250.3

-17.6

-10.4

-407.4

-1.1

160.4

-215.1

-1081.8

-124.3

20.9

-40.6

-118.0

-4.4

-242.1

60.2

52.9

150.6

171.7

103.8

134.4

-116.3

272.4

-118.6

14.6

50.5

-32.7

Begin

Year

1968.7

1966.2

1973.5

1960.5

1960.5

1969.5

1960.5

1960.8

1966.5

1961.5

1969.5

1960.5

1974.5

1960.5

1960.5

1960.5

1960.5

1964.8

1969.5

1964.7

1960.5

1964.5

1960.5

1964.8

1960.5

1965.5

1960.5

1962.7

1960.8

1960.5

1979.5

1960.5

1960.5

1960.5

1975.5

1960.5

1961.5

1968.8

1966.5

1960.5

1961.5

1960.5

1960.5

1960.6

1960.5

1960.5

1964.5

1960.5

1960.5

1960.5

1960.5

1960.5

1960.5

1960.5

•1960.5

1960.5

1960.5

1960.5

1960.5

1960.5

1964.8

1960.5

1960.5

1979.5

1966.5

1960.5

1960.5

1960.5

End

Year

1980.5

1972.5

1982.5

1977.5

1974.5

1979.5

1979.5

1982.5

1968.5

1981.5

1981.5

1981.5

1981.5

1972.5

1981.5

1975.5

1980.5

1968.5

1973.5

1966.5

1981.5

1980.5

1978.5

1976.2

1964.5

1982.5

1977.5

1981.5

1979.5

1980.5

1981.5

1977.5

1981.5

1982.5

1981.5

1971.5

1966.5

1980.5

1981.5

1981.5

1967.5

1980.5

1974.5

1970.5

1971.0

1981.5

1976.5

1980.5

1971.5

1979.5

1975.5

1980.5

1980.5

1979.5

1980.5

1969.5

1979.2

1978.5

1980.5

1981.5

1981.5

1981.5

1979.5

1981.5

1977.5

1980.5

1980.5

1981.3
TABLE 6 (Cont.)

Observatory Latitude

VICTORIA

VOSTOK

VOYEYKOVO

VYSOKAY DUBRAVA

WHITESHELL

WIEN KOBENZL

WILKES

WINGST

WITTEVEEN

YAKUTSK

YANGI-BAZAR

YELLOW-KNIFE

YUZHNO SAKH II

YUZHNO SAK III

ZAYMISHCHE

(Deg)

48.52

-78.45

59.95

56.73

49.75

48.26

-66.25

53.74

52.81

62.02

41.33

62.40

46.95

46.95

55.83

Longitude

(Deg)

-123

106

30

61

-95

16

110

9

6

129

69

-114

142

142

48

.42

.87

.71

.07

.25

.32

.58

.07

.67

.72

.62

.50

.72

.72

.85

Altitude X Bias Y Bias Z Bias Begin

(Km)

0

3

0

0

0

0

0

0

0

0

0

0

0

0

0

.19

.49

.06

.28

.0

.39

.0

.04

.01

.09

.80

.18

.06

.06

.07

(nT)

28.8

-5.5

93.4

-266.0

194.6

38.0

656.4

69.5

35.6

70.3

-271.2

416.6

51.2

-73.0

-276.0

(nT)

-6.4

124.7

17.0

-117.3

-241.7

-2.5

-289.0

43.4

1.8

-1181.3

46.8

-212.0

-146.7

-56.5

-307.6

(nT)

-327.7

85.8

-276.2

-512.2

-237.5

15.3

19.5

-67.7

-81.9

105.4

-115.0

129.6

-159.6

99.2

-258.7

Year

1960.

1960.

1960.

1960.

1977.

1960.

1960.

1960.

1960.

1960.

1964.

1975.

1960.

1970.

1960.

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

End

Year

1980.5

1979.5

1981.5

1976.5

1980.4

1980.5

1966.5

1980.5

1980.5

1979.5

1981.5

1980.5

1969.5

1980.5

1981.5
TABLE 7. STATISTICS OF MODEL DATA SET RESIDUALS RELATIVE TO GSFC (10/84)

POGO

rms

mean

sigma

MAGSAT

rms

mean

sigma

Annual Means

(with biases)

rms

mean

sigma

Annual Means

(without biases)

rms

mean

sigma

B

6.8

-0.3

6.7

7.3

-0.1

7.3

X

6.3

-0.1

6.3

15.8

-0.1

15.8

268.5

-3.8

268.5

Y

7.9

1.3

7.8

14.8

-0.0

14.8

330.2

-36.7

328.2

Z

7.2

-3.6

6.2

26.0

-0.1

26.0

457.6

-30.8

456.5
Figure 1

GSFC 9/84-0 Z = -Br COMPONENT AT CMB AT 1980

UNITS: GAUSS (10^ nT)

