Download Wave equation inversion of skeletalized geophysical data

Geophys. J . Inf. (1991) 105,289-294
Wave equation inversion of skeletalized geophysical data
Yi Luo and Gerard T. Schuster
Department of Geology and Geophysics, University of Utah, Salt Lake City, UT 84112, USA
Accepted 1990 September 19. Received 1990 July 27; in original form 1990 March 16
SUMMARY
Inversion methods based on gradient optimization techniques require the directional
derivative of the data with respect to the model parameters. Unfortunately, the data
(e.g., pressure seismograms) are usually restricted to that explicitly given in the
fundamental governing equation (e.g., wave equation). This limited choice of data
type may lead to misfit functions that are pathologically non-linear with respect to
the model parameters. We present a methodology which allows for the calculation
of directional derivatives for skeletalized data sets, yet still uses the fundamental
governing equations without the need for approximations. Skeletalized data are
defined as a reduced data set derived from the original data which retains the
important information about the model parameter of interest. The motivation for
working with skeletalized data rather than raw data is that the skeleton data may be
strongly influenced by only one type of model parameter and lead to a quasi-linear
misfit function.
Examples of skeletalized data sets include first arrival traveltimes picked from
CDP seismograms for velocity inversion, amplitudes of transmitted earthquake
SH-waves for earthquake moment inversion, or pulse width measured from first
arrivals for attenuation inversion. As a n example we devise a traveltime inversion
method based on the wave equation and free of any high-frequency approximations.
Results show that wave equation traveltime (WT) inversion is superior to ray traced
traveltime inversion for complicated velocity models. It is also shown that WT
inversion converges quickly for starting velocity models that are far from the actual
model.
Key words: seismic, skeletal, tomography, traveltime, wave equation inversion.
INTRODUCTION
Inversion methods are usually limited to those in which
there exists a fundamental governing equation which
explicitly relates the fundamental model parameters, m, to
the fundamental data deal, i.e.,
(la)
A@, d C d= 0 ,
where A is an operator and dca,can be considered a function
of m. An example might be to assign the homogeneous
wave equation as A , the calculated pressure seismograms as
deal, and acoustic velocities as m.
The optimization problem can be defined as finding
the optimal model m* in R"' which minimizes the misfit
function
E
E(m) = 1/2(dcal(m) - dabs, dcal(m) - dabs)
9
(Ib)
assume that d belongs to R" and that ( d , d ) represents the
squared Euclidean norm of d. A popular optimization
procedure is a gradient technique, which searches for m* in
directions parallel to the gradient vector
-6E(m)6mi
(y
, dcal(m)-
dabs),
where Sdca,(m)/Gmiis the directional derivative of d along
the ith component of m and dca,(m)-dobs is the data
residual vector. Assuming A is linear with respect to the
data and perturbing the fundamental governing equation
with respect to model perturbations
Sm,
dcal(m)+ A(m) 6dcal(m) - 0,
6mi
~
we can derive the directional derivative operator
where dobs is the observed data and ( ) denotes the inner
product. For convenience, the following discussion will
289
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Y. Luo and G. T. Schuster
a
Sciamogramc for 20% Velocity C o n t r a i t Modal
I
I00
80
b
60
40
20
0
200
0
600
400
(ms)
Time
C
q
,
5
0
2
2
I3
I
,
I
40
0
20
Percentage of Velocity Contrast
Figure 1. (a) Acoustic cylinder velocity model with a cylinder
diameter equal to about five source wavelengths. The homogeneous
background velocity is 2.5 km s-' which is also that for the starting
models for all W T inversions. (b) Pressure seismograms for a source
on one edge and receivers on the opposite edge. (c) Misfit functions
for transmission traveltimes (solid line) and pressure seismograms
(dashed lines) as a function of velocity change in the Fig. l(a)
cylinder. The fundamental and skeletal data are, respectively,
pressure seismograms and traveltimes of first arrivals.
Examples of geophysical inversion which employ
equations similar to equations (1) include; (a) full wave
seismic inversion (Tarantola 1987; Johnson & Tracy 1983)
which assigns the wave equation as A to relate the, say
pressure field, to the unknown velocity parameters, (b)
standard traveltime inversion which associates the traveltime
integral with A to relate the observed body wave traveltimes
(Dines & Lytle 1979; Lines 1988) to the velocities, and (c)
electromagnetic inversion methods which assign Maxwell's
equations as A to relate the conductivity parameters to the
electric or magnetic field data (Nabighian 1987).
