iva

-

N86-2CSS7

Oaclas

G3/46 05679

12/3
BUSIMSSAND TECHNOLOGICAL SYSTEMS,

BTS03-85-51/kl

1080

November 1985

GEOMAGNETIC MAIN FIELD MODELING USING

NAGNETOHYDRODYNAMIC CONSTRAINTS

by

Ronald H. Estes

prepared for

NASA/GOODARD SPACE FLIGHT CENTER

Greenbelt, Maryland 20771

under

Contract NAS5-28671

prepared by

BUSINESS AND TECHNOLOGICAL SYSTEMS, INC.

Aerospace Building, Suite 440

10210 Greenbelt Road

Seabrook, Maryland 20706
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Acknowledgement

The work performed under this contract is the result of a joint but

separately funded effort between Edward Benton of the University of Colorado,

Ronald Estes of BTS and Robert Langel at the NASA/Goddard Space Flight

Center. Theoretical aspects of the problem are funded under NASA/Goddard

Contract NAS5-28670 to the University of Colorado under the direction of Dr.

Benton while algorithmic development and computation are funded under NASA/

Goddard Space Flight Center Contract NAS5-28671 to Business and Technological

Systems, Inc. Dr. Langel serves as a co-investigator to the joint project.
BUSINESS ,*VD TECHNOLOGICAL SYSTEMS, L\c

TABLE OF CONTENTS

Acknowledgement

PAGE

1.0 INTRODUCTION 1

2.0 PHYSICAL CONSTRAINTS FOR MAIN FIELD MODELS 5

3.0 BAYESIAN LEAST SQUARES PARAMETER ESTIMATION 8

4.0 EVALUATION OF A FLUX CONSTRAINED GEOMAGNETIC

FIELD MODEL 11

5.0 FUTURE RESEARCH PLANS 15

6.0 REFERENCES 16
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1.0 INTRODUCTION

The objective of the study described in this report is to investigate the influence of physical constraints which may be approximately satisfied by the Earth's liquid core on models of the geomagnetic main field and

its secular variation. A previous report (Estes, 1984) describes the methodology used to incorporate non-linear equations of constraint into the main

field model, and will be referred to as Report I. This report describes the

application of that methodology to the GSFC 12/83 field model (Langel and

Estes, 1985a) to test the frozen-flux hypothesis and the usefulness of incorporating magnetohydrodynamic constraints for obtaining improved geomagnetic

field models.

The geomagnetic field is a highly complex phenomenon with significant spatial and temporal variations. Global satellite surveys such as POGO

and MAGSAT have led to important advances in our understanding of this global

geophysical phenomenon and its applications in geology, tectonophysics, navigation, communications, and many other fields. The geomagnetic field is composed of three major components:

The core field is the major part of the geomagnetic field

and is believed to be caused by fluid motion of conducting

material in the Earth's outer core. The field variations

have large amplitudes, broad spatial features (i.e., wavelengths on the order of 1000 km and longer at the earth's

surface), and significant low-frequency temporal

variation.

The crustal anomaly field forms a relatively small part of

the geomagnetic field, but is significant since it is

caused by geologic structure and the state of crustal

minerals. The spatial variations of the crustal field

occur with characteristic wavelengths ranging up to a few

thousand km.
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The external field is caused by electrical currents in the

ionosphere and magnetosphere. Magnetic field variations

due to the external field occur on all spatial and

temporal scales. The external field accounts for the most

complex and dynamic part of the geomagnetic field.

Models of the main, or core field are required for a number of uses,

including the production of declination charts for navigation, the removal of

background fields from aeromagnetic and ship-borne magnetic surveys, satellite

attitude control and determination, the probing of various fluid motions in

the Earth's core, and calculations of field lines and conjugate points for

ionospheric and magnetospheric studies. The geomagnetic secular variation has

been demonstrated to be correlated with decade fluctuations in the Earth's

rotation, and improved models could lead to better understanding of the nature

of the core/mantle coupling. Moreover, knowledge of the secular variation of

the core field is crucial in tying together regional aeromagnetic and marine

surveys taken at different epochs. The secular variation is one of the few

sources of information regarding the Earth's core, where the internal field

originates.

