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Journal ArticleDOI

Fast reduction of potential fields measured over an uneven surface to a plane surface

01 May 1994-IEEE Transactions on Geoscience and Remote Sensing (IEEE)-Vol. 32, Iss: 3, pp 508-512

TL;DR: The present work is aimed at rapid reduction of the gravity and magnetic fields observed over an uneven surface to a horizontal plane by estimate the Fourier transform of the potential field over an imaginary horizontal plane lying entirely above the ground surface and impose boundary conditions.
Abstract: The present work is aimed at rapid reduction of the gravity and magnetic fields observed over an uneven surface to a horizontal plane. The approach suggested is to estimate the Fourier transform of the potential field over an imaginary horizontal plane lying entirely above the ground surface and impose boundary conditions; namely, the solution must satisfy the observed field over the ground surface and vanish over an infinite hemisphere. The desired Fourier transform is obtained in an iterating manner. A 2D FFT algorithm can considerably reduce the computational burden. The FFT approach cannot be used unless the discrete data is available on a rectangular grid. If the observations are scattered, interpolation to the nearest grid point will have to be carried out. Interpolation introduces marginal increase in the rms error. The iterating approach is about 10 times faster than the least squares approach. >
Topics: Horizontal plane (56%), Fast Fourier transform (55%), Interpolation (55%), Fourier transform (54%), Plane (geometry) (53%)

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508
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING,
VOL 32.
NO
3,
MAY
1994
Fast Reduction
of
Potential Fields Measured Over an
Uneven Surface to a
Plane
Surface
P.
S.
Naidu and M.
P.
Mathew
Abstract-The present work is aimed at rapid reduction of
the gravity and magnetic fields observed over an uneven sur-
face to a horizontal plane. The approach suggested here is to
estimate the Fourier transform
of
the potential field over an
imaginary horizontal plane lying entirely above the ground sur-
face and impose boundary conditions; namely, the solution must
satisfy the observed field over the ground surface and vanish
over an infinite hemisphere. The desired Fourier transform is
obtained in an iterating manner.
A
2-D
FFT algorithm can con-
siderably reduce the computational burden. The FFT approach
cannot be used unless the discrete data is available on a rectan-
gular grid. If the observations are scattered, interpolation to
the nearest grid point will have to be carried out. Interpolation
introduces marginal increase in the rms error. The iterating
approach is about
10
times faster than the least squares ap-
proach.
I.
INTRODUCTION
N
geophysical surveys, and in particular, in gravity and
I
magnetic surveys, the observation stations are often
controlled by topography, accessibility, and the distance
to the base station to which one has to return for frequent
calibration. These factors naturally control the distribu-
tion of the observation stations; mostly along the existing
motorable tracks. In aerial low-altitude mineral surveys
the aircraft is often flown at a constant height (e.g.,
50
m) above the topographic surface. Consequently, the
flight lines, specially in hilly terrain, are undulating in the
vertical plane, though practically parallel in the horizontal
plane. Thus, the observation stations may be considered
as unevenly distributed on an irregular topographic sur-
face or on flight lines lying on an irregular surface parallel
to the topographic surface below. Such observations of
the gravity and magnetic field are inadequate for modern
quantitative methods that require discrete observations
over a square grid on a horizontal plane. The earliest work
on the subject of the continuation of the potential field
measured on a curved line to a horizontal line above the
curve line was by Strakhov and Devitsyn
[l],
who used
the method of successive approximations for the solution
of an integral equation of first order. Their method works
well when the surface undulations are weak. Tsirul’skiy
[2] reduced the problem to solving of an integral equation
Manuscript received November
3
1,
1993.
P.
S.
Naidu is with the Department
of
Electrical Communication Engi-
M.
P.
Mathew is with Airborne Mineral Survey and Exploration, Geo-
IEEE
Log Number
9400439.
neering, Indian Institute
of
Science, Bangalore
560012,
India.
logical Survey
of
India, Bangalore
560001,
India.
of the second kind in the frequency domain. An algorithm
to solve the integral equation by a method of successive
approximation is given in
[3].
Bhattacharya and Chen
[4]
have formulated the problem in terms of the Fredholm
integral equation and gave a series solution in the space
domain
[5].
Hansen and Miyazaki
[6]
modified the method
given in
[4]
for better accuracy. Henderson and Cordell
[7]
have used finite harmonic series expansion of the ob-
served data and then, using the estimated coefficients, they
were able to reduce the field to a plane surface. The
French workers used the Backus and Gilbert inverse prob-
lem formalism for the continuation of the potential field
from an uneven surface to a plane surface
[8],
[9].
Parker
and Klitford
[
101 have used the Schwartz-Christoffel
transformation to map an uneven track of a bottom-towed
magnetometer to a straight line.
The present paper addresses itself to the problem of re-
ducing the field observed over an irregular surface onto a
horizontal plane above the surface of observation. The ap-
proach adopted here is similar to that in
[l]
and [2], but
it
is
valid both for two- and three-dimensional problems
and it uses a fast computational algorithm. The compu-
tations require the repeated application of a
2-D
fast Fou-
rier transform, thus making the task of reduction very fast
compared
to
the space-domain methods described in
[4]-
[lo]
and currently used in industry. However, the major
limitation of the present approach lies in the requirement
that the
x
and
y
coordinates of a measurement station must
lie on a square grid, though the
z
coordinate may take any
arbitrary value (see Fig.
1).
Only then can one exploit the
high speed of the FFT algorithm. From a practical point
of view this does not pose a serious problem as it is a
common practice to generate gridded data from the scat-
tered observations. Using such an approach we have ob-
tained accurate reduction even when the observation sta-
tions were scattered, but a sufficiently large number of
them were available. The present paper is divided into
five sections, In the next section we briefly review the
least-squares inversion method and study its performance
when the available data is randomly scattered. In Section
I11 we describe the theory of an iterating FFT-based
method and in Section IV we provide a few illustrations
as well as a synthetic example. In Section V we provide
a practical example. We shall also compare the accuracy
and speed of the present method with that of the least-
squares method.
0196/2892/94$04.00
0
1994
IEEE

