GEOPHYSI(‘S. VOL. 53. KO. X (AIJGIIST 19X8): P. 1096-1108. I2 FIGS., I TABLE

Application of three-dimensional interactive modeling

in gravity and magnetics

ABSTRACT

Three-dimensional (3-D) interactive modeling permits integrated processing and interpretation of gravity and magnetic data, yielding an improved geologic interpre- tation. 3-D model bodies are constructed from poly- hedra of suitable geometry and physical parameters (density and susceptibility). input on an interactive graphics terminal that is tied to a host computer. The method is especially designed for concurrent processing and interpretation in an interactive mode.

The effect on gravity of a homogeneous polyhedron is calculated by transforming a volume integral into a sum of line integrals. Magnetic effects can be modeled by using either Poisson’s theorem or a slight modification

of the formulas derived for gravity modeling. The inter- active modeling program allows the user to change the geometry as well as the density and/or susceptibility of the elementary polyhedra and to observe results quickly during the course of processing. This capabi!i+v enables the interpreter to decide immediately if and where a tentative geologic structure must be changed for the modeled effect to fit that of a field survey. He is able to drive the device-dependent process by clear menu func- tions without any knowledge of the rather complicated data structure and the interaction between the main program and its many subroutines. In addition, appli- cation of this method requires considerably less com- puting time than conventional methods based on the direct evaluation of volume integrals.

INTRODUCTION

“Interactive computer graphics is a field, whose time has come.”

J. D. Foley and A. van Dam (1982)

Until a few years ago, the field of computer graphics was a

research area reserved for specialists who required expensive hardware and substantial computer resources. Only with the development of new computer equipment and corresponding soitware did graphic data processing begin to estabiish itseif. Interactive graphics has already become an important part of data processing, for example CAD or CAM, in several areas of the natural sciences and in engineering.

Geophysics, too, is increasingly concerned with the pos- sibilities provided by interactive computer graphics in data interpretation, e.g., Jordan et al. (1979): Feagin (1981), or Denham and N&on (1986). Grant (1972) recognized sixteen years ago as part of an invited paper for the SEG that the continuing development of modern interpretive techniques in gravity and magnetics. especially in data processing, would be

linked to the special application of interactive computer graphics. Below, we present the development of interpretive techniques of this kind.

The development of modeling techniques in gravity and magnet& can bc clearly shown by means of topographic re- duction calculations. This development began with manual methods such as templates (Hammer, 1939; Sandberg, 1959) and nomographs. This type of calculation is laborious and time-consuming because it is (a) station-dependent and (b) re- quires considerable simplifications. Later, with the advent of larger computers. it became possible to program more elabor- ate formuias, e.g., the formuia for the attracTion ofa~ rectangu- Jar prism (Nagy. 1966a, b). Reduction techniques as a path- Iinder of modeling became independent of the measured grav- ity stations (Ehrismann et al., 1966; Kantas and Zych, 1967; Stacey and Stephens, 1970).

At present, work is in progress at the Bundesamt fur Eich- und Vermessungswesen, Wien (Federal Weights and Measures Office, Vienna) on an elevation data bank for use by geophysi- cists and geodesists in the Eastern Alps. It will contain heights with a spacing of approximately 50 m in the central areas and 160 m at the margin. It will then be possible, for the first time

Manuscript received by the Editor February 13, 1987: revised manuscript received January 19. 1988. *Institut fuer Geophysikalische Wissenschaften, Malteserstr. 74-100, D-1000, Berlin 46, West Germany :[Institut fuer Geophysik, Arnold Sommerfeldstr. I, D-3392 Clausthal-Zellerfeld. (’ 1988 Society of Exploration Geophysicists. All rights reserved.

1096

3-D Interactive Modeling 1097

to establish a precise presentation of the difficult mountainous

topography and thereby increase the accuracy of the gravity

reduction calculations. In this context, it is of no consequence

whether forward modeling is used as in this paper. or whether

fast Fourier transform techniques are applied (Tsuhoi, 1983;

Hildenbrand. 1983; Sideris, 1984).

Great demands are also made on the data base manage-

ment programs, since large amounts of data are characteristic

of 3-D modeling, no matter what the geophysical surveying

method. The organization of the computer program and its

data files. as well as of a user-friendly presentation of the

calculated results, are the keys to the system’s development.

The programming of mathematical algorithms in itself re-

quires relatively little effort.