As indicated above, the anomaly portion of the field arises from

variably magnetized rocks lying between the Earth's surface and the Curie

isotherm (the "magnetic crust"), and is the part of the magnetic field having

direct geologic significance. The magnetic anomaly fields, which are essentially time independent for our purpose, have long been used to study nearsurface geology and structure. They have had an important role in the initial

evidence for sea-floor spreading that led to our present unifying concepts of

plate tectonics and they are routinely used in resource exploration. Models

of the sources of magnetic anomalies are the basis for geologic interpretati on.

The modeling of the main and anomalous fields with high resolution

is of major importance for earth science applications. In order to achieve

this resolution, techniques for developing core and crustal models must be

improved. Separating the core field from the crustal field is a major

problem. Conventional approaches to representing the main field by a
4M> TECHNOLOGICAL SYSTEMS, IMC

truncated spherical harmonic series suffer from aliasing due to the crustal

field above approximately degree and order twelve (Langel and Estes, 1982).

This problem is described in detail by Langel (1985). A consequence of this

is that extrapolation of main field models to the core/mantle boundary (CMB)

is severly contaminated by the attenuated higher harmonic coefficients which

are aliased by crustal influences. New approaches to this problem by Gubbins

(1983, 1984) and Gubbins and Bloxham (1985) using a priori spectral assumptions and by Shure et. al. (1982) using harmonic splines minimizing a smoothness norm for the field at the CMB are important contributions. Currently,

Langel and Estes (1985b) are investigating the separability problem through a

correlated measurement noise weight matrix approach.

In addition to the ambiguity inherent in the separation process

which imposes a fundamental limitation on accuracy, there is further difficulty in resolving the non-static nature of the main field. The main field

changes in time on scales ranging from seconds to hundreds of thousands of

years. Magnetic polarity reversals, for example, can take place over a period

of a few thousand years, while tens to hundreds of thousands of years elapse

between reversals. The interest of our study, however, is on magnetic variations over the scale of several decades and less. The secular variation (SV),

even over those time scales, is less well known than the main field. This is

due primarily to data availability. While the dense global vector magnetic

survey provided by Magsat (Langel et. al, 1980) has permitted an excellent

global "snapshot view" of the main field at epoch 1980, secular variation

models depend principally on the data from the global network of permanent

magnetic observatories which are few in number and poorly distributed. An

additional difficulty is that magnetic features of smaller spatial scale are

probably capable of more rapid temporal variation so that higher degree terms

of the spherical harmonic series representation of the secular variation may

be more important and convergence of the series weaker than that for the

constant part of the main field. The result is that with current methods of

modeling the main field and its secular variation even the large scale components of the geomagnetic field cannot be predicted into the future by more than

a few years beyond the data span which defined the model. Two approaches are

possible to improve the situation; to incorporate some statistical information
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describing the SV process into the model (Gibbs and Estes, 1982, 1984), or to

insert physical constraints provided by the electromagnetism and fluid dynamics of the source region, earth's outer core.

The work presented in this report describes the research accomplished in the second year of a multi-year study of the influence of magnetohydrodynamic constraints from the core on geomagnetic models, which is a joint

effort between Edward Benton of the University of Colorado and Business and

Technological Systems, Inc. Some theoretical aspects of the problem are presented by Benton (1984, 1985) in separate reports. As described in Report I,

the BTS contribution in the first year was devoted to establishing appropriate

geomagnetic data bases, establishing algorithms for incorporating non-linear

constraints into the estimation procedure, and developing and verifying

software based on these algorithms for computation. The demonstration that

geomagnetic observations are consistent with the field being frozen to the

core fluid has great importance for computing properties of the fluid core

velocities, and for constructing improved predictive models. Gubbins (1984)