NAIDU
AND
MATHEW:
REDUCTION
OF
FIELDS
OVER
UNEVEN
TO
Plane of
reduction
A
Number
of
2500 2000
observations.
rms
error
(%)
1.29 1.49
CPU
Time (min)
125
105
on
Vax
750.
PLANE
1500
1000
750
600 450
1.78 1.81 3.20 6.10 127
89 62
50
42 33
Fig.
1.
Profile
of
observation stations over
a
hilly terrain.
The
observed
field
is
now reduced to a horizontal plane just above the highest point in
the area
of
survey.
11.
LEAST-SQUARES
INVERSION
We shall assume that all observation points are scat-
tered in a three-dimensional space but that they all lie on
a smooth plane representing a topographic surface. The
data points when projected onto a horizontal plane are
scattered over the plane. Let
f(x,
y,
H)
be the potential
field on a horizontal plane
H
units above the mean obser-
vation surface and
F(u,
v,
H)
be its
2-D
Fourier trans-
form (see Fig.
1).
We shall continue the unknown field
on the horizontal plane downward to every point of ob-
servation,
f(xi,
yi,
H
+
Azi)
.
lrca
nm
d~
dv,
i
=
0,
1,
-
*
*
N
-
1.
(1)
.
e
jWi
+
Uyi)
In above equation the unknown quantity is
F(u,
v,
H),
which may
be
discretized over a finite domain with the
assumption that it vanishes outside this domain. For the
sake of simplicity we shall assume the domain is rectan-
gular,
2a
X
2b,
and it is divided into
(2p
+
1)
x
(2q
+
1)
cells. We shall discretize the integral in
(1)
and write
f@i,
yi,
H
-
Azi)
lP
=-
C
C
F
PQ
k=
-p
I=
-4
where
P
=
2p
+
1
and
Q
=
2q
+
1.
Define following
matrices:
f
=
CO~
{f(~;,
yi,
H
-
Azi),
i
=
1,
2,
3
*
N}
Size:
N
x
1
1
k= -p, -p
+
1
**'p
-
1,p
l=-q,-q+l*..
9-
17q
Size:
N
x
(2p
+
1)(2q
+
1)
SURFACE
509
Table
I
and
k=-p,-p+l.**p-l,p
I=
-9,
-q+
1
**.
4-
1,q
Size:
(2p
+
1)(2q
+
1)
Equation
(2)
in matrix version becomes
f
=
AF.
We
should be able to solve the above system
of
linear equa-
tions for the unknowns contained in vector
F.
Naturally,
we must have
N
1
(2p
+
1)
(2q
+
1);
in the presence of
noise a reliable solution is obtained only when
N
>>
(2p
+
1)
(2q
+
1).
A
least-squares solution of
(2)
is given by
F
=
(AHA)-'AHf
(3)
where the superscript
H
stands for the hermitian trans-
pose. We have evaluated
(3)
on a Vax
750.
The compu-
tation time required for different data sizes is shown in
Table
I.
A.
Numerical
Example
A ground magnetic survey over an area of
140
X
140
units (arbitrary units) over an undulating terrain was sim-
ulated. The field at randomly scattered stations uniformly
covering the entire area was calculated. The magnetic field
is caused by four semiinfinite prisms. A map of the un-
dulating surface including the stations and an outline of
magnetic targets are shown in Fig.
2.
The undulating sur-
face was generated mathematically using the equation
1
X2
[Jx2
+
y2
+
16
A.z(x,
y)
=
4.0
+
2.0
COS
Jx'
+
y2
+
16
y2
+
4
+
sin
-
The height of the observation stations varied from
1.1
to
7.0
units. First, the scattered field measurements were
used for reduction. The least-squares inversion approach
was used to reconstruct the field at a height of
7.5
units,
just above the highest point. The mean-square error be-
tween the reconstructed field and the actual (computed)
field is tabulated in Table I.
The error is expressed in percentage relative to the peak-
to-trough anomaly amplitude of
65.5
gamma. The
rms
er-
ror is found to increase rapidly when the number of sta-
tions is reduced below
600.