Forward modeling

Before discussing the ovcrail framework for the applied

analysis, we provide a short reminder of forward modeling. In

general, we assume that the interpretation of gravity and mag-

netic anomalies is based on determining plausible shapes,

positions. and physical parameters for the geologic structures

which cause these fields. This problem of data inversion in its

broadest sense requires conversion of the information which

has been obtained by measurements into geologic models.

Basically, an indirect modeling process is the calculation of

the efrect of a simple elementary body which approximates the

geologic situation in the investigated area. followed by match-

ing the model curve with the observed curve by trial and error or graphic-interactive tools.

Within this arrangement, it is possible to distinguish be-

tween tr~lrrlyrictrl ctr/c!rlu‘;. in which the gravitational effects of

simply formed models such as spheres. mass-points, and lines.

etc. can be calculated and [or which precise mathematical ex-

pressions can still he given, and clrltrl~ficc~l techniques, which

calculate the fields of complicated models whose geometry is

approximated by a greater number of elementary bodies

(polygons, prisms. or polyhedrons).

The above-mentioned techniques are suitable for carrying

out gravity and magnetic model calculations of varying de-

grees ol difficulty. In correspondence with the elementary

bodies used. a diKcrentiation is made between two-

dimensional and three-dimensional processes. A model is

called 2-D when the model extends infinitely along the y-axis.

Thus. all cross-sections of these bodies which are perpendicu-

lar to the ),-axis are identical and the density (magnetization)

is constant. A very well-known 2-D technique was developed

by Tnlwani et al. (1959) and improved by Won and Bevis

(1987). For 2-D techniques to be applicable to situations

found in nature, the length of the geologic structure must be

approximately five times that of its maximum width. Thus, the

for-mula given by Talwani et al (1959) is a very effective aid in

modeling.

The program developed by Talwani and Ewing (1960) for

3-D modeling could not establish itself, because the model

siruciurc, even of rciativeiy simpic geologic bodies, proved to

bc complicated and confused. In addition to the above-

mcntioncd work, several other 3-D model calculations have

become known in which the approximation of natural rock

complexes was carried out with rectangular prisms, e.g., Nagy

(I966a. b); Cordell and Henderson (1968). There are three dis-

advantages which make modeling with rectangular prisms la-

borious:

(11 The formula for the attraction effect of a cube is

extraordinarily awkward. It is both computer time and

computer memory intensive, especially since large

models can consist of prisms.

(2) .4pproximating the often complicated suhterra- nean structure is time-consuming and input data inten-

si\e. Were the Earth to be spherically modeled, then

neither the convergence of its meridians nor that of the

radii in deeper structures could be taken into consider-

ation

(3) Intcractivc modeling is, in fact. possible but it has

not been achicvcd so far.

With this background of available model techniques, we shall

now present a modern concept for easy handling of gravity

and magnetic modeling.

Simulation modeling

Rather than the actual mathematical formulas for calculat-

ing an elementary body’s attraction, the organization of the

calculation process, the graphic presentation of the results,

and the interactive control of model matching are the major

parts da computer program for simulation of gravity or mag-

netic data. These activities are grouped together under the

heading “simulation modeling.” From the viewpoint of user-

friendly organization, simulation models should (1) provide a

data structure which can be stored and which contains the

object components themselves, their interrelationship, and one

or more of the process-algorithms; (2) be computer-oriented

and include active computer graphics which improve the com-

munication between the user and the simulation model; (3)

have the characteristics of the system described by exact quan-

titative correlation, such as with an equation; (4) have fixed

geometry of the structure or layout; (5) store a gating ability;

(6) contain specific application model parameters, e.g., density

or susceptibility; and (7) he adaptable to reality. In order to

achieve the last requirement, comparative values (measure-

ments) must be available in the data structure. The availability

of all the information on the model in a data structure implies

compromises in the modeling. The requirements of data struc-

ture clarity, excellent user comfort through interaction, search

and sort algorithms, and elaborate processing algorithms are

counterbalanced by limited computer resources with regard to

CPU time and memory; hence. by a limitation on the calcula-

tions and often a limitation on the precision of numbers, as

well as a limitation on the rapid graphic display of the mod-

eled results.

Below, we discuss the design of such an application pro-

gram for simulation modeling. Figure I shows the program’s

basic structure.

RASIC l+:I.EMENTS I;OR PROGRAMMING AN INTERACTIVE GRAPHIC PROCESS

Figure 2 presents the simplified relationships among graph- ics, design, and calculation in order to differentiate between

interactive graphics and passive graphics. Pussit~~ computrr

1098 G&e and Lahmeyer

yruphics comprise the creation and storage in the computer of images of certain objects (pictures). Acti~ computer graphics involve a user who, by means of interactive elements such as a keyboard, mouse, etc., can dynamically influence the form, size, and content of the image on the monitor. The calculation by the process algorithms is immediately modified. Thus the operational process is event-driven.