A problem with the above approach is that the misfit
function in equation (lb) can be a highly non-linear function
of several types of model parameters, such as velocities and
densities. An example is given in Fig. l(c) (dashed line)
which depicts the logarithm of the misfit function E of
pressure seismograms (Fig. lb) as a function of cylinder
velocity contrast for receivers and sources surrounding an
acoustic cylinder model (Fig. la). In this case, the pressure
seismograms deal are calculated for a homogeneous model,
subtracted from the actual seismograms for a cylinder
model, and the normed difference is given by E. Fig. l(c)
suggests that the seismogram misfit function E is non-linear
for moderate ( > l o per cent) to large perturbations in the
cylinder velocity (Gauthier, Virieux & Tarantola 1986). In
this case a gradient method may tend to get stuck in local
minima if the starting model is moderately far from the
actual model. Unlike traveltime data, the seismogram
amplitudes are strongly dependent on density variations.
Are there reduced or skeletal features in the observed
data which strongly depend on only one parameter type and
might also yield a more linear misfit function? If these
skeletal data, $, are extant why not use them to form the
misfit function? The benefit might be a quasi-linear (rather
than non-linear) misfit function which is decoupled from all
but one parameter type. Unfortunately, there is usually no
fundamental governing equation, such as the wave equation,
which relates these skeletal data
to a single type of model
parameter. This means that the directional derivative
S 4 1 S r n i in equation (Id) cannot be easily calculated.
An example of skeletalization is from traveltime
tomography where first arrival times are picked or
skeletalized from a common shot point (CSP) gather. This is
beneficial because a traveltime misfit function is decoupled
from densities and depends on velocities in a quasi-linear
manner (e.g., solid line in Fig. lc). The penalty, however, in
ray-based traveltime inversion is a high-frequency approximation to the data, which can restrict its utility.
This paper describes a methodology for constructing
directional derivatives 6d,,/Smi of important features or
skeletalized features of the data with respect to the most
influential parameters of the model, without using any
approximations (except for iterative linearization). The key
idea is that a Connective functional (or equation) is
constructed which relates the important or skeletal data, $,
to the fundamental data d,
F ( 4 , d) = F ( 4 , d(m)) = 0,
(le)
which will define d,, as a function of m. The functional F
might be constructed with a statistically robust pattern
matching operator which matches the observed data with
the calculated data (e.g., cross-correlation). For example, if
d represents a calculated seismogram, then d,, can be the lag
time which maximizes the cross-correlation F(d,,, d(m))
between the observed and synthetic seismogram. In this
case, 4 can also be interpreted as the traveltime difference
between the first arrivals of the observed and synthetic
seismograms.
From equation (le) the directional derivative of the
skeletalized data Sd,,lSmi can be calculated using the rule
for an implicit function derivative (Gillespie 1960)
(If)
where 6d/Srni in the numerator can be computed from the
fundamental governing equation (equation Id) and the
denominator 6 F ' / 6 d , can be computed from the connective
equation (le). For traveltime inversion, this means that the
directional derivative of traveltimes with respect to velocity
perturbations can be computed using the wave equation; no
high-frequency approximations are needed. This methodology can also be applied to other geophysical data types as
Inversion of skeletal data
long as equations (Id), (le) and (If) can be formulated, and
the conditions for the implicit function theorem are satisfied
(Gillespie 1960). This paper will illustrate this new
methodology in the context of wave equation traveltime
(WT) inversion.
291
where fi = 3p(x, t ; x , ) / d t . Equation (3) serves as the
connective function in equation (le) which connects the
fundamental data (pressure seismograms) with the skeletal
data (traveltime residuals).
Misfit function
THEORY
In this section the new inversion methodology will be
applied in inverting traveltime data with the wave equation,
henceforth designated as W T inversion. The key steps are to
(1) define a connective function (equation l e ) which defines
the traveltime residual (skeletal data) as a function of the
pressure seismograms (fundamental data). This step allows
for the derivation of the directional derivative (equation If),
(2) define a misfit function (hopefully, quasi-linear) of
traveltime residuals [squared normed difference between
observed and synthetic transmission traveltimes, equation
(lb)] which mainly depends on velocity, not density,
parameters, (3) determine the gradient of the misfit function
for use in a gradient optimization algorithm.