and Bloxham and Gubbins (1985) have investigated the consistency of the

observed data with frozen-flux constraints by two methods; they developed

models that were forced to satisfy the flux constraints and then examined for

acceptability the resulting misfit to the data used in the model, and they

computed the flux integrals from unconstrained models at different epochs and

compared their differences with formal error estimates from the models at the

core surface. Our approach in this study uses the methodology of Report I to

produce a constrained geomagnetic model based on GSFC (12/83) using the

constraint equations as additional "data" with an associated uncertainty, and

compares the misfit errors for data beyond the time interval defining the

model (prediction errors) with those of the unconstrained model.
BVSIXESS .4ND TECHNOLOGICAL SYSTEMS, Ivc

2.0 PHYSICAL CONSTRAINTS FOR MAIN FIELD MODELS

In the earth's core the geomagnetic field satisfies the induction

equation

B = V x (VXB ) + nV2B (1)

where £ is the vector field, V^ is the flow velocity and n is the

magnetic diffusivity. On the time scale of a few decades, it has been shown

to be a good approximation to ignore magnetic diffusion in the core (Backus,

1968), so that the core fluid is assumed to behave like a perfect conductor

with the magnetic flux frozen into the fluid motion. Moreover, it is a

reasonable first order approximation to treat the mantle as a perfect

insulator. Horizontal components of B_ may be discontinuous across the CMB,

but the radial component, B , must be continuous. Bondi and Gold (1950)

introduced the scalar relation for the absolute unsigned magnetic flux, or

"pole-strength"

M(r, t) = if /J |Br(r, 8, X, t)| r2 sine d9dx (2)

as a measure of the number of field lines which cross the spherical surface of

radius r at time t . They noted that the number of field lines crossing

the top surface of a perfectly conducting core (CMB) is a constant of the

motion. Then

M(b, t) = MQ = constant (3)

is a scalar constraint for the frozen-flux hypothesis, where b is the radius

of the CMB. Voorhies and Benton (1982) show that this constraint appears to

be well satisfied, which provides support for the perfect conducting

core/insulating mantle hypothesis over short time scales. Moreover, Backus

(1968) has shown that the total absolute flux linking the CMB is the sum of

contributions arising from separate "null-flux" patches, which are regions on

the CMB bounded by contours of zero radial field. Denoting the contour of the

i null-flux curve as r. , then
BUSINESS 4-vo TECHNOLOGICAL SYSTEMS, INC.

M,(b, t) = / /_ B (b, e, x, t) b2 sine dedx (4)

i r r

= constant

provide additional scalar integral constraints. Unfortunately, the null

curves r, are non-linear functions of the geomagnetic Gauss coefficients

and, moreover, depend strongly on the truncation level of the spherical

harmonic series for B .

Benton (1984, 1985) has pursued additional physical constraints on

secular variation arising from the coupling of non-dissipative fluid dynamics

to the frozen-flux electromagnet!'sm of earth's core. He shows that the signed

flux through any patch which is bounded by a curve which moves with the fluid

should be conserved in the limit of perfect core conductivity, and examines

possible candidates. He demonstrates that under some assumptions, for

example, lines of absolute vorticity move with the fluid, and the contours on

which the vertical absolute vorticity vanishes move with the fluid. A

consequence of this is that under the stated assumptions (spherical, perfectly

conducting inviscid core free of vertical Lorentz torque at the CMB) the

geomagnetic flux through the northern geographic hemisphere on the core-mantle

boundary (CMB) should be constant. With the assumption that the mantle is an

insulator, the resulting constraint equation

LE = /2ir /J/2 Br(b, 9, <(,, t) b2 sine

n

+

2

is linear in the Gauss coefficients g . Here a is the Earth's radius, b

. nm

is the radius of the CMB, and P is the Scf

nomial of degree n and order 1. Denoting

.