If this event-dependent control is to be highly efficient, then a series of human factor principles must be considered. One such factor is the use of simple and logical interactive se- quences. English et al. (1967) showed that processing timedifferences of up to 100 percent are possible for identical ac- tivities because of clumsy interactivity. Other factors demand no overloading for communication with the program and op- portunities for feedback from the user. The user has either syntactic or semantic feedback at his disposal. The last of the human factors is support of the user in error search within the data structure after erroneous interaction. The corresponding program routines for error identification comprise approxi- mately 20 percent of the total length of the application pro- gram presented here.

Figure 3 shows the most important elements of the program for interactive processing of gravity and magnetic model cal- culations. The application program contains the mathematical formulas and the organizational structure for interactive pro- cessing. The interactive graphics are implemented with the Graphical Kernel System. The graphic system tools are the following:

Seyments: serve the collection of basic primitives, thus enabling the graphic results to be printed out at specific work stations via special drivers in a post-mortem pro- cedure ;

Codinure systems : a user-input coordinate system, normalized virtual system, and driver-dependent two- dimensional coordinate system;

LirWtrr trarzsfiwmatiorzs for which windowing and

viewport activities are required; input/output primitives for the interaction between the user and his program via a menu table on the graphical screen;

Attrihutv.s of the input/output functions; Zooming; and

Error ident$cation routines exclusively for the graph- ic system.

flaving discussed the external frame for interactive processing above, we introduce the process algorithms, moving us toward the geophysical principles of the evaluation process.

Design and Decision

FIG. 2. The relationship for the definition of computer graph- ics after Green (1970). The innermost part deals with inter- active computer graphics.

4~ User Program B Metafile

---Jp

I cl User

Frc;. 1. The structure of an interactive application program and the interrelationships among the data structure, the graphic system, and the graphic display. A = building, modification, and manipulation procedures; B = traversal for display; C = traversal for analysis; D = display and interaction dialogue handler; E = organization of computer resources (not dealing with modeling). +, = data pipe-lines. Redrawn after Foley and van Dam (1982).

3-D Interactive Modeling 1099

PROCESS ALGORITHMS

First of all, an elementary body which approximates geolog- ic phenomena well must be found. This elementary body must fulfill the conditions of the whole range of geologic-tectonic features: compact small volume structures. such as are found in engineering problems of applied gravity, have to be just as depictable as the global forms and dimensions of a subduction zone. Any polyhedron with plane surfaces can ensure the re- quired conformity; its physical material parameters are den- sity and/or magnetization. Note that in the following, the term “gravity” will be the attraction effect of a stationary ele- mentary body only.

In general, the calculation of the attraction of homogeneous polyhedrons at a station P is based on the evaluation of the

USW Program

USW

n

Data structure

Graphic System

GKS k 6 Plotter 2

fEiz Printer

3

Ftc;. 3. An extension of Figure 1 shows the graphic system’s communication with the available hardware resources, the data files, and the user program. Drivers are hardware- dependent computer programs which convert graphic infor- mation of the calculation process to connected work stations.

VA v5 St i fi & t Z'

1 !

,.,‘:i

,,J.-.-._.__& \ VI

lo /-_ ‘.

X _ ,-

*’ ,,*: 4 \ \\, CX’

‘\ ‘II ‘\i <\\

v9& Q

;?a

‘\. /’ ‘\ \’ , L -ii. ‘\ Y z

_ ’ ’ ‘\ dv_.‘--

n2 ;+; \ A=-

/’ *\

Ve ;;.

63 VS

s2 3,’

s3

I 1

FE. 4. Presentation of an arbitrarily structured polyhedron. l$ = vertices; S; = polyhedron surfaces area, j = I,~ ~. ,.m; nj = surface normal ‘of Sj,

with U(P) = potential at the station P, R = distance between P and dm, dm = p dv = p dx dy dz, and f = gravitation con- stant. Taking the derivative of the potential with respect to the vertical e-component leads to the gravity g(P):

su ;i- (PI = g(P) =fP jjjg (;) du. (2)

P”lY.