The following analysis assumes that the propagation of
seismic waves honours the 2-D acoustic wave equation
The WT method attempts to determine the velocity model
C ( X ) which predicts seismograms p ( x , , t ; xs)calthat minimize
the following traveltime misfit function:
(4)
where the summation is over source and receiver locations,
At is defined by equation (2b) and the factor 112 is
introduced for subsequent simplifications. This criterion, of
course, can be generalized to take into account the
estimated observation errors or a priori information in
model space.
Misfit gradient
A gradient method can be used to solve equation (4)for the
c ( x ) which minimizes S. For simplification we will discuss
(24
where p ( x , , t; x , ) , ~ ~denotes the observed pressure seismograms at receiver location x , due to a line source excited at
time t = 0 and at location x s , p(x,) is the density, s ( t ; x,) is
the source function and c(x,) is the material velocity.
Connective hrnction
The degree in which the synthetic and observed
seismograms match each other can be estimated by the
cross-correlation function:
the steepest descent method. To update the velocity model,
the steepest descent method gives
c(x)k+I
= -C@)k
+4X)kY(X)k,
(5)
where y(x), is along the gradient direction of the misfit
is a step length and k denotes the kth
function S,
iteration. The central problem is how to get y ( x ) based on
the wave equation. To obtain y ( x ) , construct the directional
derivative of S with respect to the velocity model c ( x ) :
where we have replaced the deltas with partial differentials.
Using equation (3) and the rule for an implicit function
derivative we get
where A ( ~ , ; x , ) is, ~the
~ maximum absolute amplitude of
p ( x , , t;x,)obs and t is the time shift between synthetic and
real seismograms. The divisor A@,;x , ) , ~ ~normalizes the
observed seismograms to a maximum amplitude of 1.
We seek a t in which a synthetic seismogram must be
shifted so that it 'best' matches the observed seismogram.
The criteria for 'best' match is defined as the traveltime
residual A t which maximizes the cross-correlation function,
f(xr, t;xS), i.e.,
f ( x , , A t ; x,) = max
W,, t;x,)
1 t E [-T,
TI),
(2b)
where T is the estimated maximum traveltime difference
between the observed and calculated seismograms. It is easy
to see that the derivative of f(x,, q x , ) with respect to t
should be zero at At unless its maximum is at an endpoint
At=TorAt=-T:
(3)
where, g(x, t; x ' , t') is the Green's function for equation
(2a); that is, the pressure field at point x and time t due to
the impulse line source 6 ( x - x') 6(r - r'). The symbol *
represents time convolution. Substituting equation (7c) into
292
Y.Luo and G. T. Schuster
equation (7b),
5%
- 2600
%
>
a f t e r 5 iterations
pyq
2400
I00
2 00
2400
u o
100
200
2000
- o
I00
200
10%
after 5 iterotions
20%
ofter 7 iterations
40%
one can rewrite equation (8a) as
(9)
where
p ’ ( x , t ; x,) =
C g ( x , -2;
r
x,, 0) * 6 t ( x , , t ; xs)t
and p ( x , t ; ~ , is) the
~ ~ pressure
~
field calculated (using a
finite difference method) for the current velocity model c ( x )
and p ’ ( x , t ; x , ) is the field computed by reverse time
propagation of the quasi residual 6t(x,, t ; x , ) acting as a
source at receiver location x,. This result is similar to that of
full wave inversion except 6t is used instead of Sp. In full
wave inversion Sp is defined as
SP =P(xr,
t ; ~ s ) o b s - ~ ( ~ t;xs)cal,
r,
(10)
where the subscripts ‘obs’ and ‘cal’ denote, respectively,
o f t e r 10 i t e r a t i o n s
Figure 2. Velocity models reconstructed by the WT method (using
transmission traveltimes) for different velocity perturbations in the
Fig. l(a) cylinder. The percentages indicate the per cent velocity
contrast of the cylinder from the background velocity. The left
column of figures depict the final reconstructed velocity model, and
the right column of figures depict the reconstructed velocity profile
(dashed lines) along a vertical cross-section through the model’s
centre.