I
510
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING,
VOL. 32,
NO.
3,
MAY
1994
Fig.
2.
A
map
of
the undulating surface including the observation stations
(+)
and an outline
of
magnetic targets.
111.
FFT-BASED METHOD
Letfo(x,
y,
Az)
be the potential field (gravity or mag-
netic) observed over a known but arbitrary surface
Az(x,
y).
We shall assume that the
x
and
y
coordinates of each
observation station lie on a regular grid; however, the
z
coordinate is arbitrary. Where this is not true we shall
assume that through interpolation, gridded digital mag-
netic data has been prepared. The data points, when
pro-
jected onto a horizontal plane, fall on a regular grid of
points. Only the height of a data point is irregular. Let
f(x,
y,
h)
be the desired field on a horizontal plane Hunits
above the mean observation surface
[H
>
Az
(x,
y)].
Now
we shall continue the field downward to the observation
surface and require that the downward continued field
must satisfy the observed field
(4)
.
ej(u+vY)
du dv
where
F(u,
u,
h)
is the Fourier transform of the desired
fieldf(x,
y,
h)
and
s
=
v.
Note that the repre-
sentation of the potential field given in
(4)
satisfies the
Laplace equation and the vanishing boundary condition at
+
00.
Next, we express
Az
(x,
y)lh
=
€17
(x,
y)
where
E
<
1
and
r]
(x,
y)
is a normalized surface, that is,
1
xy
-
c
c
$(x,
y)
=
1,
x,
Y
-+
00
4XY
-x
-Y
Use the following expansions:
k=O
k!
IXI
F(u,
u,
h)
=
c
eiFi(u,
u,
h)
i=O
in
(4).
Express the right-hand side
of
(4)
as a polynomial
in
E.
Then,
(4)
becomes
1
h(x,
Y,
Az)
=
47~2
1:
A@,
u,
r](x,
Y))
.
esh
e
l(u
+
VY)
du dv
(5)
r]
(x,
Y)
*s2FI(~,u,h)
+-
1
*
sF2(u,
u,
h)
-
F3(u,
u,
h)
I
+
-
*
-
.
We set the coefficients
of
E
to zero giving us the following
set
of
interconnected equations from which we can obtain
Fk(u,
u,
h),
k
=
0,
1,
2,
-
.
For example,
Fo(u,
v,
h)
=
e-sh
c
XY
cfo(x,
y,
Az)e-i(ux+vy)
(.+
PT
Fo(u,
u,
h)sesheei("x+vy)du du
s-
s,
(6)
and
so
on. The above system
of
iterating equations is eas-
ily evaluated using the FFT algorithm. The convergence
is rapid, usually requiring about
10
iterations, depending
upon the amplitude of the surface undulations.
IV.
COMPUTER SIMULATION
For the purposes of testing the new method of reduction
to
a horizontal surface we have a synthetically generated
field along a set
of
16
nearly parallel lines placed on the