Corresponding to the relevant theorems of vector analysis, one obtains

l?u ,:, (PI = g(P) =fp # cos (4 z(i) dS, (3)

Surface

where the surface integral in equation (3) has to be calculated for the whole polyhedron surface, and the cosine term CDS (n, Z) determines the direction of the surface element dS

with regard to the Cartesian coordinate system (Figure 4). Since, according to Figure 4, cos (nj, z) = constant for any

polyhedron surface S, (j = 1, _, m, the number of surfaces), the attraction effect of a polyhedron by the superposition of the gravity effects of its individual surfaces Sj can be expressed as

In order to maintain the simplest possible mathematical ex- pressions in the evaluation of the surface integral in equation (4), a transformation of the coordinate system is required, whereby the new system xi, y’, Z’ should be surface-oriented (Figure 4). Thus, the x’-axis runs parallel to V,V, , the L-axis runs parallel to the respective surface normals nj, and the #axis is orthogonal to the x’ and Z’ axes.

The transformed coordinates are calculated for each surface S, as follows :

(x’, Y', 2’) = T(x, Y> 4

with

(S)

being the transformation matrix. (x, y, z) = T’-(x’, y’, z’), where 1’ is the transpose of T. The directional cosines of T are determined from the determinants of the first three vertex co- ordinates (V,, V, , VJ:

x - XI Y- Yl 2 - z,

.x2 - x, y2-y1 z* -z* =o x3 - x1 Y, - Yl 23 - 21

and the relationship which the transformed coordinate axes have with the polyhedron surface S,.

The corners v are input in the positive direction. All other

1100 G&e and Lahmeyer

calculations are carried out with the transformed coordinates of equation (5).

The next step is the conv*ersion of the surface integral (4) into a linear integral via polygon Pj, which limits surface S, (after Gauss-Ostrogradski); the solution of the linear integral is very time-consuming and will not be described in detail here (see e.g.. Giitze 1984). As a final formula, the easily program- mable formula for the gravitational effect of a polyhedron is obtained:

,

(6) with

6= 0

if p* 6 ST I’ E L$’

E = factor.

Correspondingly, the vertical gradient of gravity is the second derivative of the potential

f’ 2 II (7 I i=’ (P) = Vc;(P) = fi, cos (n, 3) ii_ - dS,

0 7 R (7)

or corresponding to equation (4)

VG(P) =$, [cos (Mj> =) jJ$ (k) ds,]. (8)

In accordance with the coordinate transformation of equation (5). the differentiation in the old coordinate system is

From equation (8). we have:

The equation for calculating the vertical gradient can be derived after applying Green’s theorem, and several other con- versions :

VG(P) =.lp t cos (n,i. :) 5 (U,)jPI* i 1 1 L i 1

@,),a* In 'j.i + ‘V,.i-1 +

a ,, I + c. i

In h,, +Fi++,

ui, j + PVj. i

- hi PP;(YJj -

lh,l IPPJ*l -

x arctan Llj, i / PPY /

I hi I Fy, i

PP* ‘I

+ 2x6&(yJi -&J-- ‘i IPf’TI

(10)

Equations (6) and (IO) represent the processing algorithm for the interactive system for processing density models.

For magnetic model calculations, the Poisson theorem will be applied when pole-reduced magnetic data are used. It is generally expressed as

where

I/ = magnetic potential,

I! = gravity potential.

I = homogeneous magnetization of body,

f= gravitation constant,

p = homogeneous density,

0 = direction of magnetization, and

.$ = arbitrary dilferentiation direction.

In the case of pole reduction. 0 = 90‘, and the differentiation runs along the :-axis (s = z). Thus for reduction to the pole,

(12)

where the expression on the right-hand side is the vertical gradient of gravity from equation (IO). When r,j; and p are given. the magnetic field of a polyhedron can be calculated using the vertical gradient of gravity, as long as pole-reduced data are used.

If the field does not incline at 90”, the following repre- sentation of the magnetic potential is assumed:

= -[ 11 (+)I.& jjj($V.M]. (13)

surface Poly

When 1 = constant, V . I = 0, and

V(P) = -

because I . (is = In ds for the magnetic field,

H(P) = -V, V(P) = V,

(14)

(15)

In accordance with the earlier developments, for the field of a polyhedron with planes,

or, transforming coordinates,

The integral in equation (17) was already solved for equation

3-D Interactive Modeling 1101

(10). In conclusion, one obtains for the field H in P:

and -

hi PP* -

ci= 2 arctan aj, i I ppT I

lhil IPPj*l I hi I Fy. ;

_ arctan bj. i I”? I > Ih,IPi,i-,

Equation (18) shows that to calculate components of magnetic fields, no integrals other than those that solved foi equation (10) need to be solved. The expressions Ai, Bi, and Ci are merely multiplied by different directional cosines. Thus, the processing algorithm has also been derived for magnetic cases. Nothing more stands in the way of a simultaneous calculation of gravity and magnetic models.