Cylinder model
Gauthier et al. (1986) presented a cylinder model (Fig. l a )
composed of a cylindrical velocity perturbation five
wavelengths in diameter superimposed on a homogeneous
medium with velocity 2500ms-’. The starting model is
assumed to be a homogeneous medium with velocity
2500 m s-l, the dominant source frequency is 25 Hz, and
there are a total of 380 receivers and eight sources equally
observed and calculated pressure seismograms. Combining
spaced along the model edges.
equations (9) and (5) yields an iterative method to invert for
a velocity model c ( x ) from traveltime residuals. It can be
shown that in the high-frequency limit (Luo & Schuster
1991) equation (9) reduces to that of ray traced traveltime
tomography.
Gauthier et al. (1986) show that the full wave inversion
method will fail for cylinder velocity perturbations larger
than 10 per cent. Fig. 2 shows the success of WT inversion
for velocity cylinder perturbations up to 40 per cent. The
reason for this success is suggested in Fig. l(c) which
demonstrates that the normed traveltime misfit function has
a quasi-linear behaviour with respect to increasing velocity
contrast. On the other hand, the normed pressure
seismogram misfit function will be non-linear (Jannane et al.
1989).
NUMERICAL EXAMPLES
The WT method is tested for three different models; a
cylinder model, the Langan velocity model (Langan et al.
1988) and a fault model having steam injection. The cylinder
model is used to verify that WT inversion is more robust
than full wave inversion for incorrect starting models. The
Langan velocity model and the fault model are used to show
that WT inversion succeeds when ray tracing fails.
Langan velocity model
Figure 3(b) shows a sonic log velocity profile from Kern
River, California (Langan et al. 1988) in which ray tracing
(subject to a high-frequency approximation) fails to
Figure 4. (a) Earth velocity models, with the pre-steam injection model vpre(x,z ) at the left, the post-steam injection model vFst(x, z ) in the
middle, and the subtracted vposr(x,z ) - vp&, z) at the right. The steam injection lowers the velocity by 10 per cent. Coldest (warmest)
colour represents a velocity of 2300 m s-l (4000 m s-l) and the wells are 90 m apart and 200 m deep. (b). Reconstructed velocities using the
wave equation traveltime method are given directly below each model in Fig. 4(a).
"
01
-
-
I
5 200
a
D
True Velocity Model
30001
I
11
Inversion of skeletal data
I
2 2000
-
>
:
4ooo
o
o
o
l
100
200
300
Horizontal Distance (m)
0
200
Depth (m)
400
b
0
Cross-Well Seismograms f o r Longon Velocity Model
293
compute the first arrival times for energy travelling from the
source well to the receiver well. The observed data are
taken to be first arrival traveltimes from the synthetic
seismograms (Fig. 3c) computed by a finite difference
method (200 by 150 point grid and 700 time
steps/seismogram) for the acquisition system shown in Fig.
3(a) and the I-D velocity profile in Fig. 3(b). Eight sources
are equally distributed in the source well, and 83 receivers
with a 4.8m spacing are located in the receiver well. Figs
3(e) and (f) show the successful velocity reconstructions
obtained by the WT inversion method using a steepest
descent method. Each iteration consumed less than 10 CPU
min on a Stardent 2000 computer.
Some practical details that expedited rapid convergence
included application of a window function which suppresses
all energy but the transmitted arrivals. This mitigates
problems with multiple stationary points associated with the
cross-correlation function in equation (3).
Faulted steam injection model
A fault model is shown in Fig. (4a) and the crosswell
I00
200
Time (ms)
geometry consists of 21 sources evenly distributed in the
source well and 32 receivers distributed in the receiver well.
The leftmost model shows the pre-steam injection model,
the middle model is the post-steam injection model (steam
with 10 per cent velocity contrast injected at left well) and
the rightmost model represents the difference between these
two models. The dominant source frequency of the Ricker
wavelet is 80 Hz and the reconstructed velocity model after
20 iterations is given in Fig. (4b). The rightmost figure
shows that the steam injection zone is faithfully
reconstructed.
300
C
A f t e r 10 i t e r a t i o n s
I n i t i a l Velocity Model
3000 I
1000
I
0
200
3000 I
400
200
0
Depth ( m )
Oepth ( m )
d
e
400
A f t e r 20 I t e r a t i o n s
CONCLUSION
A methodology is described which shows how to compute
directional derivatives of skeletalized data (data types not
explicitly contained in the fundamental governing equations)
with respect to the model perturbations. This is achieved by
forming a connective equation which connects the skeletal
data with the fundamental data and using the rule for an
implicit function derivative. The connective equation can be
constructed via some pattern matching operator, such as
cross-correlation. No approximations are employed, and the
benefits include an enhanced ability to work with
skeletalized data sets and noisy data, and the capability to
match the important parts of the data (e.g., traveltimes) that
are mainly influenced by one model parameter (e.g.,
velocities).