-75fc

-90

-180 -150 -120 -90 -60 -30 0 60 90 120 150 180
Figure 2

60

50

40

ox

20

10

VARIATION OF ox WITH TIME (UNWEIGHTED) FOR OBSERVATORY

ANNUAL MEANS INCLUDING MODEL ANOMALIES

UNITS: nT

e

a

A GSFC 12/83

O GSFC 9/84-0

D GSFC 9/84-2

O GSFC 9/84-2C

GSFC 9/84-3 SYMBOL FALLS ON TOP OF D

GSFC 9/84-3C SYMBOL FALLS ON TOP OF O

e

1

YEAR 75

# OF OBSERVATORIES

USED IN STATISTICS

75

1 , 1

76 77

MODEL DATA INTERVAL

1,1.1, 1

78

75 77

79

79

80 81

I

82

79 80 75

i

I

I

I

I

69

1 , 1

83 84

59

__i —

9

1

85
Figure 3

60

50

40

30

20

10

VARIATION OF OY WITH TIME (UNWEIGHTED) FOR OBSERVATORY

ANNUAL MEANS INCLUDING MODEL ANOMALIES

UNITS: nT

a

8

8

A GSFC 12/83

O GSFC 9/84-0

D GSFC 9/84-2

O GSFC 9/84-2C

GSFC 9/84-3 SYMBOL FALLS ON TOP OF D

GSFC 9/84-3C SYMBOL FALLS ON TOP OF

8

8

1

YEAR 75

# OF OBSERVATORIES 75

USED IN STATISTICS

1

76

75

1

77

MODEL DATA INTERVAL

1,1,1, 1

78

77

79

79

80 81

I

82

79 80 75

I

I

1

1

69

1 , 1

83 84

59 9

85
Figure 4

60

50

40

oz 30

20

10

VARIATION OF oz WITH TIME (UNWEIGHTED) FOR OBSERVATORY

ANNUAL MEANS INCLUDING MODEL ANOMALIES

UNITS: nT

A GSFC 12/83

Q

O GSFC 9/84-0

D'GSFC 9/84-2

O GSFC 9/84-2C

GSFC 9/84-3 SYMBOL FALLS ON TOP OF D

GSFC 9/84-3C SYMBOL FALLS ON TOP OF <

Q

8 Q Q

O

a

0

# OF OBSERVATORIES 75 75

USED !N STATISTICS

77 79 79 80 75 69 59

/\

;CO

D

A

1.1, 1

YEAR 75 76 77

MODEL DATA INTERVAL

1 , 1 ,1, 1

78 79 80 81

l

82

l

! i

| 83

l , l

84 8
Figure 5

60

50

40

oB 30

20

10

VARIATION OF OB WITH TIME (UNWEIGHTED) FOR OBSERVATORY

ANNUAL MEANS INCLUDING MODEL ANOMALIES

UNITS: nT

A GSFC 12/83

O GSFC 9/84-0

D GSFC 9/84-2

O GSFC 9/84-2C

GSFC 9/84-3 SYMBOL FALLS ON TOP OF D

£, GSFC 9/84-3C SYMBOL FALLS ON TOP OF O

Cs

$

YEAR

1

75

1

76

1

77

MODEL DATA INTERVAL

1 , 1 ,1, 1

78 79 80 81

I

I 1

82 1

1

83

I

84

1

85

# OF OBSERVATORIES 75

USED IN STATISTICS

75 77 79 79 80 75 69 59
30

25

hc

g

K- 20

oc

5

 1S

o

LLJ

CO

o 10

X

LL

HI

N

DC

O

T

Figure 6

—I

-5

2 3 4 5 6 7 8 9 1011121 3

n, DEGREE OF SPHERICAL HARMONIC

Normalized Flux of Secular Variation Through the Northern

Geographic Hemisphere of the Core-Mantle Boundary

TT

 sin 8ded<l)' in nT/yr>

as a Function of Spherical Harmonic Degree According to

GSFC 10/84.
Appendix A. Derfvatton of an Analytic. Linear Constraint

on Secular Variation

In the absence of cross-equatorial flow Implied by (9), the geographic

equator on the CMB always consists of the same fluid parcels (Benton, 1985,

Backus and LeMouel, 1986). Hence, (6) shows that the total (not absolute)

magnetic flux, M0, out of, say, the northern geographic hemisphere of the CMB

is conserved:

M0 = /27r/

7r/2Br(b,e,<t>

,t)b2sineded<t> = constant . (Ai)

o o

If the mantle is an insulator then (c.f. Chapman & Bartels, 1940)