2-D and 3-D model calculations

If one compares the 3-D formulas in equations (6), (lo), and (18) which were derived for polyhedrons with the formulas presented by Talwani et al. (1959) it is clear that the com- plexity of 3-D calculations is only minimally different from that of 2-D calculations. Thus, we know that the derived for- mulas are rapid and powerful, and furthermore offer a more realistic approximation to geologic-tectonic structures by gaining a third dimension.

Data acquisition should also be adapted to these new op- portunities, so that horizontal derivatives in all directions are measured. The formulas do not allow us to dispense with the “principle of equivalency” of the potential methods-that principle can be derived from theoretical considerations. How- ever, the procedural scope in modeling is strictly limited be- cause the total horizontal gradient has to be modeled within the measured gravitational distribution. This means that where narrow or wide isoline distances occur in the Bouguer gravity map. they must also be recognizable in the map of the modeled gravity.

THE BASIC ELEMENTS OF USER DATA STRUCTURES 4ND THE PROCESS SEQUENCE IN

INTERACTIVE MODELING

In creating the data structure of a model, the model struc- ture must be comprehensible to the user and not overloaded with details. The program should also incorporate the usual geoscientific requirements, such as geologic vertical sections, seismic cross-sections. and depth contouring.

FIG. 5. Example of the presentation of a simple two-layer substructure in a data structure with S, S,, = triangles, EB, ... EB, = vertical sections, L, L, = lines, and VI VI 3 = model vertices. Each vertical section is both a cross-section and a working area on the graphical terminal.

1102 G&e and Lahmeyer

I ) MODEL )

I + Layer Boundary 1

FIG. 6. Interrelation in model 1 for layer I. TRIANGLE is synonymous with triangle facet and RHO is synonymous with density or susceptibility. The triangles correspond to the sur- fazes S in Figure 5.

It is. therefore, sensible to divide the whole model into verti- cal sections (so-called planes) as in Figure 5. All the other elements of the data structure are to be found in these planes, which are in their totality carriers of geometric and material parameters and are interactively manipulatable on the graphic monitor.

The main elements of the data structure are the following:

Plunes: The planes are vertical and should be placed parallel to y = a constant. They should be oriented ver- tically to the main strike direction. They do not, how- ever, necessarily have to be equidistant, but rather should be adapted by the user to the area of investi- gation.

I,ines: Within individual planes, the substructure is marked by a series of interconnected points K (Figure 5). The series of vertices domprises a line and, as a poly- gon section, it marks the intersection of a layer bound- ary with the respective plane. There may be open and closed lines, as well as lines which have only one point V.

MAIN PROGRAM HP 1

I MODEL DATA FILE I

4

,& A

,

DIGRAF

4

I

SCREEN / DATA FILE

HP 1 builds the model data structure from user input data. HP 1 tests logical links (sequences) in the data structure and tests the triangle formation within the layer boundaries. The prepared data fields are placed on an intermediate data file.

The intermediate data file serves the actual interactive activ- ities with the user data structure, i.e., modeling. The inter- mediate data file is permanent and can be printed out for checking or archival purposes.

The interactive program contains the formulas derived in “process aigorithm” (for m~agnetics a&gravity); i-iP 2; -with its pick functions, has access via the menu table to the graphics package DIGRAF. It can also access long-term data files which store calculation results and graphic presentations.

Graphic program package: complies with the American CORE-Standard.

The input/output medium used was a Tektronix monitor 4014. For long-term storage, the graphic entities can also be printed.

FIG. 7. The interactive flow between the main programs HP1 and HP2.

3-D Interactive Modeling 1103

Layer boundaries: The basic elements thus far de- scribed were presented in the 2-D x, z coordinate system of the ,r = a constant plane. The layer boundaries stabil- ize the 3-D model structure by encompassing the lines of neighboring planes which are separated by identical density (susceptibility) complexes. Thus, in Figure 5 the lines L,, L,, and L, are brought together into a layer boundary surface. Each layer boundary consists of triangles which have the v as their vertices. The data structure used allows the modeling of all structures found in nature including outcrops, intrusions, folds, thin plates, steep layers, and dikes, as well as flat and uncomplicated deposits.

Triangle planes: The vertices which are encompassed by the basic element layer lines are joined into triangles within a layer. The program automatically undertakes

the design of the triangle net. The basis of triangle for- mation can be found in the work by Akima (1978), who used Lawson’s algorithm (Lawson, 1972) for the opti- mization of triangle networks.