NAIDU AND
MATHEW:
REDUCTION
OF
FIELDS
OVER
UNEVEN
TO
PLANE
SURFACE
511
Fig.
3.
The contoured maps
of
measured field are shown in (a) (parallel
flight lines) and
(b)
(scattered stations). The results
of
field reduction to a
horizontal plane at
7.5
units above using the new method are shown in (c)
and (d). Compare the reduced field with the theoretical field shown in Fig.
4
Fig.
4.
The theoretically computed field on the plane
of
reduction.
undulating surface, as well as at randomly scattered
(2500)
points (see Fig.
2).
The parallel lines would correspond
to undulating flight paths of an aircraft over a hilly terrain.
On each line
160
samples were collected, giving a total
of
2560
samples. The data thus created was gridded using
a commercially available software package (e.g., Data
Plotting Services Inc., Toronto, Ont., Canada). The con-
toured maps are shown in Fig. 3(a) and (b). The result of
field reduction to a horizontal plane at
7.5
units above
using the new method is shown in Fig. 3(c) and (d). The
theoretically computed field on the plane of reduction is
shown in Fig.
4.
Table I1 gives the
rms
difference be-
?.I'
.d
??*
*.'
Fig.
5.
Aeromagnetic field over a hilly terrain flown at
low
altitude (nom-
inal height
50
m above ground). Contour interval is
207.
Table
I1
Parallel
Profiles
Grid data 1.44
tween the reduced and the theoretical field. We note that
in comparison with the least-squares methods there
is
a
marginal increase in the
rms
error; however, there is a
considerable savings in computer time, by a factor of
10
to
12.
The increase in the
rms
error
is
largely due to the
interpolation error. In support of this statement we have
carried out the reduction to a horizontal plane using data
on a regular grid requiring no gridding. From the last row
in Table
I1
we note that the
rms
error has come down and
is
of
the same magnitude as in the least-squares approach.
V.
APPLICATION
TO
REAL
DATA
We have applied our new method of reduction to a real
data set obtained in a low altitude (nominal flight height
50
m) aeromagnetic survey carried over a part of the dia-
mond-bearing rocks, consisting of granites and granite
gneisses belonging to the south Indian shield. The survey
was carried out by the
AMSE
division
of
the Geological
Survey
of
India using the Scintrex system mounted on a
Twin Otter aircraft. In addition to the total magnetic field,
the system provided continuous measurements of baro-
metric height and air column thickness, which were used
I

