Computers & Geos~ten~e~ Vol 15, No 6, pp 889-903. 1989
0098-301M,89 $3 00 + 0 00
Pergamon Press plc
Printed m Great Britain
A COMPUTER PROGRAM TO PERFORM THE UPWARD
CONTINUATION OF POTENTIAL FIELD DATA BETWEEN
ARBITRARY SURFACES
MARCELLOCIMI\~,LE* and MARIA"~OLODDO
D~part~mento dl Geologm e Geofistca, Campus Unlvers~tarto. 70100 Bart, Italy
(Received 27 January 1988, revtsed 10 4ugust 1988, recetved for publ, canon 26 January 1989)
Abstract--A FORTRAN 77 computer program has been developed for performing the upward contmuatton of potenttal field data between general surfaces The mathemaucal background ~s based on an
eqmvalent source algorithm and on the consequent Fredholm integral equation of the second kind. A
graphical test to illustrate the performance of the program demonstrates the program's abthty to reduce
s~gmficant topographic interference with the signal
Key Words" Potential field data, Equwalent source d~stnbutton, Arbttrary surfaces, Upward contmuauon
INTRODUCTION
The upward continuaUon of potential field data Is
used widely by geophysictsts for improved resoluuon
of the magnetic or gravity signal associated with extended and deep bodies, especially when a masking
effect of shallower sources is considerable. At present
this transformation is performed by computer programs based on the solution of Dirichlet's problem
(Baranov, 1975) with the condition that measurements are collected on a horizontal plane (Gibert and
Galdeano, 1985). Gravity or magnetic data are acquired on an uneven topographic surface which requires a more complex solution.
Moreover, if topography is characterized by a high
relief or by abrupt variations in altitude, some spunous anomalies or s~gnificant distoruons ma~ be
generated in the useful signal.
Many authors have faced this problem following
different approaches (Parker and Klitgord, 1972;
Syberg, 1972; Courtillot, Ducruix, and Le Mouel,
1974; Le Mouel, Courtillot, and Ducrmx, 1975, Hansen and Myazaki, 1984).
The computer program described here and based
on the approach suggested by Bhattacharyya and
Chart (1977) fully solves the problem and allows the
execution of the upward continuation between arbttrary surfaces.
MATHEMATICAL BACKGROUND
In the article of Bhattacharyya and Chan (1977),
the problem of reduction of potential field data observed on an arbitrary surface S is stud~ed with equivalent source representation at the points of observation. It is shown that the analytical relationship between the total magnetic field and the equivalent magnetization distribution on S is given by the following
Fredholm integral equation of the second kind.
*Present address. Dipartimento di Sclenze delia Terra, Unwerslt6 della Calabria, 87030 Cosenze, Italy
CAOl~'O 15o6-D
T~(~', fl', ~') =
×
2n/~(~', if, ~') - ff I~(~, ~. ~,)
-f,(zt' - ~ t ) - f , ( f l ' - fl) + (z' - z)~ ~,,
:
+
+ (< : ,71
o o. (1)
where 7", is the observed total field over S, p is the
spatial gradient of magneuzation m the direction of
the magnetic field, S' is the arbitrary surface S without
the point (~t', fl', z'), (~t, fl, 3) is a point on S'. By
representing the vertical height ~ of these points by a
functtonfso that z = f(~, fl),f, andf~ are the derivauves off(~t, fl) along :t and fl axes respectively.
In a similar manner the Fredholm integral equation that relates the gravity field to the equivalent
density is
g,(zt', fl', ~') = 2nn'o'(~t,
'
' fl', r') - I I o'(zt, fl, 1:)
T - T
x [(~' _ a)" + (fl' -- fl)" + (z' - T)2]32dztdfl (2)
where g, and a are the observed gravity field and the
equivalent density distribution over S respectively and
n = 11[1 + f2 + f~]l 2 is the direction cosine of the z
axis w~th respect to the unit vector normal to the
surface at the point (~t, fl, 3).
After determining the value of/z or ~ by resolving
Equations (1) or (2) respectively it is possible to compute the potential field T or g at points (x, y, z) on a
surface external to S.
The following double integrauon allows the upward continuation in the magnetic situation:
TCx, y, z) =
- f~ /~C~, fl, r)
$
-f,(x
~
~) -
' -
~) + (: -
~)~
~
890
M CIMINALEand M LODDO
Inpu
t
da t a
reading
and
ex tr aPol
a rio
n
Derivatives
dlrec
and
t/on
compu
cosines
ta tton
Equivalent
MAIN
sources
computation
PROGR
AM
ex
trapol
and
a tlon
Upward
con
ttnua
Special
tton
CaSe
Figure I. Simphfied structure chart of program UPWARD.F.