 3nm

is the radius of the CMB, and P is the Schmidt normalized Legendre poly-
BUSINESS AMD TECHNOLOGICAL SYSTEMS, INC

N

 n L£ = I Fn g[J(t) = constant

m=l

we have

(n-2) ,a. p.

TrHT V hn-2

where FQ = 0 and F^ = 1. Note that this represents a constraint equation

for secular variation

which may be readily tested against existing models or easily incorporated as

a linear constraint into a model.

Under the assumptions which established the flux linking the

northern geographic hemisphere as a constant, additional flux constraints may

be established for areas bounded by curves defined by the intersection of the

geographic equator with the null curve along the geomagnetic equator on the

CMB. These constraints, like those due to the frozen flux through the null

curves, are non-linear functions of the Gauss coefficients and must be

implemented numerically.
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3.0 BAYESIAN LEAST SQUARES PARAMETER ESTIMATION

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 represent the random

noise vector with zero mean and covariance matrix R . Then the observation

equation is

y = F(x) + v . (7)

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

y = Ax + v . (8)

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

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

a

x = (AV^ + ft ~

1

)"1 [AV^ + ft^xl . (9)

o a a

This result may be obtained by minimizing the norm

J,c(x ) = (y - Ax)T R'^y - Ax) + (x - xa )T n'^x - xJ (10)

LO a a a

with respect to x , so that 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(xQ) = A(x Q )(x - xQ) + (11)

8
BUSINESS 4/vo TECHNOLOGICAL SYSTEMS, INC

A Gauss Iteration procedure yields the equation for the (n+1)

to the solution estimate where

approximation

l

6xn+i = (AT(xn) R-1A(xn) - xn]j

and

-y - A(xn)

The rate of convergence of the procedure depends on how good the nominal 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 Gubbins, 1984) or

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

constraint equation is represented as

6(x) = g

s

then the (n+l) iteration relation is

(13)

C

T

U IA(x ) — Ri\"

where

Cf Mx ) = — V 8x

(15)
TECHNOLOGICAL SYSTEMS, L\c

The weight matrix R reflects the stiffness of the constraint, where R

c c

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 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, some of the

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

T. 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 must be used to compute the matrix

for each iteration, n = 0, 1, 2, ... , until convergence is attained.

10
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4.0 EVALUATION OF A FLUX CONSTRAINED GEOMAGNETIC FIELD MODEL

To investigate the influence of physical constraints which may be

satisfied by Earth's liquid core on geomagnetic field models, we apply the

non-linear algorithms described in Section 3.0 to a variation of the GSFC

12/83 model (Langel and Estes, 1985a). This model utilized quiet Magsat data

and observatory annual means for a selected set of observatories for the years

1977 through 1982. The internal Gauss coefficients were expanded to degree 13

for the main terms and degree 10 for the secular variation terms. In addition

to the internal Gauss coefficients, GSFC 12/83 solved for an external field of

spherical harmonic degree one, and vector biases for each observatory. Moreover, parameters for linear D . variation in g,0 and the external coefficients are included in the solution. Internal coefficients which were not

determined in the inversion of the data to the 95% confidence level were

forced to be zero in the final solution. Because the incorporation of dynamic

core constraints could improve the observability of such suppressed coefficients, the GSFC 12/83 model was recomputed with all coefficients retained in

the solution and all partial derivatives included in the normal matrix

A R A. This model is designated GSFC 9/84-0 and is summarized in Table 1.