I
512
IEEE TRANSACTIONS
ON
GEOSCIENCE AND REMOTE SENSING,
VOL.
32,
NO.
3,
MAY
1994
*
Id
50‘
Fig. 6. Reconstructed flight surface from the barometric data along with
the flight paths. Contour interval is 20 m.
Fig. 7. Total field reduced to a horizontal plane at a height of 678 m above
MSL, The plane of reduction
is
just above the highest point in the survey
area.
to reconstruct the actual flight surface. The navigation
system consisted of Doppler as well as a visual method
based on aerial photostrips. It is believed that the posi-
tional accuracy is better than
5
m. The total measured
field after the application of the diurnal and instrumental
drift corrections and subtracting a base level of
41
loOy
is
shown in Fig.
5
and the reconstructed flight surface is
shown in Fig.
6.
On the flight surface we have also su-
perposed actual flight lines. The contour values are in me-
ters, representing the height above
MSL
plus
297
m. In
Fig.
7
we show the reduced field, which now corresponds
to what would have been observed on a horizontal plane
at a height of
678
m above
MSL
or
381
m above the local
base level.
ACKNOWLEDGMENT
We wish to thank the Director-General
of
the Geolog-
ical Survey
of
India, Calcutta, for permission to use their
aeromagnetic data. We also wish to thank
M.
R. Nair,
Deputy DG,
AMSE,
for his interest in this work and
M.
V.
Ramanmurthy, R. Sharma, Dr.
S.
V.
Anand,
P.
V.
Ani1 Kumar,
M.
Singh, and
B.
Singh, all from
AMSE,
for their help in preparing the maps.
REFERENCES
[l]
V. N. Strakhov and V. M. Devitsyn, “The reduction of observed
values of a potential field to values at a constant level,”
Bull.
(Izv.)
Acud.
Sci.
(I.S.S.R.,
Earth Physics, no. 4, pp. 256-261, 1965.
[2] A. V. Tsirul’skiy, “On the reduction of observed values of potential
fields to a single level,”
Bull.
(Izv.)
Acud.
Sci.
U.S.S.
R.,
Earth Phys-
ics, no.
3,
1968.
[3] A. V. Tsirul’skiy and L. Ya. Ospishcheva, “An algorithm for the
reduction of observed potential field values to a common level,”
Bull.
(Izv.)
Acud.
Sci.
U.S.S.R.,
Earth Physics, no. 4, pp. 266-270, 1968.
[4] B.
K.
Bhattacharya and
K.
C. Chan, “Reduction of magnetic and
gravity data on an arbitrary surface acquired in a region of high to-
pographic relief,”
Geophys.,
vol. 42, pp. 141 1-1430, 1977.
[5] J. C. Wynn and B. K. Bhattacharya, “Reduction of terrain induced
aeromagnetic anomalies by parallel surface continuation: A case his-
tory from southern San Juan mountains of Colorado,”
Geophys.,
vol.
42, pp. 1431-1449, 1977.
[6]
R.
0.
Hansen and Y. Miyazaki, “Continuation of potential fields be-
tween arbitrary surfaces,”
Geophys.,
vol. 49, pp. 787-795, 1984.
[7] R.
G.
Henderson and L. Cordell, “Reduction of unevenly spaced
potential field data to a horizontal plane by means of finite harmonic
series,”
Geophys.,
vol. 36, pp. 856-866, 1971.
[8]
V. Courtillot,
J.
Ducruix, and
J.
L.
Le Mouel, “Le prolongement
d’un champ de potentiel d’un contour quelconque sur un contour hor-
izontal: Une application del la method de Backus et Gilbert,”
Ann.
Geophys.,
vol. 29, pp. 361-366, 1973.
[9]
J.
Ducruix,
J.
L. Le Mouel, and V. Courtillot, “Continuation of three
dimensional potential fields measured on an uneven surface,”
Geo-
phys.
J.
R.
Asrr.
Soc.,
vol. 38, pp. 299-314, 1974.
[lo]
R.
L.
Parker and
K.
D. Klitgord, “Magnetic upward continuation
from an uneven track,”
Geophys.,
vol. 37, pp. 662-668, 1972.
P.
S.
Naidu
was born in 1937. He received the
M.S.
degree from the Indian Institute of Technol-
ogy, Kharagpur, and the Ph.D. degree from the
University of British Columbia, Vancouver, Can-
ada.
Currently, he is a Professor of Electrical Com-
munication Engineering at the Indian Institute
of
Science, Bangalore. He has published over sev-
enty research papers in geophysical and under-
water signal processing.
Dr.
Naidu was a reciuient
of
a Humboldt Fel-
lowship (1979) and an NRC Senior Associate (1988).
M.
P.
Mathew
was born in 1947. He received the
M.S. degree in mathematics and statistics from the
Kerala University.
He joined the Geological Survey of India in
1970 and performed gravity and magnetic
sur-
veys. Currently, he is with the Airborne Mineral
Survey and Exploration wing of the
GSI
where he
is engaged in aeromagnetic data processing.
Citations
More filters

Journal ArticleDOI
Prabhakar S. Naidu1, M. P. Mathew2Institutions (2)
Abstract: In correlation filtering we attempt to remove that component of the aeromagnetic field which is closely related to the topography The magnetization vector is assumed to be spatially variable, but it can be successively estimated under the additional assumption that the magnetic component due to topography is uncorrelated with the magnetic signal of deeper origin The correlation filtering was tested against a synthetic example The filtered field compares very well with the known signal of deeper origin We have also applied this method to real data from the south Indian shield It is demonstrated that the performance of the correlation filtering is superior in situations where the direction of magnetization is variable, for example, where the remnant magnetization is dominant