If gravity data are processed the equivalent equation is:
g(x, y , : )
=
-
ffJ a(~t, fl, 3)
Z--'f
×
[(x - ~)2 + (y _ #), + (z - ~)2]3/2"d'~d/~" (4)
It can be seen that in the special situation when the
surface S is horizontal, Equations (3) and (4) can be
reduced to the solution of Dirichlet's integral. This is
easy to see if it is kept in mind that in this situation
f , = fp -- 0 , 3" - r - - 0 and the quantities tt or o
obtained from (1) or (2) are simply the observed field
divived by the constant 2n.
Finally, it should be noted that all the equations
have been derived by assuming a rectangular Cartesian coordinate system (,,, p, 3) with ~ axis positive
downward.
NUMERICAL PROCEDURE
It is assumed that potential field data collected
over the surface S are distributed on a regular grid
having square meshes. At the nodes of this grid the
topographic level also is known. If not, the data must
be gridded using an appropriate gridding algorithm.
To solve Equations (1) or (2) the numerical iterative
method of successive approximations has been adopted. At present, the program here proposed attains
only the first-order approximation. It is easy to extend
the iterations until the difference between two successive approximations lies within an acceptable limit;
nevertheless, in many synthetic and practical situations the first-order approximation has been considered satisfactory.
Considerations made by Bhattacharyya and Chan
are useful for determining the limits of integration.
Because of the particular properties of the kernel
Upward continuation of potential field data
891
Figure 2 Topographic model. Vertical axis is posttwe dov,nward. Contour interval ts 100 m
functions, it is possible to center on the fixed point (~',
/~', 3') a rectangular window of dimensions (2, ideeal+ 1,2 .]deeal+ I) covering an area smaller than S
in which the kernel functions rapidly converge. The
double integrations (3) and (4) have been solved by a
bicubic spline quadrature using as domain of integration the same rectangular window now centered on
the fixed point (,,, ~, 3). To avoid a loss ofinformation
at the edges of the surveyed area, a problem that is
encountered whenever data are transformed, a suitable method for extrapolating has been employed. It
is based on the recursive filters derived from Burg's
Maximum Entropy Spectral Analysis algorithm
(Burg, 1975) and it has been used by the authors
(Ciminale and others, 1985) with good results. This
algorithm has been examined by Gibert and Galdeano (1985) who also present the subroutines that
perform the extrapolation. In addition to topography
and potential field data, the matrix of equivalent sources must be extrapolated [see Eqs. (3) and (4)].
In order to obtain a matrix containing data of
continued field with dimensions equal to those of the
matrix with original data the extrapolated rows, both
on the top and on the bottom of the matrix, will be
equal to idecai, whereas jdecal will be the number of
extrapolated columns both on the left and on the right
of the matrix. In Equations (1)-(3) the derivatives of
the vertical height r = f(~, fl,) occur along • and fl
axes; they have been computed by evaluating the
coefficients of a bicubic spline at the nodes of the
regular grid where the topographic elevations are
known.
PROGRAM DESCRIPTION
The program here presented is named UPWARD.F. It runs on an IBM 6150/25 microcomputer
provided with a RAM memory of 4 Mbyte and with
a VRM program which allows the usage, of 16 Mbyte
as virtual memory. It is written in IBM RT PC FORT R A N 77 which differs slightly from ANSI Standard
F O R T R A N 77. All the computauon is executed in
double precision.
Some IMSL Inc. routines have been used (IMSL,
1984). They are discussed briefly (see also the Appendix). More details about their purpose and usage and
the description of their arguments are available in the
IMSL's library manual (1984).
892
M CIMINALEand M Loooo
N
O
2
A
8km
~
J
Figure 3. Total magnetm field resulting from three pnsmauc bodies computed on topographic surface
shown m Figure 2. Outhncs of boches arc shown. Contour interval is 2 nT
As shown in Figure 1 U P W A R D . F is divided in
four parts: (i) input data reading and extrapolation;
(ii) derivatives and direction cosines computation; (iii)
equivalent sources computation and extrapolation,
and (iv) upward continuation. The special situation
referring to the upward continuation between horizontal planes is treated separately.
Input data reading and extrapolation
U P W A R D . F is not a conversational program.
The input/output parameters required for the computatfon are placed in a file termed pilotupw which is
easy to handle. Some of these parameters are explained briefly (for a full description see the Appendix. namel, name2, and name3 indicate the topographic data file, the potential field (gravity or magnetic) data file and the upward continued field (output
file) respectively; deltaz is either the altitude of the
horizontal plane where potential field data must be
continued or the terrain clearance.
All data are organized in a regular square grid and
stored columnwise. In the special situation when the
surface S is horizontal instead of namel, the uniform
hetght of S qdepar must be introduced in pilotupw. If
it ts required to compute the potential field on an
uneven surface external to S, deitaz must be substituted by a file name4 containing this surface. Once
piiotupw is closed, the actual reading of the input data
(topography and potential field) and their extrapolation is executed by employing the subroutines lectur
and jupe2 from Glbert and Galdeano (1985). They
have been mo&fied slightly to allow reading in a
formatted file using F O R T R A N 77 language and
computation in double precision.