With the input of the point coordinates,

C.Y, I L’J E q; i = number of model vertices

and the material parameters

r RHO,. RHO,

LKAPPA,, KAPPA, 1 E sj;

.i = number of model surfaces,

the model is completely defined. Figure 6 shows the inter- relation in the data structure. The data structure defined so far

Table 1. Program functions of the main program HP 2. These program activities may be chosen by a pick device (cross-hairs of the graphic screen) from the menu table which is permanently presented on the graphic terminal.

Program activities of HP 2

Shifting of corner points

Shifting of Lines

Insertion of corner points

Gravity calculation of modified models

Change of gravity (susceptibility) values

3-D perspective view of the model

Any vertical cross-section through the model with cor-

Menu table

SHIFT

SHIFTLINE

INSERT

GRAVITY

DENSITY

3-D PICTURE

CROSS-SECTION responding measured and calculated field curves

Termination of program run

Depth contour lines of layer surfaces

Contour line maps of the modelled and measured fields

Calculation of the optimal material parameter by means of inversion techniques

END PROGRAM

ISOSCH

ISOLINES

INVERSION

Choosing a new vertical section with special presenta- tion of the results (differential or curve)

Zooming (enlarging or reducing the model, extract pre- sentation)

NEW FACE

WINDOW

Returning window to original boundaries

Active (passive) setting of metafile of the GKS program

Input/output of model data base of model data file of

ORIGINAL WINDOW

METON/METOFF

INPUT,‘OUTPUT MODEL IGAS

Output of intermediate models to return to HP 1 OUT2

Fixed scale for all graphic pictures FIXED SCALE

Scale automatically calculated by program NEW SCALE

1104 G&e and Lahmeyer

has been implementated in two Fortran programs, both of which are linked by a permanent model data file (Figure 7).

The main task of HP2 is to revise interactively the intcr- mediate data file on a graphic monitor until the model results conform to the measured fields from gravity and magnetics. The model geometry or material parameter (e.g., density or magnetization) can be changed and, with interactive pro- cessing, the results are immediately visible to the user; and it is possible for the user to influence the modeling directly.

In the case of gravity and magnetics, the programs offer the opportunity for calculating optimal material parameters for any given geometry and measured comparative fields by means of inversion (generalized matrix inversion).

Apart from the various possibilities for the presentation of the model structures and model results (representation graph- ics), intermediate models can also be separated and stored to be reprocessed later. Similarly, every passive graphic can be shown on various types of printout equipment after interactive processing has ceased (e.g., on a plotter for storage in the archives).

Table 1 shows the individual program activities of HP2 and the corresponding commands in the menu table. Below, we demonstrate a short application of the program package to complex subterranean conditions, whereby the high degree of user comfort in the interactive computer program will become clear.

APPLICATION EXAMPLES

The aim of the model calculations presented here was to create a plausible density structure model for an area of north- ern Germany. We expected 3-D gravity model calculations to help answer questions about any possible geometry and its estimated depth. We also intended to make use of as many partial results from other geophysical measuring methods as possible in the creation of the model. Since the area under examination lies beneath the Zechstein horizon, and thus has to be considered a poor reflection area, it seemed admissible to first draw together the results obtained by gravity, mag-

netics, magnetotellurics, geothermics, and geology into a com- bined interpretation (Dohr, 1976; Giebeler, 1983; Giebeler- Degro, 1986).

Many aspects of the formation and structure of northern Germany within the tectonic frame of the northwest European Basin are still unclear. The area is one of repeated geophysical exploration for hydrocarbon deposits, whereby geologic- tectonic studies as well as geophysical examination have been carried out (Dohr, 1976; Bartenstein and Schmidt, 1980; Zie- gler, 1981, 1982). A summary cannot be given here; rather, Figure 8 (after Bender and Hedemann, 1983) conveys an im-

FIG;. 9. Contour map of the gravity from the final model. Contour interval = 1 mGa1. S4 and SX mark the positions of the vcrtjcal model cross-sections presented in Figure 11.

ssw wiihen - gebirge Weser Aller Elbe Fehmam

Belt

NE

100 km

0 Oil Field

* Gas Field

FIG. 8. Schematic profile of the northern German Basin with some important subterranean structures. After Bender and Hedcmann (1983). The modeled area is situated in the southwest part of this vertical section.

3-D Interactive Modeling 1105

pression of the complicated structure of the area under exami- nation.

