Download Geophysical Data Inversion

And reas Vogel
Abu K. M. Sarwar
Rudolf Gorenflo
Ognyan I. Kounchev (Eds.)
Theory and Practice of
Geophysical Data Inversion
Proceedings of the 8th International
Mathematical Geophysics Seminar on Model
Optimization in Exploration Geophysics 1990
Iterative Simultaneous Inversion of Gravity and Seismic
Traveltime Data: I - Formulas
J. Svancara, J. HaW
GeofyzikaS. p. Brno,JeEnA 29a, 61246 Brno 12, Czechoslovakia
Abstract
Formulas described in this paper were designed for the
use in least-squares tomographic inversion methods with improved flexibility of parametrizing the density-velocity
model. The gravity effect of the considered volume fl is the
sum of gravity effects of 2-3 or 2 1/2-D bodies of polygonal
cross-sections which form the disjunctive decomposition of
fi. For this geometrical arrangement analytical and numerical ray tracing algorithms were developed assuming either
linear or parabolic dependence of quadratic slowness (square
of reciprocal wave speed) in Cartesian coordinates. Partial
derivatives of calculated gravity and traveltime with respect
to geometrical parameters, which are necessary for the control of the first derivative inverse algorithms, are given
in an explicit form. Two expressions for the traveltime derivative with respect to vertex coordinates were obtained :
one for the straight rays based on Fermat's principle and the
other based on the geometrical approximation not affected by
the curvature of the ray. Results of the numerical experiment
based on the formulas are presented.
1. Introduction
The goal of simultaneous inversion of gravity and seismic
data is the estimation of a subsurface density-velocity model
the response of which is consistent with these two independent
sefa of geophysical data Lines et al. (1988), Starostenko et al.
(1988). The joint inversion includes the coupling between den-
sity and velocity in the form of linear relationship. For the
solution of this problem it is necessary to define algorithms
for gravity and seismic modelling using the same parametrization of the geologic model. Our paper was primarily designed
for interpretation of regional refraction data, but we suppose
that the obtained results can be used for solving of other problems encountered in exploration seismology.
Quantitative interpretation of gravity data is usually
accomplished by 2 1/2-D bodies of polygonal cross-section,
which makes it possible to retain the simplicity of the 2-D
modelling (Talwani et al., 1959). The solution of the direct
and inverse gravity problem for 2 1/2-D geometry was described
in many papers, including those of Rasmussen, Pedersen (1979),
Enmark ( 1981) and Svancara, Halif (1987).
Initial value ray tracing in heterogeneous media is often
based on analytically computed ray segments within suitable
cells with a simple velocity law (see Berryman (1989), Bishop
et al. (1985), Langan et al. (1985)). These elementary cells
in 2-D case may be triangular or rectangular. In areas of complex geological structure this method needs many elementary
cells.
We present explicit formulas for the tracing of rays in
2-D bodies of general polygonal cross-section. The velocity
distribution within individual bodies is approximated by quadratic slowness linear or parabolic function of the Cartesian
coordinates. Rays are bent according to reflection/transmission
laws at the polygon boundaries and travel along a curved path
within the polygons. Inverse modelling of seismic arrivals is
often based on the first derivative least-squares algorithms
which require the knowledge of partial derivatives of the model
response with respect to model parameters. Twn approximative
formulas for the traveltime derivative with respect to vertex
coordinates were obtained.
This paper presents formulas necessary for joint inversion
of gravity and seismic data while our method permits construction of a complex model without subdividing them into a large
number of cells.
2. Gravity modelling
Calculating the gravity effect of volume l2, we sum the
effects of its density homogeneous parts which form the disjunctive decomposition of I L
In the 2 1/2-D approach (Fig. l.) these elementary parts are
homogeneous bodies with polygonal cross-section and finite
strike-length. According to Rasmussen, Pedersen (1979) for a
set of 2 1/2-D bodies it holds
where
m
-
is the body index
Nm
-
number of polygon vertices
Si,m n i -m,
Z
6,
Fig. l.
the body density
the i-th face
the outward directed unit vector normal to Si,,
the downward directed unit vector.