Derivattves and direction cosines computatton
This is the second part of the program UPWARD.F. In this section the derivatives off(,,, r)
along :t and fl axes are calculated. This step is performed by the subroutine ealcder. In reality the actual
computation ts carried out by the IMSL routine i k ccu. This subroutine allows the computation of the
bicublc sphne two-dimensional coefficients. At the
nodes of the grid, the values of the bicubic spline are
equal to those of the topographm elevations and thus
893
Upward continuation of potential field data
Table I Character~ of prtsm~
N
T(km)
8(km)
M(A m)
De
D;
1
2
3
20
30
15
70
70
60
I
2
I
45°W
5°E
10°E
30~'N
72°N
60°N
N = prism numerahon, T = depth o f the top, B = depth o f the
bottom, M = intensity o f magnetization, De ~ charactenshc dechnatlon. Dt = eharacterlst~c inclination
some of its coefficients represent the f~ andf~ functions
respectwely
If gravity data are processed the subroutine ealcdir
allows the computation of direction cosines n starting
from the relationship which links n to the ~ and 13
derivatives (see section on mathematical background).
of singular points ~s present m fredh (see the Appendix): it has been mtroduced because the double mtegrauon Is extended over S' which is the surface S
wtthout the point (:c',/3', r')
In order to avoid a loss of mformatmn at the edges
of the continued field because of the double integration m Equations (3) and (4), the equivalent sources
obtamed by fredh must be extrapolated. Instead of
lectur the subroutine rtest is used to border the grid
having actual values w~th fictitious values (test value);
this operation Is required for prepanng the mput to
jupe2.
Upward contmuatwn
The regular grid defining the surface over which
the field will be continued has to be created or read
before performing the upward continuation.
Equivalent sources computation and extrapolatton
Thts operation foresees three situations: (a) a horIn this third part, Equations (1) and (2) are solved tzontal plane having a level deltaz, created by the
by the method of successive approximations; only the ' subroutine plane; (b) a surface parallel to S (in this
first-order approximation ~s attained.
s~tuatmn deltaz is the terrain clearance), created by
The routine which performs this step ts termed the subroutine topar; and (c) a given topographic
fredh; a nonexecutable statement for the eliminauon relief higher than S read by lectur. The upward con-
/
0
2
A
/
\
8;
Figure 4 Upward continuation to le~,el of 4200 m abo~e mean seale~el of total magneuc field shown m
Figure 3 Contour inter~,al ts 2 nT
894
M CIMINALEand M. LODDO
/
l'q
,t
0
!
2
F~gure 5. Total magneuc field resulting from three pnsmat=c bodies shown m Figure 3 computed at level
of 4200 m above mean sealevel. Contour interval ts 2 nT.
tinuation ~s executed by the subroutine reduc; the
integrating function is evaluated at every node of the
grid and the limits of integration are determined. For
the actual computation reduc calls the IMSL routine
dbcqdu. This routine evaluates the approximate
double integral for the given table of data using a
natural bicubic spline interpolant.
Once computed, the output matrix farr(i, j) is
stored by the subroutine store.
Special situation
In the special situation when the surface S is horizontal the computation is faster. There is no reading
of topography and therefore leetur, jupe2, calcder, or
calcdir are not called. To evaluate the equivalent sources, it is not necessary to call fredh; this operation is
performed easily by the subroutine dirieh. This subroutine first creates the matrix of the uniform topography, then reads potential field data using lectur
and finally gives out the extrapolated matrix of equivalent sources by calling jupe2.
In this situation there is no analytic difference m
upward continuing gravity or magnetic data (see sec-
tlon on mathematical background). Therefore, in
order to avoid unnecessary calculations of matrices f,
and f# (in this situation all the elements are equal 0) a
control statement in the subroutine reduc directs the
program to follow the gravity option.
CONCLUSIONS
The most interesting and practical situation ts the
upward continuation to a horizontal plane of potential field data acquired on an uneven surface that has
been processed for magnetic data. The magnetic data
are stored in a 33 x 33 regular grid having a sampling
interval equal to 1000m; topography is prepared in
the same way. A window (21,21) has been used for
extrapolating and therefore idecal ~ecal----10.
Figure 2 shows the high topographic relief processed. It concerns the well known Mt Etna (Sicily).
Figure 3 shows the total magnetic field (computed
on the topographic surface) which is the result of three
prismatic bodies. It was calculated by assuming a
normal magnetic field having an inclination
Io = 65°N and a declination Do = 20°E. The par-
Upward continuation of potentml field data
ameters which identify the prisms are reported in
Table i. This synthetic field was upward continued to
an altitude o f 4200m above mean sealevel (Fig. 4).