This methodology is applied to develop a new seismic
inversion method which reconstructs velocities from
traveltimes based on the wave equation. No high-frequency
approximations are needed, velocities are decoupled from
densities, and the computer time is about the same as full
wave inversion. Compared to a ray based inversion code,
the WT computer code is much more general, and demands
less effort to write. For complicated velocity structures, the
CPU time required by the WT method is competitive with
that of our ray trace inversion code. Synthetic tests show
that successful reconstructions can be achieved with model
velocity contrasts greater than 2:l. This is an improvement
7
-.3000
2600
"7
E
2200
21
c
-
.-
5
I800
>
1400
1000
0
100
200
Oepth ( m )
300
4 00
f
(a) 2-D acquisition cross-well geometry for the Langan
acoustic velocity model. (b) 1-D Langan velocity profile. (c)
Acoustic seismograms for a source at depth 300 m in the source well
with offset 50 m, and receivers in the receiver well with offset 250 m.
Due to shadow zones and waveguide effects, ray tracing cannot
correctly compute all of the first arrival times at the receiver well.
(d) Initial velocity model (dashed line) employed by the WT
method. (e-f). The velocity model reconstructed by the WT
method (dashed line) compared to the true velocity model (solid
line).
Figure 3.
294
Y. Luo and G. T. Schuster
over full wave inversion which can fail for similar models
with little more than 10 per cent velocity contrast and over
ray based traveltime tomography when the velocity
distribution is too complex for the use of ray theory. In
addition, traveltime picking or event identification may
sometimes not be necessary, and diffraction, reflection and
refraction traveltimes can be incorporated into the inversion
process. This procedure is naturally suited for hybrid
inversion schemes (Luo & Schuster 1990).
Possible applications of this methodology include the
following.
(1) Radar tomography where the fundamental data are
the time-domain radargrams, the model parameter is the
electromagnetic propagation velocity distribution and the
skeletal data might be the time lag between the observed
and computed first arrivals. The connective functional is the
time cross-correlation between the observed and computed
time-domain radargrams. In addition, wavelet broadening
or amplitude might serve as skeletal data for conductivity
inversion in both EM and radar inversion.
(2) Surface wave inversion in the slowness-frequency
domain where the fundamental data consist of the
transformed seismograms s ( p , o),the model parameter is
the shear velocity distribution and the skeletal data are the
slowness lags between the observed and computed values of
the dispersion curve for a specified mode. The connective
equation is the slowness cross-correlation between the
observed and computed dispersion curves.
(3) Post-critical reflection inversion in the t-p domain
where the fundamental data consist of the transformed
seismograms s ( t , p ) , the model parameter is the compressional velocity distribution and the skeletal data are the
slowness lags between the observed and computed values of
the post-critical reflection envelope in the t - p domain. The
connective equation is the slowness cross-correlation
between the observed and computed post-critical reflection
envelopes.
ACKNOWLEDGMENTS
We thank the Gas Research Institute for sponsoring this
research under contract 5089-215-1872. Neither GRI,
members of GRI, nor any person acting on behalf of GRI
assumes any liability in the use of information in this report.
We are also grateful for the support of the 1989 University
of Utah tomography consortium members; Amoco, Arco,
British Petroleum, Conoco, GRI, Marathon, Mobil,
Phillips, and Texaco. The authors are very grateful to
Robert G. Keys for his comments and insights about
directional derivatives.
REFERENCES
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tomography, Proc. IEEE, 67, 1065-1072.
Gauthier, O., Virieux, J. & Tarantola, A., 1986. Two-dimensional
nonlinear inversion of seismic waveforms: numerical results,
Geophysics, 51, 1387-1403.
Gillespie, R. P., 1960. Parrial Differentiation, Interscience
Publishers, New York.
Jannane, M. et al., 1989. Wavelengths of earth structure that can be
resolved from seismic reflection data, Geophysics, 54, 906-910.
Johnson, S. & Tracy, M.L., 1983. Inverse scattering solutions by a
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Luo, Y. & Schuster, G. T., 1990. Wave equation traveltime and
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Luo, Y. & Schuster, G. T., 1991. Wave equation traveltime
inversion, Geophysics, in press.
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