Geometry of the 2 1/2-D body and description
of the polygonal cross-section.
For simplicity of notation we drop the subscript on quantites
associated with the m-th body and set
= ( 0 , 0 , 0 ) . The analytical expression for the gravity effect is given in Rasmussen,
Pedersen (1979) and Svancara, HalE (1987). It holds
if i = N then x ~ =+x1~, Z N + ~= Z1
For the solution of the inverse gravity problem we apply first
derivative algorithms, hence it is necessary to derive relations for partial derivatives of gravity with respect to variable parameters. For the partial derivative of Ag t with respect
to the X-th and z-th coordinate of the i-th vertex it holds :
I
where
and for i = 1 the functions
xo r XN and z0 E Z~ '
2i-1,pi-1,
%-l are taken with
The partial derivative of gravity with respect to density is
elementary because of their linear relationship.
The inverse gravity problem is defined by the task to find
such geometrical and density parameters that the L2 norm of
difference between the measured and calculated gravity is minimized and the apriori information about the geological structure is respected. We use alternativelly one o f the three wellMarquardt known gradient iterative algorithms : Levenberg
Meyer, quasi - Newton method with Davidon - Fletcher - Powell
update and the conjugate gradient method.
-
3. Analytical ray tracing in polygonal bodies
In some media with simple velocity distribution the ray
tracing may be performed analytically, which is the simplest
and fastest solution. Particulary simple formulas are obtained
(Eenen9, 1987) for the ray tracing in media with constant gradient of the quadratic slowness. Let us assume quadratic slowness linear in Cartesian coordinates X, z
and define variable 6 monotonic along the ray by the formula
than the ray tracing system yields the following polynomial
solution
X
xo + pxo6
=
+ 1/4 Bx d2
(3.3)
zo + pz06 + 1/4 ~~6~
z =
(3.4)
with the slowness vector g(px, pz)
P,
--
Pxo + 112 Bxd
pz
=
p,
and ,
p
=
(3.5)
+ 1/2 ~~6
sin &/v,,
pzo = cosd/vo, v
(3.6)
=
A
+
Bxxo + BZzo
(3.7)
where xo, zo are Cartesian coordinates of the initial point
( 6 =0, T 0) a n d d is the angle between the coordinate axis
z and the slowness vector at the initial point,
=1
For the traveltime T it holds (Eerven9, 1987) :
Assume now a more general case of the quadratic slowness
parabolic in Cartesian coordinates X, z
C
Pm1
Fig. 2.
366
Analytical ray tracing in a medium with
constant gradient of quadratic slowness.
'
than the ray tracing system reads
Solution of those nonhomogeneous second order linear diferential equations c m be found in analytical form. (In the next
we consider only eq. (3.11) only because (3.10) a (3.11) are
analogical.) It holds :
for CZ > 3 and
~ ( 6=) -
3 + ~ Z c o s ( b ~ +) ~ ~ , s i n ( 6 E )
2Cz
for CZ
(3.13)
0.
From the initial conditions (3.7) we obtain formulas for conIt holds :
stants K1,, Kg, and MIZ,
a,.
and
The traveltime T( 6 ) is given by the intenral
2
[km1
Fig. 3. Analytical ray tracing in a medium with quadratic
slowness parabolic in Cartesian coordinates given
by eq. (3.12) for CZ>O.
Fig. 4. Analytical ray tracing in a medium with quadratic
slowness parabolic in Cartesian coordinates given
by eq. (3.13) for Cz< 0.
and can be evaluated analytically for both Cx(CZ) > 0 and
0.
The solutions (3.12)and (3.13) of the ray tracing system
yield suitable tools for examination of effects of both lowvelocity and high-velocity zones and even the waveguide phenomenon. Examples of analytical ray tracing in media with the
above described velocity distributions are given in Figures 2,
3 and 4.
Assume that the whole model is subdivided into a set of
2D bodies of arbitrary polygonal cross-sections. Analytical determination of the intersection point of the ray with the body
boundary is possible for the velocity law (3.1) i.e. constant
gradient of quadratic slowness. For the ray intersections with
the straight line z = ax + b forming the boundary (Fig. 5.) it
holds :
Because we do not know in advance which side of the body will
be intersected by the ray, it is necessary to find intersections with all sides. The actual intersection point of the ray
with the body boundary corresponds to the smallest positive
value of 6
.