Figure 5 shows the total field which is the result of the
same three prismatic bodies computed at a horizontal
plane 4200 m above mean sealevel. A comparison of
Figures 4 a n d 5 confirms the high performance of
program U P W A R D . F .
Acknowledgments--This work was supported financially by
MPI (40%/87) and CNR (Contract No, 87.1085.02)
REFERENCES
Baranov, W., 1975, Potential fields and their transformations in applied geophysics: Geoexploration Mono,
Series 6, Gebruder Borntraeger, Stuttgart, 121 p.
Bhattacharyya, B. K., and Chan, K. C., 1977, Reduction of
magnetic and gravity data on an arbitrary surface acquired in a region of high topographic relief: Geophysics,
v. 42, no. 7, p. 141 !-1430.
Burg, J. P., 1975, Maximum entropy spectral analysis: unpubl doctoral dissertation, Stanford Univ., 168 p.
Ciminale, M., Galdeano, A , Gibert, D., Loddo, M., Peeon-
895
m. G., and Zlto. G , 1985, Magnetic survey m the Campldano Graben (Sardinia, Italy) description and ,nterpretatton. Boll Geot'. Teor Appl, v 27 no 107, p.
221-235
Courtdlot. V, Ducrulx, J, and Le Mouel, J. L.. 1974. A
solution of some reverse problems in geomagnetism and
gravimetry Jour Geophys. Res., v. 79, no. 32, p. 49334940.
GIbert, D., and Galdeano, A., 1985, A computer program to
perform transformations ofgravimetnc and aeromagnettc survey. Computers & Geosctences, v I1, no. 5, p
553-588
Hansen, R. O . and Mlyazakl. Y, 1984, Continuauon of
potentml fields between arbitrary surfaces: Geophysics,
v. 49, no. 6, p. 787-795.
IMSL's hbrary manual, 1984, FORTRAN subroutines for
mathematics and statistics: IMSL Inc Houston, Texas,
unpaginated.
Le Mouel, J L, Courttllot, V., and Ducruix, J., 1975, A
solution of some problems m potenual theory: Geophys.
Jour. Roy Astr. Soc, v. 42, no. I, p. 251-272.
Parker, R L.. and Khtgord, K. D., 1972, MagneUc upward
continuation from an uneven track. Geophysics, v. 37,
no 4, p 662-668.
Syberg, F. J. R., 197l Potential field contmuauon between
general surfaces: Geophys Prosp, v. 20, no I, p 267282
APPENDIX
Program Listing
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U P W A R D
. F
This program, w r i t t e n on a IBM 6150/25 computer,
p e r f o r m s the u p w a r d c o n t i n u a t i o n of p o t e n t i a l
field d a t a b e t w e e n a r b i t r a r y s u r f a c e s
Principal
.....
.....
.....
parameters description:
nlig ........
noel ........
idecal ......
number o f l i n e s
number o f c o l u m n s
number o f e x t r a p o l a t e d
lines
b o t h on t h e t o p
and on the b o t t o m of the m a t r i x
..... Jdecal ...... n u m b e r of e x t r a p o l a t e d c o l u m n s b o t h on t h e left
and on the right of the m a t r i x
..... ioptl - l . . . m a g n e t i c case
- 2 . . . g r a v i t y case
..... iopt2 - l...the surface w h e r e p o t e n t i a l field d a t a are c o n t i n u e d
is • h o r i z o n t a l p l a n s
- 2.,.the surface where potential field data are continued
is p a r a l l e l to t h e s u r f a c e w h e r e d a t a ere a c q u i r e d
- 3 . . . t h e surface w h e r e p o t e n t i a l field d a t a are c o n t i n u e d
is a g i v e n relief
..... iopt3 - l . . . t h s s u r f a c e w h e r e p o t e n t i a l field data ere a c q u i r e d
is a h o r i z o n t a l plane
- 2 . . . t h e surface w h e r e p o t e n t i a l field data are acquired
is a g i v e n relief
..... step ........ s a m p l i n g interval b e t w e e n lines and c o l u m n s
..... d e l t a z ...... a l t i t u d e of the h o r i z o n t a l p l a n e w h e r e p o t e n t i a l
field data are c o n t i n u e d ( if iopt2 - i )
.... t e r r a i n c l e a r a n c e ( if i o p t 2 - 2 )
. . . . . q d e p a r ...... a l t i t u d e of t h e h o r i z o n t a l p l a n s w h e r e p o t e n t i a l
field
data are acquired
..... p i l o t u p w .... p i l o t i n g filename
..... namel ....... t o p o g r a p h i c data filename
( if iopt3
- 2 )
896
M CIMtNALEand M LODDO
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..... name2 ....... magnetic or gravity data filename
..... name3 ....... output data filename
..... name4 ....... topographic data filename ( if iopt2 - 3 )
..... iform ....... input output data format
..... lunit0 ...... FortTan unit number of the file - 'pilotupw'
..... lunitl ...... Fortran unit number of the file - name1
..... lunit2 ...... Fortran unit number of the file - name2
..... 1unit3 ...... Fortran unit number of the file - name3
..... lunit4 ...... Fortran unit number of the file - name4
..... lunit5 ...... Fortran unit number of the screen
Remarks:
a) In this program, a rectangular cartesian coordinate system
with x and y axes along the geographic north and east
directions respectlvely and the z axis pointing vertically
downward was selected.