5 citations


Journal ArticleDOI
Mark Dransfield1Institutions (1)
Abstract: Accurate terrain corrections are important for all gravity surveying. In airborne surveys, a digital model of the terrain is constructed and terrain corrections are calculated at each airborne measurement point. Airborne gravity gradiometry is of high spatial resolution and is particularly sensitive to nearby topographic variations, placing particular requirements on the terrain corrections. A combination of mathematical analysis and simulation studies has led to quantification of the requirements: current on-shore, low-level gradiometer surveys require sub-metre accuracy in navigation and in digital terrain model heights; cell sizes (and therefore also topographic sampling) in the terrain model should be about one-third of the ground clearance. The choice of terrain correction density depends on the application and it is important that the interpreter of the corrected gravity data has the ability to test the impact of changes in this density. Accurate calculation of the gravity gradient field at the airborne sampling points may be achieved by a wide variety of either spatial or harmonic domain methods. Calculation in the harmonic domain is fast but assumes the data represent a periodic function on a planar surface. Padding methods for periodic extension and piecewise continuation away from a plane both add error and slow the calculation. Spatial domain methods are slower but can be sped up by the use of various approximations. In both cases, a clear understanding of accuracy requirements is essential for making an appropriate tradeoff between accuracy and speed.

1 citations


References
More filters

Journal ArticleDOI
Abstract: The problem of reduction of magnetic and gravity data, when observed on an arbitrary surface in a region of high topographic relief, is studied with equivalent source representation at the points of observation. It is shown that the analytical relationship between the total magnetic field or the gravity effect and equivalent magnetization or density on an arbitrary observational surface is given by a Fredholm integral equation of the second kind. A rapidly convergent iterative scheme is described for the solution of the integral equation, yielding the surface distribution of magnetization or density. With this distribution, the field at any other surface can be easily computed. Then it has been demonstrated with model examples that the gravity or magnetic field observed on a rough terrain can be accurately reduced to a horizontal plane for processing and interpretation.A new method has been suggested for minimization of terrain-induced anomalies on a magnetic or gravity map. This method is based on the concept that when the anomalous field observed on an arbitrary surface is continued to a surface parallel to the topography, the terrain effect in the continued field is sharply reduced relative to the field created by bodies of finite extent in the crust. Model examples are presented to show the accuracy and reliability of the method.

79 citations


Journal ArticleDOI
Abstract: An equivalent source algorithm is described for continuing either one‐ or two‐dimensional potential fields between arbitrary surfaces. In the two‐dimensional case, the dipole surface is approximated as a set of plane faces with constant moments over each face. In the one‐dimensional case, the plane faces of the dipole surface reduce to straight line segments. Application of the algorithm to model and field examples of aeromagnetic data shows the method to be effective and accurate even when the terrain has strong topographic relief and is composed of highly magnetic volcanic rocks.

75 citations


"Fast reduction of potential fields ..." refers methods in this paper

  • ...Hansen and Miyazaki [ 6 ] modified the method given in [4] for better accuracy....

    [...]


Journal ArticleDOI
Abstract: Conventional reductions of gravity and magnetic data do not lead to values that are effectively on the same horizontal plane, although it is common practice to regard them so. In regions of high topographic relief, failure to take into account local differences in vertical gradients can result in appreciable error. In this study a method is developed for reducing to a common level gravity or magnetic anomaly data observed at unevenly spaced stations at various elevations above a reference plane. The reduction is effected by means of finite harmonic series approximations in which the coefficients are determined by matrix methods and least squares. Traditional Fourier methods are not applicable because uneven station spacing and relative vertical displacement of stations preclude the use of the orthogonality properties of the trigonometric functions. The number of terms required to represent the data adequately is discussed in terms of “cutoff” wavenumbers empirically determined from residual variance estim...

53 citations


"Fast reduction of potential fields ..." refers methods in this paper

  • ...Henderson and Cordell [ 7 ] have used finite harmonic series expansion of the observed data and then, using the estimated coefficients, they were able to reduce the field to a plane surface....

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Journal ArticleDOI
Abstract: A new method for continuing two‐dimensional potential data upward from an uneven track is developed with special emphasis on solving a particular practical problem, that of magnetic data taken near the bottom of the ocean. The method is based on the use of the Schwarz‐Christoffel transformation, which maps the original, irregular track into a horizontal straight line. It has been found to be very fast computationally and to suffer none of the restrictions found in some earlier two‐dimensional algorithms.

48 citations


Journal ArticleDOI
Abstract: Summary This paper describes a method for the continuation of three-dimensional potential fields measured on an uneven surface; this method is based on a technique proposed earlier by the authors : first we represent potential functions as a sum of elementary interpolating functions; then an inversion technique gives the continued field as a very simple linear combination of the observed values. The method is of interest in several geophysical problems: for example, aeromagnetic surveys made at different altitudes can be joined with no edge effect. This case and others, both theoretical and real, are presented in the paper.

25 citations


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20101
19941