A contour map of gravity from the final processing (Figure 9) is presented as the result of the model calculations (also, refer to Figure 11). Two gravity highs are clearly recognizable, one caused by basic intrusions from the lower crust area (structure left) and the other resulting from the raised position of the lower crust in the modeling area (structure right). Mea- sured gravity field and modeled gravity conform satisfactorily.

The computer program which was made available for the calculations clearly shows good agreement in the modeled and measured gravity fields when the command “contour map of

FE. IO. Contour map of the differences of measured gravity and calculated gravity from the final model. SZ-SI 1 are the positions of vertical cross-sections. Numbers are differences in mGa1.

differences” was given. In Figure 10, the differences are pre- sented together with the positions of modeled sections S, to S, I. It is shown that the deviation at no time exceeded + 2 mGal, and, furthermore, that the adaptation was satisfactory for the total measuring area within the explored region. The terminal presentation of two central cross-sections of the mod- eled underground of the northwest German Basin is shown in Figure I I. The vertical cross-sections (S,) and (S,) show that the upper crust of Lower Saxony is structured by deep gra- bens and intrusions of basic composition (density of 2.75-2.95 g:cm”) which arise from lower crust levels. The strike of these intrusions is tied to the generally obtained Hercynian direc- tion.

The difference from 2-D model calculations becomes clear here. With 2-D model calculations, adaptation can be ob- tained only because the model changes in 3-D processing always atfcct all model gravity stations. Of course, as always, the equivalency principle is still valid; however, 3-D modeling strictly limits the possibilities of free modeling.

ACKNOWLEDGMENTS

h4any of ?he ideas presented in this paper wereborn during stimulating discussions with Prof. 0. Rosenbach and Dr. M. Gicbclcr-Degro. In addition, M. Giebeler-Degro spent her time with us in front of the graphic terminal during the phases of development and. sometimes, despair. MSc. Guy Moore attended to the translation and Dr. S. Schmidt prepared the manuscript. This project was hnancially supported by the German Science Foundation (Deutsche Forschungsgemein- schaft): the modeling of the northwest German Basin was sup-

FK. 11. Presentation of the vertical cross-sections S, and S, of the density model. At the top, the graphic contains both the measured and modeled gravity profiles. The named extrema of the gravity field correspond with Figure 9. The intrusions (RHO = 2.75 and 2.92 g/cm3) arise from the level of the Conrad discontinuity (depth = 20 km). The surface of the crystalline basement (RHO = 2.80 gjcm3) is shaped by various horsts and grabens, separated by steep deep faults.

1106 Giitze and Lahmeyer

ported by the PREUSSAG AG. Hannover. We are very grate-

ful to all of them.

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APPENDIX

THE FORTRAN SUBROUTINE NEWTON

The Subroutine NEWTON calculates the gravity effect of a triangle. That statement means the gravity effect of a

vertical prism, infinite in z-direction and with a triangle as upper boundary. The geometry of this body is shown in

Figure A-l. More complicated polyhedra may be constructed as the sum of elementary bodies like this.

IP2

FIG. A-l. Geometry of the body whose gravity effect can be calculated with subroutine NEWTON.

SUBROUTINE NBYTON~SXYZ,NST,NST~,TXYZ,NPH,IPI,IP2,IP3,GRAV,IFAIL~ IFAIL= C VERGL--l.D-8 C SUBROUTINE "NEWTON" CALCULATES THE GRAVITY EFFECT IN WGAL OF THE CONST=8.'ATAN(DBLE(l.))

C

HILF=QS(PX,PY.PZ) BETl-PXIHILF BET2=PY/HILF BET3=PZ/HILF

CONTINUE

R;41=X(l) Y(4)=Y(l) X(5)=X(2) Y(s)=Y(z)

DO 11 J=L.3 AJ=X(J+lI-X(J) BJ=YIJ+l)-YIJ) HILF=SQRTOrJ*AJ+BJ*BJ) AST(Jl=AJ/HILF BST(J)=BJ/HILF IF (ABS(AST(J)).LE.VERGL) GOT0 31 IF (ABS(BSTIJ)).LE.VERGL) GOT0 32 GOT0 33 AST(JI=DBLE(O.) GOT0 33 BST(JI=O. CONTINUE CONTINUE AST(I)=AST(l) BST(4)=BST(l) DO 12 J=2,4 HILFl=X(J)-X(J-1) HILF2=Y(J)-Y(J-1) HILF =SQRT&iILF1*HILF1+HILF2*HILF2) ASST=HILFl/HILF BSST=HILF2/HILF HFAKTZ(J)=0.5-(ACOS(AST(J)*ASST+BST(J)tBSST))/CONST CONTINUE HFAKT2(1)=HFART2(4)