b) Generally the dimensions of arrays are:
top((nlig+2*idecal)*(ncol+2*jdecal))
f((nlig+2*idecal)*(ncol+2*Jdecal))
farr((nlig+2*idecal)*(ncol+2*Jdeoal))
cosdir((nlig+2*idecal)*(ncol+2*jdecal))
sigma((nlig+2*idecal)*(nool+2*Jdecal))
sigmal((nlig+2*idecal)*(ncol+2*Jdecal))
sigma2((nlig+2*idscal)*(ncol+2*Jdecal))
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zarr((nlig+2*idecal)*(ncol+2*jdecal))
derx((nlig+2*idecal)*(ncol+2*jdecal))
dery((nlig+2*idecal)*(ncol+2*Jdecal))
aux(ncol+2*Jdecal)
x(nlig+2*idecal)
y(ncol+2*jdecal)
xx(2*idecal+l)
yy(2*jdecal+l)
wk(2*(nlig+2*idecal)*(ncol+2*Jdecal)+2*max(nlig+2*idecal,ncol+2
*Jdecal))
wkl((nlig+2*idecal)*(ncol+2*jdecal))
wk2((ncol+2*jdecal+5)*(nlig+2*idecal)+(ncol+2*Jdecal)-l+
max(5*(nlig+2*idecal)-4,(ncol+2*Jdecal)-2))
wk3((nlig+2*idecal)*(ncol+2*jdecal))
c(2*(nlig+2*idecal)*2*(ncol+2*Jdecal))
implicit double precision (a-h,o-z)
dimension top(40000),f(4OO00),aux(200)
dimension farr(40000),zarr(40000)
dimension x(200),y(200),xx(200),yy(200)
dimension sigma(40000),slgmai(40OO0),sigma2(40000)
dimension c(160000),wk(804OO),wkl(40000),wk2(80000),wk3(40000)
dimension derx(4OOOO),dery(40000),coedir(40000)
character*80 namel,name2,name3,nams4
character*80 iform
common /radl/idecal,Jdecal
common /rad/test
common /range/mi,mf,li,lf
equivalence ( w k ( 1 ) , d e r x ( 1 ) ) , ( w k ( 4 0 0 0 1 ) , d e r y ( 1 ) )
equivalence (sigma(1),siqma2(1),c(1))
equivalence (sigmal(1),c(40001))
equivalence (cosdir(1),c(80001),wk1(1))
equivalence (zarr(1),c(120001))
equivalence (wk2(1),c(40001))
equivalence (f(1),farr(1))
data xmin/O.d0/
data ymin/0.d0/
data test/l.d+4/
lunitO-lO
lunit1-11
lunit2-12
lunit3-13
lunit4-14
lunltS-6
***
piloting file reading
open(lunit0,status-'old',file-'pilotupw')
rewind(lunitO)
read(lunitO,*)nlig,ncol,idecal,Jdecal,step,ioptl,iopt2,iopt3
if(iopt3.eq.1)then
read(lunit0,*)qdepar
Upward contmuauon of potenual field data
else
rsad(lunit0,100)namel
endif
read(lunit0,100)name2
read(lunit0,100)name3
rsad(lunit0,100)iform
if(iopt2.sq.3)then
read(lunit0,100)name4
else
read(lunit0,*)dsltaz
endif
close(lunit0)
i00 format(aSO)
c
c
c
npx-nlig+2*idscal
npy-ncol+2*jdecal
first order equivalent sources computation and extrapolation
for the case iopt3 - 1
if(iept3.eq.l)then
call dirich(npx,npy,lunit2,1unit5,name2,iform,qdepar,
ltop,sigma2.wkl,aux)
go to 130
else
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routines
isctur and Jupe2
after Gibert and Galdeano (1985)
slightly modified to allow reading in a formatted file and
computation in double precision
Usage:
Purpose:
call lectur(f,m,n,lunit,name,iform)
this subroutine reads data in a formatted file
and prepares the matrix for the extrapolation
Parameters description:
f
m by n array containing data
lunit
Fortran unit number of the file containing
data
name
data filename
iform
data format
Usage
Purpose:
call Jupe2(a,m,n,aux,tsst,iterv,lunit)
this subroutine extrapolates values with a
maximum entropy prediction filter derived
from Burg's algorithm
Parameters description:
a
m by n matrix containing values to be
extrapolated
aux
work vector
test
a(i,J)-test indicates that this value
of matrix a must he extrapolated
iterv
work vector
lunit
Fortran unit number of the sreen
***
topographic data reading
call lectur(top,npx,npy,lunitl,namel,iform)
endif
***
topographic data extrapolation
call Jupe2(top,npx,npy,aux,test,wkl,lunit5)
***
field data reading
call lectur(f,npx,npy,lunit2,name2,iform)
c
c
c
***
field data extrapolation
call jupe2(f,npx,npy,aux,test,wkl,lunitS)
897
1X/[ CIMINALEand M LODOO