Fig. 5.
Geometry considered in the calculation of traveltime derivative with respect to vertex coordinates.
The behaviour of ray on a plane interface between two
bodies is determined by the reflection/transmission laws.
We denote g(nx, nZ) the unit vector normal to the boundary,
W'lx' plz) the incident slowness vector and 92(~2x, pZz) the
slowness vector of the reflected or transmited ray. According
to Born, Wolf (1964) and Eervenf (1987) we can write
=
where
I'
The upper sign in (3.20) holds for reflected waves, the lower
sign for transmitted waves.
Formulas (3.1) to (3.20) permit initial value ray tracing
and corresponding traveltime calculation for 2-B bodies of a
general polygonal cross-section.
For the solution of the inverse seismic kinematic problem
an iterative Gause - Newton type algorithm is usually constructed that produces a velocity model which minimizes the differ
ence between traveltimee generated by ray tracing through the
model and traveltimes selected from the field data. This algorithm requires the knowledge of model traveltime derivatives
with respect to model parameters. For a ray with fixed endpoints, the first variation of the traveltime for a perturbation
of medium slowness is given by (Nowak, Lyslo, 1989)
which is computed along the unperturbed raypath. Using this
simplification we can for the velocity law (3.1) write
dT
dA
G;.
The traveltime derivative with respect to vertex coordinate
(Fig. 5.) is given by
P
X
and analogical expression for z coordinate is valid. ExpresP
sion (3.23) can be calculated analytically using equations
(3.8) and (3.191, but the resulting formula is rather complicated. Therefore we present a simpler approximative formula
based on geometrical considerations shown in Fig. 5 . We assume
that the traveltime perturbation can be approximated by
L
L
where W' is distance between ray intersections with the velocity boundary in initial and perturbed position. The lower sign
in (3.24) holds for the transmitted ray, the upper sign holds
for the reflected unconverted ray where v2 = vl. Under this
assumption we can easily derive :
and
where Q(x
Q' z Q ) is the ray intersection point, P1(xl, z1) and
P(xp, zp) are the fix and variable boundary vertices and
ar = pZ/px is tangent to the raypath at point Q. The sign convention is the same as for equation (3.24). On the synthetic
example in Fig. 7.the traveltime perturbation caused by boundary displacement is compared with the value predicted 07 the
basis of traveltime derivative approximation (3.25). Relative
error in the estimation of traveltime perturbation is 8 %.
Under the assumption that the raypath can be approximated
by straight line segments a more exact derivation of formulas
Fig. 6.
Fig. 7.
Geometry used by the calculation of traveltime derivative with respect to vertex coordinates using
Fernat's principle.
AT^^^-^^^^^ p. =
11.3 m s
A TOpprox,
=
1 0 . 4 ins
bTapprox,
= 12.0 m r
El
E,=
-4.0 Vo
6.2 %
Comparison of traveltime perturbation ( Ttwo-point p,)
caused by boundary displacement with the value predicted either by the formula 3 . 2 6 . ( Tapprox. 1) Or by
the formula 3 . 2 8 . ( Tapprox. 2).
for traveltime derivative with respeut to vertex coordinates
is possible.
Using the Permat's principle, for the geometry shown in Fig. 6.
we obtain
and
where S(xo, zo) and R(xR, zR) are the source and reciever
coordinates. The test on the synthetic example in Fig. 7.shows
relative error 6 % in prediction of the traveltime perturbation caused by boundary displacement.
4. Conclusions
Presented formulas enable construction of an iterative
algorithm for cooperative inversion of gravity and seismic
traveltime data. Our approach was primarily designed for velocity inversion of refraction data, but the presented formulas describe also properties of the reflected rays because
the reflectors can be associated with the body boundaries.
For the cooperative inversion of seismic and gravity data we
intend to adopt a special separable minimization algorithm,
which respects the fact that the vector of unknown parameters
consists of linear (density, velocity) and nonlinear (geometrical) parameters.
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