C TRIANGLE DEFINED BY THE POINTS "IPl","IP2" AND "IP3" AND THE C C COORDINATES IN "TXYZ", FOR ALL STATIONS IN “SXYZ”. THE ASSUMED DENSITY X(l)=TXYZ(IPl,l) C CONTRAST IS 1 GR/CCH. X(2)=TXYZ(IP2.1)

X(3)=TXYZ(IP3;1) C

C

C AUTHORS: HANS-JIJERGEN GOETZE C INSTITUT F. GEOPHYSIKALISCHE YISSENSCHAFTEN C FU BERLIN c BALTESERSTR. 74-100 C D-1000 BERLIN 46 C C BERND LABM’YER C INSTITUT FUER GEOPHYSIK C TU CLAUSTHAL C ARNOLD SOHHERFELDSTR. 1 C D-3392 CLAUSTHAL-ZELLERFELD

C

Y(l)=TXYZLIP1.2) Y(Z)=TXYZ(IPZ,?.) Y(31=TXYZ(IP3,21 211)=TXYZ(IP1,3) Z(2)=TXYZ(IP2.3) Z(3l=TXYZ(IP3,3)

C DXZl=X(Z)-XIli

C

C

C C (JANUARY 3988) C C C INPUT: C c X(I).Y(I),Z(Il C C C SXYZ(I,l),SXYZ(I,2),SXYZ(I,3) C C NST C C NSTI4 C C TXYZ(I,1),TXYZ~I,2),TXYZII,3) C C C NPK

DZ21=2(2)-Z(1) DZ31=Z(3)-Z(l) DZ32=2(31-Z(2)

C 8888 C

c C ALL POINTS OF TRIANGLE DIFFERENT? C

1

C 2

C

C

C C

C

C

C

IF(~QS(DX21.DY21.DZ21).CT.VERGLJ.AND. f (QS(DX31,DY31,DZ31).GT.VERGL).AND. l lQS(DX32,DY32,DZ3Z).GT.'IERGL)) GOT0 2 YRITE~*.'~A)'l ' TRIANGLE HAS IDENTICAL POINTS' IFAIL=l RETURN

A=DY21*DZ31 - DZ21*DY31 B=DZZl*DX31 - DX21'DZ31 C=DXZl*DY31 - DY21*DX31 D=X(l)*A + Y(l)*8 + Z(l)'C

HILF=QSIA,B,C) COAL=A/HILF COBE=B/HILF COGA=C/RILF IF (ABS(COGA-DBLE(l.)).LT.VERGL) COGA=1.0 COGAX=COGA*6.67

P = - D/HILF

IF (ABS(COGA).LT.(VERGL)) GOT0 9999 IF (coGA.EQ.l.0) GOT0 8888

HILF=QSIDX21,DY21,DZ21) ALP1 = DX21/HILF ALP2 = DY21lHILF ALP3 = DZZl/HILF

COORDINATES OF I-TH POINT OF A TRIANGLE (IN KM1

COORDINATES OF I-TH STATION (KM1

NUUBER OF STATIONS

UAXIUUII NUHBER OF STATIONS

COORDINATES OF I-TH POINT FOR TRIANGLES (KH)

NAXIHUK NUMBER OF POINTS IN TXYZ

POINTS OF TRIANGLES TO BE CALCULATED (COORDINATES IN TXYZ)

PX=DX31+BB*DXZl PY=Dy3lfBB*DYZl PZ=DZ31+BB*DZ21

31 c C IPl,IP2,IP3 C C C

C OUTPUT: C C GRAVII) C C C C IFAIL C

32 33 11

CALCULATED GRAVITY FOR DENSITY- CONTRAST 1 GR/CCH AT STATION I OfGAL)

.EQ.B : NO ERROR DETECTED

.EQ.l : TRIANGLE HAS IDENTICAL POINTS 12 C

C ItiPLICIT INTEGER (I-K.&N) IMPLICIT LOGICAL IL) IHPLICIT REAL*8 (A-H,O-Z)

C DIMENSION X~5~,Y~5~,2~5~,SXYZ~NSTH,3l,TXYZ~NPK,3~,GRAV~NSTBl

C DIHENSION HFAKTO),ASTI4),BST(4).HFAKT2(4).ISIG(3)

C

DO 20 I=l,NST C C LOOP OVER STATIONS c

GGRAV=DBLE(B.) REXS=SXYZ(I,lI REYS=SXYZ(I,2) REZS=SXYZ~I,3)

(II