898
130 continue
do 110 i=l,npx
110 x(i)=xmin+float(i-1)*step
do 120 J=l,npy
120 y(j)-ymin+float(j-l)*step
if(iopt3.eq.l)go to 140
ic:npx
***
IMSL routine
ibcccu
Usage:
call ibcccu(top,x,npx,y,npy,c, ic,wk, ier)
Purpose: This subroutine computes the coefficients
of a bicubic spllne
Parameters description:
top
npx by npy matrix containing the topographic data
x
vector of length npx (abscissae)
y
vector of length npy (ordinates)
c
2*npx*2*npy array of spline coefficients
ic
row dimension of matrix top and second dimension
of array c exactly as specified in the dimension
statement
wk
work vector of length 2*npx*npy+2*max(npx,npy)
ier
error parameter
call ibcccu(top,x,npx,y,npy,c,ic,wk, ier)
***
partlal derivatives computation
call caloder(x,npx,y,npy,c,derx,dery)
C
C
***
direction cosines computation
C
if(ioptl.eq.2)call calcdir(npx,npy,derx,dery,cosdir)
***
1
first order equivalent sources computation
for the case iopt3 - 2
call fredh(f,top,x,y,derx,dery,coedir, eigma,sigmal,npx,npy,step,
ioptl)
***
first order equivalent sources extrapolation
call rtest(sigma2,sigmal,npx,npy)
call Jupe2(sigma2,npx,npy,aux,test,wkl,lunitS)
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C
C
C
C
C
***
reading or creation of the surface where potential field
data are continued
140 continue
If(iopt2.ne.3)then
if(iopt2.eg.2)goto170
call plane(zarr,deltaz,npx,npy)
goto 190
170 call topar(zarr,top,deltaz,npx, npy)
190 continue
else
call leotur(zarr,npx,npy,lunit4,name4,iform)
endif
***
upward continuation
call reduc(top,farr,zarr,derx,dery,npx,npy,x,y,etep,
lioptl,sigma2,iopt3,wk2,wk3,xx,yy)
***
output file storage
call store(farr,npx,npy,lunit3,name3,iform)
stop
end
Upward c o n t m u a u o n o f p o t e n t l a l f i e l d data
subroutine dirich(npx,npy,lunit2,1unitS,name2,iform,qdepar,
ltop,sigma2,iterv,aux)
This subroutine is called only if the surface where data are
acquired is a horizontal plans having an altitude equal to qdepar.
First it creates this plane, then it computes and extrapolates the
first order equivalent sources.
Parameters description:
lunit2
see main program
lunit5
s e e main program
name2
s e e main program
name5
s e e main program
iform
see main program
qdepar
s e e main program
top
horizontal surface at which data are acquired
sigma2
first order equivalent sources
double precision top(npx,npy),aux(1),iterv(1),
isigma2(npx,npy),teet
character*80 name2,iform
common /radl/idecal,Jdecal
common /rad/test
do i0 J-l,npy
do 10 i-l,npx
i0 top(i,J)-qdepar
dpi-8.d0*datan(l.d0)
call lectur(sigma2,npx,npy,lunit2,name2,iform)
il-idecal+l
i2-npx-idecal
Jl-Jdecal+l
J2-npy-Jdecal
do 20 J-Jl,J2
do 20 i-il,i2
20 sigma2(i,J)-sigma2(i,J)/dpi
call Jupe2(sigma2,npx,npy,aux,test,iterv,lunit5)
return
end
subroutine calcder(x,nx,y,ny,c,dsrx,dsry)
C
C
O
This subroutine draws out the x and y derivatives of topography
from the array of spline coefficients
C
C
C
C
C
C
C
C
C
C
C
C
Parameters description:
x
vector of length nx (abscissae)
y
vector of length ny (ordinates)
c
array of epline coefficients (output of
ibcccu IMSL routine)
derx
matrix containing the x derivative
of topography
defy
matrix containing the y derivative
of topography
implicit
double precision(a-h,o-z)
dimension
x(nx),y(ny)
dimension
c(2,nx,2*ny)
dimension
derx(nx,ny),dery(nx,ny)
nn-2*ny
nny-2*ny-1
do 100 i-l,nx
k-O
do 100 J-l,nny,2
k'k+l
I00 derx(i,k)-c(2,i,J)
do 200 i-l,nx
k-0
do 200 J~2,nn,2
k-k+l
200 dery(i,k)-c(l,i,J)
return
end
899
M CIMINALEand M LODDO
g00
subroutine calcdir(nx,ny,derx,dery,cosdir)