The GSFC 9/84-0 model, with the utilization of high quality Magsat

data, very accurately represents the geomagnetic field at and above the

Earth's surface at epoch 1980.0. Figure 1 shows the radial component of the

field from GSFC 9/84-0 at the core-mantle boundary (CMB) at 1980.0, using all

Gauss coefficients to degree and order thirteen. There are eleven separate

null flux curves resulting from this model evaluation. Because the higher

order Gauss terms, which are less well determined, become more dominant in the

field evaluation at the CMB, the resulting computations are subject to large

uncertainty. Benton, et.al. (1982) and Report I examined the effect of

truncation level on geomagnetic properties at the CMB, and demonstrated

considerable variation of null-curve location and null-patch flux as a

function of degree and order. For this study, we have selected the five

null-curves indicated in Figure 1 for which to impose flux constraints because

these curves qualitatively demonstrated stability of form and location from

truncation level nine through thirteen. Moreover, the consideration of only

11
BUSINESS .AVD TECHNOLOGICAL SYSTEMS,

five null-patches meets the objectives of the study while keeping the

computational burden within reasonable bounds.

The methodology for computing the null-patch flux and the partial

derivatives of the null-patch flux with respect to the model parameters (Gauss

coefficients) in equation (15) is described in Report I. A numerical

derivative is taken for each parameter using a 1° x 1° grid overlay of the

null-patch and varying the nominal Gauss coefficient value. In this

approximation the field is computed at the center of each 1° x 1° cell and

assumed to be constant within the cell. Note that the flux depends on the

constant and secular variation Gauss coefficients both through the field

evaluation and the null-curve boundary, r..

For each of the five null-curves selected, we impose the "data"

M.(b, t) = constant

from equation (4) at the times 1977.5, 1980.0, and 1982.5. Because the GSFC

9/84-0 model accurately represents the field at epoch 1980.0, we used the

1980.0 computed flux value for each null-patch to determine the constants. We

are thus adding fifteen non-linear observations to the GSFC 9/84-0 data set

which will influence the solution in such a way as to attempt to maintain the

constancy of the flux through the five null-curves at the given epochs. The

elements of the diagonal weight matrix R were set to represent an error

-5 measure of 10 MWb for the "observed" flux. This value was chosen

primarily by numerical experimentation to provide a stable inversion of the

normal matrix.

The solution was taken through three iterations using equation (14)

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

iteration one. No a priori information was assumed in the solution for the

nominal model, so that n is zero in equation (14). Each iteration was

<1

performed in two stages. The first stage processed the Magsat and observatory

data set as published for GSFC 12/83 and accumulated the normal matrix

12
BUSINESS ,4VD TECHNOLOGICAL SYSTEMS, L\c

A (xn) R" A(x n ) and the vector A (xn) R~ 6yn< From this information an

"unconstrained" model was generated for the iteration. These models are

denoted GSFC 9/84-1, GSFC 9/84-2, and GSFC 9/84-3 for iterations one, two and

three, respectively. Note that for iteration one, GSFC 9/84-1 is identical to

GSFC 9/84-0 since the nominal model has already been converged using the GSFC

12/83 data set. The second stage for each iteration computed the flux and

flux partial derivatives for the fifteen "observations", formed the matrix

C (xn) R~

 c(*n) and the vector C (xn) R~ 6gn , and added these to the

first stage quantities to obtain the "constrained" solution for the

iteration. The constrained models are denoted GSFC 9/84-1C, GSFC 9/84-2C and

GSFC 9/84-3C for iterations one, two, and three, respectively.

The flux at 1977.5, 1980.0 and 1982.5 and flux rate at 1980.0

through the five null-patches for all of the models through three iterations

are presented in Table 2. The maximum variation of the flux over the 1977.5 -

1982.5 interval for each patch is also given, as well as the RSS of the flux

rate over all five patches. For a perfectly constrained solution, the flux

rate through each null-patch should be zero. There is a clear trend showing a

reduced total flux variation for the five year data interval as the solution

converges. The poorest improvement is with null-curve 2, which has a maximum

variation of 30 MWb for GSFC 9/84-0 and 19 MWb for GSFC 9/84-2C. Marked

improvement is shown for null-curves 3, 4 and 5. The RSS of the flux rate at

1980.0 through all five patches displays a reduction from 16.1 MWb/yr to 5.9

MWb/yr.