N n

Br (b,e,<t>,t) =

n=1 m=0 b

m m m

lgn(t) cos m<|> * hn(t) sin m<|>l Pn(8) (A2)

where a = 637 1 .2 is the earth's radius, gn

m

, hn

m

 are the time-dependent Gauss

coefficients of order m, degree n, Pn

m

(0) is the Schmidt quasi-normalized

associated Legendre function and N is the degree of truncation. Insertion of

(A2) into (Al ) followed by integration with respect to <)> leads to
N

n0 = 27tb2 £ (n*t) (2)

n

*

2

 g° J1 P°0i)dji. p = cose . (A3)

n=l b n o n

The Integral can be carried out readily In terms of the differential definition of

Pn° (Chapman & Bartels, 1940):

n

P°0i) = —

!— ^— I0i2

-l)

 n] (A4)

n

 2r\n!) dpn

Thus,

(A5)

0 n

The quantity in curly brackets here can be Identified with the coefficients of

degree n-1 in Taylor series expansions of the function

(A6)

First expand f(u) in powers of u2-!, and then expand in powers of u. In this

way one finds that

d

 Kp2-OnJ L, =0 .

n-1

n-1

(n»Xn-f)l

. for n odd

-)!1

2 2

= 0 . , f or n even (A7)
Accordingly, (A3) can be written as

ItL

= 27tb2 Z IdL2 -- fa® - (a)

n

*

2

 9° = constant (A8)

n=l,3,5....

 n

-

 1^ b n

Division by 2TT(a/b) followed by time differentiation then leads to the

expression In (1 1) of the text, when the multiplicative constants are expressed

recursively.
Appendix B. A New Unconstrained Field Model. GSFC(IO/84)

The GSFC( 10/84) geomagnetic field model is a spherical harmonic

representation of the earth's field based on data spanning the years 1960

through 1982. It is essentially a remake of G5FC(9/80), Langel et al. 1982,

using an improved MAG5AT data set, additional observatory data and a

time-dependent representation for external field effects. The earth's field is

assumed to be derivable from the gradient of a magnetic scalar potential

N

l n

V(r,e,4>,t) = a £ Z (a)

n+1

 [gm(t)cos m<j> + hm(t)sin m<j>] • Pm(e)

n=lm=0 r n n n

N

2 n

* a £ Z (L)

n

 fqm(t)(cos m<(> + srn(t)sin m<t>]Pm(e) - (B1)

n=lm=0a n n n

where Nj, N2 are truncation levels for the expansions of the internal and

external fields. The Schmidt normalized Internal Gauss coefficients, gn

m

, hn

m

,

for G5FC( 10/84) are presented in Table 4 at epoch 1980 to order and degree Nj

= 13 for the main field and its first time derivative, with N| - 6 and 4,

respectively, for the second and third time derivatives. In contrast to

GSFC(9/80) but in common with the GSFC( 12/83) model, Langel and Estes

(1985), the first degree external terms and corresponding induced internal

terms are modeled as linear functions of Dst, while fixed biases, or anomalies,

are estimated for each observatory in the solution, as described by Langel et al.
(1982).

The observations selected for this model are comprised of POGO and

MAGSAT data plus observatory annual means. The POGO data consist of 35,780

scalar measurements obtained by selecting every other observation from the

time-ordered POGO data set used for GSFC(9/80). These data span the interval

from December 1964 through June 1971. The MAGSAT data consist of the

54,813 corrected, quiet day vector and scalar measurements used in deriving

the GSFCC12/83) model. The annual means data set spanned the years from

1960.5 to 1982.5 for a total of 3196 vector measurements (9588 total

observations from 206 observatories selected out of the NOAA National

Geophysical and Solar Terrestrial Data Center, Release 31 data set). Table 5

lists the number of observatories used in the solution by year, while Table 6

names and locates each observatory together with its anomaly bias from the

solution and its overall data interval used. Stations whose records were not

continuous in time were given unique names for each interval of continuity (e.g.

Baker Lake, Baker Lake II, Baker Lake III, etc) and each was treated as an

independent observatory in the solution. Observatory data were converted from

any three independent scalars (i.e. D,H,B etc) to X,Y,Z components in a geodetic

coordinate system assuming an equatorial radius of 6398.165 km and a

reciprocal flattening of 298.25. Annual averages of Dst were used in

processing annual means data.
The external field coefficients for G5FC( 10/84) are

q, 0= 18.48-0.64Dst(nT) (B2)

q,1 =-1.25 +0.02 Dst(nT) (B3)

s,1 =-3.31 + 0.15Dst(nT) (B4)

The corresponding gj° internal coefficient, including the effects of currents

induced by the time-varying external field, is

g1 ° =-29,992.1 +0.33 q,°(nT) . (B5)

Induced contributions to QJ ', hj' , considered negligible, were not computed.

Table 7 gives the unweighted statistics of the misfit between the satellite and

observatory data used in the model and the model itself. For the observatories,

the statistics are presented both with and without the biases.