This subroutine, called only if gravity data are processed,
computes the direction cosine of the z axis with respect to
the unit vector normal to the topographic surface at the
given point (i,k)
Parameters description:
derx
x derivative of topography
dery
y derivative of topography
cosdir
matrix containing direction cosines
300
double precision derx(nx,ny),dery(nx,ny),cosdir(nx,ny)
do 300 i-l,nx
do 300 k-l,ny
cosdir(i,k)-i/((l+derx(i,k)**2+dery(i,k)**2)**0.5)
return
end
1
C
C
C
subroutine fredh(f,z,x,y,dx,dy,cosdir,sigma,sigmal,npx,npy,step,
ioptl)
This subroutine computes the matrix
order equivalent sources
of the first
C
C
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C
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C
C
C
C
C
C
C
C
Parameters description:
f
extrapolated potential field data
z
extrapolated topographic data
x
abscissae
y
ordinates
dx
x derivative of topography
dy
y derivative of topography
cosdir
matrix containing direction cosines
sigma
zero order equivalent sources
sigml
first order eqluvalent sources
step
see main program
ioptl
see main program
implicit double precision (a-h,o-z)
dimension f(npx,npy),z(npx,npy),x(npx),y(npy)
dimension sigma(npx,npy),sigmal(npx.npy)
dimension dx(npx,npy),dy(npx,npy),cos~ir(npx,npy)
common /radl/id,jd
dpi-8.d0*datan(1.d0)
dxdy-step*step
nlig-npx-2*id
ncol-npy-2*Jd
if(ioptl.eq.2)goto 190
...... zero order approximation for magnetic data ......
do 100 i-l,npx
do 100 J-l,npy
I00 slgma (i, J )-f (i, J )/dpi
...... first order approximation for magnetic dnta .....
do 140 J-Jd+l,npy-Jd
do 140 i-id+1,npx-id
somma-O.d0
do 130 1-i-ld,i+id
do z~0 =-J-Jd,J+Jd
...... elimination of singular points ......
if(m.eq.J.and.l.eq.i)goto 130
siniz-(sigma(l,m)*(-dx(l,m)*(x(i)-x(1))-dy(l,m)*(y(J)
1-y(m))+(z(i,J)-z(l,m))))*dxdy/((x(i)-x(1))**2
l+(y(J)-y(m))**2+(z(i,J)-z(l,m))**2)**l.5
somma-aomma+siniz
130 continue
frint-(1.d0/dpi)*somma
sigmal(i,J)-sigma(i,J)+frint
140 continue
do 150 i-l,nlig
do 150 J-l,nool
150 sigmal (i, J )-sigmal (i+id, J+Jd)
goto
299
Upward continuation of potenttal field data
...... zero order approximation for gravity data .......
190 do 200 i-l,npx
do 200 J-l,npy
200 sig~a(i,J)-f(i,J)/(dpi*(cosdir(i,j)**2))
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C
C
...... first order approximation for gravity data ......
do 240 j-jd+l,npy-jd
do 240 i-id+l,npx-id
somma-0, dO
do 230 l-i-id,i+id
do 230 m-J-Jd,J+Jd
...... elimination of singular points ......
if(m.eq.J.and.l.eq.i)go to 230
siniz-(sigma(l,m)* (z (i,J)-z (l,m)) )*dxdy/( ((x(i)-x(1))
l**2+(y(J)-y(m) ) **2+ (z (i,J)-z (l,m))*,2)**i. 5)
somma-somma+s inl z
230 continue
frint- (l.dO/(dpi* (cosdir (i, j ) **2) ) )*somma
sigmal (i, J )-sigma (i, J )+ frint
240 continue
do 250 i-l,nlig
do 250 J-l,ncol
250 sigmal (i, j )-sigmal (i+id, J+J d)
299 return
end
subroutine rtest(f,sigmal,m,n)
This subroutine prepares the sigma1 matrix for the extrapolation
using the subroutine Jupe2
Parameters description:
f
matrix containing both the actual and factitious
(test) values of the first order equivalent
sources
sigmal
matrix containing the actual values of the
first order equivalent sources
I0
implicit double precision (a-h,o-z)
dimension f(m,n),sigmal(m,n)
common /radl/id,Jd
common /red/test
do 10 J-l,n
do I0 i-l,m
f(i,J)-tsst
il-id+l
i2-m-id
Jl=Ja+l
J2=n-Jd
do 20 J=J1,J2
20
do 20 i-il,i2
f(i,J)-sigmal(i-id,J-Jd)
return
end
*********************************************************************
subroutine plene(zarr,dsltez,npx,npy)
c
o
c
c
c
c
c
c
This subroutine creates the horizontal surface where