The cost function being minimized by the estimation procedure is

J

LS

 = <*l

 R

~

J

 % + 69n

 Rc! 6*n •

The first term represents the weighted sum of squares of the data misfit from

the Magsat and observatory data, which we denote by Q. Table 3 presents the

misfit to the Magsat and observatory data for the iterated solutions. As

expected, the misfit increases with the inclusion of constraint "data". While

the solution has apparently converged with respect to the Magsat data, the

situation is not as clear with respect to the observatory data which is more

13
TECHNOLOGICAL SYSTEMS,

strongly coupled with the constraint "data" through the secular variation.

Table 4 displays the compliance of the iterated solutions to the linear

vorticity constraint given in equation (6).

A stern test for the validity of the null-flux approximation and the

utility of including physical constraints into geomagnetic field models is the

demonstration of improved model predictive capability beyond the data interval

defining the model. Tables 5-8 display the standard deviation of fit to

observatory data for the 91 observatories used in the model on a yearly basis

for B, X, Y, and Z for data from 1975.5 through 1984.5. The observatory

biases recovered with the solutions were included in the model evaluations,

and the standard deviations were computed unweighted. The constrained models,

particularly for X and Z, show a clearly improved predictive capability

beyond the interval 1977.5 - 1982.5 in both directions. Within the resolution

of the plots, the symbols for GSFC 9/84-3 and GSFC 9/84-3C fall directly on

GSFC 9/84-2 and GSFC 9/84-2C, respectively, and consequently are not

displayed. The solid vertical line at 1982.5 is to emphasize that only eight

observatories were used in the solution for that year, so that the statistics

for 1982.5 primarily represent predictive errors. The statistics for 1984.5

are based on a sample of only nine observatories and must be considered

unreliable. The improved model prediction capability demonstrated

statistically in Tables 5 through 8 has been achieved without changing the

mathematical model of the secular variation. Only additional "data" have been

included. This makes a strong statement for the potential usefulness of

including physical constraints in field modeling.

The Gauss coefficient differences between GSFC 9/84-0 and GSFC

9/84-2C are presented in Table 9. As expected, the effect of adding flux

constraints is manifest almost totally in the secular variation coefficients,

with the largest influence principally in the sectorial terms. Differences

between all GSFC 9/84-3C and GSFC 9/84-2C Gauss coefficients are small and

show no obvious trends.

14
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5.0 FUTURE RESEARCH PLANS

The comparison of the predictive capability of constrained and

unconstrained models described in Section 4.0 demonstrates a positive improvement achieved by incorporating physical null flux constraints. This tends to

support the consistency of the hypothesis that the field is frozen to the core

fluid with geomagnetic observations. These results, however, are based on

utilizing a limited number of null flux constraints and a limited time interval overwhich the Gauss coefficients were modeled as linear in time. To complete the analysis begun using the GSFC 12/83 model, a converged constrained

model will be developed which includes the available observatory data to date

(1977.5 - 1984.5) and a complete set of null flux constraints. This model

will be compared with IGRF candidate models for 1985.0.

To more fully examine the influence of physical constraints in

modeling secular variation, a goal for future study will be to model the main

field over a longer time interval utilizing additional null flux constraint

equations, and to examine the predictive error of constrained and unconstrained models. Moreover, examination of Benton's (1985) constraints on secular

variation based on electromagnetism and fluid dynamics considerations offers a

linear and a number of additional non-linear constraints which should be

tested for compliance in unconstrained models and implemented into constrained

models. Alternate representations to polynomials in time for secular variation should also be investigated.

15
BUSINESS AND TECHNOLOGICAL SYSTEMS, INC

6.0 REFERENCES

Backus, G.E. "Kinematics of secular variation in a perfectly

conducting core," Phil. Trans. Roy. Soc. Lond., A263, 239 (1968).

Benton, E.R. "On fluid circulation around null-flux curves as

Earth's core mantle boundary," Geophys. Astrophys. Fluid Dyn., 11,

323 (1979).

Benton, E.R. "Inviscid, frozen-flux velocity components at the top

of Earth's core from magnetic observations at Earth's surface: Part

1. A new methodology," Geophys. Astrophys. Fluid Dyn., 18, 157

(1981).

Benton, E.R. "Constraints on geomagnetic secular variation modeling

from electromagnetism and fluid dynamics of earth's core," First

year report for NASA/Goddard Space Flight Center contract

NAS5-27671, University of Colorado, 1984.

Benton, E.R. "On the coupling of fluid dynamics and electromagnetism

at the top of the earth's core," submitted to Geophysical and

Astrophysical Fluid Dynamics (1985).

Benton, E.R., Estes, R.H., Langel, R.A., and Muth, L.A. "Sensitivity

of selected geomagnetic properties to truncation level of spherical

harmonic expansions," Geophys. Res. Lett., 9, 254 (1982).

Benton, E.R., Muth, L.A., and Stix, M. "Geomagnetic contour maps at

the core mantle boundary," J. Geomag. Geoelec., _31_, 615 (1979).

Bloxham, J. and Gubbins, D. "Geomagnetic field analysis IV: testing

the frozen flux hypothesis," Geophysical Journal (in press, 1985).

Bondi, H. and Gold, T. "On the generation of magnetism by fluid

motion," Mon. Not. R. Astro. Soc.. 110, 607 (1950).

16
BLS/HESS AND TECHNOLOGICAL SYSTEMS, INC

Estes, R.H. "FIT program description and user's guide: Version

8305," Final Report for Contract NAS5-26486, Business and

Technological Systems, Inc., BTS07-83-78, June 1983.

Estes, R.H. "Utilization of magnetohydrodynamlc constraints in

geomagnetic core field modeling," BTS, Inc., Final Report, NASA

Contract NAS5-27672, October 1984.

Gibbs, B.P. and Estes, R.H. "Geomagnetic modeling by optimal recursive filtering," Final report for NASA/Goddard Space Flight Center

contract NAS5-26250, Business and Technological Systems, 1982.

Gibbs, B.P. and Estes, R.H. "Geomagnetic field modeling errors and

optimization of recursive estimation - phase I final report," NSF

grant EAR-8360496, Business and Technological Systems, 1984.

Gubbins, D. "Geomagnetic field analysis - I. Stochastic inersion,"

Geophys. J. Roy. Astro. Soc., _73_, 641 (1983).

Gubbins, D. "Geomagnetic field analysis - II. Secular variation

consistent with a perfectly conducting core," Geophys. J. Roy.

Astro. Soc.. 77, 753 (1984).

Gubbins, D. "Finding core motions from magnetic observations,"

Phil. Trans. Roy. Soc. Lond. A., 306, 247 (1982).

Gubbins, D. and Bloxham, J. "Geomagnetic field analysis III:

magnetic fields on the core mantle boundary," accepted for

publication in the Geophys. J. Roy. Astro. Soc. (1985).

Hide, R. "How to locate the electrically-conducting core of a planet

from external magnetic observations," Nature, 271, 640 (1978).

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BUSINESS AND TECHNOLOGICAL SYSTEMS,

LaBrecque, J.L., Cande, S.C. and Jarrard, R.D. "The intermediate

wavelength magnetic anomaly field of the North Pacific and possible

source distributions," final report for NASA/Goddard Space Flight

Center grant NAS5-25891, Lamont-Doherty Geological Observatory,

1982.

Langel, R.A. "The main field", Chapter 3 of Geomagnetism. J. Jacobs,

editor, Academic Press, in press (1985).

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19
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