data must be upward continued
Parameters description:
zarr
matrix containing horizontal
surface data
dsltaz
horizontal surface altitude
c
double precision zarr(npx,npy),dsltaz
do i0 J-l,npy
do 10 i-l,npx
901
M. CIMINALE and M
902
10
LODDO
zarr(i,J)-deltaz
continue
return
end
subroutine topar(zarr,top,deltaz,npx,npy)
C
C
This subroutine creates a surface parallel to the surface
where potential field data are acquired
C
C
C
C
C
C
C
C
Parameters description:
zarr
matrix containing the new surface
top
matrix containing the surface where
potential field data are acquired
deltaz
terrain clearance
I0
double precision zarr(npx,npy),top(npx,npy),deltaz
do 10 j-l,npy
do 10 i-l,npx
zarr(i,J)-top(l,J)+dsltaz
return
end
*********************************************************************
subroutine reduc(top,farr,zarr,derx,dery, npx,npy,x,y,
&step,ioptl,eigma2,iopt3,wk2,f,xx,yy)
This subroutine executes the upward continuation to
an arbitrary surface
Parameters description:
top
extrapolated topographic data
farr
upward continuated data
zarr
surface at which field data are continued
x
see fredh
y
see fredh
step
see main program
ioptl
see main program
sigma2
extrapolated first order equivalent sources
iopt3
see main program
implicit double precision(a-h,o-z)
dimension f(npx,npy),top(npx,npy),farr(npx,npy)
dimension wk2(1),xx(1),yy(1)
dimension x(npx),y(npy),sigma2(npx,npy),zarr(npx,npy)
dimension derx(npx,npy),dery(npx,npy)
common /radl/iwindi,iwindJ
common /range/m1,m2,11,12
ml-iwindJ+1
m2-npy-iwindJ
11-iwindi+l
12-npx-iwindi
nx-2*iwindi+l
ny-2*iwindJ+l
do 140 m-ml,m2
do 140 1-11,12
liml-l-iwindi
lim2-iwindi+l
lim3-m-iwindJ
lim4-iwindJ+m
k-0
do 130 J-lim3,1im4
k-k+1
n-0
do 130 i-liml,lim2
n-n+l
if(iopt3.eq.1)go to 120
go to(l10,120),ioptl
ii0 f(i,J)-(sigma2(i,J)*(-dsrx(i,J)*(x(1)-x(i))&dery(i,J)*(y(m)-y(J))+(zerr(l,m)-top(i,J))))/
&(((x(1)-x(i))**2+(y(m)-y(J))**2+(zarr(l,m)-top(i,J))**2)**l.5)
f(n,k)-f(i,J)
go to 130
120 f(i,J)-(sigma2(i,J)*(zarr(l,m)-top(i,J)))/
&(((x(1)-x(i))**2+(y(m)-y(J))**2+(zarr(l,m)-top(i,J))**2)**l.5)
f(n,k)-f(i,J)
130 continue
ifd-npx
Upward continuation of potential field data
xx(1)-x(liml)
do 160 i-2,nx
xx(i)-xx(i-l)+stsp
~y(1)'y(lim3)
dO 170 i-2,ny
yy(i)-yy(i-l)+step
a-xx(1)
160
170
b,,xx(nx)
c-~,(1)
d-~(ny)
C
C
C
C
C
C
***
IMSL routine
abcgdu
Usage:
call dbcgdu(f,ifd,xx,nx,yy,ny,a,b,c,d,q,wk2,ier)
Purpose: This subroutine executes a bicubic spline quadrature
Parameters description:
f
nx by ny matrix containing the function values
ifd
row dimension of the matrix f exactly as specified
in the dimension statement in the calling program
xx
vector of length nx (abscissae)
yy
vector of length ny (ordinates)
a,b
x-direction limits of integration
c,d
y-direction limits of integration
q
double integral from a to b and c to d
wk2
work vector whose minimum length is (ny+5)*nx+ny-l+
max(5*nx-4,5*ny-2)
ler
error parameter
C
C
C
C
¢
C
C
C
C
C
C
¢
C
140
call dbcqdu(f, ifd,xx,nx,yy,ny,a,b,c,d,q,wk2,ier)
farr(l,m)--q
continue
return
end
*********************************************************************
subroutine store(farr,npx,npy,lunit3,name3,iform)
C
C
C
C
C
C
C
C
C
This subroutine stores the upward continued field
Parameters description:
farr
matrix containing the upward continued field
1unit3
see main program
name3
see main program
Iform
see main program
double precision farr(npx,npy)
character*80,name3,iform
common /range/Jl,J2,il,i2
opsn(lunit3,status-'nsw',file-nams3)
do 10 J-Jl,J2
I0 write(lunit3,iform)(farr(i,J),i-il,i2)
close(lunit3)
• return
end
903